Transshipment problems form a subgroup of transportation problems, where
transshipment
Transshipment, trans-shipment or transhipment is the shipment of goods or containers to an intermediate destination, then to another destination.
One possible reason for transshipment is to change the means of transport during the journey (e.g ...
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
Freight transport, also referred as ''Freight Forwarding'', is the physical process of transporting commodities and merchandise goods and cargo. The term shipping originally referred to transport by sea but in American English, it has been ext ...
of
goods
In economics, goods are items that satisfy human wants
and provide utility, for example, to a consumer making a purchase of a satisfying product. A common distinction is made between goods which are transferable, and services, which are not t ...
or
containers
A container is any receptacle or enclosure for holding a product used in storage, packaging, and transportation, including shipping.
Things kept inside of a container are protected on several sides by being inside of its structure. The term ...
to an intermediate destination, and then from there to yet another destination. One possible reason is to change the
means of transport Means of transport are transport facilities used to carry people or cargo.
__NOTOC__
Examples for means of transport
for land transport
* Automobiles
* Bicycles
* Carriages
*Pack animals
*Riding animals
* Rickshaws
* Trains
* Trucks
*Vehicles
*Wa ...
during the journey (for example from
ship transport
Maritime transport (or ocean transport) and hydraulic effluvial transport, or more generally waterborne transport, is the transport of people ( passengers) or goods (cargo) via waterways. Freight transport by sea has been widely used throu ...
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 hub
A transport hub is a place where passengers and cargo are exchanged between vehicles and/or between transport modes. Public transport hubs include railway stations, rapid transit stations, bus stops, tram stops, airports and ferry slips. F ...
s. Much international transshipment also takes place in designated
customs area
Border control refers to measures taken by governments to monitor and regulate the movement of people, animals, and goods across land, air, and maritime borders. While border control is typically associated with international borders, it a ...
s, 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:
,
* 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
*
: time of transportation from node ''r'' to node ''s''
*
: goods available at node ''i''
*
: demand for the good at node ''(m+j)''
*
: actual amount transported from node ''r'' to node ''s''
Mathematical formulation of the problem
The goal is to minimize
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
time-minimizing problem, in each case using the remained
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
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
which minimize the objective function:
,
such that
*
*
*
*
.
Corollary
*
and
need to be excluded from the model; on the other hand, without the
constraint the optimal path would consist only of
-type loops which obviously can not be a feasible solution.
* Instead of
,
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
in order to separate it from the times introduced during the first stage.
In mathematical form
The goal is to find
which minimize
,
such that
*
*
*
This problem is easy to be solved with the method developed by
Prakash. The set
needs to be partitioned into subgroups
, where each
contain the
-s with the same value. The sequence
is organized as
contains the largest valued
's
the second largest and so on. Furthermore,
positive priority factors are assigned to the subgroups
, with the following rule:
for all
. With this notation the goal is to find all
which minimize the goal function
such that
*
*
*
*
Extension
Some authors such as Das et al (1999) and Malakooti (2013) have considered multi-objective Transshipment problem.
References
*R.J Aguilar, Systems Analysis and Design. Prentice Hall, Inc. Englewood Cliffs, New Jersey (1973) pp. 209–220
*H. L. Bhatia, K. Swarup, M. C. Puri, Indian J. pure appl. Math. 8 (1977) 920-929
*R. S. Gartinkel, M. R. Rao, Nav. Res. Log. Quart. 18 (1971) 465-472
*G. Hadley, Linear Programming, Addison-Wesley Publishing Company, (1962) pp. 368–373
*P. L. Hammer, Nav. Res. Log. Quart. 16 (1969) 345-357
*P. L. Hammer, Nav. Res. Log. Quart. 18 (1971) 487-490
*A.J.Hughes, D.E.Grawog, Linear Programming: An Emphasis On Decision Making, Addison-Wesley Publishing Company, pp. 300–312
*H.W.Kuhn, Nav. Res. Log. Quart. 2 (1955) 83-97
*A.Orden, Management Sci, 2 (1956) 276-285
*S.Parkash, Proc. Indian Acad. Sci. (Math. Sci.) 91 (1982) 53-57
*C.S. Ramakrishnan, OPSEARCH 14 (1977) 207-209
*C.R.Seshan, V.G.Tikekar, Proc. Indian Acad. Sci. (Math. Sci.) 89 (1980) 101-102
*J.K.Sharma, K.Swarup, Proc. Indian Acad. Sci. (Math. Sci.) 86 (1977) 513-518
* W.Szwarc, Nav. Res. Log. Quart. 18 (1971) 473-485
* Malakooti, B. (2013). Operations and Production Systems with Multiple Objectives. John Wiley & Sons.
*Das, S. K., A. Goswami, and S. S. Alam. “Multiobjective Transportation Problem with Interval Cost, Source and Destination Parameters.” European Journal of Operational Research, Vol. 117, No. 1, 1999, pp. 100–112
Transport economics
Transportation planning
Freight transport
Mathematical optimization in business