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

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

Algorithm Engineering in Robust Optimization

Robust optimization is a young and emerging field of research having received a considerable increase of interest over the last decade. In this paper, we argue that the the algorithm engineering methodology fits very well to the field of robust optimization and yields a rewarding new perspective on both the current state of research and open research directions. To this end we go through the algorithm engineering cycle of design and analysis of concepts, development and implementation of algorithms, and theoretical and experimental evaluation. We show that many ideas of algorithm engineering have already been applied in publications on robust optimization. Most work on robust optimization is devoted to analysis of the concepts and the development of algorithms, some papers deal with the evaluation of a particular concept in case studies, and work on comparison of concepts just starts. What is still a drawback in many papers on robustness is the missing link to include the results of the experiments again in the design

A spatiotemporal Data Envelopment Analysis (S-T DEA) approach:the need to assess evolving units

One of the major challenges in measuring efficiency in terms of resources and outcomes is the assessment of the evolution of units over time. Although Data Envelopment Analysis (DEA) has been applied for time series datasets, DEA models, by construction, form the reference set for inefficient units (lambda values) based on their distance from the efficient frontier, that is, in a spatial manner. However, when dealing with temporal datasets, the proximity in time between units should also be taken into account, since it reflects the structural resemblance among time periods of a unit that evolves. In this paper, we propose a two-stage spatiotemporal DEA approach, which captures both the spatial and temporal dimension through a multi-objective programming model. In the first stage, DEA is solved iteratively extracting for each unit only previous DMUs as peers in its reference set. In the second stage, the lambda values derived from the first stage are fed to a Multiobjective Mixed Integer Linear Programming model, which filters peers in the reference set based on weights assigned to the spatial and temporal dimension. The approach is demonstrated on a real-world example drawn from software development