Bi-objective optimization problem

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. 

Since most of this book is devoted to evolutionary methods for multiobjective optimization, we here only wish to discuss some differences between EMO approaches and scalarization based approaches. As mentioned before, EMO approaches are a posteriori type of methods and they try to generate an approximation of the Pareto optimal set. In bi-objective optimization problems, it is easy to plot the objective vectors produced on a plane and ask the DM to select the most preferred one. While looking at the... 

The above bi-objective optimization problem can be transformed into a SOO problem by the -constraint method via making OFj as an additional constraint. Besides this method, goal programming and the weighted-sum method can also be used to convert the MOO problem into a SOO problem. Interested readers are referred to Chapter 4 for more details... 

For illustrating the application of e-constraint method, consider the following generic bi-objective optimization problem. 

In contrast to single-objective problems where optimization methods explore the feasible search space to find the single best solution, in multi-objective settings, no best solution can be found that outperforms all others in every criterion (3). Instead, multiple best solutions exist representing the range of possible compromises of the objectives (11). These solutions, known as non-dominated, have no other solutions that are better than them in all of the objectives considered. The set of non-dominated solutions is also known as the Pareto-front or the trade-off surface. Figure 3.1 illustrates the concept of non-dominated solutions and the Pareto-front in a bi-objective minimization problem. 

As far as software is concerned, the interactive nature of the solution process naturally sets its own requirements (Hakanen, 2006). First of all, a good graphical user-interface (GUI) is needed in order to enable the interaction between the DM and the method. In addition, visualizations of the solutions obtained must be available for the DM to compare and evaluate the solutions generated. With interactive methods, more than three objective functions can easily be considered, which sets more requirements on the visualization when compared to, e.g., visualizing the Pareto optimal set for bi-objective 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). 

This chapter is organized as follows Section 8.2 provides a review of the literature. Section 8.3 describes the problem and the formulation of the objectives and presents a bi-criteria network design model for the CLSC network. In Section 8.4, we describe a bi-criteria interactive optimization algorithm for solving the model by involving the decision maker (DM) in the process. In Section 8.5, we present an illustrative example to show the applicability of the proposed approach. Section 8.6 presents some conclusions on the work and future research directions. 

First, we presented and analyzed the MILP model with tiie profit maximization objective using an illustrative example. Then, we proposed an interactive optimization algprithm to solve tiie proposed bi-criteria MILP model. The algorithm was illustrated using an example. The results showed the ability of the method to take the DM s preferences and systematically solve the bi-criteria problem. Also, the method posed less cognitive burden on tiie DM because the DM only had to compare two solutions and give his preference. 

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