Transshipment problem

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:
Mathematical formulation of the problem.
The goal is to minimize formula_7 subject to:
Solution.
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 formula_16 time-minimizing problem, in each case using the remained formula_17 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.
Phase 1.
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 formula_18 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 formula_19 which minimize the objective function:
formula_20,
such that
Phase 2.
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:
In mathematical form.
The goal is to find formula_32 which minimize
formula_33,
such that
This problem is easy to be solved with the method developed by Prakash. The set formula_37 needs to be partitioned into subgroups formula_38, where each formula_39 contain the formula_31-s with the same value. The sequence formula_39 is organized as formula_42 contains the largest valued formula_31's formula_44 the second largest and so on. Furthermore, formula_45 positive priority factors are assigned to the subgroups formula_46, with the following rule:
formula_47
for all formula_48. With this notation the goal is to find all formula_49 which minimize the goal function
formula_50
such that
Extension.
Some authors such as Das et al (1999) and Malakooti (2013) have considered multi-objective Transshipment problem.