Multi-objective multi-item solid transportation problem in fuzzy environment

A multi-objective multi-item solid transportation problem with fuzzy coefficients for the objectives and constraints, is modeled and then solved by two different methods. A defuzzification method based on fuzzy linear programming is applied for fuzzy supplies, demands and conveyance capacities, including the condition that both total supply and conveyance capacity must not fall below the total demand. First, expected values of the fuzzy objective functions are considered to derive crisp values. Another method based on the concept of “minimum of fuzzy number” is applied for the objective functions that yields fuzzy values instead of particular crisp values for the fuzzy objectives. Fuzzy programming technique and global criterion method are applied to derive optimal compromise solutions of multi-objectives. A numerical example is solved using above mentioned methods and corresponding results are compared.     

A bicriteria solid transportation problem with fixed charge under stochastic environment

In this paper, a bicriteria solid transportation problem with stochastic parameters is investigated. Three mathematical models are constructed for the problem, including expected value goal programming model, chance-constrained goal programming model and dependent-chance goal programming model. A hybrid algorithm is also designed based on the random simulation algorithm and tabu search algorithm to solve the models. At last, some numerical experiments are presented to show the performance of models and algorithm.     

Fractional Goal Programming for Fuzzy Solid Transportation Problem with Interval Cost

In this paper, we study a solid transportation problem with interval cost using fractional goal programming approach (FGP). In real life applications of the FGP problem with multiple objectives, it is difficult for the decision-maker(s) to determine the goal value of each objective precisely as the goal values are imprecise, vague, or uncertain. Therefore, a fuzzy goal programming model is developed for this purpose. The proposed model presents an application of fuzzy goal programming to the solid transportation problem. Also, we use a special type of non-linear (hyperbolic) membership functions to solve multi-objective transportation problem. It gives an optimal compromise solution. The proposed model is illustrated by using an example.     

A fuzzy approach to the transportation problem

The transportation problem with fuzzy supply values of the deliverers and with fuzzy demand values of the receivers is analysed. For the solution of the problem the technique of parametric programming is used. This makes it possible to obtain not only the maximizing solution (according to the Bellman-Zadeh criterion) but also other alternatives close to the optimal solution.     

A solid transportation problem with mixed constraint in different environment

In this paper, we have introduced a Solid Transportation Problem where the constrains are mixed type. The model is developed under different environment like, crisp, fuzzy and intuitionistic fuzzy etc. Using the interval approximation method we defuzzify the fuzzy amount and for intuitionistic fuzzy set we use the ($\alpha,\beta$)-cut sets to get the corresponding crisp amount. To find the optimal transportation units a time and space based with order of convergence $O (MN^2)$ meta-heuristic Genetic Algorithm have been proposed. Also the equivalent crisp model so obtained are solved by using LINGO 13.0. The results obtained using GA treats as the best solution by comparing with LINGO results for this present study. The proposed models and techniques are finally illustrated by providing numerical examples. Degree of efficiency have been find out for both the algorithm.     

Vehicle routing problems with alternative paths: An application to on-demand transportation

The class of vehicle routing problems involves the optimization of freight or passenger transportation activities. These problems are generally treated via the representation of the road network as a weighted complete graph. Each arc of the graph represents the shortest route for a possible origin–destination connection. Several attributes can be defined for one arc (travel time, travel cost, etc.), but the shortest route modeled by this arc is computed according to a single criterion, generally travel time. Consequently, some alternative routes proposing a different compromise between the attributes of the arcs are discarded from the solution space. We propose to consider these alternative routes and to evaluate their impact on solution algorithms and solution values through a multigraph representation of the road network. We point out the difficulties brought by this representation for general vehicle routing problems, which drives us to introduce the so-called fixed sequence arc selection problem (FSASP). We propose a dynamic programming solution method for this problem. In the context of an on-demand transportation (ODT) problem, we then propose a simple insertion algorithm based on iterative FSASP solving and a branch-and-price exact method. Computational experiments on modified instances from the literature and on realistic data issued from an ODT system in the French Doubs Central area underline the cost savings brought by the proposed methods using the multigraph model.     

Interactive fuzzy programming approach to Bi-level quadratic fractional programming problems

In this paper we propose an interactive fuzzy programming method for obtaining a satisfactory solution to a “bi-level quadratic fractional programming problem” with two decision makers (DMs) interacting with their optimal solutions. After determining the fuzzy goals of the DMs at both levels, a satisfactory solution is efficiently derived by updating the satisfactory level of the DM at the upper level with consideration of overall satisfactory balance between both levels. Optimal solutions to the formulated programming problems are obtained by combined use of some of the proper methods. Theoretical results are illustrated with the help of a numerical example.     

Stochastic optimization models for a single-sink transportation problem

In this paper, we study a single-sink transportation problem in which the production capacity of the suppliers and the demand of the single customer are stochastic. Shipments are performed by capacitated vehicles, which have to be booked in advance, before the realization of the production capacity and the demand. Once the production capacity and the demand are revealed, there is an option to cancel some of the booked vehicles against a cancellation fee; if the quantity shipped from the suppliers using the booked vehicles is not enough to satisfy the demand, the residual quantity is purchased from an external company. The problem is to determine the number of vehicles to book in order to minimize the total cost. We formulate a two-stage and a multistage stochastic mixed integer linear programming models to solve this problem and test them on a real case provided by Italcementi, the primary Italian cement producer and the fifth largest cement producer in the world. We test the influence of different scenario-tree structures on the solutions of the problem, as well as sensitivity of the results with respect to the cancellation fee.     

A geometric approach for solving fuzzy linear programming problems

In this paper we first recall some definitions and results of fuzzy plane geometry, and then introduce some definitions in the geometry of two-dimensional fuzzy linear programming (FLP). After defining the optimal solution based on these definitions, we use the geometric approach for obtaining optimal solution(s) and show that the algebraic solutions obtained by Zimmermann method (ZM) and our geometric solutions are the same. Finally, numerical examples are solved by these two methods.     

On Sharma-Swarup algorithm for time minimising transportation problems

In this note, the fallacy in the method given by Sharma and Swarup, in their paper on time minimising transportation problem, to determine the setS _hkof all nonbasic cells which when introduced into the basis, either would eliminate a given basic cell (h, k) from the basis or reduce the amountx _hkis pointed out.     

Max-min sum minimization transportation problem

A non-convex optimization problem involving minimization of the sum of max and min concave functions over a transportation polytope is studied in this paper. Based upon solving at most (g+1)(< p) cost minimizing transportation problems with m sources and n destinations, a polynomial time algorithm is proposed which minimizes the concave objective function where, p is the number of pairwise disjoint entries in the m× n time matrix {t _ij } sorted decreasingly and T ^g is the minimum value of the max concave function. An exact global minimizer is obtained in a finite number of iterations. A numerical illustration and computational experience on the proposed algorithm is also included. We are thankful to Prof. S. N. Kabadi, University of New Brunswick-Fredericton, Canada, who initiated us to the type of problem discussed in this paper. We are also thankful to Mr. Ankit Khandelwal, Ms. Neha Gupta and Ms. Anuradha Beniwal, who greatly helped us in the implementation of the proposed algorithm.     

Uncertain models for single facility location problems on networks

In practical location problems on networks, the vertex demand is usually non-deterministic. This paper employs uncertainty theory to deal with this non-deterministic factor in single facility location problems. We first propose the concepts of satisfaction degree for both vertices and the whole network, which are used to evaluate products assignment. Based on different network satisfaction degree, two models are constructed. The solution to these models is based on Hakimi’s results, and some examples are given to illustrate these models.     

A new method for solving fuzzy transportation problems using ranking function

In the literature, several methods are proposed for solving transportation problems in fuzzy environment but in all the proposed methods the parameters are represented by normal fuzzy numbers. [S.H. Chen, Operations on fuzzy numbers with function principal, Tamkang Journal of Management Sciences 6 (1985) 13–25] pointed out that in many cases it is not to possible to restrict the membership function to the normal form and proposed the concept of generalized fuzzy numbers. There are several papers in the literature in which generalized fuzzy numbers are used for solving real life problems but to the best of our knowledge, till now no one has used generalized fuzzy numbers for solving the transportation problems. In this paper, a new method is proposed for solving fuzzy transportation problems by assuming that a decision maker is uncertain about the precise values of the transportation cost, availability and demand of the product. In the proposed method transportation cost, availability and demand of the product are represented by generalized trapezoidal fuzzy numbers. To illustrate the proposed method a numerical example is solved and the obtained results are compared with the results of existing methods. Since the proposed method is a direct extension of classical method so the proposed method is very easy to understand and to apply on real life transportation problems for the decision makers.     

A review of urban transportation network design problems

This paper presents a comprehensive review of the definitions, classifications, objectives, constraints, network topology decision variables, and solution methods of the Urban Transportation Network Design Problem (UTNDP), which includes both the Road Network Design Problem (RNDP) and the Public Transit Network Design Problem (PTNDP). The current trends and gaps in each class of the problem are discussed and future directions in terms of both modeling and solution approaches are given. This review intends to provide a bigger picture of transportation network design problems, allow comparisons of formulation approaches and solution methods of different problems in various classes of UTNDP, and encourage cross-fertilization between the RNDP and PTNDP research.     

Combining linear programming and automated planning to solve intermodal transportation problems

When dealing with transportation problems Operational Research (OR), and related areas as Artificial Intelligence (AI), have focused mostly on uni-modal transport problems. Due to the current existence of bigger international logistics companies, transportation problems are becoming increasingly more complex. One of the complexities arises from the use of intermodal transportation. Intermodal transportation reflects the combination of at least two modes of transport in a single transport chain, without a change of container for the goods. In this paper, a new hybrid approach is described which addresses complex intermodal transport problems. It combines OR techniques with AI search methods in order to obtain good quality solutions, by exploiting the benefits of both kinds of techniques. The solution has been applied to a real world problem from one of the largest spanish companies using intermodal transportation, Acciona Transmediterránea Cargo.     

A unified approach to infeasible-interior-point algorithms via geometrical linear complementarity problems

There are many interior-point algorithms for LP (linear programming), QP (quadratic programming), and LCPs (linear complementarity problems). While the algebraic definitions of these problems are different from each other, we show that they are all of the same general form when we define the problems geometrically. We derive some basic properties related to such geometrical (monotone) LCPs and based on these properties, we propose and analyze a simple infeasible-interior-point algorithm for solving geometrical LCPs. The algorithm can solve any instance of the above classes without making any assumptions on the problem. It features global convergence, polynomial-time convergence if there is a solution that is smaller than the initial point, and quadratic convergence if there is a strictly complementary solution.This research was performed while the first author was visiting the Institute of Applied Mathematics and Statistics, Würzburg University as a Research Fellow of the Alexander von Humboldt Foundation.     

An algorithm for a fuzzy transportation problem to select a new type of coal for a steel manufacturing unit

Manufacturing of steel involves thermal energy intensive processes with coal as the major input. Energy generated is a direct function of ash content of coal and as such it weighs very high as regards the choice of coal. In this paper, we study a multiobjective transportation problem to introduce a new type of coal in a steel manufacturing unit in India. The use of new type of coal serves three non-prioritized objectives, viz. minimization of the total freight cost, the transportation time and the ratio of ash content to the production of hot metal. It has been observed from the past data that the supply and demand points have shown fluctuations around their estimated values because of changing economic conditions. To deal with uncertainties of supply and demand parameters, we transform the past data pertaining to the amount of supply of the ith supply point and the amount of demand of the jth demand point using level (λ,ρ) interval-valued fuzzy numbers. We use a linear ranking function to defuzzify the fuzzy transportation problem. A transportation algorithm is developed to find the non-dominated solutions for the defuzzified problem. The application of the algorithm is illustrated by numerical examples constructed from the data provided by the manufacturing unit.      

Fuzzy linear programming problems as bi-criteria optimization problems

In this paper, we consider fuzzy linear programming (FLP) problems which involve fuzzy numbers only in coefficients of objective function. First, we shall give concepts of optimal solutions to (FLP) problems and investigate their properties. Next, in order to find all optimal solutions, we define three types of bi-criteria optimization problems.     

Solving fuzzy queueing decision problems via a parametric mixed integer nonlinear programming method

This paper proposes a mathematical programming method to construct the membership functions of the fuzzy objective value of the cost-based queueing decision problem with the cost coefficients and the arrival rate being fuzzy numbers. On the basis of Zadeh’s extension principle, three pairs of mixed integer nonlinear programs (MINLP) parameterized by the possibility level α are formulated to calculate the lower and upper bounds of the minimal expected total cost per unit time at α, through which the membership function of the minimal expected total cost per unit time of the fuzzy objective value is constructed. To provide a suitable optimal service rate for designing queueing systems, the Yager’s ranking index method is adopted. Two numerical examples are solved successfully to demonstrate the validity of the proposed method. Since the objective value is completely expressed by a membership function rather than by a crisp value, it conserves the fuzziness of the input information, thus more information is provided for designing queueing systems. The successful extension of queueing decision models to fuzzy environments permits queueing decision models to have wider applications in practice.     

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.