Fractional Transportation Problem Research Articles

Application of Water Cycle Algorithm to Stochastic Fractional Programming Problems

This paper presents an application of Water Cycle algorithm (WCA) in solving stochastic programming problems. In particular, Linear stochastic fractional programming problems are considered which are solved by WCA and solutions are compared with Particle Swarm Optimization, Differential Evolution, and Whale Optimization Algorithm and the results from literature. The constraints are handled by converting constrained optimization problem into an unconstrained optimization problem using Augmented Lagrangian Method. Further, a fractional stochastic transportation problem is examined as an application of the stochastic fractional programming problem. In terms of efficiency of algorithms and the ability to find optimal solutions, WCA gives highly significant results in comparison with the other metaheuristic algorithms and the quoted results in the literature which demonstrates that WCA algorithm has 100% convergence in all the problems. Moreover, non-parametric hypothesis tests are performed and which indicates that WCA presents better results as compare to the other algorithms.

Stability of Parametric Intuitionistic Fuzzy Multi-Objective Fractional Transportation Problem

In this study, a parametric intuitionistic fuzzy multi-objective fractional transportation problem (PIF-MOFTP) is proposed. The current PIF-MOFTP has a single-scalar parameter in the objective functions and an intuitionistic fuzzy supply and demand. Based on the (α,β)-cut concept a parametric (α,β)-MOFTP is established. Then, a fuzzy goal programming (FGP) approach is utilized to obtain (α,β)-Pareto optimal solution. We investigated the stability set of the first kind (SSFK) corresponding to the solution by extending the Kuhn-Tucker optimality conditions of multi-objective programming problems. An algorithm to crystalize the progressing SSFK for PIF-MOFTP as well as an illustrative numerical example is presented.

Solving the Multiobjective Fractional Transportation Problem through the Neutrosophic Goal Programming Approach

Nowadays, the transportation problem is a multiobjective decision-making problem. It involves deciding to determine the ideal transportation setup that matches the decision maker’s preferences while taking into account competing objectives/criteria such as transportation cost, transportation time, and environmental and social concerns. This study presents a general framework of the multiobjective fractional transportation problem (MOFTP) to deal with such complex scenarios. This paper’s major goal is to propose a solution methodology to solve the MOFTP based on a neutrosophic goal programming (NGP) approach. By obtaining the optimal compromise solution using three memberships, namely, truth membership, indeterminacy membership, and falsity membership, the suggested technique gives a novel insight into solving the MOFTP. A real-world problem such as selling wind turbine blades’ problem and a numerical example are used to demonstrate the efficacy and superiority of the proposed method.

A Fuzzy Optimization Technique for Multi-Objective Aspirational Level Fractional Transportation Problem

In this research work, a soft computing optimization operating approach is developed for a multi-objective aspirational level fractional transportation problem. In the proposed technique, a mathematical model is formulated for the multi-objective aspirational level fractional transportation problem (MOFTP) based on the highest value of one and all objectives of the model. We also used the symmetry concept over our model to identify the best optimum solution based on symmetrical data. We constructed the membership grades for the set of fetched parameters having symmetry. In this work, we also used the concept of ranking function in our mathematical model to obtain the optimum solution of the fuzzy multi-objective fractional transportation. In this proposed algorithm, the aspiration levels are also associated with the objective function of MOFTP. We are also proposing a new approach for the optimization of fractional problems in which the objectives are being optimized by using the numerator function and denominator function simultaneously. Further, a methodology is also developed to find the average cost for each fractional objective of the model. After that, we will find the ranking function for each parameter by using the defuzzification method. By this methodology, we will be able to convert the MOFTP into a bi-objective transportation problem. The provided technique is elaborated with the help of numerical computations to prove the beauty and power of the proposed technique.

Fuzzy multiple objective fractional optimization in rough approximation and its aptness to the fixed-charge transportation problem

This article presents a multiple objective fractional fixed-charge transportation problem (MFFTP) in a rough decision-making framework. A transformation procedure is modified to convert non-linear multi-objective transportation problem to its linear version. The parameters of the designed model are considered to be fuzzy. We employ separate kinds of fuzzy scale, i.e., possibility, credibility and necessity measures, to deal with the fuzzy parameters. Using the fuzzy chance-constrained rough approximation (FCRA) technique, we extract the more preferable optimal solution from our suggested MFFTP. The initial result is compared with that of the robust ranking (RR) technique. We also use the theory of rough sets for expanding as well as dividing the feasible domain of the MFFTP to accommodate more information by considering two approximations. Employing these approximations, we introduce two variants, namely, the lower approximation (LA) and the upper approximation (UA), of the suggested MFFTP. Finally, by using these models, we provide the optimal solutions for our proposedproblem. We also associate our MFFTP with a real-world example to showcase its applicability as well as performance. Our core concept of this article is that it tackles an MFFTP using two separate kinds of uncertainty and expands its feasible domain for optimal solutions. Optimal solutions of the designed model (obtained from FCRA technique) belong to two separate regions, namely, “surely region” and “possible region”. The optimal solution which belongs to the “surely region” is better (as these are minimum values) than the one in the “possible region” and other cases. An interpretation of our approach along with offers about the intended future research work are provided at last.

On characterizing solution for multi-objective fractional two-stage solid transportation problem under fuzzy environment

Abstract This article attempts to study cost minimizing multi-objective fractional solid transportation problem with fuzzy cost coefficients c ˜ i j k r {\tilde{c}}_{ijk}^{r} , fuzzy supply quantities a ˜ i {\tilde{a}}_{i} , fuzzy demands b ˜ j {\tilde{b}}_{j} , and/or fuzzy conveyances e ˜ k {\tilde{e}}_{k} . The fuzzy efficient concept is introduced in which the crisp efficient solution is extended. A necessary and sufficient condition for the solution is established. Fuzzy geometric programming approach is applied to solve the crisp problem by defining membership function so as to obtain the optimal compromise solution of a multi-objective two-stage problem. A linear membership function for the objective function is defined. The stability set of the first kind is defined and determined. A numerical example is given for illustration and to check the validity of the proposed approach.

A novel Approach for Fully Intuitionistic Fuzzy Multi-Objective Fractional Transportation Problem

This paper exhibits a novel approach for solving fully intuitionistic fuzzy multi-objective fractional transportation problem (FIF-MOFTP). Because of the change of market policies, we expect that the transportation cost, the shipped quantity, the source, and the destination parameters are not generally exact. In this paper, a theorem which demonstrate that FIF-MOFTP is ever solvable was presented. In our approach, the problem is converted into a linear one based on some transformations. Then the linearized model is reduced to a crisp multi-objective transportation problem utilizing the accuracy function for each objective. Moreover, various theorems that set up the relation among the FIF-MOFTP and its equivalent crisp model using linear, hyperbolic, and parabolic membership functions are also proofed. To approve the proposed approach, a numerical example is incorporated. Deductions and future exploration of this paper are depicted at end.

Fuzzy fractional stochastic transportation problem involving exponential distribution

This paper deals with the solution of the fuzzy fractional transportation problem in which the parameters of the transportation problem, supply, and demand, are stochastic in nature and considered as a fuzzy random variable that follows the exponential distribution with fuzzy mean and fuzzy variance. In a fuzzy fractional objective function, the costs are taken as a triangular fuzzy number. As the parameters are imprecise in nature, the obtained objective value should be a fuzzy number. To obtain the fuzzy objective value, one has to find out its lower and upper bounds, which represent the level of uncertainty. The mathematical form of bounds is expressed by applying dual formulation and variable substitution. Also, for converting the fuzzy constraints into deterministic, the chance-constrained and fuzzy programming was applied. The values of the bounds, which are calculated at different values of $$\lambda$$ , the membership function of the objective value is approximated. A numerical example illustrates the considered methodology.

Nonlinear Quadratic Fractional Transportation Problem for Optimal Solution Deduction

The manifold non-linear programming problems (NLPP) are dealt by people in their daily routines in the form of real time uses. The non-linear problem could deliver a remedies on the problems that require decision making, for instance corporate planning as well as finance, production and marketing, sales and inventory etc. this makes the fractional programing a research area of predominance. The fractional programming in transportation problem of disposing a one type of goods to various endpoint with varying quantities would enable to identify probable solution at a minimized cost and duration. The paper with the research study on the one such NLPP is coined as the fractional-quadratic transportation problem. (FQTP). The NLPP are highly popular since they deliver a supreme depictions of distribution problems for the real-life applications were the transportation cost remains changing. The proposed strides in the paper emphasis on deducing the solutions that are optimal for such difficulty. The proposed algorithm is examined with the numerical instance to demonstrate the proficiency of the algorithm and its benefits in the transportation structure belonging to different area of application

Type-2 Duality Trapezoidal Fuzzy Fractional Transportation Problem using Fuzzy Optimization Techniques

The authors focused goal programming technique for solving type-2 duality fuzzy fractional transportation problem by using interval-valued triangular intuitionistic fuzzy numbers. The proposed method solves three types of models in which variables are taken as type-2 fuzzy variables which have spring up to the fuzzy fractional transportation problem. In these models, duality is applied and a particular type of non-linear membership functions are used to resolve duality fractional transportation problem including fuzzy parameters. A numerical example for examining the performance of the proposed model is envisaged here.

An Aspect of Bilevel Fixed Charge Fractional Transportation Problem

Bilevel Programming Problem (BLPP) is a hierarchical optimization problem. Here, the constraint set of the upper level problem, called the leader, is ascertained by the lower level problem, called the follower. In this paper, a BLPP is considered where the leader’s problem is a fractional transportation problem and the follower’s problem is a fixed charge transportation problem. Hence, a bilevel fixed charge fractional transportation problem (BLFCFTP) is formulated. The problem is also explained through graphical representation. The problem comes from a food chain industry which is utilizing its used vegetable cooking oil for the production of biodiesel. The mathematical model of the problem is presented and is also explained through graphical representation. The problem pertains to a food chain industry wherein the used vegetable cooking oil is utilized for the production of biodiesel. Another example quotes Indore city where waste is converted into bio-CNG which is used to run public transport. The managerial importance of the work is that it will promote a sustainable and environment friendly practice. This will result in reduction of waste and check environmental pollution. An algorithm to solve BLFCFTP is developed. The examples explaining the algorithm are exhibited by a viable situation. These examples are subsequently deciphered in MATLAB.

An algorithm for multi objective dual hesitant fuzzy fractional transportation problem

An algorithm for multi objective dual hesitant fuzzy fractional transportation problem

Solving Multi-Objective Linear Fractional Stochastic Transportation Problems Involving Normal Distribution using Simulation-Based Genetic Algorithm

In real-life situations, we human beings faced with multi-objective problems that are conflicting and non-commensurable with each other. Especially, when goods are transported from source to locations with a goal to keep exact relationships between a few parameters, those parameters of such problems might also arise in the form of fractions which are linear in nature such as; actual transportation fee/total transportation cost, delivery fee/desired path, total return/total investment, etc. Due to the uncertainty of nature, such a relationship is not deterministic. Mathematically such kinds of mathematical problems are characterized as a multi-objective linear fractional stochastic transportation problem. However, it is difficult to handle such types of mathematical problems. It can't be solved directly using mathematical programming approaches. In this paper, a solution procedure is proposed for the above problem using a stochastic Genetic Algorithm based simulation. The parameters in the constraint of the above problem follow a normal distribution. The probabilistic constraints are handled by stochastic simulation-based GA for the solution procedure of the proposed problem. The feasibility of probability constraints is checked by the stochastic programming through the Genetic Algorithm approach, without finding the equivalent deterministic model. The feasibility is maintained all-over the problem. The stochastic simulation-based Genetic Algorithm is considered to generate non-dominated solutions for the given problem. Then, a numerical case study is provided to illustrate the method.

Genetic algorithm based fuzzy programming approach for multi-objective linear fractional stochastic transportation problem involving four-parameter Burr distribution

In real life situations, it is difficult to handle multi-objective linear fractional stochastic transportation problem. It can’t be solved directly using mathematical programming approaches. In this paper, a solution procedure is proposed for the above problem using a genetic algorithm based fuzzy programming method. The supply and demand parameters of the said problem follow four-parameters Burr distribution. The proposed approach omits the derivation of deterministic equivalent form as required in case of classical approach. In the given methodology, initially the probabilistic constraints present in the problem are tackled using stochastic programming combining the strategy adopted in genetic algorithm. Throughout the problem, feasibility criteria is maintained. Then after, the non-dominated solution are obtained using genetic algorithm based fuzzy programming approach. In the proposed approach, the concept of fuzzy programming approach is inserted in the genetic algorithm cycle. The proposed algorithm has been compared with fuzzy programming approach and implemented on two examples. The result shows the efficacy of the proposed algorithm over fuzzy programming approach.

An Optimally-Competitive Algorithm for Maximum Online Perfect Bipartite Matching with i.i.d. Arrivals

We present an optimally-competitive algorithm for the problem of maximum online perfect bipartite matching with i.i.d. arrivals. In this problem, we are given a known set of workers, a distribution over job types, and non-negative utility weights for each pair of worker and job types. At each time step, a job is drawn i.i.d. from the distribution over job types. Upon arrival, the job must be irrevocably assigned to a worker and cannot be dropped. The goal is to maximize the expected sum of utilities after all jobs are assigned. We introduce Dispatch, a 0.5-competitive, randomized algorithm. We also prove that 0.5-competitive is the best possible. When a job arrives, Dispatch first selects a “preferred worker” and assigns the job to this worker if it is available. The preferred worker is determined based on an optimal solution to a fractional transportation problem. If the preferred worker is not available, Dispatch randomly selects a worker from the available workers.

Goal programming approach to fully fuzzy fractional transportation problem

In this paper, authors devoted to study a fully fuzzy fractional multi-objective transportation problem by using goal programming approach. Also trapezoidal membership functions are applied to each objective function and constraints to describe a each fuzzy goal. A numerical example is provided to illustrate the efficiency of the multi-objective proposed approach.

Multi-Objective Multi Time Span Fractional Capacitated Transportation Problem in the Two Echelon Supply Chain with Multiform Items and Mixed Constraints

One of the most useful techniques for the transport companies is multi-level distribution to minimize the cost. This paper presents multi-objective multi time span fractional capacitated transportation problem (MOMTMIFCTP) in the two echelon supply chain with multiform items and mixed constraints. Moreover the fractional objective functions in the supply chain are important because many real life problems are based on the ratio of physical or economical values. The goal of this investigation is to minimize damage cost, labor cost, the cost related to transportation, inventory carrying cost, quality cost and packing cost, also determine an optimal inventory level for the warehouse and distribution centers using fuzzy programming approach through LINGO software.

Environmentally Liable Multi-Objective Multi Time Span Fuzzy Fractional Capacitated Transportation Problem in the Two Echelon Supply Chain with Heterogeneous Items and Mixed Constraints

In today’s world supply chain is an integral part of business. This paper proposed a model of environmentally liable multi-objective multi time span fuzzy fractional capacitated transportation problem (ELMOMTHIFFCTP) in the two echelon supply chain with heterogeneous items and mixed constraints. In this model, few objectives are considered as a linear function and few are fractional functions. Especially the objective function related to emission cost is considered as linear fractional objective function. The damage cost, labor cost, packing cost and quality cost are considered as triangular intuitionistic fuzzy numbers. This paper focuses on minimization of damage cost, labor cost, quality cost, packing cost and the ratio related to transportation cost , inventory carrying cost and emission cost of the entire supply chain. An industrial case demonstrates the feasibility of applying the proposed model to real world problem in a two – echelon supply chain under uncertain environment also determine an optimal inventory level for the warehouse and distribution centers using fuzzy programming approach through LINGO software.

Optimality of intuitionistic fuzzy fractional transportation problem of type-2

In this paper, we formulate an effective method to find an optimal solution of trapezoidal intuitionistic fuzzy fractional transportation problem[TIFFTP] of type-2. The proposed method achieve its goal successively when compared to the existing methods (Gupta and Anupum). Trapezoidal ranking method is used, which is based on the area of both membership and non membership parts of the numbers. An illustrative example is provided to demonstrate the feasibility of this method.

Goal programming approach for solving multi-objective fractional transportation problem with fuzzy parameters

In this paper, authors studied a goal programming approach for solving multi-objective fractional transportation problem by representing the parameters (γ, δ) in terms of interval valued fuzzy numbers. Fuzzy goal programming problem with multiple objectives is difficult for the decision makers to determine the goal valued of each objective precisely. The proposed model presents a special type of non-linear (hyperbolic) membership functions to solve multi-objective fractional transportation problem with fuzzy parameters. To illustrate the proposed method numerical examples are solved.