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## Meet the editors

Nodari Vakhania graduated with honors from the Faculty of Applied Mathematics and Cybernetics, Tbilisi State University, Georgia, in 1983. In 1989, he entered the PhD program in Computer Science at the Russian Academy of Sciences, Moscow, and obtained a degree in Mathematical Cybernetics in 1991. In 1992, he was a postdoctoral fellow at the Russian Academy of Sciences, and had a short-term visiting position at the University of Saa-

rbrucken, Germany. In 1995, he became a professor at the Centro de Investigación en Ciencias at the State University of Morelos, Mexico. He also has an honorary position at the Institute of Computational Mathematics of the Georgian Academy of Sciences, where he received a doctoral degree in Mathematical Cybernetics in 2004. His research interests include design and analysis of algorithms, discrete optimization, computational complexity and scheduling theory. He is an author of nearly 100 refereed research papers including more than 60 publications in highly ranked international journals. He has also worked with different scientific committees, including those at the Mexican Science Foundation CONACyT. He is an editorial board member and a referee for a number of international scientific journals. He has obtained research grants and honors in Germany, France, the Netherlands, the United States, Russia, and Mexico, and has given more than forty invited talks throughout the world.

Frank Werner studied Mathematics from 1975 to 1980 and graduated from the Technical University Magdeburg (Germany) with honors. He defended his PhD thesis in 1984 (summa cum laude) and his habilitation thesis in 1989. In 1992, he received a grant from the Alexander von Humboldt Foundation. Currently, he works as Extraordinary Professor at the Faculty of Mathematics of the Otto-von-Guericke University Magdeburg (Ger-

many). He is an author or editor of seven books, and he has published more than 280 papers. He is on the editorial board of seventeen journals; in particular, he is editor-in-chief of *Algorithms* and an associate editor of *International Journal of Production Research* and *Journal of Scheduling*. His research interests include operations research, combinatorial optimization, and scheduling.

Contents

**Section 1**

**Section 2**

**Section 3**

for Problem 1∣*rj*∣*L*max, *C*max

**Preface XI**

Traditional Approach: Pareto Optimality **1**

**Chapter 1 3**

**Chapter 2 15**

Non-Traditional Approach: Threshold Optimality **31**

**Chapter 3 33**

Applications and Overviews **45**

**Chapter 4 47**

**Chapter 5 77**

Pareto Optimality and Equilibria in Noncooperative Games

A Brief Look at Multi-Criteria Problems: Multi-Threshold

Overview of Multi-Objective Optimization Approaches in

On the Practical Consideration of Evaluators' Credibility in Evaluating Relative Importance of Criteria for Some Real-Life

Polynomial Algorithm for Constructing Pareto-Optimal Schedules

*by Vladislav Zhukovskiy and Konstantin Kudryavtsev*

*by Alexander A. Lazarev and Nikolay Pravdivets*

Optimization versus Pareto-Optimization *by Nodari Vakhania and Frank Werner*

Construction Project Management

Multicriteria Problems: An Overview

*by Maznah Mat Kasim*

*by Ibraheem Alothaimeen and David Arditi*

### Contents


Preface

Multi-criteria optimization problems naturally arise in practice when there is no single criterion for measuring the quality of a feasible solution, a solution that satisfies the restrictions of the problem. Since different criteria are contradictory, it is difficult and often impossible to find a single feasible solution that is good for all the criteria, that is, the problem cannot be addressed as a common optimization problem (with a single objective criterion). Hence, some compromise is unavoidable for the solution of multi-criteria optimization problems. A commonly accepted such compromise is to look for a Pareto-optimal frontier of the feasible solutions (i.e., a set of feasible solutions that are not dominated by any other feasible solution with respect to any of the given criteria). Although this is a reasonable compromise, it has two drawbacks. Firstly, finding a Pareto-optimal frontier is often computationally intractable, and secondly, a practitioner may be interested in solutions with some priory given (acceptable) value for each of the objective functions, not in just a set of the non-dominated feasible solutions. Hence, other effective optimality measures for the multi-criteria optimization problems are possible. Theoretical explorations of possible generalizations, relaxations and variations of standard Pareto-optimality principles may lead to robust, and at the same time, flexible and practical measures for multi-criteria optimization (with a "fair balance" for all the given objective criteria). In this book, besides the traditional Pareto-optimality approach (Section 1), we suggest one new alternative approach for the generation of an admissible solution to a multi-criteria optimization problem (Section 2). The book also presents two overview chapters on the existing solution methods for two

real-life, multi-criteria optimization problems (Section 3).

direction and suggest new solution methods.

The first chapter in Section 1 addresses multi-criteria problems that arise in game theory when each player has their own goal that does not coincide with the goal of the other players. The strategy of each player is measured by its payoff function, which value depends not only on the decisions made by the player but also on the decisions of the remaining players. To optimize their goal, the player needs to take into account possible actions of the other players. A game is traditionally referred to as noncooperative if different players cannot coordinate their actions between each other. For a noncooperative game, it is commonly accepted that the Nash equilibrium gives a reasonable solution for all the players, the so-called Nash equilibrium strategy profile. As the authors observe, two different profiles from the set of a Nash equilibrium strategy profiles might not be "equally good," that is, there may exist two different Nash equilibrium strategy profiles such that the payoffs of each player in the first strategy profile are strictly greater than the corresponding payoffs in the second one. Therefore, it is natural to look for a Nash equilibrium strategy that is Pareto optimal with respect to the rest of the Nash equilibrium strategies. The authors, in light of their earlier relevant results, expand their line of research in this

The second chapter of Section 1 considers the Pareto-optimality setting for a bi-criteria machine-scheduling problem. Broadly speaking, the scheduling problems deal with a set of jobs or orders that are to be performed by a set of

resources or machines. There is a basic, traditional resource restriction that a machine
