Abo-Sinna a, Tarek H. Ahmed Zoweil St. These fuzzy parameters are characterized as fuzzy numbers. For such problems, the a-Pareto optimality is introduced by extending the ordinary Pareto optimality on the basis of the a-level sets of fuzzy numbers. An interactive fuzzy decision making algorithm for generating a-Pareto optimal solution through TOPSIS approach is provided where the decision maker DM is asked to specify the degree a and the relative importance of objectives.

Finally, a numerical example is given to clarify the main results developed in the paper. All rights reserved. Naturally, these objec- tive functions and constraints involve many parameters whose possible values may be assigned by the experts. Abo-Sinna, T. He shows that using the fuzzy max-min operator with nonlinear membership functions, the obtained solutions are always nondominated solutions of the original MODP problems. Deng et al. Chen [4] extends the concept of TOPSIS to develop a methodology for solving multi-person multi-criteria decision-making problems in fuzzy environment.

Tzeng et al. The result shows that the hybrid electric bus is the most suitable substitute bus for Taiwan urban areas in the short and median term. But, if the cruising distance of electric bus extends to an acceptable range, the pure electric bus could be the best alternative.

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It is based upon the principle that the chosen alternative should have the shortest distance from the positive ideal solution PIS and the farthest from the negative ideal solution NIS. The single criterion of the shortest distance from the given goal or the PIS may be not enough to decision makers. A similar concept has also been pointed out by Zeleny [25]. In the following section, we will give the formulation of large scale multiple objective programming problem with fuzzy parameters in the objective functions and the right-hand side of the independent constraints LSF- MOP.

The family of dp-distance and its normalization is discussed in Section 3. We will also give a numerical example in Section 5 for the sake of illustration. Finally, concluding remarks will be given in Section 6. Thus, it seems essential to extend the notion of usual Pareto opti- mality in some sense.

If the objective function is linear as in Eq. In Section 3. Some basic concepts of distance measures The compromise programming approach [24,10] has been developed to perform multiple objective pro- gramming problem, reducing the set of nondominated solutions. The compromise solutions are those which are the closest by some distance measure to the ideal one.

Unfortunately, because of the incommensurability among objectives, it is impossible to directly use the above distance family. To obtain a compromise solution for problem 6 , the global criteria method [13] for large scale problems uses the distance family of Eq.

## Mahmoud A. Abo-Sinna - Google Scholar Citations

The problem becomes how to solve the following auxiliary problem: "" , p! In order to use the distance family of Eq. Thus, we can use membership functions to represent these individual optima. Assume that the membership functions l1 Z and l2 Z of two objective functions are linear. Then, based on the preference concept, we assign a larger degree to the one with shorter distance from the PIS for l1 Z and assign a larger degree to the one with farther distance from NIS for l2 Z. Therefore, as shown in Fig. The membership functions of l1 Z and l2 Z.

Now, by applying the max-min decision model which is proposed by Bellman and Zadeh [3] and extended by Zimmermann [26], we can resolve problem Step 2.

## Unknown error

Step 3. Step 4. Step 5. Transform problem 5 to the form of problem Step 6. Step 7. Step 8. Use Eqs. Step 9.

Transform problem 17 to the form of problem Step P Step Transform problem 20 to the form of problem 23 by using the membership functions Transform problem 23 to the form of problem Solve problem Otherwise, go to step 2. We shall use Alg-I to solve the above problem.

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Thus, problem 20 is obtained. Then, the bi-objective problem can be solved by using membership functions of fuzzy set theory to represent the satisfaction level for both criteria and obtain TOPSIS, compromise solution by a second-order compromise. Please, try again. If the error persists, contact the administrator by writing to support infona. You can change the active elements on the page buttons and links by pressing a combination of keys:. I accept. Polski English Login or register account. Masatoshi Sakawa. Source Studies in Fuzziness and Soft Computing.

Abstract Simultaneous considerations of multiobjectiveness, fuzziness and block angular structures involved in the real-world decision making problems lead us to the new field of interactive multiobjective optimization for large scale programming problems under fuzziness. Authors Close. Assign yourself or invite other person as author. It allow to create list of users contirbution.

## An Interactive Model for Fully Rough Three Level Large Scale Integer Linear Programming Problem

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