Multi-Objective Optimization Basics

In general, a MOO problem will have two or more objectives involving many decision variables and constraints. For illustration, consider an MOO problem with two objectives /i(x) and (x), and several decision variables (x). Such problems are known as two- or bi-objective optimization problems. 

Some applications may involve maximization of one or more objectives, which can he re-formulated by multiplying by -1 or taking the reciprocal (while ensuring that the denominator does not become zero) as the objective to be minimized. Hence, the above problem with two objectives to be minimized can be used for discussion without loss of generality. 

The two objectives, /i(x) and /2(x) are often conflicting. In such situations, there will be many optimal solutions to the MOO problem in equation 1.1. All these solutions are equally good in the sense that each 

Definition The set x , /i(x ) and /aCx ) is said to be a Pareto-optimal solution for the two-objective problem in equation 1.1, if and only if, no other feasible x exists such that/i(x) /i(x ) and/2(x) /2(x ) with strict inequality valid for at least one objective. 

Pareto-optimal solutions can be represented in two spaces - objective space (e.g., /i(x) versus /2(x)) and decision variable space. Definitions, techniques and discussions in MOO mainly focus on the objective space. However, implementation of the selected Pareto-optimal solution will require some consideration of the decision variable values. Multiple solution sets in the decision variable space may give the same or comparable objectives in the objective space in such cases, the engineer can choose the most desirable solution in the decision variable space. See Tarafder et al. (2007) for a study on finding multiple solution sets in MOO of chemical processes. 

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The paper is organized as follows. For completeness of the material, in Section 2 the concept of network global reliability efficiency is recalled with reference to the IEEE RTS 96 and a short introduction to the basic concepts behind the optimization procedure by Multi-Objective Genetic Algorithms is given in Section 3. In Section 4, the multi-objective optimization problem of the IEEE RTS 96 is defined and solved by the WP-MOGA approach. Conclusions on the outcomes of the study are eventually drawn in Section 5. 

In such a case, a unique HEN structure with satisfactory levels in nominal or average utility consumption and operational flexibility as well as unit numbers will be obtained. A two-phase fuzzy optimization method is proposed to find a best compromised solution for the multi-criteria HEN synthesis problem, as discussed in the next section. The basic number of constraints and variables for the multi-objective MTT.P formulation are summarized in the following. 

Fig. 11.2 Basic components of the algorithm for multi-objective emergency response optimization around chemical plants.

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