# 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:

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 , *x*_{1}*x*_{2} and log *x*_{1}. 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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