Solving a Huff-like competitive location and design model for profit maximization in the plane

European Journal of Operational Research 179 (2007) 1274–1287

www.elsevier.com/locate/ejor

Solving a Huff-like competitive location and design modelfor profit maximization in the plane q

Jose Fernandez a,*, Blas Pelegrın a, Frank Plastria b, Boglarka Toth a,1

a Department of Statistics and Operations Research, University of Murcia, Spainb BEIF—Dpt. of Management Informatics, Vrije Universiteit Brussel, Belgium

Available online 17 April 2006

Abstract

A chain wants to set up a single new facility in a planar market where similar facilities of competitors, and possibly of itsown chain, are already present. Fixed demand points split their demand probabilistically over all facilities in the marketproportionally with their attraction to each facility, determined by the different perceived qualities of the facilities and thedistances to them, through a gravitational or logit type model. Both the location and the quality (design) of the new facilityare to be found so as to maximise the profit obtained for the chain. Several types of constraints and costs are considered.

Two solution methods are developed and tested. The first is a repeated local optimisation heuristic, extending earlierproposals to the supplementary design question and the presence of locational constraints. The second is an exact globaloptimisation technique based on reliable computing using interval analysis, incorporating several novel features. An exam-ple and comparative computational results demonstrate that this difficult and very multi-modal problem can be solved bysuch techniques. The local optimisation method turns out not to be very robust in its results, even after numerous repe-titions, whereas the global optimisation method yields very useful and complete information on guaranteed near to optimalsolutions after an important but still quite acceptable computational effort.� 2006 Elsevier B.V. All rights reserved.

Keywords: Continuous location; Facility design; Competition; Global optimization; Interval analysis

0377-2217/$ - see front matter � 2006 Elsevier B.V. All rights reserved

doi:10.1016/j.ejor.2006.02.005

q This paper has been supported by the Ministry of Science andTechnology of Spain under the research project BEC2002-01026,in part financed by the European Regional Development Fund(ERDF).

* Corresponding author. Address: Facultad de Matematicas,Universidad de Murcia, 30100 Espinardo, Murcia, Spain. Tel.:+34 968364186; fax: +34 968364182.

E-mail address: [email protected] (J. Fernandez).1 On leave from the Research Group on Artificial Intelligence

of the Hungarian Academy of Sciences and the University ofSzeged, H-6720 Szeged, Aradi vertanuk tere 1, Hungary.

1. Introduction

Many factors must be taken into account whenlocating a new facility which provides goods or a ser-vice to the customers of a given area. One of the mostimportant points is the existence of competitors inthe market providing the same goods or service.When no other competitor exists, the facility to belocated will have the monopoly of the market in thatarea. However, if in the area there already exist otherfacilities offering the same goods, then the new facil-ity will have to compete for the market.

.

mailto:[email protected]

J. Fernandez et al. / European Journal of Operational Research 179 (2007) 1274–1287 1275

Many competitive location models are availablein the literature, see for instance the survey papers[14,15,40] and the references therein. They vary inthe ingredients which form the model. For instance,the location space may be the plane, a network or adiscrete set. We may want to locate just one or morethan one new facility. The competition may be sta-

tic, which means that the competitors are already inthe market and the owner of the new facility knowstheir characteristics, or with foresight, in which thecompetitors are not in the market yet but they willbe soon afterwards the new facility enters. In thiscase it is necessary to make decisions with foresightabout this competition, leading to a Stackelberg-type model. Furthermore, if the competitors canchange their decisions, then we have a dynamic

model, in which the existence of equilibrium situa-tions is of major concern. Demand is usually sup-posed to be concentrated in a discrete set ofpoints, called demand points, and it can be eitherinelastic or elastic, depending on whether the goodsare essential or inessential.

The patronising behaviour of the customers mustalso be taken into account, since the market share

captured by the facilities depends on it. In somemodels customers select among the facilities in adeterministic way, i.e, the full demand of the cus-tomer is served by the facility to which he/she isattracted most. In other cases, the customer splitshis/her demand among more that one facility,leading to probabilistic patronising behaviour. Onthe other hand, it is also necessary to specifywhich the attraction (or utility) function of a cus-tomer towards a given facility is. Usually, theattraction function depends on the distancebetween the customer and the facility, as well ason other characteristics of the facility which deter-mine its quality. Little research has been done onthis kind of problem with simultaneous decisionson location and quality in continuous space. Fora single competing facility, the problem has beenstudied under deterministic customer behaviourin [10,39], using attraction functions of gravitytype, and in [41], using different kinds of attractionfunctions. For probabilistic customer behaviour,the problem has been studied in [9], where thelocation problem is solved for a wide range ofquality values, but no optimal solution to the loca-tion and quality problem is given. An extension tomultiple facilities, considering that quality is deter-mined by a budget allocated to each new facility,is shown in [11].

In this paper, we consider a single facility loca-tion problem on the plane, with static competitionand inelastic demand, having probabilistic behav-iour, based on an attraction function dependingon both the location and the quality of the facilityto be located. These two last factors are the vari-ables of the problem. The objective is to maximizethe profit obtained by the chain, to be understoodas the income due to the market share captured bythe chain minus its operational costs.

The paper is organised as follows. The locationmodel is presented in Section 2. The following twosections give solution procedures for the problem:a Weiszfeld-like local optimization algorithm in Sec-tion 3, and an interval branch-and-bound globaloptimization algorithm in Section 4. In Section 5we present some computational results to showthe performance of the algorithms. The paper endsin Section 6 with some conclusions and directionsfor future research.

2. The model

A chain wants to locate a new single facility in agiven area of the plane, where there already exist m

facilities offering the same good or product. The firstk of those m facilities belong to the chain. Thedemand, supposed to be inelastic, is concentratedat n demand points, whose locations pi and buyingpower wi are known. The location fj and qualityof the existing facilities is also known.

The following notation will be used throughoutthis paper:

x location of the new facility, x = (x1,x2)a quality of the new facility (a > 0)n number of demand pointspi demand points, pi = (pi1,pi2) (i = 1, . . . ,n)wi demand (or buying power) at pi

w total demand (or buying power) at the re-gion, w ¼

Pni¼1wi

m number of existing facilitiesfj existing facilities (j = 1, . . . ,m)k number of existing facilities that are part of

one’s own chain (the first k of the m facilitiesare assumed in this category, 0 6 k < m)

dij distance between demand point pi and facil-ity fj

dix distance between demand point pi and thenew facility x

aij quality of facility fj as perceived by demandpoint pi

1276 J. Fernandez et al. / European Journal of Operational Research 179 (2007) 1274–1287

gi(Æ) a non-negative non-decreasing functionaij

giðdijÞ attraction that demand point pi feels forfacility fj

ci weight for the quality of x as perceived bydemand point pi

ciagiðdixÞ attraction that demand point pi feels for the

new facility x

The use of a general nonnegative and non-decreasing function gi in the attraction functionsgeneralizes the proposals found in literature, suchas giðdixÞ ¼ ðdixÞki (see [9,12,29,30,37]) or giðdixÞ ¼ekidix (see [28]), with ki > 0 a given parameter, whichmight differ between demand points. Note thatwhen g(d) = 0, which may only happen if d = 0,i.e., the facility coincides with the demand point,the attraction becomes +1, which in the currentmodel (see below) will mean that all demand willbe attracted by that facility only. Therefore wemay assume that gi(dij) > 0 "i, j, because anydemand i for which some gi(dij) = 0 would be totallylost to the new facility, so it may simply be left outof the model.

In the spirit of Huff [29], and later generalized in[30,37], we consider that the patronising behaviourof customers is probabilistic, that is, demand pointssplit their buying power among the facilities propor-tionally to the attraction they feel for them. In [9]Drezner presented a location model following Huff’sformulation, in which the quality of the new facilityis assumed to be known; profit maximization thenreduces to market share maximization. Our modelgeneralizes her work by deciding both on the loca-tion and on the quality of the facility to be located.Contrary to other attraction functions, we considerthe possibility that attractiveness depends on thedemand point (in addition to distance), which maybe reasonable when socio-economic populationcharacteristics (e.g. average age, income class, acces-sibility, . . .) are different from place to place (e.g.city, suburb, village, . . .).

By these assumptions the market share capturedby the new facility is given by

Xn

i¼1

wi

ciagiðdixÞ

ciagiðdixÞ þ

Pmj¼1

aij

giðdijÞ

and the total market share attracted by the chain is

Mðx; aÞ ¼Xn

i¼1

wi

ciagiðdixÞ þ

Pkj¼1

aij

giðdijÞcia

giðdixÞ þPm

j¼1aij

giðdijÞ.

The previous expression can be rewritten as

Mðx; aÞ ¼Xn

i¼1

wi 1�Pm

j¼kþ1aij

giðdijÞcia

giðdixÞ þPm

j¼1aij

giðdijÞ

!;

and setting

ri ¼Xm

j¼1

aij

giðdijÞ; si ¼

Xm

j¼kþ1

aij

giðdijÞ; ti ¼ wisi

we finally have

Mðx; aÞ ¼ w�Xn

i¼1

tigiðdixÞciaþ rigiðdixÞ

.

The problem to be solved is then

max Pðx; aÞ ¼ F ðMðx; aÞÞ � Gðx; aÞs:t: dix P dmin 8i;

a 2 ½amin; amax�;x 2 S � R2;

8>>>><>>>>: ð1Þ

where F(Æ) is a strictly increasing differentiable func-tion which transforms the market share into ex-pected sales, G(x,a) is a differentiable functionwhich gives the operating cost of a facility locatedat x with quality a, and P(x,a) is the profit obtainedby the chain. Note that this profit disregards theoperating costs of the existing facilities of the ownchain, since these are considered to be constant.The parameters dmin > 0 and amin > 0 are giventhresholds, which guarantee that the new facility isnot located over a demand point and that it has aminimum level of quality, respectively. The param-eter amax is the maximum value that the quality ofa facility may take in practice. By S we denote theregion of the plane where the new facility can belocated.

The function F will often be linear, F(M(x,a)) =c Æ M(x,a), where c is the income per unit of goodsold. Of course, other functions can be more suit-able depending on the real problem considered.

The function G(x,a) should increase as xapproaches to one of the demand points, since itis rather likely that around those locations the oper-ational cost of the facility will be higher (due to thevalue of land and premises, which will make the costof buying or renting the location higher). On theother hand, G should be a nondecreasing and con-vex function in the variable a, since the more qualitywe require of the facility, the higher the costs will be,at an increasing rate. We will assume G to be sepa-rable, i.e. of the form G(x,a) = G1(x) + G2(a). Possi-ble expressions for G1 may be G1ðxÞ ¼

Pni¼1UiðdixÞ,


with UiðdixÞ ¼ wi=ððdixÞui0 þ ui1Þ, ui0,ui1 > 0, or

UiðdixÞ ¼ wi= edixui0 � 1þ ui1

� �, with ui0,ui1 > 0 given

parameters. A few typical forms for G2 might beG2ðaÞ ¼ ða=a0Þa1 , a0 > 0, a1 P 1, or G2ðaÞ ¼ e

aa0þa1�

ea1 , with a0 > 0 and a1 given values.An example of the objective function for n = 100,

m = 10, k = 4 can be seen in Fig. 1 (the rest of thesettings are as in Section 5.1). As our function isthree-dimensional, we can only show it for a chosena or for a chosen location. It can be seen from theFig. 1(b) that the objective function P is neitherconvex nor concave and may have several localoptima. Fig. 1(c) illustrates the concavity of P asa function of a, which will be proven in next section.

In what follows we present two different solutionapproaches. The first one is a local optimizationmethod using a modification of the Weiszfeld-likealgorithm presented in [9], adapted to include thequality a as a decision variable, and for handlingthe constraints. The second method is an interval

(b)

(

Fig. 1. (a) Contours of the objective function P(x, 2.9) (shades of gray),points (dark gray ellipses), (b) the objective function P(x, 2.9), and (c)

branch-and-bound algorithm, which always deter-mines an enclosure of all the globally optimal solu-tions with a pre-specified precision.

3. A Weiszfeld-like algorithm

A necessary condition for a vector (x*,a*) to be alocal or global maximum of Pðx; aÞ ¼ F ðMðx; aÞÞ�Pn

i¼1UiðdixÞ � G2ðaÞ is that the partial derivativesof P at that point must vanish.

• With regard to the first variable, x1, we have that

oPox1

¼ 0 () dFdM� oMox1

�Xn

i¼1

oUi

ox1

¼ 0.

Doing some calculations the previous expressionis equivalent to

dFdM�Xn

i¼1

�acitig0iðdixÞðciaþ rigiðdixÞÞ2

� odix

ox1

�Xn

i¼1

dUi

ddix

odix

ox1

¼ 0;

(c)

a)

existing facilities (solid circles) and constraints around the demandfor P((5,5),a).


which can be rewritten asXn

i¼1

H iðyÞodix

ox1

¼ 0;

where y = (x,a) and H iðyÞ ¼ � oPodix¼ dF

dM �aci tig

0iðdixÞ

ðciaþrigiðdixÞÞ2þ dUi

ddix. Furthermore, if dix is a distance

function such that

odix

ox1

¼ x1Ai1ðxÞ � Bi1ðxÞ;

where Ai1(x) and Bi1(x) are functions of x, then

oPox1

¼ 0 () x1 ¼Pn

i¼1HiðyÞBi1ðxÞPni¼1HiðyÞAi1ðxÞ

.

• Analogously, if odixox2¼ x2Ai2ðxÞ � Bi2ðxÞ, then

oPox2

¼ 0 () x2 ¼Pn

i¼1HiðyÞBi2ðxÞPni¼1HiðyÞAi2ðxÞ

.

• Finally,

oPoa¼ 0 () dF

dM�Xn

i¼1

citigiðdixÞðciaþ rigiðdixÞÞ2

� dG2

da

¼ 0.

Notice that this last equation, when we fixx = (x1,x2), has just one variable, a. Thus wecould solve it by using any algorithm for solvingequations of a single variable. Furthermore, P,as a function of the a variable alone, is concave:regardless of the strictly increasing differentiablefunction F(Æ) considered, the second derivativeof P with regard to a is negative, since

o2Poa2¼� dF

dM�Xn

i¼1

2c2i tigiðdixÞðciaþ rigiðdixÞÞðciaþ rigiðdixÞÞ4

� d2G2

da2;

and d2G2

da2 > 0 8a > 0 for any of the expressions ofG2 previously proposed. So, any solution of theprevious equation is guaranteed to be a globalmaximum.

Among the distance functions that satisfy theconditions

odix

ox1

¼ x1Ai1ðxÞ � Bi1ðxÞ;odix

ox2

¼ x2Ai2ðxÞ � Bi2ðxÞ;

we have the inflated Euclidean distance or its re-scaled version, the l2b norm, given by

dix ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib1ðx1 � pi1Þ

2 þ b2ðx2 � pi2Þ2

q;

where b1,b2 > 0 are given parameters (see [17]). Inthis case

Ai1 ¼b1

dix;

Bi1 ¼pi1b1

dix.

Thus, the following Weiszfeld-like algorithm can beconstructed.

Algorithm 1

1. Begin with a starting vector yð0Þ ¼ xð0Þ1 ; xð0Þ2 ; að0Þ� �

.Set iteration counter r = 0.

2. Calculate the next iterate yðrþ1Þ ¼ xðrþ1Þ1 ; xðrþ1Þ

2 ;�

aðrþ1Þ�

as follows:

xðrþ1Þ1 ¼

Pni¼1H iðyðrÞÞBi1ðxðrÞÞPni¼1H iðyðrÞÞAi1ðxðrÞÞ

;

xðrþ1Þ2 ¼

Pni¼1H iðyðrÞÞBi2ðxðrÞÞPni¼1H iðyðrÞÞAi2ðxðrÞÞ

and a(r+1) as a solution of the equation

dFdM�Xn

i¼1

citigiðdixðrþ1Þ Þðciaþ rigiðdixðrþ1Þ ÞÞ2

� dG2

da¼ 0.

3. If y(r+1) is infeasible then set y(r+1) equal to apoint in the segment [y(r),y(r+1)] which is on theborder of the feasible region.

4. If the distance between the consecutive iterativevectors y(r) and y(r+1) is less than a given toler-ance or r > rmax then

[STOP:] accept y(r+1) as a potential localmaximum.

Else set r = r + 1 and go to Step 2.

Several comments on the algorithm are in order.When solving the equation for a(r+1) in Step 2, wedo not need to obtain a(r+1) exactly: an approxima-tion can serve. So, when using a solution procedurefor the equation, we do not need to wait for the con-vergence of the procedure: a few iterations may beenough. Similarly in Step 3 it is not necessary toobtain an exact boundary point; any close but feasiblepoint suffices. For the first stopping criterion in Step4, different functions can be used to measure thecloseness of two consecutive vectors. For instance,one of them may be to stop the algorithm if

xðrÞ1 ;xðrÞ2

� �� xðrþ1Þ

1 ;xðrþ1Þ2

� �� 2<�1 and ja(r) � a(r+1)j <

�2, for given tolerances �1, �2 > 0. The second stopping


criterion setting a maximum number of iterationsrmax is necessary because the convergence of the algo-rithm cannot be guaranteed.

As observed before even in the best case the algo-rithm ends at a local maximum only. Thus, in orderto have a good chance at finding the optimal solu-tion, one should apply the algorithm repeatedlyusing different starting points in Step 1, and thenselect the solution that obtains the maximum profit.However, this still does not guarantee that a globaloptimal solution has been found.

4. An interval branch-and-bound algorithm

Due to the non-concavity (and non-convexity) ofthe objective function a global optimization tech-nique is required in order to obtain its globally opti-mal solution (see [26] for a review of thosetechniques applied to location problems). In thissection, we present a branch-and-bound algorithmwhich uses interval analysis tools. Here, we brieflysummarize the fundamental concepts of intervalanalysis which are needed for understanding themethod proposed. For more details, the interestedreader is referred to [24,31,43].

Following the notation suggested by Kearfottet al. [32] as a standard, boldface will denote inter-vals, lower case will be used for scalar quantitiesand vectors (vectors are then distinguished fromcomponents by use of subscripts), and upper casefor matrices. Brackets ‘‘[Æ]’’ will delimit intervals,while parentheses ‘‘(Æ)’’ vectors and matrices. Under-lines will denote lower bounds of intervals and over-lines give upper bounds of intervals. For example,we may have the interval vector x = (x1, . . . ,xn)T,where xi ¼ ½xi; xi�. The midpoint of an interval x isdenoted by mid x ¼ xþx

2, its width by widx ¼ x� x

and its relative width by widrelat x = widx/min{jxj :x 2 x} if 0 62 x and wid x otherwise. Themidpoint of an interval vector x = (x1, . . . ,xn)T isgiven by mid x = (midx1, . . . ,mid xn)T, whereas itswidth is to be understood as widx = max{widxi : i =1, . . . ,n}. The set of intervals will be denoted by IR,and the set of n-dimensional interval vectors, alsocalled boxes, by IRn.

The interval arithmetic operations are defined by

x � y ¼ fx � y : x 2 x; y 2 yg for x; y 2 IR; ð2Þwhere the symbol * stands for +, �, Æ and /, andwhere x/y is only defined if 0 62 y. Definition (2) isequivalent to simple constructive rules (see[24,31,43]). The algebraic properties of (2) are differ-

ent from those of real arithmetic operations (for in-stance, the subtraction and division in IR are notthe inverse operations of addition and multiplica-tion, respectively), but the main properties fromthe operational point of view still hold, as for in-stance the so-called subdistributive law,

x � ðyþ zÞ � x � yþ x � z for x; y; z 2 IR;

and the inclusion isotonicity,

x � y; z � t ) x � z

� y � t ðif y � t is definedÞ for x; y; z; t 2 IR.

The inclusion isotonicity is implicitly used in theconstruction of inclusion functions, which are themain interval arithmetic tool applied to optimiza-tion methods.

Definition 1. A function f : IRn ! IR is said to bean inclusion function of f : Rn ! R provided

ff ðxÞ : x 2 xg � f ðxÞ

for all boxes x � IRn within the domain of f.

Observe that if f is an inclusion function for f

then we can directly obtain lower bounds and upperbounds of f over any box x within the domain of fjust by taking f (x) and f ðxÞ, respectively.

For a function h predeclared in some program-ming language (like sin, exp, etc.), it is not too diffi-cult to obtain a predeclared inclusion function h,since the monotonicity intervals of predeclaredfunctions are well known and then we can takeh(x) = {h(x) :x 2 x} for any x 2 IR in the domainof h. For a general function f(x), x 2 Rn, the easiestmethod to obtain an inclusion function is the natu-

ral interval extension, which is obtained by replacingeach occurrence of the variable x with a box includ-ing it, x, each occurrence of a predeclared function h

by its corresponding inclusion function h, and thereal arithmetic operators by the correspondinginterval operators. Other inclusion functions havebeen proposed in the literature (see [2,4,36,42])although it is not clear which is the best one (see[46]).

Next, we present a prototype interval branch-and-bound algorithm able to solve the competitivefacility location model presented in the previous sec-tion, in which, now, we also allow for the existenceof additional general constraints hj(x,a) 6 0,j = 1 . . . ,q. Among those constraints, we have, asin the previous sections, the inequalities

dix P dmin 8i ¼ 1; . . . ; n


and the description of S. Thus, the problem to besolved is

max Pðx; aÞ ¼ F ðMðx; aÞÞ � Gðx; aÞs:t: hjðx; aÞ 6 0; j ¼ 1; . . . ; q;

ðx; aÞ 2 ðx0; a0Þ � R2 � R:

8><>:Note that a0 = [amin,amax].

By (x0,a0) we denote an initial box containing theregion to which the search for optimal vectors canbe restricted.

Algorithm 2

1. y y0 ¼ ðx0; a0Þ; LW ;;LS ;.2. Choose coordinate directions to split y.3. Split y normal to the chosen directions, cutting

the box a given number of times in each direc-tion. Let y1, . . . ,ys be the subboxes so obtained.

4. For i = 1 to s do(a) Delete yi if it can be proven that yi contains

no optimal solution or reduce yi if it can beproven that a part of yi contains no optimalsolution.

(b) If yi is not deleted, then store it (as whole orreduced) into the working list LW.

5. If LW ¼ ; then STOP.
6. Choose a box from LW and remove it from that
list. Let y denote the chosen box.7. If y satisfies any of the stopping criteria then

enter y in the solution list LS and go to Step 5else go to Step 2.

Different methods can be derived depending onhow the steps of the algorithm are completed. Theone presented here is similar to that used in [16]for solving an undesirable facility location model(see also [18,19]). However, several changes havebeen introduced, specially in Steps 2, 3 and 4(a).

The usual rule in the literature for the subdivisionof boxes (Steps 2 and 3) is to bisect the box perpen-dicularly to the direction of maximum width. Otherstrategies are possible, and from a computationalpoint of view some of them seem to be better (see[3,7,8,34,44,45]). In this paper we propose a newpartitioning method with pruning, by combiningan adaptive multi-section rule presented in Markotet al. [34] with a new pruning test from Martınezet al. [35].

The rationale behind the use of the adaptivemulti-section is as follows. If for a given box weknew in advance the number of subboxes into which

it should be subdivided in order to be able tofathom the region covered by the box, then wecould save computational effort by dividing immedi-ately the box into that number of subboxes. Thismeans that using the same subdivision rule (bisec-tion, tetrasection, multi-section, . . .) for all the boxesleads to unnecessary computation. This fact waspointed out by Casado et al. in [3], where a heuristic‘adaptive’ multi-section rule is proposed, based onexperiences that suggest the subdivision of the cur-rent box into a larger number of pieces only if it islocated in the neighborhood of a maximizer point.Of course, an index measuring the closeness of abox to a maximizer point is needed. Let eP denotethe available best lower bound of the global maxi-mum. For constrained problems, Markot et al.[34] propose the index

pupðy; ~PÞ ¼ PðyÞ � ePwidPðyÞ

Yq

j¼1

min�hjðyÞ

widðhjðyÞÞ; 1

( );

ð3Þ

which depends on the quality of the inclusion of theobjective value over the box as well as on its feasibil-

ity degree. The higher the index value, the closer thebox to the region of attraction to a maximizer point.Our subdivision strategy is then as follows. Ifpupðy; ePÞ < c1 we subdivide y along one coordinatedirection generating at most two subboxes; ifc1 6 pupðy; ePÞ < c2 we subdivide along two direc-tions generating at most four subboxes; otherwisewe subdivide along all the three directions generat-ing at most eight subboxes (in our computationalstudies we have set c1 = 0.05, c2 = 0.27). But wehave not performed a simple bisection along the se-lected directions, as done in [34]. Instead, we haveapplied the pruning test introduced in [35], whichis briefly explained below.

A large part of the interval global optimizationliterature is devoted to Step 4(a), i.e., to tests forverifying that a box or a part of a box contains nofeasible optimal point. We next briefly discuss theones we have used here. Some of them are new,and are proposed in this paper for the first time inthe literature.

Feasibility test: We say that a box y certainly sat-

isfies the constraint hj(y) 6 0 if hjðyÞ 6 0 and thaty does certainly not satisfy it if hj(y) > 0. A boxy � y0 is said certainly feasible if it certainly satis-fies all the constraints, certainly infeasible if itdoes certainly not satisfy at least one of the con-


straints, and undetermined otherwise. A boxy � y0 is said certainly strictly feasible ifhjðyÞ < 0, j = 1, . . . ,q. The ‘feasibility test’ [43]discards boxes that are certainly infeasible. Italso provides information about the feasibilityof a box. Notice that to apply this test we justneed an inclusion function hj for each of the func-tions hj defining the constraints.Cut-off test: Every time a box y is chosen fromthe list LW, and provided that its midpointc = mid y (as a point interval) is certainly feasi-ble, we compute P(c) and update (if possible)the best lower bound eP of the global maximumP*. The ‘cut-off test’ (also called ‘midpoint test’[43]) discards boxes whose objective functionvalue at any of its points is lower than the currenteP, i.e., a box y is removed if PðyÞ < eP. Also, anew box y must only be entered into LW ifeP 6 PðyÞ is satisfied. The updating of eP canalso be done as eP :¼ maxf eP;PðyÞg, providedthat the box y contains at least one feasible point.Pruning test: This test, recently proposed in [35],uses gradient information to determine regions inthe actual box which cannot contain global opti-mizers. We briefly explain it using, for the ease offollow, a two-dimensional optimization problem,

Fig. 2. Pruning method using t

and we describe how to apply the pruning testalong the x1-direction. Consider a box x =(x1,x2), and suppose that we know upper boundsfor the value of the objective function f at (x1,x2),ðx1; x2Þ and (mid(x1),x2), and also bounds for thegradient $f ðxÞ (see Fig. 2). Then an upperbounding function of the objective function canbe constructed as the planes in Fig. 2 (similarlyto what is commonly done in Lipschitz optimiza-tion [25]). Then, using an upper bound ~f of theglobal maximum, the maximizer points in x canlie only in (a,x2) and (b,x2). Therefore, the otherparts of x can be discarded. This pruning test canbe done for any coordinate direction of the box,generating one or two new subboxes. In thissense, it can be seen as a bisection method alongthe chosen coordinate. As explained before, wehave used this pruning test during the partition-ing process of the box, and we have applied italong one, two or three directions depending onthe value of the pup index (3).Monotonicity test (for strictly feasible and unde-

termined boxes): The monotonicity test com-monly used in the interval branch-and-boundliterature [24] is used to decide whether theobjective function P is strictly monotonous in a

he gradient $f ðxÞ.


certainly strictly feasible box y � y0. Then, y can-not contain a global minimizer in its interior. If$P ¼ ð$Px1

;$Px2;$PaÞ is an inclusion function

for the gradient $P of the objective function, andfor a certainly strictly feasible box y � y0 we havethat 0 62 $PðyÞ, then y can be deleted or reducedto a respective side. However, we can only applythis test to strictly feasible boxes. We propose thefollowing extension. If y = (x1,x2,a) = (y1,y2,y3)is an undetermined box for which $PiðyÞP 0for some variable yi, and the upper-facet y 0 ofthe box y associated to yi (for instance, if yi = athen y0 ¼ ðx1; x2; aÞ) certainly satisfies the con-straints in which the variable yi appears, thenthe box y can be either discarded or reduced toy 0. To see this, notice that the feasible point(s)in y which has the best objective value lies in thefacet y 0, and therefore the box y can be reducedto it. Furthermore, if $PiðyÞ > 0, and hj(y

0) < 0for all the constraints hj(y) 6 0 in which thevariable yi appears, and also yi < y0i then thewhole box can be removed. A similar result canbe obtained if $Pi 6 0, considering the corre-sponding lower-facet. We will refer to this testas the ‘monotonicity test for undeterminedboxes’.Projected one-dimensional interval Newton

method: The ‘multi-dimensional interval Newtonstep’ (see [24]) does not seem to be useful for thiskind of problems (see [16]). However, the follow-ing use (proposed here for the first time in the lit-erature) of the ‘one-dimensional interval Newtonmethod’ turns out to be useful for discarding orreducing boxes (see Section 5). Let y = (x1,x2,a)be an undetermined (or certainly feasible) box,and suppose that y certainly satisfies the con-straints in which the variable a appears (if any).We propose to apply the one-dimensional inter-val Newton method (see [23]) to the one-dimen-sional equation oP

oa ¼ 0, considering a as theinput interval, i.e., we consider x1 and x2 as fixedvariables, with values equal to x1 and x2, respec-tively. Notice that our proposal is to perform notonly ‘one step’ (or iteration) of the interval New-ton method, as it is usually done in interval glo-bal optimization algorithms, but let the methodrun till the end (that is, till the interval cannotbe reduced more or it is discarded). Notice thatwe apply the interval Newton method to theequation oP

oa ¼ 0 because we know that P is con-cave for the variable a. This procedure allows toreduce the width of the a component of the box

y, or even to discard the whole box, since it dis-cards the a values for which in none of the loca-tions (x1,x2) 2 (x1,x2) the objective functionattains its global maximum at a.

Other general discarding tests can be found in[21,24,47–49] and discarding tests specially suitablefor location problems in [20].

The order according to which a box is enteredinto the working list LW (Step 4(b)) is usuallyrelated to the criterion used to select the next boxto be subdivided by the algorithm (Step 6). Themost widely used, and the one we have chosen, isto select the box with the maximal upper boundPðyÞ (known as Moore-Skelboe rule [43]). Berner[1] suggested that it is the most suitable box tochoose from the working list, since in this way boxeswill rarely be subdivided unnecessarily. However,other criteria [5,6,34] may be better from a compu-tational point of view.

We have used two stopping criteria (Step 7). Abox y = (y1,y2,y3) = (x1,x2,a) is sent to the solutionlist LS if

widðx1; x2Þ < �1 and wida < �2;

or if

widrelat½ ~P;PðyÞ� < �3.

The described interval branch-and-bound algo-rithm stops after a finite number of iterations. Itgives as a result a list of boxes, LS, which containsall optimal solutions. Furthermore, any vector inthe boxes has a value close to the global optimum.That region of near-optimality may be more usefulin practice to a decision maker than a single solu-tion, since they will be able to choose in that regiona final solution taking other aspects of the real-lifeproblem into account. In Fig. 3 we show the output(projected on the locational space) offered by theinterval method for the problem shown in Section2 (see Fig. 1(a)).

Notice that the method described here can beapplied to many other continuous location prob-lems, not only the one presented here. Thus, it canbe considered as an alternative to other methodsable to solve general continuous location problems,as the Big Square Small Square method [27] and itsgeneralization, the GBSSS method [38], or therecently proposed Big Triangle Small Trianglemethod [13]. See [16] for a study about the differ-ences between the BSSS and GBSSS methods andinterval branch-and-bound algorithms.

Fig. 3. Results for a problem with the B&B method.


5. Computational results

All the computational results have been obtainedunder Linux on a Pentium IV with 2.66 GHz CPUand 2 GB memory. The algorithms have been imple-mented in C++. For the Interval B&B method weused the interval arithmetic in the PROFIL/BIASlibrary [33], and the automatic differentiation ofthe C++ Toolbox library [22].

Table 1Results for the 10 problems with 50 demand points, 10 existingfacilities, chain length 2, 10�2 accuracy, and 100 runs for theWeiszfeld-like algorithm

Difference in obj (%) Times found CPU seconds

B&B Local

0.076 1 7.22 0.750.035 2 17.06 0.330.350 2 17.31 0.310.000 31 20.34 0.400.261 1 24.60 0.390.000 37 11.92 0.460.136 1 12.91 0.35

5.1. The test problems

In order to have an overall view on the perfor-mance of both algorithms, we have generated differ-ent types of problems, varying the number ofdemand points (n = 50 or 100), the number of exist-ing facilities (m = 2, 5 or 10) and the number ofthose facilities belonging to the chain (k = 0 or 1for m=2, k = 0, 1 or 2 for m = 5 and k = 0, 2 or 4for m = 10). For every type of settings 10 problemswere generated, by randomly choosing the parame-ters of the problems uniformly within the followingintervals:

• fj,pi 2 [0, 10]2,• xi 2 [1,10],• ci 2 [0.75,1.25],• aij 2 [0.5,5],• Gðx; aÞ ¼

Pni¼1UiðdixÞ þ G2ðaÞ where

– UiðdixÞ ¼ wi1

ðdixÞui0þui1with ui0 = u0 = 2, /i1 2

[0.5,2],– G2ðaÞ ¼ e

aa0þa1 � ea1 with a0 2 [7,9], a1 2 [4,4.5],

4.413 1 7.64 0.370.140 1 16.35 0.370.507 1 10.09 0.33

Av: 0.592 7.8 14.54 0.41Sd: 1.353 13.9 5.60 0.13

• c 2 [1, 2], the parameter for F(M(x,a)) =c Æ M(x,a),

• b1,b2 2 [1, 2], parameters for dix ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib1ðx1 � pi1Þ

2 þ b2ðx2 � pi2Þ2

q.

The searching space for every problem was

x 2 ½0; 10�2; a 2 ½0:5; 5�.The algorithms were run with all equal toler-

ances, either 10�2 or 10�4 as indicated. The Weisz-feld-like algorithm was compared with the IntervalB&B algorithm when the former is run from 100and 1000 random starting points.

5.2. The results

First, we give the detailed results for a chosensetting, so as to show how we generate the latersummarizing tables. In Table 1 the results corre-sponding to the 10 generated problems for the casen = 50, m = 10, k = 2, e = 10�2, and 100 runs of theWeiszfeld-like algorithm, are presented one by one.In the last two lines the average and the standarddeviation (with shortcuts Av and Sd), respectively,


are given. We show the difference between theoptimal objective value obtained by the interval algo-rithm and the best solution obtained by the Weisz-feld-like algorithm in the 100 runs, in percentage.The column ‘‘Times found’’ refers to the number oftimes that the Weiszfeld-like algorithm found the bestsolution it could find; this gives information abouthow difficult to find this approximation of the globaloptimum is. The last two columns show the CPU timespent by the interval Branch-and-Bound (B&B) andthe Weiszfeld-like (Local) algorithms, respectively.

As it can be seen from the table, the solution ofthe Weiszfeld-like algorithm does not differ muchfrom the global optimum in terms of the objectivevalue, although there are instances for which the dif-ference is much greater than the average. Further-more, that method is much faster than the intervalone; this is due to the fact that the interval B&B algo-rithm is an exact one (it has to prove that no solutionlies in other places of the feasible region) and alsofinds all the areas with close-to-optimal values.

To have a general overview of the results, it isbetter to see only the average values. Dependingon the number of existing facilities belonging to itsown chain (shortly the chain length), we can pro-duce a summarizing table, as Table 2 (for the casen = 50, m = 10, e = 10�2). Each line in it corre-sponds to a table like Table 1, showing only theaverage of the appropriate values, with the standarddeviations in brackets. Since the difference in theobjective for some problems occasionally is muchhigher than the average, we will show the maximumof the differences in the set of 10 problems too. Thefirst line of each pair of lines shows the resultsobtained when the Weiszfeld-like algorithm is run100 times, whereas the second line (in italics) is for1000 runs. In the last two lines we give the averagefor the setting, regardless of the chain length.

Table 2Results for the problems with 50 demand points, 10 existing facilities,

Chain length Difference in

obj (%) max (%)

0 0.25 (0.6) 1.850.10 (0.2) 0.66

2 0.59 (1.4) 4.410.43 (1.1) 3.64

4 0.47 (0.7) 1.850.34 (0.5) 1.27

All 0.44 (0.9) 4.410.29 (0.7) 3.64

The lines in italic correspond to the results with 1000 runs of the Weis

Notice that as the chain length is larger, the num-ber of times that the Weiszfeld-like algorithm findsits best solution becomes smaller. This behaviouris the same for every setting. Usually this factimplies that the quality of the objective value offeredby the Weiszfeld-like algorithm worsens as k

increases. Also, the time spent by the intervalmethod usually increases with the chain length,i.e., the problem becomes harder for that algorithmtoo, although this is not always the case, as can beseen in Table 2.

Finally, in Table 3, we present the results for allthe settings regardless of the chain length, with stop-ping criterion 10�2 and 10�4, and running theWeiszfeld-like algorithm 100 and 1000 times.

We can see that increasing n or m implies anincrease in the CPU time required by the intervalB&B algorithm. The Weiszfeld-like algorithm is lesssensitive to these changes.

Concerning the tolerance, the increase form 10�2

to 10�4 implies an increase about 1.5 times in theCPU time required by both algorithms. However,the quality of the objective value offered by theWeiszfeld-like algorithm usually worsens as theaccuracy increases.

Increasing the number of times that the Weisz-feld-like algorithm is run from 100 to 1000 improvesthe objective value around 50%. However, for everyone of the types of problems generated there arealways instances for which the difference betweenthe objective value offered by the Weiszfeld-likealgorithm and the optimal obtained by the intervalmethod is much greater than the average (in 18 ofthe 24 settings, the maximum is greater by 1% andin three cases is greater by 4.6%). Even doing 1000runs, the best solution found by the Weiszfeld-likealgorithm is the optimal one in only half of theproblems. As an average, the best solution found

0, 2 and 4 chain length, and 10�2 accuracy

Times found CPU seconds

B&B Local

19.0 (25.1) 17.6 (6.1) 0.7 (0.6)18.9 (23.6) 5.7 (2.9)7.8 (13.9) 14.5 (5.6) 0.4 (0.1)6.7 (12.9) 4.1 (1.5)2.3 (2.7) 13.8 (6.2) 1.1 (1.6)2.3 (4.9) 10.0 (17.1)

9.7 (16.0) 15.3 (5.8) 0.7 (0.9)9.3 (15.3) 6.6 (9.7)

zfeld-like algorithm (otherwise 100).

Table 3Summarizing table for all the computational results

Difference in Times found Time spent

obj (%) max (%) B&B Local

50 demand points 2 0.35 (0.5) 1.60 18.6 (16.8) 6.0 (3.6) 0.6 (0.5)0.10 (0.2) 0.86 18.8 (18.5) 5.7 (6.1)0.31 (0.4) 1.12 20.6 (19.1) 8.0 (4.6) 0.8 (0.5)0.19 (0.3) 1.10 19.8 (19.6) 8.5 (5.4)

5 0.20 (0.6) 3.29 19.2 (21.1) 11.3 (4.9) 0.7 (0.7)0.02 (0.1) 0.24 17.6 (20.2) 6.8 (7.1)0.14 (0.3) 1.57 19.1 (22.0) 14.9 (7.3) 1.2 (1.3)0.07 (0.2) 1.30 17.3 (21.5) 11.7 (12.4)

10 0.44 (0.9) 4.41 9.7 (16.0) 15.3 (5.8) 0.7 (0.9)0.29 (0.7) 3.64 9.3 (15.3) 6.6 (9.7)0.53 (1.1) 4.98 8.4 (14.9) 18.3 (8.3) 1.3 (1.6)0.25 (0.7) 3.53 8.2 (15.6) 11.8 (13.0)

100 demand points 2 0.13 (0.2) 0.65 12.3 (19.1) 7.3 (8.1) 0.9 (0.2)0.07 (0.1) 0.59 11.0 (19.0) 6.9 (1.6)0.16 (0.3) 1.20 11.8 (19.2) 9.7 (10.8) 1.2 (0.5)0.05 (0.1) 0.28 10.5 (19.4) 11.6 (4.8)

5 0.21 (0.5) 2.78 17.0 (18.4) 12.8 (10.1) 1.0 (0.3)0.05 (0.2) 0.89 15.0 (16.7) 8.7 (1.8)0.31 (0.9) 4.61 14.4 (18.6) 17.3 (14.1) 1.5 (0.5)0.15 (0.5) 2.62 13.6 (17.9) 14.9 (4.5)

10 0.16 (0.5) 2.63 8.8 (9.5) 32.4 (20.5) 1.2 (1.0)0.09 (0.4) 2.36 7.4 (9.5) 9.1 (3.7)0.34 (1.1) 5.59 6.9 (8.7) 37.8 (26.9) 1.6 (0.4)0.11 (0.5) 2.61 7.0 (9.3) 16.6 (5.8)

Numbers in brackets (Æ) shows the standard deviation. Lines in italic are results for 1000 runs of the Weiszfeld-like algorithm (otherwise100). Lines in boldface are results for accuracy 10�4 (otherwise 10�2).


by the Weiszfeld-like algorithm is found only in 7–20% of the runs, and for everyone of the settingsthere are instances in which the best solution pro-vided by the Weiszfeld-like algorithm is found onlyonce (see Table 1). For instance, for n = 50, m = 10and � = 10�2, an average of only 9% of the 1000runs of the Weiszfeld-like algorithm found its bestsolution. And for those problems, in a computa-tional experiment in which the Weiszfeld-like algo-rithm was run 100,000 times, this could notguarantee that the global optimum was reached.

We also want to point out that although theWeiszfeld-like algorithm produces feasible solutionswhose objective values are close to the optimal, therespective locations might be relatively far from theregion of near optimality offered by the intervalmethod. Therefore the results obtained from theWeiszfeld method have to be considered withcaution.

6. Conclusions and future research

In this paper we have introduced a new compet-itive location model for profit maximization, inwhich the costs associated to both the quality andthe location of the new facility are taken intoaccount. The patronising behaviour of the custom-ers is supposed to be probabilistic, and the attrac-tion of a customer towards a facility depends onboth the location and the quality of the facility (fol-lowing Huff’s formulation). These two factors arethe variables of the problem. The problem can alsoinclude constraints to delimit the search region. Themodel is rather general (only differentiability is sup-posed) and can be specialized for many realsituations.

Of course, its ability to model general situationscomes at the expense of the difficulty in its solution:the problem is neither concave nor convex. Thus, in


order to be able to solve it, global optimization tech-niques must be used. We have presented two differ-ent approaches. The first one is a multi-startheuristic, in which the local search is a Weiszfeld-like algorithm which is able to handle not only thelocation variables, but also the quality variableand the existence of constraints in the model. Thesecond method is an exact interval branch-and-bound algorithm, which is able to find all the globaloptima with reliability. In order to make this latteralgorithm faster, we have used a new subdivisionstrategy (which combines an adaptive multi-sectionwith a pruning test), a new discarding test (themonotonicity test for undetermined boxes) and anew use of the one-dimensional interval Newtonmethod in multi-dimensional problems.

Although the Weiszfeld-like algorithm is faster ascompared to the interval branch-and-bound method,it is important to highlight that the Weiszfeld-likealgorithm offers as a result a single point which isnot guaranteed to be a global optimum and maysometimes be quite far from such, whereas the inter-val method offers an area (a list of boxes) containingwith reliability all the possible optima, and any pointin those boxes has a value close to the global optimalone. As described in [38] the interval method may beused in a second phase, to obtain a list of boxes guar-anteed to contain all nearly optimal solutions.

The study of the influence on the global optimumof the different elements defining the model will bethe subject of another paper. Also, the design ofnew accelerating devices for the interval methodremains for future research.

Among the possible extensions of this work wemention the multi-facility model, in which the chainwants to locate more than one new facility and a bi-objective model, considering the conflicting aims ofthe chain and the actual owner of the facility whenthe chain is a franchise. Also, other patronisingbehaviours can be considered. These include thedeterministic (or binary) behaviour, in which a cus-tomer is served by the facility to which he/she isattracted most; or the partially binary behaviour,in which a customer shares his/her buying poweramong the chains proportionally to the attractionhe/she feels for the chains (probabilistic behaviourconcerning the chains), but the part of the demandto be served by a given chain is fully served by thefacility of that chain to which the customer isattracted most (deterministic behaviour for thefacilities within a chain). Here, the attraction that

a demand point feels for a given chain is to beunderstood as the maximum attraction it feels forany facility of the chain.

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http://www.mat.univie.ac.at/~neum/software/int/

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Solving a Huff-like competitive location and design model for profit maximization in the plane