facilities delocation

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Optimal shrinking of the distribution chain: the facilities delocationdecisionPradip K. Bhaumik a

a International Management Institute, New Delhi-110 016, India

Online publication date: 17 February 2010

To cite this Article Bhaumik, Pradip K.(2010) 'Optimal shrinking of the distribution chain: the facilities delocationdecision', International Journal of Systems Science, 41: 3, 271 — 280To link to this Article: DOI: 10.1080/00207720903326860URL: http://dx.doi.org/10.1080/00207720903326860

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International Journal of Systems ScienceVol. 41, No. 3, March 2010, 271–280

Optimal shrinking of the distribution chain: the facilities delocation decision

Pradip K. Bhaumik*

International Management Institute, B-10, Qutab Institutional Area, New Delhi – 110 016, India

(Received 10 July 2008; final version received 27 May 2009)

Closure of facilities is quite common among both business firms and public sector institutions like hospitals andschools. Although the facilities location problem has been studied extensively in the literature, not much attentionhas been paid to the closure of facilities. Unlike the location problem, the existing facilities and the correspondingnetwork impose additional constraints on the closure or elimination of facilities and to highlight the differencebetween the two, we have called this the facilities delocation problem. In this article, we study a firm with anexisting distribution network with known retailer and distributor locations that needs to downsize or shrink itsdistribution chain due to other business reasons. However, it is not a reallocation of demand nodes among theretained distributors. An important condition stipulates that all demand nodes must continue to get their suppliesfrom their respective current distributors except when the current source itself is delocated, and only suchuprooted demand nodes will be supplied by a different but one of the retained suppliers. We first describe thedelocation problem and discuss its characteristics. We formulate the delocation problem as an integer linearprogramming problem and demonstrate its formulation and solution on a small problem. Finally, we discuss thesolution and its implications for the distribution network.

Keywords: facilities location; distribution chain; delocation; fixed-charge problem; ILP

1. Introduction

In many sectors of the economy, favourable location offacilities is critical to the business success of the firmitself. Facility location is concerned with the placementof one or more facilities in a way that optimises certainobjectives such as minimising costs, maximising theservice level/providing equitable services to customers,maximising market share captured, minimising thetime taken to deliver emergency services or a combi-nation of these. The location decision for a new facilityis very important for most organisations – manufac-turing and service – in the private as well as the publicsectors. In fact, for many service facilities like bankbranches and retail stores, the location decision alsoaffects the demand for the service and its market shareunlike manufacturing facilities for whom the cost islargely affected.

The number and location of distributors is ofparamount importance in the design of a distributionnetwork. The distributor becomes the single point ofcontact for each of the many retailers and the closerthe distributor, the faster the response time andconsequently the higher the service level of the distri-bution network. In fact, quite often some measure ofthe distance between retailers and their respectivedistributors, e.g. mean or maximum, is used as ameasure of the service level. The distributor may

provide many services including warehousing, order-

filling, information-processing, billing and collection.

An additional distributor may increase the fixed cost of

operating the distributorship but decrease the cost of

transportation and improve the service level. The fixed-

charge problem attempts to minimise the total cost

including the fixed cost and the variable cost of

transportation. As the flow of products through the

distribution network increases, more distributors

appear in the optimum solution and the minimum

cost solution also has a higher service level. Although

this movement towards a greater number of distribu-

tors holds true in most cases, in practice the actual

distribution network is not always an optimum one

due to historical legacy, strategic and other practical

considerations. However, it is easy to see that if

product volumes fall then a reverse movement towards

fewer distributors may have to be set in motion.Globalisation has exposed firms to heightened

business risks from enhanced competition.

Consequently, the business environment of many

companies has undergone unimaginable changes

engendering new problems requiring new methodolo-

gies and new solutions. Often firms need to downsize

or shrink their distribution chain for extraneous

reasons anticipated to continue over the medium or

the long term. This could be due to a host of factors

*Email: [email protected]

ISSN 0020–7721 print/ISSN 1464–5319 online

� 2010 Taylor & Francis

DOI: 10.1080/00207720903326860

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related to competitive, technological, environmental orlifestyle changes, and as the demand for its products orservices reduce, the firm may respond by shrinking itsdistribution network along with other responses. Forexample, Coca-Cola India faced a similar situation afew years ago when the Delhi-based Centre for Scienceand Environment, a green think tank, tested its softdrinks and reported that all contained unsafe levels ofpesticide in them. Lifestyle changes among consumersin many countries brought about by the current globalrecession may require similar response from manyfirms. Although closure of facilities is quite common,surprisingly, not much attention has been paid to thisproblem in the literature and in the absence of anyformal approach to guide them, the decision maker isforced to use methods that are mostly ad hoc andintuitive in nature.

In this article, we focus on a firm that wishes todownsize its distribution network. As the flow ofproducts in the distribution network reduces, thenumber of distributors needed may have to be reduced.Given an existing distribution network with knownretailer and distributor locations, it becomes veryimportant to decide which distributors to eliminate asthe decision affects both the cost and the service levelof the modified distribution chain. It should beunderstood that while in the growth phase thecompany has to decide where to locate the distributors(the location problem), in the downsizing phase it hasto decide which distributors to eliminate or delocateand to highlight the difference, we have called this thedelocation problem.

After the short introduction, a brief review ofdelocation problems discussed in the literature ispresented in Section 2. The delocation problem isthen described and formulated as a mathematicalprogramming problem in Section 3. We next presenta small application and discuss the results in Section 4.Finally, conclusions are drawn in Section 5.

2. The delocation problem

Many variations of the location problem have beenstudied in the literature. An excellent review of locationproblems – their formulation, solution procedures andapplications is presented by Brandeau and Chiu (1989).They also refer to some earlier review articles publishedin the literature. Two books – one by Daskin (1995)and the other by Drezner (1995) also provide anoverview of different location problems and a detailedbibliography on the subject. A recent volume edited byDrezner and Hamacher (2001) provides current devel-opment in this area, while Smith, Laporte, and Harper(2009) charts the progress of locational analysis from

its earliest roots to present maturity. An extensiveliterature review of facility location models in thecontext of supply chain management is given in Melo,Nickel, and Saldanha-da-Gama (2009).

Different authors have formulated and suggestedsolution procedures for the problem of locatingfacility(ies) on a plane, on a network or on discretepoints (Liao and Guo 2008; Meyerson, Munagala, andPlotkin 2008; Nadirler and Karasakal 2008; Batanovic,Petrovic, and Petrovic 2009; Bischoff, Fleischmann,and Klamroth 2009; Boccia, Sforza, and Sterle 2009;Marin, Nickel, Puerto, and Velten 2009; Meng, Huang,and Cheu 2009; Puerto, Ricca and Scozzari 2009). Inplanar location problems, demand occurs and newfacilities can be located anywhere on a plane. Innetwork location models, demand occurs and newfacilities can be located only on the nodes or links ofthe network. In both of these, the distance between anytwo points can be expressed as a function of thecoordinates of the two points. Discrete location modelsallow for the use of arbitrary distances or costsbetween nodes and as such, the structure of theunderlying network is lost.

Although the location problem has been studied indetail even from different disciplinary areas of geo-graphy, operations research, economics, managementand mathematics, it is surprising to note the absenceof attention paid to the delocation problem in theliterature. In economics and in economic geography,the term delocation is often used to mean the first halfof relocation, i.e. closing a facility in its existinglocation so as to start operations from a new location(e.g. Midelfart-Knarvik and Overman 2002; Baldwin,Forslid, Martin, Ottavino, and Robert-Nicoud 2003).However, we use the term as an antonym of location;while in a location decision we decide where tocommence operations from a new facility, a delocationdecision involves deciding which of the existingfacilities should cease operations.

As shown by ReVelle, Murray, and Serra (2007),closure of facilities is quite common among businessfirms as well as health and educational institutionsin the public sector. Modelling the school systemconsolidation under declining enrolments, Bruno andAnderson (1982) developed an early model providingfor the closure of some schools. Diamond and Wright(1987) and Church and Murray (1993) extended thesemodels with the concepts of balancing utilisation.Murray and Wu (2003) modelled urban publictransport networks where some stops would beeliminated, whereas some others would be relocateddue to the changes in population density and networkchanges.

Klincewicz, Luss, and Yu (1988) developed a large-scale multilocation capacity planning model providing

272 P.K. Bhaumik


for plant closures in some periods in a multiperiod

setting. In a similar multiperiod framework,

Melachrinoudis and Min (2000) formulated a multiple

objective model involving the dynamic relocation and

phase-out of a hybrid, two-echelon plant/warehousingfacility. A budget-constrained location problem was

formulated by Wang, Batta, Badhury, and Rump

(2003) that simultaneously considered opening some

new branches and closing some existing branches in thebranch network of a commercial bank. Almost all of

these papers consider relocation problems where some

existing facilities would be closed while some new

facilities could be opened at other locations to accom-

modate demand changes and other factors.The only paper that focuses on the facility closure

was published as late as 2007 and models firms both

with and without competition (ReVelle et al. 2007).

For the firm facing competition, they model a market

served by the firm as well as by the competition. Theyassume that demand from each demand node will be

fully supplied from the nearest supplier – either

belonging to the firm or to its competitor and the

formulation minimises the demand lost to competition.In the second model for a firm without competition,

for example in the public sector providing healthcare

or education services, they minimise the number of

people made worse-off when some of the facilities are

closed to reduce costs.

3. Problem formulation

We consider the general case of a firm serving a spatial

market that is represented by discrete points in a

connected network. Each node in the network repre-sents a demand area, local market or a retailer

location. Customers get their respective demand met

by the retailers. Some of the nodes in the network also

represent the current location of servers or distributorsthat service the demand nodes or retailers. The

network under consideration has n demand nodes

currently served by m servers. In the changed scenario

as the demand for its products has reduced (although

not uniformly across all demand nodes), the firmwants to reduce the number of servers to r (r5m) by

delocating or eliminating [(m� r) ¼ p] distributors. We

assume that this reduction will not affect the demand

at any of the n demand nodes. However, the closure ofsome distributors is expected to reduce both the cost

and the service level. We further assume that the cost

effect can be measured through a fixed cost of

operating the distributorships and a variable cost of

transportation that varies with the product flow toeach retailer.

3.1. A simple fixed-charge formulation of thefacilities closure problem

The problem appears to have all the characteristics of a

simple fixed-charge problem and amenable to such a

formulation. It can then be formulated as an integer

linear programming (ILP) after defining the para-

meters and the decision variables as shown below.

Parameters:

i¼ index of servers, distributors or supply

nodes, in the existing distribution networkm¼ total number of existing supply nodes or

distributorsr¼ number of supply nodes to be retained

(r5m)j¼ index of demand nodes or retailers in both

the existing and the new set upn¼ total number of demand nodes or retailersfi¼ cost of operating the ith supply node per

periodMi¼ the maximum number of demand nodes

that can be served from the ith supply nodecij¼ cost of supplying the jth demand node

from the ith supply node per period

Decision variables:

xi ¼1, if the ith existing supply node is retained

0, if not

�,

i ¼ 1, 2, . . . ,m

yij ¼

1, if demand node j is served from supply

node i in new set up

0, if not

8><>: ,

i ¼ 1, 2, . . . ,m; j ¼ 1, 2, . . . , n

Minimise z ¼Xmi¼1

fixiþXmi¼1

Xnj¼1

cijyij ð1Þ

subject to:

Xmi¼1

yij ¼ 1, j ¼ 1, 2, . . . , n ð2Þ

Xnj¼1

yij �Mixi, i ¼ 1, 2, . . . ,m ð3Þ

Xnj¼1

xi ¼ r, ð4Þ

xi 2 f0, 1g

yij 2 f0, 1g

International Journal of Systems Science 273


The problem is formulated to minimise the total costper period comprising of a fixed cost of operating theretained servers and another cost of supplying fromeach retained server to each of the n demand nodes.Very high or infinite cost can be assigned to thoseroutes which are infeasible or undesirable. Constraintset (2) ensures that each demand node is supplied froma unique server, while constraint set (3) mandates thatonly retained servers can supply to any demand nodeand the total number of demand nodes supplied from aretained server i is within its capacity Mi. As describedearlier, Mi is the maximum number of demand nodesthat can be served from the ith current supplier withexisting supply infrastructure that costs fi per period.The motivation for the closure of some distributorsbeing the reduction of demand for its products, thefirm may find that for many supply nodes Mi is higherthan the number of demand nodes actually suppliedfrom the ith supply node.

It is obvious that the above simple fixed chargeformulation is independent of the existing distributionnetwork, but for the fact that the candidate servernodes that can be retained are from among the existingservers – that is, only r out of the existing m servernodes are to be retained. The existing network mayhave been optimal at some previous period but may notbe optimal at present when the product flow in thenetwork has reduced in general, but the demand at eachretailer may not have reduced by the same percentage.Constraint (4) has been added so that the exact numberof distributors can be ensured in the solution.

The pure delocation problem cannot be viewed asa simple fixed charge location problem to locate r(r5m) distributors. It is not difficult to see that thesolution to the simple fixed charge formulation abovemay reallocate some of the demand nodes to newsupply nodes to reduce costs. This may not beacceptable due to operational and practical considera-tions. This is because of the legacy of the existingdistributors who may or may not be retained. The firmis not considering starting afresh and finding anoptimum r distributor location solution due to theheavy cost of disruption of services and loss of trustand faith among current distributors. In other words,each demand node must continue to get its suppliesfrom its current supplier except when the currentsupplier is eliminated. The formulation of the deloca-tion problem becomes more complex if these restric-tions are taken into account and is presented next.

3.2. Formulation of the pure delocation problem

The pure delocation problem begins with a givennetwork with known supply nodes (i¼ 1, 2, . . . ,m) and

demand nodes ( j¼ 1, 2, . . . , n) such that each demand

node ( j) is supplied from a unique supply node. The

firm wishes to close p of these existing m supply nodes.With the closure of some servers, the service level

would deteriorate but that is expected to be compen-sated by a reduction in cost. Also, all the demand

nodes must continue to get their supplies from theirrespective existing supplier, and exceptions will be

made only if the existing supplier is eliminated. The

pure delocation problem can thus be formulated asshown below.

Additional parameters:

p¼ number of supply nodes to be eliminatedsj¼ index of the supply node for the jth

demand nodeDi¼ Set of indices of all the demand nodes for

the ith supply nodeRi¼ the maximum number of additional

demand nodes that can be served from

the ith supply node

Decision variables:

xi¼

1, if the ith existing supply node is eliminated

in the new set up

0, if not, � i:e: if the ith existing supply node

is retained in thenew set up

8>>>><>>>>:

,

i¼ 1,2, . . . ,m

yij¼

1, if demand node j is served fromadifferent

ðretainedÞ supply node i in new set up because

the current supply node is eliminated

0, if not, � i:e: if demandnode j is continued

to be served from supply node i both in

the current and the new set up or if supply

node i is eliminated

8>>>>>>>>>>>><>>>>>>>>>>>>:

,

i¼ 1,2, . . . ,m;

j¼ 1,2, . . . ,n

Minimise z¼Xmi¼1

Xnj¼1

cijyij�Xmi¼1

fixiþXmi¼1

Xj2Di

cij

!xi

" #

ð5Þ

subject to:

Xmi¼1

yij ¼ xsj, j ¼ 1, 2, . . . , n ð6Þ

Xnj¼1

yij � Rið1� xiÞ, i ¼ 1, 2, . . . ,m ð7Þ

274 P.K. Bhaumik


Xmi¼1

xi ¼ p, ð8Þ

xi 2 f0, 1g

yij 2 f0, 1g

The objective is to minimise the cost increase conse-quent to the closure of some facilities. This is theincrease due to the cost of the new links created net ofthe savings due to closure of some existing servers andlinks therefrom which would not be used in the new setup. Constraint set (6) defines that for every demandnode j, yij¼ 0 if the source node (sj) is retained (i.e.xsj¼ 0) and if the source node is eliminated, then thisuprooted demand node has to be linked to one newsource node. Similarly, the constraint set (7) requiresthat corresponding to every eliminated supply node(i.e. with xi¼ 1), all the yij’s ( j¼ 1, 2, . . . , n) should beequal to zero as new links cannot be established fromeliminated supply nodes. The same constraints alsoensure that the number of additional demand nodesconnected to the ith retained supply node (after theclosure of some other supply nodes) does not exceed itsunutilised capacity Ri. Constraint (8) specifies thenumber of supply nodes that need to be eliminated.

4. Application and results

We present a small application with 19 demandnodes ( j¼ 1, 2, . . . , 19) supplied from 5 servers or

distributors (i¼ 1, 2, 3, 4, 5) in the existing set up.

The existing distribution network is shown in

Figure 1. Table 1 provides the parameter values

for the model formulation. The cost figures per

period of operating the ith supply node ( fi) as well

as of using the link from supply node i to demand

node j for each value of i and j, (cij), are shown.

Similarly, Table 1 also lists the values for sj and Di

for each demand node and each supply node,

respectively.The cost of having a distributor at the ith supply

node ( fi) is actually a fixed cost and incorporates the

establishment costs, salaries, utilities, storage costs and

other similar expenses incurred per period. Similarly,

cij represents the cost per period of supplying demand

node j from the ith distributor and incorporates the

transportation and distribution cost. In fact, it can be

computed if the demand per period from the jth node is

known and also known is the unit cost of transporta-

tion on the route from the ith distributor to the jth

demand node.With the given data, the delocation problem can be

formulated as shown below:

Minimise z ¼X5i¼1

X19j¼1

cijyij � ½256x1 þ 124x2 þ 136x3

þ 152x4 þ 111x5�

i = 1

i = 2

i = 3

i = 4

i = 5

j = 1

j = 3

j = 2

j = 4 j = 5

j = 6

j = 7

j = 8

j = 9

j = 10 j = 11 j = 12

j = 13

j = 14

j = 15

j = 16

j = 17

j = 18

j = 19

Figure 1. Existing distribution network.



Table

1.Parameter

values

fortheexistingdistributionnetwork.

j

Supply

node

Optg

cost

Supplyingto

dem

andnodes

Maxtotal

dem

andnodes

Max.addl.

dem

andnodes

12

34

56

78

910

11

12

13

14

15

16

17

18

19

if i

Di

Mi

Ri

c ij

149

1,3,4,6,8,10

11

537

40

19

32

49

35

21

43

58

41

30

44

49

57

69

40

58

56

74

232

11,17,18

74

62

55

57

37

59

47

49

53

64

45

42

36

48

45

63

29

28

22

48

336

2,7,12

85

39

35

46

27

22

52

27

22

44

50

44

38

30

39

47

35

64

62

51

441

5,9,13,15

95

57

51

68

48

29

59

32

32

26

63

49

39

27

18

29

41

64

57

48

528

14,16,19

63

71

63

75

54

45

63

46

35

45

58

47

34

25

24

36

27

54

39

32

s j1

31

14

13

14

12

34

54

52

25

276 P.K. Bhaumik


subject to:

y1,1 þ y2,1 þ y3,1 þ y4,1 þ y5,1 ¼ x1

y1,2 þ y2,2 þ y3,2 þ y4,2 þ y5,2 ¼ x3

y1,3 þ y2,3 þ y3,3 þ y4,3 þ y5,3 ¼ x1

y1,4 þ y2,4 þ y3,4 þ y4,4 þ y5,4 ¼ x1

y1,5 þ y2,5 þ y3,5 þ y4,5 þ y5,5 ¼ x4

y1,6 þ y2,6 þ y3,6 þ y4,6 þ y5,6 ¼ x1

y1,7 þ y2,7 þ y3,7 þ y4,7 þ y5,7 ¼ x3

y1,8 þ y2,8 þ y3,8 þ y4,8 þ y5,8 ¼ x1

y1,9 þ y2,9 þ y3,9 þ y4,9 þ y5,9 ¼ x4

y1,10 þ y2,10 þ y3,10 þ y4,10 þ y5,10 ¼ x1

y1,11 þ y2,11 þ y3,11 þ y4,11 þ y5,11 ¼ x2

y1,12 þ y2,12 þ y3,12 þ y4,12 þ y5,12 ¼ x3

y1,13 þ y2,13 þ y3,13 þ y4,13 þ y5,13 ¼ x4

y1,14 þ y2,14 þ y3,14 þ y4,14 þ y5,14 ¼ x5

y1,15 þ y2,15 þ y3,15 þ y4,15 þ y5,15 ¼ x4

y1,16 þ y2,16 þ y3,16 þ y4,16 þ y5,16 ¼ x5

y1,17 þ y2,17 þ y3,17 þ y4,17 þ y5,17 ¼ x2

y1,18 þ y2,18 þ y3,18 þ y4,18 þ y5,18 ¼ x2

y1,19 þ y2,19 þ y3,19 þ y4,19 þ y5,19 ¼ x5

y1,1 þ y1,2 þ y1,3 þ y1,4 þ y1,5 þ y1,6 þ y1,7 þ y1,8

þ y1,9 þ y1,10 þ y1,11þ y1,12 þ y1,13 þ y1,14 þ y1,15

þ y1,16 þ y1,17 þ y1,18 þ y1,19 þ 5x1 � 5


þ y2,9 þ y2,10 þ y2,11 þ y2,12 þ y2,13 þ y2,14 þ y2,15

þ y2,16 þ y2,17 þ y2,18 þ y2,19 þ 4x2 � 4


þ y3,9 þ y3,10 þ y3,11 þ y3,12 þ y3,13 þ y3,14

þ y3,15 þ y3,16 þ y3,17 þ y3,18 þ y3,19 þ 5x3 � 5

y4,1 þ y4,2 þ y4,3 þ y4,4 þ y4,5 þ y4,6 þ y4,7

þ y4,8 þ y4,9 þ y4,10 þ y4,11 þ y4,12 þ y4,13 þ y4,14

þ y4,15 þ y4,16 þ y4,17 þ y4,18 þ y4,19 þ 5x4 � 5


þ y5,9 þ y5,10 þ y5,11 þ y5,12 þ y5,13 þ y5,14 þ y5,15

þ y5,16 þ y5,17 þ y5,18 þ y5,19 þ 3x5 � 3

x1 þ x2 þ x3 þ x4 þ x5 ¼ p

x1, x2, x3, x4,x5 2 f0, 1g

y1,1, y1,2, . . . , y5,19 2 f0, 1g

When solved for different values of p, using a standardpackage for ILP, the solution obtained is as given inTable 2. If one server is to be closed, the cost per periodactually decreases and the solution suggests that x3should be eliminated. In the new set up, each of theuprooted demand nodes, namely [D3¼ {2, 7, 12}], willhave to be supplied from some operating server. In theoptimum solution, demand nodes 2 and 7 are beingsupplied from supply node 1 (using new links y1,2 andy1,7) while demand node 12 is being supplied fromsupply node 5 (using new link y5,12).

We also notice that with a larger number of closureof servers, the cost first decreases and then increases. Ifthe motivation for closure of servers is cost reduction –as the service level is likely to worsen with fewer servers– then it works up to a limit (p¼ 2) and after the limit,both cost and service level may worsen. If the simplefixed charge formulation is used then the solutions fordifferent number of supply nodes retained is as shownin Table 3. Here again we notice that the cost firstdecreases and then after a point starts rising. For ourproblem this happens when three supply nodes areretained.

The difference between the delocation problemformulation and the simple fixed charge formulationcan be seen more clearly from the two diagrams shownin Figures 2 and 3, respectively. The simple fixedcharge solution of Figure 3 requires demand node 7 tobe supplied from supply node 1, whereas it is beingsupplied from supply node 3 in the current set up.While supply node 3 is retained in the new set up, itdoes not supply to its earlier demand nodes 7 and 12.This may not be acceptable in the long-term strategic

Table 2. Solution of the delocation problem.

Optimum solution

No. of serversto be eliminated (p) Cost increase Cost

Eliminatedsupply nodes New links

0 0 779 Nil Nil1 �41 738 x3 y1,2, y1,7, y5,122 �55 724 x3, x5 y1,2, y1,7, y2,12, y2,16, y2,19, y4,143 �20 759 x2, x3, x5 y1,2, y1,7, y1,11, y4,12, y4,14, y4,16, y1,17, y1,18, y1,194 No feasible solution



interest of the firm although it may be cheaper than thesolution given by the pure delocation formulation. Thedelocation problem solution, on the other hand, doesnot violate any of the existing linkages as shown inFigure 2.

A naıve argument can be advanced that thedifference in the optimal solutions of the puredelocation problem and the simple fixed charge prob-lem is because the existing distribution chain is notoptimal with five supplier nodes. In practice, even if thedistribution chain is designed to be optimal at some

stage, it usually deviates from optimality due to uneven

changes in demand at the different demand nodes.However, even if the existing distribution network isoptimal with the existing number of supply nodes, thesolutions for the two problems would be different withfewer supply nodes retained. This can be clearly seenfrom Table 3. If the existing distribution network hadfour supply nodes – these would be supply nodesnumbered 1, 2, 3 and 5. If only three supply nodes areto be retained, Table 3 suggests that supply nodesnumbered 1, 2 and 4 should be operating. Thedelocation problem would not allow the creation of anew distributor (node 4) while reducing the totalnumber of distributors. The delocation problem for-mulation and its solution is relevant for distributionchains going ahead with the closure of some of itsdistributors without annoying any of the retaineddistributors.

5. Conclusions

Closure of facilities and shrinkage of the distributionchain is not very uncommon for business units. Butthis problem has not been studied much in theliterature. With reduced volume of materials flowingthrough the distribution chain, a firm may wish toreduce the number of distributors with a minimumconsequential cost increase. However, it is not areallocation of demand nodes among the retaineddistributors. An important condition of the puredelocation problem stipulates that all demand nodesmust continue to get their supplies from their respec-tive current sources except when the current sourceitself is delocated and only such uprooted demand

Table 3. Solution of the simple fixed charge problem.

Supplynodesretained (r) Cost

Supplynode (i)

Distributing todemand nodes ( j)

5 716 1 1, 3, 6, 7, 10, 112 17, 183 2, 4, 5, 84 9, 14, 155 12, 13, 16, 19

4 706 1 1, 3, 6, 7, 10, 112 17, 183 2, 4, 5, 8, 95 12, 13, 14, 15, 16, 19

3 701 1 1, 2, 3, 4, 6, 7, 10, 112 12, 16, 17, 18, 194 5, 8, 9, 13, 14, 15

2 747 1 1, 2, 3, 4, 6, 7, 10, 11, 16, 17, 184 5, 8, 9, 12, 13, 14, 15, 19

1 No feasible solution

i = 1

i = 2

i = 3

i = 5

j = 1

j = 3

j = 2

j = 4 j = 5

j = 6

j = 7

j = 8

j = 9

j = 10 j = 11 j = 12

j = 13

j = 14

j = 15

j = 16

j = 17

j = 18

j = 19

Figure 3. Solution of the simple fixed charge problem with4 distributors retained (cost 706).

i = 1

i = 2

i = 4

i = 5

j = 1

j = 3

j = 2

j = 4 j = 5

j = 6

j = 7

j = 8

j = 9

j = 10 j = 11 j = 12

j = 13

j = 14

j = 15

j = 16

j = 17

j = 18

j = 19

Figure 2. Solution of the delocation problem with 1 distrib-utor delocated (cost 738).

278 P.K. Bhaumik


nodes will be supplied by a different but one of theretained suppliers. The pure delocation problemformulated honours the existing distribution networkwhich may not be optimal due to historical legacy,strategic and other practical considerations. We haveassumed that the elimination of some distributors willnot affect the demand at any of the demand nodes orretailers although it is expected to reduce both thecost and the level of service to the retailers. As thecustomers get their demand serviced by the retailersand there is no delocation of retailers, it is reasonableto assume that the demand at any of the demand nodeor retailer will not be affected. Aboolian, Berman, andKrass (2007) have studied the case of new facilitylocation where the customer demand is elastic,expanding with the utility of the service offered bythe facilities increases.

In this article, we focus on a firm that wishes todownsize its distribution network. As the flow ofproducts in the distribution network reduces, thenumber of distributors needed may have to be reduced.Given an existing distribution network with knownretailer and distributor locations, it becomes veryimportant to decide which distributors to eliminate asthe decision affects both the cost and the service levelof the modified distribution chain. It should beunderstood that while in the growth phase thecompany has to decide where to locate the distributors(the location problem), in the downsizing phase it hasto decide which distributors to eliminate or delocateand to highlight the difference, we have called this thedelocation problem.

We have specified the exact number of suppliers tobe eliminated in the formulation, but it can be easilymodified to specify a maximum operating cost perperiod after the shrinkage or even the minimumreduction in operating cost per period. Also, if thereare limits on the additional quantity that can behandled by a distributor, such limits could be easilyhandled by modifying the constraint set (7) to accom-modate additional quantities rather than the numberof additional supply nodes Ri. Similarly, although ourmodel uses the cost per period of using a link, it caneasily be modified to handle unit transportation costand demand per period from each demand node.

The formulation of the delocation problem and itsjustification remains valid irrespective of the size of theproblem. Depending upon the size of the existingdistribution network, the actual problem could turnout to be much bigger with hundreds or eventhousands of retailers. We have not explored thedifficulties encountered in solving the delocationproblem formulated. While the small problem demon-strated above could be solved exactly using a branch-and-bound procedure, efficient heuristics may be

needed for very large problems. In this context, the

constraint expansion approach, which has been shownto be effective in solving location problems (ReVelle1993), can be useful for delocation problems as well.Facility location problems are generally formulated aslarge integer programming problems, although in someapplications mixed integer programming formulationshave been used (Galasso, Merce, and Grabot 2008,Wu 2008). Specialised branch and cut procedure

(Marin et al. 2009) as well as the minimax regretapproach (Puerto, Rodriguez-Chia, and Tamir 2009)have also been used to solve such problems. Acar,Kadipasaoglu, and Day (2009) developed a novelapproach that allows iterative interaction with asimulation model.

Finally, although we have presented the delocationproblem in the context of the distribution chain of abusiness firm, the same formulation can be easilyextended to the case of public sector services such as

schools. Delocation of schools is becoming a seriousproblem in many countries facing low birth rates anddemographic shifts. Our formulation will ensure thatchildren from retained schools are not affected and onlychildren presently going to a school being delocatedwill be assigned to one of the retained schools.

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