Multi-criteria optimization

In practice often more than one quality criterion is relevant. In the case of the need to build in robustness, at least two criteria are already needed the quality criterion itself and its associated robustness criterion. Hence, optimization has to be done on more than one criterion simultaneously. If a simultaneous optimization technique is used then there are procedures to deal with multiple optimization criteria. Several methods for multi-criteria optimization have been proposed and recently a tutorial/review has appeared [22]. 

An introduction to one particular multi-criteria optimization method -the so called Pareto-Optimality method - is discussed in Chapter 4, where also an application of this method is given. 

For readers with no prior knowledge of optimization methods In the textbook of Box et.al. [14] the basic principles of optimization are also explained. The sequential simplex method is presented in Walters et.al. [20]. Multi-criteria optimization is presented in Chapter 4 on an introductory level. For those readers who want to know more about multicriteria optimization, see the references given in Section 1.3.4 and Chapter 4. 

In this paper, the targeted source-stream temperatures are directly treated as individual design objective, and the multi-criteria optimization approach is adopted for HEN synthesis. The minimizing utility and the maximizing operational flexibility can be simultaneously considered as two conflict objectives for synthesis of the network structure. Furthermore, other targets such as minimizing number of matches can also be considered, such as,... 

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 original multi-criteria optimization problem is now converted to the one that looks for a suitable decision variable vector that can provide the maximal degree-of-satisfaction for the multiple fuzzy objectives. 

Mixed integer nonlinear programming Multi-criteria optimization... 

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. 

Stadler, W. (1988). Multi-criteria Optimization in Engineering and in the Sciences, Plenum Press, New York. 

Vafaeyan, S., Thibault, J., and Titica, M. (2007). Multi-criteria optimization of beer quality using rough set method. IF AC 10 Conference on Computer Applications in Biotechnology, Cancun, Mexico, 4-6 June. 

Trautmann, H., Steuer, D., and Mersmann, O. rncco Multi criteria optimization algorithms and related functions, 2010. URL http //CRAN.R-project. org/package=mco. R package version 1.0.9. 

A multi-criteria optimization problem generally consists of n decision variables, m constraints, and k evaluation functions. Thereby the evaluation functions can be in conflict with each other, making it difficult to find the global optimum. To find this optimum, the solution space /I C R is created by the decision variables x = (xi, , x ) of the decision space Q with the objective function vector F O A The objective function vector F x) = (/ j(x), , / (x), X e O is optimized considering the constraints / (—x) > 0 = 1,... m x Q (Muschalla 2006). The individual fitness value of each fitness function then can be processed by the evaluation function in different ways. This can be done by a scalarization method or a Pareto dominance-based approach. 

A common methodology for evaluating a multi-criteria optimization problem is the weighted sum. Each fitness function is weighted by a certain amount, so that the sum g is defined ... 

This way, a multi-criteria optimization problem is reduced to a single-criteria problem. A disadvantage of this method is that, on the one hand, a significant improvement in one fitness function may mask deterioration of another fitness function. On the other hand, the choice of the weighting already implies knowledge of the characteristics of good solutions. 

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). 

Chapter 7 of the textbook is devoted entirely to supply chain risk management. Multi-criteria optimization models that consider profitability, customer responsiveness, and supply chain risk are discussed in this context. 

In this chapter, we discuss supplier selection models and the multiple conflicting criteria used in supplier selection. Multi-criteria ranking methods for the prequalification of suppliers are discussed in detail. Next, multi-criteria optimization models are presented to determine the optimal order alloca-hon among the shortlisted suppliers. Several variants of goal programming methods for solving multiple criteria mathematical programming models are presented with a case study. 

Risk Adjusted Multi-Criteria Optimization Model for Supplier Sourcing (Phase 2)... 

Finally, we presented a real-world application for designing a resilient global supply chain for a multinational consumer products company. A multi-criteria optimization model explicitly considered conflicting criteria that integrated financial, customer service, supply chain risk and strategic factors of the company. Model results and managerial implications were also presented. 

Presents multi-criteria optimization models, where appropriate, for supply chain decisions... 

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. 

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. 

In addition to the multi-criteria representation of the problem, relationships between projects, companies, and products should be taken into account. In such assessments, premium cost and maximal insured value can be found using Multi-Objective Decision-Making (MODM) methods and solving as a multi-criteria optimization problem (Figure 5.2) the same criteria can be reused for insurance portfolio optimization, and in the case of discrete alternatives (premium cost or insured value), the Multi-Attribute Decision-Making (MADM) approach can be used. Different risk assessment approaches can be adapted for MCDM for example, product lifecycle can be presented in detail and/or insured accidents can be presented implicitly along with business opportunities and other benefits. 

Chapters 3, 7, 9, and 10 incorporate supply chain risk as an objective in multi-criteria optimization models. 

Vitoriano B, Ortuno M, Tirado G, Montero J. (2011). A multi-criteria optimization model for humanitarian aid distribution. Journal of Global Optimization, 51, 189-208. 

We formulate a multi-criteria optimization model to make the following decisions (i) supply chain network structure, including which suppliers, manufacturing plants, and DCs to use (ii) production and distribution planning, including which plants should produce which finished products, and which plants or DCs should distribute finished products to which customers ... 

Ravindran, A. and V. Wadhwa. (2009). Multi criteria optimization models for supplier selection. In Handbook of Military Industrial Engineering, A. Badiru and M. U. Thomas (Eds.). Chapter 4, 4-1-4-35, Boca Raton, FL CRC Press. 

We have seen how planning of prevention services may involve complicated structures in terms of (1) varied prevention needs, (2) limited budget, and (3) differing behavioral targets. In this chapter, we show how planning of prevention services can be formulated using multi-criteria optimization approaches... 

Supply chain management deeisions are made under the conflicting criteria of maximizing profit and eustomer responsiveness while minimizing supply ehain risk. Multiple Criteria Deeision Making in Supply Chain Management provides a comprehensive overview of multi-criteria optimization models and methods that can be used in supply chain decision making. 

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