Equivalent Crisp Problem Research Articles

Fractional transportation problem under interval-valued Fermatean fuzzy sets

<abstract><p>The concept of an interval-valued Fermatean fuzzy set (IVFFS), an extension of Fermatean fuzzy sets, is a more resilient and reliable tool for dealing with uncertain and incomplete data in practical applications. The purpose of this paper is to define a triangular interval-valued Fermatean fuzzy number (TIVFFN) and its arithmetic operations. Fractional transportation problems (FTPs) have important implications for cost reduction and service improvement in logistics and supply management. However, in practical problems, the parameters in the model are not precise due to some unpredictable factors, including diesel prices, road conditions, weather conditions and traffic conditions. Therefore, decision makers encounter uncertainty when estimating transportation costs and profits. To address these challenges, we consider a FTP with TIVFFN as its parameter and call it an interval-valued Fermatean fuzzy fractional transportation problem (IVFFFTP). A new method for solving this IVFFFTP is proposed without re-transforming the original problem into an equivalent crisp problem. Illustrative examples are discussed to evaluate the precision and accuracy of the proposed method. Finally, the results of the proposed method are compared with those of existing methods.</p></abstract>

A Fuzzy Dynamic Programming Approach to Fuzzy Least Cost Route Problem

This paper focuses on obtaining a solution to generalized trapezoidal fuzzy least cost route problem using fuzzy dynamic programming approach. Using the fuzzy forward and backward recursive equations the fuzzy optimal solution is developed by without converting them into an equivalent crisp problem. Using appropriateranking and arithmetic operation in terms of parametric form the minimized fuzzy cost is attained. To explain the procedure an example is illustrated.

Profit maximization solid transportation problem under budget constraint using fuzzy measures

Fixed charge solid transportation problems are formulated as profit maximization problems under a budget constraint at each destination. Here item is purchased in different depots at different prices. Accordingly the item is transported to different destinations from different depots using different vehicles. Unitsare sold from different destinations to the customers at different selling prices. Here selling prices, purchasing costs, unit transportation costs, fixed charges, sources at origins, demands at destinations, conveyances capacities are assumed to be crisp or fuzzy. Budget constraints at destinations are imposed. Itis also assumed that transported units are integer multiple of packets. So the problem is formulated as constraint optimization integer programming problem in crisp and fuzzy environments. Asoptimization of fuzzy objective as well as consideration of fuzzy constraint is not well defined, different measures possibility/necessity/credibility of fuzzy event are used to transform the problem into equivalent crisp problem. The reduced crisp problem is solved following generalized reduced gradient(GRG) method using lingo software. A dominance based genetic algorithm (DBGA) and a particle swarm optimization (PSO) technique using swap sequence are also developed for this purpose and are used to solve the model. The models are illustrated with numerical examples. The results obtained using DBGA and PSO are compared with those obtained from GRG.Moreover, a statistical analysis is presented to compare the algorithms.

Modeling and optimization of multi objective non-linear programming problem in intuitionistic fuzzy environment

In dealing with real world practical optimization problems, a decision maker usually faces a state of uncertainty as well as hesitation, due to various unpredictable factors. Sometimes it is necessary to optimize several non-linear and conflicting objectives simultaneously. To deal with the uncertain parameters which arise in such situations, intuitionistic fuzzy numbers are utilized. We formulate a multiobjective non-linear programming problem in intuitionistic fuzzy environment. We propose a linear ranking function and utilize it to convert the intuitionistic fuzzy model into a crisp model. After converting the problem into equivalent crisp problem, we propose a non-linear membership function and develop various approaches for solving it by using different operators and fuzzy programming technique. We apply our methodologies for justification to a numerical problem in manufacturing systems.

Advertisement policy and reliability dependent imperfect production inventory control problem in bi-fuzzy environment

In this study, a problem for advertising policy and an imperfect production for single item are formulated with reliability parameter over a finite time horizon. The advertisement and production rates are function of time which are taken as control variables. In a competitive market on long run business, the demand face the depreciation of sale which is at a constant rate and decrease the demand rate but the advertisement policy has the positive effect to the demand rate. The reliability of the production process is an important factor to improve the quality of product and decrease the defective rate. So the quality of the product and advertisement policy play an important role to capture the customers in a competitive market. The selling price, holding cost and advertisement cost are bi–fuzzy in nature and using bi–fuzzy technique, the problem is converted into equivalent crisp problem and solved using Pontryaginn's maximum principle and generalised reduced gradient (GRG) method. Finally, numerical experiment and graphical representation are provided to illustrate the model.

EOQ model with backorders under linear combination of possibility and necessity measure

This paper considers an EOQ model with backorders and imprecise holding cost, ordering cost and shortage cost under linear combination of possibility measure and necessity measure. The objective function of this model is to minimise fuzzy total annual cost, which includes fuzzy annual holding cost, fuzzy annual ordering cost and fuzzy annual shortage penalty cost. Making use of mλ–measure, a linear combination of possibility measure and necessity measure, two fuzzy chance–constrained programming models are constructed to determine optimistic and pessimistic values of the objective function. An objective function is optimised with some predefined degree of mλ–measure and accordingly the problem is transferred to an equivalent crisp problem. An analytical approach is developed to resolve the reduced models. To analyse the characteristics of the proposed model and to find the optimal decision under different situations, numerical illustrations are presented along with a sensitivity analysis.

Advertisement Policy, Reliability Dependent Imperfect Production and Environmental Pollution Control Problem in Fuzzy-Rough Environment

In this study, a problem for advertising policy to avoid the depreciation rate of sale, an imperfect production rate and environmental pollution control is formulated with reliability and rework over a finite time horizon. The advertisement and production rates are function of time which are taken as control variables. Also, recent studies indicate that industrial solid waste (ISW) in production period has been an important contributor to greenhouse gas (GHG) emissions. The company has agreed to invest a large amount of money to manage the ISW through technological up gradation, etc that reduce the GHG in environment. The unit production cost is a function of production rate and also dependent on raw material cost, development cost due to technological up gradation and reliability and wear-tear cost. The total profit which consists of the sales proceeds, marginal selling price, production cost, operating cost, pollution cost, inventory holding cost and advertisement cost is formulated as an optimal control problem. The selling price and all inventory costs are fuzzy rough in nature and using expectation of fuzzy rough technique, the problem is converted into equivalent crisp problem and solved using Pontryagin's Maximum Principle and Generalized Reduced Gradient Method (GRG). Finally numerical experiment, sensitivity analysis for marginal selling price, defective parameter, rework and depreciation rate of sale and graphical representation are provided to illustrate the model.