Continuous OptimizationA heuristic algorithm for constrained multi-source Weber problem – The variational inequality approach
Introduction
The classical multi-source Weber problem (MWP) is to find the locations of m new facilities in order to minimize the sum of the transportation costs between these facilities and the n customers whose locations are known. We assume that the involved transportation costs are proportional to the corresponding distances. More specifically, the mathematical model of MWP is as follows:where
- (1)
is the location of the jth customer, ;
- (2)
is the location of the ith facility to be determined, ;
- (3)
is the given demand required by the jth customer;
- (4)
denotes the unknown allocation from the ith facility to the jth customer;
- (5)
is the Euclidean distance.
Due to the wide applications of MWP in operations research, marketing, urban planing, etc., see, e.g, [6], [10], [17], many authors are devoted to both promoting its theoretical development and presenting effective numerical algorithms. To mention a few, the branch-and-bound algorithm presented in [19] obtains the solution by reducing MWP with m = 2 to O(n2) pairs of Weber problems; while the algorithm in [22] contributes a linear programming approach to solving MWP by enumerating all feasible elements of a solution (non-overlapping convex hulls). Among existing effective numerical algorithms for solving MWP is the heuristic alternate location–allocation algorithm presented originally in [5], whose attractive characteristic is that each iteration consists of a location phase and an allocation phase. Let and denote the set of locations of all the customers and with and (for ) denote the disjoint partition of A at the kth iteration. At the th iteration, the location phase finds the candidates of locations of facilities by solving m single-source Weber problems (SWP) denoted by
Then the allocation phase involves an allocation or a partition, which depends on the generated by solving (1.1). More specifically, if , is the nearest facility among all facilities for each customer in , then are the desirable locations of facilities. Therefore, it is reasonable to allocate the customers in from the facility in order to minimize the total sum of transportation costs. Otherwise, the set of customers should be partitioned newly according to the heuristic of nearest center reclassification (NCR), i.e., the new partition is generated so that is the nearest facility for each customer in . Note that with the presupposition that each facility to be determined is capable of providing sufficient services for the targeted customers, the heuristic characteristic shared by all Cooper-type algorithms is that finally each custom (aj) is served only by one of the facilities (the nearest one). This observation explains the disappearance of wij in (1.1).
Then the central task of the location–allocation algorithm is to solve the involved SWP (1.1) in the location phase, and so it is meaningful to investigate effective numerical algorithms to solve (1.1). The first attractive contribution to this aspect denoted by Cooper’s algorithm was due to [5], in which the author used the Weiszfeld procedure [25] to solve the involved SWP. Recently, the so-called Newton-Bracketing (NB) method for convex minimization was utilized to solve the involved SWP and thus Cooper-NB algorithm was developed in [16]. Due that the gradients of are used in the iteration, both the Weiszfeld procedure and NB method share the common characteristic that their implementations may terminate unexpectedly when the current iterate happens to be identical with some location of the customers (which is unavoidable and uncontrollable), i.e. the singular case happens. How to improve the original Weiszfeld procedure and NB method in the singular case and make them computationally preferable become the main challenges in this study and still deserve more extensive investigations, see, e.g, [1], [3], [14], [20], [24]. For example, the effort proposed in [16] suggested to replace the gradient of with 0 whenever the singular case happens during the implementation.
This paper concentrates on presenting a new heuristic alternate location–allocation algorithm in the spirit of Cooper’s work [4], [5]. Since the new algorithm does not use the gradient of in its implementation, it is still effective even in the aforementioned singular case (which will be verified by the numerical experiments). The involved SWP (1.1) in the location phase are reformulated into some linear variational inequalities (LVI), which are solved efficiently by a new projection–contraction (PC) method. The reduced new heuristic alternate location–allocation algorithm, denoted by Cooper-PC algorithm, improves Cooper’s algorithm and Cooper-NB algorithm in computational senses. In addition, since in practice the locations of facilities are usually restricted into some specific regions, it is more practical to consider the constrained multi-source Weber problem (CMWP) which imposes additional constraints to xi . From now on we consider the CMWP whose mathematical model is as follows:where Xi ) are non-empty closed convex subsets in R2. Correspondingly, the involved SWP in CMWP are now changed to the following constrained SWP (CSWP)Note that CMWP reduces to MWP in the case that .
The rest of the paper is organized as follows. Section 2 reformulates the involved CSWP (1.2) into LVI and presents an efficient projection–contraction method for solving these LVI. Convergence of the projection–contraction method is also proved under mild assumptions in this section. In Section 3, the new heuristic alternate location–allocation algorithm is presented for solving CMWP. Numerical results including the performances of the new PC method and Cooper-PC algorithm are reported in Section 4. Finally, some conclusions are drawn in Section 5.
Section snippets
The variational inequality approach to CSWP
This section aims at solving the involved CSWP (1.2) in the location phase by a variational inequality (VI) approach. Note that the study of VI has received much attention due to its various application arising in engineering, operations research, economics, transportation, etc., see, e.g. [9]. In particular, the CSWP (1.2) are reformulated into some LVI, whose special structures motive us to present an efficient variation of the projection–contraction (PC) method originally proposed by He [11]
Cooper-PC algorithm for CMWP
In the spirit of Cooper’s work, we present a new heuristic algorithm denoted by Cooper-PC algorithm for solving CMWP. Each iteration of the new algorithm also consists of the location phase and the allocation phase. The CSWP involved in the location phase are solved by Algorithm 1; while the update of the partition of the locations of customers is also according to the spirit of nearest center reclassification (NCR) as in Cooper’s algorithm and Cooper-NB algorithm. Note that Cooper-PC algorithm
Numerical results
This section reports some numerical results to verify the theoretical assertions proved in previous sections. The first subsection mainly compares Algorithm 1 to some well-known existing methods including the NB method in [16], the PC method in [12] and the primal–dual algorithm in [15]; and thus demonstrates that the new PC method is effective for solving the involved SWP. In the second subsection, we first compare the Cooper-PC method to the Cooper-NB algorithm in [16]; and then we apply the
Conclusion
For solving constrained multi-source Weber problem, this paper presents a new heuristic alternate location–allocation algorithm, which differs from the existing related algorithms mainly in the sense that the involved constrained single-source Weber problems in the location phase are solved by a variational inequality approach. The attractive characteristics especially propitious to practical implementation make the new algorithm enrich the diversity of existing approaches to solving
Acknowledgements
The authors are grateful to the anonymous referees for their valuable and extensive comments to improve the presentation of this paper greatly.
References (25)
- et al.
Geometrical properties of the Fermat–Weber problem
European Journal of Operational Research
(1985) - et al.
A heuristic method for large-scale multi-facility location problems
Computers & Operations Research
(2004) An optimal method for solving the (generalized) multi-Weber problem
European Journal of Operational Research
(1992)- et al.
Global convergence of a generalized iterate procedure for the minisum location problem with lp distances
Operations Research
(1993) - et al.
A generalized proximal point algorithm for the variational inequality problem in a Hilbert space
SIAM Journal on Optimization
(1998) - et al.
Open questions concerning Weiszfeld’s algorithm for the Fermat–Weber location problem
Mathematical Programming
(1989) Location–allocation problems
Operations Research
(1963)Heuristic methods for location–allocation problems
SIAM Review
(1964)Facility Location: A Survey of Applications and Methods
(1995)On the basic theorem of complementarity
Mathematical Programming
(1971)
Engineering and economic applications of complementarity problems
SIAM Review
Spatial Analysis and Location–allocation Models
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This author was supported by an internal grant of Antai College of Economics and Management, Shanghai Jiao Tong University.