TRANSSHIPMENT PROBLEMS

The transportation problem is actually a special case of the transshipment problem. If items are being transported from the source through an intermediate point (called a transshipment point) before reaching a final destination, then the problem is called a transshipment problem. For example, a company might be manufacturing a product at several factories to be shipped to a set of regional distribution centers. From these centers the items are shipped to retail outlets that are the final destinations. An example follows.

Transshipment Problem

Example 1

Frosty Machines manufactures snow-blowers in factories located in Toronto and Detroit. These are shipped to regional distribution centers in Chicago and Buffalo, where they are delivered to the supply houses in New York, Phila, and Louis. Figure 1 illustrates the basic network representation of this situation. The shipping costs vary, as shown in the following table. Forecasted demands for New York, Phila, and Louis are also seen in this table, as are the available supplies of snow-blowers at the two factories. Notice that snow-blowers may not be shipped directly from Toronto or Detroit to any of the final destinations. This is why Chicago and Buffalo are listed not only as destinations but also as sources.

ToFrom

Chicago

Buffalo

N.York

Phila Louis Supply

Toronto $4 $7 - - - 800

Detroit $5 $7 - - - 700

Chicago -- -- $6 $4 $5 -

Buffalo - - $2 $3 $4 -

Demand - - 450 350 300 -

Toronto

Detroit Buffalo

Chicago

Louis

Phila

N. York

SourceTransshipment

Point

Destinations

Network representation of frosty machines problem

from these transshipment points must have arrived from either Toronto or Detroit. Therefore, Chicago and Buffalo will each have a constraint indicating this. The verbal state ment of this problem would be as follows:

Minimize cost, subject to:

1. the number of units shipped from Toronto is not more than 8002. the number of units shipped from Detroit is not more than 7003.the number of units shipped to New York is 4504.the number of units shipped to Philadelphia is 3505. the number of units shipped to St. Louis is 3006.the number of units shipped out of Chicago is equal to the number of units shipped into Chicago7.the number of units shipped out of Buffalo is equal to the number of units shipped into Buffalo

The decision variables should represent the number of units shipped from each source to .each transshipment point and the number of units shipped from each transshipment point to each final destination, as these are the decisions management must make. The decision variables are

T1 = the number of units shipped from Toronto to Chicago

T2 = the number of units shipped from Toronto to Buffalo

D1 = the number of units shipped from Detroit to Chicago

D2 = the number of units shipped from Detroit to Buffalo

C1 = the number of units shipped from Chicago to New York

C2 = the number of units shipped from Chicago to Philadelphia

C3 = the number of units shipped from Chicago to St. Louis

B1 = the number of units shipped from Buffalo to New York

B2 = the number of units shipped from Buffalo to Philadelphia

B3 = the number of units shipped from Buffalo to St. Louis

The Linear Program is: Minimize cost = 4T1+7T2+5D1+7D2 +6C1+4C2 + 5C3+2B1+3B2+4B3

Subject To:

T1 +T2<= 800 Supply from TorontoD1+D2<= 700 Supply from DetroitC1 +B1 = 450 Demand at New-yorkC2+B2 = 350 Demand at PhilaC3+B3 = 300 Demand at LouisT1+D1 = C1+C2+C3 Shipping through ChicagoT2+D2 = B1+B2+B3 Shipping through Buffalo

Solving this with QM for Windows yields the output in Program 8.7. From this we see that we should ship 650 units from Toronto to Chicago, 150 units from Toronto to Buffalo, and 300 units from Detroit to Buffalo. A total of 350 units will be shipped from Chicago to Philadelphia, 300 from Chicago to St. Louis, and 450 from Buffalo to New York. The total cost will be $9,550.

Consider a transportation problem where the origins are plants and destinations are depots. The unit transportation costs, capacity at the plants and the requirements at the depots are indicated below :

DepotPlant

X Y Z Cap.

A 1 3 15 150

B 3 5 25 300

Req. 150 150 150 450

Table 1

When each plant is also considered a destination and each depot is also considered an origin, there are altogether five origins and five destinations. Some additional cost data are also necessary. There are presented in the following Tables.

Unit transportation cost from plant to plant

To PlantFrom Plant

A B

A 0 65

B 1 0

Unit transportation cost from Depot to Depot

To DepotFrom Depot

X Y Z

X 0 23 1

Y 1 0 3

Z 65 3 0

Table 2

Table 3

Unit transportation cost from Depot to Plant

PlantDepot

A B

X 3 15

Y 25 3

Z 45 55Table 4

From Table 1, Table 2, Table 3 and Table 4 we obtain the transportation formulation of the transshipment problem

Table 5: Transshipment Table

A B X Y Z Cap.

A 0 65 1 3 15 150+450 = 600

B 1 0 3 5 25 300+450 =750

X 3 15 0 23 1 450

Y 25 3 1 0 3 450

Z 45 55 65 3 0 450

Req. 450 450 150+450=600

150+450=600

150+450=600

2700

A buffer stock of 450 which is the total capacity and total requirement in the original transportation problem is added to each row and column of the transshipment problem. The resulting transportation problem has m + n = 5 origins and m + n = 5 destinations. On solving the transportation problem presented in Table 5

we obtain x11 = 150; x13 = 300 ; x14 = 150; x21 = 300x22 = 450; x33 = 300; x35 = 150; X44 = 450 x55 = 450.

The description of the transshipment problem is given below :

1)Transport x21 = 300 units from plant B to plant A. This increase the availability at plant A to 450 units including the 150 units originally available from A.

2)From plant A transport xI3 = 300 to depot X and X14 = 150 to depot Y.

3) From 300 units available at depot X transport x35 = 150 units to depot Z.

The total transshipment cost is:1 x 300 + 3 x 150 + 1 x 300 + 1 x 150 = 1200

If, however, the consignments are transported from plants A, B to depots X, Y, Z only according to the transportation Table 1, the minimum transportation cost schedule is x13 = 150; x21 = 150; x22 = 150 with a minimum cost of 3450. Thus transshipment reduces the cost of cargo movement in this case.

Example 3

Example

Safety product Ltd. is a manufacturer of safety pins in northern India. Recently it has received an order to supply the specified quantities of safety pin in Chennai for four consecutive months starting at the end of the current month. The following details are available:

Month Order received (for no. of lots)

Maximum production Capacity

Cost of Production per lot.

Jan 2009 20 40 14

Feb. 2009 30 50 16

March 2009 50 30 15

April 2009 40 50 17

Assuming the storage cost to be Rs. 1 per lot per month, advise M/s safety products Ltd. as to the optimal production schedule such that the cost is minimized & there are no lost sales.

Try solving it yourself and then tally your answer with the solution given below.

THANKS!