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Ind. Eng. Chem. Res. 1997, 36, 4852-4863
PROCESS DESIGN AND CONTROL Optimal Layout Design in Multipurpose Batch Plants Michael C. Georgiadis, Guillermo E. Rotstein, and Sandro Macchietto* Centre for Process Systems Engineering, Imperial College of Science, Technology and Medicine, London SW7 2BY, U.K.
Plant layout is concerned with the spatial arrangement of processing equipment, storage vessels, and their interconnecting pipework. Deciding a good layout is an important activity in the design of chemical and process plants. A good layout will facilitate a correct operation of the plant. It will also provide an economic acceptable balance between the often conflicting constraints deriving from safety, environment, construction, maintenance, operation, space for future expansion, and process relationships such as those determined by gravity flow. This paper presents a mathematical formulation for addressing the problem of allocating items of equipment in a given two- or three-dimensional space. The objective function to be minimized is the total pumping, connection, and floor construction cost. Detailed cost factors are used to account for the flow direction between two connected units. The problem is formulated as a mixed integer linear programming model. Specific attention is paid to constructing a formulation which is suitable for the solution of large scale problems. The method presents the rigorous solution of problems with about 30 process equipment and of essentially unlimited size problems when the rigorous optimization is combined with simple heuristic rules. Three case studies are presented to illustrate the applicability of the proposed approach to retrofit problems in multipurpose plants. Trade-offs between capital and operating costs are captured so that the optimal number of required floors may be determined. 1. Introduction The decision of the equipment layout is an essential stage in the design of a chemical plant. The plant layout affects the supply of services and the access to the periphery of the plant for maintenance, construction, and emergency services (Mecklenburgh, 1985). The traditional method of locating equipment in chemical plants has been to allocate it over a large area, with only a secondary concern for land usage. However, this may often contradict new trends in designing plants with a greater concern for compact plots and enclosed structures. Chemical plant layout by itself has only recently been the subject of study. However, the problem of layout and location has been extensively studied by industrial engineers under the title of facility layout and location, with application to job shops and manufacturing units (Francis and White, 1974). The most relevant approach to the facility layout problem has been studied under the title of “Quadratic Assignment Problem”. This problem addresses the allocation of a number of production facilities to an equal number of locations. The cost function depends on the flow between the facilities and their respective allocations. Several formulations and algorithms based on heuristics have been proposed for the solution of these problems (Fortenberry and Cox, 1985; Bozer et al., 1994). Combined exact and heuristic methods were also presented by Bazaraa and Sherali (1980), Bazaraa and Kirca (1983), and Adams and Sherali (1986). * Author to whom all correspondence should be addressed. Telephone: +44(0)171 594 6608/9. Fax: (44)-171-5946606. E-mail:
[email protected]. S0888-5885(97)00284-4 CCC: $14.00
The general layout problem for chemical plants has recently been considered by other researchers (Gunn and Al-Asadi, 1987; Suzuki et al., 1991a; Bradley and Nolan, 1985; Amorese et al., 1991). In all these approaches a large number of heuristic rules has been proposed which may lead to non-local-optimal solutions. Recently Penteado and Ciric (1996) presented an optimization formulation that can aid in the development of safe and economical layouts. The problem was solved as a relaxed mixed integer nonlinear programming (MINLP) and illustrated with a small case study for the two-dimensional layout. In the area of batch chemical plants Jayakumar and Reklaitis (1994) proposed a graph theoretical approach to single-floor layout, establishing the analogy between distributing units on a single floor and the traditional graph partitioning problem. The same authors also considered the multilevel or multifloor layout for batch plants (Jayakumar and Reklaitis, 1996). Here, the objective of their work was not to find the optimal plant layout but the optimal allocation of units to floors, a particular case of the general layout problem. A graphical heuristic approach was presented which provides an upper bound to the true optimal value. Large scale problems were solved using this approach. A mathematical programming approach was also presented providing a lower bound which together with the upper bound “brackets” the optimal value. This approach relies on a tight linearization scheme of the initial MINLP (compared with the well-known Glover transformations). The continuous relaxation of the resulting mixed integer linear programming (MILP) problem was solved using the Lagrangean relaxation technique together with the subgradient optimization for the calculation of the required Lagrange multipliers. However © 1997 American Chemical Society
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the solution of the relaxed LP may often result in noninteger solutions (Fisher, 1981). In fact, for the small illustrative example presented, the solution in one of the cases examined was fractional. In general, for large problems, noninteger solutions cannot be converted easily to integer solutions even if the continuous optimal point is very close to the integer one. Suzuki et al. (1991b) presented a method for multifloor equipment arrangements in batch plants. Their approach assumes that the assigned floor for each equipment unit is known a priori. The objective function is based only on weighted preferences for the allocation of equipment without economic considerations. The layout problem was also considered for the case of pipeless batch plants by Realff et al. (1996). A mathematical programming formulation was presented addressing the combined design, layout, and scheduling problem. For this type of batch plants, the plant layout strongly influences the scheduling since it determines the relative positioning of the processing stations and as a consequence the transfer times for the vessels. Most of the previous work in process plant layout has mainly been focused on the use of heuristic approaches while optimization-based techniques were usually applied in small example problems. In the area of multipurpose batch plants, very little work has been done on the effect of layout considerations in the retrofit design problem. Here, layout issues are particularly relevant, since typically there is little opportunity for changing the plant boundaries which may pose significant limitation on type, size, and location of new equipment. Space constraints often impose the need for relocation and repiping of equipment items in existing plants, and this problem should also be considered (Reklaitis, 1989). The introduction of layout considerations is thus required to identify least cost configurations of the new equipment within the constraints imposed by the existing facilities. Furthermore, management considerations are often affected by the process layout. For example, in the industrial case study presented later there is a strong interaction between the desired organization of labor and the desired plant layout. This paper presents a formulation and a mathematical programming solution addressing the problem of allocating items of equipment in a given two- or threedimensional space. The objective function to be minimized is the total pumping, connection, and floor construction cost. Detailed cost factors are used to account for the flow direction between two connected units. The problem is formulated as a mixed integer linear programming (MILP) problem, with specific attention given to a formulation suitable for the solution of large scale problems. Trade-offs between the capital and operating cost are illustrated, with an industrial case study resulting in the optimal number of floors required. Furthermore, the simultaneous retrofit design/ scheduling and layout problem is considered with an example, where the effect of layout limitations on the plant structure is investigated. Finally, the applicability of the proposed approach to an industrial case study, the streaming problem in an industrial manufacturing plant, is also illustrated. 2. Problem Statement The problem addressed in this paper can be stated formally as follows:
Given: (1) a set of process equipment units, indexed i, or j ) 1, ..., n (2) a set of available locations and their coordinates, indexed k ) 1, ..., K (3) the material flowrates between connected units in kg/s (4) the maximum number of available floors for the multilevel layout (5) the purchase and installation cost of piping CCij in $/m, the floor construction cost in $/m2 (6) the retrofit plant superstructure for the retrofit design problem (7) space limitations for the retrofit problem (8) minimum safety distance between units i and j which are considered a source of hazards Determine (1) the allocation of each unit (2) the optimal number of floors (3) the optimal plant structure for the retrofit case Such that the cost of the layout is minimized In this work it is assumed that the layout area is discretized into a number of available locations (2-D or 3-D grids) which are represented by unique x, y, and z coordinates. The size of each location is determined in order to allow access for inspection, maintenance, and repair. Each unit can be allocated in all or in some of the available areas. This is determined by the unit size or other operating constraints. The same formulation is applicable to the design case. It should be emphasized that the true layout problem is a continuous nonlinear nonconvex optimization problem, because the equations needed to prevent two units from occupying the same physical space are nonconvex. This results in a very difficult mathematical problem. The discretization proposed here avoids the problems of the nonlinearities and nonconvexities by transforming into a mixed integer linear problem. This discretization is an important decision and if it is too coarse the layout that results from this approach may not be truly optimal. 3. Motivating Example Consider the instant coffee process shown in Figure 2 and first proposed by Jayakumar and Reklaitis (1996). Suppose that the connection cost between units and the floor construction cost is given. Is it possible to find the optimal allocation of units in a multifloor layout such that the total annual connection, pumping, and floor construction cost is minimized? Are there any tradeoffs between the capital and operating cost? Is there an optimal number of floors for which the above cost is minimized? These are some of the issues addresed in this paper. 4. Mathematical Model 4.1. The Objective Function. The plant layout is formulated as an optimization problem where the cost of layout is minimized. This cost is comprised by two major terms: (i) the operating cost (upwork and horizontal pumping cost) and (ii) the cost of investment (connection and floor construction). The allocation of units in different floors requires the use of different cost factors than those for the 2-D case. The additional
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piping cost for the elevation of materials to higher floors cannot be neglected and must be included in the objective function. In the case of a downward flow there is often no pumping cost due to gravity, but the connection cost ($/m) still exists. Finally where two connected units are allocated in the same floor a (low) pumping and a connection cost are taken into account. So unlike other researchers where arbitrary cost values are assigned (especially where the distances cannot be calculated explicitly), a more accurate objective function is presented in this work. The pumping cost comprises three cost parameters, depending on the flow direction between the connected units: (1) Upward cost, CUP. In this case there is a high pumping cost due to the height difference. This cost is a function of the flowrate between the connected units and of the height and given approximately by Coulson and Richardson (1985):
by
∑
min OF )
[(dijBELij + djiABij)CUP] +
((i,j),i
