SOLVING FUZZY NONLINEAR PROGRAMMING PROBLEMS WITH EXTERIOR PENALTY FUZZY VALUED FUNCTIONS

A routing optimization with risk control and reduction provides a quantitative support for an efficient organization of the hazardous materials transportation where road-rail multimodal transportation plays an important role. Oriented on the viewpoint of the multimodal operator, this study investigates a hazardous materials routing problem for their door-to-door transportation through a road-rail multimodal hub-and-spoke network. It is difficult to determine the exact demand of hazardous materials for each transportation order during the advanced routing decision making. Therefore, demand uncertainty is introduced into the problem, in which the uncertain demands are modeled by triangular fuzzy numbers. To enhance the quality of service of the transportation, the hazardous materials routing optimization in this study uses fuzzy soft time windows to optimize the last-mile delivery of hazardous materials that determines the timeliness of transportation. Accordingly, this study proposes a fuzzy multi-objective routing model that is linear and considers the constraints of the customers' preferred service levels. To solve the proposed model, its crisp reformulation is first achieved by using the fuzzy expected value method and Jimenez's fuzzy ranking method. Then, the ε-constraint method is adopted by this study to obtain the Pareto frontier for the multi-objective optimization problem. Finally, an experimental case study is carried out to verify the proposed modeling of this study. By using sensitivity analysis and fuzzy simulation, the conflicting relationship among the economic, risk, reliability and timeliness objectives that improving one objective worsens the others is clearly indicated, and the insights that helps decision makers to make effective tradeoffs among these objectives to get a most suitable hazardous materials road-rail multimodal transportation plan are also summarized.

Multi-criteria decision making method is used in such cases having multiple targets. Multi-objective linear programming problems, which deal with uncertain measurements for both objectives and constraints, are solved by fuzzy multi-objective linear programming methods. Analytic hierarchy process (AHP), one of the multi-criteria decision-making methods based on weighting of objectives, is designed to solve complex problems. In this study, multiple criteria decision making problems and decision makers’ opinions on these problems as well as their solution processes are discussed. Firstly, a fuzzy multi objective linear programming problem is solved using the approaches of Zimmerman and Hybrid. In the literature, these methods are applied using triangular fuzzy numbers based on decision makers’ opinions only in cases where the objective function is minimization. In this study, as an alternative to the triangular fuzzy numbers, trapezoidal fuzzy numbers are defined, and both minimization and maximization of objective functions are examined. Finally, a solution algorithm for a fuzzy multi-objective linear programming problem is proposed under some given conditions. The algorithm is applied to a sample problem that involves modeling of a congress organizing which aims to place attendees to accommodations under specific objectives and constraints. In the evaluation phase of the algorithm, targets are weighted by using triangular and trapezoidal fuzzy numbers. At the end, the results of the proposed algorithm are compared with the results of other methods in the literature.

This article addresses a special class of non-linear programming problems, viz., linear fractional programming problems having multiple objectives. In solving the real-life linear fractional optimization problems, the ambiguity and hesitation in the decision are inherent and ever-present, therefore, it is perfectly viable to formulate and solve these optimization models using the intuitionistic fuzzy environment. The purpose of this study is to propose a simple and computationally efficient approach to obtain the solution of multiple objective linear fractional programming problems having all the decision variables and parameters expressed in terms of triangular intuitionistic fuzzy numbers. The proposed solution algorithm is primarily based on the goal programming approach, fuzzy-based linearization technique, and a membership function strategy. The original linear fractional programming problem is first converted to its equivalent deterministic/crisp multi-objective linear fractional optimization problem using the weighted goal programming methodology along with the linear membership technique to resolve the intuitionistic fuzzy constraints into the crisp one. Finally, the variable transformation technique for the under- and over-deviational variables of the goal programming model is employed to linearize all the fractions involved in the problem so as to convert the original problem to an equivalent linear optimization problem. Further, this linear programming problem can be solved using any available commercial packages. Moreover, a numerical illustration is provided to demonstrate the steps of the proposed technique followed by the analysis and solution of an E-education set-up problem. The discussion and comparisons of the practical case establish the relevancy and usefulness of the proposed model.

The Yin-Yang bipolar fuzzy set is a powerful mathematical tool for depicting fuzziness and vagueness. We first extend the concept of crisp linear programming problem in a bipolar fuzzy environment based on bipolar fuzzy numbers. We first define arithmetic operations of unrestricted bipolar fuzzy numbers and multiplication of an unrestricted trapezoidal bipolar fuzzy number (TrBFN) with non-negative TrBFN. We then propose a method for solving fully bipolar fuzzy linear programming problems (FBFLPPs) with equality constraints in which the coefficients are unrestricted triangular bipolar fuzzy numbers and decision variables are nonnegative triangular bipolar fuzzy numbers. Furthermore, we present a method for solving FBFLPPs with equality constraints in which the coefficients and decision variables are unrestricted TrBFNs. The FBFLPP is transformed into a crisp linear programming problem, and then, it is solved to achieve the exact bipolar fuzzy optimal solution. We illustrate the proposed methodologies with several numerical examples.

Cooperative co-evolutionary algorithm for multi-objective optimization problems with changing decision variables

In this paper we develop a new algorithm for the initial fuzzy basic feasible solution to a solid transportation problem with imprecise parameters. All the decision variables are assumed to be triangular fuzzy numbers. Using parametric form of triangular fuzzy numbers such as left fuzziness index, right fuzziness index, modal value and by using proposed algorithm we obtained the initial basic fuzzy feasible solution to the problem without changing to its crisp equivalent form. We further discuss the optimality by applying the modified distribution method. A numerical example is solved to show the effectiveness of the proposed algorithm.

In this paper, we develop a new algorithm for the initial fuzzy basic feasible solution to a transportation problem with imprecise parameters. All the decision variables are assumed to be triangular fuzzy numbers or trapezoidal fuzzy numbers or real numbers. By applying the efficient ranking function and arithmetic operations the solution is obtained without changing it into its crisp equivalent form. Using the parametric form of fuzzy numbers such as left fuzziness index, right fuzziness index, modal value and by using a proposed algorithm we obtained the initial basic fuzzy feasible solution to the problem. We further discuss the optimality by applying the modified distribution method. A numerical example is solved to show the effectiveness of the proposed algorithm.

This study extends the best hybrid ant colony optimization variant developed by Liao et al. (2014) for crisp unrelated parallel machine scheduling problems to solve fuzzy unrelated parallel machine scheduling problems in consideration of trapezoidal fuzzy processing times, trapezoidal fuzzy sequencing dependent setup times and trapezoidal fuzzy release times. The objective is to find the best schedule taking minimum fuzzy makespan in completing all jobs. In this study, fuzzy arithmetic is used to determine fuzzy completion times of jobs and defuzzification function is used to convert fuzzy numbers back to crisp numbers for ranking. Eight fuzzy ranking methods are tested to find the most feasible one to be employed in this study. The fuzzy arithmetic testing includes four different cases and each case with the following operations separately, i.e., addition, subtraction, multiplication and division, to investigate the spread of fuzziness as fuzzy numbers are subject to more and more number of operations. The effect of fuzzy ranking methods on hybrid ant colony optimization (hACO) is investigated. To prove the correctness of our methodology and coding, unrelated parallel machine scheduling with fuzzy numbers and crisp numbers are compared based on scheduling problems up to 15 machines and 200 jobs. Relative percentage deviation (RPD) is used to evaluate the performance of hACO in solving fuzzy unrelated parallel machine scheduling problems. A numerical study on large-scale scheduling problems up to 20 machines and 200 jobs is conducted to assess the performance of the hACO algorithm. For comparison, a discrete particle swarm optimization (dPSO) algorithm is implemented for fuzzy unrelated parallel machine scheduling problem as well. The results show that the hACO has better performance than dPSO not only in solution quality in terms of RPD value, but also in computational time.

Combining the concepts of fuzzy set theory, entropy, ideal, and grey relation analysis, a fuzzy grey relation method for multiple criteria decision-making problems is proposed. First, triangular fuzzy numbers and linguistic values characterized by triangular fuzzy numbers are used to evaluate the importance weights of all criteria and the superiority of all alternatives versus various criteria above the alternative level. Then, the concept of entropy is utilized to solve the adjusted integration weight of all objective criteria above the alternative level. Furthermore, using the concepts of ideal, the grey ration grades of various alternatives versus ideal solution are ranked to determine the best alternative. Finally, a numerical example of selecting most appropriate company to build a new highway is used to demonstrate the applicability of proposed method. The study results show that this method is an effective means for tackling MCDM problems in fuzzy and grey environments.

In this paper, an inventory model for two-stage supply chain is investigated in the fuzzy sense. A supply chain with single vendor and single buyer is considered. We assume that shortage as a backorder is allowed for the buyer and the vendor makes the production set up every time the buyer places an order and supplies on a lot for lot basis. The decision variables are fuzzy order quantity and fuzzy shortage quantity which are expressed as triangular fuzzy numbers. After developing the mathematical model for the joint total relevant cost in fuzzy sense, its membership function is calculated and then defuzzified with the centroid method. The proposed model is solved and the amount of joint total relevant cost function in fuzzy sense is compared with the crisp one.

This paper investigates continuous review inventory models involving service level constraint in which lead time, reorder point, ordering cost and order quantity are treated as decision variables and quantity received is uncertain. The lead time can be shortened at an extra crashing cost which has two different forms. First, it can be decomposed into several components; each has a crashing cost for the reduced lead time. Second, the lead time dependent cost follows a power function. The assumption of normal distribution on lead time demand was given and the uncertainty of average demand was handled with triangular fuzzy number. The signed distance method was employed to defuzzify the average demand. To obtain the optimal policies of the proposed models in partial backorder case, we construct Lagrange function, and solution algorithms are then derived. Moreover, two different examples were used to illustrate the proposed models and solution procedures.

Circular economy is the one of the most targeted development in which remanufacturing, recycle and reuse the materials that creates job with minimised resource used. Application of circular economy is increasing day by day for economic growth and healthy business. Government encourages the use of circular economy instead of linear economy by providing subsidies through regulations and taxes. In this paper, a fuzzy economic order quantity model in circular economy is presented. Since some parameters are uncertain in nature, they are considered as triangular fuzzy number. Defuzzification is done by using graded mean integration representation method (GMIR). In this paper, ordering quantity and circularity index level are decision variables. The concavity of the average profit function is shown theoretically. The optimality of the parameters is determined analytically. It is further endorsed by numerical example with an intension to validate the proposed model. The CE gives a remarkable gain in profit, which is the focus of any industry that is concerned about environmental conservation. The financial advantage of the company is given due weight age in this model. For this, sensitivity analysis is independently carried out by deploying circular economy in inventory modelling.

In this paper, we discuss retailer’s optimal replenishment policies for non-instantaneous deteriorating items with partial backlogging under inflation over a finite horizon. The backlogging rate is a decreasing function of the waiting time up to the next replenishment. The objective of this model is to minimise the total cost of the retailer which includes the setup cost, holding cost, shortage cost and the opportunity cost due to lost sales. Here the total cost, time durations and the optimum number of replenishment are taken as decision variables. In the practical situation, it is not always possible that all the cost components in the total cost are known and have definite values. For reality, the cost components involved in the total cost of the retailer are fuzzified to a certain extent. Triangular fuzzy numbers are used to represent the mentioned costs. Signed distance method is used to defuzzify the triangular fuzzy numbers. Numerical examples are given to illustrate the model.

In the broadest sense, reliability is a measure of performance of the system under the stated conditions. The reliability–redundancy allocation problem gives a highly reliable system in the presence of optimal redundant components. This design is the most preferred by the design engineer. During the designing phase of the system, all the design data involved in the system are not very precise. Various types of uncertainties such as expert’s information character, qualitative statements, vagueness, incompleteness, unclear system boundaries, inability to evaluate the relative importance of the objectives, etc., are typical for many practical problems. Fuzzy set theory is an efficient technique to tackle such types of uncertainties in the system design problem. In this paper, the goals of the fuzzy multi-objective reliability–redundancy allocation problem are specified by various membership functions such as linear, quadratic, parabolic, and hyperbolic. An efficient multi-objective evolutionary algorithm, namely, NSGA-II is employed to solve it. The Pareto-optimal fronts for the various membership functions are shown in both the membership and objective spaces. Fuzzy ranking method then finds the best compromise solution for each membership function. Finally, the performance of membership functions is ranked by the data envelopment analysis by taking cost criteria (cost, weight, and volume) as inputs and benefit criteria (reliability and maximum satisfaction level) as outputs of the system. The effectiveness of the proposed approach is illustrated by a numerical example of the over-speed protection system for a gas turbine. A comparative analysis of the proposed approach is given with the existing approach.

Practically, reliability involved in the system design is enhanced with a reduction of other mutually conflicting objectives simultaneously such as cost, weight, volume, etc. Moreover, different types of uncertainty such as expert’s information character, qualitative statements, vagueness, incompleteness, and unclear system boundaries are typical in multi-objective decision-making of reliability. This paper proposes a hybrid NSGA-II for fuzzy multi-objective reliability optimization which comprises NSGA-II, local search strategy, and clustering technique. Local search strategy helps to update each Pareto-optimal solution after an NSGA-II simulation run which gives a better convergence near the true Pareto-optimal front while the clustering technique maintains a good diversity in the solutions set. Finally, the best compromise solution is achieved by using the fuzzy ranking method. The results obtained by the proposed methodology are then compared with popular elitist multi-objective evolutionary algorithms (MOEAs) namely NSGA-II and PESA-II as well as a multi-objective swarm technique known as MOPSO. A numerical example of the mixed series–parallel is given to show the effectiveness of the proposed approach.