Abstract
This paper extends the proposed method by Jahanshahloo et al. (2004) (a method for generating all the efficient solutions of a 0–1 multi-objective linear programming problem, Asia-Pacific Journal of Operational Research). This paper considers the recession direction for a multi-objective integer linear programming (MOILP) problem and presents necessary and sufficient conditions to have unbounded feasible region and infinite optimal values for objective functions of MOILP problems. If the number of efficient solution is finite, the proposed method finds all of them without generating all feasible solutions of MOILP or concluding that there is no efficient solution. In any iteration of the proposed algorithm, a single objective integer linear programming problem, constrained problem, is solved. We will show that the optimal solutions of these single objective integer linear programming problems are efficient solutions of an MOILP problem. The algorithm can also give subsets of efficient solutions that can be useful for designing interactive procedures for large, real-life problems. The applicability of the proposed method is illustrated by using some numerical examples.
Highlights
Multiple criteria decision making suggests, including multiple objective functions, a mathematical programming framework
Multi-objective integer linear programming (MOILP) problem is an important research area as many practical situations require discrete representations by integer variables, and many decision makers have to deal with several objectives (Ulungu and Teghem 1994)
This paper considers the recession direction to the MOILP problem and provides necessary and sufficient conditions to have unbounded feasible region and infinite values of MOILP problem and extends Jahanshahloo et al (2004) method to solve an MOILP problem
Summary
Multiple criteria decision making suggests, including multiple (two or more) objective functions, a mathematical programming framework. Numerous algorithms have been designed to solve an MOILP (Climaco et al 1997; Rasmussen 1986; Teghem and Kunsch 1986; Ulungu and Teghem 1994) and multiple objective mixed integer linear programs (Mavrotas and Diakoulaki 1998; Sylva and Crema 2007). Using a straightforward theoretical approach, Sylva and Crema's (2004) algorithm enumerates all efficient solutions of MOILP models with bounded feasible regions. Using a straightforward theoretical approach, the efficient solutions are found using a sequence of progressively more constrained integer linear programs generating new efficient solutions in any iteration. We prove that all of the optimal solutions of this single objective integer linear programming problem are efficient solutions of an MOILP problem. An MOILP problem has bounded or unbounded feasible region, which are discussed as follows
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