optimization of block layout design problems with unequal areas: a

Computers and Chemical Engineering 30 (2005) 54–69

Optimization of block layout design problems with unequal areas:A comparison of MILP and MINLP optimization methods

Ignacio Castilloa,∗, Joakim Westerlundb, Stefan Emetb, Tapio Westerlundb

a School of Business & Economics, Wilfrid Laurier University, Waterloo, Ont., Canada N2L 3C5b Process Design Laboratory, Abo Akademi University, Biskopsgatan 8, FIN-20500 Abo, Finland

Received 9 November 2004; received in revised form 26 July 2005; accepted 26 July 2005Available online 6 October 2005

Abstract

The block layout design problem with unequal areas, which was originally formulated by Armour and Buffa in the early 1960s, is a fundamentaloptimization problem encountered in many manufacturing and service organizations. In this paper, we present a new modelling framework foreffectively finding global optimal solutions for the block layout design problem with unequal areas. The most fundamental aspect of the frameworkconsists of an exact representation of the underlying area restrictions. Our computational results consistently yield optimal solutions on severalw timizationm ut solutionsc ass of haf n problems.©

K timizationm

1

ffffidIpmaplMa

d

eflectrdingated asflows. Thet andstric-ithint be

stud-antsed to

lem:blem,siblens ofitionsryingrob-

0d

ell-known test problems from the published literature. Furthermore, different mixed-integer linear and mixed-integer nonlinear opethods are compared. Our study indicates that the new modeling framework together with simple constraints to avoid symmetric layo

an be successfully used to find optimal layout solutions; therefore, seriously challenging other optimization methods on this important clrd,undamental problems. The new modeling framework may easily be applied in the context of the process plant layout and piping desig

2005 Elsevier Ltd. All rights reserved.

eywords: Facilities planning and design; Block layout design problem with unequal areas; Exact convex models; Comparison of MILP and MINLP opethods

. Introduction and motivation

The block layout design problem with unequal areas is aundamental optimization problem encountered in many manu-acturing and service organizations. The problem was originallyormulated byArmour and Buffa (1963)and is concerned withnding the most efficient arrangement of a given number ofepartments with unequal area requirements within a facility.

n the context of the process plant layout and piping designroblems, the block layout design problem and the proposedodeling framework may easily be applied to find single-floorrrangements of processing equipment and the interconnectingipework. For further details on the process plant layout prob-

em, the reader is referred toGeorgiadis, Schilling, Rotstein, andacchietto (1999),Papageorgiou and Rotstein (1998), Patsiatzisnd Papageorgiou (2002).

As defined in the literature, the objective of the block layoutesign problem is to minimize the cost associated with projected

∗ Corresponding author. Tel.: +1 519 884 0710; fax: +1 519 884 0201.E-mail address: [email protected] (I. Castillo).

interactions between departments. The interactions may rthe cost of material-handling flows or the preference regaadjacencies between departments, where the cost is calculthe rectilinear distance multiplied by the material handlingor adjacency score between the centers of department pairproblem is subject to two sets of constraints: (a) departmenfloor area requirements and (b) department locational retions; that is, departments cannot overlap, must be placed wthe facility, and some must be fixed to a location or cannoplaced in specific regions.

The block layout design problem has been extensivelyied in the literature. For detailed surveys of the different variand solution methods for the problem, the reader is referrKusiak and Heragu (1987), Meller and Gau (1996). Here, wenote that the block layout design problem is a hard probeven the variant known as the quadratic assignment prowhere the optimisation is carried over a finite set of posdepartment locations, is NP-hard. The added complicatiounequal area requirements with continuous department posthat can be anywhere in a rectangular floor area and vawidth and height dimensions make the block layout design plem extremely challenging to solve.

098-1354/$ – see front matter © 2005 Elsevier Ltd. All rights reserved.
oi:10.1016/j.compchemeng.2005.07.012

I. Castillo et al. / Computers and Chemical Engineering 30 (2005) 54–69 55

In order to handle departments with unequal areas, previousresearch into the block layout design problem has focused onthe development of solution methods based on graph theoreticalapproaches and various discrete representations that divide thefloor area into a grid of equal-sized squares. Graph theoreticalapproaches do not take into account department area require-ments: forcing the facility designer to use such information toconstruct a layout solution once a decision regarding the pairsof departments that should be adjacent has been made. (See, forexample,Foulds, 1983; Seppanen & Moore, 1970.) The layoutconstruction process is a time-consuming and difficult task thatis often achieved by trial and error. Moreover, the constructedlayout solution is not guaranteed to satisfy the prescribed adja-cencies(Heragu, 1997).

Solution methods based on discrete representations dividethe floor area into a grid of equal-sized squares and representdepartments as combinations of such squares. (See, for exam-ple,Bazaraa, 1975; Hassan, Hogg, & Smith, 1986.) Due to thediscrete representation of the floor area, the width and heightdimensions and possible positions in which departments can beplaced are often determined by the facility designer before thesolution method is executed. Moreover, the discrete represen-tation frequently causes the layout solution to contain irregularshaped departments that are not feasible for an actual implemen-tation. Some procedures, nonetheless, place restrictions on theshapes of the composite departments to make the problem moret

theuV nen(Sv sa

lay-oa post thew ala f sizw ec td urd hd ioα ump ionstth ert an be ds ot (them ncyc

c asf

BLDP0 : minN−1∑i=1

N∑j=i+1

cij(|xi − xj| + |yi − yj|) (1)

s.t. Department area constraints,

Overlappreventionconstraints,

xi + 12wi ≤ wF, i = 1, . . . , N, (2)

xi − 12wi ≥ 0, i = 1, . . . , N, (3)

yi + 12hi ≤ hF, i = 1, . . . , N, (4)

yi − 12hi ≥ 0, i = 1, . . . , N, (5)

wlowi ≤ wi ≤ w

upi , i = 1, . . . , N, (6)

hlowi ≤ hi ≤ h

upi , i = 1, . . . , N, (7)

wlowF ≤ wF ≤ w

upF , (8)

hlowF ≤ hF ≤ h

upF . (9)

Department locational restrictions are enforced via constraints(2)–(5), which require that all departments are placed withinthe facility. Also, note that the ability to specify bounds on thedesired dimensions of the floor area of the facility is availablei ndsw eflt an bee

weent thata ery,e xingt hapea them ionsw

thef ntionc

1

nti o theb trainta nstrainti( thep

4

d tol3 of

ractable and implementable.Recently, attention has shifted to the direct solution of

nderlying optimization problem. See, for example,Anjos andannelli (in press), Castillo and Westerlund (2005), Lackso1994),Meller, Narayanan, and Vance (1999),Montreuil (1990),herali, Fraticelli, and Meller (2003), Tam and Li (1991), andan Camp, Carter, and Vannelli (1991). This paper falls into thirea of research.

We begin by presenting a generic model for the blockut design problem. (The reader is referred toAppendix Aforsummary of the nomenclature used in this paper.) Sup

hat we need to find the positions within a facility andidth and height dimensions ofN departments with unequrea requirements. Let the floor area be a rectangle oF × hF and, for each departmenti, denote its position by thoordinates of its center as (xi, yi) and its width and heighimensions aswi and hi, respectively. Hence, there are foecision variables for each departmenti. Furthermore, eacepartmenti is required to have an areaai and an aspect rati ≥ 1, which, for practical reasons, delineates the maximermissible ratio between its longest and shortest dimens

hat is,αi ≥ maxwi, hi/ minwi, hi, i = 1, . . . , N. In addi-ion, we denote any valid lower and upper bounds onwi andi as (wlow

i , wupi ) and (hlow

i , hupi ), respectively. We show lat

hat the aspect ratio requirement for each department cnforced via suitable choices of the lower and upper boun

he width and height dimensions. Given the relationshipaterial-handling flow or the preference regarding adjace

ij between departmentsi and j, wherecij ∈ R, cij = cji, andii = 0, we can formulate the block layout design problemollows:

e

e

;

en

)

n BLDP0 in the sense that it allows the user to specify boulowF ≤ w

upF on the width andhlow

F ≤ hupF on the height of th

oor area. In particular, if the actual widthw and heighth ofhe floor area are known in advance, such dimensions cnforced by settingwlow

F = wupF = w andhlow

F = hupF = h.

The model above can be generalized to distinguish betwo kinds of departments: those that are mobile and thosere fixed to a location within the facility (e.g., heavy machinlevator shafts, etc.). This can be accomplished by simply fi

he coordinates of the centers of the fixed departments spriori. In addition, locational constraints can be added toodel to forbid departments to be placed in specific regithin the facility.In order to put our contribution in context, we now discuss

ormal modeling of the department area and overlap preveonstraints.

.1. Department area constraints

In principle,wihi = ai must hold true for each departme. We note, however, that most of the previous research intlock layout design problem uses a bounded perimeter conss a surrogate area constraint because the actual area co

s nonlinear (specifically, nonconvex and hyperbolic).Montreuil1990) linearizes the area constraint by placing limits onerimeter of each department as follows:√

ai ≤ 2(wi + hi) ≤ 2√

ai(1 + αi)/√

αi. (10)

Unfortunately, this area constraint linearization can leaarge errors in the final area of each department. Forαi = 2,, 4, and 5, constraint(10) is satisfied even if the final area

56 I. Castillo et al. / Computers and Chemical Engineering 30 (2005) 54–69

Fig. 1. Department area constraint.

departmenti is less thanai by 11, 25, 36, and 44%, respec-tively. Lacksonen (1994)proposes a piecewise linearization ofthe nonlinear area constraint that adds two binary variables foreach department to limit these errors. This results in area errorthat are betweenai and 1.03ai whenαi = 2.Meller et al. (1999)further propose a linearization that adds one real variable foeach department, which results in final department areas thaare constrained to be greater than 97.5, 97.5, 93.7, and 85.7%ai for αi = 2, 3, 4, and 5, respectively.

In general, the use of a bounded perimeter constraint assurrogate area constraint increases the number of feasible soltions, adds unnecessary restrictions to the problem, and oftesignificantly violates the actual area constraint, thus producingsolutions that are infeasible to the underlying optimisation prob-lem (Tompkins et al., 1996). Such a violation produces layoutsolutions that are different from the global optimal solution interms of the arrangement of departments within a facility. This,even if the areas are ‘massaged’ to reach feasibility, the layousolution would often be sub-optimal.

Consider departmenti of width wi and heighthi that isrequired to have an area ofai and an aspect ratio ofαi.Fig. 1 illustrates the combinations ofwi and hi that are fea-sible to the constraintswihi = ai, wi ≤ αihi, andhi ≤ αiwi.These width and height combinations lie on the nonconvex andhyperbolic curve between the depicted pointsP1 = (wlow

i , hupi )

andP2 = (wup, hlow), wherewlow = hlow = √

ai/αi andwup =

h

o icalla when rr n tha ntials port

are needed to reduce the area violations to acceptable levels.Castillo and Westerlund (2005)proposed a linear,ε-accuraterepresentation of the underlying nonconvex and hyperbolic arearestrictions that guarantee that, at optimality, the final area ofeach department is within anε% error of the required area. Itis clear then that existing mixed-integer linear programming(MILP) models can only approximate the nonlinear area con-straint with a given accuracy.

To our knowledge, the papers byTam and Li (1991)andvanCamp et al. (1991)are the only ones that use the actual areaconstraintwihi = ai in the published literature. Both papersproposed optimization methods that transform the underlyingnonlinear optimization problem from a constrained form into anunconstrained form using penalty function methods. Given thatthe area constraint is nonconvex and hyperbolic, penalty func-tion methods are guaranteed to find only a local minimum to theoptimization problem(Bazaraa, Sherali, & Shetty, 1993). There-fore, existing nonlinear programming models, although accuratein terms of modeling the area restrictions, have failed to use mod-elling techniques and optimization methods that guarantee thatthe computed layout solutions are globally optimal.

1.2. Overlap prevention constraints

The overlap prevention constraints are specified by projectinga department on the vertical and horizontal axes. InTam andL nc , inv s:

|∀

|∀T ters oft theirh t-m f thed hori-z raint( ontn

a n oft pp

X

X

Y

Y

X

i i i i iupi = √

aiαi.Al-Khayyal, Goetschalckx, and Van Voorhis (1997)devel-

ped a row generation branch-and-cut technique to dynamdd tangential supports of the actual area constraint as andeeded. More recently,Sherali et al. (2003)proposed a lineaepresentation of the underlying area restrictions based o

priori generation of a common given number of tangeupports per department. It is not clear how many sup

s

rt

of

au-n

t

yn

e

s

i (1991) andvan Camp et al. (1991), the overlap preventioonstraints are particularly difficult to handle. For instancean Camp et al. (1991), such constraints are given as follow

xi − xj| − 12(wi + wj) ≥ 0 if |yi − yj| − 1

2(hi + hj) < 0,

1 ≤ i < j ≤ N, (11)

yi − yj| − 12(hi + hj) ≥ 0 if |xi − xj| − 1

2(wi + wj) < 0,

1 ≤ i < j ≤ N. (12)

hese constraints state that if the distance between the cenwo departments on the vertical axis is less than the sum ofeights (i.e., if constraint(11) is active, seeFig. 2a), the deparents cannot overlap on the horizontal axis. Alternatively, iistance between the centers of two departments on theontal axis is less than the sum of their widths (i.e., if const12) is active, seeFig. 2b), the departments cannot overlaphe vertical axis. Note that constraints(11) and (12) are bothon-differentiable and nonconvex.

Anjos and Vannelli (in press)remark that constraints(11)nd (12) are also disjunctive and propose the introductio

wo new continuous variables,Xij andYij, such that the overlarevention constraints can be stated as follows:

ij ≥ 12(wi + wj) − |xi − xj|, ∀1 ≤ i < j ≤ N, (13)

ij ≥ 0, ∀1 ≤ i < j ≤ N, (14)

ij ≥ 12(hi + hj) − |yi − yj|, ∀1 ≤ i < j ≤ N, (15)

ij ≥ 0, ∀1 ≤ i < j ≤ N, (16)

ijYij = 0, ∀1 ≤ i < j ≤ N. (17)


Fig. 2. Separation illustration.

The presence of the bilinear complementarity constraints(17) means that the above constraint set is an instance of onewith equilibrium constraints. It has been observed that sequen-tial quadratic programming methods have been very successfulwhen applied to mathematical problems with equilibrium con-straints(Fletcher, Leyffer, Ralph, & Scholtes, 2002). One diffi-culty for such methods, however, is that constraints(17) implythat at any feasible solution eitherXij = 0 orYij = 0, and hencethe gradients of the active constraints, are linearly dependent forall the feasible solutions. In order to avoid computing the gra-dients, the bilinear complementarity constraints(17) could behandled numerically as inRamanaguthran and Biegler (2003),although the underlying constraints remain nonconvex.

1.3. Contribution of this paper

Our contribution is a modeling framework for effectivelyfinding global optimal solutions for the block layout design prob-lem. Our framework differs from existing MILP models in thatthe underlying nonconvex and hyperbolic area restrictions arerepresented exactly and from existing nonlinear programmingmodels in that we use modeling techniques and optimizationmethods that guarantee that the computed solutions are globallyoptimal (if the optimisation method is not prematurely termi-nated). Thus, our layout solutions are guaranteed to be optimala

us ts xacr t fund isiov etricc withs con-ta noi ven-tV l.( pro-pS les.A ticar siven ourn

a twoc qua

areas, proposes simple constraints to avoid symmetric layoutsolutions, and briefly presents the different optimization meth-ods for computing global optimal solutions that are used in ourcomparative study. In Section3, we present numerical resultsfor several well-known test problems that have appeared in thepublished literature. In this section, we present numerical resultsfor different mixed-integer linear and mixed-integer nonlinearoptimization methods. Finally, our conclusions and directionsfor future research are given in Section4.

2. Modeling framework

Suppose that we need to find the positions within a facilityand the dimensions ofN departments with unequal area require-ments. Let the floor area be a rectangle of sizewF × hF and, foreach departmenti, denote its position by the coordinates of itscenter by (xi, yi) and its width and height dimensions bywi andhi, respectively. Each departmenti is required to have an areaai. Then, given the relationship (the material-handling flow orthe preference regarding adjacency)cij between departmentsiandj, wherecij ∈ R, cij = cji, andcii = 0, we can re-formulateBLDP0 as follows:

BLDP1: minN−1∑i=1

N∑j=i+1

cij(|xi − xj| + |yi − yj|) (18)

s

w

∀

∀

∀

∀

X

Y

N -i ., an

nd feasible for an actual implementation.The use of the proposed modeling framework enables

olve several challenging test problems to optimality using eepresentations of the underlying area restrictions. The mosamental aspects of the framework consist of using (1) a decariable transformation in a convex model, and (2) symmonvex lower bounds in a relaxed convex model, togetherimple constraints that avoid symmetric layout solutions. Inrast to the approach used inLacksonen (1994)andMeller etl. (1999), our representation of the area restrictions does

nvolve any additional variables. Moreover, the overlap preion constraints, in contrast to the ones proposed byAnjos andannelli (in press); Tam and Li (1991); and van Camp et a1991), are completely linear; and in contrast to the onesosed byAl-Khayyal et al. (1997); Meller et al. (1999); andherali et al. (2003), use half as many binary decision variabspect ratio constraints, which are frequently used for prac

easons in order to guarantee that no department is excesarrow in either dimension, can be easily incorporated inew framework.

The rest of the paper is organized as follows: Section2 givesdetailed description of our modeling framework, proposesonvex models for the block layout design problem with une

ot-n

t

lly

l

.t. (2)–(9)

ihi = ai, i = 1, . . . , N, (19)

12(wi + wj) − (xi − xj) ≤ w

upF (Xij + Yij),

1 ≤ i < j ≤ N, (20)

12(wi + wj) − (xj − xi) ≤ w

upF (1 + Xij − Yij),

1 ≤ i < j ≤ N, (21)

12(hi + hj) − (yi − yj) ≤ h

upF (1 − Xij + Yij),

1 ≤ i < j ≤ N, (22)

12(hi + hj) − (yj − yi) ≤ h

upF (2 − Xij − Yij),

1 ≤ i < j ≤ N, (23)

ij ∈ 0, 1, ∀1 ≤ i < j ≤ N, (24)

ij ∈ 0, 1, ∀1 ≤ i < j ≤ N. (25)

ote that in BLDP1, asTompkins et al. (1996)remark, the facilty designer may represent an undesirable relationship (i.e


X relationship) between departmentsi andj by assigning a neg-ative value tocij sincecij ∈ R. Constraints(20)–(23)ensurethat no two departments in the layout overlap. Moreover, binaryconstraints(24) and(25) for the two new variablesXij andYij

ensure that only one of the constraints(20)–(23) is binding.Most notably, in contrast toTam and Li (1991); van Camp et al.(1991); andAnjos and Vannelli (in press), the overlap preventionconstraints(20)–(23)are completely linear. Also, in contrast tothose proposed byAl-Khayyal et al. (1997); Meller et al. (1999);andSherali et al. (2003), the overlap prevention constraints usehalf as many binary decision variables.

Now, for all 1≤ i < j ≤ N define

Φxij = 1

2

(wi + wj

wF

), (26)

Φyij = 1

2

(hi + hj

hF

), (27)

Θxij = xi − xj

wF, (28)

Θyij = yi − yj

hF, (29)

and construct the constraint set

Ωij = (Φxij, Φ

yij, Θ

xij, Θ

yij, Xij, Yij) : Φx

ij−Θxij − Xij−Yij ≤ 0,

Φx + Θx − X + Y ≤ 1,

Φ

0

Abs ,a( sg hec

fw ema

in af ral,f rri-d planl pip-i osts( 998P te-r eeno sys-tC and beinl ce isc dist etric

however, is more difficult to solve since the Euclidean distanceincludes a concave part.

Recall that the use of a bounded perimeter constraint as asurrogate area constraint increases the number of feasible solu-tions, adds unnecessary restrictions to the problem, and oftensignificantly violates the actual area constraint, thus produc-ing solutions that are infeasible to the underlying optimisationproblem. In contrast, our proposed area constraints model theunderlying nonlinear area restrictions in a consistent and exactmanner. We propose two models: (1) a convex model based on adecision variable transformation referred to as LDP1-C, and (2)a relaxed convex model based on the use of symmetric convexlower bounds referred to as BLDP1-R.

2.1. A convex model: BLDP1-C

The fundamental aspect of the convex model consists of anexact representation of the underlying nonconvex and hyperbolicarea restrictions using a decision variable transformation. In thismodel, a new continuous variableti = 1/wi is introduced foreach departmenti and the underlying area restriction given by(19) is transformed to a linear equality constraint of the formhi − aiti = 0.

The convex model BLDP1-C is as follows:

BN−1∑ N∑

s

h

x

x

B earp ari-t . Them h4 alo dN

ij ij ij ij

yij − Θ

yij + Xij − Yij ≤ 1, Φ

yij + Θ

yij + Xij + Yij ≤ 2,

≤ Φxij, Φ

yij ≤ 1, Θx

ij, Θyij ≥ 0, Xij, Yij ∈ 0, 1. (30)

lso, let Ωij be the continuous relaxation ofΩij, where theinary constraint onXij, Yij is replaced by 0≤ Xij, Yij ≤ 1. It istraightforward to see that every entry ofΩij is 0, 1, or−1. Thusny further tightening of the overlap prevention constraints(20)–23)would involve enlargingΩij with additional constraints. Aiven, constraint setΩij allows us to partially characterize tonvex hull of variablesXij andYij.

BLDP1 has a total ofN2 + 3N + 2 decision variables, ohich 4N + 2 are continuous andN(N − 1) are binary. Thodel has a total of 2N2 + 3N constraints, of which 2N2 + 2N

re linear andN are nonconvex and hyperbolic.A rectilinear distance for arranging departments with

acility is appropriate for an office layout and, in geneor organizations with facilities containing walls and coors. A rectilinear distance is also appropriate for process

ayout and piping design problems when incorporatingng, connection, pumping, and other material handling cSeeGeorgiadis et al., 1999; Papageorgiou and Rotstein, 1atsiatzis and Papageorgiou, 2002.) When using overhead ma

ial handling systems, when determining the wiring betwffice units, and, in general, for many flexible manufacturing

ems, a Euclidean distance is more realistic.Blanks (1985)andheng and Kuh (1984)show that the rectilinear and Euclideistance metrics converge as the number of departments

aid out increases. In this paper, only the rectilinear distanonsidered. Extending BLDP1 to incorporate the Euclideanance is straightforward. The model with such a distance m

t

.;

g

-,

LDP1-C: mini=1 j=i+1

cij(|xi − xj| + |yi − yj|)

(31)

.t. (3), (4), (7)–(9), (22)–(25)

i − aiti = 0, i = 1, . . . , N, (32)

i + 12ti

≤ wF, i = 1, . . . , N, (33)

i − 12ti

≥ 0, i = 1, . . . , N, (34)

1

2

(1

ti+ 1

tj

)− (xi − xj)

≤ wupF (Xij + Yij), ∀1 ≤ i < j ≤ N, (35)

1

2

(1

ti+ 1

tj

)− (xj − xi)

≤ wupF (1 + Xij − Yij), ∀1 ≤ i < j ≤ N, (36)

1

wupi

≤ ti ≤ 1

wlowi

, i = 1, . . . , N. (37)

LDP1-C is a completely convex mixed-integer nonlinrogramming (convex MINLP) model once the nonline

ies caused by the absolute value function are removedodel has a total ofN2 + 3N + 2 decision variables, of whicN + 2 are continuous andN(N − 1) are binary, and a totf 2N2 + 3N constraints, of whichN2 + 2N are linear an2 + N are convex.


Although the BLDP1-C model is convex there is one evidentdifficulty in that the number of convex constraints is quite large.It is possible, however, to relax the representation of the under-lying area constraints and formulate a more tractable model.

2.2. A relaxed convex model: BLDP1-R

The fundamental aspect of BLDP1-R consists of a relaxationof the underlying nonconvex and hyperbolic area restrictionsgiven by(19)using symmetric convex lower bounds of the form

wihi ≥ ai ⇒ −hi + ai

wi

≤ 0, (38)

⇒ −wi + ai

hi

≤ 0. (39)

This relaxation provides an exact representation of the arearestrictions since we are to minimize the expected cost ofmaterial-handling flows or the preference regarding adjacen-cies between departments. Moreover, the rectilinear distancebetween the centers of departmentsi and j (hence the objec-tive function value) will always increase (at the global optimalsolution) if at least one of the variableswi,hi,wj orhj increases.In such a case, one would only need(38) and(39) on the areaconstraint to guarantee global optimality.

In the special case where the objective function does notf oulda ena nta d) bg g theaw

t strict LP.I ricalr ea foa

B

s

Wb ritiesi ntrod

|

in the objective function(40), and adding the following con-straints to BLDP1-R:

dxij ≥ xi − xj, ∀1 ≤ i < j ≤ N, (44)

dxij ≥ xj − xi, ∀1 ≤ i < j ≤ N, (45)

dyij ≥ yi − yj, ∀1 ≤ i < j ≤ N, (46)

dyij ≥ yj − yi, ∀1 ≤ i < j ≤ N. (47)

In some situations, the facility designer may represent an unde-sirable relationship between departmentsi and j by assigninga negative value tocij. In order to account for such situa-tions in which the relationshipcij ∈ R, cij = cji, cii = 0 (ratherthat material handling flow exclusively) between departments isgiven, the nonlinearities caused by the absolute value functionare removed by introducing two new binary variablesZx

ij and

Zyij, two new continuous variablesdx

ij anddyij, setting

|xi − xj| + |yi − yj| = dxij + d

yij (48)

in the objective function(40), and adding the following con-straints to BLDP1-R:

xi − xj ≤ dxij ≤ xi − xj − 2w

upF (Zx

ij − 1), ∀1 ≤ i < j ≤ N,

(49)

x

y

y

wri-

a ncr riableB

f

wLP

m func-t aintsi bso-lfvba on-v at of

orce the area of a department to be exact, the model wllow the same objective function value with a larger departmrea, as defined bywihi = ai. However, any such departmerea can subsequently (after the optimization is completeiven final corrected width and height dimensions, satisfyinspect ratio requirement, by correctingwi andhi according tocorrectedi = √

aiwi/hi andhcorrectedi = √

aihi/wi. It is evidenthat an alternative to using the relaxation of the area reions is to use BLDP1-C, which is a completely convex MINt might, nonetheless, be worth noting that in our numeesults, there was no need to retrospectively correct the arny department as expected.

Our relaxed model BLDP1-R is formulated as follows:

LDP1-R: minN−1∑i=1

N∑j=i+1

cij(|xi − xj| + |yi − yj|)

(40)

.t. (2)–(9), (20)–(25)

− hi + ai

wi

≤ 0, i = 1, . . . , N, (41)

− wi + ai

hi

≤ 0, i = 1, . . . , N. (42)

hen the material handling flowcij ≥ 0, cij = cji, cii = 0etween departments is exclusively given, the nonlinea

ntroduced by the absolute value function are removed by iucing two new continuous variablesdx

ij anddyij, setting

xi − xj| + |yi − yj| = dxij + d

yij (43)

t

e

-

r

-

j − xi ≤ dxij ≤ xj − xi + 2w

upF Zx

ij, ∀1 ≤ i<j ≤ N, (50)

i − yj ≤ dyij ≤ yi − yj − 2h

upF (Zy

ij − 1), ∀1 ≤ i < j ≤ N,

(51)

j − yi ≤ dyij ≤ yj − yi + 2h

upF Z

yij, ∀1 ≤ i < j ≤ N,

(52)

hereZxij, Z

yij ∈ 0, 1, ∀1 ≤ i < j ≤ N.

Furthermore, it is worth noting that if the new binary vablesZx

ij andZyij are used whencij ∈ R, the overlap preventio

onstraints(20)–(23)and binary variablesXij andYij can beeplaced by a set of constraints that uses a single binary vaij for each pair of departmentsi andj, ∀1 ≤ i < j ≤ N, as

ollows:

12(wi + wj) − dx

ij ≤ wupF Bij, ∀1 ≤ i < j ≤ N, (53)

12(hi + hj) − d

yij ≤ h

upF (1 − Bij), ∀1 ≤ i < j ≤ N, (54)

hereBij ∈ 0, 1, ∀1 ≤ i < j ≤ N.We note that BLDP1-R is also a completely convex MIN

odel once the nonlinearities caused by the absolute valueion are removed. Ignoring the new variables and constrntroduced to remove the nonlinearities caused by the aute value function (i.e.,(43)–(47) for cij ≥ 0 and (48)–(54)or cij ∈ R), the model has a total ofN2 + 3N + 2 decisionariables, of which 4N + 2 are continuous andN(N − 1) areinary, and a total of 2N2 + 3N constraints, of which 2N2 + N

re linear and 2N are convex. Note that the number of cex constraints decreases substantially with respect to th


BLDP1-C: fromN2 + N to 2N. Clearly, BLDP1-R is funda-mental in nature, must be further studied, and must be seriouslyconsidered as a promising formulation for the block layoutdesign problem with unequal areas.

We now discuss the incorporation of additional constraintsto satisfy the aspect ratio requirements and to avoid symmetriclayout solutions in our modeling framework.

2.3. Incorporation of additional constraints

As stated above, it is often desirable in practice to delineatethe maximum permissible ratio between the longest and short-est dimensions of each department in order to guarantee thatno department is excessively narrow in either dimension. Suchaspect ratio requirement can be easily incorporated in our blocklayout design models via suitable choices of the lower and upperbounds on the width and height dimensions of each department(constraints(6) and(7)) by setting.

wupi = h

upi = √

aiαi, i = 1, . . . , N. (55)

Indeed, if we setwupi andh

upi using(55), then

wupi ≥ wi ⇒ aiαi ≥ w2

i

⇒ αi ≥ w2i ⇒ αi ≥ wi

, (56)

s lw wec

w

h

sto

w

h

a ente

ffortr rpo-r ons.S pti-m theo optm

seda )f etrM and2 sign

models:

x1 − x2 ≥ 0, (61)

y2 − y1 ≥ 0. (62)

That is, we force department 1 to be located to the South–East ofdepartment 2. Other criteria for selecting the two departments forthe symmetry-avoidance constraints are possible. For example,one could select the two departments with the largest interac-tion; i.e., select departmentsm andn, 1 ≤ m < n ≤ N, such thatcmn = maxi,jcij, and addxm − xn ≥ 0 andyn − ym ≥ 0 to ourmodels. We found, however, that the criterium of simply select-ing departments 1 and 2 provides on average better results forthe test problems under consideration. In addition, we note thatconstraints(61) and(62) imply thatX12 = Y12. Thus, we alsoadd the following constraint to our block layout design models:

X12 − Y12 = 0. (63)

We conclude this section by briefly presenting the global opti-mization methods for convex MINLP models of the form givenby BLDP1-C and BLDP1-R that are used in our comparativestudy.

2.4. MINLP optimization methods

The following MINLP optimization methods are used ino AndR tingP iza-t oft-w PT,M tionm

ILPs eth-o sedo inG ms,w inga

n,a o-rL odi

2

( olv-i thodd func-t anye ing as opti-m thatp ner-a ILP

ai hi

incewupi > 0. Similarly,hup

i ≥ hi ⇒ αi ≥ hi/wi. If the actuaidth w and heighth of the floor area are known in advance,an tighten constraints(55)by setting

upi = min√aiαi, w, i = 1, . . . , N, (57)

upi = min√aiαi, h, i = 1, . . . , N, (58)

incewi ≤ w andhi ≤ h. Now, given thatwihi = ai must holdrue for each departmenti, the upper bounds above (either(55)r (57)and(58)) yield the lower bounds

lowi = ai

hupi

, i = 1, . . . , N, (59)

lowi = ai

wupi

, i = 1, . . . , N, (60)

nd hence the aspect ratio requirement for each departmnforced.

In addition, it is possible to reduce the computational eequired to solve our block layout design models by incoating simple constraints that avoid symmetric layout solutiymmetric layout solutions imply the existence of multiple oal solutions, which lead to longer CPU times since allptimal symmetric nodes need to be examined before theal layout solution can be verified.There are several classes of constraints that can be u

void symmetric layout solutions. (SeeSherali and Smith (2001or various classes of constraints that can be used for symmILP models.) In this paper, we simply select departments 1and add the following constraints to our block layout de

is

i-

to

ic

ur comparative study: Branch and Bound (BB), Brancheduce Optimization Navigator (BARON), Extended Cutlane (ECP), and Outer Approximation (OA). The optim

ion methods are readily implemented in the following sare packages: Alpha-ECP, GAMS/BARON, GAMS/DICOINLPbb, and GAMS/SBB. We note that these optimisaethods represent quite different solution approaches.The ECP method is based on solving a sequence of M

ub-problems, while the nonlinear branch-and-bound mds (implemented in GAMS/SBB and MINLPbb) are ban solving NLP sub-problems. The solution approachAMS/BARON is based on solving LP and NLP sub-problehile the OA method (in GAMS/DICOPT) is based on solvn alternating sequence of MILP and NLP problems.

The reader is referred toBrooke, Kendrick, Meeraus, Ramand Rosenthal (1998)for a more detailed description of the algithms implemented in the GAMS solvers, and toFletcher andeyffer (1998)for a description of MINLPbb. The ECP meth

s described next.

.4.1. The ECP methodIn summary, the ECP method is an extension ofKelley’s

1960) cutting plane method, which was developed for sng convex nonlinear programming problems. The ECP meoes not need second order information of the objective

ion nor of the constraints, which is an advantage in mngineering applications. The method is based on solvequence of MILP sub-problems, converging to the globalal solution of a pseudo-convex MINLP problem (noteseudo-convexity implies convexity). Cutting planes are geted at feasible or optimal solutions of the corresponding M


sub-problems but in the infeasible region of the original pseudo-convex MINLP problem.

The ECP method inWesterlund and Porn (2002)is able toobtain the global optimal solution of problems of the followingform:

P : minz∈N∩L

f (z) (64)

s.t. N = z|g(z) ≤ 0, (65)

L = z|Az ≤ a, Bz = b ∩ X × Y . (66)

In problem P,a andb are vectors andA andB are matriceswith constants. The vectorzT = [x, y] consists of the vector ofcontinuous variablesx ∈ X and the vector of integer variablesy ∈ Y . The objective functionf (z) and the nonlinear constraintsg(z) are differentiable, pseudo-convex functions defined on thesetL. The setX is a compact subset ofRn and the setY is afinite discrete set inZm defined by the bounds of the variables.The feasible region of problem P is defined byN ∩ L, whichis a nonempty set. Note that the continuous relaxation ofN andN ∩ L are convex sets since the level sets of pseudo-convexfunctions are convex.

Since our block layout design models BLDP1-R and BLDP1-C only have a linear objective function and convex inequalityconstraints, the ECP method is reduced to solve a sequence ofMILP sub-problems of the following form

w

Ω

TΩ 2) at gp als s tos thes ni

ed im( ds s ag

bs m-p intem tiona ntr CPm s-s d too imaM nowi ec-t

noted by Alpha-ECP (strategy 1), while the original strategyas given inWesterlund and Pettersson (1995)is indicated byAlpha-ECP (strategy 2).

We now turn our attention to the numerical results for severalwell-known test problems that have appeared in the publishedliterature.

3. Numerical results

We note that no suitable test problems with negative valuesfor somecij ’s have been presented in the published literature.Hence, comparisons with previous results are not possible whensolving BLDP1-C or BLDP1-R in the more general situation;that is, whencij ∈ R. In addition, a pilot study showed thatthe computational effort required to solve BLDP1-C is higherthan the effort required to solve BLDP1-R. In this section wereport numerical results using BLDP1-R, the MILP models ofCastillo and Westerlund (2005)andSherali et al. (2003), as wellas using the different MINLP optimization methods. The MILPsolver is CPLEX Version 8.0. Unless indicated, we used CPLEXdefaults for all problems. The computing platform is a Pentium4, Windows-based computer.

3.1. Small test problems

s and

inms

e

es,. Asimeas

s thea-intsuta-e.theition

ervetio

atio

minzk∈Ωk

cTzk, k = 0, 1, 2, . . . , K, (67)

here the setΩk is defined by

k = L ∩ z|lj(z) ≤ 0, j = 1, 2, . . . , Jk. (68)

he iterative procedure is started withΩ0 = L. Note thatlj(z) ∈k are cutting planes generated by linearizing (41) and (4

he solution of each sub-problem.Jk is the number of cuttinlanes inΩk at iterationk. Convergence to the global optimolution is ensured when the sequence of points convergeolution in the feasible region of the problem P defined byetN ∩ L, whereN ∩ L is a subset ofΩk, such that the solutios also optimal inΩk.

The convergence properties of the method are describore detail inWesterlund, Skrifvars, Harjunkoski and Porn

1998)andWesterlund and Porn (2002). In brief, the ECP methoolves a sequence of MILP sub-problems and cutting planeenerated at each solution.

We note that although a MILP solution pointzk in (67) maye defined as an optimal MILP solution inΩk, an optimal MILPolutionzk is a strict requirement only at termination. Our coutational experience shows that the strategy to solve theediate MILP sub-problems only to an integer feasible solus inWesterlund and Porn (2002), provided the most efficieesults. It is worth mentioning, however, that in the original Eethod for convex problems as given inWesterlund and Petter

on (1995), all intermediate MILP sub-problems were solveptimality and cutting planes were only generated at optILP solution points. These two solution strategies are

mplemented in the Alpha-ECP solver which is used in Sion 3: the strategy as given inWesterlund and Porn (2002)is

t

a

n

re

r-,

l

We consider several problems fromMeller et al. (1999).These problems range in size from three to nine departmenthave common aspect ratios (i.e.,αi = α, i = 1, . . . , N) rangingfrom 2 to 5. A brief description of the problems is presentedTable 1. The model pameters for all considered test probleare given inAppendix A.

In Table 2, we consider BLDP1-R with and without thsymmetry-avoidance constraints(61)–(63). For all test prob-lems, we report the objective function values, the CPU timand the number of iterations performed by the ECP methoda parenthetical note next to the CPU time, we report the tat which the global optimal or best-known integer solution wfound. The difference between these two times representtime taken to verify the optimality of the solution (if not premturely terminated). The results indicate that by using constrathat avoid symmetric solutions, we can decrease the comptional effort in terms of CPU time by over 75% on averagNote, however, that the number of iterations performed byECP method does not necessarily decrease with the addof the symmetry-avoidance constraints. In addition, we obsfrom Table 2that problems with lower values of the aspect ra

Table 1General properties of the small test problems

Problem name Number of departments Aspect r

fO7 7 2–5fO8 8 2–5fO9 9 2–5M7 7 2–5NO7 7 2–5O7 7 2–5O8 8 4O9 9 4


Table 2Numerical results for the small test problems with and without the symmetry-avoidance constraints

Problem name Aspect ratio BLDP1-R without(61)–(63) BLDP1-R with(61)–(63) CPUs

Cost CPUs Iterations Cost CPUs Iterations Reduction (%)

fO7 2.0 24.84 189 (105) 20 24.81 54 (54) 9 712.5 23.09 152 (117) 23 23.05 68 (46) 12 553.0 22.52 875 (875) 34 22.35 84 (80) 19 904.0 20.73 1429 (804) 38 20.73 140 (132) 23 905.0 17.75 117 (3.29) 17 17.75 20 (20) 10 83

fO8 2.0 30.34 1280 (1280) 21 30.32 920 (574) 17 282.5 28.03 2325 (2325) 25 27.99 1880 (828) 17 193.0 23.91 3540 (3284) 31 23.83 76 (76) 16 984.0 22.38 1022 (467) 38 22.33 184 (184) 16 825.0 22.38 17658 (12766) 34 22.31 394 (394) 14 98

fO9 2.0 32.62 82503 (5471) 19 32.56 16394 (2698) 20 802.5 32.38a 86400 (22552) 26 32.11 10014 (1343) 15 >883.0 24.82 36710 (32122) 38 24.62 3799 (3438) 17 904.0 23.46 33575 (15312) 37 23.46 1316 (1316) 14 965.0 23.46 75170 (18008) 51 23.41 2572 (2572) 20 97

M7 2.0 190.23 9.4 (5.6) 9 190.21 6.8 (4.9) 6 282.5 143.59 4.1 (3.7) 16 143.59 0.8 (0.3) 7 803.0 143.59 20 (5.8) 17 143.59 5.9 (2.0) 10 714.0 106.76 9.2 (7.3) 19 106.76 2.3 (2.2) 11 755.0 106.46 33 (16) 19 106.46 2.8 (0.4) 6 92

NO7 2.0 107.79 433 (412) 22 107.81 80 (55) 15 822.5 107.75 1495 (1097) 26 107.81 136 (92) 16 913.0 107.80 2048 (1456) 23 107.81 355 (243) 19 834.0 98.52 855 (493) 32 98.24 325 (245) 18 625.0 90.62 428 (246) 27 90.62 218 (207) 29 49

O7 2.0 140.41 1452 (805) 21 140.41 147 (26) 10 902.5 140.41 2443 (798) 30 140.39 656 (387) 15 733.0 137.93 7963 (2102) 23 137.93 835 (835) 14 904.0 131.62 33420 (10895) 22 131.64 1195 (375) 20 965.0 116.94 6240 (4736) 22 116.95 781 (601) 19 87

O8 4.0 242.96a 86400 (67669) 47 242.73 18392 (18088) 26 >79O9 4.0 236.54a 86400 (19923) 37 236.14 83211 (4349) 33 >4

a Terminated after 24 h of CPU time. The best-known integer solution is reported.

(α = 2 and 2.5 in our case) are on average harder to solve thanproblems with larger values of the aspect ratio (α = 3, 4 and 5 inour case). As the aspect ratio increases, the optimal cost associ-ated with projected interactions between departments stays thesame or decreases; this is due to the longer and narrower depart-ments that are possible with larger aspect ratios.

We now compare the performance of our approach with theMILP models ofCastillo and Westerlund (2005)andSherali etal. (2003), which are arguably the most accurate MILP modelsin the published literature. We refer to the model ofCastillo andWesterlund (2005)as BLDP-ε and to the model ofSherali etal. (2003)as BLDP-TS. We usedε = 0.01% for BLDP-ε andgenerated 20 tangential supports per department for BLDP-TS.The models include suitable choices of the symmetry-avoidanceconstraints(61)–(63). Table 3shows how the layout solutionsusing our modeling framework compare with the solutions ofBLDP-ε and BLDP-TS. For both BLDP-ε and BLDP-TS wealso report the maximum error in department areas calculatedas maxi100(1− wihi/ai)%. This error is due approximationof the nonlinear area constraint needed in the MILP models.

Although it is well known that a reduction in the number ofbinary variables is by no means a guarantee for improved compu-tational performance, fromTable 3, we observe that 72% of thetest problems considered required less CPU time with BLDP1-Rthan with BLDP-TS. We also observe that 78% of the test prob-lems considered required less CPU time when solving BLDP1-Rwith the ECP method than solving the MILP problem BLDP-ε

with CPLEX.Sherali et al. (2003)also proposed additional constraints and

improved branching schemes for the block layout design prob-lem with unequal areas.Sherali et al. (2003)reported that themodelling enhancements reduced the computation effort by anaverage factor of 135.56 over the results byMeller et al. (1999).For future research, it might be useful to explore the use ofsimilar branching schemes in our approach. It is apparent, how-ever, that the results presented in this paper indicate that theproposed modeling framework allow us to seriously challengemixed-integer linear optimization methods.

We now compare the performance of our approach with othermixed-integer nonlinear optimization methods. We consider


Table 3Comparison with MILP models for the small test problems

Problem name Aspect ratio BLDP-TS BLDP-ε BLDP1-R

Cost CPUs Max% error Cost CPUs Max% error Cost CPUs

fO7 2.0 24.84 440 0.01 24.84 68 0.01 24.81 542.5 23.08 172 0.04 23.09 95 0.01 23.05 683.0 22.51 408 0.01 22.51 87 0.01 22.35 844.0 20.71 120 0.12 20.73 92 0.01 20.73 1405.0 17.75 64 0.01 17.75 100 0.00 17.75 20

fO8 2.0 30.34 1745 0.02 30.34 937 0.01 30.32 9202.5 28.04 1049 0.02 28.04 751 0.01 27.99 18803.0 23.91 604 0.01 23.91 375 0.01 23.83 764.0 22.38 1080 0.01 22.38 756 0.00 22.33 1845.0 22.36 665 0.12 22.38 1524 0.01 22.31 394

fO9 2.0 32.61 26097 0.03 32.62 29430 0.01 32.56 163942.5 32.24a(9.77%) 46504 0.06 32.19 17459 0.01 32.11 100143.0 24.80 2465 0.08 24.81 1125 0.01 24.62 37994.0 23.45 4895 0.13 23.46 863 0.00 23.46 13165.0 23.46 3104 0.07 23.46 6583 0.01 23.41 2572

M7 2.0 190.23 3.4 0.01 190.23 4.0 0.01 190.21 6.82.5 143.58 0.8 0.01 143.58 1.7 0.01 143.59 0.83.0 143.57 4.1 0.03 143.58 16 0.00 143.59 5.94.0 106.76 1.0 0.00 106.76 12 0.00 106.76 2.35.0 106.46 6.2 0.00 106.46 11 0.00 106.46 2.8

NO7 2.0 107.79 104 0.02 107.80 77 0.01 107.81 802.5 107.73 90 0.04 107.80 217 0.01 107.81 1363.0 107.76 391 0.07 107.81 562 0.01 107.81 3554.0 98.47 446 0.12 98.52 372 0.01 98.24 3255.0 90.59 130 0.05 90.62 498 0.00 90.62 218

O7 2.0 140.40 532 0.01 140.41 173 0.01 140.41 1472.5 140.37 657 0.05 140.41 863 0.01 140.39 6563.0 137.92 2392 0.02 137.93 1805 0.01 137.93 8354.0 131.58 4436 0.12 131.64 3383 0.01 131.64 11955.0 116.94 1905 0.01 116.95 2206 0.00 116.95 781

O8 4.0 242.93 47789 0.08 243.06 27077 0.01 242.73 18392O9 4.0 264.37a(20.93%) 37100 0.10 236.14 56648 0.01 236.14 83211

a Best-known integer solution and percentage optimality gap after stack overflow.

BLDP1-R with the symmetry-avoidance constraints(61)–(63)and four well-known optimisation methods: BARON(Sahinidis,2000), DICOPT (Duran & Grossmann, 1986), MINLPbb(Fletcher & Leyffer, 1994), and SBB(GAM, 1996). Note that allmethods are, in theory, capable of finding globally optimal solu-tions with proven optimality for convex mixed-integer nonlinearprogramming models such as BLDP1-R. All optimization meth-ods were executed using the same computing platform exceptMINLPbb, which was executed using the NEOS server(Czyzyk,Mesnier, & More, 1998). Table 4shows how the layout solu-tions using our modeling framework compare with the solutionsof the aforementioned methods.Fig. 3illustrate graphically thenumber of solutions that were proven to be optimal, the numberof integer solutions after 12 h of CPU time or after stack over-flow, and the number of non-integer solutions after 12 h of CPUtime for each of the optimization methods.Fig. 4 illustrates theoptimality gap for the integer solutions that were prematurelyterminated after 12 h of CPU time or after stack overflow.

Table 4indicates that our convex MINLP approach consis-tently yields global optimal solutions on all test problems (ifthe optimization method is not prematurely terminated). Most

notably, there is a considerable difference in the CPU timesfor the different mixed-integer nonlinear optimization methods.FromTable 4we find that the Alpha-ECP solver was clearly thefastest MINLP solver for these test problems.

Fig. 3. Number of optimal (opt), integer feasible (int), and nonfeasible (Nis)solutions for the mixed-integer nonlinear optimization methods.

64I.C

astilloetal./C

omputers

andC

hemicalE

ngineering30

(2005)54–69

Table 4Comparison with other global optimization methods for the small test problems

Problem name Aspect ratio BARON DICOPT MINLPbb SBB Alpha-ECP 1 Alpha-ECP 2

Cost CPUs Cost CPUs Cost CPUs Cost CPUs Cost CPUs Cost CPUs

fO7 2.0 24.84 4744 – 43200 33.04a 1865 24.84b 43200 24.81 54 24.83 1442.5 23.12 17402 – 43200 25.65a 2418 25.68b 43200 23.05 68 23.09 1523.0 22.52 19880 22.83b 43200 25.65a 2836 25.68b 43200 22.35 84 22.52 1504.0 21.84 33650 – 43200 24.38a 2143 23.76b 43200 20.73 140 20.73 865.0 17.75 7845 – 43200 19.62a 3948 17.75b 43200 17.75 20 17.75 33

fO8 2.0 30.96b 43200 – 43200 55.31a 1447 – 43200 30.32 920 30.34 22092.5 32.14b 43200 – 43200 46.24a 2401 56.09b 43200 27.99 1880 28.04 22153.0 28.04b 43200 23.91 12656 52.06a 3570 41.68b 43200 23.83 76 23.91 3264.0 23.91b 43200 – 43200 27.13a 3393 40.23b 43200 22.33 184 22.38 3475.0 41.01 2734 – 43200 34.95a 2020 31.96b 43200 22.31 394 22.38 1385

fO9 2.0 41.27b 43200 – 43200 32.20a 17280 – 43200 32.56 16394 32.62 317222.5 34.38b 43200 – 43200 45.95a 2688 49.88b 43200 32.11 10014 – 432003.0 50.73 12682 – 43200 45.25a 4269 66.10b 43200 24.62 3799 24.82 27224.0 24.38b 43200 23.46 26.093 41.42a 4908 – 43200 23.46 1316 23.46 35645.0 – 605 – 43200 31.76a 7887 53.62b 43200 23.41 2572 23.46 5547

M7 2.0 197.92 710 190.24b 43200 190.24 665 197.92 4395 190.21 6.8 190.22 232.5 143.58 211 143.58 2.8 143.58 355 143.59 2511 143.59 0.8 143.59 1.03.0 143.58 661 – 43200 121.05 1060 143.59 12136 143.59 5.9 143.59 164.0 106.76 519 106.76 3.2 106.76 12200 106.76 8451 106.76 2.3 106.76 4.55.0 106.46 328 – 43200 106.46 4371 106.46 20251 106.46 2.8 106.46 11

NO7 2.0 107.87 34913 107.82 362 181.56a 2779 160.16b 43200 107.81 80 107.81 1942.5 107.82b 43200 – 43200 132.44a 2922 123.94b 43200 107.81 136 107.80 4313.0 – 459 – 43200 123.91a 3339 121.82b 43200 107.81 355 107.81 16434.0 98.74b 43200 – 43200 110.45a 3505 132.69b 43200 98.24 325 98.52 16615.0 92.20b 43200 – 43200 97.29a 4514 92.42b 43200 90.62 218 90.62 836

O7 2.0 146.61b 43200 – 43200 163.25a 2699 167.03b 43200 140.41 147 140.41 5272.5 148.39b 43200 – 43200 150.05a 2992 154.29b 43200 140.39 656 140.41 14533.0 142.80b 43200 – 43200 146.28a 3458 157.88b 43200 137.93 835 137.93 33084.0 131.66b 43200 – 43200 143.92a 4003 151.75b 43200 131.64 1195 131.65 72735.0 118.86b 43200 – 43200 134.87a 3709 131.11b 43200 116.95 781 116.95 2687

O8 4.0 269.47b 43200 – 12958 344.52a 2398 299.36b 43200 242.73 18392 – 43200O9 4.0 – 43200 – 12964 370.24a 8827 – 43200 236.18b 43200 – 43200

‘–’ Non-integer solution after 12 h of CPU time.a Best-known integer solution after stack overflow.b Best-known integer solution after 12 h of CPU time.


Fig. 4. Optimality gap for integer solutions for the mixed-integer nonlinearoptimization methods.

Table 5General properties of the larger test problems

Problem name Number ofdepartments

Common width andheight lower bound

vC10 10 5Ba12 12 1Ba13 13 1

In the Alpha-ECP solver, different solution strategies canbe selected. Alpha-ECP (strategy 1) corresponds to the solution strategy as given inWesterlund and Porn (2002), wherethe cutting planes are not generated at optimal MILP solutionpoints but at the first main iteration to the first integer feasi-ble MILP solution. If this solution is also a feasible solutionfor the original MINLP problem, then the procedure contin-ues to the following integer feasible MILP solution withoutgenerating a cutting plane. On the other hand, if the solu-tion point is infeasible in the original problem, then a cuttingplane is generated and a following MILP sub-problem is gen-erated and solved. At the second main iteration, the MILPsub-problem is solved at least to the second integer feasiblsolution. The procedure will terminate when the solution pointisε-feasible in the original MINLP problem and the correspond-ing MILP sub-problem is verified to result in an optimal integersolution.

Alpha-ECP (strategy 2) corresponds to the original ECPmethod as given inWesterlund and Pettersson (1995), whereall intermediate MILP sub-problems are solved to optimalityand the cutting planes are generated only at optimal MILP solutions. FromTable 4, we find that the original ECP method

also performs fairly well. The original ECP method results ina better performance than GAMS/BARON, GAMS/DICOPT,MINLPbb, and GAMS/SBB. However, Alpha-ECP (strategy 1),where cutting planes are allowed to be generated at integer fea-sible MILP solutions is clearly the most efficient optimizationmethod for the test problems. Moreover, we note that, whenusing the Alpha-ECP solver, even if the solutions are prema-turely terminated after a certain time limit, the best-knowninteger solutions are guaranteed to be feasible for an actualimplementation due to the exact area representation.

A note may finally be given. The reasons for the poor per-formance of the branch-and-bound based MINLP optimizationmethods (GAMS/BARON, MINLPbb, and GAMS/SBB) arenot quite clear. The small test problems, however, appear tohave only a small number of feasible integer solutions; thus,branch-and-bound based methods appear to be solving mostlyinfeasible nonlinear sub-problems. Another reason may be thatthe relaxation of the nonlinear sub-problems is rather poor, giv-ing little additional information to the optimization methods.This is mainly because the integer variables are only needed tomodel the overlap prevention constraints.

In GAMS/DICOPT, the infeasible nonlinear sub-problemsresult in that only an integer cut is added to the MILP mas-ter problem after each iteration. Since the integer cut onlyexcludes a single integer variable combination, a significantamount of integer cuts may be required before a feasible solutioni lin-e ingi nlin-e rovet hats allc

3

hichaC( rob-l ture.T withtp rans-f ac unc-t

Table 6Numerical results for the larger test problems

Problem name Heuristic resulta BLDP

Cost s

2129818491

91).is rep

vC10 24445.0Ba12 11910.0Ba13 6875.0

a Three-step heuristic nonlinear optimization method(van Camp et al., 19b Terminated after 12 h of CPU time. The best-known integer solution

-

e

-

s obtained. On top of the infeasibility problem, the startingarizations were not very helpful to GAMS/DICOPT. Specify

nteger points as starting values instead of using the noar sub-problem relaxation as the starting point, could imp

he performance of GAMS/DICOPT. However, it is clear tpecifying good starting points would most likely benefitonsidered optimization methods.

.2. Larger test problems

We now present results for three larger test problems wre also well-known in the layout literature, namely thevanamp et al. (1991)10-department problem and theBazaraa

1975) 12 and 13-department problems. For these test pems, no optimal solutions have been reported in the literaherefore, we compare the performance of our approach

he heuristic results reported byvan Camp et al. (1991), whoropose a three-step heuristic optimization method that t

orms the underlying nonlinear optimization problem fromonstrained form into an unconstrained form using penalty fion methods.

1-R

CPUs Iteration

7.98 43,200 (26,637)b 420.00 43,200 (1,667)b 149.47 43,200 (6,493)b 23

orted.


Fig. 5. Heuristic result for thevan Camp et al. (1991)10-department problem.

Fig. 6. Heuristic result for theBazaraa (1975)12-department problem.

A brief description of the problems is presented inTable 5.Table 6shows how the layout solutions using our modelingframework compare with the results reported byvan Camp et al.(1991). We note that the authors reported CPU times between370 and 850 s for the three test problems. As before, for BLDP1-R, as a parenthetical note next to the CPU time, we report thetime at which the best-known integer solution was found. Note

Fig. 8. Best-known integer solution for thevan Camp et al. (1991)10-departmentproblem.

Fig. 9. Best-known integer solution for theBazaraa (1975)12-department prob-lem.

that times are also given in seconds. Although we were not ableto obtain global optimal solutions within 12 h, our best-knowninteger solutions improve upon the previously published solu-tions by approximately 13, 31, and 28%, respectively.Figs. 5–7show the heuristic results andFigs. 8–10show our best-knowninteger solutions for thevan Camp et al. (1991)10-departmentand theBazaraa (1975)12 and 13-department problems. Note

Fig. 10. Best-known integer solution for theBazaraa (1975)13-departmentp
Fig. 7. Heuristic result for theBazaraa (1975)13-department problem. roblem.


that the failure to use modeling techniques and global optimiza-tion methods favors sub-optimal (yet accurate) layout solutionswith relatively square departments that do not take advantage ofthe maximum permissible ratio between the longest and shortestdimensions of each department.

We conclude this section by noting that the results presentedhere indicate that the proposed modeling framework, its for-mulations and their exact representation of the underlying non-convex and hyperbolic area restrictions, and its solution qualityusing the ECP method allow us to seriously challenge MILP andMINLP optimization methods in this important class of hard,fundamental optimization problems.

4. Conclusions

The block layout design problem is a fundamental optimi-sation problem encountered in many manufacturing and ser-vice organizations. The problem was originally formulatedby Armour and Buffa in the early 1960s and is concernedwith finding the most efficient arrangement of a given num-ber of departments with unequal area requirements within afacility.

On one hand, we have noted that existing MILP models canonly approximate the nonlinear area constraint with a givenaccuracy. Thus, the layout solutions provided by existing MILPmodels are often infeasible to the underlying optimization prob-l feas al.O ineap elingt havf s thag optim

k fore ignp cons x anh rmat plec lay-o asilyb quipm thep

thatc imals ub-l olvef strict timew andm indi-c uset hal-l s ofh

Although the proposed modeling framework has dramaticallyreduced the computational effort required to solve the blocklayout design problem with unequal areas, further research isneeded to solve large test problems of practical interest. Futuredirections include a deeper understanding on the combinatorialstructure of the proposed modeling framework and the devel-opment of problem-specific, a priori cutting plane heuristics tofind good initial solutions that would lead to tighter bounds,enabling the early cutting of fruitless regions in the iterativeprocess of the global optimization method. The a priori gener-ation of cutting planes has been initially addressed inCastilloand Westerlund (2005). This future research, however, cannot beproperly undertaken if the underlying nonconvex and hyperbolicarea restrictions are not handled appropriately. Our modellingframework is a step forward towards that direction.

Acknowledgements

The authors would like to thank Dr. Miguel F. Anjos andDr. Ignacio Grossmann for their helpful comments on an earlierversion of this manuscript. Dr. Ignacio Castillo acknowledgesthe financial support from the Natural Sciences and EngineeringResearch Council of Canada to partially support this research.Mr. Joakim Westerlund and Ph. Lic. Stefan Emet acknowledgethe financial support from TEKES, the National TechnologyAgency of Finland.

A

Pc ef-

nts

a

w

w art-

h art-

α

N ents

Cw

x

d ,

t

BX pre-

Z re-

em. Moreover, even if the areas are ‘massaged’ to reachibility, the layout solutions would frequently be sub-optimn the other hand, we have also noted that existing nonlrogramming models, although accurate in terms of mod

he underlying nonconvex and hyperbolic area restrictions,ailed to use modeling techniques and optimization methoduarantee that the computed layout solutions are globallyal.In this paper, we have presented a modeling framewor

ffectively finding optimal solutions for the block layout desroblem. The most fundamental aspects of the frameworkist of an exact representation of the underlying nonconveyperbolic area restrictions using a decision variable transfo

ion and symmetric convex lower bounds, together with simonstraints to avoid symmetric layout solutions. The blockut design problem and the new modeling framework may ee applied to find single-floor arrangements of processing eent and the interconnecting pipework when applied torocess plant layout and piping design problems.

We have also presented computational results showingonvex optimization method consistently yielded global optolutions on several well-known test problems from the pished literature. These challenging problems have been sor the first time using exact representations of the area reions. We have accounted for a dramatic decrease in CPUith respect to the results of other mixed-integer linearixed-integer nonlinear optimization methods. Our results

ate that the new modeling framework can be successfullyo find global optimal layout solutions; therefore, seriously cenging other optimization methods on this important clasard, fundamental problems.

-

r

et-

-d-

-

a

d-s

d

ppendix A

SeeTables A.1–A.3

arametersij relationship (the material-handling flow or the pr

erence regarding adjacency) between departmeiandj

i required area of departmentiF, hF width and height of the floor arealowi , w

upi valid lower and upper bounds on the width of depmenti

lowi , h

upi valid lower and upper bounds on the height of dep

mentii aspect ratio of departmenti; that is,

αi ≥ maxwi, hi/ minwi, hinumber of departments with unequal area requirem

ontinuous variablesi,hi width and height of departmentii,yi coordinates of the center of departmentixij,d

yij distances, with respect to thex- and y-coordinates

between departmentsi andji continuous variable for 1/wi

inary variablesij, Yij variables that ensure that only one of the overlap

vention constraints is bindingxij, Z

yij, Bij alternative variables for modeling the overlap p

vention constraints whencij ∈ R


Table A.1Model parameters for all BLDP-TS problems

Problem name Linear constraints Variables

Continuous Binary

fO7 522 70 42fO8 635 88 56fO9 750 108 72M7 474 70 42NO7 523 70 42O7 523 70 42O8 877 88 56O9 1038 108 72

Table A.2Model parameters for all BLDP-ε problems

Problem name Aspect ratio Linear constraints Variables

Continuous Binary

fO7 2.0 238 72 422.5 251 72 423.0 258 72 424.0 271 72 425.0 283 72 42

fO8 2.0 304 90 562.5 320 90 563.0 328 90 564.0 344 90 565.0 356 90 56

fO9 2.0 374 110 722.5 396 110 723.0 405 110 724.0 423 110 725.0 439 110 72

M7 2.0 234 72 422.5 245 72 423.0 249 72 424.0 257 72 425.0 262 72 42

NO7 2.0 238 72 422.5 251 72 423.0 258 72 424.0 271 72 425.0 283 72 42

O7 2.0 238 72 422.5 251 72 423.0 258 72 424.0 271 72 425.0 283 72 42

O8 4.0 312 90 56O9 4.0 387 110 72

Table A.3Model parameters for all BLDP1-R problems

Problem name Constraints Variables

Convex Linear Continuous Binary

fO7 14 255 72 42fO8 16 331 90 56

Table A.3 (Continued )

Problem name Constraints Variables

Convex Linear Continuous Binary

fO9 18 417 110 72M7 14 255 72 42NO7 14 255 72 42O7 14 255 72 42O8 16 331 90 56O9 18 417 110 72vC10 20 403 132 90Ba12 26 679 210 156Ba13 26 679 210 156

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optimization of block layout design problems with unequal areas: a

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