On the Quadratic Transportation Problem

Abstract

We present a direct analytical algorithm for solving transportation problems with quadratic function cost coefficients. The algorithm uses the concept of absolute points developed by the authors in earlier works. The versatility of the proposed algorithm is evidenced by the fact that quadratic functions are often used as approximations for other functions, as in, for example, regression analysis. As compared with the earlier international methods for quadratic transportation problem (QTP) which are based on the Lagrangian relaxation approach, the proposed algorithm helps to understand the structure of the QTP better and can guide in managerial decisions. We present a numerical example to illustrate the application of the proposed method.

A simple parallelizable method for the approximate solution of a quadratic transportation problem of large dimension with additional constraints

“…Quadratic transportation problem has been studied for quite a long time [7]. In spite of that, this task remains relevant [8] and has various practical applications [9]. Efforts were made to generalize QTP for the case of convex objective function with additional constraints [10].…”

“…At every step of a particular iteration balancing of the forecast matrix either by lines or by columns is performed. It's quite clear that restrictions (7)(8) in this case are fulfilled automatically. As we start our iterative procedure from matrix A, there is hope that solutions, obtained by this heuristic, will provide the objective function ( 4) with volumes, relatively close to the minimal ones.…”

“…Problem (4-8) is quadratic transportation problem (QTP) with additional constraints (8). Without going into details of the verification procedure [6], we will further assume that the full system of restrictions is consistent.…”

Half-quadratic transportation problems

“…According to Ref. [2], the quadratic model is popular because it can approximate other cost functions. Despite such flexibility, the limitation of quadratic models has been well documented in the context of robust statistics, and its applications to image processing and computer vision [3,4].…”

A Hybrid Dynamic Programming for Solving Fixed Cost Transportation with Discounted Mechanism

“…A transportation system with a single commodity for satisfying the total demand is presented. Adlakha and Kowalski [10] proposed a solution algorithm for certain transportation problems. The concept of absolute points developed in [11] is employed and a direct analytical algorithm for transportation problems with quadratic function cost coefficients is explored.…”