We consider a class of fixed-charge transportation problems over graphs. We show that this problem is strongly NP-hard, but solvable in pseudo-polynomial time over trees using dynamic programming. We also show that the LP formulation associated to the dynamic program can be obtained from extended formulations of single-node flow polytopes. Given these results, we present a unary expansion-based...

this paper deals with a multi-period fixed charge production-distribution problem associated with backorder and inventories. the objective is to determine the size of the shipments from each supplier and backorder and inventories at each period, so that the total cost incurred during the entire period towards production, transportation, backorder and inventories is minimised. a 0-1 mixed intege...

This paper is to integrate among solid transportation problem, budget constraints and carbon emission with probable maximum profit. The range of air pollution climate change are effected by $${\text {CO}_2}$$ gas other greenhouse gases from myriad systems. Henceforth, our target minimize for pollution-free environment. Transportation system single objective hardly applicable the situation more ...

We study a multiechelon supply chain model that consists of a production level and several transportation levels, where the demands can exist in the production echelon as well as any transportation echelons. With the presence of stationary production capacity and general cost functions, our model integrates production, inventory and transportation decisions and generalizes existing literature o...

In most real world application and problems, a homogeneous product is carried from an origin to a destination by using different transportation modes (e.g., road, air, rail and water). This paper investigates a fixed charge transportation problem (FCTP), in which there are different routes with different capacities between suppliers and customers. To solve such a NP-hard problem, four meta-heur...

This paper deals with a multi-period fixed charge production-distribution problem associated with backorder and inventories. The objective is to determine the size of the shipments from each supplier and backorder and inventories at each period, so that the total cost incurred during the entire period towards production, transportation, backorder and inventories is minimised. A 0-1 mixed intege...

We introduce the Stochastic multistage fixed charge transportation problem in which a producer has to ship an uncertain load to a customer within a deadline. At each time period, a fixed transportation price can be paid to buy a transportation capacity. If the transportation capacity is used, the supplier also pays an uncertain unit transportation price. A unit inventory cost is charged for the...