Linear Programming and Network Flows

[1.11] A lathe is used to reduce the diameter of a steel shaft whose length is 36 in. from 14 in. to 12 in. The speed X\ (in revolutions per minute), the depth feed x2 (in inches per minute), and the length feed x3 (in inches per minute) must be determined. The duration of the cut is given by 36/x2x3. The compression and side stresses exerted on the cutting tool are given by 30 xj + 4500 x2 and 40 X] + 5000 x2 + 5000 x3 pounds per square inch, respectively. The temperature (in degrees Fahrenheit) at the tip of the cutting tool is 200 + 0.5 jq + 150(x2 + x3). The maximum compression stress, side stress, and temperature allowed are 150,000 psi, 100,000 psi, and 800°F, respectively. It is desired to determine the speed (which must be in the range from 600 rpm to 800 rpm), the depth feed, and the length feed such that the duration of the cut is minimized. In order to use a linear model, the following approximation is made. Since 36/x2x3 is minimized if and only if x2x3 is maximized, it was decided to replace the objective by the maximization of the minimum of x2 and x3. Formulate the problem as a linear model and comment on the validity of the approximation used in the objective function.