Optimization of a multi-constraint transport problem using a heuristic approach

A Linear transport problem can be defined as the action of transporting products from "m origins" (or units) to "n destinations" (or customers) at the lowest cost. So the solution to a transportation problem is to organize the transportation in such a way as to minimize its cost. The objective of this paper is to determine the quantity sent from each source (origin) to each destination while minimizing transport costs. Achieving this objective requires a methodology which consists in deploying an algorithm whose purpose is the search for an optimal solution, based on an initial solution. The application is made on a factory producing mechanical parts.

Tractable Relaxations of Composite Functions

In this paper, we introduce new relaxations for the hypograph of composite functions assuming that the outer function is supermodular and concave extendable. Relying on a recently introduced relaxation framework, we devise a separation algorithm for the graph of the outer function over P, where P is a special polytope to capture the structure of each inner function using its finitely many bounded estimators. The separation algorithm takes [Formula: see text] time, where d is the number of inner functions and n is the number of estimators for each inner function. Consequently, we derive large classes of inequalities that tighten prevalent factorable programming relaxations. We also generalize a decomposition result and devise techniques to simultaneously separate hypographs of various supermodular, concave-extendable functions using facet-defining inequalities. Assuming that the outer function is convex in each argument, we characterize the limiting relaxation obtained with infinitely many estimators as the solution of an optimal transport problem. When the outer function is also supermodular, we obtain an explicit integral formula for this relaxation.