make-to-order assembly management ||

Make-to-Order Assembly Management

Springer-Verlag Berlin Heidelberg GmbH

Rainer Kolisch

Make-to-Order Assem.bly Management

With 49 Figures and 44 Tables

, Springer

Professor Dr. Rainer Kolisch Technische Universităt Darmstadt Institut fiir Betriebswirtschaftslehre HochschulstraSe 1 64289 Darmstadt Germany

Als Habilitationsschrift auf Empfehlung der Wirtschafts- und Sozialwissenschaftlichen Fakultăt der Christian-Albrechts-Universităt zu Kiel gedruckt mit Unterstiitzung der Deutschen Forschungsgemeinschaft.

ISBN 978-3-642-07431-8

Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Kolisch. Rainer: Make to order assembly management: with 44 tables I Rainer Kolisch. -Berlin; HeideJberg; New York; BarceJona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer. 2001

This work is subject to copyright. Ali rights are reserved. whether the whole or part of the material is concerned. specifically the rights of translation. reprinting. reuse of illustrations. recitation. broadcasting. reproduction on microfilm or in any other way. and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9. 1965. in its current version. and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.

© Springer-Verlag Berlin HeideJberg 2001 OriginaIly published by Springer-V erlag Berlin Heidelberg New York in 200 l Softcover reprint of the hardcover Ist edition 200 I

The use of general descriptive names, registered names. trademarks. etc. in this publication does not imply. even in the absence of a specific statement. that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Hardcover-Design: Erich Kirchner. Heidelberg

SPIN 10733639 42/2202-5 4 3 2 1 O - Printed on non-aging paper

ISBN 978-3-642-07431-8 ISBN 978-3-662-04514-5 (eBook) DOI 10.1007/978-3-662-04514-5

To my father, Klaus Kolisch

Contents

1 Introd uction 1.1 Basic Problems and Decision Levels 1.2 Outline of the Book ........ .

I Preliminaries

2 General Issues 2.1 Make-to-Order Manufacturing 2.2 Assembly Management. . . . 2.3 Coordination and Integration

2.4 2.5

2.3.1 Interdependencies 2.3.2 Coordination . . . . . 2.3.3 Integration ..... . 2.3.4 Coordination and Integration for Make-to-Order

Assemblies ....... . Hierarchical Production Planning Case Descriptions ....... . 2.5.1 Ship Assembly ..... . 2.5.2 Machine Tool Assembly .

1 1 5

9

11 11 13 17 17 18 19

19 20 22 22 26

2.5.3 Synthetic Fiber Production Line Assembly 27 2.5.4 Aircraft Assembly .............. 28

3 Literature Survey and Classification 3.1 Assembly Management ....... .

3.1.1 Design for Assembly .... .

33 33 34

3.1.2 Production Planning for Assembly Systems 36

VIII CONTENTS

3.1.3 Loading and Release Planning for Assembly Sys-tems ......................... 37

3.1.4 Sequencing and Scheduling of Assembly Systems 37 3.1.5 Miscellaneous.................... 40

3.2 Make-to-Order Manufacturing . . . . . . . . . . . . . . 40 3.2.1 Make-To-Order Manufacturing with Project Net-

works . . . . . . . . . . . . . . . . . 40 3.2.2 Production Planning for Job Shops. 41

3.3 Multi-Project Scheduling .......... 42

II Decision Models

4 Hierarchical Framework 4.1 Order Selection Level. 4.2 Manufacturing Planning Level . 4.3 Operations Scheduling Level.

5 Order Selection 5.1 Outline of the Problem.

5.1.1 Performance Measure 5.1.2 Interdependencies ..

5.2 Literature Review ..... . 5.2.1 Due Date Assignment 5.2.2 Competitive Bidding . 5.2.3 Project Selection and Scheduling 5.2.4 Revenue Management . . . . . .

5.3 Model .................. . 5.3.1 Detailed Description and Notation 5.3.2 Network Representation 5.3.3 MIP Formulation . .

5.4 Discussion of the Model ... . 5.4.1 Complexity Results .. . 5.4.2 Reduction of the Problem Size 5.4.3 Release Dates and Maximal Time Lags 5.4.4 Supplier Coordination and Integration 5.4.5 Stochasticity of the Data ... 5.4.6 Time-Dependent Order Values . . . .

45

47 47 49 49

53

53 53 54 54 55 55 56 57 58 58 59 61 62 62 63 63 64 64 66

CONTENTS

5.4.7 Variable Capacity 5.4.8 What-If Analysis .

6 Manufacturing Planning 6.1 Outline of the Problem.

6.1.1 Performance Measure 6.1.2 Interdependencies ..

IX

67 67

69 69 69 72

6.2 Literature Review ...... 73 6.2.1 Multi-Level Lotsizing 75 6.2.2 Multi-Level Scheduling and Lotsizing 76 6.2.3 Multi-Level Lotsizing and Scheduling 77

6.3 Model . . . . . . . . . . . . . . . . . . . . 77 6.3.1 Detailed Description and Notation 78 6.3.2 Network Representation 80 6.3.3 MIP Formulation . . 82

6.4 Discussion of the Model . . . . 86 6.4.1 Complexity Results. . . 86 6.4.2 Regularity of the Objective Function . 87 6.4.3 Sequential vs. Integrated Manufacturing Planning 88 6.4.4 Special Cases . . . 89 6.4.5 Model Extensions. 91

7 Operations Scheduling 93 7.1 Outline of the Problem. 93

7.1.1 Performance Measure 95 7.1.2 Interdependencies .. 97

7.2 Literature Review ...... 99 7.2.1 Job Shop Scheduling under Resource Constraints 99 7.2.2 Project Scheduling under Resource Constraints . 100 7.2.3 Project Scheduling under Resource and Part Avail-

ability Constraints . . . . . . . . . . . . 101 7.2.4 Scheduling under Spatial Constraints. . 101

7.3 Model . . . . . . . . . . . . . . . . . . . . . 101 7.3.1 Detailed Description and Notation . 102 7.3.2 Network Representation 7.3.3 MIP Formulation ......... .

. 103

.105 7.4 Discussion of the Model . . . . . . . . . . . 107

7.4.1 Special Cases and Complexity Results . 108

x

7.4.2 7.4.3 7.4.4 7.4.5

CONTENTS

Left-Regularity of the Objective Function Part Pegging . . . . . . . . . . . . Further Issues of Spatial Demand . Resource and Part Assignment . .

· 109 .110 .110 .112

III Solution Methods 117

8 Order Selection Methods 119 8.1 Column-Generation Approach .............. 119

8.1.1 Knapsack Formulation ............... 119 8.1.2 Column Generation by Dynamic Programming . 122 8.1.3 LP-Based Heuristic . 123

8.2 Experimental Evaluation. . . . 8.2.1 Test Instances ..... 8.2.2 Computational Results.

· 124 · 124 · 125

9 Manufacturing Planning Methods 129 9.1 Construction Heuristics ............. . 129

9.1.1 Outline of the List Scheduling Heuristic . 130 9.1.2 List Generation. . . . . . . . . . . . . . . 130 9.1.3 Schedule Generation . . . . . . . . . . . . 131 9.1.4 Property of the List Scheduling Heuristic . 134 9.1.5 Outline of Backward Oriented Lotsizing . 134 9.1.6 Lotsizing Generation Scheme . 136 9.1.7 Cost Considerations . . . . . . . . 138

9.2 Lagrangian Relaxation . . . . . . . . . . . 141 9.2.1 Echelon-Cost MIP Formulation. . 141 9.2.2 Decomposition of Assembly Scheduling and Fab

rication Lotsizing . . . . . . . . . . . . . . . . . . 145 9.2.3 Lower Bounds for the Assembly Scheduling Prob-

lem ......................... 147 9.2.4 Lower Bounds for the Fabrication Lotsizing Prob-

lem ....................... . 151 9.2.5 A Lagrangian-Based Construction Heuristic. . 156 9.2.6 Subgradient Optimization . 160

9.3 Experimental Evaluation. . 162 9.3.1 Test Instances . 162

CONTENTS XI

9.3.2 Computational Results. .164

10 Operations Scheduling Methods 171 10.1 Construction Heuristics ................ . 171

10.1.1 Schedule Generation . . . . . . . . . . . . .. . 172 10.1.2 Property of the Schedule Generation Scheme . 176 10.1.3 List Generation . . 176 10.1.4 Priority Rules. . . . 179

10.2 Improvement Heuristics . . 180 10.2.1 Sampling Methods . 180 10.2.2 Tabu Search Based Large-Step Optimization . 182

10.3 Experimental Evaluation. . . . . 191 10.3.1 Test Instances ..... . 191 10.3.2 Computational Results. . 193

11 Research Opportunities

A Instance Generation A.1 General Concepts. A.2 Order Selection Instances . . . . . A.3 Manufacturing Planning Instances AA Operations Scheduling Instances

B Notation B.1 Notation for Order Selection ..... . B.2 Notation for Manufacturing Planning. B.3 Notation for Operations Scheduling.

List of Abbreviations

List of Figures

List of Tables

Bibliography

Index

199

201 .201 .202 .204 .205

209 .209 .211 .212

213

217

219

221

255

Chapter 1

Introd uction

1.1 Basic Problems and Decision Levels

A manufacturing system converts input to output with the output being, in general, of higher value than the input when measured in terms of market prices (cf. Zapfel [336] pp. 1). The large variety of different manufacturing systems can be characterized by the input, the output, and the way input is converted to output (cf. Dietrich [78], Zapfel [336] pp. 2). This book focuses on manufacturing systems which convert raw materials, parts, and subassemblies in a multi-level, fabrication-assembly process to customer-specific products (cf. Figure 1.1).

According to the framework of Anthony [9], there are three main issues concerned with the management of any manufacturing system: facility design, aggregate capacity planning, and production scheduling.

Facility Design encompasses long-term investment/disinvestment decisions concerning the location of the facility, the facility layout, the type and capacity of resources as well as the level and skill of employees.

Aggregate Capacity Planning is concerned with the allocation of the available resources such that the company objectives are met in the best possible way. This includes forecasting and acquisition of demand and the allocation of available capacity to the demand.

Prod uction Scheduling details the aggregated planning w.r. t. operations and machines. This entails the assignment of operations to

2 CHAPTER 1. INTRODUCTION

Purchasing .Fabrication Assembly Distribution

Figure 1.1: Multi-Level Manufacturing System for Make-to-Order Products

specific resources of a type, i.e., a certain machine or a single worker, the determination of the sequence operations are processed on a machine, and the assignment of start and finish times to operations.

We will modify this framework to be specifically suited for multilevel make-to-order manufacturing systems. We assume that the facility design issue is settled, i.e., the location and the layout of the facility as well as the capacity ofthe three main resource types of the company are determined. These resource types are the engineering department, the fabrication department, and the assembly department.

The engineering department is concerned with the construction of new products as well as the modification and customization of existing products. This entails the generation of engineering documents such as blue prints for manufacturing. The capacity of the engineering department is determined by the the count and qualification of engineers and by the availability of construction devices such as computer aided design (CAD) systems etc.

The fabrication department does the conversion of raw materials and procured parts to parts of higher value by, in general, metalworking processes. Processes are always linear, i.e., a single input part

1.1. BASIC PROBLEMS AND DECISION LEVELS 3

is converted to a single output part. The capacity of the fabrication department is defined by the number and skill of workers, the type of machinery and the layout of the system.

Within the assembly department parts are put together to subassemblies and assemblies by a mating and joining process. Assembly processes are always convergent, i.e., multiple input items are converted to a single output item. The capacity of the assembly department depends on the number and type of assemblers, the type of machines and assembly devices, the available shop floor, and the layout of the system.

The two levels 'aggregate capacity planning' and 'production scheduling' of any manufacturing system are now refined for make-to-order manufacturing systems to the three levels 'order selection', 'manufacturing planning', and 'operations scheduling'. Figure 1.2 gives the relation of Anthony's framework and the one used here in the horizontal and the way the main resources of make-to-order companies, engineering, fabrication, and assembly, are affected by the three decision levels in the vertical.

Order Selection

Manufacturing Planning

Operations Scheduling

I Engineering II Fabrication II Assembly

1 1 Facility 1 D . 1 eSlgn 1 1 _______ 1

Aggregate Capacity Planning

Production Scheduling

Figure 1.2: Managerial Levels for Make-to-Order Manufacturing Systems

The three decision levels can be detailed as follows. Order Selection. The manufacturing system is not producing for

the anonymous market but for individual customers. Hence, orders


have to be collected. Roughly this entails proposing bids to customers, rough cut capacity planning of possible orders with respect to all resources of the company, and the selection of profitable orders.

Manufacturing Planning. For given orders, all jobs to be done in the manufacturing system (fabrication and assembly) have to be properly coordinated to produce the orders for the least possible cost. In particular, raw material, parts and subassemblies have to be purchased, raw material has to be converted to parts, and parts and subassemblies have to be put together to final products.

Operations Scheduling. Once purchasing and fabrication have been initiated, detailed scheduling of the assembly operations has to take place.

Currently, there is missing coordination and integration at the three decision levels. More precisely we have the following:

At the order selection level there is neither a rough cut capacity planning of the entire order nor coordination between the planning of the order and the assembly scheduling. As a consequence, the resulting problems accumulate and become visible in the assembly as the last level in the supply chain before delivery to the customer takes place (cf. Figure 1.1). Typical problems are quality problems, late delivery of parts, wrong or unfunctional design, and changes in the course of works due to customers emerging requirements. These problems lead to late deliveries and consequently reduced revenues (cf. Bussmann [46]).

The manufacturing planning level is lacking coordination between fabrication and assembly (cf. Eversheim [99]). The consequence are late or missing parts which are delaying the assembly while, at the same time, other parts are fabricated too early and hence cause inventory holding cost.

Within the operations scheduling level the relevant interdependencies, i.e., the availability of parts, assemblers, and shop floor area (cf. Eversheim [99]), are not properly taken into account. In particular we have the following:

• Parts are pegged, i.e., a priori assigned to assemblies, and the assembly is not started before all parts (from outside supplier delivered and in-house manufactured) are available. This increases inventory holding cost and capitalized cost for additional stor-

1.2. OUTLINE OF THE BOOK 5

age space and delays the start of assemblies (cf. Eversheim [99]). Poorly coordinated fabrication and assembly causes late or missing parts (cf. Bussmann [46], Eversheim [99], Westkamper and Siinnemann [322]) and consequently interruptions of the assembly process; highly qualified assemblers have to hunt for missing parts (cf. Drexl et al. [85]) .

• Assembly planning is done without taking into account the capacity demand for assemblers. This leads to high variation of the required resource levels and thus the temporary need for overtime and subcontracting while assemblers are idle at other times (cf. Eversheim [99]).

• Also, there are no efficient tools and methods available for the spatial scheduling of project assemblies which have to be placed on the shop floor (cf. Bussman [46]). Currently the scheduling is done manually by intuition and experience of the foremen.

The deficits outlined above are widely known. Engineering and design try to remedy technical deficits by a higher degree of modularization and standardization (cf. Eversheim et al. [100]). Yet, there have not been many management science approaches to improve the situation. In particular, no approach is available which considers the coordination deficits on all three managerial levels in an hierarchically integrated way. This is the stepping stone of this book: It pursues a management science approach in order to take the relevant interdependencies of make-to-order manufacturing with high assembly content into account.

1.2 Outline of the Book

The book is structured into three main parts.

Part 1 contains two chapters. Chapter 2 addresses general issues of make-to-order manufacturing, assembly management, coordination and integration, as well as hierarchical production planning and gives case descriptions offour different make-to-order assemblies. Chapter 3 provides a classified survey of the literature in the three fields assembly


management, make-to-order manufacturing, and project scheduling. The latter area is from the modelling and methodological point of view a backbone for make-to-order assembly management.

Part 2 is concerned with the three managerial levels, viz. order selection, manufacturing planning, and operations scheduling, from a problem description and modelling point of view. An entire chapter is devoted to each managerial level. At the beginning of each chapter, we outline how the three decision levels are linked in a hierarchical top-down context. We then investigate the levels in detail. For each level we first provide an outline of the decision problem giving the managerial objectives and the inherent interdependencies. Thereby, we focus on economical rather than technical aspects. That is, whenever possible, we abstract from technical details, e.g., joining and mating techniques applied in different assemblies, in order to have a clear view on the managerial decisions. Next, we provide a focused literature review on specific topics of the decision level. Based on our problem outline and literature review, we propose a mixed-integer program (MIP) of the decision problem which is finally discussed in terms of complexity, special cases, and model extensions.

Part 3 considers solution methods for solving the MIP models developed in Part 2. Again, a chapter is devoted to each of the three managerial levels. Within each chapter we concentrate on providing straightforward methods which are versatile enough to be adapted to modifications rather than developing trimmed procedures which provide optimal performance. The rationale is that real world instances always have specific peculiarities which, by taking them into account, alter the model and thus make the performance guarantee of trimmed solution procedures obsolete or even cause these procedures to be unable to solve the modified problem. All methods are experimentally evaluated on a set of test instances. These are generated based on factorial test designs with well defined parameters representing the inherent problem structure of each model, respectively. This allows us to perform statistical analysis of the robustness of the methods as well as the influence of the problem parameters.

1.2. OUTLINE OF THE BOOK 7

Technical Considerations At the end of the introduction we detail the software systems used in the course of writing this book. Mixedinteger programs were modelled with AMPL (cf. Fourer et al. [111]) and solved with CPLEX (cf. Bixby and Boyd [31]). Algorithms were coded in GNU C or GNU C++ and run on an IBM RS 6000 workstation under AIX operating system. For statistical calculations SPSS (cf. Norusis [244]) was employed. Finally, typesetting was done with Jb.1EX (cf. Kopka [196]).

Part I

Preliminaries

Chapter 2

General Issues

This chapter is concerned with the main issues which will be of importance for the remainder of the book. We begin with the characterization of different manufacturing environments.

2.1 Make-to-Order Manufacturing

Figure 2.1 shows different 'to' manufacturing environments. We differentiate according to the point in time where the first activity takes place which is initiated by a placed customer order and not by some forecasted demand. This point, called 'customer order decoupling point' (CODP) (cf. Wouters [331]), is represented as \l in Figure 2.1.

V--MT~ ~-------ATO--------~~~

~------------MTO--------------~~ ~-------------------ETO------------------~~~

Engineering Fabrication Assembly Distribution

Figure 2.1: Manufacturing Environments

The manufacturing environment with the latest CODP is the maketo-stock (MTS) environment (cf. Balakrishnan et al. [19]). In an

12 CHAPTER 2. GENERAL ISSUES

assembly situation, the MTS environment is often termed assemble-toforecast (ATF) or assemble-in-advance (AlA) (cf. Eynan and Rosenblatt [101]). Here, the manufacturing system produces goods based on demand forecasts (cf. Tempelmeier [304] pp. 35) rather than actual orders. A customer order can then be dispatched immediately from stock. Reasons for this production strategy can be that demand can be forecasted quite accurately or that procurement and production lead time exceed the customer lead time.

In an assemble-to-order (ATO) environment, items are constructed with standard components to form a unique end item. ATO items are normally single-level items assembled with standard components and subassemblies. Major components, subassemblies, and material are held in stock awaiting the customer order. This allows some customizing within a very restricted range. Typically, customers are offered a wide variety of options on the standard set of products. Final assembly is delayed until a customer request for a specific product and option is known. A detailed survey of the implication of ATO for materials management is given by Wemmerlov [321].

In the make-to-order (MTO) case all materials and component parts have to be procured on receipt of the customer order. The customer order initiates all supply chain actions from the procurement of materials and component parts, the in-house fabrication of parts, the production of subassemblies until the final assembly. Often, the product itself or a variant of the product has been manufactured before, and hence, the bill of materials (BOM) and process plans (cf. Zapfel [338]) are available. MTO items have truly different components and attributes associated with them. Also very common to an MTO item is a multi-level characteristic. Most MTO items are built with a variety of components in a bill of materials structure that could potentially be many levels deep. A substantial time lag between the placing of an order and its acceptance is not unusual (cf. Stam and Gardiner [295]).

Finally, in the engineer-to-order (ETO) situation, items are usually manufactured in extremely small quantities, often with a lot size of one. Heavy engineering, quotation, and estimating are a very important and integral part of the process. Each order is normally handled as a job or a project. A substantial amount of engineering effort is required up front to estimate the potential cost of the finished good

2.2. ASSEMBLY MANAGEMENT 13

(cf. Hay [152]). Since each item is unique, the BOM and routing are manually constructed for the entire product. Most often, materials are purchased specifically for the project (cf. Wisner and Siferd [329]).

This book deals with manufacturing situations of the make-to-order and the engineer-to-order type.

2.2 Assembly Management

In discrete parts manufacturing systems products are processed as discrete entities. Discrete parts manufacturing systems are typical in metal-working, electric, and electronic industries. Operations involved are broadly classified in the categories fabrication and assembly. Fabrication is characterized by metal-working operations turning, drilling, milling, bending, shearing, and extrusion. The assembly process is part of the manufacturing process. The assembly is fed by the fabrication process which supplies parts which are then built together to form a complete, geometrically designed assembly or product such as a machine or an electronic circuit (cf. Nof et al. [242] p. 2). Depending on the number of levels within the process, we can differentiate single- and multi-level assembly. In the latter case parts are assembled to subassemblies, assemblies and final products. Assembly operations are characterized by two basic categories: parts mating and parts joining. In parts mating two or more parts are brought into contact or alignment to each other; parts joining means that after parts are mated, fastening is applied to hold them together (cf. Nof et al. [242] pp. 23).

The type of fastening depends on the assembly. While in electronic assembly wiring, surface mount technology, and soldering are used, in mechanical assembly fastening by screw or bolt, welding, glueing, riveting, and pressing are common (cf. Martin-Vega et al. [227]). The screw connection is one of the most frequently used connection types in the precision instrument industry, the automobile and the machine building industry (cf. Warnecke and Walther [319]). Compared to, e.g., welding the degree of automation in screwing is quite low. Riveting is, together with screwing, the most frequently applied connection technique. In aerospace manufacturing, riveting is employed most frequently (cf. Nof et al. [242] p. 60). Clinching is a joining


method combined with forming especially suited for surface-treated thin gauge sheet metal and aluminium sheet. The joint element is formed directly out of the material of the part to be joined, without the action of heat and without great power consumption (cf. Nof et al. [242] p. 72).

Assembly systems can be classified according to the facility organization. Before doing so we need to introduce the building blocks of each facility organization: assemblers, assembly tools, assembly machines and assembly stations.

• An assembler is a worker who joins parts to subassemblies, assemblies, and products. An assembler can use non-stationary tools such as screw drivers or stationary devices such as work tables with integrated part feeders.

• An assembly tool is a non-stationary device which enables or supports the assembler to perform the assembly operations. Typical assembly tools are welding machines, power-driven screw drivers, drill and riveting devices (cf. Bullen [45]).

• An assembly machine is a stationary system which, humanly controlled, performs assembly operations. Assembly objects have to be moved to the machine in order to be processed. The machine has a finite work space available for part feeders and gripper magazines.

• An assembly (work-)station is a set of assemblers and assembly machines performing manual and/or non-manual assembly (cf. N of et al. [242] p. 24).

With these building blocks we can form the four main facility organizations: assembly line, assembly cell, assembly shop, and project assembly.

• An assembly line is a stationary, unidirectional flow type production system which consists of several assembly stations. The assembly object moves in the direction determined by the line layout. Stations may be separated by finite buffers. If finite intermediate buffers exist, limited queueing between the assembly stations may occur, whereas in the line with no buffers queueing


is not allowed (cf. Sawik [269]). In assembly lines, work stations are organized unidirectionally according to the assembly sequence of the products.

A flexible assembly line (FAL), also named as mixed-model assembly line (cf. Nof et al. [242] p. 178), is a general assembly line which allows different part types to be produced simultaneously. Flexible assembly lines are typical for the assembly of cars in the automotive industry .

• An assembly cell is an assembly station with assemblers and machines assigned such that all operations needed for a complete subassembly or product can be performed.

A flexible assembly cell (FAC) , also termed flexible assembly system (FAS) or automated assembly system (cf. Sawik [267]), is an assembly station which comprises machines as well as a loading and unloading station. The machines are linked with an automated material handling system. Different product types can be assembled simultaneously. Further characteristics of an FAC are parallel processing of subassemblies as well as short processing times and hence the importance of transfer times (cf. Ho and Moodie [163]) .

• In assembly shops identical machines are grouped to stations which are linked by versatile transportation systems such as forklift trucks etc. Products have to visit the stations according to specified assembly plans. Assembly shops are common in printed circuit board assembly.

Flexible assembly systems (also called automated assembly systems, cf. Ahmadi and Tang [5]) are sets of assembly stations (single machine or identical parallel machines) and a loading/unloading station linked with an automated material handling system where different product types are assembled simultaneously. Each machine has a finite work space available for part feeders and gripper magazines (cf. Sawik [268]).

• Finally, in project assembly there are no stations with fixed positions on the shop floor. Instead, the resources, i.e., assemblers, tools and machinery, are moved to the stationary assem-


bly. This is because during the assembly process the product becomes too big to be moved. Sometimes, the final product has, after assembly and testing, to be partly disassembled in order to be shipped to the customer. This is the case, e.g., for largescale machinery. Project assembly is common for the building of ships (d. Lee et al. [212] and Subsection 2.5.1), machine tools (d. Drexl et al. [85] and Subsection 2.5.2), and subassemblies for prefabricated houses (cf. Tsubakitani and Deckro [311]). A special case of project assembly is construction project assembly where the assembly takes place at the construction site (cf. Bartusch et al. [24]) . Also, project assembly takes place within the stations (or slants) of aircraft assembly (cf. Chao and Graves [51] and Subsection 2.5.4).

The scope of this book is, as pictured in Figure 2.2, the manual fixed batch assembly with a project or shop organization of the facility.

MTS

ATO

MTO

ETO

Project Assembly

Shop Assembly

Cell Assembly

Figure 2.2: Scope of this Book

Line Assembly

Closing this section we give some further terminology relevant for describing assembly problems:

• A 'base part' is the part to which other components are added (Nof et al. [242] p. 26) .

• A 'kit' is a specific collection of parts and/or subassemblies that together (Le. , in the same container) and combined with other kits (if any) support one or more assembly operations for a given product (d. Brynzer and Johansson [44]).

2.3. COORDINATION AND INTEGRATION 17

• 'Pegging' is the task of dedicating a part to the subassembly or final product (cf. Steiner and Yeomans [298]).

2.3 Coordination and Integration

Whenever large and complex decision problems such as the management of a company are broken into smaller problems, there will be interdependencies between these subproblems. Taking them properly into account is a question of coordination and integration. The literature on coordination and integration is vast and we refrain from reviewing it. Rather, we want to give some definitions of coordination and integration as these expressions will be used in this book.

2.3.1 Interdependencies

Interdependencies are given whenever two decision problems mutually influence each other (cf. Kupper [200] p. 3). As a consequence, we cannot derive an overall optimal decision unless we take the interdependencies into account. With respect to the type of interdependencies we distinguish behavioral and factual interdependencies (cf. Kupper [200] pp. 31). Behavioral interdependencies are caused by the mutual relationship of persons while factual interdependencies are given by technical or economically reasons. We will limit our scope to factual interdependencies and will henceforth refer to them as interdependencies. They can be differentiated as follows:

• Time-based interdependencies are given whenever a decision in a period t influences the solution space in a period T > t. Examples of time-based interdependencies are, e.g., lot size decisions in single-level, single-part fabrication systems with dynamic demand (cf. Wagner and Whitin [318]). Determining the lot size in period t influences the (optimal) lot sizes in all periods T> t .

• Vertical interdependencies are given whenever technical reasons imply precedence relations between processes. An example are multi-level product structures in assembly processes where parts first have to be fabricated before they can be assembled to subassemblies and final products. Vertical interdependencies


are often crossfunctional, i.e., between different functions of the company such as sales and manufacturing or engineering and manufacturing (cf. Whang [323], Drexl et al. [88]) .

• Horizontal interdependencies exist if processes require the same capacitated resources. E.g., in job shop scheduling, two jobs are not precedence related but the operations of the jobs have to be fabricated on identical machines where each machine can only process one operation at a time (cf. Pinedo [253] pp. 125). By the joint use of identical machines, sequences of the jobs on each machine have to be determined. Horizontal interdependencies are, in general, within one function of the company (cf. Whang [323], Drexl et al. [88]).

Beside the above mentioned three interdependencies we also have goal interdependencies and risk interdependencies (cf. Kupper [200] pp. 31). Goal interdependencies are given in nonlinear objective functions where decision variables are multiplied. Risk interdependencies arise when we do not consider a deterministic but a stochastic problem with nonzero covariances. For the models and methods considered in this book we limit the scope to time-based, vertical, and horizontal interdependencies. Interdependencies necessitate either coordination or integration. We define integration as merging subproblems with their mutual interdependencies into one problem. Contrary, coordination is the separate treatment of the subproblems while taking, in whatever form, the interdependencies into account.

2.3.2 Coordination

Different approaches are available for the coordination of decisions. The classical coordination is by hierarchy (cf. Section 2.4, Kieser and Kubicek [173] pp. 96) where a superordinate level gives instructions to a subordinate level. Hierarchical coordination may be enhanced or substituted by standardization or rules, plans and schedules, mutual adjustment (cf. Thompson [310]) as well as the team concept (cf. van de Ven et al. [312]). For the specific coordination of interfaces, i.e., interfaces between functional departments such as sales and manufacturing or interfaces between projects, Brockhoff and Hauschildt [41]

2.3. COORDINATION AND INTEGRATION 19

propose a number of different means, amongst them project management (cf. Madauss [220], Meredith and Mantel [230]), market/pricemechanism, and job rotation.

2.3.3 Integration

Integration has been a major topic in recent years, both in research and practice. Supply chain management (SCM) (cf., e.g., Lee and Ng [213], Tayur et al. [303], Zapfel and Piekarz [335]) and enterprise resource planning (ERP) (cf., e.g., Olinger [246]) are new planning concepts which enlarge the scope to all relevant resources of the supply chain and the entire enterprise, respectively.

2.3.4 Coordination and Integration for Make-to-Order Assemblies

This book proposes the following approach to take the interdependencies of make-to-order assemblies into account. We employ a hierarchical framework which divides the overall decision problem into three levels. For each level we propose normative decision models and related methods which allow plans and schedules for coordination to be derived. Within each level there is, to different extents, functional integration. Id est, sales, engineering, and manufacturing are integrated at the order selection level (cf. Chapter 5) and fabrication and manufacturing are integrated at the manufacturing planning level (cf. Chapter 6). Due to the hierarchical framework, superordinate levels forward instructions to subordinate levels.

When coordinating by means of plans and schedules, a crucial point is that the underlying models, in general, consider only a part of all decision to be undertaken in acorn pany. To assure that the model takes the entire decision context of the company into account, the values of the model parameters have to be properly chosen (cf. Hax [151]). As an example, consider the operations scheduling problem (cf. Chapter 7) where subassemblies and assemblies have to be scheduled subject to different resource constraints such that the sum of the weighted tardiness is minimized. The weight assigned to each (sub-)assembly measures how important it is for the company as a whole to finish the (sub-)assembly in time.


2.4 Hierarchical Production Planning

To manage the manufacturing system of a company, a multitude of decisions have to be made: setting capacity levels, determination of the production program, deriving lot sizes, and scheduling jobs on the shop floor. A simultaneous planning approach is not possible for many reasons. First, a mixed integer program (MIP) would be far too large to be solved. Second, the cost of acquiring the data needed would be prohibitive. Third, the accuracy of the data would be insufficient. Fourth, because of a dynamic situation (cancelled orders, machine breakdowns, etc.) large portions of the solutions would be obsolete at the time they were obtained. Hence, the entire planning problem has to be broken into smaller, less complex, and thus better manageable sub-problems.

Hierarchical production planning (HPP) employs the concept of hierarchical coordination (cf. Section 2.3) in order to decompose the entire planning problem into a hierarchy of subproblems. Each subproblem should relate to a management level of the company, using data, decision variables, and objectives which are suited for the corresponding level (cf. Stadtler [293]). Originally introduced by Hax and Meal [150] for single-level flow type make-to-stock production systems, HPP has today become a planning paradigm for manufacturing management. This is most recognizable by the fact that the HPP approach has been adapted by all computerized manufacturing resource planning systems (MRP II) (cf. SchneeweiB [273]). Applications to specific types of manufacturing settings such as batch manufacturing (cf. Stadtler [292]) and flexible manufacturing systems (cf. Steven [299]) are reported in the literature. HPP approaches for make-to-order production are reported, amongst others, by Carravilla and de Sousa [49], Franck et al. [112], Neumann [240], and Schuhmacher [276]; HPP approaches for make-to-order assembly systems are due to Bussmann [46], Drexl et al. [85], and Drexl and Kolisch [87].

When decomposing the entire planning problem, interdependencies between the subproblems have to be considered. Looking at an idealized two-level decision system with a superordinate (top level) and a subordinate (base level) subproblem we find the following three types of interdependencies (cf. SchneeweiB [273]).

2.4. HIERARCHICAL PRODUCTION PLANNING 21

• Anticipation. The top level takes relevant characteristics of the base level into account (feedforward influence).

• Instruction. The top level makes decisions which influence the base level (top down influence).

• Reaction. The base level reacts to the instructions of the top level (feed back influence).

The design of a HPP relies heavily on a proper definition of the subproblems. The aim is to cut the whole problem in subproblems such that between the subproblems there are as few interdependencies as possible (cf. Kistner [181]). The majority of the HPP approaches apply the following three decision levels:

• Upper level. Timing and sizing of final production quantities of the time horizon (master schedule).

• Middle level. Determine the timing and sizing of component quantities.

• Lower level. Determination of short-term schedules and allocation of specific resources.

Once the decision levels are defined, the resulting interdependencies have to be carefully taken into account. Most important is an accurate anticipation by proper aggregation of the characteristics of the subordinate level. Typically, the time scale, resources, and products are aggregated (cf. Stadtler [293]). The time scale is aggregated such that one period of the top level, e.g., a month, covers a number of consecutive periods of the base level, e.g., 4 weeks. Products are, e.g., aggregated as follows (cf. Hax and Meal [150]): At the lowest level there are products as delivered to customers. At the next superordinate level, products are aggregated to product families, i.e., products which require the same fabrication setups. Finally, product families are aggregated to product types, i.e., group of families with similar cost and production coefficients as well as seasonal demand patterns. Resources are aggregated based on similarity indices.


2.5 Case Descriptions

In this section we describe four different types of make-to-order assemblies, namely ship assembly, machine tool assembly, assembly of synthetic fiber production lines, and airplane assembly. For each type the information available from personal visits, correspondence, and papers is organized into general information and information w.r.t. each of the three decision levels order selection, manufacturing planning, and operations scheduling. The intention is to give the reader a flavour of the type of problems encountered by make-to-order assembly management.

2.5.1 Ship Assembly

Shipbuilding is make-to-order manufacturing with flow times between 11 and 18 months for large merchant ships such as container ships and crude oil carriers (cf. Lee et al. [212]). The number of ships built differs from shipyard to shipyard. German based HDW averages 4 container ships a year while Korean based SAMSUNG assembles 40 ships per annum.

Order Selection

The customer, i.e., a shipping company, asks for a bid given a time window of about 6 weeks length. A bid entails the technical specifications, a delivery date and a price. The delivery time and the price are dependent. The capability to deliver early goes along with premium prices (cf. Harris and Pinder [143]).

The management decides with the help of so-called 'standard curves' if a delivery time proposed by the customer can be realized. A standard curve represents the cumulated amount of man-hours during the makespan of a ship. Cumulation is done in terms of time as well as different departments (engineering, fabrication, pre-assembly, final assembly) and qualification (engineers, shop floor workers, welders). The capacity requirement as given by the standard curve is subtracted from available capacity. Actions in case of capacity shortages are an earlier start of the order, rejection of the contract, overtime, and subcontracting. If the capacity demand can be met, the order could be accepted.

2.5. CASE DESCRIPTIONS 23

Next to the capacity planning, a rough estimation of the cash requirements is made. In the ship market, payments before delivery are only in the range of 10 - 20% of the contract sum. The major part of 80 - 90% is paid upon delivery. Hence, financing of the ship is common.

If orders have to be selected, this is done based on the contribution margin of the order and the reputation of the customer. For accepted orders, start and finish dates are fixed, and milestones such as finishing of engineering documents, start of fabrication, finishing of subassemblies, and end offinal assembly are determined. Careful determination of the delivery date and assignment of work to the different types of capacity is crucial for the following reasons: Subcontracting is cheaper if planned for the long term because subcontractors perform a revenue management approach where they try to keep part of their capacity with low rates busy on the long term while they reserve part of the capacity for premium prices on the short term (cf. Harris and Pinder [143]) .

Contracts include cash penalties for late delivery but concede a socalled grace window of about 3 to 4 weeks when the shipyard, although late, does not have to pay penalties. After the grace window has been exceeded substantial cash penalties are due. For a ship with a total revenue of DM 130 million the cash penalty is DM 70 thousand per day. If delivery is later than 6 weeks after the promised delivery date, the shipping company has the right to back out from the contract.


The manufacturing of a ship proceeds as follows (cf. Figure 2.3). The main task of the fabrication department is to burn steel plates of different sizes. Steel plates are then forwarded to the pre-assembly where vertical sections are welded to the steel plates in order to raise the strength of the plates. Afterwards, pre-assembled steel plates are put together to so-called blocks in the subassembly. Two types of blocks are distinguished: flat-bottomed blocks which are assembled in the panelled block assembly line and curved-bottomed blocks which are assembled in the curved block assembly shop. The assembly of flatbottomed blocks can be done automatically since there are only 90 degree angles while the curved-bottomed blocks have to be done man-


ually. Flat-bottomed and curved-bottomed blocks are assembled in the pre-erection shop to super blocks. Super blocks are then fully equipped with pipes, units etc. In the final assembly super blocks are put together in the dry dock.

5511S S~~ 53135

Figure 2.3: A Ship and its Partitioning into Blocks (courtesy HDW)

Material cost amount to about 50% of the total cost. In order to reduce material cost, parts are purchased from Asian and East-European suppliers, e.g., Diesel engines are purchased from Korea. Hatches are procured from East-European suppliers. Due to high fixed purchasing and low inventory holding cost, order sizes are large.

The following production planning problems arise at the manufacturing level.

• Within the fabrication department there is a lotsizing problem for cutting the steel plates because the machine has to be adjusted for different steel types. Types are defined by, e.g., its thickness and by its substances. Also, a 2-dimensional cutting


stock problem (cf.. e.g., Israni and Sanders [165]) arises when steel plates of given sizes have to be cut from master plates.

• Purchasing has to determine optimal order sizes due to high fixed purchasing and low inventory holding cost.

• The main problem of the subasssembly is the scarce availability of the shop fioor. Curved-bottomed blocks become quite large while being assembled but are stationary during assembly. Also, there might be limited storage space between the subassembly building and the dry docks in order to store curved- and fiatbottomed blocks.

• The final assembly faces a spatial capacity problem due to the limited number of dry docks .

• 1 , IIT1;ArtoJ,;

1j.A~ ,- t..", ..

Halle 7 Vormonlage

Figure 2.4: Placement of Curved-Bottomed Blocks on the Shop Floor (courtesy HDW)


Based on the manufacturing milestones obtained from manufacturing planning and the part supply from fabrication and outside suppliers,


the operations sequences are planned for each subassembly and final assembly. This includes detailed dynamic assignment of assemblies to the shop floor and workers to assemblies. Also the availability of parts has to be taken into account. Parts are usually pegged. The objective is to minimize overrun of milestones. In order to force suppliers to deliver in time, contracts do not specify a grace-window but penalize tardiness from the first day on. For example, a supplier of Diesel engines has to pay 6% penalty per day for a contract of DM 1 million.

2.5.2 Machine Tool Assembly

In order to stay competitive, German machine tool industry shifted from standard machinery as it is nowadays fabricated in many of the emerging economies of the world to complex machine tool systems (cf. Drexl and Kolisch [87]). This includes a high degree of customization with up to 30% order specific engineering (cf. Drexl et al. [85]).


At the manufacturing planning level, backward oriented finite scheduling of subassemblies and assemblies from the delivery date is performed in order to obtain milestones for the supply of part kits (cf. Drexl et al. [85]). The latter are partly procured and partly fabricated in-house or at plants belonging to the group. Westkamper and Siinnemann [322] report that milestones are derived by subtracting standard flow time from the delivery date. All parts have to be available at the planned start of the assembly.


At this level, detailed scheduling of operations subject to scarce resource capacity of assemblers, shop floor capacity, and part availability has to take place. The PRISMA-Leitstand allows to visualize the dynamic spatial demand of assemblies graphically and to resolve capacity conflicts manually. Part availability is taken into account via part pegging and release dates of parts (cf. Section 7.2, de Boer et al. [70]) .


2.5.3 Synthetic Fiber Production Line Assembly

NEUMAG is a medium-sized make-to-order company which produces production lines for synthetic fiber. Per year 30 to 150 machines each within a price range of DM 1 to 5 million are shipped to customers. The total annual revenue of production lines amounts to DM 150 to 180 million. The number of employees is 500. Out of these 500, 27 are within sales, 50 are within engineering, 120 are within fabrication, 80 are within in-house assembly, and 23 are within at-site assembly.

Order Selection

A bid detailing technical specification, delivery time, and price is made to the customer. The delivery times is obtained by performing a rough scheduling of the order. Thereby capacities are only taken into account for engineering and assembly while infinite capacity is assumed for fabrication. This is due to the fact that fabrication capacity can be varied due to subcontracting and acquisition offabrication orders. The obtained delivery time is forwarded to sales which is negotiating it with the customer. Depending on the outcome of the negotiating the rough planning has to be done repeatedly.

The current lead time, i.e., the time span between order acceptance and delivery to the customer, ranges between 6 and 8 months. The lead time can be roughly splitted into 2 - 3 months engineering, 3 months fabrication including subassembly, and 6 weeks final assembly.

Weekly meetings of functional department managers take place for the coordination of orders. In particular, the due dates obtained from the rough cut scheduling are negotiated between sales, engineering, fabrication, and assembly.

Manufact uring Planning

The manufacturing department comprises 5 fabrication sheds, 1 subassembly shed, and 1 final assembly shed. The 5 fabrication sheds include metal bending, welding as well as metal working (turning, milling, grinding) on computer numerically controlled (CNC) work centres. Over- and underload of the fabrication department is adjusted by subcontracting and acquisition of fabrication orders from other manufacturers.


Parts are classified according to an ABC inventory classification (cf. Hax and Candea [149] pp. 188). High-value A-parts with long lead times are fabricated in-house while low-value B- and C-parts are procured from outside suppliers. Fabrication of A- and procurement of B-parts is triggered by orders while C-parts are procured based on continuous review systems (cf. Hax and Candea [149] pp. 221). All parts have to be available at the fabrication due date calculated at the order selection level. The majority of the in-house fabricated parts can be fabricated at a single CNC work centre. That is, the fabrication process is of a single-level type. Lot sizes within fabrication are, on average, of size 10.


Operations scheduling entails the scheduling of operations for the subassembly and the final assembly. Subassembly and final assembly are placed in different sheds. Assemblers are distinguished into electric and mechanic assemblers. Parts (electric motors, roller conveyors etc.) are delivered from the fabrication and outside suppliers. Subassembly and final assembly have to be planned subject to scarce availability of assemblers and parts. Since assemblies of different orders have part commonality and parts are substitutable, assignment of available and incoming parts to assemblies is crucial.

Additionally, all final assemblies have to be placed on the shop floor where they remain stationary until the final product is finished. While being assembled, the spatial demand of the synthetic fiber production line grows; this has to be taken into account when assigning assemblies to the shop floor area. Figure 2.5 gives the placement of 10 final assemblies to the shop floor.

2.5.4 Aircraft Assembly

Airplane industry is a MTO business where every order entails a high degree of customization (cf. Greenwald [127]). The part variety is extreme high. A Boeing 747, for example, consists of about 6 million parts, of which about half are provided by suppliers from 60 different countries (d. Greenwald [127]).


I 1 II 2 18 I 4 II 5 I I 6 18 GI 9 10

Figure 2.5: Assignment of Final Assemblies to the Shop Floor

Order Selection

Selling is done by proposing bids to customers. British Airways, for example, took bids from Boeing and Airbus for 100 jets with a total value of UD $ 3.8 billion (cf. Greenwald [127]).


The assembly of airplanes is organized as a line. Hence there are no specific planning problems.


The spatial demand for the final assembly is high. The 747 assembly, for exam pIe, is done in a building with 98 acres of floor (cf. Greenwald [127]). The assembly itself is organized as a line (cf. Miese [231] p. 37) with a cycle time of about 4 days. Due to the way planes are slanted in the building, stations are called 'slants'. Each station is organized as a self-contained cell or work station with 10 assemblers on average.


Assembly is manual with the support of power tools such as drill and riveting devices (cf. Bullen [45], Chao and Graves [51]).

Figure 2.6 gives a schematic visualization of the assembly process of the Boeing 737 (cf. Greenwald [127]). The base part is the airplane fuselage which enters the line. Assemblers install wiring and insulation (1). An overhead crane lifts the fuselage to the next station where the main wing and the landing gear are assembled (2). From station (2) the airplane rolls into a slant where the vertical tail and horizontal stabilizers are fastened to the fuselage (3). In the next slant, the interior (galleys, seats, flight baggage boxes etc.) and the flight control system are assembled. In the last station, the assembly of the engines takes place (4). Afterwards, the plane enters the paint job, test flights are done, and the plane is picked up by the customer. A long flow time

1

Figure 2.6: Airplane Assembly

of up to 11 months (cf. Chao and Graves [51]) together with expensive parts and subassemblies cause high inventory holding cost. The total value added at the assembly line amounts over US $ 10 million per plane. The value added by each operation ranges between US $100 and US $1 million (cf. Rosenblatt and Lee [263]). Due to the high value, parts are ordered, received, and assembled just-in-time.

The penalties for not delivering in time are significant and thus airplane manufacturers do everything to meet the due dates. If the operations to be done within a work centre are not finished within the cylce time, the plane moves to the next slant and an extra crew is performing operations not yet done (so-called 'traveler'-jobs) in parallel with the regular operations. Traveler-jobs are about five times as ex-


finished work, coordination problems with the assemblers assigned to the regular operations, and additional labor expenses for overtime. In 1997, Boeing had US $ 1.6 billion charges relating to manufacturing problems, many of them due to lack of parts (cf. Greenwald [127]).

Chapter 3

Literature Survey and Classification

This chapter provides a general survey of literature related to management of make-to-order assemblies. Particular models and methods suited for the decision problems to be addressed in Part II will not be given in detail here but in the relevant chapters of Part II. Rather, we focus on the main streams of research in the three areas assembly management, make-to-order manufacturing, and multiproject scheduling. The relevance of the first two areas is obvious. The relevance of multi-project scheduling will be clarified below. Each of these areas has a lively and intense research. Giving a complete review of the literature would hence go beyond the scope of this book. We therefore point out the main research topics pursued in each area.

3.1 Assembly Management

We consider a contribution relevant for assembly management if it takes into account the mating and joining process of parts to (sub) assemblies. Despite this restriction there is still a vast number and variety of publications dealing with assembly management. Particular problems as assembly line balancing (ALB) problems (cf., e.g., Ghosh and Gagnon [120], Scholl [274]) or printed circuit board assembly (PCBA) problems (cf., e.g., Askin et al. [12}) are very well known areas of intensive and fruitful research. In order to give a survey we

34 CHAPTER 3. LITERATURE SURVEY

Decision Facility Organization Level Line I Shop Project

Design Design for Assembly

System Design and Evaluation

Production Multi-Level Lotsizing Planning

I Loadmg II Lme Loadmg I Shop Loadmg Sequencing Line Sequencing FAC Sequencing

Scheduling of Scheduling of Scheduling Line Scheduling Electronic Project

Assemblies Assemblies

Table 3.1: Taxonomy of Assembly Literature

depart from considering specific problems such as ALB and PCBA. Instead, we classify the literature according to the type of facility organization and the range of the decisions to be made. With respect to the facility organization we distinguish between i) line, ii) shop, and iii) project (cf. Section 2.2). With respect to the range of the decisions to be made we distinguish between i) design, ii) production planning, iii) loading, as well as iv) sequencing and scheduling. Within the framework of Anthony [9] (cf. Section 1.1), the first two decision levels correspond to facility design and aggregate capacity planning while the last two decision levels correspond to production scheduling. Table 3.1 provides the taxonomy.

3.1.1 Design for Assembly

Manufacturing cost in general and assembly cost in particular are to a great extent determined during the design phase (cf., e.g., Friedl [114]). Consequently, thorough assembly-oriented design of the product and the method of assembly are needed (cf. Nof et a1. [242], p. 84). This holds particularly for the assembly of high volume products present in facility organizations with flow line and job shop configuration. Design for assembly (DFA) approaches can be distinguished into

• Product-oriented DFA which comprises the engineering of the individual parts as well as the way they are joined together to form


the final product (cf., e.g., IPA [113], Joines and Ayoub [167]) including the generation and selection of assembly sequences (cf., e.g., Naphade et al. [238,239], Schmidt and Jackman [272]) .

• System-oriented DFA approaches are concerned with the longterm decisions of how to build up an assembly system. We will review this topic in detail below .

• Finally, integrated DFA approaches try to design the product and the system simultaneously (cf., e.g., Dh et al. [245]).

As outlined above, system-oriented DFA is concerned with the longrange decision of how to build up an assembly system. This includes the decision about the location of the system, the facility organization (line, shop, project), and the capacity of the system. With respect to the type of data employed, one can distinguish the design phase and the evaluation phase. The design phase employs mixed integer programming models together with, in general, deterministic data in order to derive the configuration of an assembly system ( decision variables) whereas the evaluation phase uses a given configuration and stochastic data (e.g., processing time distributions, machine failure distributions) in order to derive performance measures such as system throughput.

Design of Deterministic Assembly Systems Broad design questions address the configuration of multi-facility global manufacturing assembly and distribution networks (cf. Taylor [302]). A classification of design questions and a family of hierarchical decision models for single-facility design problems is given by Pinnoi and Wilhelm [254].

The most researched question for single-facility design is the flow line design problem including the classical single-model simple assembly line balancing problem (SALB) where assembly operations of a single product have to be assigned to a number of consecutive stations. The objective is to minimize the number of stations for a given cycle time or to minimize the cycle time for a predetermined number of stations (cf. Scholl [274] pp. 23). Rosenblatt and Lee [263] maximize the net revenues minus inventory holding cost and fixed line cost. This objective is important in the aerospace industry where there are high inventory holding cost due to long cycle times and expensive components such as engines and electronic parts (cf. Subsection 2.5.4).


Extensions of SALB include multi models, stochastic processing times of operations, and additional restrictions or factors such as parallel stations and zoning restrictions (cf. Ghosh and Gagnon [120], van Zante-de Fokkert and de Kok [315]).

The design of an automated assembly system incorporates the simultaneous selections of work stations and assignment of operations to stations such that given demand is satisfied and total system cost comprised by fixed station cost and variable operating cost are minimized (cf. Graves and Lamar [126]).

Evaluation of Stochastic Assembly Systems

Evaluation of stochastic manufacturing systems employs either mathematical analysis or simulation in order to derive performance measures of existing systems with stochastic data. Recent research is documented by Baker et al. [18], Di Mascolo et al. [76], Gershwin [119], Helber [155], Rao and Suri [258, 259], Tang [301], and Wilhelm et al. [327].

3.1.2 Production Planning for Assembly Systems

Production planning for assembly systems is concerned with the determination of production and inventory quantities of multi-level production systems given a dynamic demand for final assemblies and subassemblies. This problem is commonly known as multi-level lotsizing. A distinction is made whether setup cost and/or time have to be taken into account. In case of no setups classical multi-level (continuous) production planning (cf. Zapfel [336] pp. 91) arises, while in the case of setups we are entering the field of multi-level lotsizing. Early work considers the uncapacitated case (cf., e.g., Crowston and coauthors [64, 63, 65], Afentakis et al. [3]) while recent publications treat the capacitated case. For an extensive survey of relevant work we confer to Derstroff [75], Helber [153], Kimms [177] as well as Simpson and Erenguc [283].


3.1.3 Loading and Release Planning for Assembly Systems

Loading and release planning for assembly is concerned with the assignment of available assembly orders to assembly capacity. Assembly orders are available if their release dates have been reached and their component kits are available. Assembly capacity can be the period capacity of an entire assembly system or of a subsystem such as a single machine. With respect to the facility organization we can distinguish loading and release problems for assembly lines and assembly shops, respectively.

Loading and Release Planning for Assembly Lines

Work on loading and release planning for assembly lines has been, amongst others, performed by Garetti et al. [116], Kartihikeyan and Krishnaswamy [170], and van der Waart et al. [314].

Loading and Release Planning for Assembly Shops

Work in the area of loading and release planning for assembly shops is mainly concerned with PCB assembly (cf., e.g., the literature survey given by Ammons et al. [8] as well as the work of Gunther and coauthors [135, 134, 133]) and loading of FAS (cf. Sawik [269, 267, 268]).

3.1.4 Sequencing and Scheduling of Assembly Systems

Sequencing and scheduling decisions are concerned with the detailed allocation of operations to resources which might be machines or assemblers. If, due to the available capacity, not more than one operation can be processed at a time, and the performance measure is regular (cf. Definitions 6.1 and 7.1) it suffices to determine the sequence of operations. Otherwise, a schedule which details a finish time for each operation has to be obtained.

Sequencing and Scheduling of Assembly Lines

Sequencing decisions within assembly lines arise for flexible, i.e., mixedmodel, assembly lines. The problem is to find an intermixed sequence of model units which optimizes a performance measure while satisfying


the demands of all models. Typical measures are time or cost related. A survey of the decision problems arising in this context can be found in Domschke et al. [84] pp. 181, Sawik [269], and Scholl [274] pp. 98.

Recently, the problem of 'level scheduling' of the final assembly lines in the automotive industry has attracted much research effort (cf. Aigbedo and Monden [6] for a recent survey). Level scheduling is concerned with finding a sequence of models which smoothes the part demand and/or levels the product load on each station of the line.

Scheduling decision in assembly lines are, amongst others, addressed by Hariri and Potts [142], Lee et al. [211], Maimon et al. [224], Potts et al. [255], and Righter [260].

Sequencing and Scheduling of Assembly Shops

The vast amount of literature in this field can be roughly divided into sequencing of FAC and the scheduling of electronic assemblies.

The FAC sequencing problem can be described as follows. Given is an assignment of components to the workstations such that each component is exactly available at one workstation. Find a job sequence for each workstation such that the maximum of the consecutively visited workstations of all sequences is minimized (cf., e.g., Lofgren et al. [215]).

In electronic assembly, a job represents a batch of circuit boards. A job comprises several fabrication and assembly operations where each operation has to be processed on a fabrication or assembly station. Objectives are meeting due dates and throughput maximization (cf. Feo et al. [107, 108]) as well as the minimization of inventory holding and tardiness cost (cf. Faaland and Schmitt [102, 103]). Variants of this problem are addressed by many authors. Confer, e.g., Askin et al. [12], Cala et al. [47], and Ahmadi and Tang [5]. Performance evaluations of priority rules for dynamic multi-level fabrication/assembly shops have been undertaken by Goodwin and Goodwin [125], Russell and Taylor [264], as well as Lalsare and Sen [204]. Typical performance measures are mean and standard deviation of flow time, earliness, and tardiness.


Sched uling of Project Assemblies

Scheduling of project assemblies is characterized by the fact that multiple assemblies, each comprising a set of precedence related operations, have to be scheduled subject to three different types of constraints, namely capacity constraints of resources (such as assemblers, tools, and machinery), part availability, and shop floor availability constraints. Objectives are time and cost oriented. Project assembly approaches documented in the literature· are mainly concerned with machine tool manufacturing. An overview of the scheduling problems in this industry is given by Konz [195]. Agrawal et al. [4] and Anwar and Nagi [10] model the problem of just-in-time scheduling of multiple largescale batch assemblies. Anwar and Nagi [10] extend this to scheduling and batching decisions. Drexl et al. [85] and Drexl and Kolisch [87] report about a Leitstand-system which, beside taking into account precedence and resource constraints, comprises a dynamic layout component and an ordering component. The dynamic layout component enables the user to allocate assembly area supply to demand. The ordering component orders critical parts, i.e., parts which have caused delays in the past, and parts with high inventory holding cost, according to the start times of the assembly schedule. Details of the ordering component are provided by Grempe and Saretz [128]. Schlauch and Ley [271] consider the scheduling of make-to-:-order machine tool assembly subject to a dynamic shop floor demand. An application of project assembly models and methods to a scheduling problem in electronic assembly is documented in Chen and Wilhelm [52, 53]. Special emphasis is given to part kitting. The goal is to schedule the operations of customer orders such that inventory holding and tardiness cost are minimized. An extension of this problem which takes into account part substitutes is treated in Chen and Wilhelm [54].

Pick and Place Sequences

A special sequencing problem within the assembly of printed circuit boards (PCB) is to :find optimal insertion sequences (cf. Ball and Magazine [20]). Objectives are, e.g., the shortest insertion time or the shortest total travelling distance. Problems can be distinguished with regard to the fact if components have already been assigned to racks


or feeders or not. In the latter case, a simultaneous assignment and sequencing problem arises (cf. Bard et al. [22], Foulds and Hamacher [110]).

3.1.5 Miscellaneous

Disassembly

The optimal disassembly of products is an expanding area of research. The above introduced taxonomy can be employed straightforward for disassembly problems. For example, rating schemes to evaluate the ease of product disassembly (cf. Kroll et al. [197]) fall under the category DFA, disassembly programming (cf. Spengler and Rentz [288]) is a multi-level production planning problem, and the problem of generating optimal sequences for disassembly (cf. Lambert [205]) is a problem of sequencing and scheduling .

. Technical Aspects of Assembly

Technical aspects of assembly are concerned, e.g., with correctly orienting parts in automatic assembly (cf. Logendran and Sriskandarajah [216]) or the maximization of product's robustness by appropriately allocating assembly and machine tolerances (cf. Zhang and Wang [339]).

3.2 Make-to-Order Manufacturing

The literature on make-to-order manufacturing can be roughly divided into scheduling models and methodologies based on project networks and planning approaches appropriate for job shop type manufacturing.

3.2.1 Make-To-Order Manufacturing with Project Networks

The relevance of project network techniques for MTO manufacturing is, amongst many other papers, stressed in Harhalakis [141]. Table 3.2 gives a survey of the similarities between MTO manufacturing and project networks.

3.2. MAKE-TO-ORDER MANUFACTURING 41

MTO Manufacturing Project Networks order t+ project operation t+ activity bill of materials t+ precedence network machine t+ resource release date t+ minimal time lag due date t+ maximal time lag

Table 3.2: Relationship of MTO Manufacturing and Project Networks

Gunther [132] provides a methodological framework for job scheduling with operation overlapping considering fabrication as well as assembly operations. Demeulemeester and Herroelen [74] extend this line of research by taking additionally machine capacity into account. In particular, they address techniques for modelling setup times as well as process and transfer batches by using minimal time lags.

Neumann and Schwindt [241] show how make-to-order production with limited resources can be modelled as projects with minimal and maximal time lags. Amongst other topics they treat the so-called 'assignment sequencing problem' where the parts emerging from a fabrication operation have to be assigned to assembly operations such that the total makespan is minimized. Franck et al. [112] propose a capacity-oriented hierarchical approach to single-item and smallbatch production planning for make-to-order production which comprises three levels. A hierarchical scheduling system especially suited for shipbuilding which takes the shop floor as scarce resource into account is developed by Lee et al. [212].

3.2.2 Production Planning for Job Shops

A survey of production planning issues arising in make-to-order manufacturing is provided by Hendry and Kingsman [158]. Topics are the setting of delivery dates, capacity planning and control, hierarchical production planning, integration of sales and manufacturing, and the appropriateness of well known production planning systems such as manufacturing resource planning (MRP II), just-in-time (JIT), and


optimized production technology (OPT) for the specific needs of maketo-order manufacturing. An approach for managing manufacturing lead times in make-to-order companies is proposed by Kingsman et al. [178]. It consist of two levels. At the first level customer orders are selected and delivery dates are assigned (cf. Hendry and Kingsman [160]). The release of jobs to the shop floor is part of the second level (cf. Hendry et al. [159,157,161]). A production planning and loading problem for MTO manufacturing with flow lines is given by Markland et al. [225]. Here, products of multiple orders have to be assigned to time periods such that the production of all products belonging to one order is synchronized, i.e., products are produced close to an order, the lateness of order delivery is minimized, and the number of orders not produced is minimized.

3.3 Multi-Project Scheduling

It has already been mentioned that a considerable part of the MTO literature employs project network models and methods. Additionally, assembly planning problems are naturally represented by project networks. Hence, project scheduling models and methods are a backbone for management of MTO assemblies. Recent reviews of project scheduling models and methods can be found in Brucker et al. [43] as well as Kolisch and Padman [191]. Here, we want to give a short survey of the literature in the field of multi-project scheduling which has not been under investigation in other reviews. The specific importance of multi-project scheduling stems from the fact that in MTO assemblies usually many different assemblies have to be planned simultaneously. Each assembly can be depicted by a single project. Table 3.3 shows the other analogies.

The essence of multi-project scheduling is to assign start times to the activities of multiple projects such that some performance measure is optimized while precedence constraints among the activities as well as resource constraints are taken into account. Performance measures treated in the literature are related to time, cost, resources, and project value. The use of time measures is most common. Denoting with wP'

Tp, Lp, and Fp weight, tardiness, lateness, and flow time of project p,

respectively, we have the following times measures:

3.3. MULTI-PROJECT SCHEDULING 43

Project Assembly Multi-Project Scheduling assembly +7 project job / operation +7 activity assembly graph +7 precedence network assembler, tool +7 resource arrival of dedicated part +7 release time due/delivery date +7 milestone

Table 3.3: Relationship of MTO Assembly and Multi-Project Scheduling

• minimization of the sum of the total tardiness of all projects (2:: Tp),

• minimization of the sum of the weighted tardiness of all projects (2:: wp . Tp),

• minimization of the sum of the total lateness of all projects (2:: Lp),

• minimization of the sum of the weighted lateness of all projects (2:: wp . Lp),

• minimization of the maximum total fiowtime of all projects (Fmax)' and

• minimization of the sum of the total fiowtime of all projects (2:: Fp).

Table 3.4 gives a survey of the multi-project scheduling literature classified according to the pursued objectives. Beside the time measures listed above, there are measures of cost (Cost), resource usage (Res), and project value (Val).

The application of cost is a general measure which includes penalty cost for finishing a project after the due date, overtime cost for purchasing additional capacity, and holding cost which account the opportunity cost for capital tied up by the project. Relevant resource measures are the minimization of the variance of the resource demand,


the minimization of the sum of the total resource idleness, and the minimization of the sum of the weighted resource idleness. Finally, a measure of the project value is the maximization of the net present value.

Reference Objective Bock and Patterson [34] : L wp . L p, L Lp Bowers et al. [36] : Fmax

Chen [55] : L Fp , Cost Deckro et al. [72] : L Fp Kapuscinska et al. [168] : L wp . Tp Kim and Leachman [174] : L Lp Kim and Schniederjans [175] : L wp . Tp, Res, Cost Kurtulus and Davis [201] : LTp Kurtulus and Narula [202] : LTp, LWp .Tp Lawrence and Morton [210] : L wp . Tp Lee et al. [214] : Fmax

Lova et al. [218] : L Lp , Res Moccellin [232] : L wp . Tp Mohanty and Siddiq [233, 234] : L wp . Tp, L Tp, Res Norbis and Smith [243] : F max , L Fp Patterson [248] : L wp . Tp, LTp, Res Patterson [249] : L wp . Tp, LTp, Fmax

Prittsker et al. [256] : L wp . L p, L Lp, F max , L Fp Speranza and Vercellis [289] : F max , Val Tsubikatini and Deckro [311] : Fmax

Vercellis [316] Val Weiss [320] Cost Wiley et al. [326] Cost Woodworth and Willie [330] Res Yang and Sum [332, 333] : LTp, Fmax

Table 3.4: Literature Survey of Multi-Project Scheduling

Part II

Decision Models

Chapter 4

Hierarchical Framework

In this chapter we detail the 3-level approach for managing make-toorder manufacturing systems. As indicated by 'framework' it is not a fully-fledged hierarchical approach as, e.g., presented in Hax and Meal [150] and in Stadtler [292]. Rather, it is the attempt to divide the managerial problems apparent in make-to-order manufacturing systems into three quite general models and to connect these models in a hierarchical way. Depending on the practical situation, only one, two, or all three models can be applied. Possible refinements of the hierarchical approach are given in Chapter II.

Table 4.1 and Figure 4.1 specify the three decision levels. On each level the planning objects are represented as activities in a precedence network. Anticipation is done in two ways. First, by considering aggregate figures for resource demand and supply. Second, by taking minimal time lags between activities into account.

4.1 Order Selection Level

Objects of the first level are potential customer orders which are broken down into tasks. Each task relates to specific work which has to be performed by a department of the company. Example given, in Figure 4.1 we have the tasks engineering (E), procurement (p), fabrication (F), and assembly (A). Tasks of one order are related by minimal time

lags which depict, e.g., the lead time of suppliers (cf. P-4A). Resources depict functional departments of the company such as

48 CHAPTER 4. HIERARCHICAL FRAMEWORK

Decision Level

Order Manufacturing Operations Selection Planning Scheduling

maximize minimize minimize objective contribution cost tardiness

margin objects orders manufacturing (sub-) assembly

tasks jobs

t t t subobjects tasks (sub-) assembly operations

jobs assemblers, assemblers,

resources departments work centres, tools, shop floor shop floor

horizon year(s) month(s) week(s)

t t t period week(s) day(s) shift(s)

order fabrication operations schedule

decisions acceptance, lot sizes , resource assignment,

task sched ule job schedule spatial assignment, part assignment

Table 4.1: Hierarchical Framework for Make-to-Order Assemblies

design, fabrication, and assembly. To allow such highly aggregated resources, the functionality within one resource has to be homogenous.

The length of the planning horizon is determined by the latest delivery date of all orders under negotiation. The period length is 1 - 2 weeks.

Decisions to be undertaken are the acceptance of orders and the scheduling of tasks.· The objective is to maximize the contribution of accepted orders.

4.2. MANUFACTURING PLANNING LEVEL 49

4.2 Manufacturing Planning Level

At the second level we take into account the manufacturing tasks, i.e., fabrication and assembly. Each assembly task and its due date is an instruction from the order acceptance level.

Assembly tasks are refined into assembly and subassembly jobs. A job details a Leontief production function which defines how assemblies and subassemblies are produced using different resources, parts and subassemblies (cf. Fandel [104] pp. 90, Schweitzer [278] pp. 581). All jobs required to obtain a final assembly are interrelated by precedence constraints which depict technically required assembly sequences (cf. Nof et al. [242] p. 28).

Resources are assemblers, the shop floor area, and work centres for producing parts. Assemblers are distinguished according to rough qualification criteria such as electronic and mechanic assemblers.

The length of the planning horizon is set to the latest due date of all assemblies. The period length varies between a day and a week.

Decisions to be undertaken are a production plan for fabrication and a schedule for assembly jobs. The objective is to minimize holding and setup cost within the manufacturing department.

4.3 Operations Scheduling Level

The last decision level receives milestones for assemblies and final assemblies as well as the release date of parts as instruction from the manufacturing level.

The assembly and subassembly jobs are exploded into operations. Each operation is characterized by a resource and a part demand (the latter is represented by the squares in Figure 4.1) while assemblies are characterized by a shop floor demand. This implies that the time span the shop floor is required by an assembly is not known a priori. Precedence relations between operations depict technically required sequences.

Resources are types of assemblers distinguished w.r.t. specific qualifications, e.g., welder and shipbuilder in the ship assembly (cf. Subsection 2.5.1), tools, and the shop floor.

The length of the planning horizon varies between a week and a

50 CHAPTER 4. HIERARCHICAL FRAMEWORK

month and is divided in periods which represent multiples of a shift. On the last level it has to be decided on the assignment of assem

blies to the shop floor, the start times of operations, an assignment of resources to operations, and the assignment of parts to operations. The objective is to minimize the weighted tardiness of assemblies and subassemblies.

The hierarchical framework will now be detailed in Chapters 5 -10. For each level Chapters 5 - 7 will give the decision context and propose a MIP model while Chapters 8 - 10 are concerned with the proposal and assessment of solution procedures.

4.3. OPERATIONS SCHEDULING LEVEL

./' ./'

./' ./'

". ".

".

./' ./'

./'

,. ./' ,.

./' ./'

5 ,/ ".

". ".

" ,.. ,..

\

Order Selection

\ \


\ \

\ \

\ \

\ \

\ \

\ \

\

\ \

\ \

\ \

\ \

\ \

\

51

Figure 4.1: Objects and Decision Levels of the Hierarchical Framework

Chapter 5

Order Selection

5.1 Outline of the Problem

We assume a simultaneous order selection process where P :2: 0 orders have been accumulated over a specific decision period (e.g., 1 week or 1 month) (cf. Starn and Gardiner [295]). Each order p (p = 1, ... , P) is characterized by a value (such as the profit or the contribution margin), a time window where, if accepted, the delivery of the order has to take place, and a set of precedence related tasks which have to be accomplished to complete the order. Each task relates to a specific work package which has to be performed by one department of the company, e.g., engineering, fabrication, or assembly.

The general decision problem is as follows. Which orders should be accepted and how should the tasks of accepted orders be assigned to departments over time such that interdependencies are taken into account and some performance measure is optimized? In what follows we want to clarify the performance measure and the interdependencies.

5.1.1 Performance Measure

In general, the objective of the order or project selection problem is the maximization of the sum of the values associated with accepted orders or projects (cf. Subsection 5.2.3). In this context, 'value' stands for a general performance measure such as the profit (cf. Friedman [115]), the contribution margin (cf. Czeranowsky [66], Haase and Latteier [138], Jacob [166], Starn and Gardiner [295]), the (discounted) cash

54 CHAPTER 5. ORDER SELECTION

flow (cf. Aspvall et al. [13]), or another measure of commercial importance (cf. Hall and Magazine [140]). The considered value should relate to the decision context. For example, if the company wants to gain market share or wants to enter a new market, the revenue rather than the profit is a suited measure (cf. Friedman [115]). In general, the contribution margin is a proper measure. But employing the contribution margin, it should be taken into account that the pre-selected order alternatives, i.e., orders which are considered for the selection process, do have a revenue which exceeds the estimated production cost (cf. Hay [152]). An exception may hold when a company wants to come into business by obtaining a reference customer (cf. Friedman [115]).


In ETO- and MTO-companies customer orders have to be processed by different departments such as engineering, procurement, fabrication, and assembly. The capacity demand of each order in each department is, compared to the available capacity, in general high. On behalf of these two points there is a strong 'interconnectedness' amongst the orders and the functions. Between the orders we have the following three types of interdependencies (cf. Section 2.3).

Time-based interdependencies arise from the fact that an order acceptance decision now withdraws capacity in future periods and thus affects order acceptance decisions to come.

Vertical interdependencies are caused by the precedence relations amongst the tasks. Example given, the engineering task has to be accomplished before the fabrication task can begin.

Finally, horizontal interdependencies are due to the fact that tasks of different orders need capacity from the same department. For example, two different orders have to be processed by the engineering department.

5.2 Literature Review

The order selection problem outlined above comprises four prominent research areas: due date assignment, project bidding, integrated

5.2. LITERATURE REVIEW 55

project selection and scheduling, and revenue management. An additional area of research is the interconnectedness and coordination of company functions in general. For example the interface between marketing and research & development (cf., e.g., Brockhoff [40]), between marketing and manufacturing (cf., e.g., Erens and Hegge [97], Hahn et al. [139], Kingsman et al. [179], Lawrence [209], Whybark [324], Whybark and Wijngaard [325]), and between engineering and manufacturing (cf., e.g., Adler [2]).

5.2.1 Due Date Assignment

Due date assignment is dealing with the problem of giving a due date to each job about to enter the production system. The due date is a promise to the customer and an information for the scheduling system. Due date assignment problems can be classified according to the type of production system (single machine, job shop, project), the data situation (static, dynamic) and the way due dates are assigned (external, internal). The data situation is static if all jobs are available for processing at one starting time. This coincides with sequential order selection (cf. Section 5.3). In the dynamic case, jobs continually enter the system in a random manner governed by some probabilistic laws (cf. Cheng and Gupta [58]). Due dates are assigned externally when the due date is fixed upon the arrival of the job. Due dates are assigned internally when they are set by manufacturing. Performance measures for due date assignment problems are time-based, e.g., the sum of weighted tardiness or lateness (Cheng and Gupta [58]), costbased (cf. Philipoom et al. [252]), and revenue-based (cf. Duenyas and Hopp [89]). The number of publications within this field is vast. A review of due date assignment problems is given by Cheng and Gupta [58]. Table 5.1 points out some literature for the case of internal due date assignment.

5.2.2 Competitive Bidding

A bid is a commitment by the company to deliver a technically specified product at a particular point in time for a given price (cf. EImaghraby [95]). In so far, a bid is a generalized due date assignment. There are two kinds of bidding situations (cf. Friedman [115]): open


Data Type of Production System

Project Job Shop Machine

Static [34], [334] [26], [257], [219], [247] [17], [69], [68] Dynamic [90] [1], [92], [252] [17], [89]

Table 5.1: Selected Literature on Internal Due Date Assignment

bidding, i.e., auction, and closed bidding. In closed bidding two or more companies submit independent bids. Each company can only submit one bid and the bid with the lowest price is accepted. Extensions take into consideration that the acceptance probability may be greater zero for bids with a price exceeding the lowest competitor price.

In an ETO- and MTO-environment bidding is not uncommon. In making the bidding decision, the company is faced with a trade-off between potential revenue and the chance of obtaining the order (cf. Goodman and Baurmeister [124]). Assuming a risk neutral decision maker, the general idea of competitive bidding models is to calculate for each price an expected contribution margin of the order. The calculation takes the following information into account: a vector of prices submitted by competitors, for each competitor price a probability that this price will be proposed, and a conditional probability that the bid of the company will be accepted when the competitor price is proposed. Examples are provided by Edelman [91] and KuB [203]. Capacity constraints, if considered, are highly aggregated (cf. Goodman and Baurmeister [124], Stark and Mayer [297]). A literature review is given in Berndt [25] pp. 133 and Stark [296].

5.2.3 Project Selection and Scheduling

Project selection and scheduling considers a set of projects which have to be selected and whose activities have to be scheduled such that some measure of performance is optimized. This general context fits to a variety of different problems. Projects might be courses at a technical training centre (cf. Haase and Latteier [138]), space shuttle flights (cf. Hall and Magazine [140]), the extraction of petroleum deposits (cf. As-


pvall et al. [13]), research and development (R&D) projects (cf. Coffin and Taylor [61]), or manufacturing projects (cf. Czeranowsky [66]). Projects may comprise a set of activities with general precedence constraints (cf. Czeranowsky [66]), serial precedence constraints (Haase and Latteier [138]), or a single activity (cf. Hall and Magazine [140]). The performance measure may be the maximization of selected project weights (cf. Hall and Magazine [140]), the maximization of contribution margins of the selected projects (cf. Haase and Latteier [138], Czeranowsky [66]), or the net present value of selected projects (cf. Aspvall et al. [13]). Multiple objectives are proposed by Stam and Gardiner [295], Mukherjee and Bera [235], Coffin and Taylor [61].

All approaches make simplifying assumptions in order to solve the rather intricated problem. One is that only one order can be processed at a time (cf. Aspvall et al. [13], Coffin and Taylor [61], Hall and Magazine [140]). In this case the scheduling decision boils down to a sequencing decision. Another one is to generate a single schedule or a restricted set of schedules a priori instead of considering all possible schedules of each project (cf. Czeranowsky [66], Luss and Rosenwein [219]). This prevents the combinatorial explosion on the scheduling level. Finally, the dynamic (multi-period) situation may be aggregated to a static (single-period) situation which drastically reduces the number of capacity constraints (cf. Coffin and Taylor [61]).

None of the approaches for project selection and scheduling presented in the literature treats the most general case, where multiple projects comprising a set of precedence related activities have to be selected and activities have to be scheduled under multi-resource and multi-period capacity constraints.

5.2.4 Revenue Management

Revenue management is considering the problem of how much of the available capacity should be assigned to available orders and how much should be reserved for orders to come. The idea behind reserving capacity for orders to come is that customers needing orders with a close due date are price insensitive while customers able to place their orders in advance are price sensitive. So the question is how much capacity should be reserved for those late-arrival high-margin orders. A revenue management approach for an ATO-environment is proposed


by Harris and Pinder [143], a model for the ETO/MTO-case is given by Jacob [166]. There, a static (one-period) model is considered where a set of orders has to be selected such that the sum of the contribution margins is maximized. Each order requires capacity from different resources. The available capacity of each resource is divided into free capacity and capacity reserved for future orders. The use of reserved capacity is taken into account in the objective function via penalty cost. Together with the order acceptance decision it is decided how much the order uses free capacity and reserved capacity, respectively. By varying the penalty cost of reserved capacity it can be adjusted how much capacity is left for late-arrival high-margin orders.

5.3 Model

The above given literature review revealed that currently there is no model which is suited to cover order selection, due date setting, and rough cut capacity planning of involved departments in an integrated fashion. Hence, in what follows we present a decision model which stems from research documented in Kolisch [187].

5.S.1 Detailed Description and Notation

Before we can develop the model, we have to describe the order selection problem in more detail and we have to introduce the relevant notation. A survey of the notation can be found in Appendix B.l.

We assume a so-called 'sequential' order selection process where orders are accumulated over a specific decision period, e.g., 1 week or 1 month (cf. Starn and Gardiner [295]). Let P ~ 0 denote the number of accumulated customer orders. Without loss of generality we can assume that each customer order p (p = 1, ... , P) is associated with a single customer. The expectations of customer p w.r.t. technical specification, time, and cost of the order are depicted as follows.

Customer order p is associated with a value vp which can be the price the customer is willing to pay and thus the revenue for the company or, after subtracting all variable cost, the contribution margin of the order. The customer wants the order p to be delivered within the time window [4p , dp] where 4p ~ 0 denotes the earliest delivery time and

5.3. MODEL 59

dp ~ d.p denotes the latest delivery time. To process the orders, the company has R ~ 0 different resources.

With 'R = {I, ... , R} we denote the set of all resources. Each resource r E 'R depicts a functional department of the company such as engineering, fabrication, or assembly. In order to allow such highly aggregated resources, the functionality within one resource has to be homogenous. That is, e.g., all engineers must be generally able to work on all tasks requiring engineering effort. If the qualification of engineers is different such that, e.g., in machine tool companies, there are engineers for mechanic parts and electronic components, respectively, we have to depict each qualification by a separate resource type. The availability of department r in period t (t = 1, ... , T) is Gr,t ~ o. The unit of the available capacity is generally available man-hours or machine-hours per period. The time-varying capacity results from capacity reservations of already accepted customer orders, planned machine shut downs for inspection, planned vacations of assemblers, qualification courses of engineers etc.

Due to the technical specifications, order P can be split into precedence related tasks. Each task relates to specific work which has to be performed by a department of the company. Example given, consider Figure 4.1 where an order comprises the five tasks, dummy start, E, P, F, and A. The latter four tasks represent engineering (E), procurement (P), fabrication (F), and assembly (A). Let denote Jp the set of tasks of order p. Task j (j E Jp; P = 1, ... , P) has to be processed by the single department r j E 'R. The work content is defined in terms of the processing time Pj ~ 0 and the resource demand Cj,T ~ 0 for each period T (T = 1, ... , Pi) of the processing time Pi.

5.3.2 Network Representation

The technical specifications imply precedence relations among the tasks. With Pi we denote the set of immediate predecessors of task j. Without loss of generality. we assume for each order P a single start and a single terminal task which are denoted as 8 p and ep , respectively. Furthermore, we consecutively number the tasks of one order such that we have for order P tasks 8 p , ••• , ep and that the label of a task is always higher than any of its predecessor tasks. We assign the start task of the first order 81 the labell, the start task of the second order 82 the


label el + 1 etc. This way, the label of the end task of the last order ep accounts the overall number of tasks which is denoted with J.

Between the start time of task j and the finish time of each of its immediate predecessors i E Pj we have a'minimal time lag of tr:jn E Z. Figure 5.1 gives an example network with one order. Tasks are depicted by nodes and precedence relations between nodes are depicted by arcs. The weights on the arcs represent the time lags where weights of 0 are omitted.

Figure 5.1: Example Network of the Order Selection Level

Let us analyze the precedence relations more. detailed. Precedence relation P~A models a positive minimal time lag between the procurement and the assembly task. That is, after the order has been shipped to the supplier (task P), a minimal time span of 5 periods has to take place before the assembly (task A) can start. This time span accounts for the delivery time of the supplier.

Precedence relation E~F models a negative minimal time lag between the engineering and the fabrication task. Fabrication (task F) can start 2 time periods before engineering (task E) is finished. This is due to the fact that engineering is generating blue prints for fabrication and assembly. The blue prints for the fabrication can be forwarded to the shop floor while the blue prints for the assembly are still in progress.

Precedence relation F--+A models a minimal time lag of 0 between fabrication and assembly. That is, the fabrication task has to be completely finished before the assembly task can start. This restriction accounts for the fact that the entire part kit needs to be available for

5.3. MODEL 61

the assembly.

5.3.3 MIP Formulation

We assume that the processing times of tasks and the time lag between tasks are multiples of the standard period length. In this case, a task j with processing time Pi will start at the begin of period t, be finished at the end of period t+ Pj -1, and be in process during periods {t, t+ 1, ... , t + Pj - I}. We em ploy two types of binary decision variables. First, Yp, which is set to 1, if order P is accepted, 0, otherwise. Second, Xj,t = 1, if task j is started at the begin of period t, ° otherwise (cf. Pritsker et al. [256]). We can now give the following decision model:

p

Maximize Z = L vp . YP p=l

subject to

T T Lt'Xj,t - L(t+Pi) 'Xi,t ~ t~jn t=l t=l

J t

L L Cj,t-r+l . Xj,r :::; Gr,t

T

Qp • YP :::; L (t + Pep - 1) . Xep,t :::; dp • YP t=l

ep T L L Xj,t = (ep - sp + 1) . YP j=sp t=l

Xj,t E {a, I}

YP E {0,1}

j = 1, . .. ,J i E Pj

r = 1, ... ,R t= 1, ... ,T

j=l, ... ,J

P= 1, ... ,P

P= 1, ... ,P

j = 1, .. . ,J t = 1, ... ,T

P= 1, ... ,P

(5.1)

(5.2)

(5.3)

(5.4)

(5.5)

(5.6)

(5.7)

(5.8)


The objective function (5,1) maximizes the sum of all values of accepted orders. Constraints (5.2) model the minimal time lags between the tasks. Department capacity is considered in constraints (5.3) where the sum of the capacity used by the tasks in progress has to be less equal than the available capacity for each department r and each period t. Constraints (5.4) restrict each task to be started, and hence processed, not more than once. Constraints (5.5) assure that, if an order is accepted, delivery will be planned during its time window as given by the customer. Constraints (5.6) force the completion of all tasks belonging to accepted orders. Finally, the decision variables are defined in the constraints (5.7) and (5.8).

Solving the above problem gives an order acceptance vector y = (Yl, ... , yp) and for each accepted order p a schedule S = (Ssp' ... , Sep) which defines the start times of the tasks belonging to order p. From this, management can derive answers to the following questions:

• Should order p be accepted?

• What due date dp of the accepted order p shall be promised to the customer?

• What are the milestones of the the tasks of accepted orders?

• How much capacity has to be reserved in the departments for processing accepted orders?

• How much capacity for orders to come remains in the departments?

5.4 Discussion of the Model

We will now look at properties and possible extensions of the model.

5.4.1 Complexity Results

Large problem instances of the order selection problem (5.1) - (5.8) as they appear in industry cannot be solved to optimality. More technically, we can state the following.

Theorem 5.1 Problem (5.1) - (5.8) zs an NP-hard optimization problem.

5.4. DISCUSSION OF THE MODEL 63

Proof We prove Theorem 5.1 by polynomially transforming the knapsack problem which is well known to be NP-hard (cf. Garey and Johnson [117] pp. 134) to (5.1) - (5.8).

The knapsack problem can be stated as follows: Given a finite set U of cardinality U with elements u (u = 1, ... , U), for each u a size s(u) > 0 and a value v(u) > 0, a knapsack capacity B > max{s(u) I u E U}, find a subset U' ~ U such that 2:uEUI s(u) ~ B and such that 2:uEUIV(U) is as large as possible.

The transformation is straightforward. Let P = U where each order p (p = 1, ... , P) comprises a single (start and end) task sp = ep with a unit processing time of Pj = 1 and a (time constant) capacity demand of Cj,l = 1 in the single department with a capacity of Cl,l = B.D

5.4.2 Reduction of the Problem Size

The number of binary variables for the order selection problem equals T . J + P. Since the performance of MIP solvers such as CPLEX (cf. Bixby and Boyd [31]) depends heavily on the number of binary variables, we would like to reduce the count of binary variables. This can be done by introducing a time window [ESj, LSj] for each task j which defines the earliest start period (ESj) and the latest start period (LSj). Given the network representation introduced in Section 5.3, the time window for task j (j = sp, ... , ep) of order p (p = 1, ... , P) can be derived by forward recursion from 0 and backward recursion from dp (cf. Elmaghraby [94]).

If the number of variables has to be further reduced, one should employ the model proposed by Icmeli and Rom [164]. In comparison to the approach of Pritsker et al. [256] it does not limit orders to finish at discrete points in time. Instead, finish times are linear combinations of discrete adjacent time instants. The capacity constraints are only taken into account at the discrete time instants and the capacity demand of the tasks is allocated according to the (variable) weight which is for each task assigned to two adjacent time instants.

5.4.3 Release Dates and Maximal Time Lags

If release dates have to be taken into account for an order p we can model this as follows. Without loss of generality it is assumed that for


each order p the single start task sp is a dummy task. If this is not the case, we simply add this dummy task to the precedence network. Now, we introduce a minimal time lag t~inJ' between the dummy start p,

task and each immediate successor j where the weight t~inJ' equals the p,

release date. In order to reduce inventory holding cost of fabricated parts we can

impose a maximal time lag between the finish time of the fabrication task F and the start time of the assembly task A. In general, we can restrict the time span between the finish time of task i and the start time of its immediate successor j to a maximum of tijax ~ 0 periods by adding the constraints

T T ~ t . x . t - ~ (t + p.) . X· t < tr:n~x L...J J, L...J ~~, - ~,J

t=l t=l

to the order selection model (5.1) - (5.8).

j = 1, .. . ,J i E Pj

5.4.4 Supplier Coordination and Integration

(5.9)

Often ETO- and ATO-companies procure parts or even services such as design from outside suppliers. In this case, the consideration of supplier performance w.r.t. cost, technical specifications, and time is crucial for making the right order selection decision. So far we have taken the supplier implicitly into account. Consider for example the precedence network given in Figure 5.1. Here, we take the delivery time of the supplier as minimal time lag tp~ between the procurement , task P and the assembly task A into account. Contrary in Figure 5.2 where we explicitly model the supply task S with its processing time and capacity requirement w.r.t. the supplier. This implies that the capacity of the supplier and all orders of the supplier have to be captured by the model, too. From another perspective, Figure 5.1 coordinates the supplier with the MTO-company while Figure 5.2 integrates the supplier.

5.4.5 Stochasticity of the Data

Zapfel [336] pp. 150 points out that the data situation for the order acceptance process of MTO-companies is not a deterministic but a stochastic one. In particular, we have only rough estimates of cost,

5.4. DISCUSSION OF THE MODEL

-2 f::\ 31\!)I--~

Figure 5.2: Supplier Integration

65

time, and technical specifications of orders and orders have a certain winning probability which depends, c.p., on the price and the delivery time window (cf. Kingsman et al. [179]). Refering to empirical investigations, Backhaus [15] p. 476 points out that the winning probability ranges between 5% and 79%.

Using rough estimates of cost, time, and technical specifications does not pose a problem because it does not alter the decision model. One can argue that the derived solution will not be very accurate and that, due to changes in the data, we might not reserve enough capacity. But on the other hand, we use as much data as we have at the time of the decision and hence we cannot do any better.

The probability of winning an order can be employed straightforward within the model by modifying the order specific parameters vp

and Cj,t by multiplying them with the winning probability Pp of the order. The question is how the winning probability can be derived. We outline three possibilities:

• Kingsman et al. [179] introduce for MTO-companies the socalled 'strike rate matrix' which gives, based on acceptance and refusal of proposed bids, the probability of winning an order as a function of the mark-up on the cost of producing the order and the delivery time .

• Competitive bidding models (cf. Subsection 5.2.2) derive winning probabilities based on a vector of competitor prices, for each competitor price a submission probability, and a conditional acceptance probability of the companies bid when the competitor


price is proposed .

• The sales department of a MTO capital goods manufacturer derives monthly order confirmation probabilities for all orders which are still negotiated with customers by using a scoring model (cf., e.g., Samson [266] pp. 249). The scoring model takes, amongst others, the following criteria into account:

How many products have already been delivered to the customer?

Is the customer new?

Is there a competitor on the home market of the customer?

How far departs the customer specification of the product from already manufactured products?

Each criterion is valued with a number in the interval [0,1] and then multiplied with the criteria weight. The sum of all weighted values gives the confirmation probability of the order.

A problem occurs, if there are many orders with low winning probabilities. In this case, the solution of model (5.1) - (5.8) may accept many orders, which is feasible w.r.t. the resource constrains (5.3) because expected capacity demands Pp . Cj,t and not total capacity demands Cj,t are employed. But once orders are accepted, we have to take into account the total capacity demand which might exceed the available capacity. The sales department of the MTO capital goods manufacturer is taking this fact into account by considering only orders with a winning probability of Pp ~ 0.5.

5.4.6 Time-Dependent Order Values

Often, the price and hence the value of an order is time-dependent. That is, we have for every time period t = d.p , ••• , dp within the delivery time window a value Vp,t. For longer delivery times, it is common practice to give price breaks, in order not to lose customers (cf. Keskinocak et al. [172]). Formally, we have Vp,t ~ Vp,t+1 (t = d.p , ... , dp - 1). In this case the objective function has to be altered as follows.

p dp

Max Z = LYp L Vp,t (5.10) p=l t=d.p


Additionally, we can consider the time value of money by discounting the value flows with a factor a. We then obtain the following objective function.

p dp

Max Z = 2: Yp 2: Vp,t • (1 + a)-t (5.11) p=l t=4p

5.4.7 Variable Capacity

Variable capacity is a common assumption for aggregate production planning (cf., e.g., Nahmias [237] pp. 106, Hax and Candea [149] pp. 69). Typically, capacity can be altered by overtime, subcontracting, as well as hiring and firing. The latter two options can, due to labor laws, not always be considered.

Instead of hiring and firing, flexible working hours have become an option in Germany (cf. Betz [27], Gunther [131]). Here, the level of capacity can be altered between periods as long as the cumulated capacity stays constant for a number of periods, e.g., a year. If flexible working hours have to be considered, so-called partially renewable resources have to be introduced into model (5.1) - (5.8) (cf. Bottcher et al. [35]). Briefly, partially renewable resources define capacity not for a single period but for a set of periods.

In the case of subcontracting and overtime, we can use the shadow prices of constraints (5.3). If the price for additional capacity is less than the shadow price, it pays to acquire additional capacity.

5.4.8 What-If Analysis

When sales negotiates a contract with the customer it is important to know how product specification and price influence the expected profit. So far the solution of model (5.1) - (5.8) provides only the information whether the order should be accepted or not. Now, the following situation and associated questions are very common for sales .

• The customer does negotiate with a market competitor which submitted an offer for a product which, compared to our offer, is less sophisticated but has a lower price. The question is, if we cut down on the engineering effort and submit a lower price, will the order still be profitable?


• The customer wants our product but is not willing to pay the price. How far can we lower the price without forcing the order not to be selected any more by our decision model?

• Can we give a lower price to the customer by prolonging the delivery time window?

• The customer wants the order to be expedited. Can we do so without changing the price and still having a profitable order?

Briefly, sales would like to know the tradeoffs between price, technical specifications, and delivery time for orders. When employing LP decision models, these type of questions can be well answered by sensitivity analysis (cf., e.g., Domschke and Drexl [83] pp. 44, Schrage [275] pp. 35). Unfortunately, we have an MIP. So, for what-if analysis we have to alter the data of the model, re-run the solution method provided in Chapter 8, and interpret (the change in) the solution.

Chapter 6



This chapter is concerned with the manufacturing planning level. That is, we focus on the tasks to be done in the departments fabrication and assembly. Figure 6.1 shows how the two tasks fabrication (F) and assembly (A) of the order level are exploded into parts (P1 - P2), subassemblies (Sl, S2), and final assembly (A1) at the manufacturing level. The weights on the arcs give the 'quantity per' factors, i.e., the amount required of each part and subassembly in order to obtain one subassembly or assembly (cf. Hax and Candea [149] p. 442). The decision problem at this level is to schedule subassemblies and assemblies and to determine lot sizes for parts with the aim of minimizing cost while taking relevant interdependencies into account. In what follows we specify the performance measure and the interdependencies.


From the order selection decision performed in Chapter 5 we have the instruction to process P 2: 0 different customer orders such that each order p (p = 1, ... , P) is delivered not later than at its due date dp 2: O. Since the orders have been accepted, the revenue and the delivery date are set and cannot be altered. Also, we assume that the capacity situation (work force size, machine equipment) is fixed. To maximize profit, we have to minimize the cost which can be influenced by managerial decisions at the manufacturing level. Relevant cost are:

70 CHAPTER 6. MANUFACTURING PLANNING

Fabrication Assembly

5

5 PI r------r--------~--------~.

5 P2L---~--~~rt

Figure 6.1: Explosion of Manufacturing Tasks

• holding cost for inventory of raw materials, parts, subassemblies, assemblies and final products as well as

• setup cost for changing a work centre to go from the production lot of one part to the production lot of a different part.

Since the order selection decision has made an anticipation of the capacity constraints at the manufacturing planning level, orders can be delivered in time and hence, penalty cost for late deliveries do not have to be taken into account.

Holding Cost

To derive holding cost, we consider the cash flows which are determined by the decisions at the manufacturing planning level (cf. Zapfel [337], Miiller-Manzke [236]). We assume that material procured for fabrication has to arrive in the period where production takes place and that a cash outflow occurs at the beginning of that period. Material procured for assembly has to be available at the start time of the assembly, i.e., it has to be delivered during the period prior to assembly. The cash outflow associated with the delivery of the material

6.1. OUTLINE OF THE PROBLEM 71

takes place at the beginning of the prior period. A cash inflow occurs at the due date when the order is delivered to the customer. Figure 6.2 shows the cash outflows (t) and the cash inflow CD of an example where the length of each arrow gives the amount of the associated cash flow.

Helber [153] pp. 23 shows for the uncapacitated multi-Ievellotsizing problem that time-variant holding cost for the inventory of parts can be derived from the associated cash flows. The same reasoning can be undertaken in order to obtain time-variant holding cost coefficients hf,t and htt for parts and assemblies for multi-level manufacturing planning. Due to discounting, we have hft > hft+l for i = 1, ... , I , , and t = 1, ... , T - 1. The same property holds for the holding cost coefficients of assemblies. Following Helber [153] p. 30 we replace the time-variant holding cost by time-constant holding cost hf and hi for part i = 1, ... , I and assembly j = 1, ... , J. This simplification leads to a systematic error. But for two reasons the latter is considerably small. First, compared to classical investment decisions (cf., e.g., Brealey and Myers [38] pp. 100, Kruschwitz [198] pp. 64) where a period amounts a year, periods in the manufacturing planning problem amount only about a week. This causes a low discount rate per period and thus not much difference in the time-variant holding cost. Second, manufacturing planning is done in a rolling time horizon (cf. Tempelmeier [304] pp. 334) where only the production plan for the early periods is put to action. For these periods the error is the smallest.

Setup Cost

Since every product in an ETO- and MTO-environment is unique, setups cannot be saved by batching assemblies of different orders. But on the fabrication level we have common parts for different orders (cf. Eversheim et al. [100]). Examples are electric drives for tool machines (cf. Drexl et al. [85]) or steel plates of equal size for ships (Subsection 2.5.1, Lee et al. [212]). Here, setups can be saved by batching the same parts. Each time a setup takes place, setup time and setup cost are incurred. The setup time is the amount of time it takes to change a work centre to go from the last part of a production lot to the first consistently good part of the next production lot (cf. Melnyk


Assembly ----1

Fabrication

Procurement

Cash Flow

t

o 1 234 5 6 789

Figure 6.2: Cash Flows at the Manufacturing Level

and Christensen [229]). Direct setup cost accrue by cleaning material needed and by the cost for scrap parts until the first consistently good part is produced. Labor cost are not taken into account because we assume that the work force size is fixed and we do have to pay the worker regardless of the fact if he sets up the machine or not. Since we consider a so-called 'large bucket' model (cf. Eppen and Martin [96]) where periods amount a considerable large time span, i.e., a week, we do not have to consider setup times explicitly. In order to guarantee feasible production plans, we set the available period capacity not equal to the total capacity but we withdraw some capacity. This way we obtain slack capacity (cf. Helber [154]).


On the manufacturing planning level we have the following three types of interdependencies (cf. Section 2.3).

Time-based interdependencies arise from the fact that we have a dynamic problem with a planning horizon T > 1 where we cannot make a manufacturing decision for each period separately because the decision for period t will influence the solution space of decisions in


periods T > t. Vertical interdependencies are depicted in Figure 6.1. They arise

from the precedence relations between parts, subassemblies, and assemblies. We can clarify the vertical interdependencies by the bill of materials (BOM) of a hypothetical example with two orders as given in Figure 6.3. Note that order A1 stems from Figure 6.1. The BOM

is an engineering document which gives a symbolic exploded view of the part structure (cf. Hax and Candea [149]). Each pair of items connected by an arrow is vertically interdependent.

"

,., ,. \ ,;,; ,

,; \ ",; ,

,; ,. A2,. > Final Assembly ,. " ,. " ,. "

" ,. - -7--------/- ----------------,. ,;

1 .-" " " .;'-'; Subassembly

, " 'v .... "

5

P3 Fabrication

Figure 6.3: Bill of Material

Horizontal interdependencies are due to the joint use of resources by non-precedence related items. For example, in Figure 6.3 the subassembly 82 and the final product A2 are horizontal interdependent because they have to be assembled in the same department. This is indicated by the dashed box enclosing 82 and A2 (cf. Billington et al. [29]). Furthermore we have horizontal interdependencies between the parts P1, P2, P3, the subassemblies 81, 82, the final assemblies A1, A2, as well as between subassembly 81 and assembly A2.


There have been different approaches for production planning of capacitated, multi-level manufacturing systems. Before we will review


the literature in Subsections 6.2.1 - 6.2.2, let us propose a taxonomy for production planning approaches of multi-level manufacturing systems.

We distinguish the approaches according to the type of manufacturing system which is regarded at each level into

• multi-Ievellotsizing where lotsizing takes place at the upper and the lower level (lotsizing --+ lot sizing) ,

• multi-level scheduling and lotsizing where scheduling takes place at the upper and lotsizing takes place at the lower level (scheduling --+ lotsizing), and

• multi-level lotsizing and scheduling where lotsizing is done at the upper and scheduling is done at the lower level (lotsizing --+ scheduling) .

Table 6.1 gives a survey of the literature according to this taxonomy. Note, that the term 'lotsizing and scheduling' is used differently in the context of the so-called multi-level lotsizing problem (cf., e.g., Drexl and Kimms [86], Kimms [177]). There, due to small time periods, a sequencing of the lotsizes within one period is performed.

Upper Level Lotsizing Scheduling Lotsizing

t t t t Lower Level Lotsizing Lotsizing Scheduling

References [29], [30], [59], [11], [287], [67], [207], [75], [144], [286], [80], [208] [153], [154], [261]' [148], [171] , [199], [300] [177], [284], [294], [307], [306], [305]

Table 6.1: Taxonomy of Approaches for Multi-Level Manufacturing


6.2.1 Multi-Level Lotsizing

Multi-level capacitated lotsizing models depict a manufacturing system with multiple levels where on each level lotsizing rather than scheduling decisions have to be undertaken. Given are the demands for multiple parts (which are usually referred to as items) where each part is depicted by a multi-level product structure. Each part within the product structure has to be fabricated on one level of the manufacturing system. Since there is no a-priori lot size, any amount of a part can be produced as long as the capacity constraints are respected. Production of a part incurs fixed setup cost and in some cases setup times; part inventories incur variable holding cost. The problem is to obtain a cost minimal production plan which respects the scarce capacities of the resources on all levels of the production system and delivers final demand parts without backlogging.

Models and methods for this intricated problems can be distinguished w.r.t. the type of product structure allowed and w.r.t. the amount of aggregation. Regarding the product structure, general product structures and assembly product structures can be distinguished. Regarding the amount of aggregation, so-called big bucket and small bucket models can be distinguished (cf. Eppen and Martin [96]). Big bucket models such as the capacitated lotsizing problem (CLSP) have a rather high aggregation level where one period amounts about a week. Here, scheduling decisions which determine the sequence of production lots in one period are not taken into account. Contrary, in small bucket models a period embraces a smaller time span and the planning problem is to simultaneously determine lot sizes and lot sequences in each period. Multi-level big bucket problems with general product structures have been, amongst others, addressed in Derstroff [75], Helber [153, 154], Katok et al. [171], Simpson and Erenguc [284], Stadtler [294], Tempelmeier and Derstroff [305, 306] as well as Tempelmeier and Helber [307]. Helber [153] and Zapfel [337] give a cash-flow oriented model where the net present value is maximized. Work on special big bucket problems where either serial or assembly product structures are taken into account or constrained resources are only considered on one manufacturing level can be found, amongst others, in Billington et al. [29, 30], Chiu and Lin [59], Harrison and Lewis [144], and Kuik et al. [199]. Work on multi-level, small bucket


models with general product structures has been presented by Kimms [177].

6.2.2 Multi-Level Scheduling and Lotsizing

Multi-level scheduling and lotsizing problems depict a single project or multiple projects at the first level. A project comprises a set of precedence related activities where each activity requires resource capacity when processed. The activity schedule from the project level determines a time-phased part demand for the second level where uncapacitated lotsizing takes part. The literature treating this problem can be classified according to the following issues.

• Is there a single project (5) or are there multiple projects (M) at the first level.

• Are capacity constraints taken into account at the first level (C) or not (-).

• Is project scheduling performed in a forward oriented fashion where earliest start times are calculated (F) or in a backward oriented manner where latest start times (B) are calculated.

• Is there a lot-for-Iot policy (-) or is lot sizing performed either optimally or heuristically (L) at the second level.

• Is project tardiness taken into account (T) by allowing projects to finish after the due date but punishing a late project termination with tardiness cost or are projects not allowed to be late (-).

• Is project completion ahead of the due date rewarded in the objective function (E) or not (-).

• Is the crashing of activities allowed (C) or not (-).

Table 6.2 gives a categorized survey of the relevant literature. A special case which, by its assumptions, does not fit into the

proposed classification is treated by Ronen and Trietsch [261, 262]. They consider the case where parts demanded by the activities can be sourced from different suppliers. Furthermore, the processing times of the activities and the lead time of parts is not deterministic but

6.3. MODEL 77

stochastic. Each part-supplier combination is defined by a price and a lead time distribution. Inventory holding cost accrue if parts arrive ahead of the time needed by the activity. If late parts delay the project beyond its due date, tardiness cost are debited. The decision problem is to choose for each part a supplier and an ordering time such that the total expected cost are minimized. Assuming that the variability of the processing times is negligible, Ronen and Trietsch [261, 262] propose to compute order times for parts based on latest activity start times first and select a supplier for each part afterwards.

Reference Classification Aquilano and Smith [11] : S /-/-,B /-/-/-/Dodin and Elimam [80] : S /- /F ,B /L /T /E /C Hastings et al. [148] : M/C /F ,-/-/-/-/Smith-Daniels [286, 287] : S /-/-,B /L /D /-/Sum and Hill [300] : M/C /-,B /L /-/-/-

Table 6.2: Classification of Multi-Level Scheduling and Lotsizing Literature

6.2.3 Multi-Level Lotsizing and Scheduling

Multi-level lotsizing and scheduling depicts the case where multiple parts have to be lotsized on a single capacitated work centre at the first level (CLSP, cf., e.g., Drexl and Kimms [86]). At the second level each period of the first level is exploded into a job shop problem (JSP, cf., e.g., Pinedo [253]) where each part with positive lotsize is depicted by a job. This type of problem has been addressed by Dauzere-Peres and Lasserre [67], Lambrecht and Vanderveken [207] as well as Lasserre [208].

6.3 Model

From the literature review we can see that the problem of a heterogenous manufacturing system where multiple customer specific orders have to be scheduled subject to scarce capacities on the first level and


where capacitated lotsizing of multiple parts has to be done on the second level such that the relevant cost of the entire manufacturing system are minimized has not been treated so far. In what follows, we present a MIP formulation of the problem which has been recently documented in Kolisch [183].

6.3.1 Detailed Description and Notation

We will now describe the manufacturing problem in more detail, thereby introducing the notation needed for our decision model (for a survey of all indices, parameters, and variables cf. to Appendix B.2). To do so, we change from a product based view as pursued in Section 6.1 with task explosion and BOM to a job based view. Instead of a subassembly or assembly we consider the job which by means of labor and machinery transforms parts and/or subassemblies into a subassembly or assembly. Figure 6.4 shows the relationship between products and jobs. Job 4 represents the task of assembling part PI and subassemblies SI,S2 to the final assembly Al by using one resource unit for one time period. Note, that the job can be viewed as a production function f which gives the relationship between input and output, i.e., Al=f(Pl,SI,S2,1,1) (cf. Gutenberg [136] pp. 298).

Figure 6.4: Relationship between Product- and Job-Representation

The number of orders is P ~ O. Order p (p = 1, ... , P) has to be finished at its due date dp by the latest. Each order p comprises a set of Jp assembly jobs (henceforth called jobs). Jp denotes the job count of assembly p. The number of jobs from all orders is J (J = I::=1 Jp ). Due to the precedence relations in the BOM, job j

6.3. MODEL 79

(j E Jp; P = 1, ... , P) has a set Pj of immediate predecessor jobs which have to be processed before job j can start. The processing time of job j is Pj ~ o.

The assembly department comprises RA ~ 0 different resources which are assemblers distinguished by rough qualification criteria such as electronic and mechanic assemblers, expensive devices such as fixtures, power-tools, and the available assembly area. Assembly resource r (r = 1, ... , RA) has an available capacity of C!'t ~ 0 units at time instant t ~ o. While being processed, job j requires c1,r ~ 0 units from resource r (r = 1, ... , RA) during every time instant it is processed. The preemption of jobs is not allowed.

Parts required by job j are either in-house fabricated or outside procured. The number of all parts which are manufactured in-house is I ~ 0 and the amount of part i (i = 1, ... , 1) needed by job j is qj,i ~ o. The sum of the holding cost of all parts which are directly assembled by job j equals h1 per period. Note, that h1 does not account for the holding cost of subassemblies which are assembled by job j.

The in-house fabrication comprises RF ~ 0 different work centres. A work centre might be, dependent on the amount of aggregation, an entire job shop or flow shop, a set of identical machines such as numerically controlled lathes or a single machine such as a striper. Work centre r (r = 1, ... , RF) in period t has an overall period capacity of C!',t ~ 0 where period t is the time span between the time instants t - 1 and t. The length of a period amounts between one day and one week. Part i is fabricated in work centre r[ E {1, ... , RF} in a single-level production process. For the production of one unit of part i, a capacity of cf ~ 0 is needed. Within each period part i is produced, fixed setup cost of sf ~ 0 are incurred. Production of one unit in period T < t for a demand in period t incurs holding cost of hf ~ 0 'per period. Parts which are procured from outside suppliers have to be available in the period preceding the start time of the assembly. The holding cost of all parts which are procured by job j are hf per period; for reasons to be explained below, fixed ordering cost do not have to be taken into account.



A graphical representation of orders, due dates, and jobs with their technological precedence relations can be given by a precedence network where jobs are depicted by nodes and precedence relations between jobs by arcs.

The jobs of order p are consecutively labelled from 8p to ep where 8 p is the unique start job and ep the unique terminal job of order p. Without loss of generality the labels can be assigned such that each job j has a higher label than any of its predecessor jobs h, i.e., h < j (j = 1, ... , J j h E Pj). Each precedence relation between job h and its immediate successor j is depicted by an arc h --t j with weight th,~n = 0 representing a minimal time lag between the finish time of job hand the start time of job j. We introduce two additional jobs, a dummy source 0 and a dummy sink J + 1, and additional finish-start arcs from the source 0 to each start job 8 p and from each terminal job ep to the sink J + 1. The arcs from the source have weight 0 while the arcs from the terminal jobs ep to the source have the weight t~i,r:,+1 = dmax - dp

where dmax denotes the maximal due date of all orders. Based on this network representation we can obtain for each job j a time window [ESj, LSj] of earliest and latest start times by forward recursion from t = 0 and backward recursion from dmax (cf. Elmaghraby [94]). With .:r = {O, ... , J + I} we denote the set of all jobs of the network.

Example

We continue the example as introduced by the BOM in Figure 6.3. The BOM representation of the product can be converted straightforward to the precedence network given in Figure 6.5 (cf., e.g., Giinther [132] and Harhalakis [141]). Note, that only assembly but not fabrication jobs are represented as nodes in the network.

We have two orders (P = 2), a single resource within the assembly department (RA = 1), three parts (1 = 3), and a single resource within the fabrication department (RF = 1). Jobs 0 and 6 are the dummy source and the sink of the network. Order 1 comprises the jobs 1, ... ,4 and order 2 consists of the single job 5. That is, we have 81 = 1, el = 4, and 82 = e2 = 5. The due dates of the orders are d1 = 5 and d2 = 3. For each job j the processing time Pj and the capacity requirement

6.3. MODEL 81

(5,0,0)

Figure 6.5: Network Representation for the Example Instance

Ct.l w.r.t. the single resource RA = 1 are given above the nodes while the part demand vector qj = (qj,l, qj,2, qj,3) is given below the nodes. No data is given for dummy jobs with zero processing time, capacity requirement, and part demand, respectively. Also, only minimal time lags with th,~n > 0 are given as weights on the arcs of the network. Additional information is as follows. The maximal due date is 5, the capacity of the assembly resource is C~t = 3 (t = 0, ... ,5), and the capacity of the fabrication resource is eft = 10 (t = 1, ... ,5). The , production coefficients, setup and holding cost vectors of the three internal fabricated parts are cF = (1,1,1), sF = (20,10,10), and hF = (1,2,1), respectively. Parts from outside suppliers are not procured, i.e., hf = 0 for j = 1, ... ,6. According to

I

hi = hf + L h; . qj,i (6.1) i=l


the holding cost for the non-dummy jobs can be calculated to ht = 5 . 1 = 5, hf = 5 . 2 = 10, h1 = 5 . 1 = 5, and ht = 5 . 1 = 5.


We assume that all events on the manufacturing planning level, i.e., due dates and the change of available capacities, occur at discrete multiples of a standard period length and that the processing times of the jobs and minimal time lags are discrete multiples of the standard period length. In this case, the start times of the jobs will be also discrete multiples of the period length. A job j with processing time pj will start at time instant t and end at time instant t+Pj. We employ the following four types of decision variables. First, x j,t which is set to 1 if job j is started at time instant t, ° otherwise (cf. Pritsker et al. [256]). x j,O = 1 indicates that job j starts at time instant 0, ends at time instant Pj and is processed during periods 1, ... , Pj. Next, Yi,t which is set to 1 if part i is fabricated in period t, ° otherwise. Finally, we have Qi,t and li,t, the amount of part i fabricated in period t and the inventory of part i at the end of period t. Given initial inventories li,O 2: 0, we can now formulate the assembly scheduling and fabrication lotsizing problem as follows.

Minimize Z = t t hi· (dp + 1 - E t· x j,t) p=l J=Sp t=ESj

I T

+ LL (hf· li,t + sf· Yi,t) i=l t=l

subject to

LSj

L (t + Pep) . Xep,t ~ dp t=ESj

LSj

L Xj,t = 1 t=ESj

LS, LSh

L t'Xj,t- L (t+Ph)'Xh,t2: th,T t=ESj t=ESh

P= 1, ... ,P

j = 1, ... ,J

j=l, ... ,J hE Pj

(6.2)

(6.3)

(6.4)

(6.5)

6.3. MODEL

J t z: z: ct,r . X j,T ~ C~t j=l T=max{O,t-pj+l}

J

h,t-l + Qi,t - z: qj,i . Xj,t = 1i,t j=l

I z: C.f . Q;t < CF , ., _ r,t

i=llrf =r

J

Y· ~q" > Q. ',t L....J 3,' _ ',t j=l

83

r = 1, ... ,RA

t=O, ... ,T (6.6)

i = 1, ... ,1 t = 1, ... ,T

(6.7)

r = 1, ... ,RF

t=O, ... ,T (6.8)

i = 1, ... ,1 t = 1, ... ,T

(6.9)

Xj,t E {0,1} j = 1, .. . ,J (6.10) t=ESj, ... ,LSj

Yi,t E {O,l}, Qi,t? 0, 1i,t? ° i = 1, ... ,1 t = 1, ... ,T

(6.11)

(6.2) - (6.11) models a two-level manufacturing system with heterogenous production segments. The model comprises four main parts:

• the project scheduling based assembly on the first level,

• the lotsizing based fabrication on the second level,

• the interdependencies of fabrication and assembly, and

• the objective function which minimizes the sum of holding and setup cost within the entire manufacturing system.

Let us analyze these parts in more detail.

Assembly

The project scheduling based assembly is depicted by the constraints (6.3) - (6.6) and (6.10). Constraints (6.3) assure that each order is finished not later than its due date. Constraints (6.4) secure that the final assembly cannot be delivered to the customer before each job has been executed exactly once. The precedence network is given by constraints (6.5). Constraints (6.6) model the capacity constraints of the assembly resources. Beginning with time instant t = 0, the capacity check is always made at the beginning of a period. For example, for


job j with processing time Pi = 2, and start time 1, i.e., Xi,1 = 1, the left-hand side of the capacity constraint for resource r will be cf.r for time instants 1 and 2, 0 otherwise. Finally, constraints (6.10) define the binary decision variables.

Fabrication

Constraints (6.8) - (6.9) and (6.11) depict the fabrication. The capacity constraints are given in (6.8). Constraints (6.9) set the binary setup variable Yi,t = 1 when production of part type i takes place in period t. Finally, constraints (6.11) define the continuous and binary decision variables.

Interdependency between Assembly and Fabrication

Constraints (6.7) link the part flow between the fabrication and the assembly. The start of job j at time instant t imposes demand for fabricated parts at the end of period t which has to be fulfilled from the inventory available at the end of period t - 1 or from fabrication in period t.

Objective function

The cost function (6.2) minimizes the overall manufacturing cost which are made of holding cost within assembly as well as holding and setup cost within fabrication. Figure 6.6 details the accounting of the cost. The due date of the order is 9. Job j is started at time instant 5 and the required part i is fabricated in period 3. Additionally, job j requires a procured part which has to be available at the end of period 5. Within the fabrication, part i is in inventory at the end of periods 3 and 4. Within the assembly, part i and the procured part are in inventory at the end of periods 5 - 9. Setup cost occur in period 3 where part i is produced.

We end this subsection by solving the example instance introduced above with model (6.2) - (6.11). The optimal solution is depicted in Figure 6.7 where the area above the dashed line gives the assembly schedule and the area below the dashed line represents the lotsizing decisions in the fabrication department. Order due dates are depicted with the symbol I-. Within the assembly department, jobs 5 and 3 start

6.3. MODEL 85

Assembly ----I

Fabrication

Procurement

I :;. t·

o 1 2 3 4 5 6 7 8 9

Figure 6.6: Calculation of Holding Cost

at time instant 2, job 2 starts at time instant 3, and job 4 starts at time instant 4, respectively. Within the fabrication department, part 2 and part 3 are produced with a lotsize of 5 in period 2 while part 1 is produced with a lotsize of 10 in period 3. The assignment of parts to jobs is given by arcs from the north-east corner of a fabrication lot to the south-west corner of an assembly job. The overall cost of 120 comprise 40 units setup cost, 5 units fabrication holding cost, and 75 units assembly holding cost.

Order 2 Assembly

Order 1 f-

----------------

Fabrication

I ;. t 0 1 2 3 4 5

Figure 6.7: Optimal Solution of the Manufacturing Planning Instance



In this section we will look at some properties of the manufacturing planning problem.

6.4.1 Complexity Results

Problem (6.2) - (6.11) comprises two subproblems which are well known NP-hard optimization problems, the resource-constrained project scheduling problem and the capacitated lotsizing problem.

Resource-Constrained Project Scheduling Problem

Consider first the assembly scheduling problem (ASP) on the first level. The ASP is depicted by the first term of the objective function (6.2) as well as constraints (6.3) - (6.6) and (6.10). The ASP models the assembly with the objective to minimize the assembly wide holding cost. It is the 'mirror problem' of the well known resource-constrained project scheduling problem (RCPSP) when the latter has the objective of minimizing the sum of the weighted job flow times (cf. Slowinski [285], Sprecher and Drexl [290]). The RCPSP can be polynomially transformed into the ASP by giving each job j the new job number J + 1- j, reversing all arcs h -7 j to h t- j, and keeping everything else as it is. It is well known that the RCPSP is an NP-hard optimization problem (cf. Blazewicz et al. [33]). A survey of exact and heuristic solution procedures for the RCPSP with different objectives can be found in Kolisch and Padman [191].

Capacitated Lotsizing Problem

Consider now the fabrication lotsizing problem (FLP) on the second level. The FLP is depicted by the second term of the objective function (6.2) as well as constraints (6.7) - (6.9) and (6.11) where we assume that the xj,t-variables in the coupling constraints (6.7) have been determined, i.e., are parameters. The FLP models RF different work centres at the fabrication level. The objective is to minimize the fabrication wide holding and setup cost. Since each part i is produced on exactly one of the work centres, namely r[, we have RF separate


optimization problems. Each of it is the well known single-level multipart capacitated lotsizing problem (CLSP) (cf. Drexl and Kimms [86]) which is strongly NP-hard (cf. Chen and Thizy [56]). Techniques proposed to solve the CLSP to optimality are, amongst others, valid inequalities (cf. Barany et al. [21]), network flow reformulation (cf. Eppen and Martin [96]), branch-and-bound (cf. Gelder et al. [118]), cross decomposition (cf. Souza and Armentano [71]), as well as linear programming, column generation, and subgradient optimization (cf. Chen and Thizy [56]). A survey of heuristic approaches for the CLSP will be given in Subsection 9.1.5.

Since both subproblems of the manufacturing planning problem, the RCPSP and the CLSP, are known to be NP-hard optimization problems, this holds for the manufacturing planning problem, too. Moreover, the presence of deadlines makes the feasibility problem already NP-complete.

6.4.2 Regularity of the Objective Function

We will now show that the objective function (6.2) is right-regular. This will decrease the solution space considerably. We begin with

Definition 6.1 (Right-regular) Let 8 = (81, ... , 8J) be a schedule where 8j ~ 0 denotes the start time of job j (j = 1, ... , J). An objective function Z is right-regular iff the following holds. For each pair of two feasible schedules 8 = (81, •.. , 8J) and 8' = (8i,··., 8J) with Sj ~ 8j for a single job j and 8i = 8f for the remaining jobs i = 1, ... , j - 1, j + 1, ... , J holds Z(8) ::; Z(8').

Theorem 6.1 The objective function (6.2) is right-regular.

Proof Consider a feasible assembly schedule 8 = (S1, ... , 8J) where job j has the start time Sj. Changing, c.p., the start time to 8j + t with t ~ 0 will decrease the first term of the objective function by t . h1. For the second term we can reason as follows. The fact that, c.p., the part demand of job j occurs at time instant 8j + t instead of 8j enlarges the solution space (6.3), (6.4) - (6.6) of the fabrication lotsizing problem. Hence, for assembly schedule 8', the second term of the objective function must be less equal than for assembly schedule 8. 0


6.4.3 Sequential VB. Integrated Manufacturing Plan-nmg

Next, we prove that assembly and fabrication planning have to be treated in an integrated fashion if we do not want to exclude all optimal or even all feasible solutions. We begin by looking at the optimality of non-integrated solutions.

Optimality Property

Theorem 6.2 The optimal sequential solution of the ASP and the FLP, i.e., first solving ASP and thereafter FLP to optimality, does not guarantee an optimal solution of the manufacturing planning problem.

Proof We prove the theorem by showing that the optimal sequential solution of the example instance given in Section 6.3 is not an optimal solution for the manufacturing planning problem. We first solve the ASP to optimality and obtain the unique assembly schedule given in Figure 6.8 with holding cost of 70. The holding cost of 70 is 5 units less than the holding cost of the assembly schedule of the optimal solution given in Figure 6.7. The subsequent optimal solution of the remaining FLP gives rise to a production plan with setup cost of 40 and holding cost of 15 and hence overall fabrication cost of 55. Figure 6.8 illustrates the entire solution of the ASP and the FLP by means of a Gantt-chart. The total cost for the entire manufacturing system are 125 which is 5 units more than the optimal solution of the manufacturing planning problem. Note that the sequential solution has the same setup cost as the optimal solution and hence looses its optimality by higher holding cost solely. D

Feasibility Property

We now investigate on the infeasibility of non-integrated solutions.

Theorem 6.3 Solving the assembly scheduling problem to optimality might produce an infeasible fabrication lotsizing problem.

Proof We use a counterexample to prove Theorem 6.3. Consider a single order with d1 = 4, J = 4, RA = 1, RF = 1, ctt = 1 for each


Assembly

Fabrication

t o 1 2 3 4 5

Figure 6.8: Optimal Sequential Solution of the Manufacturing Planning Instance

time instant t ~ 0 and Cf.t = 10 for each period t = 1, ... ,4. The precedence relations between the jobs are 1 -+ 2, 1 -+ 3, 2 -+ 4, and 3 -+ 4. Job 1 is a dummy job with PI = 0 and Cfl = 0, jobs 2 - 4 , have a unit processing time and capacity demand, respectively, i.e., Pi = 1 and Cf,l = 1 for j = 2, ... ,4. Furthermore, we have I = 2 in-house fabricated parts and a part demand of the jobs of ql = (0,0), q2 = (5,0), q3 = (0,15), and q4 = (10,0). The part setup cost are sF = (10,10) and the part holding cost are hF = (4,1). There are no procured parts, i.e., hf = 0 for j = 1, ... ,4. Solving the ASP to optimality, we obtain the assembly schedule in part (a) of Figure 6.9. For this schedule there does not exist a feasible FLP-solution since job 2 requires 15 units of part 2 at the end of period 1 while the fabrication capacity in period 1 is only Cr.l = 10. Part (b) of Figure 6.9 gives a feasible and optimal solution for the problem. 0

6.4.4 Special Cases

There are three special cases of model (6.2) - (6.11) .

• There are no setup cost for the fabrication of parts, i.e., sf = 0 (i = 1, ... , 1). In order to obtain optimal fabrication lot sizes for a given assembly schedule it suffices to solve the (linear) production planning problem (6.8), (6.9), and (6.11) with the continuous variable Yi,t (0 ~ Yi,t ~ 1).


Sequential Solution

Assembly

(a)

Fabrication

I ....

o 1 2 3 4

Integrated Solution

Assembly

(b)

Fabrication

I ....

o 1 2 3 4

Figure 6.9: Infeasible Sequential and Optimal Integrated Solution

• If fabrication capacity constitutes no bottleneck, i.e., C!'t = 00

(r = 1, ... ,RF j t = 1, ... ,T), we obtain for a given as~embly schedule the optimal solution by solving for each part i a dynamic uncapacitated lotsizing problem (cf. Wagner and Whitin [318]).

• If for each part i we have a dedicated fabrication resource rf, we have to solve for a given assembly schedule I independent single-part capacitated dynamic lotsizing problems (cf. Subsection 9.1.5).


6.4.5 Model Extensions

At the end of this chapter we give two possible model extensions.

Consideration of Late Design and Procurement Tasks

So far we have assumed that only manufacturing tasks have to be considered. That is, we can plan jobs to start at time instant 0 and we can plan fabrication lots to be produced in period 1. But it might happen that parts ordered from outside suppliers are late, that design has yet not finished blue prints or engineering documents. This case can be considered by introducing minimal time lags tw~n > 0 between

,p

the network source 0 and the start job of assembly p. Example given, assume that parts from outside suppliers for order p will arrive at time instant 3. In this case we have tw~n = 3.

,p

Fixed Ordering Cost

We have stated in Subsection 6.3.1 that fixed ordering cost for procured parts are not taken into account by the manufacturing planning problem (6.2) - (6.11). The reason is as follows. If order cost are relevant for a procured part, we depict this part as an additional in-house fabricated part with a capacity demand of 0 on a dummy fabrication resource.

Chapter 7



Operations scheduling is the decision level at the bottom of our hierarchical planning process. Its purpose is to make detailed decisions about the start of assembly operations, the assignment of assemblies to assembly areas, and the assignment of resources as well as parts to operations. Figure 7.1 gives an overview of the decision context.

Hierarchical Planning

Due to the hierarchical planning approach operations scheduling gets the instruction from the manufacturing planning level to process P ~ 0 assemblies. Also, fabrication lot sizes have been set and parts have been ordered from outside suppliers.

Job Disaggregation

Each assembly p (p = 1, ... , P) with its due date dp is depicted as a job at the manufacturing planning level and is either a subassembly or a final assembly at the operations scheduling level. We now explode a job j from the manufacturing planning level with aggregated processing time, resource and part demand into a set of precedence related assembly operations with disaggregated processing times, resource and part demands, respectively. Figure 7.2 gives an example. Operations are represented by nodes with processing time and resource demand

94 CHAPTER 7. OPERATIONS SCHEDULING

hierarchical planning

due dates fabrication lotsizes procurement

dynamic situation ~ shop floor instructions

part arrival times operations operation start times part assignment

capacity --;:;..

scheduling -- resource assignment area assignment

i technical specifications resources requested area demand

engineering documents

Figure 7.1: The Operations Scheduling Context

given above the node. Parts are represented by squares. An arrow leading from a part to an operation represents a part demand of the amount given by the weight on the arc. Note, that for each part the part demand of all operations stemming from job j equals the part demand of job j itself. The details needed for the explosion come from engineering documents such as process plans (cf. Zapfel [338]).

Dynamic Situation

Due to a dynamic planning situation we have time-varying capacities of the resources and a dynamic arrival of parts.

Shop Floor Instructions

The managerial task at the operations scheduling level is to assign start times to operations, to assign parts and resources to operations, and to


Figure 7.2: Explosion of a Job into Operations

assign assembly areas to assemblies such that an adequate performance measure is optimized.


The question is which performance measure is suited for the operations scheduling context. Cash outflows for part procurements have been determined at the manufacturing planning level, the timing and the amount of cash inflows have been determined at the order selection level. So at first sight, the only objective is to make the above mentioned decisions such that due dates are not violated. Unfortunately, the situation at the operations scheduling level is dynamic. That is, parts may arrive late because suppliers do not deliver in time, the fabrication cannot meet the fabrication schedule, or quality problems may force parts to be shipped back to suppliers or fabrication. Due to fluctuation of assemblers as well as break down of machines and tools, the available capacity might be less than anticipated at higher decision levels. Also, technical or quality problems during the assembly process might delay assemblies and thus block the assembly area. Hence, often


due dates for assemblies are endangered or cannot even be met. Not meeting due dates is associated with the following cost:

• Due to later deliveries of assemblies, the cash inflow is delayed which causes opportunity cost for the later availability of the capital.

• Generally, contracts define a cash penalty for not meeting due dates. A typically value is 1% of the contract sum for each week (cf. Kolisch and Rinne [192]).

• Customers lose good will when the company is not able to deliver the product in time. Customer good will and company reputation for punctual delivery are extremely important in the make-to-order business. Loss of customer good will is difficult to estimate but should, by no means, be underestimated.

For final assemblies we can quantify the above mentioned cost because we have the customer due date, for subassemblies we have the milestones as determined at the manufacturing planning level. Here, it is more difficult to quantify the cost associated with exceeding the milestone because, e.g., we might have total slack time which can be assigned to different assemblies (cf. Ziegler [340]).

We aggregate all above mentioned cost for each assembly p into a constant cost coefficient wp' Figure 7.3 gives the cost function when cost of wp is incurred for each period the assembly is finished later than the due date. The objective of operations scheduling is to minimize the sum of the cost caused by exceeding the due dates of assemblies.

In scheduling it is common to use a time measure instead of a cost measure (cf. Brucker [42] pp. 1, Blazewicz et al. [32] pp. 59, Pinedo [253] pp. 8). One reason is that assembler and foreman at the shop floor can better deal with time measures than with cost measures. Hence, instead of minimizing the overall cost, we can minimize the sum of the weighted tardiness where the weight of assembly p equals the cost coefficient wp.

Note that in contrast to, e.g., Agrawal et al. [4], Chen and Wilhelm [53], and Faaland and Schmitt [102)' we do not consider any earliness cost. This is due to the short-term character of our problem where holding cost of parts and availability cost of resource levels have already been determined at higher decision levels.


Cost

t

Figure 7.3: Tardiness Cost


On the operations scheduling level, time-based, vertical, and horizontal interdependencies (cf. Section 2.3) are present.

Time-based interdependencies are given because we cannot cut the problem with a time horizon T 2 0 into T independent decision problems.

Vertical interdependencies are given by the operation sequences as pictured in Figure 7.4. Operation sequences arise from the 'working plans' as generated by the engineering department.

There are three types of horizontal interdependencies which are depicted in Figure 7.4. These interdependencies arise from the joint demand for parts, assembly resources, and assembly area .

• Operations assemble parts or subassemblies to subassemblies or assemblies. Hence, the required parts, i.e, the part kit, have to be available before the operation can start. Due to the dynamic situation there are dynamic part arrivals. Interdependencies in this context arise from the fact that parts have to be assigned to operations which are then allowed to start while operations which do not receive parts have to be delayed. In Figure 7.4 parts are represented by boxes within the lower dashed rectangle. The location of the boxes on the time axis gives the time when parts become available for assembly. For example, two parts are available at time zero and two parts become available at


time 3, 6, and 9, respectively. Arrows leading from the dashed rectangle to the operations represent part demand of operations. The number left of an arrow gives the number of parts needed by the operation .

• Next, operations require resources (assemblers, tools) in order to be processed. Therefore we have interdependencies between all non precedence related operations which require identical resources, regardless if they belong to different assemblies or not. In Figure 7.4 resource interdependencies are indicated by the dashed boxes which embrace operations 4 and 3 as well as 9 and 12 .

• Finally, we have an interdependency between the assemblies due to the fact that each assembly has to be assigned to the assembly area which is limited in size. This interdependency is represented by the dashed rectangle which encloses all three assemblies.

Assembly 1 Assembly 2 Assembly 3

2 1

:Qf OJ OJ OJ - - - 1

t :;.

0 1 2 3 4 5 6 7 8 9 10 ~ _________________________________ J

Figure 7.4: Horizontal Interdependencies within Operations Scheduling



All assembly scheduling approaches documented in the literature take time-based and vertically interdependencies into account. Differences arise w.r.t. the consideration of horizontal interdependencies and the type of production system. With respect to the horizontal interdependencies we distinguish between approaches which consider scarce capacity of assembly resources, scarce capacity of assembly resources and part availability, and scarce assembly area. With respect to the production system we distinguish project type systems with general precedence and resource constraints, job shop type systems with different machines, each capable of processing only one operation at a time, and single machine systems which can process multiple operations at a time. Table 7.1 gives a survey of the relevant literature.

Horizontal Type of Production System Interdependencies Project Shop Machine

Resources [4], [10] [102], [103], [142], [204], [211], [264], [255]

Resources and Parts [52], [53], [241]

Assembly Area [85] [192]' [212], [251], [271]

Table 7.1: Survey of Assembly Scheduling Literature

7.2.1 Job Shop Scheduling under Resource Constraints

Russell and Taylor [264] as well as Lalsare and Sen [204] investigate the efficiency of different priority rules in a dynamic fabrication/assembly shop under different objectives, among them mean flow time and mean tardiness. Potts et al. [255] consider a deterministic two-stage assembly system, a generalization of the two-machine flowshop problem, where m fabrication machines at the first stage produce m differ-


ent components which are assembled on a single assembly machine at the second stage. Faaland and Schmitt [102, 103] consider a fabrication/assembly problem where operations have to be sequenced on machines such that earliness and tardiness cost are minimized.

7.2.2 Project Scheduling under Resource Constraints

Agrawal et al. [4] present the problem of 'just-in-time' production of large-scale assemblies where final assemblies, each one consisting of a number of operations, have to be scheduled on work centres with identical machines. Each operation is processed on one machine of one work centre. The objective is to minimize the maximum makespan of all final assemblies. Anwar and N agi [10] extend the approach of Agrawal et al. [4] by integrating the scheduling decision with lotsizing. Kolisch [186] shows that this problem can be modelled and solved as resource-constrained project scheduling problem (RCPSP). Hence, the more general resource-constrained multi-project scheduling problem (RCMPSP) is suited for this type of problem, too. A survey of RCMPSP approaches is given in Section 3.3. The RCMPSP with the objective of minimizing the sum of the weighted tardiness (L: wp . Tp) is treated, e.g., by Kim and Schniederjans [175], Kurtulus and Narula [202]' Lawrence and Morton [210], Moccellin [232], Mohanty and Siddiq [233, 234], Norbis and Smith [243], and Patterson [249]. De Boer et al. [70] consider the case where part pegging is done, i.e., parts have already been assigned to operations. Then, the availability of kits can be modelled by release dates for operations.

Brimberg et al. [39] consider the problem of scheduling workers in a constricted area such that the makes pan is minimized. This problem arises for maintenance programs of military aircrafts where only a limited number of workers can work simultaneously in the cockpit. A number of non-precedence related operations, each defined by the number of workers required and the processing time, have to be executed. The aim is to schedule the operations under the constraint that the number of workers which are allowed to work simultaneously is restricted. This is a special case of the 2-dimensional packing problem (cf. Israni and Sanders [165]) and the RCPSP (cf. Hartmann [146]).

7.3. MODEL 101

7.2.3 Project Scheduling under Resource and Part Avail-ability Constraints

Chen and Wilhelm [52, 53] consider the kitting problem of multiechelon electronic assembly. A set of customer orders consists of operations which are connected in an assembly tree by precedence constraints. To be processed, an operation requires a part kit and a capacity of '1' in the shop where the operation is assembled. The availability of assembly area is not considered. The objective is to schedule operations such that the sum of order tardiness and inventory holding cost of operations is minimized. Neumann and Schwindt [241] employ activity-on-node networks with minimal and maximal time lags in order to model scheduling problems of make-to-orderproduction. Amongst other topics they treat the 'assignment sequence problem' where the parts emerging from a fabrication operation have to be assigned to assembly operations such that the overall makespan is minimized.

7.2.4 Scheduling under Spatial Constraints

Literature on scheduling under spatial constraints is rare. Drexl et al. [85] propose a Leitstand system which checks feasibility of spatial resource constraints. The approaches of Kolisch and Rinne [192]' Schlauch and Levy [271], Lee et al. [212], and Petersen [251] try to assign assemblies to the shop floor in order to optimize different timebased performance measures such as weighted delay.

7.3 Model

From the literature review it can be concluded, that the problem as outlined in Section 7.1 has not yet been treated. Therefore, we will now present a new mathematical programming formulation which has been recently proposed in Kolisch [188].

The above outlined problem can be modelled in two different ways, detailed or aggregated. Detailed modelling employs binary decision variables which explicitly assign resources and parts to operations as well as assembly areas to assemblies. Such an approach is, e.g., undertaken by Agrawal et al. [4] for modelling large-scale just-in-time


assemblies. The drawback of this approach is the great number of binary variables (cf. Kolisch [186]). Hence, we depart from this approach and propose a model where operations scheduling is done subject to aggregated availabilities of resources, assembly area, and parts, respectively. This is the same approach as done for the resource constraints in multi-project scheduling problems (cf. Section 3.3) and for part availability constraints in the assembly kitting problem (cf. Chen and Wilhelm [53]). Once the operations scheduling has been done,. the detailed assignment of resources, parts, and assembly area is a rather trivial task. We will outline this in Subsection 7.4.5.

7.3.1 Detailed Description and Notation

Before we can propose the model, we have to describe the operations scheduling problem in more detail and we have to introduce the relevant notation. A survey of the notation can be found in Appendix B.3.

The number of assemblies to be processed is P 2: o. Each assembly is exploded into a set Jp of precedence related operations (cf. Figure 7.2). For operation j (j E Jp; P = 1, ... , P) Pj denotes the set of all immediate predecessor operations which have to be processed before operation j can start. The processing time of operation j is Pj 2: O.

The entire assembly has to take place on an assembly area, which is typically the shop floor. Since the shop floor area is limited, not more than CS 2: 0 assemblies can be processed at each time instant t. Processing of assembly P begins as soon as the first operation belonging to the assembly is started and it commences until the last operation belonging to the assembly has been finished. During the entire time interval the assembly occupies part of the shop floor. Assembly preemption, i.e., the removal of a partly assembled product from the assembly area, is not allowed because of high set up cost and the risk of damage.

Assembly resources are assemblers as distinguished by specific qualifications, e.g., welder and shipbuilder in the ship assembly (cf. Subsection 2.5.1) as well as tools and machinery. Overall there are RA 2: 0 different resources. Resource r (r = 1, ... , RA) has an availability of C~t 2: 0 units at time instant t. The time varying capacity has different reasons. First, it stems from the fact that planning is based

7.3. MODEL 103

on a rolling horizon, where resources may be tied to operations which have already been started and are still being processed. Second, time varying capacity is caused by planned off-time of workers (vacation, fluctuation) and planned down-times of machines (inspection, repair). Each operation j requires ct,r ~ 0 units of resource r during every period it is processed.

Parts which are built into assemblies are distinguished in A- as well as B- and C-parts (cf. Tempelmeier [304] pp. 12, Drexl et aI. [85]). B- and C- parts are, e.g., bolts, screws, hydraulic elements, and rollers which are common to most of the products and are available from stock. A-parts are components which have been fabricated or ordered specifically for planned orders. Delayed A-parts will disrupt or delay the assembly and hence need special management attention (cf. Vaart et al. [314], Nof et al. [242]). We denote with I ~ 0 the number of different A-parts. For each part i (i = 1, ... , I) the quantity available at time 0 is ni,O ~ O. Additionally, the number of parts which will become available at time instant t is ni,t ~ O. The cumulated number of parts which will become available until time instant t is Ni,t = I:~==o ni,r' The amount and timing of incoming parts are known from vendor contracts, released orders, and production schedules ofinhouse part fabrication. Operation j requires qj,i ~ 0 units of part type i. Before an operation can be initiated, the kit of all required parts must be available.


We depict the operations scheduling problem by means of a precedence network where nodes represent operations and technological precedence relations are given by arcs. Each precedence relation between operation h and its immediate successor j is depicted by an arc h -t j with weight th,~n = 0 representing a minimal time lag between the finish time of job h and the start time of job j.

For all assemblies which do not have a unique start (terminal) operation, we insert an artificial start (terminal) operation, e.g., for the three assemblies given in Figure 7.4 we have to insert the operations 1 and 6 for assembly 1 and 2, respectively, in order to derive the network represented in Figure 7.5. In general, assemblies do have a single terminal operation which may depict the final assembly, the disassembly


of the product in order to be shipped to the customer, or the final test and quality check. The operations of assembly pare I\ow topologically ordered and consecutively labelled such that we have operations 8 p , ••• , ep where 8 p denotes the unique start operation and ep denotes the unique terminal operation. We assign the start operation of the first assembly 81 the labell, the start operation of the second assembly 82 the label el + 1 etc. This way, the label of the terminal operation of the last assembly ep accounts the overall number of operations which is denoted with J. Also, for each operation j all predecessor operations hE Pj do have a lower label than j, i.e., h < j (j = 1, ... , J; h E Pj).

We introduce a dummy source 0 and a dummy sink J + 1 of the integrated network. Also we insert minimal finish-start arcs from the source 0 to each start operation 8 p and from each terminal operation ep to the sink J + 1. The arcs from the source have weight 0 while the arcs from the terminal jobs ep to the source have the weight tr;:,i~+1 = dmax _ dp where dmax denotes the maximal due date of all assemblies.

Based on the network representation we can calculate earliest start times ESj and latest start times LSj for each operation j by forward recursion from ESo = 0 and backward recursion (cf. Elmaghraby [94]) from LSJ = T where T denotes an upper bound for the finish time of all assemblies. T can be derived with one of the heuristics presented in Chapter 10. For the time being we can apply a very simple upper bound by adding the sum of all processing times to the latest part arrival time.

Example

Figure 7.5 gives the precedence network which results from the example provided in Figure 7.4. Each node j represents an operation; the three numbers above the node give the processing time Pj, the capacity demand ctl for the single resource, and the quantity gj,l of the single part needed for assembly. The arrival quantities and times of the single part as given in Figure 7.4 are nl,O = 2, nl,3 = 2, nl,6 = 2, nl,9 = 2, and nl,t = 0 for t E {I, 2, 4, 5, 7, 8} and t ~ 10. The numbers above the arcs give the minimal time lags; all entries with value 0 have been omitted. The due dates for the assemblies are dl = 8, d2 = 6, and d3 = 7. The weights of the assemblies are WI = 2, W2 = 3, and w3=4.

7.3. MODEL 105

2,2,1 3,3,1

A Pj, Cj,l, qj,l . tmIn r?\ "j,k_ r7:\ ~

Figure 7.5: Network Representation for Operations Scheduling


We assume that events, i.e., the delivery of parts and the change of available capacity, occur at discrete multiples of a standard period length, e.g., a shift or half shift, and that processing times and minimal time lags are discrete multiples of the standard period length. In this case, all operation start times will also be discrete multiples of the standard period length. We employ two types of decision variables. First, the binary decision variable Xj,t = 1, if operation j is started at time instant t, and 0, otherwise (cf. Pritsker et al. [256]). That is, Xj,O = 1 denotes that operation j starts at time instant 0, ends at


time instant Pj and is processed during periods 1, ... , Pj. The second decision variable is Tp ~ 0, the time span order p is tardy. Denoting with wp the weight of order p, the operations scheduling problem can now be modelled as follows.

p

Minimize Z = L wp • Tp p=l

s.t.

LSj

L Xj,t = 1 t=ESj

LSj LSh

Lt. Xj,t - L (t+ Ph) 'Xh,t ~ t~~n t=ESj t=ESh

J t

L L ctr' Xj,T ::; C~t j=l T=max{O,t-pj+l}

P t

L L (XSP'T - Xep,max{O,T-Pep}) ::; C S

p=lT=O t J

L L qj,i • Xj,T ::; Ni,t T=Oj=l

LSep

L (t + Pep) • Xep,t - Tp ::; dp t=ESep

Xj,t E {O, 1}

(7.1)

j = 1, ... ,J (7.2)

j= 1, ... ,J (7.3)

hE Pj

r = 1, ... ,RA (7.4)

t = 0, .. . ,T - 1

t =O, ... ,T (7.5)

i = 1, ... ,! (7.6)

t =O, ... ,T

p=l, ... ,P (7.7)

j = 1, ... ,J (7.8)

t = ESj, ... ,LSj

p= 1, ... ,P (7.9)

The objective function (7.1) minimizes the sum of the weighted tardiness. Constraints (7.2) demand that each operation is processed exactly once with a start time within its time window. Constraints (7.3) enforce the precedence constraints as given by the network. Constraints (7.4) guarantee that the assembler capacities are not overridden. Note that capacity checks are made at the beginning of periods, i.e., at time instants 0, ... , T - 1. Constraints (7.5) ensure for every


time instant the spatial capacity imposed by the limited availability of the assembly area. Constraints (7.6) impose the part availability constraints: For each part i and each time instant t the sum of assembled units has to be less equal the quantity of the parts which has become available until t. Constraints (7.7) link for each assembly P the continuous tardiness variable with the binary start variable of the terminal operation. Whenever the finish time (t + Pep) • Xep,t of the terminal operation ep is greater than the due date dp, Tp is set to (t + Pep) • Xep,t - dp • Finally, (7.8) and (7.9) define the binary and continuous decision variables, respectively. Note, that one can model the problem with the Xj,t variables solely. But employing the Tp variables makes the objective function more handy.

Solving the example instance with model (7.1) - (7.9) we obtain the schedule S = (6,6,9,8,12,0,3,0,1,5,0,3) which is pictured as Gantt-chart in Figure 7.6. Assembly 1 is 6 periods tardy, assembly 2 is 2 periods tardy, and assembly 3 is finished ahead of the due date. Hence, the objective function value is Z = 2·6 + 3·2 + 4· ° = 18. This objective function value is optimal.

Assembly 1

8 9

11

7

12

10 I Assembly 2

Assembly 3

I I ~ t

o 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Figure 7.6: Optimal Solution of the Example Instance


We now discuss properties, special cases, and extensions of the model (7.1) - (7.9). We begin with a complexity analysis.


7.4.1 Special Cases and Complexity Results

If the constraints (7.4) - (7.6) are relaxed, the optimal solution of the operations scheduling problem is to start each operation j at its precedence feasible earliest start time ESj. If we add one of the constraints (7.4) - (7.6) to the relaxed problem (7.1) - (7.3), (7.7) - (7.9) we obtain each time a different NP-hard optimization problem, namely the resource allocation problem, the assembly area allocation problem, and the part allocation problem. Let us look at these special cases in more detail.

The Resource Allocation Problem

If we add constraints (7.4) to problem (7.1) - (7.3), (7.7) - (7.9) we obtain the resource-constrained multi-project scheduling problem (RCMPSP) which has been outlined in Section 3.3. In the RCMPSP we have to allocate for each resource r the available capacity at time instant t to the operations which are processed at t. The RCMPSP with the objective of minimizing the sum of the weighted tardiness is treated in Kim and Schniederjans [175], Kurtulus and Narula [202], Lawrence and Morton [210], Moccellin [232]' Mohanty and Siddiq [233,234]' Norbis and Smith [243], and Patterson [249]. As a generalization of the classical job shop scheduling problem, the resource-constrained multiproject scheduling problem is an NP-hard optimization problem (cf. Blazewicz et al. [33]).

The Assembly Area Allocation Problem

If we add constraints (7.5) to problem (7.1) - (7.3), (7.7) - (7.9) we obtain the parallel machine scheduling problem C S II I: wp . Tp which is considered by, e.g., Chen and Powell [57]. Here, each project p is treated as one 'job' p with a processing time ESep which equals the longest path of project p, weight wp , and due date dp . All P jobs have to be scheduled on CS identical machines where each machine can process no more than one job at a time. The objective is to schedule the jobs such that the sum of the weighted tardiness I: wp . Tp is minimized. Problem C S II I: wp . Tp is NP-hard in the strong sense (d. van den Akker et al. [313]).


The Part Allocation Problem

If we add constraints (7.6) to problem (7.1) - (7.3) and (7.7) - (7.9) we obtain the part allocation problem (cf. Balakrishnan et al. [19]). Part allocation is concerned with the allocation of currently available and incoming parts to operations. Carlier and Rinnooy Kan [48] show that the part allocation problem can be solved with a polynomially bounded algorithm if P = 1 but becomes NP-hard for P > 1.

Since each of the allocation problems is NP-hard, the operations scheduling problem (7.1) - (7.9) is NP-hard, too.

7.4.2 Left-Regularity of the Objective Function

The operations scheduling problem has an important property which reduces the size of the solution space considerably. According to Sprecher et al. [291] we give the following definition.

Definition 7.1 (Left-regular) Consider two feasible operation schedules S = (SI, .. . , SJ) and S' = (Sf, ... , SJ) for problem {7.1} - {7.9}. Sj denotes the start time of operation j. Let Sj ~ Sj for a single operation j and Si = S: for i = 1, ... , j - 1, j + 1, ... , J. The objective function of a minimization problem is left-regular, if Z(S) ~ Z(S') holds, where Z(S) denotes the objective function value associated with the schedule S.

Theorem 7.1 The objective function {7.1} of the operations scheduling problem is left-regular.

Proof Consider a feasible schedule S where a non-terminal operation j has the start time Sj. Changing, c.p., the start time to Sj-t with t ~ o cannot increase the objective function value since the latter is only affected by the start times of the terminal operations. Consider now that the start time Sep of the terminal operation ep is, c.p., decreased by t ~ 0 periods to Sep - t. If Sep ~ dp holds, the objective function value does not change while for Sep > dp , it is decreased by t . wp

units. 0


7.4.3 Part Pegging

'Pegging' is the task of dedicating parts to the assembly (cf. Section 2.2, Steiner and Yeomans [298]). The operations scheduling model (7.1) - (7.9) does not assume part pegging, Le., the assignment of parts to operations is a decision to be made by the model. If pegging takes place, this can be depicted in model (7.1) - (7.9) as follows: The part availability constraints (7.6) are dropped and ESj, the earliest start time of operation j, is modified by setting it to the maximum of the network-based earliest start time as calculated by forward recursion and the latest arrival time of the parts assigned to j. Another possibility would be to model release dates as it is done by de Boer et al. [70] for the RCMPSP.

Balakrishnan et al. [19] model the assignment problem of parts to products in an assemble-to-forecast environment.

7.4.4 Further Issues of Spatial Demand

So far we have treated the case that each assembly requires 1-dimensional equal static space on the shop floor. This case is depicted in Figure 7.7 (a). More general assumptions are

• 1-dimensional unequal static demand (cf. Figure 7.7 (a)),

• 2-dimensional rectangular static demand (cf. Figure 7.7 (b)),

• 2-dimensional polygon static demand (cf. Figure 7.7 (c)), and

• 2-dimensional polygon dynamic demand (cf. Figure 7.7 (d)).

Unequal shop floor demand arises when there are assemblies of different sizes, e.g., a subassembly needs less assembly area than the final assembly. Dynamic demand for assembly area appears, e.g., in the assembly of machine tools where the spatial demand of the assembly increases as it is processed (cf. Drexl et al. [85], Drexl and Kolisch [87]). Let us consider the four cases depicted above in more detail.

I-Dimensional Unequal Static Spatial Demand

This case can be taken into account by model (7.1) - (7.9) straightforwardly by denoting with c; the amount of spatial capacity needed

7.4. DISCUSSION OF THE MODEL

I I I I I I I I I I I I I I I I I I

I I I I I I I I I

I I I I

---------1 I

111

(a) 1-dimensional (un)equal static

(b) 2-dimensional rectangular static

-

( c) 2-dimensional polygon static

-t=2

(d) 2-dimensional polygon dynamic

Figure 7.7: Types of Assembly Area Demand

for assembly p. The spatial capacity constraints (7.5) can then be reformulated as follows.

P t

2: L~>: . (XSP'T - xep,max{o,'T-pep}) "5. C S t = 0, ... , T (7.10) p=l'T=O

2-Dimensional Rectangular Static Spatial Demand

This case has been considered by Kolisch and Rinne [192] for the case where each assembly p comprises a single operation, i.e., sp = ep (cf. the assembly area allocation problem outlined above). The model employs three types of decision variables, Sp, the start of an assembly as well as (Xp, Yp), the coordinates of the assembly on the shop floor.

2-Dimensional Polygon Static Spatial Demand

Lee et al. [212] as well as Petersen [251] have treated this case. An example of the placement of different assemblies on the shop floor of


a ship assembly is given in Figure 2.4. Petersen [251] describes the problem verbally and proposes a priority rule based algorithm which is assessed by means of simulation studies.

2-Dimensional Polygon Dynamic Spatial Demand

Drexl et al. [85] as well as Schlauch and Levy [271] propose a software system for this case. Within the approach of Drexl et al. [85], a spatial capacity check is performed after assemblies have been positioned manually. The method of Schlauch and Levy [271] tries to schedule assemblies based on simple priority rules while considering the spatial constraints. Details are not available since the system is proprietary.

7.4.5 Resource and Part Assignment

As pointed out in Section 7.3, we have modelled the problem such that the allocation of resources, parts, and assembly areas is done in an aggregated way. Model (7.1) - (7.9) assures only that for each time instant t the total number of resources, assembly areas, and parts used does not exceed the available amount. For a successful implementation of a schedule 5 = (51"'" 5J) on the shop floor we need to know precisely which of the RA resources are assigned to operation j during the processing interval 5j, ... , 5j + Pj - 1, which of the parts are assigned to operation j, and to which of the CS assembly areas assembly P is assigned. Altogether, there are RA + I + 1 independent assignment problems. Compared to the NP-hard allocation problems presented in Section 7.4, each of the assignment problems is rather trivial because operations scheduling has already been performed. In what follows, we will outline how to do the assignment for assembly resources, spatial resources, and parts, respectively.

Assignment of Assembly Resources

The resource assignment shall be clarified by Table 7.2 which reports the assignment for the example problem. As input data we need the operation start times 5j of the operations schedule 5 in increasing order. The first column of Table 7.2 gives the start times 0, 1, 3, 5, 6, 8, 9, and 12. Associated with each start time t we calculate At = {j I 5j ~ t < 5j + pj} the set of operations which are processed,


t At 1 2 k

3 4 0 {8,1l} 8 8 11 11 1 {9,1l} 9 11 11 3 {7,12} 7 12 12 12 5 {10} 10 10 6 {2,10} 10 10 2 2 8 {4} 4 4 4 9 {3,4} 4 4 4 3

12 {5} 5 5

Table 7.2: Calculation of the Resource Assignment

i.e., which are active, at t. The resource assignment begins in the first start period, which is for our example 0, and assigns the resources 1, ... , ct r to operation h which is the operation in At with the smallest , operation label. In the example, resources 1 and 2 are assigned to operation 8. Next, resources ct,r + 1, ... , ct,r + c1.r are assigned to the operation j with the second smallest label etc. When resources have been assigned to all active operations in the current period, we proceed to the next start period.

If the assembly resources have a time constant capacity, the algorithm can be refined in order to assure that to each operation the same resources are assigned for the entire processing time: Whenever a new start period is considered, operations which start in a prior period and are still active in the current period are considered first. To each of them, the same resources as in the prior period are assigned. Table 7.2 reports the resource assignment for all 8 start periods and Figure 7.8 visualizes the resulting assignment.

Note that the assignment of the same resource for the entire processing time of an operation cannot be assured in the case of variable resource capacity as it is the case for the operations scheduling problem (7.1) - (7.9). Figure 7.9 with C~ = (1,2,1) illustrates this. Operation 2 is processed by resource 2 in period 2 and by resource 1 in period 3.

114

4-

3

2

1

CHAPTER 7. OPERATIONS SCHEDULING

- --

11 2 I 3 I 12

4 8 10

9 7

I 3!0 t o 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Figure 7.8: Assembly Resource Assignment

I ~ t o 123

Figure 7.9: Assembly Resource Assignment in the Case of Variable Capacity

Assignment of Assembly Areas

The assembly area assignment can be treated as a resource assignment problem with resource capacity cf for each period t and P operations with start times Ssp, respectively. Using the above outlined algorithm we obtain the assembly area assignment given in Figure 7.10.

Assignment of Parts

Finally, the part assignment can be viewed as a transportation problem (cf., e.g., Winston [328] pp. 338 and Subsection 9.2.5) where each delivery time t with ni,t > 0 is a supply point with ni,t units of supply


CS

~i~ ___ or_de_r_3~1_-~~~o_rd_e_r1 ____ ~I--_ Order 2

i ~ t o 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Figure 7.10: Spatial Resource Assignment

and each operation j with qj,i > 0 is a demand point with demand qj,i' Supplying parts from time t to an operation j which is scheduled in Sj < t is forbidden and hence penalized with cost of 00 per unit. All other cost are set to 1 per unit. Any feasible solution of the transportation problem will provide a part assignment. Figure 7.11 gives the unique part assignment for the schedule of the example problem as represented in Figure 7.6. Arcs are leading from delivery times to the operations which are demanding parts. The weight on each arc gives the number of delivered parts.

2 2 2 2

Figure 7.11: Part Assignment

Part III

Solution Methods

Chapter 8

Order Selection Methods

In this chapter we propose a method for solving the order selection and scheduling problem proposed in Section 5.3.

8.1 Column-Generation Approach

We will propose an efficient solution method which employs a multiconstrained knapsack formulation of problem (5.1) - (5.8). For reasons to be seen below, we consider only assembly network structures where each task except the start and terminal task sp and ep , respectively, has exactly one successor and an arbitrary number of predecessors. In the scheduling literature this is referred to as an 'intree' (cf. Brucker [42] pp. 3).

8.1.1 Knapsack Formulation

Let Up ~ 0 be the number of possible schedules for order p. Schedule u (u = 1, ... , Up) uniquely defines the start time Sj for each task j = SP1"" ep of order p such that precedence constraints and time windows as given by the customer are obeyed. If order p is performed in schedule u, a capacity demand of cp,u,r,t ~ 0 units is imposed in department r and period t. An example of possible order schedules for a single project with value VI = 200 which comprises 4 tasks is given in Figure 8.1. Each task is represented as a node and has to be processed within the same (single) department. The capacity demand of each task j is given as vector (Cj,l, .•. , Cj,Pj) of length Pj above the node. Cj,n

120 CHAPTER 8. ORDER SELECTION METHODS

denotes the capacity demand in processing period n = 1, ... , Pj. Three possible schedules with capacity profiles (1,2,1,1,1,2), (1,1,2,2,0,2), and (2,3,1,2,0,0) are given below the precedence network.

I ;. I ;.

2 1-----''---1 4

3

I ;.

1 2 345 6 1 2 345 6 1 23456

Figure 8.1: Possible Order Schedules

We introduce the binary variable YP,u = 1 if order P is performed in schedule u and ° otherwise. Now, the following multi-constrained knapsack model can be formulated.

p Up

Maximize Z = L vp L YP,u p=l u=l

subject to

Up

LYp,u:::; 1 u=l P up

L L cp,u,r,t . Yp,u :::; Cr,t p=lu=l

YP,u E {O, 1}

(8.1)

P= 1, ... ,P (8.2)

r = 1, ... ,R t=l, ... ,T

(8.3)

P= 1, ... ,P u = 1, ... , Up

(8.4)

The objective function (8.1) maximizes the sum of the values of accepted orders. Constraints (8.2) restrict each order to be performed

8.1. COLUMN-GENERATION APPROACH 121

in at most one schedule and constraints (8.3) enforce that for each resource r and period t not more than the available capacity is used. Precedence constraints between the tasks are implicitly considered in the schedules and thus not represented in the model. Table 8.4 gives the MPS-matrix for the general case of P orders. The three schedules provided in Figure 8.1 are represented in columns 2 - 4.

P 1 .. P RHS

u 1 2 3 U1 .. .. .. .. Z 200 200 200 200 .. .. .. .. (8.1)

1 1 1 1 < 1 (8.2) -

< 1 .. .. -t=l 1 1 2 .. < G1,1 -t=2 2 1 3 .. < G1,2 -t=3 1 2 1 .. < G1,3 (8.3) -t=4 1 2 2 < G1,4 .. -t = 5 1 0 0 .. < G1,5

t=6 2 2 0 .. < G1,6 -1 E {0,1}

(8.4) .. E {0,1}

Table 8.1: MPS-Matrix

The problem formulation (8.1) - (8.4) poses two difficulties. First, (8.1) - (8.4) is a multi-constrained knapsack problem which is well known to be NP-hard (cf. Garey and Johnson [117] p. 247). Second, even generating the MPS-matrix of the problem may be a tedious thing to do since the number of schedules and thus columns in the matrix grows exponentially with increasing dp •

Let us assume for now that we can relax the problem (8.1) - (8.4) by letting the binary decision variables Yu,P E {O, 1} be continuous, i.e., o :S Yu,P :S 1. This way, we can solve even large sized problems with the simplex method in a reasonable amount of time. To overcome the problem of an exponential number of columns we restrain from enumerating all schedules explicitly. Instead, we generate the schedules implicitly by column generation (cf. Bradley et al. [37] pp. 540). Let us assume that a subset of the schedules are included in problem


(8.1) - (8.3) and that we have solved the linear program to optimality which gives us the variables in the basis (Yu,P > 0) as well as the shadow prices trp for the 'at-most-one-schedule-per-order' constraints (8.2) and the shadow prices "'r,t for the capacity constraints (8.3). The question is now, are there any schedules currently not represented in our LP which might improve the objective function value (8.1)? Let us assume that there is such an order P with associated schedule u. To be in the basis its reduced cost

R T ,p,u = YP - L L cp,u,r,t • "'r,t - 7rp r=lt=l

has to be greater than zero.

(8.5)

8.1.2 Column Generation by Dynamic Programming

We propose a dynamic programming approach which finds for given shadow prices trp and "'r,t the order with associated schedule which has the maximum reduced cost ,max. If ,max> 0 holds, we add this schedule to the ones which have been generated. Otherwise, i.e., for ,max ~ 0, there is no order-schedule combination which could improve the objective function value and we can stop the generation process.

We define the decision variable fi (t) to be the minimum cost of scheduling task j and all its (direct and indirect) predecessors tasks if task j has to be started until period t, i.e., can be started within the time window [ESj, min {t, LSj}]. ESj and LSj are the earliest and latest start times which can be derived by forward and backward recursion (cf. Section 5.4.3). For the start task sp of order P we initialize the recursion as follows:

Jsp(t) = 0 (8.6)

For all other tasks j = sp + 1, ... , ep of order p the recursion is:

fi(t) = T~l~, {T+f-l "'rj,s· Cj,(S-T+1) + ,L Ji (r - Pi - trjn)} J S=T tEl'j

(t = ESj, ... , LSj) (8.7)

We end up with the minimum cost of order pas Jep (LSep). We choose the schedule which maximizes the reduced cost

,max = max {vp - Jep (LSep ) - trp I p = 1, ... , p} (8.8)

8.1. COLUMN-GENERATION APPROACH 123

If ')'max ::; 0 holds, there is no further schedule which can improve objective function value (8.1). Table 8.2 shows the calculation of the fj (t)-values for the example provided in Figure 8.1 and given "'l,t. The resulting schedule S = (1,3,3,6) with minimal cost of 14(6) = 6 can be found by backtracing.

t 1 2 3 4 5 6

"'l,t 2 3 1 0 2 1 fI(t) 0 0 0 - - -

f2(t) 8 5 1 1 h(t) 6 4 3 - - -

f4(t) - - - 11 9 6

Table 8.2: Dynamic Programming Example

8.1.3 LP-Based Heuristic

What remains to do once we have generated all relevant columns of the LP (8.1) - (8.4) is to solve the MIP (8.1) - (8.4) to optimality by using standard branch-and-bound. Unfortunately, the MIP (8.1) - (8.4) is NP-hard and thus cannot be solved to optimality for large problem instances. Hence, we employ a LP-based heuristic similar to the ones proposed by Barnhart et al. [23] and Maes et al. [221] in order to select a portfolio of promising order-schedules. We first select all orders p* with schedule u* for which yp.,u. = 1 holds. Iffor none of the columns yp.,u. = 1, we choose the column (p*, u*) according to

yp. u. = max{ypu I Cpurt < Crt, r = r, ... ,R; t = 1, .. . ,T} (8.9) , (p,u) , , "- ,

Afterwards, we delete all columns and constraints (8.2) corresponding to the selected orders and adjust the right hand side of the capacity constraints (8.3). Since the LP (8.1) - (8.4) has been altered, there might be columns with positive reduced cost which have not been generated yet. Hence, we generate new columns and add them to the LP until ')'max ::; 0 holds. We stop if, after adding new columns, none of them can be chosen because of the resource constraints.


8.2 Experimental Evaluation

8.2.1 Test Instances

For evaluation purposes, a set of 180 test instances was generated with a parameter controlled instance generator which builds up on ProGen (cf. Kolisch et al. [193,194]). Details are provided in Appendix A and A.2, respectively.

Systematically Varied Problem Parameters

Four systematically varied problem parameters were employed. The instant size, the resource strength, the network restrictiveness, and the due date factor. Details of these parameters are provided in Appendix A.2.

Three different instance sizes (IS) were generated: (S)mall, (M)edium, and (L)arge. For each level, the number of orders and the number of tasks per order were set as follows: Small instances have 3 orders each with the task number in the range [4,6], medium sized instances have 5 orders with 5 to 15 tasks, and large sized instances have 10 orders with 5 to 15 tasks.

The resource strength RS E [0, 1] measures the tightness of resource constraints (5.3). The smaller the RS value is, the tighter the resource constraints are. The resource strength was set to 0.1, 0.5, and 0.9, respectively.

The network restrictiveness NR E [0, 1] indicates how close a precedence network of order p is to a serial network (NR=l) or a parallel network (NR=O). NR was set to 0.5 and 0.8.

Finally, the due date factor DF E [0,1] measures how close the earliest due date of an order p is to its earliest finish time. DF was set 2 and 3.

Non-Systematically Varied Problem Parameters

For each instance, the non-systematically varied problem parameters were set as follows.

The processing time for each task was randomly drawn from the interval [1,5]. The capacity demand Cj of activity j w.r.t. resource rj

8.2. EXPERIMENTAL EVALUATION 125

was randomly drawn out of the interval [1,5]. The value factor VFwas set to 0.5, and the time window factor TWF was set to 0.3.

Experimental Design

Realizing a full factorial design with the four above given systematically varied problem parameters and 5 replications for each level combination we obtained 2 . 3 . 2 . 3 . 5 = 180 test instances.

8.2.2 Computational Results

For each test instance we measured three variables. The percentage deviation of the solution obtained with the LP-based heuristic from the optimal solution (LPH), the optimality gap (OGAP), i.e., the percentage deviation of the optimal solution from the LP-based lower bound, and the solution gap (SGAP), i.e., the percentage deviation of the solution obtained with the LP-based heuristic from the lower bound. For solving (8.1) - (8.4) and its relaxation, we employed the CPLEX callable library [31].

Tables 8.3, 8.4, 8.5, and 8.6 report on the influence of each systematically varied problem parameter on the three variables LBH, OGAP, and SGAP, respectively. The first column gives the mean over all levels, the following columns present the mean for each level of the problem parameter under consideration. The final column 'Sig' indicates with '*' when there is a significant influence of the problem parameter under consideration on the variable at the 1% level of confidence. Testing was done with the nonparametrical Mann-Whitney test for parameters with two levels and the Kruskal-Wallis test for parameters with three levels. Nonparametrical tests were preferred because inspection of the distribution functions revealed that none of the values was normally distributed (cf. Alvarez-Valdes and Tamarit [7]).

Influence of the Resource Strength

The influence of the resource strength is given in Table 8.3. With increasing RS-level, the performance of the LP-based heuristic improves. In other words, the less problems are resource-constrained the better the heuristic solution. This effect is not significant. The optimality


gap and the solution gap decrease with increasing RS-Ievel which indicates that the lower bound improves when problems become less resource-constrained. The influence on the solution gap is significant.

Variable Mean RS Sig 0.1 0.5 0.9

LPH 9.60 10.90 9.17 8.67 SGAP 31.96 39.55 33.18 23.13 * OGAP 53.25 69.40 49.98 40.38

Table 8.3: Effect of the Problem Parameter RS

Influence of the Instance Size

Table 8.4 shows that an increasing instance size leads to a slight deterioration of the performance of the LP-based heuristic. The solution gap and the optimality gap increase significantly when the instance size is enlarged.

Variable Mean IS Sig S M L

LPH 9.60 7.59 10.69 10.51 SGAP 31.96 24.22 28.72 42.91 * OGAP 53.25 39.50 52.52 67.73 *

Table 8.4: Effect of the Problem Parameter IS

Influence of the Due Date Factor

An increasing due date factor DF gives way to slightly better results for the LP-based heuristic. As can be seen from Table 8.5, this effect is not significant. The LP-based lower bound loosens with growing DF. This can be seen by the increase in the solution and the optimality gap, respectively.


Variable Mean DF Sig 2 3

LPH 9.60 10.05 9.15 SGAP 31.96 28.70 35.20 OGAP 53.25 49.94 56.56

Table 8.5: Effect of the Problem Parameter DF

Influence of the Network Restrictiveness

Table 8.6 shows the influence of the network restrictiveness. An increasing NR-Ievelleads to a slight increase of all three variables. This is caused by a deteriorating quality of the LP-relaxation. The effect is not significant.

Variable Mean NR Sig 0.5 0.8

LPH 9.60 9.12 10.08 SGAP 31.96 27.55 36.35 OGAP 53.25 47.11 59.39

Table 8.6: Effect of the Problem Parameter NR

Chapter 9

Manufacturing Planning Methods

The focus of this chapter is on methods for solving the manufacturing planning problem. We will propose a rather simple construction heuristic first and a Lagrangian-based approach to obtain lower and upper bounds for the objective function value thereafter. All methods will be assessed on a set of systematically generated benchmark instances.

9.1 Construction Heuristics

We introduce a simple heuristic for solving the manufacturing planning problem (6.2) - (6.11). The method decomposes the problem into its subproblems, the assembly scheduling problem and the fabrication lotsizing problem. A two-level approach first determines an assembly schedule and afterwards lotsizes for fabrication. The coordination of the levels is in a strict top-down fashion with no feedback from the subordinate level. Also, there is no explicit feasibility anticipation on the first level, but only a cost anticipation (cf. Section 2.4, SchneeweiB [273]). On both levels, we perform a backward oriented planning approach.

130 CHAPTER 9. MANUFACTURING PLANNING METHODS

9.1.1 Outline of the List Scheduling Heuristic

A very popular approach to solve scheduling problems is forward oriented list scheduling (cf. Schutten [277]) which works as follows: First, the jobs of the scheduling problem are sequentially ordered in a list. Second, in the order given by the list, the jobs are scheduled at their earljest feasible start times. Here, we employ a backward oriented list scheduling heuristic (cf. Kim [176]) which schedules the jobs in the order given by the list at their latest precedence and resource feasible start times. Two issues need clarification. First, how to construct a list, and second, how to transform a list into a feasible schedule. Let us begin with the construction of a feasible list.

9.1.2 List Generation

A list 7r = 01, j2, ... , jJ] consists of the J non-dummy jobs where jg is the job at list position g. In order to transform a list 7r into a feasible schedule S = (S1, ... , SJ), the list has to respect the technological precedence constraints as imposed by the network. Recall Sj to be the set of immediate successors of job j. Then

(9.1)

states that job jg must have a greater list position than each of its direct successor jobs. By transitivity, the list position of job jg is also greater than each of its indirect successor jobs. Let £(0) = {J + I} and let £(g) = {J + l,jI, .. . ,jg} denote a set of jobs comprising the sink of the integrated network and all jobs which have been put to list positions 1, ... , g. Let further A(g) be the set of all available jobs which can be put at list position g. On account of Equation (9.1), A(g) can be defined as

A(g) = {j E .J I j ¢ £(g - 1) and Sj ~ £(g - I)} (9.2)

where .J is the set of all jobs in the network. That is, available jobs itself must not be on the list, while all their successor jobs have to be on the list. Denoting with v(j) a priority value associated with job j, we can give the list generation algorithm as follows.

I Assembly List Generation I A. Initialization: £(0) = {J + I}.

9.1. CONSTRUCTION HEURISTICS

B. Iteration: For 9 = 1 to J do

(1) Update A(g).

(2) Select jg E A(g) with v(jg) = max {v(i) liE A(g)}.

(3) Update £(g).

131

The initialization defines the set £(0) to include the sink of the integrated network only. Step (1) updates the set of available jobs. Step (2) selects one job from the available set and puts it on list position jg. In case of ties, Step (2) selects the job on the basis of 'first come first serve' (FCFS) where the job is preferred which has been first in the set A(g). Further ties are resolved by picking the job with the smaller job number. Step (3) updates the set of jobs on the list.

Priority Rule PRI PR2 PR3

v(j)

Table 9.1: Priority Rules for List Generation

To select one job from the available ones, we need to determine the priority value v(j) associated with job j. We have employed the three rules given in Table 9.1 which calculate the holding cost per period of job j itself, of job j and all its immediate predecessor jobs, and of job j and all its predecessor jobs. Pj denotes the set of all predecessors of job j. Using priority rule PR3 for the four non-dummy jobs of the example instance which has been introduced in Subsection 6.3.2, we obtain the priority values v(l) = 0, v(2) = 5, v(3) = 10, v(4) = 20, and v(5) = 5. Table 9.2 details the list generation of the example instance when these priority values are employed. The resulting list is 1r = (4,3,5,2,1].

9.1.3 Schedule Generation

We now want to transform list 1r into a feasible schedule S which gives a start time Sj for each job j such that precedence relations,


9 1 2 3 4 5 [,(g - 1) {6} {4,6} {3,4,6} {3,4,5,6} {2,3,4,5,6}

A(g) {4,5} {2,3,5} {2,5} {2} {I} jg 4 3 5 2 1

Table 9.2: List Generation for the Example Instance

the assembly resource constraints, and the order due dates are taken into account. Given the list 1l', we can simply schedule the jobs in the order of the list at their latest feasible start times. To do so, we need to know C~t(g), the available capacity of the assembly resource r at time instant t when scheduling the g-th job from the list. For the job on list position 9 = 1 the available capacity C~t(1) is equal to the initial capacity C~t for all resources r = 1, ... , RA and time instants t = 0, ... , T. For the jobs on list position 9 = 2, ... , J the available capacity is dynamically updated according to

cA (g) = CA (9-1)-{ Cjg_l,r , if t E {Sj9_l' ... , Sj9_l + Pjg-l - I} r,t r,t ° else ,

(9.3)

for all resources r = 1, ... , RA and time instants t = 0, ... , T. The schedule generation algorithm can be given as follows.

I Assembly Schedule Generation I A. Initialization: SJ+1 = max {dp I P = 1, ... , P}, T = SJ+1 - 1, C~t(l) = C~t for r = 1, ... , RA and t = 0, ... , T.


(1) Calculate C~t(g) according to (9.3).

(2) Take the next job from the list: j = jg.

(3) Determine the dynamic latest start time LSj of j w.r.t. precedence constraints: LSj = min { Sk - tr.~n IkE Sj} - Pj.

9.1. CONSTRUCTION HEURISTICS 133

(4) Determine the latest resource feasible start time Sj of j which is

~ LSj: Sj(LSj) = max {t E {ESj, ... , LSi} IC~T(g) ~ Cj,r

for all r = 1, ... , RA; T = t, ... , t + pj - I}.

The initialization sets the start time of the sink to the maximal due date, sets the planning horizon to the time instant one period prior the start time of the dummy sink, and sets the available capacity equal the initial capacity. Afterwards, Steps (1) - (4) are performed J times. At iteration 9 (g = 1, ... , J), job jg is scheduled as late as possible. Step (1) updates the available capacity. Step (2) takes the next job j from the list. Step (3) determines the dynamic latest start time LSj according to precedence relations and due dates as depicted in the network. Finally, Step (4) searches for the latest start time of job j within the time window [ESj, LSj] where ESj is the static earliest start time of job j as calculated by forward recursion (cf. Elmaghraby [94]).

Table 9.3 details the scheduling algorithm when applied with list 1r = (4,3,5,2,1] to the example instance. The second column gives the available capacity Ct,t(g) of the single resource r = 1 for time instants t = 0, ... ,4. The generated assembly schedule is S = (2,2,3,4,2). It has assembly wide holding cost of 70 and is represented as Ganttchart in Figure 9.1. Note, that the figure does only depict jobs with non-zero processing times.

9 C~t(g) jg LS'· 2 ESj Sjg

1 (3,3,3,3,3) 4 4 1 4 2 (3,3,3,3,2) 3 3 ° 3 3 (3,3,3,1,2) 5 2 ° 2 4 (3,3,2,1,2) 2 3 ° 2 5 (3,3,0,1,2) 1 3 ° 2

Table 9.3: Schedule Generation for the Example Instance


5

2

o 1 2

3

3 4

4 1)10 t

5

Figure 9.1: Assembly Schedule Generated by the List Scheduling Heuristic

9.1.4 Property of the List Scheduling Heuristic

Analogously to the definition of left-active schedules (cf. Sprecher et al. [291]), we define

Definition 9.1 (Right-active schedule) A right-active schedule is a feasible schedule where none of the jobs can be started later without forcing at least one other job to start earlier.

Theorem 9.1 The schedule generation algorithm generates schedules which are right-active.

Proof The schedule generation algorithm generates right-active schedules by construction because each job j is scheduled as late as possible. o

Since the objective function (6.2) of the manufacturing planning problem is right-regular, there exists one list 1r* which is mapped by the schedule generation algorithm into an assembly schedule which is optimal with respect to the manufacturing planning problem (6.2) -(6.11). Note, that a given list 1r might lead to an infeasible solution of the assembly scheduling problem or the manufacturing planning problem.

9.1.5 Outline of Backward Oriented Lotsizing

Given an assembly schedule S, the time-phased part demand is

J

dS - ~ q" i,t - ~ 3,t j=l\Sj=t

(9.4)


and what remains is to solve RF independent CLSP.

There have been many heuristics suggested to solve the CLSP. A survey can be found in Hindi [162] and Maes and van Wassenhove [223]. We can divide heuristic approaches in construction heuristics, improvement heuristics, and mathematical programming based heuristics.

Construction heuristics build a feasible fabrication plan by proceeding from period to period and deciding via cost values about the lot sizes of parts in the current period. Examples of such approaches are given in Eisenhut [93], Dixon and Silver [79], Gunther [130], Lambrecht and Vanderveken [206], as well as Maes and van Wassenhove [222].

Contrary to the period-by-period approach, the heuristic of Kirca and Kokten [180] performs a part-by-part approach where it is proceeded from one part to the next and for each part a fabrication plan for the whole planning horizon is constructed.

Improvement heuristics start with a not necessarily feasible fabrication plan and try to alter it first to a feasible fabrication plan and afterwards to a fabrication plan with less total cost by shifting lots. Examples of improvement heuristics are given in Dogramaci et al. [81] and Karni and Roll [169].

Finally, mathematical programming based heuristics construct for each part multiple fabrication plans. The capacity feasible and cost minimal coordination of fabrication plans is obtained by employing each fabrication plan as column in a large-scale linear program. Examples are given in Cattrysse et al. [50] and Hindi [162]. A special case are approaches where, first, optimal setup patterns are found for the uncapacitated lotsizing problem and afterwards fabrication quantities are found by solving a transportation problem (cf. Thizy and van Wassenhove [309]).

Here, we will suggest a two-phase backward oriented period-byperiod heuristic. The first phase constructs a feasible fabrication plan and the second phase tries to improve this plan by greedily shifting lots. Both phases work in a backward oriented manner. The idea of backward oriented construction heuristics to solve capacitated lotsizing problems has been first given by Haase [137] and later extended by Kimms [177] for small bucket lotsizing problems. Its advantages are twofold. First, for the manufacturing planning problem, backward


planning is more intuitive than forward planning. Second, contrary to forward oriented lotsizing methods, we do not have to employ time consuming feasibility routines.

9.1.6 Lotsizing Generation Scheme

As pointed out above, the lotsizing heuristic comprises a construction and an improvement phase. The basic idea of the construction phase is to step backward from the latest demand period T to period 1. In each period t (t = T, ... , 1), we decide which of the parts with demand in t should be produced now and for which parts demand should be shifted into the next prior demand period. We make this decision based on a cost value. The second phase is similar to the first phase but with two exceptions. First, instead of demands, it is tried to shift fabrication lots and, second, shifting is not restricted to the next prior fabrication period but to all prior fabrication periods. In order to give a formal description of the algorithm let us introduce some notation. We begin with the calculation of the time-phased demand as given in Equation (9.4). The time-phased demand unfolds a cumulated capacity demand of GDr,t w.r.t. fabrication resource r between period 1 and period t

t I

GDr,t = L L cf· dr,r' (9.5) r=l i=llr[=r

The cumulated capacity GGr,t of resource r between period 1 and t is

t

GGr,t = L C;'r (9.6) r=l

Let now I be the set of parts, i.e., I = {1, ... , I}, and It be the set of parts for which there is a demand in period t, i.e, It = {i E I I dtt > , O}. Being in period t and looking 'backward' in time, we determine for part i E It the next period Ti < t where there is demand

Ti=max({O}Umax{TIT=1, ... ,t-1: dr,r>O}). (9.7)

If there is no more part demand in periods 1, ... , t - 1, then Ti is set to O. Let denote Bt the set of parts demanded in period t which can


be shifted backwards, i.e.,

where 0i is a part specific cost saving value whose calculation will be detailed in Subsection 9.1.7. We can now outline the backward lotsizing algorithm in a more formal way. Note that for a given assembly sched ule 5, we will use the time-phased demand array dr,t in a dynamic fashion. That is, the lot sizing heuristic will alter drt by moving and , deleting demands. Whenever a change in the demand matrix takes place, It, CDr,t, and Bt as defined above are updated, respectively.

I Backward Lotsizing I

A. Initialization: Set T equal to the latest demand period.

B. Construction Phase: For t = T downto 1 do

(1) While Bt -=I- 0 do

Select one i E Bt based on the cost saving value h drr = drr + drt, drt = o. , , , , , , Update In CDrF,r' and Br for T = 1, ... , t .

•

(2) While It -=I- 0 do

Select part i E It which maximizes ~. Ci

~~:e:m~~: {th~:,~:~:}t ~f i producable in t:

, Ci'

If Qi,t < dr,t and CCr[,t_l ~ CDr[,t_l + cf . dr,t and

hf . Qi,t < sf then Qi,t = O. If t> 1 then

d~t-l = dr,t-l + dr,t - Qi,t. di,t = O. Update I r , CDrF,r' and Br for T = 1, ... , t .

•


C. Improvement Phase

After the initialization has been completed, Steps (1) - (2) are performed T times. In each iteration t (t = T, ... , 1), Step (1) iteratively selects the part i from the set of backshiftable demands which maximizes the cost measure 8i and shifts the associated demand drt back to , the prior demand period ri. When there are no backshiftable demands left, Step (2) determines fabrication lots for the remaining parts with positive demand in period t. The selection criterion is the holding cost per used capacity unit hf / cf. If a demand drt cannot be entirely pro-, duced in t but has to be splitted, it is checked whether a cost saving can be achieved by shifting the entire demand drt to the prior period , t - 1.

The improvement phase builds up on the fabrication plan obtained after the T-th iteration has been performed. It systematically checks for each part i with positive lot size in period t (t = T, ... , 2), if the total fabrication cost can be reduced by shifting the production lot Qi,t to a production period r (r = t - 1, ... ,1). Each improvement of the production schedule is realized in a greedy fashion. Parts are considered in the order of increasing index i.

Table 9.4 shows the functioning of the backward lotsizing heuristic when applied to the example instance with the demands df = (0,5,0,5), dq = (0,0,5,0), and d~ = (0,5,0,0) as derived by applying Formula (9.4) to the assembly schedule given in Figure 9.1. As cost measure we employed the one of Dixon and Silver (cf. Section 9.1.7). Note that the entries for dr,t and dr,Ti in Step (1) represent the values after the step was performed. Once the construction phase is finished, there is no improvement phase for the example because each part is produced in a single period. The obtained lotsizing policy has overall cost of 55 which gives total cost for the manufacturing planning problem of 125. The production plan, comprising assembly schedule and fabrication lotsizes, is given in Figure 9.2.

9.1.7 Cost Considerations

In order to decide if part i shall be fabricated in the current period t or shall be backshifted to the next prior demand period ri, we have employed different cost values 8i . Namely, these are adapted cost cal-


Step (1) (2)

t It 8· St d$ dS Qi,t dS S 2 Ii ~ ~,t iT 2 i,t di,t-l I I

4 {1} 1 2 2 {1} 0 10 3 {2} {} 2 5 0 0 2 {1,3} {} 1 10 0 0

3 0 5 0 1 {3} {} 3 5 0

Table 9.4: Backward Lotsizing for the Example Instance

Assembly

Fabrication

~----~----~----~----~----~~~ t

o 1 2 3 4 5

Figure 9.2: Solution Obtained by the Construction Heuristic

culations of the period-by-period approaches of Eisenhut [93], Dixon and Silver [79], Gunther [130] as well as Lambrecht and Vanderveken [206]. Originally, all of them take into account setup and holding cost in a forward oriented period-by-period approach. Here, we will adapt them to our backward oriented approach. For each part i we will only compare two alternatives: production of the demand dft in the current period t and backshifting of the demand dft to the prior demand period Ii < t. Note that the demand vector df = (df l' ... , df T) for part , , i might be sparse and hence Ii and t might not be adjacent periods, i.e., t - Ii > 1.

The cost measure of Dixon and Silver [79] employs the Silver-Meal criterion [282] for the uncapacitated LSP which compares the average


cost per period when fabricating the demand dr,t in t or in Ti. Prod uction in t incurs per period cost of sf, production in Ti incurs per period cost of

t- Ti+1 (9.9)

The per period cost saving when producing in Ti instead of t is

F sf + hf . dr,t . (t - Ti) ~ - .

t- Ti+ 1 (9.10)

Dixon and Silver generalize this measure for the capacitated case by dividing it by the number of capacity units needed to produce the demand dr,t.

F sf' +hf' ·d~ .(t-T;) s. _. • .,t (Ws =' F t-~+I (9.11)

c· . d· t , t,

Eisenhut [93] employs the total cost saving when producing the demand drt in Ti instead of t. The total cost saving equals sf - hf . dr,t' (t -'Ti). This value is divided by (t - Ti + 1)2. cf . dr,t in order to derive the following cost measure.

sf - hf . d$ . (t - T') 6~I -' ",t' (9.12)

t - (t - Ti + 1)2 . cf . dr,t

Lambrecht and Vanderveken [206] extend the cost measure of Eisenhut as follows.

sf - hf . d$t . (t - T;)2 6~v = t ", '

, cf . dr,t . (t - Ti + 1) . (t - Ti) (9.13)

The cost saving value of Groff [129] has been proposed for the uncapacitated static lotsizing problem (LSP). The general idea is that in the classical economic order quantity model (EOQ) (cf. Erlenkotter [98]) the marginal decrease of the per period setup cost equals the marginal increase of the per period holding cost. Groff transfers this idea to the dynamic uncapacitated LSP and Gunther [130] extends the approach to the capacitated lotsizing problem. Adapting the idea of Gunther to the backward oriented approach, we derive the following cost measure.

2·sf' S ( ) .. 7#-h - di t' t - Ti + 1

69U = i ' , cf . d$

, "t (9.14)

9.2. LAGRANGIAN RELAXATION 141

Additionally, we have altered cost measure (9.14) slightly to depict the cost saving (CS) per covered period divided by the capacity demand.

(9.15)

To illustrate the calculation of the different cost measures, we consider iteration t = 4 of the backward lotsizing heuristic as given in Table 9.4. Here, we have i = 1, h[ = 1, s[ = 20, t = 4, 'i = 2, dr 4 = 5, and c[ = 1. According to the formulas given above, we ob-, tain the cost measures ops = 2, op = 2/9, orv = 0, o?U = 5, and ops = 1/3.

9.2 Lagrangian Relaxation

In order to assess the performance of the construction heuristics proposed in the preceding Section we need lower bounds as well as upper bounds different from the ones obtained by the construction heuristics. Hence, we will now propose a Lagrangian relaxation approach which will generate lower and upper bounds for the manufacturing planning problem. We begin with the generation of lower bounds in Subsections 9.2.1 - 9.2.4 and thereafter turn to the generation of upper bounds in Subsection 9.2.5. Finally, Subsection 9.2.6 is devoted to subgradient optimization.

9.2.1 Echelon-Cost MIP Formulation

We employ the concept of echelon stock (cf. Clark and Scarf [60]) to reformulate the objective function (6.2) of the manufacturing planning problem. The echelon stock concept measures the entire holding cost by multiplying for each item, i.e., part, subassembly, or assembly, the echelon stock with the echelon holding cost. The echelon stock is the amount of the item in the entire manufacturing system regardless if the item can be identified as such or has already gone into another item. The echelon holding cost considered here are the opportunity cost of the cash outflow associated with the production of that item.

We assume that the material ordered for part fabrication or parts procured for assemblies are purchased just-in-time. Hence, for part i


the echelon holding cost are equal the total holding cost hr. The echelon holding cost for assembly j, i.e., the assembly which is produced by job j, are hf - ~I=l qj,i . hr = hf. That is, the echelon holding cost of assembly j equal the holding cost of the parts purchased from outside suppliers.

If we assume that all in-house fabricated parts are produced in period t = 1, we have echelon holding cost for parts of

p ep I

"""'" """'" """'" Mi' . q' . . d 666 Z J,Z p (9.16)

Depending on Qi,t, the actual production period of the parts, we obtain the correct echelon holding cost as follows:

p ep I I T

L L Lhr 'qj,i ·dp - LLhr . Qi,t . (t -1) (9.17) p=l j=sp i=l i=l t=l

Note that the lotsize Qi,t is multiplied with (t -1) because we account holding cost from the period of production onwards. Since for each part i the amount produced, ~r=l Qi,t, equals the amount requested by the assembly jobs, ~:=1 ~j~sp qj,i, we can reformulate (9.17) as follows.

Assuming that all jobs are started at time instant t

echelon holding cost for the (sub-)assemblies of

p ep

L L hf . (dp + 1) p=l j=sp

(9.18)

o we have

(9.19)

The holding cost of procured parts are multiplied with (dp + 1) since parts have to be available in the period preceding the start time of the assembly (cf. Figure 6.6). With respect to to the actual start time of the jobs we obtain the correct echelon holding cost

p ep J LSj

L L hf . (dp + 1) - L hf L (t + 1) . Xj,t (9.20) j=l t=ESj


Since L:=1 Lj~sp hf = Lf=1 hf holds, we can delete the '+ l' in both sums and we obtain

p ep J LSJ

2: 2: hf . dp - 2: hf 2: t· X j,t (9.21) p=1 j=sp j=1 t=ESj

For both echelon holding cost functions, (9.18) and (9.21), the first term is constant since the due date for orders is fixed. Writing down the entire holding cost and decomposing the constant terms we obtain

p ~ I P ~

2: 2: 2: hf . qj,i . (dp + 1) + 2: 2: hf . dp

p=1 j=sp i=1 p=1 j=sp

I T J LSj

2: 2: hf . Qi,t • t - 2: hf 2: t· Xj,t· (9.22) i=1 t=1 j=1 t=ESj

{ESj, ... , LSj} ~ {O, ... , T} holds for each j. Hence, in Equation LS T

(9.22) we can replace Lt=l!;Sj t . Xj,t by Lt=Q t . Xj,t. Furthermore,

L;=Q t . x j,t is zero for t = 0 and hence we can write "£.;=1 t . X j,t

instead of L;=Q t· Xj,t. Taking this into account and adding the setup cost we end up with the echelon holding cost based objective function of the manufacturing planning problem.

I TIT

Minimize Z = 2: 2: sf . Yi,t - I: I: hf . Qi,t . t i=1t=1 i=1 t=1 , , .. ...

(1) (2)

J T

"" "" hI:' . t . x . t + const L.J L.J J J, _____

j=1 t=1 (4)

(9.23)

.. (3)

with

p ~ I P ~

const = 2: 2: 2: hf . qj,i' (dp + 1) + I: 2: hf . dp (9.24) p=1 j=sp i=1

The new objective function (9.23) has four terms. Term (1) accounts for the setup cost. Term (2) represents the savings of echelon holding cost for fabricating parts in t 2: 1 instead of t = 1. Term (3) represents


the savings of the echelon holding cost for starting jobs later than t = O. Finally, the const-value in term (4) represents the entire holding cost when all parts are fabricated in t = 1 and all jobs are started in t = O. Note that const is an upper bound of the entire holding cost.

The new objective function (9.23) accounts holding cost for parts in the entire manufacturing system. Hence, it suffices to have due dates, job start times, and part fabrication periods. The inventory variable for physical stock is not needed any more. Following Billington [28], we can eliminate the inventory variable /i,t from our model by replacing the dynamic inventory balance constraints (6.7) with

t t J

2::: Qi,T - 2::: 2::: qj,i • x j,T ~ 0 T=1 T=1 j=1

i = 1, ... ,/ t= 1, ... ,T

(9.25)

The new constraints (9.25) assure that for each part i and each period t the cumulated part demand does not exceed cumulated production.

We can now give the echelon holding cost based MIP formulation of the manufacturing planning problem with const as defined in Equation (9.24).

I TIT

Minimize Z = 2::: 2::: sf . Yi,t - 2::: 2::: hf . t . Qi,t i=1 t=1 i=lt=1

J T - 2::: 2::: hf . t . X j,t + const

j=lt=1

subject to

LSj

2::: (t + Pep) . xep,t ~ dp t=ESJ

LSj

2::: Xj,t = 1 t=ESj

LSJ LSh

2::: t· Xj,t - 2::: (t + Ph) • Xh,t ~ th,~n t=ESj t=ESh

P= 1, ... ,P

j = 1, ... ,J

j= 1, ... ,J hE Pj

(9.26)

(9.27)

(9.28)

(9.29)


J t

L L ct,r' Xj,r ~ C~t j=1 r=max{O,t-pj+1}

r = 1, ... , RA

t =O, ... ,T (9.30)

t t J

LQi,r - LLqj,i 'Xj,r 20 r=1 r=1 j=1

i = 1, ... ,1 t = 1, ... ,T

(9.31)

I

L C7' Qi,t ~ C;'t i=1Ir[ =r

r = 1, .. . ,RF

t =O, ... ,T (9.32)

J

Y· '""" q' . > Q. ~,t L...J J,~ _ ~,t

j=1

i = 1, ... ,1 t = 1, .. . ,T

(9.33)

Xj,t E {O, I} j = 1, ... , J (9.34) t = ESj, ... ,LSj

Yi,t E {O, I}, Qi,t 2 0 i = 1, ... ,1 t = 1, .. . ,T

(9.35)

Note that constraints (9.27) - (9.30) and (9.32) - (9.34) are identical with constraints (6.3) - (6.6) and (6.8) - (6.10) of the manufacturing planning problem as formulated in Chapter 6.

9.2.2 Decomposition of Assembly Scheduling and Fab-rication Lotsizing

In the next step we decompose the integrated problem (9.26) - (9.35) into an assembly scheduling and a fabrication lotsizing problem. To do so, we have to relax the coupling constraints (9.31) which links the flow of parts from the fabrication to the assembly. We employ the Lagrangian multipliers Ai,t and add

(9.36)

to the objective function (9.26). (9.36) has two terms. The first term depends on the scheduling decision within the assembly department, the second term depends on the lotsizing decision in the fabrication department. We now want to reformulate both terms such that we end up with coefficients which can be added to the objective function (9.26) .


We begin with the second term which reflects the fabrication.

ITt

L: L: Ai,t L: -Qi,T i=1 t=1 T=1

I I

- L: Ai,1 . Qi,1 - •.• - L Ai,T • (Qi,1 + ... + Qi,T) i=1 i=1

(9.37)

We can now add (9.37) to the second term of the objective function (9.26)

and obtain the new coefficient

(9.39)

Now, we turn to the first term of (9.36) which represents the assembly.

ITt J

L L Ai,t L L: qj,i . Xj,T i=1 t=1 T=lj=1

I J I J

L Ai,1 L qj,i . Xj,1 + ... + L Ai,T L qj,i . (Xj,1 + ... + Xj,T) i=1 j=1 i=1 j=1

(9.40)

This time, we add (9.40) to the third term of the objective function (9.26) and obtain

JT JT(IT) - ~~hf·t 'Xj,t + ~~ ~~Ai'T' qj,i • Xj,t

J T LLlitt · Xj,t (9.41) j=lt=1


with the new coefficient

I T -A P LL hot = -h, ·t+ A' 'q" J, J ~,T J,~ (9.42)

i=l T=t We can now formulate the Lagrangian relaxation of the manufacturing planning problem.

I TIT

Minimize Z = L L s; . Yi,t + L Lh[t' Qi,t i=lt=l i=l t=l

J T + L Lhtt · Xj,t + const

j=lt=l

subject to (9.27) - (9.30) and (9.32) - (9.35)

(9.43)

with const as defined in Equation (9.24). Since we do not have the former coupling constraints (9.31) in the relaxed model, the latter decomposes into an assembly scheduling and a fabrication lotsizing problem. Note that both problems are indirectly coupled via the Lagrangian multipliers Ai,t given in the modified holding cost parameters -A -F hJ't and hi t, respectively. , ,

We will now consider the assembly scheduling problem in Subsec-tion 9.2.3 and the fabrication lotsizing problem in Subsection 9.2.4. By relaxing the capacity constraints for each problem we derive subproblems which can be solved in polynomial time. The const-term as defined in (9.24) will not be considered explicitly in the two problems since it is not relevant for the determination of the optimal solution. We just have to make sure that the const term is added to the sum of the objective function values of the two problems once a solution has been found. Let us begin with the assembly scheduling problem.

9.2.3 Lower Bounds for the Assembly Scheduling Problem

The assembly scheduling subproblem can be formulated as follows.

J T Minimize Z = L L litt . X j,t

j=lt=l (9.44)


subject to (9.27) - (9.30) and (9.34)

The capacity constraints (9.30) make the problem hard to solve. Hence, we use the Lagrangian multipliers (Jr,t in order to relax the capacity constraints (9.30) and add

(9.45)

to the objective function (9.44). (9.45) has two terms where only the first term depends on the

scheduling decision as imposed by the xj,t-variables. Again, we reformulate the first term of (9.45) in a way which allows us to obtain a

coefficient which can be merged with the modified cost parameter litt of the objective function (9.44).

t

L A Cj,r . Xj,r

r=l t=O j=l r=max{0,t-Pj+1}

J T (RA A min{T,Hpj-1} )

L L L Cj,r L (Jr,r' Xj,t j=l t=O r=l r=t

(9.46)

We obtain the new cost parameter.

RA min{T,Hpj-1} -A -A ~ A ~ hj,t = hj,t + L... Cj,r L... (9.47)

r=l r=t

With the cost parameter h1.t we formulate the Lagrangian relaxation of the assembly scheduling problem.

J T Minimize ZX~p = L L h1.t . x j,t - constA (9.48)

j=lt=l

subject to (9.27) - (9.29) and (9.34)

with constA = 2:~1 2:;=0 (Jr,t . eft. For given Lagrangian multipliers Ai,t and (Jr,t this problem decomposes into P different problems, where for each problem p (p = 1, ... , P) we are seeking for start times Sj of


the jobs sP' ... , ep such that each job j is started in the time window [ESj, LSj]' no job starts before its immediate predecessors have been finished, and the cost function (9.48) is minimized. This problem can be efficiently solved with dynamic programming (cf. Subsection 8.1.2, Kolisch [187], Chen and Wilhelm [53]). We define the decision variable Sj(t) to be the minimum cost of job j and all its (direct and indirect) predecessors if job j has to be started until time instant t at the latest, i.e., within time window [ESj, t]. For job 0, the source of the integrated network, we have he = 0, qO,i = 0 for all i = 1, ... , I and CO,r = 0 for all r = 1, ... ,RA and hence we can initialize

SO(t) = 0 (t = ESo, ... , LSo). (9.49)

For all other jobs j = 1, ... , J + 1 we recursively calculate Sj(t) for all t = ESj, ... , LSj as follows.

Since the jobs are topologically labelled, i.e., h < j holds for each h E Pj, we always have the Sh(t) values at hand when calculating the Sj (t) values. We end with SJ+1 (LSJ+l) which provides the minimum cost for scheduling the jobs for given Lagrangian multipliers Ai,t and Or,t. The minimum objective function value is SJ+l (LSJ+l) - constA. The corresponding start times Sj are obtained by backtracing.

Example

Consider our example problem given in Subsection 6.3 and the Lagrangian multipliers Ai,t = 1 for each part i = 1, ... ,3 and each period t = 1, ... ,5 as well as Or,t = 1 for the single resource r = 1 and each time instant t = 0, ... ,5. We begin with the calculation of the constA-value.

RA T

L L Or,t . C~t = 6 . 3 = 18 r=l t=o

Next, we calculate the htcvalues according to Equation (9.47). The

calculation of ht,l is given exemplary in Table 9.5. The obtained htt -


-A h2,1

liA + ",RA A ",min{T,t+P2- 1} () 2,1 wr=l C2,r WT=t r,T

hp ",1 ",T \ A ",min{5,t+P2-1} () - 2 • t + L.Ji=l L.JT=t /\i,T • Q2,i + C2,1 L.JT=t 1,T

hp 1 ",3 ",5 \ A ",min{5,t+P2-1} () - 2' + L.Ji=l L.JT=l /\i,T • Q2,i + C2,1 L.JT=t r,t

-0·1 + 5· (1 + 1 + 1 + 1 + 1) + 1 . 2 25+ 2 27

Table 9.5: Exemplary Calculation of ht,l

values for each j (j = 1, ... ,5) and t (t = ESj, ... , LSj) are listed on the left side of Table 9.6. The time windows [ESj, LSj] were derived by forward and backward recursion from 0 and T = 5, respectively. Additionally, we have taken into account that, due to part demand, the non-dummy jobs 2, 3, 4, and 5 cannot start before time instant 1. Non-relevant values, i.e., values which correspond to infeasible job start times, are indicated by a dash.

-A Sj (t) h' t J,

j \ t 0 1 2 3 4 5 0 1 2 3 4 5 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 - 27 22 17 - 27 22 17

3 - 27 22 17 - 27 22 17 4 - 21 16 11 - 75 60 45 5 - 26 21 - 26 21 6 0 0 - 86 66

Table 9.6: h1.t- and Sj(t)-Values for the Example Problem

The right side of Table 9.6 gives the calculation of the Sj (t)-values. The minimal cost obtained are 66. Subtracting the 18 of constA gives


an objective function value of 48. The associated assembly schedule S = (3,3,3,4,2) can be obtained by backtracing. Figure 9.3 gives the Gantt-chart. Note that dummy jobs are not represented.

2

o 1 2 3

5

3

4

4 i ... t

5

Figure 9.3: Optimal Solution of the Relaxed Assembly Scheduling Problem

9.2.4 Lower Bounds for the Fabrication Lotsizing Problem

The fabrication lotsizing problem can be put as follows.

I TIT

Min Z = LLsf . Yi,t + LLh[t' Qi,t (9.51) i=1 t=1 i=1 t=1

subject to

t t

L Qi,t ~ L df.~ 7"=1 7"=1

i = 1, .. . ,1 t = 1, .. . ,T

(9.52)

and (9.32), (9.33), (9.35)

with 7i[t as defined in equation (9.39). The valid constraints (9.52) were added to insure that fabrication takes place (cf. Derstroff [75] pp. 74). dff denotes the part demand of part i in period t incurred , when the jobs are started at their latest start times, i.e.,

J

d~s - '" q" ~,t - L...J J,~

j=l

i = 1, ... , I t = 1, .. . ,T . (9.53)

LSj=t

Not employing constraints (9.52) would lead to an optimal solution where no parts are produced, i.e., Qi,t = 0 and Yi,t = 0 for all i = 1, ... , I and t = 1, ... , T.


Problem (9.51) - (9.52), (9.32) - (9.33), and (9.35) is the well known capacitated multi-part single-level dynamic lotsizing problem (CLSP) (cf. Subsection 9.1.5). The CLSP is again hard to solve due to the capacity constraints given in (9.32). Hence, we are using the Lagrangian multipliers /Jr,t to relax constraints (9.32) and add

RF T (I ) L L /Lr,t . L cf· Qi,t - C!',t r=1 t=1 i=llr(=r

(9.54)

to the objective function (9.51). As for assembly scheduling, we have two terms in Equation (9.54)

where only the first term depends on the lotsizing decision given by the Qi,t-variables. Again, we will reformulate the first term such that we obtain a modified cost parameter hf,t. Doing so, we derive

ifrf=r , else ) Q;,.

(9.55) and can write down the modified cost parameter

RF {1i" F - f _ -~ '" . q, If r i = r h"t - h"t + L.J /Jr,t 0 else

r=1 ' (9.56)

With (9.56) we can formulate the Lagrangian relaxation of the fabrication lotsizing problem.

I TIT

Minimize z~l:P = L L sf . Yi,t + L L hf,t . Qi,t - constF (9.57) i=lt=1 i=lt=1

subject to (9.52), (9.33), and (9.35)

with constF = E~:1 Ef=1 /Jr,t . C!'t· The Lagrangian relaxation of the fabrication lotsizing problem is the well known dynamic uncapacitated lotsizing problem originally introduced by Wagner and Whitin [318] and henceforth coined as WW problem. Since there are no more capacity constraints, the parts are not linked directly any more but indirectly via the Lagrangian multipliers /Jr,t. That is, for given /Lr,tvalues, we have I different WW problems, where for each problem i


(i = 1, ... ,1) we are seeking for a dynamic fabrication plan for part i such that the demand dff is met and the sum of period dependent , fabrication cost and fixed setup cost is minimized.

Wagner and Whitin [318] have devised a dynamic programming approach with time complexity O(T2) to solve problem WW. Recently, Federgruen and Tzur [105] as well as Wagelmans et al. [317] have proposed dynamic programming approaches for the WW which have a time complexity of O(TlogT). For the sake of completeness and comprehensibility we sketch out the basic dynamic programming approach of Wagner and Whitin [318]. Our implementation uses its more efficient variant due to Wagelmans et al. [317].

Define fi(t) to be the minimum cost of a partial fabrication plan comprising periods {t, ... , Td where Ti is the latest period with a positive demand for part i. Define further ef (u, v) to be the total cost if fabrication in period u ~ 1 is exactly sufficient to meet the demands in periods {u, ... , v} with v ~ u, i.e.,

if Di,u,v > 0 else

(9.58)

where Di,u,v denotes the cumulated demand of part i for periods u through v. We initialize for the dummy period Ti + 1

(9.59)

and iteratively calculate li(t) as follows.

Ii (t) = min { ef( t, T) + Ii ( T + 1) I T = t, ... , Ti} t = Ti, ... , 1 (9.60)

We end with li(l) which gives us the minimum cost to fulfill the demand for part i. The associated fabrication plan is obtained by backtracing. Note, that we do not have explicit inventory holding cost because they are implicitly considered in the time-variant production cost coefficients. Due to time variability of the production cost coefficients we have to calculate cost minimum fabrication plans for all periods down to 1 and not only until the first demand for part i takes place.


Example

We continue with our above given example. Let the Lagrangian multipliers be Ai,t = 1 for all i = 1, ... ,3 and t = 1, ... ,5 as well as "'r,t = 1 for the single resource r = 1 and t = 1, ... ,5. The external demand values df,! are calculated according to the schedule S = (3,3,3,4,2) and Equation (9.53) to dfS = (0,0,5,5,0), dtS = (0,0,5,0,0), and dfs = (0,5,0,0,0), respectively.

The modified cost coefficients hf,t are calculated according to Equa

tion (9.56). Example given, for i = 1 and t = 1 we obtain hi 1 = -1 . 1 - (1 + 1 + 1 + 1 + 1) + 1· 1 = -5. The hf,cvalues for ~ach i = 1, ... ,3 and t = 1, ... ,5 are given in Table 9.7. For each part an entry '-' is given for the periods following the latest demand period. The value of constF accounts to 50.

-F fi(t) h· t " i \ t 1 2 3 4 5 1 2 3 4 5 1 -5 -5 -5 -5 -30 -30 -30 -5 0 2 -6 -7 -8 -30 -30 -30 0 3 -5 -5 -15 -15 0

Table 9.7: hf,t- and fi(t)-Values for the Example Problem

Table 9.8 gives the cf(u, v)-values. With these values we can now calculate the fi(t)-values. Table 9.9 details the calculation of the h (t)values. The right side of Table 9.7 provides all fi(t)-values. By backtracing, the associated optimal fabrication plan with optimal objective function value can be found.

The minimal cost are -30, -30, and -15 for parts 1, 2, and 3, respectively. By backtracing we obtain the associated fabrication plans Q1 = (0,0,10,0,0), Q1 = (0,0,5,0,0), and Q1 = (0,5,0,0,0). Figure 9.4 gives a graphical representation of the solution. Nodes depict periods, the numbers below the nodes are the fabrication quantities, and an arc from period t to period T (T 2 t) visualizes that the lot size fabricated in period t covers the demand of periods t through T. The objective function value of the optimal solution is


cf(u, v) cf(u,v) cf(u, v) u \ v 1 2 3 4 1 2 3 4 1 2 3 4 1 0 0 -5 -30 0 0 -20 - 0 -15 2 - 0 -5 -30 - 0 -25 - - -15 3 - - -5 -30 - - -30 - - - -

4 -5 - - -

Table 9.8: cf(u, v)-Values for the Example Problem

Z = fI (1) + 12(1) + h(l) - constF = -30 - 30 - 15 - 50 = -125. To obtain the lower bound for the manufacturing planning problem

(9.26) - (9.35) we have to add the objective function value of the relaxed assembly scheduling problem which is 48 and the const value. const is calculated according to (9.24).

p ~ I P ~

const L L L hf . qj,i . (dp + 1) + L: L hf . dp

p=l j=sp i=l

(0·0+ 1 ·5 + 2 ·5+ 1 ·5) ·6+ 1 ·5·4 + 0 = 140

This way we obtain a lower bound of -125 + 58 + 140 = 63.

i = 1 6)0 6)0 0 0 10 0

i = 2 Cf)@@ 0 0 0 5 Qi,t

i = 3 063 o 5

Figure 9.4: Cost Minimal Fabrication Plan


t II (t) 5 0 4 = min{cl(4,4) + 1I(5)}

= min{-5 + O} = -5

3 = min{cl(3,3) + II (4), cl(3,4) + 1I(5)} = min{ -5 - 5, -30 + O} = -30

2 = min{cl (2,2) + II (3), cl(2, 3) + II (4), cl(2,4) + II (5)} = min{O - 30,-5 - 5, -30 + O} = -30

1 = min { Cl (1, 1) + II (2) , Cl (1, 2) + II (3), Cl (1, 3) + II (4) , Cl (1, 4) + II (5)}

= min{O - 30,0 - 30, -5 - 5, -30 - O} = -30

Table 9.9: Calculation of the II (t)-Values

9.2.5 A Lagrangian-Based Construction Heuristic

With the above outlined Lagrangian relaxation scheme we are able to obtain lower bounds for the manufacturing planning problem. Associated with each lower bound, we generate an assembly schedule and a fabrication plan. Both plans are generally not resource feasible. Furthermore, they are not vertically coordinated, i.e., the parts may not be fabricated in time for the assemblies. In what follows, we show how we can exploit the information given in these plans in order to come up with a feasible solution for the manufacturing planning problem. Similar to the construction heuristic we perform a level-by-Ievel approach where we generate a resource feasible assembly schedule first and construct a resource feasible fabrication plan thereafter.

Generation of Assembly Schedule

For given Lagrangian multipliers Ai,t and Br,t we have shown in Subsection 9.2.3 how to calculate a cost minimal assembly schedule S for the relaxed assembly scheduling problem. In general, schedule S will


not be feasible w.r.t. the resource constraints. This holds especially when the resource capacity is scarce. But we can employ the information contained in schedule S in order to derive a resource feasible assembly schedule. The idea is straightforward. We construct a job list 1r = (iI,i2, ... ,j;J as employed in the construction heuristic presented in Section 9.1. Once we have generated the list, we can use the list schedule generation scheme presented in Subsection 9.1.3 in order to obtain a feasible assembly schedule. The job list is constructed by ordering the jobs 1, ... , J according to descending start times Sj as obtained by the assembly schedule for the relaxed problem. In case of ties, we select the job with the higher label.

Example

Employing the assembly schedule S = (3,3,3,4,2) which has been generated in Section 9.2.3, we obtain the job list 1r = (4,3,2,1, 5J. Applying the schedule generation algorithm given in Section 9.1 to list 1r we obtain the resource feasible schedule S(1r) = (2,2,3,4,2) represented as Gantt-chart in Figure 9.5.

5

2

o 1 2 3

3

4

4 I .. t 5

Figure 9.5: Solution of the Assembly Scheduling Problem

Fabrication Lotsizing

From the feasible assembly schedule S we obtain with Formula (9.4) the time-phased part demand dft for each part i and time period t. We now employ the demands df;, resulting from a feasible assembly schedule, instead of the demand~ df,f, obtained from the usually not (resource-) feasible latest start time schedule, in constraints (9.52) of the fabrication lotsizing problem (9.51) - (9.52), (9.32) - (9.33), and (9.35) presented in Subsection 9.2.4. We use again subgradient


optimization in order to solve the Lagrangian relaxation (9.57), (9.52), (9.33), and (9.35). This way we obtain a setup pattern Yi,t.

We now try to find a cost minimal fabrication plan where production in the periods given by the setup pattern Yi,t fulfills the demand as given by dft. The fabrication plan can be obtained by employing a , transportation problem (cf. Domschke [82] pp. 76) where we have to allot for each fabrication resource r (r = 1, ... , RF) the capacity C;'t of every setup period t (where Yi,t = 1 holds) to the demand cr . d~T (given in capacity units) with , ~ t such that the sum of the transportation cost Ci,t,T = (, - t) . hr is minimal. Since we do not know a priori if the setup pattern Yi,t allows a feasible solution, we employ the modified transportation cost

{

00,

Ci,t,T = (, - t) . hr, (, - t) . hr + M . sr,

if t > , ift :s; , and Yi,t = 1 ift :s; , and Yi,t = 0

(9.61)

The transportation cost are infinite for backorder periods, they represent the holding cost for periods where there is a setup and no backorder takes place, and they are the holding cost plus a penalty cost term for periods where there is no setup and no backorder. As penalty, we have taken the setup cost multiplied with a large number M which has been set to 10 in our computational study.

The transportation problem (TP) for fabrication resource r (r = 1, ... , RF) is now as follows.

ITT

Minimize Z = LL L Ci,t,T' Qi,t,T

subject to

I T

i=l t=l T=t

dS >0 .,T

LLcr . Qi,t,T :s; C;'t t= 1, ... ,T i=l T=t

T i = 1, ... , I ""Q' - dS L...J 't,t,T - i,T t=l

, = 1, ... , T with df T > 0 ,

(9.62)

(9.63)

(9.64)

Q't > 0 ~,,7 _ i = 1, ... , I (9.65) t = 1, ... , T; , = t, . .. , T with d~t > 0


The objective function (9.62) minimizes the sum of the transportation cost. The constraints (9.63) assure that the period capacity of the fabrication resource is respected while the constraints (9.64) force the part demands to be fulfilled. Finally, the decision variables are defined in (9.65). Note, that due to the definition of the decision variables we do exclude backordering periods and hence we do not have to define transportation cost for this case.

Solving (9.62) - (9.65) will bring forth a fabrication lotsizing schedule. Note, that due to early part demand of the assembly schedule, it may occur that there is not enough fabrication capacity and we cannot come up with a feasible solution (cf. Theorem 6.2).

Example

We continue with the above given example where we have obtained the assembly schedule S = (2,2,3,4,2). By using Formula (9.4), we obtain the time-phased part demand dr = (0,5,0,5), d~ = (0,0,5,0), and df = (0,5,0,0) for periods t = 1, ... ,4.

The demands are now considered in constraints (9.52) of the fabrication lotsizing problem (9.51) - (9.52), (9.32) - (9.33), and (9.35). We solve the Lagrangian relaxation (9.57), (9.52), (9.33), and (9.35) using subgradient optimization to come up with the setup pattern Y1 = (0,1,0,0), Y2 = (0,0,1,0), and Y3 = (0,1,0,0) for t = 1, ... ,4.

According to (9.61) we obtain the cost coefficients as given in Table 9.10. Solving the transportation problem (9.62) - (9.65) with the Ci,t,T

and dr-data, we obtain the solution 01,2,2 = 5, 01,2,4 = 5, 02,3,3 = 5, and 03,1,2 = 5 with an objective function value of 515. The solution is represented in Figure 9.6. Since the demand cannot be satisfied with the given setup pattern, an additional setup for part 3 takes place in period 1.

i_I i _ 2 i _ 3

T T T

t 1 2 3 4 2 3 2 3 4 1 200 201 202 203 100 102 104 106 100 101 102 103 2 00 0 1 2 00 100 102 104 00 0 1 2 3 00 00 200 201 00 00 0 2 00 00 100 101 4 00 00 00 200 00 00 00 100 00 00 00 100

Table 9.10: Ci t T-Values for the Example Problem , ,


10

10 5

5

10 5

10 CD 5

Cft , ~ dr.-,.

Figure 9.6: Optimal Transportation Plan

The heuristic solution for the manufacturing planning problem is given as Gantt-chart in Figure 9.7. The associated cost of 125 are comprised by setup cost of 40 and holding cost of 85.

9.2.6 Subgradient Optimization

To obtain good lower bounds, the Lagrangian multipliers have to be adjusted properly. More precisely, we have to set Ai,t, Or,t, and /-lr,t to values ~ 0 such that we obtain the maximum for ZX~p + Z~rp when solving the Lagrangian relaxation of the assembly scheduling problem (9.48), (9.27) - (9.29), (9.34) and the Lagrangian relaxation of the fabrication lotsizing problem (9.57), (9.52), (9.33), and (9.35). Methods proposed to search for good multipliers are column generation, multiplier adjustment, and subgradient optimization (cf. Fisher [109]). Subgradient optimization has been the most popular of the three methods (cf. Diaby et al. [77]). The main idea of su bgradient optimization is to move in L steps along a descent direction, iteratively

9.2. LAGRANGIAN RELAXATION

Assembly

Fabrication

o 1 2 3 4

I-

I )0 t 5

161

Figure 9.7: Solution Obtained by the Lagrangian Heuristic

approaching the optimal value of ZX~p + Z~~p, Starting at step 1 = ° with >..? t' O~ t, and J.i~ t, the Lagrangian multipliers are updated iter a-

" , tively according to Equations (9.66), (9.67), and (9.68), respectively.

{ ( t J t)}. 1 I 1+1 I l I I t = , ... , \,t =max O,>"i,t+ 8 . LLqj,i·Xj,,,.- L:Qi,7 t=l ... T

7=1 J=1 7=1 ' ,

(9.66)

r = 1, .. . ,RA

t= O, ... ,T (9.67)

{ ( I)} RF 1+1 I l F l F r = 1, ... , J.ir,t = max 0, J.ir,t + 8· . L Ci' Qi,t - Gr,t t = 1, ... , T

l=llr[=r (9.68)

81 is the scalar step size. Given the multipliers in the l-th iteration, x' is an optimal solution of the Lagrangian relaxation of the assembly scheduling problem (9.48), (9.27) - (9.29), (9.34) and Ql is an optimal solution of the Lagrangian relaxation of the fabrication lotsizing problem (9.57), (9.52), (9.33), and (9.35). The number of iterations was set to 100. The step size 81 was set as follows (cf. Held et al. [156]).


f.1 • (z - ZLR (>..1 (JI xl) _ ZLR ()."l 1/.1 Ql yl)) 81 = ASP " FLP' r' , --~-------A~+-B=-+-C=---------~ (9.69)

with

(9.70)

RA T (J t ) 2 B = L L L L Cj,r • X~'T - C~t ,

r=l t=O j=l T=max{O,t-Pj+l}

(9.71)

and

(9.72)

Z is the upper bound obtained with the Lagrangian-based heuristic outlined in Subsection 9.2.5. ZX~p(AI,(JI,xl) is the optimal objective function value of the Lagrangian relaxation of the assembly scheduling problem and Z~rp(A/, J.tl , QI, yl) is the optimal objective function value of the Lagrangian relaxation of the fabrication lotsizing problem. f.1 is initially set to 2 and reduced by a factor of 2 whenever the lower bound fails to increase in a predetermined number of iterations (cf. Diaby et al. [77]). In order to stabilize the convergence of the subgradients we additionally employed the exponential smoothing procedure proposed by Crowder [62] with a smoothing factor of 0.5.



For evaluation purposes, a set of 270 test instances was generated with a parameter controlled instance generator which builds up on ProGen (cf. Kolisch et al. [193, 194]). Details are provided in Appendix A in general and Appendix A.3 in particular.



Four systematically varied problem parameters were employed. The instance size, the assembly resource strength, the fabrication resource strength, and the time between order. Details ofthese parameters are provided in Appendix A.3.

Two different instance sizes (IS) were generated: (S)mall and (L)arge. For the small test instances, there is P = 3, Jp E [3,5], I = 4, RA = 1, RpA = 1, RF = 1, PP = 0.5, and rp E [0,10]. J p denotes the count of non-dummy jobs belonging to order p, RpA denotes the assembly resource factor for all jobs. RpA measures the density of the capacity demand array cf.r for assembly resources. Correspondingly, PP denotes the part factor which measures the density of the part demand array qj,i' RpA = 1 expresses the fact that all non-dummy jobs require each of the available resources in order to be processed. P p = 0.5 says that each non-dummy job requires a positive amount of every second part. rp denotes the release date of order p. The large problem instances are characterized by the following parameter values: P = 10, J p E [5,10], RA = 2, RF = 2, 1= 8, and rp E [0,30].

The assembly resource strength RSA E [0, 1] measures the scarcity of the assembly resource capacity given in constraint type (6.6). The smaller the RSA-value is, the tighter the assembly resource constraints are. RSA was set to 0.20, 0.35, and 0.50, respectively.

The fabrication resource strength RSF E [0, 1] measures the available capacity of the fabrication resources given in constraints (6.8). The smaller the RSF -value is, the tighter the fabrication resource constraints are. RSF was set to 0.20, 0.35, and 0.50, respectively.

The time between order TBO ~ 0 determines for each item i the ratio of holding and setup cost according to the following formula (cf. Salomon [265], Helber [153]):

(9.73)

That is, increasing the TBO increases the setup cost for given holding cost. The TBO was set to 1, 3, and 5, respectively.



The following parameters were randomly drawn from the specified intervals: Pj E [1,3], ctr E [1,3], qj,i E [10,50]' hf E [1,2]. The parameters rf and cf were set to rf = (i mod RF) + 1 and cf = 1.

To generate feasible instances, the due date factor DF was set according to the assembly and fabrication resource strength as given in Table 9.11.

I5=S I5=L RSA= 0.1 0.3 0.5 0.1 0.3 0.5 RSF =O.l 0.50 0.45 0.40 0.40 0.35 0.30 RSF=0.3 0.45 0.40 0.35 0.35 0.30 0.25 RSF =0.5 0.40 0.35 0.30 0.30 0.25 0.20

Table 9.11: DF-Setting for Small and Large Instances

Experimental Design

Realizing a full factorial design with 5 replications for each combination of the independent parameter levels IS, RSA, RSF , and TBO, 5 . 2 . 3·3·3= 270 instances were generated.


Each instance was solved with different versions of the construction heuristic proposed in Section 9.1 and with the Lagrangian-based heuristic (LBH) proposed in Subsection 9.2.5. Different versions of the construction heuristic were obtained by each combination of the 3 priority rules and the 5 cost measures. We denote each combination by 'priority rule-cost measure', e.g., 'PRI-DS' denotes the combination of priority rule PRI and the adapted cost measure of Dixon and Silver. Additional to these 15 combinations, we tested the combination of each priority rule with the lotsizing algorithm of Kirca and K8kten (KK) [180]. The latter method showed very favourable results when compared to the heuristics of Lambrecht and Vanderveken [206], Dixon and Silver [79], Maes and van Wassenhove [222], and Cattrysse et al. [50]. Finally, we


applied a so-called multi priority rule heuristic (MPRH) (cf. Kolisch and Hartmann [189]) by iteratively solving an instance with each of the 18 different single-pass heuristics and keeping the best solution. This way, altogether 20 heuristics were considered in our computational study.

The performance of each heuristic is measured by the solution gap between the heuristic under consideration and the Lagrangian relaxation based lower bound. We had to employ the solution gap because on account of the large number of binary variables the problem (6.2) - (6.11) could not be solved with standard MIP solvers like CPLEX (cf. Bixby and Boyd [31]). Two reasons led us to employ the Lagrangian-based lower bound instead of an LP-based lower bound. First, with the Lagrangian-based lower bound we obtained for all instances lower bounds while solving the MIP (6.2) - (6.11) as LP by relaxing Xj,t E {O, I} to 0 ~ Xj,t ~ 1 and Yi,t E {O, I} to 0 ~ Yi,t ~ 1 we could only derive lower bounds for the small instances. Second, for the small test instances the Lagrangian bounds were 10% better than the LP-based bounds and 6% better than the bounds we obtained by solving (6.2) - (6.11) and just relaxing the Yi,t variables.

If not stated differently, we provide means; statistical testing was done with nonparametrical tests solely because inspection of the distribution functions revealed that none of the values was normally distributed (cf. Alvarez-Valdes and Tamarit [7]). In particular we employed the following tests (cf. Golden and Steward [123]): the Wilcoxon matched-pairs signed-ranked test for 2 related samples, the Friedman test for k > 2 related samples, and the Kruskal-Wallis-H test with tie correction for k > 2 independent samples. If no level of confidence is given explicitly, the 1% level of confidence is meant.

We divide the analysis in an assessment of the performance of the 20 different heuristics and the investigation of the influence of the independent problem parameters on the best construction heuristic, the Lagrangian-based heuristic, and the multi priority rule heuristic.

Performance of the Heuristics

Table 9.12 gives the mean, the standard deviation, the minimum, and the maximum of the 20 different tested heuristics from the lower bound as obtained by the Lagrangian relaxation. The heuristics are sorted by


Heuristic Mean Std Dev Minimum Maximum Sig MPRH 31.01 34.17 0.47 155.21 * PR2-CS 33.51 34.91 0.53 162.04 * PR2-KK 33.68 35.36 0.47 162.05 * PR2-LV 34.23 35.15 0.53 162.07 PR3-CS 34.29 36.60 0.53 166.22 * PR3-KK 34.59 37.14 0.47 166.22 * PR2-EI 34.87 35.32 0.53 162.07 PR3-LV 34.97 36.78 0.53 166.23 PR2-GU 35.18 35.35 0.53 162.04 PR3-EI 35.56 37.02 0.53 166.23 PR3-GU 35.87 37.04 0.53 166.22 * LBH 38.60 38.50 0.11 204.61 PR2-DS 38.69 36.67 0.53 162.04 PR3-DS 39.07 38.25 0.53 166.22 PR1-CS 39.31 41.28 0.53 182.17 * PRI-KK 39.47 41.60 0.47 182.17 * PRI-LV 39.94 41.46 0.53 182.20 * PRI-EI 40.62 41.68 0.53 182.20 * PRI-GU 41.06 41.75 0.53 182.17 * PRI-DS 44.55 43.26 0.53 182.17

Table 9.12: Average Deviation of Heuristics from Lower Bound

descending mean. The means range from 31.01% for MPRH to 44.55% for PRI-DS. We have compared each heuristic with the next best one as given in Table 9.12 using the Wilcoxon signed-rank test. A significant difference in performance at the 1% level of confidence is marked with '*'. The best performance is due to the multi priority rule heuristic MPRH. The three best single-pass heuristics are PR2-CS, PR2-KK, and PR2-LV. All of them employ the priority rule PR2. The performance of the Lagrangian-based heuristic LBH ranges in the middle. The worst performance is obtained by the single-pass heuristic PRI-DS.

Table 9.13 compares the average performance of the scheduling and lotsizing strategies for the single-pass heuristics. Here, e.g., PRI gives the average deviation from the lower bound for the heuristics PRl-


CS, PRI-KK, PRI-LV, PRI-EI, PRI-GU, and PRI-DS while CS gives the average deviation from the lower bound for the heuristics PRI-CS, PR2-CS, and PR3-CS. Again, we have tested if a strategy provides significantly better results than the next best strategy by pairwise comparison. The best scheduling strategy is priority rule PR2 closely followed by priority rule PR3. No significant difference between PR2 and PR3 was detected at the 1% level of confidence. The best lotsizing strategy is the cost saving method CS. It provided significant better results than the next best lotsizing method which is the one of Kirca and Kokten (KK).

Scheduling Lotsizing Strategy Mean Sig Strategy Mean Sig

PR2 35.63 CS 35.96 * PR3 36.63 * KK 36.13 * PRI 41.30 LV 36.64 *

EI 37.27 * GU 37.63 * DS 40.98

Table 9.13: Comparison of Scheduling and Lotsizing Strategies

Effect of the Problem Parameters

We now focus our attention on three selected heuristics, namely the multi priority rule heuristic MPRH, the best single-pass heuristic PR2-CS, and the Lagrangian-based heuristic LBH. Tables 9.14 - 9.17 show the influence of the four systematically varied problem parameters assembly resource strength RSA , fabrication resource strength RSF ,

time between order TBO, and instance size IS. A ,*, in the last column indicates a significant influence of the problem parameter on the performance of the heuristic at the 1 % level of confidence.

Assembly Resource Strength

The assembly resource strength has a significant influence on all three heuristics. The more the assembly is resource-constrained, the higher


the deviation from the lower bound gets. The explanation is straightforward. Lower capacities do imply, c.p., that jobs cannot be scheduled close to their due date but have to be scheduled early in time which causes an increase of holding cost. Furthermore, the time-phased part demand of the jobs is spread over the entire planning horizon. This leads, c.p., to more setups and thus lligher setup cost.

Mean R§4. Sig 0.20 0.35 0.50

MPRH 31.01 54.52 29.42 9.30 * PR2-CS 33.51 57.80 31.99 11.56 * LBH 38.60 63.38 38.05 14.08 *

Table 9.14: Effect of RSA on Heuristics

Fabrication Resource Strength

The effect of the fabrication resource strength is noticeable but not significant. The average performance declines with increasing resource availability within the fabrication department. At first glance, this might be surprising. But the explanation can be given as follows. When resources in the fabrication are scarce, batching cannot be done to a great extend because we have to fabricate just-in-time. This results in small lot sizes. Note that the manufacturing planning model (6.2) - (6.11) does not take into account setup times explicitly because of the reasons outlined in Section 6.1. Therefore, small lot sizes do not go along with a greater capacity demand. Low capacities imply a solution space where not much batching can be done. Hence, heuristics cannot do many wrong decisions. The opposite holds for the case with medium capacity available. Here, the solution space comprises fabrication plans with small and large lotsizes. Hence, heuristics can make the wrong decisions leading to solutions with higher cost. The absolute value of the coefficient is lowest for the Lagrangian-based heuristic which indicates that this procedure is the most insensitive one w.r.t. the availability of the fabrication resource capacity. A rationale might be that for a given setup pattern optimallotsizes are


determined.

Mean RSF Sig 0.20 0.35 0.50

MPRH 31.01 28.57 30.12 34.38 * PR2-CS 33.51 30.95 32.89 36.80 * LBH 38.60 37.09 38.70 40.01 *

Table 9.15: Effect of RSF on Heuristics

Time Between Order

Increasing time between order levels leads to a deterioration of all three heuristics. This effect is for the multi priority rule and the Lagrangianbased heuristic significant at the 5% level of confidence. The influence of TBO is as follows. With higher setup cost, bad lotsizing decisions do lead, c.p., to an increase of the solution gap. The Lagrangian-based heuristic is more sensitive to changes in the TBo-values.

Mean TBO Sig 0 1 2

MPRH 31.01 28.75 30.61 33.68 PR2-CS 33.51 31.02 32.60 36.89 LBH 38.60 34.40 38.76 42.67

Table 9.16: Effect of TBO on Heuristics

Instance Size

The effect of the instance size as indicated by the Friedman test is for all three heuristics significant. With larger problem instances, the solution gap increases sharply. This is because the solution space enlarges and thus the difference between lower and upper bounds increases.


Mean IS Sig S L

MPRH 31.01 10.94 50.18 * PR2-CS 33.51 13.08 52.56 * LGH 38.60 16.22 60.80 * Table 9.17: Effect of IS on Heuristics

Chapter 10

Operations Scheduling Methods

We have shown in Section 7.4 that the operations scheduling problem (7.1) - (7.9) is NP-hard. Hence, optimal algorithms are not applicable for industrial applications, where hundreds to thousands of operations have to be scheduled within minutes of CPU-time. We will therefore consider two types of heuristics: construction heuristics in Section 10.1 (cf. Kolisch [188]) and improvement heuristics in Section 10.2 (cf. Kolisch and HeB [190]).

10.1 Construction Heuristics

As for the assembly scheduling problem (cf. Section 9.1.1) we employ a list scheduling heuristic which schedules the operations in the order prescribed by the list. Compared to the assembly scheduling problem, the operations scheduling problem is different w.r.t. two aspects. First, the objective function (7.1) is left- instead of right-regular (cf. Theorem 7.1). Second, additionally to the resource constraints (7.4), there are spatial constraints (7.5) and part availability constraints (7.6). These differences necessitate a modified list scheduling approach. Due to the different objective function, we perform forward oriented list scheduling (cf. Schutten [277]) where the operations are scheduled at their earliest feasible start times. The additional constraints require both, a modified list generation and a modified schedule generation. In

172 CHAPTER 10. OPERATIONS SCHEDULING METHODS

what follows we will first outline the schedule generation in Subsection 10.1.1, give then properties of the schedule generation in Subsection 10.1.2, and finally turn to the list generation in Subsection 10.1.3.

10.1.1 Schedule Generation

Let 7r = (iI,i2, .. . ,jJ] be a list of the J non-dummy operations belonging to the operations scheduling problem. jg is the operation at position 9 and 7r(j) is the list position of operation j. Let us assume for now that we have such a list 7r and want to transform it into a feasible schedule S. A feasible schedule gives a start time Sj for each operation j (j = 1, ... , J) such that precedence relations, part availability, assembly resource, and spatial resource constraints are obeyed. In order to schedule the g-th operation on the list, we need to know the material availability Ni,t(9), the assembly capacity 6~t(g), and the spatial resource capacity 6f (g) after the first 9 - 1 operations have been scheduled. This can be calculated with the following recursions for 9 = 2, .. . ,J:

For all i = 1, ... , I and t = 0, ... , T we first initialize Ni,t(l) = Ni,t and set

(10.1)

afterwards.

Similar, for the capacity of assembly resources we set 6~i1) = C~t for r = 1, ... , Rj t = 0, ... , T, and update the available capacity

CA( ) =CA( -1)-{ ct_l,r , ift E {Sj9_ll· .. ,Sj9_l +Pj9-l -I} r,t 9 r,t go, else

(10.2) for r = 1, ... , RA and t = 0, ... , T.

Finally, for the capacity of spatial resources, we set 6f (1) = C S

for t = O, ... ,T. We denote with S = {sp !P= 1, ... ,P} and £ = {ep ! P = 1, ... , P} the set of start and terminal operations of all as-semblies, respectively. Now, updating for each t = 0, ... , T can be


done by

1 ,if jg-l E £ and t E {Sj9_l + Pjg-l , ... , T} -1 ,if jg-l E Sand t E {Sj9_l"'" T}

o ,otherwise (10.3)

Assume that all list predecessors jl, ... , jg-l of operation j = jg have been scheduled and that we want to start j as early as possible. If we only take precedence constraints (PC) into account we obtain Src , the precedence-constrained start time.

(lOA)

With respect to the part availability (PA) we get Sr A, the part availability-constrained start time.

PA . { -Sj = mm t E {ESj, ... , LSj} I Ni,T(g) :2: qj,i

for all i=l, ... ,I; r=t, ... ,T} (10.5)

Considering only the assembly capacity (AC) and some (not yet specified) dynamic earliest start time ESj :2: ESj, we obtain Sfc, the assembly resource-constrained start time.

SfC(ES;) = min{t E {ES;, ... ,LSj} I t~Ag):2: ct,r for all r = 1, ... , R; r = t, ... , t + Pj - 1} (10.6)

Taking only the spatial capacity (SC) into account, we have for start operations the spatial-constrained start time Sfc.

(10.7)

We now can proceed to the schedule generation algorithm.

I Operations Schedule Generation I A. Initialization: So = O.


(1) Calculate Ni,t(g), 6~t(g), and 6f(g).


(2) Take the next operation from the list: j = jg.

(3) Determine the dynamic earliest start time ESj of j w.r.t. precedence, part availability, and spatial constraints: If J' E S then ES'· = max {SfC Sf A S$C}

J J ' J ' J

else ES'· = max {SfC Sf A} J J 'J .

(4) Determine the earliest resource feasible start time of j which is 2: ESj:

AC ' Sj = Sj (ESj ).

Step (1) updates the part availability and the resource availability of assembly and spatial resources according to (10.1), (10.2), and (10.3), respectively. After Step (2) has selected the next operation from the list, its dynamic earliest start time w.r.t. precedence constraints, part availability constraints, and spatial constraints is calculated in Step (3). Note that for non-start operations, no spatial constraints have to be taken into account. Step (4) determines the earliest resource feasible start time within time window [ESj, ... , LSj].

Table 10.1 reports the schedule generation for the example problem and the list IT = (11,12,1,2,3,4,5,6,7,8,9,10]. The resulting schedule S = (0,0,3,6,9,5,9,11,12,14,0,3) with an objective function value of 39 is depicted as Gantt-chart in Figure 10.1. Note that only operations with non-zero processing times are represented.

Q~~I 3 4 5 Assembly 1

Assembly 2 7 1 81 9 10

11 12 Assembly 3

I > t

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Figure 10.1: Initial Solution


N1,t(g) Sf A Jg

-A (t = 0, ... ,20) jg Sfc S· = S!<c 9 C1,t(g) Jg Jg Jg

ef(g) S$C J~

(2,2,2,4,4,4,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8) ° 1 (4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4) 11 ° ° (2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2) ° (1,1,1,3,3,3,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7) ° 2 (2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4) 12 3 3 (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1) (1,1,1,3,3,3,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7) ° 3 (2,2,2,1,1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4) 1 ° ° (1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2) ° (1,1,1,3,3,3,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7) ° 4 (2,2,2,1,1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4) 2 ° ° (0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1) (0,0,0,2,2,2,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6) 3

5 (0,0,2,1,1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4) 3 ° 3 (0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1) (0,0,0,0,0,0,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4) 6

6 (0,0,2,0,0,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4) 4 2 6 (0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1) (0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3) ° 7 (0,0,2,0,0,3,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4) 5 9 9 (0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1) (0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3) ° 8 (0,0,2,0,0,3,1,1,1,2,2,4,4,4,4,4,4,4,4,4,4) 6 ° 5 (0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2) 5 (0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3) 9

9 (0,0,2,0,0,3,1,1,1,2,2,4,4,4,4,4,4,4,4,4,4) 7 ° 9 (0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1) (0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1) 6

10 (0,0,2,0,0,3,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4) 8 5 11 (0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1) (0,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0) ° 11 (0,0,2,0,0,3,1,1,1,1,1,2,4,4,4,4,4,4,4,4,4) 9 12 12 (0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1) (0,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0) ° 12 (0,0,2,0,0,3,1,1,1,1,1,2,3,3,4,4,4,4,4,4,4) 10 14 14 (0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1)

Table 10.1: Generation of the Operations Schedule


10.1.2 Property of the Schedule Generation Scheme

According to Sprecher et al. [291] we define.

Definition 10.1 (Active schedule) An active schedule is a feasible schedule where none of the operations can be started earlier without delaying some other operation.

Theorem 10.1 The operations scheduling algorithm generates feasible schedules which are active in the case of C S 2 P and which might be non-active for C S < P.

Proof For C S 2 P, the operations schedule generation algorithm generates active schedules by construction. For CS < P consider the following example with P = 3, C S = 2, I = 1, and RA = 0. Each assembly comprises two precedence related operations. All operations j = 1, ... ,6 have a processing time of 1. Operation 4 requires a single unit of part type 1 which becomes available at time instant 4. List IT = (3,1,4,5,6,2] results in the feasible but not active schedule S = (0,1,0,4,5,6) given in Figure 10.2 where assembly 1 is processed on the first shop floor area unit and assembly 2 and 3 are assembled one after the other on the second shop floor area unit. Assembly 3 cannot be placed on the first shop floor area unit because by the time operation 5 is scheduled, assembly 1 is not finished since operation 2 has not yet been scheduled. Note that the active schedule S = (0,1,0,4,2,3) will be obtained with list IT = (3,1,2,5,6,4]. Here, assembly 3 can be scheduled behind assembly 1 on the first shop floor area unit. D

10.1.3 List Generation

We now turn to the problem of generating a list IT = (il,h, .. . ,jJ]' In order to transform a list IT into a feasible schedule S = (SI,"" SJ), two properties corresponding with constraints (7.3) and (7.5) have to be met. The first property corresponds to the precedence constraints (7.3). It requires that the list position of an operation j must be greater than the list position of each of its predecessors (cf., e.g., Hartmann [145])

(10.8)

10.1. CONSTRUCTION HEURISTICS

CS

2

1 ITill ITI~~~~~~liJ

Assembly 1

Assembly 2

~ Assembly 3 ~-r--~~--T-~--~~~~ t

01234567

Figure 10.2: Non-Active Schedule

177

The second property concerns the spatial resource constraints given in (7.5). Consider the available spatial capacity given in (10.3). Whenever a start operation sp is scheduled, the available spatial capacity is red uced by 1 from the start of that operation to the end of the planning horizon. Scheduling a terminal operation ep adds 1 capacity unit from the finish time of the operation to the end of the planning horizon. Hence, starting with an available capacity of CS at iteration 1 of the schedule generation algorithm, no more than CS start operations can be on list positions 1, ... , 9 without a terminal operation in-between. Let £(g) = {il," .,ig } denote the set of operations which have been assigned to list positions 1, ... , g. 6s (g), the spatial capacity available at the g-th position of the list, is

(10.9)

Sand E denote the set of all start and terminal operations, respectively. Now, 6 s(g) ~ 0 has to hold for each list position 9 (g = 1, ... ,J).

Let further A(g) be the set of all available operations which can be put at list position 9 (g = 1, .. . ,J). A(g) is the union of two disjoint


sets of operations, AS (g), the available start operations of orders,

AS (g) = {j E S I j rf- £ (g - I)} (10.10)

and An(g), the available non-start operations,

An (g) = {j E J \ S I j ~ £ (g - 1), Pj r;. £ (g - I)} . (10.11)

Both, start and non-start operations respect the precedence constraint property (10.8). The former because they do not have any (nondummy) predecessors, the latter by definition. Denoting with v(j) a priority value associated with operation j, we can give the operations list generation algorithm as follows:

I Operations List Generation I A. Initialization: £(0) = 0.


(1) Update 68 (g), An(g), and AS(g).

(2) If 6 8 (g) :2: 1 then A(g) = AS(g) U An(g) else A(g) = An(g).

(3) Choose jg E A(g) with v(jg) = miniEA(g) v(i)

(4) Update £(g).

The initialization assigns the set of operations which are in the list to be empty. Step (1) updates the spatial capacity and the set of start and non-start operations, respectively. Step (2) defines the set of available operations. If there is spatial capacity, i.e., 68 (g) :2: 1, then the set comprises start and non-start operations, otherwise, i.e., 68 (g) = 0, the set comprises non-start operations only. In Step (3) we select one operation j from the set of available operations A(g) with minimum priority value v(j). Finally, Step (4) updates the set of operations assigned to list positions.

Table 10.3 reports the list generation for the example problem introduced in Section 7.3 when the priority values as given in Table 10.2 are employed.


j 1 2 3 4 5 6 7 8 9 10 11 12 v(j) 1.5 2.5 3.5 3 4.5 0.3 1.3 0.6 1.3 2.3 1 1.5

Table 10.2: Priority Values of Operations

9 7r C"S(g) A(g) jg 1 ( ] 2 {I, 6, 11} 6 2 (6] 1 {I, 11,7,8} 8 3 (6,8] 1 {I,ll, 7, 9} 11 4 (6,8,11] 0 {7,9,12} 7 5 (6,8,11,7] 0 {9,12} 9 6 (6,8,11,7,9] 0 {12,10} 12 7 (6,8,11,7,9,12] 1 {1,10} 1 8 (6,8,11,7,9,12,1] 0 {10,2,3} 10 9 (6,8,11,7,9,12,1,10] 1 {2,3} 2

10 (6,8,11,7,9,12,1,10,~ 1 {3,4} 3 11 (6,8,11,7,9,12,1,10,2,3] 1 {4} 4 12 (6,8,11,7,9,12,1,10,2,3,4] 1 {5} 5 7r = (6,8,11,7,9,12,1,10,2,3,4,5]

Table 10.3: Operations List Generation

10.1.4 Priority Rules

Table 10.4 lists different priority rules which have been successfully utilized for multi-project scheduling problems (cf. Lawrence and Morton [210] and Section 3.3). p(j) denotes the assembly operation j belongs to and wp is the weight of assembly p. The latest finish and start times, LFj and LSj, are derived by backward recursion from the assembly specific due dates dp • The rule 'shortest processing time' (SPT) has been added for comparison purposes. In case of ties, we have employed two tie breaking rules. The first one is 'first come first serve' (FCFS) where the operation is selected which has been first in the set A(g). As second tie breaker we use the smallest operation number.


Acronym EDD LFT SLK SPT WEDD WLFT WSLK WSPT

Priority Rule Earliest Due Date Minimum Latest Finish Time Minimum Slack Shortest Processing Time Weighted Earliest Due Date Weighted Minimum Latest Finish Time Weighted Minimum Slack Weighted Shortest Processing Time

v(j) dp(j) LFj LSj - ESj

Pj

dp(j)/wp(j) LFj/wp(j) (LSj - ESj )/wp(j)

pj/wp(j)

Table 10.4: Priority Rules for Operations Scheduling

10.2 Improvement Heuristics

Building upon the list scheduling algorithm presented above, we can now turn our attention to more advanced solution methodologies, namely biased random sampling and two different tabu search procedures. All of them employ the fact that each solution can be represented as a feasible list 7r. Hence, we will consider from now on only lists. The mapping of a list 7r into a schedule S is then straightforward as given in Subsection 10.1.1.

10.2.1 Sampling Methods

The essence of sampling methods is to employ the schedule generation scheme n 2:: 1 times in a probabilistic way. I.e., instead of selecting the operation from the set of available operations with minimum priority value, we choose the operation based on selection probabilities. This way we draw a sample of at most n different schedules from the solution space of all feasible schedules and pick the schedule with associated best objective function value. Note that the solution space is determined by the properties of the schedule generation algorithm (cf. Theorem 10.1). Dependent on how the selection probabilities of the available operations are computed we can distinguish random sampling and biased random sampling (for a detailed differentiation cf. Kolisch and Hartmann [189]).

10.2. IMPROVEMENT HEURISTICS 181

Random Sampling

Random sampling assigns each operation j in the available set A(g) the same probability p(j), i.e.,

p(j) = 1 A~g) I' (10.12)

Biased Random Sampling

Biased random sampling calculates selection probabilities based on the priority values. Biased random sampling methods have been mainly used to solve resource-constrained project scheduling problems (cf. Kolisch [184]) and job shop scheduling problems (cf. Baker [16]). A recent survey of biased random sampling methods for the resourceconstrained project scheduling problem is given in Kolisch and Hartmann [189]. New biased random sampling approaches are developed in Schirmer and Riesenberg [270]. Feo et al. [106] have introduced a similar solution method called Greedy Randomized Adaptive Search Procedure (GRASP) which has been used to solve, amongst other optimization problems, different type of scheduling problems including the scheduling of printed wiring board assembly (cf. Feo et al. [107]).

The selection probability of operation j E A(g) is calculated as follows. First, we determine for each operation j in the available set A(g) the regret value r(j) which is the absolute difference between the priority value v(j) of the operation under consideration and the worst priority value of all operations in the decision set, i.e., r(j) = max{v(i) 1 i E A(g)} - v(j). Next, we modify the regret value by adding the constant 1, i.e., r'(j) = r(j) + 1. Now, we calculate the selection probability

( .)_ r'(j) PJ -~ 1(')

L.."iE.A(g) r z (10.13)

Employing the modified regret value assures two things. First, the denominator cannot be zero and hence for each operation j the selection probability p(j) is always defined. Second, each operation j in the available set A(g) has a selection probability of p(j) > O. Hence, each list 7r can be generated and the search space includes the optimal solution.


In Kolisch [184, 185] it has been experimentally shown that the performance of biased random sampling methods relies significantly on the quality of the employed priority rule. When used with the best deterministic priority rules, biased random sampling will also show the best results. Hence, we employed the weighted earliest due date (WEDD) rule which showed in the experimental evaluation (cf. Section 10.3) the best results.

10.2.2 Tabu Search Based Large-Step Optimization

We now introduce two different tabu search procedures. Both are embedded in a large-step optimization method. Large-step optimization methods have been first introduced by Martin et al. [226]; an application of this method to the job shop scheduling problem is given by Louren<;o [217]. The general idea is as follows. One starts with an initial solution and tries to improve this solution by employing a local search algorithm. From the obtained local optimal solution it is then proceeded by a large step to a new solution which again is the starting point for a new local search. That is, the large-step method visits only local optimal solutions which reduces the solution space. The application of large steps secures that the method does not get trapped in local optimal solutions. There are three elements of a large-step optimization algorithm which have to be detailed.

• The representation of a solution,

• a specification of 'large steps', and

• the specification of the local search procedure.

We will sketch out these elements before detailing them below.

Solution Representation As solution representation we employ the operations list 7r. From 7r we can additionally derive the information ofthe assembly sequence >.. >. = (aI, ... , ap] gives the sequence assemblies are placed on the assembly area. As an example consider Figure 10.1 where we have the assembly sequence>. = (3,1,2]. The assembly sequence>. can be directly obtained from the operations sequence 7r because whenever we have 7r(Sh) < 7r(sp) (h,p = 1, ... ,P,h =J p) then >.(h) < >.(p) holds.


Local Search Procedure The local search method performs a tabu search on the operations sequence 7r where the assembly sequence A is fixed. For example, in Figure 10.3 the solution space for order sequence AU is U. The dotted arrow u ~ u' shows the path from the initial solution u to the local optimal solution u'.

Large Steps The large steps perform a tabu search on the assembly sequence A. In Figure 10.3 the solid arrow u' ~ v shows the large step from the local optimal solution u' within the solution space U defined by order sequence AU to the new solution v within solution space V defined by the new order sequence AV.

Figure 10.3: Large-Step Optimization

Principles of Tabu Search

Tabu search (TS) has been originally developed by Glover [121, 122]. It is essentially a steepest descent/mildest ascent method. That is, it evaluates all solutions of the neighborhood and chooses the best one, from which it proceeds further. This concept, however, bears the possibility of cycling, that is, one may always move back to the same local optimum one has just left. In order to avoid this problem, a tabu list is set up as a form of memory for the search process. Usually, the tabu list is employed to forbid those neighborhood moves that might cancel the effect of lately performed moves and might thus lead back to a recently visited solution. Typically, such a tabu status is overrun


if the corresponding neighborhood move would lead to a new overall best solution (aspiration criterion). It is obvious that TS extends the simple steepest descent search, often called best fit strategy, which scans the neighborhood and then accepts the best neighbor solution, until none of the neighbors improves the current objective function value.

Key issues in the design of tabu search procedures are the selection of a suitable solution representation, the definition of a neighborhood, and the definition of a tabu list.

Local Search of Operation Sequences

We will now detail the tabu search procedure for finding local optimal operation sequences. We use two different neighborhoods, a simple adjacent pairwise interchange neighborhood and a more elaborated so-called 'critical neighborhood'.

Adjacent Pairwise Interchange Neighborhood The first neighborhood is an adjacent pairwise interchange (API) neighborhood (cf., e.g., Della Groce [73]). A neighbor of sequence 1r is each precedence feasible sequence 1r' with A' = A, jh = jh+b and jh+1 = jh for one h (h = 1, ... , J - 1) and jg = j; for all 9 E {1, ... , J} \ {h, h + 1}. A precedence feasible list 1r' is only derived, if operation jh is no predecessor of operation jh+1, i.e., jh ~ Pjh+l' Figure 10.4 illustrates the API-neighborhood. The new neighbor 1r' is derived by the move M(jh+1, jh) where operation jh+1 is shifted in front of operation jh. The backward move MUh' jh+d is set tabu.

Figure 10.4: API-Neighborhood with Move MUMb jh)


Critical Neighborhood The drawback of the API-neighborhood is the fact that it does not employ any knowledge which is given in the schedule S associated with the incumbent list 11". In contrast, the so-called 'critical neighborhood' exploits problem specific knowledge given in the incumbent schedule S = (SI, ... ,SJ). The general idea is to find for an operation j a set of blocking operations Bj which hampers j from being started one period earlier. A similar idea for the resourceconstrained project scheduling problem has been proposed by Baar et al. [14]. The neighbor of 11" w.r.t. j is obtained by shifting j (and all its predecessors) in front of that operation h E Bj which has the smallest list position. More detailed, the neighborhood is as follows. For operation j we define ESfA, the earliest feasible start time of operation j w.r.t. part availability.

ESfA = min {t E {ESj, ... ,LSj} I Ni,t ~ qj,i + ~ %,i hE1'j

Vi = 1, ... , I} (10.14)

An operation j is considered to be left shiftable, if Sj > max{ Sfe , ESfA} holds. Sfe is the earliest precedence feasible start time as defined in Equation (lOA). A left shiftable operation does neither start at its precedence nor part availability based earliest start time. Hence, there must be a set of operations containing at least one operation which prevents j from starting earlier than Sj.

Denoting with 6f,t and Ni,t the available assembly capacity and the material availability w.r.t. the current schedule, we can define the capacity shortage ct,r and the part availability shortage qt,i' respectively, which hampers operation j from being started at Sj - 1.

(10.15)

qt,i = max {O, qj,i - Ni,Sj-l} (10.16)

The set of resource and part types for which there is a capacity or part availability shortage for operation j at time instant Sj - 1 is defined as set of critical resources Rj and critical parts Ij, respectively.

Rj = {r 11 ~ r ~ R, ct,r > 0 } (10.17)


Ii = {i I 1 ~ i ~ I, qJ:i > 0 } (10.18)

The set of resource critical operations Cf contains all operations which are causing some, not necessary all, capacity shortage at time instant Sj - 1.

Cf = {h E :r I h rf- i\ and 7r(h) < 7r(j)

and Sh < Sj ~ Sh + Ph and :3 r E nj: ct,r > 0 } (10.19)

The set of part critical operations CJ contains all operations which are causing some, not necessarily all, part availability shortage at time instant Sj - 1.

Cf = {h E :r I h rf- Pj and 7r(h) < 7r(j) and Sh < Sj

and :3 i E Ii: qh,i > 0 } (10.20)

The set of critical operations Cj = CJ U Cf is a set of operations where each operation is causing some, not necessary all, capacity or part shortage at Sj - 1.

Let a blocking set 8j be a minimal subset of the critical operations Cj for which the following holds:

Vr E nj : L ct,r 2 ctr and Vi E Ii: L qh,i 2 qJ:i (10.21) hEBj hEBj

'Minimal' means that a blocking set must not contain any other subset for which condition (10.21) holds. In particular a blocking set does not contain any other blocking set as a subset.

Depending on the set of critical operations, there is more than one blocking set. Consider for example the left shiftable operation 4 within the schedule given in Figure 10.5 (cf. also Table 10.5).

Associated with operation 4 are the set of critical operations C4 = {3, ll} and the two blocking sets 81 = {3} and 8~ = {ll}, respectively. We determine for each left shiftable operation j exactly one blocking set. Thereby we are seeking for the blocking set with lowest cardinality. Starting with cardinality 1 we are looking for a blocking set with a single operation h E Cj. In case of more than one possible operation h, we select the operation with the latest start time Sh, in case of ties we choose the operation with the highest list position 7r (h) . Is


L~I~I 3 4 5 Assembly 1

Assembly 2 7 1 81 9 10

11 12 Assembly 3

I .. t 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Figure 10.5: Initial Solution

there no blocking set with cardinality one, we search for a blocking set with cardinality two. We start with the operation h E Cj, with latest start time (highest list position) and search for a second operation 9 =f:. h, 9 E Cj, with latest start time (highest list position). Table 10.5 gives the blocking sets which have been determined for the schedule depicted in Figure 10.1.

A list rr' where in the corresponding schedule S' operation j shall start earlier than Sj has to put operation j at a list position rrj which is smaller than the list positions rr~ of all operations h in a blocking set Bj.

The corresponding move M (j, h) is to shift j and all its predecessors which are at list positions between rr(h) + 1 and rr(j) - 1 between operation j1r(h)-l' the immediate list predecessor of h, and h itself. Figure 10.6 illustrates the move M(ji,jb) with if being predecessor of k Table 10.6 provides the four moves associated with the blocking sets of Table 10.5 and the corresponding neighbor lists with their objective function values.

Figure 10.6: Critical Neighborhood with Move (ji, jb)


For all operations k which have been in list 7r in front of j and are in list 7r' behind j, i.e., 7r(k) < 7r(j) and 7r'(j) < 7r'(k), we set the move M(k, j) tabu. That is, k is not allowed to be shifted in front of j. Table 10.6 shows the moves and the associated tabu moves for the example. A move leads always to a precedence feasible list but the new list might not be feasible w.r.t. the spatial capacity. Spatial infeasible lists are discarded. In order to obey the assembly sequence A we set for all order pairs hand p (h, p = 1, ... , P, h =J. p) the move M(sp, Sh) tabu if A(h) < A(p) holds.

j 1 - 2 3 4 5-6 7 8 9 - 12 :<:i 0 0 0 1 c'l ~ 2 1 1 0 qj,l

Cj {2,1l} {3,1l} {2,3,4,1l} {5,7} Hj {2,1l} {3} {3} {7}

Move M(3,1l) M(4,3) M(7,3) M(8,7)

Table 10.5: Blocking Sets for the Example

Figure 10.1 gives the schedule of the list 7r = (11,12, 1,2,3,4,5,6,7, 8,9,10] with objective function value 39 and Figure 10.7 gives the schedule of the best neighbor 7r' = (11,12,1,2,4,3,5,6,7,8,9,10] with objective function value 30.

Move 7r Z Tabu (11,12,1,2,3,4,5,6,7,8,9,10] 39

M(3,11) (1,3,11,12,2,4,5,6,7,8,9,10] 47 M(2,3), M(11,3), M(12,3)

M(4,3) (11,12,1,2,4,3,5,6,7,8,9,10] 30 M(3,4) M(7,3) (11,12,1,2,6,7,3,4,5,8,9,10] 48 M(3,7), M(4,7)

M(5,7) M(8,7) (11,12,1,2,3,4,5,6,8,7,9,10] 33 M(7,8)

Table 10.6: Moves for the Example


r--~-I~~~~~~~-~I --~14----3~1 _+-J----:-~-i Assembly 1

r -8-1 ~ I - - - - - --.

Assembly 2 10

11 12 Assembly 3

I 30 t

o 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Figure 10.7: Solution after Move M(4, 3)

Large-Step Optimization of Assembly Sequences

We have employed the same large-step method for the API-neighborhood and the critical neighborhood. The method can be given as follows.

Let for the incumbent local optimal solution p* be the assembly with the highest objective function contribution which is not at the first position of the assembly sequence. The new assembly sequence >..' = (a~, ... , a~] is obtained by moving p* to the first position of the sequence, i.e., a~ = p*. Any order sequence which has been evaluated is set tabu for the rest of the procedure.

Let us illustrate the large step with the help of an example. Consider the solution given in Figure 10.7. The assembly with the highest objective function value contribution is 2 with W2 • T2 = 3 . 8 = 24. We are now changing the assembly sequence). = (3,1,2] to the new assembly sequence>..' = (2,3,1] by moving assembly 2 to the first list position. A solution for the new assembly sequence is obtained by first constructing a feasible operation sequence which is then mapped into a schedule. The construction of the operation list is straightforward. In the order given by >..' we sequence the operations of each assembly by ascending operation number, i.e., 7r = (sal"'" eal , ••• , Sap, ••• , eap].

This way we obtain for the assembly sequence>..' = (2,3,1] the operation sequence 7r = (6,7,8,9,10,11, 12, 1,2,3,4,5]. The associated


schedule with an objective function value of 50 is pictured in Figure 10.8.

After the large step has been performed, the local search of operation sequences starts anew. Employing the critical neighborhood search, we have the possible moves M(2, 12), M(8, 7), M(l1, 7), and M(12,10). The move which leads to a solution with smallest objective function value is M(11,7). The new operation sequence is 7r = (6,11,7,8,9,10,12,1,2,3,4,5]. By application of the schedule generation algorithm this sequence is mapped into the schedule given in Figure 10.9 with the optimal objective function value of 18.

11 I~~~~~~]~J Assembly 1

I I --------------.

3 Assembly 2

I I 2 4 L. ____

I -I 81 7 9 10 Assembly 3 -

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Figure 10.8: Solution after Application of a Large Step

Assembly 1

Is I 9

I 11

7

12

10 I Assembly 2

Assembly 3

I

5

I >-

17

I I ~ t

o 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Figure 10.9: Optimal Solution

t


Parameter Adjustments

Large-Step Optimization with API-Neighborhood The length of the tabu list was set according to preliminary experiments to 20. The local search for a given order list is stopped whenever 20 consecutive moves have not given an improved solution. Each visited order list is set tabu until the entire solution procedure commences. The aspiration level criterion was used. The number of evaluated operation lists is set to a maximum of 5000.

Large-Step Optimization with Critical Neighborhood For the critical neighborhood we set the length of the tabu list to 30, the entire number of evaluated solutions is set to a maximum of 5000. The aspiration level criterion was used. The local search for a given order list A is stopped whenever z~, the incumbent best objective function value associated with order list A, exceeds after iter accepted operation lists the so far best objective function value Z by at least E %, i.e., (Z~ - Z) / Z ·100 > E %. According to preliminary computational results we choose E = 0 and iter = 5. That is, for a given order sequence A, the tabu search of operation sequences is stopped after 5 iterations if one has not found a solution which is at least as good as the so far best found objective function value Z.


This section reports the results obtained when evaluating the heuristics proposed in Sections 10.1 and 10.2.


For evaluation purposes, a set of 270 test instances comprising 135 small and 135 large instances was generated. For the generation we employed a parameter controlled instance generator which builds up on ProGen (cf. Kolisch et al. [193, 194]). Details are provided III

Appendix A in general and Appendix A.4 in particular.



Three systematically varied problem parameters were employed. The resource strength, the part strength, and the spatial resource strength. Details of these parameters are provided in Appendix A.4.

The assembly resource strength RSA E [0,1] measures the scarcity of the assembly resource capacity given in constraints (7.4). The smaller the RSA-value is, the tighter the assembly resource constraints are. RSA was set to 0.1, 0.3, and 0.5 for the small instances and to 0.1, 0.2, and 0.3 for the large instances, respectively.

The spatial resource strength RSs E [0, 1] measures the available capacity of the assembly area. For RSs = 0 there is just one assemble area available, while for RSs = 1, there is enough capacity to process all assemblies in parallel. RSs was set to 0.1. 0.5, and 1.0 for the small instances and to 0.3, 0.5, and 1.0 for the large instances, respectively.

The part strength PS E [0,1] measures how much the arrival times of parts delay operations to start behind the earliest start times. For PS = 0 there is no delay, while for PS = 1 operations are delayed until their latest start times. For the small instances, P S was set to 0.7, 0.8, and 0.9, while for the large instances, it was set to 0.8, 0.9, and 1, respectively.


RFA, the resource factor for assembly resources, was set for all instances to 1 and PF, the part factor, was set for all instances to 0.7. RFA measures the density of constraints (7.4). Correspondingly, PF measures the density of constraints (7.6). RpA = 1 and PF = 0.7 express the fact that for each non-dummy operation j there is ct,r > 0 and for 7 out of 10 non-dummy operations we have qj,i > O. In case of part demand, qj,i was randomly drawn out of the interval [1,3]. In case of capacity demand, ct,r was randomly drawn from the interval [1,3]. Pj, the processing time of operation j, and w p , the weight of assembly p, were randomly drawn from the interval [1,3] and [1,5], respectively.

The generation of the following problem parameters is dependent on the instant size (IS). For the small instances there is P = 3, Jp E [3,5], 1= 1, and rp E [0,5]. Jp denotes the count of operations belonging to assembly p, The part variance was set to PV = 0.3. PV measures how


much the part arrival spreads around the arrival time as measured by the part strength (cf. Appendix AA). The due date dp was calculated as follows: dp = rp+ESep where rp denotes the release date of assembly p. The following parameter values were set differently for the large test instances: P = 10, Jp E [5,10], RA = 2, I = 2, rp E [0,20], and PV = 004.

Experimental Design

Realizing a full factorial design with 5 replications for each combination of the independent parameter levels, 5 . 33 = 135 instances were generated for the small and the large instance set, respectively.


Each instance was solved with each of the 8 different single-pass construction heuristics proposed in Section 10.1, the two sampling approaches, random sampling (RS) and biased random sampling (BRS) introduced in Subsection 10.2.1, and the two large-step optimization variants CN-LSO and API-LSO presented in Subsection 10.2.2. Additionally, we applied a multi priority rule heuristic (MPRH) (cf. Kolisch and Hartmann [189] and Subsection 9.3.2) by iteratively solving an instance with each of the 8 different single-pass heuristics and keeping the best solution. This way, altogether 13 different heuristics were considered in our computational study.

We divide the presentation of the computational results in an assessment of the heuristic performance and the analysis of the effect of the problem parameters on the heuristics. As performance measure we have chosen the average deviation of the heuristic under consideration from a lower bound of the objective function value. In particular, we have solved each small instance with the MIP model (7.1) - (7.9) and each large instance with the LP relaxation of (7.1) - (7.9). Modelling was done with AMPL (cf. Fourer et al. [111]) and as solver we used CPLEX (cf. Bixby and Boyd [31]). Note that for the small instances the term 'lower bound' coincides with the optimum.

If not stated differently, we provide means; statistical testing was done with nonparametrical tests solely because inspection of the distribution functions revealed that none of the values was normally


distributed (cf. Alvarez-Valdes and Tamarit [7]). In particular we employed the following tests (cf. Golden and Steward [123]). The Wilcoxon matched-pairs signed-ranked test for 2 related samples, the Friedman test for k > 2 related samples, and the Kruskal-Wallis-H test with tie correction for k > 2 independent samples. If no level of confidence is given explicitly, the 1% level of confidence is meant.

Performance of the Heuristics

Tables 10.7 and 10.8 give for the 13 tested heuristics the mean, the standard deviation, the minimum, and the maximum, of the percentage deviation from the optimum and the lower bound, respectively. The heuristics are sorted by descending mean. The last but one column provides the number of iterations (It) of each heuristic. Different number of iterations are caused by the stopping criteria for the two large-step optimization methods (cf. Subsection 10.2.2).

For the small instances, the means range from 0.10% for the two sampling approaches BRS and RS to 84.28% for the single-pass construction heuristic with SPT. We have compared each heuristic with the next best one using the Wilcoxon signed-rank test. A '*' in the column significance (Sig) indicates a significant difference of the two heuristics at the 1% level of confidence. That is, we have a significant difference between sampling heuristics and large-step optimization approaches, between the latter and the multi priority rule heuristic, and between the latter and the WEDD heuristic. WEDD is again significant better than the next best single-pass heuristic, namely WLFT. Between the single-pass heuristics WLFT, WSLK, EDD, WSPT, SLK, LFT there is no significant difference when compared pairwise in the order of descending mean. But LFT does again differ significantly from SPT. Note the different number of iterations. The sampling heuristics perform 5000 iterations while the large-step optimization heuristics do less than 5000 iterations. This stems from the stopping criterion as outlined in Subsection 10.2.2. For both, the small and the large instances, the critical neighborhood needs less iterations than the API one. That is, after a large step has been performed, it needs less iterations in order to reach a local optimal solution.

The mean deviation of the heuristics from the lower bound on the large instances ranges from 43.63% for CN-LSO to 174.55% for SPT.


We see that the performance of the sampling heuristics deteriorates dramatically when the problem size and hence the solution space increases. This effect is well known from other problem classes, e.g., the resource-constrained project scheduling problem (cf. Hartmann and Kolisch [147]). Also, we now have a significant difference between the two large-step optimization approaches and the two sampling heuristics, respectively. This demonstrates that employing more knowledge for the definition of the neighborhood and the calculation of the selection probabilities pays in terms of the performance.

Rules which take into account the assembly weight perform significantly better than their counterparts which are based on operation information only. This holds for the small and the large instances.

We can summarize that the large-step optimization heuristic with the critical neighborhood provides the best results but, compared to single-pass heuristics, needs a large number of iterations. The best single-pass construction heuristic is WEDD. Without too much effort its performance can be increased by employing the multi priority rule heuristic.

Heuristic Mean Std Dev Minimum Maximum It Sig BRS 0.10 0.79 0.00 7.14 5000 RS 0.10 0.80 0.00 7.14 5000 * CN-LSO 1.43 4.32 0.00 25.00 86 API-LSO 2.18 7.15 0.00 41.67 264 * MPRH 6.00 10.61 0.00 50.00 8 * WEDD 19.18 28.30 0.00 150.00 1 * WLFT 31.50 40.46 0.00 212.50 1 * WSLK 49.97 65.96 0.00 276.92 1 EDD 54.39 70.29 0.00 450.00 1 WSPT 57.02 66.88 0.00 455.56 1 SLK 70.72 87.64 0.00 450.00 1 LFT 71.27 74.30 0.00 450.00 1 * SPT 84.28 82.00 0.00 575.00 1

Table 10.7: Deviation from Optimum for Small Instances


Heuristic Mean Std Dev Minimum Maximum It Sig CN-LSO 43.63 23.68 8.31 128.28 2411 * API-LSO 49.40 26.62 12.06 148.48 3832 * MPRH 57.65 31.81 13.41 186.27 8 * WEDD 61.93 34.92 13.63 186.27 1 BRS 65.98 36.66 17.24 224.61 5000 * RS 66.26 36.78 17.30 224.62 5000 * WLFT 88.22 50.87 13.57 375.75 1 WSLK 90.34 51.15 14.97 321.68 1 * EDD 109.74 65.81 29.24 458.27 1 * WSPT 122.91 58.23 32.95 364.55 1 SLK 128.60 72.92 32.27 497.82 1 * LFT 138.77 74.01 42.87 522.45 1 * SPT 174.55 83.74 48.48 591.79 1

Table 10.8: Deviation from Lower Bound for Large Instances

Effect of the Problem Parameters

We now focus our attention on the six best performing heuristics, namely the two large-step optimization approaches, the two sampling approaches, the multi priority rule heuristic, and WEDD, when applied to the large sized instances. Tables 10.9, 10.10, and 10.11 give the mean of the percentage deviation from the lower bound for each heuristic w.r. t. the different levels of the systematically varied problem parameters spatial resource strength, assembly resource strength, and part strength, respectively. The column 'Sig' indicates with ,*, if there is a significant influence of the problem parameter on the performance of the heuristic at the 1% level of confidence.

Influence of the Spatial Resource Strength

Table 10.9 shows the influence of the spatial resource strength on the selected heuristics. The performance of the two sampling methods, RS and BRS, is significantly influenced by the spatial resource strength. With increasing RSs-levels, i.e., more available assembly area, the average deviation of both methods increases. The explanation is that a


larger spatial capacity causes a greater number of available operations within each iteration of the list generation algorithm because orders can be processed in parallel. The greater number of available operations in turn lowers the selection probability of the operations which lead to the best schedule. The performance of WEDD and MPRH are almost insensitive for different strengths of the spatial resource while the performance of the two large-scale optimization approaches declines for RSs-values of 1.0. But this effect is not significant. It is caused by the stopping criterion which limits the number of moves after a new assembly sequence has been obtained.

Mean RSs Sig 0.3 0.5 1.0

CN-LSO 43.63 42.15 41.86 46.87 API-LSO 49.39 48.65 48.07 51.46 MPRH 57.64 59.08 56.03 57.82 WEDD 61.92 62.45 61.51 61.81 BRS 65.97 48.96 57.59 91.38 * RS 66.25 49.24 57.85 91.67 *

Table 10.9: Effect of RSs on Heuristics for Large Instances

Influence of the Assembly Resource Strength

Table 10.10 shows the influence of the assembly resource strength on the heuristics. The effect is for all heuristics highly significant. The average deviation increases when the capacity of the assembly resources becomes more scarce.

Influence of the Part Strength

Table 10.11 shows the influence of the part strength on the heuristics. There is a highly significant effect on all heuristics. As the part strength increases, the average deviation of the algorithms deteriorates.


Mean R§4. Sig 0.1 0.2 0.3

CN-LSO 43.63 58.98 39.68 32.22 * API-LSO 49.39 66.33 44.75 37.10 * MPRH 57.64 77.60 52.73 42.59 * WEDD 61.92 82.94 56.76 46.07 * BRS 65.97 87.98 58.83 51.11 * RS 66.25 88.25 59.12 51.38 *

Table 10.10: Effect of RSA on Heuristics for Large Instances

Mean PS Sig 0.3 0.5 1.0

CN-LSO 43.63 29.66 37.99 63.23 * API-LSO 49.39 33.83 43.24 71.12 * MPRH 57.64 38.48 50.42 84.03 * WEDD 61.92 41.63 55.00 89.14 * BRS 65.97 48.16 60.23 89.54 * RS 66.25 48.32 60.45 89.99 *

Table 10.11: Effect of PS on Heuristics for Large Instances

Chapter 11

Research Opportunities

There are three areas for research opportunities: a refinement of the hierarchical approach, an adaption of models and methods to specific real world problem instances, and the development of more efficient solution methods.

Refinement of the Hierarchical Framework

The hierarchical framework employed in this book is rather simple. In order to develop a fully fledged hierarchical approach there are four areas of work to be done. First, the anticipation mechanism has to be refined. Second, a feedback influence has to be considered. Third, the concept of revised planning has to be implemented. Finally, the entire approach has to be experimentally tested.

Adaption of Models and Methods to Specific Real World Problems

As stated in the introduction, real world problems do, in general, differ from the decision models proposed in this book. The difference might be such that the decision model takes constraints into account which are not relevant for the problem at hand, or the other way around, the problem at hand might contain peculiarities which are not covered by the decision model. For example: If a company outsources the part fabrications and procures all parts from suppliers, the manufacturing planning problem (6.2) - (6.11) can go without the capacity

200 CHAPTER 11. RESEARCH OPPORTUNITIES

constraints in the fabrication given by constraints (6.8) and the capacitated dynamic lotsizing problem turns into an uncapacitated dynamic ordering problem.

The assembly of specific capital goods such as automatic letter sorting machines is characterized by the fact that there is an increasing demand of shop floor area during the time span of the final assembly (cf. Schlauch and Levy [271]). In the case of scarce shop floor capacity, it is crucial to take into account the dynamic demand for the assembly area which is characterized as '2-dimensional polygon dynamic spatial demand' in Section 7.4. This feature is yet not incorporated in the operations scheduling model (7.1) - (7.9).

Development of More Efficient Solution Methods

The following solution methodologies seem to be promising in order to obtain better solutions to the decision problems proposed in Part II.

• The multi-constrained knapsack model of the order selection problem (8.1) - (8.4) could be solved to optimality with the branch-and-price approach proposed by Barnhart et al. [23].

• The manufacturing planning problem (6.2) - (6.11) could be attacked by metaheuristics which, similar to the operations scheduling problem, encode a solution as ajob list which is then mapped into a schedule by employing the list scheduling heuristic proposed in Section 9.1. This way different metaheuristics such as tabu search, genetic algorithms, and simulated annealing could be employed.

• Instead of the tabu search procedure outlined in 10.2.2, we could employ other metaheuristics for the operations scheduling problem (7.1) - (7.9). Two metaheuristics which have been applied successfully to intricate scheduling problems are simulated annealing and genetic algorithm (cf., e.g., Mattfeld [228], Pesch [250]). Especially, genetic algorithms have recently shown the best results for solving the resource-constrained project scheduling problem which is the backbone of the operations scheduling problem (cf. Hartmann [146], Kolisch and Hartmann [147, 189]).

Appendix A

Instance Generation

In this appendix we document the instance generator employed for the various problems discussed in Part II. The methodology builds on ProGen, an instance generator for a broad class of project scheduling problems (cf. Kolisch et al. [193, 194]). Concepts introduced in detail there will be only briefly reported here. Following the terminology there, we will speak in Appendix A.l of projects and activities. Depending on the decision level a project depicts an order or assembly while an activity depicts a task, a job, or an operation. To reduce the computational burden of instance generation and computational testing, we assumed for all experimental investigations constant resource demand and availability, as well as minimal finish-start time lags of o.

A.I General Concepts

Network Restrictiveness

The network restrictiveness measures for each project p how precedence relations between activities restrict the number of precedence feasible activity sequences. For a network restrictiveness of 0, there are no restrictions and hence there are nd~p-l) sequences, where np

denotes the number of non-dummy activities of project p. For a network restrictiveness of 1 there is only one activity sequence. N R, an estimator for the network restrictiveness, has been proposed by Thesen [308]. Schwindt [279, 280, 281] employed this concept and

202 APPENDIX A. INSTANCE GENERATION

made it available for the instance generation of scheduling problems. NR E [0,1] is calculated as

NR = Ej~~;+1 ISj I -(Jp - 2) (Jp - 2) . (Jp - 1) (A.l)

where Sj denotes the set of all (direct and indirect) successors of activity j and Jp is the count of activities of project p. The denominator is the maximal number of direct and indirect precedence relations in a project network with Jp activities. Note that sp and ep are the unique (but not necessarily dummy) start and terminal activities of project p. The numerator counts the given number of direct and indirect precedence relations of the network.

N R has been embedded into the general network generation routine developed by Kolisch et al. [193, 194].

Constraint Factor and Strength

In all three decision models we have constraints of the form I: Cj,r •

Xj,t ~ Gr,t: the resource constraints (5.3) in the order selection problem, the assembly resource constraints (6.6) and the fabrication resource constraints (6.8) in the manufacturing planning problem as well as the assembly resource constraints (7.4), the spatial resource constraints (7.5), and the part availability constraints (7.6) in the operations scheduling problem. Constraints of this type can be described by the two parameters constraint factor and constraint strength (for the specific application of these two parameters to constraints of the type as given in (6.6) cf. to Kolisch et al. [193, 194]). The constraint factor measures the density of the cj,r-matrix. The constraint strength measures the level of the right hand side as a convex combination of a lower and and upper bound of the left hand side.

A.2 Order Selection Instances

Time Windows

With respect to the constraints (5.5) we have to specify the time window [4p, dp] of order p. The time window is determined by two factors, the due date factor and the time window factor. The due date factor

A.2. ORDER SELECTION INSTANCES 203

DF E [0,1] defines how close the earliest due date is to the precedencebased lower bound of the finish time of an order. The time window factor TWF E [0,1] determines how close earliest and latest due date are. More precisely we calculate the earliest due date

!1.p = (1 + DF) . LPsp,ep p= 1, ... ,P (A.2)

where LPh,j denotes the longest path from task h to task j. For DF= 0 (1) there is a tight (loose) earliest due date. Using the time window factor we calculate the latest due date.

p=1, ... ,P (A.3)

For TWF 0 the time window reduces to the single-time instant !1.p ,

for TWP- 1 the length of the time window equals the precedencebased lower bound of the order duration.

Resource Constraints

With respect to the resource constraints (5.3) we have to determine R, the number of different resources, rj, the resource requested by j, Cj,t, the capacity demand of task j, and Cr,t the available capacity of resource r.

For each task j of an order p we calculate its rank rankj (cf. Kli1lvstad [182]). We begin by assigning each terminal task ep of an order p rankep = O. The rank of the remaining tasks j = €p - 1, ... , sp is calculated recursively as

rankj = max {rankk IkE Sj} + 1 (AA)

where Sj denotes the set of all successors of task j. If the network of each order is interpreted as a bill of materials (BOM), the rank of each task equals the low-level code number of the associated node in the network. We now assign to each rank, except the highest rank, one resource. The reason is that the highest rank corresponds with the dummy start activity of the order network which does not have a resource demand. Hence, the number of resources is

R = max {ranksp I p = 1, ... , P} - 1. (A.5)

Task j has to be processed at resource rj = rankj.


For the calculation of available capacity of each resource r the resource strength is employed. The lower bound of the resource demand is

C r = max {Cj,t I j = 1, .. . ,J with rj = rj t = 1, .. . ,Pj} (A.6)

while the upper bound C r is the peak demand of the latest start schedule S = (LSI, ... ,LSJ) w.r.t. resource r. The latest start times are obtained by backward recursion from the earliest due dates Qp •

Order Values

With respect to the objective function (5.1) we have to determine the value vp of order p. We assume that the value of an order depends on the sum of the capacity demand. Hence, by employing VF, the value factor, we calculate for each P = 1, ... , P

Vp = rand [(1- VF) ;~,~c;,n (1+ VF) ;~,~c;,rl (A.7)

where rand[l, u] denotes that we randomly draw a discrete number out of the set {l, 1 + 1, ... , u}.

A.3 Manufacturing Planning Instances

Due Date Constraints

The due date dp of order p (p = 1, ... , P) is calculated as follows. First, rp , the release date of order p, is randomly drawn from the interval [r, 'f]. Afterwards, d, an upper bound of the completion time of all orders if started at time instant t = 0 is computed by summing up the processing times of jobs, i.e., d = Ef=l Pj. Now, dp , the due date of

order P is computed to dp = rp + LPsp,ep + DF . (d - LPsp,ep) where LPsp,ep denotes the longest path from the start job sp to the terminal job ep of order P and DF E [0,1] denotes the due date factor.

Assembly Resource Strength

The assembly resource strength RSA determines the availability of the assembly resources as a convex combination of a lower and an

A.4. OPERATIONS SCHEDULING INSTANCES 205

upper bound of the capacity demand. The lower bound is C r = max {cf,r I j = 1, ... , J}. The upper bound is the peak capacity demand of schedule S = (LSI, . .. , LSJ) w.r.t. to resource f.

Fabrication Resource Strength

The fabrication resource strength RSF determines the scarceness of the fabrication resources. Here, the lower and the upper bound of the available fabrication capacity is calculated, respectively, as follows:

First we calculate the part demands as induced from the latest start schedule S = (LSI, ... , LSJ)

J

df.! = L qj,i (A.8) j=IILSj-I=t

To obtain an upper bound, we calculate cost-minimal production quantities Qi,t for part i to fulfill the demands dfs by solving the uncapacitated Wagner-Whitin problem. The capacity required to realize these quantities is

C~ = max{t cr . Qi,t I t= 1, ... ,T} t=I

(A.9)

A lower bound of the fabrication capacity can be calculated as follows:

C F - L.."T=I L.."t=I t t,T I t - 1 T { ",t "'! cf'. dJ;.,s }

-r - max t - , ... , (A.10)

A.4 Operations Scheduling Instances

Part Strength and Variability

The part strength (PS) measures the timing parts are made available for assembly.

First, earliest and latest operation start times, ESj and LSj, are calculated by performing a forward and backward recursion for each assembly, respectively. For assembly p (p = 1, ... , P) forward recursion is performed from f p • The backward recursion determines latest start times such that each assembly is finished at the planning horizon T = 2::1=1 Pj by the latest.


From the earliest and latest start times we can derive lower bounds Jli,t and upper bounds ni,t for the number of units of part i which have to become available at time t.

and

J

Jli,t = L qj,i j=lIESj=t

J

(A. H)

ni,t = L qj,i' (A.12) j=lILSj=t

The cumulated quantity of part i which becomes available until time instant t is then

and

t Not-"'"'n o -'t, - L...J -Z,T

T=O

t

Ni,t = L ni,T" T=O

(A.13)

(A.14)

The earliest and latest time instant where n (n = 1, ... , Ni,T) units of part i are available is

(A.15)

and

Ii n = ;{in {t In> Ni t}. (A.16) 't=l -,

Employing the part strength PS E [0,1] we can calculate ti,n, the time instant until n units of part i have arrived as follows.

(A.17)

We now introduce the part variability PV E [0,1] which enforces that the timing of part supply does not deterministically depend on PS and the precedence-based earliest and latest start times. _With the part variability we enlarge the point ti,n to the range [ti,n, ti,n]. Out of the latter we randomly draw the time instant until n units of part i have arrived. The left and the right border of the range are calculated as

to - max tot ° - -. to - to - {- PV (_ )} -'l.,n - -'t,n, 't,n 2 't,n -'t,n (A.18)

A.4. OPERATIONS SCHEDULING INSTANCES 207

and "'" {- - PV (- )} ti,n = min ti,n, ti,n + 2' ti,n - t.i,n . (A.19)

Note, that for PV = 0 we get ti,n = ii,n and for PS = 0.5 and PV = 0 we obtain ti,n = 0.5· (ti,n - t.i,n)' For PS = 0.5 and PV = 1, ti,n =

rand [t.i,n, ti,nJ is calculated. Figure A.l illustrates the calculation of

the range [t, f] for PS = 0.8 and PV = 0.6.

[ ]

I ...

o 1 2 3 4 5 6 7 8 9 10 11

i

Figure A.l: Part Availability Generation

Spatial Resource Strength

The spatial resource strength RSs E [0,1] measures the scarceness of the spatial resource constraints (7.5). The lower bound is CS = 1 since at least one assembly has to be processed at a time. The upper bound is C S = P where all assemblies can be processed in parallel.

Appendix B

Notation

B .1 Notation for Order Selection

Indices p = orders, p = 1, ... , P J = tasks, j = 1, .. . ,J t = periods, t = 1, ... , T r = departments, r E R = {I, ... , R} Parameters Pj = processing time of task j Cj,t = capacity demand of task j in processing period t

(t=l, ... ,pj)

rj = department where task j has to be processed Pj = set of immediate predecessors of task j t':lljn = minimal time lag between the finish time of task i and the Z,J

start time of task j trr = maximal time lag between the finish time of task i and the start time of task j

sp = start task of order p

ep = terminal task of order p

Jp = set of tasks belonging to order p

Cr,t = capacity of department r in period t

d.p = earliest delivery time for order p

dp = latest delivery time for order p

vp = value (e.g., revenue, contribution margin) of order p

210 APPENDIX B. NOTATION

Variables YP = 1, if order p is accepted, 0 otherwise Xj,t = 1, if task j is started at the begin of period t, 0 otherwise

B.2. NOTATION FOR MANUFACTURING PLANNING 211

B.2 Notation for Manufacturing Planning

Indices p ~

j r

t

= orders, p = 1, .. . ,P = parts i = 1, .. . ,1 = jobs, j E J = {O, ... , J + I} = resources, r = 1, ... , RA and r = 1, ... , RF = time instants, t = 0, ... , T = periods, t = 1, ... , T

Parameters Pj = processing time of job j ESj (LSj) = earliest (latest) start time of job j ht (hf) = per period holding cost of job j (part i) hf = per period holding cost of procured parts assembled

tmin h,j

Sp (ep )

Jp Jp

o (J + 1)

Variables

by job j = quantity of part i assembled by job j = setup cost when fabricating part i = fabrication resource where part i is processed = set of immediate (all) predecessors of job j = set of immediate successors of job j = minimal time lag between the finish time of job hand

the start time of job j = due date of order P = maximal due date of all orders = start (terminal) job of order P = set of jobs belonging to order p = count of jobs of order p = source (sink) of the precedence network = capacity demand of job j w.r.t. assembly resource r

during processing = capacity needed to produce one unit of part i = capacity of assembly (fabrication) resource r at

time instant t (in period t)

Xj,t = 1, if job j is started at time instant t, 0 otherwise Yi,t = 1, if part type i is produced in period t, 0 otherwise Qi,t = production quantity of part i in period t

Ii,t = inventory of part i at the end of period t

212 APPENDIX B. NOTATION

B.3 Notation for Operations Scheduling

Indices p

J r t

= assemblies, p = 1, ... , P = parts, i = 1, ... , I = operations, j E J = {O, ... , J + 1} = resources, r = 1, ... , RA = time instants, t = 0, ... , T

Parameters pj = processing time of operation j ESj (LSj) = earliest (latest) start time of operation j qj,i = quantity of part i assembled by operation j Pj = set of immediate predecessors of operation j Sj = set of immediate successors of operation j tmin = minimal time lag between the finish time of h,j

dp

dmax

sp (ep)

Jp Jp

o (J + 1) A

Cj,r

Variables

operation h and the start time of operation j = due date of assembly p = maximal due date of all assemblies = start (terminal) operation of assembly p

= set of operations belonging to assembly p = count of operations of assembly p

= source (sink) of the precedence network = capacity demand of operation j w.r.t. resource r

during processing = capacity of resource r at time instant t = number of assemblies which can be placed on the

assembly area = quantity of part i which becomes available at time

instant t = cumulated quantity of part i which becomes available

until time instant t

x j,t = 1, if operation j is started at time instant t, 0 otherwise Tp = tardiness of assembly p


AlA AMPL API ASP ATF ATO BRS BOM CAD cf. CLSP CN CNC CODP c.p. DFA DM EDD e.g. EOQ ERP ETO FAC FAS FCFS FLP GRASP HDW

assem ble-in-ad vance a modeling language for mathematical programming adjacent pairwise interchange assembly scheduling problem assemble-to-forecast assemble-to-order biased random sampling bill of materials computer aided design confer capacitated lotsizing problem critical neighborhood computer numerically controlled customer order decoupling point ceteris paribus design for assembly Deutsche Mark earliest due date example given economic order quantity enterprise resource planning engineer-to-order flexible assembly cell flexible assembly system first come first serve fabrication lotsizing problem greedy randomized adaptive search procedure Howaldtswerke - Deutsche Werft AG

214

HPP

I.e. IPA

JIT

JSP LBH LFT

LP LPH LR

LSO LSP MIP

MRPH

MRP MRP II MTO OGAP

OPT PCBA

R&D

RCPSP RCMPSP

RHS RS SALB

SGAP SLK

SPT TBO TP TS

WEDD

WLFT WSLK

WSPT

w.l.o.g.

hierarchical production planning id est


Institut flir Produktionstechnik und Automatisierung just-in-time job shop problem Lagrangian-based heuristic latest finish time lin~ar program linear program based heuristic Lagrangian relaxation large-step optimization dynamic lot sizing problem mixed-integer program multi priority rule heuristic material requirements planning manufacturing resource planning make-to-order optimality gap optimized production technology printed circuit board assembly research and development recource-constrained project scheduling problem recource-constrained multi-project scheduling problem right-hand side random sampling simple assembly line balancing problem solution gap minimum slack shortest processing time time between order transportation problem tabu search weighted earliest due date weighted latest finish time weighted minimum slack weighted shortest processing time without loss of generality

List of Abbreviations 215

WW Wagner-Whitin

List of Figures

1.1 Multi-Level Manufacturing System for Make-to-Order Products. . . . . . . . . . . . . . . . . . . . . . . . . .. 2

1.2 Managerial Levels for Make-to-Order Manufacturing Systems. . . . . . . . . . . . 3

2.1 Manufacturing Environments 11 2.2 Scope of this Book . . . . . . 16 2.3 A Ship and its Partitioning into Blocks (courtesy HDW) 24 2.4 Placement of Curved-Bottomed Blocks on the Shop

Floor (courtesy HDW) . . . . . . . . . . . . . . . . 25 2.5 Assignment of Final Assemblies to the Shop Floor 29 2.6 Airplane Assembly . . . . . . . . . . . . . . . . . . 30

4.1 Objects and Decision Levels of the Hierarchical Frame-work. . . . . . . . . . . . . . . . . . . . . . . . 51

5.1 Example Network of the Order Selection Level 5.2 Supplier Integration ....... .

6.1 Explosion of Manufacturing Tasks 6.2 Cash Flows at the Manufacturing Level 6.3 Bill of Material ............. .

60 65

70 72

73 6.4 Relationship between Product- and Job-Representation 78 6.5 Network Representation for the Example Instance ... 81 6.6 Calculation of Holding Cost . . . . . . . . . . . . . . .. 85 6.7 Optimal Solution of the Manufacturing Planning Instance 85 6.8 Optimal Sequential Solution of the Manufacturing Plan-

ning Instance . . . . . . . . . . . . . . . . . . . . . .. 89 6.9 Infeasible Sequential and Optimal Integrated Solution 90

218 LIST OF FIGURES

7.1 The Operations Scheduling Context 94 7.2 Explosion of a Job into Operations . 95 7.3 Tardiness Cost . . . . . . . . . . . . 97 7.4 Horizontal Interdependencies within Operations Schedul-

ing . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 7.5 Network Representation for Operations Scheduling . 105 7.6 Optimal Solution of the Example Instance . 107 7.7 Types of Assembly Area Demand . . . . . . . . . . . 111 7.8 Assembly Resource Assignment . . . . . . . . . . . . 114 7.9 Assembly Resource Assignment in the Case of Variable

Capacity. . . . . . . . . . . . . 114 7.10 Spatial Resource Assignment . 115 7.11 Part Assignment . . . . . . 115

8.1 Possible Order Schedules. . 120

9.1 Assembly Schedule Generated by the List Scheduling Heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

9.2 Solution Obtained by the Construction Heuristic .... 139 9.3 Optimal Solution of the Relaxed Assembly Scheduling

Problem . . . . . . . . . . . . . . . . . . . . . . 151 9.4 Cost Minimal Fabrication Plan . . . . . . . . . 155 9.5 Solution of the Assembly Scheduling Problem . 157 9.6 Optimal Transportation Plan . . . . . . . . . . 160 9.7 Solution Obtained by the Lagrangian Heuristic . 161

10.1 Initial Solution ..... . · 174 10.2 Non-Active Schedule .. . · 177 10.3 Large-Step Optimization · 183 10.4 API-Neighborhood with Move M(jh+1,jh) · 184 10.5 Initial Solution ............. . · 187 10.6 Critical Neighborhood with Move (ji, jb) . · 187 10.7 Solution after Move M(4, 3) ....... . · 189 10.8 Solution after Application of a Large Step · 190 10.9 Optimal Solution ...... . · 190

A.1 Part Availability Generation. .207

List of Tables

3.1 Taxonomy of Assembly Literature ............ 34 3.2 Relationship of MTO Manufacturing and Project Net-

works ............................ 41 3.3 Relationship ofMTO Assembly and Multi-Project Schedul-

ing . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.4 Literature Survey of Multi-Project Scheduling ..... 44

4.1 Hierarchical Framework for Make-to-Order Assemblies. 48

5.1 Selected Literature on Internal Due Date Assignment.. 56

6.1 Taxonomy of Approaches for Multi-Level Manufacturing 74 6.2 Classification of Multi-Level Scheduling and Lotsizing

Literature . . . . . . . . . . . . . . . . . . 77

7.1 Survey of Assembly Scheduling Literature 99 7.2 Calculation of the Resource Assignment .113

8.1 MPS-Matrix ............. .121 8.2 Dynamic Programming Example .. .123 8.3 Effect of the Problem Parameter RS .126 8.4 Effect of the Problem Parameter IS. .126 8.5 Effect of the Problem Parameter DF . 127 8.6 Effect of the Problem Parameter NR .127

9.1 Priority Rules for List Generation . 131 9.2 List Generation for the Example Instance .132 9.3 Schedule Generation for the Example Instance .133 9.4 Backward Lotsizing for the Example Instance .139 9.5 -A Exemplary Calculation of h2,1 •.•••••••• .150

220 LIST OF TABLES

9.6 ht,c and Sj(t)-Values for the Example Problem. . 150 9.7 hft- and fi(t)-Values for the Example Problem . 154 9.8 cf(u, v)-Values for the Example Problem '" . 155 9.9 Calculation of the II (t)-Values . . . . . . . 156 9.10 Ci,t,r-Values for the Example Problem . . . 159 9.11 DF-Setting for Small and Large Instances . 164 9.12 Average Deviation of Heuristics from Lower Bound . 166 9.13 Comp~rison of Scheduling and Lotsizing Strategies . 167 9.14 Effect of RSA on Heuristics . 168 9.15 Effect of RSF on Heuristics . 169 9.16 Effect of TBO on Heuristics . 169 9.17 Effect of IS on Heuristics . . 170

10.1 Generation of the Operations Schedule . 175 10.2 Priority Values of Operations . . . . . . 179 10.3 Operations List Generation .. . . . . . 179 10.4 Priority Rules for Operations Scheduling. . 180 10.5 Blocking Sets for the Example ...... . 188 10.6 Moves for the Example. . . . . . . . . . . . 188 10.7 Deviation from Optimum for Small Instances . 195 10.8 Deviation from Lower Bound for Large Instances . 196 10.9 Effect of RSs on Heuristics for Large Instances . 197 1O.lOEffect of RSA on Heuristics for Large Instances . 198 10.11Effect of PS on Heuristics for Large Instances . . 198

Bibliography

[1] N.R. Adam, J.W.M. Bertrand, D.C. Morehead, and J. Surkis. Due date assignment procedures with dynamically updated coefficients for multi-level assembly job shops. European Journal of Operational Research, 68:212-227, 1993.

[2] P.S. Adler. Interdepartmental interdependence and coordination: The case of the design/manufacturing interface. Organization Science, 6(2):147-167, 1995.

[3] P. Afentakis, B. Gavish, and U.S. Karmarkar. Computationally efficient optimal solutions to the lot-sizing problem in multistage assembly systems. Management Science, 30(2):222-239, 1984.

[4] A. Agrawal, G. Harhalakis, I. Minis, and R. Nagi. 'Just-in-time' production of large assemblies. lIE Transactions, 28:653-667, 1996.

[5] R.H. Ahmadi and C.S. Tang. An operation partitioning problem for automated assembly system design. Operations Research, 39(5):824-835,1991.

[6] H. Aigbedo and Y. Monden. A parametric procedure for multicriterion sequence scheduling for just-in-time mixed-model assembly lines. International Journal of Production Research, 35(9):2453-2564, 1997.

[7] R. Alvarez-Valdes and J .M. Tamarit. Heuristic algorithms for resource-constrained project scheduling: A review and an empirical analysis. In R. Slowinski and J. W~glarz, editors, Advances in project scheduling, pages 113-134. Elsevier, Amsterdam, 1989.

222 BIBLIOGRAPHY

[8] J.C. Ammons, M. Carlyle, L. Cranmer, G. de Puy, K. Ellis, L.F. Mc Ginnis, C.A. Tovey, and H. Xu. Component allocation to balance workload in printed circuit card assembly systems. lIE Transactions, 29:265-275, 1997.

[9] R.N. Anthony. Planning and Control Systems: A Framework for Analysis. Harvard University Press, Cambride, Mass., 1965.

[10] M.G. Anwar and R. Nagi. Integrated lot-sizing and scheduling for just-in-time production of complex assemblies with finite set-ups. International Journal of Production Research, 35(5):1447-1470, 1997.

[11] N.J. Aquilano and D.E. Smith. A formal set of algorithms for project scheduling with critical path scheduling/material requirements planning. Journal of Operations Management, 1(2):57-67, 1980.

[12] R.G. Askin, M. Dror, and J. Vakharia. Printed circuit board family grouping and component allocation for multimachine, openshop assembly cell. Naval Research Logistics, 41:587-608, 1994.

[13] B. Aspvall, S.D. Fh1m, and K.P. Villanger. Selecting among scheduled projects. Operations Research Letters, 17:37-40, 1995.

[14] T. Baar, P. Brucker, and S. Knust. Tabu-search algorithms and lower bounds for the resource-constrained project scheduling problem. In S. Voss, S. Martello, I. Osman, and C. Roucairol, editors, Meta-Heuristics: Advances and Trends in Local Search Paradigms for Optimization, pages 1-8. Kluwer Academic Publishers, Boston, 1998.

[15] K. Backhaus. Industriegiitermarketing. Vahlen, Munchen, 5. edition, 1997.

[16] K.R. Baker. Introduction to sequencing and scheduling. Wiley, New York, 1974.

[17] K.R. Baker and J.W.M. Bertrand. A comparison of due-date selection rules. AIlE Transactions, 13(2):123-131, 1981.

BIBLIOGRAPHY 223

[18] K.R. Baker, S.G. Powell, and D.F. Pyke. Optimal allocation of work in assembly systems. Management Science, 39(1):101-106, 1993.

[19] A. Balakrishnan, R.L. Francis, and S.J. Grotzinger. Bottleneck resource allocation in manufacturing. Management Science, 42(11):1611-1625,1996.

[20] M.O. Ball and M.J. Magazine. Sequencing of insertions in printed circuit board assembly. Operations Research, 36(2):192-201,1988.

[21] 1. Barany, T.J. van Roy, and L.A. Wolsey. Strong formulations for multi-item capacitated lot sizing. Management Science, 30(10):1255-1261,1984.

[22] J.F. Bard, R.W. Clayton, and T.A. Feo. Machine setup and component placement in printed circuit board assembly. The International Journal of Flexible Manufacturing Systems, 6:5-31,1994.

[23] C. Barnhart, E.L. Johnson, G.L. Nemhauser, M.W.P. Savelsbergh, and P.H. Vance. Branch-and-price: Column generation for solving huge integer programs. Operations Research, 46(3):316-329, 1998.

[24] M. Bartusch, R.H. Mohring, and F.J. Radermacher. Scheduling project networks with resource constraints and time windows. Annals of Operations Research, 16:201-240, 1988.

[25] R. Berndt. Marketing fur offentliche Auftriige. Vahlen, Miinchen, 1988.

[26] J.W.M. Bertrand. The effect of workload dependent due-dates on job shop performance. Management Science, 29(7):799-816, 1983.

[27] S. Betz. Konsequenzen der Arbeitszeitflexibilisierung fUr Produktion und Logistik. Betriebswirtschaftliche Forschung und Praxis, 50(3):344-362, 1998.

224 BIBLIOGRAPHY

[28] P.J. Billington. Multi-level lot-sizing with a bottleneck work center. PhD thesis, Cornell University, 1983.

[29] P.J. Billington, J .0. McClain, and L.J. Thomas. Mathematical programming approaches to capacity-constrained MRP systems: Review, problem formulation and problem reduction. Management Science, 29(10):1126-1141,1983.

[30] P.J. Billington, J .0. McClain, and L.J. Thomas. Heuristics for multilevel lot-sizing with a bottleneck. Management Science, 32(8):989-1006,1986.

[31] N. Bixby and E. Boyd. Using the CPLEX callable library. CPLEX Optimization Inc., Houston, 1996.

[32] J. Blaiewicz, K.H. Ecker, E. Pesch, G. Schmidt, and J. Weglarz. Scheduling computer and manufacturing processes. Springer, Berlin, 1996.

[33] J. Blaiewicz, J.K. Lenstra, and A.H.G. Rinnooy Kan. Scheduling subject to resource constraints: Classification and complexity. Discrete Applied Mathematics, 5:11-24, 1983.

[34] D.B. Bock and J .H. Patterson. A comparison of due date setting, resource assingment, and job preemption heuristics for the multi-project scheduling problem. Decision Sciences, 21:387-402,1990.

[35] J. Bottcher, A. Drexl, R. Kolisch, and F. Salewski. Project scheduling under partially renewable resource constraints. Management Science, 45(4):543-559, 1999.

[36] M.R. Bowers, K. Groom, and R. Morris. A practical application of a multi-project scheduling heuristic. Production and Inventory Management Journal, 37(4):19-25, 1996.

[37] S.P. Bradley, A.C. Hax, and T.L. Magnanti. Applied mathematical programming. Addison-Wesley, Reading, Massachusetts, 1977.

[38] R.A. Brealey and S.C. Myers. Principles of corporate finance. McGraw-Hill, New York, 5. edition, 1996.

BIBLIOGRAPHY 225

[39] J. Brimberg, W.J. Hurley, and R.E. Wright. Scheduling workers in a constricted area. Naval Research Logistics, 43: 143-149, 1996.

[40] K. Brockhoff. Management organisatorischer Schnittstellen - unter besonderer Berucksichtigung der Koordination von Marketingbereichen mit Forschung und Entwicklung. Technical Report 2, Joachim Jungius-Gesellschaft der Wissenschaften e.V., Hamburg, 1994.

[41] K. Brockhoff and J. Hauschildt. Schnittstellen-Management -Koordination ohne Hierarchie. ZeitschriJt fur Fuhrung und Organisation, 62(6), 1993.

[42] P. Brucker. Scheduling Algorithms. Springer, Berlin, 2. edition, 1998.

[43] P. Brucker, A. Drexl, R. Mohring, K. Neumann, and E. Pesch. Resource-constrained project scheduling: Notation, classification, models, and methods. European Journal of Operational Research, 112(1):3-41, 1999.

[44] H. Brynzer and M.l. Johansson. Design and performance of kitting and order picking systems. International Journal of Production Economics, 41:115-125, 1995.

[45] G.N. Bullen. The mechanization/automation of major aircraft assembly tools. Production and Inventory Management Journal, 38(3):84-87, 1997.

[46] J. BuBmann. Rechnerunterstutzte Montagesteuerung im Werkzeugmaschinenbau. ZeitschriJt fur wirtschaJtliche Fertigung, 89(4):188-190, 1994.

[47] M. Cala, K. Srihari, and R. Emerson. Process and production planning for circuit board assembly. IIE Transactions, 27:90-98, 1995.

[48] J. Carlier and A.H.G. Rinnooy Kan. Scheduling subject to nonrenewable-resource constraints. Operations Research Letters, 1(2):52-55, 1982.

226 BIBLIOGRAPHY

[49] M.A. Carravilla and J.P. de Sousa. Hierarchical production planning in a make-to-order company: A case study. European Journal of Operational Research, 86:43-56, 1995.

[50] D. Cattrysse, J. Maes, and L.N. van Wassenhove. Set partitioning and column generation heuristics for capacitated dynamic lot sizing. European Journal of Operational Research, 46:38-47, 1990.

[51] J.S. Chao and S.C. Graves. Reducing flow time in aircraft manufacturing. Production and Operations Management, 7(1):38-52, 1998.

[52] J.F. Chen and W.E. Wilhelm. An evaluation of heuristics for allocating components to kits in small-lot, multi-echelon assembly systems. International Journal of Production Research, 31(12):2835-2856, 1993.

[53] J.F. Chen and W.E. Wilhelm. Optimizing the allocation of components to kits in small-lot, multi-echelon assembly systems. Naval Research Logistics, 41:229-256, 1994.

[54] J.F. Chen and W.E. Wilhelm. Kitting in multi-echelon, multiproduct assembly systems with part substitutable. International Journal of Production Research, 35{1O):2871-2897, 1997.

[55] V.Y.X. Chen. A 0-1 goal programming model for scheduling multiple maintenance projects at a copper mine. European Journal of Operational Research, 76:176-191, 1994.

[56] W.-H. Chen and J.-M. Thizy. Analysis of relaxations for the multi-item capacitated lot-sizing problem. Annals of Operations Research, 26:29-72, 1990.

[57] Z.-L. Chen and W.B. Powell. Network representation, column generation and branch and bound for parallel machine scheduling with multiple job families. Technical Report 98-02, Department of Systems Engineering, University of Pennsylvania, 1998.

[58] T .C.E. Cheng and M.C. Gupta. Survey of scheduling research involving due date determination decisions. European Journal of Operational Research, 38:156-166, 1989.

BIBLIOGRAPHY 227

[59] H.N. Chiu and T.M. Lin. An optimal lot-size model for multistage series/assembly systems. Computers & Operations Research, 15(5):403-415, 1988.

[60] A.J. Clark and H. Scarf. Optimal policies for multi-echelon inventory problems. Management Science, 6:475-490, 1960.

[61] M.A. Coffin and B.W. Taylor. R&D project selection and scheduling with a filtered beam search approach. lIE Transactions, 28:167-176, 1996.

[62] H. Crowder. Computational improvements for subgradient optimization. Symposia Mathematica, 19:357-372, 1976.

[63] W.B. Crowston and M.H. Wagner. Dynamic lot size models for multi-stage assembly systems. Management Science, 20(1):14-21,1973.

[64] W.B. Crowston, M.H. Wagner, and A. Henshaw. A comparison of exact and heuristic routines for lot-size determination in multi-stage assembly systems. AIlE Transactions, 4(4):313-317, 1972.

[65] W.B. Crowston, M.H. Wagner, and J.F. Williams. Economic lot size determination in multi-stage assembly systems. Management Science, 19(5):517-527,1973.

[66] G. Czeranowsky. Ein Losungsansatz zur simultanen Programmund Ablaufplanung bei Einzelfertigung. ZeitschriJt fur BetriebswirtschaJt, 45(6) :353-370, 1975.

[67] S. Dauzere-Peres and J .B. Lasserre. Integration of lotsizing and scheduling decisions in a job-shop. European Journal of Operational Research, 75:413-426, 1994.

[68] P. De, J .B. Ghosh, and C.E. Wells. Optimal due-date assignment and sequencing. European Journal of Operational Research, 57:323-331, 1992.

[69] P. De, J.B. Ghosh, and C.E. Wells. Solving a generalized model for CON due date assignment and sequencing. International Journal of Production Economics, 34:179-185, 1994.

228 BIBLIOGRAPHY

[70] R. de Boer, A. van Harten, and W.H.M Zijm. Scheduling multiprocessor tasks with precedence constraints and release and due dates. Technical report, Faculty of Mechanical Engineering, University of Twente, 1997.

[71] K.X.S. de Souza and V.A. Armentano. Multi-item capacitated lot-sizing by a cross decomposition based algorithm. Annals of Operations Research, 50:557-574, 1994.

[72] R.F. Deckro, E.P. Winkofsky, J.E. Hebert, and R. Gagnon. A decomposition approach to multi-project scheduling. European Journal of Operational Research, 51:110-118, 1991.

[73] F. Della Croce. Generalized pairwise interchanges and machine scheduling. European Journal of Operational Research, 83:310-319,1995.

[74] E.L. Demeulemeester and W.S. Herroelen. Modelling setup times, process batches and transfer batches using activity network logic. European Journal of Operational Research, 89:355-365,1996.

[75] M.C. Derstroff. Mehrstufige LosgrofJenplanung mit Kapazitiitsbeschriinkungen. Physica, Heidelberg, 1995.

[76] M. Di Mascolo, R. David, and Y. Dallery. Modelling and analysis of assembly systems with unreliable machines and finite buffers. lIE Transactions, 23:315-330, 1991.

[77] M. Diaby, H.C. Bahl, M.H. Karwan, and S. Zionts. A lagrangian relaxation approach for very-large-scale capacitated lot-sizing. Management Science, 38(9):1329-1340, 1992.

[78] B.L. Dietrich. A taxonomy of discrete manufacturing systems. Operations Research, 39(6):886-902, 1991.

[79] P.S. Dixon and E.A. Silver. A heuristic solution procedure for the multi-item, single-level, limited capacity, lot-sizing problem. Journal of Operations Management, 2(1):23-39, 1981.

[80] B. Dodin and A.A. Elimam. Integrated project scheduling and material planning with variable activity durations and rewards.

BIBLIOGRAPHY 229

Technical report, The Gary Anderson Graduate School of Management, University of California, Riverside, 1998.

[81] A. Dogramaci, J.E. Panayiotopoulos, and N.R. Adam. The dynamic lot sizing problem for multiple items under limited capacity. AIlE Transactions, 13(4):294-303, 1981.

[82] W. Domschke. Logistik: Transport. Oldenbourg, Miinchen, 4. edition, 1995.

[83] W. Domschke and A. Drexl. Einfuhrung in Operations Research. Springer, Berlin, 4. edition, 1998.

[84] W. Domschke, A. Scholl, and S. VoB. ProduktionsplanungAblauforganisatorische Aspekte. Springer, Berlin, 2. edition, 1997.

[85] A. Drexl, W. Eversheim, R. Grempe, and H. Esser. CIM im Werkzeugmaschinenbau - Der PRISMA-Montageleitstand. ZeitschriJt fur betriebswirtschaJtliche Forschung, 46(12) :279-295, 1994.

[86] A. Drexl and A. Kimms. Lot sizing and scheduling - survey and extensions. European Journal of Operational Research, 99:221-235, 1997.

[87] A. Drexl and R. Kolisch. Assembly management in machine tool manufacturing and the PRISMA-Leitstand. Production and Inventory Management Journal, 37(4):55-57,1996.

[88] A. Drexl, R. Kolisch, and A. Sprecher. Koordination und Integration im Projektmanagement. ZeitschriJt fur BetriebswirtschaJt, 68(3):275-295, 1998.

[89] I. Duenyas and W.J. Hopp. Quoting customer lead times. Management Science, 41(1):43-57,1995.

[90] J. Dumond and V.A. Mabert. Evaluating project scheduling and due date assignment procedures: An experimental analysis. Management Science, 34(1):101-118, 1988.

230 BIBLIOGRAPHY

[91] F. Edelman. Art and science of competitive bidding. Harvard Business Review, 43(4):53-56, 1965.

[92] S. Eilon and L.G. Chowdhury. Due dates in job shop scheduling. International Journal of Production Research, 14(2):223-237,1976.

[93] P.S. Eisenhut. A dynamic lot sizing algorithm with capacity constraints. AIlE Transactions, 7(2):170-176, 1975.

[94] S.E. Elmaghraby. Activity networks: Project planning and control by network models. Wiley, New York, 1977.

[95] S.E. Elm agh raby. Project bidding under deterministic and probabilistic activity durations. European Journal of Operational Research, 49:14-34, 1990.

[96] G.D. Eppen and R.K. Martin. Solving multi-item capacitated lot-sizing problems using variable redefinition. Operations Research, 35(6):832-848, 1987.

[97] F.J. Erens and H.M.H. Hegge. Manufacturing and sales coordination for product variety. International Journal of Production Economics, 37:83-99, 1994.

[98] D. Erlenkotter. Ford Whitman Harris and the economic order quantity model. Operations Research, 38:937-946, 1990.

[99] W. Eversheim. Die Einzel- und Kleinserienmontage komplexer Produkte. In Neue Strategien zur Rationalisierung der Montage, pages 1-19. Verein Deutscher Ingenieure - Gesellschaft Produktionstechnik, 1987.

[100] W. Eversheim, U. Ungeheuer, and H. Peffekoven. Montageorientierte Erzeugnisstrukturierung in der Einzel- und Kleinserienproduktion - ein Gegensatz der funktionsorientierten Erzeugnisgliederung? VDI-Zeitschrijt, 125(12):475-479, 1983.

[101] A. Eynan and M.J. Rosenblatt. Assemble to order and assemble in advance in a single-period stochastic environment. Naval Research Logistics, 42:861-872, 1995.

BIBLIOGRAPHY 231

[102] B. Faaland and T. Schmitt. Scheduling tasks with due dates in a fabrication/assembly process. Operations Research, 35(3):378-381,1987.

[103] B. Faaland and T. Schmitt. Cost-based scheduling of workers and equipment in a fabrication and assembly shop. Operations Research, 41(2):253-268, 1993.

[104] G. Fandel. Produktion I - Produktions- und Kostentheorie. Springer, Berlin, 5. edition, 1996.

[105] A. Federgruen and M. Tzur. A simple forward algorithm to solve general dynamic lot sizing models with n periods in o( n log n) or o(n) time. Management Science, 37(8):909-925, 1991.

[106] T.A. Feo, J.F. Bard, and R.F. Claflin. An overview of GRASP methodology and applications. Technical report, Department of Mechanical Engineering, University of Texas at Austin, 1991.

[107] T.A. Feo, J.F. Bard, and S.D. Holland. Facility-wide planning and scheduling of printed wiring board assembly. Operations Research, 43(2) :219-230, 1995.

[108] T.A. Feo, J.F. Bard, and S.D. Holland. A GRASP for scheduling printed wiring board assembly. lIE Transactions, 28:155-165, 1996.

[109] M.L. Fisher. The lagrangian relaxation method for solving integer programming problems. Management Science, 27(1):1-18, 1981.

[110] L.R. Foulds and H.W. Hamacher. Optimal bin location and sequencing in printed circuit board assembly. European Journal of Operational Research, 66:279-290, 1993.

[111] R. Fourer, D.M. Gay, and B.W. Kernighan. AMPL - A modeling language for mathematical programming. Boyd & Fraser, Danvers, 1993.

[112] B. Franck, K. Neumann, and C. Schwindt. A capacity-oriented hierarchical approach to single-item and small-batch produc-

232 BIBLIOGRAPHY

tion planning using project scheduling methods. OR Spektrum, 19:77-85, 1997.

[113] Frauenhofer-Institut fur Produktionstechnik und Automatisierung. Erfolgreiche Rationalisierung durch montagegerechte Produktgestaltung, Stuttgart, 1989.

[114] B. Friedl. Kostenorientierte Produktionsplanung und -steuerung. In H. Corsten and R. Gossinger, editors, Dezentrale Produktionsplanungs- und -steuerungs-Systeme, pages 259-288. Kohlhammer, Stuttgart, 1998.

[115] L. Friedman. A competitive-bidding strategy. Operations Research, 4:104-122, 1956.

[116] M. Garetti, C. Lanza, and A. Pozetti. An algorithm for the weekly production planning of flexible assembly systems. International Journal of Computer Integrated Manufacturing, 7(4):249-259,1994.

[117] M.R. Garey and D.S. Johnson. Computers and Intractability -A guide to the theory of NP-Completeness. W.H. Freeman and Company, New York, 1979.

[118] L.F. Gelders, J. Maes, and L.N. van Wassenhove. A branch and bound algorithm for the multi item single level capacitated dynamic lotsizing problem. In S. Axsater, C. Schneeweiss, and E. Silver, editors, Multi-stage production planning and inventory control, pages 92-108. Springer, Berlin, 1986.

[119] S.B. Gershwin. Assembly / dissambly systems: An efficient decomposition algorithm for tree-structured networks. lIE Transactions, 23:302-314, 1991.

[120] S. Ghosh and R.J. Gagnon. A comprehensive literature review and analysis of the design, balancing and scheduling of assembly systems. International Journal of Production Research, 27(4):637-670, 1989.

[121] F. Glover. Tabu search - Part 1. OR SA Journal on Computing, 1(3):190-206,1989.

BIBLIOGRAPHY 233

[122] F. Glover. Tabu search - Part II. ORSA Journal on Computing, 2(1):4-32, 1990.

[123] B.L. Golden and W.R. Steward. Empirical analysis of heuristics. In E.L. Lawler and J .K. Lenstra, editors, The traveling salesman problem, pages 207-249. Wiley, New York, 1985.

[124] D. Goodman and H. Baurmeister. A computational algorithm for multi-contract bidding under constraints. Management Science, 22(7):788-798, 1976.

[125] J .S. Goodwin and J .C. Goodwin. Operating policies for scheduling assembled products. Decision Sciences, 13:585-603, 1982.

[126] S.C. Graves and B.W. Lamar. An integer programming procedure for assembly system design problems. Operations Research, 31(3):522-545, 1983.

[127] J. Greenwald. Is Boeing out of its spin? Time, 152(2):53-55, 1998.

[128] R. Grempe and B. Saretz. Material in der Montage auftragsbezogen bereitstellen. Zeitschrift fur wirtschaftliche Fertigung, 87(1):51-54,1992.

[129] G.K. Groff. A lot sizing rule for time-phased component demand. Production and Inventory Management, 20(1):47-53, 1979.

[130] H.-O. Gunther. Planning lot sizes and capacity requirements in a single stage production system. European Journal of Operational Research, 31:223-231, 1987.

[131] H.-O. Gunther. Produktionsplanung bei ftexibler Personalkapazitiit. Poeschel, Stuttgart, 1989.

[132] H.-O. Gunther. Netzplanorientierte Auftragsterminierung bei offener Fertigung. OR Spektrum, 14:229-240, 1992.

[133] H.-O. Gunther, M. Gronalt, and F. Piller. Component kitting in semi-automated printed circuit board assembly. International Journal of Production Economics, 43:213-226, 1996.

234 BIBLIOGRAPHY

[134] H.-O. Gunther and M. Grunow. Die Bildung von Auftragsserien bei der Montage von Leiterplatten - Teil 1: Die Problemstellung. Wirtschaftswissenschaftliches Studium, 25(1):49-52, 1996.

[135] H.-O. Gunther, M. Grunow, and C. Schorling. Workload planning in small lot printed circuit board assembly. OR Spektrum, 19:147-157,1997.

[136] E. Gutenberg. Grundlagen der Betriebswirtschaftslehre - Die Produktion, volume 1. Springer, Berlin, 1979.

[137] K. Haase. Lotsizing and scheduling for production planning, volume 478 of Lecture Notes in Economic and Mathematical Systems. Springer, Berlin, 1994.

[138] K. Haase and J. Latteier. Deckungsbeitragsorientierte Lehrgangsplanung bei der Lufthansa Technical Training GmbH. OR Spektrum, 20(2):135-142, 1998.

[139] C.K. Hahn, E.A. Duplaga, and K.Y. Kim. Production/Sales interface: MPS at Hyundai Motor. International Journal of Production Economics, 37:5-17, 1994.

[140] N.G. Hall and M.J. Magazine. Maximizing the value of a space mission. European Journal of Operational Research, 78:224-241, 1994.

[141] G. Harhalakis. The use of project scheduling for the manufacture of custom made products. In R. Slowinski and J. Weglarz, editors, Advances in project scheduling, pages 355-368. Elsevier, 1989.

[142] A.M.A. Hariri and C.N. Potts. A branch and bound algorithm for the two-stage assembly scheduling problem. European Journal of Operational Research, 103:547-556,1997.

[143] F.H. Harris and J.P. Pinder. A revenue management approach to demand management and order booking in assemble-to-order manufacturing. Journal of Operations Management, 13:299-309, 1995.

BIBLIOGRAPHY 235

[144] T.P. Harrison and H.S. Lewis. Lot sizing in serial assembly systems with multiple resource constraints. Management Science, 42(1):19-36, 1996.

[145] S. Hartmann. A competitive genetic algorithm for resource-constrained project scheduling. Naval Research Logistics, 45:733-750, 1998.

[146] S. Hartmann. Project Scheduling under Limited Resources: Models, Methods, and Applications, volume 478 of Lecture Notes in Economics and Mathematical Systems. Springer, 1999.

[147] S. Hartmann and R. Kolisch. Experimental evaluation of state-of-the-art heuristics for the resource-constrained project scheduling problem. European Journal of Operational Research. Forthcoming.

[148] N.A.J. Hastings, P. Marshall, and R.J. Willies. Schedule based M.R.P.: An integrated approach to production scheduling and material requirements planning. Journal of the Operational Research Society, 33:1021-1029,1982.

[149] A.C. Hax and D. Candea. Production and Inventory Management. Prentice-Hall, Englewoord Cliffs, New Jersey, 1984.

[150] A.C. Hax and H.C. Meal. Hierarchical integration of production planning and scheduling. In M. Geisler, editor, TIMS Studies in Management Science, Vol. 1, Logistics, pages 53-69. North Holland / American Elsevier, New York, 1975.

[151] H. Hax. Bewertungsprobleme bei der Formulierung von Zielfunktionen fur Entscheid ungsmodelle. Zeitschrijt fur betriebswirtschaftliche Forschung, 19:749-761, 1967.

[152] H. Hay. Planungs- und Kontrollrechnung im Anlagengeschaft. In J. Funk and G. LaBmann, editors, Zeitschrift fur betriebswirtschaftliche Forschung - Sonderheft 20, pages 81-100. Handelsblatt-Verlag, Dusseldorf, 1986.

[153] S. Helber. Kapazitiitsorientierte LosgrojJenplanung in PPS-Systemen. M&P Verlag fUr Wissenschaft und Forschung, Stuttgart, 1994.

236 BIBLIOGRAPHY

[154] S. Helber. Lot sizing in capacitated production planning and control systems. OR Spektrum, 17:5-18, 1995.

[155] S. Helber. Decomposition of unreliable assembly/disassembly networks with limited buffer capacity and random processing times. European Journal of Operational Research, 109:24-42, 1998.

[156] M. Held, P. Wolfe, and H.P. Cowder. Validation of subgradient optimization. Mathematical Programming, 6:62-88, 1974.

[157] L. Hendry and B. Kingsman. A decision support system for job release in make-to-order companies. International Journal of Operations (3 Production Management, 11(6):6-16,1991.

[158] L.C. Hendry and B.G. Kingsman. Production planning systems and their applicability to make-to-order companies. European Journal of Operational Research, 40:1-15, 1989.

[159] L.C. Hendry and B.G. Kingsman. Job release: Part of a hierarchical system to manage manufacturing lead times in make-toorder companies. Journal of the Operational Research Society, 42(10):871-883,1991.

[160] L.C. Hendry and B.G. Kingsman. Customer enquiry management: Part of a hierarchical system to control lead times in make-to-order companies. Journal of the Operational Research Society, 44(1):61-70, 1993.

[161] L.C. Hendry, B.G. Kingsman, and P. Cheung. The effect of workload control (WLC) on performance in make-to-order com panies. European Journal of Operational Research, 16:63-75, 1998.

[162] K.S. Hindi. Solving the CLSP by a tabu search heuristic. Journal of the Operational Research Society, 47(1):151-161, 1996.

[163] Y.-C. Ho and C.L. Moodie. A heuristic operation sequencepattern identification method and its application in the design of a cellular flexible assembly system. International Journal of Computer Integrated Manufacturing, 7(3):163-174,1994.

BIBLIOGRAPHY 237

[164] O. kmeli and W. Rom. Solving the resource constrained project scheduling problem with optimization subroutine library. Computers (3 Operations Research, 23(8):801-817, 1995.

[165] S. Israni and J. Sanders. Two-dimensional cutting stock problem research: A review and a new rectangular layour algorithm. Journal of Manufacturing Systems, 1(2):169-181,1982.

[166] H. Jacob. Zur optimalen Planung des Produktionsprogramms bei Einzelfertigung. ZeitschriJt fur BetriebswirtschaJt, 41 (8) :495-516, 1971.

[167] S. Joines and M.A. Ayoub. Design for assembly: An ergonomic approach. Industrial Engineering, (1):42-46, 1995.

[168] T.T. Kapuscincka, T.E. Morton, and P. Ramnath. Highintensity heuristics to minimize weighted tardiness in resourceconstrained multiple dependent project scheduling. Technical report, Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, 1998.

[169] R. Karni and Y. Roll. A heuristic algorithm for the multi-item lot-sizing problem with capacity constraints. IIE Transactions, 14(4):249-256, 1982.

[170] G. Karthikeyan and K.N. Krishnaswamy. Assembly manpower allocation under proportionality constraints. European Journal of Operational Research, 44:39-46, 1988.

[171] E. Katok, H.S. Lewis, and T.P. Harrison. Lot sizing in general assembly systems with setup costs, setup times, and multiple constrained resources. Management Science, 44(6):859-877, 1998.

[172] P. Keskinocak, R. Ravi, and S. Tayur. Algorithms for reliable lead time quotation. Technical report, Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, 1997.

[173] A. Kieser and H. Kubicek. Organisation. Walter de Gruyter, Berlin, 3. edition, 1992.

238 BIBLIOGRAPHY

[174] S.-Y. Kim and R.C. Leachman. Multi-project scheduling with explicit lateness costs. IIE Transactions, 25(2):34-44, 1993.

[175] S.O. Kim and M.J. Schniederjans. Heuristic framework for the resource constrained multi-project scheduling problem. Computers (3 Operations Research, 16(6):541-556, 1989.

[176] Y.-D. Kim. A backward approach in list scheduling algorithms for multi-machine tardiness problems. Computers (3 Operations Research, 22(3):307-319, 1995.

[177] A. Kimms. Multi-level lot sizing and scheduling - Methods for capacitated, dynamic, and deterministic models. Physica, Heidelberg, 1997.

[178] B.G. Kingsman, I.P. Tatsiopoulos, and L.e. Hendry. A structural methodology for managing manufacturing lead times in make-to-order companies. European Journal of Operational Research, 40:196-209, 1989.

[179] B.G. Kingsman, L. Worden, L. Hendry, A. Mercer, and E. Wilson. Integrating marketing and production planning in maketo-order companies. International Journal of Production Economics, 30-31:53-66, 1993.

[180] O. Kirca and M. Kokten. A new heuristic approach for the multi-item dynamic lot sizing problem. European Journal of Operational Research, 75:332-341, 1994.

[181] K.-P. Kistner. Koordinationsmechanismen in der hierarchischen Planung. ZeitschriJt fur BetriebswirtschaJt, 62(10):1125-1146, 1992.

[182] P. Kl0vstad. The technological network: Macro and micro structure. In H.J .M. Lombaers, editor, Project planning by network analysis, pages 441-450. North-Holland, Amsterdam, 1969.

[183] R. Kolisch. Integration of assembly and fabrication for maketo-order production. International Journal of Production Economics. Forthcoming.

BIBLIOGRAPHY 239

[184] R. Kolisch. Project scheduling under resource constraints - Efficient heuristics for several problem classes. Physica, Heidelberg, 1995.

[185] R. Kolisch. Serial and parallel resource-constrained project scheduling methods revisited: Theory and computation. European Journal of Operational Research, 90:320-333, 1996.

[186] R. Kolisch. 'Just-in-time' production of large-scale assemblies: An integrative perspective. Technical Report 437, Manuskripte aus den Instituten fUr Betriebswirtschaftslehre der Universitat Kiel, Kiel, 1997.

[187] R. Kolisch. Integrated production planning, order acceptance, and due date setting for make-to-order manufacturing. In P. Kischka, H.-W. Lorenz, U. Derigs, W. Domschke, P. Kleinschmidt, and R. Mohring, editors, Symposium on Operations Research, pages 492-497. Springer, Berlin, 1998.

[188] R. Kolisch. Integrated scheduling, assembly area-, and partassignment for large scale make-to-order assemblies. International Journal of Production Economics, 64:127-141, 2000.

[189] R. Kolisch and S. Hartmann. Heuristic algorithms for the resource-constrained project scheduling problem: Classification and computational analysis. In J. W~glarz, editor, Project Scheduling - Recent Models, Algorithms and Applications, pages 147-178. Kluwer Academic Publishers, Boston, 1999.

[190] R. Kolisch and K. HeB. Efficient methods for scheduling maketo-order assemblies under resource, assembly area, and part availability constraints. International Journal of Production Research, 38(1):207-228,2000.

[191] R. Kolisch and R. Padman. An integrated survey of deterministic project scheduling. OMEGA International Journal of Management Science. Forthcoming.

[192] R. Kolisch and A. Rinne. Assembly scheduling in a make-toorder environment with shop floor constraints. In P. Kleinschmidt, A. Bachem, U. Derigs, D. Fischer, U. Leopold-

240 BIBLIOGRAPHY

Wildburger, and R.H. Mohring, editors, Symposium on Operations Research, pages 376-381. Springer, 1996.

[193] R. Kolisch, A. Sprecher, and A. Drexl. Characterization and generation of a general class of resource-constrained project scheduling problems: Easy and hard instances. Technical Report 301, Manuskripte aus den Instituten fUr Betriebswirtschaftslehre der Vniversitat Kiel, 1992.

[194] R. Kolisch, A. Sprecher, and A. Drexl. Characterization and generation of a general class of resource-constrained project sched uling problems. Management Science, 41(10):1693-1703, 1995.

[195] H.-J. Konz. Steuerung der Standplatzmontage komplexer Produkte. PhD thesis, RWTH-Aachen, 1989.

[196] H. Kopka. YT.FJ(Einfiihrung. Addison-Wesley, Reading, Massachusetts, 1994.

[197] E. Kroll, B. Beardsley, and A. Parulian. A methodology to evaluate ease of disassembly for product recycling. IIE Transactions, 28:837-845, 1996.

[198] L. Kruschwitz. Investitionsrechnung. de Gruyter, Berlin, 6. edition, 1995.

[199] R. Kuik, M. Salomon, L.N. van Wassenhove, and J. Maes. Linear programming, simulated annealing and tabu search heuristics for lotsizing in bottleneck assembly systems. IIE Transactions, 25(1):62-72, 1993.

[200] H.-V. Kupper. Controlling - Konzeption, Aufgaben und Instrumente. In H.-V. Kupper, editor, Controlling. Schaffer-Poeschel, Stuttgart, 1995.

[201] I.S. Kurtulus and E.W. Davis. Multi-project scheduling: Categorization of heuristic rule performance. Management Science, 28(2):161-172, 1982.

[202] I.S. Kurtulus and S.C. Narula. Multi-project scheduling: Analysis of project performance. IIE Transactions, 17:58-66, 1985.

BIBLIOGRAPHY 241

[203] A. KuB. Competitive-Bidding-Modelle. Zeitschrift fur betriebswirtschaftliche Forschung - Kontaktstudium, 29:63-70, 1977.

[204] P. Lalsare and S. Sen. Evaluating backward scheduling and sequencing rules for an assembly shop environment. Production and Inventory Management Journal, 36(4):71-77, 1995.

[205] A.J.D. Lambert. Optimal disassembly of complex products. International Journal of Production Research, 35(9):2509-2523, 1997.

[206] M.R. Lambrecht and H. Vanderveken. Heuristic procedures for the single operation, multi-item loading problem. IIE Transactions, 11(4):319-326, 1979.

[207] M.R. Lambrecht and H. Vanderveken. Production scheduling and sequencing for multi-stage production systems. OR Spektrum, 1:103-114, 1979.

[208] J .B. Lasserre. An integrated model for job-shop planning and scheduling. Management Science, 38(8):1201-1211, 1992.

[209] S.R. Lawrence. Negotiating due-dates between customers and producers. International Journal of Production Economics, 37:127-138, 1994.

[210] S.R. Lawrence and T .E. Morton. Resource-constrained multiproject scheduling with tardy costs: Comparing myopic, bottleneck, and resource pricing heuristics. European Journal of Operational Research, 64:168-187, 1993.

[211] C.-Y. Lee, T.C.E. Cheng, and B.M.T. Lin. Minimizing the makespan in the 3-machine assembly-type flowshop scheduling problem. Management Science, 39:616-625, 1993.

[212] J .K. Lee, K.J. Lee, H.K. Park, J .S. Hong, and J .S. Lee. Developing scheduling systems for Daewoo shipbuilding: DAS project. European Journal of Operational Research, 97:380-395, 1997.

[213] L.H. Lee and S.M. Ng. Introduction to the special issue on global supply chain management. Production and Operations Management, 6(3):191-192, 1997.

242 BIBLIOGRAPHY

[214] S.M. Lee, O.E. Park, and S.C. Economides. Resource planning for multiple projects. Decision Sciences, 9:49-67, 1978.

[215] C.B. Lofgren, L.F. McGinnis, and C.A. Tovey. Routing printed circuit cards through an assembly cell. Operations Research, 39(6):992-1004, 1991.

[216] R. Logendran and C. Sriskandarajah. A mathematical modelling approach for the process of part orientation in batch assembly automation. International Journal of Computer Integrated Manufacturing, 10(5) :335-345, 1997.

[217] H.R. Louren~o. Job-shop scheduling: computational study of local search and large-step optimization methods. European Journal of Operational Research, 83:347-364, 1995.

[218] A. Lova, C. Maroto, and P. Tormos. A multicriteria heuristic to improve resource allocation in multiproject scheduling. Technical report, Department of Statistics and Operational Research, Universidad Politecnica de Valencia, 1998.

[219] H. Luss and M.B. Rosenwein. A due date assignment algorithm for multiproduct manufacturing facilities. European Journal of Operational Research, 65:187-198, 1993.

[220] B.-J. Madauss. Handbuch Projektmanagement. Poeschel, Stuttgart, 6. edition, 2000.

[221] J. Maes, J.O. McClain, and L.N. van Wassenhove. Multilevel capacitated lotsizing complexity and LP-based heuristics. European Journal of Operational Research, 53:131-148, 1991.

[222] J. Maes and L.N. van Wassenhove. A simple heuristic for the multi item single level capacitated lotsizing problem. Operations Research Letters, 4(6):265-273, 1986.

[223] J. Maes and L.N. van Wassenhove. Multi-item single-level capacitated dynamic lot-sizing heuristics: A general review. Journal of the Operational Research Society, 39(11):991-1004, 1988.

BIBLIOGRAPHY 243

[224] O.Z. Maimon, E.M. Dar-EI, and T.F. Carmon. Set-up saving schemes for printed circuit boards assembly. European Journal of Operational Research, 70:177-190, 1993.

[225] R.E. Markland, K.H. Darby-Dowman, and E.D. Minor. Coordinated production scheduling for make-to-order manufacturing. European Journal of Operational Research, 45:155-176, 1990.

[226] O. Martin, S.W. Otto, and E.W. Felten. Large-step markov chains for TSP incorporating local search heuristics. Operations Research Letters, 11:219-224, 1992.

[227] L.A. Martin-Vega, H.K. Brown, W.H. Shaw, and T.J. Sanders. Industrial perspective on research needs and opportunities in manufacturing assembly. Journal of Manufacturing Systems, 14(1):45-58, 1995.

[228] D.C. Mattfeld. Evolutionary search and the job shop. Physica, Heidelberg, 1996.

[229] S.A. Melnyk and R.T. Christensen. Understanding the nature of setups. APICS, (3):77-78,1997.

[230] J.R. Meredith and S.J. Mantel. Project Management - A managerial approach. Wiley, New York, 3. edition, 1995.

[231] M. Miese. Systematische Montageplanung in Unternehmen mit Kleinserienproduktion. Giradet, Essen, 1976.

[232] J. V. Moccellin. A two-stage approach to resource-constrained multi-project scheduling problems. Policy and Information, 13(1):1-18, 1989.

[233] R.P. Mohanty and M.K. Siddiq. Multiple projects - Multiple resource-constrained scheduling: A multi-objective analysis. Engineering Costs and Production Economics, 18:83-92, 1989.

[234] R.P. Mohanty and M.K. Siddiq. Multiple projects - multiple resource-constrained scheduling: Some studies. International Journal of Production Research, 27(2):261-280, 1989.

244 BIBLIOGRAPHY

[235] K. Mukherjee and A. Bera. Application of goal programming in project selection decision - A case study from the Indian coal mining industry. European Journal of Operational Research, 82:18-25, 1995.

[236] U. Miiller-Manzke. Die Bedeutung der relevanten KapitalBindung in PPS-Systemen. ZeitschriJt fur betriebswirtschaJtliche Forschung, 9:902-913, 1992.

[237] S. Nahmias. Production and Operations Analysis. Irwin, Burr Ridge, Illinois, 4. edition, 1993.

[238] K.S. Naphade, R.H. Storer, and S.D. Wu. A graph-theoretic framework for correct generation of assembly precedence graphs. Technical Report 97T-008, Lehigh University, Departement of Industrial and Manufacturing Systems Engineering, 1997.

[239] K.S. Naphade, R.H. Storer, and S.D. Wu. A branch-and-bound method for optimal selection of assembly plans. Technical report, Lehigh University, Departement ofIndustrial and Manufacturing Systems Engineering, 1998.

[240] K. Neumann. A capacitated hierarchical approach to make-toorder production. In Proceedings Book I - International Conference on Industrial Engineering and Production Management, pages 110-115, Lyon, 1997.

[241] K. Neumann and C. Schwindt. Activity-on-node networks with minimal and maximal time lags and their application to maketo-order production. OR Spektrum, 19(3):205-217, 1997.

[242] S.Y. Nof, W.E. Wilhelm, and H.-J. Warnecke. Industrial Assembly. Chapman & Hall, London, 1997.

[243] M.1. Norbis and J.M. Smith. Two level heuristic for the resource constrained scheduling problem. International Journal of Production Research, 24(5):1203-1219, 1989.

[244] M.J. Norusis. The SPSS guide to data analysis. SPSS Inc., Chicago, 1990.

BIBLIOGRAPHY 245

[245] J.S. Oh, P. O'Grady, and R.E. Young. A constraint network approach to design for assembly. lIE Transactions, 27:72-80, 27.

[246] C. Olinger. The issues behind ERP acceptance and implementation. APICS, 8(6):44-49, 1998.

[247] L. Ozdamar and T. Yazgac;. Capacity driven due date settings in make-to-order production systems. International Journal of Production Economics, 49:29-44, 1997.

[248] J .H. Patterson. Alternate methods of project scheduling with limited resources. Naval Research Logistics Quarterly, 23:767-784,1973.

[249] J .H. Patterson. Project scheduling: The effects of problem structure on heuristic performance. Naval Research Logistics Quarterly, 20:95-123, 1976.

[250] E. Pesch. Learning in automated manufacturing. Physica, Heidelberg, 1994.

[251] U. Petersen. Produktionsplanung und Belegung von Mon-tageftiichen. Gabler, Wiesbaden, 1992.

[252] P.R. Philopoom, 1. Wiegmann, and L.P. Rees. Cost-based duedate assignment with the use of classical and neural-network approaches. Naval Research Logistics, 44:21-46, 1997.

[253] M. Pinedo. Scheduling - Theory, algorithms, and systems. Prentice Hall, Englewood Cliffs, 1995.

[254] A. Pinnoi and W.E. Wilhelm. A family of hierarchical models for the design of deterministic assembly systems. International Journal of Production Research, 35(1):253-280, 1997.

[255] C.N. Potts, S.V. Sevastjanov, V.A. Strusevich, L.N. van Wassenhove, and C.M. Zwaneveld. The two-stage assembly scheduling problem: Complexity and approximation. Operations Research, 43(2):346-355, 1995.

246 BIBLIOGRAPHY

[256] A.A.B. Pritsker, L.J. Watters, and P.M. Wolfe. Multiproject scheduling with limited resources: A zero-one programming approach. Management Science, 16:93-107, 1969.

[257] W.H.M. Raaymakers, J.W.M. Bertrand, and J.C. Fransoo. Identification of aggregate job set characteristics for predicting job set makespan. Technical report, Department of Operations Planning Control, Eindhoven University of Technology, 1998.

[258] P.C. Rao and R. Suri. Approximate queueing network models for closed fabrication/assembly systems. Part I: Single level systems. Production and Operations Management, 3(4):244-275, 1994.

[259] P.C. Rao and R. Suri. Approximate queueing network models for closed fabrication/assembly systems. Part II: Multi level systems. Production and Operations Management, 3(4):242-273, 1995.

[260] R. Righter. A generalized Johnson's rule for stochastic assembly systems. Naval Research Logistics, 44:211-220, 1997.

[261] B. Ronen and D. Trietsch. A decision support system for purchasing management of large projects. Operations Research, 36(6):882-890, 1988.

[262] B. Ronen and D. Trietsch. Optimal scheduling of purchasing orders for large projects. European Journal of Operational Research, 68:185-195, 1993.

[263] M.J. Rosenblatt and H.L. Lee. The effect of work-in-process inventory costs on the design and scheduling of assembly lines with low throughput and high component costs. IIE Transactions, 28:405-414, 1996.

[264] R.S. Russell and B.W. Taylor III. An evaluation of sequencing rules for an assembly shop. Decision Sciences, 16(1):196-212, 1985.

[265] M. Salomon. Deterministic lotsizing models for production planning. Springer, Berlin, 1991.

BIBLIOGRAPHY 247

[266] D. Samson. Managerial decision analysis. Irwin, Homewood, Illinois, 1988.

[267] T. Sawik. An interactive approach to bicriterion loading of a flexible assembly systems. Mathematical Computer Modelling, 25(6):71-83, 1997.

[268] T. Sawik. A lexicographic approach to bi-objective loading of a flexible assembly system. European Journal oj Operational Research, 103:101-113,1997.

[269] T. Sawik. Production planning and scheduling in flexible assembly systems. Springer, Berlin, 1999.

[270] A. Schirmer and S. Riesenberg. Parameterized heuristics for project scheduling - Biased random sampling methods. Technical Report 456, Manuskripte aus den Instituten fUr Betriebswirtschaftslehre der U niversitat Kiel, 1997.

[271] R. Schlauch and W. Ley. Flachenorientierte Termin- und Kapazitatsplanung in der Anlagenmontage. Zeitschrift Jur wirtschaftliche Fertigung, 83:223-227, 1988.

[272] L.C. Schmidt and J. Jackman. Evaluating assembly sequences for automatic assembly systems. IIE Transactions, 27:23-31, 1995.

[273] C. SchneeweiB. Hierarchical structures in organisations: A conceptual framework. European Journal oj Operational Research, 86:4-31, 1995.

[274] A. Scholl. Balancing and Sequencing oj Assembly Lines. Physica, Heidelberg, 1995.

[275] L. Schrage. LINDO - An optimization modeling system. The Scientific Press, San Francisco, 4. edition, 1991.

[276] S. Schuhmacher. PPS-SystemeJur Unternehmen der Klein- und MittelserienJertigung. Physica, Heidelberg, 1994.

[277] J .M.J. Schutten. List scheduling revisited. Operations Research Letters, 18:167-170, 1996.

248 BIBLIOGRAPHY

[278] M. Schweitzer. Industrielle Fertigungswirtschaft. In M. Schweitzer, editor, Industriebetriebslehre, pages 561-696. Vahlen, Miinchen, 1990.

[279] C. Schwindt. Progen/max: A new problem generator for different resource-constrained project scheduling problems with minimal and maximal time lags. Technical Report 449, Institut flir Wirtschaftstheorie und Operations Research, Universitat Karlsruhe, 1995.

[280] C. Schwindt. Generation of resource-constrained project scheduling problems with minimal and maximal time lags. Technical Report 489, Institut flir Wirtschaftstheorie und Operations Research, Universitat Karlsruhe, 1996.

[281] C. Schwindt. Verfahren zur Losung des ressourcenbeschriinkten Projektdauerminimierungsproblems mit planungsabhiingigen Zeitfenstern. Shaker, Aachen, 1998.

[282] E.A. Silver and H.C. Meal. A simple modification of the EOQ for the case of a varying demand rate. Production and Inventory Management, 10:51-55, 1969.

[283] N.C. Simpson and S.S. Erenguc. Multiple stage production planning research: History and opportunities. International Journal of Operations & Production Management, 16(6):25-40, 1996.

[284] N.C. Simpson and S.S. Erenguc. Improved heuristic methods for multiple stage production planning. Technical report, School of Management, State University of New York at Buffalo, 1997.

[285] R. Slowinski. Multiobjective project scheduling under multiplecategory resource constraints. In R. Slowinski and J. W~glarz, editors, Advances in Project Scheduling, pages 151-167. Elsevier, Amsterdam, 1989.

[286] D.E. Smith-Daniels and V.L. Smith-Daniels. Maximizing the net present value of a project subject to materials and capital constraints. Journal of Operations Management, 7:33-45, 1987.

BIBLIOGRAPHY 249

[287] D.E. Smith-Daniels and V.L. Smith-Daniels. Optimal project scheduling with materials ordering. IIE Transactions, 19(2):122-129, 1987.

[288] T. Spengler and O. Rentz. Betriebswirtschaftliche Planungsmodelle zur Demontage und zum Recycling komplexer zusammengesetzter Produkte. Zeitschrift fur Betriebswirtschaft, (Erganzungsheft 2):79-96, 1996.

[289] M.G. Speranza and C. Vercellis. Hierarchical models for multiproject planning and scheduling. European Journal of Operational Research, 64:312-325, 1993.

[290] A. Sprecher and A. Drexl. Multi-mode resource-constrained project scheduling by a simple, general and powerful sequencing algorithm. European Journal of Operational Research, 107(2) :431-450, 1998.

[291] A. Sprecher, R. Kolisch, and A. Drexl. Semi-active, active, and non-delay schedules for the resource-constrained project scheduling problem. European Journal of Operational Research, 80:94-102, 1995.

[292] H. Stadtler. Hierarchische Produktionsplanung bei losweiser Fertigung. Physica, Heidelberg, 1988.

[293] H. Stadtler. Hierarchische Produktionsplanung. In W. Kern and J. Weber, editors, Handworterbuch Produktion, pages 631-641. Poeschel, 1996.

[294] H. Stadtler. Mixed integer programming model formulations for dynamic multi-item multi-level capacitated lotsizing. European Journal of Operational Research, 94:561-581, 1996.

[295] A. Starn and L.R. Gardiner. A multiple objecitve marketingmanufacturing approach for order (market) selection. Computers (3 Operations Research, 19(7):571-583, 1992.

[296] R.M. Stark. Competitive bidding: A comprehensive bibliography. Operations Research, 19:484-490, 1971.

250 BIBLIOGRAPHY

[297] R.M. Stark and R.H. Mayer. Some multi-contract decisiontheoretic competitive bidding models. Operations Research, 19:469-483, 1971.

[298] G. Steiner and J .S. Yeomans. Optimal level schedules in mixedmodel, multi-level JIT assembly systems with pegging. European Journal of Operational Research, 95:38-52, 1996.

[299] M. Steven. Hierarchische Produktionsplanung. Physica, Heidelberg, 2. edition, 1994.

[300] C.-C. Sum and A.V. Hill. A new framework for manufacturing planning and control systems. Decision Sciences, 24(4) :739-760, 1993.

[301] C.S. Tang. Controlling inventories in an acyclic assembly system. Management Science, 38(5):743-750, 1992.

[302] G.D. Taylor. Design for global manufacturing and assembly. lIE Transactions, pages 585-597, 1997.

[303] S. Tayur, M. Magazine, and R. Ganeshan. Quantitative Models for Supply Chain Management. Kluwer Academic Publishers, Boston, 1998.

[304] H. Tempelmeier. Material-Logistik - Modelle und Algorithmen fur die Produktionsplanung und -steuerung und das Supply Chain Management. Springer, Berlin, 4. edition, 1999.

[305] H. Tempelmeier and M.C. Derstroff. Mehrstufige MehrproduktLosgroBenplanung bei beschrankten Ressourcen und generellen Erzeugnisstrukturen. OR Spektrum, 15:63-73, 1993.

[306] H. Tempelmeier and M.C. Derstroff. A lagrangian-based heuristic for dynamic multilevel multiitem constrained lotsizing with setup times. Management Science, 42(5):738-747, 1996.

[307] H. Tempelmeier and S. Helber. A heuristic for dynamic multiitem multi-level capacitated lotsizing for general product structures. European Journal of Operational Research, 75:296-311, 1994.

BIBLIOGRAPHY 251

[308] A. Thesen. Measures of the restrictiveness of project networks. Networks, 7:193-208, 1977.

[309] J.M. Thizy and L.N. van Wassenhove. Lagrangean relaxation for the multi-item capacitated lot-sizing problem: A heuristic implementation. IIE Transactions, 17(4):308-313, 1985.

[310] J.D. Thompson. Organizations in Action. McGraw-Hill, New York, 1967.

[311] S. Tsubakitani and R.F. Deckro. A heuristic for multi-project scheduling with limited resources in the housing industry. European Journal of Operational Research, 49:80-91, 1990.

[312] A.H. van de Ven, A.L. Delbecq, and R. Jr. Koenig. A task contingent model of work-unit structure. American Sociological Review, 41:322-338, 1976.

[313] J.M. van den Akker, J.A. Hoogeven, and S.L. van de Velde. Parallel machine scheduling by column generation. Operations Research, 47(6):862-872, 1999.

[314] J.T. van der Vaart, J. de Vries, and J. Wijngaard. Complexity and uncertainty of materials procurement in assembly situations. International Journal of Production Economics, 46-47: 137-152, 1996.

[315] J.1. van Zante-de Fokkert and T.G. de Kok. The mixed and multi modell line balancing problem: A comparison. European Journal of Operational Research, 100:399-412, 1997.

[316] C. Vercellis. Constrained multi-project planning problems: A Lagrangian decomposition approach. European Journal of Operational Research, 78:267-275, 1994.

[317] A. Wagelmans, S. van Hoesel, and A. Kolen. Economic lot sizing: An 0 (n log n) algorithm that runs in linear time in the WagnerWhitin case. Operations Research, 40:145-156, 1992.

[318] H.M. Wagner and T.M. Whitin. Dynamic version of the economic lot size model. Management Science, 5:89-96, 1958.

252 BIBLIOGRAPHY

[319] H.-J. Warnecke and J. Walther. Automatisches Schrauben mit Industrierobotern. Zeitschrift fur industrielle Fertigung, 74(3):137-140, 1984.

[320] E.N. Weiss. An optimization based heuristic for scheduling parallel project networks with constrained renewable resources. IIE Transactions, 20(2) :137-143, 1988.

[321] U. Wemmeri6v. Assemble-to-order manufacturing: Implications for material management. Journal of Operations Management, 4(4):347-368, 1984.

[322] E. Westkamper and F. Siinnemann. Endmontageleittech-nik starkt Wettbewerbsfahigkeit eines ostdeutschen Schienenfahrzeugherstellers. Zeitschrift fur wirtschaftliche Fertigung, 88(1):10-13, 1993.

[323] S. Whang. Coordination in operations: A taxonomy. Journal of Operations Management, 12:413-422, 1995.

[324] C.D. Whybark. Marketing's influence on manufacturing practices. International Journal of Production Economics, 37:41-50, 1994.

[325] C.D. Whybark and J. Wijngaard. Manufacturing-sales coordination. International Journal of Production Economics, 37:1-4, 1994.

[326] V.D. Wiley, R.F. Deckro, and J.A. Jackson Jr. Optimization analysis for design and planning of multi-project programs. European Journal of Operational Research, 107(2):492-506,1998.

[327] W.E. Wilhelm, S. Saboo, and R.A. Johnson. An approach for designing capacity and managing material flow in small-lot assembly lines. Journal of Manufacturing Systems, 5(3):147-160, 1986.

[328] W.L. Winston. Operations Research - Applications and Algorithms. Duxbury, Belmont, 3. edition, 1994.

BIBLIOGRAPHY 253

[329] J.D. Wisner and S.P. Siferd. A survey of U.S. manufacturing practices in make-to-order machine shops. Production and Inventory Management Journal, 36(1):1-7, 1995.

[330] B.M. Woodworth and C.J. Willie. A heuristic algorithm for resource levelling in multi-project, multi-resource scheduling. Decision Sciences, 6:525-540, 1975.

[331] M.J.F. Wouters. Economic evaluation ofleadtime reduction. International Journal of Production Economics, 22:111-120, 1991.

[332] K-K Yang and C.-C. Sum. A comparison of resource allocation and activity scheduling rules in a dynamic multi-project environment. Journal of Operations Management, 11:207-218, 1993.

[333] K-K Yang and C.-C. Sum. An evaluation of due date, resource allocation, project release, and activity scheduling rules in a multi-project environment. European Journal of Operational Research, 103:139-154, 1997.

[334] C. Yau and E. Ritchie. A linear model for estimating project resource levels and target completion times. Journal of the Operational Research Society, 39(9):855-862, 1988.

[335] G. Zafpel and B. Piekarz. Supply Chain Controlling. Ueberreuter, Wien, 1996.

[336] G. Zapfel. Produktionswirtschaft - Operatives ProduktionsManagement. de Gruyter, Berlin, 1982.

[337] G. Zapfel. Produktionslogistik - Konzeptionelle Grundlagen und theoretische Fundierung. ZeitschriJt fur BetriebswirtschaJt, 61(2):209-235,1991.

[338] G. Zapfel. Stiicklisten, Verwendungsnachweise, Arbeitsplane und Prod uktionsfunktionen. Wirtschaftswissenschaftliches Studium, 20(7):340-346, 1991.

[339] C. Zhang and H.-P. Wang. Robust design of assembly and machine tolerance allocations. IIE Transactions, 30:17-29,1998.

254 BIBLIOGRAPHY

[340] H. Ziegler. Minimal and maximal floats in project networks. Engineering Costs and Production Economics, 9:91-97, 1985.

Index

'quantity per' factor, 69

ABC inventory classification, 28 activity, 40, 42

rank, 203 aggregate capacity planning, 1 aircraft assembly, 28 anticipation, 21, 47

mechanism, 199 aspiration criterion, 184 assemble

-in-advance, 12 -to-forecast, 12, 110 -to-order, 12

assembler, 14, 42, 79 assembly, 13, 42

area allocation problem, 108 classification, 34 department, 3 graph, 42 list generation, 130 machine, 14 management, 5, 33 schedule, 156 schedule generation, 132 scheduling problem, 86, 129 sequence, 182 station, 14 tool, 14

assembly cell, 15

flexible, 15 scheduling, 38 sequencing, 38

assembly line, 14 balancing, 33, 35 flexible, 15, 37 level scheduling, 38 loading and relase planning,

37 mixed-model, 37 scheduling, 37 sequencing, 37

assembly shop, 15 multi-level, 38 loading and relase planning,

37 assembly system

flexible, 15 loading and release planning,

37 scheduling, 37 sequencing, 37

assignment assembly area, 114 assembly resource, 112 part, 114 problem, 112

assignment sequence problem, 101

backtracing, 123, 149, 153

256

backward recursion, 205 base part, 16, 30 batch

process, 41 transfer, 41

batching, 168 best fit strategy, 184 bid, 27,55 bill of materials, 12,40, 73, 203 block, 23

curved-bottomed, 23 fiat-bottomed, 23

blocking operation, 185

capacitated lotsizing problem, 75, 87, 135, 152

clinching, 13 column generation, 121, 160 competitive bidding, 55 contract sum, 96 contribution margin, 23, 53 coordination, 4, 5, 18

cost

by hierarchy, 18 hierarchical, 20 of interfaces, 18 vertical, 156

fixed ordering, 91 inventory holding, 4, 38, 70

time variant, 71 setup, 71 tardiness, 38

critical operations, 186 parts, 185 resources, 185

customer order decoupling point, 11

INDEX

cutting stock problem, 25

decision level, 21 lower, 21 middle, 21 upper, 21

decision model, 19 delivery date, 42 demand forecast, 12 design, 34

assembly-oriented, 34 design for assembly, 34

design phase, 35 evaluation phase, 35 integrated, 35 product-oriented, 34 system-oriented, 35

disassembly, 40 dry dock, 25 due date, 40, 42

assignment, 55 dynamic programming, 122, 149

echelon stock, 141 economic order quantity, 140 engineer-to-order, 12 engineering department, 2 enterprise resource planning, 19

fabrication department, 2 fabrication lotsizing problem, 86,

129 facility design, 1 fiowtime, 43 forward recursion, 133, 205 fuselage, 30

grace window, 23

INDEX

greedy randomized adaptive search procedure, 181

heuristic backward lotsizing, 137 list scheduling, 130 LP-based, 123 multi priority rule, 165 schedule generation, 132

hierarchical approach, 47 production planning, 5, 20

insertion sequence, 39 instruction, 21 integration, 5, 19 interdependency, 4, 17, 54

behavioral, 17 factual, 17 goal, 18 horizontal, 18, 54, 97 risk, 18 time-based, 17, 54, 72, 97 vertical, 18, 54, 73, 97

intree, 119 inventory balance constraints, 144

job, 42 assembly, 49 rotation, 19 subassembly, 49

job shop production planning, 41 scheduling problem, 108

just-in-time, 30, 42, 100, 141, 168

kit, 16 kitting problem, 101

257

knapsack problem, 63 multiple constrained, 119

Lagrangian multipliers, 145 Lagrangian relaxation, 141, 148,

152 large-step optimization, 182, 189 lead time, 27 left regular, 109 left shiftable, 185 Leitstand, 39, 101 line, 34 list, 130 list scheduling, 130

backward oriented, 130 forward oriented, 171

local search, 182 longest path, 108, 203, 204 lot-for-Iot, 76 lotsizing

backward oriented, 134 generation scheme, 136 multi-level, 36, 74, 75 policy, 138 problem, 24

lotsizing and scheduling, 74, 77 lotsizing problem

dynamic capacitated, 135 dynamic uncapacitated, 152 static uncapacitated, 140

low-level code number, 203

machine, 40 machine tool assembly, 26 make-to-order, 5, 12 make-to-stock, 11 managerial level, 6 man ufacturing

258

environment, 11 make-to-order, 40 planning, 4, 69

integrated, 88 sequential, 88

resource planning, 20,42 system, 1

stochastic, 36 market/price-mechanism, 19 mean flow time, 99 mean tardiness, 99 milestone, 26,42 mixed-integer program, 6 move, 187 multi-level assembly, 13 multi-project scheduling prob-

lem, 179 multiplier adjustment, 160 mutual adjustment, 18

neighborhood adjacent pairwise interchange,

184 critical, 184

net present value, 44 network restrictiveness, 201

operation, 40, 42 operations scheduling, 4, 19 operations sequence, 182 optimized production technology,

42 order, 40 order selection, 3, 4, 53

sequential, 58 simultaneous, 53

parallel machine scheduling problem, 108

INDEX

part in-house fabricated, 79 joining, 13 kitting, 39 mating, 13 outside procured, 79 pegging, 110

part allocation problem, 109 pegging, 17, 110 plan, 18 planning

backward oriented, 129 forward oriented, 76, 136

precedence network, 40, 42, 80 printed circuit board assembly,

33,39 priority rule

first come first serve, 131, 179

smallest job number, 131 smallest operation number,

179 tie breaking, 179 weighted earliest due date,

182 priority value, 130 product, 21

family, 21 type, 21

production function, 78 planning, 34, 36

multi-level, 36 planning and loading, 42 planning problem, 89 scheduling, 1

ProGen, 124, 162, 201

INDEX

project, 34, 40, 42 assembly, 15, 39 management, 19 network, 40, 42 scheduling, 42 selection, 53

reaction, 21 regret value, 181 regular performance measure, 37 release

date, 40 time, 42

resource, 40, 42 allocation problem, 108 constrained project schedul

ing problem, 86 partially renewable, 67

revenue, 35 management, 23, 57

revised planning, 199 right-regular, 87 riveting, 13 rule, 18

sampling biased random sampling, 181 method, 180 random sampling, 181

schedule, 18 active, 176 right-active, 134

scheduling, 34 just-in-time, 39

scheduling and lotsizing, 74, 76 scoring model, 66 screw connection, 13 search space, 181

section, 23 sensitivity analysis, 68 sequencing, 34 setup

pattern, 158 time, 41

shadow price, 67, 122 ship assembly, 22 shop, 34 simplex method, 121 single-level assembly, 13 slant, 29

259

solution representation, 182 spatial demand, 110 standard curve, 22 standardization, 18 start time

earliest, 206 latest, 206

statistical test Friedman, 165, 194 Kruskal-Wallis, 125 Kruskal-Wallis-H, 165, 194 Mann-Whitney, 125 nonparametrical, 165, 193 Wilcoxon matched-pairs signed

ranked, 165, 194 subassembly, 13 subgradient optimization, 141,

160 supply chain, 4

management, 19

tabu list, 183 tabu search, 180, 182, 183 tardiness

total, 43 weighted, 43, 108

260

team concept, 18 time between order, 163 time lag

maximal, 40, 41 minimal, 40, 41, 201

tool, 42 transportation problem, 114, 135,

158 traveler-jobs, 30

valid constraints, 151 value added, 30

Wagner-Whitin problem, 90,205 what-if analysis, 67 winning probability, 65 work package, 53

INDEX

make-to-order assembly management ||

Documents