Bulanık küme teorisi ve doğrusal programlamada kullanımı: Karşılaştırmalı bir analiz

Geleneksel matematiğin belirlilik ile sınırlı dünyasını, derecelendirme mekanizması ile belirsizliğe doğru genişleten Zadeh, özellikle Uzak Doğu'dan yoğun bir ilgi gören ve başarılı uygulamaları sayesinde tüm dünyaya yayılan bir paradigma değişimi gerçekleştirmiştir. Klasik Küme Teorisinden daha geniş bir çerçeve yaratan Bulanık Küme Teorisi, karar vericiye daha geniş bir hareket alanı sağlayarak, doğrusal programlamanın gerçek dünyayı yansıtma becerisine ve uygulanabilirliğine katkıda bulunmuştur. Çalışma kapsamında, Bulanık Küme Teorisi tanıtılmış, uygulama alanlarına değinilmiş ve teorinin temel kavram ve işlemleri anlatılmıştır. Bulanık Doğrusal Programlamaya bir giriş yapılarak, literatürde sunulan yaklaşımlar incelenmiş, önerilen metotların bulanıklıkla olan ilgileri analiz edilmiş, doğrusal programlamaya ve birbirlerine göre getirdikleri yaklaşım farkları vurgulanmıştır. Son olarak örnek bir üretim planlama çalışması sunulmuştur.

Fuzzy sets theory and its uses in linear programming: A comparative analysis

Zadeh who extends the certainty limited world of mathematics through uncertainty by degree mechanism, has'created a paradigm shift which has taken great deal of attention first in the Far East then all over the world. Fuzzy Sets Theory providing a more widely frame than Classic Sets Theory, has been contributing^ to capability of reflecting real world and applicability of linear programming by supplying a wide moving area to decision maker. In this study, Fuzzy Sets Theory is introduced, its application areas are reviewed and basic concepts and operations of theory are told: After an introduction to fuzzy linear programming, the approaches presented in literature are surveyed, interest of methods proposed with fuzziness is analysed, differences of approaches to linear programming and each other is emphasized. Finally, a sample production planning study is presented.

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Bazaraa, M. S., Jarvis, J, J., Sherali, H. D.,1990, Linear programming and Network Flows, 2n Ed., John Wiley&Sons, New York.
Bector, C. R., Chandra, S., 2002, On duality in linear programming under fuzzy environment, Fuzzy Sets and Systems, Vol 125, No 3, pp317-325.
Bellman, R. E., Zadeh, L. A., 1970, Decision Making in a Fuzzy Environment, Management Science, 17, ppl41-164.
Carlsson, C, Korhonen, P., 1986, A Parametric Approach to Fuzzy Linear Programming, Fuzzy Sets and Systems, 20, ppl7-30.
Chalam, G. A., 1994, Fuzzy Goal Programming approach to a stochastic transportation problem under budgetary constraint, Fuzzy Sets and Systems, Vol 66, pp293-299.
Chanas, S., 1983, The Use of Parametric Programming in Fuzzy Linear Programming, Fuzzy Sets and Systems, 11(3), pp243-251.
Chanas, S., Kolodziejczyk, W., Machaj, A., 1984, A Fuzzy Approach to the Transportation Problem, Fuzzy Sets and Systems, Vol 13, pp211-221.
Chanas, S., Kuchta, D., 1998, Fuzzy integer transportation problem, Fuzzy Sets and Systems, Vol 98, pp291-298.
Chiang, J., 2001, Fuzzy Linear Programming based on statistical confidence interval and interval valued fuzzy set, European Journal of Operational Research, Vol 129, No 1, pp65-86.
Cross, V., De Cabello, M., 1995, Fuzzy interactive multiobjective optimization on borrowing/lending problems, Proceedings of 3rd Int. Symposium on Uncertainty Modeling and Analysis and Annual Conf. of the North American Fuzzy Inf. Processing Soc, pp513-518.
Deporter, E. L., Ellis, K. P., 1990, Minimization of Project Networks with Goal Programming and Fuzzy Linear Programming, Computers and Industrial Engineering, V 9, n 1-4, pp500-504.
Gupta, A. P., Harboe, R. and Tabucanon, M. T., 2000, Fuzzy multiple-criteria decision making for crop area planning in Narmada river basin, Agricultural Systems, Vol 63, No 1, ppl-18.
Herrera, F., Verdegay, J. L, 1995, Three models of fuzzy integer linear programming, European Journal of Operational Research, Vol 83, No 3, pp581-593.
Hota, P. K., Chakrabarti, Rv Chattopadyay, P. K., 2000, Economic emission load dispatch through an interactive fuzzy satisfying method, Electric Power Systems Research, Vol 54, No 3, ppl51-157.
Huang, G. H., Baetz, B. W., Patry, G. G., 1995, Grey Fuzzy Integer Programming: An Application to Regional Waste Management Planning under Uncertainty, Sodo-Economic Planning Sciences, Vol 29, No 1, ppl7-38.
Inuiguchi, M., Ichihashi, H., Kume, Y., 1990, A Solution Algorithm For Fuzzy Linear Programming with Piecewise Linear Membership Functions, Fuzzy Sets and Systems, 34, ppl5-31.
Jamison, K. D., Lodwick, W. A., 2001, Fuzzy Linear Programming using a penalty method, Fuzzy Sets and Systems, Vol 119, No 1, pp97-110.
Jimenez, F., Verdegay, J. L., 1998, Uncertain solid transportation problems, Fuzzy Sets and Systems, Vol 100, pp45-57.
Jimenez, F., Verdegay, J. L., 1999, Solving fuzzy solid transportation problems by an evolutionary algorithm based parametric approach, European Journal of Operational Research, Vol 117, pp485-510.
Kikuchi, S., 2000, A method to defuzzify the fuzzy number: transportation problem application, Fuzzy Sets and Systems, Vol 116, pp3-9.
Kubokawa, J., Okubo, Y., Sasaki, H., Yokoyama, R., 2000, Study of an interactive fuzzy multi-objective optimal power flow, Electrical Engineering in Japan, Vol 130, No 1, pp59-67.
Lai, Y. J., Hwang, C. L., 1992, Fuzzy Mathematical Programming; Methods And Applications, Springer-Verlag, Berlin.
Li, L., Lai, K. K., 2000, A fuzzy approach to the multiobjective transportation problem, Computers & Operations Research, Vol 27, pp43-57.
Liu, X., 2001, Measuring the satisfaction of constraints in fuzzy linear programming, Fuzzy Sets and Systems, Vol 122, No 2, pp263- 275.
McNeill, F. M., Thro, E., 1994, Fuzzy Logic, A Practical Approach, Academic Press, London.
Miller, W. A., Leung, L. C., Azhar, T. M. and Sargent, S., 1997, Fuzzy production planning model for fresh tomato packing, International Journal of Production Economics, Vol 53, No 3, pp227-238.
Negoita, C. V., Sularia, M., 1976, On Fuzzy Programming and Tolerances in Planning, Economic Computation and Economic Cybernetics Studies and Research, Vol 1, pp 3-15.
Orlovsky, S. A., 1978, Decision-making with a Fuzzy Preference Relation, Fuzzy Sets and Systems, 1(3), ppl55-167.
Östermark, R., 1996, A fuzzy control model (FCM) for dynamic portfolio management, Fuzzy Sets and Systems, Vol 78, No 3, pp243-254.
Pendharkar, P. C., 1997, A fuzzy linear programming model for production planning in coal mines, Computers & Operations Research, Vol 4, No 12, ppll41-1149.
Ramesh, R., Karwan, M. H., Zionts, S., 1989, "Interactive multicriteria linear programming: an extension of the method of zionts and wallenius", Naval Research Logistics, Vol.36, pp321-335.
Ross, T. J., 1995, Fuzzy Logic with Engineering Applications, Me Graw-Hill, New York.
Sakawa, M., Sawada, K., 1994, "Interactive fuzzy satisficing method for large-scale multiobjective linear programming problems with block angular structure", Fuzzy Sets and Systems, 67,1, pp5-17.
Sakawa, M., Yano, H., Yumine, T., 1987, Interactive fuzzy satisficing method for multiobjective linear programming problems and its applications, IEEE Trans. Syst. Man. Cybern., Vol 17, No 4, pp654-661.
Shih, L. H., Cement transportation planning via fuzzy linear programming, International Journal of Production Economics, Volume 58, Issue 3,25 January 1999, pp277-287.
Tanaka, H., Asai, K., 1984a, Fuzzy Linear Programming Problems with Fuzzy Numbers, Fuzzy Sets and Systems, 13, ppl-10.
Tanaka, H., Asai, K., 1984b, Fuzzy Solutions in Fuzzy Linear Programming Problems, IEEE Transactions on Systems, Man and Cybernetics, 14, pp325-328.
Tanaka, H., Guo, P., Zimmerman, H. J., 2000, Possibility distributions of fuzzy decision variables obtained from possibilistic linear programming problems, Fuzzy Sets and Systems, Vol 113, No 2, pp323-332.
Tanaka, H., Okuda, T., Asai, K., 1974, On Fuzzy Mathematical Programming, Journal of Cybernetics, 3, pp37-46.
Tapia, C. G., Murtagh, B. A., 1991, Interactive fuzzy programming with preference criteria in multiobjective decision making, Comp. Operations Research, Vol 18, No 3, pp307-316.
Teng, J. Y., Tzeng, G. H., 1998, Transportation investment project selection using fuzzy multiobjective programming, Fuzzy Sets and Systems, Vol 96, pp259-280.
Teodorovic, D., 1999, Fuzzy Logic Systems for transportation engineering: the state of the art, Transportation Research, Part A, 33, pp337-364.
Venkatesh, B., Sadasivam, G., Khan, M. A., 2001, An efficient multiobjective fuzzy logic based successive LP method for optimal reactive power planning, Electric Power Systems Research, Vol 59, No 2, pp89-102.
Verdegay, J. L., Fuzzy Mathematical Programming, in: Eds. Gupta, M. M. and Sanchez, E., Fuzzy Information and Decision Processes, North-Holland, Amsterdam.
Verma, R., Biswal, M. P., Biswas, A., 1997, Fuzzy programming technique to solve multiobjective transportation problems with some non-linear membership functions, Fuzzy Sets and Systems, Vol 91, pp37-43.
Vlaisavljevic, D., Djukanovic, M. B., 1999, Fuzzy Linear Programming Based Optimal Power System Rescheduling Including Preventive Redispatch, IEEE Transactions on Power Systems, Vol 14, 2, pp525-532.
Wahed, W. F. A., 2001, A multiobjective transportation problem under fuzziness, Fuzzy Sets and Systems, Vol 117, pp27-33.
Wang, L. X., 1994, Adaptive Fuzzy Systems and Control, Design and Stability Analysis, PTR Prentice-Hall, New Jersey.
Wang, R. C., Fang, H. H., 2001, Aggregate production planning with multiple objectives in a fuzzy environment, European Journal öf Operational Research, 133, pp521-536.
Werners, B., 1987, An Interactive Fuzzy Linear Programming System, Fuzzy Sets and Systems, 23, pp131-147.
Wu, S. M., Huang, G. H., Guo, H. C., 1997, Interactive inexact-fuzzy approach for multiobjective planning of water resource systems, Water Science and Technology, 36, 5, pp235-242.
Yu, C. S., Li, H. L., 2001, An algorithmic generalized fuzzy binary linear programming problems, European Journal of Operational Researcn, Vol 133, No 3, pp496-511.
Zadeh, L. A., 1965, Fuzzy Sets, Information and Control, Vol 8, pp338-353..
Zadeh, L. A., 1972, A Fuzzy Set Theoretic Interpretation of Linguistic Hedges, International Cybernetics, 2, pp4-34.
Zimmermann, H. J., 1976, Description and Optimization of Fuzzy Systems , International Journal of General Systems, 2, pp209-215.
Zimmerman, H. J., 1983, Fuzzy Mathematical Programming, Computers & Operations Research, Vol 10, No 4, pp291-298.
Zimmermann, H. J., 1992, Fuzzy Sets Theory And Its Applications, Kluwer A. P., Massachusetts.