Objective Integer Linear Programming Problems Research Articles

An improved algorithm to compute the $$\omega $$-primality

In this paper, we present an improved methodology to compute the $$\omega $$ -primality of a numerical semigroup. The approach is based on exploiting the structure of the problem on a resolution method for optimizing a linear function over the set of efficient solutions of a multiple objective integer linear programming problem. The numerical experiments show the efficiency of the proposed technique compared to the existing methods.

An L1-norm method for generating all of efficient solutions of multi-objective integer linear programming problem

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.

The augmented weighted Tchebychev norm for optimizing a linear function over an integer efficient set of a multicriteria linear program

AbstractIn this paper, we propose a new exact algorithm, using an augmented weighted Tchebychev norm, for optimizing a linear function on the efficient set of a multiple objective integer linear programming problem. This norm is optimized progressively by improving the value of the linear criteria and going through some efficient solutions. The method produced not only the best efficient solution of the linear objective function but also a subset of nondominated solutions that can help decision makers to select the best decision among a large set of Pareto solutions.

A method for optimizing over the integer efficient set

In this paper, we are interested in optimizing a linear function onthe set of efficient solutions of a Multiple Objective Integer LinearProgramming problem ($MOILP $). We propose an exact algorithm formaximizing a linear function denoted $ \phi $ on the set ofefficient solutions of a $MOILP$ problem without having toenumerate explicitly all the elements of this set. Two techniquesare used: the first is to reduce progressively the admissibledomain by adding more constraints eliminating all the dominatedpoints by the current solution; the second, when the new solutionobtained by maximizing the function $\phi $ in the reduced area isnot efficient, an exploration procedure is applied over the edgesincident to this solution in order to find new alternativeefficient solutions if they exist. The algorithm produces not onlyan optimal value of the linear function but also a subset ofnon-dominated solutions in the direction of $\phi$ that can behelpful in the practice.

An Algorithm For Solving Multiple Objective Integer Linear Programming Problem

In the present paper a complete procedure for solving Multiple Objective Integer Linear Programming Problems is presented. The algorithm can be regarded as a corrected form and an alternative to the method that was proposed by Gupta and Malhotra. A numerical illustration is given to show that this latter can miss some efficient solutions. Whereas, the algorithm stated bellow determines all efficient solutions without missing any one.

A reference direction approach to multiple objective integer linear programming

We propose the use of a reference direction/reference point approach to solving multiple objective integer linear programming problems. The reference direction/reference point is determined by the aspiration levels for the criteria that the decision-maker wants to improve. Within this framework, two methods are developed, viz., a pure integer method that operates entirely with integer solutions and a continuous/integer method that works with continuous solutions and finds an integer solution closest to the continuous solution in terms of the achievement scalarizing function. Obviously, the pure integer method is time consuming. We therefore propose a decision support system that combines the two methods. This way the adBANtages of each method can be used to their fullest extent. We illustrate the proposed decision support system with a numerical example.

An interactive algorithm for solving multiple objective integer linear programming problems

We propose a reference direction based interactive algorithm to solve multiple objective integer linear programming (MOILP) problems. At each iteration of the solution procedure, the algorithm finds (weak) nondominated solutions to the relaxed MOILP problem. Only at certain iterations, if the DM so desires, an additional mixed integer programming problem is solved to find an integer (weak) nondominated solution which is close to the current continuous (weak) nondominated solution to the relaxed MOILP problem. In the proposed algorithm, DM has to provide only the reference point at each iteration. No special software is required to implement the proposed algorithm. The algorithm is illustrated with an example.

A Reference Direction Algorithm for Solving Multiple Objective Integer Linear Programming Problems

In this paper, we propose a reference direction approach and an interactive algorithm to solve the general multiple objective integer linear programming problem. At each iteration, only one mixed integer linear programming problem is solved to find an (weak) efficient solution. Each intermediate solution is integer. The decision maker has to provide only the reference point at each iteration. No special software is required to implement the proposed algorithm. The algorithm is illustrated with an example.

A Reference Direction Algorithm for Solving Multiple Objective Integer Linear Programming Problems

A Reference Direction Algorithm for Solving Multiple Objective Integer Linear Programming Problems

A Reference Direction Algorithm for Solving Multiple Objective Integer Linear Programming Problems

A Reference Direction Algorithm for Solving Multiple Objective Integer Linear Programming Problems

An interactive procedure for multiple objective integer linear programming problems

There are many practical problems where the objective functions and the constraints can be represented by linear functions but the decision variables are restricted to only integer values. Our objective is to present an interactive procedure for the multiple objective integer linear programming problem that is fast, does not place too many demands on the decision maker and can be implemented without requiring any special software. We illustrate the proposed procedure with a numerical example and report some computational experience.

An algorithm for the multiple objective integer linear programming problem

A technique is presented for solving the multiple objective integer linear programming problem. The technique can be used to identify some or all efficient solutions. While the technique is applicable with any integer programming algorithm, it is well suited for implementation using integer postoptimality techniques. Such an implementation, based on Balas' Additive algorithm, is described for problems with zero-one variables.