a lagrangean/surrogate heuristic for the maximal covering ... - CiteSeerX

A LAGRANGEAN/SURROGATE HEURISTIC FOR THE MAXIMAL COVERING LOCATION PROBLEM USING HILLSMAN´S EDITION Luiz A. N. Lorena*, Marcos A. Pereira#. Laboratório Associado de Computação e Matemática Aplicada - LAC. Instituto Nacional de Pesquisas Espaciais - INPE. PO Box 515, São José dos Campos, 12201-970, Brazil. Email: {*lorena, #marcos}@lac.inpe.br The Maximal Covering Location Problem (MCLP) deals with the location of facilities in order to attend the largest subset of a population within a service distance. Many successful heuristic approaches have been developed to solve this problem. In this work we use the Unified Linear Model developed by Hillsman to adapt the distance coefficients of a p-median problem to reflect the demand information of a population. This transformation permits the application of a Lagrangean/surrogate heuristic developed for solving p-median problems to solve the MCLP. In previous works this heuristic proved to be very affordable, providing good quality solutions in reduced computational times. Computational tests for random generated scenarios ranging from 100 to 900 vertices and GIS-referenced instances of São José dos Campos city (Brazil) were conducted, showing the effectiveness of the combined approach. Significance: In addition to the economic relevance of decisions related to facility location problems, applications in computer network design and flexible manufacturing systems can benefit with the use of MCLP models. Keywords: Maximal covering location problems, p-Median problems, Hillsman´s edition, Lagrangean and surrogate relaxation, Heuristic solution, Geographic Information Systems. (Received

; Accepted

.)

1. INTRODUCTION Location-allocation problems deals with decisions of finding the best (or optimal) configuration for the installation of one or more facilities in order to attend the demand of a population (Daskin (1995), Drezner (1995)). In the private sector the term facility can be replaced by plants, warehouses, telecommunication antennas etc. The applications in the public sector can be divided between public services (schools, libraries, hospitals, bus stops) and emergency services (fire and police stations, ambulance posts). Facility location analysis can be improved if geo-referenced data, as provided by Geographic Information Systems (GIS), is available. Despite the possible different nature of the applications, location-allocation models present the same basic structure. Based in the p-median models of Hakimi (1965) and ReVelle and Swain (1970), Hillsman (1974) developed an Unified Linear Model (ULM) that can be adapted to model other location-allocation problems. Given a network, the p-median problem (pMP) is the problem of locating p facilities minimizing the sum of the distances of each demand point to its nearest facility. The maximal covering location problem (MCLP) is the problem of locating p facilities on a network such that the maximal population is attended (or covered) within a given service distance (Church and ReVelle (1974)). In his work, Hillsman proposed a change in the distance coefficients of a pMP to obtain a new set of coefficients, based in the population information and the service distance of a MCLP. Since no changes are made in the structure of the pMP model, existent solution procedures for solving pMP’s can be applied to the new data set and obtain solutions for the corresponding MCLP. This paper assess the quality of the combined approach using the Lagrangean/surrogate heuristic of Senne and Lorena (2000) for solving pMP’s in problem instances with both random generated and real world data. In the next section we present the unified linear model and the proper change in problem coefficients to model a maximal covering location problem from data of a p-median problem. Section 3 presents the Lagrangean/surrogate relaxation of Senne and Lorena for solving the p-median problem. Section 4 contains an interchange algorithm for improvement of primal solutions. In section 5 we report the computational tests for random generated and real world data, ranging from 100 to 900 vertices. The integration of the heuristic to ArcView, a GIS software developed by Environmental Systems Research Institute Inc., is also presented, using geo-referenced data of São José dos Campos city, Brazil. Section 6 presents the conclusions and future extensions.

2. THE UNIFIED LINEAR MODEL FOR THE P-MEDIAN PROBLEM AND THE MAXIMAL COVERING LOCATION PROBLEM For a n vertex network and a symmetric distance matrix D = [dij]n×n, the ULM adapted for the pMP can be stated as the following binary integer programming problem: n

n

v( pMP) = min ∑∑ d ij xij

... (1)

i =1 j =1

n

(pMP) s.t.

∑x

ij

= 1 , ∀j ∈ N .

... (2)

ii

=p

... (3)

i =1 n

∑x i =1

xij ≤ xii , ∀i, j ∈ N .

... (4)

xij ∈ {0,1} , ∀i, j ∈ N .

... (5)

Constraint (3) is obtained assuming k = p and the equality relation in the generalized form of the corresponding inequality of the ULM: n ≤ xii = k ... (6) ≥ i =1 The variables xij, i, j ∈ N = {1, ..., n}, indicate if node j is served by the facility located in candidate node i (xij = 1) or not (xij = 0) and if candidate node i is chosen for the installation of a facility (xii = 1) or not (xii = 0). The objective function (1) represents the total distance from every node to its nearest facility. Constraints (2) and (4) specify that every node must be served by only one installed facility. Constraint (3) indicates that exactly p nodes are to be chosen as candidates for installation of facilities. The binary condition over the variables are given in constraint (5). An optimal solution to the model (1)-(5) is a solution that yields the minimum value of equation (1) for some matrix of coefficients C = [cij]n×n. When population information is available for every node j ∈ N, say wj, then a new set of coefficients can be calculated from the distance matrix D of the pMP in the following way:  0, if d ij ≤ S ; cij =  ... (7) w j , if d ij > S .

∑

If these coefficients are used in equation (1), then the problem switches to determine the best candidates for the installation of p facilities, minimizing the unattended population of the nodes that are more than S distance units away from any facility or, equivalently, maximizing the attendance of the population of the nodes within S distance units of any facility. The model (1)-(5), with coefficients calculated as in (7), is the ULM correspondent of MCLP. If D contains information about time, then S should be chosen accordingly to represent the limit time to reach a served node from an installed facility. In both problems, xij represents the solution of the location-allocation problem, with xii = 1 representing the chosen candidates for the installation of the facilities. Although constraint (2) forces every node to be allocated to exactly one facility, the fact that xij = 1 in the MCLP does not guarantee attendance: only the nodes within S distance (or time) units from a facility will be considered covered. Another particularity of this transformation is that the value v(pMP) of the objective function using the coefficients from matrix C gives the total unattended population: the value of the corresponding solution to the MCLP is calculated as: n

v(MCLP) = ∑ w j − v( pMP)

... (8)

j =1

3.THE LAGRANGEAN/SURROGATE RELAXATION To shorten the notation, we refer to the model defined as in (1)-(5), with coefficients taken from the calculated matrix C, as P. Problem P can be solved using relaxation heuristics. Narciso and Lorena (1999) developed a Lagrangean/surrogate heuristic to approximately solve problem P. As proposed by Glover (1968), for a given ∈ R+n , a surrogate relaxation of P can be defined by:

n

n

v(SP ) = min ∑∑ cij xij

... (9)

i =1 j =1

n

λ

(SP )

n

n

∑∑

s.t.

j xij = ∑

j =1 i =1

... (10)

j

j =1

and (3)-(5). λ The optimal value of v(SP ) is less than or equal to v(P), and results from the solution of the dual surrogate problem λ max {v(SP )} . Problem SP is an integer linear problem with no special structure to explore. In addition, the surrogate ≥0

λ

function s : R+n → R , (λ, v(SP )) has some properties that make it difficult to find a dual solution. Methods for find approximated solutions of the surrogate dual were proposed by Karwan and Rardin (1979) and Dyer (1980). λ Due to the difficulties with relaxation SP we proposed to relax again the problem, now in the Lagrangean way. For a given t ≥ 0, constraint (10) is relaxed, and the Lagrangean/surrogate relaxation is given by: n

n

n

v(L t SP ) = min ∑∑ (cij − t

j ) xij + t ∑

j =1 i =1

j

... (11)

j =1

λ

(3), (4) and (5). (LtSP ) s.t. λ λ λ For given t ≥ 0 and ∈ R+n , v(LtSP ) ≤ v(SP ) ≤ v(P). Problem LtSP is solved considering implicitly constraint (3) and decomposing for index i, obtaining the following n problems: n

min ∑ (cij − t

j

... (12)

) xij

j =1

s.t. (4) and (5). that can be easily solved calculating: n

i

[

]

= ∑ min{0, cij − t j } , ∀i ∈ N, j =1

... (13)

and defining I as the index set of the p smallest βi (here constraint (3) is considered implicitly). Then, a solution xij to λ

problem LtSP is: 1, xii =  0  , and for i ≠ j:  1, xij =  0,

if i ∈ I otherwise

... (14)

if i ∈ I and cij − t otherwise

j