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The Transshipment problem has a long and rich history. It has its origins in medieval times when trading started to become a mass phenomenon. Firstly, obtaining the minimum-cost transportational route had been the main priority, however technological development slowly gave place to minimum-durational transportation problems. Transshipment problems form a subgroup of transportational problems, where transshipment is allowed, namely transportation can, or in certain cases has to shipped through intermediate nodes.
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.
A few initial assumptions are required in order to formulate the transshipment problem completely:
The goal is to minimize \sum\limits_{i=1}^m \sum\limits_{j=1}^n t_{i,j} x_{i,j} subject to:
Since in most cases an explicit expression for the objective function does not exist, an alternative method is suggested by Rajeev and Satya. The method uses two consecutive phases to reveal the minimal durational route from the origins to the destinations. The first phase is willing to solve n\cdot m time-minimizing problem, in each case using the remained n+m-2 intermediate nodes as transshipment points. This also leads to the minimal-durational transportation between all sources and destinations. During the second phase a standard time-minimizing problem needs to be solved. The solution of the time-minimizing transshipment problem is the joint solution outcome of these two phases.
Since costs are independent from the shipped amount, in each individual problem one can normalize the shipped quantity to 1. The problem now is simplified to an assignment problem from i to m+j. Let x'_{r,s}=1 be 1 if the edge between nodes r and s is used during the optimization, and 0 otherwise. Now the goal is to determine all x'_{r,s} which minimize the objective function:
T_{i,m+j}=\sum_{r=1}^{m+n}\sum_{s=1}^{m+n}{t_{r,s}\cdot x'_{r,s}},
such that
During the second phase, a time minimization problem is solved with m origins and n destinations without transshipment. This phase differs in two main aspects from the original setup:
The goal is to find x_{i,m+j}\geq 0 which minimize
z=max\left\{t'_{i,m+j}: x_{i,m+j}>0\;\; (i=1\ldots m,\; j=1\ldots n)\right\}, such that
This problem is easy to be solved with the method developed by
Twistlock, Iso 6346, Container ship, Weathering steel, Bureau International des Containers
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