tabu search based heuristics for multi-floor facility layout

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Tabu search basedheuristics for multi-floorfacility layoutSue Abdinnour-Helm & Scott W. HadleyPublished online: 14 Nov 2010.

To cite this article: Sue Abdinnour-Helm & Scott W. Hadley (2000) Tabusearch based heuristics for multi-floor facility layout, International Journal ofProduction Research, 38:2, 365-383

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int. j. prod. res., 2000, vol. 38, no. 2, 365± 383

Tabu search based heuristics for multi-¯ oor facility layout

SUE ABDINNOUR-HELM{* and SCOTT W. HADLEY{

A pair of two-stage heuristics, GRASP/TS and FAF/TS, for solving the multi-¯ oor facility layout problem are given. In both heuristics, the objective of the ® rststage is to obtain a layout with minimal inter-¯ oor ¯ ow. Tabu search is used inthe second stage to re® ne the initial layout based on total inter/intra-¯ oor costs.GRASP/TS applies a GRASP to obtain the initial layout. FAF/TS uses an exactprocedure FAF from the open literature to obtain an initial layout with minimalinter-¯ oor ¯ ow. Computational tests indicate that GRASP/TS comparesfavourably with other heuristics that do not rely on exact algorithms. FAF/TSis shown to outperform all other heuristics in the open literature.

1. Introduction

One of the most in¯ uential factors a� ecting the e� ciency of a production facilityis its layout. A poor layout implies that, regardless of other factors, the facility willbe ine� cient. The interactions between each pair of departments must be taken intoaccount in order to obtain the most e� cient layout. In a production facility we mayassume that these interactions are the ¯ ow of items between departments. A surro-gate measure for e� ciency can be based on the total cost of transporting the itemsthroughout the facility (e.g. between di� erent departments). The transport costbetween a given pair of departments is a function of the ¯ ow of items and thedistance between the departments.

Multi-¯ oor layouts may not be as common as single-¯ oor layouts, but as pointedout by Bozer et al. (1994) and Meller and Bozer (1997) there is an increase in thenumber of such facilities as ® rms consider renovating older buildings to save money.Meller and Bozer (1997) mention that they have visited numerous new and reno-vated multi-¯ oor facilities in the USA in the past ® ve years. They observed that, inJapan, multi-¯ oor facilities are often used in warehousing as well as manufacturing.Material Handling Engineering (1988a) stated that most of the older buildings in theUSA are multi-¯ oor, and may be obtained at bargain prices in the city and otherplaces where redevelopment is desired. Renovating such buildings has a costadvantage compared to erecting a new building. King and Johnson (1983) estimatethat a refurbished industrial plant is likely to cost one-tenth as much per square footof building as a new factory. They gave an example of a successful implementationby Hendrix Electronics. In their need to expand capacity for future growth, andgiven the limited ® nancial resources, Hendrix chose to remodel an older multi-¯ oormillyard building over building a new facility. Not all multi-¯ oor buildings are olderbuildings that have been renovated. The Gertude Hawk Chocolate Company

International Journal of Production Research ISSN 0020± 7543 print/ISSN 1366± 588X online # 2000 Taylor & Francis Ltdhttp://www.tandf.co.uk/journals/tf/00207543.htm l

Revision received April 1999.{ Barton School of Busines, Wichita State University, Wichita, KS 67260-0077, USA.{ Numetrix Limited, 655 Bay Streeet, Suite 1200, Totonto, Ontario, M5G 2K4, Canada.* To whom correspondence should be addressed. e-mail: [email protected]

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recently constructed a 170 000-ft2 three-story integrated facility to accommodate newbusiness (Industrial Engineering 1990a). Production occupied the second and third¯ oors while storage and o� ces were on the ground ¯ oor. The company saved onland and construction by having three ¯ oors rather than one. An important factor toconsider in all this is the advances in vertical conveyor technology that have mademulti-¯ oor layouts more feasible (Material Handling Engineering 1988b, IndustrialEngineering 1990b).

Over the past three decades a signi® cant amount of e� ort has been expended indevising tools and techniques for determining layouts of facilities that attempt tominimize inter-departmental transportation costs. For a recent review of theliterature, see Meller and Gau (1996) and Meller and Bozer (1996). Most of thee� ort in the past has focused on single-¯ oor facilities with limited work focusedon multi-¯ oor facilities. In general practice, additional costs and constraints mustbe addressed when considering multi-¯ oor facilities. These include: (i) the cost oftransporting an item one distance unit in a horizontal direction (e.g. on the same¯ oor) usually di� ers from the cost of transporting that same item one distance unit ina vertical direction; (ii) there are usually a speci® ed set of lift facilities (e.g. elevators),with ® xed locations, where items can be transported vertically; and (iii) in general adepartment cannot be split across several ¯ oors.

Below we describe several multi-¯ oor heuristics that address all of the additionalcosts and constraints that are noted above. Perhaps the ® rst such heuristic, namedMULTIPLE was introduced by Bozer et al. (1994). MULTIPLE exploits space® lling curves to: (i) de® ne the shape (e.g. footprint on the facility ¯ oor) of eachdepartment; (ii) permit quick and e� cient checking of area constraints on each¯ oor; and (iii) facilitate quick and e� cient cost computations for di� erent layoutscenarios. MULTIPLE starts with a random initial feasible layout and then usespairwise interchange to move from one feasible layout to a lower cost feasible layout.MULTIPLE terminates at local optimality when there is no pairwise interchangethat can improve the current layout. Abdinnour-Helm and Hadley (1995) introduceda two-stage heuristic, which was the ® rst two-stage heuristic in the literature basedon GRASP and tabu search. As inter-¯ oor (e.g. vertical) transportation costs areusually more expensive than intra-¯ oor (e.g. horizontal) transportation costs, the® rst stage GRASP generated an initial feasible assignment of departments to ¯ oorsthat attempted to minimize total vertical transportation costs. The second stage(tabu search) embodied both pairwise interchange and shift moves (e.g. moving asingle department). The tabu search component provided means for the embeddedheuristic to escape locally optimal solutions. Meller and Bozer (1996) introducedanother single-stage heuristic, SABLE, which addresses the issue of termination atlocal optimality. SABLE begins with a random feasible layout and employs thesimulated annealing technique to ® nd a solution. Meller and Bozer (1997) introducedtwo more two-stage simulated annealing based heuristics, named STAGES andFLEX. The ® rst stage (FAF) obtained an optimal initial department-to-¯ oor assign-ment that minimizes the total vertical transportation by solving a mixed integerlinear programming problem using branch & bound. Random initial layouts thatsatisfy the optimal department-to-¯ oor assignment are then used as the startingpoint for the second stage. For STAGES, the second stage was a modi® ed versionof SABLE that did not allow the department-to-¯ oor assignments obtained by FAFto change. For FLEX, the second stage was SABLE, which permits the reassignmentof departments to di� erent ¯ oors. It should be noted, that in order to improve

366 S. Abdinnour-Helm and S. W . Hadley

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robustness, MULTIPLE and SABLE are applied to a number of initial solutions,and the best ending solution is chosen. This di� ers from all the two-stage heuristicintroduced above, all of which start from a single initial solution.

Other heuristics, e.g. SPACECRAFT (Johnson 1982), ALDEP (Seehof andEvans 1967), BLOCPLAN (Donaghey and Pire 1990), SPS (Liggett and Mitchell1981), and MSLP (Kaku et al. 1988), have been developed to solve the multi-¯ oorlayout problem but are not considered in this study since they all have at least one ofthe following modelling shortcomings: (i) departments can be split over multiple¯ oors; (ii) multiple lift locations are not considered; (iii) all departments are assumedto have equal area requirements; and (iv) once departments are assigned to ¯ oorsthere is no consideration of inter-¯ oor dependencies.

In this paper we introduce a pair of two-stage heuristics, both of which employthe same tabu search procedure in the second stage. The ® rst stage of each heuristicfocuses on obtaining an initial layout with minimal inter-¯ oor ¯ ow. The ® rst heur-istic, denoted GRASP/TS, uses a computationally scaleable and straightforwardGRASP procedure. Further, the GRASP procedure is easy-to-implement, customizeand maintain, which is very attractive from a practitioner’s perspective. The secondheuristic, denoted FAF/TS, utilizes an exact algorithm for an NP-hard mixed integerprogramming problem to obtain the initial layout.

The paper is structured as follows. In the following section we motivate thisstudy. In } 3 the problem is described in a manner suitable to illustrating the newheuristics. An illustrative example is given that will be used throughout the paper.Section 4 provides a description of the models and techniques that are used in theheuristics proposed in this paper. The heuristics are presented in a consolidatedfashion in } 5. Section 6 presents computational results on a benchmark suite ofproblems from the literature. Finally, } 7 provides recommendations for futureresearch.

2. Motivation

We now describe our motivation to introduce GRASP/TS and FAF/TS as two-stage interchange based heuristics for the multi-¯ oor layout problem.

MULTIPLE is an interchange based heuristic that starts with a random initialfeasible layout. It is known, however, that when applying interchange techniques thequality of the ® nal solution is strongly correlated to the quality of the solution fromwhich the interchange is initiated (see e.g. Hadley et al. 1992, Meller and Bozer 1997,Pothen et al. 1990). Further, Hadley et al. (1992) observed that improvement heur-istics tend to converge and terminate more quickly when started from a good qualityinitial solution. In order to achieve some level of robustness MULTIPLE wouldapply its interchange techniques to a number of di� erent initial random solutions.Another aspect of multi-¯ oor layouts is that the cost of transporting an item verti-cally (i.e. between ¯ oors) is, in general, relatively more expensive than moving thesame item the same distance on the same ¯ oor. Thus one would expect that a goodinitial layout would be one which has limited inter-¯ oor ¯ ow of items.

We exploit the above two observations to obtain an initial feasible layout for ourGRASP/TS heuristic by creating a model based on graph partitioning. A modi® ca-tion of a simple, yet very e� ective technique known as GRASP (Greedy RandomizedAdaptive Search Procedure, Feo and Resende 1994), is used to obtain good (notprovably optimal) solutions to the resulting graph partitioning problem. In a similarvein FAF/TS, STAGES and FLEX ® nd an optimal solution by using FAF, which

367Multi-¯ oor facility layout using tabu search

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solves a mixed integer linear program. The department-to-¯ oor assignment gener-ated by FAF and used in STAGES and FLEX will in general, be higher quality thanthose determined by GRASP. However there is a trade-o� in the computationale� ort required to obtain these solutions (Meller and Bozer 1997). One may arguethat facility layout decisions are very costly decisions that are infrequent, so time isnot a problem. This is a valid argument, which led the authors to explore FAF/TS.The point to remember though, is that FAF is an optimal procedure and GRASP isa heuristic. As the number of departments grows, an optimal procedure may facemore di� culty in ® nding a solution in reasonable time (or ® nding one at all) whencompared to a good heuristic approach. This is why we think there is need to haveboth GRASP/TS and FAF/TS available in the literature.

Another characteristic of MULTIPLE, and other pairwise interchange basedtechniques, is that it is impossible to alter the number of departments on anygiven ¯ oor. Thus, the number of departments on each ¯ oor in ® xed as a result ofthe initial layout. This limitation has the potential of excluding a large number ofcost-e� ective layouts. Bozer et al. (1994) indicate in the conclusion of their paper thatone can overcome this limitation in MULTIPLE by introducing `dummy depart-ments’ (e.g. departments with very small areas and zero ¯ ow). However, implement-ing this alternative is not straightforward for the following reasons. Firstly, one mustdetermine how many `dummy’ departments to add. By observing that if there is nounused ¯ oor capacity after all real’ and `dummy’ departments have been included inthe layout (e.g. `dummy’ departments are added to ® ll all excess ¯ oor capacity) it iseasy to see that inter¯ oor exchanges can only take place with departments of equalsize. In general this excludes an exchange with a real’ department and a `dummy’department. Thus one has to determine how many `dummy’ departments to add tothe problem, and conversely how much ¯ oor capacity should still be unused afterdummy departments are assigned. Secondly, increasing the number of departmentswould decrease the e� ciency of MULTIPLE since the algorithm would have toconsider many more potential exchanges. For these reasons it is our view that theintroduction of `dummy’ departments is not a practical way to overcome the issue ofnot being able to change the number of departments on any ¯ oor. Due to the natureof the heuristics (e.g. not based on interchange) SABLE, FLEX, and STAGES donot encounter the problem of being unable to alter the number of departments onindividual ¯ oors.

One of the major contributions of this paper is introducing interchange basedheuristics, GRASP/TS and FAF/TS, that overcome the limitation of MULTIPLEby introducing the concept of a shift move. A shift move allows one to modify thesolution by shifting the position of a single department rather than being restrictedto interchanging pairs of departments.

Finally, standard interchange techniques, including MULTIPLE terminate at alocally optimal solution. In recent years there has been signi® cant development oftechniques, often called meta-heuristics, that allow one to escape’ from locallyoptimal solutions. For combinatorial problems, one of these techniques is tabusearch, denoted TS, which was introduced by Glover (1989, 1990a). TS usuallyincorporates any of the standard interchange techniques at a lower level and hasthe feature that it is relatively easy to implement. Simulated annealing is anothermeta-heuristic that has been applied to combinatorial and other optimizationproblems. Meller and Bozer (1996, 1997) have introduced SABLE, STAGES andFLEX, which are e� ective heuristics for the multi-¯ oor layout problem, based on


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simulated annealing. Tabu search is the basis for the second stage of our proposedheuristics GRASP/TS and FAF/TS. By comparing FAF/TS with STAGES andFLEX, we are in essence comparing the e� ectiveness of tabu search and simulatedannealing in solving the second stage of the three approaches.

In summary, this paper introduces interchange based heuristics that extend andgeneralize MULTIPLE in two ways. Firstly, e� ort is expended on ® nding a goodinitial layout before starting the interchange procedure. Secondly, a new shift moveis introduced that allows one to change a layout by moving only one, rather than apair, of departments. Finally, the extended interchange procedure is embedded in atabu search framework and compared to the performance of three simulated anneal-ing based approaches, SABLE, FLEX, and STAGES.

3. An illustrative example

We now present an example that will be used in the remainder of the paperfor illustrative purposes. This example was introduced by Bozer et al. (1994).The problem involves determining an e� cient layout of 15 departments in a three-¯ oor facility. The dimensions of all three ¯ oors are identical, each being 5 units by15 units. There are six lifts available for transporting goods: one at the mid-point of each short’ side and two each uniformly placed on each long’ side.The horizontal and vertical cost to travel a distance unit is assumed to be $1.00and $5.00, respectively, for all pairs of departmentsÐ with the exception ofdepartment 15 (receiving/shipping) which assumes costs of $0.25 and $1.25,respectively, for each distance unit (horizontal or vertical). Finally, between adjacent¯ oors the distance is assumed to be 10 units. The relevant data about thedepartments and the ¯ ow of products amongst them is provided in tables 1 and 2.(The entries in table 2 indicate the ¯ ow originating at the column’ department andterminating at the row’ department.) Alternatively, the ¯ ow matrix can berepresented by the graphical model illustrated in ® gure 1. Note that due tosymmetry in the cost structure and distances we can disregard the direction of¯ ow between each pair of departments.


Department Current area Minimum area

1 15 122 10 73 9 64 7 55 9 76 25 227 25 228 15 139 10 7

10 25 2211 10 912 15 1313 6 414 19 1715 25 25

Table 1. Area requirements.

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4. Models and techniques

In this section we will describe the various models and techniques that comprisethe essential elements of GRASP/TS. We begin by describing GRASP and how weuse it to obtain an initial assignment of departments to ¯ oors. We then review theconcept of space ® lling curves as they relate to both the MULTIPLE heuristic andour heuristic. We conclude with a description of our tabu search implementation.

4.1. The graph partitioning problemThe graph partitioning problem is a problem studied often in the ® eld of combi-

natorial optimization. Before we state the problem, we ® rst provide some de® nitionsand notation.

Let G ˆ … V ; E† denote a graph on node-set V and edge-set E. Further assumethat each node has a size and the each edge has a weight. A k-partition of the


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð 2402 240 Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð3 Ð Ð Ð 1200 Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð4 Ð Ð Ð Ð Ð Ð Ð Ð Ð 1200 Ð Ð Ð Ð Ð5 Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð 600 Ð6 Ð Ð Ð Ð Ð Ð Ð 480 Ð Ð Ð Ð Ð Ð Ð7 Ð Ð Ð Ð Ð Ð Ð 480 Ð Ð Ð Ð Ð Ð Ð8 Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð 1209 Ð Ð Ð Ð Ð Ð Ð Ð Ð 600 Ð Ð Ð Ð Ð

10 Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð 600 Ð Ð Ð11 Ð Ð Ð Ð Ð Ð 480 Ð Ð Ð Ð Ð Ð Ð Ð12 Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð 60013 Ð Ð Ð Ð Ð Ð 480 Ð Ð Ð Ð Ð Ð Ð Ð14 Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð 600 Ð Ð Ð15 Ð 10 25 Ð 25 40 Ð Ð 25 Ð 40 Ð 20 Ð Ð

Table 2. Flow matrix.

Figure 1. Graphical representation of ¯ ows.

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nodes is de® ned as k (possibly empty) subsets, say V 1; V 2; . . . ; V k, of the nodessuch that each node is contained in exactly one subset. The size of a subset isthe sum of the sizes of the nodes it contains. A k-partition is said to be feasibleif the size of V i is less than some pre-speci® ed maximum size, say max-sizei foreach i. An edge is said to be cut by a k-partition if the nodes (endpoints) of theedge are in di� erent (node) subsets.

A generalized version of the graph partitioning problem can now be stated as:

Given a graph G ˆ … V ; E† , an integer k, and max-sizei for i ˆ 1 . . . k, Find thefeasible k-partition which minimizes the weight of edges cut by the partition.

In our case, we can use the above version of the graph partitioning problem tomodel the problem of assigning departments to ¯ oors such that the inter-¯ oor ¯ owof items is minimized. We do this by representing each department by a node, whosesize is the space requirement of the department. For each pair of departments havingnon-zero ¯ ow between them we generate the corresponding edge with weight equalto that total ¯ ow. The integer k is set to be the number of ¯ oors in the facility, andmax-sizei corresponds to the space availability of the ith ¯ oor.

4.1.1. GRASP and a modi® cationGRASP is a simple yet e� ective heuristic for obtaining high quality solutions to

problems for which it is easy to construct a feasible solution. For a survey onGRASP see Feo and Resende (1994).

GRASP can be described as an iterative randomized sampling technique in whicha solution to a problem at hand is generated in a `greedy’ way during each iteration.The best solution over all iterations is then presented as the ® nal’ solution. There areessentially two phases within each GRASP iteration: (i) a solution construction(based on a randomized greedy approach) and (ii) solution improvement, wherethe solution of the ® rst phase is improved using local search.

In the construction phase, a solution is iteratively constructed, one element at atime. At each construction iteration, each element is assigned a value based on theperceived bene® t of including this element in the solution. All elements are thenranked in order of these values and a restricted set of elements is determined. Anelement is then selected at random from this restricted subset and is then added tothe solution under construction. This phase continues until a true solution to theproblem at hand is attained.

In the improvement phase, one tries to improve the solution from theconstruction phase using any local search technique (e.g. interchange).

An important feature of GRASP is that it is, in general, easy to accommodateconstraints that do not make creating a feasible solution exceptionally di� cult. Anexample of these type of constraints are those which restrict the ¯ oor, or a portion ofa ¯ oor, to which a particular department may, or may not, be assigned to. Thisfeature, thus is not restrictive when one desires to implement the proposed techniquein a practical environment.

4.1.1.1. A modi® cation. In most GRASP applications, it is required that avery large number of iterations be performed before the current best candidate isclose to the optimal solution. In order to control the CPU expense in obtaining


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the initial layout we have modi® ed the GRASP concept by altering the greedyselection criteria.

Since we want to obtain our solution in fewer iterations it is natural to try tomodify the selection criteria to increase the greediness while still maintaining somedegree of randomness. In the standard GRASP the probability of selecting aparticular move is uniform over the restricted subset of all moves. To increase thegreediness, one can decrease the size of the speci® ed subset of candidate moves,however, this can have signi® cant negative impact on the randomness of the search.

We propose an alternative method to intensify the greedy behaviour of the searchwithout restricting the subset size by generalizing the selection criteria. We do this bymaintaining the greedy ranking of the moves. Then instead of randomly selecting amove from a restricted subset, we traverse the ranked list of moves in order, selectingthe ith move with probability pi , until a move is selected. This new selection criteriaincreases greediness but does not a� ect randomness as severely as would occur bylimiting the number of candidate moves in the subset. It is interesting to note that forp1 ˆ 1 then the algorithm becomes the well-known greedy algorithm. If the subsetsize is denoted by s, then setting pi ˆ 1=… s ‡ 1 ¡ i† is equivalent to the standardGRASP selection criteria (that is, each move in the subset has equal probability ofbeing selected).

We have also modi® ed the `standard’ GRASP approach by not performing theimprovement phase at each iteration. The rational for this is that, in reality we arenot solving the graph partitioning problem. Rather, we are using the graph parti-tioning problem as a surrogate in order to obtain a good starting point for a localsearch procedure applied to the complete layout problem.

4.1.1.2. A GRASP for graph partitioning. One way to approach the graph parti-tioning problem is from the point-of-view of maximizing the weight of uncut edgesby a partition.

Assume that at some point in the construction phase of a GRASP we have a setof subsets of nodes, where each node of the original graph is contained in exactly oneof these subsets. Now, construct a graph whose nodes correspond to the subsets, andwhose edges are determined as illustrated in the following example. Assume two ofthe subsets are: S1 ˆ 5; 10 and S2 ˆ 12; 14. In the original graph (see ® gure 1) thereare two edges containing one node in each of S1 and S2 (i.e. (5, 14) and (10, 12)) bothwith weight 600. Thus in the new graph we will introduce one edge between S1 andS2 with a weight of 1200… ˆ 600 ‡ 600† .

The moves that we consider as candidates during the construction phase aremergers of two subsets into one. The value/bene® t of a merger of two subsets isthe weight on the edge joining them. Thus, in the previous example, the bene® t ofmerging S1 and S2 is 1200. That is, the weight of the edge that must be `uncut’ as theresult of the merger. In the graphical sense this merging of the two nodes is oftendescribed as contracting an edge’.

Each iteration of our construction phase can be described as follows:

Step 1. Construct a weighted graph G ˆ … V ; E† as follows.Create a node v (in V ) for each department.Let size (v) denote the minimum area requirement of the department corre-sponding to node v.


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For each pair of nodes u and v, create an edge … u; v† with weight equal to thesum of ¯ ow from u to v and also v to u.Set max-size(i) as the capacity of ¯ oor i’ .

Step 2. Rank the edges in order of non-increasing weight.Step 3. (An edge-contraction is feasible if the number of merged subsets with size

max-size(i) is less than or equal to the number of ¯ oors with max-size(k)size)Considering only feasible edge-contractions:IF no feasible contractions THEN

Construction phase failedRestart construction phase

ELSEREPEAT (in order of ranking)

Select the edge with probability p;UNTIL (an edge is selected);

Step 4. IF no edge selected THENrandomly select a feasible edge-contraction;

Step 5. Contract selected edge. Iteration complete.

Since at each iteration in the construction phase, the total number of subsets (e.g.nodes in the graph of Step 1) decreases by one, it is easy to see that after jV j ¡ kiterations a k-partition is obtained. Thus, to create an assignment of j Dj departmentsto k ¯ oors will require exactly jDj ¡ k iterations.

For our running example, GRASP (with an iteration limit of 50, pi ˆ 0:5 for all i)obtained the following assignment of departments to ¯ oors: Floor 1 ˆ (1, 5, 12, 14,15), Floor 2 ˆ (2, 3, 4, 9, 10), and Floor 3 ˆ (6, 7, 8, 11, 13) with space requirements74, 47, and 70 units respectively. This assignment yields a vertical cost (i.e. inter-¯ oorcost) of $48875, compared with the vertical cost of $94 000 of the ® nal solutionobtained by MULTIPLE. This represents a reduction in the cost of vertical trans-port of roughly 48%. (In fact, in the 50 iterations, GRASP obtained two uniqueassignments with the same vertical cost.) Note that the above are not to be inter-preted as layout sequences, rather they only represent the assignment of departmentto ¯ oors. Determining layout sequences for each ¯ oor is done at the next stage.

4.2. A space ® lling curve representation of a layoutIn MULTIPLE, SABLE, STAGES, and FLEX, space ® lling curves are generated

for each ¯ oor of the facility. Given a particular curve and the space requirements ofeach department, any particular sequence of the departments gives rise to a uniquelayout for the corresponding ¯ oor. Thus, for a k-¯ oor facility, a layout can bespeci® ed as a collection of k layout-sequences (one for each ¯ oor). Each elementin a layout sequence corresponds to a particular department. Further eachdepartment must be represented in exactly one sequence. A layout sequence isfeasible if there is su� cient space in the corresponding ¯ oor to satisfy the spacerequirements of the applicable departments.

A space ® lling curve can be thought of as a special way of ordering of grid cells.For instance, if we consider a rectangular grid of dimension 5 15, we can ordereach of the 75 cells as is shown in ® gure 2. Assume that we wish to layoutdepartments 1± 6 using the following layout sequence; (1, 5, 3, 4, 2, 6). (We


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assume the minimum area requirement for each department. See table 1.) We cannow determine a layout in the following way:

Assign department 1 to cells 1± 12, department 5 to the next 7 cells (i.e. cells 13±19) and so on, until all six departments are assigned. The resulting layout is illu-strated in ® gure 3.

4.3. Tabu searchTabu search (TS) is one of the successful arti® cial intelligence techniques used for

solving combinatorial optimization problems (Glover 1986). The fundamental prin-ciples underlying TS are fully described by Glover (1989, 1990a). A tutorial and a listof successful applications is given by Glover (1990b).

Brie¯ y described, TS is an iterative search procedure that moves from one fea-sible solution to another. After a move is made, it is classi ® ed as a forbidden, i.e.tabu’ , for a certain number of iterations in the future. The primary purpose ofassigning a tabu’ status is to prevent cycling. The set of tabu moves is recordedby means of a tabu list (which has a given tabu_size). The key idea in TS is that themove chosen at each iteration may or may not be an improving one (i.e. a cost savingmove). The selection of non-improving moves is what drives the TS strategy awayfrom getting trapped in locally optimal solutions. If, after making a move, a solutionis found which is better than all solutions found in prior iterations then the newsolution is saved as the `best’ solution.

In the following we describe our implementation of TS to the multi-¯ oor layoutproblem.

4.3.1. Allowable movesAs described above, by assigning one space ® lling curve to each of the k ¯ oors of

the facility, all layouts that we consider are uniquely determined by one set of klayout sequences. The set of moves that we consider in our interchange method canbe described as either:

(i) an exchange of a pair of elements, or(ii) a shift (i.e. removal and subsequent insertion) of a single element.

For example, consider a 2-¯ oor/7-department case with current layout de® ned bythe pair of layout sequences L 1 ˆ … 1; 2; 3; 4† and L 2 ˆ … 5; 6; 7† . A move of the ® rst


Figure 2. A space ® lling curve.

Figure 3. Layout sequence (1, 5, 3, 4, 2, 6) using minimal area requirements.

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type is demonstrated by exchanging elements `1’ and 6’ giving rise to the new layoutL 1 ˆ … 6; 2; 3; 4† and L 2 ˆ … 5; 1; 7† .

From this new layout we can demonstrate a move of the second type by shiftingelement 3’ to be the ® rst element of the second sequence giving the new layoutL 1 ˆ … 6; 2; 4† and L 2 ˆ … 3; 5; 1; 7† .

Shift moves are very important in the case where the current layout is feasible buttotal transportation cost could be lowered further if one department, for example, is(1) placed in a di� erent position on the same ¯ oor or (2) moved to another ¯ oorcompletely. In the ® rst case, the situation may only get recti® ed after a series ofexchange moves. In the second case, the situation will never change if only exchangemoves are used. Only with shift moves can one change the number of departments ineach individual layout sequence.

Note that both an exchange and a shift move can also be made on the same ¯ oor.Making such moves will always be feasible (i.e. the ¯ oor capacity constraint satis-® ed), whereas moves of both types between di� erent ¯ oors need to be checked forfeasibility ® rst.

Similar to the other multi-¯ oor heuristics a move is evaluated by summing thehorizontal transportation cost on each ¯ oor and the vertical transportation costbetween ¯ oors. The distance is computed as the centroid± centroid distance. Whendepartments are on di� erent ¯ oors the minimal cost is used (e.g. the lift that mini-mizes the total distance is selected).

4.3.2. Tabu listsAfter a move is chosen and executed in a typical iteration of TS, the tabu list of

forbidden moves is updated. This list is based on the past history of moves. In ourimplementation, and given that we have two types of moves (exchange and shift) thatare quite di� erent in nature, it was decided to keep a tabu list for each type of move.Initially, a tabu_size is set for each list. When a move is made, say an exchange, thetabu list value of the move is updated:

tabu_list_exchange[move x]ˆ iteration # + tabu_size_exchange,

which will prevent the exchange move x from being repeated again for tabu_sizenumber of iterations in the future. Of course, the tabu lists are initialized to zero atthe start of the TS procedure.

The only way that it is possible to make a move that is tabu’ , is for the moveto satisfy some `aspiration level’ criterion. In our implementation we employ thefollowing simple criterion, a move on the tabu list satis® es the aspiration level’criterion if the resulting cost value is better than the best cost value found thus far.

4.3.3. The TS methodThe TS method is summarized below. The notation used includes iter (iteration

number); max_iter (maximum number of iterations); random (a randomly generatednumber between 0 and 1); and prob_exchange (probability to choose exchangemoves).Step 1. Use the initial layout from phase I (see } 5 below) as the starting solution.

current_costˆ best_cost ˆ cost of initial layoutStep 2. IF iter < iter_max

Go to step 3


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ELSEReport best_cost solution and STOP.

Step 3. random ˆ rand(0, 1)IF random < prob_exchange

Find the best exchange move by evaluating all possible feasible pairwiseexchanges of departments on the same ¯ oor as well as between ¯ oors. Thebest exchange move is the one with the lowest total transportation cost.best move ˆ best exchange moveIF best exchange move is non-improving

Find the best shift movebest move ˆ min(best exchange move, best shift move)

ELSEFind the best shift move by evaluating all possible feasible repositioning ofdepartments on the same ¯ oor as well as on another ¯ oor. The best shiftmove is the one with the lowest total transportation cost.best move ˆ best shift move

IF best shift move is non-improvingFind the best exchange movebest move ˆ min(best exchange move, best shift move)

Step 4. Execute the best movecurrent_costˆ cost of layout resulting from applying the best move

Step 5. IF current_cost < best_costbest_cost ˆ current_cost

Step 6. iter ˆ iter ‡ 1; Go to step 2.

In step 3, if prob_exchange is equal to one (zero), then every iteration of TS willonly consider exchange (shift) moves. If prob_exchange takes a value between zeroand one then this implies that a combination of both types of moves is considered.

5. The layout heuristics

The ® rst proposed heuristic, GRASP/TS, is a two-stage heuristic. The output ofthe ® rst phase is an initial layout that will then be improved in the second stage. Thetwo stages can be summarized as follows.

Stage 1. Obtain an initial layout using GRASP

Step 1. Generate a graph G ˆ … V ; E† . The nodes V represent the departments.For each pair of departments that have non-zero ¯ ow between them, thereis an edge … e 2 E† representing the total (e.g. the sum of the ¯ ows in eachdirection) ¯ ow.

Step 2. Apply the modi® ed-GRASP algorithm as described in } 4.1.1 to ® nd aninitial partition of the departments. At this point we know that all depart-ments in a given subset of the partition will appear on the same ¯ oor in theinitial layout of stage 2.

Step 3. The subsets (e.g. groups of departments on the same ¯ oor) will now beassigned to speci® c ¯ oors. This can be done by enumerating the k! di� erentpossibilities and assessing the total inter-¯ oor ¯ ows’ where the ¯ ow is theproduct of the number of units ¯ ow times the number of ¯ oors traversed.(Since the number k is small this step requires only a small number of costevaluations.)


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Step 4. For each ¯ oor, determine the initial layout sequence as follows.Select a department, say v, that is incident with the largest ¯ ow with anotherdepartment on the same ¯ oor.Initialize the layout sequence by assigning v to the layout sequence.Repeat until all departments on the ¯ oor are assigned.

Select an unassigned department, say d, with the largest ¯ ow with anyalready assigned department. Let u denote the assigned department withthe largest ¯ ow with d.Assign d to the end of the layout sequence closest to u.

This simple heuristic crudely attempts to generate a layout sequence withminimal inter-¯ oor ¯ ow.

Stage 2. Improve the layout using tabu searchApply the tabu search procedure as described in } 4.3 to try to improve the initial

layout resulting from Stage 1.The second proposed layout heuristic, FAF/TS, is also a two-stage heuristic.

Whereas GRASP ® nds an approximate solution, the ® rst stage FAF, obtainsan exact solution to the graph partition problem using mixed integer linear program-ming. The FAF formulation is thoroughly described by Meller and Bozer (1997).The second stage applies the same tabu search procedure as is done in GRASP/TS.

6. Computational experiments

In order to assess the e� ectiveness of the new approach, experiments were con-ducted on the benchmark suite used in Meller and Bozer (1996, 1997). The testproblems consist of 11, 12, 21, and 40 departments. Since there may be more thanone problem with the same number of departments, a hyphenated notation is used.For example 21-4-1 means it is the ® rst data set with 21 departments and four ¯ oors.The data sets are used with varying ratios of vertical (V) to horizontal (H) unithandling cost. Initially, the V/H ratio used is 5:1. Later on in the analysis, a lowerV/H ratio of 1:1 (indicated by the letter L), and a higher V/H ratio of 20:1 (indicatedby the letter H) is evaluated. For more detailed description of the data sets and theparameters used, please refer to Meller and Bozer (1997). The test data is availableon Meller’s web site at: http://www.eng.auburn.edu/~rmeller or can be obtained fromthe authors.

The purpose of conducting the experiments is as follows. First, we want tocompare the performance of our proposed heuristic GRASP/TS to SABLE, thebest known single-stage approach to date. Comparison to MULTIPLE will not bemade since Bozer and Meller (1996) established that SABLE outperformedMULTIPLE. Second, we want to compare the performance of FAF/TS toSTAGES and FLEX, the only two-stage approaches to date. Bozer and Meller(1997) have shown that STAGES outperforms SABLE and FLEX, given equalruntimes and unit vertical handling costs that are greater than or equal to horizontalunit handling costs. The reason for comparing FAF/TS to FLEX, even though it isoutperformed by STAGES, is to assess the value of allowing the initial departmentsto ¯ oor assignments to change. STAGES does not allow changing the initial depart-ments to ¯ oor assignments obtained by FAF, whereas FAF/TS and FLEX allowsuch changes. Finally, we compare the performance of FAF/TS to GRASP/TS.


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In both GRASP/TS and FAF/TS the tabu search parameters were: numberof iterations ˆ 200, shift : exchange ˆ 50 : 50, and tabu list size ˆ 20. For all datasets the following GRASP parameters were use: number of iterations ˆ 50 andprobability of acceptance ˆ 0.30. The probability of acceptance was chosen as 0.3in order to generate su� cient randomness of solutions over the 50 iterations. Highervalues would generate fewer unique samples, and values less than 0.3 lower theprobability that a good choice will be selected. The value of 0.3 was chosenempirically motivated by the above mentioned points.

As was mentioned earlier in this paper, facility layout decisions are madeinfrequently and usually incur high costs. The focus of this study was the qualityof the ® nal solution (total transportation cost) as opposed to the time required toreach that solution. Nevertheless, times were recorded in table 3 for the record, andto give a general idea of the amount of computational e� ort needed for both stages 1and 2 of the GRASP/TS heuristic.

6.1. Analysis of resultsThe results for running SABLE, FLEX, STAGES, GRASP/TS, and FAF/TS for

all the data sets are given in tables 4 and 6. The percentages over the best knownsolution to date (i.e. lowest total transportation cost) for each data set are calculatedin tables 5 and 7. Best known solutions were either found at http://www,eng.auburn.edu/~rmeller or were obtained as a result of running GRASP/TS or FAF/TS. The entries in tables 4 and 6 that result in the best known solution are in bold.The entries marked by an * denote new best known solutions that were obtainedeither by GRASP/TS or FAF/TS. Table 8 provides a comprehensive summary list ofall the data sets, the best known solution for a given data set, and the heuristicapproaches that achieved the best known solution. The best known solution isrepresented by the cost column, which re¯ ects the total transportation cost, and


Number of departments

Phase 11 12 21 40

GRASP 0.04 0.04 0.37 2.48TS 0.44± 0.60 0.55± 0.60 4.34± 7.20 54.73± 62.25

Total 0.48± 0.64 0.59± 0.64 4.71± 7.57 55.21± 64.73

Table 3. Computing times in seconds for GRASP/TS.

Data set SABLE (best of 5) FLEX STAGES GRASP/TS FAF/TS

11-2-1 8477.3 8539.0 8477.3 8477.3 8477.3

11-2-2 2501.8 2501.8 2582.4 2572.2 2572.212-3 1513.2 1513.2 1513.2 1513.2 1513.2

21-4-1 16 188.5 15 895.5 14 505.5 15 653.8 14 505.5

21-4-2 11 971.0 12 656.3 11315.0 11 694.3 11 259.0*

21-4-3 10 835.8 10 966.8 10678.2 10 910.5 10 608.2*

21-4-4 8834.8 8658.3 8556.3 8892.3 8497.7*

40 20 988.5 22 330.0 19807.9 20 158.2 19 184.7*

Table 4. Solution values for 5:1 cost ratio.

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the layout column, which represents the layout sequence of each ¯ oor. For example,data set 21-4-1 with V/H ratio of 5:1 achieved the best known solution (lowest cost)with STAGES and FAF/TS. The total transportation cost was 14505.5. The layoutsequences for the ¯ oors are arranged from ® rst to fourth, so that for the ® rst ¯ oor itis 20, 19, 21 and for the fourth ¯ oor it is 10, 12, 7, 6. An alternate layout for thefourth ¯ oor (as obtained by FAF/TS) is 10, 6, 7, 12.

Golden and Stewart (1985) suggested the use of the Wilcoxon signed-rank test,which is a non-parametric test, to compare empirical data obtained from heuristics.We used the test in comparing the heuristics on all the data sets. Statistical signi® -cance was determined at % ˆ 0.05. The results are presented below.



11-2-1 Ð 0.73 Ð Ð Ð11-2-2 0.32 0.32 3.55 3.14 3.1412-3 Ð Ð Ð Ð Ð21-4-1 11.60 9.58 Ð 7.92 Ð21-4-2 6.32 12.41 0.50 3.87 Ð21-4-3 2.15 3.38 0.66 2.85 Ð21-4-4 3.97 1.89 0.69 4.64 Ð40 9.40 16.39 3.25 5.07 Ð

Table 5. Percentage over best known solution value for 5:1 cost ratio.


21-4-1 H 27 298.0 31 162.0 26 456.0 32 373.8 26 768.121-4-2 H 20 918.0 19 410.0 18 054.7 19 719.3 18 234.021-4-3 H 23 305.7 24 076.0 22 777.7 23 923.0 22 645.7*

21-4-4 H 22 297.0 21 783.7 20 375.0 19 242.3* 19 972.740 H N/A N/A N/A 38 300.2 37 067.7*

21-4-1 L 11 269.5 11 708.5 11 275.0 11 305.5 11 230.5

21-4-2 L 9948.7 11 673.0 9399.0 9446.0 9399.0

21-4-3 L 6986.2 7369.5 7546.2 6785.5* 7025.521-4-4 L 5466.3 6615.0 5500.3 5515.5 5285.3*

40 L N/A N/A N/A 15 263.9 13 837.9*

Table 6. Solution values for high and low ratio data sets.


21-4-1 H 3.18 17.79 Ð 22.37 1.1821-4-2 H 15.86 7.51 Ð 9.22 0.9921-4-3 H 2.91 6.32 0.58 5.64 Ð21-4-4 H 15.87 13.21 5.89 Ð 3.8040 H N/A N/A N/A 3.32 Ð21-4-1 L 0.35 4.26 0.40 0.67 Ð21-4-2 L 5.85 24.19 Ð 0.50 Ð21-4-3 L 2.96 8.61 11.21 Ð 3.5421-4-4 L 3.42 25.16 4.07 4.36 Ð40 L N/A N/A N/A 10.31 Ð

Table 7. Percentage over the best known solution for high and low ratio data sets.

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Data set Cost Layout Heuristic

11-2-1 8477.3 9,6,2,5,1,10,11/ SABLE, STAGES,3,4,8,7 GRASP/TS, FAF/TS

11-2-2 2493.9 9,7,3,11/ Unknown4,8,1,10,6,5,2

12-3 1513.2 7,6,10,1,12/ All4,5,8,2/(alternateÐ 4,5,2,8) (alternate ˆ GRASP/TS)11,9,3

21-4-1 14 505.5 20,19, 21/ STAGES, FAF/TS1,11,5,8,9,13,17/ (alternateÐ 1,5,11,8,17,13,9) (alternate ˆ FAF/TS)2,4,3,14,16,15,18/ (alternateÐ 2,4,3,15,16,14,18)10,12,7,6 (alternateÐ 10, 6, 7, 12)

21-4-2 11 259.0 18,17,12,20,21/ FAF/TS2,16,14,6,9,11/3,5,19,7,15/10,1,8,13,4

21-4-3 10 608.2 11,17,1,4,21/ FAF/TS12,19,14,18,10,13/6,16,20,15,9,7/8,2,5,3

21-4-4 8497.7 10,5,21/ FAF/TS13,20,17,16,19,14,11/18,12,4,6,8,3/9,2,15,1,7

40 19 184.7 28,10,9,1,4,3,30,40/ FAF/TS2,36,11,5,34,29,6,23,8 /27,25,31,37,32,33,39,38,17,15,24,16,7,26/35,22,

21,20,12,14,13,18,1921-4-1 H 26 456.0 N/A STAGES21-4-2 H 18 054.7 N/A STAGES21-4-3 H 22 645.7 11,17,1,4,21/ FAF/TS

12,19,14,18,10,13/6,16,20,15,9,7/8,2,5,3

21-4-4 H 19 242.3 10,5,14,21/ GRASP/TS19,20,16,17,8,13 /18,11,12,4,15,6,3 /7,1,2,9

40 H 37 067.7 28,10,9,1,4,3,30,40/ FAF/TS2,36,11,5,34,29,6,23,8,12/27,25,31,37,32,33,39,38,17,15,24,16,7,26/35,18,22,21,20,14,13,19

21-4-1 L 11 230.5 21/ FAF/TS1,5,8,13,9,17/11,4,3,14,16,15,2,18/10,6,7,12,19,20

21-4-2 L 9399.0 18,17,12,20,21/ STAGES, FAF/TS2,16,14,6,9,11/3,5,19,7,15/10,1,8,13,4

21-4-3 L 6785.5 5,3,10,21/ GRASP/TS12,14,18,4,19,13 /6,16,20,11,17,1 /15,9,7,8,2

21-4-4 L 5285.3 10,5,21/ FAF/TS9,13,20,17,16,14,19/18,11,12,8,6,4/2,1,15,7,3

40 L 13 837.9 10,4,1,9,12,3,40/ FAF/TS35,2,11,5,15,23,6,29,8 /7,24,25,37,32,34,33,39,38,36,27,16,17,26/20,21,22,31,30,28,14,13,19,18

Table 8. Best known solution summary.

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(1) In comparing SABLE with GRASP/TS on 16 problems, the Wilcoxon testconcludes that both heuristics give equally accurate results (costs) i.e. SABLEdoes not outperform GRASP/TS and GRASP/TS does not outperformSABLE. The average of the percentage over the best known solution is5.26 for SABLE and 4.39 for GRASP/TS. The standard deviation of thepercentage over the best known solution is 5.28 for SABLE and 5.61 forGRASP/TS. So the di� erence in average and standard deviation is small.A potential advantage of GRASP/TS over SABLE, however, is the compu-tational e� ort needed to run large problems with SABLE. Since this is not thefocus of this paper, readers are referred to Meller and Bozer (1997), who listaverage runtimes of SABLE for all problems. An important point to remem-ber is that several runs of SABLE had to be made to remove the bias of theinitial layout. Tables 4 and 6 list the best of ® ve runs. This implies that theaverage runtime must be multiplied by 5 to re¯ ect a fair comparison ofSABLE with GRASP/TS in terms of computational e� ort.

(2) In comparing FAF/TS with STAGES on 16 problems, the Wilcoxon testconcludes that FAF/TS outperforms STAGES. The average of the percen-tage over the best known solution is 0.79 for FAF/TS vs 1.92 for STAGES.The standard deviation of the percentage over the best known solution is 1.39for FAF/TS vs 3.08 for STAGES. The average and standard deviation forSTAGES are more than twice those of FAF/TS.

(3) In comparing FAF/TS with FLEX on 16 problems, the Wilcoxon test con-cludes that FAF/TS outperforms FLEX. The average of the percentage overthe best known solution is 0.79 for FAF/TS vs 9.48 for FLEX. The standarddeviation of the percentage over the best known solution is 1.39 for FAF/TSvs 8.12 for STAGES. The average and standard deviation for FLEX aremuch higher than those of FAF/TS. Note that both FAF/TS and FLEXcan alter the optimal department to ¯ oor assignments obtained by FAF.

(4) In comparing FAF/TS with GRASP/TS on 18 problems (the 40-L and 40-Hare included), the Wilcoxon test concludes that FAF/TS outperformsGRASP/TS. This was expected, since FAF obtains optimal department to¯ oor assignments yielding a better initial layout than one that is obtainedheuristically by GRASP. The average of the percentage over the best knownsolution is 0.7 for FAF/TS vs 4.66 for GRASP/TS. The standard deviation ofthe percentage over the best known solution is 1.33 for FAF/TS vs 5.46 forGRASP/TS.

Other interesting observations include the following:

(5) In the 16 tests (excluding 40-L and 40-H) FAF/TS found the best knownsolution a total of 11 times, vs six for STAGES, one for FLEX, two forSABLE, and four for GRASP/TS. For six of the 12 lower ratio tests (i.e.1:1 and 5:1, and excluding 40-L) FAF/TS was the only heuristic to ® nd thebest known solution. For two of the four high ratio (i.e. 20:1) tests STAGESwas the only heuristic to ® nd the best known solution. One possible explana-tion for the improved relative performance of STAGES is that the higher theV/H ratio, the more likely the optimal layout corresponds to a layout wherethe overall vertical transportation cost is minimized. STAGES restricts itselfto only considering layouts with this property, whereas FLEX and FAF/TS


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consider a larger search space by permitting departments to move to other¯ oors.

(6) FAF/TS was never more than 3.8% from the best known solution. STAGESwas also robust except for two cases e.g. 5.89% (21-4-4H) and 11.21%(21-4-3 L). FLEX was the least robust resulting in several problems withlarge percentages e.g. 12.41% (21-4-2), 16.39% (40), 17.79% (21-4-1-H),13.21% (21-4-4-H), 24.19% (21-4-2-L), and 25.16% (21-4-4-L).

(7) In a `head-to-head’ comparison between FAF/TS and STAGES, FAF/TShad a 10-2-4 `won-lost-tied’ record.

(8) FAF/TS gave better results than GRASP/TS for all 40 department data sets.

7. Conclusions

In this paper a pair of two-stage heuristics, GRASP/TS and FAF/TS, for themulti-¯ oor facility layout problem were introduced. The ® rst stage generates aninitial layout that is used in the second stage, which is based on tabu search.

Computational results indicate that these heuristics are competitive with otherpublished heuristics. FAF/TS obtained several new best known solutions on abenchmark suite of data sets. The bene® t of ® nding an initial layout that minimizesinter-¯ oor transportation costs has been demonstrated by the superiority of two-stage approaches (FAF/TS and STAGES) over single-stage approaches (SABLE).Statistical results show that FAF/TS outperforms STAGES. As the second stage ofthese heuristics start with the same department to ¯ oor assignments, one may con-clude that tabu search is more successful than simulated annealing at reaching a ® nalsolution.

In the ® rst stage of FAF/TS and STAGES, the computational e� ort required byan exact algorithm (i.e. FAF) may become prohibitive as the number of departmentsincreases. An area of potential research is to develop other e� cient heuristics, suchas GRASP, to ® nd assignments of departments to ¯ oors such that inter-¯ oor trans-portation costs are minimized.

Acknowledgement

We would like to thank the referees for their careful review of the paper and theirhelpful suggestions.

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