Article Preview
Top1. Introduction
The transportation problem (TP) is a special type of linear programming problem which is considered to determine an optimal solution for distributing certain types of goods from different supply points (sources) to different demand points (destinations). The cost of transportation between distinct sources and destinations is related to the number of items sent in the transportation problem. However, in real-world scenarios, we may encounter issues such as fixed costs that are independent of the amount transported, as well as a variable cost. For each rout that is opened or used to send the demands, FCTP imposes a fixed cost or price.
Many real-world transportation and distribution issues can be modelled as fixed- charge transportation issues (FCTP). Two kinds of costs are considered in the fixed charge transportation problems: a variable cost that s linearly increases with the amount transported between a source and a destination, and a fixed charge, which is incurred whenever a route is opened or used to send the demands to destination.
The fixed price could be, among other things, the cost of renting a car, tolls on a public road, airport entry fees, machine setup costs in a developed environment, the cost of creating roads in shipping systems, or the time it takes to locate a file in a distributed database system. In a wide spectrum of production and shipping systems, fixed charge concerns can be found.
In many practical applications of FCTP, the parameters cannot be determined in a precise form. Due to globalization of market, FCTP with imprecise coefficients are highly associated for transporting system.
Many researchers have used fuzzy variables/numbers (especially triangular fuzzy numbers, trapezoidal fuzzy numbers, interval numbers) to express these imprecise data, linguistic terms etc. Hence many real-world shipping and distribution problems can be represented as a fixed-charge transportation problem (FCTP) with fuzzy or interval parameters.
There are many techniques for solving these types of problems in the literature. Hirschn et al (1954) proposed a fixed charge problem. Balinski (1961) has created a fixed-charge transportation problem. Ishibuchi et al. (1990) proposed a method for optimising the interval objective function using multi-objective programming. Adlakha et al. (2003, 2006, 2010, 2014) have discussed different types of algorithms for the solution of FCTP. Ganesan et.al (2005, 2007) developed arithmetic operations on interval numbers and discussed some properties of interval matrices. Li et al. (2011,2012,2013,2018,2020) have discussed different types of algorithms for the Alfa-cut based linear programming methodology for constrained matrix games with payoffs of trapezoidal fuzzy numbers. Ramesh et.al (2011) have proposed an interval linear programming with generalized interval arithmetic. Safi et.al (2013) also discussed fixed charge transportation problems with interval parameters. Roy et.al (2018) have introduced a Multi-objective fixed-charge transportation problem with random rough variables Gholian-Jouybari et. al (2018) utilized three types of Electromagnetism-like Algorithms (EM), Genetic Algorithm (GA) and Simulated Annealing (SA) for solving fuzzy fixed charge transportation problems. Yousefi et.al (2018) proposed a new heuristic along with well-known meta-heuristics to solve the FCTP with discount supposition on both fixed and variable charges. Fei et.al (2018) have introduced an approach to computing interval-valued discounted Shapley values for a class of cooperative games under interval data. Kumar (2018) proposed a method for solving intuitionistic fuzzy solid transportation problems in which only the transportation costs are represented in terms of intuitionistic fuzzy numbers and the remaining parameters such as supply, demand and conveyance capacity, are all considered as crisp numbers. Liang and Li. (2019) have proposed a direct method of interval Banzhaf values of interval cooperative games. Midya et al. (2020) used rough programming for solving multi-objective fixed-charge transportation problems where the parameters of the MOFCTP are treated as rough intervals. Ye and Li. (2021) have solved a direct approach to compute triangular fuzzy Banzhaf values of cooperative games with coalitions’ values represented by triangular fuzzy numbers. Fuzzy programming method and linear weighted-sum method are used to obtain Pareto-optimal solution of the MOFCTP. Singh et.al (2021) have solved fixed-charge transportation problems using a modified particle swarm optimization algorithm.