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Decision makers usually face multiple, conflicting objectives and the complicated fuzzy-like environments in the real world. What are the fuzzy-like environments? How do we model the multiple objective decision making problems under fuzzy-like environments? How do you deal with these models?

In order to answer these questions, this book provides an up-to-date methodology system for fuzzy-like multiple objective decision making, which includes modelling system, model analysis system, algorithm system and application system in structure optimization problem, selection problem, purchasing problem, inventory problem, logistics problem and so on. The difficulty of the problem originates from the presence of more than one criterion. There is no longer a unique optimal solution to an MCDM problem that can be obtained without incorporating preference information.

The concept of an optimal solution is often replaced by the set of nondominated solutions. A nondominated solution has the property that it is not possible to move away from it to any other solution without sacrificing in at least one criterion.

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Therefore, it makes sense for the decision-maker to choose a solution from the nondominated set. Generally, however, the set of nondominated solutions is too large to be presented to the decision-maker for the final choice. Hence we need tools that help the decision-maker focus on the preferred solutions or alternatives. Normally one has to "tradeoff" certain criteria for others.

MCDM has been an active area of research since the s. There are different classifications of MCDM problems and methods. A major distinction between MCDM problems is based on whether the solutions are explicitly or implicitly defined.

Whether it is an evaluation problem or a design problem, preference information of DMs is required in order to differentiate between solutions. The solution methods for MCDM problems are commonly classified based on the timing of preference information obtained from the DM.

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There are methods that require the DM's preference information at the start of the process, transforming the problem into essentially a single criterion problem. These methods are said to operate by "prior articulation of preferences". Methods based on estimating a value function or using the concept of "outranking relations", analytical hierarchy process, and some decision rule-based methods try to solve multiple criteria evaluation problems utilizing prior articulation of preferences.

Similarly, there are methods developed to solve multiple-criteria design problems using prior articulation of preferences by constructing a value function. Perhaps the most well-known of these methods is goal programming. Once the value function is constructed, the resulting single objective mathematical program is solved to obtain a preferred solution. Some methods require preference information from the DM throughout the solution process.

These are referred to as interactive methods or methods that require "progressive articulation of preferences". Multiple-criteria design problems typically require the solution of a series of mathematical programming models in order to reveal implicitly defined solutions.

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For these problems, a representation or approximation of "efficient solutions" may also be of interest. When the mathematical programming models contain integer variables, the design problems become harder to solve. Multiobjective Combinatorial Optimization MOCO constitutes a special category of such problems posing substantial computational difficulty see Ehrgott and Gandibleux, [15] , for a review. The MCDM problem can be represented in the criterion space or the decision space. Alternatively, if different criteria are combined by a weighted linear function, it is also possible to represent the problem in the weight space.

Below are the demonstrations of the criterion and weight spaces as well as some formal definitions.

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Let us assume that we evaluate solutions in a specific problem situation using several criteria. Let us further assume that more is better in each criterion. Then, among all possible solutions, we are ideally interested in those solutions that perform well in all considered criteria. However, it is unlikely to have a single solution that performs well in all considered criteria. Typically, some solutions perform well in some criteria and some perform well in others. Finding a way of trading off between criteria is one of the main endeavors in the MCDM literature.

If Q is defined explicitly by a set of alternatives , the resulting problem is called a multiple-criteria evaluation problem. If Q is defined implicitly by a set of constraints , the resulting problem is called a multiple-criteria design problem. The quotation marks are used to indicate that the maximization of a vector is not a well-defined mathematical operation.

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This corresponds to the argument that we will have to find a way to resolve the trade-off between criteria typically based on the preferences of a decision maker when a solution that performs well in all criteria does not exist. The decision space corresponds to the set of possible decisions that are available to us. The criteria values will be consequences of the decisions we make. Hence, we can define a corresponding problem in the decision space. For example, in designing a product, we decide on the design parameters decision variables each of which affects the performance measures criteria with which we evaluate our product.

Mathematically, a multiple-criteria design problem can be represented in the decision space as follows:.

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A well-developed special case is obtained when X is a polyhedron defined by linear inequalities and equalities. If all the objective functions are linear in terms of the decision variables, this variation leads to multiple objective linear programming MOLP , an important subclass of MCDM problems. There are several definitions that are central in MCDM. Two closely related definitions are those of nondominance defined based on the criterion space representation and efficiency defined based on the decision variable representation.

Definition 1. Roughly speaking, a solution is nondominated so long as it is not inferior to any other available solution in all the considered criteria. Definition 2. If an MCDM problem represents a decision situation well, then the most preferred solution of a DM has to be an efficient solution in the decision space, and its image is a nondominated point in the criterion space. Following definitions are also important.

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Definition 3. Definition 4. Weakly nondominated points include all nondominated points and some special dominated points.

The importance of these special dominated points comes from the fact that they commonly appear in practice and special care is necessary to distinguish them from nondominated points. If, for example, we maximize a single objective, we may end up with a weakly nondominated point that is dominated. The dominated points of the weakly nondominated set are located either on vertical or horizontal planes hyperplanes in the criterion space.

Ideal point : in criterion space represents the best the maximum for maximization problems and the minimum for minimization problems of each objective function and typically corresponds to an infeasible solution.

A multi-objective decision-making approach to the journal submission problem

Nadir point : in criterion space represents the worst the minimum for maximization problems and the maximum for minimization problems of each objective function among the points in the nondominated set and is typically a dominated point. Liang H. Ren J. Gao Z. Gao S. Luo X.