On the structure of the fixed charge transportation problem

The fixed-charge transportation problem (FCTP) is still a challenging problem in the field of mathematical programming. In this paper, we consider fixed-charge transportation problem with logarithmic objective function. In the absence of any suitable algorithm to obtain the solution of this type of nonlinear transportation problem, we discuss the advantage of polynomial approximation. There exists a major difference between the two problems that the variables in the polynomial transportation problem have no upper bound but in the logarithmic transportation problem they are bounded. Using the expansion of logarithm we show the resemblance between the structural behaviour of linear and fixed-charge transportation problems. We illustrate a numerical example in support of the developed method.

Solving fixed charge transportation problem with interval parameters

In this paper, an effective ant lion optimisation (ALO) algorithm is proposed for two stage supply chain network associated with fixed charge transportation problem (FCTP) which is strongly NP-hard. In a FCTP, fixed cost is incurred for every route, along with the variable cost that is proportional to the amount shipped. In some circumstances, the variable cost is associated with quadratic variables in which the cost function will be nonlinear. The aim of this paper is to determine the least cost transportation plan that minimises the total variable and fixed costs while satisfying the supply and demand requirements of each plant and customer. The performance of the proposed ALO is compared in terms of total cost with other algorithms addressed in the literature such as genetic algorithm, simulated annealing and ant colony optimisation. The computational results reveal that the proposed ALO provides better solutions.

The fixed-charge transportation problem is an extension of the classical transportation problem in which a fixed cost, sometimes called a setup cost, is incurred for every origin. In this paper, an algorithm is presented to find the optimum time-cost trade-off pair in a fixed-charge bi-criterion transportation problem giving the same priority to both time and cost. The algorithm identifies efficient time-cost trade-off pairs and thereby arrives at the conclusion that among the trade-off pairs, the optimum trade-off pair has the minimum D1-distance from the ideal point. A numerical example is given to illustrate the developed algorithm

The single-sink, fixed-charge transportation problem has a variety of applications, including supplier selection, product distribution/fleet selection, and process selection. In this paper we present a dynamic programming algorithm for solving this important problem that is very easy to implement and that improves considerably in terms of computational attractiveness on the best methods in the literature.

The primary focus of this article is dedicated to a thorough investigation of the Fixed Load Transportation Problem (FCTP) and the proposition of an exceedingly efficient resolution method, with a specific emphasis on the achievement of optimal transportation plans within practical time constraints. The FCTP, recognized for its intricate nature, falls into the NP-complete category, notorious for its exponential growth in solution time as the problem's size escalates. Within the realm of combinatorial optimization, metaheuristic techniques like the Dragonfly algorithm and genetic algorithms have garnered substantial acclaim due to their remarkable capacity to deliver high-quality solutions to the challenging FCTP. These techniques demonstrate substantial potential in accelerating the resolution of this formidable problem. The central goal revolves around the exploration of groundbreaking solutions for the Fixed Load Transportation Problem, all while concurrently minimizing the time investment required to attain these optimal solutions. This undertaking necessitates the adept utilization of the Dragonfly algorithm, an algorithm inspired by natural processes, known for its adaptability and robustness in solving complex problems. The FCTP, functioning as an optimization problem, grapples with the multifaceted task of formulating distribution plans for products originating from multiple sources and destined for various endpoints. The overarching aspiration is to minimize overall transportation costs, a challenge that mandates meticulous considerations, including product availability at source locations and demand projections at destination points. The proposed methodology introduces an innovative approach tailored explicitly for addressing the Fixed Charge Transport Problem (FCTP) by harnessing the inherent capabilities of the Dragonfly algorithm. This adaptation of the algorithm's underlying processes is precisely engineered to handle large-scale FCTP instances, with the ultimate objective of unveiling solutions that have hitherto remained elusive. The numerical results stemming from our rigorous experiments unequivocally underscore the remarkable prowess of the Dragonfly algorithm in discovering novel and exceptionally efficient solutions. This demonstration unequivocally reaffirms its effectiveness in overcoming the inherent challenges posed by substantial FCTP instances. In summary, the research represents a significant leap forward in the domain of FCTP solution methodologies by seamlessly integrating the formidable capabilities of the Dragonfly algorithm into the problem-solving process. The insights and solutions presented in this article hold immense promise for significantly enhancing the efficiency and effectiveness of FCTP resolution, ultimately benefiting a broad spectrum of industries and logistics systems, and promising advancements in the optimization of transportation processes

The fixed charge transportation problem is an extension of the traditional transportation problem in which two kinds of costs, say direct cost and fixed charge, are taken into consideration. The direct cost is the cost associated with each source-destination pair, and the fixed charge occurs when the transportation activity takes place in the corresponding source-destination pair. In this paper, the fuzzy fixed charge transportation problem is modeled as chance-constrained programming by using the possibility measure and credibility measure firstly. Then a heuristic genetic algorithm is designed.

Recently, research on logistics caught more and more attention. One of the important issues on logistics system is to find optimal delivery routes with the least cost for products delivery. Numerous models have been developed for that reason. However, due to the diversity and complexity of practical problem, the existing models are usually not very satisfying to find the solution efficiently and convinently.In this paper, we treat a real-world logistics case with a company named ABC Co. ltd., in Kitakyusyu Japan. Firstly, based on the natures of this conveyance routing problem, as an extension of transportation problem (TP) and fixed charge transportation problem (fcTP) we formulate the problem as a minimum cost flow (MCF) model. Due to the complexity of fcTP, we proposed a priority-based genetic algorithm (pGA) approach to find the most acceptable solution to this problem. In this pGA approach, a two-stage path decoding method is adopted to develop delivery paths from a chromosome. We also apply the pGA approach to this problem, and compare our results with the current logistics network situation, and calculate the improvement of logistics cost to help the management to make decisions. Finally, in order to check the effectiveness of the proposed method, the results acquired are compared with those come from the two methods/ software, such as LINDO and CPLEX.

In this paper, Glowworm swarm optimisation (GSO) is proposed to solve nonlinear fixed charge transportation problem (NLFCTP) in a single stage supply chain network. In a fixed charge transportation problem, fixed cost is incurred for every route, along with the variable cost that is proportional to the amount shipped. In some cases, the variable cost is associated with quadratic variables in which the cost function will be nonlinear. This problem is more difficult to solve due to presence of the fixed costs that result in discontinuities in the objective function. The objective of this paper is to determine the least cost transportation plan that minimises the total variable and fixed costs while satisfying the supply and demand requirements of each plant and customer. The performance of GSO is compared in terms of total distribution cost with a spanning tree-based genetic algorithm. The comparison reveals that GSO provides better solutions.

Fixed Charge Transportation Problem (FCTP) is known as an NP-hard problem. Therefore, algorithms are developed to find the approximate solution of this problem. The success of these algorithms depends on the choice of initial feasible solutions. In this study, a new algorithm is proposed for the initial feasible solutions of FCTP. In the proposed algorithm, firstly, for each cell in the matrix of the given inputs of the problem, when the entire product corresponding to this cell is transported, its cost is calculated and a new cost matrix is created. Then, the rows or columns of this cost matrix are randomly selected and the least cost cell in the selected row or column is found and the value of the variable corresponding to this cell of the initial solution is determined. The proposed algorithm is explained on the example.

One of the popular variants of fixed charge transportation problem is the non-linear fixed charge transportation problem (NFCTP). It has a ‘variable cost’ proportional to the quadratic of the amount of goods shipped from supplier to customer, and the other is ‘fixed cost’ independent of the amount shipped. This paper aims at providing an optimal technique for solving single-objective NFCTP. Based on the series hybridization of Jaya search and quadratic approximation operator, an efficient evolutionary algorithm is developed as a solution approach in this work. This new approach takes the advantage of well-balanced exploration and exploitation mechanism and exhibits good performance with respect to solution quality and computational run times. At the end, numerical instances are solved to illustrate the robustness of the proposed algorithm.KeywordsNon-linear fixed cost transportation problemEvolutionary algorithmJaya searchQuadratic approximationMulti-criteria decision-making

In the past years several evolutionary algorithms have been applied to the fixed charge transportation problem (FCTP). Although suffering from a lack of locality, the Prüfer number representation has recently been suggested for the FCTP, and — to our surprise — it has been reported to yield good results. Since this disagrees with the common intuition that locality is an important prerequisite of successful evolutionary search, we analyse the Prüfer number representation in greater detail. The results show that the Prüfer number representation is superior to random search but clearly inferior to the permutation representation with respect to solution quality, which is explained by our locality analysis.KeywordsSpan TreeMutation OperatorRandom SearchPermutation RepresentationEvolutionary SearchThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Recent network design techniques using evolutionary algorithms

A fixed charge transportation problem with damageable items under uncertain environment

Nonlinear fixed charge transportation problem by minimum cost flow-based genetic algorithm