The Pennsylvania State University

The Graduate School

The Harold and Inge Marcus

Department of Industrial and Manufacturing Engineering

A MULTI-OBJECTIVE MODEL FOR MULTI-SUPPLIER SELECTION FOR MULTI

PRODUCTS

A Thesis in

Industrial Engineering and Operations Research

by

Hong Ren

2013 Hong Ren

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Master of Science

August 2013

The thesis of Hong Ren was reviewed and approved* by the following:

M. Jeya Chandra

Professor of Industrial Engineering

Thesis Adviser

Terry Harrison

Professor of Supply Chain and Information System

Thesis Reader

Paul Griffin

Professor of Industrial Engineering

Head of Industrial and Manufacturing Department

*Signatures are on file in the Graduate School

iii

ABSTRACT

In today’s competitive environment, it is important that decision makers select

appropriate suppliers for multiple products in effective supply chain management.

Multiple suppliers can reduce cost, decrease production lead time, increase customer

satisfaction and strengthen corporate competitiveness. The objective of this thesis is to

solve a multiple-objective model of multiple-supplier selection and inventory

optimization. Three objectives are considered which are minimization of total cost which

consists of purchasing cost, fixed cost for choosing specific suppliers, and inventory cost,

maximization of total product quality, and minimization of total number of late delivery

products. In the model of this thesis, each supplier has limited capacity to supply

products; purchasing budget and storage capacity of retailers are also considered. The

retailer faces deterministic demand and lead time for each product from each supplier.

Inventory control is an important part of the object. It is guided by the continuous (r, Q)

policy, and shortage is allowed. Inventory cost in this model includes ordering cost,

holding cost and shortage cost. Non-preemptive goal programming is used to solve the

multiple-objective problem. A numerical example is given to illustrate the use of the

developed model.

iv

TABLE OF CONTENTS

List of Figures .............................................................................................................. v

List of Tables ............................................................................................................... vi

Acknowledgements ...................................................................................................... vii

Chapter 1 Introduction ................................................................................................. 1

1.1 Literature Review................................................................................................... 2

1.1.1 Multiple-supplier selection for multiple-product in supply chain ............... 3

1.1.2 Inventory management in supply chain ....................................................... 5

1.2 Contribution ........................................................................................................... 8

1.3 Summary of this thesis ........................................................................................... 9

Chapter 2 Model Formulations .................................................................................... 11

2.1 Model description .................................................................................................. 11

2.2 Notations ................................................................................................................ 12

2.3 Model assumptions ................................................................................................ 15

2.4 Objective Function ................................................................................................. 16

2.4.1 Total cost of the retailer ............................................................................... 16

2.4.2 Product quality ............................................................................................. 25

2.4.3 Late-delivery product ................................................................................... 25

2.5 Constraints ............................................................................................................. 26

2.5.1 Supplier capacity ......................................................................................... 26

2.5.2 Storage capacity ........................................................................................... 26

2.5.3 Purchasing budget ........................................................................................ 27

v

2.5.4 Last delivery exceeds reorder point ............................................................. 27

2.6 The optimization model ......................................................................................... 28

2.7 Goal programming ................................................................................................. 29

Chapter 3 Numerical Example ..................................................................................... 31

3.1 Parameter Setting ................................................................................................... 31

3.2 Numerical analysis ................................................................................................. 34

Chapter 4 ...................................................................................................................... 38

Conclusions and Future Work ..................................................................................... 38

4.1 Conclusions ............................................................................................................ 38

4.2 Future works .......................................................................................................... 39

References .................................................................................................................... 41

vi

LIST OF FIGURES

Figure 2.1.1 Process of the model……………………………………………..12

Figure 2.4.1 Inventory level without shortage per cycle ……………………...20

Figure 2.4.2 Inventory level with shortage per cycle ………………………….20

Figure 3.2.1 Distribution of each supplier for products ……………..………....35

Figure 3.3.2 Comparison between multi-supplier and single supplier..…...…..37

vii

LIST OF TABLES

Table 3.1.1 Demand data for product i (in units)……………………………...31

Table 3.1.2 Capacity data for product i from supplier j (in units)……………..32

Table 3.1.3 Lead time data (in days)…………………………………………...32

Table 3.1.4 Purchasing cost data (in $)…………………………………………32

Table 3.1.5 Inventory data (in $)………………………………………………..33

Table 3.1.6 Fixed cost data (in $)……………………………………………….33

Table 3.1.7 Product quality and late-delivery product proportion……………....33

Table 3.2.1 Ideal values and target values of multi-supplier…………………….34

Table 3.2.2 Optimal solution of the model……………………………………....34

Table 3.2.3 Ideal values and target values of single supplier…………………….36

Table 3.2.4 Optimal solution for single supplier………………………………....36

viii

ACKNOWLEDGEMENTS

Foremost, I would like to express my sincere gratitude to my advisor Prof.

Chandra for the continuous support of my study and research, for his patience,

motivation, enthusiasm, and immense knowledge. His guidance helped me in all the time

of research and writing of this thesis. Besides my advisor, I would like to thank the rest of

my thesis committee: Prof. Griffin and Prof. Harrison, for their encouragement, insightful

comments, and hard questions. Last but not the least, I would like to thank my family: my

parents, Yinlu Ren and Qingfen Meng, for giving birth to me at the first place and

supporting me spiritually throughout my life.

1

Chapter 1

Introduction

In the past few decades, many companies have considered supplier selection as a

significant problem in supply chain management (SCM). Companies have to work with a

large number of suppliers to complete their business activities (Demirtas 2008). Proper

supplier selection is important for companies, because the cost of raw materials and

component parts represents the largest percentage of the total product costs in many

industries (Chopra 2007, Mendoza 2010).

In additional to total product costs, supplier selection also has an influence on

reducing risks. In today’s global market, there are two kinds of risks that companies need

to face; one is routine operational problems, and the other is major disruptions such as

earthquakes, fires, hurricanes and labor strikes. Many researches have been done on

reducing operational risks, but there are only a few models developed for mitigating

disruption risks. These risks have caused major damages to various companies in

different business segments (Bilsel 2011). “Under a competitive environment of global

sourcing, core-competence outsourcing strategy, supply base reduction, strategic buyer-

supplier relationship, cross-functional purchasing program, Internet and e-commerce and

so forth, the supplier selection problem is becoming more and more complicated”

(Ouyang 2002, Rezaei 2008).

2

For many years, inventory management has also been very active in supply chain

area, and inventory risk is one of the main risks in supply chain management. Excess

inventory may influence financial performance and cause other problems, like increasing

holding cost and probability of products damage. Simultaneously, low inventory levels

may lead to backorders, increase shortage costs and longer lead time. Hence, it is very

important to keep inventory at an appropriate level (Slack et al. 2010).

At the same time, companies strive to reduce total cost, without sacrificing

product quality or customer service (Banerjee 1994). In this thesis, a multiple-objective

optimization model is formulated to solve the supplier selection problem and inventory

optimization. The selection of competent suppliers to minimize the total costs is

considered, product quality and delivery are also selected as critical factors to evaluate

the competence of potential suppliers (Keskin 2010). This thesis aims at solving the

problems of both supplier selection and inventory optimization in a multiple-objective

model.

1.1 Literature Review

In recent several decades, a large amount of research has been done related with

multiple-supplier selection for multiple products, and inventory control. In this chapter,

the literature review mainly discusses the research that is important for the development

of the model in this thesis.

3

1.1.1 Multiple-supplier selection for multiple-product in supply chain

Companies select multiple suppliers to fulfill the demand, and replenishment

order quantity is split into different portions for each supplier at the same time. It is

advantageous for companies to use several suppliers, for example: 1) Multiple suppliers

can reduce the risk of supply and increase competition, which reduce costs and improve

product quality; 2) It is necessary as suppliers have limited capacities to fulfill the

demand; 3) Multiple suppliers can also reduce the average inventory and holding and

shortage costs (Wang 2008).

From previous study, basically, there are two types of supplier selection problem.

In the first type of supplier selection, a single supplier can fulfill the entire buyer’s

demand. Only one decision should be made in this situation: which supplier is the best. In

the other type of supplier selection, there exists no single supplier who can satisfy the

entire buyer’s needs. In this situation, the buyer has to split order quantities among

suppliers for having a stable environment of competitiveness (Demirtas 2008). Sculli

(1990)’s research showed that using two suppliers instead of one with normal lead times

reduce the effective lead times (time at which the first supply arrives). Ramasesh et al.

(1993) were the first to evaluate the costs and profits associated with the use of multiple

suppliers. With the assumption of deterministic demand and either exponential or

uniform lead times, the results showed that dual sourcing decreased inventory costs.

Chiang and Benton (1994) extended this dual sourcing study by using random demand

and any stochastic lead time distribution with unequal order quantity. The conclusion also

showed that dual sourcing worked better than single supplier (R. Ganeshan 1999).

4

Sedarage and Fujiwara (1999) considered multiple-supplier single-product

inventory systems, where lead times of suppliers and demand arrival were random

variables, and shortage was allowed. The replenishment took place when the inventory

level reached the reorder level and the order was split among different suppliers. The lead

times may have different distributions. It is found that it is economical to select suppliers

with higher lead time and standard deviation and higher unit purchasing costs. In Wang

and Jiang (2008)’s research, a single-item multiple-supplier model under constant

demand and lead times was formulated. This integer linear programming model was

developed to choose the best set of suppliers, figure out the optimal order quantity for

each supplier and reorder level which can minimize the total inventory cost and satisfy

the constraints of supplier capacity, quality and demand. An algorithm combining the

branch-bound algorithm and enumeration algorithm was provided to solve the problem.

The results showed that inventory decreased when using multiple suppliers.

In this thesis, research is not only focused on single product, but also on multiple

products. Literature review showed that a lot of study has been done to analyze multiple-

product supplier selection problem. A supplier selection model by Benton (1991)

provided a heuristic algorithm to select one supplier under conditions of multiple-

products, multiple-supplier and quantity discounts. The algorithm aimed at calculating an

optimal order quantity for all products for each supplier and choosing the best supplier to

obtain the lowest costs. Davari (2008) presented a multiple suppliers and multiple

products model. There were three objectives to achieve, minimizing purchasing cost,

rejected units and late delivered units. Wadhwa (2010) introduced a multipe-objective

multiple-supplier selection model for low risk and cost products. The first objective was

5

to minimize the total purchasing cost, which concluded total variable cost, fixed cost,

inventory holding cost and the bundling discounts. The second objective was to minimize

the reject units under supplier capacity constraint. Shortage was not allowed and the

multi-objective model was solved by preemptive goal programming.

Several researches discussed the problem of selecting proper suppliers and

optimizing the allocation of each product simultaneously (Dai and Qi 2007, Rezaei and

Davoodi 2005). Rezaei and Davoodi (2011) presented two multiple-objective mixed

integer non-linear models to compare the results with shortage and without shortage. The

model was developed for multiple-period allocation problem for multiple-product and

multiple-supplier selection. Each model had three objectives including cost, product

quality and service level. After the comparison, it was found out that allowing shortages

was better.

1.1.2 Inventory management in supply chain

In recent years, different innovative manufacturing approaches have been figured

out to improve competitive. Reducing inventory is one major focus among these

approaches (Banerjee 1994). In the process of supply chain, holding products in stock

does not make profits. “Storage activity requires handling that may damage items, buying

resources that are costly and immobilizing items that consequently cannot be sold. Thus

storage increases the production cost” (Sethi 2005). However, inventory can deal with

uncertainties like stochastic demands and lead times, influence of natural disasters,

product quality problems, etc.

6

Inventory exists through all the supply chain management in different types for

various reasons. “At any manufacturing point, inventory may exist as raw materials, work

in progress, or finished goods”. Ganeshan (1999) concluded that the proportion of

inventory costs in the total annual cost was between 20 and 40%. Although holding

inventory is very necessary to improve product quality and reduce costs and risks,

keeping inventory at the lowest optimum levels makes economic sense.

Bahl et al (1987) classified inventory lot-splitting into four categories: 1) single –

level unconstrained resources; 2) single-level constrained resources; 3) multiple-level

constrained resources; 4) multiple-level unconstrained resources. “Levels here refer to the

different levels in a bill of material structure where dependency of requirements exists,

and constrained resources refer to production capacity limitations” (Basnet 2005). The

model used in this thesis belongs to the second category; cause level dependencies are not

considered and capacity is limited.

In inventory model, there are several costs that could be considered. As usual, the

following costs are taken into account: holding cost, ordering cost and shortage cost

(backlogging cost). Holding cost is incurred for keeping products in stock; ordering cost

is the cost that occurs each time products are ordered, including administrative costs,

transportation costs, costs incurred in the case of promotions and costs incurred for

changing production; shortage cost is charged when a product is in demand and cannot be

delivered due to a shortage (Sethi 2005).

There are two fundamental questions that combined to form an inventory policy,

the first question is when to place the order, it is either at the end of a fixed period or as

soon as the inventory level falls to a specific value; the second one is the order quantity

7

which may be a fixed value or determined as the difference between a value and the

inventory position. Continuous inventory (r, Q) policy, a fixed quantity Q is ordered as

soon as inventory level falls below reorder point r. This policy reflects that the supplier

just offer fixed discrete order batches of size Q or an integer multiple of Q. As for the (r,

Q) policy, instant review as well as the possibility that an order could always be placed is

required for the application of this policy.

Extensive work has been done in inventory management area; beginning from the

work of Basnet (2005), his research presented a multi-period inventory model for

multiple suppliers and multiple products. The demand of multiple products was known

over a finite horizon. Every product could be fulfilled from a set of potential suppliers; a

supplier-dependent transaction cost for each period was added to the total costs. The

objective was to decide the order quantity for each product in different periods in order to

minimize the total costs including purchasing cost, transaction cost and holding cost.

Backorders were not allowed. In Tsai and Yeh (2008)’s model, there were three objective

functions, minimizing costs, maximizing inventory turnover ratios, and maximizing

inventory correlation. Many large companies, such as Dell and HP, have to face with

inventory backorders (Gollner 2008, Walsh 2010). The reason why there exists shortage

including part variations, mis-operation, inventory reduction (Jiang et al.2010). Zhu

(2009) presented an inventory optimization model to minimize the total inventory cost,

which consisted of ordering cost, holding cost and shortage cost. It aimed at selecting one

best supplier from a set of potential suppliers for multiple items under storage capacity

constraint and purchasing budget constraint. Suppliers have unlimited capacity in his

model. Mendoza and Ventura (2009) developed a mixed integer non-linear programming

8

model to allocate order quantities to the selected suppliers while considering the

purchasing cost, holding cost and transportation cost under suppliers’ capacity and

product quality constraints. Mendoza (2012) introduced a decision model and technique

to select multiple suppliers to single item and determine the appropriate allocation of

order quantities which aimed at minimizing the sum of ordering, holding and purchasing

cost. Suppliers’ limited capacity and quality constraints were considered. No previous

work a continuous review inventory policy has considered a multiple-objective model,

multiple-supplier selection for multiple-product, deterministic demand and lead times,

combined with capacity limitation, storage space and purchasing budget constraints at the

same time.

1.2 Contribution

This thesis is focused on solving a multiple-objective optimization model. In this

model, there are three objectives: (1) minimize the total costs; (2) maximize the total

product quality; (3) minimize the total late-delivery products. It considers a multiple-

supplier selection for multiple-product under a multiple-constraint environment. The goal

of this thesis is to find the appropriate suppliers to fulfill the multiple-product, order

quantities for each product from specific suppliers and reorder level for each product in

inventory environment. At last, a goal programming is provided to solve the multiple-

objective problem.

In addition to purchasing cost, inventory cost is considered as a major part in total

costs, which includes ordering cost, holding cost and shortage cost. Shortage is allowed

9

and backorders immediately appear when demands cannot be fulfilled. Fixed cost is also

considered in this thesis, which means the total cost of managing suppliers. Fixed cost

will be charged at most once for one supplier that is selected to meet the products.

In previous research, most works are done on single supplier selection for

multiple-product or multiple-supplier selection for single product in procurement process

or inventory system, respectively. In this thesis, multiple-supplier can be selected for

multiple-product under a supply chain environment which concludes both purchasing

process and inventory system.

Multiple-objective model with inventory optimization is also an innovation in this

thesis. Reducing the total cost is obviously a significant objective to be considered.

However, mitigating the risk and increasing the product quality are becoming more and

more important in real business environment. In order to solve this three-objective model,

non-preemptive goal programming is provided, which can be easily implemented and

find out the optimal solution.

1.3 Summary of this thesis

Chapter 1 is an introduction to the previous research and describes the

contribution and summary of this thesis. The remainder of this thesis is organized as

follows: Chapter 2 gives the problem formulation of a multiple-objective optimization

model with multiple-supplier selection for multiple-product under inventory system. Goal

programming is provided to solve this problem. In chapter 3, a numerical example is

10

described to illustrate the approach to get the optimal solution. Finally, conclusions and

some future works are summarized in chapter 4.

11

Chapter 2

Model Formulations

2.1 Model description

In this thesis, a multiple- objective model for multiple-supplier selection and

inventory optimization is developed for multiple-product. For a single retailer, multiple

products are procured through several suppliers. More than one supplier may be chosen

to supply one specific product so that the objectives are realized and constraints are

satisfied. The process of the model is shown in figure 2.1.1.

There are three objectives in this model: (1) minimize the total cost of the retailer,

consist of fixed cost for choosing specific suppliers, purchasing cost, and inventory cost;

(2) maximize the product quality; (3) minimize the total number of late-delivery items. In

this model, demand and lead time for each product is a deterministic value and different

from chosen suppliers. The inventory review policy is the continuous (r, Q) policy and

shortage is allowed in this model. There are three items included in inventory cost which

are ordering cost, holding cost and shortage cost. Several constraints are also illustrated

in this chapter: each supplier has limited capacity to replenish the retailer; a limited

storage capacity for each product should be satisfied and a purchasing budget is

considered. The decision variables are the selection of suppliers for each product, order

quantity allocation and reorder point for all products. In the end, the goal programming is

12

used in order to solve the multiple-objective model for choosing the best combination of

suppliers for each product and to gain the optimal decision variable for inventory control.

2.2 Notations

The following listed notations are used in this thesis to formulate the model:

Index

m Number of products

n Number of suppliers

q Number of selected suppliers

Decision variables

Number of selected suppliers for product i, i=1…m

The amount of product i purchased from supplier j, i=1…m; j=1…n

Order quantity of product i from th delivery, =1…

Binary variable which taken on a value 1 if order for product i is placed with

supplier j, otherwise, equals 0, i=1…m; j=1…n

Supplier 1 Supplier 2 ….. …..

Supplier j Supplier n

….. ….. Product 1 Product 2 Product i Product m

Figure 2.1.1. Process of the model

13

Binary variable that equals 1 if any order is placed with supplier j; equals zero

otherwise, j=1…n

Reorder point for product i, i=1…m

Parameters

Purchasing cost per unit per cycle of product i from supplier j, i=1…m; j=1…n

Unit time demand for product i, i=1…m

Capacity per order for product i from supplier j, i=1…m; j=1…n

Fixed cost for selecting supplier j, j=1…n

Proportion of non-defective items of product i from supplier j, i=1…m; j=1…n

Proportion of late-delivery items of product i from supplier j, i=1…m; j=1…n

Lead time of product i from supplier j, i=1…m; j=1…n

Lead time of product i from the th delivery supplier, k=1…

On-hand inventory level of product i before the th delivery, equals positive

value when there is no shortage, equals zero when shortage occurs, k=1…

On-hand inventory level of product i after the th delivery, k=1…

Total holding cost for product i to the retailer per unit time, i=1…m

Total ordering cost for product i from supplier j to the retailer per unit time,

i=1…m; j=1…n

Total shortage cost for product i to the retailer per unit time, i=1…m

Holding cost of product i between the th and ( )th deliveries, i=1…m;

k=1…

14

Holding cost of product i between the th delivery in a cycle and the first

delivery in the next cycle

Shortage quantity before the th delivery of product i, i=1…m; k=1…

Holding cost per unit per unit time for product i, i=1…m

Ordering cost per order for product i from supplier j, i=1…m; j=1…n

Shortage cost per unit per unit time for product i, i=1…m

Storage space per unit for product i, i=1…m

Maximum storage space allocated to product i, i=1…m

The length of one order cycle for product i, i=1…m

Time interval between th delivery and when inventory drops to zero for

product i, i=1…m; k=1…

TIC Total inventory cost per unit time

TC Total cost per unit time

M Total purchasing budget limit for the retailer

Numerical weights assigned for objective , =1, 2, 3

Positive deviation from target value for objective , =1, 2, 3

Negative deviation from target value for objective , =1, 2, 3

Target value for objective , =1, 2, 3

15

2.3 Model assumptions

In order to develop the appropriate model in this thesis, there are some

assumptions made throughout the model building and solution process.

1. The unit time demand for product i, , is a constant value.

2. Each supplier has its limited production capacity, and the capacity is a constant

value for each supplier.

3. Lead time for product i from supplier j, , is a constant value.

4. Time value of cash flow is not considered in this thesis.

5. Each item is allocated to its specific location with storage space and available

total storage space is limited by W.

6. Suppliers may not be able to fulfill the demand of the retailer which means that

shortage or backordering occurs.

7. A fixed cost for choosing suppliers is included in calculating the total cost in

order to minimize the total number of suppliers, which is different from the fixed

ordering cost in inventory cost.

8. The fixed cost for choosing specific supplier is a constant value.

9. Only one delivery is permitted from each selected supplier within each cycle time.

10. There is no replenishment order during the order cycle even if the inventory level

drops to the reorder point.

11. When the replenishment of the last supplier arrives, the product on-hand

inventory level should exceed the reorder point.

16

2.4 Objective Function

A multiple-objective model is developed based on the above assumptions. Several

performance measures are used to evaluate the effectiveness of supply chain

management.

2.4.1 Total cost of the retailer

The total cost per unit time is the most considered measure as decision makers of

the retailer want to make the profit as large as possible. The objective is to minimize the

sum of fixed cost, transportation cost and inventory cost.

Purchasing cost

Purchasing cost is the sum of all products purchased from all selected suppliers

per cycle. It is assumed that the purchasing price for product i is not included in the

inventory ordering cost.

∑∑

(2.1)

where is the unit purchasing cost per unit time of product i from supplier j, is the

amount of product i purchased from supplier j, and is the binary variable which is

equal to 1 if an order for product i is placed with supplier j, otherwise, equals to 0.

Since the length of one cycle time, denoted as , is expressed as below:

∑

(2.2)

where is the demand per unit time for product i, i=1…m.

17

The number of order cycles per unit time is ⁄ Hence, purchasing cost per

unit time is expressed as follows:

∑[(∑

)

∑

]

(2.3)

Fixed costs

Fixed costs are the costs of selecting specific suppliers. For each product, it is

assumed that the retailor has to pay a deterministic value of money for choosing the

supplier. The fixed cost is just paid once if the particular supplier is selected regardless of

how many products are purchased from this supplier.

∑

(2.4)

and

{

∑

∑

(2.5)

where is the fixed cost associated with supplier j, is the binary variable which is

equal to 1 if an order is placed with supplier j, and is equal to zero if no order is placed

with supplier j.

Fixed cost per unit time is expressed as below:

18

∑[(∑

)

∑

]

(2.6)

Inventory cost

The inventory cost that will be introduced in this thesis is based on the continuous

review(r, Q) inventory policy.

1. Ordering cost for product i

The ordering cost is the cost of placing an order for one product. In this thesis,

ordering cost just includes the fixed ordering cost when the retailer orders products from

the suppliers. Ordering cost per cycle for product i from supplier j is expressed as below:

(2.7)

where is the ordering cost per order for product i from supplier j, i=1…m; j=1…n.

Therefore, ordering cost per unit time for product i from supplier j is expressed as

below:

∑

(2.8)

where is the ordering cost for product i from supplier j to the retailer per unit time,

i=1…m; j=1…n.

Hence, the total ordering cost per unit time for product i is expressed as below:

(∑

)

∑

(2.9)

19

2. Inventory holding cost for product i

The inventory holding cost per unit time is the sum of the costs associated with

the storage of the inventory until it is sold or used. It is calculated by multiplying the

holding cost per unit per unit time and the average inventory for one product.

(2.10)

where is the holding cost per unit per unit time for product i.

20

Figure 2.4.1. Inventory level without shortage per cycle

Figure 2.4.2. Inventory level with shortage per cycle

Inventory level

0

Time 0

th delivery

th delivery

Inventory level

Time

0

th delivery

th delivery

21

Firstly, two intermediate variables should be identified. is the on-hand

inventory level before the th delivery, equals positive value when there is no shortage,

equals zero when shortage occurs, k=1,… . In figure 2.4.1, when there is no shortage

occurred before the th delivery, is equal to the beginning inventory level which is

equal to reorder point, , plus the order quantity of the sum of the first th

deliveries, equals to ∑ , minus the demand during lead time , which is

. In figure 2.4.2, is equal to zero when shortage occurs. Hence, it is

expressed as below:

{

∑

∑

(2.11)

where is the reorder point for product i, is the order quantity of product i from th

delivery, =1,… , is lead time of product i from the th delivery supplier,

k=1,… , and is the unit time demand of product i.

is the on-hand inventory level before the th delivery, it is expressed below:

∑

(2.12)

From the figure 2.4.2, the time taken to drop the on-hand inventory level from

to zero is less than ( ). By assuming that the on-hand inventory

depletes linearly, the time interval between the th delivery and when inventory

drops to zero for product i is estimated as , and is given as below:

22

[( ∑

) ∑

] (2.13)

Therefore, the inventory holding cost between the th and th deliveries of

product i, denoted by is given as:

=

{[( ∑

) ( ∑

)]

[( ∑ ) ∑

]

} (2.14)

Similarly, the inventory holding cost between the reorder point and the first

delivery in the cycle, and inventory holding cost after the th delivery in a cycle can also

be calculated as below:

(

)

{

[( ∑

)(∑

)]

[ ]

} (2.15)

where is the number of selected suppliers for product i, i=1…m, which can be

expressed as below:

∑ (2.16)

Therefore, the total inventory holding cost per unit time, denoted as , is

expressed as below:

23

∑

∑

∑

{∑

{

[( ∑ ) ( ∑

)]

[( ∑ ) ( ∑

)]

}

{

[( ∑

)(∑

)]

[ ]

}} (2.17)

3. Inventory shortage cost for product i

Inventory shortage cost per unit time is incurred when suppliers cannot meet the

demand of the retailer; it is calculated by multiplying the shortage cost per unit and

average shortage for one product.

(2.18)

where is the shortage cost per unit for product i.

The shortage quantity just before the th delivery is denoted as . When

shortage occurs, is equal to the lead time demand minus the summation of

beginning inventory level, , and the order quantity of the th delivery, ∑ .

It is expressed as:

{ ∑

∑ )

(2.19)

The time of shortage just before the th delivery is denoted as below:

∑

(2.20)

24

Therefore, the total inventory shortage cost for product i per unit time, , is

expressed as below:

∑

∑

∑

∑ [ ( ∑ ) ]

(2.21)

The total inventory cost per unit time, which is denoted by TIC, is expressed as:

∑ {

∑

{∑

{

[( ∑ ) ( ∑

)]

[( ∑ ) ( ∑

)]

}

{

[( ∑

)(∑

)]

[ ( )]

}

∑ [ ( ∑

) ]

∑

}} (2.22)

Hence, the objective function of minimizing the total cost of the retailer is given

as below:

∑ {

∑

{∑

∑ ∑

{

[( ∑ ) ( ∑

)]

[( ∑ ) ( ∑

)]

}

25

{

[( ∑

)(∑

)]

[ ( )]

}

∑ [ ( ∑

) ]

∑

}} (2.23)

2.4.2 Product quality

The product quality can be measured by the number of non-defective products

divided by the total number of products delivered. It is assumed that the retailer has the

history data of product quality from each supplier for each product, and accepts all

products purchased from suppliers, no matter whether products are in good condition or

not. This objective function is shown in the following formula,

Max ∑ ∑

(2.24)

where is the proportion of non-defective items of product i from supplier j.

2.4.3 Late-delivery product

Late-delivery products are the total products that are delivered late by the

suppliers. It is calculated by multiplying the proportion of late-delivery items of product i

and the amount of products from supplier j. The objective function is to minimize the

total number of late-delivery products and is shown as below:

26

Min ∑ ∑

(2.25)

where is the proportion of late-delivery items of product i from supplier j, i=1…m;

j=1…n.

2.5 Constraints

2.5.1 Supplier capacity

Each supplier has limited capacity for product i. This constraint indicates that the

number of products i ordered from supplier j should be equal to or less than the supplier’s

capacity to deliver this product.

(2.26)

where is capacity per order of supplier j for product i.

2.5.2 Storage capacity

In this constraint, it is assumed that each product is allocated to its specific

location with limited storage space. Hence, there are m independent inequalities for m

types of products.

The maximum of inventory level is equal to the maximum of on-hand inventory

after each delivery, denoted as . Therefore, the storage capacity should be

equal to or less than the maximal inventory storage for product i.

27

∑ , i=1…m (2.27)

where is the storage space per unit for product i, is the maximum storage space

allocated to product i.

2.5.3 Purchasing budget

The purchasing budget is the total cost of purchasing cost, fixed cost and

inventory ordering cost. Hence, the sum of these three costs should be equal to or less

than the purchasing budget,

∑∑( ) ∑

∑

(2.28)

where M is the total purchasing budget limit for the retailer per unit time.

2.5.4 Last delivery exceeds reorder point

The last delivery of one cycle is equal to , and it is assumed that when the

replenishment of the last supplier arrives, the product on-hand inventory level should

exceed the reorder point. It is expressed as:

∑

(2.29)

28

2.6 The optimization model

The resulting multiple-objective mixed integer-programming model is shown

below:

Min

∑ {

∑

{∑

∑ ∑

{

[( ∑ ) ( ∑

)]

[( ∑ ) ( ∑

)]

}

{

[( ∑

)(∑

)]

[ ( )]

}

∑ [ ∑

]

∑

}}

Max ∑ ∑

Min ∑ ∑

Subject to the following constraints:

,

∑ ,

∑ ∑ ( ) ∑

∑

,

∑ ,

∑ ,

0 or 1 for all i and j,

29

for all i and j, integer.

2.7 Goal programming

The optimization model is a multiple-objective problem. There exist some

techniques to solve this kind of problems. Goal programming is one of these techniques

and has four different methods, which is preemptive, non-preemptive, Min-Max and

fuzzy goal programming. In this thesis, non-preemptive goal programming is used.

Non-preemptive goal programming can transform multiple-objective to single

objective problem by setting numerical weights to each objective. After assigning

weights, the non-preemptive goal programming minimizes the deviations from some

target values associated with each objective. In this thesis, the target values are set at 10%

of ideal values. Ideal value for an objective is its optimal values in the multiple-objective

model ignoring other objectives. Since multiple objectives conflict with each other, the

ideal value can never be reached for all objectives simultaneously in a multiple-objective

optimization problem. It should be noted that the objective functions need to be scaled for

proper implementation of non-preemptive goal program. Scaling is achieved through

dividing each objective by its ideal value. The non-preemptive goal programming

formulation is as follows:

Min

Subject to:

(2.30)

30

where , and are the numerical weights assigned for objectives. The variables,

, and are target values for objectives, and and

are negative and positive

deviations from target values for objectives. All other constraints are included when

calculating in programs.

The non-preemptive objective function in equation (2.27) minimizes the weighted

sum of the deviations from the target values specified in the additional constraints

appended above. It should be noted that and

are assigned to and

respectively since both the cost and lead time objectives are to minimize (therefore it is

aimed at minimizing the positive deviations from targets). The deviation , on the other

hand, is assigned to since the quality objective is to maximize (Bilsel 2011).

`

31

Chapter 3

Numerical Example

In this chapter, a numerical model is developed to verify the optimal solution in

chapter 3. Firstly, several parameters are set to simulate the situation. Then, the problem

will be calculated using software Matlab and GAMS and obtain the optimal solution.

Finally, results will be displayed and analyzed.

3.1 Parameter Setting

In this numerical example, a scenario with three products is considered (m=3).

The daily demand for product is shown in table 3.1.1. It is assumed that unit time is one

month (thirty days).

Table 3.1.1. Demand data for product i (in units)

Product

1 2 3

40 60 75

Each product can be supplied by three suppliers (n=3). Table 3.1.2 shows the

capacity data for product i from supplier j, and table 3.1.3 shows the lead time data for

each product from all suppliers.

32

Table 3.1.2 Capacity data for product i from supplier j (in units)

Supplier Product

1 2 3

1 65 80 90

2 90 95 30

3 75 70 130

Table 3.1.3 Lead time data (in days)

Supplier Product

1 2 3

1 5 11 9

2 7 8 12

3 9 10 7

Purchasing cost for product i from supplier j, , fixed cost for supplier j, , and

all the inventory data are shown respectively in tables 3.1.4, 3.1.5 and 3.1.6.

Table 3.1.4 Purchasing cost data (in $)

Supplier Product

1 2 3

1 24 33 26

2 27 28 32

3 23 34 23

33

Table 3.1.5 Inventory data (in $)

Product Supplier M

1

1 120

4 26 1.4 60

4500

2 135

3 145

2

1 130

2 17 3 180 2 165

3 120

3

1 160

0.7 14 2 270 2 110

3 180

Table 3.1.6 Fixed cost data (in $)

Supplier

1 2 3

2000 1500 2500

The product quality and late-delivery product data for all products from all

suppliers are shown in table 3.1.7 as below:

Table 3.1.7. Product quality and proportion of late-delivery products

Product Supplier

1

1 96% 94%

2 99% 92%

3 98% 96%

2

1 98% 97%

2 97% 93%

3 99% 90%

3

1 99% 91%

2 94% 97%

3 96% 89%

34

3.2 Numerical analysis

After setting all the parameters to simulate the model, non-preemptive goal

programming is used to solve this multiple-objective problem. Firstly, decision makers

will assign weights to these three objectives, in this thesis; =0.42, =0.32 and

=0.26. The ideal values and target values are calculated and the results are shown in

table 3.2.1. Achievements are the optimal values for this model.

Table 3.2.1. Ideal values and target values of multi-supplier

Objective Ideal value Target value Achievement

Total cost 5639.23 6203.15 5987.22

Number of good quality products 542.37 493.06 527.11

Late delivery product 649.24 714.16 669.90

The optimal solution of this multiple-supplier selection and multiple-product

problem is shown in table 3.2.2. The optimal solution shows that for product 1, supplier 2

is the only best supplier to fulfill the demand, but for product 2, there are two suppliers,

which are suppliers 1 and 2, and for product 3, suppliers 1 and 3 are the proper suppliers.

Table 3.2.2. Optimal solution of the model

Product Supplier

1

1 0

163 2 427

3 0

2

1 307

127 2 203

3 0

3

1 146

171 2 0

3 508

35

The distribution of suppliers for products 1, 2 and 3 are given in figure 3.2.1.

Figure 3.2.1. Distribution of each supplier for products

Although the optimal solution is obtained by using the formulations developed in

chapter 2, it is important to figure out whether having multiple suppliers for some

products is better than having a single supplier for some products. In order to compare the

results between choosing multiple suppliers and choosing single supplier, equation

∑ =1, which constraints that only one supplier is selected for each product, is added

to the set of constraints. The results in tables 3.2.3 and 3.2.4 show the optimal solution

and distribution of each product with a single supplier.

0 100 200 300 400 500 600 700

product 1

product 2

product 3

supplier 1

supplier 2

supplier 3

36

Table 3.2.3. Ideal values and target values of single supplier

Objective Total cost Product quality Late delivery product

Achievement 6033.78 500.54 690.56

Table 3.2.4. Optimal solution for single supplier

Product Supplier

1

1 0

156 2 458

3 0

2

1 407

143 2 0

3 0

3

1 0

161 2 0

3 486

Figure 3.2.2 (total cost is scaled down by dividing by 10) shows the comparison

of multiple-supplier selection and single supplier selection. From the results, it can be

seen that the total costs of these two methods are almost the same; the difference is less

than 1%. However, for product quality factor, choosing multiple suppliers is 8% better

than single supplier, and for the total number of late delivery products, having multiple

suppliers is 6% better than having a single supplier. In real environment, most companies

have more than three products and choosing multiple suppliers may be advantageous

because of large profits and lower risks.

37

Figure 3.2.2. Comparison between multi-supplier and single supplier

0

100

200

300

400

500

600

700

800

scaled cost quality late delivery

multi-supplier

single supplier

38

Chapter 4

Conclusions and Future Work

In this chapter, conclusions of the model developed in this thesis will be made and

some suggestions will be mentioned for future work.

4.1 Conclusions

In today’s global market, rather than considering supplier selection and inventory

policy separately, linking these two becomes more and more important as correct

strategies may bring large profits and lower management risks in supply chain

management. Many researches have been done in this area. In this thesis, a multiple-

objective model is developed to do research on situations when multiple suppliers exist

for multiple products in order to optimize inventory.

In this multiple-objective model, three objectives are used: (1) minimize the total

cost of the retailer, consisting of fixed cost for choosing specific suppliers, purchasing

cost, and inventory cost; (2) maximize the product quality; and (3) minimize the total

number of late-delivery items. Inventory is the main factor considered in this model.

Continuous (r, Q) policy is used and shortages are allowed. Ordering cost, holding cost

and shortage cost are the three elements of the total inventory cost. In this model,

demands and lead times for products from suppliers are deterministic values. Several

39

assumptions and constraints are also used. Capacities of suppliers are limited and are

different from each other. Each product has limited storage capacity as well, and a

purchasing budget is considered. This model is formulated to figure out the set of

selected suppliers for each product, order quantities and reorder points for all products.

At last, goal programming is implemented to solve the multiple-objective model

in order to obtain optimal solutions. The numerical example in chapter 3 illustrates the

use of the model. The optimal solution shows the approach to determining the order

quality and reorder level. From the comparison of multiple-supplier selection and single

supplier selection, it is obviously that multiple-supplier selection is better, because it

minimizes cost and risks and maximizes product quality. Some suggestions for further

research are shown in the next section.

4.2 Future works

The multiple-objective model in this thesis could be improved in the future in

several aspects:

Firstly, the current model in this thesis is deterministic, which has constant unit

time demand and lead time. In reality, it is almost impossible to guarantee that demand

and lead time stay the same all the time. It may be necessary to consider demand and lead

time as random variables. The lead time of each product from different suppliers may

vary from each other, following Poisson distribution, normal distribution etc. The whole

model can be developed as a stochastic model.

40

Secondly, multiple-period horizon can be considered instead of single-period. The

multiple-period model can offer opportunities to change suppliers for each product from

one period to another. It makes the model much more flexible when there are some

unforeseen disruptions in suppliers some suppliers, and retailers can adjust the model

timely to change suppliers to deliver products and reduce risks.

Finally, several algorithms can also be considered to solve the multiple-objective

model. Goal programming is implemented by setting weights by decision makers to

objectives, which is not always acceptable in reality as it mainly depends on decision

makers’ judgments. In addition to goal programming, genetic algorithm (GA) and fuzzy

algorithm are widely used in solving complex, large, and multiple-objective models. The

model may be further improved by using these algorithms.

41

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