Abstract
This paper considers bi-objective knapsack problem with fuzzy weights, says bi-objective fuzzy knapsack problem (BOFKP). Here we introduce an index which gives the possibility of choosing the item (weights and knapsack availability are fuzzy in nature) for knapsack with crisp capacity such that both the objective value are optimized. A methodology using dynamic programming technique has been introduced in this paper with an algorithm which gives the optimal solution for single objective fuzzy knapsack problem (FKP) with some possibility. Using this methodology an algorithm is given to find the Pareto frontier in case of bi-objective fuzzy knapsack problem. Compromise ratio method for decision-making under fuzzy environment has been used to find the compromise solution. The possibility index gives an idea to choose the solution according to decision-maker’s choice. An illustrative example is given to demonstrate the methodology.
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References
Bellman, R.: Dynamic programming and lagrange multipliers. Proc. Nat. Acad. Sci. U.S.A. 42(10), 767 (1956)
Bellman, R.E., Zadeh, L.A.: Decision-making in a fuzzy environment. Manag. Sci. 17(4), B-141 (1970)
Dubois, D., Prade, H.: Possibility Theory. Springer (1988)
Guha, D., Chakraborty, D.: Compromise ratio method for decision making under fuzzy environment using fuzzy distance measure, a a. 1, 2 (2008)
Horowitz, E., Sahni, S., Rajasekaran, S.: Computer algorithms C++: C++ and pseudocode versions. Macmillan (1997)
Kasperski, A., Kulej, M.: The 0–1 knapsack problem with fuzzy data. Fuzzy Optim. Decis. Making 6(2), 163–172 (2007)
Lin, F.-T., Yao, J.-S.: Using fuzzy numbers in knapsack problems. Eur. J. Oper. Res. 135(1), 158–176 (2001)
Martello, S., Toth, P.: Knapsack Problems. Wiley, New York (1990)
North, D.W.: A tutorial introduction to decision theory. IEEE Trans. Syst. Sci. Cybern. 4(3), 200–210 (1968)
Okada, S., Gen, M.: Fuzzy multiple choice knapsack problem. Fuzzy Sets Syst. 67(1), 71–80 (1994)
Sengupta, A., Pal, T.K.: On comparing interval numbers. Eur. J. Oper. Res. 127(1), 28–43 (2000)
Toth, P.: Dynamic programming algorithms for the zero-one knapsack problem. Computing 25(1), 29–45 (1980)
Yoshida, Y.: Dynamical aspects in Fuzzy decision making, Vol. 73, Springer (2001)
Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)
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Authors are grateful to the anonymous reviewers for their constructive comments and valuable suggestions.
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Singh, V.P., Chakraborty, D. (2015). A Dynamic Programming Algorithm for Solving Bi-Objective Fuzzy Knapsack Problem. In: Mohapatra, R., Chowdhury, D., Giri, D. (eds) Mathematics and Computing. Springer Proceedings in Mathematics & Statistics, vol 139. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2452-5_20
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DOI: https://doi.org/10.1007/978-81-322-2452-5_20
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