Qualitative Characteristics and Quantitative Measures of Solution's Reliability in Discrete Optimization: Traditional Analytical Approaches, Innovative Computational Methods and Applicability

The purpose of this thesis is twofold. The first and major part is devoted to sensitivity analysis of various discrete optimization problems while the second part addresses methods applied for calculating measures of solution stability and solving multicriteria discrete optimization problems. Despite numerous approaches to stability analysis of discrete optimization problems two major directions can be single out: quantitative and qualitative. Qualitative sensitivity analysis is conducted for multicriteria discrete optimization problems with minisum, minimax and minimin partial criteria. The main results obtained here are necessary and sufficient conditions for different stability types of optimal solutions (or a set of optimal solutions) of the considered problems. Within the framework of quantitative direction various measures of solution stability are investigated. A formula for a quantitative characteristic called stability radius is obtained for the generalized equilibrium situation invariant to changes of game parameters in the case of the H¨older metric. Quality of the problem solution can also be described in terms of robustness analysis. In this work the concepts of accuracy and robustness tolerances are presented for a strategic game with a finite number of players where initial coefficients (costs) of linear payoff functions are subject to perturbations. Investigation of stability radius also aims to devise methods for its calculation. A new metaheuristic approach is derived for calculation of stability radius of an optimal solution to the shortest path problem. The main advantage of the developed method is that it can be potentially applicable for calculating stability radii of NP-hard problems. The last chapter of the thesis focuses on deriving innovative methods based on interactive optimization approach for solving multicriteria combinatorial optimization problems. The key idea of the proposed approach is to utilize a parameterized achievement scalarizing function for solution calculation and to direct interactive procedure by changing weighting coefficients of this function. In order to illustrate the introduced ideas a decision making process is simulated for three objective median location problem. The concepts, models, and ideas collected and analyzed in this thesis create a good and relevant grounds for developing more complicated and integrated models of postoptimal analysis and solving the most computationally challenging problems related to it.Siirretty Doriast

Stability Analysis in Multicriteria Discrete Portfolio Optimization.

Almost every problem of design, planning and management in the technical and organizational systems has several conflicting goals or interests. Nowadays, multicriteria decision models represent a rapidly developing area of operation research. While solving practical optimization problems, it is necessary to take into account various kinds of uncertainty due to lack of data, inadequacy of mathematical models to real-time processes, calculation errors, etc. In practice, this uncertainty usually leads to undesirable outcomes where the solutions are very sensitive to any changes in the input parameters. An example is the investment managing. Stability analysis of multicriteria discrete optimization problems investigates how the found solutions behave in response to changes in the initial data (input parameters). This thesis is devoted to the stability analysis in the problem of selecting investment project portfolios, which are optimized by considering different types of risk and efficiency of the investment projects. The stability analysis is carried out in two approaches: qualitative and quantitative. The qualitative approach describes the behavior of solutions in conditions with small perturbations in the initial data. The stability of solutions is defined in terms of existence a neighborhood in the initial data space. Any perturbed problem from this neighborhood has stability with respect to the set of efficient solutions of the initial problem. The other approach in the stability analysis studies quantitative measures such as stability radius. This approach gives information about the limits of perturbations in the input parameters, which do not lead to changes in the set of efficient solutions. In present thesis several results were obtained including attainable bounds for the stability radii of Pareto optimal and lexicographically optimal portfolios of the investment problem with Savage's, Wald's criteria and criteria of extreme optimism. In addition, special classes of the problem when the stability radii are expressed by the formulae were indicated. Investigations were completed using different combinations of Chebyshev's, Manhattan and Hölder's metrics, which allowed monitoring input parameters perturbations differently.Siirretty Doriast

On one type of stability for multiobjective integer linear programming problem with parameterized optimality

A multiobjective problem of integer linear programming with parametric optimality is addressed. The parameterization is introduced by dividing a set of objectives into a family of disjoint subsets, within each Pareto optimality is used to establish dominance between alternatives. The introduction of this principle allows us to connect such classical optimality sets as extreme and Pareto. The admissible perturbation in such problem is formed by a set of additive matrices, with arbitrary H\"{o}lder's norms specified in the solution and criterion spaces. The lower and upper bounds for the radius of strong stability are obtained with some important corollaries concerning previously known results.</p

About one discrete analog of Hausdorff semi-continuity of suitable mapping in a vector combinatorial problem with a parametric principle of optimality ("from Slater to lexicographic")

multicriteria linear combinatorial problem is considered, principle of optimality of which is defined by a partitioning of partial criteria onto groups with Slater preference relation within each group and the lexicographic preference relation between them. Quasistability of the problem is investigated. This type of stability is a discrete analog of Hausdorff lower semicontinuity of the many-valued mapping that defines the choice function. A formula of quasistability radius is derived for the case of metric $l_\infty.$ Some conditions of quasistability are stated as corollaries

Stability measure for a generalized assembly line balancing problem

AbstractA generalized formulation for assembly line balancing problem (GALBP) is considered, where several workplaces are associated with each workstation. Thus, all tasks assigned to the same workstation have to be partitioned into blocks: each block regroups all tasks to be performed at the same workplace. The product items visit all workplaces sequentially, therefore, all blocks are proceeded in a sequential way. However, the tasks grouped into the same block are executed simultaneously. As a consequence, the execution of a block takes only the time of its longest task. This parallel execution modifies the manner to take into account the cycle time constraint. Precedence and exclusion constraints also exist for workstations and their workplaces. The objective is to assign all given tasks to workstations and workplaces while minimizing the line cost estimated as a weighted sum of the number of workstations and workplaces. The goal of this article is to propose a stability measure for feasible and optimal solutions of this problem with regard to possible variations of the processing time of certain tasks. A heuristic procedure providing a compromise between the objective function and the suggested stability measure is developed and evaluated on benchmark data sets

A Scenario-Based Parametric Analysis of Stable Marriage Approaches to the Army Officer Assignment Problem

This paper compares linear programming and stable marriage approaches to the assignment problem under conditions of uncertainty. Robust solutions should exhibit reduced variability in the presence of one or more additional constraints. Several variations of each approach are compared with respect to solution quality, as measured by the overall social welfare among Officers and Assignments, and robustness as measured by the number of changes after a number of randomized perturbations. We examine the contrasts between these methods in the context of assigning Army Officers among a set of identified assignments. Additional constraints are modeled after realistic scenarios faced by Army assignment managers, with parameters randomized. The Pareto efficient approaches, relative to these measures of quality and robustness, are identified and subjected to a regression analysis. The coefficients of these models provide insight into the impact the different scenarios under study, as well as inform any trade-off decisions between Pareto-optimal approaches

Advances in Evolutionary Algorithms

With the recent trends towards massive data sets and significant computational power, combined with evolutionary algorithmic advances evolutionary computation is becoming much more relevant to practice. Aim of the book is to present recent improvements, innovative ideas and concepts in a part of a huge EA field