Continuous-Domain Mathematical Models for ... - ACS Publications

Ind. Eng. Chem. Res. 1998, 37, 3631-3639

3631

Continuous-Domain Mathematical Models for Optimal Process Plant Layout Lazaros G. Papageorgiou and Guillermo E. Rotstein* Centre for Process Systems Engineering, Imperial College of Science, Technology and Medicine, London SW7 2BY, U.K.

This paper presents mathematical programming models for the optimal process plant layout based on a continuous dimensional space. Two alternative formulations are proposed which can accommodate rectangular equipment footprints of arbitrary size. Then, one of the formulations is extended to account for a common designer objective: to organize the layout into well-defined production sections. All models are formulated as mixed-integer linear programming (MILP) problems which can be solved to global optimality by using commercial optimization software. The applicability of the proposed mathematical models is demonstrated by a number of illustrative examples. 1. Introduction Designing the layout of a chemical plant involves decisions concerning the spatial allocation of equipment items and the required connections among them.1 Traditionally, decisions concerning the plant layout have not received significant attention during the design or retrofit of chemical plants. However, increased competition is leading contractors and chemical companies to look for potential savings at every stage of the design process. There is, therefore, a need to develop computer-aided methods to support engineers in the rapid generation of alternative chemical process plant layouts. In general, the process plant layout problem may be characterized by a number of cost, management, and/or engineering drivers such as the following: (a) Connectivity cost: involving cost of piping and other required connections between equipment items. In addition, other related network operating costs such as pumping may be taken into account. (b) Safety: introducing constraints with respect to the minimum allowable distance between specific equipment items. (c) Retrofit: fitting new equipment items within an existing plant layout. (d) Construction cost: leading to the design of compact plants particularly significant in cases such as offshore platforms. The trade-off between the cost of occupied area (land) and height (multifloor plants) must also be considered. (e) Production organization: facilitating the movements of goods and operators through the plant. Frequently, the accommodation of a specific manufacturing pattern (e.g., the organization of the workforce into teams, working in well-defined plant sections) may also be of great importance. The facility layout has been investigated by industrial engineers for several years (see for comprehensive reviews refs 2 and 3). Here, we concentrate on the chemical plant layout problem, which has recently received attention from the research community. Initial approaches were based on heuristics.4-6 Although * Author to whom all correspondence should be addressed. E-mail: [email protected].

heuristic approaches may be efficient from the computational point of view, they do not offer any guarantee on the optimality of the solution obtained. Alternatively, the problem of organizing the equipment items into sections and formulating it as a graph-partitioning problem was analyzed in ref 7. A mathematical programming model-based approach was proposed in ref 8 resulting in mixed-integer nonlinear programming (MINLP) models where particular attention was given to the safety aspects of the layout problem. The multifloor arrangement of batch plants was considered in ref 9 for a given assignment of equipment items to floors and with use of an objective function representing “preferences” rather than economic considerations. A method to aid decisions concerning the assignment of equipment items to floors in three-dimensional arrangements has been proposed in ref 10 with no consideration of the detailed layout within each floor. The method relied on a graphical heuristic approach, providing an upper bound to the optimal value and a mathematical programming formulation representing a lower bound. Layout considerations have been found to be of particular significance in pipeless plants11 since the location of the processing stations determines the transfer times for the moving vessels. These considerations were captured in a mathematical programming formulation addressing simultaneously the design, layout, and scheduling problems in pipeless plants. Finally, a mixed-integer linear programming (MILP) formulation was presented in ref 12, considering the allocation of equipment items to floors and the detailed layout of each floor. The formulation was based on space discretization into a set of candidate locations, with each equipment item occupying one and only one location. A detailed cost function was optimized by taking into account piping, floor construction, and operating costs. This formulation was recently generalized in ref 13, adopting “finer” discretization (block sizes) to account for equipment items with different sizes, allowing each item to occupy, potentially, more than one block. The main limitation of these formulations was the adopted discretization of the available space which constitutes a convenient approach from the modeling point of view but often leads to suboptimal solutions.

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3632 Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998

Figure 1. Flowsheet of the motivating example.

in any orientation 12 or small enough to allow each equipment item occupying more than one potential block 13. In the first case, the block sizes are given by the two largest footprint dimensions (in this example, 14.4 m × 13.8 m), resulting in a suboptimal solution. However, in the second case, a global optimum can be obtained if a 0.1 m × 0.1 m grid is adopted. Of course, a coarser discretization may be used, compromising the optimality of the solution obtained. It is worth noting that the fine grid mathematical model is much larger in size than the coarse one. The second alternative is based on a nonuniform grid and, therefore, restricts the potential allocation of some equipment items to “suitable” blocks (i.e., those that are large enough for the corresponding footprint). Overall, both alternatives (uniform and nonuniform) may lead to suboptimal solutions unless very fine discretization is adopted,13 and therefore there is a significant tradeoff between solution quality (optimality) and tractability of the resulting mathematical model. In order to alleviate the deficiencies raised with respect to the quality of the solution and/or possible excessive computational times due to the discretization of the available space, a more appropriate approach needs to be developed by taking into account the exact sizes of the equipment. The development of suitable mathematical programming formulations to support such an approach is the subject of this work. 3. Problem Statement

Figure 2. Units footprints and sizes of the motivating example.

The paper is structured as follows: in the next section, the motivations for this paper are discussed through an example. In section 3, the process plant layout problem in a two-dimensional space is stated. Section 4 presents two alternative mathematical programming formulations, both of them based on a continuous-domain representation. The applicability of the resulting mixed-integer linear programming models is demonstrated in section 5 through a number of illustrative examples, and some comparative results between the two alternative formulations are presented. Section 6 considers the extension of these formulations to design layouts that are organized into a well-defined production section. Finally, some concluding remarks are made in section 7. 2. Motivating Example The motivating example used here was introduced in ref 10 and then considered in ref 12. The characteristics of the problem are summarized in Figures 1 and 2. The objective is to obtain a process plant layout that minimizes the connection cost between equipment items. In general, if we adopt a discretization-based approach (e.g., refs 12 and 13), then two alternatives exist: uniform and nonuniform space discretization. The first one defines a uniform block, which can be either large enough to accommodate any equipment footprint

In the formulations presented in this paper, we assume that the equipment items to be allocated in the available space are described by rectangular shapes. Such an assumption is in accordance with current industrial practice, where the starting point for the layout design is often a set of rectangular modules comprising a central processing item(s) as well as the related equipment and instrumentation.14 The layout design for a typical gas separation plant involves the allocation of 10-20 of these modules. Another important issue of the proposed formulations is concerned with the cost of equipment connections. Here, since the piping and instrumentation network is usually built in either aerial or underground corridors,14 rectilinear distances are assumed to give a more realistic estimate of piping costs (e.g., ref 12) than direct connections give (e.g., ref 8). The process plant layout problem is stated as follows: Given: (a) A set of N equipment items and their dimensions. (b) Cost of piping and other connections between equipment items. (c) Space and equipment allocation limitations. (d) Minimum distance constraints between equipment items (e.g., safety considerations). (e) Production sections. Determine: (f) The allocation of each equipment item (i.e., coordinates and orientation); so as to minimize the process plant layout cost. 4. Mathematical Formulation The indices and parameters associated with the layout problem are listed below:

Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998 3633 Indices i, j ) equipment items Parameters Ri, βi ) dimensions of item i Cij ) connection cost between items i and j xmax ) maximum x coordinate ymax ) maximum y coordinate The formulation is based on the following key variables: Binary Variables oi ) 1 if length of item i (parallel to the x axes) is equal to Ri; 0 otherwise zxij ) 1 if item i is strictly to the right of item j; 0 otherwise zyij ) 1 if item i is strictly above item j; 0 otherwise 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 items i and j list

4.1. Objective Function. The objective function used is the minimization of the total connection cost as follows:

min

CijDij ∑i ∑ j*i

(1)

taking into account piping costs and other related operating expenses such as pumping (see more comprehensive cost functions in ref 12). 4.2. Equipment Orientation Constraints. The values of the li and di variables depend on decisions concerning the orientation of the equipment items in the plane. The effect of equipment orientation can be captured as follows:

li ) Rioi + βi(1 - oi) di ) Ri + βi - li

∀i ∀i

(2) (3)

4.3. Distance Constraints. Constraints similar to those proposed in ref 12 are used although no binary variables are included here:

Rij - Lij ) xi - xj

∀ i ) 1, ..., N - 1, j ) i + 1, ..., N (4)

Aij - Bij ) yi - yj

∀ i ) 1, ..., N - 1, j ) i + 1, ..., N (5)

objective function, thus minimizing Rij + Lij + Aij + Bij summations. Consequently, only one variable at most of each pairsi.e., (Rij, Lij) and (Aij, Bij)sis guaranteed to be nonzero at the optimal solution of each linear programming (LP) problem during the branch-andbound procedure used for the MILP solution. 4.4. Nonoverlapping ConstraintssModel LAYOUT1. In order to avoid situations where two equipment items i and j occupy the same physical location, overlapping of the projections of each equipment footprint, either in the x or in the y dimension, should be prohibited.15 This constraint is clearly depicted in Figure 3 and is guaranteed to hold if at least one of the following inequalities is active:

xi - xj g

x j - xi g

yi - yj g

y j - yi g

li + lj 2

∀ i ) 1, ..., N - 1,

li + lj 2

∀ i ) 1, ..., N - 1,

j ) i + 1, ..., N (7)

j ) i + 1, ..., N (8)

di + dj 2

∀ i ) 1, ..., N - 1,

di + dj 2

∀ i ) 1, ..., N - 1,

j ) i + 1, ..., N (9)

j ) i + 1, ..., N (10)

For instance, in case (a) of Figure 3 inequality (8) is active while in case (b) inequality (9) is active. These nonoverlapping disjunctive conditions can be modeled in a mixed-integer linear form by using “big M” constraints (see, for instance, refs 16 and 17) as follows:

li + lj 2

∀ i, j * i

di + dj 2

∀ i, j * i (12)

xi - xj + M(1 - zxij) g yi - yj + M(1 - zyij) g

(11)

where M is a suitable upper bound on the distance between two equipment items. Finally, overlapping between equipment item pairs is avoided by forcing at least one of the corresponding constraints (11) or (12) to be active:

zxij + zxji + zyij + zyji g 1 ∀ i ) 1, ..., N - 1, j ) i + 1, ..., N (13) along with the following constraints

zxij + zxji e 1

∀ i ) 1, ..., N - 1, j ) i + 1, ..., N (14)

and Thus, the distance between items i and j is given by

Dij ) Rij + Lij + Aij + Bij

∀ i ) 1, ..., N - 1, j ) i + 1, ..., N (6)

It should be noted that the above constraints (4)-(6) are written only for those pairs of items whose relative distance appears in the objective function (i.e., terms with nonzero connection costs, Cij * 0). The problem being solved involves minimization of a distance-related

zyij + zyji e 1

∀ i ) 1, ..., N - 1, j ) i + 1, ..., N (15)

4.5. Additional Layout Design Constraints. In this section we discuss how various aspects of the layout problem can be modeled. Lower bound constraints on the coordinates of the geometrical center of each item (i.e., xi, yi) should be written in order to avoid intersection of items with the


Figure 3. Avoiding equipment overlapping.

More specifically, for the case described in Figure 4, we prefix (xA, yA, lA, dA) and (xC, yC, lC, dC). In a similar fashion, the case where the location of specific items (e.g., plant services) has been fixed before the layout design can easily be accommodated. The layout optimization MILP model can then be summarized as

model LAYOUT1 min

Figure 4. Optimal layout for the motivating example with space restrictions.

origin of axes, achieved by

xi g

li 2

∀i

(16)

yi g

di 2

∀i

(17)

and

Similarly, upper bound constraints may force equipment to be allocated within a prespecified rectangular space defined by the corners (0, 0) and (xmax, ymax):

xi +

li e xmax 2

∀i

(18)

and

∀i

(19)

Finally, an interesting case to be considered is when specific space restrictions are imposed, thus resulting in a nonrectangular available space. An example is shown in Figure 4, where area B is the available space for equipment allocation while areas A and C represent either nonavailable or preallocated space. In terms of our formulation, the “restricted” areas can be treated as additional pseudo-items with fixed sizes and locations.

(20)

subject to constraints (2)-(6) and (11)-(19). All continuous variables in the formulation are defined as non-negative (i.e., g0). It is often necessary, due to safety considerations, to keep the distance between specific items i and j above a prespecified minimum value, Dmin ij . In order to accommodate these extra rectilinear distances between the equipment items, the right-hand sides of constraints (11) and (12) can be augmented appropriately by involving additional terms (parameters). For instance, constraint (11) should be replaced by

xi - xj + M(1 - zxij) g

li + l j + Dmin ij 2

∀ i, j * i (21)

Furthermore, due to operational and/or control reasons, it may also be necessary to keep the distance between specific items i and j below a certain value, Dmax ij . This case can simply be captured, for instance, for the x-direction, by adding the following constraint:

xi - xj e di yi + e ymax 2

CijDij ∑i ∑ j*i

li + lj + Dmax ij 2

∀ i, j * i

(22)

4.6. Alternative Nonoverlapping Constraintss Model LAYOUT2. The use of the binary variables zxij and zyij in constraints (11) and (12) is appealing from the modeling point of view since they have a clear physical meaning. However, the number of binary variables and constraints required can be reduced by using an alternative formulation. We can adopt a representation similar to the one used in ref 18 suitable for palletization in warehouses. Apart from the original orientation binary variables, oi, two additional ones need

Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998 3635

to be introduced, namely, E1ij and E2ij, where each pair of values (0 or 1) to these variables determines which constraint from 7 to 10 is active. For every i, j such that j > i (i.e., ∀ i ) 1, ..., N - 1, j ) i + 1, ..., N), we have

If constraint (7) is active then E1ij ) 0

E2ij ) 0


E2ij ) 0 Figure 5. Optimal layout for the motivating example without space restrictions.


E2ij ) 1


E2ij ) 1

Then, inequalities (11)-(15) can be replaced in model LAYOUT1 by the following set:

li + lj 2 ∀ i ) 1, ..., N - 1, j ) i + 1, ..., N (23)

xi - xj + M(E1ij + E2ij) g

li + l j 2 ∀ i ) 1, ..., N - 1, j ) i + 1, ..., N (24)

xj - xi + M(1 - E1ij + E2ij) g

di + dj 2 ∀ i ) 1, ..., N - 1, j ) i + 1, ..., N (25)

yi - yj + M(1 + E1ij + E2ij) g

di + dj 2 ∀ i ) 1, ..., N - 1, j ) i + 1, ..., N (26)

yj - yi + M(2 - E1ij - E2ij) g

In conclusion, the number of binary variables required is reduced by approximately 50%. Also, the number of constraints is reduced significantly as constraints (13)(15) are not included in the mathematical model. However, it is still relevant to compare the computational performance of models LAYOUT1 and LAYOUT2 (see the next section) as a smaller model size does not guarantee enhanced performance.19 The summary of the alternative layout optimization MILP model is outlined by

model LAYOUT2 min

CijDij ∑i ∑ j*i

(27)

subject to constraints (2)-(6), (21)-(19), and (23)-(26). Of course, safety, operational, and/or control considerations, as described in the previous subsection, can easily be incorporated. 5. Computational Results In this section, the proposed formulations are applied to three previously published applications of process

layout optimization. All examples were modeled using the GAMS modeling system20 coupled with the CPLEX V4.0.8 mixed-integer linear programming optimization package.21 All the computational experiments have been performed on a Sun Ultra workstation with a 5% margin of optimality. Comparative results concerning the effectiveness of the two models are shown before introducing the concept of production “sections”. 5.1. Motivating Example Revisited. Here, we consider again the motivating example presented in section 2. The problem was solved using both models (i.e., Layout1 and Layout2), assuming the same cost, 1 rmu/m (rmu ≡ relative money units), for all connections. The optimal layout, obtained with model Layout2, had a cost of 18.2 rmu and is depicted in Figure 5, illustrating relative positioning of equipment items rather than actual coordinates (all corresponding actual coordinates of the following examples are given in tables). Finally, it is interesting to analyze the effect of space restrictions/considerations as described in section 4.5 and also how the new solution compares to the one obtained before (without space restrictions). More specifically, we examine the motivating example with space restrictions/considerations as shown in Figure 4 along with the following additional data:

xmax ) 20, ymax ) 20, lA ) 8, dA ) 5, lC ) 13, dC ) 9 The new optimal solution is 21.5 rmu, representing an 18% increase compared to that obtained without space restrictions. The plant layout obtained is illustrated in Figure 4. It should be noted that in all of the following examples it is assumed that unlimited plant area is available to locate the equipment. 5.2. Example 1. This example is derived from the case study presented in ref 8, considering the layout design for an ethylene oxide plant. Here, the total piping cost for the given flowsheet (see Figure 6) is minimized. The connection costs are given in Table 1, while the sizes of the footprints for each equipment item and the resulting optimal locations are shown in Table 2. An optimal solution is shown in Figure 7 with a total layout cost of 9978.5 rmu. Note that although equipment items 6 and 7 are connected, they are quite distant from each other in the optimal solution obtained. This is reasonable since the connection cost for items 6 and


Figure 6. Flowsheet for example 1.

Figure 8. Flowsheet for example 2. Table 3. Connection Costs for Example 2 connection

cost [rmu/m]

connection

cost [rmu/m]

(V1, 1a) (V2, 1b) (V2, R2) (1a, 2a) (1a, R4) (1b, 2a)

1 20 5 10 1 20

(R2, R4) (2a, V5) (2a, V6) (R4, V5a) (R4, V6a)

5 10 10 1 1

Table 4. Footprint Sizes and Optimal Locations for Example 2 dimensions optimal locations optimal orientations

Figure 7. Optimal layout for example 1. Table 1. Connection Costs for Example 1 connection

cost [rmu/m]

connection

cost [rmu/m]

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

346.0 416.3 118.0 111.0

(4, 5) (5, 6) (5, 7) (6, 7)

85.3 86.3 82.8 6.5

Table 2. Footprint Sizes and Optimal Locations for Example 1 dimensions

optimal location

equipment

Ri

βi

xi

yi

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

14.19 14.19 4.64 4.24 14.19 19.33 9.15

10.29 18.61 18.61 10.53 3.84 3.84 3.84

7 is low compared to the cost of other connections (i.e., (5, 6) and (5, 7)). In addition, we can see that the equipment items 2 and 3 are center-aligned, but the same does not hold for items 3 and 4. Once again, the same reasoning of different connection costs (see Table 2) applies. 5.3. Example 2. The optimal flowsheet for a batch plant proposed in ref 12 is studied. The plant flowsheet

equipment

Ri

βi

xi

yi

li

di

V1 V2 1a 1b 2a R2 R4 V5 V6 V5a V6a

5.0 6.0 6.0 5.0 6.0 4.5 5.0 5.0 6.0 2.0 3.0

3.0 6.0 6.0 5.0 6.0 4.5 5.0 3.0 6.0 1.0 2.0

3.00 9.00 3.00 9.00 9.00 14.25 14.50 13.50 9.00 15.50 18.00

9.50 3.00 14.00 8.50 14.00 3.00 7.75 14.00 20.00 11.25 4.25

5.0 6.0 6.0 5.0 6.0 4.5 5.0 3.0 6.0 1.0 3.0

3.0 6.0 6.0 5.0 6.0 4.5 5.0 5.0 6.0 2.0 2.0

is illustrated in Figure 8, and the problem data are presented in Tables 3 and 4. The flowsheet depicts a division of the plant into three main sections (i.e., raw materials, production, and products). This represents the way in which the plant management envisions that the site should be organized. However, at this stage, the problem was solved without taking into account this organizational goal. The optimal solution with a cost of 470 rmu is depicted in Figure 9, and the actual equipment coordinates and orientations are summarized in Table 4. We can notice that the connections related to reactors 1b and 2a are particularly expensive and therefore the items connected with them are placed quite close. On the other hand, inexpensive connections such as the one between items 1a and R4 lead to their placement relatively far apart from each other. It should be recalled that all examples are solved within a 5% margin of optimality. Consequently, every solution may further be improved within that margin. For instance, here, the solution obtained can be improved by placing items V5a and/or V6a closer to reactor R4. This will not alter significantly the layout cost due to the related low connection costs (see Table 3).


Figure 9. Optimal layout for example 2. Table 5. Comparison of Models LAYOUT1 and LAYOUT2 LP nodes motivating example 1 example 2

CPU

LAYOUT1

LAYOUT2

LAYOUT1

LAYOUT2

600 87451 6263

300 35100 1868

1.2 202.4 34.1

0.5 54.9 6.3

It is worth mentioning that the resulting layout does not fulfill the original management goal of organizing the site into three well-defined production sections as this goal was not explicitly included in the mathematical model. In section 6 we describe how this aspect of the problem can be formally incorporated into the proposed formulations. 5.4. Comparative Results. The three examples presented before have been solved using both models LAYOUT1 and LAYOUT2, while the computational statistics are shown in Table 5. In all of the cases, the second model (LAYOUT2) provided a better computing performance, in terms of both the LP nodes explored and the total CPU time required. This is mainly due to the more efficient integer representation of the second model by using fewer binary variables than that in the first model, thus reducing the combinatorial nature of the problem required. Additionally, the size of each LP being solved at each node is also smaller in the second model because of the reduced number of constraints and binary variables. In the next section, we extend model LAYOUT2 so as to incorporate aspects of production organization into sections. 6. Organization into Production Sections Often, considerations such as safety, efficient material handling, or workforce management necessitate the organization of the plant layout into well-defined pro-

duction sections or “subplants”.7 The boundaries of these sections are drawn by walls or by corridors which facilitate the movement of materials and operators. The organization of the plant layout into sections can be formally incorporated into the layout models presented above. The basic assumption is that the equipment items are partitioned into subsets by using either rule-based/intuitive techniques (designer-based) or algorithmic approaches (e.g., ref 7). Each equipment subset constitutes “a section”, and then each section is enclosed in a well-defined rectangular shape. The aim of the optimization is then to determine the locations of both the sections and the equipment items within each section so that the total layout cost is minimized. For the present study, model LAYOUT2 has been extended to account for the constraints concerning the organization of the process plant layout into sections. New constraints must be introduced to guarantee that no overlapping occurs: (a) among different sections and (b) among the footprints within each section. A similar extension can easily be developed for model LAYOUT1. The main idea of the proposed model is that since each section is characterized by a rectangular shape (i.e., a shape identical with that adopted for each equipment footprint), the nonoverlapping constraints in model LAYOUT2 are to be instantiated at two levels: 1. Plant level (i.e., overlapping among sections). 2. Section level (i.e., overlapping among equipment footprints). Additional notation is introduced to account for the above two levels: Indices s, t ) sections Sets Ks ) set of equipment items involved in section s Pi ) single-element set denoting the section to which equipment item i belongs; Pi ) {s: i ∈ Ks} I ) set of pairs of equipment with nonzero connection cost and belonging to different sections; I ) {(i, j): Cij * 0 and Pi * Pj}

All of the variables introduced in model LAYOUT2 still hold and have exactly the same meaning, except for xi and yi, which now represent relative rather than absolute coordinates within each section. New continuous variables are introduced to describe each section as follows: Continuous Variables Xs, Ys ) absolute coordinates of geometrical center of section s L ˆ s ) length of section s D ˆ s ) depth of section s

To avoid overlapping of sections, new binary variables are introduced, namely, S1ij and S2ij, which are similar to the ones described in section 4.6 but now refer to sections rather than to equipment items. The relevant


constraints that guarantee nonoverlapping of sections are the following:

L ˆs + L ˆt 2 ∀ s ) 1, ..., S - 1, t ) s + 1, ..., S (28)

Xs - Xt + M(S1st + S2st) g

ˆt L ˆs + L 2 ∀ s ) 1, ..., S - 1, t ) s + 1, ..., S (29)

Xt - Xs + M(1 - S1st + S2st) g

D ˆs + D ˆt 2 ∀ s ) 1, ..., S - 1, t ) s + 1, ..., S (30)

Ys - Yt + M(1 + S1st - S2st) g

Figure 10. Flowsheet for example 3.

D ˆs + D ˆt 2 ∀ s ) 1, ..., S - 1, t ) s + 1, ..., S (31)

Yt - Ys + M(2 - S1st - S2st) g

where S is the number of production sections. In addition, the following bounding constraints are imposed:

Xs +

L ˆs e xmax 2

∀s

(32)

Ys +

D ˆs e ymax 2

∀s

(33)

L ˆ s g xi +

li 2

∀ s, i ∈ Ks

(34)

Figure 11. Optimal layout for example 3. Table 6. Connection Costs for Example 3

di D ˆ s g xi + 2

∀ s, i ∈ Ks

(35)

Finally, constraints (4) and (5) are now written only for items i and j belonging to the same section. If they belong to different sections with nonzero connection cost between them, then constraints (4) and (5) should be modified appropriately:

(

Rij - Lij ) Xs -

(

Aij - Bij ) Ys -

) (

)

) (

)

L ˆs L ˆt + xi - Xt - + xj 2 2 ∀ (i, j) ∈ I (36) D ˆs D ˆt + yi - Yt + yj 2 2 ∀ (i, j) ∈ I (37)

The summary of the resulting MILP model is as follows:

model SECTIONS min

CijDij ∑i ∑ j*i

(38)

subject to plant level constraints (28)-(37); and section level constraints (2)-(6), (21)-(19), and (23)-(26). 6.1. Example 3. The example considers the layout design for a plant manufacturing cosmetic-grade isopropyl alcohol first studied in ref 7. They suggested an optimal partition of the flowsheet into two distinct production sections illustrated in Figure 10. Here, model SECTIONS is used to design the plant layout while taking into account the proposed partitioning. The costs for the connections are obtained from

connection

cost [rmu/m]

connection

cost [rmu/m]

(1, 2) (1, 4) (1, 5) (1, 12) (2, 3) (2, 4) (4, 5) (4, 6) (4, 12)

1965 100 400 100 3930 1965 500 1565 200

(6, 7) (7, 8) (7, 11) (8, 9) (8, 11) (9, 10) (10, 11) (11, 12)

1565 1450 100 1250 200 1000 200 300

ref 7 and are given in Table 6. In addition, we assume equal square footprint sizes for all equipment items. The optimal layout obtained is depicted in Figure 11, where the two sections are clearly delimited. It is interesting to note that items that are connected but belong to different sections, i.e., (4, 6) and (11, 12), are placed close to the boundary of these sections, thus minimizing the relevant connection cost. 6.2. Example 2 Revisited. Consider now the optimal layout of the example presented in section 5.3 when the management goal to organize the plant into three distinct sections (depicted in Figure 8) is explicitly taken into account. With this purpose in mind, model SECTIONS is again applied. The optimal layout obtained is shown in Figure 12, where the clear delineation of the three sections can be observed. The cost of this layout is 12% higher than the one obtained when the organization into sections was not considered. This is reasonable since through introduction of additional layout requirements the problem becomes more constrained. This difference in the objective function is the cost that management has to pay in order to achieve the desired organization of the production environment.


nonorthogonal restrictions on the available space, (d) multifloor plant layout, and (e) simultaneous design/ retrofit and layout. Literature Cited

Figure 12. Optimal layout with sections for example 2.

7. Concluding Remarks In this paper, the optimal process plant layout problem in a two-dimensional continuous space has been considered. Two alternative MILP models were proposed, determination of the optimal location (i.e., coordinates) and orientation for each equipment item so as to optimize a given performance criterion. Different designer objectives can easily be accommodated within the proposed model such as space restrictions and/or safety considerations (e.g., minimum distance between items). Finally, the MILP model was generalized to account for a common plant management goal: the organization of the process plant layout into production “sections”, each of them containing a predefined subset of equipment items. The resulting model determines simultaneously the optimal layout of both sections and equipment items within each section. The applicability of the proposed models was demonstrated by a number of examples. The presented formulations seem promising since they incorporate many interesting features of the process plant layout problem. However, they could further benefit by considering other important aspects such as (a) detailed safety and risk analysis, (b) cost of land, (c)

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Received for review March 3, 1998 Revised manuscript received June 12, 1998 Accepted June 12, 1998 IE980146V

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