Joining points of equal output on the four production process rays creates a set of isoquant curves. Figure 9.2 illustrates isoquants for Q = 1, 2, 3, 4, and 5. These isoquants have the same interpretation as those developed in Chapter 8. Each isoquant represents combinations of input factors L and K that can be used to produce a given quantity of output. Production isoquants in linear programming are composed of linear segments connecting the various production process rays. Each of these isoquant segments is parallel to one another. For example, line segment A1B1 is parallel to segment A2B2; isoquant segment B3C3 is parallel to B2C2.
Points along each segment of an isoquant between two process rays represent a combination of output from each of the two adjoining production processes. Consider point X in Figure 9.2, which represents production of 4 units of Q using 25 units of L and 16 units of K. None of the available production processes can manufacture Q using L and K in the ratio of 25 to 16, but that combination is possible by producing part of the output with process C and part with process D. In this case, 2 units of Q can be produced using process C and 2 units using process D. Production of 2 units of Q with process C uses 15 units of L and 6 units of K. For the production of 2 units of Q with process D, 10 units each of L and K are necessary. Although no single production system is available that can produce 4 units of Q using 25 units of L and 16 units of K, processes C and D together can produce that combination.
All points lying along production isoquant segments can be interpreted in a similar manner. Each point represents a linear combination of output using the production process systems that bound the particular segment. Point Y in Figure 9.2 provides another illustration. At Y, 3 units of Q are produced, using a total of 38.5 units of L and 4.3 units of K.2 This input/output combination is possible through a combination of processes A and B. This can be analyzed algebraically. To produce 1 unit of Q by process A requires 15 units of L and 1 unit of K. Therefore, to produce 1.7 units of Q requires 25.5 (1.7 X 15) units of L and 1.7 (1.7 X 1) units of K. To produce a single unit of Q by process B requires 10 units of L and 2 units of K, so 1.3 units of Q requires 13 (10 X 1.3) units of L and 2.6 (2 X 1.3) units of K. Thus, point Y calls for the production of 3 units of Q in total, 1.7 units by process A and 1.3 units by process B, using a total of 38.5 units of L and 4.3 units of K.
2 Another assumption of Linear programming is that fractional variables are permissible. In many applications, this assumption is not important. For example, in the present illustration, we might be talking about labor hours and machine hours for the inputs. The solution value calling for L = 38.5 merely means that 38.5 hours of labor are required.
In some cases, however, inputs are large (whole plants, for example), and the fact that linear programming assumes divisible variables is important. In such cases, linear programming as described here may be inappropriate, and a more complex technique, integer programming, may be required.
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