A filter-and-fan algorithm for the capacitated minimum spanning tree problem

Computers & Industrial Engineering 60 (2011) 187–194

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A filter-and-fan algorithm for the capacitated minimum spanning tree problem César Rego ⇑, Frank Mathew School of Business Administration, University of Mississippi, University, MS 38677, USA

a r t i c l e

i n f o

Article history: Received 24 February 2010 Received in revised form 4 October 2010 Accepted 5 October 2010 Available online 14 October 2010 Keywords: Capacitated minimum spanning tree Compound neighborhoods Strategic oscillation Variable-depth neighborhood search Filter-and-fan

a b s t r a c t The capacitated minimum spanning tree (CMST) is a notoriously difficult problem in combinatorial optimization. Extensive investigation has been devoted to developing efficient algorithms to find optimal or near-optimal solutions. This paper proposes a new CMST heuristic algorithm that effectively combines the classical node-based and tree-based neighborhoods embodied in a filter-and-fan (F&F) approach, a local search procedure that generates compound moves in a tree search fashion. The overall algorithm is guided by a multi-level oscillation strategy used to trigger each type of neighborhood while allowing the search to cross feasibility boundaries. Computational results carried out on a standard set of 135 benchmark problems show that a simple F&F design competes effectively with prior CMST metaheuristics, rivaling the best methods, which are significantly more complex. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction The capacitated minimum spanning tree (CMST) problem is fundamental to the design of communication networks and has been widely studied for its importance in practical applications. A classical application consists of finding a minimum cost design of a capacitated centralized processing network where a central node of limited capacity must be linked via a tree topology to a number of remote terminals with a specified demand (see e.g. Kawatra & Bricker, 2000; Woolston & Albin, 1988). In the context of teleprocessing system design (Chandy & Russell, 1972; Esau & Williams, 1966; Kershenbaum & Boorstyn, 1983) the remote terminals are data terminals and the central node is the data processing (or control) center. The terminals communicate with the center along communication links of limited capacity. The capacity of a link is the maximum traffic the link can carry while maintaining acceptable response times. The tree topology stems from the fact that traffic from a single terminal cannot be simultaneously transmitted along several links without additional synchronizing circuitry. The problem is a special case of the so-called Telpak problem, which is a fundamental design problem in fiber-optic local access networks (Gavish, 1991). The CMST also finds application in a variety of other settings in distribution, transportation, and logistics (see Amberg, Domschke, & Voß, 2000; Gavish, 1982, 1991). In addition, the problem provides a relaxation of the classical capacitated vehicle routing problem, which is central to many

⇑ Corresponding author. E-mail addresses: [email protected] (C. Rego), [email protected] (F. Mathew). 0360-8352/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2010.10.003

other complex problems, including the design of communications networks with topological ring structures (Klincewicz, Luss, & Yan, 1998). The CMST problem has been widely studied over the last four decades with proposals of a variety of solution approaches. In this paper we are interested in heuristic algorithms, focusing on local search methods and guiding strategies superimposed by metaheuristic approaches. Local search defines a general class of heuristic methods that explore the solution space by iteratively generating new solutions derived from a neighborhood structure, and whose effectiveness is typically amplified by embedding them in metaheuristic strategies. Earlier proposals for metaheuristic strategies applied to the CMST problem are due to Amberg, Domschke, and Voß (1996) who introduced a number of variants of simulated annealing and tabu search algorithms. In this work the authors consider two basic types of neighborhoods: a shift neighborhood that transfers a node from one sub-tree to another, and a swap neighborhood that interchanges nodes between sub-trees. Sharaiha, Gendreau, Laporte, and Osman (1997) proposed another tabu search approach based on a sub-tree neighborhood structure, which splits the current spanning tree into two sub-trees and reconnects them by adding an arc different from the one that had been deleted in the original tree. These neighborhoods that modify at most two sub-trees are generally called two-exchange neighborhoods. Patterson, Pirkul, and Rolland (1999) proposed an adaptive reasoning technique drawing on principles of adaptive memory programming of the type used in tabu search and joining them with constructive neighborhood search processes. Their approach iteratively executes the classical constructive heuristic of Esau and Williams (1966) under the guidance of tabu restrictions, which are probabilistically modified at each iteration of the method.

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The state-of-the-art metaheuristic algorithms for the CMST are provided by a sequence of two papers by Ahuja, Orlin, and Sharma (2001,2003). In the initial paper, the authors proposed two multiexchange neighborhood structures by generalizing the node-based and tree-based two-exchange neighborhood structures previously proposed by Amberg et al. (1996) and Sharaiha et al. (1997), respectively. The generalization of these methods considers larger neighborhoods of each type that may propagate over all sub-trees in the solution, the size of which grows exponentially with problem size, making them extremely costly to evaluate even for problems of modest dimension. To overcome this limitation, the authors consider a reduced neighborhood using the concept of an improvement graph. An arc in the improvement graph with respect to a feasible solution represents an elementary move that transfers a node (or sub-tree) to join another sub-tree, with the cost of the arc being the change in cost associated with the move. Hence, depending on the type of neighborhood used a sequence of arcs tracing a node-simple path or cycle in the improvement graph can represent multi-exchanges of either nodes or sub-trees of the original graph. To evaluate the best multi-exchange move a technique based on shortest path constructions is used. The authors implemented tabu search as well as a greedy randomized adaptive search procedure (GRASP) for both neighborhoods and report the results from all four algorithms. Between the four different approaches, the authors obtained the best known solutions for all the benchmark problems. In the second paper, the authors proposed a unified neighborhood that subsumes both the node-based and tree-based multi-exchange neighborhoods to form a composite neighborhood along with a GRASP procedure. Interestingly, these multi-exchange neighborhood structures also proved effective in a local search procedure developed by Gamvros, Golden, and Raghavan (2006) for the Telpak problem. Our present paper integrates selected components of previous neighborhood search strategies for the CMST with a variable-depth multi-exchange neighborhood search procedure based on the filterand-fan (F&F) approach (Glover, 1998). Our main purpose is to investigate the effectiveness of the F&F procedure in this setting while keeping the algorithm design relatively simple. As a supplementary goal, we seek to uncover relevant insights to the design of more advanced algorithms for solving other challenging network design problems. The remainder of this paper is organized as follows. The CMST is formally defined in Section 2, followed by describing the filter-andfan algorithm in Section 3. Section 4 presents the computational analysis and conclusions are given in Section 5.

lem are due to Gavish (1982) and Gouveia (1995). A generalization of Gouveia’s formulation for the more general heterogeneous-demand CMST discussed here is as follows (Rego, Mathew, & Glover, in press):

ðCMSTÞ Minimize

di n X n Q X X

cij Z ijq

ð1Þ

i¼0 j¼1 q¼dj

subject to

di n Q X X

Z ijq ¼ 1 j ¼ 1; . . . ; n

ð2Þ

i¼0 q¼dj di n Q X X i¼0 q¼dj

qZ ijq 

dj n Q X X

qZ jiq ¼ dj j ¼ 1; . .. ; n

ð3Þ

i¼1 q¼di

Z ijq 2 f0; 1g i ¼ 0; .. . ; n j ¼ 1; . . . ;n; q ¼ dj ;. . . ; ðQ  di Þ ð4Þ The three dimensional binary variable Zijq indicates the existence (Zijq = 1) or absence (Zijq = 0) of a specific quantity of flow q on an arc (i, j) associated with the solution for the CMST. The objective (1) is to select a CMST having minimum cost, i.e., that minimizes the sum of the costs of arcs included in the solution. The constraints of (2) ensure that each node j e V is sourced by exactly one arc (i, j) from some node i e V0, while conservation of flow is ensured by the constraints of (3). 3. The filter-and-fan algorithm Our proposed algorithm comprises three main components: a constructive method to generate an initial feasible solution, a filter-and-fan procedure to explore the solution space in the vicinity of the current best solution, and an oscillation strategy aimed at guiding the search beyond local optimality while exploiting tradeoffs between intensification and diversification. The filter-and-fan procedure makes use of both node-based and tree-based neighborhoods embedded in two different search strategies: (1) an oscillation strategy used in the local search procedure that drives the search by alternating between the two types of neighborhoods and (2) a composite neighborhood search strategy in the tree search that combines the two types of neighborhoods according to filter and fan candidate lists. The following subsections provide the details of these algorithmic components. 3.1. Initial solution

2. Problem definition The CMST problem can be stated in reference to a complete undirected graph G = (V0, A), whose vertex (node) set is represented by V0 = {0, 1, . . ., n} and whose arc set is represented by A ¼ fði; jÞji; j 2 V; i – jg. Node 0 denotes a special node named root, and each node i for i 2 V ¼ V 0 n f0g has a specified demand di. A matrix C = (cij) is associated with A, where cij is a non-negative weight (distance or cost) for arc (i, j) if there is link between nodes i and j. Otherwise cij is infinity. The CMST problem consists in finding a minimum cost of a tree T spanning all nodes of G, so that the sum of the demands in each sub-tree off the root node does not exceed a fixed arc capacity Q. When all the nodes i e V have the same demand the problem is referred to as the homogenous demand CMST problem. Analogously, when the nodes have different demand the problem is referred to as the heterogeneous-demand CMST problem. The CMST problem is NP-Complete (Papadimitriou, 1978). Two classical formulations for the homogeneous-demand CMST prob-

A feasible solution for the CMST can be obtained simply by introducing an arc linking the root node to each terminal node, corresponding to a centralized point-to-point network. Alternatively, a constructive heuristic can be used to produce more cost-effective centralized multipoint networks. The greedy savings heuristic of Esau and Williams (1966) has been a standard to create feasible solutions for the CMST. Due to its simplicity and ability to quickly create reasonably good solutions, the procedure has often been adopted to provide initial solutions for more advanced heuristics (Ahuja, Orlin, & Sharma, 2001, 2003; Amberg et al., 1996) as well as to implement constructive neighborhoods for adaptive search methods (Patterson et al., 1999). To facilitate the comparisons with other methods we also used Esau-Williams (EW) heuristic to create starting solutions for our filter-and-fan algorithm. Starting from a solution with all nodes directly connected to the root, the method operates by disconnecting a node from the root at each step and reconnecting it to another node so as to obtain a maximum possible savings in cost. The process is repeated until no further improvement is possible.

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3.2. Neighborhood structures A brief review of some standard terminology is useful as a prelude to describing our method. In the context of local search, neighborhood structures may be classified as simple or compound depending on whether the generation of neighboring solutions is carried out by a single or multiple elementary operations, respectively. If more than one type of neighborhood is considered in a compound neighborhood, the resulting is a composite neighborhood. Moreover, if the number of elementary operations in a compound neighborhood can vary in the course of the algorithm, it is called a variable-depth neighborhood. The F&F framework provides a means for creating such a variable depth design. In our algorithm elementary operations are defined by simple moves obtained by either a node-based or a tree-based neighborhood. Formally, we define these neighborhoods by a general function kðu; v ; wÞ specifying the elementary moves carried out by either a node-based ðk ¼ NÞ or a tree-based ðk ¼ TÞ neighborhood that close an arc (u, v) and open an arc (v, w) with proper direction of flow. Fig. 1 gives an example for N(4, 7, 1) and T(2, 6, 9), where dotted lines in diagram (A) represent arcs that are opened by the move and arcs crossed by two parallel segments are those to be closed. The resulting solution after the application of these moves is shown in diagram (B). 3.3. The filter-and-fan procedure The basic filter-and-fan approach may be viewed as a method that melds a local search procedure with a tree search procedure, both operating on the same type of neighborhood structure, and alternating between them until the tree search fails to improve a local optimum found by the local search procedure. The tree search expands each level based on two fundamental processes: (1) a fan process that generates a number of neighboring solutions for each solution at the current level and (2) a filter process that prunes the tree by keeping only the most promising solutions among those generated by the fan process. Advanced forms of the method incorporate the use of specialized filter and fan candidate lists coupled with tabu search adaptive memory. In this case the local search procedure may simply consist of a method that develops singlethread search paths as opposed to multi-thread searches as developed by the tree search procedure. For the original proposal and extensions of the F&F method see Glover (1998) and Rego and Glover (2002). In our implementation we keep the use of adaptive memory at very rudimentary levels, limiting it to basic short-term memory components and simple strategic oscillation. The single-thread search procedure oscillates between simple node-based and tree-

node-based move

11

5

1

5

11

8

1

2

7

3

(A) Moves: N(4, 7, 1) and T(2, 6, 9)

0

6

4

9

tree-based move

8 2

7

6

0

4

10

based neighborhoods whenever one neighborhood fails to produce an improvement. On the other hand, the multi-thread search procedure implements variable-depth composite neighborhoods by combining both types of simple neighborhoods throughout the various paths of the search tree. In both procedures, tabu search strategic oscillation is used to allow for these neighborhoods to temporarily accept infeasible solutions. The oscillation between feasible and infeasible solution spaces is controlled by adding a penalty term to the objective function whose value depends on both the degree of violation of the solutions (with respect to the capacity constraints) and the transition time (or cycle) of the oscillation process (measured in terms of the number of successive iterations performed before the method selects a solution in a different solution space). Whenever no transition occurs between feasible and infeasible spaces within a predefined number of iterations, the penalty value is appropriately adjusted to promote oscillation. More details about the oscillation strategies used in the algorithm are provided later. In brief, our filter-and-fan procedure works as follows. Given an initial solution S and a starting neighborhood N 1 the algorithm executes a single-thread tabu search that attempts to find improved solutions in a process that oscillates between neighborhoods N 1 and N 2 until no improvement is found in two successive singlethread searches with different neighborhoods. In order to speed up the neighborhood search the procedure makes use of two types neighbor lists: a nearest-neighbor list, which stores for each node the p nodes that are closest to it; and a smallest-angle list which stores for each node the q nodes that form the smallest angle between the lines that link the node and its neighbor to the root node. During the course of the search the procedure maintains a list M of moves that have led to the g0 best solutions evaluated throughout the search. At the end of the procedure, the best of these g0 solutions is set to be the starting solution for a multi-thread tabu search using the filter-andfan strategy. We select the starting solution S(0) to be the root node of the filter-and-fan tree. The best g1 legitimate moves for S(0) in M define the candidate list Mð0Þ used to create the first level (k = 1) of the neighborhood tree with solutions Si(1) (i = 1, . . . , g1). The next levels (k = 2, . . . , L) are created by filter and fan processes. A fan process selects a subset of g2 legitimate moves from M associated with each solution Si(k) to generate g = g1  g2 trial solutions for level k + 1 (as a result of applying g2 moves to each of the g1 solutions at level k). A filter process prunes the tree by keeping only the best g1 moves from among the g moves that have just been generated by the fan process. The application of the fan and filter processes results in a candidate list MðkÞ of moves for level k. Once the maximum number of levels L is reached (or if no more legitimate candidate moves remain to be evaluated) the method

3

10

(B) New Solution

Fig. 1. Node-based and tree-based neighborhood structures.

9

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Table 1 Cost matrix for an instance of the CMST. 0

0

0 1 2 3 4 5 6 7 8 9 10 11

– 4 4 6 3 5 6 5 8 6 8 8

1

2 4

– 5 7 5 4 9 4 9 12 9 5

4 5 – 7 7 3 4 9 4 9 12 9

3 6 7 7 – 4 8 4 9 8 4 9 11

4

5

3 5 7 4 – 8 9 5 11 9 3 8

5 4 3 8 8 – 8 8 7 12 12 7

6 6 9 4 4 9 8 – 12 4 4 13 13

7

8

9

10

11

5 4 9 9 5 8 12 – 13 13 6 4

8 9 4 8 11 7 4 13 – 11 16 12

6 12 9 4 9 12 4 13 11 – 12 16

8 9 12 9 3 12 13 6 16 12 – 11

8 5 9 11 8 7 13 4 12 16 11 –

switches back to the single-thread search starting with the best trial solution encountered in the tree search. The procedure is repeated as long as an improved solution has been found in the single-thread or in the multi-thread search. To illustrate the process consider an instance of a homogenous demand CMST problem with n = 12, Q = 3, and a cost matrix C of elements cij as in Table 1. The graphs in Fig. 2 correspond to two feasible solutions A and B (for the instance just defined), where solution B is obtained from solution A through the application of the moves specified on the graph. For the sake of illustration assume that solution A is given as an entry solution to a filter-and-fan search. Fig. 3 illustrates the filter-and-fan tree used to perform a compound move from solution A (the root node) to the enhanced solution B obtained after removing the doubly-crossed arcs and adding the dotted arcs associated with the move, as shown in diagram B of Fig. 2. Nodes in the F&F tree represent solutions with the corresponding objective function values. A node is gray or black if the solution is feasible and is white if the solution is infeasible with respect to capacity constraints. Black nodes specifically denote local optimal solutions. Arcs denote the transition between solutions carried out by either type of elementary neighborhood as defined in Section 3.2. Strategic oscillation between feasible and infeasible regions is illustrated using a penalty of 3 (added to the objective function) for every unit of capacity violation created by the associated move. The two black nodes in the F&F tree of Fig. 3 show that the entry solution of cost 46 was improved by a composite neighborhood obtained from a sequence of one tree-based (T) and two node-based (N) elementary neighborhoods, giving rise to a new local optimal solution of cost 45. The application of this composite move to the problem graph is depicted in diagram B of Fig. 2, wherein a move kðu; v ; wÞ in the associated neighborhood ðk ¼ N or TÞ is specified in the cycle formed when arc (v, w) is added to the current span-

5

11 1

8

ning tree before removing the associated arc (u, v). The infeasibility of the solution at the first level of this improving tree-search path illustrates the relevance of crossing feasibility boundaries to effectively find improved solutions. Likewise, the combination of neighborhoods within a filter-and-fan tree adds another degree of flexibility to the search process by allowing for the creation of composite moves of various depths. As previously mentioned the interaction between the two types of neighborhoods is made simpler in the single threaded local search procedure as the search with one type of neighborhood is discontinued when the search with the other type of neighborhood starts. However, this type of strategic oscillation was also found beneficial in overcoming local optimality in our algorithm. We now provide a detailed description of our implementation of the filter-and-fan procedure in a way that permits it to be used with any number of neighborhood structures and embodied in a variety of advanced oscillation strategies. In the general case, the oscillation between different neighborhoods need not follow a specific order but may be determined dynamically by an adaptive memory strategy. In that event, it is necessary to keep track of the neighborhoods that have already been used and that have successively failed to improve the current best solution. This process can be controlled by locking neighborhoods as they fail and unlocking all previously locked neighborhoods whenever one alternative neighborhood succeeds in improving the best solution. Under this framework, a single-thread search stops when all neighborhoods are locked. The filter-and-fan procedure is as shown in Fig. 4. We make use of the following additional definitions: S⁄ S ~ S N L

3.4. Oscillation strategies for diversification and intensification We have illustrated the merit of allowing the search to cross feasibility boundaries in order to reach solutions that are not readily available through neighborhood search paths hemmed in the feasibility space. To implement such an oscillation process we have simply added a penalty term to the objective function to compensate for violations of capacity constraints. The choice for penalty values can be critical in order to effectively explore tradeoffs between intensification and diversification of the search. Low penalty

5

11

2

7

N(1,5,2)

1 7

6

0

The best solution found The input solution for a single-thread or a multi-thread search The best solution found in a single-thread or a multithread search The set of neighborhoods of cardinality |N | The set of current locked neighborhoods

8 2

N(6,8,2)

0

6 T(2,6,9)

4 3

10

4

9

3

9

10

Fig. 2. Spanning trees representing two feasible solutions: (A) entry solution at the root node of the F&F tree and (B) improved solution found in the F&F search.

C. Rego, F. Mathew / Computers & Industrial Engineering 60 (2011) 187–194

46

K=0 T(2,6,9) K=1

T(0,3,4)

49

48

52

N(6,8,2) T(9,6,0)

N(1,5,2) N(1,5,0)

N(3,9,0) N(4,7,0)

K=2

48

46 N(1,5,2) T(0,2,5)

K=3

N(4,7,1)

45

51

47

46

49

49

N(2,6,9) N(2,6,3) T(0,2,5) T(2,6,9) 46

47

48

49

Fig. 3. Example of a filter-and-fan tree (g1 ¼ 3; g2 ¼ 2; L ¼ 3).

values can allow the search to wander too far away from the feasible space and thus prevent the method from reaching possible improving solutions. Conversely, high penalty values may bring the search back to the feasible space too quickly, thus not providing an opportunity for sufficient diversification. In a process similar to that used in Gendreau, Hertz, and Laporte (1994) for the capacitated vehicle routing problem we use a simple rule to exploit

191

these tradeoffs by regularly adjusting penalties as needed. Given an initial penalty value u associated with each unit of capacity violation and a specified transition cycle t, if all moves performed in the last t iterations involve feasible solutions u is halved; similarly, if all moves involve infeasible solutions u is doubled; otherwise, if not all moves led to feasible or infeasible solutions u remains unchanged for another cycle of t iterations. 4. Computational analysis The performance of our F&F algorithm is evaluated on a set of classical benchmark instances obtained from the OR-library (Beasley, 1990) and compared against alternative algorithms from the literature. Test instances consist of 90 homogenous demand problems with 41 and 81 nodes and 45 heterogeneous demand problems with 50, 100 and 200 nodes. The 41-node problems consider arc capacities of 3, 5 and 10 while the 81-node problems have arc capacities of 5, 10 and 20. The capacities for heterogeneous demand problems are 200, 400 and 800. The algorithm was implemented in C++ and the computational experiments were carried out on a computer with a 2 GHz Pentium 4 processor and 2 GB of memory. Table 2 summarizes the results of the F&F method on the complete testbed. We report the number of best solutions found (NBSF)

Fig. 4. The filter-and-fan procedure.

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Table 2 Results for the filter-and-fan algorithm on 135 instances. Algorithm

Homogeneous NBSF

F&F APRD ACPU

Heterogeneous 40(60)

80(30)

Total

46

NBSF

Overall

50(15)

100(15)

200(15)

Total

0.94 19.0

3.13 93.4

5.9 526.6

3.33 213

6 0.17 3.3

0.48 27.4

0.28 11.3

52 1.29 78.5

Table 3 Relative performance of various computers. Reference

Algorithm

Platform

1s

This paper Amberg et al. (1996) Sharaiha et al. (1997) Patterson et al. (1999) Ahuja et al. (2001)

F&F TS-STS, TS-CMS, TS-REM, SA TS ART GRASP-1, GRASP-2, TS-1, TS-2

2 GHz Intel P4 66 MHz Intel 486 100 MHz R4000 SGI Indigo 266 MHz Pentium II DEC Alpha

1317 2.4 15 69 412

Table 6 Comparisons with Patterson et al. (1999) over 60 problems.

Table 4 Comparison with Amberg et al. (1996) over 70 instances. Algorithm

Algorithm

Homogeneous NBSF

TS-STS APRI

–

TS-CSM APRI

–

TS-REM APRI

–

SA APRI

–

F&F APRI ACPU

37

40(40)

80(30)

NBSF

Overall

3.22

4.46

3.75

3.22

4.51

3.77

3.20

4.26

3.65

3.22

4.31

3.69

4.36 2.5

4.49 27.4

4.42 13.7

Homogeneous

over all problem sets, the average percentage deviation (APRD) over the optimum or best known solution and the associated average of execution times (ACPU) in seconds for each group of problems with the corresponding totals (Total) for each class of problems and over all (Overall) problems. In addition, we provide in parenthesis the number of instances tested for each problem size. A similar format is used for tables showing comparisons with other algorithms from the literature. To make suitable comparisons with prior algorithms the following tables present results on the same problem instances attempted by these algorithms. Also, since the various algorithms have been tested on very different computer systems we provide in Table 3 approximate performance factors as reported in Dongarra (2009). Because these factors were obtained using standard

ART APRD ACPU

18

F&F APRD ACPU

26

40(30)

80(30)

Overall

0.29 11.6

1.22 84

18 0.76 53.6

0.23 3.2

0.48 27.43

26 0.36 15.32

software for solving linear equations, the reader should bear in mind possible discrepancies when using these values in estimates of computer-independent times for algorithms requiring different types of computations. Amberg et al. (1996) tested a simulated annealing algorithm (SA) and multiple variants of tabu search that differ by the method used to manage tabu restrictions, namely, the static tabu search (STS) method, the cancellation sequence method (CSM), and the reverse elimination method (REM) as introduced in Glover (1990). The various approaches were tested on 70 homogeneous-demand instances and results are provided as average percent improvement (APRI) over the Esau-Williams heuristic for each group of problems. In the original paper, the authors provide results for 12 runs of tabu search using STS, four runs using REM, and 20 runs using CSM. Similarly, results of the simulated annealing algorithm are given for nine runs with different settings. The computations were performed on a computer with a 66 MHz Intel 486 processor by running each problem instance for 600 s. In a general analysis with the full set of experiments reported in the original paper we observe that the results obtained from independent runs of the

Table 5 Comparison with Sharaiha et al. (1997) over 105 problems. Algorithm

Homogeneous NBSF

TS APRD ACPU

12

F&F APRD ACPU

26

40(30)

Heterogeneous 80(30)

Total

NBSF

Overall

50(15)

100(15)

200(15)

Total

5.57 65.9

2.83 571.5

7.69 4509.3

5.36 1715.6

16 3.44 833.83

0.94 19.0

3.13 93.4

5.9 526.6

3.33 213

32 1.63 100.04

4 1.30 50

2.71 295

2.00 172.5

0.23 3.2

0.48 27.43

0.36 15.32

6

193

C. Rego, F. Mathew / Computers & Industrial Engineering 60 (2011) 187–194 Table 7 Comparison with Ahuja et al. (2001) over 105 problems. Algorithm

Homogeneous NBSF

GRASP-1 APRD ACPU

39

TS-1 APRD ACPU

49

GRASP-2 APRD ACPU

21

TS-2 APRD ACPU

27

F&F APRD ACPU

33

Heterogeneous 40(30)

80(30)

Total

NBSF

Overall

50(15)

100(15)

200(15)

Total

0.40 200

4.76 200

6.89 500

4.02 300

44 1.82 214.29

0.25 200

4.45 200

7.57 500

4.09 300

56 1.76 214.29

0.56 200

1.25 200

2.49 500

1.43 300

31 0.99 214.29

0.55 200

1.1 200

1.95 500

1.20 300

37 0.82 214.29

0.94 19

3.13 93.4

5.92 526.6

3.33 213

39 1.59 99.81

5 0.07 100

0.26 200

0.17 150

0.00 100

0.01 200

0.01 150

0.31 100

1.01 200

0.66 150

0.19 100

0.86 200

0.53 150

0.10 2.4

0.48 27.43

0.29 14.92

7

10

10

6

algorithms on each group of problems show that the F&F algorithm produces solutions that are on average better than any one run of all these alternative algorithms, on the same groups of problems. Additionally, when the best solution averages of all runs on each problem group are compared, F&F yields the best overall average for each of the corresponding groups. For a more specific analysis, we establish comparisons of a single run of our (deterministic) F&F algorithm with the best run of each tabu search and simulated annealing algorithms. Table 4 summarizes these results. Although, computational times are not directly comparable, we should note that solutions of tabu search and simulated annealing algorithms were produced within a limit of 600 s in a much slower computer than the one used for our filter-and-fan algorithm. Remarkably, when factoring the differences in speed between the two processors, it appears that these TS and SA algorithms have managed to find their best solutions in a time that is on average one order of magnitude smaller than that used by the F&F algorithm. However, the substantially superior quality of the solutions produced by the F&F algorithm clearly compensates for the differences in running times, especially since these times are still very small. The tabu search (TS) procedure developed in Sharaiha et al. (1997) is tested on a 100 MHz Silicon Graphics Indigo machine (R4000). Table 5 gives the results for comparison with our F&F algorithm. When compared on the same problem instances we find this TS algorithm performs slightly better than F&F on only one group of instances. On all the other four groups of homogenous and heterogeneous instances F&F is significantly more effective than the Shariaha et al. TS algorithm. Overall, the F&F method provides a better average solution quality and finds twice as many best known solutions, although using longer computation times on average. Table 6 provides a comparison with the adaptive reasoning technique (ART) of Patterson et al. (1999), which was tested on a computer with a 266 MHz Pentium II processor and 96 MB of RAM. The results show that ART found its best solutions in a relatively shorter time than F&F. In terms of solution quality, F&F slightly edges out ART for the smaller instances and significantly outperforms this algorithm on the larger instances. In particular, F&F matches the best known solution in a much larger number of instances than ART and likewise provides a higher average solution quality. The most competitive approaches are due to Ahuja et al. (2001) and Rego, Mathew and Glover (in press). Ahuja, Orlin and Sharma made a breakthrough by proposing two very large-scale neighborhood search (VLSN) approaches rooted in multi-exchanges of node-based and tree-based neighborhoods, respectively. These

neighborhoods are used to create two different variants of tabu search and GRASP algorithms. The four algorithms were run on a DEC Alpha computer with a specified time limit for each type of instance and size. A summary of the results of all four algorithms along with those of our F&F algorithm on the corresponding instances are presented in Table 7. GRASP-1 and TS-1 represent the algorithms for the node-based neighborhood while GRASP-2 and TS-2 represent those using tree-based neighborhoods. Results on the two groups of problems of different sizes and characteristics indicate that the F&F algorithm performs better than GRASP-2 and TS-2 variants for the homogeneous instances and better than GRASP-1 and TS-1 for the heterogeneous instances. Overall, F&F is better than GRASP-1 and TS-1, but not as good as GRASP-2 or TS-2. Based on the differences in the computer systems, computational times for F&F seem to be relatively larger on average than for these other algorithms. The GRASP implementation that uses a composite very large search neighborhood (VLSN) proposed in Ahuja et al. (2003) is one of the state-of-the-art algorithms for the CMST problem. The results reported were obtained on the computer with a Pentium 4 processor and 512 MB of RAM. It found the best known solutions for all problems it was tested on and it improved upon 36% of these problems, establishing itself as the most effective known approach. However due to the method’s complexity, the algorithm requires more than four times as much effort as our F&F algorithm (on a similar computer) to find solutions of equal or higher quality. The RAMP algorithm recently proposed in Rego, Mathew and Glover (in press) competes on a par with this best GRASP/VLSN algorithm and exhibits superior performance on large instances. However, this RAMP algorithm is likewise appreciably more complex than the present F&F algorithm and requires a computational time of the same order as that of GRASP/VLSN. Consequently, in spite of its simple design, our F&F approach ranks not far behind these two methods that dominate all others, while performing more effectively than remaining methods, which in most instances are somewhat more complex.

5. Conclusion We developed a variable-depth multi-exchange neighborhood search procedure based on the filter-and-fan (F&F) approach. The algorithm combines node-based and tree-based neighborhoods and uses strategic oscillation to explore tradeoffs between search intensification and diversification. Extensive computational analysis shows significant advantage of the proposed F&F algorithm over

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