A new fuzzy multi-objective programming: Entropy based geometric programming and its application of transportation problems

In this paper, we introduce a fuzzy mathematical programming with generalized fuzzy number as objective coefficients. We also examine a transportation problem with additional restriction. There is an additional entropy objective function in the transportation problem besides transportation cost objective function. Using new fuzzy mathematical programming, this multi-objective entropy transportation problem with generalized trapezoidal fuzzy number costs has been reduced to a primal geometric programming problem. Pareto optimal solution of the transportation model is found. Numerical examples have been provided to illustrate the problem.     

The Economic Distribution of Coal Supplies in the Gas Industry: An Application of the Linear Programming Transport Problem

The North Western Gas Board bases its allocation of coal supplies from coal mines to gas works, on the solution to a linear programming transportation problem which is calculated at regular intervals on an electronic computer. A complete solution to the problem would require a more general linear programming model but, due to practical limitations with regard to both data and computer, development has been in the relatively simple transportation form. A variety of restrictions have been built into the programme, however, so as to avoid technical production difficulties which might otherwise arise, and in practice the transportation model has proved extremely versatile.     

Initial Solutions to Sets of Related Transportation Problems

Where sets of similar transportation problems are solved independently a great deal of information is wasted. A routine which has been used successfully is described; it is for use in conjunction with the Ford and Fulkerson method for solving the transportation problem, and utilizes the information provided by the optimum solution to one problem to give a good initial solution to the later problems.A similar routine could be easily developed for use with the Stepping Stone method.     

Capacitated transportation problem with bounds on RIM conditions

Motivated by dead-mileage problem assessed in terms of running empty buses from various depots to starting points, we consider a class of the capacitated transportation problems with bounds on total availabilities at sources and total destination requirements. It is often difficult to solve such problems and the present paper establishes their equivalence with a balanced capacitated transportation problem which can be easily solved by existing methods. Sometimes, total flow in transportation problem is also specified by some external decision maker because of budget/political consideration and optimal solution of such problem is of practical interest to the decision maker and has motivated us to discuss such problem. Various situations arising in unbalanced capacitated transportation problems have been discussed in the present paper as a particular case of original problem. In addition, we have discussed paradoxical situation in a balanced capacitated transportation problem and have obtained the paradoxical solution by solving one of the unbalanced problems. Numerical illustrations are included in support of theory.     

Multiobjective transportation problem with interval cost,source and destination parameters

In this paper, we focus on the solution procedure of the multiobjective transportation problem (MOTP) where the cost coefficients of the objective functions, and the source and destination parameters have been expressed as interval values by the decision maker. This problem has been transformed into a classical MOTP where to minimize the interval objective function, the order relations that represent the decision maker's preference between interval profits have been defined by the right limit, left limit, centre, and half-width of an interval. The constraints with interval source and destination parameters have been converted into deterministic ones. Finally, the equivalent transformed problem has been solved by fuzzy programming technique. Numerical examples have been provided to illustrate the solution procedure for three possible cases of the original problem.     

Production and shipment lot sizing in a vendor–buyer supply chain with transportation cost

Previous research on the joint vendor–buyer problem focused on the production shipment schedule in terms of the number and size of batches transferred between the two parties. It is a fact that transportation cost is a major part of the total operational cost. However, in most joint vendor–buyer models, the transportation cost is only considered implicitly as a part of fixed setup or ordering cost and thus is assumed to be independent of the size of the shipment. As such, the effect of the transportation cost is not adequately reflected in final planning decisions. There is a need for models involving transportation cost explicitly for better decision-making. In this study we analyze the vendor–buyer lot-sizing problem under equal-size shipment policy. We introduce the complete solution of the problem in an explicit and extended manner that has not existed in the literature. We incorporate transportation cost explicitly into the model and develop optimal solution procedures for solving the integrated models. All-unit-discount transportation cost structures with and without over declaration have been considered. Numerical examples are presented for illustrative purpose.     

An improved disaggregation method for transportation problems

A method is given to disaggregate the solution to an aggregated transportation problem. The resulting solution to the original problem is feasible, all-integer, and has lower cost that those of solutions produced by earlier methods.     

Routing and scheduling in a liquefied natural gas shipping problem with inventory and berth constraints

Liquefied natural gas (LNG) is natural gas that has been transformed to liquid form for the purpose of transportation, which is mainly done by specially built LNG vessels travelling from the production site to the consumers. We describe a real-life ship routing and scheduling problem from the LNG business, with both inventory and berth capacity constraints at the liquefaction port. We propose a solution method where the routing and scheduling decisions are decomposed. The routing decisions consist of deciding which vessels should service which cargoes and in what sequence. The scheduling decisions are then to decide when to start servicing the cargoes while satisfying inventory and berth capacity constraints. The proposed solution method has been tested on several problem instances based on the real-life problem. The results show that the proposed solution method is well suited to solve this LNG shipping problem.     

基于遗传算法与贪婪策略的多港口集装箱配载研究

在物流运输行业中,集装箱运输已经成为我国长江沿岸各大港口的主要运输业务。集装箱的处理流程,尤其是集装箱的配载过程直接影响着班轮的运输效率,配载方案的制定对班轮运输起着至关重要的作用。本文针对多港口集装箱船的配载情况,利用CPLEX对该线性规划问题进行求解,并设计遗传算法和贪婪算法对长江沿岸多港口集装箱船配载情形进行对比。通过仿真实验,在小规模时遗传算法与CPLEX求解的精确解相同,验证了遗传算法的有效性。并且在大规模运输情形下,遗传算法得出的结果明显优于贪婪策略,进一步说明了遗传算法是行之有效的。得出的解决方案降低了班轮公司的运输成本,提高了港口的工作效率,对我国长江沿岸港口集装箱配载计划的制定具有一定的指导作用。     

满意度优化运输问题

传统运输问题只考虑配送方案的效率, 而不考虑参与者对配送方案的满意度. 通过引入参与者对配送方案的满意度这一概念, 提出了满意度优化运输问题, 构建了以最大化相对公平为目标的满意度优化运输模型, 并证明了: (1) 当运输问题的可行域不空时, 新模型的解集非空; (2) 从满意度的角度来看, 新模型的解是唯一的. 另外, 还给出了新模型的求解方法. 研究结果进一步丰富了运输问题的类型, 可为解决其他类型运输问题提供借鉴.     

Multi-Item Scheduling by Benders' Decomposition

Scheduling the production of several items requires the determination of production quantities in different periods in the presence of resource constraints. Several approximate and heuristic algorithms have been proposed to solve this problem. However, no method for finding an optimal solution has as yet been developed. It is shown that the problem may be solved advantageously using Benders' decomposition. The subproblem in Benders' decomposition is shown to be a transportation problem, and some strategies for solving the master problem are indicated. The paper concludes with a sample problem demonstrating the application of the method.     

A Note on Resolution of Degeneracy in Transportation Problems

Degeneracy in transportation problems is a frequent phenomenon. In this paper, a method of obtaining optimal solutions to degenerate transportation problems has been suggested. The method is a modification of the already-known Modified Distribution (MODI) method and consists in proceeding with the non-zero cells of the basis and a dual solution corresponding to these cells—without attempting to complete the basis. It appears that the method proposed here would be more effective than the well-known perturbation method, especially when the problem is highly degenerate.     

Fuzzy goal programming for multiobjective transportation problems

Several fuzzy approaches can be considered for solving multiobjective transportation problem. This paper presents a fuzzy goal programming approach to determine an optimal compromise solution for the multiobjective transportation problem. We assume that each objective function has a fuzzy goal. Also we assign a special type of nonlinear (hyperbolic) membership function to each objective function to describe each fuzzy goal. The approach focuses on minimizing the negative deviation variables from 1 to obtain a compromise solution of the multiobjective transportation problem. We show that the proposed method and the fuzzy programming method are equivalent. In addition, the proposed approach can be applied to solve other multiobjective mathematical programming problems. A numerical example is given to illustrate the efficiency of the proposed approach.     

需求区间型运输问题的求解算法

为了便于建立与需求区间型运输问题有关的决策支持系统，本给出了一个求解需求区间型运输问题的数值算法，证明了算法的理论依据，并举例说明算法的应用，该算法能求得问题的最优解，并具有易于编程实现、收敛性好等优点，大量数值实验表明该算法有较高的计算效率。     

Equitable Demand Adjustment for Infeasible Transportation Problems

We present a modelling/solution procedure for adjusting demands to obtain an ‘equitably infeasible’ solution for an infeasible transportation problem. The infeasibility may be due to total supply not being equal to total demand, or inadmissible routes (arcs). We show that the problem can be modelled and solved as a pre-emptive, multicriteria, capacitated transportation problem, whose objective is to minimize the maximum deviation between the fractional undersupply to the demand nodes or, equivalently, to minimize the fractional undersupply of the demands. Further, we show that an optimal solution can be obtained by solving either a single-criterion transportation problem (by choosing sufficiently large weights to aggregate the criteria) or a sequence of single-criterion transportation problems.     

An Exact Solution Algorithm for a Class of Production Planning and Scheduling Problems

This paper presents a formulation and an exact solution algorithm for a class of production planning and scheduling problems. The problem is one of optimally specifying production levels for each product in each period of the planning horizon. The objective is to minimize the sum of the set-up, regular time production, overtime and inventory holding costs. The problem has been formulated as a variation of fixed charge transportation problem. The problem discussed here is NP-hard in computational complexity. A numerical example is presented for better understanding of the algorithm.     

A decentralized two-level transportation problem in a housing material manufacturer: Interactive fuzzy programming approach

In this paper, we deal with a transportation problem in a housing material manufacturer and derive a satisfactory solution to the problem by taking into account not only the degree of satisfaction with respect to objectives of the housing material manufacturer but also those of two forwarding agents to which the housing material manufacturer entrusts transportation of products. The two forwarding agents handle the regular transportation and the small lot transportation, and each of them assigns work force, the transformation tasks properly. There are two objectives in the housing material manufacturer: one is to minimize the transportation cost and the other is to minimize the opportunity loss with respect to transportation time. The objectives of the two forwarding agents are to maximize their profits. We formulate the transportation planning and work force assignment problem as a decentralized two-level integer programming problem, and derive a satisfactory solution by applying an interactive fuzzy programming method.     

Two level hierarchical time minimizing transportation problem

A two level hierarchical balanced time minimizing transportation problem is considered in this paper. The whole set of source-destination links consists of two disjoint partitions namely Level-I links and Level-II links. Some quantity of a homogeneous product is first shipped from sources to destinations by Level-I decision maker using only Level-I links, and on its completion the Level-II decision maker transports the remaining quantity of the product in an optimal fashion using only Level-II links. Transportation is assumed to be done in parallel in both the levels. The aim is to find that feasible solution for Level-I decision maker corresponding to which the optimal feasible solution for Level-II decision maker is such that the sum of shipment times in Level-I and Level-II is the least. To obtain the global optimal feasible solution of this non-convex optimization problem, related balanced time minimizing transportation problems are defined. Based upon the optimal feasible solutions of these related problems, standard cost minimizing transportation problems are constructed whose optimal feasible solutions provide various pairs for shipment times for Level-I and Level-II decision makers. The best out of these pairs is finally selected. Being dependent upon solutions of a finite number of balanced time minimizing and cost minimizing transportation problems, the proposed algorithm is a polynomial bound algorithm. The developed algorithm has been implemented and tested on a variety of test problems and performance is found to be quite encouraging.     

Multi-item single source ordering problem with transportation cost: A Lagrangian decomposition approach

As a part of supply chain management literature and practice, it has been recognized that there can be significant gains in integrating inventory and transportation decisions. The problem we tackle here is a common one both in retail and production sectors where several items have to be ordered from a single supplier. We assume that there is a finite planning horizon to make the ordering decisions for the items, and in this finite horizon the retailer or the producer knows the demand of each item in each period. In addition to the inventory holding cost, an item-base fixed cost associated with each item included in the order, and a piecewise linear transportation cost are incurred. We suggest a Lagrangean decomposition based solution procedure for the problem and carry out numerical experiments to analyze the value of integrating inventory and transportation decisions under different scenarios.     

运输问题求解的一种网络算法

本着重探讨了在网络图上求运输问题的初始解的方法，并指出在求解受时间约束的运输问题时得到的初始解，在很大程度就是该问题的最优解，通过实例说明了该算法。