A Reference Direction Algorithm for Solving Multiple Objective Integer Linear Programming Problems

In this paper, we propose a reference direction approach and an interactive algorithm to solve the general multiple objective integer linear programming problem. At each iteration, only one mixed integer linear programming problem is solved to find an (weak) efficient solution. Each intermediate solution is integer. The decision maker has to provide only the reference point at each iteration. No special software is required to implement the proposed algorithm. The algorithm is illustrated with an example.     

Using cutting planes in an interactive reference point approach for multiobjective integer linear programming problems

We propose an interactive approach for multiple objective integer linear programming (MOILP) problems that combines the use of the Tchebycheff metric with cutting plane techniques. At each interaction, the method computes the nondominated solution for the MOILP problem that is closest to a reference point according to the Tchebycheff metric. The information provided by the decision maker in each dialogue phase is used to adjust the next reference point through a sensitivity analysis stage. Cutting plane techniques enable the method to take advantage of computations performed at previous iterations to solve the next scalarizing integer program. We address both theoretical issues and the computational implementation.     

Fuzzy approaches for multiple objective linear fractional optimization

The problem of finding a solution to a multiple objective linear fractional program arises in several real world situations.In this paper we advocate that fuzzy sets theory provides a basis for solving this problem with sufficient consistency and rigorousness.After representing imprecise aspirations of the decision maker by structured linguistic variables or converting the original problem via approximations or change of variables into a multiple objective linear program, techniques of fuzzy linear programming may be used to reach a satisfactory solution.It is shown that under reasonable restrictions, this solution is efficient (Pareto optimal) for the original problem. Numerical examples are also included for illustration.     

Finding multiple solutions to general integer linear programs

Integer linear programming (ILP) problems occur frequently in many applications. In practice, alternative optima are useful since they allow the decision maker to choose from multiple solutions without experiencing any deterioration in the objective function. This study proposes a general integer cut to exclude the previous solution and presents an algorithm to identify all alternative optimal solutions of an ILP problem. Numerical examples in real applications are presented to demonstrate the usefulness of the proposed method.     

An Interactive Multiple Criteria Approach to Ranking Alternatives

In this paper we are concerned with ranking various orderings of a set of alternatives to a composite order as a multiple criteria problem. The orderings (called preference orderings) can be real preference orderings or any natural orderings. The objective is to find the most preferred order of the decision maker using the preference orderings as criteria.In principle, the problem can be formulated as a multiple objective linear programming problem using the model of Bowman and Colantoni and then solved with the interactive method proposed by Zionts and Wallenius. However, the fact that we are dealing with integer variables prohibits us from applying this approach as such. We discuss the problem formulation and propose a modified approach to that of Zionts and Wallenius for solving the problem.     

Comparative studies in interactive multiple objective mathematical programming

In the past two decades, there has been a significant increase in the number of interactive algorithms proposed for solving multiple objective mathematical programming (MOMP) problems. Most of these procedures have neither been tested in real decision making situations, nor compared to each other. In this study, we emphasize the importance of comparative studies of interactive MOMP procedures and present a state of the art review. Our scope is limited to the comparisons of interactive procedures for solving deterministic, linear, integer or nonlinear constrained multiple objective optimization problems involving a single decision maker.     

The range of the efficient frontier in multiple objective linear programming

Because a rational decision maker should only select an efficient alternative in multiple criterion decision problems, the efficient frontier defined as the set of all efficient alternatives has become a central solution concept in multiple objective linear programming. Normally this set reduces the set of available alternatives of the underlying problem. There are several methods, mainly based on the simplex method, for computing the efficient frontier. This paper presents a quite different approach which uses a nonlinear parametric program, solved by Wolfe's algorithm, to determine the range of the efficient frontier.     

求解多层线性规划的模糊规划法

本文用模糊集理论中的隶属函数描述多层线性规划的各层目标,在第一层给定最小满意水平下,通过求解相应层次的模糊规划来确定各层的最小满意度,从而最终得到问题的一个满意解。提出的方法只需求解一系列线性规划问题,具有较好的计算复杂性和可行性,最后的算例进一步验证了方法的有效性。     

Computing the Pareto frontier of a bi-objective bi-level linear problem using a multiobjective mixed-integer programming algorithm

In this article, we study the bi-level linear programming problem with multiple objective functions on the upper level (with particular focus on the bi-objective case) and a single objective function on the lower level. We have restricted our attention to this type of problem because the consideration of several objectives at the lower level raises additional issues for the bi-level decision process resulting from the difficulty of anticipating a decision from the lower level decision maker. We examine some properties of the problem and propose a methodological approach based on the reformulation of the problem as a multiobjective mixed 0–1 linear programming problem. The basic idea consists in applying a reference point algorithm that has been originally developed as an interactive procedure for multiobjective mixed-integer programming. This approach further enables characterization of the whole Pareto frontier in the bi-objective case. Two illustrative numerical examples are included to show the viability of the proposed methodology.     

An interactive method for multiple-objective mathematical programming problems

An interactive method is developed for solving the general nonlinear multiple objective mathematical programming problems. The method asks the decision maker to provide partial information (local tradeoff ratios) about his utility (preference) function at each iteration. Using the information, the method generates an efficient solution and presents it to the decision maker. In so doing, the best compromise solution is sought in a finite number of iterations. This method differs from the existing feasible direction methods in that (i) it allows the decision maker to consider only efficient solutions throughout, (ii) the requirement of line search is optional, and (iii) it solves the problems with linear objective functions and linear utility function in one iteration. Using various problems selected from the literature, five line search variations of the method are tested and compared to one another. The nonexisting decision maker is simulated using three different recognition levels, and their impact on the method is also investigated.     

DIS-CARD: a new method of multiple criteria sorting to classes with desired cardinality

In this paper, we present a new preference disaggregation method for multiple criteria sorting problems, called DIS-CARD. Real-life experience indicates the need of considering decision making situations in which a decision maker (DM) specifies a desired number of alternatives to be assigned to single classes or to unions of some classes. These situations require special methods for multiple criteria sorting subject to desired cardinalities of classes. DIS-CARD deals with such a problem, using the ordinal regression approach to construct a model of DM’s preferences from preference information provided in terms of exemplary assignments of some reference alternatives, together with the above desired cardinalities. We develop a mathematical model for incorporating such preference information via mixed integer linear programming (MILP). Then, we adapt the MILP model to two types of preference models: an additive value function and an outranking relation. Illustrative example is solved to illustrate the methodology.     

Solving multiple criteria problems by interactive decomposition

An interactive decomposition method is developed for solving the multiple criteria (MC) problem. Based on nonlinear programming duality theory, the MC problem is decomposed into a series of subproblems and relaxed master problems. Each subproblem is a bicriterion problem, and each relaxed master problem is a standard linear program. The prime objective of the decomposition is to simplify and facilitate the process of making preference assessments and tradeoffs across many conflicting objectives. Therefore, the decision-maker's preference function is not assumed to be known explicitly; rather, the decision maker is required to make only limited local preference assessments in the context of feasible and nondominated alternatives. Also, the preference assessments are of the form of ordinal paired comparisons, and in most of them only two criteria are allowed to change their values simultaneously, while the remaining (l–2) criteria are held fixed at certain levels.     

A kind of fuzzy linear programming problems based on interval-valued fuzzy sets

Abstract. The objective of this paper is to deal with a kind of fuzzy linear programming problem based on interval-valued fuzzy sets (IVFLP) through the medium of procedure that turns IVFLP into parametric linear programming via the mathematical programming. Some useful results for the benefit of solving IVFLP are expounded and proved,developed and discussed. Furthermore,that the proposed techniques in this paper allow the decision-maker to assign a different degree of importance can provide a useful way to efficiently help the decision-maker make their decisions.     

Cutting plane algorithms for the inverse mixed integer linear programming problem

We present cutting plane algorithms for the inverse mixed integer linear programming problem (InvMILP), which is to minimally perturb the objective function of a mixed integer linear program in order to make a given feasible solution optimal.     

Stackelberg Solutions to Multiobjective Two-Level Linear Programming Problems

In this paper, we consider a multiobjective two-level linear programming problem in which the decision maker at each level has multiple-objective functions conflicting with each other. The decision maker at the upper level must take account of multiple or infinite rational responses of the decision maker at the lower level in the problem. We examine three kinds of situations based on anticipation of the decision maker at the upper level: optimistic anticipation, pessimistic anticipation, and anticipation arising from the past behavior of the decision maker at the lower level. Mathematical programming problems for obtaining the Stackelberg solutions based on the three kinds of anticipation are formulated and algorithms for solving the problems are presented. Illustrative numerical examples are provided to understand the geometrical properties of the solutions and demonstrate the feasibility of the proposed methods.     

Interactive bicriterion decision support for a large scale industrial scheduling system

In this paper we develop an interactive decision analysis approach to treat a large scale bicriterion integer programming problem, addressing a real world assembly line scheduling problem of a manufacturing company. This company receives periodically a set of orders for the production of specific items (jobs) through a number of specialised production (assembly) lines. The paper presents a non compensatory approach based on an interactive implementation of the ε-constraint method that enables the decision maker to achieve a satisfactory goal for each objective separately. In fact, the method generates and evaluates a large number of non dominated solutions that constitute a representative sample of the criteria ranges. The experience with a specific numerical example shows the efficiency and usefulness of the proposed model in solving large scale bicriterion industrial integer programming problems, highlighting at the same time the modelling limitations.     

An interactive fuzzy satisficing method for structured multiobjective linear fractional programs with fuzzy numbers

In this paper, by considering the experts' vague or fuzzy understanding of the nature of the parameters in the problem formulation process, multiobjective linear fractional programming problems with block angular structure involving fuzzy numbers are formulated. Using the a-level sets of fuzzy numbers, the corresponding nonfuzzy a-multiobjective linear fractional programming problem is introduced. The fuzzy goals of the decision maker for the objective functions are quantified by eliciting the corresponding membership functions including nonlinear ones. Through the introduction of extended Pareto optimality concepts, if the decision maker specifies the degree a and the reference membership values, the corresponding extended Pareto optimal solution can be obtained by solving the minimax problems for which the Dantzig-Wolfe decomposition method and Ritter's partitioning procedure are applicable. Then a linear programming-based interactive fuzzy satisficing method with decomposition procedures for deriving a satisficing solution for the decision maker efficiently from an extended Pareto optimal solution set is presented. An illustrative numerical example is provided to demonstrate the feasibility of the proposed method.     

Using multiple objectives to approximate normative models

This paper discusses the rationale for the use of additive models involving multiple objectives as approximations to normative analyses. Experience has shown us that organizations often evaluate important decisions with multiple objective models rather than reducing all aspects of the problem to a single criterion, dollars, as many normative economic models prescribe. We justify this practice on two grounds: managers often prefer to think about a problem in terms of several dimensions and a multiple objective model may provide an excellent approximation to the more complex normative model. We argue that a useful analysis based on a multiple objective model will fulfill both conditions—it will provide insights for the decision maker as well as a good approximation to the normative model. We report several real-world examples of managers using multiple objective models to approximate such normative models as the risk-adjusted net present value and the value of information models. The agreement between the approximate models and the normative models is shown to be quite good. Next, we cite a portion of the behavioral decision theory literature which establishes that linear models of multiple attributes provide quite robust approximations to individual decision-making processes. We then present more general theoretical and empirical results which support our contention that linear multiple attribute models can provide good approximations to more complex models.     

Interactive scheme for a MOLP problem given two partial orders: One on variables and one on objectives

An interactive approach for solving a multiple objective linear programming (MOLP) problem is presented. This approach assumes that the decision maker (in contrast to the analyst) can establish a (preferential) partial order P_v on the decision variables and a partial order P_o on the objective functions. The solution approach presented in this paper requires minimal additional information in its interactive stage. Results are presented for a number of tests: both conceptual development and applicability in solving specific problems have been successfully tested.     

A concept of a robust solution of a multicriterial linear programming problem

A new concept of a robust solution of a multicriterial linear programming problem is proposed. The robust solution is understood here as the best starting point, prepared while the preferences of the decision maker with respect to the criteria are still unknown, for the adaptation of the solution to the preferences of the decision maker, once they are finally known. The objective is the total cost of the initial preparation and of the later potential adaptation of the solution. In the starting robust solution the decision variables may have interval values. The problem can be solved by means of the simplex algorithm. A numerical example illustrates the approach.