Transshipment problems form a subgroup of transportation problems, where transshipment is allowed. In transshipment, transportation may or must go through intermediate nodes, possibly changing modes of transport. The Transshipment problem has its origins in medieval times when trading started to become a mass phenomenon. Obtaining the minimum-cost route had been the main priority. However, technological development slowly gave priority to minimum-duration transportation problems.
Overview Transshipment or Transhipment is the shipment of goods or containers to an intermediate destination, and then from there to yet another destination. One possible reason is to change the means of transport during the journey (for example from ship transport to road transport), known as transloading. Another reason is to combine small shipments into a large shipment (consolidation), dividing the large shipment at the other end (deconsolidation). Transshipment usually takes place in transport hubs. Much international transshipment also takes place in designated customs areas, thus avoiding the need for customs checks or duties, otherwise a major hindrance for efficient transport.
Formulation of the problem A few initial assumptions are required in order to formulate the transshipment problem completely:
The system consists of m origins and n destinations, with the following indexing respectively: i = 1 , … , m {\displaystyle i=1,\ldots ,m} , j = 1 , … , n {\displaystyle j=1,\ldots ,n}
One uniform good exists which needs to be shipped The required amount of good at the destinations equals the produced quantity available at the origins Transportation simultaneously starts at the origins and is possible from any node to any other (also to an origin and from a destination) Transportation costs are independent of the shipped amount The transshipment problem is a unique Linear Programming Problem (LLP) in that it considers the assumption that all sources and sinks can both receive and distribute shipments at the same time (function in both directions)
Notations
t r , s {\displaystyle t_{r,s}} : time of transportation from node r to node s
a i {\displaystyle a_{i}} : goods available at node i
b m + j {\displaystyle b_{m+j}} : demand for the good at node (m+j)
x r , s {\displaystyle x_{r,s}} : actual amount transported from node r to node s
Mathematical formulation of the problem The goal is to minimize ∑ i = 1 m ∑ j = 1 n t i , j x i , j {\displaystyle \sum \limits _{i=1}^{m}\sum \limits _{j=1}^{n}t_{i,j}x_{i,j}} subject to:
x r , s ≥ 0 {\displaystyle x_{r,s}\geq 0} ; ∀ r = 1 … m {\displaystyle \forall r=1\ldots m} , s = 1 … n {\displaystyle s=1\ldots n}
∑ s = 1 m + n x i , s − ∑ r = 1 m + n x r , i = a i {\displaystyle \sum _{s=1}^{m+n}{x_{i,s}}-\sum _{r=1}^{m+n}{x_{r,i}}=a_{i}} ; ∀ i = 1 … m {\displaystyle \forall i=1\ldots m}
∑ r = 1 m + n x r , m + j − ∑ s = 1 m + n x m + j , s = b m + j {\displaystyle \sum _{r=1}^{m+n}{x_{r,m+j}}-\sum _{s=1}^{m+n}{x_{m+j,s}}=b_{m+j}} ; ∀ j = 1 … n {\displaystyle \forall j=1\ldots n}
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