# American Institute of Mathematical Sciences

July  2018, 14(3): 1157-1178. doi: 10.3934/jimo.2018004

## Solving the interval-valued optimization problems based on the concept of null set

 Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 802, Taiwan

Received  April 2016 Revised  August 2017 Published  July 2018 Early access  January 2018

We introduce the concept of null set in the space of all bounded closed intervals. Based on this concept, we can define two partial orderings according to the substraction and Hukuhara difference between any two bounded closed intervals, which will be used to define the solution concepts of interval-valued optimization problems. On the other hand, we transform the interval-valued optimization problems into the conventional vector optimization problem. Under these settings, we can apply the technique of scalarization to solve this transformed vector optimization problem. Finally, we show that the optimal solution of the scalarized problem is also the optimal solution of the original interval-valued optimization problem.

Citation: Hsien-Chung Wu. Solving the interval-valued optimization problems based on the concept of null set. Journal of Industrial and Management Optimization, 2018, 14 (3) : 1157-1178. doi: 10.3934/jimo.2018004
##### References:
 [1] A. K. Bhurjee and G. Panda, Efficient solution of interval optimization problem, Mathematical Methods of Operations Research, 76 (2012), 273-288.  doi: 10.1007/s00186-012-0399-0. [2] J. R. Birge and F. Louveaux, Introduction to Stochastic Programming, Physica-Verlag, NY, 1997. [3] G. R. Bitran, Linear multiple objective problems with interval coefficients, Management Science, 26 (1980), 694-706.  doi: 10.1287/mnsc.26.7.694. [4] Y. Chalco-Cano, W. A. Lodwick and A. Rufian-Lizana, Optimality conditions of type kkt for optimization problem with interval-valued objective function via generalized derivative, Fuzzy Optimization and Decision Making, 12 (2013), 305-322.  doi: 10.1007/s10700-013-9156-y. [5] S. Chanas and D. Kuchta, Multiobjective programming in optimization of interval objective functions --a generalized approach, European Journal of Operational Research, 94 (1996), 594-598.  doi: 10.1016/0377-2217(95)00055-0. [6] A. Charnes, F. Granot and F. Phillips, An algorithm for solving interval linear programming problems, Operations Research, 25 (1977), 688-695.  doi: 10.1287/opre.25.4.688. [7] J. W. Chinneck and K. Ramadan, Linear programming with interval coefficients, The Journal of the Operational Research Society, 51 (2000), 209-220. [8] M. Delgado, J. Kacprzyk, J. -L. Verdegay and M. A. Vila (eds. ), Fuzzy Optimization: Recent Advances, Physica-Verlag, NY, 1994. [9] M. Inuiguchi and J. Ramík, Possibilistic linear programming: A brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem, Fuzzy Sets and Systems, 111 (2000), 3-28.  doi: 10.1016/S0165-0114(98)00449-7. [10] A. Jayswal, I. Stancu-Minasian and I. Ahmad, On sufficiency and duality for a class of interval-valued programming problems, Applied Mathematics and Computation, 218 (2011), 4119-4127.  doi: 10.1016/j.amc.2011.09.041. [11] P. Kall, Stochastic Linear Programming, Springer-Verlag, NY, 1976. [12] R. Osuna-Gomez, Y. Chalco-Cano, B. Hernandez-Jimenez and G. Ruiz-Garzon, Optimality conditions for generalized differentiable interval-valued functions, Information Sciences, 321 (2015), 136-146.  doi: 10.1016/j.ins.2015.05.039. [13] A. Prékopa, Stochastic Programming, Kluwer Academic Publishers, Boston, 1995. [14] R. S lowiński (ed. ), Fuzzy Sets in Decision Analysis, Operations Research and Statistics, Kluwer Academic Publishers, Boston, 1998. [15] R. S lowiński and J. Teghem (eds), Stochastic Versus Fuzzy Approaches to Multiobjective Mathematical Programming under Uncertainty, Kluwer Academic Publishers, Boston, 1990. [16] I. M. Stancu-Minasian, Stochastic Programming with Multiple Objective Functions, D. Reidel Publishing Company, 1984. [17] S. Vajda, Probabilistic Programming, Academic Press,, NY, 1972. [18] H.-C. Wu, The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function, European Journal of Operational Research, 176 (2007), 46-59.  doi: 10.1016/j.ejor.2005.09.007. [19] H.-C. Wu, On Interval-valued nonlinear programming problems, Journal of Mathematical Analysis and Applications, 338 (2008), 299-316.  doi: 10.1016/j.jmaa.2007.05.023. [20] H.-C. Wu, Wolfe duality for interval-valued optimization, Journal of Optimization Theory and Applications, 138 (2008), 497-509.  doi: 10.1007/s10957-008-9396-0. [21] H.-C. Wu, Duality theory for optimization problems with interval-valued objective functions, Journal of Optimization Theory and Applications, 144 (2010), 615-628.  doi: 10.1007/s10957-009-9613-5.

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##### References:
 [1] A. K. Bhurjee and G. Panda, Efficient solution of interval optimization problem, Mathematical Methods of Operations Research, 76 (2012), 273-288.  doi: 10.1007/s00186-012-0399-0. [2] J. R. Birge and F. Louveaux, Introduction to Stochastic Programming, Physica-Verlag, NY, 1997. [3] G. R. Bitran, Linear multiple objective problems with interval coefficients, Management Science, 26 (1980), 694-706.  doi: 10.1287/mnsc.26.7.694. [4] Y. Chalco-Cano, W. A. Lodwick and A. Rufian-Lizana, Optimality conditions of type kkt for optimization problem with interval-valued objective function via generalized derivative, Fuzzy Optimization and Decision Making, 12 (2013), 305-322.  doi: 10.1007/s10700-013-9156-y. [5] S. Chanas and D. Kuchta, Multiobjective programming in optimization of interval objective functions --a generalized approach, European Journal of Operational Research, 94 (1996), 594-598.  doi: 10.1016/0377-2217(95)00055-0. [6] A. Charnes, F. Granot and F. Phillips, An algorithm for solving interval linear programming problems, Operations Research, 25 (1977), 688-695.  doi: 10.1287/opre.25.4.688. [7] J. W. Chinneck and K. Ramadan, Linear programming with interval coefficients, The Journal of the Operational Research Society, 51 (2000), 209-220. [8] M. Delgado, J. Kacprzyk, J. -L. Verdegay and M. A. Vila (eds. ), Fuzzy Optimization: Recent Advances, Physica-Verlag, NY, 1994. [9] M. Inuiguchi and J. Ramík, Possibilistic linear programming: A brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem, Fuzzy Sets and Systems, 111 (2000), 3-28.  doi: 10.1016/S0165-0114(98)00449-7. [10] A. Jayswal, I. Stancu-Minasian and I. Ahmad, On sufficiency and duality for a class of interval-valued programming problems, Applied Mathematics and Computation, 218 (2011), 4119-4127.  doi: 10.1016/j.amc.2011.09.041. [11] P. Kall, Stochastic Linear Programming, Springer-Verlag, NY, 1976. [12] R. Osuna-Gomez, Y. Chalco-Cano, B. Hernandez-Jimenez and G. Ruiz-Garzon, Optimality conditions for generalized differentiable interval-valued functions, Information Sciences, 321 (2015), 136-146.  doi: 10.1016/j.ins.2015.05.039. [13] A. Prékopa, Stochastic Programming, Kluwer Academic Publishers, Boston, 1995. [14] R. S lowiński (ed. ), Fuzzy Sets in Decision Analysis, Operations Research and Statistics, Kluwer Academic Publishers, Boston, 1998. [15] R. S lowiński and J. Teghem (eds), Stochastic Versus Fuzzy Approaches to Multiobjective Mathematical Programming under Uncertainty, Kluwer Academic Publishers, Boston, 1990. [16] I. M. Stancu-Minasian, Stochastic Programming with Multiple Objective Functions, D. Reidel Publishing Company, 1984. [17] S. Vajda, Probabilistic Programming, Academic Press,, NY, 1972. [18] H.-C. Wu, The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function, European Journal of Operational Research, 176 (2007), 46-59.  doi: 10.1016/j.ejor.2005.09.007. [19] H.-C. Wu, On Interval-valued nonlinear programming problems, Journal of Mathematical Analysis and Applications, 338 (2008), 299-316.  doi: 10.1016/j.jmaa.2007.05.023. [20] H.-C. Wu, Wolfe duality for interval-valued optimization, Journal of Optimization Theory and Applications, 138 (2008), 497-509.  doi: 10.1007/s10957-008-9396-0. [21] H.-C. Wu, Duality theory for optimization problems with interval-valued objective functions, Journal of Optimization Theory and Applications, 144 (2010), 615-628.  doi: 10.1007/s10957-009-9613-5.
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