Comparison of exact and approximate methods of solving the uncapacitated plant location problem

JOURNAL

OF OPERATIONS

Vol. 6, No. I, November

MANAGEMENT

1985

Comparison of Exact and Approximate Methods of Solving the Uncapacitated Plant Location Problem JEAN-MICHEL

THIZY*

LUK N. VAN WASSENHOVE BASHEER M. KHUMAWALA**

EXECUTIVE

SUMMARY

This study was prompted by a recently published article in this journal on facility location by D. R. Sule. We show that the claim made by Sule of a novel and extremely simple algorithm yielding optimum solutions is not true. Otherwise, the algorithm would represent a breakthrough in decision-making for which a number of notoriously hard problems could be efficiently recast as location problems and easily solved. In addition, several variants of common location problems addressed by Sule are reviewed. In the last twenty years, many methods of accurately solving these problems have been proposed. In spite of their increased sophistication and efficiency, none of them claims to be a panacea. Therefore, researchers have concurrently developed a battery of fast but approximate solution techniques. Sule’s method was essentially proposed by Kuehn and Hamburger in 1963, and has been adapted many times since. We exhibit several examples (including the one employed by Sule) in which Sule’s algorithms lead to nonoptimal solutions. We present computational results on problems of size even greater than those utilized by Sule, and show that a method devised by Erlenkotter is both faster and yields better results. In a cost-benefit analysis of exact and approximate methods, we conclude that planning consists of generating an array of corporate scenarios, submitting them to the “optimizing black box,” and evaluating their respective merits. Therefore, much is to be gained by eliminating the vagaries of the black box-that is, by using an exact method-even if the data collection and the model representation introduce sizable inaccuracies. Ironically, large problems (those that typically require most attention) cannot be solved exactly in acceptable computational times. Pending the imminent design of a new generation of exact algorithms, the best heuristics are those that guarantee a certain degree of accuracy of their solutions. INTRODUCTION

The success of long-term plans often hinges on a judicious choice of sites to locate some facilities, particularly in the design of distribution systems such as

* Princeton University, Princeton, New Jersey. ** University of Houston, Houston, Texas.

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23

l l l l l l l

Warehouse and plant location Air, railroad, and mass transportation Design of telephone, television and computer networks, siting of broadcasting Layout of electric power systems Construction of oil and gas pipelines Planning of health centers, public services, police and fire stations Location of lock-boxes, collection centers

stations

Such situations occur in other managerial problems: production planning, capital budgeting, aggregation of database information, optimal constitution of portfolios, or in seemingly unrelated problems such as the clinical detection of glaucoma [59]. The characteristics of locational problems can be extremely diverse-the set of potential sites may be finite or infinite; for example, only major cities can be eligible in a nationwide distribution network. On the other hand, it may be possible to choose any point in a twoor three-dimensional space, as for the location of machines or control panels in a building or in a mine. If the set of potential sites and that of demand points are both finite, the problem is said to be discrete. The distribution may involve several products or several stages. The economic criterion may vary from the minimization of the distribution cost to the minimization of the penalty incurred by the least-favored area serviced, or the aggregation of each user’s satisfaction [7]. The demand for services may be stochastic (location of fire stations), or deterministic, static or dynamic; in particular, the choice of a location for a plant must be made bearing in mind possibilities of expansion. A survey of the dynamic location models has recently been published by D. Erlenkotter [29]. For some of these models, the optimizing goal itself becomes elusive. Besides the profuse bibliography of Lea [65], some particularly relevant surveys are listed below in chronological order: Balinski and Spielberg [4]; ReVelle and Swain [82]; Eilon, Watson-Gandy, and Christofides [25]; Hansen and Kaufman [48]; Elshafei, Haley, and Lee [27]; Francis and Goldstein [33]; Francis and White [34]; Kaufman [53]; Geoffiion [39]; Guignard and Spielberg [45]; Salkin [84]; Jacobsen and Pruzan [50]; Cornuejols, Fisher, and Nemhauser [ 141; Krarup and Pruzan [60,62]; Erlenkotter [29]; Wong [96,97]; Handler and Mirchandani [46]; Van Roy [93]; and the recent book on discrete location problems by Francis and Mirchandani [35]. The host of methods proposed rarely addresses the problem facing the decision maker. Fortunately (owing to its physical nature), the problem at hand can often be decomposed or simplified into common models. This is not an academic exercise because the simpler and well-explored formulations can be accommodated to recover the specificity of the problem. A series of successful implementations warrants the comforting conclusion that the efficacy of the decomposition generally depends on its economic justification. The Uncapacitated Plant Location Problem, also known as the Simple Plant Location Problem, constitutes a fundamental building block of the varied models proposed. It is a discrete, static, deterministic, one-product, one-stage model to minimize the distributor’s costs. We address the following questions: l l l

24

When can the problems be solved optimally? How well do simple approximate methods perform? For which type of problems is either approach best suited?

APES

THE

UNCAPACITATED

PLANT

LOCATION

PROBLEM

“The simple plant location problem is one of the simplest mixed integer problems which exhibits all the typical combinatorial difficulties of mixed (0,1) programming and at the same time has a structure that invites the application of various specialized techniques.” [45] The uncapacitated plant location problem consists of selecting facilities to be open among a finite set J of potential sites in order to minimize the cost of serving a specified number of customers. A cost cij is incurred to satisfy the totality of the demand of customer i from location j if a facility is kept open there; the facility can be maintained at this site at a cost 4. Since no capacity constraints on the sites are considered, this problem is referred to as an uncapacitated location problem; its mathematical formulation is then Z =

min

C C iE1 j63

2

Xij

=

1

CijXij +

2

fjyj

(1)

ja

all i E I

(2)

all i E I, j E J

(3)

all j E J

(4)

.EJ OTZXijlYjll Yj

integer

The uncapacitated plant location problem occupies a central position among popular variants; a common additional requirement is to select exactly p facilities [ 151, which means the addition of the following constraint: (5)

CYj’P ZJ

The p-Median Problem is a special case of the former problem. Among its many applications is the clustering analysis performed by Mulvey, Crowder, and Beck [73, 741 where there can be thousands of items. On the other hand, the p-Plant Location Problem stipulates only a limit on the maximum number of facilities to be chosen [ 141: CYjsP ja

(6)

The addition of this requirement (6) generalizes the uncapacitated plant location problem, because it can be waived by setting the upper limit p equal to the total number of potential sites. Solution methods for the preceding problems overlap each other to a considerable extent; we attempt to present a unified and selective survey. When necessary, we use the term uncapacitated plant location problem in a generic fashion, whether the additional constraints (5) and (6) are enforced or not. The following two sections provide some background information and can be omitted in a first reading. SOME APPROXIMATE METHODS PLANT LOCATION PROBLEM

FOR THE

UNCAPACITATED

The initial procedures designed to obtain some solutions to the uncapacitated plant location problem were based on simple approximation schemes. Since then, a flurry of approximate methods have been proposed. First and foremost is the bump-and-shift routine of Kuehn and Hamburger [64]: in the add routine, locations are selected one at a time on the basis of the minimum incremental cost incurred; subsequently, in the bump routine,

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25

each location is temporarily removed to see if any savings can be achieved; finally, in the shift routine, each location not selected can be interchanged with one selected if this lowers the total distribution cost. Kuehn and Hamburger’s approximate method was seminal in generating a large number of variations, extensions, and computational tests: Manne [67]; Feldman, Lehrer, and Ray [31]; Levy [66]; Teitz and Bat-t [6, 911; Jarvinen, Rajala, and Sinervo [5 11; Diehr [21]; Hansen and Kaufman [47, 481; and Eilon and Galvao [26]. Cornuejols, Fisher, and Nemhauser [ 141 decisively branded the routines greedy (interchange) heuristics. A Dynamic Programming Heuristic incrementing the number of selected sites is proposed by Baker [I]. The natural idea of aggregating the customers in groups and finding a best facility for each group, represented by Cooper [ 12, 131 and Maranzana [68], has had less success. Among other approximate methods introduced are those of Drysdale and Sandiford [22], Rosing and ReVelle [83] and Hochbaum [49]. All the preceding approximate methods can construct a solution from scratch, whether independently or as an initial guess pending refinement by other approximate or optimal methods. They distinguish themselves for the heuristic principles that can accelerate an enumerative algorithm, such as the approximate methods of Khumawala [54, 55, 561. EXACT

METHODS

The design of efficient algorithms to solve the uncapacitated plant location problem must be founded on three basic findings emerging from two decades of successive improvements. First, the direct application of general optimization techniques such as branch-and-bound (Efroymson and Ray [24]), Benders’ or Dantzig-Wolfe decomposition (Balinski and Wolfe [2, 31, Swain [SS]), and the group-theoretic approach (Garfinkel, Neebe, and Rao [37]), can be improved by heuristic choices in the enumeration such as those devised by Spielberg [86,87], Khumawala [54,55,56], Jarvinen, Rajala, and Sinervo [5 11, Hansen and Kaufman [47, 481, and El-Shaieb [28]. Second, the structure of the problem is so specific that it favors the use of streamlined algorithms (Marsten [69] for the p-Median Problem, and Schrage [85] for the Uncapacitated Plant Location Problem). Finally, the dual formulations introduced by Diehr [21] and Geoffrion [38] can be implemented by fast algorithms based on subgradient optimization (Neebe and Rao [76], Narula, Ogbu, and Samuelson [75], Christofides and Beasley [ 1 l] for the p-Median Problem; Cornuejols, Fisher, and Nemhauser [ 141 for the p-Plant Location Problem), or on local ascent search (Bilde and Krarup [8, 91, Galvao [36] for the p-Median Problem). This strain culminates in the dual adjustment procedure, best described by Erlenkotter’s own characterization of his algorithm [30, p. 10081: “. . . this solution approach may be close to the ultimate in efficiency for the problems solved.” The procedure has been recently improved by Van Roy and Erlenkotter [94]. For the p-Plant Location Problem, Erlenkotter’s algorithm is applicable because it is generally possible to find a (uniform) set of fixed charges that ensures the opening of the desired number p of facilities. This technique is studied by Mavrides 1701 and used in computational experiments by Christofides and Beasley [ 111; it has been successful with all the tests carried out on the three problem sets reported below. A direct solution based on subgradient optimization has been implemented by Cornuejols, Fisher, and Nemhauser [ 141. For the p-Median Problem, Galvao [36] introduces a dual-adjustment procedure, and

26

APES

TABLE 1 Total Cost Matrix Facility Demand Location 1 2 3 4 5 Total Cost

Christofides optimization SULE’S

Location

1

2

3

500 150 750 400 900

300 250 300 200 600

200 100 0 1600 1200

800 300 150 400 0

2700

1650

3100

1650

4

5 500 350 0 600 1200 2650

and Beasley [ 1 l] display a graph with 200 nodes for which their subgradient is faster than Erlenkotter’s algorithm.

METHOD:

ANALYSIS

AND

COMPUTATIONAL

EXPERIMENTS

This study was prompted by the article of D. R. Sule [88] who has claimed to have developed novel methods for the uncapacitated facility location/allocation problems. However, for the p-Plant Location (with no fixed costs), his method is simply the add-and-bump routine of Kuehn and Hamburger [64]. For the Uncapacitated Facility Problem (with fixed costs), it differs from the add-and-bump routine only in a criterion reminiscent of Jarvinen, Rajala, and Sinervo [5 11: When choosing the first site, instead of the transportation costs, consider for each customer the opportunity savings (or cost differentials) offered by the best distribution channel over the second best as if all facilities were opened. Sule makes claims of optimality for this method, which is not true. For instance, consider Sule’s example for the p-Plant Location Problem ([88], Table 1, p. 2 17) reproduced in Table 1, for p = 2. By arbitrarily breaking a tie in the initial selection, Sule would find a final solution consisting of Locations 3 and 4. Breaking the tie in the other way, the final solution consists of Locations 2 and 4, which is clearly not optimal. Therefore the method can lead to a nonoptimal solution (a slight perturbation of the data would eliminate the tie and make Locations 2 and 4 the only solution found by the procedure). For the Uncapacitated Plant Location Problem, consider the problem summarized in Table 2. Table 3 shows the Minimum Savings (Step 2 of Sule’s method). Location 1 is

TABLE 2 Cost and Demand Data Possible Location Demand 1 2 Fixed Cost

Journal of Operations

Management

1

2

3

0 100 99

100 0 99

1 1 99

27

TABLE 3 Minimum Savings Table Possible Location Demand

1

1 2 Fixed Cost Savings Net Savings

2

1

3

-

-

I -99 -98

-

-99 -98

-99 -99

chosen (Location 2 could also be selected; it is a symmetric case). Table 4 shows the possible savings once Location 1 has been chosen; Location 2 is then chosen. No additional savings can be obtained by opening Location 3; neither Location 1 nor 2 can be dropped. Sule’s solution has a cost of 198; but by opening Location 3 only, the two demands can be satisfied at a cost of 101. We have also compared Sule’s method with the optimum method of Erlenkotter [30] on 30 problems cited by Sule that possess as many potential sites as customers: 33 and 57 major U.S. cities in two sets of problems initially proposed by Karg and Thompson [52], and 100 points randomly selected in a rectangle for the third set [63]. For each set, ten different levels of uniform fixed charges are tested. These problems, solved in [85, 14, 30, 171, are deemed to be good representatives of uncapacitated plant location problems. The solutions obtained by Sule’s method, the optimal values and the number of facilities selected appear in Tables 5, 6, and 7. Sule’s solutions are rarely optimal. Moreover, the computation times required by Erlenkotter’s algorithm to provide optimum solutions are often significantly smaller than those of Sule’s approximate method. Note that, contrary to Sule’s method, the length of Erlenkotter’s exact resolution is not directly related to the number of facilities selected. Khumawala [54] reports that an optimal solution seems easier to obtain when only a few sites are chosen or when some facilities are placed in almost all potential locations. In the latter case, it is advisable to model the approximation strategy after the Stingy Heuristic of Feldman, Lehrer, and Ray [31], which selects all locations initially and discards them one at a time if proven economical. TABLE 4 Savings Matrix-l Possible Location Demand

28

1

2

3

1

-

-

2

100

99

Fixed Cost Savings Net Savings

-99 1

-99 0

APICS

TABLE 5 Results for Some 33 X 33 Problems

Comparative

Erlenkotter

Sule

Number of Locations Chosen

CPU Time

(ms)

Optimal Value

6 9 6 6 10 11 13 22 47 86

27414 25414 23474 22127 20363 17832 14832 11267 8673 6024

2 2 2 3 4 6 6 10 17 31

1 7 8 9 46 8 4 3 3 4

Fixed Charge

Objective Value

Number of Locations Chosen

CPU Time

5000 4000 3000 2500 2000 1500 1000 500 295 184

21474 25414 23474 22414 21166 18391 15764 11734 8710 6024

2 2 2 2 4 5 6 10 18 31

(ms)

CPU Times for IBM 308 I CMS 2.0 FORTHX

A practical conclusion is that the optimal solution is generally obtained in a fraction of a second, a sufficient speed to allow an interactive exploration of various managerial scenarios without breaking the natural flow of thought of the analyst. The authors can attest that some optimal algorithms such as the one made available by D. Erlenkotter have been used successfully in various didactic and professional situations. Contending that the use of such algorithms presupposes an advanced mathematical maturity is no more tenable than claiming that it takes an extensive knowledge of electronics to use a pocket-calculator.

TABLE 6 Results for Some 57 X 57 Problems

Comparative Sule

Fixed Charge

Objective Value

5000 4000 3000 2500 2000 1500 1000 500 200 50

38742 36366 32559 30160 27660 24700 20815 15348 925 1 2821

Erlenkotter

Number of Locations Chosen

CPU Time

2 3 4 5 5 6 9 13 29 55

Number of Locations Chosen

CPU Time

(ms)

Optimal Value

13 14 22 19 23 24 50 62 163 393

38547 35547 32136 30022 21222 23943 20307 15261 9142 2821

3 3 4 5 6 7 9 13 29 55

62 38 40 24 23 19 37 28 7 8

(ms)

CPU Times for IBM 308 1 CMS 2.0 FORTHX

Journal of Operations

Management

29

Comparative

T

TABLE 7 Results for Some 100 X 100 Problems

Sule

1

Erlenkotter

Fixed Charge

Objective Value

Number of Locations Chosen

CPU Time

8000 7000 6000 5000 4000 3000 2900 2000 1150 1000

90579 86579 82579 78127 72599 63176 62133 52056 39637 37435

4 4 4 5 6 8 8 10 14 15

Number of Locations Chosen

CPU Time

(ms)

Optimal Value

40 39 40 48 60 94 99 113 196 188

87889 83720 78720 73073 66407 59407 58613 50103 38479 35965

4 5 5 6 7 7 8 12 16 17

226 372 337 273 221 2248 1604 147 141 79

(ms)

CPU Times for IBM 308 I CMS 2.0 FORTHX

APPROXIMATE

METHODS

VERSUS

EXACT

ALGORITHMS

There is no known algorithm that could solve all instances of the uncapacitated plant location problem in a time polynomially related to the size of the data (for our purposes the number of potential sites). In oversimplified terms, the time required by existing solution methods grows exponentially with the size of the problem. The conjecture is that no such algorithm can be devised (in computer parlance, the problem is called NP-hard). Otherwise, it could be used to solve a large number of reputedly difficult optimization problems. “There exist telecommunication network problems which could use algorithms handling problems with thousands of “plants” and “destinations”. These can only be tackled by heuristics at present.” [45] Are fast (polynomial) approximations the best recourse left to decision makers? Geoffrion [40] has pointed out that because most managerial decisions entail the comparison of various alternatives evaluated in several computations, approximate methods present some serious difficulties: “Not only does an optimizing capability enhance the value of most individual runs, but it also provides the opportunity to make valid comparisons between the results of different runs. This is extremely important because the conclusions reached by the planning project typically rely far more heavily on comparisons between computer runs than on runs considered individually. With “quasi-optimizing” programs, such as so-called cost calculators or simulators fitted with heuristics, one never knows whether different results are due to different inputs or to the vagaries of the computer program.” For example, we have seen that Sule’s method would find two different results depending on how a tie is broken. Also, in Table 5, for different levels of fixed charges f = 1500 and f = 1000, the approximate method yields two different solutions, whereas the optimum locations remain unchanged. Faced with an increase of fixed cost the managers might have relocated their facilities because they relied on a simple approximation!

30

APES

The design of efficient algorithms to solve large-scale locational largely motivated by these considerations. Two general approaches l

l

problems optimally have been explored:

is

Some polynomial algorithms can be designed to yield an optimum with a high degree of probability [32, 161. Other polynomial algorithms can solve some special classes of uncapacitated plant location problems exactly: -The Uncapacitated One-product Lot-sizing Problem [6 l] -The Tree Location Problem [lo, 57, 58, 71, 901, where the distances are measured on a tree spanning the demand and location points -The Tree Partitioning Problem, subsuming the preceding two, where a maximum weight subtree partition is sought [5, 191

Although the above problems encompass many practical situations, more general classes need to be found. This research is intertwined with the design of difficult uncapacitated plant location problems [20]. For several combinatorial optimization problems, the best algorithms that exist today use information about the facial structure of the integer polytope (the convex hull of the feasible solutions). The matching problem [23,8 l] is an example where the integer polytope has been fully described by means of its facets. The traveling salesman problem [ 18, 4 1,42, 43, 44, 801 and the vertex packing problem [77, 78, 79, 92, 951 are examples of NP-hard problems where the partial description of the integer polytope has proven useful in developing new algorithms. An algorithm selectively adding constraints based on this description promises to yield decisive advances in the domain of large-scale locational problems. A first implementation is given in [72]; note that such algorithms are essentially of dual nature, a possible explanation of the success of the dual approach.

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APES

44.

45.

46.

47.

48.

49.

50.

5 1.

52.

53.

54.

55.

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