Some Optimality Criteria of Interval Programming Problems
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Some Optimality Criteria of Interval Programming Problems Emrah Karaman1 Received: 14 December 2019 / Revised: 17 August 2020 / Accepted: 28 August 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this work, interval programming problems are considered for financial investment and optimality criteria. An order relation, defined by Ishibuchi and Tanaka for maximization problems, is used to obtain the solution of the problems. A real-life example, related to investment, and its solution are given. Necessary and sufficient optimality criteria including weak and strongly solution for interval programming problems are introduced via basic calculus rule, scalarization, and vectorization. Keywords Interval programming problem · Scalarization · Vectorization · Optimality criteria Mathematics Subject Classification 90C26 · 90C29 · 90C30
1 Introduction One of the most encountered problems in our life are optimization problem (mathematical programming problem). Translating this problem into mathematical language, give us an objective function. Optimization problems are called according to objective functions. For example, an interval programming problem (interval-valued optimization problem) arises when the objective function is the interval-valued function. Interval programming problems are a special form of the set-valued programming problems because the interval-valued function is a special case of the set-valued map. Also, interval programming problems are a generalization of scalar programming problems. The purpose of a mathematical programming problem is to find the best among the options. Naturally, it has been attracted the attention of scientists, who have been
Communicated by Anton Abdulbasah Kamil.
B 1
Emrah Karaman [email protected] Department of Mathematics, Faculty of Science, University of Karabük, Karabük, Turkey
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E. Karaman
working in mathematics, engineering, economics, and many other disciplines, for many years. There are many methods to solve programming problems as scalarization [5,14,25], vectorization [8,12,16,17], derivative [13], subdifferential [6,10,11], duality [21,24] etc. Scalarization is an effective tool for solving mathematical programming problems by using scalar-valued functions. This method replaces a mathematical programming problem with a scalar programming problem, which can be solved by using known methods. Solutions of scalar programming problems obtained via these methods are also solution of mathematical programming problems. Xu and Li [25], Karaman et al. [14], and Hernández and Rodríguez-Marín [5] have obtained optimality criteria for set-valued programming problems with the aid of scalarization. Vectorization is frequently used to get solutions and optimality conditions of set and vector programming problems. Vectorization reduces a programming problem to a vector programming problem as well as scalarization. In the vectorization, mathematical programming problems are characterized by vector programming problems
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