Operations Research Letters. A generalized Weiszfeld method for the multi-facility location problem

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1 Operations Research Letters 38 ) 7 4 Contents lists available at ScienceDirect Operations Research Letters journal homepage: A generalized Weiszfeld method for the multi-facility location problem Cem Iyigun a,, Adi Ben-Israel b a Department of Industrial Engineering, Middle East Technical University, Ankara, Turkey b RUTCOR Rutgers Center for Operations Research, Rutgers University, NJ, USA a r t i c l e i n f o a b s t r a c t Article history: Received 5 May 9 Accepted 5 October 9 Available online 6 November 9 Keywords: Fermat Weber location problem Multi-facility location problem Decomposition method Probabilistic assignments Weiszfeld method An iterative method is proposed for the K facilities location problem. The problem is relaxed using probabilistic assignments, depending on the distances to the facilities. The probabilities, that decompose the problem into K single-facility location problems, are updated at each iteration together with the facility locations. The proposed method is a natural generalization of the Weiszfeld method to several facilities. 9 Elsevier B.V. All rights reserved.. Introduction The Fermat Weber location problem also single-facility location problem) is to locate a facility that will serve optimally a set of customers, given by their locations and weights, in the sense of minimizing the weighted sum of distances traveled by the customers. A well known method for solving the problem is the Weiszfeld method [3], a gradient method that expresses and updates the sought center as a convex combination of the data points. The multi-facility location problem MFLP) is to locate a given) number of facilities to serve the customers as above. Each customer is assigned to a single facility, and the problem also called the location allocation problem) is to determine the optimal locations of the facilities, as well as the optimal assignments of customers assignment is absent in the single-facility case.) MFLP is NP hard, [6]. We relax it by replacing rigid assignments with probabilistic assignments, as in [4,3] and [8]. This allows a decomposition of MFLP into single-facility location problems, coupled by the membership probabilities that are updated at each iteration.. The problem Notation., K : {,,..., K}. x denotes the Euclidean norm of a vector x R n. The Euclidean distance dx, y) x y is used throughout. Corresponding author. addresses: iyigun@ie.metu.edu.tr C. Iyigun), adi.benisrael@gmail.com A. Ben-Israel) /$ see front matter 9 Elsevier B.V. All rights reserved. doi:.6/j.orl.9..5 Let X : {x i : i, N} be a set of N data points in R n, with given weights {w i > : i, N}. Typically, the points {x i } are the locations of customers, the weights {w i } are their demands. Given an integer K < N, the MFLP is to locate K facilities, and assign each customer to one facility, so as to minimize the sum of weighted distances min c,c,...,c K x i C k w i dx i, c k ) L.K ) where {c k } are the locations or centers) of the facilities, and C k is the cluster of customers that are assigned to the kth facility. For K, one gets the Fermat Weber location problem: given X and {w i : i, N} as above, find a point c R n minimizing the sum of weighted distances, min c R n w i dx i, c), L.) see [,3,4,3] and their references. If the points {x i } are not collinear, as is assumed throughout, the objective function of L.) f c) w i dx i, c) ) is strictly convex, and L.) has a unique optimal solution. The gradient of ) is undefined if c coincides with one of the data points {x i }. For c X, f c) w i x i c x i c, )

2 8 C. Iyigun, A. Ben-Israel / Operations Research Letters 38 ) 7 4 and the optimal center c, if not in X, is characterized by f c ), expressing it as a convex combination of the points x i, c λ i x i, with weights λ i w i / x i c that depend on c. w m / x m c m This circular result gives rise to the Weiszfeld iteration, [3], c + : Tc) 3) where c + is the updated center, c is the current center, and w i / x i c Tc) : x i, if c X; w m / x m c m c, if c X. In order to extend f c) to all c, Kuhn [] modified it as follows: f c) : Rc), where f c), if c X; Rc) : max {, R j R j w j } R j, if c x 5) j X, where R j : i j 4) w i x i x j x i x j ) 6) is the resultant force of N forces of magnitude w i and direction x i x j, i j. The following properties of the mappings R ), T ), the optimal center c and any point x j X were proved by Kuhn []: c c Rc). 7a) c conv X the convex hull of X). If c c then Tc) c. Conversely, if c X, Tc) c then c c. If Tc) c then f Tc)) < f c). 7b) 7c) 7d) x j c w j R j. 7e) If x j c, the direction of steepest descent of f at x j is R j / R j. If x j c there exists δ > such that 7f) < c x j T s c) x j > δ for some s. 7g) lim c x j Tc) x j c x j Rj w j. 7h) For any c, if no c r : T r c ) X, then lim c r c. 7i) r These results are generalized in Theorem to the case of several facilities. Remark. Another claim in [], that T r c ) c for all but a denumerable number of initial centers c, was refuted by Chandrasekaran and Tamir [6]. Convergence can be assured by modifying the algorithm 3) and 4) at a non-optimal center that coincides with a data point x j. Balas and Yu [] proposed moving from x j in the direction R j 6) of steepest descent, assuring a decrease of the objective, and non-return to x j by 7c) and 7d). Vardi and Zhang [9] guaranteed exit from x j by augmenting the objective function with a quadratic of the distances to the other data points. Convergence was also addressed by Ostresh [7], Eckhardt [3], Drezner [], Brimberg [5], Beck et al. [], and others. 3. Probabilistic assignments For < K < N, the problem L.K ) is NP hard, [6]. It can be solved polynomially in N for K, see [], and possibly for other given K. We relax the problem by using probabilistic or soft) assignments, with cluster membership probabilities, p k x) : Prob {x C k }, k, K, assumed to depend only on the distances {dx, c k ) : k, K} of the point x from the K centers. A reasonable assumption is assignment to a facility is more probable the closer it is and a simple way to model it, p k x) dx, c k ) Dx), k, K, 8) w where w is the weight of x, and D ) is a function of x, that does not depend on k. There are other ways to model assumption A), but 8) works well enough for our purposes. Model 8) expresses probabilities in terms of distances, displaying neutrality among facilities in the sense of the Choice Axiom of Luce, [5, Axiom ], see [9, Appendix A]. Other issues such as attractiveness, introduced in the Huff model [5,6], see also [9], are ignored. Using the fact that probabilities add to one, we get from 8), dx, c j ) p k x) /dx, c k) /dx, c l ) l j k l m l A), k, K, 9) dx, c m ) interpreted as p k x) if dx, c k ), i.e., x c k. In the special case K, dx, c ) p x) dx, c ) + dx, c ), dx, c ) p x) dx, c ) + dx, c ). From 8), we similarly get Dx) w K dx, c j ) j l m l ), ) dx, c m ) which is up to a constant) the harmonic mean of the distances {dx, c j ) : j, K}. In particular, Dx) w dx, c )dx, c ), for K. ) dx, c ) + dx, c ) The function ) is called the joint distance function JDF) at x. Abbreviating p k x) by p k, Eq. 8) is an optimality condition for the extremum problem { } min w p dx, k c k) : p k, p k, k, K 3) with variables {p k }. The squares of probabilities in 3) are explained as a device for smoothing the underlying objective, min { x c k : k, K}, see the seminal article by Teboulle [8].

3 C. Iyigun, A. Ben-Israel / Operations Research Letters 38 ) L.K ) can thus be approximated by the minimization problem min dx i, c k ) P.K ) s.t. p k x i ), i, N, p k x i ), k, K, i, N, with two sets of variables, the centers {c,..., c K } and probabilities {p k x i ) : k, K, i, N}, corresponding, respectively, to the centers and assignments of the original problem L.K ). Remark. Given a solution of P.K ), if each customer frequents the centers with the corresponding probabilities, then the expected sum of weighted distances is p k x i ) w i dx i, c k ), 4) which is no better than if each customer is assigned to the nearest facility, i.e. the one with the highest probability. Therefore, even if an optimal solution of P.K ) is also optimal for L.K ), the value 4) is just an upper bound on the optimal value of L.K ). Another way to see this is by rewriting 4), using 8) and ), p k x i ) w i dx i, c k ) K Dx i ) w i Dx i ) min {dx i, c k )}. 5) k,k The inequality follows since the harmonic mean is no less than the minimum. The larger K is, the better is the upper bound 4), because the harmonic mean is then closer to the minimum. 4. Probabilities and centers The objective function of P.K ) is denoted f c,..., c K ) : dx i, c k ). 6) A natural approach to solving P.K ), see e.g. [7], is to fix one set of variables, and minimize P.K ) with respect to the other set, then fix the other set, etc. We thus alternate between ) the probabilities problem, i.e. P.K ) with given centers, and ) the centers problem, P.K ) with given probabilities, and update their solutions as follows: Probabilities update. With the centers given, and the distances dx i, c k ) computed for all centers c k and data points x i, the minimizing probabilities are given explicitly by 9), dx i, c j ) p k x i ) j k l m l, k, K. 7) dx i, c m ) Centers update. Fixing the probabilities p k x i ) in P.K ), the objective function 6) is a separable function of the cluster centers, f c,..., c K ) : f k c k ), 8) where f k c) : x i c, k, K. 9) The centers problem thus decomposes into K problems of type L.), coupled by the probabilities. The gradient of 9), wherever it exists, is f k c) x i c x i c), k, K. ) Zeroing ), we get the optimal centers c k λ k x i ) x i, k, K, a) as convex combinations of the data points, with weights λ k x i ) given by λ k x i ) p k x i) w i / x i c k, k, K, i, N. b) p k x j) w j / x j c k j Eqs. a) and b) induce K mappings T k : c T k c), k, K, T k c) : p k x i) w i / x i c x i p k x j) w j / x j c j a) for c different than the data points {x j : j, N}, and by continuity, T k x j ) : x j, for all j, N. 5. Territories of facilities b) In the MFLP L.K ), each facility serves a cluster of customers, in a certain territory see the discussion by Huff [5]). These territories are given naturally by the membership probabilities of the approximate problem P.K ). For any two centers c j, c k the locus of the points x with p j x) p k x) is by 7) represented by the equation x c j x c k 3) and is thus a hyperplane line in R ). The portion of the hyperplane 3), where p j x) p k x) p m x), m j, k, 4) is either empty, bounded or unbounded. We call it the common boundary of the clusters C j and C k. The common boundaries constitute a Voronoi diagram of the centers. Each Voronoi cell is a polyhedron, and is the natural territory of its facility, see e.g. Figs. b) and b). 6. Contour approximation and uncertainty The JDF at a point was given in ). The JDF of the set X {x i : i, N} is defined as the sum, DX) : w i Dx i ). 5) It depends on the K centers, and has the following useful property: for optimal centers, most of the data points are contained in its lower level sets, see [3,8]. We call this property contour approximation. The JDF of the set X, 5), is thus a measure of the proximity of customers to their respective facilities: the lower the value of DX), the better is the set of centers {c k }. The JDF ) has the dimension of distance. Normalizing it, we get the dimensionless function

4 C. Iyigun, A. Ben-Israel / Operations Research Letters 38 ) a) Data points b) Probability level sets. Fig.. Illustration of Example a) Data points b) Probability level sets. Fig.. Illustration of Example. Ex) / /K K w K Dx) dx, c j )), 6) j with / interpreted as zero. Ex) is the harmonic mean of the distances {dx, c j ) : j, K} divided by their geometric mean, and can be written, using 9), as the geometric mean of the probabilities up to a constant), /K K Ex) K p j x)). 7) j It follows that Ex), with Ex) if any dx, c j ), i.e. if x is a cluster center, and Ex) if and only if the probabilities p j x) are all equal. In particular, for K, dx, c )dx, c ) Ex) dx, c ) + dx, c ) p x)p x). 8) Ex) represents the uncertainty of classifying the point x, indeed it can be written as { Ex) exp I px), K )}, 9) where I px), ) is the Kullback Leibler distance, [], between the distributions K px) p x), p x),..., p K x)) and K K, K,..., ). K The CUF of the data set X {x i : i, N} is defined as EX) : N Ex i ), 3) and is a monotone decreasing function of K, the number of clusters, decreasing from EX) for K ), to EX) for K N, the trivial case where every data point is its own cluster). The right number K of facilities to serve the given customers if not given) is in general determined by economic considerations operating costs, etc.) An intrinsic criterion for determining the optimal K is provided by the rate of decrease of EX), see e.g. Fig. 3d). 7. Optimality conditions and convergence The gradient ) is undefined /) if c coincides with any of the data points. We modify the gradient, following Kuhn [] [], and denote the modified gradient by R k. If a center c k is not one of the data points, we copy ) with a change of sign, R k c k ) : x i c k x i c k ). 3) Otherwise, if a center c k coincides with a data point x j then x j belongs with certainty to the kth cluster and, by 7), p k x j ), p m x j ) for all m k. 3) In this case, define The latter distribution,, is of maximal uncertainty. Therefore, K Ex) px). We call Ex) the classification K uncertainty function, abbreviated CUF, at x. R k x j ) : max { R j k w j, } Rj k R j k, where 33a)

5 C. Iyigun, A. Ben-Israel / Operations Research Letters 38 ) a) Data points b) Level sets of the JDF c) Level sets of the CUF. classification uncertanity number of clusters d) Validation. Fig. 3. Illustration of Example 3. R j k i j x i x j x i x j ). 33b) Therefore, if R j k < w j then R k x j ) ; otherwise, R k x j ) is a vector with magnitude R j k w j and direction R j k. For K, 33a) and 33b) reduce to their single-facility counterparts 5) and 6), respectively. The results 7a) 7i) are reproduced, for several facilities, in parts a) i) of the following theorem. Theorem. a) Given the data {x i : i, N} and {w i : i, N}, let {c k : k, K} be arbitrary points, and let the corresponding KN probabilities {p k x i )} be given by 7). Then, the condition R k c k ), for all k, K, 34) is necessary and sufficient for the points {c,..., c K } to minimize f c,..., c K ) of 8). b) The optimal centers {c k : k, K} are in the convex hull of the data points {x,..., x N }. c) If a center c is optimal, then it is a fixed point of T k of a)), T k c) c, for some k, K. 35) Conversely, if c {x j : j, N} and satisfies 35), then c is optimal. d) If T k c) c, then f k T k c)) < f k c), k, K. 36) e) If the data point x j is an optimal center, then w j R j. f) If the data point x j is not an optimal center, the direction of steepest descent of f at x j is Rj R j. g) Let x i be a data point that is not an optimal center c k, for a given k, K. Then, there exist δ > and an integer s > such that h) < x i c δ implies x i T s k c) δ and x i T s k c) > δ x j T k c) lim Rj k c x j x j c p k x for k, K, j, N. j) w j i) Given any c k, k, K, define the sequence {cr k T r k c) : k r,,...}. If no c r k is a data point, then lim r c r k c k, for some optimal centers {c,..., c K }. 8. A generalized Weiszfeld method for the multi-facility location problem The above results are implemented in an algorithm for solving P.K ). Algorithm. A generalized Weiszfeld method for several facilities Data: D {x i : i,n} data points locations of customers), {w i : i,n} weights, K the number of facilities ɛ > stopping criterion) Initialization: K arbitrary centers {c k : k,k}, Iteration: Step compute distances {dx, c k ) : k,k} for all x D Step compute probabilities {p k x) : x D, k,k} using 7)) Step 3 update the centers {c + : k T kc k ) : k,k}

6 C. Iyigun, A. Ben-Israel / Operations Research Letters 38 ) 7 4 Step 4 using a) and b)) if dc +, k c k) < ɛ stop return to step Writing a) as T k c) j j we get, from 3), x i c p k x j) w j x j c x i c p k x j) w j x j c x i c + c x i c) + c T k c) c + h k c) R k c) 37) with h k c) j p k x j) w j x j c 38) showing that Algorithm is a gradient method, following the direction of the resultant R k c) with step of length h k c) R k c), except for the data points {x j : j, N} which are left fixed by T k. Remark 3. a) Convergence can be established as in the Weiszfeld algorithm, by forcing a move away from non-optimal data points where the algorithm gets stuck by b)), using ideas of Balas and Yu [], or Vardi and Zhang [9], see Remark. b) The probabilities {p k x i ) : k, K, i, N} are not needed explicitly throughout the iterations. These probabilities are given by the distances {dx i, c k )}, see 7), and therefore the centers update in Step 3 requires only the distances. Step may thus be omitted. c) The final probabilities following the stop in Step 4) are needed for assigning customers to facilities. In the absence of capacity constraints, each customer is assigned to the nearest center, which by 8) has the highest probability. d) The algorithm is robust, since the weights in b) are inversely proportional to the distance, and as a result outliers are discounted, having little influence on the facility locations. e) Algorithm can be modified to account for capacity constraints on the facilities, as in [7]. This is necessary in capacitated MFLP where such constraints may force the customers to split their demands among several facilities, and travel farther than in the unconstrained case. Capacitated MFLP is handled in a sequel article. f) Algorithm can handle other models for assumption A), for example, 8) replaced by, p k x) φdx, c k )) Dx), k, K, w where φ ) is an increasing function, such as d α, α > or e d, see [3, Sections.8 and 3]. 9. Numerical examples The optimal solution has value 43.9, and the value of the worst stable solution is Algorithm was tried times on this P.3), using random initial centers, and a tolerance ɛ.. The optimal solution was found in 6% of the trials, the worst solution in.7% of the trials. The average number of iterations was 3. The optimal centers are shown in Fig. b), together with some level sets of the membership probability for each cluster. The three half-lines are the common boundaries of the clusters, a Voronoi diagram of the centers, partitioning the plane into territories served by each facility. Example. This example uses data from Eilon et al. [4, p. 83]. It is required to locate 5 facilities to serve the 5 customers shown in Fig. a). For the original problem L.5), different solutions were reported in [4, Table 4.], with an optimal value of 7.54, and the objective value at the worst solution is.99. We tested Algorithm on these data, times, with random initial centers and ɛ.. The average number of iterations was. An optimal solution with value of 7.5, slightly lower than the one reported in [4]) was found in % of the, trials. In.35% of the cases, the algorithm found the worst solution. Fig. b) shows the optimal centers found by Algorithm ), some probability level sets for each cluster, and the common boundaries of the centers. The Voronoi diagram has 4 segments and 4 half-lines common boundaries are empty.) Example 3. Fig. 3a) shows a simulated data of 45 customers, organized in 3 equal clusters. Problem P.3) was solved. Fig. 3b) shows the centers, and some level sets of the JDF ), illustrating the contour approximation property, namely that the lower level sets of the JDF capture the data points. Fig. 3c) shows some level sets of the CUF 7), with darker shades indicating higher uncertainty. Uncertainty is minimal Ex) ) if x is one of the centers. Note the patch of maximal uncertainty Ex).99) in the middle, where one is indifferent between the three centers. The right number of clusters known a priori in this simulated example) is determined by solving P.K ) for values of K,,..., calculating the CUF EX) of the whole dataset 3), and stopping when the marginal decrease of uncertainty is negligible. The results are plotted in Fig. 3d), confirming K 3 as correct. If the data are amorphous with no clear number of clusters, e.g. the data shown in Fig. a), then the graph of EX) does not give a clue. Acknowledgements Helpful comments and suggestions by Jack Brimberg, Zvi Drezner, Arie Tamir, Marc Teboulle and a referee are gratefully acknowledged. Appendix. Proof of Theorem Proof of part a). If c k is not one of the data points, then R k c k ) is the gradient ) at c k, and 34) is both necessary and sufficient for a minimum, by the convexity of f k. If c k coincides with a data point x j, consider the change from x j to x j + t z where z. Then, d dt f kx j + t z) p k x j) w j R j k z. 39) t The greatest decrease of f k is for z along R j k, i.e., when Example. This example uses the data of Cooper [7]. It is required to locate 3 facilities to serve the 5 customers shown in Fig. a). z Rj k R j k, proving part f)).

7 C. Iyigun, A. Ben-Israel / Operations Research Letters 38 ) Therefore, c k that coincides with x j ) is a local minimum if and only if, p k x j) w j Rj k Rj k R j k, which is equivalent to R j k p k x j) w j, or R k c k ), by 33a). Proof of part b). If c k is one of the data points, then it is trivially in the convex hull. Otherwise, the condition R k c k ), see 34), results in a) and b) as above. Proof of part c). Follows from part a), since for c X, c T k c) R k c), while for c X, c T k c) for all k. Proof of part d). The two functions f k ) in 36) are different because the probabilities changed. We prove the inequality 36) for the original probabilities, noting that the updated probabilities 7) are optimal, and result in further decrease. c is not one of the data points, since T k c) c. It follows then from a) that T k c) is the center of gravity of weights / x i c placed at the data points x i. By elementary calculus, T k c) is the unique minimum of the strictly convex function gy) Since c T k c), gt k c)) x i c x i y. On the other hand, x i c x i T k c) < gc) x i c x i c f k c). p k gt k c)) x i) w [ )] i x i c + x i T k c) x i c x i c ) f k c) + f k T k c)) f k c) + Combining these results, f k T k c)) + proving that f k T k c)) < f k c). p k x i) w [ ] i x i T k c) x i c x i c p k x i) w [ i x i T k c) x i c ] < fk c) x i c Proof of part e). Follows from part a), the definition R k x j ) : max { R j k p k x j) w j, } Rj k R j k, and p k x j ), since x j c k. Proof of part f). This was shown in part a). 33a) Proof of part g). T k c) x i c + h k c) R k c) x i p k h k c) x j) w j x j c x j c) + j i hk c) x i c Since x i is not optimal, we have p k x j) w j x j x i x j x i ) > p k x i) w i. j i ) x i c). Hence, there exist δ > and ɛ > such that p k x j) w j x j c x j c) + ɛ) p k x i) w i j i for x i c δ. By the definition of h k, we have h k c)p k lim x i) w i. c x i x i c Hence, there exists δ > such that h k c) x i c < ɛ for < x i c δ. + ɛ) Set δ minδ, δ ). For < x i c δ, we have x i T k c) > h k c) + ɛ)p k x ɛ i) w i + ɛ) x i c ) ɛ > + ɛ) x i c + ɛ) ɛ + ɛ) x i c + ɛ) x i c. Since x i c >, + ɛ) t x i c > δ for some positive integer t and hence x i T s k c) > δ for some positive integer s with x i T s k c) δ. Proof of part h). For c not a data point, p k T k c) x / ) i) w i N p k x m) w m x i c x m c i j x i c m x i x j ) + x j m T k c) x j p k x m) w m x m c i j T kc) x j x j c i j w j + x j c p k x j) w j x i c x i p k x / i) w i N x i c x p k i x j ) x m) w m x m m c, / x i c x i x j ) i j p k x ) i) w i. x i c Taking the limits of the lengths of both sides, lim c x j x j T k c) x j c Rj k p k x j) w j. p k x j)

8 4 C. Iyigun, A. Ben-Israel / Operations Research Letters 38 ) 7 4 Proof of part i). With the possible exception of c k, the sequence {c r k } lies in the convex hull of vertices, a compact set. By the Bolzano Weierstrass theorem, there exits a convergent subsequence {c l : l,,.. k.}, such that lim l c l k c k. We prove that c k c k. If c l+ k T k c l) k cl k for some l, then the sequence repeats from that point and c k c l k. Since cl k is not a data point, c k c k by Theorem c). Otherwise, by Theorem d), f k c ) > k f kc ) > > k f kc r ) > > k f kc k ). 4) Hence lim fk c l ) k f kt k c l ))) k. 4) l The continuity of T k implies lim T kc l ) k T kc ) k 4) l we have f k c ) k f kt k c k )). 43) x j c l k Therefore, by Theorem d), c k T k c k ). If c k is not a data point, then c k c k by Theorem c). In any event, c k lies in the finite set of points {x,..., x N, c k }, where c k may be a data point. The only case that remains is c k x j for some j. If x j c k, we first isolate x j from the other data points and c k if it is not a data point) by a δ-neighborhood that satisfies Theorem g). Then, it is clear that we can choose our subsequence c l k x j such that x j T k c l) > δ k for all l. This means that the ratio x j T k c l k ) is unbounded. However, this contradicts Theorem h). Hence, x j c k and the theorem is proved. References [] E. Balas, C.-S. Yu, A note on the Weiszfeld-Kuhn algorithm for the general Fermat problem, Tech. Report, Carnegie Melon University, September 98. [] A. Beck, M. Teboulle, Z. Chikishev, Iterative minimization schemes for solving the single source localization problem, SIAM J. Optim. 9 8) [3] A. Ben-Israel, C. Iyigun, Probabilistic distance clustering, J. Classification 5 8) 5 6. [4] J.C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms, Plenum, 98. [5] J. Brimberg, Further notes on convergence of the Weiszfeld algorithm, Yugoslav J. Oper. Res. 3 3) [6] R. Chandrasekaran, A. Tamir, Open questions concerning Weiszfeld s algorithm for the Fermat Weber location problem, Math. Program ) [7] L. Cooper, Heuristic methods for location allocation problems, SIAM Rev ) [8] T. Drezner, Z. Drezner, The gravity p median model, European J. Oper. Res. 79 7) [9] T. Drezner, Z. Drezner, S. Salhi, Solving the multiple competitive facilities location problem, European J. Oper. Res. 4 ) [] Z. Drezner, The planar two-center and two-median problems, Transport. Sci ) [] Z. Drezner, A note on the Weber location problem, Ann. Oper. Res. 4 99) [] Z. Drezner, K. Klamroth, A. Schöbel, G.O. Wesolowsky, The Weber problem, in: Z. Drezner, H.W. Hamacher Eds.), Facility Location: Applications and Theory, Springer,. [3] I.N. Eckhardt, Weber s problem and Weiszfeld s algorithm in general spaces, Math. Program. 8 98) [4] S. Eilon, C.D.T. Watson Gandy, N. Christofides, Distribution Management: Mathematical Modelling and Practical Analysis, Hafner, 97. [5] D.L. Huff, Defining and estimating a trading area, J. Marketing 8 964) [6] D.L. Huff, A programmed solution for approximating an optimum retail location, Land Econom ) [7] C. Iyigun, A. Ben-Israel, Probabilistic distance clustering adjusted for cluster size, Probab. Engrg. Info. Sci. 8) 9. [8] C. Iyigun, A. Ben-Israel, Contour approximation of data: a duality theory, Lin. Algeb. Appl. 43 9) [9] C. Iyigun, A. Ben-Israel, Probabilistic semi-supervised clustering and the uncertainty of classification, in: A. Fink, B. Lausen, W. Seidel, A. Ultsch Eds.), Advances in Data Analysis, Data Handling and Business Intelligence, Springer, 9. [] H.W. Kuhn, On a pair of dual nonlinear programs, in: J. Abadie Ed.), Methods of Nonlinear Programming, North-Holland, 967, pp [] H.W. Kuhn, A note on Fermat s problem, Math. Program ) [] S. Kullback, R.A. Leibler, On information and sufficiency, Ann. Math. Statist. 95) [3] Y. Levin, A. Ben-Israel, A heuristic method for large-scale multi-facility location problems, Comput. Oper. Res. 3 4) [4] R.F. Love, J.G. Morris, G.O. Wesolowsky, Facilities Location: Models and Methods, North-Holland, 988. [5] R.D. Luce, Individual Choice Behavior, Wiley, 959. [6] N. Megiddo, K.J. Supowit, On the complexity of some common geometric location problems, SIAM J. Comput ) [7] L.M. Ostresh, On the convergence of a class of iterative methods for solving the Weber location problem, Oper. Res ) [8] M. Teboulle, A unified continuous optimization framework for center-based clustering methods, J. Machine Learning 8 7) 65. [9] Y. Vardi, C.-H. Zhang, A modified Weiszfeld algorithm for the Fermat Weber location problem, Math. Program. A9 ) [3] E. Weiszfeld, Sur le point par lequel la somme des distances de n points donnés est minimum, Tohoku Math. J ) [3] G.O. Wesolowsky, The Weber problem: Its history and perspectives, Location Sci. 993) 5 3.