IIE Transactions (2006) 38, 53–65Copyright C© “IIE”ISSN: 0740-817X print / 1545-8830 onlineDOI: 10.1080/07408170500208370

Assembly system facility design

YOSSI BUKCHIN1, RUSSELL D. MELLER2,∗ and QI LIU2

1Industrial Engineering Department, Tel Aviv University, Tel Aviv, IsraelE-mail: bukchin@eng.tau.ac.il2Department of Industrial and Systems Engineering, Virginia Polytechnic Institute and State University, Blacksburg,VA 24061, USAE-mail: rmeller@vt.edu; qliu@vt.edu

Received September 2004 and accepted June 2005

This paper addresses the design of an assembly system facility consisting of multiple assembly lines of different shapes. In such adesign problem there are two conflicting objectives: (i) to minimize the overall area of the facility; and (ii) to maximize the efficiencyof the material handling transportation system. We first address the optimization problem of objective (ii) when replacing objective(i) with a constraint on the facility area. We propose a mixed-integer linear program to determine the layout of a facility with givendimensions and with given assembly line areas and shapes (that cannot be changed due to technological considerations). In the layoutmodel, the physical placement of each line within the facility is a decision variable. The objective function of the layout model is tominimize the distances traveled by material flow. Our performance analysis provides an indication of the maximal problem size thatcan be solved in a reasonable amount of time and we examine the effect of the problem parameters on the solution run time. Thislayout model is then incorporated into an efficiency frontier approach for facility design to address both objectives. Examples arepresented to illustrate the use of the proposed facility design model.

1. Introduction

In designing a manufacturing facility, a facilities plannertypically has competing objectives. An example of thesecompeting objectives is apparent when one considers thesize of the facility and the efficiency of the material handlingtransportation system. Clearly, minimizing the overall sizeof the facility is the least costly option in terms of invest-ment in the building itself. However, in so doing, the designof the facility layout is severely constrained, which has anadverse impact on the efficiency of the material handlingtransportation system.

In this paper we consider the facility design problemas a dual-objective problem and employ the well knownefficiency frontier approach. In the efficiency frontier ap-proach, a systematic search mechanism is used to considerthe various combinations of weights that can be given to thevarious objectives. Our mechanism is discussed in detail inSection 5. Until that point in the paper, we concentrate onsolving the primary sub-problem: the facility layout designproblem for a fixed facility area.

The traditional layout design of manufacturing facilitiesis based on the concept of planning departments that areeither process-based or product-based entities. The depart-ments typically are constrained in terms of area, but only

∗Corresponding author

loosely constrained in terms of shape (e.g., an aspect ratio ofless than 4:1 must be maintained). Although the decision-maker is asked to specify an interaction matrix (e.g., a quan-titative from-to flow matrix or a qualitative REL chart), thedetailed layout is assumed to follow the block layout design.As a result, the traditional facility layout design formula-tion treats the departments as malleable objects, with de-partment shape refinement to follow based on user-basedmassaging.

There are many examples of manufacturing layout facil-ities where most, if not all, of the department shape lay-outs are fixed due to technological constraints. One class ofexamples is assembly systems, where the component, sub-assembly and final assembly lines are located within thesame facility (another class is flexible manufacturing facili-ties). The result is a complex system consisting of many dif-ferent lines feeding one another. For example, the final as-sembly line may be fed by several sub-assembly lines, whichmay, in turn, be fed by component assembly lines, etc. Thephysical layout of such a system can have a large impacton both the fixed costs associated with the construction ofsuch a system, including the automatic conveyancing sys-tems, and the operational costs associated with the materialhandling cost in the facility. Moreover, the management ofsuch a system is greatly aided by “line-of-sight” manage-ment, which in layout terms, means that the output pointof one line is as close as possible to the input point for that

0740-817X C© 2006 “IIE”

54 Bukchin et al.

sub-assembly or component on the next line. The assem-bly lines may have different shapes, such as, an I-shape, anL-shape, a U-shape, etc. The U-shaped line, for example,is currently very popular in industry and has replaced theI-shaped line in many environments. The reason for this isthat the U-shaped line enables good communication amongthe workers and locates each worker relatively close to allthe stages of the process performed in the line.

The primary sub-problem in such a facility design prob-lem is how to locate a given set of lines in a given facilitysuch that the distances between output points and inputpoints are minimized. As is common in layout design, rec-tilinear distances are applied. The main drawback of thisdistance measure is the fact that in some cases it assumesthat material can be transported through the area of an-other department. Still, these cases are relatively rare andusing rectilinear distances is more appropriate here thanusing Euclidean distances.

We assume that the shape of each line is given, and theplacement (i.e., location and orientation) of the line in thefacility is to be determined (Savas et al., 2002). In so doing,we assume that each line has four possible orientations, de-termined by rotating the shape by 90◦, 180◦ and 270◦. Inorder to construct a general formulation that is not depen-dant on specific line shapes, we assume that each line canbe divided into rectangles, as explained later.

In Fig. 1 we demonstrate the problem’s characteristicsvia an example. We can see an assembly system consistingof several lines of different shapes. Each line is divided intoseveral rectangles, denoted as sub-lines, except for I-shaped

Fig. 1. An example assembly system.

lines, which are rectangular to begin with. The notation (i, j)refers to the rectangular sub-line j of line i. The thin arrowsdenote the assembly process flow for each line. The thickarrows show the flow of material within the facility. First,raw material/components are transported from the inputpoints of the facility, denoted by FIk, to the input pointsof the lines (sub-lines), denoted as Ii(j). After the assemblyprocess on the line, the material is transported from theoutput point of the line (Oi(j)) to the input point of the nextline in sequence. At the end, the sub-assemblies are movedto the final assembly line and from there to the output pointof the facility (FOk).

The remainder of the paper is organized as follows. Theliterature review focusing on reviewing exact models forthe general manufacturing layout problem is presented inthe next section. In Section 3 a formal presentation of theproblem is given, along with a mixed-integer programmingformulation. The results from a performance analysis of theformulation in solving problems with different characteris-tics is given in Section 4, along with an illustrative example.In Section 5 we present our facility design model that incor-porates the efficiency frontier approach described earlier. InSection 6 we conclude the paper and outline future researchavenues.

2. Literature review

Most of the research dealing with the design of assem-bly systems focuses on the operational side of the buildingblocks of the system, the assembly lines. A lot of research

Assembly system facility design 55

has been conducted on the line balancing problem, wherea single model or several models of the same product are tobe assembled on the line. When a mixed-model line is con-cerned, an additional problem of determining the sequenceof the different models to the line arises. For recent surveysof the various types of assembly line balancing problems seeErel and Sarin (1998), Becker and Scholl (2004) and Scholland Becker (2004).

To our knowledge, there is no existing research that ad-dresses the layout of a system of assembly lines. The lay-out of a flexible manufacturing system, a related prob-lem, has been addressed in Das (1993) in which the layoutis represented in a discrete fashion and it is determinedheuristically. However, note that both of these problemsare specialized versions of the general manufacturing fa-cility layout problem, which has been extensively studied(see Meller and Gau (1996) for the latest survey that in-cludes over 100 references on manufacturing layout re-search in the 10 years surveyed). In particular, Montreuil(1990) provided a general formulation for the layout ofcontinuously-represented rectangular-shaped departments.In addition, Heragu and Kusiak (1991) presented a modelthat specifies each department’s length and width assum-ing a rectangular-shaped department with a fixed orien-tation. Likewise, Sarkar et al. (2005) examined the prob-lem of inserting a department of given area, but unknowndimensions into an existing layout via a graph-theoreticapproach.

We propose to adapt the general mixed-integer program-ming model first developed by Montreuil (1990), and laterenhanced by other researchers, to our problem setting. Wenow review these mixed-integer programming models.

The mixed-integer programming (MIP) formulation forthe floor layout problem (FLP) was originally presentedby Montreuil (1990). This model uses a distance-based ob-jective, but is not based on the traditional discrete repre-sentation employed in the quadratic assignment problem.Instead, the MIP-FLP utilizes a continuous representationof a layout and considers departments with unequal areas.In this model, the locations and dimensions of departmentsare decision variables. A number of binary integer variablesare used in the model to avoid overlapping departments.Montreuil’s model is commonly referred to as FLP0.

One of the problems in FLP0 is that in lieu of the exactnon-linear (specifically non-convex and hyperbolic) areaconstraint, a bounded perimeter constraint is used to lin-earize the model. However, using a bounded perimeter con-straint instead of an exact area constraint can lead to errorsin the final area of each department.

A modified MIP-FLP model based on FLP0 was pre-sented by Meller et al. (1998) to improve the model accu-racy and approach efficiency. This model is commonly re-ferred to as FLP1. The bounded perimeter constraint fromFLP0 is modified in FLP1 with a linear approximation tothe area constraint, which improves area accuracy. The bi-nary variables that ensure departments do not overlap in

the facility were also redefined. FLP1 considers every all-rectangular-department layout solution, but becomes dif-ficult to solve for small instances of n ∼= 7 (where n is thenumber of assembly lines) even when valid inequalities areadded in order to eliminate some infeasible solutions fromthe solution space and to improve the algorithm’s efficiency(Meller et al., 1998).

A further enhanced MIP-FLP model based on FLP1was presented by Sherali et al. (2003), which we refer toas FLP2. The first enhancement is an improved represen-tation of the nonlinear area constraint based on a novelpolyhedral outer approximation scheme. The new surrogatearea constraint provides as tight a representation as desired(unlike the approximation used in FLP0 and FLP1) by us-ing a large enough number of discretization points for thetangential supports.

Another important enhancement in FLP2 concerns pre-venting department overlapping by two alternative formu-lations, DJ1 and DJ2. In addition, a new class of validinequalities, called UB inequalities, was discovered by ex-ploring DJ2. DJ1 and DJ2 possess certain partial convexhull properties and provide increased tightness for the MIP-FLP model, but they also lead to a substantial increase insolution time as compared with FLP1 because of their size.Thus, three different strategies were implemented in Sheraliet al. (2003) to impart the tightness from DJ1 and DJ2, withany increase in the problem size, being limited.

Other enhancements in FLP2 consist of a symmetry-breaking constraint and some branching priorities for thebranch-and-bound search. Combing all of these enhance-ments leads to greatly reduced algorithm run time and im-proves the tractable problem size n ∼= 9. Thus, some chal-lenging test problems from the literature were solved for thefirst time by FLP2. However, the problem size is still limitedand not applicable for most industrial applications, whichcan range from 30–50 departments.

As we can see from prior research on MIP-FLP mod-els, the resulting model is difficult to solve. However, theMIP-FLP models provide a good framework for modelingmanufacturing layout problems and we will specialize theMIP-FLP for assembly system layout design in the nextsection.

3. Problem description and model formulation

An assembly system consists of multiple assembly lines ofdifferent shapes. We assume that each line can be dividedinto several rectangles, denoted as sub-lines. Without lossof generality, we assume that there is an orientation of theline, called the standard orientation, consisting of verticalsub-lines which are ordered from left to right. Next, we cantake each pair of adjacent sub-lines as a basic block to formthe constraints that connect the two sub-lines together. Theconnectivity constraints of all the pairs constructing a linepreserve the shape of the line. The fundamental blocks of

56 Bukchin et al.

Fig. 2. Four orientations of an adjacent sub-line pair.

any line (i.e., the two adjacent rectangular sub-lines) cantake one of the four possible orientations in the layout,which are shown in Fig. 2. The most left orientation is thestandard orientation, and then we assume that the shapecan be rotated by 90◦, 180◦ and 270◦.

We give the general MIP formulation for a layout of theassembly system.

Parameters:

Ls = length of the facility in direction s;n = number of assembly lines;N = assembly line set (N = {1, . . . , n});mi = number of sub-lines for assembly line i;Mi = sub-line set for assembly line i(Mi =

{1, . . . , mi});i(i′) = sub-line i′ of assembly line i(i′ ∈ Mi);hi(i′) = height of sub-line i′ for assembly line i in the

standard orientation;wi(i′) = width of sub-line i′ for assembly line i in the stan-

dard orientation;�h

i(i′)(j′) = the height difference of sub-lines i′ and j′(j′ =i′ + 1) centroids for assembly line i in the stan-dard orientation;

�wi(i′)(j′) = the width difference of sub-lines i′ and j′(j′ = i′ +

1) centroids for assembly line i in the standardorientation;

asi(i′) = the relative location in direction s from the out-

put point of sub-line i′ of assembly line i to thecentroid of i′ of i;

bsi(i′) = the relative location in direction s from the in-

put point of sub-line i′ of assembly line i to thecentroid of i′ of i;

fi(i′)j(j′) = the material flow from the output point of sub-line i′ of assembly line i to the input point ofsub-line j′ of assembly line j;

f Ii(i′)k = the material flow from the input point k of the fa-

cility to the input point of sub-line i′ of assemblyline i;

f Oi(i′)k = the material flow from the output point of sub-

line i′ of assembly line i to output point k of thefacility;

FIsk = the location of the input point k of the facility;

FOsk = the location of the output point k of the facility;

FIsk = the location of the input point k of the facility.

Decision variables:

csi(i′) = centroid of sub-line i′ of assembly line i in direc-

tion s;lsi(i′) = side length of sub-line i′ of assembly line i in

direction s;Is

i(i′) = input point of sub-line i′ of assembly line i indirection s;

Osi(i′) = output point of sub-line i′ of assembly line i in

direction s;zs

i(i′)j(j′) = Binary decision variables, which denote relativelocations of sub-line i′ of i and j′ of j with re-spect to direction s and are used to ensure thecorrect connection of sub-lines in the same as-sembly line and prevent the overlapping of thesub-lines from different assembly lines.

The definition of zsi(i′)j(j′) is given as follows:

Given i �= j or i′ �= j′:

zsi(i′)j(j′) =

⎧⎨⎩

1 if sub-line i′of i proceeds sub-line j′ of jin direction s;

0 otherwise.

For the assembly lines composed of multiple sub-lines,we use the relative location variables zs

i(i′)i(i′+1) to denote theassembly line orientation. Note that since the sub-lines ofeach line are ordered in one direction, as explained above,the value of the zs

i(i′)i(i′+1) variable of any pair of sub-linesof that line also denotes the orientation of the line. Forexample, considering the line in Fig. 2, if zx

i(i′)i(j′) = 1, thenwe know that the standard orientation is applied. However,for the I-shaped assembly line consisting of a single rectan-gle, zs

i(i′)i(i′+1) is clearly not valid. Instead, we use a new setof binary variables r t

i , t ∈ {I, II, III, IV} to denote the fourorientations, as show in Fig. 3, of the assembly line.

The definition of r ti is given as follows:

r ti =

{1 if I-shaped assembly line i takes orientation t ;0 otherwise.

Assembly system facility design 57

Fig. 3. Illustration of different orientations of an I-shaped assem-bly line.

The MIP formulation of the assembly system layoutproblem (MIP-ASLP1) is given next.MIP-ASLP1:

min∑

i

∑j>i

∑i′

∑j′

fi(i′)j(j′)(∣∣Ox

i(i′) − Ixj(j′)

∣∣ + ∣∣Oyi(i′) − Iy

j(j′)

∣∣)

+∑

i

∑i′

∑k

f Ii(i′)k

(∣∣FIxk − Ix

i(i′)

∣∣ + ∣∣FIyk − Iy

i(i′)

∣∣)

+∑

i

∑i′

∑l

f Oi(i′)l

(∣∣Oxi(i′) − FOx

l

∣∣ + ∣∣Oyi(i′) − FOy

l

∣∣) (1)

subject to

lsi(i′) ≤ cs

i(i′) ≤ Ls − lsi(i′), ∀i, i′, s, (2)

cxi(i′) = cx

i(i′+1) + (zx

i(i′+1)i(i′) − zxi(i′)i(i′+1)

)�w

i(i′+1)i(i′)

+ (zy

i(i′+1)i(i′) − zyi(i′)i(i′+1)

)�h

i(i′+1)i(i′), ∀i, i′, (3)

cyi(i′) = cy

i(i′+1) + (zx

i(i′)i(i′+1) − zxi(i′+1)i(i′)

)�h

i(i′)i(i′+1)

+ (zy

i(i′+1)i(i′) − zyi(i′)i(i′+1)

)�w

i(i′)i(i′+1), ∀i, i′, (4)

zsi(i′)i(j′) = zs

i(i′)i(i′+1), ∀i, i′, s; j′ > i′, (5)

zsi(i′)i(j′) = zs

i(i′)i(i′−1), ∀i, i′, s; j′ < i′, (6)∑s

(zs

i(i′)j(j′) + zsj(j′)i(i′)

) = 1, ∀i �= j or, i′ �= j′, (7)

∑t

r ti = 1, ∀i ∈ {j|mj = 1}, (8)

csi(i′) + ls

i(i′) ≤ csj(j′) − ls

j(j′) + Ls(1 − zsi(i′)j(j′)

),

∀i �= j; ∀i′, j′, s, (9)lxi(i′) = (

zxi(i′)i(i′+1) + zx

i(i′+1)i(i′))wi(i′) + (

zyi(i′)i(i′+1)

+ zyi(i′+1)i(i′)

)hi(i′), ∀i, 1 ≤ i′ < mi, (10)

lxi(i′) = (

zxi(i′)i(i′−1) + zx

i(i′−1)i(i′))wi(i′) + (

zyi(i′)i(i′−1)

+zyi(i′−1)i(i′)

)hi(i′), ∀i, i′ = mi > 1, (11)

lxi(i′) = (

r Ii + r III

i

)wi(i′) + (

r IIi + r IV

i

)hi(i′)

∀i, mi = 1, (12)lxi(i′) + ly

i(i′) = wi(i′) + hi(i′), ∀i, i′, (13)

Oxi(i′) = cx

i(i′) + (zx

i(i′)i(i′+1) − zxi(i′+1)i(i′)

)ax

i(i′) + (zy

i(i′+1)i(i′)

− zyi(i′)i(i′+1)

)ay

i(i′), ∀i, i′, 1 ≤ i′ < mi, (14)

Oxi(i′) = cx

i(i′) + (zx

i(i′−1)i(i′) − zxi(i′)i(i′−1)

)ax

i(i′) + (zy

i(i′)i(i′−1)

−zyi(i′−1)i(i′)

)ay

i(i′), ∀i, i′, i′ = mi > 1, (15)

Oyi(i′) = cy

i(i′) + (zx

i(i′)i(i′+1) − zxi(i′+1)i(i′)

)ay

i(i′) + (zy

i(i′)i(i′+1)

− zyi(i′+1)i(i′)

)ax

i(i′), ∀i, i′, 1 ≤ i′ < mi, (16)

Oyi(i′) = cy

i(i′) + (zx

i(i′−1)i(i′) − zxi(i′)i(i′−1)

)ay

i(i′) + (zy

i(i′−1)i(i′)

−zyi(i′)i(i′−1)

)ax

i(i′), ∀i, i′, i′ = mi > 1, (17)

Ixi(i′) = cx

i(i′) + (zx

i(i′)i(i′+1) − zxi(i′+1)i(i′)

)bx

i(i′) + (zy

i(i′+1)i(i′)

−zyi(i′)i(i′+1)

)by

i(i′), ∀i, i′, 1 ≤ i′ < mi, (18)

Ixi(i′) = cx

i(i′) + (zx

i(i′−1)i(i′) − zxi(i′)i(i′−1)

)bx

i(i′) + (zy

i(i′)i(i′−1)

−zyi(i′−1)i(i′)

)by

i(i′), ∀i, i′, i′ = mi > 1, (19)

Iyi(i′) = cy

i(i′) + (zx

i(i′)i(i′+1) − zxi(i′+1)i(i′)

)by

i(i′) + (zy

i(i′)i(i′+1)

−zyi(i′+1)i(i′)

)bx

i(i′), ∀i, i′, 1 ≤ i′ < mi, (20)

Iyi(i′) = cy

i(i′) + (zx

i(i′−1)i(i′) − zxi(i′)i(i′−1)

)by

i(i′) + (zy

i(i′−1)i(i′)

−zyi(i′)i(i′−1)

)bx

i(i′), ∀i, i′, i′ = mi > 1, (21)

Oxi(i′) = cx

i(i′) + (r I

i − r IIIi

)ax

i(i′) + (r II

i − r IVi

)ay

i(i′),

∀i, i′, mi = 1, (22)Oy

i(i′) = cyi(i′) + (

r Ii − r II

i

)ay

i(i′) + (r IV

i − r IIi

)ax

i(i′),

∀i, i′, mi = 1, (23)Ix

i(i′) = cxi(i′) + (

r Ii − r III

i

)bx

i(i′) + (r II

i − r IVi

)by

i(i′),

∀i, i′, mi = 1, (24)Iy

i(i′) = cyi(i′) + (

r Ii − r III

i

)by

i(i′) + (r IV

i − r IIi

)bx

i(i′),

∀i, i′, mi = 1. (25)

The objective function of problem (MIP-ASLP1), Equa-tion (1), minimizes the total distances traveled by the inter-nal material flows between the Input/Output (I/O) pointsof assembly lines and the external material flows and be-tween the I/O points of the facility and the I/O points ofassembly lines. The boundaries of sub-lines are constrainedwithin the facility by Equation (2). The sub-lines of the sameassembly line are connected by constraining the distancebetween centroids of adjacent sub-lines from the same as-sembly line in Equations (3) and (4). The relative locationof non-adjacent sub-lines from the same assembly line isconstrained in Equations (5) and (6). In Equation (7), wedenote that for any pair of sub-lines there is only one rel-ative location relationship. In Equation (8), we ensure thatfor each I-shaped line, only one of the four different ori-entations can be taken. We prevent the overlapping of thesub-lines from the different assembly lines through Equa-tion (9). The side length of the sub-lines are determinedin Equations (10)–(13), where we use Equation (10) and(11) to determine the side length of sub-lines for multiplerectangles lines and Equation (12) to determine the sidelength of I-shaped assembly lines. The complementary di-mension of each sub-line is determined in Equation (13).The I/O points of sub-lines of multiple rectangles lines aredefined according to the relative location of the I/O pointsto the centroids of sub-lines in Equations (14)–(21). The I/Opoints of the sub-lines of the I-shaped assembly lines aredefined according to the relative location of the I/O points

58 Bukchin et al.

to the centroids of the assembly lines in Equations (22)–(25). The non-linear absolute value in the objective func-tion can be easily addressed by the following replacement:|a − b| = ab+ + ab−, where a, b, ab+, ab− ≥ 0.

4. Assembly system layout formulationperformance analysis

The following numerical tests are based on problem (MIP-ASLP1), Equations (1)–(25), given in Section 3. Through-out the testing, each problem is solved with the CPLEX 7.0platform. Although it is clear that the run time for solvingthe formulation directly increases as an exponential func-tion of the problem size, it is interesting to see what problemsize can be solved in a reasonable time, and to identify em-pirically the factors that most affect the run time.

Four main types of problems were solved. The first typeconsisted of only I-shaped lines, the second, L-shaped lines(two connected sub-lines in each line), the third, U-shapedlines (three connected sub-lines in a line), and the fourth,mixed-shaped lines. For each problem type, several in-stances were solved with different characteristics, such as,the number of lines, the size of the rectangular facility, andthe number and location of the I/O points. The run time ofeach instance was recorded. The results are listed in Table 1.Columns 1 to 4 denote the data set, the number of assemblylines, the type of the lines (I-shaped, L-shaped, U-shapedor mixed) and the total number of sub-lines, respectively.Columns 5 to 7 refer to the facility size while denoting thedimensions of the facility and the area utilization, namely,the percentage of the facility area occupied by the assemblylines. In columns 8 to 10 we can see the number of inputpoints, the number of output points, and the CPU run time,respectively.

In order to examine how the problem parameters affectthe run time, a multiple regression analysis was conductedwith four independent variables: (i) the total number oflines; (ii) the total number of sub-lines; (iii) the facility areautilization; and (iv) the total number of I/O points. Thebest model was obtained by taking log10 (run time) as thedependent variable with p-values of 3.04 × 10−10.

The model shows that the run time increases significantly(p-value smaller than 0.05) with the number of sub-lines aswell as with the area utilization and with the number of I/Opoints. The former is easily understood since the numberof sub-lines is directly related with the problem size. Thelast two effects can be explained by noting that the problembecomes more constrained as the facility area in which thelines are located decreases and as the number of I/O pointsincreases. For a highly constrained problem, an optimal so-lution is more difficult to achieve. Note that the variabledenoting the number of lines was not found to be signifi-cant. The reason for this is that this variable alone ignoresthe number of sub-lines constructing each line, and there-fore, the information regarding the number of lines alone

Table 1. Solution CPU run times of the MIP formulation for theassembly system layout problem

Dataset

Numberof lines Type

Numberof

sub-linesFacX

FacY

Areautil.

FIsize

FOsize

CPUtime(sec)

1–1 12 I 12 50 40 7.2 1 1 0.411–2 12 I 12 40 30 12.0 1 1 0.641–2 12 I 12 30 25 19.2 1 1 22.771–4 12 I 12 30 25 19.2 1 1 1.471–5 12 I 12 30 25 19.2 2 1 20.272–1 15 I 15 50 16 30.0 2 1 1.052–2 15 I 15 43 16 34.9 2 1 10.663–1 20 I 20 50 30 21.2 2 2 18.503–2 20 I 20 43 20 37.0 2 2 207.804–1 25 I 25 50 50 15.1 2 2 2197.524–2 25 I 25 43 40 22.0 2 2 17 085.005–1 5 L 10 50 40 6.6 1 1 0.065–2 5 L 10 40 30 11.0 1 1 0.645–3 5 L 10 30 25 17.6 1 1 0.255–4 5 L 10 25 20 26.4 1 1 1.085–5 5 L 10 20 15 44.0 1 1 2.635–6 5 L 10 20 10 66.0 1 1 6.396–1 8 L 16 50 40 10.4 2 1 6.786–2 8 L 16 40 30 17.3 2 1 78.096–3 8 L 16 30 30 23.1 2 1 12.366–4 8 L 16 20 20 52.0 2 1 10 194.626–5 8 L 16 15 30 46.2 2 1 21 804.667–1 10 L 20 50 50 10.2 2 2 3.807–2 10 L 20 40 40 15.9 2 2 6.447–3 10 L 20 35 35 20.7 2 2 33.347–4 10 L 20 35 31 23.4 2 2 40.367–5 10 L 20 30 30 28.2 2 2 12 924.318–1 12 L 24 45 45 15.8 2 2 42 857.958–2 12 L 24 40 30 26.7 2 2 15 788.319–1 15 L 30 60 60 11.0 2 2 11 717.23

10–1 5 U 15 50 50 8.7 1 1 0.4810–2 5 U 15 35 35 17.8 1 1 0.5910–3 5 U 15 25 25 34.9 1 1 13.0611–1 7 U 21 50 50 13.1 1 1 1.7711–2 7 U 21 35 35 26.8 1 1 2.4412–1 8 U 24 50 50 15.8 2 1 315.0912–2 8 U 24 40 40 24.6 2 1 840.5813–1 10 U 30 50 50 18.6 2 1 496.2013–2 10 U 30 40 40 29.0 2 1 13 264.1614–1 11 U 33 50 50 20.0 2 1 844.7715–1 12 U 36 50 50 22.6 2 2 9124.6317–1 9 Mixed 18 50 50 8.2 2 1 13.0017–2 9 Mixed 18 50 30 13.6 2 1 20.0917–3 9 Mixed 18 35 30 19.4 2 1 17.2518–1 12 Mixed 24 50 50 10.9 2 1 2.5918–2 12 Mixed 24 35 30 25.9 2 1 12 111.50

is insufficient to indicate the problem size. Consequently,another regression model was conducted, this time withoutthe number of lines as a dependent variable. The ANOVAtable is presented in Tables 2(a) and 2(b). One can see thatthe p-value of the model has improved, and is now equal to5.10 × 10−11.

Assembly system facility design 59

Table 2a. ANOVA table for the run time performance

df SS MS F Significance F

Regression 3 86.264 617 52 28.754 872 51 32.458 372 13 5.103 56 × 10−11

Residual 42 37.207 800 83 0.885 900 02Total 45 123.4724 183

The regression equation is then given by the followingexpression: log10(run time) = −4.0767 + 0.136 (no. of sub-lines) + 0.05 (area util.) + 0.697 (no. of I/O points). In areal-world situation, the number of I/O points is usually rel-atively low. Therefore, the most influencing and interestingfactors are the number of sub-lines and the area utilization.In Fig. 4 we can see three graphs demonstrating the run timeas a function of the number of sub-lines, each for a differentlevel of area utilization. We can first see the exponential na-ture of the graphs with respect to the area utilization level.In addition, one can see that given a relatively low area uti-lization, on the average, problems up to around 50 sub-linescan be solved to optimality in a reasonable amount of time.This number goes down to around 28 sub-lines for highervalues of area utilization.

One should note that estimating the run time based onthe average values obtained by the regression model maybe quite risky due to the high variance of the run time. Forexample, the average run time of a configuration of 30 sub-lines, 50% utilization level and two I/O points is 2.23 hours.However, looking at worst-case scenarios based on the 90%confidence interval of each of the coefficients (keeping theother coefficients at their average values), the expected runtime becomes much larger. Run time values of 56.3, 21.3and 13.2 hours are obtained for the worst case of the coef-ficient of the number of sub-lines, area utilization and thenumber of I/O points, respectively. Looking at the worst-case scenario of this problem based on the 90% confidenceinterval of all the coefficients simultaneously, one would geta run time value of 35 613 hours. This value is of course aresult of the compounding of the three coefficients and isnot expected to occur; however, it does give an indicationof the large variability of the run time.

As we realized that the model is highly affected by the areautilization, and since we assume that in a real-world envi-ronment, the facility area is well utilized, we tested the runtime over several instances of very-high-facility-area uti-

Table 2b. Further ANOVA data on the run time performance

StandardCoefficients error t Stat P-value

Intercept −4.076 70 0.624 07 −6.532 47 6.805 43 × 10−8

No.of sub-lines 0.135 97 0.027 78 4.894 86 1.496 65 × 10−5

Area util. 0.050 18 0.011 64 4.310 81 9.618 97 × 10−5

No.of I/O 0.696 88 0.229 72 3.033 56 0.004 133 706points

lization. The largest problem that could be solved in a rea-sonable amount of time (12.3 hours) was of seven (mixed)assembly lines (13 sub-lines). Such a problem represents areasonably large assembly facility. Details on this exampleproblem are presented next.

The material flow within the example assembly systemis given via the precedence diagram shown as Fig. 5. Eacharrow denotes a flow of material, the circles are the I/Opoints of the facility and within each rectangle we can seethe line number and its shape in parentheses. We locatedthe assembly system in a facility with dimensions of 15 × 13and the area utilization is 80%. The optimal layout solutionthat minimizes the distance flow, is shown in Fig. 6, with anobjective value of 42 distance units.

5. A cost model for the Assembly System Facilitydesign problem

Problem (MIP-ASLP1) locates the set of assembly lines ina given facility area and minimizes the weighted distancesbetween the input and output points of the facility and theassembly lines. Where an automated material handling sys-tem exists to move material between the input and outputpoints, the total cost (investment plus operational costs) isassumed to increase with the distance traveled. On the otherhand, where such a system does not exist, the weighted dis-tances are associated with the operational cost of trans-portation. For real-world data, the net present value of theoperational costs has to be evaluated based on a planninghorizon in order to be compared with the investment cost.In either case, we assume that the total material handlingcosts increase as the distances between the input and outputpoints increase.

Another important aspect to consider when designing(or constructing) an assembly system facility is the cost dueto the size (area) of the facility. That is, since in practice acost is associated both with material handling and with thearea of the facility, a mechanism is needed to analyze thetrade-off between the two. To that end, a design model isneeded to examine the impact of changes in the facility areaon the resulting changes in material handling cost.

In developing a model for the design or construction ofa facility, we may ignore the placement of some or all ofthe facility I/O points since such points would typically bedetermined after the lines have been placed (otherwise itwould be very difficult to assign lines to I/O points with-out knowing both the location of the lines and the inputpoints). Still, the location of one point, the output point of

60 Bukchin et al.

Fig. 4. Run time as a function of the number of sub-lines and area utilization.

the facility, can generally be determined and we use it toorient the general direction of the material flow. By con-sidering only one facility output point (and ignoring thefacility input points in the experiments) it will also make itmuch easier to standardize our test problems to isolate theimpact of facility size versus material handling distance.

In general, note that the minimal material handlingtransportation costs will be achieved for a facility that islarge enough so as to not limit the placement of the as-sembly lines. For any problem, the maximum side lengthneeded, Lmax, is equal to

∑ni=1 max{∑mi

i′=1 hi(i′),∑mi

i′=1 wi(i′)}.Of course, a facility measuring Lmax × Lmax would be veryunder-utilized. However, as the facility dimensions are re-duced to decrease the area of the facility and to increasethe facility utilization, the placement of the assembly linesis constrained, and as a result, the material handling trans-portation costs will rise in general (and certainly can-not decrease). In general, due to department shapes thathave been determined a priori, it is not possible to calcu-late the minimum feasible facility dimensions, but clearly∑n

i=1

∑mii′=1 hi(i′)wi(i′) serves as a lower bound to the facility

area (Lx × Ly).We define the bi-objective Assembly System Facility De-

sign problem (ASFDP) as follows:(ASFDP):

min{w1(Transportation) + w2Lx′Ly ′ }, (26)

Fig. 5. Material flow within the example assembly system.

where w1 and w2 are the cost of one-unit of transporta-tion and one-unit of area, respectively, and Transporta-tion is defined by Equation (1), the objective to problem(MIP-ASLP1). Such a formulation has reduced a multi-objective optimization problem to a single-objective prob-lem. Still, two problems arise here. First, the values of theweights are usually difficult to obtain, and some times othernon-quantitative considerations are involved. To that end,we prefer to avoid the setting of the weights in the ob-jective and instead adopt an efficiency frontier approachto study the different possible feasible solutions prior toselecting the preferred one. The second problem involvesthe non-linearity of the objective function, Equation (26),due to the facility area term. This problem is addressedbelow.

Fig. 6. Optimal layout of the example problem.

Assembly system facility design 61

Let us first look at the extreme case of the design problem,in which the material handling cost is minimized subject toconstrained dimensions of the facility, while ignoring thecost of the area. Although solving problem (MIP-ASLP1)will provide the optimal solution, this solution may besomewhat inefficient, since the same transportation costmay be possible for a smaller area than the one defined bythe constraint. Hence, we modify problem (MIP-ASLP1)(resulting in problem (MIP-ASLP2)) with a new objectivefunction to incorporate the above aspect of facility size; i.e.:(MIP-ASLP2):

min{M(Transportation) + Lx′Ly ′ }, (27)

Subject to Equations (2)–(25),

Ls ′ ≤ Ls ∀s . (28)

Since M is a large number, the transportation cost becomesthe primary objective and the facility area is the secondaryobjective, which allows us to find the minimum transporta-tion cost for particular facility side lengths. Note that elim-inating Equation (28) from problem (MIP-ASLP2), wewould get the formulation for finding the minimum trans-portation cost along with the largest possible area beyondwhich there is no further reduction in the transportationcost. Similarly, if in addition we change Equation (27)to min{(Transportation) + MLx′

Ly ′ }, the smallest possiblearea will be obtained, beyond which placement of the linesbecomes infeasible.

One can see that problem (MIP-ASLP2) has a non-linearobjective function. In order to make a linear approxima-tion, we replace the area component in Equation (27) bythe perimeter of the facility, assuming that in general thetwo components are highly correlated (Meller et al., 1988),resulting in problem (MIP-ASLP3).MIP-ASLP3:

min{M(Transportation) + 2(Lx′ + Ly ′)} (29)

Subject to Equations (2)–(25)

Ls ′ ≤ Ls ∀s . (30)

Thus, solving problem (MIP-ASLP3) will then providean approximation to the minimal area solution for the

Fig. 7. Shapes of the lines in the example problem.

lowest transportation cost given the maximal values ofthe facility’s dimensions. However, our original facility de-sign problem is to consider the trade-off between mate-rial handling costs and facility area costs. To that end, weadopt the efficiency frontier approach (Steuer, 1986). Ac-cording to this approach, all or some non-dominated (ef-ficient) solutions have to be generated, and the preferredsolution is to be selected later by the decision-maker. Asolution i is defined as efficient if there is no other so-lution, j, which is at least as good as solution i in bothobjectives. The efficiency frontier approach gives a full pic-ture of the trade-off between the two objectives as we showbelow.

One can see that solving problem (MIP-ASLP3) withEquation (29) yields an approximation to the extreme pointof the efficiency frontier, where the transportation costs areconsidered as a primary objective and the area cost as a sec-ondary objective. One way to generate the efficiency fron-tier is by changing gradually the values of w1 and w2 andusing Equation (26) in problem (MIP-ASLP2) (or a mod-ified version with a perimeter component instead of thearea component). Since we do not know how the efficientsolutions are distributed along the efficiency frontier, thisapproach may consume a large number of iterations, andworse, skip efficient solutions. Instead, we suggest usingthe following algorithm to generate the approximated effi-ciency frontier. The algorithm is based on constraining onecomponent of the objective function (the perimeter valuein our case) while minimizing the other component (thetransportation distance).

Algorithm EF

Step 1. Solve MIP-ASLP-3.Step 2. Set i, the iteration index, equal to one; i.e., i = 1.Step 3. Calculate the facility perimeter resulting from

Step 1, Pi.Step 4. Set Pi+1 = Pi − �, where � is a resolution param-

eter.Step 5. Solve problem (MIP-ASLP-3) with the following

additional constraint, 2(Lx′ + Ly ′) ≤ Pi+1.

Step 6. If no feasible solution obtained, go to Step 7. Oth-erwise, set i = i + 1, keep the obtained solution Si,

62 Bukchin et al.

Fig. 8. Material flow in the example problem.

with a perimeter value of Pi and a distance value ofdistance Di, and go to Step 4.

Step 7. For each solution i: (i) calculate the facility areaAi; (ii) if there is any solution, Sj, where Dj ≤ Diand Aj ≤ Ai, eliminate solution Si as a dominatedsolution. The remaining set of solutions is the ap-proximated efficiency frontier.

Note that in Step 5, one could use the constraint 2(Lx′ +Ly ′

) < Pi instead of the current constraint. In this case, eachtime a solution with Pi was obtained, the above constraintincluding the strict inequality would have been used in thenext iteration. This approach would guarantee finding allefficient solutions with regard to the perimeter as one ofthe objectives. The reason for preferring the suggested con-straint is that the problem is mixed integer, the solution ispartially continuous, and consequently, the efficiency fron-tier may contain an infinite number of solutions. In theproposed approach, we can control the number of efficientsolutions in the set by changing the resolution parameter, �.

The following example demonstrates the generation ofthe approximated efficiency frontier. The considered assem-bly system consists of five lines and nine sub-lines, as canbe seen in Fig. 7. The material flow is shown in Fig. 8. Thefigures in the parentheses represent the sub-lines compos-ing each line. The facility output point was forced to thebottom of the facility (y = 0) with the x-value being left asa variable. Such an approach is flexible in terms of the lay-

Fig. 9. Approximated efficiency frontiers for the example problem: (a) the approximated efficiency frontiers associated with the perime-ter and transportation components; and (b) the approximated efficiency frontier associated with the total area and transportationcomponents.

Table 3. The algorithm results of the example problem

Solution Lx Ly Perimeter Area Utilization z

1 14.5 27.0 83 391.5 0.378 1.02 14.0 27.0 82 378.0 0.392 1.53 20.0 19.0 78 380.0 0.389 2.04 20.5 13.0 67 266.5 0.555 3.05 20.0 13.0 66 260.0 0.569 3.56 14.5 18.0 65 261.0 0.567 4.07 19.0 13.0 64 247.0 0.599 4.58 14.5 17.0 63 246.5 0.600 5.09 12.0 19.0 62 228.0 0.649 5.5

10 13.0 17.5 61 227.5 0.651 6.011 13.0 17.0 60 221.0 0.670 6.512 13.0 16.5 59 214.5 0.690 8.013 13.0 16.0 58 208.0 0.712 8.514 15.5 13.0 57 201.5 0.734 13.015 15.0 13.0 56 195.0 0.759 13.516 14.0 13.5 55 189.0 0.783 15.517 14.0 13.0 54 182.0 0.813 16.518 15.0 11.5 53 172.5 0.858 26.519 15.0 11.0 52 165.0 0.897 27.520 N/A N/A N/A N/A N/A N/A

out algorithm as well as the facility design algorithm sincea particular facility output point may constrain the size ofthe facility.

Without loss of generality, Algorithm EF was startedwithout limiting the initial area and the value of � was setto one-unit of distance. Twenty iterations were performedand 19 solutions were obtained, as is shown in Table 3. Incolumns 2 and 3 we can see the dimensions of each layout.Then, the associated perimeter and area are presented incolumns 4 and 5, respectively. The area utilization is pre-sented in column 6 and the total transportation distancepresented in column 7. We can see that the first solutionis the solution of problem (MIP-ASLP3) with a perimeter

Assembly system facility design 63

Fig. 10. Total distance traveled against the area utilization.

value of 83 distance units, an area of 391.5 distance unitssquared, and an objective function of one distance unit. Wecan see that no feasible solution was obtained for a perime-ter smaller than or equal to 51 distance units. Another inter-esting observation concerns solutions 3 and 6. These solu-tions were found to be dominated by other solutions, sincein their iteration, the obtained area was increased althoughthe perimeter was decreased. These solutions were elimi-nated as specified by Step 7 of Algorithm EF.

Fig. 11. Three solutions of the example problem: (a) 14.5 × 27 = 391.5 (z = 1); (b) 13 × 17 = 221 (z = 6.5); and (c) 15 × 11 = 165(z = 27.5).

In. Fig. 9(a) we can see the approximated efficiency fron-tier associated with the perimeter and transportation com-ponents. In this graph we can see that all 19 solutions areefficient. Note that the algorithm guarantees that each newsolution will be an efficient one with regard to the perimeterobjective. The reason for this is that each new solution is su-perior to the previous solutions with regard to the perimeter(due to Steps 4 and 5 of the algorithm) and inferior with re-gard to the transportation cost (otherwise, the value of thetransportation cost would be obtained in a previous itera-tion while solving a less-constrained problem). In Fig. 9(b)we see the approximated efficiency frontier associated withthe total area and transportation components is presented,before eliminating the two dominated solutions, which aremarked by arrows. Here we can see that the facility perime-ter is a relatively good surrogate measure for the facilityarea, since 17 out of the 19 perimeter-efficient solutions arealso area-efficient solutions.

Another perspective on the results can be obtained bylooking at Fig. 10, and considering the relationship be-tween the transportation distances and the area utilization.One can see that the optimal solution with regards to thetransportation distances is obtained for an area utilizationof 37.8%. The utilization is then increased by decreasingthe facility perimeter until a maximal value of 89.7% isreached (with an associated transportation distance of 27.5

64 Bukchin et al.

distance units). Again, we can see the two dominated solu-tions, marked by the arrows that should be omitted fromthe efficient set.

Another insight can be obtained by looking at the threelayouts presented in Fig. 11(a–c). In Fig. 11(a), we can seethe first solution obtained by Algorithm EF, which is opti-mal with regard to the transportation distance. In Fig. 11(c),the last solution obtained by the algorithm, with the small-est area and the highest utilization, is presented. In between,we can see a solution that was taken from the middle areaof the efficiency frontier (i.e., a solution that appears to rep-resent a good trade-off of the two objectives). In fact, thissolution, Si, was selected based on the following measure,

i = arg mini

[max

(Ai − Amin

Amax − Amin,

zi − zmin

zmax − zmin

)],

namely, the solution with the smallest maximal relativedistance from either of the two objectives. This solution rep-resents a compromise between the two components of theobjective function and is relatively good with respect to bothcomponents, with a total distance of 6.5 distance units andan area utilization of 67%. Note that in some cases this solu-tion may be preferred over the other solutions if the weights,representing the costs, are relatively close. However, if thereis one weight which is much higher that the other, a solu-tion much closer to one edge of the efficiency frontier will bechosen.

Such an efficiency frontier approach will provide quitea bit of information to the decision-makers as they designtheir assembly system facility. In addition to providing thedesign that represents the best trade-off between materialhandling cost and area utilization, the chosen design willalso imply the values of w1 and w2 in Equation (26). Do-ing so may then allow the solution of the assembly systemfacility design problem directly in future iterations.

6. Concluding remarks

In this paper we considered a facility design problem for afacility that consists of a system of assembly lines. In solvingthis problem, we introduced a new facility layout problem.To solve this layout problem, we proposed a MIP formula-tion that models each line as a set of rectangular sub-lines.Our experimental results indicate that small-to-moderate-sized problems can be solved with this formulation depend-ing on the number and types of lines considered. We alsodiscovered that the percentage of facility area that was oc-cupied and the number and location of I/O points also hasan impact on the run time.

Due to our performance analysis of the assembly systemlayout formulation, we believe that many assembly systemfacilities can be solved in a reasonable amount of com-putational effort with our approach. Still, there may existproblems that are larger than the ones shown here. In such a

case, a heuristic will be needed. Development of a heuristicis left for future research.

To solve the facility design problem, the efficiency frontierapproach was applied to analyze the trade-off between thefacility area and the transportation distance. An algorithmable to generate a set of efficient solutions was presented.On the practical side, the efficiency frontier can be usedby designers to solve the facility design problem enablingthem to see the whole picture and consider the complextrade-off between facility area costs and material handlingcosts.

References

Becker, C. and Scholl, A. (2004) A survey on problems and methodsin generalized assembly line balancing. European Journal of Opera-tional Research (forthcoming).

Das, S.K. (1993) A facility layout method for flexible manufacturing sys-tems. International Journal of Production Research, 31(2), 279–297.

Erel, E. and Sarin, S.C. (1998) A survey of the assembly line balancingprocedures. Production Planning & Control, 9(5), 414–434.

Heragu, S.S. and Kusiak, A. (1991) Efficient models for the facilitylayout problem. European Journal of Operational Research, 53, 1–13.

Meller, R.D. and Gau, K.-Y. (1996) The facility layout problem: recentand emerging trends and perspectives. Journal of Manufacturing Sys-tems, 15(5), 351–366.

Meller, R.D., Narayanan, V. and Vance, P.H. (1998) Optimal facility lay-out design. Operations Research Letters, 23(3–5), 117–127.

Montreuil, B. (1990) A modelling framework for integrating layout designand flow network design, in Proceedings of the Material HandlingResearch Colloquium, pp. 43–58.

Sarkar, A., Batta, R. and Nagi, R. (2005) Planar area location/layoutproblem in the presence of generalized congested regions with therectilinear distance metric. IIE Transactions, 37(1), 35–50.

Savas, S., Batta, R. and Nagi, R. (2002) Finite-size facility placement inthe presence of barriers to rectilinear travel. Operations Research,50(6), 1018–1031.

Scholl, A. and Becker, C. (2004) State-of-the-art exact and heuristic solu-tion procedures for simple assembly line balancing. European Jour-nal of Operational Research (forthcoming).

Sherali, H.D., Fraticelli, B.M.P. and Meller, R.D. (2003) Enhanced modelformulations for optimal facility layout. Operations Research, 51(4),629–644.

Steuer, R.E. (1986) Multiple Criteria Optimization: Theory, Computationand Application, Wiley, New York, NY.

Biographies

Yossi Bukchin is a faculty member in the Department of Industrial En-gineering at Tel Aviv University. He received his B.Sc., M.Sc., and D.Sc.degrees in Industrial Engineering from the Technion Israel Institute ofTechnology. He is a member of the Institute of Industrial Engineer-ing (IIE) and on the Editorial Board of IIE Transactions. Dr. Bukchinhas held a visiting position in the Grado Department of Industrial &Systems Engineering at Virginia Tech. His papers have been publishedin IIE Transactions, European Journal of Operational Research, Interna-tional Journal of Production Research, Annals of the CIRP and otherjournals. His main research interests are in the areas of assembly systemsdesign, assembly line balancing, facility design, operational scheduling,multi-objective optimization as well as work station design with respectto cognitive and physical aspects of the human operator.

Assembly system facility design 65

Russell D. Meller is Professor of Industrial and Systems Engineering atVirginia Tech in the Grado Department of Industrial & Systems En-gineering. He joined Virginia Tech after seven years on the faculty atAuburn University. He received his B.S.E., M.S.E., and Ph.D. in Indus-trial and Operations Engineering from The University of Michigan. Hisdissertation on facility layout was awarded the 1994 Institute of Indus-trial Engineers Outstanding Dissertation Award and First Prize in the1993 College on Location Analysis Dissertation Prize Competition fromThe Institute of Management Sciences. In 1996 he received a CAREERDevelopment Grant from the National Science Foundation. In 2002, Dr.Meller was named the Outstanding Young Industrial Engineer from theInstitute of Industrial Engineers. Dr. Meller’s professional experience in-cludes consulting with the SysteCon Division of Coopers & Lybrand,General Electric, Cross Creek Apparel, and the Russell Corporation. Hisresearch interests include facilities layout and location, automated ma-terial handling systems, and operations research applications in forestry.He has published his research in IIE Transactions, Manufacturing & Ser-

vice Operations Management, Operations Research, Management Science,Journal of Manufacturing Systems, IJPR, and other journals. In addition,he is a department editor for IIE Transactions and is on the editorialboard of Journal of Manufacturing Systems. He is a member of IIE andhas served as faculty advisor for over ten years. He is also a member ofINFORMS and Alpha Pi Mu, and is currently serving as President ofthe College-Industry Council for Material Handling Education.

Qi Liu is a Senior Analyst for Capital One in Richmond, Virginia. Hereceived his Ph.D. from the Grado Department of Industrial and SystemsEngineering from Virginia Tech in 2004. He received both his M.S. andB.S. in Industrial Automation from East China University of Science andTechnology. In 2001 he commenced his Ph.D. studies at Virginia Tech inthe Department of Industrial & Systems Engineering. His research wassponsored by the National Science Foundation. While working towardsthe Ph.D. degree, he received the Robert R. Reisinger Honor Scholarshipfrom the Material Handling Education Foundation.