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The optimal retrofit of multiproduct batch plants
Computers and Chemical Engineering 27 (2003) 1277 /1290 www.elsevier.com/locate/compchemeng
The optimal retrofit of multiproduct batch plants Jorge Marcelo Montagna Facultad Regional Santa Fe, INGAR, Instituto de Desarrollo y Disen˜o, Universidad Tecnolo´gica Nacional, Avellaneda 3657, 3000 Santa Fe, Argentina Received 6 February 2003; accepted 7 February 2003
Abstract This paper presents new alternatives in the retrofit model of multiproduct batch plants. Besides the duplication of the batch units implemented in previous works, the model considers the inclusion of intermediate storage tanks. These tanks can be plainly added or replace existing units that can be sold. The allocation of intermediate storage tanks is not considered in the previous retrofit works, even if this alternative is not new in the design of multiproduct batch plants. This option allows getting more efficient and real world solutions, although it requires working with a more complex model. # 2003 Elsevier Science Ltd. All rights reserved. Keywords: Retrofit; Multiproduct batch plants; Storage tanks
1. Introduction The design of multiproduct batch plants is an optimization problem that has been studied by several authors in the past years (Grossmann & Sargent, 1979; Knopf, Okos & Reklaitis, 1982; Modi & Karimi, 1989; Ravemark, 1995; Montagna & Vecchietti, 1998 etc.). The goal is to determine the size and number of batch units so that they can meet production requirements in the provided time horizon. Products produced by this kind of plant fall into the category of chemical specialties, drugs, pharmaceutical specialties, etc. Those are high value products with a short life cycle. Research on new products is intensive and it is frequent that new products replace the old ones. As new processes are to be implemented, the batch plant structure is changed in order to meet the new production requirements. The retrofit problem is an optimization problem whose objective is to obtain the optimal new structure for a batch plant starting from the old one, so as to maximize the benefits subject to a new demand pattern. In general, the retrofit problem solution comes up with a new plant configuration where useless units are sold and new ones are allocated and adjusted to work with old ones to configure the new process.
E-mail address: [email protected] (J.M. Montagna).
Multiproduct batch plants manufacture a set of products using the same equipment operating in the same sequence. Since products differ from one another, each unit is shared by all products but they do not use their total capacity for all of them. Each unit works at full capacity only when processing the products for which this stage is size-constraining. By the same token, the stages do not work all the time for all products. A stage works all the time (charging a new batch just after finishing with the previous one) only when processing the products for which this stage is a time-bottleneck, defining the cycle time for that product. The idea is to optimize the production rate for each product in the plant, which is a function of the batch size and the cycle time. The unit with minimum capacity limits the batch size while the limiting cycle time is fixed by the stage with the longest processing time. In order to reduce the investment cost, several alternatives are possible (Ravemark, 1995). The first one is the introduction of parallel units out-of-phase. In this case the cycle time is reduced if the unit has the longest operating time. Another option is to add a parallel unit in-phase to increase the operating capacity of the stage. This is a useful alternative for stages that are limiting the batch size. Finally the allocation of intermediate storage tanks allows increasing the equipment utilization (Modi & Karimi, 1989). Two objectives are reached: the reduction of idle time by increasing the
0098-1354/03/$ - see front matter # 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0098-1354(03)00052-8
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Nomenclature Bij bij dij Fij F1ij Gj H M NOLD j NTj NTj P pi Pri pri Qi qi QU i Sij STij tij Tij TLi Vijkg Vjk VTjk yijkg yijg yjk ytjk yttj Zj Subscripts G I J K Superscripts H L U Greek letters gij F f
J.M. Montagna / Computers and Chemical Engineering 27 (2003) 1277 /1290
batch size of product i at stage j (kg) transformed variable for the batch size of product i at stage j parameter in the expression of Tij constant for a big-M constraint constant for a big-M constraint maximum number of groups allowed at stage j time horizon (h) number of batch stages in the plant number of existing units in the plant at stage j total number of units at stage j considering old and new ones total number of existing intermediate storage tanks in position j number of products in the multiproduct batch plant net profit per unit of product i (USA dollars) production rate of product i (kg/h) transformed variable for production rate of product i production of product i (kg) transformed variable for production of product i production demand of product i (kg) size factor for product i for a batch unit at stage j (m3/kg) size factor for product i for an intermediate storage tank in position j (m3/kg) parameter in the expression of Tij operation time for product i at stage j [h] limiting cycle time for product i [h] volume of unit k at stage j, taking part in group g, for product i (m3) volume of unit k at stage j (m3) volume of the intermediate storage tank k in position j (m3) binary variable for the inclusion of unit k at stage j for product i in group g binary variable for the existence of group g at stage j for product i binary variable for the use of unit k at stage j binary variable for the use of intermediate storage tank k in position j binary variable for the existence of some storage tank in position j maximum number of batch units to be added at stage j group product batch stage and position to allocate an intermediate storage tank batch unit or intermediate storage tank subprocess lower bound upper bound parameter in the expression of Tij maximum ratio between two consecutive batch sizes log F
working time of the stages, and the increased equipment utilization. When an intermediate storage tank is allocated, the original process is divided into two subprocesses, each one presenting its own batch size and limiting cycle time. Productivity of both subprocesses must be the same to avoid accumulation in the storage tank.
The first retrofit paper (Vaselenak, Grossmann & Westerberg, 1987) considered the addition of batch units to an existing plant to face an increase in the products demands. The model was a mixed integer nonlinear program (MINLP), whose objective was to maximize the benefit considering the incomes from the sales of the products minus the cost of the new units. Each product
J.M. Montagna / Computers and Chemical Engineering 27 (2003) 1277 /1290
has an upper bound in the demand based on the sales projection. The model of Vaselenak et al. (1987) assumed that the units in parallel either in-phase or out-of-phase must be used in the same way for every product. Fletcher, Hall and Johns (1991) modified the Vaselenack’s approach by allowing the new units to operate in different ways for each product. Better solutions were obtained in this last case. Yoo, Lee, Ryu and Lee (1999) presented the most updated complete retrofit model for multiproduct batch plants. No condition was imposed on the use of the batch units. The new and the old units can be used inphase and out-of-phase, forming groups that can be different for each product. The model also contemplates that the old units that are no longer used in the new structure can be sold. Van den Heever and Grossmann (1999) presented a general disjunctive model for the multiperiod retrofit problem. They used logic and disjunctive programming to pose all considered alternatives, obtaining a great performance for large problems. However, the available options for each period in the model are reduced because they worked with the problem formulation by Fletcher et al. (1991). No previous work has considered intermediate storage tanks for the retrofit problem. In this work, an extension of the explicit model by Yoo et al. (1999) is presented adding a new degree of difficulty by including the allocation of intermediate storage tanks. There is no doubt that the number of feasible alternatives will increase and, therefore, the possibility of obtaining better solutions exists. The previous assumptions were maintained? Especially the capability for changing the configuration of every batch unit for each product considered in the plant and the possibility of selling the useless old units. Several examples were solved with this model in order to show its effectiveness.
/1, . . ., NTj are The sizes of new units Vjk, k /NOLD j obtained with the problem solution. The NTj units of stage j can be grouped in different ways for each product i. According to Yoo et al. (1999), we can have groups where all units in the group operate in parallel and in-phase. The different groups at the stage operate in parallel and out-of-phase. The allocation of an intermediate storage tank decouples the process into two subprocesses. Each one operates with a different batch size and cycle time, but maintaining the same production rate for product i, Pri, to avoid accumulation of material in the tank. Considering intermediate storage tanks requires working with new variables. Now the batch sizes or the limiting cycle times in each subprocess could be different if a tank is allocated. There are M /1 possible locations for the storage tank (j /1, M /1), where the jth location is between batch stages j and j/1. In this model we assume that in some locations we can have storage tanks of different sizes from previous retrofits. At location j there are NTj storage tanks of dimension VTjk with k /1, 2, . . ., NTj. Only one tank can be added in position j with size VTj,NTj1. Unlike batch stages, intermediate storage tanks can not be grouped in different ways. Therefore, the allocation of several tanks is not a feasible option, except when an upper bound over its size is reached. This last alternative can be easily added to the model, and it is not included here to simplify this presentation. /
3. Problem formulation The objective of the problem is to maximize the annual benefit of the plant considering incomes from the product sales plus useless unit sales minus new units costs: P X
2. Problem definition In a multiproduct batch plant, P products (i /1, 2, . . ., P) are processed. For each product demand QU i is known, which is an upper bound on the amount to be produced of product i. This demand must be produced over time horizon H. Since this is a multiproduct batch plant, every product follows the same production sequence over M batch stages (j /1, 2, . . ., M) of the plant. At each stage we have a set of NOLD batch units. The size of unit k at j stage j is Vjk (k /1, 2, . . ., NOLD ). Taking into account j that these units could have been added in previous retrofits, their sizes can be different. Zj units can be added at stage j so that NTj /NOLD / j Zj is the maximum possible number of units at stage j.
1279
OLD
pi Qi
M N j X X
Rjk (1 yjk )
j 1 k 1
i 1
NTj X
RTjk (1 ytjk )
j 1 M X
NJT X
r
(Kj yjk cj Vjkj )
j 1 k NOLD 1 j
k 1
M 1 X
M 1 X
rt
j (KTj ytj;NTj1 ctj VTj;NT ) j1
(1)
j 1
The first term of the objective function is the annual benefit corresponding to the product sales. Variable Qi corresponds to total production of product i. Parameter pi is the total benefit per unit of product i. The second and third terms correspond to the incomes from useless batch units and storage tanks, respectively. Binary variable yjk indicates if unit k at stage j is included in the new plant structure (yjk /1) or
J.M. Montagna / Computers and Chemical Engineering 27 (2003) 1277 /1290
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not (yjk /0). For this last case, the unit is sold at price Rjk. In a similar way we proceed with the useless storage tank using binary variable ytjk and sale price RTjk. The fourth and fifth terms correspond to the investment cost of the new batch units and storage tanks, respectively. Kj, cj and rj are cost parameters for the batch units. Kj corresponds to a fixed cost independent of the unit size, while cj is proportional to the unit size. Previous papers work with rj /1. The fifth term is for storage tanks like the fourth is for batch units, with cost parameters KTj, ctj and rtj. The difference is that only one storage tank can be incorporated at location j. If a storage tank exists, the number and size of the batches up and downstream of the tank are different for each product. Variable Bij is introduced corresponding to the batch size of product i at stage j. The number of batches for product i at stage j is Nij. In practice, this variable must be integer. However, considering Nij as a continuous variable, we do not introduce a significant error for long production horizons. Production at each stage limits the total production for each product, then the following equation is applied: Qi 5Nij Bij
i 1; . . . ; P; j 1; . . . ; M
(2)
In this way, the problem tries to maximize Qi to increase the benefits and to reduce Bij to decrease the new unit sizes. The following constraints are applied, relating batch sizes of consecutive stages (Ravemark, 1995):
B 1 1 yttj 5 ij 51(F 1)yttj F Bi;j1 1
(3)
i 1; . . . ; P; j 1; . . . ; M 1 where F is a parameter corresponding to the maximum difference allowed between two consecutive batch sizes. Binary variable yttj is used to define if a storage tank is located at position j (yttj /1) or not (yttj /0). The following constraints are applied for this situation: yttj 5
NT j 1 X
ytjk
j 1; . . . ; M 1
(4)
k 1
yttj ] ytjk j 1; . . . ; M 1; k 1; . . . ; NTj 1
(5)
If no intermediate storage tank exists, yttj is fixed at 0. If at least one tank is located yttj is one. It is important to note that from Eqs. (4) and (5) and using an upper bound equal to 1, yttj can be 0 or 1 without considering it as a binary variable. If no intermediate storage tank exists between j and j/1 then Bij is forced to be equal to Bij1. Otherwise, they are different and must satisfy the following constraint:
Fig. 1. Conforming groups at a stage j.
1 B 5 ij 5 F F Bi;j1
i 1; . . . ; P; j 1; . . . ; M 1
(6)
Several binary variables are introduced to determine the plant structure. Since the units can be grouped in different forms at each stage we use binary variable yijg. The value of this variable is 1 if group g is generated for product i at stage j; otherwise, the value is zero. Group g is generated if at least one unit is assigned to it. Binary variable yijkg is equal to one if unit k of stage j is assigned to group g for product i, otherwise the variable is equal to zero (Yoo et al., 1999). Fig. 1 illustrates how units can be arranged to conform the groups. In this example, there are four units (NTj ) in a stage: two old units, the white ones (NOLD ), and two new units, the gray ones (Zj). In this j way, up to four groups of one unit could be conformed. There are several possible combinations of the units to determine groups. In this example units have been arranged to conform only two groups. Therefore, variables yij1 and yij2 are equal to one, and yij3 and yij4 are equal to zero. Units 1 and 3 form group 1 and operate in-phase. Units 2 and 4 conform group 2 and operate in-phase too. Then variables yij11, yij31, yij22 and yij42 are equal to one and all the other variables yijgk are equal to zero. Both groups 1 and 2 operate out-of-phase. Each unit k at stage j can be assigned at most to one group for product i: Gj X
yijkg 5 1
(7)
g 1
i 1; . . . ; P; j 1; . . . ; M; k 1; . . . ;
NTj
Gj is the maximum number of groups allowed at stage j. Group g exists at stage j only if at least one unit is assigned to the group:
J.M. Montagna / Computers and Chemical Engineering 27 (2003) 1277 /1290 g
T
yijg 5
Nj X
(tij dij Bijij )Pri
yijkg
(8)
k 1
Bij
5
Gj X
1281
yijg
(15)
g 1
i 1; . . . ; P; j 1; . . . ; M; g 1; . . . ; Gj
i 1; . . . ; P; j 1; . . . ; M
If unit k is assigned to the group, the group must exist:
The plant production is limited to time horizon H, then
yijkg 5 yijg
P X Qi
i 1; . . . ; P; j 1; . . . ; M; g 1; . . . ; Gj ; k
(9)
1; . . . ; NTj and the upper bound for binary variable yijg is: yijg 5 1
i 1; P; j 1; M; g 1; Gj
(10)
From Eqs. (8) /(10), it can be seen that binary variable yijg can be converted to a continuous variable. If unit k is assigned to group g at stage j for product i, the unit must exist: yijkg 5 yjk i 1; . . . ; P; j 1; . . . ; M; g 1; . . . ; Gj ; k 1; . . . ;
(11)
NTj
The maximum time between two consecutive batches in subprocess h must be considered to determine the limiting cycle time for h. This maximum time is given by the division between operation time Tij and the number of groups out-of-phase for product i, considering all the stages included in subprocess h: TLhi ]
Tij Gj X yijg
(12)
g 1
i 1; . . . ; P; Öh; Öj Subprocess h where Tij is a function of the batch size, with tij,, dij and gij fixed parameters: g
Tij tij dij Bijij
i 1; . . . ; P; j 1; . . . ; M
(13)
We do not know a priori which units conform subprocess h. This is a result of the mathematical program and depends on the location of the storage tank. The expression Eq. (12) can be modified with the assumption that in every subprocess, production rate for product i, Pri, must be the same: Pri
Bhi TLhi
i 1; . . . ; P; Öh
(14)
Considering expression Eq. (3) where the value of the batch size for subprocess h (Bhi ) can be adjusted, using Eq. (13) and replacing TLhi from Eq. (14) into Eq. (12), we have:
i 1
Pri
(16)
5H
Batch size Bij is determined by the lowest capacity between the groups for product i at stage j, then the following constraint can be applied to get the Bij value, with Sij the size factor for product i at stage j: T
Sij Bij 5
Nj X
Vjk yijkg (1 yijg )Fij
(17)
k 1
i 1; . . . ; P; j 1; . . . ; M; g 1; . . . ; Gj Constraint Eq. (17) is a Big-M type that guarantees that batches can be processed if group g exists, otherwise the constraint is redundant because of the large value of Fij. The value of Fij can be calculated by (Yoo et al., 1999): Fij
OLD N j X
Vjk Zj VU j
(18)
k 1
i 1; . . . ; P; j 1; . . . ; M where VU j is the upper bound of the batch units to be added at stage j. Constraint Eq. (17) is nonlinear in the binary variable yijkg, which adds difficulty to the convergence. To overcome this problem Yoo et al. (1999) replaced the product Vjkyijkg by the new continuous variable Vijkg, then Eq. (17) is replaced by the following constraints set: T
Sij Bij 5
Nj X
Vijkg (1 yijg )
k 1
OLD NX j
Vjk Zj VU j
k 1
(19)
i 1; . . . ; M; g 1; . . . ; Gj Vijkg 5Vjk i 1; . . . ; P; j 1; . . . ; M; k 1; . . . ; NTj ; j
(20)
1; . . . ; Gj Vijkg 5VU j yijkg i 1; . . . ; P; j 1; . . . ; M; k 1; . . . ; NTj ; j
(21)
1; . . . ; Gj OLD . where VU j is equal to Vjk for k /1, . . ., Nj If the unit is selected, then the following constraints are used to guarantee that the unit size is between its bounds:
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J.M. Montagna / Computers and Chemical Engineering 27 (2003) 1277 /1290
Vjk 5VU j yjk j 1; . . . ; M; k NOLD 1; . . . ; NTj j Vjk ]VLj yjk j 1; . . . ; M; k NOLD 1; . . . ; NTj j VU j
(22)
(23)
VLj
where and are the upper and lower bounds for the new units at stage j. We use the following expression adapted from Modi and Karimi (1989) to determine the size of the intermediate storage tank: NT j 1 X
VTjk ]STij (Bij Bi;j1 )F1ij (yttj 1)
k 1
(24)
i 1; . . . ; P; j 1; . . . ; M 1 This constraint also belongs to a Big-M type and it takes into account all intermediate storage tanks located at that position. STij are the size factors for the tanks for product i at stage j. The value of F1ij can be calculated by: F1ij STij
1
OLD NX j
Sij
k 1
1 Vjk Zj VU j Si;j1
each unit k. The order of the group is obtained by adding the weights of all units in the group. Finally, we must add bound constraints for the demand. The lower bound on the demand for product i is set when the production is already sold. Qi 5QU i
i 1; . . . ; P
(29)
Qi ]QLi
i 1; . . . ; P
(30)
4. Problem convexification The model for the retrofit of a multiproduct batch plant is defined by maximizing Eq. (1) subject to constraints Eqs. (2)/(5), (7) /(11), (15), (16), (19) /(24), (26) /(30). This is a MINLP problem that presents non convex terms in several constraints and in the objective function. Thus, it is not possible to assure that the global optimum can be reached. In a similar way as Vaselenak et al. (1987), the following transformations are introduced to avoid non-convex terms: bij ln Bij
i 1; . . . ; P; j 1; . . . ; M
nij ln Nij
i 1; . . . ; P; j 1; . . . ; M
i 1; . . . ; P; j 1; . . . ; M 1
pri ln Pri
i 1; . . . ; P
We must also consider the bounds on the tanks to be added by the following constraints:
qi ln Qi
VTjk 5VTU jk ytjk
Using these transformations, the objective function results:
OLD NX j1
Vj1;k Zj1 VU j1
(25)
k 1
j 1; . . . ; M 1; k NTj 1
(26)
Min
VTjk ]VTLjk ytjk j 1; . . . ; M 1; k NTj 1
T
Nj X k 1
T
2
NTj k
yijkg ]
Nj X
T
2Nj k yijk;g1
k 1
(28)
i 1; . . . ; P; j 1; . . . ; M; g 1; . . . ; Gj This constraint order the different groups. For example, if there are four units at a stage, one solution is to conform group 1 by units 1 and 2 and group 2 by units 3 and 4. However, the same solution is attained assigning units 3 and 4 to group 1 and units 1 and 2 to group 2. This constraint avoids these assignments by T ordering the group through a weight 2Nj k assigned to
pi exp(qi )
NTj X
RTjk (1 ytjk )
k 1
M N j X X
Rjk (1 yjk )
j 1 k 1
i 1
(31)
OLD
P X
(27)
L where VTU jk and VTjk are the upper and lower bounds, respectively, that must be considered if the tanks exist. Redundant assignation to a group with the same value for the objective function are avoided by the following constraint (Yoo et al., 1999):
i 1; . . . ; P
M X
M 1 X j 1
NTj X
(Kj yjk cj Vjk )
j 1 k NOLD 1 j
M 1 X
(KTj ytj;NTj1 ctj VTj;NTj1 )
(32)
j 1
where we have considered the minimization of Eq. (1) with a change of sign. Also, constraints Eqs. (2), (3), (15), (16), (19), (24), (29) and (30) are modified to obtain the final formulation: i 1; . . . ; P; j 1; . . . ; M qi 5 nij bij bij bi;j1 5fyttj i 1; . . . ; P; j 1; . . . ; M 1 bij bi;j1 ] fyttj i 1; . . . ; P; j 1; . . . ; M 1 where f /ln F.
(33) (34) (35)
J.M. Montagna / Computers and Chemical Engineering 27 (2003) 1277 /1290
tij exp(pri bij )dij exp[(gij 1)bij pri ]5
Gj X g 1
i 1; . . . ; P; j 1; . . . ; M P X exp(qi pri )5H
yijg
(36)
i 1; . . . ; P i 1; . . . ; P i 1; . . . ; P; j 1; . . . ; M
KTj
Product A
(37) (38) (39) (40)
Following Vaselenak et al. (1987), constraint Eq. (40) is added to reduce the number of nonlinear constraints. Working with bij, constraints Eqs. (19) and (24) should be non-linear. Restriction Eq. (40) is satisfied trivially at the optimal solution because by minimizing Eq. (32), qi assume the greatest possible value that is limited by Bij. However, this model presents difficulties. The first term in the objective function is concave. To overcome this problem, Vaselenak et al. (1987) have proved that the linearization of the negative exponential functions in the first term of Eq. (32) can be approximated by a system of piecewise linear underestimators. This approximation overestimates the objective function so that it can be employed to find the global solution of this model.
5. Model resolution The final model minimizes Eq. (32) subject to constraints Eqs. (33) /(40), (4), (5), (7)/(11), (17), (19) /(24), (26) /(28). This MINLP problem is solved using the algorithm of Duran and Grossmann (1986), afterwards completed by Viswanathan and Grossmann (1990) with the OA/ER/AP algorithm and implemented in DICOPT. /
Table 1 Example 1 data added to Yoo et al. (1999) STij
i 1
qi 5ln QU i qi ]ln QLi exp(bij )5Bij
1283
/
6. Examples All the examples from previous papers have been solved with this model. In all cases the inclusion of the intermediate storage tanks depends on the cost and size factors of those units. Here appropriate values have been selected values to show the potential applications of this approach. The main objective of this section is to show the possibility of obtaining better solutions, taking into account that the number of feasible solutions has been increased. However, it is difficult to compare the solution obtained with previous approaches since the optimal solution will greatly depend on the values chosen for the cost coefficients.
Position 1
ctj
Product B
1
1
10 000
10
Table 2 Results corresponding to Example 1 Yoo et al. (1999)
This approach
Product A
Product B
Product A
Product B
Qi (kg)
1 200 000
1 000 000
1 200 000
1 000 000
New units Stage 1 (m3) Stage 2 (m3) Storage 1 (m3) Profit ($)
1358 / / 3 125 236
/ / 3333 3 156 667
6.1. Example 1 This example has been solved by Vaselenak et al. (1987), Fletcher et al. (1991) and Yoo et al. (1999). Table 1 presents the example data added to Example 1 of Yoo et al. (1999). Table 2 shows the results obtained using the model by Yoo et al. (1999) without intermediate storage tanks and the results with the model presented in this paper including tanks. In the first case, one unit is added at stage 1 operating in-phase for product A and out-of-phase for product B. In the last case, no parallel units are in the plant but one intermediate storage tank is located between both stages. The objective function is reduced by 1.0%. This reduction depends on the costs proposed for the units. The whole demand is produced for both products. Cost reduction comes from the difference in cost between parallel unit 1 ($74 800) of the first approach against the cost of the intermediate storage tank ($43 300) with the new approach. Note that the improvement in the objective function strongly depends on the relationship between products incomes and units costs. 6.2. Example 2 Vaselenak et al. (1987), Fletcher et al. (1991) and Yoo et al. (1999) have solved this example. Table 3 presents the data about the intermediate storage tank to be added to the original data. Table 4 shows the results of both approaches. In the stages with two units, it shows the operating policy: in-phase or out-of-phase. In this problem, the production targets are not achieved in the optimal solution with the previous approach (product D). However, with the new model all the demands
J.M. Montagna / Computers and Chemical Engineering 27 (2003) 1277 /1290
1284
Table 3 Example 2 data added to previous works STij
Position 1 Position 2 Position 3
Product A
Product B
Product C
Product D
7.913 2.0815 5.2268
0.7891 0.2871 0.2744
0.7122 2.5889 1.6425
4.6730 2.3586 1.6087
KTj
ctj
10 10 10
1 1 1
Table 4 Results corresponding to Example 2 Yoo et al. (1999)
Qi/1000 (kg) New units Stage 1 (m3) Stage 2 (m3) Stage 3 (m3)
A
B
C
D
A
B
C
D
268.2
156.0
189.7
158.1
268.2
156.0
189.7
166.1
In
In
Out
In
In
In
/ / / Out 3000 In / 521 780
Stage 4 (m3) Storage 3 (m3) Profit ($)
This approach
In
Out
Out
In
Out
Out
/ / / Out 2252 Out 3667 528 000
Table 5 Example 3 data added to Example 3 from Yoo et al. (1999) STij
Position 1
Product A
Product B
Product C
Product D
2.4
1.8
1.95
4.15
KTj
ctj
22
0.27
Table 6 Results corresponding to Example 3 Yoo et al. (1999)
This approach
A
B
C
D
A
B
C
D
Qi/1000 (kg)
290
300
350
140
290
300
350
140
New units Stage 1 (m3) Stage 2 (m3) Storage 1 (m3) Profit ($)
/ 1698 / 616 275
are satisfied. The increase in the objective function is only 1.2%. The optimal solution shows a new unit at stage 4, which is smaller than the unit added with the previous approach and an intermediate storage tank before this stage.
/ 855 6440 624 760
6.3. Example 3 Fletcher et al. (1991) and Yoo et al. (1999) have solved this example. Table 5 shows the modified problem data from the previous works. Table 6 com-
J.M. Montagna / Computers and Chemical Engineering 27 (2003) 1277 /1290 Table 7 Example 4 data added to Example 4 from Yoo et al. (1999) STij
KTj
Product A Position 1
Table 10 Results Example 5 ctj
Product B
4
2
0
100
Qi/1000 (kg) New units Stage 1 (m3) Stage 2 (m3)
Table 8 Results corresponding to Example 4
Yoo et al. (1999)
This approach
Product A
Product B
Product A
Product B
2000
4000
2000
4000
1000 (u2, u3) / (u2)
Storage 1 (m3) Yoo et al. (1999) and this approach
Qi/1000 (kg) New units Stage 1 (m3)
Product A
Product B
2000
4000
2000 (u1, u2, u3) 1500 /2 (u1, u3) /(u2, u4) / 5 300 000
Stage 2 (m3) Storage 1 (m3) Profit ($)
1285
Sold units Stage 1 (m3) Stage 2 (m3) Profit ($)
u1 u1, u3 752 000
(u2) /(u3) (u2)
/ (u2) / (u2) 4800
(u2) (u2)
u1 u1, u3 758 200
(u1, u3) /(u2) (u1, u2, u3, u4)
Table 9 Example 5 data added to Example 5 from Yoo et al. (1999) STij Product A Position 1
4
KTj
ctj
1000
1
Product B 1
pares the results between Yoo et al. (1999) and this approach. The solution without intermediate storage tanks has a parallel unit at stage 2 operating in-phase for products B and C, and out-of-phase for products A and D. All product demands are satisfied. The cost of the new equipment is $32 000. Considering intermediate storage tanks, the objective function improves by 1.4%. There is a new unit in parallel at stage 2, operating in-phase for all products and an intermediate storage tank is located between both stages. The cost reduction in equipment is $23 500, 27% lower. Fig. 2. The optimal structure for Example 5 by the Yoo et al. (1999) formulation.
6.4. Example 4 Table 7 presents the data added to Example 4 from Yoo et al. (1999). Table 8 shows the same solution for both approaches. The optimal structure is obtained by adding one unit at stage 1 and two units at stage 2. As shown in Table 8, different arrangements of the units are proposed for each product. In this table, units between parenthesis are included in the same group. The symbol u1 refers to unit 1, and in the same way for the others units. For product A, the three units at stage 1 are
grouped, while, for product B, two groups are generated, one with units 1 and 3 and the other with unit 2. In the same way, two groups are held at stage 2 for product A, and only one group with four units for product B. 6.5. Example 5 The last example presented by Yoo et al. (1999) is solved. The input data added to the previous model and results are listed in Tables 9 and 10, respectively. In this
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two assessed alternatives. All the times were obtained with a Intel Celeron 650 MHz processor. In the first option, the constraints presented until now were included. In the last option, the following new constraints were added to reduce the number of options to be considered in the resolution. If a unit is incorporated for one product, it is available for the other products without increasing the cost of the solution. Therefore, the following constraint is posed: T
Nj Gj X X
T
yi j;k;g
g 1 k 1
Nj Gj X X
yi j;k;g
(41)
g 1 k 1
Ö i ; i 1; . . . ; P; i "i ; j 1; . . . ; M Fig. 3. The optimal structure for Example 5 by the proposed formulation.
example and in both formulations, old units are sold. Figs. 2 and 3 show the final plant structure for both formulations, where gray units correspond to added equipment to the original structure. Each unit includes its number and its capacity. Fig. 2 shows different configurations for both products: for product A units 2 and 3 operate in-phase, and, for product B, they operate out-of-phase. Unit 3 at stage 1 is added in the retrofit of the plant. Overlapped units correspond to in-phase operation. Fig. 3 corresponds to the solution of the proposed formulation, where a new intermediate storage tank is added between stages 1 and 2. The same structure for both products has been obtained. Though both solutions have a small difference in the optimal objective function values (that depends on the cost coefficients of the added tanks), the found structures look different.
The following constraint determines that if unit k at stage j exists, it must be used at least in one group for one product: yjk 5
Gj P X X
yijkg
(42)
i 1 g 1
Ö j 1; . . . ; M; k 1; . . . ;
NTj
If unit j is allocated at stage k, it can be included in only one group: Gj X
yijkg 5 yjk
(43)
g 1
Ö i 1; . . . ; P; j 1; . . . ; M; k 1; . . . ; NTj Groups must be generated following an order: yij;g1 5yijk Ö i 1; . . . ; P; j 1; . . . ; M; g 1; . . . ; Gj
(44)
For each product at stage j, one unit at least must be allocated in one group: T
Nj X Gj X
7. Computational performance Table 11 shows different information about the resolution of the examples. The first column corresponds to the number of binary variables of the model and the following two columns present the number of total variables and constraints. All these values are considered before piecewise linearization of the objective function. Columns 4 and 5 present the elapsed time of
yijkg ] 1
Ö i 1; . . . ; P; j 1; . . . ; M (45)
k 1 g 1
Table 11 shows the reduction in the elapsed time after adding constraints Eqs. (41) /(45). In order to compare this approach with the previous work by Yoo et al. (1999), Table 12 is included. It shows information about the same examples of Table 11, but now solved using the previous formulation. All exam-
Table 11 Computational performance of this approach
Example Example Example Example Example
1 2 3 4 5
Number of binary variables
Total number of variables
Number of constraints
Option 1 CPU time (s)
Option 2 CPU time (s)
37 87 73 51 101
118 294 216 148 260
265 724 511 330 575
15 112 818 247 1331
3 60 21 97 144
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Table 12 Computational performance of Yoo et al. (1999) approach Number of binary variables Example Example Example Example Example
1 2 3 4 5
Total number of variables
36 84 72 50 100
109 253 205 141 253
Table 13 Data for Example 6 Stage 1
Stage 2
Stage 3
Stage 4
6.3822 6.7938
4.7393 6.4175
8.3353 6.4750
3.9443 5.4382
7.913 0.7891
2.0815 0.2871
5.2268 0.2744
4.9523 3.3951
STij A B VOLD (m3) j VLj (m3) 3 VU j (m )
2 1 4000 1000 4000
1 1 4000 1000 4000
1 1 3000 /2 1000 3000
3000 1000 3000
Case 1 Kj cj KTj ctj
6000 4.5 2000 2
14 000 12 2000 2
18 000 15 2000 2
4000 4
Case 2 Kj cj KTj ctj Product
60 000 45 20 000 20 pi ($/kg)
140 000 120 20 000 20 QUP (kg) i
180 000 150 20 000 20
40 000 40
A B
1.1 1.5
500 000 400 000
Product /Tij A B Sij A B
ples also included constraints Eqs. (41) /(45). Table 12 shows reduced times in respect to Table 11, which is a logical conclusion taking into account that in the first table more alternatives must be assessed.
8. Impact of unit costs Previous examples were selected because they were considered in the literature on this area. However, all of them have a common characteristic: incomes from product sales are greater than unit costs. As a consequence, products demands are fulfilled. Therefore, if in both approaches the total demand is covered, the difference in the objective function is only due to the reduction in the cost of the new units, a small percentage
Number of constraints 250 647 488 317 562
Option 2 CPU time (s) 3 44 4 12 33
of the total objective function. In a new example, the impact of storage tanks is considered in several scenarios. Table 13 presents the problem data of Example 6. The plant has four stages to produce two products. At all stages there is only one unit, except at stage 3 that has two units. We considered two cases for the cost of the equipment to be added. In the first one, the tanks and units costs are cheaper than in the second one. Table 14 presents results for the first case. Two units at stages 1 and 4 are added in the option without storage tanks. A storage tank between stages 2 and 3 and a new unit at stage 1 are allocated in the solution considering intermediate storage. The demands of both products are satisfied for both options. Allowing storage tanks improves equipment costs by 34%. However, in this case, as we are considering low units costs, the total objective function only improves by 1%. Figs. 4 and 5 show the optimal structure of the plant for both options. In the second case, units and storage tanks costs are more expensive than in case 1. Table 14 also shows the results for this case. In the solution without intermediate storage tanks, no unit has been added (Fig. 6). The cost of the new units is so expensive that no unit has been incorporated. Production of A (which has the lower benefit) is reduced and its demand is not fulfilled. When intermediate storage is allowed, all demands are satisfied (Fig. 7). We have one unit in parallel at stage 1 operating in different ways for each product: inphase for product A and out-of-phase for product B. An intermediate storage tank is allocated between stages 2 and 3. In this way, production capacity is increased at stage 1 for product A operating both units in-phase and reducing the limiting cycle time for product B. The storage tank uncouples the original process into two subprocesses reducing the batch sizes for stage stages 3 and 4 and allowing a higher productivity for both products. The added equipment cost is $227 900, that is justified by the increase of $294 800 in the value of production A. Allocating a tank between stages 2 and 3 improves the total benefits by 8%. In this particular example, allowance of intermediate storage tanks had a really important impact.
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Table 14 Results for Example 6 Option 1 Solution without intermediate storage tanks Product A Qi (kg) 500 000 Profit ($) 1 115 100 Stage 1 New units (m3) 3838 Solution considering intermediate storage tanks Product A Qi (kg) 500 000 Profit ($) 1 127 200 Stage 1 New units (m3) 2327 Position 1 Storage (m3) /
Option 2
B 400 000 2 /
3 /
A 232 000 854 300 1 /
4 1905
B 400 000 2 / 2 2161
3 / 3 /
9. Conclusions A new model is presented for the retrofit of multiproduct batch plants. The main difference between this model and the previous ones is that it admits intermediate storage tanks. Although this option is usually taken into account in the design problem, it is not used in the retrofit problem. We consider the possibility of having storage tanks in the old plant, which can be used in the new one or not. A new storage tank can be
A 500 000 922 100 1 2327 1 /
4 /
B 400 000 2 /
3 /
4 /
B 40 000 2 / 2 2161
3 / 3 /
4 /
allocated in each position. The model also considers the capacity of having parallel units in-phase and out-ofphase as in the previous approaches. The availability of new alternatives allows meeting product demands better than previous methods, as is shown in the last considered example. The proposed model improved the previous solution obtained by the other authors in the analyzed examples. DICOPT performance was good to reach the solution for all solved examples. /
/
Fig. 4. Option 1 of Example 6. Solution without intermediate storage tanks.
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Fig. 5. Option 1 of Example 6. Solution with intermediate storage tanks.
Fig. 6. Option 2 of Example 6. Solution without intermediate storage tanks.
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Fig. 7. Option 2 of Example 6. Solution with intermediate storage tanks.
Acknowledgements The author would like to acknowledge financial support received from Foundation VITAE within the Cooperation Program among Argentina /Brazil /Chile under the grant Project B-11487/10B006, and from CONICET under grant PIP 4802.
References Duran, M. A., & Grossmann, I. E. (1986). An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Mathematical Progress 36 , 307. Fletcher, R., Hall, J. A., & Johns, W. R. (1991). Flexible retrofit design of multiproduct batch plants. Computers and Chemical Engineering 12 , 843. Grossmann, I. E., & Sargent, R. W. H. (1979). Optimal design of multipurpose chemical plants. Industrial Engineering and Chemical Process Design and Development 18 , 343. Knopf, F. C., Okos, M. R., & Reklaitis, G. V. (1982). Optimal design of batch/semicontinuous processes. Industrial Engineering and Chemical Process Design and Development 21 , 79.
Modi, A. K., & Karimi, I. A. (1989). Design of multiproduct batch processes with finite intermediate storage. Computers and Chemical Engineering 13 , 127. Montagna, J. M., & Vecchietti, A. R. (1998). Alternatives in the optimal allocation of intermediate storage tank in multiproduct batch plants. Computers and Chemical Engineering 22 , S801 /S804. Ravemark, D. (1995). Optimization models for design and operation of chemical batch processes. Ph.D. thesis, Swiss Federal Institute of Technology, Zurich. Van den Heever, S. A., & Grossmann, I. E. (1999). Disjunctive multiperiod optimization methods for design and planning of chemical process systems. Computers and Chemical Engineering 23 , 1075. Vaselenak, J. A., Grossmann, I. E., & Westerberg, A. W. (1987). Optimal retrofit design of multipurpose batch plants. Industrial Engineering and Chemical Research 26 , 718. Viswanathan, J. V., & Grossmann, I. E. (1990). A combined penalty function and outer approximation method for MINLP optimization. Computers and Chemical Engineering 14 , 769. Yoo, D. J., Lee, H., Ryu, J., & Lee, I. (1999). Generalized retrofit design of multiproduct batch plants. Computers and Chemical Engineering 23 , 683.
The optimal retrofit of multiproduct batch plants Jorge Marcelo Montagna Facultad Regional Santa Fe, INGAR, Instituto de Desarrollo y Disen˜o, Universidad Tecnolo´gica Nacional, Avellaneda 3657, 3000 Santa Fe, Argentina Received 6 February 2003; accepted 7 February 2003
Abstract This paper presents new alternatives in the retrofit model of multiproduct batch plants. Besides the duplication of the batch units implemented in previous works, the model considers the inclusion of intermediate storage tanks. These tanks can be plainly added or replace existing units that can be sold. The allocation of intermediate storage tanks is not considered in the previous retrofit works, even if this alternative is not new in the design of multiproduct batch plants. This option allows getting more efficient and real world solutions, although it requires working with a more complex model. # 2003 Elsevier Science Ltd. All rights reserved. Keywords: Retrofit; Multiproduct batch plants; Storage tanks
1. Introduction The design of multiproduct batch plants is an optimization problem that has been studied by several authors in the past years (Grossmann & Sargent, 1979; Knopf, Okos & Reklaitis, 1982; Modi & Karimi, 1989; Ravemark, 1995; Montagna & Vecchietti, 1998 etc.). The goal is to determine the size and number of batch units so that they can meet production requirements in the provided time horizon. Products produced by this kind of plant fall into the category of chemical specialties, drugs, pharmaceutical specialties, etc. Those are high value products with a short life cycle. Research on new products is intensive and it is frequent that new products replace the old ones. As new processes are to be implemented, the batch plant structure is changed in order to meet the new production requirements. The retrofit problem is an optimization problem whose objective is to obtain the optimal new structure for a batch plant starting from the old one, so as to maximize the benefits subject to a new demand pattern. In general, the retrofit problem solution comes up with a new plant configuration where useless units are sold and new ones are allocated and adjusted to work with old ones to configure the new process.
E-mail address: [email protected] (J.M. Montagna).
Multiproduct batch plants manufacture a set of products using the same equipment operating in the same sequence. Since products differ from one another, each unit is shared by all products but they do not use their total capacity for all of them. Each unit works at full capacity only when processing the products for which this stage is size-constraining. By the same token, the stages do not work all the time for all products. A stage works all the time (charging a new batch just after finishing with the previous one) only when processing the products for which this stage is a time-bottleneck, defining the cycle time for that product. The idea is to optimize the production rate for each product in the plant, which is a function of the batch size and the cycle time. The unit with minimum capacity limits the batch size while the limiting cycle time is fixed by the stage with the longest processing time. In order to reduce the investment cost, several alternatives are possible (Ravemark, 1995). The first one is the introduction of parallel units out-of-phase. In this case the cycle time is reduced if the unit has the longest operating time. Another option is to add a parallel unit in-phase to increase the operating capacity of the stage. This is a useful alternative for stages that are limiting the batch size. Finally the allocation of intermediate storage tanks allows increasing the equipment utilization (Modi & Karimi, 1989). Two objectives are reached: the reduction of idle time by increasing the
0098-1354/03/$ - see front matter # 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0098-1354(03)00052-8
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Nomenclature Bij bij dij Fij F1ij Gj H M NOLD j NTj NTj P pi Pri pri Qi qi QU i Sij STij tij Tij TLi Vijkg Vjk VTjk yijkg yijg yjk ytjk yttj Zj Subscripts G I J K Superscripts H L U Greek letters gij F f
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batch size of product i at stage j (kg) transformed variable for the batch size of product i at stage j parameter in the expression of Tij constant for a big-M constraint constant for a big-M constraint maximum number of groups allowed at stage j time horizon (h) number of batch stages in the plant number of existing units in the plant at stage j total number of units at stage j considering old and new ones total number of existing intermediate storage tanks in position j number of products in the multiproduct batch plant net profit per unit of product i (USA dollars) production rate of product i (kg/h) transformed variable for production rate of product i production of product i (kg) transformed variable for production of product i production demand of product i (kg) size factor for product i for a batch unit at stage j (m3/kg) size factor for product i for an intermediate storage tank in position j (m3/kg) parameter in the expression of Tij operation time for product i at stage j [h] limiting cycle time for product i [h] volume of unit k at stage j, taking part in group g, for product i (m3) volume of unit k at stage j (m3) volume of the intermediate storage tank k in position j (m3) binary variable for the inclusion of unit k at stage j for product i in group g binary variable for the existence of group g at stage j for product i binary variable for the use of unit k at stage j binary variable for the use of intermediate storage tank k in position j binary variable for the existence of some storage tank in position j maximum number of batch units to be added at stage j group product batch stage and position to allocate an intermediate storage tank batch unit or intermediate storage tank subprocess lower bound upper bound parameter in the expression of Tij maximum ratio between two consecutive batch sizes log F
working time of the stages, and the increased equipment utilization. When an intermediate storage tank is allocated, the original process is divided into two subprocesses, each one presenting its own batch size and limiting cycle time. Productivity of both subprocesses must be the same to avoid accumulation in the storage tank.
The first retrofit paper (Vaselenak, Grossmann & Westerberg, 1987) considered the addition of batch units to an existing plant to face an increase in the products demands. The model was a mixed integer nonlinear program (MINLP), whose objective was to maximize the benefit considering the incomes from the sales of the products minus the cost of the new units. Each product
J.M. Montagna / Computers and Chemical Engineering 27 (2003) 1277 /1290
has an upper bound in the demand based on the sales projection. The model of Vaselenak et al. (1987) assumed that the units in parallel either in-phase or out-of-phase must be used in the same way for every product. Fletcher, Hall and Johns (1991) modified the Vaselenack’s approach by allowing the new units to operate in different ways for each product. Better solutions were obtained in this last case. Yoo, Lee, Ryu and Lee (1999) presented the most updated complete retrofit model for multiproduct batch plants. No condition was imposed on the use of the batch units. The new and the old units can be used inphase and out-of-phase, forming groups that can be different for each product. The model also contemplates that the old units that are no longer used in the new structure can be sold. Van den Heever and Grossmann (1999) presented a general disjunctive model for the multiperiod retrofit problem. They used logic and disjunctive programming to pose all considered alternatives, obtaining a great performance for large problems. However, the available options for each period in the model are reduced because they worked with the problem formulation by Fletcher et al. (1991). No previous work has considered intermediate storage tanks for the retrofit problem. In this work, an extension of the explicit model by Yoo et al. (1999) is presented adding a new degree of difficulty by including the allocation of intermediate storage tanks. There is no doubt that the number of feasible alternatives will increase and, therefore, the possibility of obtaining better solutions exists. The previous assumptions were maintained? Especially the capability for changing the configuration of every batch unit for each product considered in the plant and the possibility of selling the useless old units. Several examples were solved with this model in order to show its effectiveness.
/1, . . ., NTj are The sizes of new units Vjk, k /NOLD j obtained with the problem solution. The NTj units of stage j can be grouped in different ways for each product i. According to Yoo et al. (1999), we can have groups where all units in the group operate in parallel and in-phase. The different groups at the stage operate in parallel and out-of-phase. The allocation of an intermediate storage tank decouples the process into two subprocesses. Each one operates with a different batch size and cycle time, but maintaining the same production rate for product i, Pri, to avoid accumulation of material in the tank. Considering intermediate storage tanks requires working with new variables. Now the batch sizes or the limiting cycle times in each subprocess could be different if a tank is allocated. There are M /1 possible locations for the storage tank (j /1, M /1), where the jth location is between batch stages j and j/1. In this model we assume that in some locations we can have storage tanks of different sizes from previous retrofits. At location j there are NTj storage tanks of dimension VTjk with k /1, 2, . . ., NTj. Only one tank can be added in position j with size VTj,NTj1. Unlike batch stages, intermediate storage tanks can not be grouped in different ways. Therefore, the allocation of several tanks is not a feasible option, except when an upper bound over its size is reached. This last alternative can be easily added to the model, and it is not included here to simplify this presentation. /
3. Problem formulation The objective of the problem is to maximize the annual benefit of the plant considering incomes from the product sales plus useless unit sales minus new units costs: P X
2. Problem definition In a multiproduct batch plant, P products (i /1, 2, . . ., P) are processed. For each product demand QU i is known, which is an upper bound on the amount to be produced of product i. This demand must be produced over time horizon H. Since this is a multiproduct batch plant, every product follows the same production sequence over M batch stages (j /1, 2, . . ., M) of the plant. At each stage we have a set of NOLD batch units. The size of unit k at j stage j is Vjk (k /1, 2, . . ., NOLD ). Taking into account j that these units could have been added in previous retrofits, their sizes can be different. Zj units can be added at stage j so that NTj /NOLD / j Zj is the maximum possible number of units at stage j.
1279
OLD
pi Qi
M N j X X
Rjk (1 yjk )
j 1 k 1
i 1
NTj X
RTjk (1 ytjk )
j 1 M X
NJT X
r
(Kj yjk cj Vjkj )
j 1 k NOLD 1 j
k 1
M 1 X
M 1 X
rt
j (KTj ytj;NTj1 ctj VTj;NT ) j1
(1)
j 1
The first term of the objective function is the annual benefit corresponding to the product sales. Variable Qi corresponds to total production of product i. Parameter pi is the total benefit per unit of product i. The second and third terms correspond to the incomes from useless batch units and storage tanks, respectively. Binary variable yjk indicates if unit k at stage j is included in the new plant structure (yjk /1) or
J.M. Montagna / Computers and Chemical Engineering 27 (2003) 1277 /1290
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not (yjk /0). For this last case, the unit is sold at price Rjk. In a similar way we proceed with the useless storage tank using binary variable ytjk and sale price RTjk. The fourth and fifth terms correspond to the investment cost of the new batch units and storage tanks, respectively. Kj, cj and rj are cost parameters for the batch units. Kj corresponds to a fixed cost independent of the unit size, while cj is proportional to the unit size. Previous papers work with rj /1. The fifth term is for storage tanks like the fourth is for batch units, with cost parameters KTj, ctj and rtj. The difference is that only one storage tank can be incorporated at location j. If a storage tank exists, the number and size of the batches up and downstream of the tank are different for each product. Variable Bij is introduced corresponding to the batch size of product i at stage j. The number of batches for product i at stage j is Nij. In practice, this variable must be integer. However, considering Nij as a continuous variable, we do not introduce a significant error for long production horizons. Production at each stage limits the total production for each product, then the following equation is applied: Qi 5Nij Bij
i 1; . . . ; P; j 1; . . . ; M
(2)
In this way, the problem tries to maximize Qi to increase the benefits and to reduce Bij to decrease the new unit sizes. The following constraints are applied, relating batch sizes of consecutive stages (Ravemark, 1995):
B 1 1 yttj 5 ij 51(F 1)yttj F Bi;j1 1
(3)
i 1; . . . ; P; j 1; . . . ; M 1 where F is a parameter corresponding to the maximum difference allowed between two consecutive batch sizes. Binary variable yttj is used to define if a storage tank is located at position j (yttj /1) or not (yttj /0). The following constraints are applied for this situation: yttj 5
NT j 1 X
ytjk
j 1; . . . ; M 1
(4)
k 1
yttj ] ytjk j 1; . . . ; M 1; k 1; . . . ; NTj 1
(5)
If no intermediate storage tank exists, yttj is fixed at 0. If at least one tank is located yttj is one. It is important to note that from Eqs. (4) and (5) and using an upper bound equal to 1, yttj can be 0 or 1 without considering it as a binary variable. If no intermediate storage tank exists between j and j/1 then Bij is forced to be equal to Bij1. Otherwise, they are different and must satisfy the following constraint:
Fig. 1. Conforming groups at a stage j.
1 B 5 ij 5 F F Bi;j1
i 1; . . . ; P; j 1; . . . ; M 1
(6)
Several binary variables are introduced to determine the plant structure. Since the units can be grouped in different forms at each stage we use binary variable yijg. The value of this variable is 1 if group g is generated for product i at stage j; otherwise, the value is zero. Group g is generated if at least one unit is assigned to it. Binary variable yijkg is equal to one if unit k of stage j is assigned to group g for product i, otherwise the variable is equal to zero (Yoo et al., 1999). Fig. 1 illustrates how units can be arranged to conform the groups. In this example, there are four units (NTj ) in a stage: two old units, the white ones (NOLD ), and two new units, the gray ones (Zj). In this j way, up to four groups of one unit could be conformed. There are several possible combinations of the units to determine groups. In this example units have been arranged to conform only two groups. Therefore, variables yij1 and yij2 are equal to one, and yij3 and yij4 are equal to zero. Units 1 and 3 form group 1 and operate in-phase. Units 2 and 4 conform group 2 and operate in-phase too. Then variables yij11, yij31, yij22 and yij42 are equal to one and all the other variables yijgk are equal to zero. Both groups 1 and 2 operate out-of-phase. Each unit k at stage j can be assigned at most to one group for product i: Gj X
yijkg 5 1
(7)
g 1
i 1; . . . ; P; j 1; . . . ; M; k 1; . . . ;
NTj
Gj is the maximum number of groups allowed at stage j. Group g exists at stage j only if at least one unit is assigned to the group:
J.M. Montagna / Computers and Chemical Engineering 27 (2003) 1277 /1290 g
T
yijg 5
Nj X
(tij dij Bijij )Pri
yijkg
(8)
k 1
Bij
5
Gj X
1281
yijg
(15)
g 1
i 1; . . . ; P; j 1; . . . ; M; g 1; . . . ; Gj
i 1; . . . ; P; j 1; . . . ; M
If unit k is assigned to the group, the group must exist:
The plant production is limited to time horizon H, then
yijkg 5 yijg
P X Qi
i 1; . . . ; P; j 1; . . . ; M; g 1; . . . ; Gj ; k
(9)
1; . . . ; NTj and the upper bound for binary variable yijg is: yijg 5 1
i 1; P; j 1; M; g 1; Gj
(10)
From Eqs. (8) /(10), it can be seen that binary variable yijg can be converted to a continuous variable. If unit k is assigned to group g at stage j for product i, the unit must exist: yijkg 5 yjk i 1; . . . ; P; j 1; . . . ; M; g 1; . . . ; Gj ; k 1; . . . ;
(11)
NTj
The maximum time between two consecutive batches in subprocess h must be considered to determine the limiting cycle time for h. This maximum time is given by the division between operation time Tij and the number of groups out-of-phase for product i, considering all the stages included in subprocess h: TLhi ]
Tij Gj X yijg
(12)
g 1
i 1; . . . ; P; Öh; Öj Subprocess h where Tij is a function of the batch size, with tij,, dij and gij fixed parameters: g
Tij tij dij Bijij
i 1; . . . ; P; j 1; . . . ; M
(13)
We do not know a priori which units conform subprocess h. This is a result of the mathematical program and depends on the location of the storage tank. The expression Eq. (12) can be modified with the assumption that in every subprocess, production rate for product i, Pri, must be the same: Pri
Bhi TLhi
i 1; . . . ; P; Öh
(14)
Considering expression Eq. (3) where the value of the batch size for subprocess h (Bhi ) can be adjusted, using Eq. (13) and replacing TLhi from Eq. (14) into Eq. (12), we have:
i 1
Pri
(16)
5H
Batch size Bij is determined by the lowest capacity between the groups for product i at stage j, then the following constraint can be applied to get the Bij value, with Sij the size factor for product i at stage j: T
Sij Bij 5
Nj X
Vjk yijkg (1 yijg )Fij
(17)
k 1
i 1; . . . ; P; j 1; . . . ; M; g 1; . . . ; Gj Constraint Eq. (17) is a Big-M type that guarantees that batches can be processed if group g exists, otherwise the constraint is redundant because of the large value of Fij. The value of Fij can be calculated by (Yoo et al., 1999): Fij
OLD N j X
Vjk Zj VU j
(18)
k 1
i 1; . . . ; P; j 1; . . . ; M where VU j is the upper bound of the batch units to be added at stage j. Constraint Eq. (17) is nonlinear in the binary variable yijkg, which adds difficulty to the convergence. To overcome this problem Yoo et al. (1999) replaced the product Vjkyijkg by the new continuous variable Vijkg, then Eq. (17) is replaced by the following constraints set: T
Sij Bij 5
Nj X
Vijkg (1 yijg )
k 1
OLD NX j
Vjk Zj VU j
k 1
(19)
i 1; . . . ; M; g 1; . . . ; Gj Vijkg 5Vjk i 1; . . . ; P; j 1; . . . ; M; k 1; . . . ; NTj ; j
(20)
1; . . . ; Gj Vijkg 5VU j yijkg i 1; . . . ; P; j 1; . . . ; M; k 1; . . . ; NTj ; j
(21)
1; . . . ; Gj OLD . where VU j is equal to Vjk for k /1, . . ., Nj If the unit is selected, then the following constraints are used to guarantee that the unit size is between its bounds:
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Vjk 5VU j yjk j 1; . . . ; M; k NOLD 1; . . . ; NTj j Vjk ]VLj yjk j 1; . . . ; M; k NOLD 1; . . . ; NTj j VU j
(22)
(23)
VLj
where and are the upper and lower bounds for the new units at stage j. We use the following expression adapted from Modi and Karimi (1989) to determine the size of the intermediate storage tank: NT j 1 X
VTjk ]STij (Bij Bi;j1 )F1ij (yttj 1)
k 1
(24)
i 1; . . . ; P; j 1; . . . ; M 1 This constraint also belongs to a Big-M type and it takes into account all intermediate storage tanks located at that position. STij are the size factors for the tanks for product i at stage j. The value of F1ij can be calculated by: F1ij STij
1
OLD NX j
Sij
k 1
1 Vjk Zj VU j Si;j1
each unit k. The order of the group is obtained by adding the weights of all units in the group. Finally, we must add bound constraints for the demand. The lower bound on the demand for product i is set when the production is already sold. Qi 5QU i
i 1; . . . ; P
(29)
Qi ]QLi
i 1; . . . ; P
(30)
4. Problem convexification The model for the retrofit of a multiproduct batch plant is defined by maximizing Eq. (1) subject to constraints Eqs. (2)/(5), (7) /(11), (15), (16), (19) /(24), (26) /(30). This is a MINLP problem that presents non convex terms in several constraints and in the objective function. Thus, it is not possible to assure that the global optimum can be reached. In a similar way as Vaselenak et al. (1987), the following transformations are introduced to avoid non-convex terms: bij ln Bij
i 1; . . . ; P; j 1; . . . ; M
nij ln Nij
i 1; . . . ; P; j 1; . . . ; M
i 1; . . . ; P; j 1; . . . ; M 1
pri ln Pri
i 1; . . . ; P
We must also consider the bounds on the tanks to be added by the following constraints:
qi ln Qi
VTjk 5VTU jk ytjk
Using these transformations, the objective function results:
OLD NX j1
Vj1;k Zj1 VU j1
(25)
k 1
j 1; . . . ; M 1; k NTj 1
(26)
Min
VTjk ]VTLjk ytjk j 1; . . . ; M 1; k NTj 1
T
Nj X k 1
T
2
NTj k
yijkg ]
Nj X
T
2Nj k yijk;g1
k 1
(28)
i 1; . . . ; P; j 1; . . . ; M; g 1; . . . ; Gj This constraint order the different groups. For example, if there are four units at a stage, one solution is to conform group 1 by units 1 and 2 and group 2 by units 3 and 4. However, the same solution is attained assigning units 3 and 4 to group 1 and units 1 and 2 to group 2. This constraint avoids these assignments by T ordering the group through a weight 2Nj k assigned to
pi exp(qi )
NTj X
RTjk (1 ytjk )
k 1
M N j X X
Rjk (1 yjk )
j 1 k 1
i 1
(31)
OLD
P X
(27)
L where VTU jk and VTjk are the upper and lower bounds, respectively, that must be considered if the tanks exist. Redundant assignation to a group with the same value for the objective function are avoided by the following constraint (Yoo et al., 1999):
i 1; . . . ; P
M X
M 1 X j 1
NTj X
(Kj yjk cj Vjk )
j 1 k NOLD 1 j
M 1 X
(KTj ytj;NTj1 ctj VTj;NTj1 )
(32)
j 1
where we have considered the minimization of Eq. (1) with a change of sign. Also, constraints Eqs. (2), (3), (15), (16), (19), (24), (29) and (30) are modified to obtain the final formulation: i 1; . . . ; P; j 1; . . . ; M qi 5 nij bij bij bi;j1 5fyttj i 1; . . . ; P; j 1; . . . ; M 1 bij bi;j1 ] fyttj i 1; . . . ; P; j 1; . . . ; M 1 where f /ln F.
(33) (34) (35)
J.M. Montagna / Computers and Chemical Engineering 27 (2003) 1277 /1290
tij exp(pri bij )dij exp[(gij 1)bij pri ]5
Gj X g 1
i 1; . . . ; P; j 1; . . . ; M P X exp(qi pri )5H
yijg
(36)
i 1; . . . ; P i 1; . . . ; P i 1; . . . ; P; j 1; . . . ; M
KTj
Product A
(37) (38) (39) (40)
Following Vaselenak et al. (1987), constraint Eq. (40) is added to reduce the number of nonlinear constraints. Working with bij, constraints Eqs. (19) and (24) should be non-linear. Restriction Eq. (40) is satisfied trivially at the optimal solution because by minimizing Eq. (32), qi assume the greatest possible value that is limited by Bij. However, this model presents difficulties. The first term in the objective function is concave. To overcome this problem, Vaselenak et al. (1987) have proved that the linearization of the negative exponential functions in the first term of Eq. (32) can be approximated by a system of piecewise linear underestimators. This approximation overestimates the objective function so that it can be employed to find the global solution of this model.
5. Model resolution The final model minimizes Eq. (32) subject to constraints Eqs. (33) /(40), (4), (5), (7)/(11), (17), (19) /(24), (26) /(28). This MINLP problem is solved using the algorithm of Duran and Grossmann (1986), afterwards completed by Viswanathan and Grossmann (1990) with the OA/ER/AP algorithm and implemented in DICOPT. /
Table 1 Example 1 data added to Yoo et al. (1999) STij
i 1
qi 5ln QU i qi ]ln QLi exp(bij )5Bij
1283
/
6. Examples All the examples from previous papers have been solved with this model. In all cases the inclusion of the intermediate storage tanks depends on the cost and size factors of those units. Here appropriate values have been selected values to show the potential applications of this approach. The main objective of this section is to show the possibility of obtaining better solutions, taking into account that the number of feasible solutions has been increased. However, it is difficult to compare the solution obtained with previous approaches since the optimal solution will greatly depend on the values chosen for the cost coefficients.
Position 1
ctj
Product B
1
1
10 000
10
Table 2 Results corresponding to Example 1 Yoo et al. (1999)
This approach
Product A
Product B
Product A
Product B
Qi (kg)
1 200 000
1 000 000
1 200 000
1 000 000
New units Stage 1 (m3) Stage 2 (m3) Storage 1 (m3) Profit ($)
1358 / / 3 125 236
/ / 3333 3 156 667
6.1. Example 1 This example has been solved by Vaselenak et al. (1987), Fletcher et al. (1991) and Yoo et al. (1999). Table 1 presents the example data added to Example 1 of Yoo et al. (1999). Table 2 shows the results obtained using the model by Yoo et al. (1999) without intermediate storage tanks and the results with the model presented in this paper including tanks. In the first case, one unit is added at stage 1 operating in-phase for product A and out-of-phase for product B. In the last case, no parallel units are in the plant but one intermediate storage tank is located between both stages. The objective function is reduced by 1.0%. This reduction depends on the costs proposed for the units. The whole demand is produced for both products. Cost reduction comes from the difference in cost between parallel unit 1 ($74 800) of the first approach against the cost of the intermediate storage tank ($43 300) with the new approach. Note that the improvement in the objective function strongly depends on the relationship between products incomes and units costs. 6.2. Example 2 Vaselenak et al. (1987), Fletcher et al. (1991) and Yoo et al. (1999) have solved this example. Table 3 presents the data about the intermediate storage tank to be added to the original data. Table 4 shows the results of both approaches. In the stages with two units, it shows the operating policy: in-phase or out-of-phase. In this problem, the production targets are not achieved in the optimal solution with the previous approach (product D). However, with the new model all the demands
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Table 3 Example 2 data added to previous works STij
Position 1 Position 2 Position 3
Product A
Product B
Product C
Product D
7.913 2.0815 5.2268
0.7891 0.2871 0.2744
0.7122 2.5889 1.6425
4.6730 2.3586 1.6087
KTj
ctj
10 10 10
1 1 1
Table 4 Results corresponding to Example 2 Yoo et al. (1999)
Qi/1000 (kg) New units Stage 1 (m3) Stage 2 (m3) Stage 3 (m3)
A
B
C
D
A
B
C
D
268.2
156.0
189.7
158.1
268.2
156.0
189.7
166.1
In
In
Out
In
In
In
/ / / Out 3000 In / 521 780
Stage 4 (m3) Storage 3 (m3) Profit ($)
This approach
In
Out
Out
In
Out
Out
/ / / Out 2252 Out 3667 528 000
Table 5 Example 3 data added to Example 3 from Yoo et al. (1999) STij
Position 1
Product A
Product B
Product C
Product D
2.4
1.8
1.95
4.15
KTj
ctj
22
0.27
Table 6 Results corresponding to Example 3 Yoo et al. (1999)
This approach
A
B
C
D
A
B
C
D
Qi/1000 (kg)
290
300
350
140
290
300
350
140
New units Stage 1 (m3) Stage 2 (m3) Storage 1 (m3) Profit ($)
/ 1698 / 616 275
are satisfied. The increase in the objective function is only 1.2%. The optimal solution shows a new unit at stage 4, which is smaller than the unit added with the previous approach and an intermediate storage tank before this stage.
/ 855 6440 624 760
6.3. Example 3 Fletcher et al. (1991) and Yoo et al. (1999) have solved this example. Table 5 shows the modified problem data from the previous works. Table 6 com-
J.M. Montagna / Computers and Chemical Engineering 27 (2003) 1277 /1290 Table 7 Example 4 data added to Example 4 from Yoo et al. (1999) STij
KTj
Product A Position 1
Table 10 Results Example 5 ctj
Product B
4
2
0
100
Qi/1000 (kg) New units Stage 1 (m3) Stage 2 (m3)
Table 8 Results corresponding to Example 4
Yoo et al. (1999)
This approach
Product A
Product B
Product A
Product B
2000
4000
2000
4000
1000 (u2, u3) / (u2)
Storage 1 (m3) Yoo et al. (1999) and this approach
Qi/1000 (kg) New units Stage 1 (m3)
Product A
Product B
2000
4000
2000 (u1, u2, u3) 1500 /2 (u1, u3) /(u2, u4) / 5 300 000
Stage 2 (m3) Storage 1 (m3) Profit ($)
1285
Sold units Stage 1 (m3) Stage 2 (m3) Profit ($)
u1 u1, u3 752 000
(u2) /(u3) (u2)
/ (u2) / (u2) 4800
(u2) (u2)
u1 u1, u3 758 200
(u1, u3) /(u2) (u1, u2, u3, u4)
Table 9 Example 5 data added to Example 5 from Yoo et al. (1999) STij Product A Position 1
4
KTj
ctj
1000
1
Product B 1
pares the results between Yoo et al. (1999) and this approach. The solution without intermediate storage tanks has a parallel unit at stage 2 operating in-phase for products B and C, and out-of-phase for products A and D. All product demands are satisfied. The cost of the new equipment is $32 000. Considering intermediate storage tanks, the objective function improves by 1.4%. There is a new unit in parallel at stage 2, operating in-phase for all products and an intermediate storage tank is located between both stages. The cost reduction in equipment is $23 500, 27% lower. Fig. 2. The optimal structure for Example 5 by the Yoo et al. (1999) formulation.
6.4. Example 4 Table 7 presents the data added to Example 4 from Yoo et al. (1999). Table 8 shows the same solution for both approaches. The optimal structure is obtained by adding one unit at stage 1 and two units at stage 2. As shown in Table 8, different arrangements of the units are proposed for each product. In this table, units between parenthesis are included in the same group. The symbol u1 refers to unit 1, and in the same way for the others units. For product A, the three units at stage 1 are
grouped, while, for product B, two groups are generated, one with units 1 and 3 and the other with unit 2. In the same way, two groups are held at stage 2 for product A, and only one group with four units for product B. 6.5. Example 5 The last example presented by Yoo et al. (1999) is solved. The input data added to the previous model and results are listed in Tables 9 and 10, respectively. In this
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two assessed alternatives. All the times were obtained with a Intel Celeron 650 MHz processor. In the first option, the constraints presented until now were included. In the last option, the following new constraints were added to reduce the number of options to be considered in the resolution. If a unit is incorporated for one product, it is available for the other products without increasing the cost of the solution. Therefore, the following constraint is posed: T
Nj Gj X X
T
yi j;k;g
g 1 k 1
Nj Gj X X
yi j;k;g
(41)
g 1 k 1
Ö i ; i 1; . . . ; P; i "i ; j 1; . . . ; M Fig. 3. The optimal structure for Example 5 by the proposed formulation.
example and in both formulations, old units are sold. Figs. 2 and 3 show the final plant structure for both formulations, where gray units correspond to added equipment to the original structure. Each unit includes its number and its capacity. Fig. 2 shows different configurations for both products: for product A units 2 and 3 operate in-phase, and, for product B, they operate out-of-phase. Unit 3 at stage 1 is added in the retrofit of the plant. Overlapped units correspond to in-phase operation. Fig. 3 corresponds to the solution of the proposed formulation, where a new intermediate storage tank is added between stages 1 and 2. The same structure for both products has been obtained. Though both solutions have a small difference in the optimal objective function values (that depends on the cost coefficients of the added tanks), the found structures look different.
The following constraint determines that if unit k at stage j exists, it must be used at least in one group for one product: yjk 5
Gj P X X
yijkg
(42)
i 1 g 1
Ö j 1; . . . ; M; k 1; . . . ;
NTj
If unit j is allocated at stage k, it can be included in only one group: Gj X
yijkg 5 yjk
(43)
g 1
Ö i 1; . . . ; P; j 1; . . . ; M; k 1; . . . ; NTj Groups must be generated following an order: yij;g1 5yijk Ö i 1; . . . ; P; j 1; . . . ; M; g 1; . . . ; Gj
(44)
For each product at stage j, one unit at least must be allocated in one group: T
Nj X Gj X
7. Computational performance Table 11 shows different information about the resolution of the examples. The first column corresponds to the number of binary variables of the model and the following two columns present the number of total variables and constraints. All these values are considered before piecewise linearization of the objective function. Columns 4 and 5 present the elapsed time of
yijkg ] 1
Ö i 1; . . . ; P; j 1; . . . ; M (45)
k 1 g 1
Table 11 shows the reduction in the elapsed time after adding constraints Eqs. (41) /(45). In order to compare this approach with the previous work by Yoo et al. (1999), Table 12 is included. It shows information about the same examples of Table 11, but now solved using the previous formulation. All exam-
Table 11 Computational performance of this approach
Example Example Example Example Example
1 2 3 4 5
Number of binary variables
Total number of variables
Number of constraints
Option 1 CPU time (s)
Option 2 CPU time (s)
37 87 73 51 101
118 294 216 148 260
265 724 511 330 575
15 112 818 247 1331
3 60 21 97 144
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Table 12 Computational performance of Yoo et al. (1999) approach Number of binary variables Example Example Example Example Example
1 2 3 4 5
Total number of variables
36 84 72 50 100
109 253 205 141 253
Table 13 Data for Example 6 Stage 1
Stage 2
Stage 3
Stage 4
6.3822 6.7938
4.7393 6.4175
8.3353 6.4750
3.9443 5.4382
7.913 0.7891
2.0815 0.2871
5.2268 0.2744
4.9523 3.3951
STij A B VOLD (m3) j VLj (m3) 3 VU j (m )
2 1 4000 1000 4000
1 1 4000 1000 4000
1 1 3000 /2 1000 3000
3000 1000 3000
Case 1 Kj cj KTj ctj
6000 4.5 2000 2
14 000 12 2000 2
18 000 15 2000 2
4000 4
Case 2 Kj cj KTj ctj Product
60 000 45 20 000 20 pi ($/kg)
140 000 120 20 000 20 QUP (kg) i
180 000 150 20 000 20
40 000 40
A B
1.1 1.5
500 000 400 000
Product /Tij A B Sij A B
ples also included constraints Eqs. (41) /(45). Table 12 shows reduced times in respect to Table 11, which is a logical conclusion taking into account that in the first table more alternatives must be assessed.
8. Impact of unit costs Previous examples were selected because they were considered in the literature on this area. However, all of them have a common characteristic: incomes from product sales are greater than unit costs. As a consequence, products demands are fulfilled. Therefore, if in both approaches the total demand is covered, the difference in the objective function is only due to the reduction in the cost of the new units, a small percentage
Number of constraints 250 647 488 317 562
Option 2 CPU time (s) 3 44 4 12 33
of the total objective function. In a new example, the impact of storage tanks is considered in several scenarios. Table 13 presents the problem data of Example 6. The plant has four stages to produce two products. At all stages there is only one unit, except at stage 3 that has two units. We considered two cases for the cost of the equipment to be added. In the first one, the tanks and units costs are cheaper than in the second one. Table 14 presents results for the first case. Two units at stages 1 and 4 are added in the option without storage tanks. A storage tank between stages 2 and 3 and a new unit at stage 1 are allocated in the solution considering intermediate storage. The demands of both products are satisfied for both options. Allowing storage tanks improves equipment costs by 34%. However, in this case, as we are considering low units costs, the total objective function only improves by 1%. Figs. 4 and 5 show the optimal structure of the plant for both options. In the second case, units and storage tanks costs are more expensive than in case 1. Table 14 also shows the results for this case. In the solution without intermediate storage tanks, no unit has been added (Fig. 6). The cost of the new units is so expensive that no unit has been incorporated. Production of A (which has the lower benefit) is reduced and its demand is not fulfilled. When intermediate storage is allowed, all demands are satisfied (Fig. 7). We have one unit in parallel at stage 1 operating in different ways for each product: inphase for product A and out-of-phase for product B. An intermediate storage tank is allocated between stages 2 and 3. In this way, production capacity is increased at stage 1 for product A operating both units in-phase and reducing the limiting cycle time for product B. The storage tank uncouples the original process into two subprocesses reducing the batch sizes for stage stages 3 and 4 and allowing a higher productivity for both products. The added equipment cost is $227 900, that is justified by the increase of $294 800 in the value of production A. Allocating a tank between stages 2 and 3 improves the total benefits by 8%. In this particular example, allowance of intermediate storage tanks had a really important impact.
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Table 14 Results for Example 6 Option 1 Solution without intermediate storage tanks Product A Qi (kg) 500 000 Profit ($) 1 115 100 Stage 1 New units (m3) 3838 Solution considering intermediate storage tanks Product A Qi (kg) 500 000 Profit ($) 1 127 200 Stage 1 New units (m3) 2327 Position 1 Storage (m3) /
Option 2
B 400 000 2 /
3 /
A 232 000 854 300 1 /
4 1905
B 400 000 2 / 2 2161
3 / 3 /
9. Conclusions A new model is presented for the retrofit of multiproduct batch plants. The main difference between this model and the previous ones is that it admits intermediate storage tanks. Although this option is usually taken into account in the design problem, it is not used in the retrofit problem. We consider the possibility of having storage tanks in the old plant, which can be used in the new one or not. A new storage tank can be
A 500 000 922 100 1 2327 1 /
4 /
B 400 000 2 /
3 /
4 /
B 40 000 2 / 2 2161
3 / 3 /
4 /
allocated in each position. The model also considers the capacity of having parallel units in-phase and out-ofphase as in the previous approaches. The availability of new alternatives allows meeting product demands better than previous methods, as is shown in the last considered example. The proposed model improved the previous solution obtained by the other authors in the analyzed examples. DICOPT performance was good to reach the solution for all solved examples. /
/
Fig. 4. Option 1 of Example 6. Solution without intermediate storage tanks.
J.M. Montagna / Computers and Chemical Engineering 27 (2003) 1277 /1290
Fig. 5. Option 1 of Example 6. Solution with intermediate storage tanks.
Fig. 6. Option 2 of Example 6. Solution without intermediate storage tanks.
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Fig. 7. Option 2 of Example 6. Solution with intermediate storage tanks.
Acknowledgements The author would like to acknowledge financial support received from Foundation VITAE within the Cooperation Program among Argentina /Brazil /Chile under the grant Project B-11487/10B006, and from CONICET under grant PIP 4802.
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Modi, A. K., & Karimi, I. A. (1989). Design of multiproduct batch processes with finite intermediate storage. Computers and Chemical Engineering 13 , 127. Montagna, J. M., & Vecchietti, A. R. (1998). Alternatives in the optimal allocation of intermediate storage tank in multiproduct batch plants. Computers and Chemical Engineering 22 , S801 /S804. Ravemark, D. (1995). Optimization models for design and operation of chemical batch processes. Ph.D. thesis, Swiss Federal Institute of Technology, Zurich. Van den Heever, S. A., & Grossmann, I. E. (1999). Disjunctive multiperiod optimization methods for design and planning of chemical process systems. Computers and Chemical Engineering 23 , 1075. Vaselenak, J. A., Grossmann, I. E., & Westerberg, A. W. (1987). Optimal retrofit design of multipurpose batch plants. Industrial Engineering and Chemical Research 26 , 718. Viswanathan, J. V., & Grossmann, I. E. (1990). A combined penalty function and outer approximation method for MINLP optimization. Computers and Chemical Engineering 14 , 769. Yoo, D. J., Lee, H., Ryu, J., & Lee, I. (1999). Generalized retrofit design of multiproduct batch plants. Computers and Chemical Engineering 23 , 683.