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Model Building in Mathematical Programming, 5th Edition by H. Paul Williams

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Chapter 7

Non-linear models

7.1 Typical applications

It has already been pointed out that many mathematical programming models contain variables representing activities that compete for the use of limited resources. For a linear programming model to be applicable the following must apply:

1. there must be constant returns to scale;
2. use of a resource by an activity is proportional to the level of the activity;
3. the total use of a resource by a number of activities is the sum of the uses by the individual activities.

These conditions are clearly applied in the product mix example of Section 1.2 and the result was the linear programming model given there. All the expressions in that model are linear. Nowhere do we get expressions such as , x1x2 and log x1. Suppose, however, that the first of the above conditions did not apply. Instead of each unit of PROD 1 produced contributing £550 to profit, we suppose that the unit profit contribution depends on the quantity of PROD 1 produced. If this unit profit contribution increases with the quantity produced we are said to have increasing returns to scale. For decreases in the unit profit contribution there is said to be decreasing returns to scale. These two situations together with the case of a constant return to scale are illustrated in Figures 7.1-7.3 respectively.

In our product mix ...

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