Designing a distribution network in a supply chain system: Formulation and eﬃcient solution procedure

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Production, Manufacturing and Logistics

Designing a distribution network in a supply chain

system: Formulation and eﬃcient solution procedure

Ali Amiri

Department of MSIS, Oklahoma State University, 210 College of Business, Stillwater, OK 74078, USA

Received 19 November 2001; accepted 6 September 2004

Available online 2 November 2004

Abstract

This paper addresses the distribution network design problem in a supply chain system that involves locating pro-

duction plants and distribution warehouses, and determining the best strategy for distributing the product from the

plants to the warehouses and from the warehouses to the customers. The goal is to select the optimum numbers, loca-

tions and capacities of plants and warehouses to open so that all customer demand is satisﬁed at minimum total costs of

the distribution network. Unlike most of past research, our study allows for multiple levels of capacities available to the

warehouses and plants. The paper presents a computational study to investigate the value of coordinating production

and distribution planning. We develop a mixed integer programming model and provide an eﬃcient heuristic solution

procedure for this supply chain system problem.

Keywords: Distribution; Facility planning and design; Supply chain system; Heuristics

1. Introduction

It is quite common nowadays to see manufacturers and retailers like Proctor & Gamble and Wal-Mart

joining eﬀorts to eﬃciently handle the ﬂow of products and to closely coordinate the production and supply

chain system. An important strategic issue related to the design and operation of a physical distribution

network in a supply chain system is the determination of the best sites for intermediate stocking points,

or warehouses. The use of warehouses provides a company with ﬂexibility to respond to changes in the mar-

ketplace and can result in signiﬁcant cost savings due to economies of scale in transportation or shipping

costs. In this paper, we consider the problem of designing a distribution network that involves determining

doi:10.1016/j.ejor.2004.09.018

Tel.: +1 405 744 8649; fax: +1 405 744 5180.

E-mail address: amiri@okstate.edu

European Journal of Operational Research 171 (2006) 567–576

www.elsevier.com/locate/ejor

simultaneously the best sites of both plants and warehouses and the best strategy for distributing the prod-

uct from the plants to the warehouses and from the warehouses to the customers.

A common objective in designing such a distribution network is to determine the least cost system design

such that the demands of all customers are satisﬁed without exceeding the capacities of the warehouses and

plants. This usually involves making tradeoﬀs inherent among the cost components of the system that in-

clude: (1) costs of opening and operating the plants and warehouses, and (2) the inbound and outbound

transportation costs.

Many researchers have extensively studied facility and demand allocation problems. Previous research

studies are well surveyed by Francis et al. [5], Aikens [1], Brandeau and Chiu [4], and Avella et al. [2]. More

recently, Jayaraman [8] studied the capacitated warehouse location problem that involves locating a given

number of warehouses to satisfy customer demands for diﬀerent products. Pirkul and Jayaraman [9] extended

the previous problem by considering locating also a given number of plants. They formulated the problem as

a mixed integer model and developed a Lagrangean based heuristic solution procedure. The procedure was

tested using problem instances with up to 100 customers, 20 potential warehouses and 10 potential plants.

Tragantalerngsak et al. [11] considered a two-echelon facility location problem in which the facilities in

the ﬁrst echelon are uncapacitated and the facilities in the second echelon are capacitated. The goal is to

determine the number and locations of facilities in both echelons in order to satisfy customer demand of

the product. They developed a Lagrangean relaxation based branch and bound algorithm to solve the prob-

lem and reported results of computational tests with up to 100 customers and 15 facilities. Gourdin et al. [7]

studied a particular type of the uncapacitated facility location problem where two customers allocated to

the same facility are matched. They developed several methods to solve the problem after deriving valid

inequalities, and optimality cuts for the problem.

One major drawback in most of past research studies like [7–9,11] is that they limit the number of capac-

ity levels available to each facility to just one. However, as it is the case in practice, there exist usually sev-

eral capacity levels to choose from for each facility. The use of diﬀerent capacity levels makes the problem

more realistic and, at the same time, more complex to solve. Another major drawback in some previous

studies like [8,9] is that they limit the number of facilities to open to a pre-speciﬁed value. Moreover, these

studies fail to describe how this value can be determined in advance.

Our current study represents a signiﬁcant improvement over past research by presenting a uniﬁed model

of the problem that includes the numbers, locations, and capacities of both warehouses and plants as var-

iables to be determined in the model and develops at the same time the best strategy for distributing the

product from the plants to the warehouses and from the warehouses to the customers. An eﬃcient heuristic

solution procedure based on Lagrangean relaxation of the problem is developed and extensive computa-

tional tests with up to 500 customers, 30 potential warehouses, and 20 potential plants are reported.

The remainder of this paper is organized as follows. In Section 2, a mathematical formulation of the dis-

tribution design problem is presented. A Lagrangean relaxation of the problem is proposed in Section 3. An

eﬃcient solution algorithm based on this relaxation is developed in Section 4. Computational results are

reported in Section 5. A summary of the work presented in this paper is given in Section 6.

2. Model formulation

The following notation is used in the formulation of the model.

Nindex set of customers/customer zones

Mindex set of potential warehouse sites

Lindex set of potential plant sites

Rindex set of capacity levels available to the potential warehouses

568 A. Amiri / European Journal of Operational Research 171 (2006) 567–576

Hindex set of capacity levels available to the potential plants

cost of supplying one unit of demand to customer zone ifrom warehouse at site j

Cjk cost of supplying one unit of demand to warehouse at site jfrom plant at site k

jﬁxed cost per unit of time for opening and operating warehouse with capacity level rat site j

kﬁxed cost per unit of time for opening and operating plant with capacity level hat site k

demand per unit of time of customer zone i

jcapacity with level rfor the potential warehouse at site j

kcapacity with level hfor the potential plant at site k

The decision variables are:

= fraction (regarding a

) of demand of customer zone idelivered from warehouse at site j

jk = fraction (regarding br

j) of shipment from plant at site kto warehouse at site jwith capacity level r

j=1 if a warehouse with capacity level ris located at site j

0 otherwise



k=1 if a plant with capacity level his located at site k

0 otherwise



In terms of the above notation, the problem can be formulated as follows.

Problem DistriNet:

Zp ¼Min X

i2NX

j2M

CijaiXij þX

r2RX

j2MX

k2L

Cjk br

jYr

þX

j2MX

r2R

jUr

jþX

k2LX

h2H

kVh

kð1Þ

subject to

j2M

Xij ¼18i2N;ð2Þ

i2N

aiXij 6X

r2R

jUr

j8j2M;ð3Þ

r2R

j618j2M;ð4Þ

i2N

aiXij 6X

k2LX

r2R

jYr

jk 8j2M;ð5Þ

j2MX

r2R

jYr

jk 6X

h2H

kVh

k8k2L;ð6Þ

h2H

k618k2L;ð7Þ

Xij P08i2Nand j2M;ð8Þ

j2ð0;1Þ8j2Mand r2R;ð9Þ

jk P08k2L;j2Mand r2R;ð10Þ

k2ð0;1Þ8k2Land h2H:ð11Þ

The model minimizes total costs made of: the costs to serve the demands of customers from the ware-

houses, the costs of shipments from the plants to the warehouses, and the costs associated with opening

and operating the warehouses and the plants. Constraint set (2) ensures that the demands of all customers

A. Amiri / European Journal of Operational Research 171 (2006) 567–576 569