Single objective optimization

Searching for Pareto-optimal solutions can be computationally very expensive, especially when too many objectives are to be optimized. Therefore, it is very appealing to convert a multiobjective optimization problem into a much simpler single-objective optimization problem by combining the multiple objectives into a single objective function as follows (53-55) ... 

Consider the multi-criteria optimization problem defined in Eq. (11). Because of the fact that these objective functions usually conflict with each other in practice, the optimization of one objective implies the sacrifice of other targets it is thus impossible to attain their own optima, Js, s e <5 = [1,..., 5], simultaneously. Therefore, the decision maker (DM) must make some compromise among these goals. In contrast to the optimality used in single objective optimization problems, Pareto optimality characterizes the solutions in a multi-objective optimization problem [13]. 

The main focus of optimization of chemical processes so far has been optimization for one objective at a time (i.e., single objective optimization, SOO). However, practical applications involve several objectives to be considered simultaneously. These objectives can include capital cost/investment, operating cost, profit, payback period, selectivity, quality and/or recovery of the product, conversion, energy required, efficiency, process safety and/or complexity, operation time, robustness, etc. A few of these will be relevant for a particular application for example, see Chapter 2 for the objectives (typically 2 to 4) used in each of the numerous chemical engineering applications summarized. 

NSGA-n Kheawhom and Hirao (2004) proposed and used a two-layer methodology. Inner layer consists of single objective optimization to minimize operating cost. The outer layer involves multiobjective optimization. Kheawhom and Hirao (2004)... 

Fed-batch bioreactors for (a) lysine and (b) protein by recombinant bacteria The objectives are (1) maximization of both productivity and yield of lysine, and (2) maximization of amount of protein produced while minimizing volume of inducer added. NSGA-11 The two applications were solved as single objective optimization problems in the earlier studies. Sarkar and Modak (2005)... 

Simultaneous minimization of (1) reaction time, (2) polydispersity index for desired value of monomer conversion and (3) degree of polymerization. Weighting method The resulting single objective optimization problem was solved by SQP, GA and a hybrid of the two. Based on more than 100 optimization runs, all the three methods were concluded to be trustworthy. Curteanu et al (2006)... 

The single-objective optimization problem involving two decision variables that we have chosen for illustration is given by... 

The strength of evolutionary approaches is their wide applicability to, e.g., nondifferentiable and nonconvex problems. We wish to emphasize that this positive feature can be combined with scalarization based approaches by using evolutionary algorithms (i.e., not EMO but single objective optimizers) for solving the scalarized problem. 

There are different single-objective optimizers available for solving the scalarized problems formed and the user can decide after each classification which optimizer to use or use the default one. The proximal bundle method (Makela and Neittaanmaki, 1992) is a local optimizer and needs initial values for variables as well as (sub)gradients for functions. (The system can generate the latter automatically.) Alternatively, it is possible to use two variants of (global) real-coded genetic algorithms that differ from each other... 

As said, traditionally, this type of a problem has been formulated as a single objective optimization problem hiding the interrelationships between the objectives. Then, monetary values have to be assigned a priori to en-... 

The above model equations are validated by reproducing the results in Pintaric and Kravanja (2006) for single objective optimization, using NSGA-II-aJG. For this, the 4 decision variables are V, 7) and t]. For... 

Table 10.1 Optimal values for single objective optimization obtained using NSGA-II-aJG in the present study and reported values by Pintaric and Kravanja (2006) in brackets.

Table 10.8 Single-objective optimization of IE with 6 plants for profit, lEcP and lEvP separately.

To have a quantitative comparison between both Cases A and B, specific optimal solutions were chosen from Case A and Case B that have the same lEvP value, which is chosen to be the average of the maximum and minimum EvP values obtained in both cases (= 1.253). The two solutions with EvP value of 1.253 for both cases are summarized in Table 10.12. It is expected that Case A has a higher EcP and lower Profit than Case B due to the objective functions in each case. As a result, values of decision variables are different. As discussed above for single objective optimization, maximization of profit is achieved by the maximization of the capacities of each plant. So, capacities of all plants are higher in Case B than in Case A except for the capacity of plant 2 (X2). Note that maximum profit is not achieved as yet and X2 will eventually increase to its maximum limit when higher profits are achieved at the expense of higher EvP. 

As the Excel Solver is only for single objective optimization, use the e-constrained method and the Excel Solver.xls on the CD to optimize the 4-plant IE for the two objectives as in Cases A and B. For the Solver to work reliably, number of decision variables should be limited. Thus, it is recommended to set Z21 = Z32 = 0 and Z22 = 1 = 1 for Z = a, b, c and d. This would leave the capacities of the 4 plants (Xj) as the decision variables. Treat lEvP as the constraint and vary it in the range 1.213-1.419 for Case A and 1.220-1.321 for Case B, and observe the trends of the decision variables and the objective. Do they follow similar trends as the IE for 6 plants ... 

Generally, two or more objective functions are defined for gene expression profiling and gene network analysis. Usually, these objectives are conflicting in nature. Use of traditional single objective optimization techniques to solve these multi-objective optimization problems suffer from many drawbacks. Single objective problems either use penalty function approach or use some of the objectives as constraints. Both of these approaches have user-defined biases. Thus, multi-objective optimization techniques are definitely needed to model and solve these and similar other problems. 

Problem Type Differentiable/nondifferentiable multiobjective/single objective optimization problems with nonlinear/linear constraints... 

In contrast to single-objective optimization, in multi-objective optimization two goals have to be achieved ... 

While the first goal comes along with single-objective optimization, the second goal is characteristic for multi-objective optimization problems. Hence, multi-objective optimization procedures try to spread the pool of non-dominated configurations along the Pareto front. To do so, the diversity of non-dominated configurations has to be incorporated in the solution procedure. 

Regeneration unit Single-objective optimization Water splitter... 

Subject to the constraints (7.27 through 7.32) and (7.34 through 7.38). Because of the use of numerical weights, the objectives have to be scaled properly. However, only a single objective optimization problem had to be solved here. 

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