E-Constraint method

In this work, the e-constraint method (2) is employed for generating a trade-off curve (or non-inferior set), and the surrogate worth trade-off method (9) for selecting the preferred decision. A milk evaporation process is optimized for illustration. 

The e-constraint method used here is based on the Kuhn-Tucker condition for non-inferior decision ( 2 ). The first equation of the Kuhn-Tucker condition can be rewritten as... 

The optimal value of J. is the minimum value of e. Although the e-constraint method is somewhat intricate compared with the weighting methods, it is widely used because of its applicability to non-convex problems. 

According to the e-constraint method, this problem can be formulated as (see Appendix B) ... 

The non-inferior solution set for the two-objective problem can be solved by various methods (5,6). In this paper the e-constraint method is used by taking into account its applicability to the non-convex problems. 

Trade-Off Between Two Objectives. By increasing e a family of e-constraint problems is solved successively using the max-sensitive method ( ). Using the usual e-constraint method the problems are solved separately for the various values of e. However with this method the optimal solution for the new value of e is pursued into the neighbourhood of the optimal solution for the last value by examining the sensitivities for all parameters. Incidentally, the Lagrange multiplier for the e-constraint is obtained from among the sensitivities for the parameters. ... 

In the e-constraint method, the MOO problem is transformed into a SOO problem by retaining only one of the objectives and converting all others into inequahty constraints. For example, the MOO problem in equation 1.1 is transformed as ... 

Optimize the alkylation process for two objectives (cases A and/or B) using the e-constraint method and Solver tool in Excel. Are the results comparable to those in Figures 1.5 and 1.6 ... 

Optimize the alkylation process for two objectives (cases A and/or B) using the weighting method. One can use the Solver tool in Excel for SOO. Try different weights to find as many Pareto-optimal solutions as possible. Compare and comment on the solutions obtained with those obtained by the -constraint method (Figures 1.5 and 1.6). Which of the two methods - the weighting and the e-constraint method, is better ... 

Solvent selection for acetic acid recovery Maximization of acetic acid recovery and process flexibility, and minimization of environmental impact based on lethal-dosage (LDa) and lethal-concentration (LC50) Constraint multiobjective programming (MOP) method Aspen Plus was employed to simulate the process, and uncertainty was also considered. The proposed MOP method is similar to the e-constraint method. Kim and Diwekar (2002)... 

Reactor-separator-recycle system Minimization of total cost and maximization of controllability. e-constraint method Eigen-value optimization approach was used along with the e-constraint method, to solve the design problem for two different control strategies. Blanco and Bandoni (2003)... 

Solvent for acrylic acid-water separation by extraction Simultaneous minimization of total annualized cost and Eco-indicator 99. e-constraint method Results show fiiat solvent substitution improves both the process economics and environmental impact of file entire plant despite its adverse effect on the extractor unit alone. Hugo et al. (2004)... 

Maximization of final product quality and minimization of drying time. e constraint method with SQP Optimal trajectories of air temperature and relatiye humidity for drying paddy rice were determined. Olmos el al. (2002)... 

Maximization of throughput and minimization of desorbent consumption. e constraint method A superstructure optimization problem for SMB process is considered. An interior point optimizer (IPOPT) is used to solve the single objective subproblems. Kawajiri and Biegler (2006)... 

Aspergilllus niger fermentation for catalase and protease production Two cases (a) maximization of catalase enzyme while minimizing protease enzyme, and (b) maximization of protease enzyme while minimizing catalase enzyme. e-constraint method along with differential evolution (DE) Penalty function approach was used for constraint handling. Mandal et al. (2005)... 

Vinyl chloride monomer (VCM) manufacture Maximization of VCM production and minimization of environmental burden, environmental impact and operating cost simultaneously. e-constraint method A design methodology consisting of 4 steps was proposed and applied to VCM plant The steps are (1) life cycle analysis of the process, (2) formulation of the design problem, (3) MOO, and (4) multi-criteria decision-making to find best compromise solutions. Khan et al. (2001)... 

Scheduling of refinery processes environmental impact. e-constraint method along with a MILP method Scheduling of refinery processes was modeled as a mixed-integer linear programming (MILP) model. Song et al. (2002)... 

Batch free radical polymerization of styrene Two cases (a) minimization of both the deviations from the desired conversion and number-average molxular weight, and (b) minimization of the initiator concentration in the product and the deviation from the desired conversion. e-constraint method Three problems, exh with a different set of desired values for conversion and number-average molecular weight, were solved. Merquior et al. (2001)... 

If one of the objective functions is selected to be optimized and the others are converted into constraints, we get the scalarization of the e-constraint method (Haimes et al., 1971 Chankong and Haimes, 1983) ... 

The solution of problem (6.3) is always weakly Pareto optimal and Pareto optimal if it is unique. On the other hand, x G S is Pareto optimal if and only if it solves (6.3) for every I = 1,..., k, where Sj = for j = 1,..., k, j 1. Thus, ensuring Pareto optimality means either solving k problems or obtaining a unique solution (which is not necessarily easy to verify in practice). What is positive when compared to the weighting method is that the e-constraint method can find any Pareto optimal solution even for nonconvex problems. 

In order to accelerate the process optimization, Kawajiri and Biegler (2006b) have developed an efficient full discretization approach combined with a large-scale nonlinear programming method for the optimization of SMBs. More recently, they have extended this approach to a superstructure SMB formulation and used the e-constraint method to solve the bi-objective problem, where throughput and desorbent consumption were optimized (Kawajiri and Biegler, 2006a). 

Use the e-constraint method to generate Pareto optimal solutions for this problem. Try using different upper bounds and generate four different Pareto optimal solutions. 

Lee et al. (2005b) evaluated the correlation between maximum biomass and succinic acid production for various combinatorial gene knockout strains. This sets the stage for the simultaneous maximization of biomass and succinic acid production using the e-constraint method. In this method, an MOO problem is converted into an equivalent single... 

A number of approaches are available for solving MOO problems (Rangaiah, 2009). One common approach (known as scalarization) is to transform the MOO problem into a SOO problem, which can then be solved using established SOO methods. Solution of the resulting SOO problem gives one Pareto-optimal solution, and hence the SOO problem has to be solved many times for obtaining other Pareto-optimal solutions. Two techniques for the scalarization approach are weighted sum and e-constraint methods. 

Figure 12.2 shows the results obtained for two objectives, using the e-constraint method and different limits [that is, e values in Equation (12.21)] on the total flow rate through... 

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