Heuristic solutions to multi-depot location-routing problems.pdf

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*Corresponding author. Tel.:#886-4-8528469 x 2230; fax: #886-4-8520781.

E-mail address: [email protected] (T.-H. Wu).

Computers & Operations Research 29 (2002) 1393}1415

Heuristic solutions to multi-depot location-routing problems

Tai-Hsi Wu*, Chinyao Low, Jiunn-Wei Bai

Department of Industrial Engineering, Da-Yeh University, Changhua, Taiwan

Received 1 November 1999; received in revised form 2 February 2001

Abstract

This paper presents a method for solving the multi-depot location-routing problem (MDLRP). Since

several unrealistic assumptions, such as homogeneous#eet type and unlimited number of available vehicles,

are typically made concerning this problem, a mathematical formulation is given in which these assumptions

are relaxed. Since the inherent complexity of the LRP problem makes it impossible to solve the problem on

a larger scale, the original problem is divided into two sub-problems, i.e., the location-allocation problem,

and the general vehicle routing problem, respectively. Each sub-problem is then solved in a sequential and

iterative manner by the simulated annealing algorithm embedded in the general framework for the problem-

solving procedure. Test problems from the literature and newly created problems are used to test the

proposed method. The results indicate that this method performs well in terms of the solution quality and

run time consumed. In addition, the setting of parameters throughout the solution procedure for obtainingquick and favorable solutions is also suggested.

Scope and purpose

In many logistic environments managers must make decisions such as location for distribution centers

(DC), allocation of customers to each service area, and transportation plans connecting customers. The

location-routing problems (LRPs) are, hence, de"ned to"nd the optimal number and locations of the DCs,

simultaneously with the vehicle schedules and distribution routes so as to minimize the total system costs.

This paper proposes a decomposition-based method for solving the LRP with multiple depots, multiple #eet

types, and limited number of vehicles for each di!erent vehicle type. The solution procedure developed is very

straightforward conceptually, and the results obtained are comparable with other heuristic methods. Inaddition, the setting of parameters throughout the solution procedure for obtaining quick and favorable

solutions is also suggested. 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Location-routing; Simulated annealing; Location-allocation; Vehicle routing

0305-0548/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.

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1. Introduction

In many logistic environments managers must make decisions such as (1) location for facto-ries/warehouses/distribution centers (DC), referred to as depots, (2) allocation of customers to each

service area, and (3) transportation plans connecting customers, raw materials, plants, warehouses,and channel members. These decisions are important in the sense that they greatly a!ect the level ofservice for customers and the total logistic system cost. For determining the depot locations, many

mathematical models have been developed to solve the problems. However, in these models thetransportation costs between depot and customers were incorrectly assumed and performed ona straight-and-back basis (the moment sum equation [1]). The fact that several customers can be

served on a single route, provided that the total demand does not exceed the tour capacity, revealsthe necessity for unifying the location and routing problems. The interdependence of these two

problems was not recognized until the 1970s [2]. The location-routing problems (LRPs) can, hence,

be de"ned as vehicle routing problems (VRPs) in which the optimal number and locations of thedepots are to be determined simultaneously with the vehicle schedules and distribution routes so asto minimize the total system costs. Recent surveys of LRPs can be found in Laporte [3]. Research

on LRPs is quite limited compared with the extensive literature on pure location problems, VRPsand their variants. Since VRPs have long been recognized as NP-hard problems due to the

embedded traveling salesman problem, not to mention the di$culty of the more complicatedLRPs, it was considered impractical to incorporate the VRPs into the location problems until thelate 1970s. In addition to the problem complexity, some sensitivity analyses such as the changes invehicle capacity, depot capacity, and number of potential depot locations, bring valuable informa-

tion to decision makers and, thus, require repetitive problem solving. It is, hence, not surprisingthat approximation methods have been much more widely used than exact methods for solving the

LRPs. Several studies, for instance, Golden et al. [4], Or and Pierskalla [5], Jacobsen and Madsen[6], Srikar and Srivastava [7], Perl [8], Perl and Daskin [2], have proposed formulations andalgorithms for the general LRPs and LPRs under some side constraints such as capacity limit andmaximum cost/tour-length restriction. In terms of the procedure for solving the LRP, it can be

viewed as a compilation of the following three sub-problems: (1) facility location (2) demandallocation and (3) vehicle routing. Separate optimization of these sub-problems always leads to

a non-optimal decision. An incorporation of all the sub-problems together, however, is computa-tionally impractical. For solving the LRP more e$ciently, two types of sequential methods arecommonly used: (1) location-allocation-routing (LAR), for example, [5,6] and (2) allocation-

routing-location (ARL), [6,7]. Other than heuristic algorithms, Laporte et al. made a series of

signi"cant contributions in the presentation of exact methods [3,9}12]. In Laporte [11] a methodwas presented which transforms the multi-depot VRP (MDVRP) and LRP into an equivalent

constrained assignment problem through the use of graphical representation. The problem wasthen solved by the branch-and-bound method. The LRP with 80 nodes was solvable withina reasonable amount of time. In Laporte et al. [12] a stochastic LRP model was introduced in

which the demands are known only when vehicles reach the customers. In the "rst stage LRPdecisions are still made without the actual demand information. It is not surprising that therestriction of route capacities might be violated. If that happens, corrective recourse action is taken

accompanied by a penalty in the second stage. The problem is formulated and solved to optimality,but only solutions for problems with up to 30 nodes are provided. Due to the inherent

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characteristics of the optimization approach, the number of depots to be established and the

number of customer nodes in the exact models must be very restricted. Thus these exact formula-tions can be applied to only very small-sized problems. Perl and Daskin [2] "rst formulated thewarehouse location-routing problem (WLRP). After making some modi"cations to the WLRP,

they presented a solution method which solved the MWLRP (modi"ed WLRP) in a sequentialmanner. Hansen et al. [13] further proposed a more e$cient method for the MWLRP. Srivastavaand Benton [14] investigated several environmental factors that might in#uence the design of

a distribution system. Chien [15] proposed an approximate approach for the LRP, in which tworoute length estimators are used in calculating the routing cost. Salhi and Fraser [16] proposed aniterative method that alternates between the location phase and routing phase until a suitable

stopping criterion is met. In that study a more practical version of the LRP is addressed, where thevehicles may have di!erent capacities. Nagy and Salhi [17,18] adopted the concept of nested

methods to treat the routing element as a sub-problem within the larger problem of location when

solving the LRP. Min et al. [19] synthesized the past research and suggested some future researchdirections for the LRP.

Though the close relationship between location problems and the VRP and the correct way for

calculating depot}customer distance have been recognized and presented by both academics andpractitioners [2], research in the past seems to have ignored the fact that most companies own

delivery #eets of di!erent capacities. To our best knowledge, the study by Salhi and Fraser [16]seems to be the only one in which the vehicles involved in the LRP do not necessarily have thesame capacities. But as typically assumed in the literature, the number of vehicles for eachtype is unlimited in their study, an assumption that does not agree with the real situation. In

this paper, we address an extended and more practical version of the LRP, i.e., an LRP consideringmultiple depots, multiple #eet types, and limited number of vehicles for each di!erent vehicle

type. This setting allows one to re#ect reality more accurately and, in addition, analyze the e!ectof#eet type and size. The solution of problems under the assumption of no restrictions on #eettype and size can be obtained by setting the #eet type to one, and the #eet size to a very largenumber.

This paper proposes a heuristic method, which decomposes the LRP into a location-allocationproblem (LAP) and a vehicle routing problem (VRP). Both LAPs and VRPs are di $cult to solve.

Hence, heuristic methods for the LAP and VRP, respectively, were developed. Since simulatedannealing (SA) has been applied to a number of combinatorial problems with fairly good results[20], it was selected as the basis for developing search methods for both the LAP and VRP. SA can

be viewed as a process which attempts to move from the current solution to its neighborhood

solutions resulting in better objective values. However, for solutions with worse objective values,they are accepted with a speci"ed probability mainly to escape from the local optima in its search

for the global optima.This paper is organized as follows. Section 2 de"nes the multi-depot location-routing problem

and gives a corresponding mathematical formulation. Section 3 presents a heuristic model in which

the LRP model is decomposed into three sub-problems. The heuristic solution for each sub-problem is also provided in Section 3. Computational results of test problems from Perl [8] andsimulated problems are presented and discussed in Section 4. Some managerial issues regarding

integrating both location and routing problems are discussed in Section 5. Section 6 summarizesthe conclusions of this study.

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2. Multi-depot location-routing problem

The multi-depot location-routing problem is "rst de"ned. In a logistic system assume that thenumber, location, and demand of customers, the number, and location of all potential depots, as

well as the #eet type and size are given. The distribution and routing plan must be designed so that:

(1) the demand of each customer can be satis"ed,

(2) each customer is served by exactly one vehicle,(3) the total demand on each route is less than or equal to the capacity of the vehicle assigned to

that route, and

(4) each route begins and ends at the same depot.

The problem is to simultaneously determine the number, locations of depots, assignment of

customers to depots, vehicle types to routes, and the corresponding delivery routes, so that the totalcosts consisting of depot-establishing cost, transportation cost, and dispatching cost for vehiclesare minimized. The proposed mathematical programming formulation below di!ers slightly from

Perl's MWLRP model [2] on three points:

1. Another set of sub-tour elimination constraints resulting in a much lower number of constraintsis introduced;

2. The variable warehousing cost is not considered in the objective function while adding thedispatching cost for vehicles assigned;

3. In addition to the homogeneous#eet type, the#exibility of allowing the #eet to have di!erentcapacities in the problems is added.

For analyzing the close relationship between location and routing problems in the LRP, weconsider only the depot establishing cost, transportation, and vehicles' dispatching costs in theobjective function of the proposed model. Other costs such as the variable warehousing cost

appearing in Perl's formulation, proportional to the quantity of throughput at each depot, areignored in this study.

For reducing computational complexity, Perl and Daskin [2] assumed that the delivery #eetconsists of standard vehicles and the #eet size is unlimited. It is, thus, unnecessary to consider the"xed cost of the vehicles in their model. However, the addition of the vehicles ' "xed cost is

important because it provides the capability for exploring the e!ect of a limited#eet size. The sets,

parameters and variables used in the mathematical model are de"ned below:

Sets:

I set of all potential DC sitesJ set of all customers

K set of all vehiclesParameters:

N number of customers

C

distance between points i and j. i, j3IJ

G

"xed costs of establishing depot i

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F

"xed costs of using vehicle k

Heuristic solutions to multi-depot location-routing problems.pdf

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