Page 155 - Compact Numerical Methods For Computers

144 Compact numerical methods for computers
Example 12.1. Function minimisation-optimal operation of a public lottery
Perry and Soland (1975) discuss the problem of deciding values for the main
variables under the control of the organisers of a public lottery. These are p, the
price per ticket; u, the value of the first prize; w, the total value of all other prizes;
and t, the time interval between draws. If N is the number of tickets sold, the
expected cost of a single draw is
K 1 + K N
2
that is, a fixed amount plus some cost per ticket sold. In addition, a fixed cost per
time K is assumed. The number of tickets sold is assumed to obey a Cobb-
3
Douglas-type production function

where F is a scale factor (to avoid confusion the notation has been changed from
that of Perry and Soland). There are a number of assumptions that have not been
stated in this summary, but from the information given it is fairly easy to see that
each draw will generate a revenue

R = Np – (v + w + K + K N + K t).
3
2
2
Thus the revenue per unit time is
R/t = –S = [(p – K )N – (v + w + K )]/ t – K .
2
1
3
Therefore, maximum revenue per unit time is found by minimising S(b) where

Example 12.2. Nonlinear least squares
The data of table 12.1 are thought to approximate the logistic growth function
(Oliver 1964)

g(x) = b /[1 + b exp(xb )] (12.9)
3
2
1
for each point x=i. Thus the residuals for this problem are
f = b /[1 + b exp(ib )] – Y . (12.10)
i 1 2 3 i1
Example 12.3. An illustration of a system of simultaneous nonlinear equations
In the econometric study of the behaviour of a market for a given commodity, the
following relationships are observed:
a
quantity produced = q = Kp
where p is the price of the commodity and K and a are constants, and
-b
quantity consumed or demanded = q = Zp