Global Approaches for Facility Layout and VLSI Floorplanninge- Approaches for Facility Layout and VLSI Floorplanning 3 cialized version of facility layout. The objective is typically

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  • Global Approaches for Facility Layout andVLSI Floorplanning

    Miguel F. Anjos1 and Frauke Liers2

    1 Department of Mathematics and Industrial Engineering, Ecole Polytechnique deMontreal, Montreal QC, H3C 3A7, Canada anjos@stanfordalumni.org

    2 Universitat zu Koln, Institut fur Informatik, Koln, Germanyliers@informatik.uni-koeln.de

    1 Introduction

    The facility layout problem consists of partitioning a rectangular facility ofknown dimensions into departments with a given (fixed) area so as to min-imize the total cost associated with the (known or projected) interactionsbetween these departments. This cost is modeled as the weighted sum of thecenter-to-center distances between all pairs of facilities, where the pairwiseweights reflect transportation costs and/or adjacency preferences between de-partments. If the height and width of the departments vary, then finding theiroptimal (rectangular) shapes is also a part of the problem. This is a hard prob-lem, in particular because any desirable layout must have no overlap amongthe areas of the different departments. Versions of the facility layout problemoccur in many environments, such as flexible manufacturing and service centerlayout, as well as in other engineering applications, such as the design of VeryLarge Scale Integration (VLSI) circuits. Nearly all of the resulting problemsare known to be NP-hard.

    Two types of approaches capable of yielding provably optimal solutionshave been proposed in the literature. The first type are graph-theoretic ap-proaches which assume that the desirability of locating each pair of facilitiesadjacent to each other is known. Initially, the area and shape of the depart-ments are ignored, and each department is simply represented by a node in agraph. Adjacency relationships between departments can now be representedby arcs connecting the corresponding nodes in the graph. The objective is thento construct a graph which maximizes the weight on the adjacencies betweennodes. We refer the reader to Foulds [27] for more details. The second typeof these approaches are mathematical programming formulations with objec-tive functions based on an appropriately weighted sum of centroid-to-centroiddistances between departments. Exact mixed-integer linear programming for-mulations were proposed in Montreuil [54], Meller et al. [53], and Sheraliet al. [64]. Non-linear programming formulations include the work of Castillo

  • 2 Miguel F. Anjos and Frauke Liers

    et al. [18, 19]. Most recently, Meller et al. [51] solved problems up to 11 de-partments to global optimality.

    Thus, most of the approaches in the literature that tackle realisticallysized problems are based on heuristics with no guarantee of optimality. Theseinclude genetic algorithms, tabu search, simulated annealing, fuzzy logic, andmany others. Mathematical-programming-based heuristics can also be used toprovide high-quality solutions for large problems with 20 or more departments[9, 17]. We refer the reader to the extensive bibliographies in the survey papersof Meller and Gau [52], Mavridou and Pardalos [50], and Singh and Sharma[67].

    The survey of Meller and Gau [52] divides the research papers on facilitylayout into three broad areas. The first is concerned with algorithms for tack-ling the general layout problem as defined above. The second area is concernedwith extensions of the problem in order to account for additional issues whicharise in applications, such as designing dynamic layouts by taking time depen-dency issues into account; designing layouts under uncertainty conditions; andachieving layouts which optimize two or more objectives simultaneously. Thethird area is concerned with specially structured instances of the problem.One such special case that has been extensively studied occurs when all thefacilities have equal dimensions and the possible locations for the departmentsare given a priori; this is the quadratic assignment problem (QAP) formulatedby Koopmans and Beckman [42]. Since the possible locations are fixed, theproblem reduces to optimize a quadratic objective over all possible assign-ments of departments to locations. The QAP is NP-hard, and is in general adifficult problem to solve. (Refer to chapter(s) on QAP/Comb Problems.)

    In this chapter we will be concerned with two layout problems to whichcone optimization approaches have been successfully applied. The first one isa specially structured instance of facility layout, namely the single-row facilitylayout problem (SRFLP). The SRFLP consists of arranging a given number ofrectangular facilities next to each other along a line so as to minimize the totalweighted sum of the center-to-center distances between all pairs of facilities.This problem is also known in the literature as the one-dimensional spaceallocation problem; see, e.g., [58]. The SRFLP also has several interestingconnections to other known combinatorial optimization problems such as themaximum-cut problem, the quadratic linear ordering problem, and the lineararrangement problem. We explore some of these connections in Section 2.3.Note that for the SRFLP, numerous approaches based on linear programming(LP) have been proposed in the literature. Using integer LP methods, themost effective such approach for determining exact solutions is a branch-and-cut algorithm based on the recent model by Amaral [5].

    The second problem arises from the application of facility layout to fixed-outline floorplanning in VLSI circuit design. Fixed-outline floorplanning con-sists of arranging a set of rectangular modules on a rectangular chip area sothat an appropriate measure of performance is optimized, and is hence a spe-

  • Global Approaches for Facility Layout and VLSI Floorplanning 3

    cialized version of facility layout. The objective is typically to minimize totalwire length.

    The impact to date of the cone optimization approaches for facility layoutproblems, excluding its impact for the QAP addressed elsewhere in this book,can be summarized as follows:

    Anjos and Vannelli [10] used a semidefinite programming (SDP) relaxationto provide globally optimal solutions for SRFLP instances in the literaturewith up to 30 departments that had remained unsolved for nearly 20 years.

    Hungerlander and Rendl [38] show that the SDP relaxation, when aug-mented with valid inequalities and optimized using a suitable combinationof algorithms, can provide global optimal solutions for SRFLP instanceswith up to 40 departments. Their approach can solve larger instances tooptimality than the approach by Amaral [5].

    Anjos and Yen [11] obtained tight global bounds for SRFLP instanceswith up to 100 facilities using an alternate SDP relaxation. Their approachconsistently achieves optimality gaps not greater than 5%.

    Takouda, Anjos and Vannelli [69] provided non-trivial lower bounds for theVLSI macrocell floorplanning problem. Furthermore, their second-ordercone programming (SOCP) formulations (described in Section 3.1) for thearea and aspect ratio constraints were key to the success of the two-stageconvex optimization-based methodology for floorplanning of Luo et al. [49]and of the facility layout model of Jankovits [39], both of which providecomputational results outperforming those previously reported in the lit-erature.

    This chapter provides an overview of the SOCP/SDP models underpinningthis impact, and concludes with a summary of further research directions.

    2 Introduction to the Single-Row Facility LayoutProblem

    An instance of the SRFLP is formally defined by n one-dimensional facilitieswith given positive lengths `1, . . . , `n, and pairwise non-negative weights cij .The objective is to arrange the facilities so as to minimize the total weightedsum of the center-to-center distances between all pairs of facilities. If all thefacilities have the same length and all the non-zero weights are equal, theSRFLP becomes an instance of the linear arrangement problem, see e.g. [46],which is itself a special case of the QAP; see e.g. [20]. (For a survey on thelinear arrangement and other graph layout problems, we refer to [26].) Sev-eral practical applications of the SRFLP have been identified in the literature,such as the arrangement of departments on one side of a corridor in super-markets, hospitals, or offices [65], the assignment of disk cylinders to files[58], the assignment of airplanes to gates in an airport terminal [68], and thearrangement of machines in flexible manufacturing systems, where machines

  • 4 Miguel F. Anjos and Frauke Liers

    within manufacturing cells are often placed along a straight path travelled byan automated guided vehicle [36]. We refer the reader to the book of Heragu[34] for more information.

    Several heuristic algorithms for the SRFLP have also been proposed. Wepoint out the early work of Hall [30], the application of non-linear optimiza-tion methods by Heragu and Kusiak [37], the simulated annealing algorithmsproposed independently by Romero and Sanchez-Flores [61] and Heragu andAlfa [35], a greedy heuristic algorithm proposed by Kumar et al. [43], andthe use of both simulated annealing and tabu search by de Alvarenga et al.[23]. However, these heuristic algorithms do not provide a guarantee of globaloptimality, or an estimate of the distance from optimality.

    Simmons [65] was the first to state and study the SRFLP, and proposeda branch-and-bound algorithm. His subsequent note [66] mentioned the pos-sibility of extending the dynamic programming algorithm of Karp and Held[40] to the SRFLP, which was done by Picard and Queyranne [58]. Differentmixed-integer LP models have been proposed [2, 3, 37, 48], and the first poly-hedral study of the so-called distance polytope for SRFLP was carried outby Amaral and Letchford [6]. Most recently, the IP-based cutting-plane algo-rithm of Amaral [5] can compu