Ind. Eng. Chem. Res. 1993,32, 345-362
346
Retrofitting a General Multipurpose Batch Chemical Plant Savoula Papagemgaki and Gintaras V. Reklaitis* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907
The objective of this paper is to investigate the optimal retrofit design of multipurpose batch chemical plants. The approach taken builds on our earlier developments for the grass-roots synthesis and design of multipurpose plants. The retrofit problem is formulated as a mixed integer nonlinear programming (MINLP) model which accommodates changes in the product demands, revisions in the product slate, addition of new units in and out of phase, elimination of old ine€ficient units,and batch size dependent processing times. A decompoeition strategy is developed which permits solution of the proposed MINLP optimization model in reasonable computation time. T h e effectiveness of the proposed model and solution strategy are illustrated with three test examples.
Introduction Retrofit design is defined as the redesign of an existing facility to accommodate revisions in the product slate and/or readjustments in product demands and feedstock availability, aa well as to improve the operability of the procese by means of increases in the process flexibility and reduction of the operating costs and the energy consumption. This problem is an important one to process operations because it ariees with great frequency. Moreover, redesign will become increasingly even more important in the future because of the need to respond to variations in the availability and prices of feedstocks and energy, the short life cycles of many specialty chemicals, and the continuing preesure to develop and accommodate new products. The objective of this paper is to investigate the optimal retrofit design of multipurpose batch chemical planta in view of changes in the product demands and/or revisions in the product slate.
6. the status (stable or unstable) and the transfer rules for the intermediates produced between tasks 7. resource utilization levels or rates and changeover times between producta with their associated coats 8. inventory availability and costa 9. a suitable performance function involving capital and/or operating costa, sales revenue and inventory costa Determine (a) a feasible equipment c o w a t i o n which will be used for the manufacture of each product in the plant (new and old) and (b) the sizes of new processing unita and intermediate storage vessels and the number of unita required for each equipment type (new and old), 80 aa to optimize the given performance function.
Aspects of the Retrofit Design Problem The retrofit problem can be considered in two ways: retrofitting to expand the capacity of the existing plant or retrofitting to accommodate a revised product slate. In both instances,the existing equipment unita impose batch size,processing rate, and poseibly connectivity limitations which serve to constrain the configuration and selection of new equipment. Retroflt/Expansion. In this case, the plant equipment inventory needs to be expanded in view of changes in the product demands over one or more time periods. Since the product slate remains the same, the recipe structure remains unchanged and thus the determination of the task partitions for each product need not be revisited. The operating strategy is typically not revised, only the campaign lengths, since products may progress from short to intermediate to long campaigns, and thus the principal design decisions remainconfined to structure syntheak and equipment sizing. Important consideratione for thistype of application include the selection of the product tasks that will need more capacity and the introduction of nonidentical prmsaing unita operating in and out of phase. Retrofit/Revised Product Slate. In this case, the design must be modified to accommodate changes in the product slate which might occur in the form of addition of new producta, removal of some existing producta, or modification of the recipe structure of old products. Important new considerations for this type of application include the decisions of whether to add new equipment unite and whether to remove existing equipment. The decieion to retire old equipment may be important because of the batch size limitations or inefficient processing times/ratee which the old equipment impose on the operation. Alternatively, the decision to add new equipment introduces an additional level of complexity to the problem, because, as in the retrofit/expansion case, the new
Problem Statement The deterministic retrofit design problem for a general multipurpose batch facility can be defined as follows: Given 1. a set of products, the current production requirementa for each product and its selling price, and an initial configuration of equipment item used for the manufacture of these products 2. a set of changes in the demands and prices of the existing products over a prespecified time period (horizon) and/or a set of modifications in the product slate in the form of addition of new products, elimination of old products, or modification of the product recipe of existing producta 3. a eet of available equipment item classified according to their function into equipment families, the items of a particular family differing in size or proceesing rate. Items that are members of the same equipment family and have the same size belong to the same equipment t y p e . 4. recipe information for each product (new and old) which includes the task precedence relationship, a set of proceesing bes/ratea and a correeponding eet of aize/duty fadore,both associated with every feasible teek-equipment pair. In general, the processing time may be specified as a function of the equipment capacity. 6. the eet of feasible equipment items for each product task To whom correspondence should be addressed. 0SSS-58S5/93/2632-0345$04.o0/0
Q
1993 American Chemical Society
346 Ind. Eng. Chem. Res., Vol. 32, No. 2,1993
processing units that may be assigned to the various product tasks can be nonidentical and can be operated either in or out of phase. Literature Review Retrofit in process synthesis has been systematically approached only during the past 4-6 years. Most of the attention has been directed at the retrofit in process synthesis problems that deal with energy systems. The majority of publications have come from the heat exchanger network (HEN) domain, separation systems applications, and process retrofit to improve the flexibility, operability, controllability, and economics of a plant (Gundersen, 1990). The retrofit problem in batch processes, however, has been only sparingly investigated. The limited work reported to date only addressed the multiproduct plant with moat of the attention directed at the addition of new units. Specifically, Yeh and Reklaitis (1985) considered the retrofitting of a multipurpose facility with batch and/or semicontinuousequipment and fixed number of parallel units in each stage. The processing units assigned to each batch stage are assumed to be identical and can only be used in the out-of-phase mode, and no parallel units are allowed in the semicontinuousstages. The objective of this retrofit application is to meet new demand levels at minimum capital cost. Since it is assumed that some of the equipment is already in place and not subject to optimization, limitations on the allowable batch size and the cycle time are imposed by the fixed equipment. The sizing of the new equipment units is performed through a heuristic procedure which is ala0 applicable to the grass-roots design case and which yields very good, suboptimal solutions without requiring the solution of nonlinear programming (NLP) formulations. Vaselenak et al. (1987) studied the pure batch multiproduct plant in which the number of parallel units in each stage is treated as a variable. They assumed fixed processing times, given product recipes and no intermediate storage. They proposed a mixed integer nonlinear programming (MINLP) formulation which employs a superstructure that embeds all possible alternatives for equipment addition, where the new equipment can be utilized to decrease the limiting cycle time by operating out of phase or to increase the batch size by operating in phase. The net profit is chosen as the objective function of the resulting retrofit problem formulation, with the production levels selected so as to maximize the balance between the income resulting from increased production and the annualized capital cost of the new equipment. The p r o p e d MINLP model can be effectively convexified through suitable variable transformations and approximations and then solved using the outer approximation method of Duran and Grossmann (1986), which guarantees attainment of the global optimal solution for convex models. Finally, &puna and Puigjaner (1989) developed a heuristic strategy for the more general case of the retrofitting of a batch/semicontinuous multiproduct plant with variable number of parallel units in each stage. The net profit is again chosen as the objective function. The product recipe structure is assumed to be fixed, in- and out-ofphase modes of operation are both permitted in batch stages, only in-phase operation of pardel units is allowed in the semicontinuousstages, and no intermediate storage is considered. In addition, the processing times are allowed to be batch size dependent. The proposed solution procedure combines gradient computations with heuristic considerations to obtain better values of the variable set
at each iteration. This solution strategy can be used for both grass-roots and retrofit design applications. The key conceptual limitations of the p r o p e d retrofit design procedures are the requirement that all old units must be retained in the plant and that products must use all units,new and old; that is, the structural variables have no product dependence. In addition, a common feature of the last two solution strategies is the requirement that the same batch size be employed for each production path, although opportunity arises for path-dependent batch sizes, since both solution strategies allow the addition of unequal parallel units. Model Formulation In this section a mixed integer nonlinear programming (MINLP) formulation for the deterministic retrofit design of a general multipurpose plant is developed. In our approach to this problem, we make the following assumptions: 1. No intermediate storage is available, but the batch (if stable) can be held in the current processing unit until the next appropriate unit is available. 2. Determination of the task partitions for each product is made a priori. 3. There may be different equipment families that are feasible for the performance of each product task. 4. Use of the same equipment type by two nonconsecutive tasks is disallowed. 5. The size factors associated with every pair of product task-equipment item are constant, and the corresponding processing times are batch size dependent. 6. The processing units assigned to the various tasks are arranged into equipment groups. 7 . The units within a group belong to the same equipment family and may differ in size. 8. Each equipment group is used by one product at a time. 9. Equipment groups only retain their identity for a particular product task in a particular production line. 10. The production horizon is divided into a preapecified masimum number of campaigns, each campaign consisting of a number of production lines used to produce the same or different products. 11. Each product may appear in several campaigns. 12. The initial equipment and campaign confiiation may be modified; that is, old equipment units may be reassigned to different product tasks or removed from the plant and old products may be reallocated to different campaigns. 13. The individual batch sizes and the cycle times are campaign structure dependent, and thus, their values may be altered from one campaign to another. 14. The paths through a production line have the same capacity, and consequently,all batches processed through this line have the same size. 15. No changeover costs, inventory charges, or resource utilization levels are considered. 16. The coet of semicontinuousequipment is considered negligible compared to the cost of batch equipment and, thus, is excluded from analysis. Notice that, in the retrofit design case, the processing time is specified as a function of the batch size, whereas in the grass-roots design case (Papageorgaki and Reklaitis, 1990a), the processing time has been assumed to be constant. The difference stems from the fact that the information required to establish processing times as functions of the batch size is normally not well developed when a new product is introduced (grass-rootsdesign),but it may
Ind. Eng. Chem. Res., Vol. 32, No. 2, 1993 347 well be available after an extended period of operation (retrofit design). In addition, the formation of nonidentical equipment groups is poseible in the retrofit problem, whereas only formation of identical equipment groups has been allowed in the gramroots design problem, since in this case it is generally preferable to design for simple operation. Relaxation of this reatriction in the retrofit case introduces an additional degree of freedom to the optimization problem. In addition to the above aasumptions, the following sets must be defined For each product i, TAi (mitask m is member of the task sequence of product i) For each task m of product i , Pi, = (elequipment e is feasible for task m of product i ) For each equipment type e, Ue = ((i,m)i(i,m)can be performed in unit type e} For each equipment family f , L, = (ele belongs to family f l In the grass-roots design problem, the capital cost of equipment has been used as the optimization criterion. In the retrofit design problem, however, a more appropriate objective function is the maximization of net profit. The net profit is calculated by subtracting the capital cost of the new equipment and the operating cost associated with the new and old equipment from the sales profit resulting from the increased production of the old products and the production levels of the new products over a given period of time. Assuming that the plant is used to manufacture N products and that there are NEQ new equipment types available in the plant, the objective function can be formulated as follows: N
max
i= 1
N
NEB
Cpipi- C
a3e(Ve)be-
e=l
C C
C
i = l mETA, e E P , ,
OimeQime
or, in minimization form
where Pi is the production amount of product i and p i its selling price, Veand Ne are the size and number of units of equipment type e, a, and be are cost coefficients associated with equipment type e, Qim is the amount of product i produced in equipment type e during the execution of task m, and uimeis the associated operating cost. The operating cost coefficient oimeis defined as the operating cost per unit of product i that is produced in equipment type e and includes costs such as cost of energy consumption, cost of manpower, and maintenance cost. The constraint set of the grass-roots design problem developed in Papageorgaki and Reklaitis (1990a) can be extended to describe the retrofit design problem. The constraint set consists of seven principal subsets: (1)assignment and connectivity constrainte; (2) production demand constzaintq (3) cycle time and horizon constraintq (4) equipment size and equipment number constrainte; (5) batch size and batch number constraints; (6) direct and derived variable bounds; (7) degeneracy reduction constraints. These constraint sets are presented in the following sections. Assignment and Connectivity Constraints. The structural decisions that must be made at the design stage
are handled through the introduction of the following binary variables: x j m=
1'
11task rn of produd I IS pedormed in unit type e in equipment group g and campnign k othernlse
0
The assignment and connectivity constraints of the grass-roots design model (Papageorgaki and Reklaitis, 1990a) can be easily extended to the retrofit design case. Furthermore, since the equipment groups can be nonidentical, an additional constraint must be added that requires the equipment groups to be assigned to each product task in indicial order: Ztmgk 2 Zim(g+l)k v i, m E TAi g = 1, ..., NG% - 1 where the continuous variable z,,& represents the assignment of group g to task m of product i in campaign k. This variable is constrained to binary values with the following constraints: zlwk2 Xgmegk V i, k m E TA, g = 1, ..., NGkT - 1 e, q E P,,
C
ZlWk
Xlmegk
V i, k m
E TA,
e€P,,
g = 1, ...,N G S Production Demand Constraints. In the grass-roots design problem, lower bounds on the production demands for the various products are assumed to be given, but the production levels are not explicitly treated as variables. In the retrofit design w e , however, the production levels for the various products must be treated as variables with prespecified upper and lower bounds. The upper bounds will be fixed to an estimated value over the available time horizon, whereas the lower bounds will be equal to the old production levels for the old products that remain in the plant and to some predetermined value for the new products that have been added to the product set. As already mentioned, a sales revenue term will be included in the objective function along with the capital cost of the new equipment in order to allow the model to decide the optimal production level for each product in the plant. Let Ykdenote the amount of product i that is produced in campaign k, P, denote the production amount of product i, and pp"" denote the upper bound on the production amount. Then the following constraints must be satisfied: C Q,me 2 P, V i m E TA, e€p,,
K
C Y,k = P, k=l
V i, k
In addition, let B S m g k denote the batch size of product
i in campaign k which is produced in unit type e and in equipment group g during the execution of task m, Nuvmk denote the number of units of type e that are contcuned in equipment group g which is assigned to task m of product i in campaign k , and nplWkdenote the number of batches of produst i that are being processed by equipment group g in stage rn and in campaign k. Then, the following constraints must be satisfied: K NG,dk=l
g=l
NUrmegkBSimegknPimgk
5
Qime
Vi m
E TA, e E Plm Vi,k mETAi
348 Ind. Eng. Chem. Res., Vol. 32, No. 2, 1993
Cycle Time and Horizon Constraints. As noted earlier, the cycle time of product i may vary from one campaign to another depending on the number of outof-phase units that are assigned to the cycle time limiting task in each campaign. The corresponding constraints from the grass-roots design model can used in the retrofit case. The difference, however, is that, in the retrofit problem, the processing time is assumed to be a function of the batch size and, consequently, the group processing time ti,,,&can be defined as follows: timgk
+%neBSimegkB‘me V i, k m E TAi e E Pi, g = 1, ..., NGgY
e€P,nl
TI, 1 - V i, k m E TAi NGimk
NG,d-
nik 1
V i, k m E TAi e E Pi, g = 1, ..., NGgY
1 SimaSimegk
In addition, the total number of units of type e that are used in the plant is determined by the following constraint .
Vi m
E TAi
e
E Pi,
Furthermore, the following constraint defines the number of equipment groups NGimk: NG,d-
NGimk =
C
g=1
gwimgk
V i, k m E TAi
where wimgk is a continuous variable that takes the value of 1if exactly g equipment groups are assigned to task m of product i in campaign k and the value of 0 otherwise. This variable is restricted to binary values through the following constraints: Wimgk = Zimgk - Zim(g+l)k Vi,k mETAi g = 1, ..., NGFa Wimgk = Zimgk NG,dWimgk
V i , m, e, g, k
BSimegk 2 Bi*Ximegk npimgk 1 nim*Zimgk
V i , m, g, k
npimgk 5 nvzimgk V i, m, 8,k where nim*and Bi* are suitable lower bounds. Variable Bounds. Appropriate upper and lower bounds must be impaed on each member of each variable set. The majority of the Variables have zero lower bounds, because they are switched on and off depending on the valuea of the binary assignment variables. Consequently, a set of “pseudo-lowerSbounds which will only be used when the corresponding variables become positive must also be derived. The set of direct and derived upper and lower limita are given in Appendix I. Notice that, for the existing equipment, = =Vp ve =
e..
and Clearly, the sizes and number of the old proceasing units are not optimized. They are used, however, to impose limits on the batch sizes in the case where the corresponding old equipment units are retained in the plant. Degeneracy Reduction Constraints. As in the grass-roots design problem, degeneracy arises from the equivalence of the production schedules which occurs as a result of the possibility of formation of identical campaigns and the interchangeability of two campaigns. Additional degeneracy arises from the structural equivalence of each equipment family and the equivalence of the equipment group assignments. The f i t two forms of degeneracy can be partly eliminated by the introduction of the campaign ordering and size ordering constraints from the gram-roots design model and the following constraints used to exclude the formation of identical camP&W
V i, k m E TAi g = NGf”m”K
In addition, the following constraint must be satisfied.
C
V i, k m E TAi
npimgk
Ne I @lm = NEld
NG.-..
g=l
Lastly, the following bounding constraints are introduced in the model:
g = 1, ..., NGKY
where NGw denotes the number of equipment groups that are assigned to task m of product i in campaign k. Equipment Size and Equipment Number Constraints. These constrainta are used to determine the size and number of the equipment units as well as the arrangement of these units into equipment groups. Let Sim denote the size factor of product i in equipment type e during the execution of task m. Then, the capacity of processing unit e must accommodate the maximum proces~ingvolume for all the products using that unit, namely
g= 1, ..., NGKY
In addition, the total number of batches of product i produced in campaign k,nik,is determined by the following inequality:
t!nJirnegk
where aimO and BimOare given constants. The cycle time constraint is then formulated as follows:
Ve
produced in each equipment group assigned to each product task cannot exceed the maximum batch size C NUimegkBSimcgk IBYU V i, k m E TAi
5: 1
V i, k m E TAi
g-1
Finally, an additional set of bounding constraints on NUimgkfrom the grass-roots design model is also included in the retrofit design formulation (see Appendix I). Batch Size and Batch Number Constraints. Let Br- denote the maximum batch size of product i that can be produced in any campaign k. Then, the batch size
21@j- 1
V
@j
C fl (1.52)
where PRikare continuous varibles which denote the assignment of product i in campaign k and are constrained to integer (0 or 1)values with constraints 1.48and 1.49 (see Appendix I). In addition, lajjl is the cardinality of set aj, fl denotes the set of products that are being produced in the plant, @j C 0, \k. = fl - aj,and k # k’. The third form of degeneracy can be partly eliminated with the group ordering constraints introduced in the section Aaaignment and Connectivity Constraints and the
Ind. Eng. Chem. Res., Vol. 32, No. 2, 1993 949 following constrainta which are impoaed on the unit types within each equipment family: Ximgk = Xirn(e+l)hk = Xim(e+F)pk= 0 V i, k rn E TAi e, e 1, ..., e F E L, n Pim F - 1 g 2, ..., NGET h 3, ..., NGE, q = NGET (1.53)
+
+
To illustrate, assume that unit types R1 and R2 belong to the same equipment family and that they can be allocated to two parallel groups. Then, the assignmenta Rl-R2 and R2-R1 (where Rl-R2 denotes the assignment of R1 and R2 to group 1and 2, respectively) are equivalent. The above constraint does not permit R1 to be assigned to group 2, and thus, it excludes the equivalent assignment R2-R1 from Consideration. The complete model for the retrofit design of a general multipurpose batch plant is shown in Appendix I. The resulting mixed integer nonlinear (MINLP) formulation has a nonconvex objective function and a nonconvex set of feaeible solutions. Furthermore, it cannot be convexifed through exponential or other suitable variable transformations. As a result, the attainment of the globally optimal solution cannot be guaranteed. In addition, the number of binary variables that are required by this formulation is controlled by the number of product tasks,the number of equipment types, the number of campaigns, K, and the maximum number of equipment groups that are allowed for each product task, N G S . Consequently, it is important that K and N G S be as small as possible to reduce the problem size and hence the solution time. In practical applications, however, the number of binary variables can become quita large. Rather than attempting to solve the proposed model directly, a formulation specific decomposition of the original MINLP will be developed.
Solution Procedure The solution procedure embodies the general concept of the decomposition strategy developed in the grass-roots design case of multipurpose plants (Papageorgaki and Reklaitis, 1990b). Specifically, the original MINLP problem will be decomposed into two subproblems which are relaxations of the original model and which will be solved alternately until one of the termination criteria is met. The forms of the relaxed subproblems, however, will be different, since the retrofit design MINLP model differs from the corresponding grass-roots design model. In addition, the algorithm per se will be different, because the decomposition of the problem will be based on the natural partition of the variables into integer and continuous, whereas in the grass-roots design case the binary (integer) variables and the campaign lengths (continuous variables) were selected as the complicating variables. The selection of a different set of complicating variables for the retrofit problem will be obvious after the presentation of the master problem and the upper bound subproblem. Master Problem The maater problem is a broad relaxation of the original MINLP problem. The corresponding formulation (11)is shown in Appendix 11. Integer cuts corresponding to infeasible assignments identified by the upper bound subproblem (described below) or cuts corresponding to previously identified aseignments may also be included in the formulation. The following proposition describes the sufficient condition under which the maater problem provides a valid lower bound on the optimal solution of the original MINLP.
Table I. Available Equipment Itemr for Example 1 equipment item capacity range (L) Ny RlnR2,R3 4000,” 500-4000 1,” 1,1 Ll,”L2,L3 4000,” 500-4000 1,” 1,1 Fl,”F2,”F3,F4, F5 3000,” 3000,” 500-3000 1,” 1,” 1,1,1 Gl,”G2,G3 3000,’500-3000 1,“ 1,1 a
Existing units.
Proposition 1. The master problem is a relaxation of the original MINLP model and provides a valid lower bound on the optimal solution of the original MINLP, if be I1for all e. The proof appears in Appendix 111. The continuous variables VCimskdenote the capacity of equipment type e in group g that is assigned to task m of product i in campaign k, namely VCimegk = NUimegkVe In addition, the continuous variables CAP, denote the total capacity of equipment type e, namely CAP, = Vfle The master problem is nonconvex due to the nonlinear terms involved in constraints 11.2 and 11.4. This form of the problem cannot be convexified, since most of the variables involved in the formulation have zero as their natural lower bound. As a first step toward the convexification of the master problem, we assume that the lower bounds on the variables CAP,, VCimgk, Qim, Tk,timgk, Yh, nik,and NGimkare set equal to t instead of 0, where t is a very small positive number. Then the following is true: Proposition 2. If the zero lower bound on the variables C u e , VCimegk, Qirne, Tk,timgk, NGimk, nik, and Yik is Substituted with e, then (i) the optimal solution to the master problem will not change and (ii) the optimal profit will be modified by a term o(t), provided that t is a sufficiently small positive number. Proof of this proposition appears in Appendix IV. A practical guide to selecting the value of t is to choose a value which is much smaller than the values of the cost coefficients in the objective function. After this substitution, the bounding constraints 1.13, 11.6,11.8, and 11.9 take a slightly different but equivalent form to account for the nonzero lower bound: K NG,d’-
Qime
5
(e” - €1 C
k=l
C
g=l
Ximegk
+
V i m E TAi e timgk 2
E Pi,
(1.13*)
+ ~ ( 1Ximegk) V i, k m E TAi g = 1, ..., NGf”3 (11.6*)
t?meXtmegk
VCimegk 5
(We”- c)XimeSk + t I(pp”- t)PRik
+
V i, m,e, g, k (II.8*)
(II.9*) V i, k Now the problem can be convexified through exponential variable transformations. Specifically, formulation I1 can be modified by defining and substituting in the model the following variables: Yik exp(syik) Yik
t
Tk = e x p ( 4 nik = exp(snik) NGimk
exp(ngi,k)
timgk = exp(stimgk)
3SO Ind. Eng. Chem. Res., Vol. 32, No. 2, 1993 Table 11. Si- Facton (L/(kg* batch)) and Processing Times (h/batch)O size factors and processing times for equip. type product R1, R2, R3 L1, L2, L3 Fl-F5 2.0815 (4.7393) 5.2268 (8.3353) A 7.9130 (6.3822) 0.2871 (6.4175) 0.2744 (6.4750) B 0.7891 (6.7938) D 0.7122 (1.0135) 2.5889 (6.2699) 1.6425 (5.3713) E 4.6730 (3.1977) 2.3586 (3.0415) 1.6087 (3.4609)
G1. (32. G3 4.9523 (3.9443) 3.3951 (4.4382) 3.5903 (11.9213) 2.7879 (3.3047) .
I
Values in parentheses are processing times.
Table 111. Upper Bounds on Demands and Selling Prices for Example 1 product projected demand (kg/yr) price ($/kg1 A 268 200 1.114 B 156000 0.535 D 189 700 0.774 E 166 100 0.224 Table IV. Cost Coefficients for Example 1 cost coeff unit tm Y. A. 0.1627 15280 R1, R1, R3 0.4068 38200 L1, L2, L3 0.4881 45840 F1, F2, F3, F4, F5 0.1084 44573 G1, G2, G3 Table V. Solution for Example 1 (Only Nonzero Values) unit type Ve Ne G2 3000 1 product n, TL,(h) P,(kg) A 530.6 6.382 268 200 B 88.3 6.794 156000 122.8 D 11.921 189700 E 166.6 3.305 142600 ~~~
~~
~~
net profit: $516 100
After this substitution, constraints 1.11,1.18,11.2,11.3, and 11.4 take the following form: K
Eexp(syik) = pi
k=l
Table VI. Decomposition Algorithm Performance for Example 1 no. of CPU eqs/vars/O-1 obj time" ( 8 ) iteration subproblem function V m 0.64 1 MINLPl -536 200 13517719 -496 900 NLPl 184/114 0.65 0.93 -521 700 2 MINLP2 13617719 NLP2 1841114 0.62 -483 400 2.1 3 MINLP3 13717719 -521 000 1841114 -504 900 0.84 NLP3 -518 200 0.68 4 MINLP4 13817719 1841114 -516100 0.78 NLP4 infeasible 5 MINLP5 13917719 0.77 total 8.01 IBM 3090. Table VII. Available Eauipment Items for Example 2 equipment item capacity range (L) FO,' FN 1000,' 500-3000 1,n 3 2000: 500-2000 1," 1 RO," RN P 25000 1" z 500-2000 3
elu
(I
Existing units.
Table VIII. Size Factors (L/(kg*batch)) for Example 2 size factors for equip. type task FO, FN RO, RN P 2 A1 3.5 A2 2.0 A3 4.6 3.6 B1 2.5 B2 3.2 B3 1.9
linearly approximated ae follows: K k=l
(6iksyik + Bik) 2 pi
where Tk
1 Stimgk
bik
+ anik- ngimk
of which the last four are convex constraints and the first is a nonlinear equality that causes nonconvexity of the resulting formulation. To remedy the situation, this constraint will be replaced by two equivalent constraints: K
Zexp(syik) 5 pi
k=l
K
Zexp(syik) 2 pi
k=1
the first of which is convex and will be retained in the formulation, and the second is nonconvex and it will be
exp(ayfp'")- exp(syp) S Y P-S Y P
and exp(syfg=) - exp(syP) SYP - exp(syt%9 S Y Y - SYtg'" Notice that a piecewise linear overestimation (Garfiikel and Nemhauser, 1972) to more closely approximate this constraint can ale0 be constructed at the cost of introducing additional binary variables in the model. The new formulation (111) is shown in Appendix V. Clearly, formulation I11 is a relaxation of formulation I1 due to the linear underestimation of the exponential terms in the above constraint. 6ik
Ind. Eng. Chem. Res., Vol. 32, No. 2,1993 351 MINLP Formulation Fix K
Update K 4
Fix PR
Fix X
New PR
New X
4
i Upper Bound RU
Subproblem
Lower Bound RL
1
MINLP Master Problem
1
Infeasible Master ?
NLP Subproblem
f Lower Bound R~
MINLP Master Problem
No
11
Infeasible Master ? K=Kmx ?
Yes
No No
Yes
STOP
Figure 1. Decomposition algorithm for the retrofit problem.
Upper Bound Subproblem
The upper bound subproblem corresponds to the original MINLP with the values of the integer variables PRgk fmed. The value of the objective function of this problem provides an upper bound on the optimal solution of the original MINLP. The upper bound subproblem remains a MI"model, but contains fewer binary variables than the original MINLP, since the product-campaign assignment is fmed and, thus, several sets of binary variables can be eliminated from the model. However, the problem formulation is nonconvex and cannot be convexified through variable transformations. Decomposition Algorithm
The decomposition scheme which we propose for the solution of the retrofit problem will be somewhat different from the corresponding decomposition strategy proposed for the grass-roots design problem (Papageorgaki and Reklaitis, 1990b). Specifically, in the retrofit design case the variables will be partitioned into integer and continuous variables. Hence, the integer variables PRikwill serve as the complicating variables for this problem, whereas in the grass-roots design case the binary variables xi,,,& and the campaign lengths Tkconstituted the complicating variable set. This choice is based on the fact that the number of binary variables in the retrofit problem is larger than in the grass-roots case and the selection of these variables as complicating variables would partition the original MINLP into a large set of NLP subproblems, one for each feasible integer solution of the MINLP. Furthermore, the selection of the campaign lengths will not necessarily help the decomposition procedure because of the form of the master problem in the retrofit design case. Specifically, the master problem (11)contains two constraint sets, namely constraints 11.2 and 11.4, which involve nonlinear terms. As evident from the form of these constraints, firing the campaign lengths Tkdoes not remove
any of the nonlinearity and nonconvexity of the corresponding formulation, as o p p o d to the grass-roots design problem. In the latter case, the only nonlinearity in the master problem occurs in one constraint set which contains bilinear terms. For values of the campaign lengths, Tk, temporarily fixed, this constraint becomes linear and, consequently, the master problem becomes a mixed integer linear programming (MILP)problem which can be solved for its corresponding global optimal solution. Since the master problem in the retrofit ca8e has been effectively convexified through exponential variable transformations, ita resulting form may be readily solved for the optimal solution. In addition, our computational experience has shown that the use of the number of campaigns K as parameter throughout the iteration procedure reduces the total computation time required by the algorithm. The reason for this is that, as the value of K increases, product-campaign assignments as well as their permutations that have been already examined for lower values of K can be excluded from analysis through the use of integer cuts. Therefore, the problem contains less degeneracy for the current value of K, and, thus, less computational effort is required. A flowchart of the algorithm is given in Figure 1. The overall algorithm terminates when all feasible values of K have been examined. For each K,the algorithm terminates when the master problem is infeasible. Let RU denote the best upper bound provided by the upper bound subproblem. Then, the algorithm can be summarized as follows: Step 1: Set K = Pin and
N
c Pie'"
i=l
Step 2: Solve the master problem. If feasible, go to step 3; otherwise, go to step 4.
352 Ind. Eng. Chem. Res., Vol. 32, No. 2, 1993 Table
IX. Processing Time Coefficients
(e,.
alme, B,,,)
for Example 2 processing time coeff for equip. type RO,RN P
FO, FN
task
A1 A2
7.5, 1.073 X 0.823 5, 6.86 X 0.865
A3 9.3, 0, 0
4.2, 0, 0
Table X. Production Requirements and Selling Prices for Examsle 2 product projected demand (kglyr) selling price ($/kg) A 700 000 0.3 B 500000 2.25 Table XI. Capital Cost Coefficients (a,) for Example 2" unit tMe a, unit tswe a. 2189.6 P 3602.86 FO, FN 700.66 2 1130.62 RO,RN ~~
Table XIV. Solution of Example 2 Obtained with the P r o w d Algorithm unit type Ve Ne unit type V, Ne RO 2000 1 P 2500 1 RN 5 0 0 1 Z 1190 1 product-campaign n$k B-1 1050.8 product B
Table XII. Operating Cost Coefficients, q,,, for Example 2 oDer cost coeff for eauia tswe task F0,FN R0,RN P Z A1 0.37 A2 0.53 0.33 0.35 A3 0.15 B1 B2 0.05 0.07 B3 ~~
Table XIII. Solution for Example 2 Obtained with DICOPT unit type V, Ne unittype V, FO 1000 1 P 2500 RO 2000 1 Z 641 RN 5 0 0 1
++
nik
product-campaign A-1 B-2
10.0 12.9 1307.4 product A B
TLik
Ne 1 2.5
Yik
12.04 1400 9.3 4900 4.65 495 100 demand (Pi) 1400 500000
Tk net profit: $824000
Step 3 Solve the upper bound subproblem. If feasible, create integer cuta to exclude previously identified assignmente. T h e integer cuta will have the following form introduced by Crowder et al. (1983) PRikPRitkt 5 121- 1 (11.14) i'P3, k'Eb2,.
where 121 is the cardinality of set 2, rj,ej C a, fl= (ilproduct i is being produced in the plant) A l i = {&(producti is processed in campaign &)
A2it
(klproduct i'is not processed in campaign k) In addition, if the current upper bound is greater than RU,
yik
5.9 500000 demand (Pi) 500000 6200
objective function value: $881 700 Table XV. Decomposition Algorithm Performance for Example 2 no. of obj eqs/vars/O-l CPU VLv8 time" (8) iteration sU bP oblem function 203J 106J 15 3 1 1d[IN 9 1 a b -886400 12616616 5.4 1d[IN dPlbb -881700 382/183/30 3.6 -942600 2 1d:IN ,P2a 282J 143J 15 6.6 -879 400 1d:IN aP2b 7.3 384/183/30 3 1d:IN dP3a -866400 19.4 417/209/24 1d:IN dP3b -503 300 5.9 386/183/30 4 1d:IN dP4a -866400 21.5 3841193121 1d[IN 9 4 b -824400 4.1 5 1d[IN AP5a infeasible 388J 183130 total 76.8
--
"IBM 3090. b a and b denote the master problem and upper bound subproblem, respectively. Table
XVI. Available Equipment Familier for Example 3 equip. item capacity range (L) W" FO,O FN R1, R2, R3 GO," GN El,"E2 P"
120 6080
T1 T2
i€r, k € b l ,
TL,h
Tk Tl
Power coet coefficient be = 0.6.
B-1
6.7, 0, 0 5.9, 0, 0
B1 B2 B3
a
Z
12, 1.058 X 0.737
ZP a
1000," 500-3000 2000,500-2000 2000: 600-2000 2500,' 500-3000 25004 2000," 500-2000
1," 3 1, 1, 1 1: 3 1,a 1 1" 1"
Existing unite.
tha update R U and go to step 2. otherwise,create integer cuta of the above form to exclude the infeasible aesignment from consideration and go to step 2. Step 4: If all values of K have been examined, S T O P . Otherwise, set K = K + 1 and go to step 2. Note that although the master problem has been convexified so that the corresponding relaxed NLP haa a unique optimal solution, the upper bound subproblem remains a nonconvex MINLP due to the nonconvexity of the original M I N L P model. Consequently, the decomposition algorithmcannot theomtically guarantee attainment of the global optimal solution. Thia problem can be partly corrected by the use of an alternative decomposition strategy based solely on the selection of the binary variaas the complicating variables. An algorithm bles Ximegk
Ind. Eng. Chem. Res., Vol. 32, No. 2,1993 353 2groupo
lgroup
PRODUCT A
(8)
PRODUCT B 2 groups
added unit (1)
Fimre 3. Final design confwation for example 1.
baaed on this decomposition will partition the original MINLP into a MINLP master problem (identical to the one described above) and a NLP subproblem corresponding to the original MINLP with the values of the binary variables fiied. Then, the following iterative procedure can be employed Step 1': Set K = I W and RU = RF. Step 2.: Solve the master problem. If feasible, go to step 3.; otherwise, go to step 4'. Step 3.: Solve the NLP subproblem. If feasible, create integer cub to exclude previously identified assignments
where SH = l(i,m,e&)lXwk = 11, SL = l(i,m,e&z)lX, k = 01 and ISHI is the cardinality of the index set SH.% addition, if the current upper bound is greater than RU, then update Ru and go to step 2.. Otherwise, create integer cuts of the same form to exclude the infeasible assignment from consideration and go to step 2. Step 4': If all valuea of K have been examined, STOP. Otherwise, create integer cuts of the form of inequality 11.14, set K = K + 1 and go to step 2.. A flowchart of the algorithm is shown in Figure 2. Notice that the master problem provides a valid lower bound on the optimal solution of the original MINLP, since it contains constraints which result from aggregation and relaxation of constraints in the original MINLP, it has been effectivelyconve&ed, and it ia solved independently of the NLP subproblem. Therefore, the algorithm is guaranteed to reach the optimal solution with respect to the values of the binary variables. The NLP subproblem, however, remains nonconvex, and consequently, no t h e retical guarantee of attainment of the optimal values of the continuous variables can be given. A problem that can be encountered during the application of the proposed algorithm is that the linear overestimation of the exponential terms in constraint 111.4 contained in the final convexified form of the master problem may cause a sufficiently large gap between the lower and upper bounds on the optimal Solution provided by the master problem and the NLP subproblem, respectively. As a result, the algorithm may be forced to examine a large fraction of the feasible integer solutions with the consequent increase in the computational effort. As an alternative, a heuristic procedure can be devised to introduce additional hounding constraints into the master problem. This procedure will be based on the examination of the amounts of different pmduds that are produced in each campaign, namely the values of the variable Y,kprovided by the NLP subproblem. Specifically, if the value of YGk for some product i in some campaign k is very close to zero or sufficiently small when compared to the values of Yhfor other products, then the corresponding product-paign assignment can be elim-
added units (2)
0
0
retai;;
units
remove; units
Pyvs 4. Deaign confiiatim for example 2 (a) initial and (b) h d obtained with the propxed decomposition algorithm.
hated !?om further d d e r a t i o n . This statement is based on the intuitive argument that since the current equipment configuration allocated to product i for this pmduct-cam-
paign assignment resulted in the production of insignificant amount of this product in some campaign, any alternative equipment configuration will lead to the same result. Therefore, all such configurations should be excluded from further aualysis. The correspondingbounding constrainta that must be included into the master problem involve the continuous variable PRikand are identical to constraints 11.14 introduced in step 4 of the proposed algorithm. Accordingly, only step 3 of the algorithm w i l l be modified as follows: Step 3.: Solve the NLP subproblem. If feasible, check the values of variable Y,. If for some i and some k Y, is in~igniiicant,then create a bounding constraint of the form of constraint 11.14 to exclude the current product-campaign assignment. Otherwise, create integer cuts of the form of constraint 11.15 to exclude the current productequipment assignment. Then, go to step 2. If the NLP subproblem is infeasible, create integer cuts of the same form to exclude the infeasible product-equipment assignment from consideration and go to step 2. If the current lower bound provided by the master problem is greater than the best upper bound provided by the NLP subproblem, go to step 4. Notice that the incorporation of this heuristic procedure in the decomposition algorithm may force the master problem to cut off the global optimal solution, because information from the nonconvex NLP subproblem is now used to create the bounding constraints that are included into the master problem. Nevertheless, our computational experience with several teat problem has shown that thia procedure works well in identifying redundant configurations. Also note that the master problem can be solved via the outer approximation/equality relaxation (OA/ER) algorithm (Duran and Grossmann, 1986; Kocis and Grossmann, 1988) which guarantees the global optimal solution when the corresponding MINLP is convex, whereas the M I " upper bound subproblem in the first version of the algorithm can be effectively solved via the
354 Ind. Eng. Chem. Res., Vol. 32, No. 2, 1993 Table XVII. Size Factors (L/(ka*batch)) for ExamDle 3 size factors for equip. type task F0,FN Rl,R2,R3 GO,GN El,E2 P ZO A1 2.5 1.7 A2 1.6 A3 B1 2.4, 4.5 3.6 2.5 B2 1.9 B3 1.4 c1 1.2, 1.5 1.7 c2 D1 1.34, 1.5 2.2 D2 0.6 D3 Table XVIII. Constant Processing Times (6,h/batch) for Example 3 const processing times for equip. item task F0,FN Rl,R2,R3 GO,GN E l , E 2 P ZO A1 12 9.7 A2 5 A3 B1 19.72, 8.9 5.9 B2 5.5 B3 4.2 c1 4.5 c2 9.5, 4 15.5 D1 6.7, 4 D2 6.5 D3 9.7
AP/OA/ER algorithm (Viswanathan and Grossmann, 1990)and the NLP upper bound subproblem of the second version of the algorithm can be solved using MINOS 5.0 (Murtagh and Saunders, 1983). Examples
Three test examples will be solved in this section to illustrate the effectiveness of the proposed model and the decomposition solution procedure. DICOPT (software implementation of the OA/ER algorithm) was used to solve the convex MINLP master problems utilizing MPSX to solve the MILP subproblems and MINOS 5.0 for the NLP subproblems of the corresponding MINLP. In addition, DICOPT++ (software implementation of the AP/OA/ER algorithm) was used for the solution of the nonconvex MINLP upper bound subproblems in the first version of the proposed decomposition strategy and MINOS 5.0 was used for the solution of the NLp upper bound subproblems in the second version of the proposed algorithm. Both codes were executed under the modeling language GAMS (Brook et al., 1988) on an IBM 3090. Example 1. A multiproduct plant analyzed by Vaselenak et al. (1987)is considered in this example. This multiproduct facility involves four products and four stages. Since it is assumed that there is a one to one correspondence between stages and equipment families, four different equipment families are available in the plant. An initial equipment configuration involving one unit in each of the stages 1,2,and 4 and two out-of-phase units in stage 3 is given. For each of the existing units, the addition of a single unit in and out of phase is considered. Consequently, the resulting maximum number of singleunit equipment groups that are allowed in each stage is two for stages 1,2,and 4 and three for stage 3. In addition, there are 14 equipment types available in the plant that are detailed in Table I. Note that since no upper bounds on the equipment sizes have been explicitly given by Vaselenak at al., the sizes of the existing equipment will be used as upper bounds. In addition, since the proposed
model requires nonzero lower bounds on the equipment capacities, a minimum capacity of 500 has been assumed for each equipment type. The unit processing times (assumed to be constant in this example) and size factors and the upper bounds on the annual production requirements and selling prices are given in Tables 11and IJI. Vaselenak et al. approximated the capital cost of equipment by a fixed-charge model which is incorporated into our formulation in the following equivalent form: NEQ
C (revae + W
e= 1
e )
The cost coefficients Y~ and A, are given in Table IV. Notice that since CAP = V Y , in the master problem, the values of coefficients Y~ and A, will be used for coefficients ce and de. Also notice that the value of coefficient Xci has been corrected from the value of 10180 to the value of 44 573 to agree with the reported results. Additional assumptions that further simplify the model are that all products must use all units in the plant, and thus, there is no product dependence of the structural variables, no operating costs are considered, and the old units must be retained in the plant. As a consequence, a two-subscript binary variable suffices for the representation of the structural decisions that must be made at the design stage:
xe0=
t
1
0
If unit type e is assigned in equipment group g otherwise
The rest of the variables and the constraints in the model are simplified accordingly, and thus, the original MINLP involves 64 constraints and 46 (9binary) variables. The second version of the proposed decomposition algorithm required five major iterations to attain the optimal solution, which suggests the purchase of one unit that must operate in phase in stage 4. The resulta obtained are shown in Table V. Figure 3 shows the design configuration at the optimum. Details on the problem size and the computation time for the master problem and the NLP subproblem during each of the iterations are shown in Table VI. Note that a slightly better value of the objective function has been attained with the proposed model (profit of $516 100 compared to profit of $513300 in Vaselenak et al.). Nevertheless, the optimal solution with respect to the binary variables was obtained with both the proposed decomposition algorithm and the model proposed by Vaselenak et al. The computation times required by both models are comparable. This example shows that the proposed model and decomposition stategy can lead to the global optimal solution even though no theoretical guarantee for attainment of the global optimum can be given. Example 2. This example will be solved to illustrate the performance of the proposed decomposition sheme when compared to the MINLP solver DICOPT++. We consider a single-product plant which produces product A wing the equipment configuration depicted in Figure 4. The transformation of this single-product plant into a multipurpose facility is considered with the possible addition of a new product B. The problem data are detailed in Tables VII-XII. The complete MINLP formulation involves 259 (30 binary) variables and 523 constraints. DICOPT++ terminated after 51.9 CPU 8, converging to the design configuration depicted in Figure 5 with a profit of $824O00. This solution suggests the transformation of the initial single-product plant into a multipurpose plant that manufactures both products A and B in two sequential cam-
Ind. Eng. Chem. Res., Vol. 32, No.2,1993 355 for Example 3 proceasing time coeff for equip. item GO, GN El, E2
Table XIX. Processing Time Coefficients (a,,, & ),
tank A1 A2 A3 B1 B2 B3 c1 c2 D1 D2 D3
FO, FN
R1, R2, R3
P
7A
0.0
0,o
4.41 X
0,o
0,o
0.633 1.943 X
lo-’, 0.785
9.865 X
lo4, 0.730
0.027, 1 1.85 X 1W6, 1.5 1.257, 0.9 3.88 X I I I
PRODUCT A
PRODUCT B
I campaign 1
I I
campaign 2
Figure 5. Final design configuration for example 2 obtained with DICOFT++. I
PRODUCTB
esmplgn I
I
I
I I I I I I
PRODUCTC
I I
I
PRODUCTD
I I I I
3.26 X 1W’. 2
Table XX. Production Requirements and Selling Prices for Example 3 current projected profit demand demand ($/unit product (kg/yr) (kg/yr) production) 700ooO 2.67 A B 200ooO 500 ooo 1.18 380W 0.865 C 16OooO D 400 ooo 0.34 Table XXI. Operating Cost Coefficients (a;,) for Example 3 oper cost coeff for equip, type task FO,FN Rl,R2,R3 G0,GN El,E2 P A1 0.035, 0.01 A2 0.013 0.023 A3 B1 0.075. 0.035 0.15 0.01 B2 0.001 B3 0.015 Cl
7.0
0.137 0.121
I
campaign 2
lo4, 0.83
campaign 3
I
Figure 6. Initial design configuration for example 3.
paigns. The resulting sizes and number of uNts and the projected production levels are given in Table XIII. Notice here that the resulting valuea of YA1and Ysl would be considered insignificant by the heuristic procedure mentioned above (used to modify step 3 of the algorithm). Consequently, the resulting product-paign configuration would be excluded from further analysis, and, the decomposition algorithm would continue to identify a more “efficient”design configuration as will be seen below. Alternatively, the solution obtained with the fmt version of the proposed decomposition algorithm suggests that the plant remains a single-product plant used to produce product B instead of A. The corresponding design configuration is depicted in Figure 4. Notice that the profit attained with this solution is $881700,which constitutes an improvement of approximately 1% over the profit obtained with DICOPT++. The corresponding final values of the variables are given in Table XIV. The decomposition algorithm required five major iterations to obtain
the “optimal” design configuration. The computation times are listed in Table XV. This example indicates that a general MINLP solver such as DICOPT++ cannot always identify a relatively good solution, because of the nonconvexities involved in optimization models; however, integration of this solver within the framework of a formulation specific decomposition strategy can lead to significant improvement in the quality of the attained solution. Example 3. This example will be solved to illustrate the flexibdity of the proposed model to handle more general multipurpose plants. Consider a plant which is currently used to produce three produds B, C, and D. The initial equipment configuration which is used to manufacture these p d u c t a is shown in Figure 6. The available equipment families are detailed in Table XVI. The possible addition of a new product A and the removal of old product D from the product slate are considered. Furthermore, it is assumed that the production demands of products B and C can be increased to values obtained from market research data and that the operating costs for some of the existing units have been increased due to the inefficient operation of these units (increased processing times and/or decreased size factors). Consequently, the possibility exists of retiring some of the old units. The feasible equipment families for each product task and the associated size factors and unit processing times (constant part, t;,, coefficients aimand &,,,J are given in Tables XW-XM. The current and projected production
356 Ind. Eng. Chem. Res., Vol. 32, No. 2, 1993 Table XXII. Capital CMt Cosfficients (ae) for E-ple 3’ unit t.. we a. unit t .. we a. FN 880.33 E2 2598 Rl,R2,R3 690 ZN 3699 GN 2804 DPowermst coefficient b. = 0.6.
I
I I I
PRODUCT A 2graups
zgroupa
PRODUCT B 2 groups
Table XXIII. Solution of Example 3 (Only Nonzero
Values)
unittwe .. FO FN R1
V.
N.
1000 3000 2000
1 2 1
product-campaign njk A-2 595 B-1 451.9 c-2 375.8 product A B C
unittwe _. R2
P El
V.
N.
2000 2500 2500
1
Tr, 6
5.82 9.5 demand (P,) 700 000 301 300 313 200
PRODUCT C
1 1 ylk
700000 301 300 313200
I
campaign 1
I I
campaign 2
I
m
Figure 1. Final design confwation for example 3. net profit: $2069200
requirements for each product and the selling prices per unit production are given in Table XX. In addition, the operating cost coeficienta and the capital cost coefficients are given in Tables XXI and Xxn, respedively. The value of is used for c in the master problem. Lastly, since we are interested in multipurpose operation of the plant, we restrict the maximum number of campaigns . - to three (totalnumber of products - 1). The mmnlete MINLP model involves 625 variables (79 biary) anh 1324 co&traints. The first version of the proposed decomposition strategy was used to solve this problem. The algorithm required 12 major iterations to attain the ‘optimal” solution which suggesta the f d d e sign configuration depicted in Figure 7 with the values of the continuous variables listed in Table XXIII. Details on the problem size and the computation time for the
IBM 3090.
master problem and the upper bound subproblem during each of the iterations are shown in Table XXIV. The initial configuration has a profit of $470800. The corresponding production schedule consists of three sequential single-product campaigns involving production of producta B, C, and D. In the f d plant configuration, product D has been removed from the product slate because of ita inefficient production and it has been replaced by product A which is the most profitable product. In addition, units u)and GO have been removed and the new units FN, R1, and R2 along with the old units El,P, and FO are now used to form the new production lines. The final production schedule consists of two campaigns with the f i t involving a single production line used to produce product B and the Beeond involving two parallel production lines for products A and C. The equipment units within the production lines have been arranged in equipment groups which contain either single units as in tasks B1 and
and b denote the master problem end the upper bound subproblem, respectively.
Ind. Eng. Chem. Res., Vol. 32, No. 2,1993 357 A1 or nonidentical units as in task A2. Notice that multiple equipment groups were assigned to tasks B1, All and A2 in an attempt to directly reduce the cycle times of products A and B. The final plant has a profit of $2 069 200. Clearly, the transformation of the initial multiproduct plant into a mult.ipurpoee facility resulted in a dramatic increase in the profit. Furthermore, the production levels of products A, B, and C are a t their upper bounds in the “optimal” solution. In general, the former result is expected to hold when the investment cost for additional equipment and the cost due to the plant operation are offset by the increased revenue due to the new production targets.
Conclusions The retrofit design problem for a general multipurpose plant is posed as a nonconvex mixed integer nonlinear program (MINLP) which accommodates changes in the product demands, revisions in the product slate, addition and/or elimination of equipment units, and batch size dependent processing times. The proposed model is developed as an extension of the correspondingmodel for the grass-roots design of a general multipurpose plant (Papageorgaki and Reklaitis, 1990a). The basic differences between the two models can be summarized as follows: 1. In the grass-roots design problem, the production demands are assumed to be constant and they are given as lower bounds on the actual production levels. In the retrofit problem, however, the production demands are treated as variables with predetermined upper and lower bounds. 2. In the grass-roots design problem, only the capital cost of equipment is used as the optimization criterion, whereas in the retrofit problem, the objective function is a maximization of net profit which involves capital costs, operating costs, and sales profit. 3. In the grass-roots design case the processing times are assumed to be constant, whereas in the retrofit design case they are specified as a function of the batch size. 4. Only formation of identical equipment groups is allowed in the grass-roots design case, whereas formation of nonidentical equipment groups is possible in the retrofit problem. 5. In the retrofit problem an initial equipment configuration is available, and consequently, the existing units will have fixed size and number, whereas in the grass-roots problem, sizing is performed on all the units that are selected for the plant. 6. In the grass-rook case the product slate is fixed, whereas in the retrofit case the product slate may be revised. Notice, however, that the proposed optimization model can be also used to describe the grass-roots design of a multipurpose plant. In fact, this model can be viewed as a generalization of the MINLP formulation proposed in Papageorgaki and Reklaitis (1990a), since some of the assumptions used for the original model have been relaxed. The complexity of the proposed model makes the problem computationally intractable for direct solution using existing MI” solution techniques. Consequently, a formulation specificdecomposition strategy is developed, which builds on our earlier developments for the grassroots synthesis and design problem. Two versions of this solution procedure were proposed. In the fmt version, the original MINLP is partitioned into a MINLP master problem which determines the allocation of products to campaigns and provides a lower bound on the optimal solution and a MINLP upper bound problem which de-
termines the assignment of equipment units to product
tasks and performs equipment sizing. In the second version, the same MINLP master problem is used to determine both the product-campaign allocation and the product-equipmont assignment and the upper bound subproblem, which is now a NLP model, is used to perform continuous optimization. The proposed solution strategy cannot guarantee the global optimal solution due to the nonconvexity of the upper bound subproblems. Reasonable computation times are required to determine the optimal design configurations for an example multiproduct plant and an example multipurpose plant. Acknowledgment This work was supported in part by E. I. du Pont de Nemours & Co. and the Procter and Gamble Foundation. Nomenclature N = number of products E = number of batch equipment types NEQ = number of new equipment types F = number of equipment families K = maximum number of campaigns H = total available production time i = index on products m = index on tasks e = index on equipment types g = index on equipment groups k = index on campaigns TAi = set of tasks for product i Pi,! = set of feasible equipment types for task m of product 1
U,= set of tasks that can be executed by equipment type e L,= set of equipment types that belong to equipment family f Yik = amount of product i produced in campaign k Qi = yearly production requirement for product i Sime = size factor of task m of product i in equipment type e
tiWk= group proceasing time of task m of product i in group g and in campaign k t:me aim,,Bime = processing time Coefficients a,,
be = cost coefficients for equipment type e
c,, de = cost coefficients for equipment type e used in master
problem
pi = unit profit for product i qme = operating cost equipment type e
coefficient for task m of product i in
V, = size of units of equipment type e Ne = number of units of equipment type e Pi = production demand for product i Qime = amount of product i produced during task m in equipment type e Ximegk = 0-1 assignment variable for task m of product i in equipment type e in group g and in campaign k n,k = number of batches of product i produced in campaign
k npi k =. number of batches of product i processed by group g3uring task m in campaign k BSiTegk= split batch size produced during task m of product t in equipment type e in group g and in campaign k NUimegk = number of units of type e that are contained in group g assigned to task m of product i in campaign k NGimk= number of equipment groups assigned to task m of product i in campaign k TL,, = limiting cycle time of product i in campaign k Tk = length of campaign k CAP, = total capacity of equipment type e (=VY,) PRik= priority index denoting assignment of product i in campaign k
358 Ind. Eng. Chem.Res.,Vol. 32, No. 2, 1993
Appendix I: MINLP Formulation N
NEQ
min C
e=l
afle(ve)*e
+C C
N
C
i=l ,ETA, eEP,,
WirneQirne
- C Pipi i=l
(1.1) subject to K NG,d-
C
C
k=l
g=l
C
Ximegkzl
eEP,,
i = l , ..., N m E T A i (1.2)
Tk I nikTLik V i, k Ve I SimeBSimegk V i, k
(1.19)
m E TAi e E Pi, g = 1, ..., NGKT (1.20)
NG.,."
NG,d-
I3
Wimgk
g=1
I1
V i , k m E TAi
(1.27)
"Lk-
nik 1
C npimgk
g= 1
V i, k m E TAi (1.29)
K NG,d" NUimegkBSirnegknPimgk k=l
Qime
g=l
V i m E TAi e E Pi, (1.12)
e i n ~ V e ~ e Y" e
(1.33)
O I N e I ~V e
(1.34)
NG,imY
C
g=l
C
NUirnegkBSimegknPimgk
2
Yik
eEP,,
V i, k m E TAi (1.14)
0 INGimkImas e€P,,
TL,k2
timgk
NGirnk
{ear) V i, k
m E TAi (1.37)
V i, k m E TAi e E Pi, g = 1, ..., NGET (1.15)
0 INUimegk4
V i, k m E TAi e E Pim g = 1, ..., N G E i (1.38)
Ind. Eng. Chem. Res., Vol. 32, No. 2, 1993 369
V k
O I T k I H
N
(1.39)
NC.4-
Inner
(1.42)
0 In i k , npimgkI-
Vi,k mETAi
G h i
g = 1, ..., NG&T (1.43) Inn"
0 IVCimegk I
w i n
Bi* = min min
'e
-
mETAi eEP,, S i m e
(1.45)
V i
V i, m,e, g, k
(11.12) (11.13)
RL IR U
+ (1.2)-(1.11), (L13),(1.18),(1.24)-(1.27),(1.34)-(1.37),(1.39), (1.41)-(1.43), (1.46)-(r.49), (1.51)
Appendix 11: Master Problem Formulation (11)
cppi (11.1)
i=l
Appendix 111: Proof of Proposition 1 The master problem (11) is a relaxation of the original MINLP (I) problem and consequently provides a valid lower bound on the optimal solution of the original formulation if: (i) F(1) F(II) which means that every feasible solution to the original problem (I) is feasible to problem (11) (ii) u(I1) I41) where F(P)and u(P)denote the set of feasible solutions and the objective value of problem P, respectively. (i) We must show that any feasible assignment and corresponding feasible values of the continuous variables for problem I are feasible in problem II. First, any feasible assignment of (I) is feasible in (111, because (11) contains all the asaignment and connectivity constraintsof (I)which involve the binary variables. Feasibility with respect to the values of the continuous variables will be shown by aggregating constraints in (I) to yield constraints in (11). Let v e , Ne, Qime, Yikt p i , n i k , npimgk, B S i m e k, N U i m e kt N G i m k , ti&, TL.,, and T k denote a feasible sodution to (1) with objective function value ~ ( 1 ) .Constraints 1.2-1.11, 1.13,1.18,1.24-1.27,1.34-1.37,1.39,1.41-1.43,1.46-1.49, and 1.51 of (I) are retained in (11). Next, by aggregating constraints 1.14,1.20, and 1.29 we get constraint 11.2 in (11).
subject to Wd-
z
8-1
z -VCimegk L-nikYik
eEP,,
V i, k m E TAi
(11.2)
-
(11.3)
where VCimek = NUimegkVe.Furthermore, aggregation of constraints f.14, 1.28, and 1.29 leads to constraint 11.3. Yik
g
= 1, ..., NGgy (11.4)
I C N U i m e g k B S i m e g k n P i m g k IcBf"a'nPimgk I B e
8
360 Ind. Eng. Chem. Res., Vol. 32, No. 2, 1993
B y taking this constraint one step further, we get bounding constraint 11.9. Yik 5 Bt"nik 5 BF"nF"PRik 5 p"PRik
Next, aggregation of constraints 1.15 and 1.19 yields constraint 11.4. X I , x2, bl,
Furthermore, aggregation of constraint 1.21 with the use of definition CAP, = vy,
..',x ,
10
..., b k , bk+l', ..., b,'
LO
Clearly, formulation F2 has the same optimal solution x* as the original NLP model (Fl). Now, by defining
- x,*,
ax, = x ,
r =p
yields constraint 11.5,
r = 1, ..., p, ax, = x ,
+ 1, ..., q,
bxt = xt
- x,*, - xt*, r = q +1,...,g
the Taylor series expansion around x* of the system of nonlinear equations in the original NLP model can be written as n
or
f ( x ) = Cajixi+ CCCdjrat[xr*xa*xt* + xr*xa*bxt i=l
)3Ccvcimegk5 Neve = CAP,
VCimegk
= NUimegkVe
a
t
+
+
+
Let us assume that the lower bound on variables xr, r = 1, ..., ul, (ul Ip ) , is increased from 0 to e > 0. Then, by defining the vector of variables y such that yr = x r - e, r = 1, ..., ul
einXimegk
yi = xi, i = u l
Furthermore, constraints 1.50 and 1.51 can be combined to yield constraint 11.10 in (11). CAP, = V Y , 1 V,+iN,+i CAP,+l (ii) This condition states that the objective function value of problem I1 must be less than the corresponding value of problem I. Proof appears in Appendix I1 in Papageorgaki and Reklaitis (1990b).
+
x,*zt*bx, xs*xt*6xr + 1/2(x,*bx,bxt X,*GX,bxt x t * 6 x ~ x s )+'/Gbxrbxsbxt]
i m 8
Next, relaxation of constraint 1.16 yields constraint 11.6 since aimand BSimgkare positive. In addition, constraint 1.22 and bound 1.33 of (I) can be combined to yield constraint II.7 in (11)and constraint 1.23 and bounds 1.33 and 1.34 of (I) can be used to yield constraint 11.8.
r
+ 1, ..., n
the Taylor series expansion of the nonlinear function f ( x ) becomes n
= Cajgi + C C C d j r s t [ x r * x s * Y t + xr*xt*Ys +
f(y)
ill
r s t
+ %(xr*Qsbyt + xs*@JYt + Xt*bYrbYa) +
xa*xt*Yr
I
+
'/Gbyrbysbyt- ~X,*X~*X~*] EA
where ul
Appendix I V Proof of Proposition 2 For fixed values of the binary variables in the master problem (II), the corresponding NLP can be written in the following general form: max Ccixi
A=
Coir+
r=l
r=l a t
Now, the original NLP model can be written in the following equivalent form with respect to variables y: n
io1 n
is1
gajixi +
ill
(
I
i=l
J' = 1, ..., k
Cajixi= bj
D
max Cciyi
2 2 2 djrs,z,.xpt= bj r-1 a=n+l t=u+l j' = k X I , x2, bl,
+ 1, ..., m
(Fl)
ul
ill
r=l
n
r
s
t
B
62, ..., b, L 0 bl,
+ 1,..., m
(F3)
...,x , L 0
..., b k , bk+l', ..., b,'
2O
s
t
Y d y r 6JY ~ tI
ill
j' = 1,
x2,
J' = k
where B C ~ ~ d j r ' t [ ' / z ( x r * b ~ *+b Za*bJrbYt ~t + Xt*4Yr6Ya) + r
Cajixi = bj
+ d = bj'
x,,
max Ccixi n
j = 1, ..., k
Cajm + CCCdjrst(Xr*2E*Yt + X r * X t * Y a + X , * X t * Y r ) +
ill
..., x n 1 0
The functions in the above NLP are continuous and smooth. Let us assume that x* is the optimal solution to the master problem (11). Then, linearization around x* of the nonlinear constraints provides the following linear model:
i=l
n
Cajiyi= bj - Zaire
P
..., k
In the vicinity of the optimum, say a neighborhood of radius c around x*, 1 = 1, ..., n by, = y I - yl* = t
Ind. Eng. Chem. Res., Vol. 32, No. 2, 1993 361
and thus, A and B become constanta A
lll
ul
r-1
r-1
Cajr + CCCdjr8t[zs*~t* + '/zt(~,*+ ~ t
8
+ %e2]
t * )
+ let*) + %€31
B = 7 r 7 , d j r 8 t [ 7 2 e 2 (+ ~r* r s t
Notice that model F3 corresponds to model F2 with different right-hand-side (RHS) coefficients. Since both models are linear, the basis will remain the same if the 100% rule for RHS coltstants (Reklaitis et al., 1983) is satisfied, namely ul
-CUjr€ k r-1 E-+ Pj
j-1
2 B- 5+1 EA Pj
j-k+l
where Aj is the maximum decrease allowed by sensitivity analyeis. Thus,for Q sufficiently small such that the above inequality is satisfied, the basis and thus the optimal solution will remain the same. The optimal values of the variables will be yr* = ler* - e, r = 1, ..., u l
i = u l + 1, ..., n The optimal value of the objective function will become yi*
lei*,
I
u1
n
r=l
i=ul+l
u* = C cryr* + C
C&*
From the definition of the vector-norm we get d(x,y) = Ix* - y*l I )x,* - yr*I + Ini* - yi*l
g = 1,
Now, since
- nr*l = t,
Lyr*
+ 1, ...,p
as c becomes smaller (lim c = 01, we get lim d(x,y) = lim c = 0
Literature Cited
c-0
t-0
n
lim u* = Ccilei* c-0
i=l
From the above analysis it becomes clear that the optimal values of the variables and the objective function of the perturbed problem (F3) attain the values of the variables and the objective function of problem F2-r equivalently, of problem F1-for arbitrarily small values of t. Appendix V Convedfied Master Problem Formulation (111)
C p p , (111.1) i-1
subject to
V i, k m E TAi (111.2) K
Eexp(syik)I pi
k-1
vi
(I11.13)
V i, k m E TAi (111.14) ngimkI In N G Z + (1.2)-(LlO), (1.13*), (1.25)-(1.27), (1.34)-(1.36), (1.41), (1.46)-(I.49), (1,51), (ILS), (11.7), (11.8*), (11.10)-(11.13)
r = 1, ..., u l
pi* - X i * l = 0, i = u l
..., NGKy
(111.3)
Brook, A.,; Kendrick, D.; Meeraus, A. GAMS,A User's Guide;Scientific Press: Redwood City, CA, 1988. Crowder, H.; Johnson, E. L.; Padberg, M. Solving Large-Scale Zero-One Linear Programming Problems. Oper. Res. 1983,31, 803-834. Duran, M. A.; Grwmann, I. E. An Outer-ApproximationAlgorithm for a c h of Mixed-Integer Nonlinear Programs. Math. Prog. 1986,36,307-339. Espuna, A.; Puigjaner, L. One the Solution of the Retrofitting Problem for Multiproduct Batch/Semicontinuous Chemical Planta. Comput. Chem. Eng. 1989,13 (4/5),483-490. Garfinkel, R. 5.;Nemhauear, G. L. Integer Programming; Wiley: New York, 1972. Gundereen, T. Retrofit Process Design: Research and Applications of Systematic Methods. In Foundations of Computer-Aided Process Design; Siirola, J. J., Grossmann, I. E., Stephanopouloe, G., Eds.; Elsevier: New York, 1990. pp 213-240. Kocis, G. R.; Groeemann, I. E. Global Optimization of Nonconvex MINLP problems in Process Syntheais. Presented at the AIChE Annual Meeting. Znd. Eng. Chem. Res. 1988,27,1407-1421. Murtagh, B. A;Saundere, M. A. MZNOS 5.0 User's Cui& Technical R e ~ o r SOL t 83-20Stanford Universitv- Svstems ODtimization Laboratory, 1983. ' Papageorgaki, S.;Reklaitis, G. V. Optimal Design of Multipurpoee Batch Planta. 1. Problem Formulation. Znd. Eng. Chem. Res. 1990a,29,2054-2062. Papageorgaki, S.; Reklaitie, G. V. Optimal Design of Multipurpose Batch Planta. 2. A Decomposition Solution Strategy. Znd. Eng. Chem. Res. 1990b,29,2062-2073. Reklaitis, G. V.;Ravindran,A.; FLagdell, K. M. Linear Programming. In Engineering Optimization: Methods and Applications; Wiley New York, 1983;Chapter 4.
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Vaaelenak, J. A.; Croeemann, I. E.; Westerberg, A. W. Optimal Retrofit D e s i of Multiproduct Batch Plants. Ind. Eng. Chem. Res. 1987,26,718-726. Viswanathan, J.; Crossmann, I. E. A Combined Penalty Function and Outer Approximation Method for MINLP Optimization. Comput. Chem. Eng. 1990,14,769-782. Yeh, N. C.; Reklaitis, G. V. Synthesis and Sizing of Batch/Semi-
continuous Processes. Presented at the AIChE Annual Meeting, Chicago, IL, 1985; paper 35a.
Received for review May 28, 1992 Revised manuscript received October 19, 1992 Accepted November 12, 1992
GENERALRESEARCH Simple Activity Coefficient Model for the Prediction of Solvent Activities in Polymer Solutions Georgios M. Kontogeorgis and Aage Fredenslund* Znstitut for Kemiteknik, The Technical University of Denmark, DK-2800 Lyngby, Denmark
Dimitrios P . Tassios Department of Chemical Engineering, National Technical University of Athens, Zographos, 15773, Greece
A new simple activity coefficient model for the prediction of solvent activities in polymer solutions is presented. The model consists of two terms, a recently proposed "combined" combinatorialfree-volume term by Elbro et al. and the residual term of UNIFAC. The linear temperature-dependent parameter table for UNIFAC, recently developed by Hansen e t al., is used. The ability of the new model to predict solvent activity coefficients a t infinite dilution for many different polymer-solvent systems is shown. Considering the significant errors often found in the experimental measurements of these coefficients and the scatter of the experimental data, the obtained results are very good: they compare favorably with the UNIFAC-FV model by Oishi and Prausnitz and they are better than the two rather complicated and recently developed equations of state proposed for polymer solutions by High and Danner and Chen et al.
Introduction Since Flory (1970)developed his well-known equation of state for polymer solutions, much work has been carried out in order to establish a model for accurate predictions-and not simply correlations-of solvent activities in polymer solutions. It has been demonstrated that UNIFAC in its original form (Fredenslund et al., 1977)cannot be used for this purpose, since it generally leads to underestimation of solvent activities. This is due to the Staverman-Guggenheim combinatorial term used in W A C , which does not account for the significant free-volume differences between solvents and polymers that occur in most polymemolvent solutions. Free-volume differences explain the frequent occurrence of partial miscibility at high temperatures for polymer-solvent solutions and the existence of a lower critical solution temperature. The modified UNIFAC model of Larsen et al. (1987)cannot be used either, since it always leads to a large overestimation of solvent activities. Thia is a general deficiency of the models that have a combinatorial term of the empirical "exponential" form: di di In 7;= In - + 1- Xi
xi
where diis a modified segment fraction of component i:
Both activity coefficient models and equations of state have been used for the thermodynamic modeling of polymer solutions. Two somewhat different kinds of activity coefficient models have been proposed: (1)the Flory-Huggins (FH) model, i.e., the FH(x) approach (Flory, 1953); (2) the UNIFAC-FV models introduced by Oishi and Prausnitz (1978)and Iwai and Ami (1989). A similar but simpler and theoretically more sound free-volume expression has been recently proposed by Elbro et al. (19901,and it will be the basis of the model presented here. 1. The Flory-Huggins Model. From the FloryHuggins theory the following expression for the solvent activity in a binary polymer solution can be derived: dV0l
In al = In dyol+ 1 - - + x(d~"1)~ 1
X1
(3)
where x is the FH parameter. Although this parameter was initially introduced to account for the energetic interactions between the polymer and the solvent alone, it was recognized early that acceptable resulta are obtained only if we assume that it can be regarded as a free-energy term with an enthalpic and an entropic part (denoted by the subscripts h and s, respectively):
x = Xh + x s
(4)
The Flory-Huggins approach can qualitatively describe a number of phenoma occurring in polymer solutions, but 0 1993 American Chemical Society
