Sebastian Sitarz

Ant algorithms and simulated annealing for multicriteria dynamic programming

The paper presents a comparison of ant algorithms and simulated annealing as well as their applications in multicriteria discrete dynamic programming. The considered dynamic process consists of finite states and decision variables. In order to describe the effectiveness of multicriteria algorithms, four measures of the quality of the nondominated set approximations are used.

Mean value and volume-based sensitivity analysis for Olympic rankings

This article describes two methods of creating Olympic rankings based on number of medals won. One method is based on the weighted mean value, which we will show is equivalent to the compromise programming known in Multi-Criteria Decision Analysis (MCDM). The other method uses volume-based sensitivity analysis. Both methods presented in this paper can also be used to construct rankings that include more than just the three top positions.

Postoptimal analysis in multicriteria linear programming

The aim of the paper is to present the postoptimal analysis of a chosen extreme efficient point in multicriteria linear programming. There are three cases considered: one objective function coefficient change, objective function addition and objective function removal. The proven theorems allow us to create methods based on the analysis of a simplex tableau.

The medal points’ incenter for rankings in sport

Abstract This work addresses the issue of finding a system of points for medals in Olympic ranking and scoring systems in sport. The incenter of a convex cone is used to obtain this system. The work deals also with the scoring systems. In applications, we present a ranking of countries in the 2010 Olympic Games and a ranking of drivers in the 2011 Formula One World Championship.

Standard sensitivity analysis and additive tolerance approach in MOLP

We consider sensitivity analysis of the objective function coefficients in multiple objective linear programming (MOLP). We focus on the properties of the parameters set for which a given extreme solution is efficient. Moreover, we compare two approaches: the standard sensitivity analysis (changing only one coefficient) and the additive tolerance approach (changing all coefficients). We find the connections between these two approaches by giving a theorem describing the upper bound on the maximal additive tolerance.

Hybrid methods in multi-criteria dynamic programming

Abstract The paper presents multi-criteria discrete dynamic process consisting of finite states and decision variables. The outcomes of the period criteria function are in the vector valued space. The hybrid algorithms which use ant and genetic algorithms together with the Bellman’s principle of optimality are presented. There are numerical results presented for six proposed algorithms in order to verify the developed algorithms.

Maximal and supremal tolerances in multiobjective linear programming

Let a multiobjective linear programming problem and any efficient solution be given. Tolerance analysis aims to compute interval tolerances for (possibly all) objective function coefficients such that the efficient solution remains efficient for any perturbation of the coefficients within the computed intervals. The known methods either yield tolerances that are not the maximal possible ones, or they consider perturbations of weights of the weighted sum scalarization only. We focus directly on perturbations of the objective function coefficients, which makes the approach independent on a scalarization technique used. In this paper, we propose a method for calculating the supremal tolerance (the maximal one need not exist). The main disadvantage of the method is the exponential running time in the worst case. Nevertheless, we show that the problem of determining the maximal/supremal tolerance is NP-hard, so an efficient (polynomial time) procedure is not likely to exist. We illustrate our approach on examples and present an application in transportation problems. Since the maximal tolerance may be small, we extend the notion to individual lower and upper tolerances for each objective function coefficient. An algorithm for computing maximal individual tolerances is proposed.

SENSITIVITY ANALYSIS OF WEAK EFFICIENCY IN MULTIPLE OBJECTIVE LINEAR PROGRAMMING

This paper, studies the sensitivity analysis of weakly efficient extreme solutions in multiple objective linear programming (MOLP). The aim of the paper is to compute the set of the parameters (corresponding to one coefficient) for which a given extreme point is a weakly efficient solution. We also focus on the properties of the parameters set by proving convexity and closeness of this set. Moreover, we compare the results of the sensitivity analysis of efficiency and of weak efficiency in MOLP.

Dynamic programming with ordered structures: Theory, examples and applications

The paper presents a dynamic, discrete optimization model with returns in ordered structures. It generalizes multiobjective methods used in vector optimization in two ways: from real vector spaces to ordered structures and from the static model to the dynamic model. The proposed methods are based on isotone homomorphisms. These methods can be applied in dynamic programming with returns in ordered structures. The provided numerical example shows an application of fuzzy numbers and random variables with stochastic dominance in dynamic programming. The paper also proposes applications in the following problems: a problem of allocations in the market model, a location problem, a railway routing problem, and a single-machine scheduling problem.

Pareto optimal allocations and dynamic programming

The aim of the paper is to show the relations between dynamic programming (DP) and the Pareto optimal allocations (PAO) problem. Moreover, the paper shows how to use DP methods in order to find the Pareto optimal allocations at a particular point in time.