# Designing an efficient method for tandem AGV network design problem using tabu search

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3000 chemin de la Cote-Sainte-Catherine, Montreal, Canada H3T 2A7b Department of Industrial Engineering, Amir Kabir University of Technology, Tehran, Iran

c Supply Chain Management Research Group, Tehran, Iran

Keywords: AGV; Tandem conguration; Tabu search

not new. A number of algorithms for AGV guide-path design have been developed over the past 20 years[2]. The AGV guide-path congurations discussed in previous research include Conventional/Traditional

* Corresponding author.E-mail address: farahan@aut.ac.ir (R.Z. Farahani).

Applied Mathematics and Computation 183 (2006) 14101421

www.elsevier.com/locate/amc0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved.1. Introduction

The design of handling systems is one of the most important decisions in facility design activities. Materialhandling operations cover nearly 2050% of the overall operational costs [40]. An automated guided vehicle(AGV) is a driverless vehicle used for the transportation of goods and materials within a production plantpartitioned into cells (or departments), usually by following a wire guide-path. One of the most importantissues in designing AGV systems is the guide-path design. The problem of guide-path design for AGVs isAbstract

A tandem AGV conguration connects all cells of a manufacturing area by means of non-overlapping, single-vehicleclosed loops. Each loop has at least one additional P/D station, provided as interface between adjacent loops. This studydescribes the development of a tabu search algorithm for the design of tandem AGV systems. Starting from an initial par-tition generated by a k-means clustering method, the tabu search algorithm partitions the stations into loops by minimizingthe maximum workload of the system, without allowing the paths of loops to cross each other. The new algorithm and thepartitioning algorithm presented by Bozer and Srinivasan are compared on, randomly generated problems. Results showthat in large scale problems, the partitioning algorithm often leads to infeasible congurations with crossed loops in spiteof its shorter running time. However the newly developed algorithm avoids infeasible congurations and often yields betterobjective function values. 2006 Elsevier Inc. All rights reserved.Designing an ecient method for tandem AGV networkdesign problem using tabu search

Gilbert Laporte a, Reza Zanjirani Farahani b,*, Elnaz Miandoabchi b,c

a Canada Research Chair in Distribution Management and GERAD, HEC Montreal,doi:10.1016/j.amc.2006.05.149

G. Laporte et al. / Applied Mathematics and Computation 183 (2006) 14101421 1411[17,18,22,41,33,25,32,38,15,14,21,24], Tandem [8,9,27], Single loop [39,34,5,26,4,3,13,12], bi-directional shortestpath [23,11,28] and segmented ow [3537,6].

The tandem conguration, which is the concern of this paper, was introduced by Bozer and Srinivasan [7,8]and is based on the divide-and-conquer principle. A tandem conguration is obtained by partitioning all theworkstations into single-vehicle, non-overlapping zones. Additional pick-up/delivery (P/D) points are pro-vided between adjacent zones to serve as transfer points. This conguration oers some advantages such asthe elimination of blocking and congestion, simplicity of control, and exibility due to system modularity.It also has some disadvantages including the need for handling a load by two or more vehicles and thus longerload movement times, extra oor space and cost requirements, resulting from the use of additional P/D pointsand conveyors.

There are relatively few papers published on tandem AGV systems compared with the number of publica-tions on other systems. These studies have focused on the design of routes, routing and control of AGVs, per-formance comparison with conventional systems and modeling by Petri nets. Our literature review mostlycovers route design.

Tandem paths initially were initially proposed by Bozer and Srinivasan [7,8] who presented an analyticalmodel to compute the workload of a single loop. They developed a heuristic method for the partitioningthe stations in loops [9]. Hsieh and Sha [19] proposed a design process for the concurrent design ofmachine layout and tandem routes. Aarab et al. [1] used hierarchical clustering and tabu search to determinetandem routes in a block layout. Yu and Egbelu [43] presented a heuristic partitioning algorithm for atandem AGV system, based on the concept of variable path routing. Later Bozer and Lee [10] consideredeliminating the conveyers by using an existing station as a transfer point. Ventura and Lee [42] studied tan-dem congurations with the possibility of using more than one AGV in the loops. Huang [20] proposed anew design concept of tandem AGV based on using of a transportation center to link the transfer points inloops.

To test the viability of tandem congurations, Farling et al. [16] have performed a simulation study to eval-uate the impact of system size, machine failure rate and unload/load time on the performance of three AGVcongurations, namely traditional (parallel unidirectional ows), the tandem ow-path and the tandem loop.In a tandem loop ow-path there exists an express loop which connects each loop see [30]. The authors con-clude that traditional layouts perform better in small systems, and tandem loop congurations perform betterin large systems.

As can be seen, most previous studies have proposed heuristic algorithms for the design of tandem routes,and only in one of the papers [1] tabu search (TS) has only been used as a subroutine in designing tandemcongurations. Thus, the application of this metaheuristic as an overall designing process in tandem systemshas not yet been considered.

The aim of this paper is to develop a TS algorithm for the design of routes in a tandem conguration. Givena grid layout, a route sheet and the number of designed loops as inputs, the problem is to assign workstationsto loops without allowing any overlap among loops, the maximum workload of the system is minimized. Theproposed algorithm will be compared to the partitioning heuristic algorithm of Bozer and Srinivasan [9],referred to as the base algorithm, using randomly generated test problems. This paper is organized as follows.A brief description of the problem and its assumptions are presented in Section 1. The TS algorithm and itscharacteristics are discussed in Section 3, and Section 4 reports computational results. Conclusions and guide-lines to future work are presented in Section 5.

2. Problem denition

Tandem AGV systems were rst introduced by Bozer and Srinivasan [7,8]. Although the tandem congu-ration can be used both in warehousing and manufacturing, it was mainly dened and developed for the lattercase. The system proposed by Bozer and Srinivasan [9] is dened on a grid layout (Fig. 1). Each workstation ispresented as a single point and may represent a machine, or a group of machines, such as a cell or adepartment.

The problem of conguring a tandem AGV system consists of partitioning a set of N workstations into

several independent single AGV loops (zones). Additional P/D stations called transfer points are introduced

1412 G. Laporte et al. / Applied Mathematics and Computation 183 (2006) 14101421Fig. 1. A typical grid layout ([9]).to provide an interface between adjacent loops. Transfer points are connected to each other by conveyors.Fig. 2 illustrates a typical tandem conguration.

The aim is to partition the workstations of a layout in such a way that each station is assigned to only oneloop, the workload of the AGVs associated with the material ow within and between the loops does notexceed the AGV capacity, and the workload is evenly distributed among all loops. The workload factor ofeach AGV, denoted by x, is the proportion of time a vehicle is busy, either loaded or empty. This is the build-ing block of the system and must be calculated for each loop.

The assumptions made by Bozer and Srinivasan [9] are as follows: there are two types of workstations, therst type is input/output station and the second type is process station where actual processing takes place.Transfer points also are considered as I/O stations. Every station has an I/O queue. A bidirectional single loadAGV is used in each loop. When loaded, the AGV follows the shortest path to the destination station, andwhen empty it uses the FEFS (rst encountered rst served) empty dispatching rule, which will never leavethe AGV idle. Additional limitations are as follows. Intersections and overlaps are forbidden between loops;the number of loops must be at least 2; the number of loops can be determined as an input, or can be obtainedthrough the designing process. Here number of loops is given at the beginning of the algorithm.

3. Tabu search algorithm

In this section, the developed algorithm based on TS is described.

Fig. 2. A typical tandem conguration ([9]).

3.1. F

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loop.

G. Laporte et al. / Applied Mathematics and Computation 183 (2006) 14101421 14133.2. Neighborhood structure

The neighborhood of a solution is simply obtained by removing a station from one loop and adding itto another one, provided that it does not create intersection and the workload of loops does not exceed 1.A current solution corresponding to a partition of the set of workstations in L loops can be represented asfollows:

S fP 1; . . . ; P i; . . . ; PLg; i 1; . . . ; L: 1Consider the station s from the set of stations in loop Pi, where st 2 Pi. Also consider solution S 0 in the

neighborhood of solution S:

S0 P 01; . . . ; P 0i; . . . P 0L

; i 1; . . . ; L: 2Solution S 0 is obtained by moving station s from loop Pi to loop Pj in solution S. In other words,

P 0i P i fsg; 3The fworkby ap

AFig. 4

3.3. O

Thprogrload

ThworkFig. 3 shows a typical infeasibility in the presence of overlapping loops.The algorithm proposed by Bozer and Srinivasan [9], does not have a specic mechanism to check the over-laps among the loops. The TS algorithm checks the intersection of loops for every move. For the sake of sim-plicity, the initial routes of loops are considered as the Euclidean traveling salesman route of the stations in theto one loop, and (3) the workloads are less than 1 (or a reasonable value less than 1).

A solution is called feasible if (1) no overlap or intersection exists among loops, (2) each station is assignedeasibility conditions2

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Fig. 3. A typical infeasible solution.P 0j Pj [ fsg; j 6 i: 4easible move mij is characterized by the transmission of station s from loop i to loop j, subject to theload constraint. The neighborhood of S is the set of all feasible solutions that can be reached from Splying moves mij.typical neighborhood structure is shown in Fig. 4. Fig. 4(a) shows the possible moves for station 11, and(b) shows the new solutions resulting from possible moves of station 11 to 3 loops.

bjective function and evaluation of neighborhood

e evaluation method of neighborhood structure is based on the objective function of the integer linearamming model presented by Bozer and Srinivasan [9]. The objective is to minimize the maximum work-of all loops.us the basis of the search procedure is to select a loop with maximum workload and trying to reduce itsload. The workload reduction is obtained by moving one of the stations of the selected loop to another

1414 G. Laporte et al. / Applied Mathematics and Computation 183 (2006) 14101421loop, which causes an increase in the workload of the second loop. The best move is one causing the largestdecrease in the workload of the selected loop and the smallest increase in the workload of the second loop. Soa simple phrase is used as the evaluation criterion: workload reduction of the selected loop, minus workloadincrease of the second loop.

3.4. Number of loops

In the heuristic algorithm of Bozer and Srinivasan [9], the number of loops is given as an input. This alsoapplies to the TS algorithm.

3.5. Generation of feasible solutions

The procedure of generating initial solutions consists of three phases:

(a) Clustering the stations into L groups by means of k-means clustering method.(b) If necessary, removing workload infeasibilities.(c) Reducing the number of singleton stations.

In the rst stage, a k-means clustering method is applied ([31]). Applying the k-means clustering method togenerate the initial partition, ensures that no intersections will occur among the created loops, since distance isused as the closeness measure. On the other hand, the workstations in a loop should be reasonably close toeach other, so that unnecessary vehicle trips are avoided.

The initial cluster centers are chosen randomly and the rectilinear distance is used as the closeness measure,due to the rectilinear shape of the nal routes of the loops. The resulting clusters are then checked for work-load feasibility. In case of infeasibility, a simple search method is used to reduce the workload of infeasibleloops by moving some stations, following the dened neighborhood evaluation method and based on theobjective function. The search method will stop as soon as all infeasible workloads are reduced to less than

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Fig. 4. A typical neighborhood structure.1. In the last phase, the conguration is checked for the presence of singleton loops and if more than oneis present, another algorithm is employed to reduce the number of singletons by adding a station from adja-cent loops to them.

3.6. Denition and selection of moves

According to the neighborhood denition, a move consists of moving a station from one loop to anotherone. Based on minimax objective function, the evaluation of possible moves in each of the iterations isrestricted to the loop with maximum workload.

Not all loops must be considered when looking for possible moves since in most cases applying the movewill lead to overlaps. As a rule of thumb, usually just a few adjacent loops are worth checking. In order toprevent unnecessary calculations and to improve eciency, we only consider the closest adjacent six or sevenloops, according to the distance of their geometrical centers to the geometrical center of the selected loop. It

tabu move leads to a solution better than the best solution found so far, its tabu status is revoked.

probable local minima. In our implementation, when no feasible move exists, the tabu list is emptied and the

changed. In our case this value was set to 20.

G. Laporte et al. / Applied Mathematics and Computation 183 (2006) 14101421 14153.11. Controlling singleton stations

A control is imposed in order to limit the number of singleton stations in the conguration. This is done intwo stages: in the generation of initial solution and in the main body of the algorithm. After generating aninitial feasible solution, the number of singleton stations is checked and if there is more than one, another su...