A Construction-Based Approach to Process Plant Layout Using Mixed

Nov 30, 2006 - This paper presents a novel solution approach for addressing large-scale single-floor process plant layout problems. Based on the ...
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Ind. Eng. Chem. Res. 2007, 46, 351-358

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RESEARCH NOTES A Construction-Based Approach to Process Plant Layout Using Mixed-Integer Optimization Gang Xu and Lazaros G. Papageorgiou* Centre for Process Systems Engineering, Department of Chemical Engineering, UniVersity College London (UCL), Torrington Place, London, WC1E 7JE, United Kingdom

This paper presents a novel solution approach for addressing large-scale single-floor process plant layout problems. Based on the mixed-integer linear programming (MILP) representation proposed by Papageorgiou and Rotstein in 1998, the optimal layout (i.e., coordinates and dimensions) is determined through constructionbased schemes using mixed-integer optimization. The applicability of the proposed approach is finally demonstrated through four illustrative examples by investigating flowsheets with up to 36 equipment items. 1. Introduction The purpose of designing the layout of a process plant is to consider the spatial arrangement of equipment items and the required connections among them.1 Because of the increasing competition of process industries, industrial engineers are eager to seek every practical scenario to build their plants reasonably and economically in the design stage of a new plant or during the improvement phase of current flowsheets. Therefore, the layout aspects of a process plant are of vital importance, because of the economic point of view, as well as the safety, engineering, and management considerations. In regard to comparing process plant layouts, the facility layout problem has been studied extensively during the last two decades. Given a number of candidate locations with equal or unequal dimensions, the task of facility layout is to assign items to some of these positions, to minimize the material handling costs. Facility layout problems were initially formulated as quadratic assignment problems (QAPs)2 that address only equalsized items. Later, graph theory was applied, considering the items and connections as nodes and arcs and maximizing the adjacencies among nodes. Furthermore, stochastic optimization techniques including simulated annealing3 and genetic algorithms4-6 were widely applied. Finally, there have been some attempts to solve facility layout problems using mathematical programming (MP). First, a mixed-integer programming (MIP) model for the problem was proposed.7 Optimality can be guaranteed for cases with less than five items. Meller et al.8 gave another formulation with tighter constraints; however, problems of more than 10 items could not be solved optimally by the branched-and-bound algorithm. Also, mixed-integer linear programming (MILP) models for designing industrial facilities in two- and three-dimensional continuous space were presented.9,10 Equipment items orientations, distance restrictions, equipment connectivity inputs and outputs, and rectangular and irregular equipment shapes were considered. Later, two convex MIP models were proposed and symmetry breaking constraints were applied to avoid equivalent layout solutions.11 The computational results indicated that the proposed framework * To whom all correspondence should be addressed. E-mail address: [email protected].

yielded optimal solutions on several well-known test problems. Finally, a new MIP formulation for facility layout design using flexible bays was addressed.12 Examples with up to 14 units were tackled using the proposed model. Here, we focus on the process plant layout problem, which has attracted more and more attention from the research community, because of the particular production, environmental, and safety considerations of process industries. Initially, most published work was based on heuristics.13,14 Although they are efficient from a computational point of view, optimality of the solutions cannot be guaranteed. Graph theory was also applied to partition the plant into sections by aisles and corridors.15 The applicability of stochastic optimization techniques, including simulated annealing and genetic algorithms,16 were also demonstrated. Finally, several mathematical programming approaches have been presented for single and multi-floor process plant layout problems. An MINLP (mixed-integer nonlinear programming) formulation was developed, integrating the safety and economic considerations.17 Grid-based MILP models were proposed18,19 to discretize the land to several small candidate areas and each item occupies one or more of these grids. Furthermore, general MILP models for single- and multi-floor process plant layout problems were addressed,20,21 determining the land area, floor location of each process item, and detailed layout simultaneously. Finally, the combination of the process plant layout with safety issues22 and design/operation of batch pipeless plants23 was also investigated. Apart from various mathematical models, several solution approaches have been proposed to examine larger flowsheets. In facility layout problems, Kim presented a two-stage graphbased solution algorithm.24 An initial layout was obtained by constructing a planar adjacency graph and the current solution was improved by exchanging the adjacent vertices iteratively. Caccetta and Kusumah reviewed several constructive graph theoretical heuristics and proposed a new algorithm.25 In the process plant layout problem, Patsiatzis and Papageorgiou26 presented two efficient solution methods in multi-floor problems, based on the continuous-domain MILP model in ref 20. The first method is a rigorous decomposition approach, and the second method is an iterative approach. Computational experiments show their ability to examine flowsheets with up to 16

10.1021/ie060843k CCC: $37.00 © 2007 American Chemical Society Published on Web 11/30/2006

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Table 1. Computational Results of the Proposed Approach and the LAYOUT Model Example 1

c

Example 2

Example 3

Example 4

algorithm

OBJ

CPU

OBJ

CPU

OBJ

CPU

OBJ

CPU

C_2_2 C_2_3 C_2_4 C_3_2 C_3_3 C_3_4 C_4_2 C_4_3 C_4_4 C_5_2 C_5_3 C_5_4 C_6_2 C_6_3 C_6_4 LAYOUTb

9978.50 9948.03 9948.03 9948.03 9948.03 9948.03 11278.42 11278.42 11278.42 9948.03 9948.03 9948.03 9948.03 9948.03 9948.03 9948.03

0.30 0.39 0.53 0.41 0.37 1.14 0.58 0.56 0.47 1.06 1.41 0.92 1.28 1.25 1.30 2.86

17990.00 17990.00 17990.00 18390.00 17590.00 18790.00 17190.00 17590.00 17590.00 18190.00 17190.00 17190.00 18590.00 18390.00 17990.00 17190.00

0.54 0.59 0.69 0.45 0.64 1.23 0.42 0.78 0.50 0.61 0.69 0.62 1.48 1.61 3.64 10000c

35012.50 32522.50 32645.00 37580.00 33067.50 33067.50 35012.50 34792.50 34292.50 32377.50 32522.50 32070.00 33217.50 32690.00 32645.00 32550.00

2.65 5.84 43.62 1.90 3.81 6.62 2.61 5.05 56.58 3.17 14.39 16.47 7.50 13.06 180.19 10000c

4282.91 4264.14 4437.42 4312.06 4402.69 4470.06 4277.67 4291.40 4516.60 4608.96 4377.68 4276.63 4260.31 4489.14 4196.07 6058.38

12.51 14.35 70.62 13.79 35.92 450.28 19.30 47.95 1105.01 12.35 111.36 2204.50 9.19 51.73 12556.95a 10000c

a Iteration terminated by the maximum CPU limit (10 000 s); integer solution obtained. b Solve LAYOUT model for all units in the flowsheet simultaneously. Maximum CPU limit (10 000 s).

Table 2. Gap for Each Algorithm Over All Four Examplesa Gap [%] algorithm

example 1

example 2

example 3

example 4

average

C_2_2 C_2_3 C_2_4 C_3_2 C_3_3 C_3_4 C_4_2 C_4_3 C_4_4 C_5_2 C_5_3 C_5_4 C_6_2 C_6_3 C_6_4 LAYOUT

0.31 0.00 0.00 0.00 0.00 0.00 13.37 13.37 13.37 0.00 0.00 0.00 0.00 0.00 0.00 0.00

4.65 4.65 4.65 6.98 2.33 9.31 0.00 2.33 2.33 5.82 0.00 0.00 8.14 6.98 4.65 0.00

9.18 1.41 1.79 17.18 3.11 3.11 9.18 8.49 6.93 0.96 1.41 0.00 3.58 1.93 1.79 1.50

2.07 1.62 5.75 2.76 4.92 6.52 1.94 2.27 7.64 9.84 4.33 1.92 1.53 6.98 0.00 44.38

4.05 1.92 3.05 6.73 2.59 4.74 6.12 6.62 7.57 4.16 1.44 0.48 3.31 3.97 1.61 11.47

a The gap for each run is defined as the relative distance between the solution obtained and the best result overall schemes on the same column in Table 1.

units. Although the aforementioned solution approaches are able to examine examples with modest scales, current industrial practice needs more efficient and robust approaches to solve larger flowsheets. The aim of the paper is to extend the work of Patsiatzis and Papageorgiou26 by more-detailed investigation of the insertion scheme of the iterative approach, to solve larger problems without compromising the solution quality. The rest of the paper is structured as follows. In section 2, the general problem of single-floor process plant layout is stated. A constructive solution approach is proposed in section 3, and its applicability is demonstrated through four illustrative examples in section 4. Finally, some concluding remarks are made in section 5.

Figure 1. Flowchart for Algorithm C_M_N.

dimensions, and (ii) the connection costs among equipment items, determine the allocation of each equipment item (i.e., coordinates and orientations), to minimize the total connection cost. Because of practices in the process industry, chemical units are simplified to be rectangular shapes and connections among units are calculated as rectilinear distances. In this work, we adopt a continuous-domain MILP formulation20 (referenced as LAYOUT; see Appendix A for a brief description) for the single-floor process plant layout problem. 3. Iterative Solution Approach

2. Problem Statement The single-floor process plant layout problem can be described as follows: GiVen (i) a set of equipment items and their

Based on the MILP formulation for single-level process plant layout problems, flowsheets with up to 11 units have been solved

Table 3. Average Results of 10 Trials for Insertion Schemes with Random Selection Example 1

Example 2

Example 3

Example 4

algorithm

OBJ

CPU

OBJ

CPU

OBJ

CPU

OBJ

CPU

R_5_4 R_5_3 R_6_4 R_2_3 R_3_3

11475.20 11475.19 10056.35 10749.52 10526.00

0.54 0.53 1.52 0.28 0.32

19532.50 19910.50 18758.00 21463.00 19913.50

0.94 0.45 0.60 0.59 0.47

41805.50 44681.50 43905.00 45398.25 42922.75

11.28 7.605 134.86 9.94 9.77

7188.21 7250.37 6887.67 7248.95 6956.02

3487.50 2427.78 3183.08 1366.45 364.01

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Figure 2. Flowsheet for the ethylene oxide plant (example 1). Table 4. Gap of Insertion Schemes with Random Selectiona Gap [%] algorithm

example 1

example 2

example 3

example 4

average

R_5_4 R_5_3 R_6_4 R_2_3 R_3_3

15.35 15.35 1.09 8.06 5.81

13.63 15.83 9.12 24.86 15.85

30.36 39.32 36.91 41.56 33.84

71.38 72.79 64.15 72.76 65.77

32.68 35.82 27.82 36.81 30.32

a The gap for each run is defined as the relative distance between the solution obtained and the best result overall schemes on the same column in Table 1.

as Algorithm C_M_N, where M is defined as the number of units selected in the initial step and N is defined as the number of added units. Although the global optimality of the final solution cannot be guaranteed, the reduced MILPs involve fewer variables and constraints; therefore, significant computational savings are expected. It is believed that the insertion rule of the approach is of great importance to the final solution quality. Here, we propose a cost-based insertion rule that is determined by an MILP model (referenced as SELECT; see Appendix B for details). Selection priorities are given to the items that have expensive connections to other units. Next, the proposed iterative algorithm is outlined (see also corresponding flowchart in Figure 1). [Algorithm C_M_N] can be described as follows:

Step 1: Initialization (i) Set ∆ ) φ, k ) 1, SU ) M. Step 2: Selection of units (i) Solve SELECT. (ii) Update ∆ ) {i|Vi ) 1}. Free all Zij. Step 3: Determine plant layout (i) Solve LAYOUT for all i ∈ ∆. (ii) If all units are inserted (i.e., ∆ ) I), STOP. Figure 3. Best layout obtained for example 1.

to optimality. In this section, a novel iterative solution approach is proposed to solve larger process plant layout problems. In this approach, an initial set of items is selected and the reduced subproblem is solved for these items only. After fixing the nonoverlapping binary variables (E1ij and E2ij in constraints A6-A9 of Appendix A), the next subproblem will be addressed by inserting new units from the flowsheet, according to some rules. Therefore, the resulting subproblem involves only the newly inserted items with other existing items selected during previous iterations. We repeat the insertion scheme until all units in the flowsheet are included. The proposed approach is named

Step 4: Insertion of new units (i) k ) k + 1, SU ) |∆| + N. (ii) Fix E1ij, E2ij for all i,j ∈ ∆. (iii) Fix Vi ) 1 for i ∈ ∆, fix Zij ) 0 if both i and j ∉ ∆. (iv) Go to Step 2. To find the best insertion rule, several variants of Algorithm C_M_N are attended by tuning the values of M and N. Because

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Figure 4. Flowsheet for the isopropyl alcohol manufacturing plant (example 2).

Figure 6. Flowsheet for the industrial multipurpose batch plant (example 3).

Figure 5. Best layout obtained for example 2.

of the complexity of the problem, 2-6 units are selected initially and 2-4 units are added subsequently, so that the proposed approach can terminate within a reasonable computational time. 4. Computational Results In this section, the applicability of the proposed approach is demonstrated through four process plant layout examples. All examples and solution schemes are implemented in GAMS modeling systems27 with the CPLEX mixed-integer optimization solver. All runs are performed on a Hewlett Packard HP Pavilion laptop with 0% margin of optimality and a 10 000 s CPU limit. All input data for each example are included in Appendix C. Table 1 lists all computational results of the proposed approach, and the best results obtained for each example is highlighted in bold. For each example, the relative gaps between result obtained by the particular algorithm and the best result overall algorithms

are calculated. The average relative gap for each algorithm is then reported (see Table 2) and the minimum average gap is shaded. To demonstrate the efficiency of the cost-based insertion rule of our approach, we provide some comparative results in Tables 3 and 4 by applying a random-based insertion rule to the first five best construction algorithms. The random-based algorithm is similarly referenced as Algorithm R_M_N. Every random selection scheme is repeated 10 times, and the average objective function value and average CPU times are reported. It should be added that the first seven-unit example can be solved to optimality using the LAYOUT model, whereas all the remaining examples cannot converge to optimality within a given computational limit (see Table 1 for computational details). The first example we consider is a seven-unit ethylene oxide plant (see Figure 2) that has been introduced by Penteado and Ciric.17 The optimal solution obtained using the LAYOUT model is 9948.03 rmu within 2.68 s. [Here, the unit rmu denotes relative monetary units.] When using the construction algorithms with different values of M and N, 11 out of 15 achieve the optimal solution. Although the insertion schemes with random

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gas processing installation. Figure 8 shows a simplified representation of the flowsheet. The LAYOUT model results in an objective function of 6058.38 rmu within 10 000 s. The construction approach with random selection schemes reaches an average value of 6887.67 rmu using 3183.08 s (see Table 2). In comparison, Algorithm C_6_4 outperforms other schemes, obtaining an objective function of 4196.07 rmu (see Figure 9 for the detailed layout). 5. Concluding Remarks

Figure 7. Best layout obtained for Example 3.

selection also receive promising computational savings, our construction approach finds better objective function values. The layout obtained from Algorithm C_5_4 is plotted as Figure 3. The second example studied is a 12-unit cosmetic-grade isopropyl alcohol manufacturing plant that was introduced by Jayakumar and Reklaitis.15 The flowsheet includes 12 units, as shown in Figure 4. We assume equal square footprint sizes for all equipment items. It is indicated that the LAYOUT model cannot solve it to optimality within the prespecified CPU limit. The best feasible solution obtained is 17 190 rmu. Comparison with the LAYOUT model and random-based selection algorithms (see Tables 1 and 3) reveals that Algorithms C_4_2, C_5_3, and C_5_4 obtain the best solution, with a value of 17 190 rmu (see Figure 5 for the detailed layout). Note that all proposed schemes solve this example with significant lower CPU times than the LAYOUT model (see Table 1), thus illustrating the efficiency of the proposed approach. Example 3 considers an 18-unit industrial multipurpose batch plant (see Figure 6) that was investigated by Barbosa-Povoa and Macchietto.28 The connection costs are given by Georgiadis et al.19 Within 10 000 s, the LAYOUT model cannot terminate to optimality. The best result obtained is 32 070 rmu through Algorithm C_5_4 (see Figure 7 for the final layout), which is 1.50% better than the LAYOUT model. The last example is a 36-unit process, which is a simplified representation of the compression section on a crude oil and

Figure 8. Flowsheet for the crude oil and gas processing section (example 4).

In this paper, a novel iterative solution approach for the singlefloor process plant layout problem has been proposed. According to the MILP representation,20 the proposed approach determines the detailed layout (centroid positions and dimensions) of the flowsheet through iterative insertion schemes. According to the computational results, significant computational reductions and better solutions have been achieved when compared with the single-level MILP model (LAYOUT) and insertion schemes with random selections. The C_5_4 construction algorithm (select five units initially and insert up to four items during each of the iterations with the maximum connection costs) has achieved the minimum average gap, thus illustrating its effectiveness in regard to obtaining good-quality solutions. Note that inserting three or four units in the construction steps seem reasonable. However, it is expected that improved quality solutions can be obtained for higher values of M and N, provided that sufficient computational resources are available. Finally, the proposed approach shows great potential to examine larger flowsheets with modest computational requirements. Appendix A: Summary of the MILP Model by Papageorgiou and Rotstein20 The MILP model for the single-floor process plant layout problem in ref 20 (referenced here as LAYOUT) is briefly described next. A.1. Objective Function. The objective function considered in this model is the minimization of the total connection costs:

Min

CijDij ∑i ∑ j*i

A.2. Equipment Orientation Constraints. The following two constraints determine li and di, based on the decisions

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A.5. Additional Layout Design Constraints. Additional layout constraints are as follows:

Xi g

li ∀i 2

(A10)

Yi g

di ∀i 2

(A11)

Appendix B: Unit Selection The MILP model (referenced here as SELECT) for unit selection in step 2 of the construction-based approach (Algorithm C_M_N) is formulated as follows. B.1. Objective Function. The objective function considered in this model is the maximization of the total connection costs among selected equipment items:

Max

∑i j,C∑*0 CijZij ij

B.2. Logical Constraints. First, the connection costs between unit i and j will be attributed to the objective function if both units are selected: Figure 9. Best layout obtained for example 4.

concerning the orientations of each equipment item in the space:

li ) RiOi + βi(1 - Oi) ∀i

(A1)

di ) Ri + βi - li ∀i

(A2)

Zij eVi ∀ (i,j): Cij * 0

(B1)

Zij eVj ∀ (i,j): Cij * 0

(B2)

Finally, at most, SU units are selected:

∑i Vi e SU

(B3)

Appendix C: Input Data Tables A.3. Distance Constraints. The distance constraints include the following:

Rij - Lij ) Xi - Xj ∀ i ) 1, ..., NU - 1, j ) i + 1, ..., NU (A3) Aij - Bij ) Yi - Yj ∀ i ) 1, ..., NU - 1, j ) i + 1, ..., NU (A4) Dij ) Rij + Lij + Aij + Bij ∀ i ) 1, ..., NU - 1, j ) i + 1, ..., NU (A5) A.4. Nonoverlapping Constraints. To avoid the overlapping of the two items, nonoverlapping constraints for both the xand y-directions, using two types of binary variables (E1ij , E2ij) are proposed:

li + lj Xi - Xj + M(E1ij + E2ij) g 2 ∀ i ) 1, ..., NU - 1, j ) i + 1, ..., NU (A6) li + lj 2 ∀ i ) 1, ..., NU - 1, j ) i + 1, ..., NU (A7)

Xj - Xi + M(1 - E1ij + E2ij) g

di + d j 2 ∀ i ) 1, ..., NU - 1, j ) i + 1, ..., NU (A8)

Yi - Yj + M(1 + E1ij - E2ij) g

di + d j 2 ∀ i ) 1, ..., NU - 1, j ) i + 1, ..., NU (A9)

Yj - Yi + M(2 - E1ij - E2ij) g

The input data (dimensions and connection costs) for all illustrative examples studied are listed in Tables C1-C4. Table C1. Input Data for the Ethylene Oxide Plant (Example 1) item

width [m]

length [m]

connection

cost [rmu/m]a

1 2 3 4 5 6 7

5.22 11.42 7.68 8.48 7.68 2.60 2.40

5.22 11.42 7.68 8.48 7.68 2.60 2.40

(1,2) (1,5) (2,3) (3,4) (4,5) (5,6) (5,7) (6,7)

346.0 416.3 118.0 111.0 85.3 86.3 82.8 6.5

a

Here, rmu represents relative monetary units.

Table C2. Input Data for the Isopropyl Alcohol Manufacturing Plant (Example 2) item

width [m]

length [m]

connection

cost [rmu/m]

1 2 3 4 5 6 7 8 9 10 11 12

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

(1,2) (1,4) (1,5) (1,12) (2,3) (2,4) (4,5) (4,6) (4,12) (6,7) (7,8) (7,11) (8,9) (8,11) (9,10) (10,11) (11,12)

1965 100 400 100 3930 1965 500 1565 200 1565 1450 100 1250 200 1000 200 300

Ind. Eng. Chem. Res., Vol. 46, No. 1, 2007 357 Table C3. Input Data for the Industrial Multipurpose Batch Plant (Example 3) item

width [m]

length [m]

connection

cost [rmu/m]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0

4.0 4.0 6.0 4.0 6.0 8.0 8.0 6.0 4.0 6.0 6.0 6.0 4.0 6.0 4.0 4.0 6.0 2.0

1-18 18-7 2-7 3-6 6-7 7-10 6-11 13-10 10-6 8-17 9-12 3-7 4-9 5-6 5-9 6-8 7-14 11-13 13-15 8-16 16-8 9-8 12-8

200 240 230 400 230 270 280 170 300 250 170 230 160 250 160 170 270 300 170 250 140 175 175

Table C4. Input Data for the Crude Oil and Gas Processing Section (Example 4) item

width [m]

length [m]

connection

cost [rmu/m]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

20.0 20.0 3.0 3.0 8.0 8.0 4.0 1.0 8.0 8.0 8.0 3.0 3.0 3.0 8.0 8.0 8.0 4.0 4.0 4.0 8.0 8.0 8.0 3.0 3.0 3.0 8.0 8.0 8.0 4.0 4.0 4.0 5.0 1.0 8.0 8.0

5.0 5.0 3.0 3.0 3.0 3.0 4.0 1.0 2.5 2.5 2.5 3.0 3.0 3.0 8.0 3.0 3.0 4.0 4.0 4.0 2.5 2.5 2.5 3.0 3.0 3.0 3.0 3.0 3.0 4.0 4.0 4.0 5.0 1.0 2.5 2.5

(1,8) (1,2) (4,2) (4,6) (6,8) (5,8) (2,7) (2,3) (3,5) (8,9) (8,10) (8,11) (9,12) (12,15) (12,18) (18,33) (15,21) (21,24) (24,30) (30,33) (24,27) (32,33) (27,34) (34,35) (35,36) (10,13) (13,19) (19,33) (13,16) (16,22) (22,25) (25,31) (31,33) (25,28) (28,34) (11,14) (14,20) (20,33) (14,17) (17,23) (23,26) (26,29) (26,32) (29,34)

50.00 50.00 16.67 16.67 16.67 16.67 16.67 16.67 16.67 27.78 27.78 27.78 27.78 13.89 13.89 13.89 13.89 13.89 6.95 6.95 6.95 6.95 6.95 10.42 10.42 27.78 13.89 13.89 13.89 13.89 13.89 6.95 6.95 6.95 6.95 27.78 13.89 13.89 13.89 13.89 13.89 6.95 6.95 6.95

Nomenclature Indices i,j ) equipment items Sets I ) the set of plant equipment items considered ∆ ) the set of items considered by the subproblem Parameters Ri, βi ) dimensions of item i Cij ) connection cost between items i and j M ) number of initial units selected N ) number of units added NU ) total number of units SU ) number of selected units k ) iteration number Binary Variables Oi ) if the length of item i (parallel to the x-axis) is equal to Ri, Oi ) 1; otherwise, Oi ) 0 E1ij, E2ij ) nonoverlapping binary variables Vi ) if unit i is selected, Vi ) 1; otherwise, Vi ) 0 Zij ) if both unit i and j are selected, Zi ) 1; otherwise, Zi ) 0 Continuous Variables li ) length of item i di ) depth of item i Xi, Yi ) coordinates of the geometrical center of item i Rij ) relative distance in x coordinates between items i and j, if i is to the right of j Lij ) relative distance in x coordinates between items i and j, if i is to the left of j Aij ) relative distance in y coordinates between items i and j, if i is above j Bij ) relative distance in y coordinates between items i and j, if i is below j Dij ) total rectilinear distance between listed items i and j Literature Cited (1) Mecklenburgh, J. C. Process Plant Layout; Institution of Chemical Engineers: London, 1985. (2) Koopmans, T. C.; Beckman, M. Assignment problems and the location of economic activities. Econometrica 1957, 25, 53. (3) Chwif, L.; Marcos, R.; Barretto, P; Moscato, L. A. A solution to the facility layout problem using simulated annealing. Comput. Ind. 1998, 36, 125. (4) Mak, K. L.; Wong, Y. S.; Chan, F. T. S. A genetic algorithm for facility layout problems. Comput. Integr. Manuf. 1998, 11, 113. (5) Moghaddain, R. T.; Iris, E. S. Facility layout design by genetic algorithms. Comput. Ind. Eng. 1998, 35, 527. (6) Lee, K. Y.; Roh, M. I.; Jeong, H. S. An improved genetic algorithm for multi-floor facility layout problems having inner structure walls and passages. Comput. Oper. Res. 2005, 32, 879. (7) Montreuil, B. A modeling framework for integrating layout design and flow network design. In Proceedings from the Material Handling Research Colloquium, Hebron, KY, 1990; Vol. 2, p 95. (8) Meller, R. D.; Narayanan, V.; Vance, P. H. Optimal facility layout design. Oper. Res. Lett. 1999, 23, 117. (9) Barbosa-Povoa, A. P.; Mateus, R.; Novais, A. Q. Optimal twodimensional layout of industrial facilities. Int. J. Prod. Res. 2001, 39, 2567. (10) Barbosa-Povoa, A. P.; Mateus, R.; Novais, A. Q. Optimal 3D layout of industrial facilities. Int. J. Prod. Res. 2002, 40, 1669. (11) Castillo, I.; Westerlund, J.; Emet, S.; Westerlund, T. Optimization of block layout design problems with unequal areas: A comparison of MILP and MINLP optimization methods. Comput. Chem. Eng. 2005, 30, 54.

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ReceiVed for reView June 30, 2006 ReVised manuscript receiVed October 27, 2006 Accepted November 15, 2006 IE060843K