Multi-objective Optimization Problem Formulation

Reduces salts to 20 ppm Reduces organics to 50 ppm Reduces H2S and NH3 to 5 and 30 ppm, respectively 

The MOO problem for the water network is presented below Equations (12.13)-(12.19). 

In this optimization problem, decision variables are fresh water flow rate to each process, water flow rates through interconnections between processes and/or regeneration units, and contaminant concentrations in water streams at the inlet and outlet of processes and regeneration units. Equations (12.16) and (12.17), respectively, limit the maximum inlet and outlet concentrations of contaminants in water streams for different processes, whereas Equation (12.18) limits the concentrations of contaminants in the waste water stream. Minimum flow rate in each interconnection is maintained by Equation (12.19). Values of outKz in Table 12.1, whereas the value of is assumed to be 1.0 

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 

Decision variables and all other constraints are the same as those in the MOO problem. The SOO problem is a MINLP problem this problem for water network design (Section 12.5.1) has 396 continuous variables, 182 integer variables and 586 constraints. Its solution for one value of e gives one Pareto-optimal (non-dominated) solution, and so the SOO problem has to be solved for different values of e to find many Pareto-optimal solutions, as discussed in Chapter 4. 

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Several prior publications formulate non-linear, multi-objective optimization problems and solve them with some success [5], but these algorithms typically exhibit super-linear runtime complexity and do not scale well enough to optimize an entire modem VLSI design at once. Other approaches focus on a handful of gates, and apply more time-consuming algorithms to relocate several gates at once. 

ABSTRACT This work proposes a robust optimization criterion of mechanical parameters in the design of linear Tuned Mass Dampers (TMD) located at the top of a main structural system subject to random base accelerations. The dynamic input is modelled as a stationary filtered white noise random process. The aim is to properly consider non-uniform spectral contents that happen in many real physical vibration phenomena. The main structural system is described as a single linear degree of freedom, and it is assumed that uncertainty affects the system model. The problem parameters treated are described as random uncorrelated variables known only by the estimation of their means and variances. Robustness is formulated as a multi-objective optimization problem in which both the mean and variance of a conventional objective function (OF) are minimized simultaneously. Optimal Pareto fronts are obtained and results show a significant improvement in performance stability compared to a standard conventional solution. 

In this chapter, a multi-objective optimization problem is formulated in terms of the distribution of both device and sensor and interstory drift responses of structures. The maximum drift is a normalized measure (Barroso 1999, Barroso et al. 2002)... 

General Formulation of the Multi-Objective Optimization Problem... 

In this chapter, the integrated design and control of bioprocesses was considered as a multi-objective optimization problem subject to non-linear differential-algebraic constraints. This formulation has a number of advantages over the traditional sequential approach, not only because it takes into account the process dynamics associated to a particular design, but also it provides a set of possible solutions from which the engineer can choose the most appropriate to his/her requirements. However, these problems are usually challenging to solve due to their non-convexity, which causes the failure of procedures based on local (e.g. SQP) NLP solvers. 

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. 

As mentioned in the introduction, we here assume that a DM is able to participate in the solution process. (S)he is expected to know the problem domain and be able to specify preference information related to the objectives and/or different solutions. We assume that less is preferred to more in each objective for him/her. (In other words, all the objective functions are to be minimized.) If the problem is correctly formulated, the final solution of a rational DM is always Pareto optimal. Thus, we can restrict our consideration to Pareto optimal solutions. For this reason, it is important that the multi-objective optimization method used is able to find any Pareto op>-timal solution and produce only Pareto optimal solutions. However, weakly Pareto optimal solutions are sometimes used because they may be easier to generate than Pareto optimal ones. A decision vector x G S (and the corresponding objective vector) is weakly Pareto optimal if there does not exist another x G S such that /i(x) < /i(x ) for alH = 1,..., A . Note that Pareto optimality implies weak Pareto optimality but not vice versa. 

We can say that interactive methods have not been used to optimize SMB processes and, usually, only one or two objective functions have been considered. The advantages of interactive multi-objective optimization in SMB processes has been demonstrated in Hakanen et al. (2008, 2007) for the separation of fructose and glucose (the values of the parameters in the SMB model used come from Hashimoto et al. (1983) Kawajiri and Biegler (2006b)). In Hakanen et al. (2008, 2007), the problem formulation consists of four objective functions maximize throughput (T, m/h ), minimize consumption of solvent in the desorbent stream (D, m/h ), maximize product... 

More recently, Lavan and Dargush (2009) examined a multi-objective seismic design optimization in which the maximum interstorey drift andmaximiun acceleration were considered as the primary control parameters. The multi-objective problem was formulated in Pareto optimal sense (Pareto 1927) and a genetic algorithm based approach was adopted to identify the Pareto front. The endresultofthis multi-objective optimization is a family of Pareto front solutions providing the decisionmakers with an opportunity to understand the tradeoff between the drift and acceleration. 

Design and synthesis of chemical processes are multi-objective optimisation problems in nature where several objectives are required to be optimised, maximised or minimised, simultaneously within a specified range of constraints. In this section the proposed integrated framework is introduced. The approach formulates the decision maker preferences into mathematical forms and then integrates these goals as a multi-objective optimiser into an overall stepwise procedure to cover all aspects of optimal design considerations, i.e. economical, environmental, heat integration as well as operability/controllability issues. 

The environmentally conscious process selection problem for the design of chemical SCs is addressed by Hugo and Pistikopoulos (2005). They present an MILP for the explicit inclusion of LCA as part of the strategic investment decisions. By jointly considering multiple environmental concerns and traditional economic criteria, the planning is formulated as a multi-objective optimization. 

In this paper, we extend the work of [10] by simultaneously considering minimization of the total utility consumption, maximization of operational flexibility to source-stream temperatures, and even minimum number of matches as multiple design objectives. The flexible HEN synthesis problem is thus formulated as the one of multi-objective mixed-integer linear programming (MO-MILP). This formulation also assumes that the feasible region in the space of uncertain input parameters is convex, so that the optimal solution can thus be explored on the basis of the vertices... 

This chapter considers the practical problem of feed optimization of the FCC process in a local refinery. A thorough study of the FCC process and the formulation of the feed optimization problem will be presented. Accounting for the coirflicting nature between the various objectives, the FCC feed optimization is solved by a multi-objective evolutionary toolbox developed by the authors (Tan et al, 2001a). Finally, for the actual implementation of the FCC process, three major key performance indexes are adopted to rate the various solutions evolved, so as to facilitate the decision-making process. 

Unlike optimization models with a single objective function, the interest is on finding a set of solutions that describe how the improvement of a single objective function value impacts the value of the other objectives. This set is commonly known as the Pareto-optimal set and each of its elements as a Pareto optimal solution. In this respect, a general formulation of a multi-objective problem is ... 

To solve the above optimization problem, a Multiple-Objective Evolutionary Algorithms (MOEA) is embraced here. MOEA is a term employed in the Evolutionary Multi-criteria Optimization field to refer to a family of evolutionary algorithms formulated to deal with MO. MOEA are able to deal with non-continuos, non-convex and/or non-linear objectives/constraints, and objective functions possibly not explicitly known (e.g. the output of Monte Carlo simulation tuns). 

The formulated optimization model defines relationships between the physical and information flows and takes into account QoS requirements for efficient information processing. Preliminary experimental results show that the information flows indeed affect selection of appropriate physical supply chain units. However, the QoS requirements have minor impact of the supply chain configuration decisions for the test supply chain analyzed in the chapter. An alternative approach to including QoS criteria directly in the objective model would be the specification of minimum quality requirements in the form of constraints. That would also alleviate the problem of selecting appropriate weights for multi-criteria optimization. The QoS characteristics also have impact on customer demand which could also be represented in the optimization model. 

The problem is formulated as a multie-objective optimisation model and is solved using Goal Programming technique. The optimal solution includes one oil boiler (B3), one... 

In rare problems it is admissible to limit the optimization to one single criterion, for example, a structure of minimum cost or a material of maximum strength may be considered as an appropriate solution. In general, such a formulation has somewhat academic character and may be used only as a simplified example for preliminary explanation of the problem. In most cases the existence and necessity of several criteria is obvious, though often they are considered in an indirect way, which means by appropriate constraints. Multi-objective or multi-criteria optimization is the next step, presented below, in which several criteria are directly considered. 

As mentioned earlier, the main idea of this study is to formulate the problem of the definition of an optimal ANN for a specific dataset in a multi-objective fashion. For example, in order to maximize the accuracy, a Multi-Objective Evolutionary Algorithm can be used to minimize the validation error while minimizing the test error. However, depending on the requirements one could also include different optimization criteria in the problem formulation, such as a measure of complexity of the classifier ANN. We will show this in the next section. 

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