Fixed Charge Transportation Problem Research Articles

An approach to solve a fuzzy bi-objective multi-index fixed charge transportation problem

An approach to solve a fuzzy bi-objective multi-index fixed charge transportation problem

Offline learning-based competitive swarm optimizer for non-linear fixed-charge transportation problems

The fixed-charge transportation (FCT) problem, an extension of the classical transportation problem, holds significance in practical logistics scenarios where fixed charges play a crucial role. Fixed charges categorize it as an NP-hard problem, posing challenges for conventional methods due to their inefficiency, high computational costs, and susceptibility to local optima. This paper introduces an enhanced evolutionary algorithm, offline learning-based competitive swarm optimization (OLCSO), tailored to address the FCT problems efficiently. OLCSO draws inspiration from offline learning, where particles benefit from the experiences of high performers. The algorithm incorporates (i) a ranking-based mechanism where losers learn from peer-to-peer winners based on their ranks, (ii) a two-phase cooperative evolutionary mechanism enhancing searchability and convergence of the algorithm. In the first phase, losers learn from personal best experiences and the shifted center of the population, boosting diversity. In the second phase, losers learn from winners and the center of the population, improving exploitation, (iii) a mutation strategy is embedded to update winner locations in each evolutionary procedure, and (iv) to ensure a feasible solution to FCT problems, the paper also introduces negative and fractional repair mechanisms. OLCSO is applied to solve two sets of FCT problems, linear and non-linear. Experimental results are compared with various heuristic and metaheuristic techniques, commonly used for these problems, using diverse metrics. A comparative analysis of OLCSO’s design elements is also conducted. The results demonstrate OLCSO’s superiority over existing algorithms.

Distributionally robust disaster relief planning under the Wasserstein set

We study a two-stage natural disaster management problem modeled as a stochastic program, where the first stage consists of a facility location problem, deciding where to open facilities and pre-allocate resources such as medical and food kits, and the second stage is a fixed-charge transportation problem, routing resources to affected areas after observing a disaster. Our model has binary variables present in both stages. Due to the lack of data, classical stochastic programming approaches may be ill-suited, and we propose a two-stage distributionally robust formulation with a Wasserstein ambiguity set, where we consider distributions consistent with historical data and a tunable parameter to control the level of risk aversion. We develop a tailored column-and-constraint generation (CCG) algorithm to solve an extensive reformulation, where scenarios are iteratively generated. We handle the presence of binary variables in the second stage by leveraging the structure of our support set and second-stage problem, and provide conditions under which the optimal value of the latter is concave with respect to the intensity of the disaster, leading to an efficient scenario generation procedure. We also show that our results extend to the case where the second stage is a fixed-charge network flow problem. We perform extensive computational experiments demonstrating the computational advantage of our method over classical CCG implementations on synthetic instances, and illustrate the benefits of our approach on a popular case study from the literature of hurricane threats on the Gulf of Mexico in the United States.

A dragonfly algorithm for solving the Fixed Charge Transportation Problem FCTP

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.

Fixed charge 4-dimensional transportation problem for breakable incompatible items with type-2 fuzzy random parameters under volume constraint

Real-life problems are full of uncertainties combined with impreciseness and randomness. Transportation problems become difficult when its parameters are type-2 fuzzy random type. This paper presents a 4-dimensional fixed charge transportation problem for incompatible and compatible breakable items with type-2 fuzzy random parameters under the availability, demand, conveyance’s weight and volume capacity constraint. The model is developed for usual transportation cost minimization objective, where transportation costs, fixed charges, availabilities, demands, conveyances’ weights and volume capacities are type-2 fuzzy random. A type-2 fuzzy random parameter is converted to type-1 fuzzy random one by critical value-based reduction method and then converted to a random parameter using two different methods-generalized credibility measure and centroid method. The random parameter is converted to crisp parameter using probability chance constraint method for constraints and expectation for objective. A detailed conversion procedure from type-2 fuzzy random to crisp is also presented. The reduced deterministic problem is solved by the Generalized Reduced Gradient method using Lingo 11.0 and numerically illustrated. A real-life example is introduced. Particular models are deduced for non-breakable items or different pairwise incompatible items. The sensitivity analyses against different generalized credibility levels of different parameters are presented for total transportation cost. The numerical results show the efficiency of the proposed critical value reduction-generalized credibility-chance constraint, and expectation method. The importance of uncertainty, multi-path and vehicles’ volume capacity in transportation problems is demonstrated.

A novel approach and hybrid parallel algorithms for solving the fixed charge transportation problem

This article is dedicated to the efficient resolution of the fixed charge transport problem (FCTP) with the goal of identifying optimal solutions within reduced timeframes. FCTP is a combinatorial and NP-complete problem known for its exponential time complexity relative to problem size. Metaheuristic methods, including genetic algorithms, represent effective techniques for obtaining high-quality FCTP solutions. Consequently, the integration of parallel algorithms emerges as a strategy for expediting problem-solving. The proposed approach, referred to as the parallel genetic algorithm (PGA), entails the application of a genetic algorithm across multiple parallel architectures to tackle the FCTP problem. The primary aim is to explore fresh solutions for the fixed charge transportation problem using genetic algorithms while concurrently optimizing the time required to achieve these solutions through parallelism. The FCTP problem is fundamentally a linear programming challenge, revolving around the determination of optimal shipment quantities from numerous source locations to multiple destinations with the overarching objective of minimizing overall transportation costs. This necessitates consideration of constraints tied to product availability at the sources and demand dynamics at the destinations. In this study, a pioneering approach to addressing the Fixed Charge Transportation Problem (FCTP) using parallel genetic algorithms (PGA) is unveiled. The research introduces two distinct parallel algorithms: The Master-Slave Approach (MS-GA) and the Coarse-Grained Approach (CG-GA). Additionally, investigation into the hybridization of these approaches has led to the development of the NMS-CG-GA approach. The numerical results reveal that our parallelism-based approaches significantly improve the performance of genetic algorithms. Specifically, the Master-Slave (MS-GA) approach demonstrates its advantages in solving smaller instances of the FCTP problem, while the Coarse-Grained (CG-GA) approach exhibits greater effectiveness for larger problem instances. The conclusion reached is that the novel hybrid parallel genetic algorithm approach (NMS-CG-GA) outperforms its predecessors, yielding outstanding results, particularly across diverse FCTP problem instances.

A heuristic approach for fuzzy fixed charge transportation problem

The fixed charge transportation problem is a nonlinear programming problem in which a fixed charge is incurred if a distribution variable assumes a nonzero value. Its structure is almost identical to that of a linear programming problem, and indeed it can be written as a mixed-integer linear program. In this paper, we consider the fixed charge transportation problem under uncertainty, particularly when the direct and fixed costs are the generalized trapezoidal fuzzy numbers. The first step it transforms into the classical fuzzy transportation problem. The next we obtain the best approximation fuzzy on the optimal value of the fuzzy fixed-charge transportation problem. This method obtains a lower and upper bound both on the fuzzy optimal value of the fuzzy fixed-charge transportation problem which can be easily obtained by using the approximation solution. Conclusion: we propose a new method as the best approximation method, with representation both of the transportation cost and the fixed cost, to find a fuzzy approximation solution close to the optimal solution for fuzzy fixed charge transportation problem.

A Transportation Problem Considering Fixed Charge and Fuzzy Shipping Costs

This paper considers a fixed-charge transportation problem with fuzzy shipping costs. Whereas the shipping costs of routes are fuzzy intervals, with increasing linear membership functions, fixed costs, supplies, and demands are deterministic numbers. By defining a membership function associated with the objective values and utilizing Bellman-Zadeh’s max-min criterion instead of the main problem, a new crisp nonlinear mixed-integer programming problem is formulated. To solve this non-linear programming problem, at first, we find optimality conditions for a feasible solution, then, using these optimality conditions, reformulate the non-linear programming problem as a linear mixed-integer fractional programming problem, and finally transform the fractional programming problem as a mixed-integer linear programming problem, which can be solved using the existence methods. An illustrative example is solved to explain the presented details.

Solving non-linear fixed-charge transportation problems using nature inspired non-linear particle swarm optimization algorithm

The main objective of this study is to formulate and solve the model of non-linear fixed-charge transportation problem (NFCTP), which is one of the NP-hard problems. This type of problem is challenging to solve using conventional methods because these methods are usually inefficient, costly to compute, and may stuck at the local optimum solution. To overcome this issue, a non-linear particle swarm optimization (NPSO) algorithm with new non-linear acceleration parameters is proposed in this study. Also, to preserve the feasibility condition (non-negative integer solution) of the transportation problem, two novel negative repair and fraction repair strategies have been incorporated into the proposed NPSO. The efficiency of the NPSO algorithm is tested on both the small-scale and large-scale NFCTPs. The dataset are considered from the existing studies. The obtained results are compared with those obtained by the spanning tree-based genetic algorithm (st-GA), priority-based genetic algorithm (pb-GA), and minimum cost flow-based genetic algorithm. The comparative study reveals that for all the considered problems, the proposed NPSO provides better feasible solutions in lesser computational time. Furthermore, the effectiveness of the NPSO algorithm is shown by comparing it to other seven existing variants of PSO and is found to outperformed these variants for the small scale as well as large scale NFCTPs.

Genetic algorithm with immigration strategy to solve the fixed charge transportation problem

This paper is about improving the performance of genetic algorithm (GA) to solve the fixed-charge transportation problem (FCTP). Several approaches have been developed, based on adaptation and improvement of genetic operators. We propose a new genetic algorithm adopting an immigration strategy to maintain the diversity in the population and then overcome the stagnation of the values of the objective function. Thereby, we applied two types of immigration, random immigration and memory-based immigration. The numerical results obtained with several standard instances of the FCTP problem demonstrate the effectiveness of these strategies in improving the performance of the GA. Especilly, for the second strategy.

Improved Lagrangian-relaxation based approaches for multi-period multi-stage fixed charge transportation problem

This paper considers the Fixed-Charge Transportation Problem (FCTP), which spans multiple time periods and includes multiple stages, termed as Multi-Period, Multi-Stage Fixed-Charge Transportation Problem (MPMS-FCTP) in a supply chain. A generalised mathematical model which minimises the fixed and variable transportation, storage and backlog costs is proposed for the problem under study. Subsequently, two improved Lagrangian-Relaxation (LR)-based approaches are developed to efficiently solve the MPMS-FCTP. In addition, heuristics based on the improved LR approaches are proposed to obtain good upper bounds. The experimental results show that the proposed LR approaches perform better than the traditional LR approach with the sub-gradient optimisation. The lower bounds obtained using the proposed approaches are always non-decreasing in successive iterations of LR, providing tighter lower bounds. In addition, the proposed LR-based heuristics yield better solutions than the existing heuristics in the literature. Finally, the convergence to optimality of the proposed approaches is also discussed.

A new algorithm for optimal solution of fixed charge transportation problem

A new algorithm for optimal solution of fixed charge transportation problem

A novel reformulation for the single-sink fixed-charge transportation problem

A novel reformulation for the single-sink fixed-charge transportation problem

Evolutionary algorithm based approach for solving transportation problems in normal and pandemic scenario

In recent times, COVID-19 pandemic has posed certain challenges to transportation companies due to the restrictions imposed by different countries during the lockdown. These restrictions cause delay and/ or reduction in the number of trips of vehicles, especially, to the regions with higher restrictions. In a pandemic scenario, regions are categorized into different groups based on the levels of restrictions imposed on the movement of vehicles based on the number of active cases (i.e., number of people infected by COVID-19), number of deaths, population, number of COVID-19 hospitals, etc. The aim of this study is to formulate and solve a fixed-charge transportation problem (FCTP) during this pandemic scenario and to obtain transportation scheme with minimum transportation cost in minimum number of trips of vehicles moving between regions with higher levels of restrictions. For this, a penalty is imposed in the objective function based on the category of the region(s) where the origin and destination are situated. However, reduction in the number of trips of vehicles may increase the transportation cost to unrealistic bounds and so, to keep the transportation cost within limits, a constraint is imposed on the proposed model. To solve the problem, the Genetic Algorithm (GA) has been modified accordingly. For this purpose, we have designed a new crossover operator and a new mutation operator to handle multiple trips and capacity constraints of vehicles. For numerical illustration, in this study, we have solved five example problems considering three levels of restrictions, for which the datasets are generated artificially. To show the effectiveness of the constraint imposed for reducing the transportation cost, the same example problems are then solved without the constraint and the results are analyzed. A comparison of results with existing algorithms proves that our algorithm is effective. Finally, some future research directions are discussed.

Interval-Fuzzy Fixed Charge Transportation Problems

The fixed charge transportation problem (FCTP) is a variant of the well-known transportation problem in which the price of conveying products from a source to a sink is made up of a set cost and a continuous cost proportionate to the number of goods sent. The transportation system and FCTP with uncertain parameters are inextricably linked. Fuzzy numbers or interval numbers can be used to successfully handle these imprecise values. The cost coefficients, supplies, and demands are all represented as interval numbers in this paper. A modified interval version of Vogel's Approximation process is used to obtain the initial basic feasible solution (IBFS) of the interval fixed charge transportation problem (IFCTP), which is then optimized using interval version of MODI method. A numerical example is provided to show the efficiency of the proposed method.

Neutrosophic multi-objective green four-dimensional fixed-charge transportation problem

Neutrosophic multi-objective green four-dimensional fixed-charge transportation problem

Paradox in The d-Dimensional Fixed Charge Transportation Problem and Algorithm for Finding The Paradox

The d-dimensional fixed charge transportation problem is a generalization of fixed charge transportation. This problem has d-type of constraints so that can be applied to more complex problem. In transportation problem, sometimes there is a cases when increase the product in shipping, the number of costs incurred is less than before the increase of product. This problem is called the transportation paradox. In this research, it will be explained about the model of d-dimensional fixed charge transportation problem and sufficient condition for the occurrence of the paradox. Furthermore an algorithm is given in finding the paradox in the d-dimensional fixed charge transportation problem with example to support the theory presented.

An approach to solve an unbalanced fully rough multi-objective fixed-charge transportation problem

An approach to solve an unbalanced fully rough multi-objective fixed-charge transportation problem

An Uncertain Programming Model for Fixed Charge Transportation Problem with After-Sale Service

This paper constructs an uncertain mathematical model for the fixed charge transportation problem in which several kinds of items are transported from different sources to various destinations via different methods. In addition, after-sale service such as product return and exchange that help to raise the customer’s satisfaction is considered during the trade to meet the reality. Meanwhile, some indeterministic factors may occur during the transportation process; probability theory is not the best option due to the lack of adequate historical data. In the light of this statement, some parameters like supplies, demands, return rate, and exchange rate are all defined as uncertain variables to formulate the uncertain programming model. After that, the equivalent deterministic models are derived with the help of uncertainty theory. Finally, some numerical experiments are implemented via three algorithms; the optimal values are displayed and compared to show the application of this problem. The study of this paper provides an applicable method for the decision maker.

A study of multi-objective restricted multi-item fixed charge transportation problem considering different types of demands

In this paper, we formulate a multi-objective fixed charge transportation problem (FCTP) for multiple items, considering availability of multiple modes of transport along each pair of origin and destination. The variable cost of each item, the fixed cost and the transport time are considered to be different for each mode of transport. It is also considered that some items are mutually incompatible and cannot be transported in the same mode of transport. Two models of the multi-item FCTP are presented, considering the demands of items as crisp and interval numbers, respectively. The transportation problem is then posed as a multi-objective optimization problem (MOOP), in which the objectives are to minimize the total cost and total transport time. For the model with interval demands, the instances are only solved, in which, for each item, the sum total of demands of an item at different destinations is at least the sum total of availabilities of the item at different origins. For each model, a set of four numerical examples are solved using two multi-objective evolutionary algorithms (MOEAs), namely, Non-dominated Soring Genetic Algorithm-II (NSGA-II) and Strength Pareto Evolutionary Algorithm 2 (SPEA2). A comparative study of the computational results are made in terms of four performance index, namely, RNI value, Hyper-volume (HV), Inverted Generational Distance (IGD) and Spread. The computational results clearly indicates that NSGA-II outperforms SPEA2 for all the numerical examples, in terms of generating better approximate Pareto front with respect to dominance, diversity and spread.