This note describes some recent results on the regularity of optimal transport maps. As we shall see, in general optimal maps are not globally smooth, but they are so outside a “singular set” of measure zero. 1. The optimal transportation problem The optimal transportation problem, whose origin dates back to Monge [19], aims to find a way to transport a distribution of mass from one place to an...

Consider the problem of transporting a set of objects between the vertices of a tree by a vehicle that travels along the edges of the tree. The vehicle can carry only one object at a time, and it starts and finishes at the same vertex of the tree. It is shown that if each object must be carried directly from its initial vertex to its destina.tion then finding a minimum cost transportation is NP...

In this article we try to quantify the potential environmental benefits of horizontal collaboration in transportation and to study the sensibility of CO2 emissions to transportation cost reduction. We adopted an approach based on bi-objective mathematical modeling to minimize both total transportation cost and total environmental effect by simultaneously combining facility location and routing ...

Previous research on the joint economic lot size (JELS) problem was focused on the decisions of lot sizes and batches delivered among players in the supply chain, assuming that transportation cost was independent of shipment amounts. Nevertheless, the transportation cost accounts for a major part of supply chain cost, especially when transportation companies offer freight rate discount schedule...

In this paper, a socially responsible supply chain consisting of one supplier and one retailer is proposed. The supplier decides on replenishment cycle multiplier and the retailer invests in corporate social responsibility (CSR) and decides on the order-up-to-level under a periodic review replenishment policy. The retailer’s decisions impact on the supplier’s probability as well as the supply c...

Transportation networks play a vital role in modern societies. Structural optimization of a transportation system under a given set of constraints is an issue of great practical importance. For a general transportation system whose total cost C is determined by C = Sigma(i<j)C(ij)(I(ij)), with C(ij) (I(ij)) being the cost of the flow I(ij) between node i and node j, Banavar and co-workers [Phys...

The presence or absence of loops in the emergent transportation networks, that are characterized by a minimum overall cost, is shown to depend on the convexity of the cost function for the local transportation of material. Our results are directly applicable to a variety of situations across disciplines.

The fixed charge (fixed cost) may present the cost of renting a vehicle, landing fees in an airport, setup cost for machines in a manufacturing environment, etc. In this paper, we discuss fixed charge capacitated in a non-linear transportation problem. Thereby, we establish local optimum condition of this problem. Next we establish an algorithm for solving this transportation problem. Also, we ...

Purpose The impact of transportation on the supplier selection has received a very scant attention in the literature. This is a great limitation because splitting orders across multiple suppliers will lead to smaller transportation quantities which will likely imply larger transportation cost. Moreover, transportation and inventory elements are highly interrelated and contribute most to the tot...