Subdrop: A modified drop heuristic for location problems

Lomlion

Pergamon SO96&8349(%)OOOOS4

SUBDROP:

A MODIFIED

DROP HEURISTIC PROBLEMS

Scirncr, Vol. 3, No. 4. pp. 261-213. 1995 Copyright @ 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0966.8349/96 SI5.00 + 0.00

FOR LOCATION

S. SALHI and R. A. ATKINSON School of Mathematics and Statistics, University of Birmingham, Edgebaston B15 2TT. (Received

11January 1995)

Abstract-A modification of the DROP heuristic is presented. This modification utilizes randomly selected subsets of the candidate facility set instead of the entire set as has been the practice in the past. Computational experiments indicate that the modification results in better solutions and reduced computation time when compared to the standard implementation of the drop heuristic. Copyright @ 1996 Elsevier Science Ltd

Keywords: Heuristics, computational

experiments, location.

1. INTRODUCTION

Facility location models have many applications and consequently have been formulated in numerous ways; but unfortunately, most of these problems are difficult to solve optimally. Consequently, heuristic procedures have been developed to solve them. One of the oldest and most computationally efficient of these is the drop heuristic (Feldman et al., 1966). Interesting models and algorithms for location problems exist in the literature (Erlenkotter, 1978; Krarup and Pruzan, 1983; Mirchandani and Francis, 1990) and their applications arise in both public and private sectors (Brandeau and Chiu, 1989; Golden and Eiselt, 1992). Typically, drop heuristics start by opening a facility at every potential site and then close (i.e. drop) a facility at each iteration until a stopping criterion is satisfied. In our modification, we start with a subset of all the potential sites open and drop facilities until the stopping criterion is met. This is executed with several different starting subsets of facilities and the best solution from these various subsets is selected. Computational tests indicate that this modification not only decreases computational time but also improves solution quality (i.e. identifies better solutions) when compared to the traditional approach of the drop heuristic. In addition, our heuristic has the advantage of producing more than one single solution. Our study is based on heuristics as they can be the only way to solve this sort of large combinatorial problem. These heuristic search procedures are based on common sense, logic and experience. In addition, they have the advantage of being easy to understand, to modify, and are capable of producing more than one single solution to the problem. It is also easy to introduce improvement modules if needed. Lastly, they usually produce good (if not optimal) solutions fairly quickly. Although they do not guarantee optimality, their performance is usually assessed using worst case analysis, probabilistic analysis, empirical testing, lower bound based concepts, etc. Although our heuristic is applicable to any location problem for which the drop heuristic is applicable, we present it in a p-median problem framework (Hakimi, 1965). In the next section, we describe the modification. In the third section, we present results of the computational tests and in the last section a summary and conclusions are given. 267

S. SALHI and R. A. ATKINSON

268

2. THE HEURISTIC

2.1. Notation

N, M, K:

the number of customers, the number of potential facility sites and the number of replications of the drop heuristic respectively (for simplicity we consider M = N)

I: E: P: Dij: Ai: Fi: m: k:

c mk’* c,: m*:

set of all potential sites (I = (1, . . . , M}). the current set of open facilities, ( E ( cardinality of E. the number of facilities required by the user (the p-median problem). the cost (time or distance) to serve customer j E (1, . . . , N} from facility located at node iEE set of customers served by a facility located at node i E E (Ai = &(l, . . . . N} st D, < D,VIE E, I # i} the fixed cost of a facility at node iE E. the number of potential sites from which to start the drop heuristic (mE{p + 1, . . . . M - 1)). the kth repetition (k = 1,. . . , K) when starting the drop heuristic from a selected number m of potential sites chosen randomly from the set I. the final cost obtained when using the drop heuristic on the kth repetition with m potential sites. the minimum cost obtained over K trials (C, = Min(C,; k = 1, . , . , K}) the value of m such that C,. = Min{C,; m = p + 1, . . . , M - l}.

2.2. The p-median problem The p-median problem is to locate p facilities, which have to service a set of customers, at a minimum cost. The cost includes the fixed cost of the facilities and the total transportation cost. In most practical circumstances, company management will specify the value of p in advance. However, if this is not the case p can be treated as a parameter. Then marginal analysis can be used to determine the value of p and gain greater insight into the nature of the problem. 2.3. Description

of the drop heuristic

The idea of the drop heuristic is to start with a large number of open facilities and remove one facility at a time. The removal of a facility, at a given step, is carried out as follows: each facility is in turn temporarily removed, and each of its customers are re-allocated to their nearest (cheapest) open facility. The change in the cost of the objective function is evaluated and the facility that produces the largest reduction in total cost is selected for removal. This process is repeated until either the required number of depots is obtained (e.g. p in the p-median problem), or no reduction in total cost can be made (e.g. the uncapacitated plant location). The main steps of the algorithm, which we denote by DROP, are described below. The heuristic DROP: Step 0: Set E = I, Step 1: Determine A,ViE E, compute COST = XieEFi + XisEXjGA,Dij Step 2: For each i E:E compute Ai = Fi - Xj&Dsj - Dii> where s is the second cheapest facility to serve customer j (i.e. Dij < D, < D,, VI # i, s).

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269

Step 3: Select i* s.t. Ai, = MaxiA,, A, > 0 and iEE} Step 4: Set E = E - ii*>; COST = COST - Ai*; update A,ViEE.

If IE( = p go to step 5,

else go to step 2. Step 5: Set C, = COST and record the solution (E and all Ai) Note that the decision to remove a facility is based only on the change in total cost, rather than in recomputing the whole objective function from scratch. In this way the computing time is reduced, as no unnecessary calculations are performed. 2.4. A description of the modified drop heuristic Basic idea. Given a specific starting set of sites, the final solution, as found using the DROP heuristic, is uniquely determined. In general however, this solution will be sub-optimal. The idea of this paper is to generate a number of solutions. Each solution is generated by picking a number (m < M) of sites at random from the full set of sites, and using the drop heuristic on this reduced starting set. This process is repeated several times, say K, for each value of m. The choice of the value of m and K will be discussed in the next section. For each value of m the following steps of our algorithm which we denote by SUBDROP are used. The heuristic SUBDROP: Step 0: Set k = 1, choose m, K (m > p); Select randomly m nodes from the set I to form the set E of open facilities. Steps 14: as in DROP Step 5: Set C,, = COST, k = k + 1, ??

??

if k < K select m nodes randomly from the set I, define the set E of open depots and go to step 1, else compute C, and record the solution associated with it (E and all AJ

2.5. Computational comparisons The traditional drop heuristic (DROP) requires approximately NM’p - NMp’ steps whereas our modification (SUBDROP) uses only KNm2p - KNmp2 steps. When M >>p, as is usually true for larger problems, the time complexity of DROP is O(NM2p) and of speed reduction of SUBDROP is O(KNm2p). In other words, an approximate 1 - K(E)’ + K(z) can be obtained. For instance, if M = 200, p = 5, m = 0.25M and K = 10 we have a reduction in computing time of around 43%. In practice, the reduction in computing time could be slightly greater for larger problems, because less time is spent in the updating process. 3. COMPUTATIONAL

EXPERIMENTS

We tested SUBDROP on 7 networks taken from the literature (Christofides et al., 1979). These problems vary in size from 50 to 199 nodes. Problems l-5 are randomly generated, whereas problems 6 and 7 take into account clusters to represent better real life situations. For each network we solved 5 problems (p = 1, 2, 3, 4, 5), giving a total of 35 problems. For convenience in this study we used M = N and we assigned a large positive constant to all the fixed costs. Although our investigation is based on the p-median problem, it can be extended without any difficulties to other location problems.

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S. SALHI and R. A. ATKINSON

The computer program was coded in Fortran 77 and executed on a VAX 4000-500 computer at the University of Birmingham. We have carried out two experiments: in the first one we use the SUBDROP heuristic for all values of m E (p+ 1, . . . , A4 - l} and K = 10. This full investigation is done to: (i) show that starting the drop method from a smaller subset of open facilities yields better results than starting from the full set of potential sites; (ii) provide detailed information that enables us to construct empirically a suitable rule for determining values of m and K. Such a criterion needs to maintain the quality of solution found in (i) while reducing the computing time. In the second experiment we test our heuristic SUBDROP on the same problem using those values of m and K obtained in (ii). Table 1. Relative performance (%) between SUBDROP, DROP and the optimal solution method Experiment 1 SUBDROP vs DROP

SUBDROP vs OPTIMAL

No

N

p Subset size (m*)

1

50

1 2 3 4 5

5 12 27 13 32

8.08 4.14 8.64 8.98 7.31

0.00 0.00 0.00 0.16 0.74

2

75

1 2 3 4 5

6 13 44 28 34

10.35 5.39 7.06 7.98 4.65

1.17 0.07 0.00 0.54 0.49

Experiment 2 SUBDROP vs DROP

SUBdROP vs OPTIMAL

CP” time

3.41 1.92 3.48 3.00 0.68

5.07 2.32 5.65 6.58 7.95

-66 -66 -66 -66

9.81 5.32 6.31 0.69

1.78 4.27 1.87 2.37 4.66

33 23 -8 0 0

1.42

0

3

100

1 2 3 4 5

7 32 37 27 35

13.69 3.49 3.86 4.89 1.73

0.36 0.00 0.48 0.94 0.65

9.19 1.64 2.36 2.48 0.27

5.58 1.92 2.05 3.48 2.32

18 21 21 22 32

4

150

1 2 3 4 5

8 38 51 21 95

4.11 5.02 7.78 11.99 7.22

0.22 0.15 0.14 0.22 0.98

- 1.26 3.81 6.19 7.94 5.04

5.83 1.42 1.87 4.83 3.35

49 50 48 54 54

5

199

1 2 3 4 5

9 17 40 19 21

6.11 3.85 4.16 9.54 8.45

0.00 0.19 0.32 0.17 0.94

0.00 2.61 -0.37 4.79 5.60

6.51 1.48 5.05 5.42 4.07

57 59 59 59 50

6

100

1 2 3 4 5

4 11 41 21 36

5.48 1.91 1.75 4.98 6.38

0.00 0.00 0.54 1.51 2.06

1.39 1.59 0.82 3.57 4.80

4.32 0.32 1.49 3.02 3.77

25 20 19 28 30

7

120

1 2 3 4 5

6 24 30 25 19

15.18 16.93 9.69 11.19 9.84

0.00 0.22 0.23 0.60 2.02

9.75 16.20 7.19 8.23 7.60

6.39 1.10 2.99 3.94 4.55

36 36 39 43 44

SUBDROP

271

3.1. First experiment

For each mE{p + 1, . . . . M - l} we pick m from M sites at random. We then applied the drop heuristic starting from this subset of sites. For each m this process is repeated K times. Thus we obtain a set of costs {C,,). The number of replications is arbitrary, but 10 seemed sufficient to obtain satisfactory results. We shall denote the cost associated with the solution obtained by DROP by C,. This will be our bench mark, with which we will compare any alternative solutions obtained. Table 1 summarizes our results. Columns l-3 give the problem number, number of nodes and value of p. Column 4 gives the value of rn* (the subset size) which produced the least overall cost. Column 5 compares SUBDROP solutions with that of DROP in %. Solutions are presented in terms of the average percentage improvement of SUBDROP over DROP. SUBDROP identifies a better solution than DROP in all 35 problems and produces an average improvement of 7.29%. The greatest improvement over C, observed was nearly 17% (problem 7, p = 2) and the smallest improvement was 1.73% (problem 3, p = 5). It is worth noting that the large improvements occur in the largest problems. Having established that a modified drop heuristic obtains better solutions than the original drop heuristic it is important to find out how close these experimental solutions are to the optimal solutions. The optimal solutions were found using Lagrangian relaxation heuristics with a duality gap of less than 0.5%. This is similar to the work given by Beasley (1993). These optimal solutions were also found by complete enumeration technique for all problems except the two largest problems (when p = 5) where computing time was excessively large. Column 6 compares the SUBDROP solutions with the optimal ones in %. Solutions are presented in terms of average percentage excess of SUBDROP over the optimal. The last three columns will be described in the next subsection. As can be seen from column 6 in Table 1, the experimental results are quite good. SUBDROP produces 9 optimal solutions and on average the solutions obtained were 0.85% above the optimal solutions, with the worst solution being about 2% sub optimal. Our worst solutions occurred in clustered problems. For large problems without clusters, our method seems to yield good quality solutions. In addition, SUBDROP does not have the disadvantage of DROP where p - 1 solution locations are always a subset of p solution sites. For instance, for the 50 customer problem SUBDROP found the following sites: p = 1 (site at node 46), p = 2 (1, 17), p = 3 (16, 17, 48), p = 4 (9, 17, 48, 20) and for p = 5 (3, 9, 37, 41, 23) whereas DROP generated the following sites: p = 1 (site at node 47), p = 2 (47, 50), p = 3 (47, 50, 23), p = 4 (47, 50, 23, 20) and p = 5 (747, 50, 23, 20, 41). The bold numbers represent optimal sites. 3.2. Second experiment The simulations we have performed confirm that the solution found when starting with all M potential sites is almost always inferior to the overall best solution obtained when starting from a number m < M of potential sites. Even though each solution picked was the best on a large number of possible solutions, these results strongly suggest that some modification of the drop heuristic, generating a small number of solutions and picking the best of them would be a good idea. From the detailed experimental results and some

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S. SALHI and R. A. ATKINSON

preliminary testing on some larger problems we have identified some potential guidelines. These are: ??

If p < M/3 choose m = max{0.25M, 3p). If m = 0.25M then K = 10, otherwise K = 5.

??

else choose either m = OS(M + p) and K = 2 or 3, or simply apply the original drop heuristic.

We tested this rule on the same 35 problems that we experimented upon. The results are given in Table 1. Column 7 indicates the percentage improvement of SUBDROP over DROP and column 8 shows how suboptimal SUBDROP solutions are with respect to the optimal ones. Negative percentages in column 7 correspond to solutions where DROP is better. The last column reports on the computing time gained in % terms when using SUBDROP instead of DROP (negative % in this column means that the DROP heuristic is faster). On average, we found that SUBDROP produced solutions which are 4.5% better than the ones found by DROP, and 3.6% above the optimum. Our heuristic finds better solutions than DROP in 33 out of the 35 problems tested. Furthermore, when it fails the solutions were at most 1.26% worse. In addition, SUBDROP requires on average over 50% less computational time than DROP, especially in the larger problems, and around 25% for the other ones except for the smallest problem. Note that for the smaller problems the computing effort for both methods is negligible. The average differences were 21% for p = 1, and 20, 16, 20, and 30% for p = 2 to 5 respectively. In order to test this heuristic further we generated 7 new larger problems based on the 7 sets of data used previously. The new problems vary in size from 200 to 796 nodes and were tested for p = 10, 20, 30, and 40, totalling 28 test problems. When tested on these larger problems SUBDROP yielded an average of 3.5% improvement over DROP. The reduction in computing time was of the order of 60%. This substantial reduction in cpu time could be used to introduce refining modules for further improvements of the solution. For brevity, the detailed results and the generation of these data sets are not reported here. It is obvious that as p increases in size (tending towards M) then any random starting set is almost M in size. Hence computing time for our heuristic that finds more than one solution will obviously be larger than the original drop heuristic. Note that for small values of p, smaller values of m could also be used. In that case, K could be increased above 10, without significantly increasing comptiting time. 4. CONCLUSION

In this paper we introduced a modification of the original drop heuristic that starts from a smaller number of potential sites, randomly chosen from the entire set of potential sites. This approach was shown to produce better solutions with an important reduction in computational time. To assess the performance of our proposed heuristic we compared it with the DROP heuristic and Lagrangian relaxation heuristic. The solutions obtained by SUBDROP were encouraging both in solution quality and in computing effort. Clearly, any gain in computational time could be used to search for even better solutions, either by increasing the value of K, or by using the time saved to refine the existing solutions.

SUBDROP

273

It is worth noting that our heuristic SUBDROP overcomes the handicap of DROP since our best solution found with “p” facilities is not necessarily an extension of the “p - 1” facility solution, as observed in the original implementation of the drop heuristic. Intuitively we would not expect the p solution to be an extension of the p - 1 solution. The SUBDROP strategy can be used in any drop heuristic, or as a component of a metaheuristic like tabu search or simulated annealing which incorporates drop schemes. In addition, it is worth noting that by using random subsets our heuristic increases flexibility by producing a number of different solutions. Each of these solutions could be used as a starting solution in any suitable location heuristic. We have presented a rule to determine m and K that seems to work satisfactorily for all the problems we tested. Clearly this rule is not the final answer to the question of the best starting subset size, but it would seem to be a step in the right direction. One avenue of future research would be to carry out more substantial simulation and statistical analysis. It may also be desirable to make the number of replications (K) and the subset size (m) a function of p, M and N. The present work can be enhanced further by considering some kind of probabilistic rule of thumb for sampling the number of points (m). A concept similar to the greedy randomized adaptive search procedure given by Achabal, Gorr and Mahajan (1982) and Klincewicz (1992) may be a good way forward. Acknowledgements-The authors are grateful to the referees, the Associate editors and Professor John Current for their helpful suggestions and comments which improved the presentation of the paper. Thanks also to Mark Agar for providing us with the solutions found by Lagrangian relaxation heuristics. REFERENCES Archabal, D. D., Gorr, W. L. & Mahajan, V. (1982) Multiloc: a multiple store location decision model. Journal of Retailing, 58, 5-25.

Beasley, J. (1993) Lagrangian heuristics for location problems. European Journal of Operational Research, 65, 383-399.

Brandeau, M. L. & Chiu, S. S. (1989) An overview of representative problems in location research. Management Science, 35, 645-674.

Christofides, N., Mingozzi, A. & Toth, P. (1979) The vehicle routing problem. In N. Christofideg A. Mingozzi, P. Toth & C. Sandi (Eds), Combinatorial optimization (pp. 315-338). Chichester: Wiley. Erlenkotter, D. (1978) A dual-based procedure for uncapacitated facility location. Operations Research, 26, 992-1009.

Feldman, E., Lehrer, F. A. & Ray, T. L. (1966) Warehouse location under continuous economies of scale. Management Science, 12, 670-684.

Golden, B. L. & Eiselt, H. A. (Eds) (1992) Location modelling in practice: applications, theory and history. American Journal of Mathematics and Management Sciences, 12, l-94.

Hakimi, S. L. (1965) Optimal distribution of switching centers in a communications network and some related graph theoretic problems. Operations Research, 13, 462-475. Kincewicz, J. G. (1992) Avoiding local optima in the p-hub location problem using tabu search and GRASP. Annals of Operations Research, 40, 283-302.

Krarup, J. & Pruzan, P. M. (1983) The simple plant location problem: survey and synthesis. European Journal of Operational Research, 12, 36-81.

Mirchandani, P. B. & Francis, R. L. (Eds) (1990) Discrete location theory. Chichester: Wiley.