Fixed Charge Transportation Problem Research Articles

Multi-objective fixed-charge transportation problem using rough programming

This paper analyses the multi-objective fixed-charge transportation problem (MOFCTP) using rough programming. Due to globalisation of market, the parameters of MOFCTP may not be defined precisely, so the parameters of the MOFCTP are treated as rough intervals. Expected value operator is used to convert rough MOFCTP to deterministic MOFCTP. Fuzzy programming method and linear weighted-sum method are used to obtain Pareto-optimal solution from deterministic MOFCTP. A comparative study is made between the obtained solutions extracted from the methods; and thereafter we perform a procedure to analyse the sensitive analysis of the parameters in MOFCTP. Finally, in order to show the applicability of our proposed study, an example on MOFCTP is included in this paper.

The noncooperative fixed charge transportation problem

We introduce the noncooperative fixed charge transportation problem (NFCTP), which is a game-theoretic extension of the fixed charge transportation problem. In the NFCTP, competing players solve coupled fixed charge transportation problems simultaneously. Three versions of the NFCTP are discussed and compared which differ in the treatment of shared social costs. This may be used from central authorities in order to find a socially balanced framework which is illustrated in a numerical study. Using techniques from generalized Nash equilibrium problems with mixed-integer variables we show the existence of Nash equilibria for these models and examine their structural properties. Since there is no unique equilibrium for the NFCTP, we also discuss how to solve the Nash selection problem and, finally, propose numerical methods for the computation of Nash equilibria which are based on mixed-integer programming.

Sustainable transportation planning for a three-stage fixed charge multi-objective transportation problem

In the recent past, sustainability has become a major concern for transportation policies and planning in both developed and developing countries. This paper focuses on transportation sustainability for a three-stage fixed charge transportation problem. The major components of transportation sustainability considered include economical issues, social concerns, environmental concerns, and transportation system efficiency. Another important issue considered from a social point of view is the interrelationships between various customers of an end product, which has several benefits culminating in a healthier bottom line. The approach adopted in this paper consists of two phases, wherein the efficiency of vehicles is evaluated independently on all three parameters of sustainability using the data envelopment analysis technique in the first phase. The second phase consists of optimizing an integrated multi-objective optimization model that utilizes efficiency of the vehicles obtained from the first phase in a benefit criterion, considering the interrelationships among customers in terms of minimizing the independence values, and maximizing total profits along with many real-world constraints. Numerical illustration of a real-world case is included in order to demonstrate the utility of the proposed approach.

Extending the solid step fixed-charge transportation problem to consider two-stage networks and multi-item shipments

This paper develops a new mathematical model for a capacitated solid step fixed-charge transportation problem. The problem is formulated as a two-stage transportation network and considers the option of shipping multiple items from the plants to the distribution centers (DC) and afterwards from DCs to customers. In order to tackle such an NP-hard problem, we propose two meta-heuristic algorithms; namely, Simulated Annealing (SA) and Imperialist Competitive Algorithm (ICA). Contrary to the previous studies, new neighborhood strategies maintaining the feasibility of the problem are developed. Additionally, the Taguchi method is used to tune the parameters of the algorithms. In order to validate and evaluate the performances of the model and algorithms, the results of the proposed SA and ICA are compared. The computational results show that the proposed algorithms provide relatively good solutions in a reasonable amount of time. Furthermore, the related comparison reveals that the ICA generates superior solutions compared to the ones obtained by the SA algorithm.

An uncertain two-echelon fixed charge transportation problem

In the present study, a two-echelon fixed charge transportation problem is investigated under uncertainty. Due to the existence of considerable amount of uncertainties, the demands, supplies, availabilities, fixed charges and transported quantities in this problem are assumed as uncertain variables. The aim is to maximize the total profit under uncertain environments. The expected value model, chance-constrained model and measure chance model are developed, and the deterministic equivalent forms of these models are obtained by inverse uncertainty distribution. Genetic algorithm and particle swarm optimization are proposed to solve the equivalent forms of the models based on the structure of the problem. To verify the effectiveness of these proposed approaches, numerical experiments are performed.

Multi-objective non-linear fixed charge transportation problem with multiple modes of transportation in crisp and interval environments

This paper aims to propose an approach based on NSGA-II for solving multi-objective non-linear fixed charge transportation problem with multiple modes of transport in crisp and interval environments. Certain modifications need to be made in the existing NSGA-II configuration to calculate the crowding distance of a solution in the interval environment. Besides, a crossover and a mutation scheme suitable for multiple modes of transportation are developed. In the end, a set of test problems are solved in both environments and some comparative studies are performed restricting the problem to only one mode of transport at a time. Finally, the results of the proposed algorithm are compared with SPEA2.

Solving the Fixed Charge Transportation Problem by New Heuristic Approach

The fixed charge transportation problem (FCTP) is a deployment of the classical transportation problem in which a fixed cost is incurred, independent of the amount transported, along with a variable cost that is proportional to the amount shipped. Since the problem is considered as an NP-hard, the computational time grows exponentially as the size of the problem increases. In this paper, we propose a new heuristic along with well-known metaheuristic like Geneticalgorithm (GA), simulated annealing (SA) and recently developed one, Keshtel algorithm (KA) to solve the FCTP. Contrary to previous works, we develop a simple and strong heuristic according to the nature of the problem and compare the result with metaheuristics. In addition, since the researchers recently used the priority-based representation to encode the transportation graphs and achieved very good results, we consider this representation in metaheuristics and compare the results with the proposed heuristic. Furthermore, we apply the Taguchi experimental design method to set the proper values of algorithms in order to improve their performances. Finally, computational results of heuristic and metaheuristics with different encoding approaches, both in terms of the solution quality and computation time, are studied in different problem sizes.

New algorithmic framework for conditional value at risk: Application to stochastic fixed-charge transportation

This paper introduces a new algorithmic scheme for two-stage stochastic mixed-integer programming assuming a risk averse decision maker. The focus is the minimization of the conditional value at risk for a hard combinatorial optimization problem. Some properties of a mixed-integer non-linear programming formulation for conditional value at risk are studied as well as their algorithmic implications. This yields to a procedure for obtaining lower and upper bounds on the optimal value of the problem that may lead to an optimal solution. The new developments are applied to a fixed-charge transportation problem with stochastic demand, and they are computationally tested. The corresponding results are thoroughly presented and discussed.

A model for two-stage fixed charge transportation problem with multiple objectives and fuzzy linguistic preferences

In this paper, a multi-objective model for two-stage fixed charge transportation planning problem is studied. The transportation process is considered to occur from manufacturing plants to the distributers and then from distributers to the customers. The availabilities at the manufacturing plants, capacities of the distributers and demand of the customers, all are considered to be fuzzy numbers. The proposed model is formulated with three conflicting goals or objective functions. The first objective is to minimize the total transportation cost involved in the whole transportation process. The second objective is to maximize the total quantity of the products to be transported, whereas minimizing the total deterioration that occurred during the transportation process is considered to be the third objective function. Fuzzy linguistic relations or preferences among the three objective functions are studied. A linear membership function is used to represent the fuzzy relative preferences between the objective functions. For solving the multi-objective problem, fuzzy goal programming technique is adopted with some linear and nonlinear membership functions. Finally, the proposed model is illustrated and solved for some simulated numerical data and some sensitivity analysis for the problem is also discussed. The best results for the solved numerical problem are found when hyperbolic membership functions are considered to model the aspiration levels for objective functions, whereas comparatively less significant results are found when linear membership functions are used to model the aspiration levels for objective functions.

Time-cost trade-off in a multi index bi-criteria fixed charge transportation problem

This paper is motivated by the necessity of finding efficient time-cost trade-off pairs for multi index bi-criteria fixed charge transportation problem wherein time criteria assumes greater significance than cost criteria and is therefore minimised first. So far no study on finding optimal time-cost pairs of the problem has been undertaken, a gap which the newly developed algorithm aims to fulfil. Results of a numerical illustration are obtained by the proposed algorithm by applying extended Vogel's approximation method and extended modified distribution method; it has been observed that the proposed algorithm is a far better alternative in problems in which time minimisation holds priority over cost minimisation.

Time-cost trade-off in a multi index bi-criteria fixed charge transportation problem

Time-cost trade-off in a multi index bi-criteria fixed charge transportation problem

Multi-Product Multi-Period Fixed Charge Transportation Problem: an Ant Colony Optimization Approach

Most of the practical applications of a transportation network, in addition to the variable cost, there incurs a fixed charge. This work formulates a Multi-Stage Multi-Period Fixed Charge Transportation Problem for a multi-product scenario. The problem is modeled using an optimization modeling tool, ‘A Mathematical Programming Language’ and its solution is obtained in BONMIN solver. The exact algorithms mostly require longer computational time to find an optimal solution for large problem size that are in practice. In these operational problems, in which process speed is as important as the solution quality, an Ant Colony Optimization based heuristic is proposed. Finally, the solution obtained from proposed heuristic is compared with that of exact methods using randomly generated data sets. The comparative analysis shows the competitiveness of the proposed heuristic.

An efficient iterated local search heuristic algorithm for the two-stage fixed-charge transportation problem

This paper concerns the two-stage transportation problem with fixed charges associated to the routes and proposes an efficient multi-start Iterated Local Search (ILS) procedure for the total distribution costs minimization. Our heuristic approach constructs an initial solution, uses a local search procedure to increase the exploration, a perturbation mechanism and a neighborhood operator in order to diversify the search. Computational experiments were performed on two sets of instances: one that consists of 20 benchmark instances available in the literature and a second one that contains 10 new randomly generated larger instances. The achieved computational results prove that our proposed solution approach is highly competitive in comparison with the existing approaches from the literature.

Multi-Objective Fixed-Charge Transportation Problem with Random Rough Variables

This paper considers a multi-objective fixed-charge transportation problem (MOFCTP) in which the parameters of the objective functions are random rough variables, while the supply and the demand parameters are rough variables. In real-life situations, the parameters of a multi-objective fixed-charge transportation problem may not be defined precisely, because of globalization of the market, uncontrollable factors, etc. As such, the multi-objective fixed-charge transportation problem is proposed under rough and random rough environments. To tackle uncertain (rough and random rough) parameters, the proposed model employs an expected value operator. Furthermore, a procedure is developed for converting the uncertain multi-objective fixed-charge transportation problem into a deterministic form and then solving the deterministic model. Three different methods, namely, the fuzzy programming, global criterion, and ϵ-constrained methods, are used to derive the optimal compromise solutions of the suggested model. To provide the preferable optimal solution of the formulated problem, a comparison is drawn among the optimal solutions that are extracted from different methods. Herein, the ϵ-constrained method derives a set of optimal solutions and generates an exact Paretofront. Finally, in order to show the applicability and feasibility of the proposed model, the paper includes a real-life example of a multi-objective fixed-charge transportation problem. The main contribution of the paper is that it deals with MOFCTP using two types of uncertainties, thus making the decision making process more flexible.

Heuristic approaches to solve the fixed-charge transportation problem with discount supposition

ABSTRACTThe fixed-charge transportation problem (FCTP) is one of important and classical transportation problems with many real-world applications in the area of logistics and supply chain management. Due to nature complexity of this problem, the literature has seen a large number of heuristics and meta-heuristics to solve the FCTP. This paper proposes a new heuristic along with well-known meta-heuristics to solve the FCTP with discount supposition on both fixed and variable charges. In addition, two models with all-units discount and incremental discount are firstly introduced in this study to apply the discount mechanism. As such, since the previous researchers mainly used spanning tree-based and priority-based representations, this study utilizes both of these methods and compared the results. Finally, a comprehensive discussion based on the computational results of heuristic and meta-heuristics with different encoding approaches has been investigated through different problem sizes.

Prioritized K-mean clustering hybrid GA for discounted fixed charge transportation problems

Abstract The problem of allocating different types of vehicles for transporting a set of products in an existing transportation network, to minimize the total transportation costs, is considered. The distribution network involves a heterogeneous fleet of vehicles each with the given capacity and with a variable transportation cost and a fixed cost with a discounting mechanism. Due to nonlinearity of the discounting mechanism, a nonlinear mathematical programming model is developed. A prioritized K-mean clustering encoding is introduced to designate the distribution depots distances, their demands, and the vehicles’ capacity. Using this priority clustering, a heuristic routine is developed by which heavy capacity vehicles are assigned to the longer distance depots. Then the proposed heuristic is incorporated into a new GA to construct a hybrid GA. Through an extensive computational experiment, first the algorithm parameters are tuned using “factorial experimental designs”, and then the efficiency of the proposed algorithm is compared with the powerful package of the CPLEX solver (OPL 12.3.0.1 Model) and two existing algorithms. The results are revealed that the proposed algorithm can provide superior solutions with the minimal computational effort.

A Heuristic Approach for Solving the Fixed Charge Transportation Problems

A Heuristic Approach for Solving the Fixed Charge Transportation Problems

English

English

The fixed charge transportation problem: a strong formulation based on Lagrangian decomposition and column generation

A new and strong convexified formulation of the fixed charge transportation problem is provided. This formulation is obtained by integrating the concepts of Lagrangian decomposition and column generation. The decomposition is made by splitting the shipping variables into supply and demand side copies, while the columns correspond to extreme flow patterns for single sources or sinks. It is shown that the combination of Lagrangian decomposition and column generation provides a formulation whose strength dominates those of three other convexified formulations of the problem. Numerical results illustrate that our integrated approach has the ability to provide strong lower bounds. The Lagrangian decomposition yields a dual problem with an unbounded set of optimal solutions. We propose a regularized column generation scheme which prioritizes an optimal dual solution with a small $$l_1$$ -norm. We further demonstrate numerically that information gained from the strong formulation can also be used for constructing a small-sized core problem which yields high-quality upper bounds.

A matheuristic for the two-stage fixed-charge transportation problem

This paper addresses the two-stage fixed-charge transportation problem which involves the distribution of a commodity from plants to customers through intermediate depots, while minimizing the overall costs incurred. There are two costs associated with each arc: a fixed cost for the use of the arc, and a variable cost proportional to the number of units sent along the arc. First, we prove some theoretical properties which extend well-known results of the fixed-charge transportation problem. Then, we present a matheuristic that uses an evolutionary algorithm and exploits these properties to guide the algorithm towards better solutions. The chromosome of the evolutionary algorithm controls the arcs that can be used in the delivery. Its fitness is computed as the objective function value of a feasible solution of the problem, which is obtained by applying optimization techniques. The computational results show the effectiveness of the algorithm.