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Performance of an ADD-DROP-INTERCHANGE Heuristic for the Capacitated Facility Location Problem

Pritibhushan Sinha

6, A J C Bose Road - Thakurpukur

Kolkata – 700063 West Bengal INDIA E-mail: p_sinha@live.com

Abstract: In this article, we present a heuristic method for solving instances of the capacitated

facility location problem. In the method, first, a linear programming relaxation, a

transportation problem, is solved to obtain an initial solution. This also gives a lower bound of

an optimal solution of the capacitated facility location problem instance. The initial solution is

improved in subsequent iterations. In each iteration all possible add, drop or interchange of

the facilities are considered. Without solving the resultant transportation problems exactly, an

approximate method is used for the purpose. Some asymptotic conditions when the method

would give an optimal solution are given. Average performance of the method, with respect to

the quality of the solutions, has been investigated with benchmark and random instances of

the problem. An optimal solution is obtained for most of the benchmark instances. For

random instances also, performance of the method is satisfactory.

Keywords: Capacitated facility location problem, heuristic.

1. Introduction

Facility location problems are some of the most well-known optimization models. Those have

a great scope of application. Such applications include location of facilities such as hospitals,

schools, fire service stations; supply chain design; production planning; purchase planning;

locating computer servers, and many others. There is a number of variants of facility location

problems. Two major variants are the un-capacitated and the capacitated facility location

problems. The capacitated facility (synonymously, plant) location problem (CFLP) may be

described as follows. In an m × n CFLP we have m possible locations for setting up some

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facilities from where an item would be supplied to n customers or demand points. There is a

fixed cost to locate a facility at the i-th location and it is given by fi. Operational cost of

supplying one unit from a facility, set up at i-th location, to the j-th customer is cij. A facility

at i-th location has a capacity of Si and j-th customer has a demand of Dj. The problem is to

find an optimal solution, specifying the number and locations of facilities and supply quantity

from each facility to each customer, minimizing the total of fixed and operational variable

costs. We may consider a problem of production planning to exemplify the parameters in the

problem. An item may be produced at different periods, to meet the demands of such periods.

In every period, the item may be produced in the regular time shift or over-time shift or both.

Every period is considered to be a demand point with some demand (Dj), whereas every shift

in which production can happen is seen as a possible location of a facility, with a capacity

(Si). Per unit production cost, including an inventory cost, may vary from period to period and

this is considered in the variable costs (cij). Apart from the variable costs, there may also be

some fixed costs (fi). If an over-time shift is used in any period, there may be some such costs,

which may, for example, represent additional wages to the operators. Given such conditions,

the decision-maker needs to find out which production shifts are to be used to produce what

quantity, to meet the total demand, minimizing total of variable and fixed costs. The CFLP

may be written, with the preceding notation, as the following mixed integer linear program

(P1):

mi iiij

nj ij

mi yfxcMinimize 111 (1)

Subject to,

;,...,1,1 miySx iinj ij (2)

;..., ,1=∀ ,=∑1= njDx j

mi ij (3)

yi = 1 or 0, ∀i = 1, …, m; (4)

xij ≥ 0, ∀i, ∀j. (5)

If all Si = ∞, or equivalently, greater than equal to the total demand ∑j Dj, then we have an un-

capacitated facility location problem (UFLP) or, as sometimes called, a simple facility / plant

location problem. Both of CFLP and UFLP are NP-complete problems. There can be many

variants and extensions of the CFLP, considering conditions such as nonlinear variable costs,

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supply capacities also as decision variables (see, for example, Wu et al. (2006)), etc. In this

article, we shall be concerned about the CFLP, and more specifically, about heuristic solution

methods for the problem.

Considerable research effort has been spent on CFLP. Different methods have been

suggested in the literature for solving it. We may divide such methods into three types – exact

methods, approximation methods and heuristics. Exact methods (see, for example, Beasley

(1988), Ramos and Saez (2005)) give a correctly optimal solution. But such methods are not

practical for prohibitive time requirement for large instances of the problem, which often do

arise in actual situations. Approximation methods give solutions with a guaranteed upper

bound for a measure of deviation from optimality. The time requirement may be bounded by a

polynomial function of the input size and degree of approximation. An example of such a

method is found in Korupolu et al. (2000). But such methods have been possible only for

some special cases of the problem. Moreover, deviation from the optimal value can be much

more than what is realistically allowable. Heuristic methods do not have many theoretical

properties, but are useful to get a “near optimal” solution within a reasonable amount of time,

for many real-life instances of the problem. There are no theoretical measures of non-

optimality. Yet such methods may be the only way to solve many such instances of the

problem. A survey of the solution methods for CFLP is given by Sridharan (1995).

Heuristic methods suggested for the problem in the literature are of the types as

described in the following. Many heuristics (Geoffrion and McBride (1978), Barcelo and

Casanovas (1984), Beasly (1993), Barahona and Chudak (2005), Avella et al. (2008), among

others) are based on solving Lagrangean relaxation problems, which is often an approach for

solving other combinatorial problems also. A different type of heuristics, of the nature of local

search, are called as add / drop heuristics (for example, Kuehn and Hamburger (1963),

Jacobsen (1983), Domschke and Drexl (1985)). In such heuristics facilities are successively

placed in or deleted from locations, dependent on some measures of suitability. In this way,

different combinations of selected locations are checked in the search procedure. In drop

heuristics, one starts with a solution using all the locations. In add heuristics, a set of

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locations, which would give a good initial solution, is used initially. Domschke and Drexl

(1985) give some rules to select such a set. Solutions obtained with add or drop heuristics

may be improved by subsequent interchange heuristics which try to get a better solution by

perturbations, making occupied locations unoccupied and vice versa. Performance of such

methods has been reported to be quite good, particularly for small-scale instances. But a

drawback of add / drop/ interchange heuristics is that, a large number of transportation

problems has to be solved in every iteration. This makes such methods highly time-taking for

larger instances. Researchers have also applied meta-heuristic methods such as tabu search,

simulated annealing to solve the problem (for example, Grolimund and Ganascia (1997),

Bornstein and Azlan (1998)). One limitation of meta-heuristic methods is that, these are rather

general in approach and fail to make use of the specific structure for a particular type of a

problem. In Cornuejols et al. 1991, the authors compared some heuristic methods for CFLP

and indicated the superiority in average performance of Lagrangean relaxation based

methods. But large instances and instances for many different values of the parameters of the

problem had not been considered there.

We present a heuristic method to solve CFLP instances where we solve a linear

programming (LP) relaxation problem of (P1). This gives a feasible solution. Locations are,

then, checked for suitability for adding, dropping or being interchanged. In every iteration, all

possible drops, additions and interchanges are verified and the best possible move, decided

according to an approximate criterion, is made. The present method resembles the one

discussed by Korupolu et al. (2000). But they assume a metric CFLP, in which variable costs

(cij) are non-negative, symmetric and satisfy triangle inequality. Also, all locations should

have same supply capacity. The authors show that, for this subset, a similar method when

add/ drop/ interchange decisions are made solving the transportation problems exactly, gives a

solution scheme with an optimality factor of (8 + ε), ε > 0. Starting with any feasible solution,

the method is guaranteed to give a solution which has cost less than, equal to (8 + ε) times the

optimal cost. Solution time is bounded by a polynomial of the input size and 1/ε. The method

proposed by us is a heuristic method, devised for a general CFLP. In it, an initial solution is

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obtained solving a LP relaxation problem; a change in a solution is done based on an

approximate criterion, without solving each transportation problem exactly. Appropriateness

of the criterion makes the method fast, at the same time leading to very good solutions for

most of the instances.

This article is organized in the following manner. We describe the heuristic solution

method and some of its properties in the next section. Results of numerical experiments on

the method are given in Section 3. This is followed by concluding remarks and discussions.

2. A Heuristic for the CFLP

Consider the transportation problem that is obtained if fixed costs are omitted and a subset of

the locations is considered. Let the set of all possible locations be I = {1, 2, ..., m} and the set

of customers be as J = {1, 2, ..., n}. Denote the problem as (P2).

xcMinimize ijJj ijKi (6)

Subject to,

KiSx inj ij ,1 (7)

JjDx jmi ij ,1 (8)

xij ≥ 0, ∀i∊ K, ∀j ∊ J, (9)

with K ⊆ I as a set of some possible locations. A feasible solution of (P2) is also feasible for

(P1). In the heuristic method, which is described next, (P2) is solved to get an initial solution

which is then improved in subsequent iterations. It is assumed that, fi ≥ 0, Si > 0, ∀ i; and,

.> ∑∑j ji i DS

2.1. Method

A near optimal solution is obtained by solving an instance in the following two phases.

Phase 1

An initial solution is obtained with solving problem (P2) with K = I and costs modified as,

,+=/ Sfcc iiijij , ∀ i, ∀ j. A CFLP solution, i.e., a solution of (P1), is obtained by considering

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the locations which have nonzero supply to at least one demand point, in the optimal solution

obtained for (P2). K is updated accordingly to include only such locations.

Phase 2

The solution obtained in Phase 1 is tried to be improved through adding/ dropping/

interchanging locations. Let I1 and I2 be the sets of used and unused locations, corresponding

to a CFLP solution, at the beginning of an iteration. The following steps are carried out in an

iteration.

Step 1. (i) Try dropping the used locations in the following manner. For ∀ k ∈ I1, the steps, as

given next, are performed.

(a) Initialize as, ri = Si, − ∑∈Jj ijx , ∀ i ∈ I1 \ k; Dk = 0, p = 0.

(b) If p < n, p = p + 1; R = xkp and go to (c). Else, stop.

(c) If R > 0, go to (d), else go to (b).

(d) Let }.0> ,\∈ | min{= 1* rIcc iippi ki If such i* exists then, z = },min{ *riR , Dk = Dk + z

× ( *c pi − ckp), R = R – z, = ** rr ii − z and go to (c). Else, location k cannot be dropped, stop

with Dk = ∞.

Let C(drop) = max {fk − =}∈| *1 fID kk k − . *Dk

(ii) Try adding the unused locations in the following manner. For ∀ l ∈ I2, the subsequent

steps are carried out.

(a) Initialize as, R = Sl, Dl = 0, A = ∅.

(b) If R = 0, stop. Else go to (c).

(c) Let ),(,∈ ,∈,0> ,> | max{= 1** jiJji Ixcccc ijljijijji ∉ A}. If such i

* and j

* exist then, z =

},min{ ** Rx ji and Dl = Dl + z × ( **c ji − clj), A = A + (i*,j

*), R = R – z, and go to (b). Else, stop.

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Let C(add) = max{Dl − =}∈| *2 DIf lll − . *f

l

(iii) Try to interchange an unused location with a used location, with ∀ k ∈ I1, ∀ l ∈ I2. In

interchanging, first, k is dropped, considering l is also used, as in (ii). Then, if all supply

capacity of l has not been used, it is attempted to be added, as in (i).

Let C(interchange) = max{fk − Dk − =}∈,∈|+ *21 fIIDfkll

lk −Dk* − .+ ** Df ll

Step 2. Calculate }. , ,max{= e)interchang(drop)(add)( CCCC If 0>C , add, drop or interchange

the corresponding locations k*, l

* to update K, and solve (P2) with costs cij. Return to Step 1.

Else, i.e., 0≤C , stop with the current solution, with used locations in K.

2.2. Some Properties of the Method

The following observations about the method are straightforward. Optimal objective function

value of (P2), with K = I and costs modified as in Phase 1, gives a lower bound of the same of

(P1). This is so, as any feasible solution of CFLP (P1) is a feasible solution for the

corresponding (P2). For the same solution, (P2) cost is less than, equal to (P1) cost. Denote

the lower bound as L*. But this bound may not be very good, as discussed later.

As the set of the locations used may be changed in Phase 2, improvement is found

with respect to a feasible solution, not an optimal solution, with the changed set of locations.

0>C implies that, an improved solution is possible. A better solution may exist even if

0≤C . Since there is a finite number of solutions of (P1), the method terminates in finite

number of iterations.

For example, if total supply, without anyone facility is less than total

demand, ∑∑ <≠ , j jkii i DS , the method, clearly, would give an optimal solution. For some

other conditions also, we would obtain an optimal solution, as shown in a proposition, as

follows. Define the following parameters of the problem.

Capacity ratio (CR) = Total supply (i iS ) / Total demand ( j jD )

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Fixed cost ratio (FCR) = (Total of fixed costs (i if ) / Total supply (i iS )) /

Average of variable costs per unit ( i j ijc / mn).

We also note the fact that, (P2) is bounded in minimization or maximization. Next,

we have a result about the lower bound L*. We shall use the same notation as in (P1).

Proposition 2.1: Let C* be the cost of an optimal solution of an instance of (P1) and L

* a

lower bound of the same, obtained as given earlier. The difference (C* − L

*) and the ratio

(C* − L

*) / L

* can be arbitrarily large.

Proof: Consider (P1) instances with, i. m ≥ n; ii. Si ≥ ∑ 1=nj jD , ∀ i. iii. fi = f > 0, ∀ i; iv. cij = 0,

if i = j; cij > 0, otherwise. Clearly, L* = f. ( ∑

1=nj jD /∑ 1=

mi iS ) and C

* ≥ f. Thus, with

sufficiently large f, and small ( ∑1=

nj jD / ∑ 1=

mi iS ), we see that the proposition holds. ⎕

The next proposition identifies some conditions when the proposed heuristic would yield an

optimal solution.

Proposition 2.2: (i) Let m ≤ n. Suppose that, (a) every location has the unique minimum

variable cost for at least one demand point, cil < ckl, i, k = 1, 2,..., m; i ≠ k; l = l(i); (b) FCR is

sufficiently small, with each fixed cost fi sufficiently small. The heuristic method gives an

optimal solution of the CFLP instance.

(ii) Let, for a CFLP instance, m* be the minimum number of locations to be used to meet the

total demand. If (a) FCR is sufficiently high, with every fixed cost being sufficiently different

from other fixed costs, i.e., min i, k (i ≠ k) {|fi – fk|} is sufficiently high, and, (b) supply capacities

are nearly equal, min i {Si}/ max i {Si} > (1 – 1/m*), the heuristic method gives an optimal

solution of the CFLP instance.

(iii) Suppose that, (a) CR is sufficiently high, with every supply capacity greater than, equal

to total demand, Si ≥ ∑ Dn

1=j j , i = 1, 2, ..., m; (b) FCR is also sufficiently high, with each

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fixed cost sufficiently high, the heuristic method gives an optimal solution for the CFLP

instance.

Proof: (i) Consider an optimal solution of the corresponding (P2) for the problem, costs not

modified. All locations must be used in this solution. Let α be the difference between the

objective function values of an optimal solution and a solution with least cost, among

solutions in which all locations are not used. Dropping a location either makes the problem

infeasible or worsens optimal solution value. So, α > 0. Let ∑i i

f < α. Then, an optimal

solution of (P1) is obtained if and only if all locations are used, which is the case for an

optimal solution of (P2). The same holds also if (P2) is solved with costs modified. Thus, in

the first iteration only 0≤C and an optimal solution of (P1) is obtained.

(ii) Let α be the minimum objective function value and β be the maximum objective function

value for (P2), costs not modified, and min i,k (i ≠ k){|fi – fk|} > (β – α). An optimal solution of

(P1) would be given by m* locations with

least fixed costs. If a solution of (P1) has more than

m*

used locations, dropping one location improves the solution and also CC ,drop)( > 0 in an

iteration in Phase 2. So, consider a solution, which has m* used locations. If it is not a solution

with the least fixed cost locations, interchange of such a location with one of least cost

locations would give CC ,e)interchang( > 0, demand being possible to be satisfied with any

m* locations. Hence, the method yields an optimal solution of (P1).

(iii) Let α and β be as before and min i {fi} > (β – α). If a solution of (P1) has more than one

location with nonzero supply, C(drop) > 0. So, consider solutions with exactly one location with

nonzero supply. If a solution is not optimal for (P1), the location used would be interchanged

with one which gives an optimal solution for (P1). ⎕

The conditions included in the above proposition can be verified with polynomial

computations. The algorithm may also give very inefficient solutions for instances, e.g., one

facility has very high supply capacity exceeding the total demand, at the same time has very

high fixed cost. But these are some asymptotic conditions. No comment can be made, in

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general, about efficiency of the solutions, theoretically. Numerical experiments have been

carried out to check the average performance of the method, for more realistic data.

3. Numerical Experiment and Observations

Performance of the method has been investigated with some benchmark and some randomly

generated instances of the problem. Random instances have been generated with CR as a

given parameter and in some cases also controlling, approximately, FCR.

The computer routine for solving transportation problems in the method and also to

calculate lower bounds is based on the commonly known primal-dual method (discussed, for

example, by Hadley (1969)) in which check for optimality and improvement is done through

finding a feasible solution for the dual variables. The computer routine has been validated

comparing with solutions with Excel Solver. The heuristic method and generation of random

instances have been implemented with a computer program written in MS Visual FoxPro for

carrying out the numerical experiments. It has been run on a Pentium 4 personal computer,

with 2800 MHz processor and 256 Mb RAM (random access memory). Operating system for

the computer is Windows XP Professional. The following comparisons are made for the

solutions, as given by the method.

3.1. Benchmark Instances

We have used the (16 (m) × 50 (n)), (25 × 50) and (50 × 50) instances given in the “OR

Library” in the website page people.brunel.ac.uk/~mastjjb/jeb/info.html as benchmarks. Such

instances, based on real-life situations, were first used in Kuehn and Hamburger (1963) and

later in many studies. In these instances, fixed costs and capacities are uniform for the

facilities; variable cost is proportional to the railroad distance between a demand point and a

facility; demand is proportional to the population of a city, a demand point.

We calculate Efficiency of a solution as,

Efficiency = (1 − (Cost of the solution – Cost of an optimal solution) / Cost of an

optimal solution) ×100%,

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assuming cost of an optimal solution is strictly positive. An optimal solution has Efficiency of

100% and it decreases as cost of the solution increases. The results are shown in Table 1. An

optimum solution is obtained for majority of these instances, Efficiency being above 99% for

almost all the cases. Maximum time taken to solve a (50 × 50) instance has been 15.1 second

(s). It is possible that, solution time may be further improved by better implementation of the

method, particularly using a more efficient code for solving the transportation problems.

3.2. Random Instances

Two types of random instances are solved with the heuristic. Instances of the first type have

been generated with the method outlined by Barahona and Chudak (2005), used also in other

investigations. The second type of instances follows a different pattern, enlarging the variety

of the problem instances checked.

3.2.1. Random Instances – Type 1

The way of generating these instances has been as follows. In these instances, m = n.

i. Input m, CR.

ii. Dj = u, a random deviate following a uniform distribution in (5, 35), j = 1, ..., n;

iii. Si = u, a random deviate following a uniform distribution in (10, 160), i = 1, ..., m;

iv. The demand and facility points are generated uniformly at random in (0, 1) × (0, 1). Each

point so generated is a demand point and also a location. The Euclidean distance, between

two points, multiplied by 10, gives unit variable cost.

v. Fixed cost fi is given as, fi = U(0,90) + U(100,110) √Si.

vi. Capacities Sis are rescaled, each multiplied with the same factor, so that CR is fixed at the

selected value.

Instances of size 250 × 250 have been solved for capacity ratio 5, 10 and 15. For one

CR, 10 instances are solved. As the problem size is large for these instances, solution values

are compared with the lower bound L*, as given in Section (2.1). Efficiency Lower Bound

(ELB) is calculated as,

ELB = (1 − (Cost of the Solution – L*) / L

*) × 100%,

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assuming that L* > 0.

A solution would have Efficiency, as defined earlier, greater than equal

to ELB. ELB would decrease as cost of the solution increases. But, in the light of the

Proposition 2.1, ELB may be computed to be very bad even for optimal solutions.

3.2.2. Random Instances – Type 2

For this type, the way of generating such instances is, using same notation as before, may be

given as:

i. Get m, n, CR as inputs;

ii. cij = u, for i = 1, 2, …, m; j = 1, 2, …, n;

iii. ,100/CR 5 umnS i rounded to the nearest integer, for i = 1, 2, …, m;

iv. ,)+5.0( ××04.0= 2uSf iifor i = 1, 2, …, m;

v. Get total demand A = i iS /CR, rounded to the nearest integer;

vi. Dj = 5 + (A − 5n) × ( uj / nl lu1 ), rounded to the nearest integer, for j = 1, 2, …, n−1; Dn =

A − 11

nj jD .

Here each u denotes an independent, uniform random deviate in (0, 1). For these instances,

fixed cost is roughly proportional to supply capacity. FCR lies in the range 1.4 – 2.1.

In this case also, size of the instances is (250 × 250), CR fixed at 5, 10 and 15. In

each case 10 instances have been solved. Comparison has been made with a lower bound as

obtained in the previous case and ELB has been calculated in the same way.

Performance of the heuristic is seen to be better for the second type of the random

instances. For the first type of instances also, except for CR = 15, ELB is satisfactory. It

should be noted, however that, a lower bound of Efficiency is calculated, not Efficiency itself,

for the random instances. For most of the instances, 28 out of 30, of the second type, the

method gives a solution as obtained with solving the relaxation problem (P2). The findings of

the experiment for the random instances are shown in Table 2.

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4. Discussions

We have proposed a heuristic method to solve the CFLP. In this method, a relaxation

problem, a transportation problem, is solved to get an initial solution. The costs of the

transportation problem are derived from the supply capacities, fixed costs and variable costs

of the facility location problem. The initial solution is improved by add/ drop/ interchange of

the locations. Although, our method combines some of the approaches, suggested earlier by

other authors, such a method, in totality, has not been applied for the CFLP.

We have derived some properties of the method. These give some insights about the

working the method, but do not, however, make it possible to comment about the

performance of the method for all types of instances. The performance of the method also has

been investigated with benchmark and random instances. It has been very much noteworthy

for the benchmark instances. The benchmark instances used are of smaller size relatively;

although instances of such size may arise sometimes in the practical situations. Random

instances are of larger size and for these a lower bound of the efficiency measure is

calculated. According to this, performance of the method is largely satisfactory, except a few

situations. Such instances have mostly arisen for fixed cost approximately proportional to

square root of supply capacity (Random instances, Type I), with large capacity ratio.

It may be concluded that, although the method presented in this article has limitations

being a heuristic one, it would be quite suitable to get an efficient solution for some practical

instances of the problem, particularly large-scale instances, within a reasonable time limit. It

may also be used to get an efficient initial solution/ upper bound, for using the same in branch

and bound and other exact methods, if viable, for the problem. It is very much possible that,

different heuristic methods would give efficient solutions for different types of instances of

the CFLP. It would be preferable that, a number of promising heuristic methods is considered

and it is identified for which type of instances a particular method performs better than others.

The present heuristic method is a prospective one, to be considered as such. It may also be

possible to find some further desirable properties of the method for different special cases of

the problem. This will be a worthwhile research direction related to CFLP. We may also note

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that, the CFLP model considered here may not be applicable in various situations and we may

need to consider nonlinear variable costs, fixed cost dependent on the supply capacity decided

for a location, multi-item case where more than one item may be supplied from the facilities,

and such conditions. It would also be of high practical value, if the present heuristic, with

possible modifications, is considered for such models to yield desired solutions.

Table 1: Performance of the Heuristic Method for Benchmark Instances

Obsvn.

No.

Problem

Identifier

Problem

Size (m × n) CR FCR Efficiency (%)

1 Cap-41 16 × 50 1.37 0.01 100.00

2 Cap-42 ” ” ” 100.00 3 Cap-43 ” ” ” 100.00

4 Cap-44 ” ” ” 100.00

5 Cap-51 ” 2.74 0.005 99.79

6 Cap-61 ” 4.11 0.0033 100.00 7 Cap-62 ” ” ” 100.00

8 Cap-63 ” ” ” 100.00

9 Cap-64 ” ” ” 100.00 10 Cap-71 ” 16.0 0.0009 100.00

11 Cap-72 ” ” ” 100.00

12 Cap-73 ” ” ” 99.82 13 Cap-74 ” ” ” 100.00

14 Cap-81 25 × 50 2.14 0.01 99.02

15 Cap-82 ” ” ” 98.94 16 Cap-83 ” ” ” 98.25

17 Cap-84 ” ” ” 98.73

18 Cap-91 ” 6.43 0.0033 99.89 19 Cap-92 ” ” ” 100.00

20 Cap-93 ” ” ” 100.00

21 Cap-94 ” ” ” 99.75

22 Cap-101 ” 25.0 0.0009 99.89 23 Cap-102 ” ” ” 100.00

24 Cap-103 ” ” ” 99.97

25 Cap-104 ” ” ” 100.00

26 Cap-111 50 × 50 4.29 0.01 99.72

27 Cap-112 ” ” ” 100.00 28 Cap-113 ” ” ” 99.92

29 Cap-114 ” ” ” 99.45

30 Cap-121 ” 12.87 0.0033 99.89

31 Cap-122 ” ” ” 100.00 32 Cap-131 ” 50.0 0.0009 99.89

33 Cap-132 ” ” ” 100.00

34 Cap-133 ” ” ” 100.00 35 Cap-134 ” ” ” 100.00

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Table 2: Performance of the Heuristic for Random Instances

Obsvn.

No. CR FCR

Max. Solution

Time (s) Min. ELB (%) Avrg. ELB (%)

Random Instances – Type 1

1 5 2.02 − 2.62 1372.2 95.96 97.42

2 10 1.19 – 1.30 1090.5 82.58 85.73

3 15 0.80 – 0.87 1192.6 66.26 70.30

Random Instances – Type 2

1 5 1.95 – 2.04 2217.2 97.4 98.19

2 10 1.65 – 1.69 1003.0 91.4 94.65

3 15 1.41 – 1.45 585.1 81.96 88.78

References

Avella, P., Boccia, M., Sforza, A .and Vasil‟ev, I. (2008) „An effective heuristic for

large-scale capacitated facility location problems,‟ Journal of Heuristics (published on-line on

date, DOI: 10.1007/s 10732-008-9078-y).

Barahona, F., Chudak, F. (2005) „Near-optimal solutions to large-scale facility

location problems,‟ Discrete Optimization, vol. 2, pp. 35-50.

Barcelo. J., Casanovas, J. (1984) „A heuristic Lagrangean algorithm for the

capacitated plant location problem,‟ European Journal of Operational Research, vol.15; pp.

212-226.

Beasley, J. E. (1988) „An algorithm for solving large capacitated warehouse location

problems,‟ European Journal of Operational Research, vol. 33, pp. 314-325.

Beasley, J. E. (1993) „Lagrangian Heuristics for Location Problems,‟ European

Journal of Operational Research, vol. 65, pp. 383-399.

Bornstein, C. T., Azlan, H. B. (1998) „The use of reduction tests and simulated

annealing for the capacitated plant location problem,‟ Location Science, vol. 6, pp. 67-81.

16

Cornuejols, G., Sridharan, R. and Thizy, J. M. (1991) „A comparison of heuristics and

relaxations for the capacitated plant location problem,‟ European Journal of Operational

Research, vol. 50, pp. 280-297.

Domschke, W., Drexl, A. (1985). „ADD-heuristics‟ starting procedures for

capacitated plant location problems,‟ European Journal of Operational Research, vol. 21, pp.

47-53.

Geoffrion, A. M., McBride, R. (1978) „Lagrangean relaxation applied to capacitated

facility location problem.,‟ AIIE Transactions, vol. 10, pp. 40-47.

Grolimund, S., Ganascia, J. G. (1997) „Driving Tabu Search with case-based

reasoning,‟ European Journal of Operational Research, vol. 103, pp. 326-338.

Hadley, G. (1969) Linear Programming, Addison-Wesley, Reading, MS (USA).

Jacobsen, S. K. (1983) „Heuristics for the capacitated plant location problem,‟

European Journal of Operational Research, vol. 12, pp. 253-261.

Korupolu, M., Plaxton, C. and Rajaraman, R. (2000) „Analysis of a local search

heuristic for facility location problems,‟ Journal of Algorithms, vol. 37, pp. 146-188.

Kuehn, A. A. and Hamburger, M. J. (1963) „A Heuristic Program for Locating

Warehouses,‟ Management Science, vol. 9 (9), pp. 643-666.

Ramos, M. T. and Saez, J. (2005) „Solving capacitated facility location problems by

Fenchel cutting planes,‟ Journal of the Operational Research Society, vol. 56, pp. 297 – 306.

Sridharan, R. (1995) „The capacitated plant location problem,‟ European Journal of

Operational Research, vol. 87, pp. 203-213.

Wu, L. Y., Zhang, X. S. and Zhang, J. L. (2006) „Capacitated facility location

problem with general setup cost,‟ Computers & Operations Research, vol. 33, pp. 126 – 1241.

OR Library – <http://people.brunel.ac.uk/~mastjjb/jeb/info.html>.

(*** This is a version of the article, Performance of an Add-Drop-Interchange Heuristic

for the Capacitated Facility Location Problem. International Journal of Applied Management

17

Science, 1(4), pp. 388-400 (2009). Some typos are corrected in it. For citations, the journal

article (which is also more readable) may be used.)

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