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Ind. Eng. Chem. Res. 2001, 40, 5040-5041
CORRESPONDENCE Comments on “An Improved RTN Continuous-Time Formulation for the Short-term Scheduling of Multipurpose Batch Plants” M. G. Ierapetritou† and C. A. Floudas* Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544-5263
Sir: In a recent paper Castro et al.1 made several incorrect statements and misleading interpretations of the novel continuous-time formulation that we proposed in 1998.2 The primary objective of this paper is to address and clarify these issues. The format of this paper consists of first mentioning the points made and subsequently providing responses. Point 1. Castro et al.,1 in the last sentence of the abstract and the fifth paragraph of the Introduction, point out that the approach presented by Ierapetritou and Floudas “is less accurate, as it violates time horizon constraints”. This statement is incorrect and misleading. In our approach2 all of the time horizons are always satisfied precisely, and hence there are no violations. The different results obtained by Castro et al.1 are due to the use of different parameter data. In Table 1, we provide detailed information on the precise data that we used (column 2), the data that Castro et al.1 used (column 3), and the most accurate data that can be used (column 4). The reader can observe that using a different set of parameters in the duration constraints or more precisely a different approximation of the ratios involved results in slightly different values of the objective function. Note also that the horizon requirement is met exactly. An important observation is that the different parameter sets result in schedules that have the same sequences of tasks. It is interesting to point out that Castro et al.1 did not notice that the schedule they obtained was identical in the sequence of tasks to the one obtained by the approach of Ierapetritou and Floudas.2 Point 2. In example 2 (section 4.2), the authors state that the solution reported by Ierapetritou and Floudas “exceeds the time horizon of 8 h by 0.07 h”. This statement is incorrect. The difference in the objective function $1503.15 compared to $1480.06 is due to the use of different duration parameters. The time horizon is precisely 8 h (see Table 1). Point 3. This point is very similar to point 2 because Castro et al.1 made the same misinterpretation for example 3. In particular, they stated that the formulation of Ierapetritou and Floudas2 has “an optimum of $58.81; however, as in example 2, the time horizon is exceeded, now by 0.12 h”. * To whom all correspondence should be addressed. Tel: (609) 258-4595. Fax: (609) 258-0211. E-mail: floudas@ titan.princeton.edu. † Current address: Department of Chemical and Biochemical Engineering, Rutgers University, New Brunswick, NJ.
The above statement is incorrect because in example 3 Castro et al.1 use different round-off values for the duration parameters. Table 2 provides all of the relevant data used (see columns 2-4). Point 4. Section 6 of the paper is devoted to the comparison of the proposed approach with the approach of Ierapetritou and Floudas.2 In particular, Castro et al.1 stated that “the formulation presented by Ierapetritou and Floudas ... produces very intriguing results for both examples 2 and 3”. They also stated that “our belief is that these constraints (sequence constraints), in addition to the fact that a different time grid is used for each unit, make this STN formulation less accurate”, and they again restate that the approach of Ierapetritou and Floudas leads to the violation of time horizon constraints. It is apparent from the responses to points 1-3 that these statements are incorrect and misleading. As was shown, (a) the differences in the objectives are only due to the use of different duration constraint parameters and (b) both formulations produce the same schedules in terms of the sequence of tasks. The sequence constraints introduced by Ierapetritou and Floudas2 do not cause any infeasibilities because the schedules obtained satisfy the material balances and time limitations. Moreover, the formulation of Ierapetritou and Floudas2 results in a more efficient way to solve the short-term scheduling problem as shown in Tables 5 and 7 of the paper by Castro et al.1 Point 5. In section 7, the authors use example 2 to show a limitation of the procedure of determining the optimal number of event points. What is proposed is an iterative procedure where one increases the number of event points by one until no improvement in the objective function can be achieved. We solved example 2 using the data provided by Castro et al.1 [i.e., approximate values for the duration constraints parameters (column 3 in Table 1)] and obtained the results shown in Table 3. The reader can observe that for five event points the objective value is $1480.06, for six event points it is $1480.13, and for seven event points it is $1480.13, which is the signal for the termination of the iterative process leading to the correct solution. Consequently, the results that Castro et al.1 presented in Table 8 using their formulation are not consistent with our results that use a true continuous-time formulation. Moreover, on the basis of the results presented in Table 8 of the paper by Castro et al.,1 one should notice the inefficiency of their proposed approach with the increased number of event points. For seven
10.1021/ie010555i CCC: $20.00 © 2001 American Chemical Society Published on Web 09/21/2001
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001 5041 Table 1. Different Approximations of the Duration Parameters for Example 2 Ierapetritou and Floudas
Castro et al.
exact duration parameters
0.666 + 0.00666B 1.333 + 0.0266B 1.333 + 0.0166B 1.333 + 0.0266B 1.333 + 0.0166B 0.666 + 0.0133B 0.666 + 0.00833B 1.333 + 0.00666B 1503.156 8
0.667 + 0.00667B 1.333 + 0.027B 1.333 + 0.017B 1.333 + 0.027B 1.333 + 0.017B 0.667 + 0.0133B 0.667 + 0.00833B 1.333 + 0.00667B 1480.063 8
2/3 + (1/150)B 4/3 + (2/75)B 4/3 + (1/60)B 4/3 + (2/75)B 4/3 + (1/60)B 2/3 + (1/75)B 2/3 + (1/120)B 4/3 + (1/150)B 1498.185 8
heating reaction 1, reactor 1 reaction 1, reactor 2 reaction 2, reactor 1 reaction 2, reactor 2 reaction 3, reactor 1 reaction 3, reactor 2 separation objective time horizon (h)
Table 2. Different Approximations of the Duration Parameters for Example 3 Ierapetritou and Floudas
Castro et al.
exact duration parameters
17.333 + 0.866B 2.667 + 0.133B 2.667 + 0.133B 4 + 0.2B 5.333 + 0.266B 5.333 + 0.266B 58.81 76
17.33 + 0.87B 2.67 + 0.13B 2.67 + 0.13B 4 + 0.2B 5.33 + 0.27B 5.33 + 0.27B 58.55 76
52/3 + (13/15)B 8/3 + (2/15)B 8/3 + (2/15)B 4 + 0.2B 16/3 + (4/15)B 16/3 + (4/15)B 58.755 76
reaction mixing, mixer 1 mixing, mixer 2 filtering stripping, strip 1 stripping, strip 2 objective time horizon (h)
Table 3. Effect of the Number of Event Points no. of event points objective CPU s
3
4
5
6
7
520 0.04
866.67 0.13
1480.06 0.23
1480.13 2.44
1480.13 21.88
event points, their proposed formulation required 41 024 nodes to obtain the optimal solution compared to relatively few nodes and less CPU time required by the approach by Ierapetritou and Floudas.2 Thus, the Castro et al.1 statement “because our formulation gives rise to mathematical problems that are solved faster” is also incorrect. Point 6. Castro et al.1 conclude in section 8 that although the formulation of Ierapetritou and Floudas2 gives “rise to smaller mathematical problems that are solved in slightly less time than those resulting from our formulation, their STN formulation is less accurate, generating better objective values because of the violation of time horizon constraints”. This conclusion is incorrect as we showed with the results provided in the responses of points 1-4.
In summary, all of the aforementioned speculative arguments of Castro et al.1 are incorrect, and the differences in the objective function for examples 2 and 3 are due to the use of different parameter data. The novel approach of Ierapetritou and Floudas2 meets precisely the horizon constraints and does not lead to any infeasibilities. Furthermore, the approach of Ierapetritou and Floudas2 results in a mathematical model of fewer binary variables, continuous variables, and constraints that exhibits a smaller integrality gap and requires less computational effort. Literature Cited (1) Castro, P.; Barbosa-Po´voa, A. P. F. D.; Matos, H. An Improved RTN Continuous-Time Formulation for the Short-term Scheduling of Multipurpose Batch Plants. Ind. Eng. Chem. Res. 2001, 40, 2059-2068. (2) Ierapetritou, M. G.; Floudas, C. A. Effective ContinuousTime Formulation for Short-Term Scheduling. 1. Multipurpose Batch Processes. Ind. Eng. Chem. Res. 1998, 37, 4341-4359.
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