Customer segmentation in retail facility location planning
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Abstract
In this contribution, we discuss a facility location model to maximize firms’ patronage, while demand is determined by a multinomial logit model (MNL). We account for customer segmentation based on customer characteristics. Hence, we are able to reduce the bias to the objective, which is due to constant substitution patterns of the MNL. Numerical studies show that averaging customer characteristics yield a bias of more than 15 % of the objective function value compared to segmentation. Using GAMS/CPLEX, we are able to solve problem sets with 2 segments, 500 demand points and 10 potential locations to optimality in 1 h computation time. If we consider 50 potential locations, the gap reported by CPLEX is \(<\)8 % in 1 h. We present an illustrative case example of a furniture store company in Germany (data are available as electronic supplementary material to this article). The corresponding problem is solved to optimality in a few minutes.
Keywords
Multinomial logit model Facility location Maximum capture problem Customer choice Heterogeneous demand Substitution patternsJEL Classification
C61 C35 C44 R321 Introduction
In this paper, we consider a situation where companies (retail store chains, for example) compete for their market share. Suppose for example that a firm wants to locate new shops in a geographical market. The decision variable under control is only where to locate the new facilities. The way customers make their choices is to be taken into account, too (Serra and Colome 2001). The reaction of possible competitors (price, locations) is not considered here.
We discuss a model—based on the maximum capture problem—for the optimal location of \(K\) facilities. Customers’ choices are modeled according to a specific discrete choice model, namely the multinomial logit model (MNL). Other demand models (the Huffmodel, for example) might be used instead of the MNL. Our approach is valid for such kind of models as well. However, we do not consider this here. In general, discrete choice models are the workhorse for the analysis of individual choice behavior (McFadden 1973, 2001). In literature, we find several applications of discrete choice models for spatial choice situations (Timmermans et al. 1992; Dellaert et al. 1998). Inspite of their longterm and widespread use, we find only few references in the operations research literature on facility location that account for discrete choice models. One reason may be the mathematical sophistication of the choice models. For example, de Palma et al. (1989), Benati (1999) and Marianov et al. (2008) discuss nonlinear model formulations for discrete locational decisions. To the best of our knowledge, Benati and Hansen (2002) are the first who proposed a linear reformulation of the nonlinear MNL. Their approach results in a hyperbolic sum integer problem. Haase (2009) uses constant substitution patterns of the MNL to find a linear integer reformulation. ArosVera et al. (2013) apply this approach to the planning of parkandride facilities. Finally, Zhang et al. (2012) propose an alternative approach similar to Benati and Hansen (2002). Haase and Müller (2014a) show that a variant of the model of Haase (2009) seems to be superior to the formulations of Benati and Hansen (2002) and Zhang et al. (2012).
The MNL exhibits the wellknown independence from irrelevant alternatives property (IIA). Roughly speaking, this property implies that each choice alternative (facility location) is an equal substitute to every other alternative. Unfortunately, it is empirically evidenced that this core property is unlikely to hold in spatial choice context (Bhat and Guo 2004; Hunt et al. 2004). The linear reformulations of the MNL already introduced in the literature are all based on the assumption that customers of a given demand point are homogenous in their observable characteristics (age and income, for example). In this contribution, we show that, if customers of a given demand point are portioned into homogenous subgroups according to their characteristics, the predictive bias due to the IIA might be reduced (Sect. 2). Of course, simply considering average characteristics are not sufficient as the following illustrative example shows (see Fig. 1).
In this paper, we present an elucidating model formulation to account for customer segmentation within a mixedinteger program that enables to consider customer choice behavior by an MNL that accounts for customer characteristics (Sect. 3.1). Moreover, we present a simple lower bound and objective cuts for our problem (Sect. 3.2). We demonstrate the usefulness of our approach in extensive numerical studies (Appendix). Finally, we present an illustrative case example to show how our approach might be applied to support decision making for the management of a globally operating furniture store retail chain (Sect. 4).
2 A probabilistic choice model
Let us consider the following problem statement:
Find\(K\)facility locations from all potential locations\(J\)such that the total patronage for the\(K\)facilities is maximized.
 \(I\)

demand nodes representing zones, like census blocks etc., that contain the customers,
 \(M_i\)

locations (existing and potential ones) from which the customers located in \(i \in I\) choose exactly one location. \(M_i\) may include a nochoicealternative, indicating that customers might not occupy any facility. Hence, the nochoice alternative (a dummy facility, for example) reflects the proportion of customers who do not consume (services or products) at any facility. We might consider a special case such that \(M_i = M \ \forall \ i \in I\).
 \(J\)

potential locations for the facilities a decision maker (a firm, for example) has to decide on: \(J \subseteq \bigcup _{i \in I} M_i\). Note \(M_i \setminus J\) may include facility locations of competitors and/or the nochoicealternative. That is, \(\left\{ M_i \setminus J \right\} \) comprises locations that are not influenceable by the decision maker. Further, \(J_i = M_i \cap J\).
 \(R_i\)

is a set of choice alternatives faced by the customers of \(i \in I\) that denotes the number, type, and/or the amount of purchases conducted by the customers. Hence, the choice set faced by customers located in \(i \in I\) is \(\left\{ M_i \times R_i\right\} \). Consider exemplarily a customer located in a given demand node \(i=1\) who chooses to make a purchase of €10, €20, or €30 at any opened facility within a given time period. So \(R_1 = \left\{ 10, 20, 30\right\} \). Let us further assume there are only two facilities, i.e., \(M_1 = \left\{ A, B\right\} \), then the choice set is \(\left\{ (A,10),\ldots , (B,10),\ldots ,(B,30)\right\} \). A choice of \((A, 20)\) means that the customer chooses to make a purchase of €20 at facility \(A\). Note, the choice set must be exhaustive and the choice alternatives have to be mutually exclusive. Roughly speaking, all alternatives the customers actually face have to be included in the choice set. The generation of \(\left\{ M_i\times R_i\right\} \) is a sophisticated issue. We refer to Swait (2001) for further details.
 \(h_i\)

number of customers located in node \(i \in I\), and
 \(v_{ijr}\)

as the deterministic utility of customers located in \(i \in I\) patronizing \(j \in M_i\) making a purchase denoted by \(r \in R_i\). This could be a measure of generalized cost etc.
 \(K\)

number of facilities to be located, with \(0 < K < \left J\right \).
 \(y_j\)

= 1, if location \(j \in J\) provides a facility (0, otherwise), and
 \(x_{ijr}\)

as the choice probability of customers of node \(i \in I\) who makes a purchase denoted by \(r \in R_i\) at a facility located at \(j \in J_i\). If we assume that the choice probability is given by the MNL, \(x_{ijr}\) is defined as
We assume in the following that \(\left R_i\right = 1 \ \forall \ i \in I\) simplifying \(v_{ijr}\), (1), and (2) for convenience reasons. Of course, all formulations of the subsequent sections are valid for \(\left R_i\right > 1 \ \forall \ i \in I\) as well.
2.1 The independence from irrelevant alternatives property
2.2 Aggregation issues
The MNL and hence (1) is based on the theory of utility maximization behavior of individuals. That is, each individual chooses the location that maximizes its utility. Given our problem statement of Sect. 2 and the corresponding model (2)–(4), we are interested in aggregate measures (market shares, total patronage etc.) instead of individual choice probabilities. Data on customer demand are usually given as an aggregate measure (number of customers, for example). Now, the question arises how we should compute the choice probability of all customers (individuals) located in a given demand point \(i \in I\)? The answer depends on the specification of the utility \(v_{ij}\). If \(v_{ij}\) does not contain characteristics of the customers (age, income, and so forth) then the choice probability \(x_{ij}\) applies to all customers in \(i \in I\) in the same way and thus, (2) is a proper formulation. In contrast, the incorporation of customer characteristics in \(v_{ij}\) will improve the accurateness of \(x_{ij}\) (Koppelman and Bhat 2006, pp 21–23 and pp 41–46). However, aggregation is more tedious in such a case.
Example 1
 1.
we use the average income of \(n=1\) and \(n=2\) (i.e., the average income of demand node \(i'\)) denoted by \(\overline{q}_{i'} = (q_1 + q_2)/2\) to compute \(\overline{v}_{i'j}\) and thus \(\overline{x}_{i'j}\), or
 2.
we first compute the choice probabilities for each customer \(x_{nj}\) and then we determine the average choice probability of customers located in \(i'\) as \(\tilde{x}_{i'j} = (x_{n=1,j} + x_{n=2,j})/2\).
Aggregation, choice probabilities and the IIA property
\(j=A\)  \(j=B\)  \(j=C\)  \(x_{i'A}/x_{i'C}\)  

\(g_{i'j}\)  1  2  4  
\(q_1\)  9  9  9  
\(q_2\)  1  1  1  
\(\overline{q}_{i'}\)  5  5  5 
Solution I: \(y_j\)  1  1  1  

\(\overline{x}_{i'j}\) using \(\overline{q}_{i'}\)  0.422  0.346  0.232  1.822 
\(x_{n=1,j}\)  0.383  0.343  0.274  1.396 
\(x_{n=2,j}\)  0.705  0.259  0.035  20.085 
\(\tilde{x}_{i'j}\)  0.544  0.301  0.155  3.516 
Solution II: \(y_j\)  1  0  1  

\(\overline{x}_{i'j}\) using \(\overline{q}_{i'}\)  0.646  0  0.354  1.822 
\(x_{n=1,j}\)  0.583  0  0.417  1.396 
\(x_{n=2,j}\)  0.953  0  0.047  20.085 
\(\tilde{x}_{i'j}\)  0.768  0  0.232  3.302 
There are two lessons learned so far: First, the more customer characteristics are included in \(v_{ij}\) in an appropriate way, the better are the forecast properties of MNL, \(x_{ij}\), respectively. Second, by applying segmentation to our model (2)–(4) as outlined in (2), we are able to reduce the bias of \(x_{ij}\) and \(F\) due to the IIA of (5) to some extent. In applications, one would be interested in how to classify customers, and how many customer segments are appropriate for a given application. Of course, segmentation makes sense only if the deterministic part of utility contains factors that vary over choice makers. Usually, such factors are socioeconomic factors like age, gender, income, occupation, car ownership, and so forth. In empirical studies, socioeconomic factors that are continuous measures (age and income, for example) are usually considered as categorical measures. For example, a proband is asked whether his/her age is (a) below 20 years, (b) between 20 and 40 years, (c) between 40 and 60 years, or (d) older than 60 years. Now consider a deterministic utility function with only two socioeconomic factors: gender and age. Gender, of course, consists of only two categories: female and male. So, we end up with eight customer segments: the four age levels for each of the two genders. Considering many socioeconomic factors with many levels yields a large number of segments. How many segments are appropriate and tractable could not be said in the abstract. It rather depends on the application, in particular, the empirically specified choice model. See BenAkiva and Lerman (1985), pp 131–153 for a detailed discussion of aggregation and segmentation.
3 A probabilistic choice model with customer segmentation
In Sect. 2, we have demonstrated that the IIA may yield biased values of \(x_{ij}\) of (1) and hence a biased objective function value \(F\) of (2). Moreover, a partition of the population of a demand point \(i \in I\) into homogenous subpopulations (i.e., segmentation) enables us to reduce the bias due to the IIA. In this section, we propose how to explicitly account for segments of customers (heterogeneous customer demand) in a linear mixedinteger model formulation of (2)–(4).
3.1 Mathematical formulation
 \(S_i\)

segments of the customers located in demand node \(i \in I\); for example high and low income or male and female or a combination of income and gender.
 \(\widetilde{h}_{is}\)

number of customers according to segment \(s \in S_i\) located in node \(i \in I\),
 \(\widetilde{v}_{isj}\)

as the deterministic utility of customers of segment \(s \in S_i\) located in \(i \in I\) patronizing \(j \in M_i\),
 \(\pi _{isj}\)

choice probability of customers of segment \(s \in S_i\) at node \(i \in I\) who access service at a facility located at \(j \in J_i\) given that all \(m \in J\) are established, i.e., \(\pi _{isj} = \mathrm{e}^{\widetilde{v}_{isj}}/ \sum _{m \in M_i}\mathrm{e}^{\widetilde{v}_{ism}}\),
 \(\varphi _{isj}\)

choice probability of customers of segment \(s \in S_i\) at node \(i \in I\) who access service at a facility located at \(j \in J_i\) given that \(j \in J_i\) is the only facility location established, i.e., \(\varphi _{isj} = \mathrm{e}^{\widetilde{v}_{isj}}/ (\mathrm{e}^{\widetilde{v}_{isj}} + \sum _{m \in M_i \setminus J}\mathrm{e}^{\widetilde{v}_{ism}})\), and
 \(\zeta _{is}\)

cumulative choice probability of customers of segment \(s \in S_i\) at node \(i \in I\) who access service at competing facilities given that all potential facilities \(j \in J\) are located, i.e., \(\zeta _{is} = \sum _{l\in M_i \setminus J} (\mathrm{e}^{\widetilde{v}_{isl}} / \sum _{m \in M_i}\mathrm{e}^{\widetilde{v}_{ism}})\). Therefore,
 \(\widetilde{x}_{isj}\)

as the MNL choice probability of customers of segment \(s \in S_i\) at node \(i \in I\) who access service at a facility located at \(j \in J_i\), and
 \(z_{is}\)

as the cumulative choice probability of customers of segment \(s \in S_i\) at node \(i \in I\) who do not access any facility of the considered firm.
3.2 Lower bound and objective cuts
We are interested in the impact of the number of segments, the lower bound, the objective cuts, the number of competitors on the solution and the solvability of our approach. The corresponding numerical examples can be found in the Appendix. The major findings of these numerical examples are that (1) segmentation has significant impact on the computational effort, (2) the lower bound (22) provides a quite good solution (it deviates \(<\)1 % from the optimal solution), and (3) the use of the objective cut OC1 (23) is particularly appealing if we do not expect to find an optimal solution within a given time. Further, we solve problem sets with 2 segments, 500 demand points and 10 potential locations to optimality in 1 h computation time. If we consider 50 potential locations, the gap reported by CPLEX is \(<\)8 % in 1 h.
4 Illustrative case example: furniture store location in Germany
In this section, we apply our model of Sect. 3.1 to a hypothetical—but still realistic— branchextension of a large furniture store company in Germany. Figure 2a shows the already existing facility locations and the potential facility locations of the considered firm, as well as the locations of the main competitors in the market. The firm already runs 46 stores in the year 2012 with a market share of 12.5 % and 46 million customers yielding 3.7 billion Euro revenue. The firm aims to massively expand in the market in the near future. It is intended to establish 5–15 new facilities until 2020. The task is to find out the optimal locations for a given number of new facilities (\(K^+\)) from 50 potential facility locations and the corresponding expected market share of the firm.
Since our parameters do not stem from a unique study on furniture store customer behavior in Germany, we first investigate the sensitivity of the solution to parameter variations. The locational decision variables \(y_j\) are fixed to one for the already existing facility locations (i.e., \(j<47\)). We solve our model of Sect. 3 for various parameter settings and for different distance thresholds \(\delta \) of (27). We are interested in MS’s dependence on \(K^+\). We have implemented our model in GAMS 23.7 and we use CPLEX 12.2 on a 64bit Windows Server 2008 with 4 Intel Xeon 2.4 GHz processors and 24 GB RAM for all studies. All problems considered in this section are solved to optimality within minutes. The results of Fig. 3 show a piecewise linear increase of the market share in \(K^+\). The slope is nearly 0.35 indicating that with each additional facility, the total market share of the firm increases by 0.35 % points. Note, the underlying function is not necessarily concave. The sensitivity analysis indicates that the market share is independent from the distance threshold \(\delta > 50\) and the weight of the income \(\beta ^{\text {inc}}\). In contrast, the scale of the market share heavily depends on the distance parameters (\(\beta _s^{\text {dist}}\)). This finding stresses the need for firms to employ the estimates based on unique choice studies (see Street and Burgess 2007; Müller et al. 2008; Louviere et al. 2000 for how to design studies and experiments for discrete choice analysis).
Example 2
We expect the more the two segments differ, the larger is the predictive bias of the MNL and thus the larger is the bias of the objective function value if segmentation is neglected. Due to the specification of the deterministic part of utility in (29), the difference in choice probabilities between the two segments corresponds to the difference between \(\beta ^{\text {dist}}_{s=1}\) and \(\beta ^{\text {dist}}_{s=2}\).To evaluate the impact of neglected segmentation, we first consider \(\beta ^{\text {dist}}_{s=1} =\beta ^{\text {dist}}_{s=2} = \beta ^{\text {dist}}\) with \(\beta ^{\text {dist}} = (\beta ^{\text {dist}}_{s=1} + \beta ^{\text {dist}}_{s=2})/2\) in (29). This corresponds to a simple average of utilities as described in (1) of Sect. 2.2. The corresponding solution in terms of selected locations is denoted by \(\overline{J} = \left\{ j \in J\left y^*_j=1\right. \right\} \). Based on \(\overline{J}\), we compute the MNL choice probabilities using segmentation, i.e., we use \(\beta ^{\text {dist}}_{s=1}\) and \(\beta ^{\text {dist}}_{s=2}\) instead of \(\beta ^{\text {dist}}\) in (29). The corresponding objective function value is denoted as \(\overline{F}\) and the corresponding market share is given by MS(\(\overline{F}\)).
We consider \(\beta ^{\text {dist}}_{s} \in \left\{ \right. 1, 0.1, 0.01, 0.001, 0.0001\left. \right\} \), \(\beta ^{\text {inc}}=0.015\), and \(\delta = 150\). Further, we consider two scenarios: \(K^+ = 5\) and \(K^+ = 10\). The results are given in Fig. 5. The patterns for the total deviation \(\overline{F} \widetilde{F}\), relative deviation \(100 \times (\overline{F} \widetilde{F})/\widetilde{F}\), and the deviation of the market shares \(\text {MS}(\overline{F})\text {MS}(\widetilde{F})\) are similar. The most eyecatching bias occurs if \(\beta ^{\text {dist}}_{s=1} = 1\). Consider exemplarily \(\beta ^{\text {dist}}_{s=1} = 1\) and \(\beta ^{\text {dist}}_{s=2} = 0.1\), i.e., segment \(s=1\) evaluates each additional kilometer ten times as negative as segment \(s=2\) (i.e., \(\beta ^{\text {dist}}_{s=1}/\beta ^{\text {dist}}_{s=2} = 10\)). In case that segmentation is neglected, the corresponding distancecoefficient is \(\beta ^{\text {dist}} =0.55\). As a consequence, a large part of customers (recall that, \(\sum _i \widetilde{h}_{i, s=1} / \sum _i \widetilde{h}_{i, s=2} =0.163\)) evaluates distance more than five times as negative as this would be the case with segmentation. Of course, the corresponding deviation is remarkable (\(\)8.9 % for \(K^+=5\) and \(\)12.5 % for \(K^+=10\)). The asymmetric pattern in Fig. 5 is due to the uneven distribution of population over the two segments (the population of segment 2 is larger than the population of segment 1): the more the true coefficient of the large part of the population (segment 2) deviates from the average coefficient the larger is the expected predictive error. In contrast, a large deviation of the true coefficient of segment 1 has impact only on a small part of the population and the corresponding expected predictive error is comparably small. Obviously, the extent of the error heavily depends on the scale of the coefficients. Consider, for example, \(\beta ^{\text {dist}}_{s=1} = 1\) and \(\beta ^{\text {dist}}_{s=2} = 0.1\). The corresponding ratio is 10 and the expected error for \(K^+ = 5\) is \(\)8.88 %. Now, for \(\beta ^{\text {dist}}_{s=1} = 0.1\) and \(\beta ^{\text {dist}}_{s=2} = 0.01\) the corresponding ratio is 10 again. However, the corresponding error is only \(\)0.18 %. This pattern is due to the nonlinear relationship between distance (deterministic utility) and the choice probabilities (i.e., a sshaped probability function). As the coefficients (weighting of travel distance) get larger (i.e., approaching 0) the probabilities of choosing to patronize a facility approach the largest possible value. For these values of the deterministic utility the difference in the corresponding choice probabilities between the two segments become small.
The bias found in our study is comparable to those reported in studies on spatial aggregation (Andersson et al. 1998; Daskin et al. 1989; Current and Schilling 1987; Murray and Gottsegen 1997). In literature, ratios of segmentspecific coefficients larger than 50 are reported (Müller et al. 2012; Koppelman and Bhat 2006, pp 133–134). However, the difference between segmentspecific distancecoefficients used in our application is small. We have considered parameter settings that yield a ratio \(\beta ^{\text {dist}}_{s=1}/\beta ^{\text {dist}}_{s=2} = 0.91\) (see Fig. 3). As a consequence, the expected bias is below 1 % if we neglect segmentation in our application. Nevertheless, the consideration of segments yields valuable insights, because the utility function (29) and the corresponding coefficients are arbitrarily chosen. As stated before, for a real application, the company is expected to specify utility functions and estimate the corresponding coefficients on unique choice data. The firm may use such a numerical study to make assumptions about worstcase scenarios.
5 Summary
By an intelligible example, we demonstrate that the independence from IIA of the MNL may yield false predictions. This finding is well founded on empirical studies. When the MNL is used in a mathematical program to incorporate customer choice behavior, the model outcomes are very likely to be biased as well. Although the MNL is founded on individual choice behavior, in facility location planning we are interested in the share of customers of a demand point patronizing a certain facility. If we assume the customers of a demand point are homogenous, i.e., they exhibit the same observable characteristics, then there is no need for segmentation. If we assume the customers to be heterogeneous then segmentation of the customers according to their characteristics (income and age, for example) should be employed. By proper segmentation, we are able to reduce the predictive bias of the MNL in terms of market shares.
In this contribution, we present a model formulation for the maximum capture problem that explicitly allows for customer segmentation using the MNL to find optimal shopping facility locations. Moreover, we propose an intelligible approach to derive a lower bound for our model. Extensive computational studies show the impact of proper segmentation as well as the efficiency of our approach: using aggregate customer characteristics instead of proper segmentation may yield a predictive bias of the objective function value of more than 15 % deviation from the optimal objective function value. Our lower bound is found in \(<\)1 s and deviates \(<\)1 % from the optimal solution. Problems with 2 segments, 50 potential locations and 500 demand points can be solved to a gap \(<\)8 % within 1 h using GAMS/CPLEX. Based on our numerical studies concerning the quality of the lower bound, it is reasonable to assume that the true gap is remarkably smaller than 8 %. We apply our approach in an illustrative case example of a globally operating furniture store company that intends to increase its market share in Germany by branch expansion. This problem can be solved to optimality within few minutes. Our example shows how the novel approach can be used for management decision support.
Based on our findings, several possible directions of future research appear. It is of interest to find analytically bounds on the bias of the objective function value due to missing segmentation under various segmentation patterns and specifications of utility. Further, the explicit consideration of substitution patterns, i.e., correlation between facility locations, is a very important issue to be analyzed. Efficient solution methods are necessary to account for larger problem sets. Finally, our approach is useful to other areas of operations research; assortment optimization, for example Kök and Fisher (2007).
Notes
Acknowledgments
The very helpful comments and suggestions of three anonymous reviewers are gratefully acknowledged. They made significant contributions in order to improve the paper. We further thank the editors for helpful suggestions. Finally, we thank Sonja Bröning for copy editing. However, the responsibility for any remaining error is with the authors.
Supplementary material
References
 Andersson, G., R.L. Francis, T. Normark, and B.M. Rayco. 1998. Aggregation method experimentation for largescale network location problems. Location Science 6(4): 25–39.CrossRefGoogle Scholar
 ArosVera, F., V. Marianov, and J.E. Mitchell. 2013. pHub approach for the optimal parkandride facility location problem. European Journal of Operational Research 226(2): 277–285.CrossRefGoogle Scholar
 BenAkiva, M., and S. Lerman. 1985. Discrete choice analysis, theory and applications to travel demand. Cambridge: MIT Press.Google Scholar
 Benati, S. 1999. The maximum capture problem with heterogeneous customers. Computers and Operations Research 26(14): 1351–1367.CrossRefGoogle Scholar
 Benati, S., and P. Hansen. 2002. The maximum capture problem with random utilities: problem formulation and algorithms. European Journal of Operational Research 143(3): 518–530.CrossRefGoogle Scholar
 Bhat, C.R., and J. Guo. 2004. A mixed spatially correlated logit model: formulation and application to residential choice modeling. Transportation Research Part B Methodological 38(2): 147–168.CrossRefGoogle Scholar
 Casado, E., and J.C. Ferrer. 2013. Consumer price sensitivity in the retail industry: latitude of acceptance with heterogeneous demand. European Journal of Operational Research 228(2): 418–426.CrossRefGoogle Scholar
 Current, J.R., and D.A. Schilling. 1987. Elimination of source a and b errors in pmedian location problems. Geographical Analysis 19(2): 95–110.CrossRefGoogle Scholar
 Daskin, M., A. Haghani, M. Khanal, and C. Malandraki. 1989. Aggregation effects in maximum covering models. Annals of Operations Research 18(1): 113–139.CrossRefGoogle Scholar
 Dellaert, B.G., T.A. Arentze, M. Bierlaire, A.W. Borgers, and H.J. Timmermans. 1998. Investigating consumers’ tendency to combine multiple shopping purposes and destinations. Journal of Marketing Research 35(2): 177–188.CrossRefGoogle Scholar
 Fotheringham, S., and R. Trew. 1993. Chain image and storechoice modeling: the effects of income and race. Environment and Planning A 25(2): 179–196.CrossRefGoogle Scholar
 Goldman, A. 1976. Do lowerincome consumers have a more restricted shopping scope? Journal of Marketing 40(1): 46–54.CrossRefGoogle Scholar
 Haase, K. 2009. Discrete location planning. Tech. Rep. WP0907. Institute for Transport and Logistics Studies, University of Sydney.Google Scholar
 Haase, K., and S. Müller. 2013. Management of school locations allowing for free school choice. Omega 41(5): 847–855.CrossRefGoogle Scholar
 Haase, K., and S. Müller. 2014a. A comparison of linear reformulations for multinomial logit choice probabilities in facility location models. European Journal of Operational Research 232(3): 689–691.CrossRefGoogle Scholar
 Haase, K., and S. Müller. 2014b. Insights into clients’ choice on preventive health care facility location planning. OR Spectrum. doi 10.1007/s0029101403676
 Haynes, K., and S. Fotheringham. 1990. The impact of space on the application of discrete choice models. The Review of Regional Studies 20(1): 39–49.Google Scholar
 Hunt, L., B. Boots, and P. Kanaroglou. 2004. Spatial choice modelling: new opportunities to incorporate space into substitution patterns. Progress in Human Geography 28(6): 746–766.CrossRefGoogle Scholar
 Kök, G., and M.L. Fisher. 2007. Demand estimation and assortment optimization under substitution: methodology and application. Operations Research 55(6): 1001–1021.CrossRefGoogle Scholar
 Koppelman, F.S., and C. Bhat. 2006. A self instructing course in mode choice modeling: multinomial and nested logit models. Prepared for US Department of Transportation Federal Transit Administration.Google Scholar
 Louviere, J., D. Hensher, and J. Swait. 2000. Stated choice methods : analysis and applications. Cambridge: Cambridge University Press.Google Scholar
 Marianov, V., M. Rfos, and M.J. Icaza. 2008. Facility location for market capture when users rank facilities by shorter travel and waiting times. European Journal of Operational Research 191(1): 30–42.CrossRefGoogle Scholar
 McFadden, D. 1973. Conditional logit analysis of qualtitative choice behaviour. In Frontiers of econometrics, ed. P. Zarembka, 105–142. New York: Academic Press.Google Scholar
 McFadden, D. 2001. Economic choices. American Economic Review 91(3): 351–378.CrossRefGoogle Scholar
 Müller, S., S. Tscharaktschiew, and K. Haase. 2008. Traveltoschool mode choice modelling and patterns of school choice in urban areas. Journal of Transport Geography 16(5): 342–357.CrossRefGoogle Scholar
 Müller, S., K. Haase, and S. Kless. 2009. A multiperiod school location planning approach with free school choice. Environment and Planning A 41(12): 2929–2945.CrossRefGoogle Scholar
 Müller, S., K. Haase, and F. Seidel. 2012. Exposing unobserved spatial similarity: evidence from German school choice data. Geographical Analysis 44(1): 65–86.CrossRefGoogle Scholar
 Murray, A.T., and J.M. Gottsegen. 1997. The influence of data aggregation on the stability of pmedian location model solutions. Geographical Analysis 29(3): 200–213.CrossRefGoogle Scholar
 de Palma, A., V. Ginsburgh, M. Labbe, and J.F. Thisse. 1989. Competitive location with random utilities. Transportation Science 23(4): 244–252.CrossRefGoogle Scholar
 Ray, P. 1973. Independence of irrelevant alternatives. Econometrica 41(5): 987–991.CrossRefGoogle Scholar
 Sener, I., R. Pendyala, and C. Bhat. 2011. Accommodating spatial correlation across choice alternatives in discrete choice models: an application to modeling residential location choice behavior. Journal of Transport Geography 19(2): 294–303.CrossRefGoogle Scholar
 Serra, D., and R. Colome. 2001. Consumer choice and optimal locations models: formulations and heuristics. Papers in Regional Science 80(4): 439–464.CrossRefGoogle Scholar
 Sheppard, E. 1978. Theoretical underpinnings of the gravity hypothesis. Geographical Analysis 10(4): 386–402.CrossRefGoogle Scholar
 Street, D., and L. Burgess. 2007. The construction of optimal stated choice experiments. New York: Wiley.Google Scholar
 Suarez, A., I.R. del Bosque, J.M. RodriguezPoo, and I. Moral. 2004. Accounting for heterogeneity in shopping centre choice models. Journal of Retailing and Consumer Services 11(2): 119–129.CrossRefGoogle Scholar
 Swait, J. 2001. Choice set generation within the generalized extreme value family of discrete choice models. Transportation Research Part B Methodological 35(7): 643–666.CrossRefGoogle Scholar
 Timmermans, H., A. Borgers, and P. van der Waerden. 1992. Mother logit analysis of substitution effects in consumer shopping destination choice. Journal of Business Research 23(2): 311–323.Google Scholar
 Train, E. Kenneth. 2009. Discrete choice methods with simulation. Cambridge: Cambridge University Press.Google Scholar
 Zhang, Y., O. Berman, and V. Verter. 2012. The impact of client choice on preventive healthcare facility network design. OR Spectrum 34(2): 349–370.CrossRefGoogle Scholar
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