Multi-Objective Optimization

Randomly initialize the population and evaluate objective functions and constraints of each individual in the population by calling HEN model .  

Select two individuals from the current population by binary tournament.  

Generate two new individuals by performing crossover and mutation operations.  

Check violation of decision variable bounds randomly re-initialize the violated decision variables within the bounds. 

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Chen C, Lee W (2004) Multi-objective optimization of multi-echelon supply chain networks with uncertain product demands and prices. Computers Chemical Engineering 28 (6-7) 1131-1144... 

Molecular Library Design Using Multi-Objective Optimization Methods... 

Instead, due to the multi-objective nature of drug discovery, other factors, such as absorption, distribution, metabolism, excretion, toxicity (ADMET), selectivity and cost, molecular screening libraries need to be carefully planned and a number of design objectives must be taken into account (8). In recent times, MLD efforts have been exploring the use of multi-objective optimization (MOOP) techniques capable of designing libraries based on a number of properties simultaneously (9). 

MOPs are often characterized by vast, complex search spaces with various local optima that are difficult to explore exhaustively, largely due to the competition among the various objectives. In order to decrease the complexity of the search landscape, MOPs have traditionally been simplified, either by ignoring all objectives but one or by aggregating them. Multi-objective optimization (MOOP) methods enable the simultaneous optimization of... 

Key words Combinatorial library design, computer algorithms, product properties, multi-objective optimization. 

An optimization problem is a mathematical model which in addition to the aforementioned elements contains one or multiple performance criteria. The performance criterion is denoted as objective function, and it can be the minimization of cost, the maximization of profit or yield of a process for instance. If we have multiple performance criteria then the problem is classified as multi-objective optimization problem. A well defined optimization problem features a number of variables greater than the number of equality constraints, which implies that there exist degrees of freedom upon which we optimize. If the number of variables equals the number of equality constraints, then the optimization problem reduces to a solution of nonlinear systems of equations with additional inequality constraints. 

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]. 

C.L. Chen, B.W. Wang, W.C. Lee, Multi-objective Optimization for Multi-enterprise Supply Chain Networks, Ind. Eng. Chem. Res. 42 (2003) 1879-1889. 

Sakawa, M., Fuzzy Sets and Interactive Multi-Objective Optimization, Plenum Press, New York, 1993. 

In previous work (Filipe et al. 2007) the multi-objective optimization of a distillation column was performed and the Pareto front relating the total number of stages, reactive holdup and cost, identified. In this work a study on how the Pareto optimal designs could be adapted for real implementation is presented. Different design details, such as reactive holdup and feed quality, are investigated and the sensitivity of the solutions assessed to quantify the effect on the column expected performance. 

Filipe, R. M., A. Q. Novais, et al. (2006). Multi-objective optimization of reactive distillation columns using feasible regions. 17 International Congress of Chemical and Process Engineering, Prague, Czech Republic. 

Generically the models considered have a clear, quantitative way to compare feasible solutions. That is, they have single objective functions. In many applications single objectives reahstically model the true decision process. Decisions become much more confused when the problem arises in a complex engineering design, where more than one objective may be relevant. For such cases, as referred above, a multi-objective optimization model is required to capture all the possible perspectives. This is the case of the design of batch plants where two objectives are under consideration - one that maximizes the revenues (that is, production) and the other that minimizes the cost. 

The sizes of the units in the system are calculated using a the Queuing Multi Objective Optimizer (qmoo) developed at the Energy Systems Laboratory at the EPFL (Leyland [3]) in combination with a linear programming problem as described by in Weber et al. [2]. The sizes of the units considered are shown on Table 1. 

Keywords BSMl, multi-objective, optimization, wastewater. 

As already mentioned, two optimization objectives were chosen in this study the effluent quality and the energy consumption. In the case of multi-objective optimization, two solutions are possible. The first one consists in choosing a weighting scheme that will aggregate the different objectives in a single criteria. This technique is acceptable when it is possible to find a common measure or unit for all objectives. In our case, the effluent quality is already an aggregation, in unit of kilograms of pollution... 

To conclude, the technique developed in this study proved to be reliable for the optimization of a complex control law, as well as the determination of its robustness. Further work will focus on the comparison of different usual control schemes. Their multi-objective optimization will help us to have a clear insight into their optimal performances. Other objectives formulations will also be studied that may lead to better understanding of these results by the decision maker. Typically, at the industrial scale, a WWTP manager does not want to discharge pollutant loads as low as possible but his objective is to have the insurance to meet the quality standards (usually in term of... 

Multi-objective optimization (MOO), also known as multi-criteria optimization, particularly outside engineering, refers to finding values of decision variables which correspond to and provide the optimum of more than one objective. Unlike in SOO which gives a unique solution (or several multiple optima such as local and global optima in case of non-convex problems), there will be many optimal solutions for a multiobjective problem the exception is when the objectives are not conflicting in which case only one unique solution is expected. Hence, MOO involves special methods for considering more than one objective and analyzing the results obtained. 

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