Efficiency and Optimality in Minisum,Minimax 0-1 Programming Problems

Inspired by an increasing interest in multicriteria 0-1 programming problems in general and by a recent result on the reducibility of minimax to minisum problems in particular, we consider properties of efficient and optimal solutions to two-criteria (minisum and minimax) 0-1 programming problems with any constraint set.A solution procedure is suggested for solving problems whose objective functions are a convex combination of these criteria. The solution properties are illustrated with examples mainly within the context of locational decision problems.     

Solving Fractional Multicriteria Optimization Problems with Sum of Squares Convex Polynomial Data

This paper focuses on the study of finding efficient solutions in fractional multicriteria optimization problems with sum of squares convex polynomial data. We first relax the fractional multicriteria optimization problems to fractional scalar ones. Then, using the parametric approach, we transform the fractional scalar problems into non-fractional problems. Consequently, we prove that, under a suitable regularity condition, the optimal solution of each non-fractional scalar problem can be found by solving its associated single semidefinite programming problem. Finally, we show that finding efficient solutions in the fractional multicriteria optimization problems is tractable by employing the epsilon constraint method. In particular, if the denominators of each component of the objective functions are same, then we observe that efficient solutions in such a problem can be effectively found by using the hybrid method. Some numerical examples are given to illustrate our results.     

A multicriteria approach to the location of public facilities

We argue that practical problems involving the location of public facilities are really multicriteria problems, and ought to be modeled as much. The general criteria are those of cost and service, but there exist several distinct criteria in each of those two categories. For the first category, fixed investment cost, fixed operating cost, variable operating cost, total operating cost, and total discounted cost are all reasonable criteria to consider. In terms of service, both demand served and response time (or distance traveled) are appropriate criteria, either agglomerated or considered on the basis of the individual clients. In this paper we treat such multicriteria questions in the framework of a model for selecting a subset of M sites at which to establish public facilities in order to serve client groups located at N distinct points. We show that for some combinations of specific criteria, parametric solutions of a generalized assignment problem (GAP) will yield all efficient solution. In most other cases the efficient solutions can be found through parametric solution of a GAP with additional constraints of a type which can be incorporated into an existing algorithm for the GAP. Rather than attempting to find all efficient solutions, however, we advocate an interactive approach to the resolution of multicriteria location problems and elaborate on a specific interactive algorithm for multicriteria optimization which for the present model solves a finite sequence of GAP's or GAP-type problems. Finally, some similar aspects of private sector location problems are discussed.     

Generalization of domination structures and nondominated solutions in multicriteria decision making

The concepts of domination structures and nondominated solutions are important in tackling multicriteria decision problems. We relax Yu's requirement that the domination structure at each point of the criteria space be a convex cone (Ref. 1) and give results concerning the set of nondominated solutions for the case where the domination structure at each point is a convex set. A practical necessity for such a generalization is discussed. We also present conditions under which a locally nondominated solution is also a globally nondominated solution.     

Multicriteria identification sets method

A multicriteria identification and prediction method for mathematical models of simulation type in the case of several identification criteria (error functions) is proposed. The necessity of the multicriteria formulation arises, for example, when one needs to take into account errors of completely different origins (not reducible to a single characteristic) or when there is no information on the class of noise in the data to be analyzed. An identification sets method is described based on the approximation and visualization of the multidimensional graph of the identification error function and sets of suboptimal parameters. This method allows for additional advantages of the multicriteria approach, namely, the construction and visual analysis of the frontier and the effective identification set (frontier and the Pareto set for identification criteria), various representations of the sets of Pareto effective and subeffective parameter combinations, and the corresponding predictive trajectory tubes. The approximation is based on the deep holes method, which yields metric ε-coverings with nearly optimal properties, and on multiphase approximation methods for the Edgeworth–Pareto hull. The visualization relies on the approach of interactive decision maps. With the use of the multicriteria method, multiple-choice solutions of identification and prediction problems can be produced and justified by analyzing the stability of the optimal solution not only with respect to the parameters (robustness with respect to data) but also with respect to the chosen set of identification criteria (robustness with respect to the given collection of functionals).     

A multiobjective decomposition-coordination framework with an application to vehicle layout and configuration design

This short paper addresses both researchers in multiobjective optimization as well as industrial practitioners and decision makers in need of solving optimization and decision problems with multiple criteria. To enhance the solution and decision process, a multiobjective decomposition-coordination framework is presented that initially decomposes the original problem into a collection of smaller-sized subproblems that can be solved for their individual solution sets. A common solution for all decomposed and, thus, the original problem is then achieved through a subsequent coordination mechanism that uses the concept of epsilon-efficiency to integrate decisions on the desired tradeoffs between these individual solutions. An application to a problem from vehicle configuration design is selected for further illustration of the results in this paper and suggests that the proposed method is an effective and promising new solution technique for multicriteria decision making and optimization. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The set of all nondominated solutions in linear cases and a multicriteria simplex method

In this note we are interested in the properties of, and methods for locating the set of all nondominated solutions of multiple linear criteria defined over a polyhedron. We first show that the set of all dominated solutions is convex and that the set of all nondominated solutions is a subset of the convex hull of the nondominated extreme points. When the domination cone is polyhedral, we derive a necessary and sufficient condition for a point to be nondominated. The condition is stronger than that of Ref. [1] and enables us to give a simple proof that the set of all nondominated extreme points indeed is connected. In order to locate the entire set of all nondominated extreme points, we derive a generalized version of simplex method—multicriteria simplex method. In addition to some useful results, a necessary and sufficient condition for an extreme point to be nondominated is derived. Examples and computer experience are also given. Finally, we focus on how to generate the entire set of all nondominated solutions through the set of all nondominated extreme points. A decomposition theorem and some necessary and sufficient conditions for a face to be nondominated are derived. We then describe a systematic way to identify the entire set of all nondominated solutions. Through examples, we show that in fact our procedure is quite efficient.     

A Transportation model with multiple criteria and multiple constraint levels

In this paper, a transportation model with multiple criteria and multiple constraint levels (MC²) is formulated by using the framework of MC² linear programming. An algorithm is developed to solve such MC² transportation problems. In this algorithm, the traditional northwest corner rule is adopted to find an initial basic feasible solution for a given MC² transportation problem. Then the MC²-simplex method is applied to locate the set of all potential solutions over possible changes of the objective coefficient parameter and the supply and demand parameter for the MC² transportation problem. A numerical example is illustrated to demonstrate the applicability of the algorithm in solving the MC² transportation problems.     

Ersatz function method for minimizing a finite-valued function on a compact set

A method is proposed for solving optimization problems with continuous variables and a function taking a large finite set of values. Problems of this type arise in the multicriteria construction of a control rule for a discrete-time dynamical system whose performance criteria coincide with the number of violations of requirements imposed on the system. The rule depends on a finite set of parameters whose set of admissible values defines a collection of admissible control rules. An example is the problem of choosing a control rule for a cascade of reservoirs. The optimization method is based on solving a modified problem in which the original function is replaced by a continuous ersatz function. A theorem on the relation between the average-minimal values of the original and ersatz functions is proved. Optimization problems are solved with power-law ersatz functions, and the influence exerted by the exponent on the quality of the solution is determined. It is experimentally shown that the solutions produced by the method are of fairly high quality.     

Deriving crisp and consistent priorities for fuzzy AHP-based multicriteria systems using non-linear constrained optimization

Fuzzy optimization models are used to derive crisp weights (priority vectors) for the fuzzy analytic hierarchy process (AHP) based multicriteria decision making systems. These optimization models deal with the imprecise judgements of decision makers by formulating the optimization problem as the system of constrained non linear equations. Firstly, a Genetic Algorithm based heuristic solution for this optimization problem is implemented in this paper. It has been found that the crisp weights derived from this solution for fuzzy-AHP system, sometimes lead to less consistent or inconsistent solutions. To deal with this problem, we have proposed a consistency based constraint for the optimization models. A decision maker can set the consistency threshold value and if the solution exists for that threshold value then crisp weights can be derived, otherwise it can be concluded that the fuzzy comparison matrix for AHP is not consistent for the given threshold. Three examples are considered to demonstrate the effectiveness of the proposed method. Results with the proposed constraint based fuzzy optimization model are more consistent than the existing optimization models.     

Hyper-Sparsity in the Revised Simplex Method and How to Exploit it

The revised simplex method is often the method of choice when solving large scale sparse linear programming problems, particularly when a family of closely-related problems is to be solved. Each iteration of the revised simplex method requires the solution of two linear systems and a matrix vector product. For a significant number of practical problems the result of one or more of these operations is usually sparse, a property we call hyper-sparsity. Analysis of the commonly-used techniques for implementing each step of the revised simplex method shows them to be inefficient when hyper-sparsity is present. Techniques to exploit hyper-sparsity are developed and their performance is compared with the standard techniques. For the subset of our test problems that exhibits hyper-sparsity, the average speedup in solution time is 5.2 when these techniques are used. For this problem set our implementation of the revised simplex method which exploits hyper-sparsity is shown to be competitive with the leading commercial solver and significantly faster than the leading public-domain solver.     

Criteria and dimension reduction of linear multiple criteria optimization problems

Two of the main approaches in multiple criteria optimization are optimization over the efficient set and utility function program. These are nonconvex optimization problems in which local optima can be different from global optima. Existing global optimization methods for solving such problems can only work well for problems of moderate dimensions. In this article, we propose some ways to reduce the number of criteria and the dimension of a linear multiple criteria optimization problem. By the concept of so-called representative and extreme criteria, which is motivated by the concept of redundant (or nonessential) objective functions of Gal and Leberling, we can reduce the number of criteria without altering the set of efficient solutions. Furthermore, by using linear independent criteria, the linear multiple criteria optimization problem under consideration can be transformed into an equivalent linear multiple criteria optimization problem in the space of linear independent criteria. This equivalence is understood in a sense that efficient solutions of each problem can be derived from efficient solutions of the other by some affine transformation. As a result, such criteria and dimension reduction techniques could help to increase the efficiency of existing algorithms and to develop new methods for handling global optimization problems arisen from multiple objective optimization.     

Theory of convex cones in multicriteria decision making

We develop the theory of convex polyhedral cones in the objective-function space of a multicriteria decision problem. The convex cones are obtained from the decision-maker's pairwise judgments of decision alternatives and are applicable to any quasiconcave utility function. Therefore, the cones can be used in any progressively articulated solution procedure that employs pairwise comparisons. The cones represent convex sets of solutions that are inferior to known solutions to a multicriteria problem. Therefore, these convex sets can be eliminated from consideration while solving the problem. We develop the underlying theory and a framework for representing knowledge about the decision-maker's preference structure using convex cones. This framework can be adopted in the interactive solution of any multicriteria problem after taking into account the characteristics of the problem and the solution procedure. Our computational experience with different multicriteria problems shows that this approach is both viable and efficient in solving practical problems of moderate size.     

A provably convergent heuristic for stochastic bicriteria integer programming

We propose a general-purpose algorithm APS (Adaptive Pareto-Sampling) for determining the set of Pareto-optimal solutions of bicriteria combinatorial optimization (CO) problems under uncertainty, where the objective functions are expectations of random variables depending on a decision from a finite feasible set. APS is iterative and population-based and combines random sampling with the solution of corresponding deterministic bicriteria CO problem instances. Special attention is given to the case where the corresponding deterministic bicriteria CO problem can be formulated as a bicriteria integer linear program (ILP). In this case, well-known solution techniques such as the algorithm by Chalmet et al. can be applied for solving the deterministic subproblem. If the execution of APS is terminated after a given number of iterations, only an approximate solution is obtained in general, such that APS must be considered a metaheuristic. Nevertheless, a strict mathematical result is shown that ensures, under rather mild conditions, convergence of the current solution set to the set of Pareto-optimal solutions. A modification replacing or supporting the bicriteria ILP solver by some metaheuristic for multicriteria CO problems is discussed. As an illustration, we outline the application of the method to stochastic bicriteria knapsack problems by specializing the general framework to this particular case and by providing computational examples.     

A o(n logn) algorithm for LP knapsacks with GUB constraints

A specialization of the dual simplex method is developed for solving the linear programming (LP) knapsack problem subject to generalized upper bound (GUB) constraints. The LP/GUB knapsack problem is of interest both for solving more general LP problems by the dual simplex method, and for applying surrogate constraint strategies to the solution of 0–1 Multiple Choice integer programming problems. We provide computational bounds for our method of o(n logn), wheren is the total number of problem variables. These bounds reduce the previous best estimate of the order of complexity of the LP/GUB knapsack problem (due to Witzgall) and provide connections to computational bounds for the ordinary knapsack problem.We further provide a variant of our method which has only slightly inferior worst case bounds, yet which is ideally suited to solving integer multiple choice problems due to its ability to post-optimize while retaining variables otherwise weeded out by convex dominance criteria.     

On Optimization over the Efficient Set in Linear Multicriteria Programming

The efficient set of a linear multicriteria programming problem can be represented by a reverse convex constraint of the form g(z)≤0, where g is a concave function. Consequently, the problem of optimizing some real function over the efficient set belongs to an important problem class of global optimization called reverse convex programming. Since the concave function used in the literature is only defined on some set containing the feasible set of the underlying multicriteria programming problem, most global optimization techniques for handling this kind of reverse convex constraint cannot be applied. The main purpose of our article is to present a method for overcoming this disadvantage. We construct a concave function which is finitely defined on the whole space and can be considered as an extension of the existing function. Different forms of the linear multicriteria programming problem are discussed, including the minimum maximal flow problem as an example. The research was partly done while the third author was visiting the Department of Mathematics, University of Trier with the support by the Alexander von Humboldt Foundation. He thanks the university as well as the foundation.     

On Analyses of the Solution to a Linear Programming Problem

Many complex problem situations in various contexts have been represented in recent years by the linear programming model. The simplex method can then be used to give the optimal values of the variables corresponding to a given set of values of the parameters. However, in many situations it is useful to have the solution to many other related problems which differ from the original problem only in the values of some of the parameters. This paper presents procedures by which the solutions to the changed problems can be derived from the simplex solution tableau corresponding to the original problem. The method will be illustrated by means of an example problem, and it will be shown how quantitative information obtained from such analyses can aid management in decision making.     

Multicriteria equilibrium programming: Extragradient method

Multicriteria equilibrium programming includes as its particular cases mathematical programming, saddle point calculation, the multicriteria search for Pareto solutions, minimization with an equilibrium choice of the feasible set, etc. An extragradient method is proposed for the numerical solution of the multicriteria equilibrium programming problem, and the convergence of this method is examined.     

Definition of the feasible solution set in multicriteria optimization problems with continuous, discrete, and mixed design variables

Engineering optimization problems are multicriteria with continuous, discrete, and mixed design variables. Correct definition of the feasible solution set is of fundamental importance in these problems. It is quite difficult for the expert to define this set. For this reason, the results of searching for optimal solutions frequently have no practical meaning. Furthermore, correct definition of this set makes it possible to significantly reduce the time of searching for optimal solutions. This paper describes construction of the feasible solution set with continuous, discrete, and mixed design variables on the basis of Parameter Space Investigation (PSI) method.