A relationship between the gravity model for trip distribution and the transportation problem in linear programming

In this paper, a transportation problem with an objective function as the sum of a linear and linear fractional function is considered. We present an algorithm to solve the above transportation problem. In addition, we consider two special cases of the problem, where the transportation flow is either restricted or enhanced. The first case deals with when one wishes to keep reserve stocks at the sources for emergencies, thereby restricting the total flow to a known specified level. The second case deals with when extra demand in the market compels some of the factories to increase their production, thereby enhancing the total flow to meet the customer demands fully. For the above special cases, we formulate a related linear plus linear fractional transportation problem. We present a numerical example to illustrate the proposed algorithm for the different cases. We show that the solution obtained is a local minimum occurring at an extreme point of the convex set of feasible solutions.

Logistics, supply chain management, and resource allocation are just a few of the many real-world contexts in which the Multi-Objective Linear Transportation Problem presents itself as a formidable optimization issue. In this paper, we seek to solve a multi-objective, linear and non-linear transportation problem by suggesting the most efficient solution possible. The unique Maximum divide and minimum allotment (MDMA) approach was proposed to find a workable answer to a particular transportation issue. It also calculates the solution to the problem using a number of different transportation methods and compares the result to MDMA methods. It is determined that the MDMA approach offers a practical algorithm and solution for the designated transportation problem.

This article addresses a special class of non-linear programming problems, viz., linear fractional programming problems having multiple objectives. In solving the real-life linear fractional optimization problems, the ambiguity and hesitation in the decision are inherent and ever-present, therefore, it is perfectly viable to formulate and solve these optimization models using the intuitionistic fuzzy environment. The purpose of this study is to propose a simple and computationally efficient approach to obtain the solution of multiple objective linear fractional programming problems having all the decision variables and parameters expressed in terms of triangular intuitionistic fuzzy numbers. The proposed solution algorithm is primarily based on the goal programming approach, fuzzy-based linearization technique, and a membership function strategy. The original linear fractional programming problem is first converted to its equivalent deterministic/crisp multi-objective linear fractional optimization problem using the weighted goal programming methodology along with the linear membership technique to resolve the intuitionistic fuzzy constraints into the crisp one. Finally, the variable transformation technique for the under- and over-deviational variables of the goal programming model is employed to linearize all the fractions involved in the problem so as to convert the original problem to an equivalent linear optimization problem. Further, this linear programming problem can be solved using any available commercial packages. Moreover, a numerical illustration is provided to demonstrate the steps of the proposed technique followed by the analysis and solution of an E-education set-up problem. The discussion and comparisons of the practical case establish the relevancy and usefulness of the proposed model.

In classical fractional transportation programming problem, we want to optimize the objective function in the form of one or several ratios subject to some linear constraints. If in multi-objective transportation problem, objective function is in ration of two linear function under some linear restrictions, then the problem is called multi-objective linear fractional transportation problem (MOLFTP). In this paper we propose a new method to solve multi-objective linear fractional transportation problem which is extension of Nomani et al. (Int J Manag Sci Eng Manag (2016) [9]). Two numerical problems are presented to validate the proposed algorithm.

Chapter 19 - Transportation problems

The fuzzy set theory is being applied massively in many fields these days. One of these is linear programming problems. Kheirfam and Hasani (Sensitivity analysis for fuzzy linear programming problems with fuzzy variables, Advanced Modeling and Optimization, 12 (2010) 257-272), proposed a method for the sensitivity analysis for fuzzy linear programming (FLP) problems with fuzzy variables. There are some important cases that are not considered by Kheirfam and Hasani. In this paper, those cases are considered and numerical examples are solved.

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.

ABSTRACTThis paper shows that the so‐called “transportation problem” possesses the four characteristics which define linear programming (LP) type problems. Thus, the transportation problem can always be solved by using the Simplex Method, a well‐known but tedious technique for dealing with any linear programming problem. A special procedure for solving the transportation problem is the so‐called “Transportation Method,” which involves three steps. However, due to intricacies in steps 2 and 3, this Method can, like Simplex, become quite tedious and time‐consuming.A short‐cut approach to solving the transportation problem is the Vogel Approximation Method (VAM), which is a very simple means of performing step 1 of the Transportation Method. Application of VAM to a given problem does not guarantee that an optimal solution will result. However, a very good solution is invariably obtained, and is obtained with comparatively little effort. For many purposes, using VAM to carry out step 1 of the Transportation Method eliminates (or all but eliminates) the need for performing steps 2 and 3. The mechanics of the Vogel Approximation Method are illustrated with reference to a particular transportation problem.

The problem to find a vector x* = (x 1 * , x 2 * ,..., x n * )⊤ such that its components satisfy the conditions $$ \begin{gathered} {\alpha _{11}}{x_1} + {\alpha _{12}}{x_2} + \ldots + {\alpha _{1n}}{x_n} \leqslant {\alpha _1} \hfill \\ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \hfill \\ {\alpha _{r1}}{x_1} + {\alpha _{r2}}{x_2} + \ldots + {\alpha _{rn}}{x_n} \leqslant {\alpha _r} \hfill \\ {\beta _{11}}{x_1} + {\beta _{12}}{x_2} + \ldots + {\beta _{1n}}{x_n} \geqslant {\beta _1} \hfill \\ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \hfill \\ {\beta _{s1}}{x_1} + {\beta _{s2}}{x_2} + \ldots + {\beta _{sn}}{x_n} \geqslant {\beta _s} \hfill \\ {\gamma _{11}}{x_1} + {\gamma _{12}}{x_2} + \ldots + {\gamma _{1n}}{x_n} = {\gamma _1} \hfill \\ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \hfill \\ {\gamma _{t1}}{x_1} + {\gamma _{t2}}{x_2} + \ldots + {\gamma _{tn}}{x_n} = {\gamma _t} \hfill \\ \end{gathered} $$ and a given objective function z(x) = c ⊤ x+c 0 = c 1 x 1+c 2 x 2+...+c n x n +c 0 attains its smallest value (minimum problem) or its greatest value (maximum problem) under all vectors x = (x 1, x 2,..., x n )⊤ fulfilling these conditions is called a linear programming (or optimization) problem. The conditions posed above are called the constraints or the restrictions of the problem. A vector x = (x 1,..., x n )⊤ satisfying all constraints is said to be feasible. A variable x i for which the relation x i ≥ 0 (non-negativity requirement) fails to occur among the constraints is referred to as a free or unrestricted variable.

In this paper, we propose a new model for linear fractional programming problem with absolute value functions. The major contribution of this paper is that transformation of linear fractional programming problem into separate linear programming problems with some theorems and then solution of these problems by popular algorithm. This work also used to simplex type algorithms to arrive at an optimal solution for a linear programming problem with absolute value. Moreover, we compare this method with an existing method. Numerical experiments are also given to illustrate the assertions.

Fully intuitionistic fuzzy multi-level linear fractional programming problem

Elementary Linear Programming with Applications

Using different dominance criteria in stochastic fuzzy linear multiobjective programming: A case of fuzzy weighted objective function

In this paper, we propose a method of solving the fully fuzzy linear fractional programming problems, Express all the parameters and variables are triangular fuzzy numbers. Convert all the triangular fuzzy numbers in their parametric form, we convert the fractional programming problem in to a single objective linear programming problem in parametric form. We put new fuzzy arithmetic and fuzzy ranking, we obtain the optimal solution the given fully fuzzy linear fractional programming problem without converting to its equivalent crisp linear programming problem. A numerical example is provided to illustrate the efficiency of the proposed method.

The levelling process of a vast area of land, for irrigation or any other development plan, is usually done in two stages. The first stage starts by determining the levels of the terrain, while the second stage solves the earthwork allocation for the predetermined levels. The most recent attempt, reported in the literature, to find global optimality for this problem was a trial-and-error approach which solves a linear transportation problem for every possible terrain's level and selects the level that results in the lowest transportation cost. This approach, however, being an iterative procedure, can only come closer to the optimum solution depending on how small the iteration steps are, but does not guarantee global optimality. The paper presented here offers a linear programming model that combines the levelling of terrains and the associated transportation in a single linear programming problem, thus guaranteeing global optimality.