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A Decomposition Based Algorithm for the Design of Multipurpose Batch Facilities Using Economic Assessments
10th International Symposium on Process Systems Engineering - PSE2009 Rita Maria de Brito Alves, Claudio Augusto Oller do Nascimento and Evaristo Chalbaud Biscaia Jr. (Editors) © 2009 Elsevier B.V. All rights reserved.
429
A Decomposition Based Algorithm for the Design of Multipurpose Batch Facilities Using Economic Assessments Tânia Rute X. M. Pinto1,2, Ana Paula F. D. Barbosa-Povoa2 and Augusto Q. Novais1 1
DMS ± Departamento de Modelação e Simulação, INETI, Lisboa, Portugal Estrada do Paço do Lumiar, 1649-038 Lisboa, Portugal E-mail address: [email protected] 2 CEG- Centro de Estudos de Gestão, IST, Lisboa, Portugal
Abstract In this paper we propose a multi-objective decomposition algorithm for the design of multipurpose batch plants, which avoids the direct solution of the fully detailed MILP model. Most of such design problems involve the maximization of the total revenue, as well as the minimization of the total cost. The way to deal with these two terms simultaneously is either to combine them into a single criterion (e.g., profit), or to define the efficient frontier that offers the optimal solutions by multi-objective optimization. In this work the latter approach, while more elaborate, was adopted, since the exploration of this frontier enables the decision maker to evaluate different alternative solutions. A combination of the proposed decomposition algorithm and the -constraint method is employed, which supports the application of this approach to perform economic assessments. The proposed algorithm allows the identification of the plant topologies, scheduling, equipment design and storage policies, subject to WKH SODQW¶s cost minimization and revenue maximization. A comparative analysis is presented between the detailed model proposed by Pinto et al. (2008b) and the proposed algorithm. Keywords: Scheduling, Design, -constraint, Multipurpose batch facilities.
1. Introduction In multipurpose batch facilities a wide variety of products can be produced via different processing recipes, by sharing all available resources, such as equipment, raw material, intermediates and utilities. In order to ensure that any resource in the design can be utilized as efficiently as possible, an adequate representation is necessary in order to address such type of problems without creating ambiguities in the process/plant mathematical model. The Resource-Task Network (RTN) is one of the possible adequate representations to describe the design of multipurpose batch plants as suggested by Pinto et al. (2008a). Like most real-world problems, the design of multipurpose batch facilities involves multiple objectives, while most of the existing literature on the design problem has been focused on single objectives (Barbosa-Povoa 2007). Therefore, the multi-objective optimization of such problems requires the development of an adequate modelling approach, as a pre-requisite for the resulting models to be useful as decision making tools where trade-offs among objectives can be investigated. In order to guarantee
430
T.R.X.M. Pinto et al.
optimal solutions, most of the published mathematical design formulations consider a large number of potential equipment items, out of which a selection is made of those that will be incorporated into the optimal plant configuration. This factor and the diversity of SURGXFWV¶recipes, gives rise to large MILP problems and consequently to an increasing computational burden. For this reason, effective solution tools are still an open area of research in the design of batch plants. In this paper, we aim to overcome some of these computational difficulties and also to avoid the direct solution of the fully detailed design and scheduling MILP model. With that aim, we propose a decomposition algorithm that exploits the hierarchical structure of the problem: the original detailed design and scheduling model is decomposed into an upper-level (UL) and a lower-level (LL) models; the UL uses profit maximization as the objective function and selects the so-called complicated variables, which in the current case are the design variables responsible for equipment selection; the LL comprises both the equipment design and the scheduling problem. Applying the -constraint method (Chankong and Haimes, 1983), the procedure iterates until the Pareto-optimum surface is completed. This allows the identification of a range of plant topologies, design facilities and storage policies that minimize the total cost of the system while maximizing revenue, subject to product demands and operational restrictions.
2. Decomposition Algorithm To avoid the direct solution of the MILP model, we propose a decomposition algorithm (DA) that exploits the hierarchical structure of design/scheduling models, and employ it together with the constraint method. As referred to previously, the original detailed design and scheduling model is decomposed into one UL design problem and one LL design plus scheduling problem. The UL determines the necessary equipment to satisfy the demand along the entire time horizon, with the design constraints being omitted, leading to a relaxation of the original, DM, or detailed problem and therefore to an upper-bound on profit. In the LL, the DM is solved by fixing the equipment obtained from the UL. The LL corresponds to a sub-problem of the original MILP problem, in a reduced space and at iteration i. It considers the selected subset of equipment obtained from the UL, producing a lower-bound on profit. The procedure iterates until the difference between the upper-bound and the lower-bound is less than a specified tolerance. To expedite the search, integer cuts are added to the UL to exclude previous solutions. The DA is subject to an objective constraint function that defines the maximum cost available for the optimal solution. The initial restriction takes the value of the necessary cost to generate the maximum revenue (Cmax). This restriction is decreased by small values, , and successive optimization steps (L «Q) undertaken until the intended range is covered and the Pareto front defined. 2.1. Integer Cut If the upper-bound obtained from the UL and the lower-bound obtained from the LL do not lie within a pre-defined tolerance, it is necessary to obtain a new solution from the UL. Then an integer cut on the binary design variable,
UL r ,
where r characterizes the
equipment obtained from the UL model, is added to the UL problem so as to avoid the solution in the previous iteration i. For iteration s > k, this is defined as: r U1k
UL r
r U 0k
UL r
U1k
1
(1)
A Decomposition Based Algorithm for the Design of Multipurpose Batch Facilities Using Economic Assessments
Where
U 0k
r|
UL r
0 and U1k
r|
431
1 . The U 0k and U1k are obtained
UL r
from the optimal UL solution, in terms of the assignment variable in iteration k. 2.2. -constraint In order to obtain the Pareto front, we used the -constraint method combined with the decomposition algorithm. The -constraint requires a cost constraint which is activated in both models (UL and LL) and imposes limits on the objective functions. The detailed model, LL, uses the objective functions FO1, which defines the revenue, and FO2, which reflects the cost associated with both equipment and operational tasks. To obtain : each point in the Pareto front, the cost constraint is activated and decreased by
F 02 Cmax i
L «Q.
(2)
The final decomposition algorithm is illustrated in Figure 1. i=0;k=0
Add cost constraint e-constraint
k
No Display solution
Upper-level - Design Solve UL assign the binary variables and get an upper-bound (UB)
Add integer cut k=k+1
Fixing the binary variable for the equipment selected by the UL
Lower-level - Scheduling and Design Solve the design and scheduling for the LL to get a lower-bound (LB) for the subset of equipments defined by the UL
No
i=i+1
(UB-LB)/LB <
yes Display solution
Figure 1: Flow chart for the decomposition algorithm.
2.3. Algorithm Steps An overview of the proposed decomposition follows: 1. The indices k and i control, respectively, the inner loop for estimating individual points on the Pareto front and the outer loop for generating successive points. Initially these are set at k=0 and i=0, while the upper-bound at UB= , the lower-bound at LB=- and a value assumed for the optimality tolerance.
432
T.R.X.M. Pinto et al.
2. Within the inner loop the cost constraint is activated; the MILP aggregated model solved for the upper-bound; the following sets defined:
U 0k
r|
UL r
0 , U1k
r|
UL r
1 ; and
r
UL r
for
k
r U1 ;
3. Still within the inner loop, the LL is solved; an optimal solution obtained, which is a lower-bound of the problem that defines the equipment design, scheduling and storage policy for each point of the Pareto front. is tested. 4. The convergence criterion UB LB LB If it is satisfied the solution corresponding to the lower-bound is the optimal solution and the outer loop is entered, where i defines the next point on the Pareto front. 5. If this convergence criterion is not satisfied an integer cut given by r U1k
UL r
r U 0k
UL r
U1k
1 is added to the UL and the inner loop controlled by k is
again calculated.
3. Example results In this example, the design of a multipurpose batch plant for a non-periodic mode of operation is performed. A production of [0; 170] tons of products S5, [0; 166] tons of S9 and S10, [0; 270] tons of products S6 and [0; 143] tons of products S11 is defined. Three raw materials, S1, S2 and S7, are used over the horizon of 24 h. The materials S5 and S6 are both intermediate and final products. S3, S4 and S8 are intermediate material. S3 and S8 are unstable and S4 is storable in V4 if necessary (Pinto et al. 2008b). There are available six main reactors (R1 to R6) and nine dedicated vessels. In terms of equipment suitability, only reactors R1 and R2 may carry out the two processing tasks, T1 and T2, while each storage vessel and reactors R3, R4, R5 and R6 are dedicated to a single State/Task. T1 may process S1 during 2 hours in R1 or R2; T2 may processes S2 during 2 hours in R1 or R2; T3 may process 0.5 of S3 and S4 during 4 hours in R3; T4 process 0.5 of S3 and S4 during 2 hours in R4; T5 process S6 during 1 hour to produce 0.3 of the final product S11 and 0.7 of S8 in R5, and finally T6 processes S8 during 1 hour in reactor R6 to produce the final products S9 and S10. The connections capacity range is assumed between 0 to 200 [m.u./m2] at a fix/variable cost of 0.1/ 0.01 [103c.u.]. The capacity of R1, R2, R5 and R6, range from 0 to 150 [m.u./m2], while the others range from 0 to 200 [m.u./m 2] ( m.u. and c.u. are, respectively, mass and currency units). 3.1. Comparative analysis for the Example In this section a comparative analysis of the performance is made, between the DM previously presented (Pinto et al. 2008b) and the currently proposed algorithm. The optimality gap used for the UL was 7% followed by an additional 5000 seconds. For the LL and DM it was used a 5% of optimality gap and an additional 5000 seconds time for solution polishing. The results obtained for these specifications correspond to an approximation to the optimal set. The models characteristics are presented in Table 1. Points A, B, C, D and E are those where there is a change caused by the addition of one or more main equipment units to the previous topology (Pinto et al. 2008b). Table 2 presents the optimal design of the main equipment for each point in terms of capacities, while Table 3 presents the final product and their quantity.
A Decomposition Based Algorithm for the Design of Multipurpose Batch Facilities Using Economic Assessments
433
Table 1: Model characteristics. Model
Bin.Var.
Tot.Var.
Equat.
DM
612
2582
4378
UL
228
1809
2478
LL
606
2583
4379
Table 2: Design for the main equipment using the decomposition algorithm. Equipment
A
B
C
D
E
R1
76.2
93.3
103.4
141.2
120.3
R3
76.2
62.2
103.4
141.2
180.5
R4
-
155.5
129.3
140.5
159.1
R5
-
-
-
21.8
138.2
R6
-
-
-
15.3
96.8
V4
-
-
-
-
120.3
V5
76.2
-
51.72
170
170
V6
-
155.5
258.6
270
270
V9/V10
-
-
-
7.6
145.1
V11
-
-
-
6.5
124.4
Table 3: Quantities produced for each final product using the decomposition algorithm. Final Products
A
B
C
D
E
S5
76.2
-
-
170
170
S6
-
155.5
258.6
270
270
S9
-
-
-
7.6
145.1
S10
-
-
-
7.6
145.1
S11
-
-
-
6.5
124.4
A comparative analysis between results obtained for the DM model (Pinto et al. 2008b) versus the DA model is shown in Table 4. The first column characterizes the different points on the Pareto curve. There were topology changes only in points A to E. The second column quantifies the relative performance of the two models. The one with the highest performance is identified on the third column. From the gap column it can be noted that the execution for the DM and the LL ends when the optimality gap reaches (5%). In run 3, the DM model reached the maximum CPU time available without attaining the optimal gap. Looking at the best performance column, we verify that in most runs the DA model shows a better performance than DM, except for runs 1 and 9, where the latter took around 12% and 10%, i.e. less CPU time than DA. For all runs the DA converged in the first iteration. Despite the computational results, the equipment design and amounts of final products remained equal for both models.
434
T.R.X.M. Pinto et al. Table 4: Comparative analysis.
Run/Point
(CPU DA CPU DM )
CPU DA
100
Best performance
DM Cpu (s)
DM Gap (%)
DA Cpu (s)
LL Gap (%)
Iteration
1
12.145
DM
2.17
2.15
2.47
3.6
1
2/E
-261.27
DA
461.92
4.99
127.86
4.99
1
3
-256.691
DA
5000
5.77
1401.77
4.99
1
4
-168.921
DA
210.27
4.99
78.19
4.99
1
5
-324.125
DA
227.84
4.99
53.72
4.99
1
6/D
-102.481
DA
1843.45
4.99
910.43
5.0
1
7
-131.622
DA
596.45
5.0
257.51
5.0
1
8/C
-87.147
DA
415.0
4.99
221.75
4.99
1
9/B
10.271
DM
103.75
4.99
115.63
4.99
1
10/A
-99.78
DA
217.78
4.99
109
4.99
1
4. Conclusions A multi-objective decomposition algorithm is proposed for the design of multipurpose batch plants. The model aims to overcome some performance difficulties from the direct solution of the fully detailed MILP model. A combination of the decomposition constraint is applied allowing the definition of a range of algorithm with the topologies, schedules, design facilities and storage profiles in the vicinity of the efficient frontier. The performance of the proposed methodology is illustrated with an example. As seen in Table 4 the proposed model presents an all-round better performance than the detailed one. Despite these results some future work must be undertaken to improve the optimality gap and the model performance.
References Barbosa-Povoa, A. P. (2007). "A Critical Review on the Design and Retrofit of Batch Plants." Computers & Chemical Engineering 31(7): 833-855. Pinto, T., A. Barbosa-Povoa and A. Q. Novais (2008a). "Design of Multipurpose Batch Plants: A Comparative Analysis between the Stn, M-Stn, and Rtn Representations and Formulations." Industrial & Engineering Chemistry Research 47(16): 6025-6044. Pinto, T. R., A. Barbosa-Povoa and A. Q. Novais (2008b). Multi-Objective Design of Multipurpose Batch Facilities Using Economic Assesments. 18th European Symposium on Computer Aided Process Engineering (ESCAPE-18), Lyon, France. Chankong, V.; Haimes, Y. (1983). Multiobjective Decision Making Theory and Methodology. Elsevier. New York.
429
A Decomposition Based Algorithm for the Design of Multipurpose Batch Facilities Using Economic Assessments Tânia Rute X. M. Pinto1,2, Ana Paula F. D. Barbosa-Povoa2 and Augusto Q. Novais1 1
DMS ± Departamento de Modelação e Simulação, INETI, Lisboa, Portugal Estrada do Paço do Lumiar, 1649-038 Lisboa, Portugal E-mail address: [email protected] 2 CEG- Centro de Estudos de Gestão, IST, Lisboa, Portugal
Abstract In this paper we propose a multi-objective decomposition algorithm for the design of multipurpose batch plants, which avoids the direct solution of the fully detailed MILP model. Most of such design problems involve the maximization of the total revenue, as well as the minimization of the total cost. The way to deal with these two terms simultaneously is either to combine them into a single criterion (e.g., profit), or to define the efficient frontier that offers the optimal solutions by multi-objective optimization. In this work the latter approach, while more elaborate, was adopted, since the exploration of this frontier enables the decision maker to evaluate different alternative solutions. A combination of the proposed decomposition algorithm and the -constraint method is employed, which supports the application of this approach to perform economic assessments. The proposed algorithm allows the identification of the plant topologies, scheduling, equipment design and storage policies, subject to WKH SODQW¶s cost minimization and revenue maximization. A comparative analysis is presented between the detailed model proposed by Pinto et al. (2008b) and the proposed algorithm. Keywords: Scheduling, Design, -constraint, Multipurpose batch facilities.
1. Introduction In multipurpose batch facilities a wide variety of products can be produced via different processing recipes, by sharing all available resources, such as equipment, raw material, intermediates and utilities. In order to ensure that any resource in the design can be utilized as efficiently as possible, an adequate representation is necessary in order to address such type of problems without creating ambiguities in the process/plant mathematical model. The Resource-Task Network (RTN) is one of the possible adequate representations to describe the design of multipurpose batch plants as suggested by Pinto et al. (2008a). Like most real-world problems, the design of multipurpose batch facilities involves multiple objectives, while most of the existing literature on the design problem has been focused on single objectives (Barbosa-Povoa 2007). Therefore, the multi-objective optimization of such problems requires the development of an adequate modelling approach, as a pre-requisite for the resulting models to be useful as decision making tools where trade-offs among objectives can be investigated. In order to guarantee
430
T.R.X.M. Pinto et al.
optimal solutions, most of the published mathematical design formulations consider a large number of potential equipment items, out of which a selection is made of those that will be incorporated into the optimal plant configuration. This factor and the diversity of SURGXFWV¶recipes, gives rise to large MILP problems and consequently to an increasing computational burden. For this reason, effective solution tools are still an open area of research in the design of batch plants. In this paper, we aim to overcome some of these computational difficulties and also to avoid the direct solution of the fully detailed design and scheduling MILP model. With that aim, we propose a decomposition algorithm that exploits the hierarchical structure of the problem: the original detailed design and scheduling model is decomposed into an upper-level (UL) and a lower-level (LL) models; the UL uses profit maximization as the objective function and selects the so-called complicated variables, which in the current case are the design variables responsible for equipment selection; the LL comprises both the equipment design and the scheduling problem. Applying the -constraint method (Chankong and Haimes, 1983), the procedure iterates until the Pareto-optimum surface is completed. This allows the identification of a range of plant topologies, design facilities and storage policies that minimize the total cost of the system while maximizing revenue, subject to product demands and operational restrictions.
2. Decomposition Algorithm To avoid the direct solution of the MILP model, we propose a decomposition algorithm (DA) that exploits the hierarchical structure of design/scheduling models, and employ it together with the constraint method. As referred to previously, the original detailed design and scheduling model is decomposed into one UL design problem and one LL design plus scheduling problem. The UL determines the necessary equipment to satisfy the demand along the entire time horizon, with the design constraints being omitted, leading to a relaxation of the original, DM, or detailed problem and therefore to an upper-bound on profit. In the LL, the DM is solved by fixing the equipment obtained from the UL. The LL corresponds to a sub-problem of the original MILP problem, in a reduced space and at iteration i. It considers the selected subset of equipment obtained from the UL, producing a lower-bound on profit. The procedure iterates until the difference between the upper-bound and the lower-bound is less than a specified tolerance. To expedite the search, integer cuts are added to the UL to exclude previous solutions. The DA is subject to an objective constraint function that defines the maximum cost available for the optimal solution. The initial restriction takes the value of the necessary cost to generate the maximum revenue (Cmax). This restriction is decreased by small values, , and successive optimization steps (L «Q) undertaken until the intended range is covered and the Pareto front defined. 2.1. Integer Cut If the upper-bound obtained from the UL and the lower-bound obtained from the LL do not lie within a pre-defined tolerance, it is necessary to obtain a new solution from the UL. Then an integer cut on the binary design variable,
UL r ,
where r characterizes the
equipment obtained from the UL model, is added to the UL problem so as to avoid the solution in the previous iteration i. For iteration s > k, this is defined as: r U1k
UL r
r U 0k
UL r
U1k
1
(1)
A Decomposition Based Algorithm for the Design of Multipurpose Batch Facilities Using Economic Assessments
Where
U 0k
r|
UL r
0 and U1k
r|
431
1 . The U 0k and U1k are obtained
UL r
from the optimal UL solution, in terms of the assignment variable in iteration k. 2.2. -constraint In order to obtain the Pareto front, we used the -constraint method combined with the decomposition algorithm. The -constraint requires a cost constraint which is activated in both models (UL and LL) and imposes limits on the objective functions. The detailed model, LL, uses the objective functions FO1, which defines the revenue, and FO2, which reflects the cost associated with both equipment and operational tasks. To obtain : each point in the Pareto front, the cost constraint is activated and decreased by
F 02 Cmax i
L «Q.
(2)
The final decomposition algorithm is illustrated in Figure 1. i=0;k=0
Add cost constraint e-constraint
k
No Display solution
Upper-level - Design Solve UL assign the binary variables and get an upper-bound (UB)
Add integer cut k=k+1
Fixing the binary variable for the equipment selected by the UL
Lower-level - Scheduling and Design Solve the design and scheduling for the LL to get a lower-bound (LB) for the subset of equipments defined by the UL
No
i=i+1
(UB-LB)/LB <
yes Display solution
Figure 1: Flow chart for the decomposition algorithm.
2.3. Algorithm Steps An overview of the proposed decomposition follows: 1. The indices k and i control, respectively, the inner loop for estimating individual points on the Pareto front and the outer loop for generating successive points. Initially these are set at k=0 and i=0, while the upper-bound at UB= , the lower-bound at LB=- and a value assumed for the optimality tolerance.
432
T.R.X.M. Pinto et al.
2. Within the inner loop the cost constraint is activated; the MILP aggregated model solved for the upper-bound; the following sets defined:
U 0k
r|
UL r
0 , U1k
r|
UL r
1 ; and
r
UL r
for
k
r U1 ;
3. Still within the inner loop, the LL is solved; an optimal solution obtained, which is a lower-bound of the problem that defines the equipment design, scheduling and storage policy for each point of the Pareto front. is tested. 4. The convergence criterion UB LB LB If it is satisfied the solution corresponding to the lower-bound is the optimal solution and the outer loop is entered, where i defines the next point on the Pareto front. 5. If this convergence criterion is not satisfied an integer cut given by r U1k
UL r
r U 0k
UL r
U1k
1 is added to the UL and the inner loop controlled by k is
again calculated.
3. Example results In this example, the design of a multipurpose batch plant for a non-periodic mode of operation is performed. A production of [0; 170] tons of products S5, [0; 166] tons of S9 and S10, [0; 270] tons of products S6 and [0; 143] tons of products S11 is defined. Three raw materials, S1, S2 and S7, are used over the horizon of 24 h. The materials S5 and S6 are both intermediate and final products. S3, S4 and S8 are intermediate material. S3 and S8 are unstable and S4 is storable in V4 if necessary (Pinto et al. 2008b). There are available six main reactors (R1 to R6) and nine dedicated vessels. In terms of equipment suitability, only reactors R1 and R2 may carry out the two processing tasks, T1 and T2, while each storage vessel and reactors R3, R4, R5 and R6 are dedicated to a single State/Task. T1 may process S1 during 2 hours in R1 or R2; T2 may processes S2 during 2 hours in R1 or R2; T3 may process 0.5 of S3 and S4 during 4 hours in R3; T4 process 0.5 of S3 and S4 during 2 hours in R4; T5 process S6 during 1 hour to produce 0.3 of the final product S11 and 0.7 of S8 in R5, and finally T6 processes S8 during 1 hour in reactor R6 to produce the final products S9 and S10. The connections capacity range is assumed between 0 to 200 [m.u./m2] at a fix/variable cost of 0.1/ 0.01 [103c.u.]. The capacity of R1, R2, R5 and R6, range from 0 to 150 [m.u./m2], while the others range from 0 to 200 [m.u./m 2] ( m.u. and c.u. are, respectively, mass and currency units). 3.1. Comparative analysis for the Example In this section a comparative analysis of the performance is made, between the DM previously presented (Pinto et al. 2008b) and the currently proposed algorithm. The optimality gap used for the UL was 7% followed by an additional 5000 seconds. For the LL and DM it was used a 5% of optimality gap and an additional 5000 seconds time for solution polishing. The results obtained for these specifications correspond to an approximation to the optimal set. The models characteristics are presented in Table 1. Points A, B, C, D and E are those where there is a change caused by the addition of one or more main equipment units to the previous topology (Pinto et al. 2008b). Table 2 presents the optimal design of the main equipment for each point in terms of capacities, while Table 3 presents the final product and their quantity.
A Decomposition Based Algorithm for the Design of Multipurpose Batch Facilities Using Economic Assessments
433
Table 1: Model characteristics. Model
Bin.Var.
Tot.Var.
Equat.
DM
612
2582
4378
UL
228
1809
2478
LL
606
2583
4379
Table 2: Design for the main equipment using the decomposition algorithm. Equipment
A
B
C
D
E
R1
76.2
93.3
103.4
141.2
120.3
R3
76.2
62.2
103.4
141.2
180.5
R4
-
155.5
129.3
140.5
159.1
R5
-
-
-
21.8
138.2
R6
-
-
-
15.3
96.8
V4
-
-
-
-
120.3
V5
76.2
-
51.72
170
170
V6
-
155.5
258.6
270
270
V9/V10
-
-
-
7.6
145.1
V11
-
-
-
6.5
124.4
Table 3: Quantities produced for each final product using the decomposition algorithm. Final Products
A
B
C
D
E
S5
76.2
-
-
170
170
S6
-
155.5
258.6
270
270
S9
-
-
-
7.6
145.1
S10
-
-
-
7.6
145.1
S11
-
-
-
6.5
124.4
A comparative analysis between results obtained for the DM model (Pinto et al. 2008b) versus the DA model is shown in Table 4. The first column characterizes the different points on the Pareto curve. There were topology changes only in points A to E. The second column quantifies the relative performance of the two models. The one with the highest performance is identified on the third column. From the gap column it can be noted that the execution for the DM and the LL ends when the optimality gap reaches (5%). In run 3, the DM model reached the maximum CPU time available without attaining the optimal gap. Looking at the best performance column, we verify that in most runs the DA model shows a better performance than DM, except for runs 1 and 9, where the latter took around 12% and 10%, i.e. less CPU time than DA. For all runs the DA converged in the first iteration. Despite the computational results, the equipment design and amounts of final products remained equal for both models.
434
T.R.X.M. Pinto et al. Table 4: Comparative analysis.
Run/Point
(CPU DA CPU DM )
CPU DA
100
Best performance
DM Cpu (s)
DM Gap (%)
DA Cpu (s)
LL Gap (%)
Iteration
1
12.145
DM
2.17
2.15
2.47
3.6
1
2/E
-261.27
DA
461.92
4.99
127.86
4.99
1
3
-256.691
DA
5000
5.77
1401.77
4.99
1
4
-168.921
DA
210.27
4.99
78.19
4.99
1
5
-324.125
DA
227.84
4.99
53.72
4.99
1
6/D
-102.481
DA
1843.45
4.99
910.43
5.0
1
7
-131.622
DA
596.45
5.0
257.51
5.0
1
8/C
-87.147
DA
415.0
4.99
221.75
4.99
1
9/B
10.271
DM
103.75
4.99
115.63
4.99
1
10/A
-99.78
DA
217.78
4.99
109
4.99
1
4. Conclusions A multi-objective decomposition algorithm is proposed for the design of multipurpose batch plants. The model aims to overcome some performance difficulties from the direct solution of the fully detailed MILP model. A combination of the decomposition constraint is applied allowing the definition of a range of algorithm with the topologies, schedules, design facilities and storage profiles in the vicinity of the efficient frontier. The performance of the proposed methodology is illustrated with an example. As seen in Table 4 the proposed model presents an all-round better performance than the detailed one. Despite these results some future work must be undertaken to improve the optimality gap and the model performance.
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