Posynomial Geometric Programming Research Articles

Strong solvability of restricted interval systems and its applications in quadratic and geometric programming

We consider interval systems of linear equations and inequalities with a restriction to some a priori given set. We focus on a characterization of strong solvability, that is, solvability for each realization of interval values, and we compare this with an existence of a strong solution defined analogously. The motivation comes from the area of interval-valued optimization problems, where strong solvability means guaranteed feasibility of any realization of the problem. Strong solvability with strict inequalities implies the robust Slater condition, which ensures that standard optimality conditions can be used. We apply the issues particularly in two optimization classes, convex quadratic programming with quadratic constraints and posynomial geometric programming. For the former, we also utilize the presented result to improve a characterization of the worst case optimal value. Eventually, we state several open problems that emerged while deriving the results.

Solving a posynomial geometric programming problem with fully fuzzy approach

In this paper we have investigated a class of geometric programming problems in which all the parameters are fuzzy numbers. In fact, due to impreciseness of the cost components and exponents in geometric programming with their inherently behavior as in economics and many other areas, we have used fuzzy parametric geometric programming. Transforming the primal problem of fuzzy geometric programming into its dual and using the Zadeh's extension principle, we convert the dual form into a pair of mathematical programs. By applying the ?-cut on the objective function and r-cut on the constraints in dual form of geometric programming, we obtain an acceptable (?, r) optimal values. Then, we further calculate the lower and upper bounds of the fuzzy objective with emphasize on modification of a method presented in [14, 32]. Finally, we illustrate the methodology of the approach with a numerical example to clarify the idea by drawing the different steps of LR representation of Z?r.

Inner approximation algorithm for generalized linear multiplicative programming problems

An efficient inner approximation algorithm is presented for solving the generalized linear multiplicative programming problem with generalized linear multiplicative constraints. The problem is firstly converted into an equivalent generalized geometric programming problem, then some magnifying-shrinking skills and approximation strategies are used to convert the equivalent generalized geometric programming problem into a series of posynomial geometric programming problems that can be solved globally. Finally, we prove the convergence property and some practical application examples in optimal design domain, and arithmetic examples taken from recent literatures and GLOBALLib are carried out to validate the performance of the proposed algorithm.

Single machine scheduling problem with convex multi-resource dependent processing times and job deadlines

In this paper, the single machine scheduling problem with convex multi-resource dependent processing times, subject to meeting job deadlines is considered. The objective is to minimize the total cost, including the resource allocation costs and the fixed costs. We assume that the actual processing time of each job (task) is a function of the amount of resources allocated. Therefore, the decision variables of the model are: 1) resources allocated to the jobs, 2) total consumed resources, 3) processing times of the jobs, and 4) start/completion times of the jobs. We reformulate and solve the problem using a posynomial geometric programming model. In the proposed exact solution method based on the geometric programming, the original problem of any size is reduced to a two-variable unconstrainted optimization problem which can be easily solved by a simple grid search.

An Alternative Approach to the Solution of Multi-Objective Geometric Programming Problems

The aim of this study is to present an alternative approach for solving the multi-objective posynomial geometric programming problems. The proposed approach minimizes the weighted objective function comes from multi-objective geometric programming problem subject to constraints which constructed by using Kuhn-Tucker Conditions. A new nonlinear problem formed by this approach is solved iteratively. The solution of this approach gives the Pareto optimal solution for the multi-objective posynomial geometric programming problem. To demonstrate the performance of this approach, a problem which was solved with a weighted mean method by Ojha and Biswal (2010) is used. The comparison of solutions between two methods shows that similar results are obtained. In this manner, the proposed approach can be used as an alternative of weighted mean method.

Posynomial geometric programming problem subject to max–min fuzzy relation equations

We discuss a class of posynomial geometric programming problem(PGPF), aimed at minimizing a posynomial subject to fuzzy relational equations with max–min composition. By introducing auxiliary variables, we convert the PGPF into an equivalent programming problem whose objective function is a non-decreasing function with an auxiliary variable. We show that an optimal solution consists of a maximum feasible solution and one of the minimal feasible solutions by an equivalent programming problem. In addition, we introduce some rules for simplifying the problem. Then by using a branch and bound method and fuzzy relational equations (FRE) path, we present an algorithm to obtain an optimal solution to the PGPF. Finally, numerical examples are provided to illustrate the steps of the procedure.

Solving geometric programming problems with fuzzy random variable coefficients

Geometric programming problem is a special class of nonlinear programming problem which deals with posynomial type functions. However, the parameters of the problem are sometimes uncertain in nature. Two kinds of uncertainty, fuzziness and randomness, do exist simultaneously in real decision situation. One of the way to model such situation with fuzzy random variable. In this paper, posynomial geometric programming problem with fuzzy random variable coefficients is studied. A fuzzy two stage stochastic programming technique has been developed to handle fuzzy random variable. A numerical example is solved to illustrate the methodology.

New Method to Posynomial Geometric Programming of Trapezoidal Fuzzy Numbers

This paper presents a method for solving posynomial geometric programming with fuzzy coefficients. By utilizing comparison of fuzzy numbers with a method, the programming with fuzzy coefficients is reduced to the programming with constant coefficients. Then the programming with fuzzy coefficients can be solved by using a method for posynomial geometric programming. Finally, one comparative example is used to illustrate advantage of the new method.

The more-for-less paradox in fuzzy posynomial geometric programming

The more-for-less (MFL) problem in fuzzy posynomial geometric programming (FPGP) is advanced in this paper. The research results presented here focus primarily on the nonconvex FPGP in both objective functions and constraint functions. Convexification, quasiconvex, or pseudoconvex, is extended in the sense of an MFL paradox by consolidating the necessary and sufficient conditions. Since the FPGP is equivalent to fuzzy linear programming correspondingly, there exists a solution to the FPGP. Furthermore, the duality or strong duality theorem, the equivalent condition of the MFL paradox and its condition under expansion are examined in detail. It is well known that the fundamental understanding of problems on MFL paradox is of paramount importance to applications of resource allotments and optimal resource management, and correspondingly that the information science and technology advancement play a rule to resource allotments and resource option in management problems. In fact, they are dependent and interwinded.

A Posynomial Geometric Programming Restricted to a System of Fuzzy Relation Equations

A posynomial geometric optimization problem subjected to a system of max-min fuzzy relational equations (FRE) constraints is considered. The complete solution set of FRE is characterized by unique maximal solution and finite number of minimal solutions. A two stage procedure has been suggested to compute the optimal solution for the problem. Firstly all the minimal solutions of fuzzy relation equations are determined. Then a domain specific evolutionary algorithm (EA) is designed to solve the optimization problems obtained after considering the individual sub-feasible region formed with the help of unique maximum solution and each of the minimal solutions separately as the feasible domain with same objective function. A single optimal solution for the problem is determined after solving these optimization problems. The whole procedure is illustrated with a numerical example.

A joint lot-sizing and marketing model with reworks, scraps and imperfect products

In this paper, we establish an economic production quantity (EPQ) based inventory model by considering various types of non-perfect products . We classify products in four groups of perfect, imperfect, defective but reworkable and non-reworkable defective items. The demand is a power function of price and marketing expenditure and production unit cost is considered to be a function of lot size. The objective of this paper is to determine lot size, marketing expenditure, selling price, set up cost and inventory holding cost, simultaneously. The problem is modeled as a nonlinear posynomial geometric programming and an optimal solution is derived. The implementation of the proposed method is demonstrated using a numerical example and the sensitivity analysis is also performed to study the behavior of the model.

Optimal value bounds in nonlinear programming with interval data

We consider nonlinear programming problems the input data of which are not fixed, but vary in some real compact intervals. The aim of this paper is to determine bounds of the optimal values. We propose a general framework for solving such problems. Under some assumption, the exact lower and upper bounds are computable by using two non-interval optimization problems. While these two optimization problems are hard to solve in general, we show that for some particular subclasses they can be reduced to easy problems. Subclasses that are considered are convex quadratic programming and posynomial geometric programming.

Rough Posynomial Geometric Programming

A rough posynomial geometric programming is put forward by the author. This model is advantageous for us to consider questions not only from the quantity of aspect, but from the quality because it contains more information than a traditional geometric programming one. Here, a rough convex function concept is advanced in rough value sets on foundation of rough sets and rough convex sets. Besides, a knowledge expression model in rough posynomial geometric programming is established and so is a mathematical one. Thirdly, solution properties are studied in mathematical model of rough posynomial geometric programming, and antinomy of the more-for-less paradox is solved with an arithmetic in rough posynomial geometric programming given, which can be changed into a rough linear programming after monomial rough posynomial geometric programming is solved. Finally, validity in model and algorithm is verified by examples.

An Original Geometric Programming Problem Algorithm to Solve Two Coefficients Sensitivity Analysis

Problem statement: It has been noticed by Dinkel and Kochenberger tha t they developed sensitivity procedure for Posynomial Geometric Programming Problems based on making a small changes in one coefficient. Approach: This study presented an original algorithm for fin ding the ranging analysis while studying the effect of pertu rbations in the original coefficients without resol ving the problem, this proposed procedure had been trapp ed on two coefficients simultaneously. We also had developed one of the incremental strategies to make suitable comparisons. Results: Comparison results had been done between the gained result fro m the sensitivity analysis approach and the incremental analysis approach. Conclusion: In the standard Geometric Programming Problem, we obtained an original algorithm, for the first time, by changing two coefficients simultaneously in the objective function.

Determining Optimal Demand Rate and Production Decisions: A Geometric Programming Approach

In this paper a nonlinear model is presented to demonstrate the relation between production and marketing departments. By introducing some functions such as pricing cost and market share loss functions it will be tried to show some aspects of market modelling which has not been regarded before. The proposed model will be a constrained signomial geometric programming model. For model solving, after variables’ modifications an iterative technique based on the concept of geometric mean will be introduced to solve the resulting non-standard posynomial model which can be applied to a wide variety of models in non-standard posynomial geometric programming form. At the end a numerical analysis will be presented to accredit the validity of the mentioned model. Keywords—Geometric programming, marketing, nonlinear optimization, production.

Posynomial geometric programming with interval exponents and coefficients

Geometric programming provides a powerful tool for solving nonlinear problems where nonlinear relations can be well presented by an exponential or power function. In the real world, many applications of geometric programming are engineering design problems in which some of the problem parameters are estimates of actual values. This paper develops a solution method when the exponents in the objective function, the cost and the constraint coefficients, and the right-hand sides are imprecise and represented as interval data. Since the parameters of the problem are imprecise, the objective value should be imprecise as well. A pair of two-level mathematical programs is formulated to obtain the upper bound and lower bound of the objective values. Based on the duality theorem and by applying a variable separation technique, the pair of two-level mathematical programs is transformed into a pair of ordinary one-level geometric programs. Solving the pair of geometric programs produces the interval of the objective value. The ability of calculating the bounds of the objective value developed in this paper might help lead to more realistic modeling efforts in engineering optimization areas.

Geometric programming with fuzzy parameters in engineering optimization

Geometric programming provides a powerful tool for solving a variety of engineering optimization problems. Many applications of geometric programming are engineering design problems in which some of the problem parameters are estimates of actual values. When the parameters in the problem are imprecise, the calculated objective value should be imprecise as well. This paper develops a procedure to derive the fuzzy objective value of the fuzzy posynomial geometric programming problem when the exponents of decision variables in the objective function, the cost and the constraint coefficients, and the right-hand sides are fuzzy numbers. The idea is based on Zadeh’s extension principle to transform the fuzzy geometric programming problem into a pair of two-level of mathematical programs. Based on duality algorithm and a simple algorithm, the pair of two-level mathematical programs is transformed into a pair of conventional geometric programs. The upper bound and lower bound of the objective value are obtained by solving the pair of geometric programs. From different values of α, the membership function of the objective value is constructed. Two examples are used to illustrate that the whole idea proposed in this paper.

Fenchel-Lagrange Duality Versus Geometric Duality in Convex Optimization

We present a new duality theory to treat convex optimization problems and we prove that the geometric duality used by Scott and Jefferson in different papers during the last quarter of century is a special case of it. Moreover, weaker sufficient conditions to achieve strong duality are considered and optimality conditions are derived. Next, we apply our approach to some problems considered by Scott and Jefferson, determining their duals. We give weaker sufficient conditions to achieve strong duality and the corresponding optimality conditions. Finally, posynomial geometric programming is viewed also as a particular case of the duality approach that we present.

An efficient genetic algorithm for determining the optimal price discrimination

This paper proposes a genetic algorithm (GA) to solve a price discrimination model developed to determine the product’s selling price in two markets. The model considers the production as a function of price in one market and as a function of price and marketing expenditure in other market. The cost of production is also assumed to be a function of production in both markets. The objective is to maximize the benefit under restricted prices and marketing expenditure in two markets. The considered model is a posynomial geometric programming (GP) problem with continues non-linear/non-convex objective function and linear boundary constraints. The proposed model cannot be solved optimally by a common approach such as geometric programming in a reasonable amount of time. The results show that the proposed GA can become rapidly convergent to near global optimum. Some of typical numerical examples are solved by GA and results are reported. Also, we implement a sensitivity analysis on some of model parameters in order to have a better understanding of the performance and behavior of the proposed GA.

Posynomial geometric programming with parametric uncertainty

Geometric programming provides a powerful tool for solving nonlinear problems where nonlinear relations can be well presented by exponential or power function. This paper develops a procedure to derive the lower and upper bounds of the objective of the posynomial geometric programming problem when the cost and constraint parameters are uncertain. The imprecise parameters are represented by ranges, instead of single values. An imprecise geometric program is transformed to a family of conventional geometric programs to calculate the objective value. The derived result is also in a range, where the objective value would appear. The ability of calculating the bounds of the objective value developed in this paper might help lead to more realistic modeling efforts in engineering design areas.