Market capture models under various customer-choice rules - CiteSeerX

Environment and Planning tl Planning and Design \1)1)1), volume 26, pages 741 750

Market capture models under various customer-choice rules

D Sernt Department of Heonomies and Business, Univcrsilat Pompeu Fnbra, Balmes 132, Barcelona 08022. Spain; e-mail: dnnicl.serra(fi'ceon.upf.es II A Riselt Faculty of Administration, University of New Brunswick, PC) Box 4400, Fredericton, New Brunswick, Canada F3B 5A3: e-mail: HAF:isdt(UNB.eA G I.aportc Centre de recherche stir les transports, University de Montreal, Case postale 6128, succursale Centre-viile, Montreal, Canada H3C 3.17; e-mail: gilbertcnC'RT.UMontreal.CA C S RcVcIlc Department of Geography and Hnvironmcntul Fngineering, The Johns Hopkins University, Baltimore, MI) 21218-2686, USA; e-mail: revelief^jhu.edu Received 18 November 1998; in revised form 5 April 1999

Abstract. Given that a firm currently operates p facilities in a (retail) market, a competing firm considers entering this market by locating r facilities so as to maximize its market share. This problem, known as the maximum capture problem or as the (r|/Vr)-mcdianoid problem, assumes, as do most location decision problems, that consumers always patronize the closest facility regardless of ownership or proximity to alternative facilities. In this paper we relax this assumption by allowing different customer-choice rules. Two new models are proposed for the optimal location for the entering firm under different consumer decision rules. The models are solved by using an exact method and a heuristic. Solutions are then compared with those obtained by the classical maximum capture problem with the usual nearest facility allocation rule. Computational experiments suggest that the maximum capture problem provides locational patterns whose objective values, that is, captures, are very similar to those of the other two objectives. 1 Introduction Multifacility location models hold a central place in regional science. Such models require a rule that allocates customers to facilities. In general, we distinguish between two different types of model: one in which the facility planners decide which customer is served from a facility, and a second in which customers decide themselves which facility they want to patronize. The first type is typically applicable when customers receive goods from warehouses. In this case, provided the goods from the different warehouses are homogeneous, they are indifferent as to which warehouse they are served from and they do not in fact control that choice. On the other hand, customers in the retail context have full control over which facility they choose to patronize. This class of customer-choice models is the subject of this paper. Though there exist many models dealing with different aspects of multifacility customer-choice problems, most assume total homogeneity of the products. Coupled with equal prices charged at the facilities, this choice rule results in customers patronizing the nearest facility and satisfying their entire demand there. Two issues need to be addressed at this point. First, most geographers attempt to capture the fact that in many instances facilities and products are not completely homogeneous. They do so by associating an attractiveness factor with each facility. This does of course require a function that specifies a trade-off between attractiveness and customer - facility distance. Typical examples of such functions are gravity models dating back to Reilly (1929)

744

D Serra, H A Eiselt, G Laporte, C S ReVelle

and

I>, = r, xtj = Oor 1, = 0 or 1,

yj

(4) Vi € /,V/ G J ,

(5)

V/' € / ,

(6)

where y y is equal to af if dfj < dip,, and ft is firm B's facility closest to /; otherwise, ytj is set equal to 0. The constraints are the same as those of the P-median problem (see ReVelle and Swain, 1970). Constraints (2) assign each customer / to only one facility, and constraints (3) ensure that no customer is assigned to an unopened facility. Constraint (4) fixes the number of facilities to be opened by firm A. The objective defines the total capture that firm A can achieve with r facilities. A variant of this problem is the case where the number of facilities is not given a priori. Constraint (4) can then be eliminated and fixed facility costs are included in the objective function. If profits are maximized, two forces then act in opposite directions: revenues are increased by opening more facilities closer to the demand, whereas minimizing the costs will lead to a reduction in the number of opened facilities. The model will then balance the trade-off between revenues and costs. In the first new model proposed here it is assumed that customers consider the closest facility of each firm and then patronize those two facilities in proportion to the customer-facility distances. In particular, let af" and af denote the unknown amounts of the demand at of customer i captured by facilities A and B, respectively. Clearly a*

+ af = ai.

(7)

Given a customer /, denote by at the closest facility of firm A, and by ft the closest facility of firm B. The corresponding distances between i and these facilities are diaL and diPi, respectively. The proportionality assumption is then formally expressed as 4a. + difii so that the objective (maximizing capture by firm A) of the first proportional model can be written as MAXPROP1 :

maximize z = V V iel

jeJ

(

"'"'*!

\U*J ^

)xij,

(9)

u

it

subject to constraints (2) to (6). Because dtj appears in the denominator of the objective, the model will always assign, for a given /, a value of 1 to the variable xtj for which dtj is the smallest among all open facilities of A. Note that we have only changed the objective in MAXCAP, so that the mathematical structures of the two problems are identical. In the second proportional model, the demand captured by a facility of firm A is affected by the existence and locations of all facilities of both firms. The demand from customer / captured by a facility of firm A located at a given vertex j is now given by

a

'J = 4j:/j:})-

do)

In other words, firm A captures a fraction of the demand of customer / equal to the ratio of the inverse distance, between i and the facility, to the sum of inverse distances between i and all facilities of firms A and B. In equations (10), (11), and (12), to avoid dividing by zero, it is necessary to work with limits so that the fraction of the demand

Market capture models under various customer-choice rules

745

of customer /captured by firm /'approaches 1 as 0 0.26 (,) 0.06 (,) 0.2(,» 0.05 (l> 0

0 0 0 0.12 ( , ) 0 0.9 ( l ) l.45

2 3 4 3 5 3 4 4 6

2 3 4 3 5 3 4 4 6

2.4 (,) 5.1 (l>

0 0 0.3 ( l ) 0 0 0.2 0 0.5 ( , ) 0

0 0 0 0 0 0 0 0 0

30 40 70 [50, 1501

20 30 40 70

j.Qd)

0 0 4.4(0 0 0 0

Notes: The numbers in parentheses represent the nonoptimal solutions. The missing data (/; - 70, MAXPROP2) correspond to cases where the instances could not be solved to optimality. A question of particular interest and a focus of this article is whether the three models produce significantly different solutions. In general, there arc two main ways of comparing the solutions with each other. One possibility is to compare the point patterns generated by the different methods. However, most work on the subject deals with testing whether or not a given point pattern could be the result of a random process; sec for example, Boots and Getis (1988) and Cressic (1991). Yet a suitably modified version of the famed 'nearest neighbor method', described in general in Cressie's book, could be employed to compare the patterns generated by any pair of the three methods discussed in this paper. The main feature that allows the comparison of any pair of solutions is that the same number of facilities is located in both cases, thus allowing pairing. This can be accomplished by determining a perfect matching that minimizes the sum of distances between the facility locations of the two patterns. It must be mentioned though that the average nearest neighbor distance by itself has limited meaning even if related to the size of the space the facilities are located in. Its main value is that it allows comparison between other pairs of solutions. Limited experiments on some small examples, carried out in this study, suggest that the locational patterns generated by the different models do indeed differ. Another way to compare the solutions of the three problems in this paper is to compare their objective function values. The main advantage of this approach is its simplicity and the fact that the objective value, typically the cost of profit associated with a solution, is what really matters to planners. In other words, how far off would a planner be if he or she wrongly assumes that customers behave according to model ux when in fact they behave according to model w2? To answer this, we randomly generated 50 instances on a 49-vertex square grid, with p — 4, r = 4, and uniform demand distributions in [90, 100] or [50, 150]. The instances were obtained by generating several different sets ofp vertices for firm B. The results of these experiments are reported in table 3 (see over). Here, each instance is solved to


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Table 3. Comparison of the objective; value for the three models (discrete uniform point distribution). Demand

Model solved to optimality

Model used to compute the solution MAXCAP ( v = 1)

MAXPROP1 (v = 2)

MAXPROP2 (v = 3)

[90, 100]

MAXCAP (u = 1) MAXPROP1 (u = 2) MAXPROP2 (i# = 3)

1.000 1.211 1.264

1.071 1.000 1.023

1.063 1.015 1.000

[50, 150]

MAXCAP (w = 1) MAXPROP1 (u = 2) MAXPROP2 {u = 3)

1.000 1.159 1.224

1.052 1.000 1.020

1.047 1.011 1.000

optimality for each model and the objective function of the other two models is computed at the optimum. If X* denotes the optimal solution of model u, and zv (X) is the objective function value of model v corresponding to solution X, then an entry buv in table 3 is the average value over all 50 instances of the ratio zv(X*)/zu(X*). The main conclusion to be drawn from table 3 is that the MAXCAP model is more robust on average than the other two. If it is applied when in fact either proportional model should have been used, this results in a departure from optimality of up to 7% on test problems with demand generated in [90, 100], and slightly less for demand in [50, 150]. In order to assess the robustness of this result with respect to customer distribution in the plane, we generated instances with a stronger spatial structure, in which customers tend to be clustered around some fixed centers, as is often the case in practice. Here we used the generation procedure proposed by Cordeau et al (1997). It uses three parameters: (1) n, the number of customers to be generated; (2) t, the number of centers; and (3) 0 a density parameter. A larger value of 0 yields more concentration of the vertices around the centers. The procedure can be described as follows. Step 1 (Generation of t centers). Randomly generate t centers in the [—50, 50]2 square according to a continuous uniform distribution and set i: = 1. Step 2 (Generation of n customers). For / < n, randomly generate a customer i in the [—100, 100]2 square according to a continuous uniform distribution and compute its distance h to the nearest center. Let 2 be a number randomly chosen in [0, 1] according to a continuous uniform distribution. If z < e~ # , set i: = z + 1. Otherwise delete i. Put simply, for a tentatively generated customer i not to be deleted, a high value of -median algorithm, and we generated demands in [90, 100] and in [50, 150]. Fifty instances were generated for each combination of the input parameters. Average computational results presented in table 4 are consistent with those of table 3. These experiments confirm the robustness of the MAXCAP model. In other words, if the planner assumed erroneously that consumers considered the facilities and goods completely homogeneous and hence used the winner-takes-all rule whereas in fact they did not, the penalty of such a wrong assumption would be very small. In contrast, if planners assumed that customers considered the goods and facilities as (somewhat) heterogeneous and consequently used one of the proportional models whereas in fact they did not, the resulting error would be considerably larger. This clearly indicates that in case of doubt regarding customer behavior, it is safer to employ the usual winner-take-all rule. It is worth pointing out that, although the penalty for applying a model which works with the wrong behavioral assumption is small so far as the objective value is concerned,

749

Market capture models under various customer choice rules

luhle 4. ( ' o m p a r i s o n of the objective value for the three models (cxponentta point d i s t r i b u t i o n ) . Demand

(90, 100]

Density

M o d e l solved

0

to o p t i m a l l y

0.01

o.os 0.10

Itt). I5()|

0.01

0.05

0.10

M A X C A P (u MAXPROPI MAXPROP2 M A X C A P (// MAXPROP! MAXPROP2 M A X C A P (// MAXPROP1 MAXPROP2 M A X C A P (it MAXPROP1 MAXPROP2 M A X C A P (u MAXPROP! MAXPROP2 M A X C A P (// MAXPROPI MAXPROP2

M o d e l used to compute the s o l u t i o n

1) (// (u

2) 3) 1)

(u (M

2) 3) 1)

(i/ [u

2) 3) 1)

(u (u

2) 3) 1)

(// (i/

2) 3) 1)

(// (//

2) 3)

MAXCAP (v 1)

MAXPROPI (v 2)

(»•

1.000 1.117 1.184 1.000 1.143 1.181 1.000 1.147 1.145

1.050 1.000 1.020 1.031 1.000 1.014 1.044 1.000 1.016

1.037 1.012 1.000 1.036 1.009 1.000 1.030 1.008 1.000

1.000 1.117 1.191 1.000 1.163 1.169 1.000 1.167 1.157

1.049 1.000 1.019 1.037 1.000 1.016 1.053 1.000 1.016

1.046 1.013 1.000 1.036 1.012 1.000 1.037 1.104 1.000

MAXPROIM 3)

the locational pattern is a different matter. In fact, we suspect that one of the reasons for the robustness of the model with the winner-takes-all assumption appears to be its similarity to the /^median model which, as is well known, is very flat in the vicinity of the optimal solution. 4 Conclusions We have investigated two new models for the location of facilities in a competitive context. Contrary to the standard maximum capture model (MAXCAP) where customers always patroni/e the closest facility, the new models assume customer demand that is shared among the existing facilities according to some function of proximity The first of these models was shown to have a structure very similar to that of MAXCAP, whereas the second is nonlinear and more intricate. An efficient heuristic was proposed to solve these models. On test problems it consistently yields optimal or near-optimal solutions within very modest computing times. A further analysis reveals that the MAXCAP model is more robust than the other two and that this observation holds for various customer spatial distributions and demand distributions. It is recommended in the absence of accurate information on customer behavior. Competitive location is a difficult field not only because it involves rather complex mathematical models, but also because customer behavior cannot easily be transcribed into neat equations. The models we have provided can be described only as approximations and we believe this subject is not yet fully understood. For example, the question of how customers select a grocery store or a supermarket for occasional or weekly shopping is far from simple. Clearly one does not always patronize the closest nor the most attractive facility. Price and distance certainly come into play, but exactly in which way? This question becomes even more complicated if several firms occupy a market or if customers have different objectives and constraints, which is the case in practice. More research is needed on this aspect of customer behavior modeling. However, as demonstrated in this paper, the competitive medianoid models are very

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robust in the sense that the winner-takes-all assumption, while oversimplifying behavioral patterns, may be employed even though its accuracy cannot be asserted. Acknowledgements. This work was partially funded by the Natural Sciences and Engineering Research Council of Canada under grants OGP0009160 and OGP0039682. Their support is gratefully acknowledged. Thanks are due to two anonymous referees for their valuable comments. References Abernathy W J, Hershey J C, 1972, "A spatial-allocation model for regional health-services planning" Operations Research 20 629-642 Baxter M J, 1979, "The interpretation of the distance and attractiveness components in models of recreational trips" Geographical Analysis 11 311-315 Benati S, Laporte G, 1994, "Tabu search algorithms for the (r | X/3)-medianoid and (r \p)centroid problems" Location Science 2 193-204 Boots B N, Getis A, 1988 Point Pattern Analysis (Sage, Newbury Park, CA) Cordeau J-F, Gendreau M, Laporte G, 1997, "A Tabu search heuristic for periodic and multi-depot vehicle routing problems" Networks 30 105-119 Cressie N, 1991 Statistics for Spatial Data (John Wiley, New York) Eiselt H A, Laporte G, 1989, "The maximum capture problem in a weighted network" Journal of Regional Science 29 433-439 Eiselt H A, Laporte, 1998, "Demand allocation functions" Location Science 6 175 -187 Eiselt H A, Laporte G, Thisse J-F, 1993, "Competitive location models: a framework and bibliography" Transportation Science 27 44-54 Glover F, 1986, "Future paths for integer programming and links to artificial intelligence" Computers and Operations Research 13 533 - 549 Glover F, Laguna M, 1997 Tabu Search (Kluwer, Boston, MA) Goodchild M F, 1978, "Spatial choice in location-allocation problems: the role of endogenous attraction" Geographical Analysis 10 5 - 72 Hakimi S L, 1983, "On locating new facilities in a competitive environment" European Journal of Operational Research 12 29 - 35 Hakimi S L, 1986, "/^-Median theorems for competitive location" Annals of Operations Research 5 79-88 Hakimi S L, 1990, "Location with spatial interactions: competitive location and games", in Discrete Location Theory Eds R L Francis, P B Mirchandani (John Wiley, New York) pp 439-478 Hansen P, Labbe M, Peeters D, Thisse J-F, 1987, "Facility location analysis", in Systems of Cities and Facility Location Ed. R Arnott (Harwood Academic, London) pp 1 - 7 0 Hodgson M J, 1978, "Toward more realistic allocation in location-allocation models: An interaction approach" Environment and Planning A 10 1273 -1285 Huff D L, 1964, "Defining and estimating a trade area" Journal of Marketing 28 31 - 38 Murtagh B A, Saunders M A, 1991, "MINOS 5.1 User's Guide", technical report, Department of Operations Research, Stanford University, Stanford, CA Reilly W, 1929 Methods for the Study of Retail Relationships Monograph 4 Bureau of Business Research, University of Texas, Austin, TX ReVelle C S, 1986, "The maximum capture or sphere of influence problem: Hotelling revisited on a network" Journal of Regional Science 26 343 - 357 ReVelle C S, Serra D, 1991, "The maximum capture problem including relocation" Information and Operations Research 29 130 -138 ReVelle C S, Swain R, 1970, "Central facilities location" Geographical Analysis 2 31 - 4 2 Serra D, ReVelle C S, 1994, "Market capture by two competitors: the preemptive capture problem" Journal of Regional Science 34 549 - 561 Serra D, ReVelle C S, 1995, "Competitive location in discrete space", in Facility Location: A Survey of Methods and Applications Ed. Z Drezner, Springer Series in Operations Research (Springer, Berlin) pp 367-386 Serra D, Marianov V, ReVelle C S, 1992, "The hierarchical maximum capture problem" European Journal of Operational Research 62 363 - 371 Serra D, Ratick S, ReVelle C S, 1996, "The maximum capture problem with uncertainty" Environment and Planning B: Planning and Design 23 49 - 59 Stackelberg H von, 1952 The Theory of the Market Economy translated from the German by A T Peacock (William Hodge, London)

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