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 minimal—cost transportational route had been the main priority, however technological developement slowly gave place to minimal—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.

Formulation of the problem

A few initial assumptions are required in order to furmulate the transshipment problem completly:

• The system consists m origins ans n destinations, with the following indexing respectively: $i=1,\ldots ,m$, $j=n+1,\ldots ,m+n$

• One uniform good exists which is need to be shipped

• The required amount of good at the destinations equals the produced quantity available at the origins

• Transportation simultaneously starts at the originsis and is possible from any node to any other (also to an origin and from a destination)

• Transportational costs are independent of the shipped amount

Notations

• $t_{r,s}$: 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 find $x_{r,s}\leq 0$ subject to:

• $\sum _{s=1}^{m+n}{x_{i,s}}-\sum _{r=1}^{m+n}{x_{r,i}}=a_{i}$ $i=1\ldots m$

• $\sum _{r=1}^{m+n}{x_{r,m+j}}-\sum _{s=1}^{m+n}{x_{m+j,s}}=b_{m+j}$ $j=1\ldots n$

• $\sum _{i=1}^{m}{a_{i}}=\sum _{j=1}^{n}{b_{m+j}}$

Solving the Problem

Since in most cases an explicite expression for the objective function does not exists, an alternative method is suggested by Rajeev and Satya. The method uses two consequtive 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 we can normalize the shipped quantity to 1. The problem now is simplifyed to an assigment 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

• $\sum _{s=1}^{m+n}{x'_{r,s}}=1$

• $\sum _{r=1}^{m+n}{x'_{r,s}}=1$

• $x'_{m+j,i}=1$

• $x'_{r,s}=0,1$.

Corollary

• $x'_{r,r}=1$ and $x'_{m+j,i}=1$ needs to be excluded from the model, on the other hand, without the $x'_{m+j,i}=1$ constraint the optimal path would consist only $x'_{r,r}$-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 assigment 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 Prakash. The set $\left\{t'_{i,m+j},i=1\ldots m,\;j=1\ldots n\right\}$ needs to be partitioned into subgroups $L_{k},k=1\ldots q$, where each $L_{k}$ contain the $t'_{i,m+j}$-s with the same value. The sequence $L_{k}$ is organized as $L_{1}$ contains the largest valued $t'_{i,m+j}$'s $L_{2}$ the second largest and so on. Furthermore, $M_{k}$ positive priority factors are assigned to the subgroups $\sum _{L_{k}}{x_{i,m+j}}$, with the following rule:

$\alpha M_{k}-\beta M_{k+1}=\left\{{\begin{array}{cc}-ve,&if\;\alpha <0\\ve,&if\;\alpha >0\end{array}}\right.$

for all $\beta$. With this notation the goal is to find all $x_{i,m+j}$ which minimize the goal function

$z_{1}=\sum _{k=1}^{q}{M_{k}}\sum _{L_{k}}{x_{i,m+j}}$

such that

• $\sum _{i=1}^{m}{x_{i,m+j}}=a_{i}$

• $\sum _{j=1}^{n}{x_{i,m+j}}=b_{m+j}$

• $\sum _{i=1}^{m}{a_{i}}=\sum _{j=1}^{n}{b_{m+j}}$

• $\alpha M_{k}-\beta M_{k+1}=\left\{{\begin{array}{cc}-ve,&if\;\alpha <0\\ve,&if\;\alpha >0\end{array}}\right.$

Sources

• Rajeev Garg and Satya Prakash: Time Minimizing Transshipment Problem (1983)

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