an efficient tabu search procedure for the p-median problem

E L S E V I E R European Journal of Operational Research 96 (1996) 329-342

EUROPEAN JOURNAL

OF OPERATIONAL RESEARCH

T h e o r y and M e t h o d o l o g y

An efficient tabu search procedure for the p-Median Problem

Er ik R o l l a n d a, D a v i d A. Schi l l ing h,, , J o h n R. C u r r e n t b

a The A. Gary Anderson Graduate School of Management, University ofCali~brnia, Riverside, CA 92521, USA b Department of Management Sciences, Fisher College of Business, Ohio State University, 1775 College Road, 302 Hagerty Hall,

Columbus, OH 43210, USA

Received 1 August 1995; accepted 1 April 1996

Abstract

In this paper we present a new solution heuristic for the p-Median Problem. The algorithm is based on tabu search principles, and uses short term and long term memory, as well as strategic oscillation and random tabu list sizes. Our proposed procedure is compared with two other move heuristics: a well-known interchange heuristic and a recent hybrid heuristic. In computational tests on networks ranging in size up to 500 nodes the new heuristic is found to be superior with respect to the quality of solutions produced.

Keywords: Facility location; Heuristics; P-median; Tabu search

1. Introduction

Mathematical approaches to facility location decisions have received considerable attention from re- searchers in numerous disciplines (e.g., see Brandeau and Chiu, 1989). One of the classic problems in the field is the p-Median Problem (pMP) (Hakimi, 1965; ReVelle and Swain, 1970). Given a set of demand locations and potential facility sites, the object of the pMP is to identify the locations of a pre-determined number of facilities, p, in such a way as to minimize the total distance that demand must traverse to reach its nearest facility. The pMP has received widespread attention because it is appropriate for many facility location decisions (e.g., Daskin, 1995; GalvSo, 1993) and forms the basis for more complex problems

* Corresponding author. Fax: + 1-614-292-1272.

(e.g., Current and Schilling, 1994; Weaver and Church, t985).

In practice, the pMP occurs most frequently on a network such as a road or telecommunication network. Hakimi (1965) proved that for network instances of the problem, an optimal solution exists for which all of the facility locations are at nodes on the network. This discovery enabled Hakimi (1965) to formulate the network version of the pMP as a binary integer program. Unfortunately, this problem belongs to the class of problems known as NP-complete (Kariv and Hakimi, 1979). ReVelle and Swain (1970) demonstrated that optimal integer solutions to Hakimi's binary integer formulation were frequently found by relaxing the binary constraints and solving the resulting linear program. Unfortunately, even the linear programming version of the problem may be computationally intractable as the number of constraints increase exponentially with problem size.

037%2217/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PII S 0 3 7 7 - 2 2 1 7 ( 9 6 ) 0 0 1 41-5

330 E. Rolland et a l . / European Journal o f Operational Research 96 (1996) 329-342

Given both the practical importance and computational intractability of the pMP, various heuristic solution approaches have been developed to solve it. As is evidenced both by its widespread use as well as empirical comparisons with other techniques, the vertex substitution heuristic of Teitz and Bart (1968) and its variations (e.g., Goodchild and Noronha, 1983) have proven to be the most robust methods for solving pMP (Densham and Rushton, 1992). A ma- jor drawback of these heuristics is that they may converge on a local optimum rather than identify the globally optimal solution. In this paper, we introduce a tabu search (TS) heuristic for pMP in an effort to avoid termination at a local optima. Tabu search (Glover, 1986) is a domain independent heuristic procedure for solving optimization problems which has been successfully applied to several combinatorial problems. These applications include: machine scheduling (Laguna et al., 1991), the quadratic assignment problem (Skorin-Kapov, 1990a,b), hub location problems (Klincewicz, 1992), graph problems (Friden et al., 1989; Rolland et al., 1995), neural networks (deWerra and Hertz, 1989), and graph coloring (Hertz and deWerra, 1987), among others. The interested reader is referred to Glover and La- guna (1993) for a more thorough list of applications. A comprehensive tutorial on tabu search can be found in Glover (1990a).

The remainder of this paper is organized as follows. In Section 2 and Section 3 respectively, we formulate the pMP mathematically and describe the Goodchild and Noronha (1983) and Densham and Rushton (1992) heuristics. In Section 4, we describe the tabu search heuristic. Computational results of the heuristics are presented in Section 5 and Section 6 contains a summary and conclusions.

2. A mathematical formulation of the p-Median Problem

The p-Median Problem can be stated as follows: given a graph G = (V,E), we are asked to find a set of nodes, S, of size p, where S c V, such that the weighted sum of the distances from the remaining nodes {V - S} to the set S is minimized. As a binary integer program, this can be written as:

P-median:

Min E ~_,aidijxij (1) i j

subject to

~_~xij= 1, Vi, (2) J

xij < yj, Vi,j, (3)

~,,yj =p, (4) J

xij,Y j~{O,1} Vi,j, (5)

where a i =demand at node i. dij = distance from node i to node j. p = number of facilities to be sited.

1 if node i is assigned to facility j Xij =

0 otherwise. 1 if facility j is opened

y j = 0 otherwise.

The objective function (1) minimizes the weighted sum of distances associated with assigning demand nodes to facility sites. Constraint set (2) ensures that all demand nodes are assigned to exactly one facility. Constraint set (3) prohibits assignment to a facility if it is not opened. The total number of open facility sites is set to p in constraint (4) and the binary nature of the facility siting and assignment decisions is enforced by constraint set (5). If appropriate, the 0-1 restriction on the xi. / variables can easily be relaxed by redefining these variables as the fraction of demand at node i that assigns to a facility at node j. In such a case, no upper bound constraints on these variables are necessary due to constraint sets (2) and (3). This formulation and the following solution procedures assume that all nodes j ~ V are potential facility sites. We make this assumption to facilitate the discussion, because it is often the case in practice, and is the worst-case from a computational standpoint. The formulation and solution procedures can be modified readily for cases where the potential facility sites are a subset of V.

3. Move heuristics for the p-Median Problem

One way to view solutions to the pMP is to represent any given solution as two disjoint sets of

E. Rolland et aL / European Journal of Operational Research 96 (1996) 329-342 331

nodes: one which is the set of selected facility sites, S and the other which is the set of unselected potential facility sites { V - S}. Many heuristics (e.g., Teitz and Bart, 1968, Goodchild and Noronha, 1983 and Densham and Rushton, 1992) are based on moving nodes between these sets. A " m o v e " essentially is selecting a single node from either set and transferring it to the other set. If a node is moved into S, we call this an " A D D " , if a node is moved out of S, it is called a " D R O P " . If ISl~p the resulting solution is not a feasible one. A " S W A P " is defined as a one-for-one exchange of nodes in S and {V - S} (essentially a paired ADD and DROP). SWAPS form the basis of exchange or node substitution (NS) heuristics, an example of which is Good- child and Noronha (1983), which may be stated as follows:

Heuristic NS: Step 1. Select p nodes and designate as S. Calcu- late (1) for this solution and label this value as Best_Solution. Step 2. Identify the two nodes, one from S and one from { V - S}, which, when swapped, result in the greatest improvement in the feasible solution value (note, all possible pair-wise exchanges must be considered). Designate this solution value as Feas. Step 3. If Feas < Best_Solution then set Best_ Solution = Feas and repeat Step 2. Otherwise STOP, a local optimum has been identified and no improvement can be found via additional swaps.

The NS heuristic procedure is not guaranteed to find a global optimum and the quality of the best solution identified often depends upon the starting solution identified in Step 1. Consequently, in practice, the procedure is executed several times with different starting solutions.

The SWAP move in Step 2 requires O( p(IV] - p)) operations, and O([V 12) operations in the worst-case, for each iteration. To reduce computation time, the ADD, DROP, and SWAP moves have been com- bined in a recent hybrid heuristic by Densham and Rushton (1992) called the Global/Regional Inter- change Algorithm (GRIA). The name refers to the heuristic's two phases: Global and Regional. The Global phase is first and performs the DROP and

ADD moves sequentially to produce a pair-wise exchange that does not have the computational burden of the SWAP move. This DROP/ADD combi- nation is repeated until no improvements are found. Once the Global phase is completed, GRIA moves to the Regional phase. Output from the Global phase includes p facility locations and the demand nodes assigned to each facility. The Regional phase essentially decomposes the problem into p, 1-median problems. That is, for the set of demand nodes assigned to a given facility site in the Global phase, it identifies, via the NS heuristic, the best location for the facility within that set. This phase is more computationally efficient than a global SWAP search, as it requires only O(IVl) operations.

4. Tabu search heuristic

Tabu search (Glover, 1986) is a metaheuristic that is intended to overlay a core search heuristic and seeks to help heuristics break out of local optima and explore other regions of the solution space. The move heuristics of the pMP are particularly amenable to this structure. The basic tabu search employs " t abu" restrictions which inhibit certain moves, and aspiration criteria which allow very good solutions to overcome any tabu status. The tabu restrictions are generally implemented with a short term memory function to make them time-dependent. Designing tabu search heuristics involves defining what types of moves to restrict and the nature of the aspiration criteria and short term memory to utilize. In addition to these features, most tabu search designs include other strategies such as a long-term memory function to diversify the search into other areas of the solution space. In this section we describe the various components of our heuristic.

4.1. Moves and tabu restrictions

In the search heuristics used with Tabu search, each iteration of the search focuses on a "neighborhood" of the current solution (set S of open facilities). The neighborhood is defined as that set of solutions which can be reached by a single " m o v e " . Moves, as discussed earlier, can be of several different types. In the heuristic developed in this paper,

332 E. Rolland et al . / European Journal of Operational Research 96 (1996) 329-342

there are two types of moves permitted: ADDs and DROPs. Thus, there are two types of neighborhoods: "constructive" neighborhoods resulting from ADDing a node to the set of open facilities; and "destructive" neighborhoods resulting from DROP- ing a node from the set of facilities.

As these moves are performed, tabu restrictions are employed to prevent moving back to previously investigated solutions. In general, these restrictions can be associated with any or all permissible moves, however, in the heuristic developed here, they are linked only to the ADD move. That is, once added to the set of open facilities, a node is classified as tabu. By utilizing short term memory, however, the tabu status is not permanent. Rather, there is a "tabu time" (or "tabu tenure") which is the time, measured in terms of iterations, that must elapse for a node to be removed from the tabu list. If Add_Time(v) is the memory function holding the iteration at which node v was last added, and Cur- rent_Time denotes the current iteration, then node o is tabu if Add_Time(v) >_ Current_Time - Tabu_Time. The parameter TabuTime represents the number of iterations a node retains its tabu status.

In general, there are at least three possible approaches to setting the tabu time: fixed, dynamic, or random. The fixed times remain constant throughout the solution procedure. Dynamic times on the other hand are adjusted according to a function of solution process attributes, such as halving the tabu time after a given number of unproductive iterations. The third method sets the tabu time to some randomly selected number if there were no improvements in a specified number of iterations. In this study, we chose the random approach based on our empirical tests which showed the random time was more efficient than either the fixed or the dynamic method, and resulted in no loss of solution quality.

4.2. Aspiration criteria

The aspiration criteria in a tabu search are used as insurance against restricting moves which would have led to finding high quality solutions. In other words, the aspiration criteria determines when a node can be moved even if tabu. In this paper, we use the typical criteria which states that if a move produces a solution better than the best known solution (and the

resulting solution is feasible), then the tabu status is disregarded and the move is executed.

4.3. Diversification and long term memory

Diversification is utilized to escape from local optima and is achieved by using a long-term memory function. Here, this is a frequency-based function that deters the search from performing moves made too often. It is implemented by letting Freq(v) equal the number of times node v has been added to the set S. Then let /7 denote a penalty function, where H(v) = k. Freq(v), and k is a constant. The value of this function is incorporated into the objective function (1) when node v is to be added to set S, and thereby inhibits the selection of nodes that have been chosen frequently. This strategy provides a "diversification" into areas of the problem state which have not been investigated. The parameter k is chosen so that the penalty has a significant impact on the objective function. In our experiments we used k = M a x ij{dij}.

It should not noted that we sought an approach that was straightforward to implement, as a basis for developing comparisons. Advanced tabu search procedures, which have not been incorporated here but which may readily be used to supplement our approach, are a subject of future research.

4.4. Evaluation, add and drop procedures

The evaluation of any new trial solution includes the long term memory function and is performed as follows:

1.

.

Procedure EVALUATE: Given the set S, assign all nodes in the set { V - S} to their closest facility in S (i.e., set the decision variables xij). If the node v is entering S add the penalty H (v ) = k. Freq(v) to the assignment cost of v (i.e., set d,. v = div + / 7 ( v ) and compute the total cost of the assignments using Eq. (1). Set this value : = Solution.

The only moves performed in the basic heuristic are ADD and DROP and are implemented as follows:

E. Rolland et a l . / European Journal of Operational Research 96 (1996) 329-342 333

Procedure ADD:

Select a non-tabu node from { V - S } which when added to S results in the best possible value of Solution. Add this node to S, and set New_So lu t ion : = Solution

Procedure D R O P :

Select a node from S which when dropped from S results in the best possible value of Solution.

Move this node into { V - S} (dropping it from {S}). According to the aspiration criteria, a tabu node must produce the best solution yet found in order to be selected. If no tabu node produces such a result, then the best non-tabu move is selected. Set N e w _ S o l u t i o n : = Solution.

4.5. Strategic oscillation

Note that the solution value obtained from either the ADD or DROP procedure above does not necessarily produce a feasible solution. That is, ISI may not equal p. A feature of tabu search, frequently referred to as strategic oscillation (Glover, 1989, 1990b), allows the search to intentionally pass through infeasible solutions as another way to escape local optima. Moreover, permitting temporary infea- sibility may lead to a more rapid descent to better feasible solutions. To illustrate, assume that m denotes the average number of iterations needed to move from a local optima to an adjacent neighborhood, i.e., from the bottom of a "val ley" to the ridge of the adjacent valley as seen in objective

function space (see Ryan, 1992, for an estimation method for m). Then, on the average, it would require 2m tabu search moves to reach an adjacent local optima. By using strategic oscillation we may be able to reach another local optima (not necessarily adjacent) in less than 2m moves.

Strategic oscillation is implemented in the selection of the type of move to make:

Procedure C H O O S E _ M O V E :

If (IsI < p - Slack) then ADD otherwise

If (ISI > p + Slack) then DROP otherwise

Flip a fair coin. If it comes up Heads and IsI > 0 then DROP otherwise ADD

Note that the use of the Slack parameter effectively allows S to grow and shrink beyond size p + 1. That is, ADD and DROP moves are not necessarily implemented sequentially. An additional feature of the tabu search heuristic in this paper is that the value of Slack is not fixed. Instead, it is set to zero whenever an improved solution has been found, and increased by one when no improvements have been identified in a specified number of iterations.

In summary, the components of the tabu search procedure described in this paper are:

MOVE TYPE- TABU RESTRICTION- TABU TIME- ASPIRATION CRITERIA- DIVERSIFICATION STRATEGY- STRATEGIC OSCILLATION-

ADD or DROP, Nodes are assigned tabu status when ADDed, Random, Produce solution better than the current best, Frequency-based memory (penalty) function, p +_ Slack, the value of Slack changes dynamically.

Our complete tabu search procedure can now be described:

Procedure TSpMP:

Initialize: Max__Iteration : = Max{2 • Nodes, 100}, Stable l terations : = 0.2 • M a x Iteration,

Iteration : = 1,

Bes t_Solut ion : = ~c,

Slack : = O,

Add_Time(v)" = - ~c and

nodes v. s : = {0} k = " Maxi jd i j

Freq(v): = 0 for all

Step O. Generate a feasible starting solution: While IsI < p, perform procedure ADD. WHILE Iteration < Max_I tera t ion do the following

334 E. Rolland et al. / European Journal of Operational Research 96 (1996) 329-342

Step 1. Choose a candidate for moving using procedure CHOOSE_MOVE. Step 2. Iteration" = Iteration + 1 If IsI = p and NewSolu t ion <Best_Solution then

set Best_Solution:= NewSolut ion, set Slack .= O, and save the best solution configuration.

otherwise Go to Step 3.

Step 3. If no improvement has been found in a number of iterations = Stable_Iterations × 2 then

Slack '= Slack + 1. Step 4. If no improvement has been found in a number of iterations = Stable_Iterations~2 then

Set Tabu_time:= a uniformly distributed random number in the range [1,p + 1]

Step 5. If a feasible solution has been obtained, but no improvement in the best feasible solution has been found in a number of iterations = Stable__Iterations then

Iteration : = Max__Iteration otherwise Iteration: = Iteration + 1.

END WHILE Step 6. To ensure that a local optima is found, run the NS local search heuristic using BestSolut ion as a starting solution. END.

The first p iterations of the procedure (Step 0) are spent incrementally building a feasible solution in a greedy fashion using a sequence of p ADD moves.

Step 1 makes either an ADD or DROP move. Step 2 determines if the problem state is feasible, and, if so, whether the new solution is better than the best known solution. If it is, the best solution is updated, and its configuration saved. In Step 3 the strategic oscillation (Slack) parameter is increased if no improvement in solution quality has been found in the last (Stable_.Iterations × 2) iterations, and in Step 4 the tabu time is randomly reset if no improvement has been made in the last (Stable_Iterations/2) iterations. Step 5 increases the iteration count, and causes termination of the algorithm if a feasible solution has been obtained but no improvement in the best feasible solution has been found in the last

Stable_Iterations moves. All steps are then repeated if the iteration counter is less than Max Iteration.

Setting the parameters for a tabu search heuristic is a key step in its design. In this heuristic there are several such parameters: the long-term memory penalty (k, discussed above), the times when the strategic oscillation slack and the tabu tenure should be updated, and the heuristic's stopping criteria. Each was established through the typical approach of empirically testing the heuristic. In setting stopping criteria, we tested problems with 20 to 100 nodes, and allowed up to 1000 iterations. Since we found no improvement in the solutions generated beyond 89 iterations, we chose, conservatively, to set the maximum iterations = Max{2 • Nodes,lO0}.

We also used Stable__Iterations as a stopping criterion to gain speed improvements which proved to be especially useful for small problems, and also for problem instances where p was small. Empirical tests demonstrated that, after having found a feasible solution, if the search proceeded through about 1 /5 of the maximum iterations without finding any solution improvements no further improvements were uncovered. Consequently, we used this lull as a stopping criteria (shown in Step 5). For the test problems (20-100 nodes), this stopping criterion leads to a 30% reduction in average computational time.

Finally, it is important to reiterate that no swap move is employed in this basic heuristic. Our strategy is that a tabu search structure can still identify good solutions without the computational burden of node interchange. However, a true local optimum cannot be guaranteed. To ensure one is reached, Step 6 is added as a post-processing step, where we use a node substitution (NS) search. The impact of this step, and the results of our computational experience with this algorithm are reported in the next section.

5. Computational results

The tabu search heuristic, TSpMP, as well as NS and GRIA were used to solve 100 test problems. The tests were conducted on eight smaller graphs (13 to 100 nodes) and four larger graphs (200 to 500 nodes), where each node served as both a demand point and a potential facility site. The graphs were

E. Rolland et al . / European Journal of Operational Research 96 (1996) 329-342 335

randomly generated in a 100 x 100 square with demand at each node being a randomly generated integer uniformly distributed in the range 0-100. For each graph, the pMP was solved for several values of p. The results were compared in terms of solution quality and solution time to those generated by the NS (Goodchild and Noronha, 1983) heuristic and the GRIA (Densham and Rushton, 1992) heuristic. All three algorithms were coded in Pascal for the IRIX operating system version 5.3 (Silicon Graphics) and executed on a 144 MHz R4400 based SGI computer with 64 megabytes of RAM. The smaller problems (i.e., 100 nodes or less) were also solved optimally via integer programming.

The Densham and Rushton paper discusses several novel and effective data structures that improve the solution time of GRIA. We did not include these modifications, so that we could specifically focus our comparison on the different moves and process designs of the heuristics. That is, the computational tests evaluate the relative speeds of the adds, drops and swaps, and how they are sequenced, without regard to special data representations or calculation methods. In all three methods, however, the quality of the solutions produced is unaffected by the way it was implemented in our tests.

There are several alternative approaches to apply- ing the three heuristics to any given problem. The first regards the nature of the starting solution for NS and GRIA. Two alternatives are typically employed to begin these heuristics; either a randomly generated set of facility sites or a solution constructed by a greedy heuristic. When using a random starting solution, these heuristics are generally run several times in succession with different random starting points, and the best overall result is chosen as the final solution. Greedy starts, on the other hand, are run only once since the entire process is deterministic. As discussed earlier, TSpMP builds its own starting solution also using a greedy approach. However, it subsequently does have a random component in the choice of adding or dropping, so that multiple runs of TSpMP may produce different results.

In conducting computational experiments, all of the above approaches were examined. These experiments showed that when NS and GRIA started with a greedy solution the result was on average worse than that found with ten random starts (optimality

gaps were 14% larger for NS; 40% larger for GRIA), although the solution time was generally faster (about 40%) for a greedy start than a random one. When TSpMP was run twice, only about 10% of the runs improved, with an average improvement of 1.4%, indicating that the tabu search heuristic is reasonably stable. In order to keep the tables from becoming unwieldy, we present below the results from NS and GRIA using 10 random starts and the TSPMP results from a single run. The solutions presented for the NS and GRIA heuristic are the best obtained in the ten random starts for each problem. However, the run times listed for NS and GRIA are the average single run time over the 10 starts.

For all problems the run times for TSpMP include the node substitution post processing step. In only 22% of the 88 smaller problems did this post- processing step produce an improved solution. Inter- estingly, node substitution did not improve the tabu search solution for any of the twelve 200-500 node problems.

A comparison of the results of the 88 smaller problems is presented in Table 1. The optimal solution was identified by TSpMP in 66% of the problems versus 48% of the problems for GRIA and for 39% of the problems by NS. The average optimality gaps were 0.5% for TSpMP, 1.4% for GRIA and 3.1% for NS where the gap is defined as (heuristic so lu t ion- optimal solution)/optimal solution. The maximum gaps were 6.0% for TSpMP, 7.5% for GRIA, and 20.1% for NS. In terms of comparative solution quality the TSpMP solution equalled or bettered the NS solution in 97% of the problems and was worse than the NS solution in only 3 problems. The TSpMP solution was worse than the GRIA solution in only 8% of the problems. On average, the TSpMP solutions improved upon the NS result by 2.0% and the GRIA solutions by 0.5%. The maximum TSpMP improvement over a NS solution was 14% and over a GRIA solution was 7%. In the worst case, TSpMP was 6% better than both the NS solution and GRIA solutions. It should be noted here that solution quality for TspMP may be improved by using additional runs. These would replace Step 0 with randomly generated starting solutions.

In terms of average solution time for the set of 88 problems, TSpMP required 64% less time than NS and 26% less time than GRIA. In a problem-by-prob-

336 E. Rolland et a l . / European Journal of Operational Research 96 (1996) 329-342

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340 E. Rolland et al . / European Journal of Operational Research 96 (1996) 329-342

lem comparison, TSpMP was, on average, 22% faster than NS and 2% slower than GRIA. (The times for TSpMP include the post-processing time, and the NS and GRIA times are for an average single run over the ten random starts each received.)

A comparison of the 12 larger problems is presented in Table 2. The optimal solutions to these problems were not identified as they were too large to be solved by available MIP packages. (Sciconic mathematical programming package running on a Prime 9955 minicomputer under PRIMOS 22.1.0, MPSX mathematical programming package running on an IBM 3090 mainframe under MVS, or CPLEX running on a SUN Sparc 10/41, which were used to solve the smaller problems.) Consequently, the gaps for these problems were defined as [(heuristic_solu- tion - best_solution) / best_solution] where best_solution is defined as the best solution found among the three heuristics.

TSpMP found the best solution in all but one of the problems.. It was always better than the NS solution, while GRIA identified the best solution only once. On average, the TSpMP solutions improved the NS solutions by 3.7% and the GRIA solutions by 2.3%. In the one case were GRIA identified a better solution than TSpMP, the improvement was 3%. The maximum TSpMP improvement over a NS solution was 7.8% and over a GRIA solution was 9.0%. The average gaps were 0.20% for TSpMP, 1.59% for GRIA, and 4.09% for NS. The maximum gaps were 2.80% for TSpMP, 9.86% for GRIA, and 13.26% for NS.

Regarding average solution time, TSpMP solved problems in 65% less time than NS and in 24% less time than GRIA. (Again, TSpMP times include the post-processing time.) The average solution times for the twelve problems were: 4,326 CPU seconds for TSpMP, 5,660 CPU seconds for GRIA, and 12,408 CPU seconds for NS. As before, the NS and GRIA times are a single run average of the l0 runs actually made for each problem.

6. Summary and conclusions

In this paper, we introduce a new heuristic, TSpMP, for solving the p-Median Problem (Hakimi, 1965). The heuristic is based on the generic tabu

search principles developed by Glover (1986) and Hansen (1986). In the design of any heuristic, computational effort and solution quality are important criteria. Several specific strategies were employed here. First, the search only considers ADD and DROP moves; thereby, eliminating the more computationally burdensome SWAP moves included in tradi- tional exchange (node substitution) heuristics for the p-Median Problem. Secondly, in an effort to move from one local optima to another one more effi- ciently, we use strategic oscillation to allow the search path to include infeasible solutions. Thirdly, we employed a random tabu time. Finally, we used dynamic rules for determining the number of iterations performed.

TSpMP was used to solve 100 test problems ranging in size from 13 to 500 nodes. These problems were also solved using the NS solution heuristic developed by Goodchild and Noronha (1983) and the GRIA heuristic developed by Densham and Rushton (1992).

A comparison of the results indicates that TSpMP performs very favorably vis a v i s NS and GRIA, both in terms of solution time and quality. For example, for the 88 smaller pro. blems (13-100 nodes) the average solution times in CPU seconds were 13.95 for TSpMP, 19.07 for GRIA, and 39.09 for NS. For the 12 larger problems (200-500 nodes) these average times were 4,326 for TSpMP, 5,660 for GRIA, and 12,408 for NS. With respect to solution quality, on average, all three heuristics performed reasonably well when compared to the optimal solutions for the smaller problems or the best heuristic solution found for the larger problems. In general, however, TSpMP outperformed both of the other heuristics in terms of the quality of the solutions produced. This is particularly true when consid- ering the worst-case solutions (i.e., the maximum % difference between a heuristic's solution and the best known solution to a problem which is referred to here as the maximum gap). For the smaller problems these maximum gaps were measured against the optimal solutions and were 6.0% for TSpMP, 7.5% for GRIA, and 20.1% for NS. For the larger problems these maximum gaps were measured against the best of the three heuristic solutions and were 2.8% for TSpMP, 9.86% for GRIA, and 8.47% for NS.


In sum, tabu search offers an effective way to develop improved move heuristics for the pMP. The incorporation of tabu architecture and search strategies with the computationally efficient ADD and DROP moves produce a heuristic which is quite competitive in terms of speed, and superior with respect to the quality of solutions generated. Finally, as we have observed, there are additional tabu search strategies we have not incorporated that may yield an opportunity for further refinements of our procedure.

7. For further reading

Cornuejols et al., 1977, GalvSo, 1980, Glover et al., 1992, Hakimi, 1964, Narula et al., 1977, Rolland and Gupta, 1995, Taillard, 1991, Tansel et al., 1983

Acknowledgements

The authors would like to thank Dr. Fred Glover and two anonymous referees for numerous com- ments which have significantly improved this paper.

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an efficient tabu search procedure for the p-median problem

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