An Achievement Rate Approach to Linear Programming Problems with an Interval Objective Function

In this paper, we focus on a treatment of a linear programming problem with an interval objective function. From the viewpoint of the achievement rate, a new solution concept, a maximin achievement rate solution is proposed. Nice properties of this solution are shown: a maximin achievement rate solution is necessarily optimal when a necessarily optimal solution exists, and if not, then it is still a possibly optimal solution. An algorithm for a maximin achievement rate solution is proposed based on a relaxation procedure together with a simplex method. A numerical example is given to demonstrate the proposed solution algorithm

Robust optimality analysis for linear programming problems with uncertain objective function coefficients: an outer approximation approach

summary:Linear programming (LP) problems with uncertain objective function coefficients (OFCs) are treated in this paper. In such problems, the decision-maker would be interested in an optimal solution that has robustness against uncertainty. A solution optimal for all conceivable OFCs can be considered a robust optimal solution. Then we investigate an efficient method for checking whether a given non-degenerate basic feasible (NBF) solution is optimal for all OFC vectors in a specified range. When the specified range of the OFC vectors is a hyper-box, i. e., the marginal range of each OFC is given by an interval, it has been shown that the tolerance approach can efficiently solve the robust optimality test problem of an NBF solution. However, the hyper-box case is a particular case where the marginal ranges of some OFCs are the same no matter what values the remaining OFCs take. In real life, we come across cases where some OFCs' marginal ranges depend on the remaining OFCs' values. For example, the prices of products rise together in tandem with raw materials, the gross profit of exported products increases while that of imported products decreases because they depend on the currency exchange rates, and so on. Considering those dependencies, we consider a case where the range of the OFC vector is specified by a convex polytope. In this case, the tolerance approach to the robust optimality test problem of an NBF solution becomes in vain. To treat the problem, we propose an algorithm based on the outer approximation approach. By numerical experiments, we demonstrate how the proposed algorithm efficiently solves the robust optimality test problems of NBF solutions compared to a conventional vertex-listing method

Interval linear regression analysis based on Minkowski difference – a bridge between traditional and interval linear regression models

summary:In this paper, we extend the traditional linear regression methods to the (numerical input)-(interval output) data case assuming both the observation/measurement error and the indeterminacy of the input-output relationship. We propose three different models based on three different assumptions of interval output data. In each model, the errors are defined as intervals by solving the interval equation representing the relationship among the interval output, the interval function and the interval error. We formalize the estimation problem of parameters of the interval function so as to minimize the sum of square/absolute interval errors. Introducing suitable interpretation of minimization of an interval function, each estimation problem is well-formulated as a quadratic or linear programming problem. It is shown that the proposed methods have close relation to both traditional and interval linear regression methods which are formulated in different manners

An Extented Sensitivity Analysis in Linear Programming Problems

When a real world problem is formulated as a linear programming model, we are often faced with difficulties in the parameter specification. We might know the plausible values or the possible ranges of parameters, but there still remains uncertainty. The parameter values could be obtained more exactly by experiments, investigations and/or inspections. However, to make such an experiment, investigation or inspection, expenses would be necessary. Because of capital limitations, we cannot invest in all possible experiments, investigations and inspections. Thus, we have a selection problem, which uncertainty reduction is the most profitable. In this paper, we discuss an analytic approach to the problem. Because of the difficulty of the global analysis, we make a local analysis around appropriate values of parameters. We focus on giving the decision maker useful information for the selection. First, sensitivity analyses with respect to the uncertain parameters are developed. The sensitivities are available only for the marginal domain without changing the optimal basis. The domain is obtained as an interval. The difficulty of the sensitivity analysis is in the cases of degeneracy and multiplicity of the optimal solutions. A treatment of such difficult cases is proposed. Finally, a numerical example is given for illustrating the proposed approach

On Computation Methods for a Minimax Regret Solution Based on Outer Approximation and Cutting Hyperplanes

In this paper, we discuss computation methods for minimax regret solutions to linear programming problems whose objective coefficient vectors are not known exactly but guaranteed to lie in polytopes. A solution algorithm for a minimax regret problem has been proposed based on the relaxation procedure.However, in the algorithm, we should solve non-concave sub-problems sequentially. To the non-concave sub-problem, many approaches including two phase and two-level programming approaches have been proposed. As new approaches, we discuss applications of an outer approximation method and a cutting plane method to the sub-problem. Moreover, a combination of the outer approximation and cutting hyperplane methods is proposed. We compare the computational efficiency of the solution algorithms by a numerical experiment. The results show that the outer approximation method and its combination with the cutting hyperplane method are the most efficient

An inner approximation method incorporating with a penalty function method for a reverse convex programming problem

AbstractIn this paper, we consider a reverse convex programming problem constrained by a convex set and a reverse convex set which is defined by the complement of the interior of a compact convex set X. When X is not necessarily a polytope, an inner approximation method has been proposed (J. Optim. Theory Appl. 107(2) (2000) 357). The algorithm utilizes inner approximation of X by a sequence of polytopes to generate relaxed problems. Then, every accumulation point of the sequence of optimal solutions of relaxed problems is an optimal solution of the original problem. In this paper, we improve the proposed algorithm. By underestimating the optimal value of the relaxed problem, the improved algorithms have the global convergence