A NEW GRAVITY MODEL WITH VARIABLE DISTANCE DECAY Müge Sandıkcıoğlu 1, Özden Gür Ali 2, Serpil Sayın 3

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1 Internationa Conference 20th EURO Mini Conference Continuous Optimization and Knowedge-Based Technoogies (EurOT-2008) May 20 23, 2008, Neringa, LITHUANIA ISBN L. Sakaauskas, G.W. Weber and E. K. Zavadskas (Eds.): EUROT-2008 Seected papers. Vinius, 2008, pp Institute of Mathematics and Informatics, 2008 Vinius Gediminas Technica University, 2008 A NEW GRAVITY MOEL WITH VARIABLE ISTANCE ECAY Müge Sandıkcıoğu, Özden Gür Ai 2, Serpi Sayın 3 Koç University, Istanbu E-mai: msandikciogu@ku.edu.tr; 2 oai@ku.edu.tr; 3 ssayin@ku.edu.tr Abstract: Our main goa is to understand the customers store choice behavior in a grocery retai setting. We see this as a first vita step in order to make store ocation, format and product promotion decisions in the retai organization roposed modes in the iterature generate consumer utiity functions for different stores which are used in store saes estimation. For exampe, in one of its basic forms, Huff mode proposes that, utiity of a store for an individua is equa to the saes area of the store divided by a power of the individua s distance to the store. arae to this stream of research Mutipicative Competitor Interaction mode estimates og-transformed utiity functions by ordinary east squares regression. It is ess specific in terms of variabe seection compared to the Huff mode. This paper proposes a new market share mode which is a variant of the Huff mode and evauates most estabished market share modes such as Huff and Mutipicative Competitor Interaction Mode as we as a data mining method in a onebrand heterogonous size retai store setting. We observe that the Huff mode performs we in its basic form. By representing distance decay vaue as a function of the saes area of the retai store we are abe to improve the performance of the Huff mode. We propose using optimization for estimating the mode parameters in certain cases and observe that this improves the generaization abiity of the mode. Keywords: Gravity modes, Share prediction, istance decay parameter, Huff mode, Retai, Site seection.. Introduction Location is a crucia decision for service companies as it directy affects revenue generated by the store. In order to make successfu store site seection decisions, one shoud understand the consumer store patronization behaviour which is the resut of the customer's store choice. Spatia choice modes predict the probabiity of a customer choosing a store by cacuating the store s utiity reative to the others. The first mode offered by Huff (963) modeed utiity as a function of distance and store size. Let i denote the probabiity of customer i visiting store. Then, according to Huff (963), i i i, () where denotes the saes area of the retai ocation, i is the traveing cost of customer i to store, is the distance decay parameter, and the index runs over a stores. Later, Back, Ostund and Westbrook (985) added store characteristics such as discount pricing, advertising and reputation to the mode. This mode provided a higher R-square vaue for market share prediction. rezner and rezner (2002) tried a different method. They inferred attractiveness vaues from the Huff mode using buying power index and retai center saes. The attractiveness scores were ater compared to survey resuts. According to their findings, survey resuts and inferred attractiveness scores were consistent. A more comprehensive mode offered by Nakanishi and Cooper (974) is known as the Mutipicative Competitor Interaction (MCI) mode. Their mode was estimated on og-transformed variabes. Ordinary Least Squares and Generaized Least Squares are the methods they propose to be used for parameter estimation. The MCI mode represents the patronization probabiity as: 376

2 A NEW GRAVITY MOEL WITH VARIABLE ISTANCE ECAY where q δk x ki k i m q k x ki is the k th variabe describing store for customer i, and k 377 x δk ki, (2) δ is the sensitivity parameter for variabe k. According to their findings OLS provided good estimates. Ghosh, Nesin, Shoemaker (984) reported that OLS regression provides more accurate estimates than GLS. Fotheringham (988) proposes a specia kind of nested ogit mode known as competing destinations mode. He says that consumers do not necessariy evauate a competing stores because of time and other imitations. Most of the eary modes modeed the distance decay vaue as a constant which shoud be caibrated according to the nature of the ocation. Fotheringham(98) says that distance decay vaue is dependent on the spatia structure of the area which he cas accessibiity of the origin. Later, Orpana and Lampinen (2003) observed that distance decay parameters are smaer for big stores. They cassified stores as oca market, supermarket and hypermarket but they did not use saes area directy for distance decay estimation. ifferent from other approaches, distance decay parameters speciaized for customer segments can aso be used for improved resuts ( Nakaya et a., 2007). Nakaya et a. (2007) used micro simuation and spatia interaction modes together. They grouped customers into ifestye segments and estimated parameters of the competing destinations mode (Fotheringham, 988) according to each customer group. They observed that utiity functions vary according o customer segments. The advantage of the Huff mode over other modes such as MCI and the Nested ogit is that it is very simpe. Therefore it is worth exporing whether we can improve its resuts keeping its advantage. The Huff mode suggests that the distance decay vaue is the same for a stores in the seected area. However, parae to Orpana and Lampinen s (2003) observations in a retai setting, distance decay can be represented as a function of store size, as bigger stores have more attraction power. This change in the mode may improve the accuracy resuts of the Huff mode, whie keeping its practicaity. Our proposed mode described in section 2.2 is buit on this hypothesis. 2. Share prediction 2.. ata The oyaty card data provided by the Turkish market eader retaier Migros is used for evauating the modes. It incudes individua customer spending in seected Migros stores and the tota Migros spending. Customers tota grocery budgets are not avaiabe, since data incudes ony Migros spending. ata is very heterogonous in terms of stores used in this anaysis and spatia distribution of customers. The area in Istanbu that is seected for this anaysis, incudes ten stores that range between 200 and 2500 square meters in size customers ocated within the area are incuded in the anaysis. A stores are of the same brand, so service quaity and prices are homogenous. However, product assortments are quite different as a resut of the difference in the store size. Moreover, one of the big stores is ocated in a shopping ma Estimation procedure In this paper, our goa is to predict the shares of each store in the spending of a customer within the seected region. The actua share of a customer i in a store is computed as: Yi S i, (3) Y where Y i is the spending of a customer i (:6942) in store (:0) in YTL (new Turkish Lira). Our dataset is divided into two parts as the training and the test set. The training set incudes 465 customers where the test set has 2327 customers. The training set is seected from the tota dataset randomy. If a mode requires parameter estimation, the parameters are ony estimated on the training set. We compare modes in terms of their Mean Absoute Error (MAE) in predicting the shares of customers in stores, cacuated as i

3 M. Sandıkcıoğu, Ö. G. Ai, S. Sayın MAE i S i, (4) NM i where N is the number of stores and M is the number of customers. Comparisons of modes reported beow are based on MAE computations conducted on the test set. As our data is homogenous in terms of store brand and service quaity, we do not incude variabes refecting brand image and quaity in any of the modes. Moreover, since cost of traveing is not easy to cacuate, we use Manhattan distances of customers to the stores in a modes. We experiment with three variations of the Huff mode. Mode assumes that the distance decay parameter is equa to 2. Mode 2 is a variation of the Huff mode, where the distance decay parameter is estimated via optimization. We first minimize MAE over the training set. Since we use Exce Sover which finds ony oca optima, in order to improve the quaity of the soution, we use a grid search. Mode 3 is a Huff variation that adds an saes area sensitivity parameter as described beow: α i i α i, (5) where, α is the added saes area sensitivity parameter. The parameters α and are again estimated via optimization as in the previous modes. Mode 4 is an impementation of the MCI mode. We incuded distance, saes area of the store, popuation density and workpace density as variabes. opuation density is given by the number of peope iving in the neighbourhood of the seected store divided by the area of the neighbourhood. Workpace density is the percentage of office buidings within a circe of 500 meters radius around the store. arameters of the MCI mode by Nakanishi and Cooper (974), as described in Equation (2), are estimated by Ordinary Least Squares regression. Mode 5 uses the regression tree method. The impementation is performed in Weka version The input variabes incude, distance, saes area of the store, competition, popuation density, workpace density and the number of stores the customer shops at. Competition is the sum of saes areas of the stores ocated within a circe with a radius of 3000 meters originated from the store. Number of stores the customer shops at is the average of the number of stores the customer shopped at in previous years. Mode 6 is our proposed variation of the Huff mode. This mode is motivated by our observation that distance decay vaues decrease as store size increases, as dispayed in Figure. We represent distance decay vaue θ for a particuar store as a function of its saes area, as described beow: This eads to the mode: where θ i c *. (6) A θ λ i θ A i, (7) A is defined as the attractiveness of store. The constant c and the power of saes area λ are the parameters of the distance decay function. A, c and λ are estimated via optimization. Fig.. Reationship between distance decay vaue and saes area 378

4 A NEW GRAVITY MOEL WITH VARIABLE ISTANCE ECAY 2.3. Mode evauations and resuts Tabe. Mode evauations Mode MAE Comments *oca optima *oca optima *oca optima According to the paired two sampe t-test resuts, Mode 6 s error vaue is significanty ower than the other modes, at the 0.05 significance eve. An advantage of the proposed mode is that it ets the decision makers to cacuate distance and saes adusted attractiveness vaues. According to the Huff mode and its variations above, if a customer is the same distance away from two stores with simiar saes areas, the customer woud be indifferent between them. Now think about a situation, where one of these two stores is in a shopping ma. The probabiity of the customer visiting the store in the shopping ma is higher compared to other store. The aternative mode is capabe of capturing this effect unike the standard Huff mode. The new mode is more fexibe and consequenty has higher prediction power. Our approach is first estimating the attractiveness vaues and then interpreting them. As shown in Tabe 2, the estimated individua store attractiveness vaues are negativey correated with the competition in the area. Here competitor-discount is the sum of saes areas of discount stores around the store. Simiary competitor-reguar is the sum of the saes areas of reguar supermarkets around the store. Finay, competitor-tota is the number of a competitors around the store. Tabe 2. Correation of attractiveness vaues with seected factors Attractiveness Competitor tota 0.64 Competitor discount 0.69 Competitor reguar 0.4 We now briefy describe the specific vaues for the parameters associated with the modes. Mode is the standard Huff mode and its distance decay parameter is accepted as 2. Mode 2 is a variation of the Huff mode and the optima distance decay parameter vaue is equa to The saes area sensitivity parameter of Mode 3 is equa to 3.25 and its distance decay parameter is equa to In Mode 4 which is the MCI mode, variabes are distance, saes area, popuation density and workpace density. Sensitivity parameters, δ k, are -.77,.86, -0.08, -4 respectivey. Mode 5 produces 76 rues and every branch has a minimum of 20 instances. Finay the attractiveness parameters of Mode 6 are: A 0. 6, A , A 3 0, A 4 0.7, A , A , A , A 8 0.4, A , A The parameters of the function defining distance decay are for the constant c for λ. 3. Concusion Gravity modes have a comprehensive iterature going back to 960 s. The first estabished mode was proposed by Huff in 962. Later other market share modes took their pace in the iterature. Athough mode performances depend on the data set, in a retai setting where store brand reated effects are eiminated, the Huff mode performs in an acceptabe error range. Moreover, the Huff mode has a simpe structure and is practica compared to those modes such as MCI, or regression trees. Our research suggests that it is possibe to expain customer store shares with higher accuracy whie keeping the practicaity of the Huff Mode. The proposed mode represents the distance decay parameter as a function of the store saes area, and introduces individua store attractiveness coefficients. A parameters are estimated by soving an optimization mode that uses data from a training set. Evauated on a test set, the accuracy of the proposed mode is superior to the Huff and MCI modes even though the mode parameters are obtained as oca optima in this preiminary study. Another interesting finding is that the estimated individua store attractiveness coefficients can be expained by the oca competition around the individua store. 379

5 M. Sandıkcıoğu, Ö. G. Ai, S. Sayın For further research, we pan to investigate whether finding possiby better soutions for the optimization probem by using goba optimization techniques improves the performance of the mode on the test set. Aso attractiveness vaues may be further examined and interpreted. Finay, designing a decision support too that provides guidance in store ocation decisions based on the mode presented here is a question we wish to address in the future. Acknowedgement This research has been partiay supported by KUMEM research funds. KUMEM is the oint rofessiona Education Center of Koc University and Migros. References Back, W.; Ostund, L.; Westbrook, R Spatia demand modes and intraband concept, Journa of Marketing 49(3): 06. Cooper, L.; Nakanishi, M arameter estimation of a MCI mode, east squares approach, Journa of Marketing Research (3): rezner, Z.; rezner, T Vaidating the gravity-based competitive ocation mode using inferred attractiveness, Annas of Operations Research (): 227. Fotheringham, S. 98. Spatia structure and distance decay parameters, Annas of Association of American Geographers 7(3): 425. Fotheringham, S Consumer store choice and choice set definition, Marketing Science 7(3): 299. Ghosh, A.; Nesin, S.; Shoemaker, R A comparison of market share modes and estimation procedures, Journa of Marketing Research 2(2): Huff, A probabiistic anaysis of shopping center trade areas, Land Economics 39: Orpana, T.; Lampinen, J Buiding spatia choice modes from aggregate data, Journa of Regiona Science 43(2): 39. Nakaya, T.; Fotheringham, S.; Hanaoka, K.; Carke, G.; Baas,.; Yano, K Combining microsimuation and spatia interaction modes for retai ocation anaysis, Journa of Geographica Systems 9: