European Journal of Operational Research 174 (2006) 1414–1426 www.elsevier.com/locate/ejor
Discrete Optimization
Single- and multi-objective facility layout with workflow interference considerations Wen-Chyuan Chiang a
c
a,*
, Panagiotis Kouvelis
b,1
, Timothy L. Urban
c,2
The University of Tulsa, College of Business Administration, Tulsa, OK 74104-3189, United States b Washington University, Olin School of Business, St Louis, MO 63130-4899, United States The University of Tulsa, College of Business Administration, Tulsa, OK 74104-3189, United States Received 26 January 2004; accepted 1 March 2005 Available online 6 June 2005
Abstract The effect of workflow interference is a major concern in facility layout design. Yet, despite the extensive amount of research conducted on the facility layout problem, very little has been done to incorporate interference as part of an overall approach to layout design. This paper examines the impact of workflow interference considerations on facility layout analyses. Linear and nonlinear integer programming formulations of the problem are presented. The structural properties of the resulting formulations, as applied to facility design, are investigated. Finally, a multi-objective approach to facility layout design is presented, incorporating the traditional distance-based objective with that of workflow interference. 2005 Elsevier B.V. All rights reserved. Keywords: Facilities planning and design; Combinatorial optimization; Assignment
1. Introduction An extensive amount of research has been conducted on the facility layout problem, much of it based on the quadratic assignment problem (QAP) formulation. The QAP objective is to minimize the distancebased transportation cost expressed as the product of the quantity of workflow and the distance traveled.
*
Corresponding author. Tel.: +1 918 631 2939; fax: +1 918 631 2037. E-mail addresses:
[email protected] (W.-C. Chiang),
[email protected] (P. Kouvelis),
[email protected] (T.L. Urban). 1 Tel.: +1 314 935 4604; fax: +1 314 935 6359. 2 Tel.: +1 918 631 2230; fax: +1 918 631 2037. 0377-2217/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.03.007
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Since Lawler (1963) and Gilmore (1962) developed optimal solution procedures for the QAP, many heuristics have been developed due to the difficulty in finding the optimal solution to this problem, and several alternative approaches have been examined (for recent reviews of the QAP literature, see Rendl, 2002; Hahn, 2004). From a facility-layout perspective, a variety of applications and alternative formulations have also been investigated; however, relatively little has been done to incorporate workflow interference in facility layout design. Several authors have identified workflow interference as a major concern in traditional facility layout design. Tompkins and White (1984) discussed the importance of minimizing interruptions on flow paths and recognized that the effect of interruptions results in congestion and undesirable intersections. Apple (1972) acknowledged the need to arrange equipment to optimize materials flow and suggested we ‘‘be aware of cross traffic and take necessary precautions. Avoid traffic jams’’. Luggen (1991) also noted the benefits of eliminating complex material flow patterns, stating ‘‘complex material flow patterns create extensive part move and queue time, result in lost or misplaced parts, and contribute to damaged parts due to excessive movement’’. Workflow interference is also a problem with automated manufacturing systems. Concerning automated guided vehicle systems (AGVS), Egbelu and Tanchoco (1986) noted that there is a ‘‘problem of traffic control at intersections’’, such that a significant amount of cross traffic can result in considerable delays. Krishnamurthy et al. (1993) also recognized the importance of conflict-free routes on the static routing problem for an AGVS. An excessive amount of cross traffic can also impact the initial cost of an automated manufacturing system, either through the need for more complex traffic management systems (Miller, 1987), such as buffering areas or interchange ramps, or through the cost of acquiring control software to manage intersection activities (Egbelu and Tanchoco, 1986). If the number of intersections at which conflict could occur can be minimized as well as minimizing the interference at those intersections that are necessary, the cost of the system can obviously be reduced. The minimization of interference between pairs of ‘‘facilities’’ is important in other types of layout analyses as well. For example, in the layout design of multilayer IC technology, the overlap of wires between the nets (a set of terminals on the devices to be connected) is not allowed. If the devices are located in such a manner that results in a crossing (interference), the connection would be made by routing it through another layer of the IC. Therefore, we want to place the nets in such a manner to minimize the number of vias (points connecting the different layers) as they increase the fabrication cost and degrade system performance (Cong et al., 1993). To illustrate the effect of conducting a layout analysis that fails to account for workflow interference, consider an eight-department example with the workflow matrix shown in Fig. 1. The solution to this problem using traditional layout analyses (the quadratic assignment problem is solved using Euclidean distances) is provided in Fig. 1a. While this layout arrangement will minimize the total distance traveled, it is obvious that there are several points at which workflow interference could occur. An alternative layout (Fig. 1b) can provide a workflow in which there are no conflicting workflows. As shown in this figure, the flow pattern is considerably less congested, with more of a uni-directional flow clockwise from facility 6 to facility 2, resembling the U-shaped layout advocated by many researchers and practitioners (e.g., Sekine, 1992, and Miltenburg and Wijngaard, 1994, identify several advantages of using U-lines). Recently, Chiang et al. (2002) modeled workflow interference in facility layout design as a quartic assignment problem, and developed a branch-and-bound procedure and a tabu search heuristic to solve the problem. The term ‘‘quartic assignment problem’’ was used since the objective function is a fourth-degree polynomial function of the variables and the constraints are those of the assignment problem. Assignment problems with fourth-degree polynomial objectives were discussed by Lawler (1963); however, they were not presented in the context of the minimization of workflow interference. It has also been noted that this type of problem arises in the area of VLSI synthesis (Burkard et al., 1994; Burkard and C ¸ ela, 1995). A related problem models congestion in facility layout design through the use of queueing networks (Benjaafar,
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0 0 0 0 7 0 0 0 7 0 0 0 0 0 0 0
F=
0 0 0 2 0 6
8 3 0 0 0 5
0 0 0 0 3 0
7 0 0 0 0 5
0 0 0 3 0 1
0 0 0 0 0 0
0 0 4 9 0 0
6 0 0 0 4 0
4
8
7
6
3
2
1
5
3
8
4
2
7
5
1
6
(a)
(b) Fig. 1. Solution to the eight-facility example: (a) minimizing distance traveled, and (b) minimizing workflow interference.
2002), measuring congestion by the amount of work-in-process in the system in lieu of workflow interference. The paper is organized as follows. In Section 2, we analyze the structural properties of the quartic assignment problem (QrAP) as applied to facility design. In Section 3, the problem is also modeled as a modified quadratic assignment problem (MQAP) as an alternative to the quartic formulation. A multi-objective approach to facility design is presented in Section 4 by integrating the quadratic and quartic assignment problems; in particular, we focus on identifying an efficient frontier for the two objectives. We conclude with a summary of the research in Section 5.
2. The workflow interference problem Unlike the quadratic assignment problem that considers the relative location of pairs of facilities, we must consider ‘‘pairs of pairs’’ of facilities to minimize workflow interference. To measure whether or not there is interference or cross traffic, we must relate the workflow between one pair of facilities to that of another pair of facilities. Looking again at Fig. 1, we can see that due to the relative location of two pairs of facilities (7–3 and 8–5), we realize an intersection with one layout but not the other. The model must be able to measure and evaluate this type of relationship; therefore, two factors are considered: (1) the amount
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(sx, sy)
(jx, jy)
(lx, ly)
(qx, qy)
(jx, jy)
(a)
(sx, sy)
(qx, qy)
(lx, ly)
(b)
Fig. 2. Types of interference: (a) cross point and (b) overlay.
of workflow between two pairs of facilities, which affects the potential amount of interference between the facilities; and (2) whether interference exists between two pairs of locations. This interference may be a cross point—a point where two different workflows intersect each other at a single point, as shown in Fig. 2a—or an overlay—a portion of a route where two opposite workflows conflict, as shown in Fig. 2b. The potential interference of the workflow from facility i to facility k and from facility p to facility r can 0 be measured as fikpr ¼ minffik ; fpr g, where fik and fpr are the workflows between the pairs of facilities i, k and p, r, respectively. Other appropriate measures can also be used, depending on the particular application. Of course, the fact that workflows between pairs of facilities does not guarantee interference will occur; it is possible that the flow could occur at different times. However, unless specifically scheduled, a greater amount of workflow will result in a higher likelihood of experiencing interference. The presence of interference from location j to location l and from location q to location s can be expressed as 8 > < 0 if the paths do not cross or if the flow is in the same direction; d jlqs ¼ 1 if the paths cross at a cross point; ð1Þ > : x if the paths cross along an overlay and the flow is not in the same direction. The value of x will also depend on the particular application (expressed as a constant, as a function of the distance of the overlay, etc.), although it is restricted not to be a function of the workflows. If desired, a value can also be included to represent potential interference resulting from flow in the same direction. 2.1. An algorithm to generate the interference matrix In this section, we illustrate how the interference matrix can be generated from the Euclidean coordinates of any four arbitrary locations j, l, q, and s with coordinates (jx, jy), (lx, ly), (qx, qy), and (sx, sy) as shown in Fig. 2. The idea behind the algorithm is that the equation of a line passing through two given points (x1, y1) x1 y 1 1 and (x2, y2) is f ðx; yÞ ¼ x2 y 2 1 ¼ 0. This line divides a plane into three parts: points on the line which x y 1 make f(x, y) = 0, those points lying on one side of the line which make f(x, y) > 0, and those points lying on the other side of the line which make f(x, y) < 0. The value j1 determines which side of the line (qx, qy) ! (sx, sy) that point (jx, jy) is located: sx sy 1 j1 ¼ f ðjx ; jy Þ ¼ qx qy 1 ¼ ðsx qx Þðqy jy Þ ðsy qy Þðqx jx Þ. jx jy 1
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Likewise, l1 determines which side of the line (qx, qy) ! (sx, sy) that point (lx, ly) is located: l1 ¼ ðsx qx Þðqy ly Þ ðsy qy Þðqx lx Þ. If j1 · l1 < 0, then points (jx, jy) and (lx, ly) are located on different sides of the line defined by points (sx, sy) and (qx, qy). Similarly, q1 ¼ ðjx lx Þðly qy Þ ðjy ly Þðlx qx Þ s1 ¼ ðjx lx Þðly sy Þ ðjy ly Þðlx sx Þ If q1 · s1 < 0, then points (qx, qy) and (sx, sy) lie on different sides of the line defined by points (jx, jy) and (lx, ly). Therefore, the following two scenarios can be identified: Scenario 1. If j1 · l1 < 0 and q1 · s1 < 0, then there must be a crosspoint between lines (jx, jy) ! (lx, ly) and (qx, qy) ! (sx, sy); therefore, djlqs = 1. Scenario 2. If (j1 · l1 P 0 or q1 · s1 P 0) and (j1 5 0 or l1 5 0 or q1 5 0 or s1 5 0), then either: • points (qx, qy) and (sx, sy) lie on the same side of line (jx, jy) ! (lx, ly) or • points (jx, jy) and (lx, ly) lie on the same side of line (qx, qy) ! (sx, sy); therefore, there is no crosspoint between the lines, and djlqs = 0. If neither of these two scenarios is applicable, then the four points must lie on the same line. To determine whether the lines (jx, jy) ! (lx, ly) and (qx, qy) ! (sx, sy) are overlapping and/or in the same direction, the coordinates of the four points are projected onto the X-axis (unless the line is vertical—i.e., jx = lx = qx = sx—in which case the coordinates are projected onto the Y-axis). If the lines do not overlap, then jjx + lx qx sxj P jjxlxj + jqx sxj. Also, if (jx lx)(qx sx) P 0, then the workflows from (jx, jy) ! (lx, ly) and (qx, qy) ! (sx, sy) are in the same direction. Scenario 3. If the four points lie on a vertical line and either in the same direction or with no intersection— (cond 1) ^ (cond 2 _ cond 3)—or on a non-vertical line and either in the same direction or with no intersection—(not cond 1) ^ (cond 4 _ cond 5)—then djlqs = 0. Note: cond 1 ¼ ðjx ¼ sx Þ ^ ðsx ¼ qx Þ ^ ðqx ¼ lx Þ; cond 2 ¼ jjy þ ly qy sy j P jjy ly j þ jqy sy j; cond 3 ¼ ðqy sy Þðjy ly Þ P 0; cond 4 ¼ jjx þ lx qx sx j P jjx lx j þ jqx sx j; cond 5 ¼ ðqx sx Þðjx lx Þ P 0; where ^ means ‘‘and’’ and _ means ‘‘or’’. Finally, if none of the previous scenarios are applicable, then the four points lie on the same line and are overlapping in opposite directions; thus, there is an overlay, and djlqs = x. This can be summarized as followed: 8 1 ifðj1 l1 < 0Þ ^ ðq1 s1 < 0Þ; > > > < 0 if ððj l P 0 _ q s P 0Þ ^ ðj 6¼ 0 _ l 6¼ 0 _ q 6¼ 0 _ s 6¼ 0ÞÞ 1 1 1 1 1 1 1 1 d jlqs ¼ > _ððcond 1Þ ^ ðcond 2 _ cond 3ÞÞ _ ððnot cond 1Þ ^ ðcond 4 _ cond 5ÞÞ; > > : x otherwise.
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2.2. Problem characteristics We can determine a limit on the number of cross points or overlays for a problem of given size as illustrated in the following proposition. Proposition 1. The density f of interference matrix D ¼ ðd jlqs Þ 2 ðZ þ ÞN
2
N 2
is asymptotically bounded by 1/3.
Proof. Consider any four locations: j, l, q, and s. These locations can be arranged in one of four possible patterns: Quadrilateral
20
Linear
52
Triangle & line
20
Triangle & bisector
12
The total number of nonzero entries in the interference matrix for each of these patterns are shown below each of the figures; the pattern generating the greatest amount of potential interference is the linear layout (note that these values include the fixed cost associated with the bidirectional flow between two facilities— that is, situations in which there is flow from point A to B and point B to A—which accounts for 2C 2N ¼ 12 nonzero entries). As additional locations are added to the linear layout, the number of nonzero entries is also maximized compared to the other patterns; the incremental interference for a linear layout is determined by: (a) flow from each of the existing points to the new point and flow from the new point to each of the existing points, resulting in 2(N 1)2 nonzero entries; (b) flow from each of the existing to the new point with flow between the existing points in the PN 2points opposite direction, with 2 i¼1 i2 nonzero entries; and (c) flow from the new point to each the existing points with flow between the existing points in the PNof 2 opposite direction, also with 2 i¼1 i2 nonzero entries. Thus, the number of nonzero entries, g(N), for a linear layout of size N will be: " # N 2 X N ðN 1ÞðN 2 N þ 1Þ 2 2 . gðN Þ ¼ gðN 1Þ þ 2 ðN 1Þ þ 2 i ¼ 3 i¼1 Hence, the density of the interference matrix is f¼
gðN Þ ðN 1ÞðN 2 N þ 1Þ ¼ . N4 3N 3
It follows that ðN 1ÞðN 2 N þ 1Þ 1 lim ¼ . N !1 3 3N 3 This completes the proof.
h
0 A cost coefficient, cijklpqrs ¼ fikpr d jlqs , can be defined as the interference cost of locating facilities i, k, p, and r at locations j, l, q, and s, respectively. For a layout of size N, we will realize N8 cost coefficients for this problem; however, there are several properties that we can take advantage of to simplify the problem. For
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example, many of the cost coefficients are redundant as they represent the same layout arrangement. If cijklpqrs is a cost that is realized for a particular layout arrangement, then: 0 0 cijklpqrs ¼ fikpr d jlqs ¼ faðjÞaðlÞaðqÞaðsÞ d jlqs ;
ð2Þ
where a = (a(j), j = 1, . . ., N) is the assignment vector of locations to facilities. Given these assignments, we can easily identify other cost coefficients that comprise these same assignments (e.g., cpqrsijkl). It is obvious that the order of presentation of assignments in the cost coefficient is irrelevant. For example, there are 4! = 24 cost coefficients representing the same layout arrangement for the assignment of 4 distinct facilities to 4 locations. A second property that we can exploit to simplify the problem is the fact that we may ignore the cost of any bidirectional workflow as this can be considered a sunk cost. It is easy to see that there is a fixed cost associated with the bidirectional flow between two facilities: FC ¼
N X N X i¼1
fikki d aðiÞaðkÞaðkÞaðiÞ ¼
k¼1
N X N X i¼1
minðfik ; fki Þ x.
ð3Þ
k¼1
This total amount of interference (overlay) will exist no matter where any arbitrary pair of facilities i and k are located. Thus, the minimization of interference using the cost coefficient cijklpqrs is equivalent to the minimization of interference using c0ijklpqrs where cijklpqrs ¼ FC þ c0ijklpqrs . Third, there exists a set of cost coefficients that will always equal zero. For example, any workflow originating and terminating at the same location, cijijpqrs, will not interfere with any other workflows. Also, all of the workflows emanating from a particular location, ciklijqrs, will neither intersect at a cross point or conflict along an overlay; the same is true for those workflows converging on a particular location, cijklpqkl. Therefore, we can disregard those assignments that will incur no workflow interference.
3. Model formulations 3.1. Quartic assignment problem formulation Chiang et al. (2002) noted that the relationship between two pairs of facilities can be modeled by utilizing four binary variables indicating the location of given facilities: 1 if facility i is assigned to location j; xij ¼ ð4Þ 0 otherwise. The objective function will then be the cross product of four binary variables and the cost will be incurred if and only if all four of the variables are equal to one. The quartic assignment problem can then be formulated as (assuming, without loss of generality, an equal number of facilities and locations): XXXXXXXX ðQrAPÞ Minimize cijklpqrs xij xkl xpq xrs ð5Þ i
subject to
X
j
k
l
p
q
r
s
xij ¼ 1;
8j;
ð6Þ
xij ¼ 1;
8i;
ð7Þ
8i; j.
ð8Þ
i
X j
xij 2 f0; 1g;
In Appendix A, we present linearizations of this model, such that commercially available integer linear programming software can be used to find the optimal solution to the quartic assignment problem.
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3.2. An alternative formulation for QrAP Let P = {1, . . ., p, . . ., jPj} represent a set of ordered pairs of facilities (e.g., p = (3, 5) is the ordered pair of facilities 3 and 5, with facility 3 in the first position in the pair) and L = {1, . . ., l, . . ., jLj} represent a set of ordered pairs of locations (e.g., l = (1, 4) is the ordered pair of locations 1 and 4). In this case, when the binary variable, xpl, assumes the value of one, we assign the ordered pair of facilities p to the ordered pair of locations l (e.g., for p = (3, 5) and l = (1, 4), xpl implies that facility 3 has been assigned to location 1 and facility 5 to location 4). The objective function coefficient, cplqr, will include the same costs as with the quartic formulation. To ensure that we do not assign more than one facility to a location (or vice versa), we define sets: I = {1, . . ., i, . . ., jIj}, where i = (l, r) and the pairs l and r have one common facility, l 5 r, and J = {1, . . ., j, . . ., jJj}, where j = (p, q) and the pairs p and q have one common location, p 5 q. The quartic assignment problem can then be formulated as: XXXX ðMQAPÞ Minimize cplqr xpl xqr p
subject to
X
l
q
ð9Þ
r
xpl ¼ 1;
8p;
ð10Þ
xpl ¼ 1;
8l;
ð11Þ
l
X p
xpl þ xqr 6 1;
if j ¼ ðl; rÞ 62 J ; i ¼ ðp; qÞ 2 I or if j ¼ ðl; rÞ 2 J ; i ¼ ðp; qÞ 62 I or if j ¼ ðl; rÞ 2 J ; i ¼ ðp; qÞ 2 I but the common location is not in the
ð12Þ
same position in the ordered pair as that of the corresponding common facility; xpl 2 f0; 1g;
8p; l.
ð13Þ
For example, constraint set (12) would prohibit the assignment p = (3, 5), l = (1, 4), q = (3, 7), and r = (6, 2)—since j = (l, r) 5 J and i = (p, q) 2 I—keeping facility 3 from being assigned to locations 1 and 6. Similarly, it would preclude p = (3, 5), l = (1, 4), q = (3, 7), and r = (6, 4)—since j = (l, r) 2 J, i = (p, q) 2 I, and the common location, 4, is not in the same position in the ordered pair as the common facility, 3. Note, though, that the same assignment with the exception of r = (1, 2) would make this a feasible solution—and allowed by constraint set (12)—since common location 1 is now in the same position in the ordered pair as common facility 3 (so xpl and xqr both specify that facility 3 is to be assigned to location 1). Thus, the quartic assignment problem can be formulated as a quadratic assignment problem with the addition of one set of side constraints.
4. Multi-objective facility design We now illustrate how to incorporate the distance-based (QAP) objective with the objective of workflow interference (QrAP). In practice, the QAP is not always solved to minimize the dollar cost directly (due to
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the difficulty of estimating a precise cost coefficient other than simply workflow times distance); instead it is frequently used to minimize the total distance traveled. The QrAP will likely encounter a similar fate as it will likely be used to minimize the amount of interference, not necessarily the interference cost. Therefore, the facility designer will be faced with two, possibly conflicting, objectives without a common measure (e.g., dollars). Consider again the eight-facility example first presented in the introductory section. The layout arrangement provided by solving the quartic assignment problem (Fig. 1b) is obviously quite different from that of the quadratic assignment problem (Fig. 1a). A designer will need to be able to evaluate the relative benefits of each objective. A useful approach to this situation is the concept of an efficiency frontier; Rosenblatt and Sinuany-Stern (1986) discuss this approach to multiple-objective facility layout design. For any given layout design, the total amount of interference and the total amount of workflow can be determined. Only those layouts that are not dominated by others need to be considered, and a designer can then determine the tradeoffs between the remaining layout designs. Fig. 3 illustrates the efficient frontier of the example. Since the objective function values of the QAP and the QrAP are somewhat correlated—for this example, the correlation coefficient is 0.674—there are relatively few efficient layout arrangements. As shown in the figure, there are eight layouts (of the 40,320 total layout arrangements) that are not dominated by other layouts. Furthermore, if we assume a linear relationship between the two objectives (that is, if we are trying to minimize z = w1f1 + w2f2, where w1 and w2 are the weights of the two objectives, f1 and f2, respectively), then there are only four efficient layout arrangements that warrant further consideration. Unfortunately, even for a moderately-sized problem, it is computationally prohibitive to evaluate all possible layouts to identify the efficient frontier. Thus, we have developed a heuristic, based on Malakooti (1989), to identify efficient layout arrangements (Ehrgott et al., 2004, provide a recent discussion of multi-objective quadratic assignment problems). Since the two extremes of the efficient frontier will include optimal QrAP and optimal QAP solutions, this heuristic starts with those solutions. If these layouts (or a symmetrical layout) are equivalent, it is the optimal solution. As with the QAP, most rectangular layouts will have a symmetry measure of 4 for the quartic assignment problem. Furthermore, some layouts will have more symmetrical layouts for the QrAP than for the QAP. As an example, consider the following layout arrangement:
1
2
5
3
4
The layout given above (1–2–3–4–5) and the layout (3–4–1–2–5) are symmetrical for both the QAP and the QrAP. However, there are eight additional layout arrangements that are ‘‘symmetrical’’ for the QrAP but not the QAP. These are obtained by simply rotating the facilities one location at a time (1–2–3–4–5), (3– 1–4–5–2), (4–3–5–2–1), etc. In these situations, the solution of the QrAP will result in several layouts from which the best QAP solution can be utilized. If the QAP and QrAP solutions are not equivalent, we then exchange pairs of facilities to identify other efficient layout arrangements. This heuristic is not guaranteed to identify all non-dominated layouts. For example, it identified all four of the linear efficient layout arrangements and five of the eight non-dominated layouts in the example. The heuristic is described in detail as follows: Step 1. Generate optimal QrAP solution, X1. If there are symmetric QrAP layouts that are not symmetric QAP layouts, select the layout with the minimum QAP objective function value.
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20
Total conflict (QrAP)
16
12
8
4
0 90
100
110
120
130
140
Total distance (QAP)
Fig. 3. Efficiency frontier for the eight-facility example.
Step 2. Generate optimal QAP solution, X2. If this is the same, or a symmetric, layout as that generated in Step 1; stop, this is the optimal solution. Otherwise, continue with Step 3. Step 3. Set A = B={X1, X2}, where A is defined to be the set of ‘‘unexpanded’’ layouts, and B is defined to be the set of ‘‘efficient’’ layouts. Step 4. Arbitrarily select Xi 2 A. Set A = A Xi. Step 5. Generate all N(N 1)/2 adjacent layouts to Xi, call them X 0ij . For each X 0ij , perform the following tests: (i) If X 0ij is the same as or is dominated by any Xi 2 A or Xi 2 B, ignore X 0ij . (ii) If X 0 ij dominates any Xi 2 A or Xi 2 B, replace Xi with X 0ij . (iii) If X 0ij is efficient with respect to all Xi 2 A and Xi 2 B, set B ¼ B [ X 0ij . Step 6. If A = ;, stop; B is the set of efficient layouts. Otherwise, go to Step 4.
5. Conclusion The problem of interruptions and interference is frequently cited in the literature of traditional layout design, the design of automated systems, and various other layout problems. The use of a model that explicitly considers the workflow interference results in layout arrangements that have smoother material flow. This should make the operations easier to observe and control resulting in better supervisory vision, higher quality control, etc.; accomplishing much of what the Japanese U-shaped layouts gain (reference again Fig. 1). Certainly, conducting facility layout analyses that attempt to minimize workflow interference as well as minimize material movement will provide a superior layout arrangement.
Appendix A. Integer linear programming formulations Let us define a set of binary variables: y ijklpqrs ¼ xij xkl xpq xrs .
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Then we state the following integer programming formulation: ðIP1Þ Minimize
N X N X N X i¼1
subject to
N X i¼1 N X
j¼1
k¼1
N
m
l¼1
N X N X N X N X p¼1 q¼1 r¼1
ðA:1Þ
cijklpqrs y ijklpqrs
s¼1
xij ¼ 1;
8j;
ðA:2Þ
xij ¼ 1;
8i;
ðA:3Þ
j¼1
xij þ xkl þ xpq þ xrs 4y ijklpqrs P 0; 8i; j; k; l; p; q; r; s N X N X N X N X N X N X N X N X y ijklpqrs ¼ N 4 ; i¼1
j¼1
k¼1
l¼1
p¼1 q¼1 r¼1
ðA:4Þ ðA:5Þ
s¼1
xij 2 f0; 1g; 8i; j; y ijklpqrs 2 f0; 1g; 8i; j; k; l; p; q; r; s.
ðA:6Þ ðA:7Þ
Proposition 2. The quartic assignment problem (QrAP) is equivalent to the integer linear program (IP1). The validity of Proposition 2 follows immediately from Theorem 3.1 in Burkard et al. (1994). The integer linear program (IP1) contains N2x variables, N8y variables, and N8 + 2N + 1 constraints. However, as described in Section 2, many of the variables can be eliminated. For example, since a facility can be placed at only one location and a location can have only one facility, we do not need to consider the y variables in which j = a(i) and l = a(k) where i = k and j 5 l or j = l and i 5 k. Also, those variables representing fixed costs need not be considered. Finally, those variables representing no workflow (e.g., yijijpqrs) or flow from/ to a given point (e.g., yijklijrs) can be eliminated, as they will never incur a cost. Elimination of these variables would reduce the required number of y variables to: 1 ½N ðN 24
2
2
2
1ÞðN 2ÞðN 3Þ þ 12½N ðN 1ÞðN 2Þ þ 12½N ðN 1Þ þ N 2
as well as a corresponding reduction in the number of constraints. With these modifications, the right hand side of Constraint (A.5) becomes C 4N þ 3C 3N þ C 2N þ N . Let us now consider the following integer programming formulation: ðIP2Þ Minimize
N X N X N X N X N X N X N X N X i¼1
j¼1
k¼1
l¼1
p¼1 q¼1 r¼1
subject to (A.2), (A.3), (A.6) and N X N X N X y ijklpqrs ¼ xrs ; i¼1
k¼1
8j; l; q; r; s;
ðA:9aÞ
y ijklpqrs ¼ xpq ;
8j; l; p; q; s;
ðA:9bÞ
y ijklpqrs ¼ xkl ;
8j; k; l; q; s;
ðA:9cÞ
y ijklpqrs ¼ xij ;
8i; j; l; q; s;
ðA:9dÞ
r¼1
N X N X N X i¼1
ðA:8Þ
k¼1 p¼1
N X N X N X i¼1
cijklpqrs y ijklpqrs
s¼1
p¼1 r¼1
N X N X N X k¼1 p¼1 r¼1
W.-C. Chiang et al. / European Journal of Operational Research 174 (2006) 1414–1426 N X N X N X j¼1
l¼1
l¼1
q¼1
q¼1
ðA:9eÞ
y ijklpqrs ¼ xpq ;
8i; k; p; q; r;
ðA:9fÞ
y ijklpqrs ¼ xkl ;
8i; k; l; p; r;
ðA:9gÞ
y ijklpqrs ¼ xij ;
8i; j; k; p; r;
ðA:9hÞ
s¼1
N X N X N X l¼1
8i; k; p; r; s;
s¼1
N X N X N X j¼1
y ijklpqrs ¼ xrs ;
q¼1
N X N X N X j¼1
1425
s¼1
y ijijijij ¼ xij ;
8i; j;
0 6 y ijklpqrs 6 1;
ðA:9iÞ
8i; j; k; l; p; q; r; s.
ðA:10Þ
Proposition 3. The quartic assignment problem (QrAP) is equivalent to the integer linear program (IP2). Proof. Given a feasible solution to the QrAP and by taking yijklpqrs = xijxklxpqxrs, it is straightforward to show that ~x ¼ ðxij Þ and ~y ¼ ðy ijklpqrs Þ is a feasible solution to (IP2) and further that the objective function values are the same. Conversely, let ~x, ~y be a feasible solution to (IP2). In order to show equivalence, we need to show that yijklpqrs = xijxklxpqxrs. Observe that: xrs ¼ 0 ) y ijklpqrs ¼ 0
ðfrom (A.9a)Þ;
xpq ¼ 0 ) y ijklpqrs ¼ 0 xkl ¼ 0 ) y ijklpqrs ¼ 0
ðfrom (A.9b)Þ; ðfrom (A.9c)Þ;
xij ¼ 0 ) y ijklpqrs ¼ 0
ðfrom (A.9d)Þ;
Let u be the permutation of {1, . . ., N} such that xi,u(i) = 1 for i = 1, . . ., N. We need only show that yi,u(i),k,u(k),p,u(p),r,u(r) = 1. Let us nowPobserve P that P yijklpqrs = 0 if j 5 u(i), l 5 u(k), q 5 u(p), or s 5 u(r). For example, if j 5 u(i),PthenP Nk¼1P Np¼1 Nr¼1 y ijklpqrs ¼ xij ¼ 0 and, thus, from (A.10), yijklpqrs = 0 for j 5 u(i). Then, from (A.9d), Nk¼1 Np¼1 Nr¼1 y iuðiÞklpqrs ¼ y iuðiÞkuðkÞpuðpÞruðrÞ ¼ xiuðiÞ ¼ 1. This completes the proof. h The integer program (IP2) has the same number of variables as (IP1) but only 8N5 + N2 + 2N equality constraints. By aggregating the number of constraints in (IP2), we can further reduce the number of constraints as presented below in formulation (IP3): ðIP3Þ Minimize
N X N X N X N X N X N X N X N X i¼1
j¼1
k¼1
l¼1
p¼1 q¼1 r¼1
cijklpqrs y ijklpqrs
ðA:11Þ
s¼1
subject to (A.2), (A.3), (A.6) and N X N X N X N X N X N X i¼1
j¼1
k¼1
l¼1
N X N X N X N X N X N X i¼1
j¼1
k¼1
l¼1
y ijklpqrs ¼ 3Nxrs ;
8r; s;
ðA:12aÞ
y ijklpqrs ¼ 3Nxpq ;
8p; q;
ðA:12bÞ
p¼1 q¼1
r¼1
s¼1
1426
W.-C. Chiang et al. / European Journal of Operational Research 174 (2006) 1414–1426 N X N X N X N X N X N X i¼1
j¼1
p¼1 q¼1
r¼1
N X N X N X N X N X N X k¼1
l¼1
p¼1 q¼1
r¼1
y ijklpqrs ¼ 3Nxkl ;
8k; l;
ðA:12cÞ
y ijklpqrs ¼ 3Nxij ;
8i; j.
ðA:12dÞ
s¼1
s¼1
Formulation (IP3) has 4N2 + 2N equality constraints and the same number of variables as (IP2). Its equivalence to (QrAP) can be proved in a similar way to Proposition 3.
References Apple, J.M., 1972. Material Handling Systems Design. John Wiley & Sons, New York. Benjaafar, S., 2002. Modeling and analysis of congestion in the design of facility layouts. Management Science 48 (5), 679–704. Burkard, R.E., C ¸ ela, E., 1995. Heuristics for biquadratic assignment problems and their computational comparison. European Journal of Operational Research 83 (2), 283–300. Burkard, R.E., C ¸ ela, E., Klinz, B., 1994. On the biquadratic assignment problem. In: Pardalos, P.M., Wolkowicz, H. (Eds.), DIMACS Series in Discrete Mathematics and Theoretical Computer Science. American Mathematical Society, Providence, pp. 117–146. Chiang, W.-C., Kouvelis, P., Urban, T.L., 2002. Incorporating workflow interference in facility layout design: The quartic assignment problem. Management Science 48 (4), 584–590. Cong, J., Hossain, M., Sherwani, N.A., 1993. A provably good multilayer topological planar routing algorithm in IC layout design. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 12 (1), 70–78. Egbelu, P.J., Tanchoco, J.M.A., 1986. Potentials for bi-directional guide-path for automated guided vehicle based systems. International Journal of Production Research 24 (5), 1075–1097. Ehrgott, M., Stephan, T., Tenfelde-Podehl, D., 2004. A level set method for multiobjective combinatorial optimization: Application to the quadratic assignment problem. Report in Wirtschaftsmathematik 84, Fachbereich Mathematik, Universita¨t Kaiserslautern. Gilmore, P.C., 1962. Optimal and suboptimal algorithms for the quadratic assignment problem. Journal of the Society for Industrial and Applied Mathematics 10 (2), 305–313. Hahn, P., 2004. QAPLIB—A quadratic assignment problem library [online], University of Pennsylvania. Available from: . Krishnamurthy, N.N., Batta, R., Karwan, M.H., 1993. Developing conflict-free routes for automated guided vehicles. Operations Research 41 (6), 1077–1090. Lawler, E.L., 1963. The quadratic assignment problem. Management Science 9 (4), 586–599. Luggen, W.W., 1991. Flexible Manufacturing Cells and Systems. Prentice Hall, Englewood Cliffs, NJ. Malakooti, B., 1989. Multiple objective facility layout: A heuristic to generate efficient alternatives. International Journal of Production Research 27 (7), 1225–1238. Miller, R.K., 1987. Automated Guided Vehicles and Automated Manufacturing. Society of Manufacturing Engineers, Dearborn, MI. Miltenburg, G.J., Wijngaard, J., 1994. The U-line line balancing problem. Management Science 40 (10), 1378–1388. Rendl, F., 2002. The quadratic assignment problem. In: Drezner, Z., Hamacher, H.W. (Eds.), Facility Location: Applications and Theory. Springer-Verlag, Berlin. Rosenblatt, M.J., Sinuany-Stern, Z., 1986. A discrete efficient frontier approach to the plant layout problem. Material Flow 3 (4), 277– 281. Sekine, K., 1992. One-piece Flow: Cell Design for Transforming the Production Process. Productivity Press, Portland. Tompkins, J.A., White, J.A., 1984. Facilities Planning. John Wiley & Sons, New York.
