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).

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 direction and suggest new solution methods.

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 can handle at most one job at a time, and there may be additional restrictions on the ways of how the jobs can be scheduled on the machines, which define a set of feasible solutions. Moreover, we have one or more objective functions defined on the set of feasible solutions that are to be minimized or maximized, as in any other discrete optimization problem. This chapter considers a single-machine environment and aims to minimize two objective functions: the maximum job lateness and the maximum job completion time. Since the problem of finding the Pareto-optimal set of feasible solutions for this bi-criteria problem is NP-hard, the authors consider a special case when a polynomial-time solution is possible.

The second section consists of a chapter that introduces an alternative optimality measure for multi-criteria optimization problems. The new approach is motivated by the observation that, in practical circumstances, there may exist different tolerance/requirements to the quality of the desired solution for different objective criteria. In other words, for some objective criteria, solutions that are far away from an optimal one can be acceptable, whereas for some other criteria, near-optimal solutions are required. Hence, a uniform homogeneous Pareto-optimality approach may not be good from the point of view of the practical needs, and the computational point of view, since most Pareto-optimality problems are known to be intractable. Even if the Pareto-optimal set of feasible solutions is created, it may not be computationally efficient to choose an appropriate solution from the Paretooptimal set of feasible solutions. The multi-threshold optimization setting suggested in this chapter takes into account different requirements for different objective criteria. Hence, a single feasible solution with an admissible value for each objective function can be generated with the computational cost, inferior to the cost of finding the corresponding Pareto-optimal feasible set.

The last section consists of two survey chapters. The first overviews the multicriteria optimization problems that arise in construction projects. In such projects, there are different contradictory criteria and it is a challenging question to meet all of them. Hence, multi-criteria optimization methods are applied for the solution of such problems. The chapter surveys the solution methods including different metaheuristic and genetic algorithms and integer programming methods. The last chapter overviews some work that relates a practical problem of measuring an expert's credibility in evaluation of the importance of different criteria with the multicriteria optimization problems. In particular, the situation when different criteria may have different importance (which is represented by the corresponding weights) is considered. Common methods for determining the relative importance of each criteria and the corresponding feasible solutions are briefly described.

> **Nodari Vakhania** Centro de Investigación en Ciencias, UAEM, Mexico

> > **Frank Werner** Otto-von-Guericke University, Germany

Section 1

Traditional Approach:

Pareto Optimality

Section 1
