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Applications of fuzzy set theory to OR in Kyoto
260
Bulletin
[7] M. Inuiguchi and Y. Kume, A discrimination method of possibility efficient extreme points for interval multiobjective linear programming problems, Trans. Soc. Instr. Control Engrs. 25 (7) (1989) 824-826 (in Japanese). [8] M. Inuiguchi and Y. Kume, Modality goal programming problems, J. Oper. Res. Soc. Japan 32 (3) (1989) 326-351 (in Japanese). [9] M. Inuiguchi, H. Ichihashi and Y. Kume, A solution algorithm for fuzzy linear programming with piecewise linear membership functions, Fuzzy Sets and Systems 34 (1) (1990) 15-31. [10] M. Inuiguchi and Y. Kurne, Solution concepts for fuzzy multiple objective programming problems, J. Japan Soc. Fuzzy Theory and Systems 2 (1) (1990) 66-78 (in Japanese). [11] M. Inuiguchi and Y. Kume, Extensions of fuzzy relations considering interaction between possibility distributions and an application to fuzzy linear program, Trans. Inst. Systems Control Inform. Engrs, 3 (4) (1990) 93-102 (in Japanese). [12] M. Inuiguchi and H. Ichihashi, Relative rnodalities and their use in possibilistic linear programming, Fuzzy Sets and Systems 36 (3) (1990) 303-323. [13] M. Inuiguchi and Y. Kume, Goal programming problems with interval coefficients and target intervals, European J. Oper. Res., to appear. Masahiro Inuiguchi Department of Industrial Engineering University of Osaka Prefecture 804 Mozu-Umemachi 4-cho, Sakai, Osaka 591, Japan
4. Applications of fuzzy set theory to OR in Kyoto We have investigated some applications of fuzzy set theory to Operations Research (OR). Let us introduce our research. As one aspect, we considered fuzzy versions of scheduling problems. For many scheduling prolems, due-dates are usually rigid. In some situations, however, it may be natural that due-dates are not rigid. That is, a decision maker (DM) may tolerate a little lateness in the completion time for a job. Then we introduce a membership function of fuzzy set theory. The jobwise membership function decided by DM shows a satisfaction degree for the corresponding completion time. Naturally, more lateness makes the degree low. Thus, it is a non-increasing function and we call it 'fuzzy due-date'. Accordingly, a variety of scheduling problems are formulated as 'fuzzy scheduling problems'. As another application, we considered fuzzy versions of network flow problems such as transportation problems and sharing problems. In case of transportation problems, the problem is usually the determination of an optimal flow pattern on a bi-graph for given rigid supply and demand values. The more flexible problem is formulated by taking account of fuzziness of supply and demand values. Thus, there exist t w o kinds of membership functions which show a degree of satisfaction for the flux sent out from a supply node or for the flux sent into a demand node. As further research, we introduced 'fuzzy processing time' into scheduling problems. That is, we take account of the situations where the processing time is not constant because of some conditions, e.g., the condition of the machine and so on. From practical and analytic points of v i e w it may be valid that the membership function has the shape of a triangular fuzzy number (TFN). As stated above, a variety of problems of OR are generalized by introducing membership functions which show a degree of satisfaction or a TFN, Minoru Tada Faculty of Business Administr, Ryukoku University Kyoto, Japan Hiroaki Ishii Department of Information Technology Okayama University, Japan
Bulletin
[7] M. Inuiguchi and Y. Kume, A discrimination method of possibility efficient extreme points for interval multiobjective linear programming problems, Trans. Soc. Instr. Control Engrs. 25 (7) (1989) 824-826 (in Japanese). [8] M. Inuiguchi and Y. Kume, Modality goal programming problems, J. Oper. Res. Soc. Japan 32 (3) (1989) 326-351 (in Japanese). [9] M. Inuiguchi, H. Ichihashi and Y. Kume, A solution algorithm for fuzzy linear programming with piecewise linear membership functions, Fuzzy Sets and Systems 34 (1) (1990) 15-31. [10] M. Inuiguchi and Y. Kurne, Solution concepts for fuzzy multiple objective programming problems, J. Japan Soc. Fuzzy Theory and Systems 2 (1) (1990) 66-78 (in Japanese). [11] M. Inuiguchi and Y. Kume, Extensions of fuzzy relations considering interaction between possibility distributions and an application to fuzzy linear program, Trans. Inst. Systems Control Inform. Engrs, 3 (4) (1990) 93-102 (in Japanese). [12] M. Inuiguchi and H. Ichihashi, Relative rnodalities and their use in possibilistic linear programming, Fuzzy Sets and Systems 36 (3) (1990) 303-323. [13] M. Inuiguchi and Y. Kume, Goal programming problems with interval coefficients and target intervals, European J. Oper. Res., to appear. Masahiro Inuiguchi Department of Industrial Engineering University of Osaka Prefecture 804 Mozu-Umemachi 4-cho, Sakai, Osaka 591, Japan
4. Applications of fuzzy set theory to OR in Kyoto We have investigated some applications of fuzzy set theory to Operations Research (OR). Let us introduce our research. As one aspect, we considered fuzzy versions of scheduling problems. For many scheduling prolems, due-dates are usually rigid. In some situations, however, it may be natural that due-dates are not rigid. That is, a decision maker (DM) may tolerate a little lateness in the completion time for a job. Then we introduce a membership function of fuzzy set theory. The jobwise membership function decided by DM shows a satisfaction degree for the corresponding completion time. Naturally, more lateness makes the degree low. Thus, it is a non-increasing function and we call it 'fuzzy due-date'. Accordingly, a variety of scheduling problems are formulated as 'fuzzy scheduling problems'. As another application, we considered fuzzy versions of network flow problems such as transportation problems and sharing problems. In case of transportation problems, the problem is usually the determination of an optimal flow pattern on a bi-graph for given rigid supply and demand values. The more flexible problem is formulated by taking account of fuzziness of supply and demand values. Thus, there exist t w o kinds of membership functions which show a degree of satisfaction for the flux sent out from a supply node or for the flux sent into a demand node. As further research, we introduced 'fuzzy processing time' into scheduling problems. That is, we take account of the situations where the processing time is not constant because of some conditions, e.g., the condition of the machine and so on. From practical and analytic points of v i e w it may be valid that the membership function has the shape of a triangular fuzzy number (TFN). As stated above, a variety of problems of OR are generalized by introducing membership functions which show a degree of satisfaction or a TFN, Minoru Tada Faculty of Business Administr, Ryukoku University Kyoto, Japan Hiroaki Ishii Department of Information Technology Okayama University, Japan