9.3 The Transshipment Problem
In a transportation problem, if the items being transported must go through an intermediate point (called a transshipment point) before reaching a final destination, 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. Figure 9.3 provides a network representation of a transshipment problem. In this example, there are two sources, two transshipment points, and three final destinations.
Figure 9.3 Network Representation of a Transshipment Example
Linear Program for Transshipment Example
Frosty Machines manufactures snow blowers in factories located in Toronto and Detroit. These are shipped to the regional distribution centers in Chicago and Buffalo, and then delivered to the supply houses in New York, Philadelphia, and St. Louis, as illustrated in Figure 9.3.
The available supplies at the factories, the demands at the final destinations, and shipping costs are shown in the Table 9.3. Notice that snow blowers may not be shipped directly from Toronto or Detroit to any of the final destinations but must first go to either Chicago or Buffalo. This is why Chicago and Buffalo are listed not only as destinations but also as sources.
Table 9.3 Frosty Machines Transshipment Data
| TO | ||||||
|---|---|---|---|---|---|---|
| FROM | CHICAGO | BUFFALO | NEW YORK CITY | PHILADELPHIA | ST. LOUIS | SUPPLY |
| Toronto | $4 | $7 | — | — | — | 800 |
| Detroit | $5 | $7 | — | — | — | 700 |
| Chicago | — | — | $6 | $4 | $5 | — |
| Buffalo | — | — | $2 | $3 | $4 | — |
| Demand | — | — | 450 | 350 | 300 | |
Frosty would like to minimize the transportation costs associated with shipping sufficient snow blowers to meet the demands at the three destinations while not exceeding the supply at each factory. Thus, we have supply and demand constraints similar to the transportation problem, but we also have one constraint for each transshipment point, indicating that anything shipped from these to a final destination must have been shipped into that transshipment point from one of the sources. The verbal statement of this problem would be as follows:
Minimize cost subject to
The number of units shipped from Toronto is not more than 800
The number of units shipped from Detroit is not more than 700
The number of units shipped to New York is 450
The number of units shipped to Philadelphia is 350
The number of units shipped to St. Louis is 300
The number of units shipped out of Chicago is equal to the number of units shipped into Chicago
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
where
The numbers are the nodes shown in Figure 9.3, and there is one variable for each arc (route) in the figure.
The LP model is
subject to
The solution found using Solver in Excel 2016 and Excel QM is shown in Program 9.5. The total cost is $9,550 by shipping 650 units from Toronto to Chicago, 150 units from Toronto to Buffalo, 300 units from Detroit to Buffalo, 350 units from Chicago to Philadelphia, 300 units from Chicago to St. Louis, and 450 units from Buffalo to New York City.
