Multi-modal supply chain distribution problem

Supply chain networks are representation of interaction among different entities. Usually these entities are facilities which can be represented as nodes in a network and the flow of material between them can be represented as flow on arcs (paths) connecting them. These flows can be facilitated via multiple modes available to transport material from one facility to another. We discuss a multi-modal supply chain distribution problem where the aim is to minimize sum of transportation cost on various modes between facilities, inventory, backlog and lost sales costs over a time-horizon. The problem can be represented as a time-space network of nodes and arcs. Each node defines the state of a facility at a given time-period and the arcs between these nodes are either transportation, inventory or backlog carrying arcs. The time-horizon consists of discrete time-periods and the flows on transportation arcs are required to be an integer multiple of predefined lot sizes as in vehicle capacities, batch sizes, etc. Apart from this, there are certain business rules which are posed on transportation modes incoming to a facility or posed on the suppliers of a facility are to be followed. The problem stated above is first modeled as a Mixed Integer Linear Program (MILP) and solved using a MILP solver. We propose integer rounding heuristics to get a feasible solution to the problem. We report in our results that these heuristics can be used to generate an integer feasible solution quickly. Using this feasible solution as an MIP start in solver helps us in reaching optimal solution in lesser time.