This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and

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2 European Journal of Operational Researc 04 (010) Contents lists available at ScienceDirect European Journal of Operational Researc journal omepage: Production, Manufacturing and Logistics Extensions to STaTS for practical applications of te facility layout problem Daniel Scolz a, *, Florian Jaen b, Andreas Junker c a Institute of Operations Researc, Tecnisce Universität Darmstadt, 6489 Darmstadt, Germany b Institute of Information Systems, University of Siegen, Siegen, Germany c Business Department, University of Siegen, Siegen, Germany article info abstract Article istory: Received 1 Marc 009 Accepted 7 November 009 Available online 14 November 009 Keywords: Facility layout problem Layout planning Slicing trees Tabu searc We consider a very general case of te facility layout problem, wic allows incorporating various aspects appearing in real life applications. Tese aspects include loose requirements on facilities footprints, eac of wic only needs to be of rectangular sape and can optionally be restricted concerning te surface area or te aspect ratio. Compared to former approaces oter generalizations of practical relevance are multiple, not necessarily rectangular worksops, exclusion zones in worksops, predefined positions of facilities, te consideration of aisles, and te aderence of furter restrictions suc as te enforced placement of certain facilities next to an exterior wall or a minimum distance between certain pairs of facilities. Altoug different objectives could be applied, we especially focus on te most relevant one in practice, te minimization of transportation costs. We sow tat tis problem can euristically be solved using an extension of te Slicing Tree and Tabu Searc (STaTS) based approac. Te application of tis algoritm on practical data sows its effectiveness. Te paper concludes wit a step-by-step guide for te application of STaTS in practice. Ó 009 Elsevier B.V. All rigts reserved. 1. Introduction and problem description 1.1. Te facility layout problem A facility layout problem (FLP) is concerned wit te question of positioning facilities to locations. Tis very general problem is usually specified in different ways. In te past many approaces for te numerous variants of layout planning ave been developed. Due to te complexity and peculiarities of te problem, many of tese approaces are based on simplifying assumptions. However, a large number of practical applications does not fulfill all assumptions proposed by te various models. In order to overcome te disadvantages of a too specified model, let us propose a more general framework of a FLP, representing a larger number of practical specifications at te cost of te simplicity of te model. Tere are n rectangular facilities eac of wic wit given surface area and a given set of allowed sapes. Tese facilities are to be positioned in one out of several rigt-angled worksops wit given dimensions so tat no two facilities overlap. Alternatively te dimension of a worksop can also be unrestricted wic leads to an arrangement of facilities relative to eac oter. * Corresponding autor. Tel.: ; fax: addresses: scolz@bwl.tu-darmstadt.de (D. Scolz), florian.jaen@unisiegen.de (F. Jaen), andreas.junker@uni-siegen.de (A. Junker). Tis framework requires not only deciding on te position of te facilities, but also on teir exact sape. Let us formalize tis problem using te Euclidean plane R wit te Cartesian coordinate system. Te abscissa will be denoted by x and te ordinate by y. Ten te following input data as to be given for te facility layout problem. A closed, not necessarily connected set W # R describing te available space for placing facilities. W represents te worksop(s). As mentioned above, we assume eac worksop to be of rigt-angled sape.te restriction of rigt-angled worksop sapes is not very restrictive as any closed subset of R can be approximated arbitrarily accurate by rigt-angled areas. However, for practical applications te number of corners sould not exceed a certain number, say 10. So prevalent sapes suc as rectangles, L-sapes, or Z-sapes are allowed. n facilities, eac of wic as a given surface area aðiþ R, for all i {1,,...,n}. For eac facility i, tere is a set S(i) of (rectangular) sapes tat can be adopted by tis facility. Tese sapes can be denoted by te potential positions of i s top rigt corner relative to te bottom left corner. Tat means, under te assumption tat te bottom left corner is placed on te origin (0,0), eac element of S(i) describes an allowed position of te top rigt corner, i.e. SðiÞ # fðx; yþ R jx y ¼ aðiþg /$ - see front matter Ó 009 Elsevier B.V. All rigts reserved. doi: /j.ejor

3 464 D. Scolz et al. / European Journal of Operational Researc 04 (010) Optionally, tere migt be furter restrictions concerning te placement of facilities or an objective function (bot to be described below). Given tese input data, te FLP is concerned wit te question of finding a position and a concrete sape for eac facility. Te position of te facility can be caracterized by te position of its bottom left corner, denoted by p i ¼ p i x ; pi y R. Te exact sape is ten determined by te position of te top rigt corner (in relation to te bottom left corner), denoted by s i ¼ s i x ; si y SðiÞ. Te expanse covered by a facility is given by EðiÞ :¼ ðx; yþ R jp i x 6 x < p i x þ si x ; pi y 6 y < pi y þ si yg. Te FLP is now to find a mapping / : f1; ;...; ng! R R ; /ðiþ# p i ; s i ; s:t: EðiÞ # W 8i f1; ;...; ng; EðiÞ\EðjÞ ¼; 8i j; i; j f1; ;...; ng: Te first restriction ensures tat te facilities are placed witin te worksops, wereas te second restriction proibits overlapping facilities. In case of unlimited worksop space, finding a feasible solution is easy. However, even if te available space in te worksop(s) considerably exceeds te sum of te required space of all facilities, a feasible solution migt not exist, or migt be ard to obtain. Several objectives ave been formulated for te FLP. Among te most common ones are te minimization of transportation costs between facilities or te minimization of te smallest rectangle containing all facilities (especially if W = R Þ. We will focus on te minimization of transportation costs as follows. For eac ordered pair (i,j) of facilities, a certain flow f ij R and material transportation costs per unit and distance c ij R are given. Furtermore a metric d : R R! R is given, wic allows us to determine te distance d ij between two facilities i and j by means of te distance of teir central points: d ij :¼ d s i x þ pi x ; si y þ pi y ; s j x þ pj x ; sj y þ pj y Note tat d ij itself defines a pseudo-metric on te set of connected subsets of R. Using tese data, te objective of te FLP is min F ¼ n i¼1 n j¼1 i j d ij f ij c ij : To te very best of our knowledge, no paper as been presented wic proposes a solution metod for tis general FLP. For W = R, te Slicing Tree and Tabu Searc (STaTS) based approac of Scolz et al. (009) can be applied. We will extend tis approac in order to cope wit rigt-angled worksops and in order to consider te following additional restrictions tat migt appear in real life situations. 1. Some facilities must be placed in certain areas of te worksops (e.g. because of different ceiling eigts, or an inventory must be placed at a certain point). If W(i) describes te area in wic facility i may be placed, we introduce te restriction EðiÞ # WðiÞ:. All facilities and worksop doors must be connected via aisles. As doors can be modeled as facilities wit a fixed position, we can w.l.o.g. assume tat only facilities ave to be connected via aisles. Tis leads to te restriction 8i; j f1; ;...; ng9d # W n [ EðkÞ : k EðiÞ[EðjÞ[D is a connected space:. For practical applications we migt need to force tat te aisles ave a minimum widt, i.e. only tose D can be cosen tat ensure te minimum widt A (e.g. for forklifts). 3. Some facilities must be placed next to an exterior wall (e.g. because of an exaust air conduit). Let Wall # f1; ;...; ng be te set of facilities to be placed next to a wall. Ten we force inf fdððx; yþ; ðx 0 ; y 0 ÞÞjðx; yþ EðiÞ; ðx 0 ; y 0 Þ R Wg ¼ 0 8i Wall: Note tat tis restriction makes only sense if W is bounded somewere. We ave used te same term d for te metric as in te objective function. However, it would be possible to apply different metrics. 4. For some pairs of facilities, a separation distance d min as to be respected. A reason for tat could be dust (oscillation) sensitive and dust raising (oscillation causing) macines. We consider tis fact using te following restriction in wic BadNeigbors f1; ;...; ng f1; ;...; ng describes all pairs of facilities wit separation distance. min fdððx; yþ; ðx 0 ; y 0 ÞÞjðx; yþ EðiÞ; ðx 0 ; y 0 ÞEðjÞg P d min 8ði; jþ BadNeigbors: Put togeter, we are to solve te following optimization problem: min F ¼ n d ij f ij c ij ; / i¼1 j¼1 i j s:t: /ðiþ ¼ p i x ; pi y ; s i x ; si y ; s i x ; si y SðiÞ; n o EðiÞ :¼ ðx; yþ R jp i x 6 x < pi x þ si x ; pi y 6 y < pi y þ si y ;!!! d ij ¼ d s i x þ pi x ; si y þ pi y ; s j x þ pj x ; sj y þ pj y ; EðiÞ # WðiÞ 8i f1; ;...; ng; EðiÞ\EðjÞ ¼; 8i j; i; j f1; ;...; ng; 8i; j f1; ;...; ng9d # W n [ k connected space; EðkÞ : EðiÞ[EðjÞ[D is a inf fdððx; yþ; ðx 0 ; y 0 ÞÞjðx; yþ EðiÞ; ðx 0 ; y 0 Þ R Wg ¼ 0 8i Wall; min fdððx; yþ; ðx 0 ; y 0 ÞÞjðx; yþ EðiÞ; ðx 0 ; y 0 ÞEðjÞg P d min 8ði; jþ BadNeigbors: Note tat tis problem formulation is not restricted to te positioning of macines witin manufacturing worksops. It can be applied to oter applications of FLPs, e.g. wen producing printed circuit boards. In suc a case te circuit board is regarded as a worksop and transistors, resistors, etc. correspond to facilities. Circuit pats can be modeled as aisles. 1.. Literature review Koopmans and Beckmann (1957) propose a quadratic assignment problem in wic n equal sized facilities ave to be placed on n equal sized locations. Te objective is to minimize transportation costs between facilities. Tis model is often referred to be te common classical layout planning problem. It is a special case of te above-mentioned problem. Any of te n equal sized locations can be modeled as a worksop, wic as te size of exactly one of te n facilities. As te quadratic assignment problem is NP-ard, our general FLP framework is NP-ard as well. Bazaraa (1975) stated a generalized quadratic assignment problem, wic incorporates facilities wit unequal areas. In is model,

4 D. Scolz et al. / European Journal of Operational Researc 04 (010) a single facility may be represented by a given number of multiple blocks resulting in a large number of blocks and problems assuring connectivity and given sapes of te facilities. Oter models presented in literature propose grap oriented approaces for solving te FLP. Again, te main drawback of tese approaces is tat geometric constraints, e.g. unequal sizes of facilities, cannot be considered sufficiently. Te dimensions of facilities are not incorporated during optimization, but tey ave to be considered afterwards wen te layout is constructed according to te optimal grap (see e.g. Goetscalckx, 199; Pesc et al., 1999; Osman, 006). Since te nineties, researc in tis area focused mainly on te unequal area rectangular FLP. Tam (199a,b) presented te concept of slicing trees in te context of FLPs. A slicing tree is a very useful representation especially for solutions of unequal area layout problems. But te compliance wit te facility s sape restrictions is not guaranteed by Tam s layout generation procedure, a disadvantage tat can be overcome by a layout generation procedure based on bounding curves. Tis metod was primarily used in VLSI (Very Large Scale Integration) design to determine te layout of microprocessors, but it can be adapted to solve te FLP. In VLSI design, bounding curves are used by Wong and Liu (1986, 1989) and Valenzuela and Wang (00). Anoter approac to te FLP is based on mixed integer programming (MIP) wit restrictions to prevent te overlapping of facilities. Up to now MIP models of te facility location problem ave been solved to optimality for only less tan ten facilities in a reasonable amount of time (Serali et al., 003; Castillo and Westerlund, 005; Castillo et al., 005). If facilities wit variable sapes are considered, we can classify te geometric caracteristics using four not necessarily disjoint types: Type 1: Facilities wit fully free aspect ratio, i.e. SðiÞ ¼fðx; yþ R jx y ¼ aðiþg. Type : Facilities wit fixed dimensions, i.e. js(i)j = 1. Type 3: Facilities wit n restricted range of feasible aspect ratios, o i.e. SðiÞ ¼ ðx; yþ R jx y ¼ aðiþ; a 6 x 6 b; a; b R. y Type 4: Facilities wit varying surface area wic can ave one out of several given implementations, i.e. SðiÞ fðx; yþ R g. Moreover it is distinguised weter facilities are allowed to be rotated by 90 (e.g. for type tis would imply js(i)j = ). Previously, te most common solution approaces only consider a subset of te types above. E.g. types, 3 and 4 are not considered directly in layout generation procedures of many slicing tree solution approaces as in Sayan and Cittilappilly (004), Gau and Meller (1999) and Tam (199a,b). In te solution approac to be described in te next section, all four types and te rotation by 90 may occur. Te peculiarities appearing in practice as described in te previous section can rarely be found in literature. To te best of our knowledge, tere is no combined approac considering all restrictions of our framework. For example aisles are often introduced after te relative position of te facilities as been determined (see e.g. Alagoz et al., 008).. Solution approac To solve a real world FLP we use te Slicing Tree and Tabu Searc (STaTS) based approac wic as been presented in Scolz et al. (009). We enance tis approac to fulfill additional requirements occurring in practice as described in te previous section..1. Slicing Tree and Tabu Searc based approac To make tis paper self-contained we explain in brief ow facility caracteristics are considered and ow layouts are stored and improved in STaTS. For te moment let us assume tat W ¼ R and no optional restrictions ave to be considered Slicing trees and slicing structures In STaTS, layouts are considered to be slicing structures. A slicing structure results from dividing an initial rectangle eiter in orizontal or vertical direction completely from one side to te oter (so-called guillotine cut) and recursively going on wit te newly generated rectangles. A slicing tree is a binary tree wic is used to represent suc a slicing structure. For a layout problem of n facilities, te slicing tree consists of n leaf nodes and n 1 internal nodes. Eac leaf node symbolizes a site for a facility, and eac internal node contains information about te direction of a guillotine cut (: orizontal, v: vertical). Eac internal node corresponds to te area in te layout of te rectangle enfolding bot of its cildren. In Fig. 1, a slicing tree and its corresponding slicing structure are sown. Te slicing direction of te root node is orizontally. Terefore, te first cut is dividing te layout completely from left to rigt. In te layout te facilities of te rigt subtree are placed above tose of te left one. Te rigt subtree starts wit a vertical cut, and consequently facility is arranged on its left and side and facilities 5 and 6 on te rigt and side. Wit te left subtree it is te same, but tere are two consecutive cuts wit te same direction, so facilities 1, 3, and 4 are placed side by side..1.. Geometric caracteristics of departments and layout determination Facilities can be caracterized as four basic rectangular types as listed in Section 1.. Eac of tese four types can be represented by a specific bounding curve even if te facilities orientations are free. A bounding curve is te borderline between feasible and infeasible dimensions for sites of facilities. Furtermore, a slicing tree gives information about te relative location of facilities to eac oter, but it does not include exact positions and dimensions of facilities. Terefore, a layout generating procedure based on bounding curves is needed. Tis procedure starts at te bottom of a slicing tree wit an internal node wose bot cildren are leaf nodes wit given bounding curves. It calculates te bounding curve of te internal node depending on its cut direction (v or ) by adding te curves of te leaf nodes in x- orin y-direction, respectively. Regarding a wole slicing tree, te bounding curve of eac internal node is determined recursively starting at te bottom till te bounding curve of te root node is computed. Tis bounding curve gives te smallest feasible dimensions for all possible final layouts corresponding to te given slicing tree. Eac point on te curve or on its upper rigt and side corresponds to te dimen- v v v Fig. 1. Slicing tree and slicing structure

5 466 D. Scolz et al. / European Journal of Operational Researc 04 (010) sions of a feasible layout. Tey all differ in means of transportation costs and floor space requirements. After aving determined te root node s bounding curve in a bottom up way, te layout is calculated in a top down manner starting at te root node. Te layout wic was determined tat way satisfies all facilities geometric requirements and may now be evaluated to get an objective function value Tabu searc algoritm Every slicing structure can be represented by a slicing tree, so te idea is to searc for good layouts in te set of slicing trees. Starting wit an initial feasible solution, a tabu searc algoritm is provided wic performs small canges in te slicing tree corresponding to small canges in te layout Initial feasible solution. To determine an initial solution, r slicing trees are randomly generated and te one wit te best objective function value is cosen. If te worksop s dimensions are not given, tese random trees are feasible solutions, because by using te bounding curve based layout generation metod, every slicing tree corresponds to a layout wit non-overlapping facilities wic meet teir geometric requirements Neigborood. Four types of moves define te neigborood of an existing solution. Eac type of move is performed on te slicing tree and as an effect on te layout as well. Te neigborood definition in Table 1 allows te searc to reac eac arbitrary slicing tree witin a finite number of moves and ence covers te set of all slicing structures (a proof is provided in Scolz et al. (009)). So an optimal slicing structure may be reaced. Te tabu searc cecks all of te presented moves and uses te best solution found in te neigborood as a starting point for te next iteration (best-fit strategy). For large problems it can be advantageous to examine only a part of te large neigborood. In tis section, we present possibilities to enance STaTS to master tese callenges...1. Bounded and non-rectangular worksops, exclusion zones in worksops, predefined positions of facilities Basically, te feasible solution space searced by STaTS is te set of slicing trees wic corresponds to te set of slicing structures. Tese slicing structures are rectangles so te layout, respectively, te factory worksop are expected to be rectangles as well. However, an existing worksop may possess a different sape, wic as to be mapped to a slicing structure. To solve tis problem we take te smallest rectangle containing te worksop and place dummy facilities in it, so tat tey cover all regions wic do not belong to te available area of te worksop. Eac of tese dummy facilities as a fixed position and fixed dimensions (type ), is not rotatable, and as no flow interactions wit any oter facility. Of course, te dummy facilities are rectangles itself, so tis procedure will only work if all of te worksops boundaries are ortogonal to eac oter as required in te problem formulation. However, tis is not a armful restriction because buildings are usually constructed tat way. Finally, te outline of te worksop as to be converted into a slicing structure or its slicing tree representation. Note tat eac of te four boundaries of a facility as to be adjacent wit eiter a cut or a boundary of te total layout (te rectangle containing te worksop). But tere exists more tan one slicing structure wic may contain a specific set of dummy facilities. Tis is sown in te small example in Fig., were eiter te orizontal or te vertical cut may be applied first. Te sequence of te cuts can be arbitrary, but sould be cosen carefully because it affects te structure of te remaining solution space. It is important to be aware tat no facility may overlap wit a cut, so especially if te FLP contains large facilities an unfavorable selection of te cut sequence may cause few feasible positions for suc facilities or even Intensification strategy. An intensification strategy is needed to perform a more detailed searc in regions of te solution space wic are estimated to contain very good solutions. For tat reason te best solution found during te tabu searc is stored. Afterwards it is used as a starting point for a ill-climbing strategy performed on te complete neigborood witout te use of a tabu list. Tis results in a local optimum wit respect to te examined neigborood Diversification strategy. A diversification strategy is needed to guide te searc into very different regions of te solution space preventing te searc from being caugt in a region wit only a local optimum. To avoid tis situation, a multi-start procedure wit eac run starting at a randomly selected point in te solution space (as described in Section.1.3.1) is performed... Additional real world restrictions To solve a real world FLP some additional restrictions, wic ave been listed in te first section ave to be taken into account. Fig.. Worksop and two corresponding slicing structures. Table 1 Types of moves. Slicing tree Move 1 Swap two leaf nodes Swap two facilities Move Invert te label of one internal node (/v) Invert te direction of one cut (/v) Move 3 Swap one internal node including its subtree wit eiter a leaf node or anoter subtree Swap one partial layout wit eiter a facility or anoter partial layout Move 4 Move a node including its subtree and its fater into anoter edge Move one facility or partial layout Layout

6 D. Scolz et al. / European Journal of Operational Researc 04 (010) infeasibility (see Fig. 3). However, wic sequence of cuts sould be cosen is mostly indicated by te construction of a non-rectangular worksop. Usually te allocation of piers or bearers implies a reasonable cut sequence. Te selected sequence of cuts is now converted into te corresponding slicing tree. As an example te slicing tree according to te left layout in Fig. 3 is presented in Fig. 4. Tis part of te slicing tree will be fixed, and no move will be applied to any of tese nodes during te neigborood searc. Te tree as some open brances (e.g. a, b, and c) were a single facility or a subtree consisting of several facilities may be attaced. Eac open branc corresponds to a section of te worksop, wic was not initially intersected by any cut. STaTS as been modified so tat it gets te fixed part of te slicing tree as an input and generates random trees at its open brances to generate an initial solution. Afterwards te neigborood searc performs moves affecting only nodes at te open brances. Eac section of a worksop as specified dimensions but in order to find feasible solutions tey could be exceeded during te searc process. To force STaTS finding a fitting layout for eac section te excess of a section s dimension is penalized in te objective function. Te given size of eac section giv size s d of te worksop is compared to its size in te actual layout act size s d. Te amount by wic te given size is exceeded is summed up for all of te worksop s sections and bot dimensions: P 1 ¼ a 1 dfx;yg ssections max 0; act size s d giv size s d : 1 b a c 3 Fig. 3. Nested worksop and two corresponding slicing structures. v b 1 d a c 3 Te procedure described in tis section is also applied to consider exclusion zones in worksops; tey are treated and positioned like te oter dummy facilities. Facilities wit predefined positions are andled te same way, but in contrast to te dummy facilities tese ones may ave interactions wit oter facilities.... Multiple worksops If tere is more tan one worksop situated on factory premises, te facilities wic are interacting wit eac oter ave to be assigned to tese multiple worksops. We use te same idea as in Section..1 but we regard te rectangle containing te wole factory premises and place dummy facilities at all areas wic are outside of te worksops (e.g. see te practical application in Figs. 7 and 8). It as to be considered tat te rectilinear or Euclidean distances, wic in most practical applications are used in te objective function for measuring te distance between two facilities, migt not be appropriate if tere is more tan one worksop. In suc a case, te transportation pat is dependent on te positions of doors in te worksops (see Fig. 5 in wic te rectilinear distance is used). Te door-to-door distances between worksops are known and can be added to te distances of eac facility to te door of its worksop. Te information to wic worksop a facility is assigned is included in te slicing tree. It as to be cecked to wic of te open brances te facility is attaced...3. Aisles Aisles between facilities are planned to be able to perform te transport activities. In tis approac we present two ways to consider aisles of a constant widt A during layout planning. One possibility is to use every cut in te slicing structure as an aisle, wic can be integrated in te layout determination procedure. Eac time two bounding curves are summed up te constant aisle widt A is added to te x-values (y-values) according to a vertical (orizontal) cut of te fater node. Finally, in te root node s bounding curve te area requirements of all facilities and aisles are considered. Using tis approac togeter wit dummy facilities introduced in Sections..1 and.. is difficult because tese aisles will be also establised between tem. Tere are two possibilities to andle tis problem: (1) Te dimensions of te dummy departments could be sortened by te widt of adjacent aisles/ cuts. () Te widt of cuts tat correspond to nodes of te fixed part of te slicing tree could be set to zero. However, it leads to missing aisles between sections of a worksop. Te second way to preserve space for aisles is to enlarge eac facility s dimensions in x- and y-direction by te constant aisle widt A. So eac facility is surrounded by some empty space (A/ on eac side) as sown in te left part of Fig. 6. Tis is realized by adding A to te x- and y-values of all facilities bounding curves. Using tis approac te worksops are not filled as dense as wit te approac presented above. A alf aisle widt remains unoccupied at eac exterior wall of a worksop (see Fig. 6 rigt). Te layout designer can use te remaining flexibility in te final layout for manually adjustments or enlarge te worksop s dimensions given to STaTS by A so tat te worksop will be completely filled up. Tis way to consider aisle widts can be easily used in non-slicing approaces, too. b v 1 a c 3 f 1 f Fig. 4. Slicing tree corresponding to te left layout in Fig. 3. Fig. 5. Distance between facilities in different worksops.

7 468 D. Scolz et al. / European Journal of Operational Researc 04 (010) Facilities forced to be placed at an exterior wall of a worksop Due to tecnical requirements, it may appen tat some facilities must be placed directly at an exterior wall of a worksop. Te information weter in te actual solution a facility as been placed next to an exterior wall or not can be gained eiter directly from te slicing tree or from te detailed layout. A facility is positioned at te layouts left (rigt) margin if and only if it is part of te left (rigt) subtree of eac of its predecessors (i.e. parent nodes) labeled wit a vertical cut in te slicing tree. On te oter and, te facility may be attaced to any side of predecessors labeled wit a orizontal cut. Te same olds true for te upper (lower) margin if it is always part of te left (rigt) subtree of its predecessors labeled wit a orizontal cut. Using a fixed part of te slicing tree, tese conditions ave to be cecked only for tose predecessors tat are not part of te fixed input tree. Not necessarily all four margins of te rectangle corresponding to te open branc are exterior walls so te condition is cecked only for tose margins wic are. Regarding te layout, we extract te information about exterior wall positions of facilities by cecking if te position of at least one of te four margins of a facility s site is identical to te corresponding exterior wall. If multiple worksops are considered te assignment of te facility to a worksop can be cecked eiter via te coordinates in te layout or via te slicing tree as described above. Using te layout information instead of te slicing tree information we can determine not only weter a facility is adjacent wit an exterior wall but also its minimum distance to te wall. We use tis information to penalize te objective function if a facility s exterior wall restriction is not kept. We may add a fixed penalty for eac facility not being placed at te wall or a penalty depending on te sum of distances between eac facility and te nearest wall. We cose to do te latter wic is calculated in te following equation: P ¼ a min fdððx; yþ; ðx 0 ; y 0 ÞÞjðx; yþ EðiÞ; ðx 0 ; y 0 Þ R Wg: iwall Our algoritm was applied to a facility layout problem faced by an automotive supplier converting tecnical foamed plastics. Te company as used tree manufacturing worksops but due to a newly purcased fourt worksop and several new macines, te layout ad to be replanned completely. Te four worksops (A, B, C, and D) are located to eac oter as sown in Fig. 7. Furtermore, storage areas for two different groups of raw materials (RM1 and RM) are sown. Tose can be regarded as fixed but tey will be relevant for transportation costs to be described later. Te raw materials of type 1 can be placed eiter close to worksop A or close to worksop D. Besides tese two storage areas, tere are two more facilities tat ave to follow restrictive constraints. Tere is anoter storage area called RM3 and an inventory of finised goods to be placed witin te worksops. Tese facilities ave to be reaced by trucks, wic implies tat tey must be placed eiter in worksop A or in worksop D as only tere public road access is ensured. Due to lack of space, tey must not be in te same worksop. RM1 sould furtermore be placed close to RM3. Tus, we may consider two scenarios in eac of wic RM3 and te inventory of finised goods are fixed. Let us focus on tese two scenarios in more detail. Let us consider te case in wic te inventory of finised goods is placed in worksop A and RM3 in worksop D, to wic we refer as Scenario 1. According to Sections..1 and.., dummy facilities ave to be introduced in order to reproduce te sape of worksop C and in order to combine all worksops to one. Tis as been done as sown in Fig. 8. All fixed facilities (including dummy facilities) are denoted by roman numbers. As can be seen, worksop C is divided not only by a orizontal cut, but bot parts are even departed from eac oter using dummy facility I. Tis is done because of several piers placed along tat line, proibiting facilities to be placed tere. Note tat dummy facility III actually is muc wider as indicated by te dased lines. Furtermore, te size of te raw material storages is increased to simplify matters. Tis as no effect on te flows as tey are calculated from te midpoint of te facilities. Te use of fixed (dummy) facilities as proposed in Fig. 8 leads to an uncangeable part of a slicing tree as presented in Fig. 9. Horif..5. Bad neigbors Some facilities may not be placed directly adjacent to eac oter but must ave a minimum distance. We call suc a pair of facilities bad neigbors. We force STaTS to consider tis by adding penalties for eac pair of bad neigbors if te smallest Euclidian distance between tem falls below te minimum distance d min. P 3 ¼ a 3 max f0; d min min fdððx; yþ; ðx 0 ; y 0 ÞÞjðx; yþ ði;jþbadneigbors A Fig. 6. Facility surrounded by space for aisles and layout sketc. EðiÞ; ðx 0 ; y 0 ÞEðjÞgg:..6. Predefined dimensions of worksops/factory premises If te dimensions of te total layout (dimensions of te rectangle containing te total layout according to Sections..1 and..) are given, it is sufficient to generate te layout corresponding to tese dimensions from te bounding curve of te root node of te slicing tree. If te actual slicing tree does not correspond to a layout wic can be realized witin te given dimensions te bounding curve is located to te rigt and above of tis point in te coordinate plane. We generate te layout wit te dimensions according to te point on te bounding curve wic possess te smallest deviation in terms of rectilinear distance to te given dimensions. To guide te searc in direction of feasible layout dimensions we add a penalty to te objective function for eac section of a worksop wose dimensions are failed (as presented in Section..1). 3. Practical application B C RM D RM 1 A RM 1 Fig. 7. Location of te manufacturing worksops and outside raw material storages.

8 D. Scolz et al. / European Journal of Operational Researc 04 (010) v1 v v3 v4 v5 v6 v7 II I (Inv. of fin. goods) III IV B V C top C down v8 I VII (RM ) VI D VIII (RM 3) (RM 1) Fig. 8. Scenario 1 wit fixed facilities. v4 v v1 v3 v8 v6 III V C down VI v5 v7 I II IV B VII I C top VIII D Fig. 9. Te fixed part of te slicing tree for Scenario 1. zontal and vertical cuts are denoted by and v, respectively. Te vertical cuts are enumerated and labeled in Fig. 8 in order to clarify te transfer from Fig. 8 to 9. Considering Scenario, in wic RM3 is placed in worksop A and te inventory of finised goods in worksop D, leads to fixed dummy facilities as sown in Fig. 10. Note tat te fixed facility (corresponding to RM1) actually as no area. An area was only drawn in order to make it visible. Te according fixed part of te slicing tree is sown in Fig. 11. In bot scenarios, tere are ten fixed facilities and four of tem are relevant concerning material flow. Besides te fixed facilities, tere are 51 unfixed facilities to be placed witin te worksops. Tose are: Tirty-eigt macines wit a given area and a given sape. Tese macines correspond to facilities of type (see Section 1.). Eac of tese macines may be rotated by 90. Five macines ave to be placed next to a wall in order to install an air outlet conduit. Tere are four macines raising dust wic ave to be placed at least 5 m away (using te Euclidean distance) from eac of five dust sensitive macines. Two pools of congeneric macines. Tese pools ave a given area and a given minimum and maximum aspect ratio, tus belonging to facility type 3. Here te sape of te facility is to some degree flexible. Nine (interim) storage facilities wit given area and a given minimum and maximum aspect ratio. Again, tese facilities are of type 3, but compared to te macine pools, te sape of te interim storage facilities is even more flexible. Two waste containers wit given area and given sape, corresponding to facility type. Space for aisles is added to te facility sizes, following te second alternative proposed in Section..3. Te total required space for all of te 51 unfixed facilities including te space for aisles is less tan 900 m. Te available space witin te worksops is II I (RM 3) A v1 v v3 III B v4 C top C down v7 v8 v9 (RM 1) IV V v5 VII (RM ) I v6 VIII (Inventory of finised goods) VI Fig. 10. Scenario wit fixed facilities.

9 470 D. Scolz et al. / European Journal of Operational Researc 04 (010) v4 v v1 v3 v9 III V C down VI I v6 v7 II IV B v5 VIII I v8 C top VII A Fig. 11. Te fixed part of te slicing tree for Scenario. more tan 3800 m. So in general finding a feasible solution is practicable altoug a strict application of te rule based solution (to be described below) would ave lead to infeasible solutions. Especially te allocation of some large macines as to be done carefully so tat tey do not block too muc furter available space. Te objective function considered is te sum of te above-mentioned transportation costs tat may appear between te 51 unfixed facilities and four fixed facilities (RM1, RM, RM3 and te inventory of finised goods): min 55 i¼1 55 j¼1 i j d ij f ij c ij : Te distance d ij is represented by te rectilinear distance. Altoug it would ave been possible to consider doors of worksops as described in Section.., we did not do so as all worksops ave large doors facing te oter worksops. Terefore te error obtained from not considering doors seems to be negligible. Te flow of materials f ij ad to be forecasted, as te (future) flows cannot be determined to certainty. However, te ratio of different product demands faced by te company underlies only little fluctuations, at least in te long run. Tus, istorical data are meaningful. Orders from a wole year ave been evaluated and two different types of flows between facilities ave been determined: Flow of material for furter processing and flow of waste material. Te latter is regarded to be less important and tus devaluated wit factor c ij ¼ 0:1. Te oter flows are weigted wit c ij ¼ 1. Te devaluation of waste flow is argued wit its influence on te position of te waste containers, wic in turn migt ave an influence on te cycle times. Te cycle times are normally a goal, wic is complementary to te minimization of transportation costs (Domscke and Drexl, 1996) and tus not considered explicitly. Even before planning, it was obvious tat te company will ave to move almost all facilities in order to guarantee space during te reorganization process. Tus, relocation costs for single facilities ad not to be considered. Put togeter, te input for te computer program implementing STaTS consists of te fixed part of te slicing tree as depicted in Figs. 9 and 11, te size and te sape of te fixed facilities, te set of unfixed facilities including teir size, teir possible sapes and if existent te exterior wall restriction or bad neigbor restrictions, and te values f ij wit te flow of material between all unfixed and four fixed facilities. For eac scenario STaTS as been executed 10 times. In eac of tese runs, 30 random trees were generated as candidates to be used as te initial solution. A fraction of 35% of te neigborood was cecked in every tabu searc iteration and a lengt of te tabu list of as been used. Te neigborood searc was performed as long as te objective function value was not furter improved for 0 seconds. Te mean number of iterations amounted to approximately 500. Te weigts of te penalties were cosen as follows: a 1 ¼ 300; a ¼ a 3 ¼ 50. Te minimum distance ðd min Þ between bad neigbors was set to 5 m and generally all distances in te penalty terms were measured in meters. We ave compared te results of STaTS to rule based solutions. Te rules were determined wit te elp of experienced production managers of te company and tese solutions sould reflect te layout tat would ave been used witout STaTS. In order to receive a rule based solution, initially only four (non-dummy) facilities are fixed in eac scenario. Te rules for placing te remaining facilities are te following: As long as tere are unfixed facilities, determine an unfixed facility i and a fixed facility j so tat te flow between tese two facilities is maximal: arg max 16i;j6n i not fixed; j fixed f ij þ f ji ; i is te next facility to be fixed. Place i as close (using te rectilinear distance) to j as possible subject to te restrictions and regard i to be fixed. In order to do so, te sape of oter facilities may be canged according to teir sape restrictions. After all facilities are fixed try to improve te objective function troug intercanging te position of two facilities at a time.

10 D. Scolz et al. / European Journal of Operational Researc 04 (010) Note tat tese rules were not regarded to be dogmatic. Wenever te rules would obviously lead to unfavorable or even infeasible solutions, tey were disesteemed. Tis was especially te case if one worksop was almost full. E.g. te case appeared tat tere was still enoug space for a big facility or two small ones. If it is more favorable to use te space for te two small facilities, tey would be placed tere even if te big facility is to be placed next, according to te rules. Te results of our tests are sown in Table. For bot scenarios, te total transportation costs of STaTS and te rule based solution are listed as well as te relative deviation of te rule based solution compared to STaTS. In bot scenarios, te results generated by STaTS are considerably better tan te rule based approac concerning te objective function. A consideration of te runtimes leads to te same conclusion. Te mean duration of STaTS was approximately 180 seconds for eac run on a Pentium IV.4 GHz CPU, wereas it took ours in order to receive te rule based results. Te restrictions were satisfied so tat te objective obtained by STaTS did not contain any penalties P 1, P or P 3. Te calculation of tigt lower bounds for te problem at and seems to be very ard. Tis is e.g. caused by te fact tat tere is no flow conservation restriction for any facilities, wic guarantees tat te number of inbound flow units corresponds to te number of outbound flow units. However, tere are tree facilities, namely RM1, RM and RM3 not receiving any flow ( sources ), and tree facilities, namely te inventory of finised goods and te waste containers wit no outbound flow ( sinks ). For te remaining facilities, it is observed tat te outbound flow does not exceed te inbound flow f ji P j j f ij : A simple idea for generating a lower bound is te fact tat eac transport unit arriving in te inventory must ave covered at least te distance from te nearest source. As can be seen in Fig. 8, in Table Total transportation costs in unit transportation kilometers. STaTS Rule based Quotient (%) Scenario Scenario Scenario 1 tis would be te distance from facility VII to facility I. Te number of transport units arriving in te inventory of finised goods multiplied by te minimum required distance leads to lower bound of for te transportation costs. However, as te inbound flow of te inventory of finised goods exceeds te outbound flow of RM (facility VII) by far, we can determine a number of transport units aving to cover at least te distance between facility VIII and I. Tis leads to a lower bound of 9.4, wic still seems to be very loose. Te same can be applied to Scenario. We may state tat tere is no direct flow between RM and te inventory of finised goods and tus all goods ave to be transported at least to te rigt end of Worksop C. Using tis fact we obtain a lower bound of 0.7 for te transportation costs of Scenario. Furtermore, we ave analyzed te objective function value in dependency on te space requirements of te facilities. In te presented scenarios, all facilities cover approximately 75% of te available worksop space. We ave canged tis value by increasing and decreasing te space requirements of all facilities. We ave tested four furter instances in wic eac facility requires 80%, 90%, 110%, or 10% of its original area, wic corresponds to a coverage of te worksops space of 60%, 68%, 83%, and 90%, respectively. In te latter case, none of five repetitions as led to a feasible solution. If te required space of te facilities is increased by 10%, only one out of five repetitions as led to a feasible solution altoug tere are five feasible solutions for te original problem. Te effect on te objective function is rater straigt forward as can be seen in Fig. 1. A decrease of te required space leads to a moderate improvement of te objective function value. Tese results also sow tat te consideration of aisles as only little influence on te objective function score. Te disregard of aisles leads to approximately 5% less required space for facilities, wic in turn improves te objective function according to Fig. 1 by less tan 1.5%. If oter practical restrictions are omitted, a similar result appears. We ave run a test neglecting te Bad- Neigbor and te Wall restriction. In tis case, te objective function was only reduced by 0.78% on average. 4. Step-by-step guide for te practical application of STaTS Tis section is to outline in wic cases te proposed metods are applicable. Steps 1 and generally describe te assignment, Objective function value in relation to te orginal problem 105% 104% 103% 10% 101% 100% 99% 98% 97% 96% 95% 94% 93% 80% 90% 100% 110% Percentage of te original space requirement Fig. 1. Instances in wic te facilities space requirements ave been canged.

11 47 D. Scolz et al. / European Journal of Operational Researc 04 (010) Steps 3 6 consider restrictions for tis assignment, and Steps 7 9 propose possible objectives. 1. Tere ave to be one or more worksops in wic facilities can be placed. All corners of te worksops must be rigt angles, e.g. a rectangular sape, an L-sape or a Z-sape. Tis approac is not suitable for round worksops or miscellaneous worksops.. Tere as to be a set of facilities, eac of wic as to be of rectangular sape. It is not necessary to know te exact sape of te facilities in advance, but it is necessary to know all possible sapes a facility may ave. All facilities ave to be placed to te worksops. 3. Facilities must be placed in suc a way tat no two facilities overlap. 4. Optionally, certain facilities can be forced not to be placed in certain areas of te worksop. E.g. if a facility needs to be placed under an overead crane, it can be assured tat it will only be placed in according areas of te worksops. Using tis restriction, facilities can even be forced to be placed to a predefined position in a worksop. It is also possible to force facilities to be placed to an exterior wall. 5. Optionally, aisles can be integrated into te layout, so tat eac facility is reacable via tese aisles. Te widt of te aisles as to be provided in advance. In tis context, it is possible to consider given worksop doors. Note tat considering aisles implies tat all facilities are reacable via aisles. Providing access only to a subset of facilities is not possible. 6. Optionally, it is possible to provide pairs of facilities, wic must placed wit a minimum distance in between. For eac suc pair of facilities, te separation distance as to be provided in advance. 7. If desired, te assignment can be developed in suc a way tat transportation costs are minimized. In tis case, for eac pair of facilities, te amount of transported goods and an according cost parameter per distance unit as to be provided. Almost all distance measures can be applied, altoug linear (Euclidian) distance and rectilinear distance are te most common ones. 8. If Step 7 does not apply, it is possible to put all facilities as close togeter as possible. Tis objective is especially suitable if a worksop as not yet been built. In tis case, a very compact layout indicates te required size for a new worksop. Te algoritm ten tries to determine te smallest rectangle capable of placing all facilities. 9. If Step 7 and Step 8 do not apply, te algoritm simply aims at finding a feasible solution. 5. Conclusion We ave presented a very general framework of te facility layout problem. Altoug numerous intricacies are included, eac layout can easily be represented by a slicing tree at wic it is easy to observe weter te according layout is feasible or not. Furtermore, te slicing tree representation allows te efficient application of tabu searc. Tis approac as been applied to a real world problem and its effectiveness compared to rule based solutions as been sown. Future extensions to tis approac migt focus on furter generalizations suc as non-rectangular facilities or te application of te approac to oter kinds of facility layout problems, e.g. wen producing printed circuit board assemblies. Acknowledgements Te autors tank Erwin Pesc for is valuable comments, wic improved tis paper. Furtermore, we would like to tank two anonymous referees. References Alagoz, O., Norman, B.A., Smit, A.E., 008. Determining aisle structures for facility designs using a ierarcy of algoritms. IIE Transactions 40, Bazaraa, M., Computerized layout design: A branc and bound approac. AIIE Transactions 7, Castillo, I., Westerlund, J., Emet, S., Westerlund, T., 005. Optimization of block layout design problems wit unequal areas: A comparison of MILP and MINLP optimization metods. Computers and Cemical Engineering 30, Castillo, I., Westerlund, T., 005. 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