## Transformation of Flow

 A company makes three products in four plants. Because of differences in labor skills, different times are required to produce the products in the various plants. The time requirements (minutes per unit) are shown in the table below. A dash indicates that the product cannot be made at the plant. The demand for the products is also shown. The total time available in each plant is 25 hours per week. The hourly charge for labor is \$10, 12, 9, and 13 for plants 1, 2, 3, and 4, respectively. Not all the time available at a plant need be used. There is no labor charge for time not used. Our problem is to construct and solve the network model that minimizes the cost of manufacturing to meet the demand.

The example illustrates the use of gains to transform one type of flow to another. The situation is like a transportation problem except the plants have capacities measured in hours, and demand is in units. We use the gain factor transform the flow as indicated in Fig. 21.

Figure 21. Using the gain factor to transform flow

The data for this situation is given in minutes per unit, so to compute the appropriate gain factors, invert the quantities and multiply by 60 to find units per hour. The gain for plant 1 producing product A is 2 units per hour. The complete network model and the solution are shown in Fig. 22 and 23.

Figure 22. Model

Figure 23. Solution