World Library  
Flag as Inappropriate
Email this Article

Transshipment problem

Article Id: WHEBN0035852253
Reproduction Date:

Title: Transshipment problem  
Author: World Heritage Encyclopedia
Language: English
Subject: Transshipment, Transport economics, Transportation planning, Network science
Publisher: World Heritage Encyclopedia

Transshipment problem

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.

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,\ldots, m, 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


  • t_{r,s}: time of transportation from node r to node s
  • a_i: goods available at node i
  • b_{m+j}: demand for the product at node (m+j)
  • x_{r,s}: actual amount transported from node r to node s

Mathematical formulation of the problem

The goal is to minimize \sum\limits_{i=1}^m \sum\limits_{j=1}^n t_{i,j} x_{i,j} subject to:

  • x_{r,s}\geq 0 ; \forall r=1\ldots m , s=1\ldots n
  • \sum_{s=1}^{m+n}{x_{i,s}}-\sum_{r=1}^{m+n}{x_{r,i}}=a_i; \forall i=1\ldots m
  • \sum_{r=1}^{m+n}{x_{r,m+j}}-\sum_{s=1}^{m+n}{x_{m+j,s}}=b_{m+j}; \forall j=1\ldots n
  • \sum_{i=1}^{m}{a_i}=\sum_{j=1}^{n}{b_{m+j}}


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.

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 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

  • \sum_{s=1}^{m+n}{x'_{r,s}}=1
  • \sum_{r=1}^{m+n}{x'_{r,s}}=1
  • x'_{m+j,i}=1
  • x'_{r,s}=0,1.


  • x'_{r,r}=1 and x'_{m+j,i}=1 need to be excluded from the model; on the other hand, without the x'_{m+j,i}=1 constraint the optimal path would consist only of x'_{r,r}-type loops which obviously can not be a feasible solution.
  • Instead of x'_{m+j,i}=1, t_{m+j,i}=-M can be written, where M is an arbitrarily large positive number. With that modification the formulation above is reduced to the form of a standard assignment problem, possible to solve with the Hungarian method.

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:

  • Transportation is only possible from an origin to a destination
  • Transportation time from i to m+j is the sum of durations coming from the optimal route calculated in Phase 1. Worthy to be denoted by t'_{i,m+j} in order to separate it from the times introduced during the first stage.

In mathematical form

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

  • \sum_{i=1}^{m}{x_{i,m+j}}=a_i
  • \sum_{j=1}^{n}{x_{i,m+j}}=b_{m+j}
  • \sum_{i=1}^{m}{a_i}=\sum_{j=1}^{n}{b_{m+j}}

This problem is easy to be solved with the method developed by

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from World Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.