The facility layout problem: Recent and emerging trends and perspectives

Journal of Manufacturing Systems Vol. 15/No. 5

1996

Trends and Perspectives

The Facility Layout Problem: Recent and Emerging Trends and Perspectives Russell D. Meller and Kai-Yin Gau, Auburn University, Auburn, Alabama

Abstract Recent and emerging trends in the facility layout prob-

lem, covering the last 10 years of research, are presented, including new methodologies, objectives, algorithms, and extensions to this well-studied combinatorial optimization problem. The state of the art in facility layout software is compared to the state of the art in facility layout research. New developments in emerging layout research provide a perspective on what the future of the field will be like. A trend toward concurrent engineering approaches to layout and production system design is observed.

Keywords: Facility Layout, Manufacturing System Design, Production and Operations Management, Quadratic Assignment Problem

1. Introduction Determining the physical organization of a pro-

duction system is defined to be the facility layout problem. This well-studied combinatorial optimization problem arises in a variety of production facilities, including service and communications settings, but here the focus is on manufacturing facility layout. The manufacturing system is the critical com- ponent in such production systems, and thus, it must be considered when determining the layout.

The facility layout problem is concerned with finding the most efficient arrangement of m indivis- ible departments with unequal area requirements within a facility. As defined in the literature, the objective of the facility layout problem is to minimize the material handling costs inside a facility subject to two sets of constraints: (1) department and floor area requirements and (2) department location- al restrictions (departments cannot overlap, must be placed within the facility, and some must be fixed to a location or cannot be placed in specific regions). Floor loading and floor-to-ceiling clear-height constraints also exist in multiple-floor facilities, is [Note: Reference citation numbers correspond to the alphabetical reference list at the end of the paper.]

This paper supports two sets of readers. The first is comprised of readers who want to explore facility layout research since the most recent review paper. 78 These readers need an organizational tool to estab- lish the trends in the research because it is a very active area (nearly 100 papers have been published on the facility layout problem in the last 10 years; see Section 3), and due to the problem's significance in manufacturing organizations as discussed above, it is likely to remain active as well.

The second set of readers is comprised of those who want to learn more about facility layout research due to the emergence of facility layout software packages. The software packages are not reviewed here, but readers are provided with the link between the software packages and the research on which the packages are based. Together with a description of the research, this provides these readers with information not typically supplied in trade magazine software reviews.

Background for the facility layout problem is presented in Section 2 (fundamental models, assump- tions, and common notation) for those not familiar with layout research. In Section 3 the layout research literature of the last 10 years is examined, highlight- ing trends in the research. Special emphasis is placed on research that has had a direct effect on facility layout software packages. Software packages and perspectives on the issues that should be given more attention in future research are discussed in Section 4.

2. Background The output of the facility layout problem is a block

layout, which specifies the relative location of each department (see Figure la). One then can perform further work to obtain the detailed layout, which specifies exact department locations, aisle structures, input/output (I/O) point locations, and the layout within each department (see Figure lb). The detailed

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Journal of Manufacturing Systems Vol. 15/No. 5 1996

2 3

(a)

1 I I oo ,Uo • Q

O0 rid

0 • l l i 0 . . .

"IZ] I F-q

(b)

Figure 1 (a) Block Layout and (b) Detailed Layout

layout problem includes flowline layout problems, machine layout problems, and cellular manufacturing design problems, where machines are assumed to be of equal area or of fixed dimensions, s°

In the literature, the layout's efficiency is typically measured in terms of material handling costs. These costs are approximated with one or more of the following parameters: interdepartmental flows, j~ (the flow from department i to department j); unit-cost values, c o (the cost to move one unit load one distance unit from i to j); and department closeness ratings, r0 (the numerical value of a closeness rating between departments i and./'). These parameters are used in two common surrogate material handling cost functions. 34

The first of the two surrogate material handling cost functions is based on departmental adjacencies:

max E E ( r ~ j ) x 0 (1) i j

where x o equals 1 if departments i andj are adjacent, and 0 otherwise. Such an objective is based on the material handling principle that material handling costs are reduced significantly when two departments are adjacent.

The second of the two surrogate material handling cost functions is based on interdepartmental distances:

m i n E E ( f o c i j ) d O (2) i ]

where d U is the distance from department i to departmentj . This objective is based on the material han-

dling principle that material handling costs increase with the distance the unit load must travel. The distances in Eq. (2) are measured in a variety of ways; here are two ways--the most accurate and a widely used approximation.

• Distances between I/O points: This distance is measured between the specified I/O points of two departments and in some cases is measured along the aisles when traveling between two departments. ~a2 The major drawback of this accurate measure is that one does not know the location of the I/O points (or aisles) until one has developed the detailed layout, leading to the widely used centroid-to-centroid approximation.

• Centroid-to-centroid (CTC): When the input and output points of the departments are not known, the department centroid is used to represent the department I/O point. The shortcomings of CTC distances include: the optimal layout is one with concentric rectangles; ~3° an algorithm based on CTC attempts to align the department centroids as close as possible, which may make the departments very long and narrow; 129 and a department that is L-shaped may have a centroid that falls outside of the department (Francis and White, 35 pp. 132-133).

For each of the distance measures mentioned above, there are two metrics used to measure the distance between two points. Rectilinear distance is the most common distance metric used because it is

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based on travel along paths parallel to a set of per- pendicular (orthogonal) axes./a° The second distance metric is Euclidean distance, which is appropriate when distances are measured along a straight-line path connecting two points (for example, conveyor travel and air travella°). Distances are measured with the Euclidean metric in Tam and Li ~26 and van Camp, Carter, and Vannelli? 38

In the multifloor facility layout problem, one needs to consider the vertical distance in addition to the horizontal distance. ~ Multifloor problems require the user to specify data on potential lift locations and the cost to move one unit load one vertical distance unit between departments i andj (c[), as well as to specify data on the horizontal material handling costs (cU). The objective used in multifloor problems is to

N N V V min ~ ~(c/~d/~ +codij )fo (3)

i=1 j= l

where H v d u (d 0) denotes the horizontal (vertical) distance between department i and department j. Horizontal distances between departments on different floors are typically measured between department centroids via the lift that minimizes the total horizontal distance traveled to and from the lift.

Because there are advantages and disadvantages to these two objectives, 94 in some cases they are combined in a weighted criteria:

minotZZ(foco)do-(1-oOZZrox 0 (4) i j i j

where oL E [0,1 ]. Meller and Gau 94 examine how to set et and whether its exact value is critical.

2.1 Exact Procedures Two traditional approaches have been developed

to obtain, in theory, an optimal solution to the facility layout problem. The phrase, "in theory" is used because both approaches are generally unsolvable to guaranteed optimality? 7 The first is the quadratic assignment problem approach and the second is the graph-theoretic approach. Background and references are provided.

Quadratic Assignment Problem Approach Koopmans and Beckman 71 introduced the qua-

dratic assignment problem (QAP) to model the problem of locating interacting plants of equal areas. The QAP has been applied to a wide range of applica-

tions, including urban planning, control panel layout, and wiring design .12 Note that the QAP is a special case of the facility layout problem using Eq. (2) because it assumes that all departments have equal areas (or the distance from one "site" to another can be predetermined and remains the same regardless of department-to-site assignments) and that all locations are fixed and known a priori.

The QAP formulation assigns every department to one location and at most one department to each location. The cost of placing a department at a particular location is dependent on the location of the interacting departments. Such dependency leads to the quadratic objective that inspires the problem's name. A recent alternative formulation of the QAP considers assigning interdepartmental distances to department pairs. 1°9

Solution techniques for the QAP are based on variations of the branch-and-bound approaches ini- tially proposed by Gilmore 39 and Lawler. u The QAP is NP-complete, a7 which implies that, in general, it is a hard problem to solve. To date, optimal solutions to general cases of the problem can only be found for problems with less than 18 departments. 69 Recent heuristics for the QAP may be found in Section 3 and in Kaku, Thompson, and Morton ~ and Smith and MacLoed. m

According to Liao, ~ Kusiak and Heragu, 7s and others, the unequal-area facility layout problem may be modeled as a modified QAP by breaking the departments into small grids with equal area, assigning a large artificial flow between those grids of the same department to ensure that they are not split, and solving the resulting QAP. However, due to the increase in "departments" it is not possible to solve even small problems with a few unequal-area departments. Moreover, Bozer and Meller 17 show that such an approach is ineffective because it implicitly adds a department shape constraint. Expanded formulations based on the QAP representation are thorough- ly reviewed in the most recent layout review paper. 7s

Graph- Theoretic Approaches In graph-theoretic approaches, it is assumed that

the desirability of locating each pair of facilities adjacent to each other is known. 33 The area and shape of the departments are ignored (at the beginning), and each department is then represented by a node in a graph. Satisfied department adjacency relationships are represented by an arc connecting the two adjacent

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departments (nodes) in the graph. The objective function is Eq. (1), which translates to constructing a graph that maximizes the weight on the adjacencies (arcs) between department pairs (nodes).

Developing a layout in the graph-theoretic approach requires the following three steps: ~7 (1) developing an adjacency graph from department relationships (which departments are adjacent), (2) constructing the dual graph of the adjacency graph (represent departments as adjacent regions having specific boundaries), and (3) converting the dual graph into a block layout (specifying departments with regular shapes and specific areas).

The objective function of the graph-theoretic approach is maximized if all department pairs with positive flow have an arc between them. However, to facilitate step 2, it is necessary to limit the number of arcs incident at each department. Such a problem is difficult in general, and thus, heuristics must be used to construct a maximally weighted adjacency graph. Like the QAP-approach, unequal-area problems of even small size cannot be solved to guaranteed optimality with graph-theoretic approaches. A review of complete graph-theoretic results and heuristics can be found in Foulds 3° (and also in Hassan and Hogg46).

3. Recent Developments (1986-1996) and Trends

A summary of the layout research during 1986- 1996 is presented in Table I, where 9 | published papers on models and algorithms are classified by the facility layout research classification scheme presented in Figure 2 (some judgment calls were made for papers that might reasonably be assigned to multiple categories). The papers are presented in chronological, order and are alphabetical by first author within a year.

The classification scheme in Figure 2 helps estab- lish trends in facility layout research over the last I 0 years. The highest level of the scheme divides research into one of three areas. The first area is concerned with algorithms for the facility layout problem as it has been defined so far in this paper. With a few exceptions, this is the only area covered in Kusiak and Heragu. 7s The second area is concerned with extending the problem definition in some fashion, for example, adding a time element (dynamic layout), adding uncertainty (stochastic layout), or adding multiple criteria for evaluation (multicriteria, robust, or flexible layout). The third area considers

special cases of the problem, some of which are derived from their support of integrated manufacturing system designs, for example, flowlines, machine layout, and cellular layout design. Figure 2 also presents a level in the scheme differentiating research by its investigation into a model or a heuristic. Additional classification levels are possible (such as, level A.2.b could be divided into whether the research is based on the generation of the adjacency graph or the formation of!the layout from a graph), but this is best left to users of the research.

Examining Table 1 with respect to each classif i , cation category indicates t he trends in layout research, which are presented in the next section~ The following section presents a new m o ~ approach based on a mixed-integer programming, formulation (level A,l~c). Finelly, a sample of~tite heuristics listedin Table 1 is presented.

3.1 Trends in Recent R ~ e ~ e h There is not one approac& tlaat is domii~ating the

field in-classification, e a t e g ~ A (that is, there?at'e, [3 paper~in A.l,.a and A.2.a -~s, 15 papers in A;1.1k ~ 1 A.2.1~, indicating that thee is likely t o be fitt-tlaer research at~ exploratio~ ~ this area. Relal~g to gral~ta-theot'etic approacl~, much of the eurrenttalten- tiou is fo~sed on trartslnti~ the dual off the ~jaeen- cy, graph to a hock l a z a r because it is ve~, ~ c u l t for a c ~ u t e r and time consuming for~ htmaan? 2

The numl~er of papers in classification category B (exter~ions to the basic model) has: significantly increased. Due to pressures on m a ~ t u r i n g systems to adapt to change, one sees ~ emphasis on algorithms for the dynamic layout problem, which was introduced by Rosenblatt. ~3 For the stochastic layout problem, Rosenblatt and Kropp nQ show that if the objective is to minimize the expected material handling cost, then it is equivalent to solve the layout problem that results from the expected flow matrix. Such a result allows o~te to effectively use traditional layout algorithms tl~t consider one flow matrix. Combination approaches that address both - the stochastic and the dynamic layout problem have been examined by various researchers because, as pointed out in Rosenblatt and Lee, m minimizing expected layout costs may not be as important as identifying layouts that perform well under a variety of flow scenarios (termed robust layouts). One of the simplest cases of layout robustness is robustness

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Table 1 Recently Published Layout Models and Algorithms by Layout Classification Scheme

Authors Year Category Notes

Aneke and Carrie 4 1986 C. 1 .b Gupta *~ 1986 B.3.a Hassan, Hogg, and Smith 49 1986 A.2.a Rosenblatt u3 1986 B. 1.a Rosenblatt and Sinuany-Stern nl 1986 B.3.a Drezner 26 1987 A.2.b Evans, Wilhelm, and Karwowski 28 1987 A.2.d Giffin and Foulds 3s 1987 A.2.b Hassan and Hogg 46 1986 A. 1 .b Jacobs 62 1987 B.3.a Montreuil, Ratliff, and Goetschalckx 1°3 1987 A.2.b Rosenblatt and Lee 112 1987 B.3.b Urban 133 1987 B.3.b Wemmerlov and Hyer ~4z 1987 C.3.a Wilhelm and Ward 144 1987 A.2.d Co and Araar z3 1988 C.3.a Heragu and Kusiak s3 1988 C.2.a Kaku, Thompson, and Baybars 68 1988 A.2.d Smith and MacLeod n° 1988 A. 1.a Malakooti s9 1989 B.3.b Malakooti and Tsurushima 9° 1989 B.3.a Montreuil and Ratliff ~0z 1989 A. 1 .a Urban TM 1989 B.3.a Wemmerlov and HyeP 43 1989 C.3.a Heragu and Kusiak s4 1990 C.2.b Kouvelis and Kiran vs 1990 C.2.a Montreuil 9s 1990 A. 1 .c Vakharia and Wemmerlov 137 1990 C.3.a A1-Hakim ~ 1991 A.2.b Foulds 3° 1991 A. 1 .b Hassan and Hogg 4s 1991 A. 1 .b Heragu and Kusiak ss 1991 A. 1 .c Huntley and Brown ss 1991 A.2.d Kaku, Thompson, and Morton 66 1991 A.2.d Kouvelis and Kiran 73 1991 B. 1.a Logendran ss 1991 C.3.1 Montreuil and Venkatadri 96 1991 B. 1 .a Raoot and RakshiP °s 1991 A.2.a Tam and Li n6 1991 A.2.a van Camp, Carter, and Vannetli 13s 1991 A.l.a Al_Hakim 3 1992 A.2.b Balakrishnan, Jacobs, and Venkataramanan 61992 B. 1 .b Banerjee et al. s 1992 A.2.c BoswelPS 1992 A.2.b Chhajeck Montreuil, a ~ Lowe2° 1992 A. 1 .a Goetsehalckx ~ 1992 A.2.b Iqarmortoslo: and Tothero 4s 1992 B.3.b I - I ~ and Alfa s~ 1992 C. 1.b t ~ , Cohen, and Cavalier 61 1992 C.3.a Jajodia et al. 6a 1992 A.2.d Kaku and Rachamadugu 6s 1992 C. 1 .b Kouvelis and Chiang n 1992 C. 1 .b Kouvelis, Kurawarwala, and Guti6rrez 7n 1992 B.3.b Leung ss 1992 C. 1 .b Montreuil and Laforge 99 1992 B. 1 .a Palekar et al) °s 1992 B. l .a Rosenblatt and Golany 1°9 1992 A. 1.a Rosenblatt and Kropp H° 1992 B.2.a

flowline layout heuristic simulation of layout flexibility SHAPE (see Section 3.3) dynamic layout problem robust layout selection scatter diagram based on eigenvectors fuzzy logic based layout construction graph-theoretic, continuous adjacency measure graph-theoretic review multiple-criteria MATCH (see Section 3.3) robust layout evaluation for QAP weighted-objective model cellular manufacturing research overview simulated annealing for QAP cellular manufacturing layout machine layout in FMS multifloor QAP quadratic set packing problem relaxation multiobjective heuristic expert system to choose layout cut trees for layout multiobjective cellular manufacturing application survey machine layout in FMS modified QAP for manufacturing MIP-approach (see Section 3.2) cellular manufacturing design graph-theoretic construction graph-theoretic text block layout from graph MIP formulation parallel computation for QAP QAP heuristic dynamic layout models operation and machine sequence dynamic layout requirements placement procedures in heuristics HAL (see Section 3.3) nonlinear optimization block layout from graph dynamic facility layout for QAP reasoning-based construction graph-theoretic construction--TESSA flow network design SPIRAL (see Section 3.3) multiobjective QAP heuristic simulated annealing for row layout simultaneous cell formation and layout simulated annealing for QAP loop/line design of FMS simulated annealing for row layout robust layout models graph-theoretic construction dynamic facility layout stochastic-dynamic QAP models alternative QAP-approach stochastic layout problem continued

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Table 1 (Continued) Recently Published Layout Models and Algorithms by Layout Classification Scheme

Authors Year Category Notes

Sarin et al. TM 1992 B.3.a Tam ns'm 1992 A.2.a Urban 135 1992 B. 1.1 Das zs 1993 C.3.2 Ho, Lee, and Moodie 57 1993 C. 1 .b Irani, Cavalier, and Cohen n° 1993 C.3.a Liao s6 1993 C. 1 .b Montreuil, Venkatadri, and RatlifW 1993 A.2.c Shang 1~6 1993 B.3.b Suresh and Sahu TM 1993 B.3.b Tate and Smith m 1993 A.2.a Urban 136 1993 B. 1 .b Welgama and Gibson ~4° 1993 C.2.b Bozer, Meller, and ErlebacheP s 1994 A.2.a Conway and Venkataramanan 24 1994 B. 1 .b Heragu and Gupta 52 1994 C.3.b Lacksonen 79 1994 A.2.c Langevin, Montreuil, and Riopel s3 1994 C. 1 .a Sirinaovakul and Thajchayapong ~ ~7 1994 A.2.d Skorin-Kapov us 1994 A.2.d Tretheway and Foote t~z 1994 A.2.b Welgama, Gibson, and Al-Hakim ~4~ 1994 A.2.b Banerjee and Zhou 9 1995 C. 1.a Hassan s° 1995 A.2.b Kim and Kim 7° 1995 A.2.b Kouvelis, Chiang, and Yu TM 1995 C. 1 .a Sarker et al. ns 1995 C.l.a Souilah m 1995 A.2.d Benjaafar, Sheikhzadeh, and Soewito ~4 1996 B.3.a Chiang and Kouvelis 2~ 1996 A.2.d Fu and Kaku 36 1996 A. 1.a Liao et al) 7 1996 C.3.a Meller and Bozer 9z 1996 A.2.a Meller and Gau 94 1996 B.3.a Goldschmidt, Takvorian, &nd Yu 4~ 1996 A. 1 .b

decision-theoretic approach to QAP simulated annealing and genetic algorithm heuristics bounds for dynamic layout flexible manufacturing system layout multi-flowline layout layout and virtual manufacturing cells cellular manufacturing along line design skeleton for MIP hierarchical multicriterion simulated annealing, multiobjective QAP FLEX-BAY (see Section 3.3) dynamic facility layout heuristic machine layout problem MULTIPLE (see Section 3.3) genetic algorithm for dynamic QAP cellular manufacturing design dynamic layout MIP spine-based layout knowledge-based, fixed-shape departments tabu-search for QAP scatter diagram to facility layout knowledge-based graph to layout single-loop layout group technology layouts graph-theoretic, distance-based row layout models one-dimensional machine location hierarchical simulated annealing design flexible layouts tabu-search work-in-process vs. layout cellular manufacturing along line simulated annealing algorithm layout objectives study graphical representation

over adjacency-based and distance-based objectives, as in the o~-weighted objective (Eq. 4). Rosenblatt and Sinuany-Stern m present an efficient procedure to determine, for a given value of a, the best layout of a set of layouts generated by existing adjacency- based or distance-based approaches. Meller and Gau 94 indicate that even when the exact value of e~ is unknown, it is better to use an estimate of it in the layout search process than to ignore it until the layout selection stage. Heuristics for the a-weighted criteria layout problem are presented in Dutta and Sahu, 27 Fortenberry and Cox, 29 and Urban. m

There is a marked increase in the number of papers in classification category C (special cases) that are motivated by specialized layout problems (as opposed to simplifications of the block layout

problem). The treatment of these problems goes bey~2d the "practical" techniques that are discussed in textbook; ,(for example, see Tompkins and White ~3° and Suie m) or "flchniques that are modeled as variations of the QAP (see Pardalos and

\

Wolkowicz, eds. 106 for the most rece]~ t book oil *~bis problem). These problems include the l~t]5~ut ° f machines along a production line, which was fi i~ t considered by Carrie ~9 and has been considered by a number of researchers since. The general detailed (machine) layout problem is addressed by solving it with a simulated annealing algorithm m as well as. in the context of flexible manufacturing systems.~J~ Finally, one sees that most papers are focused on cellular manufacturing systems layout. It is not possible to include all of the references in this area without

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A. Facility Layout Models and Heuristics for Block Layout 1. Model

a. QAP-based b. Graph-theoretic c. Mixed-integer programming

2. Heuristic a. QAP-based b. Graph-theoretic c. Mixed-integer programming d. QAP-only (equal-area departments only)

B. Facility Layout Model Extensions I. Dynamic Layout

a. Model b. Heuristic

2. Stochastic Layout a. Model b. Heuristic

3. Multicriteria, Robust, and Flexible Layout a. Model b. Heuristic

C. Special Cases 1. Flowlines, Row, and Loop Layout

a. Model b. Heuristic

2. Machine Layout a. Model b. Heuristic

3. Cellular Layout a. Model b. Heuristic

Figure 2 Facility Layout Research Classification Scheme

significant overlap with previous papers because this research also considers how cells are formed and operated. Therefore, see Wemmerlov and Hyer ~42a43 for surveys of cellular manufacturing research motivation and cellular manufacturing applications, respectively, and Hassan s° for a review of group technology layout design.

N o t a t i o n The first six items of the notation are parameters,

while the last four items are decision variables.

1. The building is L units in the x-direction and W units in the y-direction.

2. The indices i,j will be used for departments, where i,j = 1 . . . . , N.

3. Each department i has minimum area requirements of ai.

4. The upper and lower limit on the length or width of department i is denoted by ubi and lbi, respectively.

5. The minimum and maximum perimeter of department i is denoted byp~ and P~, respectively.

6. The set of positive flows is denoted by F = {~}. That is,fj > 0 V i,j E F a n d furthermore, IF{ = M. The mth positive flow, f,,, originates from department i (m) and terminates at departmentj(m).

7. The rectilinear distance between departments i a n d j is expressed as the sum of the distance in the x-direction, do", and the distance in the y- direction, doL Note that for flow m the distances in the x and y directions are denoted as dm x and d~, respectively.

8. The location of department i is indicated by its centroid, which is denoted by (x~,y~).

9. Each department is rectangular shaped; department i has dimension 2 g in the x-direction and dimension 2 wi in the y-direction.

10. The relative location decision variables are denoted by z~u and zY~. In general, the z~/decision variables determine whether two departments are to the north, south, east, or west of one another in the layout.

3.2 Montreuil's Mixed-Integer Programming Formulation

A mixed-integer programming formulation for the facility layout problem was presented by Montreuil in 19909s at a material handling research conference (a slight modification of the model is presented in Tompkins et al., ~3~ pp. 344-350). The model uses a distance-based objective but is not based on the traditional discrete (QAP) framework. Instead, it utilizes a continuous representation of a layout. The notation and general formulation follow. Note that a specialized case of this model was developed by Heragu and Kusiak s5 where the department length, width, and orientation are specified a priori.

M o n t r e u i i ' s Formulation: Relative location decision variables:

[ 0 if i must be to the west (left) of j z x - 5 [1 if the above imposition is not enforced

ifimu t o e= a ove, 'J i f the above imposition is not enforced

min Z f , , ( d ~ + d ~ ) II1

s.t. d x >- xum ) - xj(,.)

d x >_ xj(,.) - Xi(m)

V m

Vm

(5)

(6)

(7)

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d~ > Yio,) - Yj(,,) V m (8)

d~ > Yjt,,) - Yum) V m (9)

4 < xi < L - 4 Vi (10)

w~ <- yi <- W - wl V i (11)

Ib~ <<. 2 4 < ub, Vi (12)

Ibi < 2 w~ < ub~ V i (13)

2 < z ~ + z ) ~ + z . Y . + z . y. < 3 V i , j ; i < j (14) tj i t

xl + 4 < xj - ~ + Lz'~ Vi , j (15)

xj + ~ < xi - 4 + Lz~, Vi , j (16)

y~ + wi < yj - wj + Wz~ V i,j (17)

yj + wj < y i - wi + Wzj~ Vi , j (18)

p, < 4(4 + w~) _< P~ Vi (19)

The objective (Eq. 5) is based on flow times rectilinear distance between department centroids. The standard linear programming trick is used to lin- earize the absolute values in the distance function (see Eqs. 6-9). In Eqs. (10) and (11), each department is constrained to be within the facility, while in Eqs. (12) and (13) the maximum and minimum lengths of the department rectangles are constrained. The constraint set (Eq. 14) ensures that relative department location constraints are relaxed in two or three directions. In Eqs. (15)-(18), the relative location decision variables are utilized to ensure that departments do not overlap. Finally, in Eq. (19), a bounded perimeter constraint is used as a surrogate area constraint because the actual area constraint, 44 wi -- ai, is nonlinear. Note that the formulation is easily extended to model fixed departments and other linear side constraints.

Although this mixed-integer programming approach is powerful and holds much promise, currently only problems with six or less departments can be solved optimally. Consequently, the solution approach heuristically sets the binary decision variables and solves the resulting linear program 97 (more heuristic adaptations are discussed later).

3.3 Recent Heuristics A comprehensive investigation of the facility lay-

out problem literature includes examining heuristics for this problem. However, because the applicability and performance of one heuristic over another can be

subjective and based on many factors (Tompkins and White, ~a° p. 295), a qualitative description of these algorithms is presented (arranged by the objective the algorithm is based on and the year it was published). All of the algorithms are not presented due to space constraints, but a sample was chosen that includes the algorithms that are commercially available as layout soRware packages (discussed further in Section 4). (Two older algorithms, and a few other algorithms, are added as well to give a perspective on how the state of the art has evolved.) The reader can use Table I to identify other algorithms similar to the algorithms in this sample.

Adjacency -Based A lgor i thms Adjacency-based algorithms are usually incorpo-

rated within a graph-based approach, where the objective function of the algorithm is Eq. (1). As mentioned previously, there is much research on the various stages of the graph-theoretic approach. 2as,4s,49 After presenting an archetypal graph construction- type algorithm, two of the most recent adjacency- based heuristics are described.

Deltahedron Approach. One of the most widely cited adjacency-graph construction approaches is the Deltahedron Approach (DA) of Foulds and Robinson. 33 The DA proceeds by determining the sequence that nodes will enter the graph. At any stage, a node is entered into the center of the face (a triangle formed by three nodes) in the graph that will maximize the adjacency benefits with the other departments in the face. Thus, a planar graph is always maintained in DA, which allows for an easi- er transformation to a block layout. Many heuristics have been developed in an attempt to improve on DA's performance (for example, see Al,blakim 2 and Boswell~S), and Leung s5 represents the most recent algorithm that addresses this problem. The DA has also been modified to consider a continuous relaxation of the adjacency decision variables using a shortest path approach) 8

MATCH. MATCH, developed by Montreuil, Ratliff, and Goetschalckx, 1°3 is an interactive construction-type approach that utilizes a discrete representation and integer programming to solve a b- matching problem. Their algorithm attempts to find a matching that maximizes the adjacency score while satisfying the lower and upper bound on the number of matches with each department, and the

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total number of times a department must be matched with all other departments. The algorithm considers the number of adjacent segments when computing adjacency scores. The departments generated by MATCH are all rectangular in shape and the approach is iterative, based on user input.

SPIRAL. SPIRAL, created by Goetschalckx, 4° develops an adjacency graph and then a block layout from the graph. SPIRAL utilizes the concept of "relationship tuples" to construct an adjacency graph, where tuples quantify the relationship between one department and other departments. The graph remains planar due to its hexagonal structure and is used to construct an approximate relative location diagram by fitting the unequal-area departments into a row-and-column structure. SPIRAL compares favorably to layouts generated by other approaches.

Distance-Based Algorithms The following algorithms employ a distance-

based objective like Eq. (2) and are presented in chronological order. SHAPE, 49 NLT, 13s and QLAARP s

are construction algorithms, and the others are improvement algorithms.

CRAFT (Computerized Relative Allocation of Facilities Technique). CRAFT is the archetypal improvement-type approach and was developed by Armour and Buffa s in 1963. CRAFT begins by determining the centroid of each department in the initial layout. It then performs two-way or three-way exchanges of the centroids of nonfixed departments that are also equal in area or adjacent in the current layout. For each exchange, CRAFT will calculate an estimated reduction in cost and it chooses the exchange with the largest estimated reduction (steepest descent). It then exchanges the departments exact- ly and continues until there exists no estimated reduction due to two-way or three-way exchangesl Constraining the feasible department exchanges to those departments that are adjacent or equal in area is likely to affect the quality of the solution, but it is necessary due to its exchange procedure. The exchange procedure has also been criticized because it may lead to departments with irregular shape. 34~6,13°

SHAPE. SHAPE, developed by Hassan, Hogg, and Smith, 49 is a construction algorithm that utilizes a discrete representation and an objective based on rectilinear distances between department centroids. The department selection sequence is dependent on a

ranking, which is based on each department's flows and a user-defined critical flow value. Department placement begins at the center of the layout. Subsequent department placement is based on the objective function value with the department placed on each of the layout's four sides. The algorithm is easy to implement; however, because the department shape is controlled by the objective function, the shape of departments may deteriorate toward the end.

NLT (Nonlinear optimization Layout Tech- nique). NLT, a construction algorithm developed by van Camp, Carter and Vannelli, 13s is based on nonlinear programming techniques and utilizes Euclidean distances between department centroids. In the NLT model, there are three sets of constraints: departments cannot overlap, cannot be located outside the facility, and cannot be assigned area less than required. The constrained model is transformed to an unconstrained form by an exterior point quadratic penalty function method. With a three-stage approach, successively more difficult problems are solved using the solution from the previous stage as an initial solution point. The department shapes are all rectangular.

QLAARP (Qualitative Layout Analysis using Automated Recognition of Patterns). QLAARP, a construction approach that was developed by Banerjee et al., s uses qualitative layout anomalies (QLAs) to set binary variables in Montreuil's MIP. 9s That is, the algorithm heuristically uses context-based information to reduce the solution tree. A design skeleton is used to structure the QLAs. Other reasoning-based approaches during this time period include Sirinaovakul and Thajchayapong, n7 and for graph-theoretic approaches, Welgama, Gibson, and A1-Hakim. 14~

LOGIC (named for Layout Optimization using Guillotine-lnduced Cuts). LOGIC is an improvement-type algorithm developed by Tam, ~2s where the layout is represented as a collection of rectangular partitions organized as a slicing tree. A slicing tree consists of branches and branching operators that specify whether the departments on opposite sides of a branch are to the left, right, above, or below each other. With a given slicing tree and department area values, the layout can be determined by recur- sively partitioning a rectangular area by placing the departments into the area according to the four specific branching operators. Because this approach is likely to produce long and narrow department

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shapes, two shape constraints are added as a penalty function to the objective. The algorithm uses simulated annealing in an attempt to find a better layout by two-way exchanges of branching operators. The final layout of this algorithm has all rectangular shapes, except for potentially those departments that are placed near fixed departments. Tam also utilized the same representation in a genetic algorithm search structure) 27 LOGIC appears to outperform the algorithm presented in Tam) 27

MULTIPLE (MULTI-floor Plant Layout Eval- uation). MULTIPLE, a single or multifloor improvement-type algorithm developed by Bozer, Meller, and Erlebacher, ~8 uses the objective function Eq. (3). MULTIPLE uses a discrete representation and extends CRAFT by applying spacefilling curves to single floor or multi floor facility layout problems (also see Kaku, Thompson, and Baybars 6s for another multifloor layout algorithm developed during this time period). MULTIPLE improves CRAFT by increasing the number of exchanges considered at each itera- tion. In addition, MULTIPLE can restrict the irregularity of department shapes by using an irregularity measure based on the perimeter and area of each department; however, because it uses a discrete representation, the department shapes may not be rectangular. MULTIPLE, like CRAFT, is a steepest- descent search and may be affected by the initial layout. SABLE 92 extends MULTIPLE by employing a simulated annealing based search and by generaliz- ing the department-exchange algorithm. SABLE is shown to produce lower cost layout solutions than MULTIPLE or LOGIC.

FLEX-BAY (named for FLEXible BAY structure). FLEX-BAY is an improvement-type algorithm based on a continuous representation developed by Tate and Smith. 12s A dynamic penalty function is used to evaluate the shape-constrained unequal area facility layout problem. A layout is represented by a flexible number of vertical bays of varying width, each divided into one or more rectangular departments. Encoding flexible bay layouts is a two-part representation: permutation of the departments and breakpoints for the bays. FLEX-BAY utilizes a genetic algorithm to search the solution space by varying department-to-bay assignments or by adding or removing a bay breakpoint. The algorithm generates good layouts and was shown to outperform CRAFT and NLT.

4. Perspectives on the Facility Layout Problem

A listing of facility layout software packages is presented along with the algorithms on which they are based. Then, perspectives on the future of facility layout research are presented.

4.1 Facility Layout Software Packages Certainly facility layout research has always had

some effect on the practice of facility layout problem-solving; however, this has not been as evident in the past as it is today due to the increased availabil- ity of facility layout software packages. And although these packages do not correspond to the most ambitious facility layout research (detailed next), they do represent currently available research methodologies, as presented in Section 3.3.

According to an article in the August 1995 issue of liE Solutions, 119 there are four layout software packages (available in English) that incorporate an algorithm for layout generation (in addition to layout evaluation). The attributes of each software package are comprehensively analyzed in the liE Solutions article, and therefore, the analysis will not be repeated here. However, each package is listed (in alphabetical order by distributor) with its associated algorithm. The interested reader can then refer back to Section 3.3 to review the corresponding algorithms and refer to the article for the details on the software itself.

1. FactoryOPT by CIMTECHNOLOGIE incorpo- rates a licensed version of the SPIRAL algorithm 4° as well as some CRAFT-like 5 improvement routines to provide the user with a choice of algorithms. Previous layout packages by CIMTECH- NOLOGIES (for example, FactoryPlan and FactoryFlow) were based on a computerized graphical representation of the manual systematic layout procedure (SLP) developed by Muther. 1°4

2. SPIRAL is distributed by Marc Goetschalckx and, as the name implies, is his implementation of the SPIRAL algorithm 4° with other options for improvement routines.

3. LayOPTby the Production Modeling Corp. is an implementation of MULTIPLE n and SABLE. 92

4. Factory Modeler by Syst6ms Espace Temps, Inc., implements the MIP-based approach discussed in Section 3.2. Various procedures are used to set the binary variables in the MIP (such

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as the cut-tree design skeleton approach outlined in Montreuil, Venkatadri, and RatlifPT).

4.2 Perspectives on Future Research Directions As noted previously in Section 3.3, research on

the facility layout problem is not converging--rather it is somewhat diverging, with research into three separate modeling frameworks. Also, there are more papers today on extensions to the general problem and on investigations into objective functions that include more relevant layout criteria. This indicates not only great interest in the problem but also a healthy critical self-examination of previous research. This self-examination is continued here.

Every two years, the Material Handling Institute of America, its sponsoring companies, and other sup- porters (like the National Science Foundation) spon- sor a material handling research colloquium where researchers are asked not only to present their research but also to form task groups to debate the future of material handling research. Because material handling and the facility layout problem are relat- e& many of the conclusions drawn at these colloquia hold for the layout community as well. One of the conclusions reached at the 1994 colloquium notes "a lack of concurrent engineering of material handling systems with respect to product and process design activities" (Graves et al.,4a p. 17). The analogous con- clusion with respect to facility layout research is that there is a lack of concurrent engineering of facility layout with respect to material handling system design. Taken one step further, there is also a lack of concurrent engineering of facility layout with respect to production system design.

The lack of concurrent engineering with respect to material handling system design is clear. Layout decisions are predicated on the assumption that there is a set of material handling alternatives to choose from and that some decision is made before the layout is designed as to how unit loads will be moved. For example, consider using the distance-based objective, Eq. (2). This requires the user to supply cij values before the problem is solved. Because the c o

values should be based on the material handling method that will be used for that unit load, this assumes that one can make that determination a priori. However, which material handling method is used is often dependent on the distance the load must travel, which, of course, is dependent on the

final layout. Furthermore, the size of the unit load between two operations is often dependent on the distance between those operations. These two obser- vations would imply that all three factors in Eq. (2) are decision variables, as opposed to only one of them as is assumed in almost every approach to date.

Now consider the perspective that facility layout decisions and production system design decisions must be made concurrently. With today's emphasis on supply chain analysis, increased coordination between various stages of production, and an overall increase in production system analysis, it appears that analysis tools that support understanding of proposed changes at the interface of two systems are of more value than ever. Moreover, due to the continu- ing emphasis on shorter lead times, smaller lot sizes, and increased flexibility, the production system operations must change over time in any organization. The question for the facilities research community is whether they have the tools to support facility layout design decisions to serve the production system effectively with respect to these changes. It would appear that the community does not because current facility layout software packages and most research papers are focused on finding the best layout for a given production system design.

Increased emphasis is needed at the interface of facility layout and production system design problems: determining the number and variety of machines, defining departments and manufacturing cells, determining alternative product routings, defining and determining actual production routings, determining unit load sizes and production batch sizes, defining department shapes, and so on. Solving these higher level problems, in parallel with the facility layout problem, will provide more oppor- tunities for improvement than focusing on the facility layout problem alone. These improvements will impact design costs, throughput times, inventory costs, quality issues, and so on, in addition to material handling costs. Future facility layout tools should prescribe production system designs as well as evaluate the layout for a particular production system.

Future research is needed in areas that break the sequential processing of layout and material handling system design, and layout and production system design. Concurrent engineering must be used to ensure that solving the facility layout problem is not finding the best layout for the wrong production sys-

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tem. To advance the field, research must focus on more high-level planning and emphasize metrics not associated with layout (inventory, quality, confor- mance to agile manufacturing, and so on).

4.3 Emerging Research: The Layout Community's Perspective

To this point, this paper has focused on papers that have already appeared (or are scheduled to appear) in the published refereed literature. Emerging research is now presented to indicate the future directions of layout research from the layout community's perspective.

There is interest in refining the estimation of distances between two departments. In an attempt to remove the limitations of the CTC distance measure, the expected distance (EDIST) measure has been developed by Bozer and Meller, ~7 where this measure is mathematically defined as the expected distance between any two points in each department pair. Banerjee and Mehrotra 7 have investigated solution procedures to detailed layout problems (for a given initial layout) where the distance between departments is measured with the rectilinear and/or Euclidean distance metric depending on the nature of the flow. Their work is also unique because it is the first to employ interior-point programming methods to the facility layout problem.

Attempting to refine the distance measure is a first step to concurrent engineering of the facility layout problem and the material handling system design problem. Due to the difficulty of these two problems, it is not possible to optimally solve them concurrently (unless special cases are considered, such as a "loop" material handling systemg). Nonetheless, integration attempts are emerging; examples include: Montreuil, 98 who attempts to integrate the flow path design problem into the facility layout problem; Yang and Peters, z46 who consider a similar problem in a flexible manufacturing system; Goetschalckx, 41 who attempts to represent both problems jointly by formulating one large-scale mixed-integer programming problem; Lacksonen, 81 who attempts to integrate the material flow problem into a concurrent engineering design framework; Krishnasami and Banerjee, 77 who examine material flow congestion and its impact on the manufacturing facility layout; and Ioannou and Minis, s9 who consider integrated layout and material handling decision-making in manufacturing shop design.

Integrating layout design and the production system operation is also likely to receive more attention in the future. Montreuil et al. 1°~ consider the problem of determining the facility layout in a chaotic system (a production system in which little can be predicted a priori). They term this type of layout a holographic layout because small portions of the layout con- tain the range of the facility's production processing capabilities. Montreuil and Lefranqois l°° character- ize layouts by the responsibilities assigned to organizations, departments, or machines. Their concept of a responsibility network is used as a production system/facility layout classification scheme.

Bozer and Kim ~6 illustrate the interrelationship of the production system, facility layout, and material handling system when they vary the transfer batch size. Their model illustrates, for a fixed material handling system capacity, that improvements to the layout provide an opportunity for decreasing work- in-process inventory. Fu and Kaku 36 utilize an open queuing network to evaluate changes in work-in- process inventory with respect to different facility layouts of equal-area departments.

Given the modeling capabilities of Montreuil's MIP approach, there is considerable activity in developing heuristics to set the binary variables of the model. For a case where the material handling network follows a loop design, Montreuil's MIP is heuristically solved with a genetic algorithm in Banerjee and Zhou. 9 Various interactive systems for setting the binary variables have been attempted in Banerjee et al. s,u Furthermore, a two-stage (construction, followed by improvement) genetic algorithm to solve the layout MIP with I/O points has been developed by Banerjee, Zhou, and MontreuilJ ° After using some graph-theoretic techniques to construct relative department locations, a genetic search algorithm is employed to improve the layout. Finally, the quadratic assignment problem heuristic developed by Lacksonen and Encore s° may be used to similarly set binary variables in an alternative-layout MIP formulation. 79 Lacksonen s2 also sets a subset of the binary variables in Lacksonen 79 using estimated locations, department areas, and flow costs.

Graph theory models continue to receive attention in the facility layout arena. Smith m presents a design methodology and algorithm for incorporating graph results from the Steiner Star Duals by integrating a quadratic set packing model, Delaunay tri-

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angulation, and Voronoi diagram constructions. Cimikowski and Mooney 22 study the problem of relaxing the binary adjacency measures in a similar fashion as Giffin and Foulds, 3s although neither one considers department areas, as does the approximation adopted by Foulds and Giffin. 31 W/ischer and Merker 139 evaluate heuristics for constructing the adjacency graph as well as present two new heuristics for this problem. 95

In addition to the heuristics listed in Section 3, there is emerging research into applying genetic algorithms i° and tabu search 67 to the facility layout problem. Of course, these improvement algorithms rely on initial starting points. Surprisingly, there is little recent research into determining these starting points with construction-type approaches (for example, Banerjee et al., l° Chiang and Kouvelis, 21 and Kaku and Mazzola 67 utilize random starting points). However, for multifloor problems, two-stage construction-type approaches have been developed by Abdinnour-Helm and Hadley ~ and Meller and B o z e r . 91 The multifloor layout problem relates close- ly to the multibay manufacturing layout problem as defined by Meller, 93 where a construction-type approach is presented along with its application at a manufacturing facility.

Finally, emerging research is appearing on group technology, cellular manufacturing, and flexible machine system applications (for example, Bemelman and Jarvis, ~3 Liao et al., s7 Peters and Yang, ~°7 and Yang and Petersm*5). Note that in Peters and Yang ~°7 the authors combine graph-theoretic approaches like that of Goetschalckx *° with a mixed-integer programming approach, while in Yang and Peters t4s the authors address integration of flexible machine system design with respect to production system dynamics over time.

5. Conclusions The trends of facility layout research over the past

1 0 years (since the most recent layout review paper) were presented. Ninety-one recent layout papers were identified and summarized with a new layout research classification scheme. A sample of the algorithms were presented, concentrating on those algorithms that form the foundation of computerized layout software.

Perspectives on the future research that is needed in facility layout were also presented. It was argued that advances are likely to be made only by concur-

rently addressing layout and production system design issues. The emerging research on this problem indicates that considerable attention is likely to continue on this problem.

Acknowledgments The authors would like to thank many members

of the layout community for their help in contribut- ing to the completeness and accuracy of this paper. This material is based on work supported by the National Science Foundation under Grant No. DMII-9412646 and the Russell Corporation under Auburn University Administrative Digest Nos. 1222-94CG and 878-95CG.

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Authors' Biographies Russell D. Meller is an assistant professor of industrial and systems

engineering at Auburn University. He received his BSE, MSE, and PhD in industrial and operations engineering from the University of Michigan. His dissertation was awarded the 1994 Institute of Industrial Engineers Outstanding Dissertation Award and first prize in the 1993 College on Location Analysis (in the Institute of Management Sciences) Dissertation Prize Competition. His research interests include facility layout, material handling systems, warehousing systems, operations research applications, and preventive maintenance programs. He is a member of liE, INFORMS, and Alpha Pi Mu.

Kai-Yin Gau is a PhD student in the industrial and systems engineering department at Auburn University. She holds an MIE degree from Auburn and received her BS degree from National Chiao Tung University in Taiwan. Her dissertation topic is titled "Facility Layout Design with an lterative Algorithm." She is a student member of liE and INFORMS.

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Documents

The facility layout problem: Recent and emerging trends and perspectives