Abstract
This paper considers the locations of multiple facilities in the space <svg style="vertical-align:-0.1092pt;width:18.75px;" id="M1" height="14.5875" version="1.1" viewBox="0 0 18.75 14.5875" width="18.75" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,14.387)"><path id="x1D445" d="M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117
t-115.5 -72q40 -86 66 -133q39 -68 65 -101q28 -37 73 -51zM491 483q0 67 -33.5 101t-91.5 34q-35 0 -51 -10q-13 -8 -20 -48l-45 -245h49q71 0 113 28q79 52 79 140z" /></g> <g transform="matrix(.012,-0,0,-.012,11.013,6.225)"><path id="x1D45D" d="M570 304q0 -108 -87 -199q-40 -42 -94.5 -74t-105.5 -43q-41 0 -65 11l-29 -141q-9 -45 -1.5 -58t45.5 -16l26 -2l-5 -29l-241 -10l4 26q51 10 67.5 24t26.5 60l113 520q-54 -20 -89 -41l-7 26q38 28 105 53l11 49q20 25 77 58l8 -7l-17 -77q39 14 102 14q82 0 119 -36
t37 -108zM482 289q0 114 -113 114q-26 0 -66 -7l-70 -327q12 -14 32 -25t39 -11q59 0 118.5 81.5t59.5 174.5z" /></g> </svg>, with the aim of minimizing the sum of weighted distances between facilities and regional customers, where the proximity between a facility and a regional customer is evaluated by the closest distance. Due to the fact that facilities are usually allowed to be sited in certain restricted areas, some locational constraints are imposed to the facilities of our problem. In addition, since the symmetry of distances is sometimes violated in practical situations, the gauge is employed in this paper instead of the frequently used norms for measuring both the symmetric and asymmetric distances. In the spirit of the Cooper algorithm (Cooper, 1964), a new location-allocation heuristic algorithm is proposed to solve this problem. In the location phase, the single-source subproblem with regional demands is reformulated into an equivalent linear variational inequality (LVI), and then, a projection-contraction (PC) method is adopted to find the optimal locations of facilities, whereas in the allocation phase, the regional customers are allocated to facilities according to the nearest center reclassification (NCR). The convergence of the proposed algorithm is proved under mild assumptions. Some preliminary numerical results are reported to show the effectiveness of the new algorithm.
Highlights
Due to the large number of practical applications in various fields such as marketing, urban planning, supply chain management, and transportation, the continuous facility location problem has aroused the attention of many researchers ever since the pioneering work [1, 2]
Note that the targeted CMLP (9) is an extension of the multi-source Weber problem (MWP) and it is harder than MWP, and in this paper, we focus on applying the locationallocation heuristic algorithm for solving the CMLP in the spirit of Cooper’s work
The allocation task generates a new disjoint partition of all the regional customers according to the principle of nearest center reclassification (NCR) as in the Cooper algorithm, and the location phase identifies the optimal locations for the current partition of customers via implementing the variational inequality approach for solving m constrained singlesource location problems (CSLP)
Summary
Due to the large number of practical applications in various fields such as marketing, urban planning, supply chain management, and transportation, the continuous facility location problem has aroused the attention of many researchers ever since the pioneering work [1, 2]. Regional demand arises frequently in such scenarios as uncertain demand, mobile demand, or cumbersome discrete situation whose number of demand points is extremely large. For such scenarios, many authors (e.g., [5,6,7,8,9,10,11]) promote that the regional customer, that is, the locations of customers are geometrically connected regions rather than points, should be considered. Focusing on the real-life applications with vast eyes, the regional customers and the closest distances are highly essential to be considered.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
