## Production Planning For A Single Product

Although linear programming has been widely applied in managerial decision making, it has been used most frequently in production decisions. To illustrate the method, a simple two-input/one-output problem is examined. Later sections consider more realistic and complex problems.

### Production Processes

Assume that a firm produces a single product, Q, using two inputs, L and K, which might represent labor and capital. Instead of assuming continuous substitution between L and K, as in Chapter 7, assume that Q can be produced using only four input combinations. In other words, four different production processes are available for making Q, each of which uses a different fixed combination of inputs L and K. The production processes might represent four different plants, each with its own fixed asset configuration and labor requirements. Alternatively, they could be four different assembly lines, each using a different combination of capital equipment and labor.

The four production processes are illustrated as rays in Figure 9.1. Process A requires the combination of 15 units of L and 1 unit of K for each unit of Q produced. Process B uses 10 units of L and 2 units of K for each unit of output. Processes C and D use 7.5 units of L and 3 units of K, and 5 units of L with 5 units of K, respectively, for each unit of Q produced. Each point along the production ray for process A combines L and K in the ratio 15 to 1; process rays B, C, and D are developed in the same way. Each point along a single production ray combines the two inputs in a fixed ratio, with the ratios differing from one production process to another. If L and K represent labor and capital inputs, the four production processes might be different plants employing different production techniques. Process A is very labor intensive in comparison with the other production systems, whereas B, C, and D are based on increasingly capital-intensive technologies.

Point A1 indicates the combination of L and K required to produce one unit of output using the A process. Doubling both L and K doubles the quantity of Q produced; this is indicated by the distance moved along ray A from A1 to A2. Line segment 0A2 is exactly twice the length of 