Exact Methods for Multi-Objective Integer Nonlinear Programming

An improved version of the augmented ε-constraint method (AUGMECON2) for finding the exact pareto set in multi-objective integer programming problems

This paper deals with multiobjective integer programming problems by considering fuzzy goals of the decision maker for objective functions. After determining the fuzzy goals of the decision maker, if the decision maker specifies the reference membership values, the corresponding Pareto optimal solution can be obtained by solving the augmented minimax problem which becomes an integer programming problem. For solving the problem, decoding algorithms for 0-1 programming problems are revised and ringed double strings are also introduced. Then an interactive fuzzy satisficing method is presented together with an illustrative numerical example.

In this paper, we present a multi-objective fractional probabilistic programming problem where the parameters in the objective functions and right-hand side constraints are multi-choice in nature. Multi-choice multi-objective objective fractional probabilistic programming problem is formulated. The multi-choice parameters in the right-hand side of constraints are random variables following Cauchy distribution. Multi-choice multi-objective fractional probabilistic programming problem is transformed to deterministic equivalent multi-choice multi-objective fractional programming problem using chance-constrained programming problem. Lagrange interpolating polynomial approach is proposed to deal with multi-choice parameters. The multi-choice multi-objective fractional programming problem is transformed to non-linear mixed integer multi-objective programming problem. An \(\epsilon -\) constrained method is applied to find the compromise solutions of non-linear mixed integer multi-objective programming problem using different values of \(\epsilon \). LINGO software is applied to find the optimal solution of single objective non-linear mixed integer programming problem. Finally, a numerical example is given to illustrate the proposed programming problem.KeywordsMulti-choice programming problemMulti-objective programming problemProbabilistic programming problemFractional programming problem\(\epsilon \)-constraint method

This work is geared toward a real-world manufacturing planning (MP) task, whose two objectives are to maximize the order fulfillment rate and minimize the total cost. More important, the requirements and constraints in real manufacturing make the MP task very challenging in several aspects. For example, the MP needs to cover many production components of multiple plants over a 30-day horizon, which means that it involves a large number of decision variables. Furthermore, the MP task's two objectives have extremely different magnitudes, and some constraints are difficult to handle. Facing these uncompromising practical requirements, we introduce an interactive multiobjective optimization-based MP system in this article. It can help the decision maker reach a satisfactory tradeoff between the two objectives without consuming massive calculations. In the MP system, the submitted MP task is modeled as a multiobjective integer programming (MOIP) problem. Then, the MOIP problem is addressed via a two-stage multiobjective optimization algorithm (TSMOA). To alleviate the heavy calculation burden, TSMOA transforms the optimization of the MOIP problem into the optimization of a series of single-objective problems (SOPs). Meanwhile, a new SOP solving strategy is used in the MP system to further reduce the computational cost. It utilizes two sequential easier SOPs as the approximator of the original complex SOP for optimization. As part of the MP system, TSMOA and the SOP solving strategy are demonstrated to be efficient in real-world MP applications. In addition, the effectiveness of TSMOA is also validated on benchmark problems. The results indicate that TSMOA as well as the MP system are promising.

An Interactive Fuzzy Satisficing Method for Random Fuzzy Multiobjective Integer Programming Problems through Probability Maximization with Possibility

The aim of this paper is twofold. Firstly, to define a solution concept of Pareto-optimality for a multi-objective flexible linear programming (MOFLP) problem (or multi-objective fuzzy linear programming problem) and design a method to extract a Pareto-optimal solution of MOFLP problem from the set of optimal solutions of equivalent optimization problem formulated by Dubey and Mehra (2013). Secondly, to extend this study to multi-objective linear programming problem involving hard and flexible constraints with interval uncertainty. A flexible constraint with interval uncertainty generalizes a flexible constraint by allowing preferences to be expressed in the form of intervals. An optimistic-pessimistic approach is proposed to solve multi-objective flexible linear programming with interval uncertainty (MOFLPIU) using an interval-valued fuzzy set representation and the Hurwicz optimism-pessimism criterion.

Optimization problems with multiple criteria measuring solution quality can be modeled as multiobjective programming problems. Because the objective functions are usually in conflict, there is not a single feasible solution that can optimize all objective functions simultaneously. An optimal solution is one that is most preferred by the decision maker (DM) among all feasible solutions. An optimal solution must be nondominated but a multiobjective programming problem may have, possibly infinitely, many nondominated solutions. Therefore, tradeoffs must be made in searching for an optimal solution. Hence, the DM's preference information is elicited and used when a multiobjective programming problem is solved. The model, concepts and definitions of multiobjective programming are presented and solution methods are briefly discussed. Examples are used to demonstrate the concepts and solution methods. Graphics are used in these examples to facilitate understanding.

In sample surveys, stratified sampling is useful if the strata weights are known for each stratum. If they are not known double sampling may be used by selecting a large preliminary sample to estimate the strata weights. Then a stratified sample may be selected independently or from the initial sample. If the problem of non-response is also present then the strata are to be virtually divided into two disjoint and exhaustive groups of respondents and non-respondents. A subsamples from non-respondents is then selected and second more extensive attempt is to be group is to be made for obtaining the required information. The problem of obtaining a compromise allocation for first and second phase of sampling is the formulated as a multi-objective non- linear programming problem that minimizes the sum of variances of the stratified sample mean subject to the non-linear cost constraint. The formulated problem is solved using fuzzy programming and fuzzy goal programming based on piecewise linear approximation. A numerical example is presented to illustrate the computational details. Keywords: Non-response, Multivariate Stratified Sampling, Multiobjective Integer non-linear programming problem, Fuzzy Programming, Travel cost

On computational methods for solutions of multiobjective linear production programming games

Particle swarm optimization (PSO) was proposed by Kennedy et al. as a general approximate solution method for nonlinear programming problems. Its efficiency has been shown, but there have been left some shortcomings of the method. Thus, the authors proposed a revised PSO (rPSO) method incorporating the homomorphous mapping and the multiple stretching in order to cope with these shortcomings. In this paper, we construct an interactive fuzzy satisficing method for multiobjective nonlinear programming problems based on the rPSO. Furthermore, in order to obtain better solutions in consideration of the property of multiobjective programming problems, we incorporate the direction to nondominated solutions into the rPSO. Finally, we show the efficiency of the proposed method by applying it to numerical examples

In this paper, we propose a linear programming based interactive method for multiobjective linear programming problems, in which fuzzy coefficients and random variable coefficients are involved in the objective functions simultaneously. In the proposed method, it is assumed that the decision maker has a fuzzy goal for each objective function, and such a fuzzy goal can be quantified by eliciting the membership function. Through the possibility measure and a fractile optimization model, the original problem is transformed to the well-defined multiobjective programming problem. Then, a generalized Pareto optimal concept is defined, and a linear programming based interactive algorithm is proposed to obtain a satisfactory solution from among a generalized Pareto optimal solution set.

A neural network approach to multiobjective and multilevel programming problems

In Chapter 8 we described a sampling based procedure for stochastic multiobjective mathematical programming problems. In this chapter we will present a procedure which also uses sampling but in a more structured approach. The method presented here is an interactive procedure based on Box’s complex search. It requires the decision maker to provide progressive articulation of preferences. This method has an advantage over the interactive methods described in Chapter 8 in that the decision maker’s underlying value function need not be explicitly specified. Also, the problem may have nonlinear objective functions and nonlinear constraints as well as objective function coefficients that are random variables.

Convergence of exponential penalty function method for multiobjective fractional programming problems

In this paper, an iterative technique based on the use of parametric functions is proposed to obtain the best preferred optimal solution of a multi-objective linear fractional programming problem. The decision maker ascertains own desired tolerance values for the objectives as termination constants and imposes them on each iteratively computed objective functions in terms of termination conditions. Each fractional objective is transformed into non-fractional parametric function using certain initial values of parameters. The parametric values are iteratively computed and $$\epsilon $$ -constraint method is used to obtain the pareto (weakly) optimal solutions in each step. The computations get terminated when all the termination conditions are satisfied at a pareto optimal solution of an iterative step. A numerical example is discussed at the end to illustrate the proposed method and fuzzy max–min operator method is applied to validate the obtained results.