A generalized equilibrium value-based approach for solving fuzzy programming problem

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

This paper deals with a fuzzy goal programming methodology for solving multiobjective linear fractional chance constrained programming problems containing fuzzy numbers and exponentially distributed fuzzy random variables associated with the system constraints. In model formulation process, the problem is converted into an equivalent fuzzy programming model by applying chance constrained programming technique in a fuzzily defined probabilistic decision making situation. Then using the concept of α-cut for fuzzy numbers and by considering the tolerance level of fuzzy numbers the problem reduces to an equivalent sub problem with interval coefficients. In this method the convex combination of each interval is used and the problem is reduced to a nonlinear programming problem. Finally the model is solved by converting nonlinear model into its equivalent multiobjective linear programming model by using Taylor series; and a priority based fuzzy goal programming method is used for achievement of the highest membership degree. To demonstrate the efficiency of the proposed technique an illustrative example, studied previously, is solved and the solution is compared with the existing methodology.

A genetic algorithm (GA) based approach to solve a fuzzy multi-objective nonlinear programming (FMNP) problem is presented and applied to water resources allocation. A FMNP model of regional water allocation is established originally. The objective functions of the model include three conflicting goals which are economic profits, environment goal and society goal. By turning the FMNP to an optimal level cut problem, the ldquofuzzy optimal solutionrdquo is solved by using a genetic algorithm. An example of the Dongyingpsilas water allocation strategy is provided by using the proposed method.

The concept of duality plays an important role in optimization theory. This paper discusses some relations between primal and dual nonlinear programming problems in fuzzy environment. Here, fuzzy feasible region for a general fuzzy nonlinear programming is formed and the concept of fuzzy feasible solution is defined. First order dual relation for fuzzy nonlinear programming problem is studied.

Here, we propose a mathematical model to analyze the multi-objective bimatrix game under entropy environment. In this new approach, the entropy function of each player is considered as an objective to the multi-objective bimatrix game. This model is named as multi-objective entropy bimatrix game model. The formulated model is a multi-objective non-linear programming problem. To solve this model, we have introduced a solution procedure, which determines the feasibility of multi-objective non-linear programming problem using fuzzy programming approach and revised Genetic Algorithm (GA). By a real coded GA we have obtained the bounds of the objectives of the said model. Then we have determined the solutions of our proposed model. Finally, numerical examples are presented to illustrate the methodology.

Recently, Ghanbari and Mahdavi-Amiri (Appl Math Model 34:3363–3375, 2010) gave the general compromised solution of an LR fuzzy linear system using ABS algorithm. Here, using this general solution, we solve quadratic programming problems with fuzzy LR variables. We convert fuzzy quadratic programming problem to a crisp quadratic problem by using general solution of fuzzy linear system. By using this method, the crisp optimization problem has fewer variables in comparison with other methods, specially when rank of the coefficient matrix is full. Thus, solving the fuzzy quadratic programming problem by using our proposed method is computationally easier than the solving fuzzy quadratic programming problem by using ranking function. Also, we study the fuzzy quadratic programming problem with symmetric variables. We show that, in this case, the associate quadratic programming problem is a convex problem, and thus, we able to find the global optimal.

Effective and comprehensive evaluation of cold stress is critical for healthy dairy cow breeding in the winter. Previous studies on dairy cow cold stress have considered thermal environmental factors but not physiological factors or air quality. Therefore, this study aimed to propose a multilevel fuzzy comprehensive evaluation (FCE) method for cold stress in dairy cows based on the analytic hierarchy process (AHP) and a genetic algorithm (GA). First, the AHP was used to construct an evaluation index system for cold stress in dairy cows from 3 dimensions: thermal environment (temperature, relative humidity, wind speed, and illumination), physiological factors (respiratory rate, body surface temperature), and air quality [NH3, CO2, inhalable particulate matter (PM10)]. Second, the consistency test of the judgment matrix was transformed into a nonlinear constrained optimization problem and solved using the GA. Next, based on fuzzy set theory, the comment set and membership function were established to classify the degree of cold stress into 5 levels: none, mild, moderate, high, and extreme. Then, the degree of cold stress in cows was obtained using multilevel fuzzy comprehensive judgment. To investigate the effect of illumination indicators on cold stress in dairy cows, 24 prelactation cows from the south and north sides were selected for a 117-d comprehensive cold stress evaluation. The results showed that the mean mild cold stress durations were 605.3 h (25.22 d) and 725.5 h (30.23 d) and the moderate cold stress durations were 67.2 h (2.8 d) and 96 h (4.0 d) on the south and north sides, respectively. Simultaneously, generalized linear mixed model showed that there were significant correlations between the daily cold stress duration and milk yield, feeding time, lying time, and active steps in the cows on both sides. This method can reasonably indicate cow cold stress conditions and better guide cold protection practices in actual production.

We formulate fuzzy multiobjective integer programming problems considering the vagueness or ambiguity of the decision maker as a human being and introduce an interactive fuzzy satisficing method into them. As a result, the problem to be solved turns out to be an ordinary integer programming problem. For integer programming problems, Sakawa et al. (1997) proposed an approximate solution method based on genetic algorithms using double string representation, but it calls for more improvement on accuracy and processing time. Thus, we attempt to make use of information about solutions of continuous relaxation problems in the genetic algorithm proposed by Sakawa et al., since it is expected to be useful to search the (approximate) optimal solution of the integer programming problem.

In this paper, the triangular fuzzy bilevel programming problem (TFBLPP) is investigated. We transformed TBFLPP to interval fuzzy bilevel linear programming problem by using nearest interval approximation. In the next step, three problem were obtained that by solving them solution TBFLPP obtained.

A fuzzy robust nonlinear programming model is developed for the assessment of filter allocation and replacement strategies in hydraulic systems under uncertainty. It integrates the methods of fuzzy mathematic programming (FMP) and robust programming (RP) within the mixed integer nonlinear programming framework, and can facilitate dynamic analysis and optimization of filters allocation and replacement planning where the uncertainties are expressed as fuzzy membership functions. In modeling formulation, theory of contamination wear of typical hydraulic components is introduced to strengthen the presentation of relationship between system contamination and work performance. The fuzzy decision space is delimited into a more robust one by specifying the uncertainties through dimensional enlargement of the original fuzzy constraints. The piecewise linearization approach is employed to handle the nonlinearities of problem. The developed method has been applied to a case of planning filter allocation and replacement strategies under uncertainty and the generated optimal solution will help to reduce the total system cost and failure-risk level of the FPS.

In the present paper, a fuzzy programming model with quadratic membership functions has been developed for the solution of a Multi-Objective Transportation problem. In literature, several fuzzy programming approaches exist with various types of membership functions such as linear, exponential, hyperbolic etc. These membership functions are defined, by taking the lower and upper values of the objective functions into account. In some cases, these methods fail to obtain an integer compromise optimal solution. In the present method, two coefficients of the quadratic membership functions are determined by the lower and upper values of the objective functions. The other coefficient is taken as a variable in the fuzzy programming approach. This means that the membership curve is fixed at the two end points and set free in between. Application of the method on numerical examples proved that the approach could generate integer compromise optimal solutions.

In this article a comparative study on the effectiveness of Real-coded Genetic Algorithm (RGA) approach in finding the compromised near-to-optimal solution and to by-pass the allocation rule of (m+n−1) for a Fuzzified Multi-Objective Transportation Problem (FMOTP) is discussed. For this purpose, some benchmark problems with well-defined membership function were considered and several changes have been made in the execution of Genetic Algorithm with real codes. The performance of the RGA Approach in obtaining the compromised near-to-optimal solution is compared with the traditional methods, namely as Interactive approach and Fuzzy programming approach. The obtained results reveal that, the applied RGA technique is more advantageous and effective after incorporating some changes according to the problem statement. This approach gives more than one compromised near-to-optimal solutions as well as bypasses the allocation rule of (m +n − 1), which is a great advantage in solving FMOTP. Two numerical examples have been considered and discussed to demonstrate this advent.

The downstream supply interruption of manufacturers is a disaster for the company when the demand is uncertain in the market; a fuzzy programming with fuzzy parameters model of supply interruption supply chain network is established by simulating market operation rules. The aim of the current study is to build a fuzzy chance-constrained programming method which is developed for supporting the uncertainty of demand. This method ensured that the fuzzy constraints can be satisfied at specified confidence levels, leading to cost-effective solutions under acceptable risk magnitudes. Finally, through the case of the electronic product manufacturing enterprise, the feasibility and effectiveness of the proposed model are verified by adopting a sensitivity analysis of capacity loss level and minimizing objective function. Numerical simulation shows that selecting two manufacturing centers can effectively reduce the supply chain cost and maintain business continuity.

Genetic algorithm for non-linear mixed integer programming problems and its applications

Most of the manufacturing enterprises indulge in the bonding of metals during the production process. This makes welding one of the most important processes in industries. Subsequently, due to the high usage of welding process, industrial engineers desire to optimize the parameters concerned to achieve the desired weld bead characteristics. This paper focuses on optimization of flux cored arc welding process parameters, which are used for deposition of duplex stainless steel on low carbon structural steel plates. Experiments were conducted based on central composite rotatable design and mathematical models were developed using multiple regression method. Further, optimization with objectives as minimizing percentage dilution, maximizing height of reinforcement and bead width was carried out using genetic algorithm and memetic algorithm. This problem was formulated as a multi objective, multivariable and non-linear programming problem. The algorithms were implemented using basic functions of C language making it highly reliable, adoptable, very user friendly and extendable to other welding processes such as GMAW, GTAW, robotic welding, etc. The adopted optimization techniques were further compared based on various computational factors.