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Allocation of discrete demand with changing costs
Computers & Operations Research 26 (1999) 1335}1349
Allocation of discrete demand with changing costs Zvi Drezner *, George O. Wesolowsky Department of Management Science/Information Systems, California State University-Fullerton, Fullerton, CA 92834, USA Faculty of Business, McMaster University, Hamilton, Ont., Canada L8S 4M4
Abstract This paper investigates the allocation of discrete demand among facilities by stipulating that the unit mill price charged to users by a facility is a function of the total number of users patronizing that facility. This method of allocating customers to facilities can be used, in conjunction with global search strategies, to "nd the best location for a new facility. 1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: Location; Location}allocation; Competitive
1. Introduction The location}allocation problem (usually called the p-median problem for network location problems) is very well known. The problem is to "nd p locations for p facilities servicing a given set of demand points. Each demand point gets its service from the closest facility, and the objective is to minimize the sum of weighted distances between the demand points and their servicing facility. The "rst e$cient algorithm for the solution of this problem on a network was DUALOC by Erlenkotter [1]. This algorithm was improved by Captivo [2]. The book by Ghosh and Rushton [3] provides a review of location}allocation models. Although most location}allocation models assume that customers patronize the closest facility, several modi"cations of this assumption have appeared in the literature. One such modi"cation is in hierarchical p-median problems, where it is assumed that there are several types of facilities, and customers may get their services from a facility which is farther away if its hierarchy is higher [4,5]. Another modi"cation is that weights may be assigned to facilities [6,7]. Alternatively, Drezner [8] assumes that facilities di!er in their attractiveness and customers do not
* Corresponding author. Tel.: #1-714-278-2212; fax: #1-714-278-5940. E-mail address: [email protected] (Z. Drezner) 0305-0548/99/$ - see front matter 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 5 4 8 ( 9 9 ) 0 0 0 3 9 - 8
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necessarily patronize the closest facility. They may by-pass a closer facility to patronize a farther but more attractive one. Competitive models on a network, where the facilities compete for customers, were introduced by Hakimi [9}11] and discussed in many papers; see for example, Darzentas and Bairaktaris [12], Dobson and Karmakar [13] and Hamilton et al. [14]. Chung and Tcha [15] discuss the multiple objective of expenditure minimization and supply maximization. Hotelling [16] in his classical paper, introduced the concept of price-location competition. Many recent papers consider prices as part of the competition model; see, for example, Granot [17], Hanjoul et al. [18], Logendran and Terrel [19], Thisse and Wildasin [20] and Ye and Yezer [21]. This paper uses a new approach for di!erentiating between facilities which is most appropriate for a system of public facilities. This model was "rst suggested by Drezner and Wesolowsky [22] for the special case of location on a line, and extended to continuous demand in an area in Drezner and Wesolowsky [23]. It is assumed that the price charged by a facility is dependent on the number of customers attracted to that facility and customers select the facility with the lowest delivered price. This is the basic idea of economies of scale, which is fundamental to competition. The more customers patronize a facility the lower is the mill price charged to all customers. Another approach to the application of economies of scale is presented in Brimberg and Love [24]. It assumes that the capacities of facilities are decision variables, and the setup cost of a facility is a concave function of the capacity. In their model, economies of scale enter the calculation of the setup cost directly, while in our model, costs per customer decrease because they are shared by a larger number of customers; this decrease is re#ected in the price charged to customers. In the model suggested in this paper, facilities update their prices periodically (say, monthly). The mill price charged in a given month depends on the number of customers patronizing the facility in the previous month. Customers base their selection of a facility on the total cost } the cost of transportation (proportional to the distance to the facility) and the mill price charged by the facility. By this assumption customers do not necessarily patronize the closest facility. Attractiveness of a facility is not an external given attribute but is determined by the number of customers patronizing the facility. Common economic thought assumes a sophisticated manager who tries to assess his competition. Many economic models even consider anticipated response by the competition. We assume an environment with imperfect information about the competition which concentrates on the current pro"tability of operations. Operating costs are passed on to the customers. Speci"cally, the operating costs are divided by the number of customers in the immediately preceding period, and after a pro"t margin is added, determine the price. This myopic approach is typical of many operations. This paper investigates a dynamic environment of such decision makers. The choice of the best facility for customers is now determined by price changes caused by the choices of other customers. We investigate the problem of "nding stable con"gurations, if such exist. A version of this problem, where demand is continuously spread on a line, is discussed in Drezner and Wesolowsky [22]. We now deal with some of the complications that occur when a "nite number of demand points on a plane are separated with Euclidean distances. Several analytical results are developed and then additional possibilities are explored by simulation. The problem is quite complex and many equilibrium solutions are possible.
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2. The model 2.1. Dexnitions Let E E E E E E E E E
N be the number of demand points, p be the number of facilities, n be the number of customers associated with demand point i for i"1,2, N, G d be the distance between demand point i and facility j for i"1,2, N; j"1,2, p, GH w be the weight converting distance to cost for a customer located at demand point i for G i"1,2, N, d be a 0}1 variable. In a certain partition d "1 if demand point i selects facility j, and is equal GH GH to 0 otherwise, h be the number of customers attracted to facility j for j"1,2, p. h " , d n , H H G GH G c be the marginal cost, including a "xed pro"t, of serving a customer at facility j for j"1,2, p, H and g be the setup cost shared among customers at facility j for j"1,2, p. This variable cost is H divided evenly among the customers. The mill price charged to a customer patronizing facility j is c #g /h . H H H
2.2. Finding the allocation of customers to facilities Facilities change their prices periodically (say the "rst of every month). They base their prices on the number of customers that patronized the facility in the previous month. It is assumed that a customer will switch to another facility if the delivered price by that facility is lower than the delivered price of the facility he is currently patronizing. Customer i will prefer facility j to facility k if: g g w d #c # H(w d #c # I G GI I h G GH H h I H
(1)
There are many possible allocations which are `stablea (i.e., no customer changes his selected facility). As an illustration, consider a problem with two facilities and two demand points each having one customer who purchase one unit each. The distances between the demand points and the facilities are listed in Table 1. The marginal cost at each facility is c"10, the setup cost is g"2, and w "1 ∀i. If both demand G points select facility 1, the mill price charged is 11 and the total cost is 20.3 for demand point 1 and 20.6 for demand point 2. The present mill price charged by facility 2 is in"nite, so no one changes its selection. A similar result is obtained when both demand points select facility 2. If demand point 1 selects facility 1 and demand point 2 selects facility 2, then no switching will take place because presently both facilities charge a mill price of are 12 and each customer selects its closest facility. Note that consumers base their decision on current pricing by facilities and do not anticipate price changes caused by their action. Therefore, even though a switch of demand point 2 to facility 1 will
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Z. Drezner, G.O. Wesolowsky / Computers & Operations Research 26 (1999) 1335}1349 Table 1 Distances for the example problem Demand point
Facility
Distance
1 1 2 2
1 2 1 2
9.3 9.4 9.6 9.5
reduce the delivered price for both, such a change is not considered by consumers. If demand point 2 selects facility 1 and demand point 1 selects facility 2, they will exchange selections simultaneously and stay there. In conclusion, there are three stable con"gurations. The following process is suggested for determining the partition: 1. Each demand point selects the facility for which w d #c is minimized. (Alternatively, the initial G GH H selection can be based on expected or randomly generated n for each existing facility.) 2. Mill prices are calculated and facilities determine new mill prices simultaneously. 3. Repeat until stabilization. Extensive computational experiments were performed and the algorithm always terminated. No non-convergence was encountered in over a million cases. We could not prove convergence, however the following Lemma 1 rules out the most frequent non-convergence cases. The Lemma is proved for general cost functions. Let the cost function for facility j be c (h ). The function c (h ) is H H H H monotonically decreasing in h . Let h (k) be the total number of customers patronizing facility j at H H iteration k. If h (k#1)"h (k) for all j in a speci"c iteration k, then the algorithm terminates. We H H also de"ne * (k)"!+c [h (k#1)]!c [h (k)],. Let m(k) be the facility for which * (k) is maxiH H H H H H mized. It is clear that the algorithm terminates if * (k)"0. KI Lemma 1. The algorithm cannot oscillate between two partitions. Proof. Consider facility m(k). Since * (k)'0, at least one demand point selected m(k) in iteration KI k while selecting another facility in iteration k!1. None of the customers selecting m(k) will select another facility in iteration k#1. To show this, we prove that if for demand point i w d #c [h (k)]*w d #c [h (k)], then w d #c [h (k#1)]*w d #c [h (k#1)]. G I KI KI G GH H H G I KI KI G GH H H This is clear by the maximality of * (k). Therefore, if a demand point changed its selection to m(k) KI in iteration k, it cannot select the previously selected facility in iteration k#1 and a #ip-#op is not possible. 䊐 Lemma 1 proves convergence for the two-facility case. The following two lemmas prove some properties of the procedure for the particular cost function c (h )"c #g /h . H H H H H
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Lemma 2. If all existing facilities service customers, then the minimal grand total cost for all customers is achieved when customer i selects the facility that minimizes w d #c . G GH H Proof. The grand total cost for all customers is
N , N g , N H d n w d #c # " d n [w d #c ]# g . GH G G GH GH G G GH H H H , d n G GH H H G H G H The last term of the equation is constant. Therefore, the grand total is minimized when customer i selects the facility that minimizes w d #c . 䊐 G GH H Note that such a selection may not be a stable one.
3. Computational experiments We tested 1000 randomly generated problems each with 100 demand points and 10 existing facilities in a square whose side is 10 miles. The weights w were randomly generated in [0, 1], the G marginal costs c were randomly generated in the segment [10,12], and the setup costs g were "xed H H at 5. This set of parameters provided middle ground solutions as reported in Table 2. Much smaller values for g retained all 10 facilities in most cases, and much larger values of g resulted in only one active facility in the majority of cases. See also Fig. 6 for further analysis of the sensitivity of the number of active facility as a function of g. The algorithm was started by the selection rule de"ned in Lemma 2. The algorithm converged in all 1000 cases. The number of iterations ranged between 2 and 15 with a mean of 4.859. In 442 cases the grand total cost for all customers decreased from the "rst assignment and in the other 558 cases it increased. The minimum change was !0.616% of the total cost, the maximum change was #1.086% and the average change was #0.037%. These changes are quite insigni"cant.
Table 2 Frequency of the number of active facilities for 1000 problems Number of active facilities
Frequency
1 2 3 4 5 6 7 8 9 10
0 25 232 444 247 50 2 0 0 0
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Z. Drezner, G.O. Wesolowsky / Computers & Operations Research 26 (1999) 1335}1349 Table 3 Comparing 10 partitions for each problem K
1 2 3 4 5 6 7 8 9 10
Frequency of identical partitions out of 10 K di!erent partitions
K numbers in largest group
504 281 134 56 19 5 1 0 0 0
0 0 7 29 55 73 65 81 186 504
The number of active facilities in the "nal partition was also recorded. In no case were there more than seven facilities with one or more customers. That means that at least three facilities disappeared. In all 1000 cases there were at least two facilities that remained active. The number of active facilities are reported in Table 2. Since there might be several stable con"gurations (which are terminal partitions), additional experiments were performed on this set of randomly generated problems. Values for the initial number of demand points attracted to each facility were randomly generated in [5,15] (as a continuous number). Customers initially selected the facility according to the calculated cost for each facility (based on this randomly generated number of customers for each facility.) Each of these 1000 problems was solved 10 times. The 10 resulting partitions were compared. The results are summarized in Table 3. As is clear from Table 3, in 504 cases out of 1000 all 10 partitions were identical. In 186 other cases, 9 out of 10 were identical. In 281 cases there were only two di!erent partitions. In one case, seven di!erent partitions were encountered. 3.1. Number of demand points attracted to facilities as a function of g The relationship between the number of demand points attracted to facilities when each demand point selects its closest facility (i.e., g"0), and the number of demand points attracted to each facility for a positive set-up cost g, was also investigated. We considered problems with 100 demand points located on a grid in a 10;10 square area. The square was divided into 100 small squares, and demand points were placed at the centers of these small squares forming 10 rows of 10 demand points in each row. A number of facilities, p, was randomly generated in the area, and the number of demand points attracted to each facility by the proximity rule was recorded. The partition of the demand points at equilibrium was also recorded. This process was repeated 1000 times yielding 1000p pairs of original number of demand points determined by proximity, and "nal number of
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demand points determined at equilibrium. These pairs are depicted in Figs. 1}5 for various values of g and p. These "gures indicate that for p"2 facilities, most of the pairs are near the diagonal, which means that in many con"gurations facilities retain about the same number of demand points. However, as g increases, the change in the number of attracted demand points is more signi"cant and there are more cases in which one facility captures all demand points as indicated by the number of 100 demand points or 0 demand points captured by facilities at equilibrium. For more than two facilities (Figs. 4 and 5) the original number of customers is usually low but in many cases facilities attract at equilibrium more customers than they did originally and many facilities are reduced to no customers. When the number of customers at a facility was reduced, it was usually reduced to no customers at all. In both cases, there are many instances in which one facility captures all customers at equilibrium. 3.2. The delineation of the catchment area The delineation of the catchment area as a function of the variable cost was also investigated. We used the example problem in Drezner [8], which consists of 100 demand points evenly spread in a 10;10 mile square (a demand point is located at the center of each 1;1 mile small square), and seven facilities located as depicted in Figs. 6 and 7, with coordinates given in Table 4. All the marginal costs were assumed the same for all facilities as long as they are the same, their value is irrelevant to the problem. The experiments were conducted with values of g"0, 5, 10,2, 40 for the
Fig. 1. Original vs. "nal number of customers with g"10 and 2 facilities.
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Fig. 2. Original vs. "nal number of customers with g"25 and 2 facilities.
Fig. 3. Original vs. "nal number of customers with g"40 and 2 facilities.
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Fig. 4. Original vs. "nal number of customers with g"40 and 5 facilities.
Fig. 5. Original vs. "nal number of customers with g"40 and 10 facilities.
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Fig. 6. The delineation of the catchment areas.
setup costs. Note that g"0 is the well-researched (stationary) location}allocation problem where each demand point selects the closest facility. In Table 5, the number of demand points attracted to each facility are presented. In Fig. 6, the delineations of demand points are depicted. Each demand point is at the center of a small square whose side is one mile. In Fig. 6, if a demand point patronizes a facility, all the points inside the small square are assumed to patronize the same facility. Therefore, the catchment area of a facility is depicted as a union of such small squares. As g increases, facilities 5 and 6 are `taken out of businessa. For g"20 facility 2 is eliminated and only facilities 1, 3, 4, and 7 remain in business. When g is further increased, Facility 4 is eliminated and
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Fig. 7. Best locations for a new facility obtained by a random selection.
Table 5 Demand points captured as a function of g Value of g
Table 4 Locations of existing facilities Facility
x
y
1 2 3 4 5 6 7
7 5 3 3 1 9 8
1 5 3 9 7 2 8
0 5 10 15 20 25 30 35 40
Facility 1
2
3
4
5
6
7
9 11 7 19 18 18 18 14 0
13 13 13 13 0 0 0 57 100
22 22 26 26 35 35 40 16 0
11 13 13 19 21 21 15 0 0
12 10 8 0 0 0 0 0 0
12 10 12 0 0 0 0 0 0
21 21 21 23 26 26 27 13 0
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facility 2 re-appears, attracting more than half of the demand points. When g is increased to 40, facility 2 is the only one in business attracting all the demand points. The disappearance and re-appearance of facility 2 as an attractive facility is very interesting. It should be noted that for each value of g the problem was solved from scratch. The solution will not have the same pattern if the "nal partition for a certain g is used as a starting partition for the next g. If that was the process, then a facility that goes out of business cannot be resurrected.
4. The location problem We are required to locate a new facility in the area. The objective of the new facility is to attract as many customers as possible once a stable conxguration is reached. This is the common max-cover [25] which is sometimes called the max-capture [26] objective. When adding an additional facility, two approaches for determining the initial partition are plausible: (1) allocating 1/(p#1) of the customers to the new one, and reduce the allocation of existing ones by multiplying their present allocation by p/(p#1) and (2) allocating 1/(p#1) of the customers to each facility whether existing or new. Although the resulting stable con"guration depends on the initial partition, the computational experience is expected to be similar by any rule of initial partition. The second approach was used in the computational experiments because of its simplicity. The same example problem was used for this investigation. Ten thousand possible locations were randomly generated in the square. The number of demand points attracted at each location was found. In Table 6 the frequency distribution of the number of points attracted to the new facility is given. Table 6 Number of demand points attracted by the new facility Number of points attracted
Frequency
3 4 5 6 7 8 9 10 11 12 13 14 15 16 19 20 21
203 307 832 988 640 645 573 428 237 210 62 63 15 7 10 31 63
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Fig. 8. Best locations for a new facility obtained by a grid pattern.
In Fig. 7 the 63 locations that attract 21 demand points are depicted. In Fig. 8 we depicted the area where all optimal locations exist by generating a grid of one million points and depicting the 6938 points that attract 21 or more customers (there were a few cases of 22 customers). Fig. 8 zooms in on a portion of the area where all the best locations are concentrated. The "gure has some interesting features. It seems to contain four disjoint areas. It is interesting that an empty diagonal `corridora exists between the two main areas. There are no highly attractive locations in that corridor. This example demonstrates that "nding the locus of all points that attract the most demand is a contrived problem. It is not expected that a simple approach can de"ne the area of all best locations.
5. Conclusions A modi"cation of the location}allocation model was formulated and analyzed. It is assumed that the mill price charged to customers declines as more customers patronize a facility. We designed a simple algorithm that "nds stable con"gurations. We also analyzed the problem of "nding
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a location for a new facility that maximizes the number of customers captured by the new facility. Computational experiments that demonstrate the problem and its solution are presented. The resulting partition is very sensitive to the cost parameters. As the setup cost shared by customers is increased, fewer facilities survive the competition. For a large setup cost, usually only one facility captures the whole market and the other facilities disappear. A setup cost of zero is equivalent to the standard location}allocation model in which all the facilities stay in business. Figs. 1}5 show that the "nal number of customers is positively correlated with the initial number of customers. It is intuitive that the chance for survival depends on the initial number of customers. A facility that attracts more customers initially should have a better survival chance. However, this may not hold in general. In the example problem (see Fig. 6 and Table 5), facilities 3 and 7 attract the most customers initially (22 and 21, respectively). However, for a large g, the surviving facility is facility 2 which initially attracts only 13 customers. The reason for that might be that facility 2 is better located at the center of the area. Extensions to this problem are still open for investigation. The versions of the problem de"ned on a line, and in a continuous space are already analyzed in other papers. Other environments for this setting, such as location on a network, are still open. Some analysis of the problem of "nding the best location for a new facility which attracts the most number of customers is provided. More research is required to provide a more complete analysis for this location problem.
Acknowledgements This research was supported, in part, by the Natural Sciences and Engineering Research Council of Canada.
References [1] Erlenkotter D. A dual based procedure for uncapacitated facility location. Operations Research 1978;26: 992}1009. [2] Captivo ME. Fast primal and dual heuristics for the p-median location problem. European Journal of Operational Research 1991;52:65}74. [3] Ghosh A, Rushton G, editors. Spatial analysis and location}allocation models. New York: Van Nostrand Reinhold, 1987. [4] Karkazis J. Facilities location in a competitive environment: a promethee based multiple criteria analysis. European Journal of Operational Research 1989;42:294}304. [5] Serra D, Marianov V, ReVelle C. The maximum-capture hierarchical location problem. European Journal of Operational Research 1992;62:363}71. [6] Eiselt HA, Laporte G. The maximum capture problem in a weighted network. Journal of Regional Science 1989;29:433}9. [7] Eiselt HA, Laporte G. Competitive spatial models. European Journal of Operational Research 1989;39:231}42. [8] Drezner T. Locating a single new facility among existing, unequally attractive facilities. Journal of Regional Science 1994;34:237}52. [9] Hakimi SL. On locating new facilities in a competitive environment. European Journal of Operational Research 1983;12:29}35. [10] Hakimi SL. p-Median theorems for competitive locations. Annals of Operations Research 1986;6:77}98.
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[11] Hakimi SL. Locations with spatial interactions: competitive locations and games. In: Mirchandani PB, Francis RL, editors. Discrete location theory. Wiley-Interscience Series in Discrete Mathematics and Optimization, New York: Wiley-Interscience. [12] Darzentas J, Bairaktaris P. On the set partitioning type formulation for the discrete location problem. Computers & Operations Research 1986;13:671}9. [13] Dobson G, Karmarkar US. Competitive location on a network. Operations Research 1987;35:565}74. [14] Hamilton JH, MacLeod WB, Thisse J-F. Spatial competition and the core. Quarterly Journal of Economics 1991;106:925}37. [15] Chung KH, Tcha DW. A fuzzy set-theoretic method for public facility location. European Journal of Operational Research 1992;58:90}8. [16] Hotelling H. Stability in competition. Economics Journal 1929;39:41}57. [17] Granot D. The role of cost allocation in locational models. Operations Research 1987;35:234}48. [18] Hanjoul P, Hansen P, Peeters D, Thisse J-F. Uncapacitated plant location under alternative spatial price policies. Management Science 1990;36:41}57. [19] Logendran R, Terrell MP. Capacitated plant location}allocation problems with price sensitive stochastic demands. Logistics & Transportation Review 1991;27:33}53. [20] Thisse J-F, Wildasin DE. Public facility location and urban spatial structure: equilibrium and welfare analysis. Journal of Public Economics 1992;48:83}118. [21] Ye MH, Yezer AM. Location and spatial pricing for public facilities. Journal of Regional Science 1992;32:143}54. [22] Drezner Z, Wesolowsky GO. Location}allocation on a line with demand-dependent costs. European Journal of Operational Research 1996;90:444}50. [23] Drezner Z, Wesolowsky GO. Location}allocation with continuously distributed demands and demand-dependent costs, submitted for publication. [24] Brimberg J, Love RF. A location problem with economies of scale. Studies in Locational Analysis 1994;(Issue 7). [25] Church RL, ReVelle C. The maximal covering location problem. Papers of the Regional Science Association 1974;32:101}18. [26] ReVelle C. The maximum capture or sphere of in#uence problem: hotelling revisited on a network. Journal of Regional Science 1986;26:343}57.
Allocation of discrete demand with changing costs Zvi Drezner *, George O. Wesolowsky Department of Management Science/Information Systems, California State University-Fullerton, Fullerton, CA 92834, USA Faculty of Business, McMaster University, Hamilton, Ont., Canada L8S 4M4
Abstract This paper investigates the allocation of discrete demand among facilities by stipulating that the unit mill price charged to users by a facility is a function of the total number of users patronizing that facility. This method of allocating customers to facilities can be used, in conjunction with global search strategies, to "nd the best location for a new facility. 1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: Location; Location}allocation; Competitive
1. Introduction The location}allocation problem (usually called the p-median problem for network location problems) is very well known. The problem is to "nd p locations for p facilities servicing a given set of demand points. Each demand point gets its service from the closest facility, and the objective is to minimize the sum of weighted distances between the demand points and their servicing facility. The "rst e$cient algorithm for the solution of this problem on a network was DUALOC by Erlenkotter [1]. This algorithm was improved by Captivo [2]. The book by Ghosh and Rushton [3] provides a review of location}allocation models. Although most location}allocation models assume that customers patronize the closest facility, several modi"cations of this assumption have appeared in the literature. One such modi"cation is in hierarchical p-median problems, where it is assumed that there are several types of facilities, and customers may get their services from a facility which is farther away if its hierarchy is higher [4,5]. Another modi"cation is that weights may be assigned to facilities [6,7]. Alternatively, Drezner [8] assumes that facilities di!er in their attractiveness and customers do not
* Corresponding author. Tel.: #1-714-278-2212; fax: #1-714-278-5940. E-mail address: [email protected] (Z. Drezner) 0305-0548/99/$ - see front matter 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 5 4 8 ( 9 9 ) 0 0 0 3 9 - 8
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necessarily patronize the closest facility. They may by-pass a closer facility to patronize a farther but more attractive one. Competitive models on a network, where the facilities compete for customers, were introduced by Hakimi [9}11] and discussed in many papers; see for example, Darzentas and Bairaktaris [12], Dobson and Karmakar [13] and Hamilton et al. [14]. Chung and Tcha [15] discuss the multiple objective of expenditure minimization and supply maximization. Hotelling [16] in his classical paper, introduced the concept of price-location competition. Many recent papers consider prices as part of the competition model; see, for example, Granot [17], Hanjoul et al. [18], Logendran and Terrel [19], Thisse and Wildasin [20] and Ye and Yezer [21]. This paper uses a new approach for di!erentiating between facilities which is most appropriate for a system of public facilities. This model was "rst suggested by Drezner and Wesolowsky [22] for the special case of location on a line, and extended to continuous demand in an area in Drezner and Wesolowsky [23]. It is assumed that the price charged by a facility is dependent on the number of customers attracted to that facility and customers select the facility with the lowest delivered price. This is the basic idea of economies of scale, which is fundamental to competition. The more customers patronize a facility the lower is the mill price charged to all customers. Another approach to the application of economies of scale is presented in Brimberg and Love [24]. It assumes that the capacities of facilities are decision variables, and the setup cost of a facility is a concave function of the capacity. In their model, economies of scale enter the calculation of the setup cost directly, while in our model, costs per customer decrease because they are shared by a larger number of customers; this decrease is re#ected in the price charged to customers. In the model suggested in this paper, facilities update their prices periodically (say, monthly). The mill price charged in a given month depends on the number of customers patronizing the facility in the previous month. Customers base their selection of a facility on the total cost } the cost of transportation (proportional to the distance to the facility) and the mill price charged by the facility. By this assumption customers do not necessarily patronize the closest facility. Attractiveness of a facility is not an external given attribute but is determined by the number of customers patronizing the facility. Common economic thought assumes a sophisticated manager who tries to assess his competition. Many economic models even consider anticipated response by the competition. We assume an environment with imperfect information about the competition which concentrates on the current pro"tability of operations. Operating costs are passed on to the customers. Speci"cally, the operating costs are divided by the number of customers in the immediately preceding period, and after a pro"t margin is added, determine the price. This myopic approach is typical of many operations. This paper investigates a dynamic environment of such decision makers. The choice of the best facility for customers is now determined by price changes caused by the choices of other customers. We investigate the problem of "nding stable con"gurations, if such exist. A version of this problem, where demand is continuously spread on a line, is discussed in Drezner and Wesolowsky [22]. We now deal with some of the complications that occur when a "nite number of demand points on a plane are separated with Euclidean distances. Several analytical results are developed and then additional possibilities are explored by simulation. The problem is quite complex and many equilibrium solutions are possible.
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2. The model 2.1. Dexnitions Let E E E E E E E E E
N be the number of demand points, p be the number of facilities, n be the number of customers associated with demand point i for i"1,2, N, G d be the distance between demand point i and facility j for i"1,2, N; j"1,2, p, GH w be the weight converting distance to cost for a customer located at demand point i for G i"1,2, N, d be a 0}1 variable. In a certain partition d "1 if demand point i selects facility j, and is equal GH GH to 0 otherwise, h be the number of customers attracted to facility j for j"1,2, p. h " , d n , H H G GH G c be the marginal cost, including a "xed pro"t, of serving a customer at facility j for j"1,2, p, H and g be the setup cost shared among customers at facility j for j"1,2, p. This variable cost is H divided evenly among the customers. The mill price charged to a customer patronizing facility j is c #g /h . H H H
2.2. Finding the allocation of customers to facilities Facilities change their prices periodically (say the "rst of every month). They base their prices on the number of customers that patronized the facility in the previous month. It is assumed that a customer will switch to another facility if the delivered price by that facility is lower than the delivered price of the facility he is currently patronizing. Customer i will prefer facility j to facility k if: g g w d #c # H(w d #c # I G GI I h G GH H h I H
(1)
There are many possible allocations which are `stablea (i.e., no customer changes his selected facility). As an illustration, consider a problem with two facilities and two demand points each having one customer who purchase one unit each. The distances between the demand points and the facilities are listed in Table 1. The marginal cost at each facility is c"10, the setup cost is g"2, and w "1 ∀i. If both demand G points select facility 1, the mill price charged is 11 and the total cost is 20.3 for demand point 1 and 20.6 for demand point 2. The present mill price charged by facility 2 is in"nite, so no one changes its selection. A similar result is obtained when both demand points select facility 2. If demand point 1 selects facility 1 and demand point 2 selects facility 2, then no switching will take place because presently both facilities charge a mill price of are 12 and each customer selects its closest facility. Note that consumers base their decision on current pricing by facilities and do not anticipate price changes caused by their action. Therefore, even though a switch of demand point 2 to facility 1 will
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Z. Drezner, G.O. Wesolowsky / Computers & Operations Research 26 (1999) 1335}1349 Table 1 Distances for the example problem Demand point
Facility
Distance
1 1 2 2
1 2 1 2
9.3 9.4 9.6 9.5
reduce the delivered price for both, such a change is not considered by consumers. If demand point 2 selects facility 1 and demand point 1 selects facility 2, they will exchange selections simultaneously and stay there. In conclusion, there are three stable con"gurations. The following process is suggested for determining the partition: 1. Each demand point selects the facility for which w d #c is minimized. (Alternatively, the initial G GH H selection can be based on expected or randomly generated n for each existing facility.) 2. Mill prices are calculated and facilities determine new mill prices simultaneously. 3. Repeat until stabilization. Extensive computational experiments were performed and the algorithm always terminated. No non-convergence was encountered in over a million cases. We could not prove convergence, however the following Lemma 1 rules out the most frequent non-convergence cases. The Lemma is proved for general cost functions. Let the cost function for facility j be c (h ). The function c (h ) is H H H H monotonically decreasing in h . Let h (k) be the total number of customers patronizing facility j at H H iteration k. If h (k#1)"h (k) for all j in a speci"c iteration k, then the algorithm terminates. We H H also de"ne * (k)"!+c [h (k#1)]!c [h (k)],. Let m(k) be the facility for which * (k) is maxiH H H H H H mized. It is clear that the algorithm terminates if * (k)"0. KI Lemma 1. The algorithm cannot oscillate between two partitions. Proof. Consider facility m(k). Since * (k)'0, at least one demand point selected m(k) in iteration KI k while selecting another facility in iteration k!1. None of the customers selecting m(k) will select another facility in iteration k#1. To show this, we prove that if for demand point i w d #c [h (k)]*w d #c [h (k)], then w d #c [h (k#1)]*w d #c [h (k#1)]. G I KI KI G GH H H G I KI KI G GH H H This is clear by the maximality of * (k). Therefore, if a demand point changed its selection to m(k) KI in iteration k, it cannot select the previously selected facility in iteration k#1 and a #ip-#op is not possible. 䊐 Lemma 1 proves convergence for the two-facility case. The following two lemmas prove some properties of the procedure for the particular cost function c (h )"c #g /h . H H H H H
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Lemma 2. If all existing facilities service customers, then the minimal grand total cost for all customers is achieved when customer i selects the facility that minimizes w d #c . G GH H Proof. The grand total cost for all customers is
N , N g , N H d n w d #c # " d n [w d #c ]# g . GH G G GH GH G G GH H H H , d n G GH H H G H G H The last term of the equation is constant. Therefore, the grand total is minimized when customer i selects the facility that minimizes w d #c . 䊐 G GH H Note that such a selection may not be a stable one.
3. Computational experiments We tested 1000 randomly generated problems each with 100 demand points and 10 existing facilities in a square whose side is 10 miles. The weights w were randomly generated in [0, 1], the G marginal costs c were randomly generated in the segment [10,12], and the setup costs g were "xed H H at 5. This set of parameters provided middle ground solutions as reported in Table 2. Much smaller values for g retained all 10 facilities in most cases, and much larger values of g resulted in only one active facility in the majority of cases. See also Fig. 6 for further analysis of the sensitivity of the number of active facility as a function of g. The algorithm was started by the selection rule de"ned in Lemma 2. The algorithm converged in all 1000 cases. The number of iterations ranged between 2 and 15 with a mean of 4.859. In 442 cases the grand total cost for all customers decreased from the "rst assignment and in the other 558 cases it increased. The minimum change was !0.616% of the total cost, the maximum change was #1.086% and the average change was #0.037%. These changes are quite insigni"cant.
Table 2 Frequency of the number of active facilities for 1000 problems Number of active facilities
Frequency
1 2 3 4 5 6 7 8 9 10
0 25 232 444 247 50 2 0 0 0
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1 2 3 4 5 6 7 8 9 10
Frequency of identical partitions out of 10 K di!erent partitions
K numbers in largest group
504 281 134 56 19 5 1 0 0 0
0 0 7 29 55 73 65 81 186 504
The number of active facilities in the "nal partition was also recorded. In no case were there more than seven facilities with one or more customers. That means that at least three facilities disappeared. In all 1000 cases there were at least two facilities that remained active. The number of active facilities are reported in Table 2. Since there might be several stable con"gurations (which are terminal partitions), additional experiments were performed on this set of randomly generated problems. Values for the initial number of demand points attracted to each facility were randomly generated in [5,15] (as a continuous number). Customers initially selected the facility according to the calculated cost for each facility (based on this randomly generated number of customers for each facility.) Each of these 1000 problems was solved 10 times. The 10 resulting partitions were compared. The results are summarized in Table 3. As is clear from Table 3, in 504 cases out of 1000 all 10 partitions were identical. In 186 other cases, 9 out of 10 were identical. In 281 cases there were only two di!erent partitions. In one case, seven di!erent partitions were encountered. 3.1. Number of demand points attracted to facilities as a function of g The relationship between the number of demand points attracted to facilities when each demand point selects its closest facility (i.e., g"0), and the number of demand points attracted to each facility for a positive set-up cost g, was also investigated. We considered problems with 100 demand points located on a grid in a 10;10 square area. The square was divided into 100 small squares, and demand points were placed at the centers of these small squares forming 10 rows of 10 demand points in each row. A number of facilities, p, was randomly generated in the area, and the number of demand points attracted to each facility by the proximity rule was recorded. The partition of the demand points at equilibrium was also recorded. This process was repeated 1000 times yielding 1000p pairs of original number of demand points determined by proximity, and "nal number of
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demand points determined at equilibrium. These pairs are depicted in Figs. 1}5 for various values of g and p. These "gures indicate that for p"2 facilities, most of the pairs are near the diagonal, which means that in many con"gurations facilities retain about the same number of demand points. However, as g increases, the change in the number of attracted demand points is more signi"cant and there are more cases in which one facility captures all demand points as indicated by the number of 100 demand points or 0 demand points captured by facilities at equilibrium. For more than two facilities (Figs. 4 and 5) the original number of customers is usually low but in many cases facilities attract at equilibrium more customers than they did originally and many facilities are reduced to no customers. When the number of customers at a facility was reduced, it was usually reduced to no customers at all. In both cases, there are many instances in which one facility captures all customers at equilibrium. 3.2. The delineation of the catchment area The delineation of the catchment area as a function of the variable cost was also investigated. We used the example problem in Drezner [8], which consists of 100 demand points evenly spread in a 10;10 mile square (a demand point is located at the center of each 1;1 mile small square), and seven facilities located as depicted in Figs. 6 and 7, with coordinates given in Table 4. All the marginal costs were assumed the same for all facilities as long as they are the same, their value is irrelevant to the problem. The experiments were conducted with values of g"0, 5, 10,2, 40 for the
Fig. 1. Original vs. "nal number of customers with g"10 and 2 facilities.
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Fig. 2. Original vs. "nal number of customers with g"25 and 2 facilities.
Fig. 3. Original vs. "nal number of customers with g"40 and 2 facilities.
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Fig. 4. Original vs. "nal number of customers with g"40 and 5 facilities.
Fig. 5. Original vs. "nal number of customers with g"40 and 10 facilities.
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Fig. 6. The delineation of the catchment areas.
setup costs. Note that g"0 is the well-researched (stationary) location}allocation problem where each demand point selects the closest facility. In Table 5, the number of demand points attracted to each facility are presented. In Fig. 6, the delineations of demand points are depicted. Each demand point is at the center of a small square whose side is one mile. In Fig. 6, if a demand point patronizes a facility, all the points inside the small square are assumed to patronize the same facility. Therefore, the catchment area of a facility is depicted as a union of such small squares. As g increases, facilities 5 and 6 are `taken out of businessa. For g"20 facility 2 is eliminated and only facilities 1, 3, 4, and 7 remain in business. When g is further increased, Facility 4 is eliminated and
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Fig. 7. Best locations for a new facility obtained by a random selection.
Table 5 Demand points captured as a function of g Value of g
Table 4 Locations of existing facilities Facility
x
y
1 2 3 4 5 6 7
7 5 3 3 1 9 8
1 5 3 9 7 2 8
0 5 10 15 20 25 30 35 40
Facility 1
2
3
4
5
6
7
9 11 7 19 18 18 18 14 0
13 13 13 13 0 0 0 57 100
22 22 26 26 35 35 40 16 0
11 13 13 19 21 21 15 0 0
12 10 8 0 0 0 0 0 0
12 10 12 0 0 0 0 0 0
21 21 21 23 26 26 27 13 0
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facility 2 re-appears, attracting more than half of the demand points. When g is increased to 40, facility 2 is the only one in business attracting all the demand points. The disappearance and re-appearance of facility 2 as an attractive facility is very interesting. It should be noted that for each value of g the problem was solved from scratch. The solution will not have the same pattern if the "nal partition for a certain g is used as a starting partition for the next g. If that was the process, then a facility that goes out of business cannot be resurrected.
4. The location problem We are required to locate a new facility in the area. The objective of the new facility is to attract as many customers as possible once a stable conxguration is reached. This is the common max-cover [25] which is sometimes called the max-capture [26] objective. When adding an additional facility, two approaches for determining the initial partition are plausible: (1) allocating 1/(p#1) of the customers to the new one, and reduce the allocation of existing ones by multiplying their present allocation by p/(p#1) and (2) allocating 1/(p#1) of the customers to each facility whether existing or new. Although the resulting stable con"guration depends on the initial partition, the computational experience is expected to be similar by any rule of initial partition. The second approach was used in the computational experiments because of its simplicity. The same example problem was used for this investigation. Ten thousand possible locations were randomly generated in the square. The number of demand points attracted at each location was found. In Table 6 the frequency distribution of the number of points attracted to the new facility is given. Table 6 Number of demand points attracted by the new facility Number of points attracted
Frequency
3 4 5 6 7 8 9 10 11 12 13 14 15 16 19 20 21
203 307 832 988 640 645 573 428 237 210 62 63 15 7 10 31 63
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Fig. 8. Best locations for a new facility obtained by a grid pattern.
In Fig. 7 the 63 locations that attract 21 demand points are depicted. In Fig. 8 we depicted the area where all optimal locations exist by generating a grid of one million points and depicting the 6938 points that attract 21 or more customers (there were a few cases of 22 customers). Fig. 8 zooms in on a portion of the area where all the best locations are concentrated. The "gure has some interesting features. It seems to contain four disjoint areas. It is interesting that an empty diagonal `corridora exists between the two main areas. There are no highly attractive locations in that corridor. This example demonstrates that "nding the locus of all points that attract the most demand is a contrived problem. It is not expected that a simple approach can de"ne the area of all best locations.
5. Conclusions A modi"cation of the location}allocation model was formulated and analyzed. It is assumed that the mill price charged to customers declines as more customers patronize a facility. We designed a simple algorithm that "nds stable con"gurations. We also analyzed the problem of "nding
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a location for a new facility that maximizes the number of customers captured by the new facility. Computational experiments that demonstrate the problem and its solution are presented. The resulting partition is very sensitive to the cost parameters. As the setup cost shared by customers is increased, fewer facilities survive the competition. For a large setup cost, usually only one facility captures the whole market and the other facilities disappear. A setup cost of zero is equivalent to the standard location}allocation model in which all the facilities stay in business. Figs. 1}5 show that the "nal number of customers is positively correlated with the initial number of customers. It is intuitive that the chance for survival depends on the initial number of customers. A facility that attracts more customers initially should have a better survival chance. However, this may not hold in general. In the example problem (see Fig. 6 and Table 5), facilities 3 and 7 attract the most customers initially (22 and 21, respectively). However, for a large g, the surviving facility is facility 2 which initially attracts only 13 customers. The reason for that might be that facility 2 is better located at the center of the area. Extensions to this problem are still open for investigation. The versions of the problem de"ned on a line, and in a continuous space are already analyzed in other papers. Other environments for this setting, such as location on a network, are still open. Some analysis of the problem of "nding the best location for a new facility which attracts the most number of customers is provided. More research is required to provide a more complete analysis for this location problem.
Acknowledgements This research was supported, in part, by the Natural Sciences and Engineering Research Council of Canada.
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