I ran into trouble trying to formulate this strange problem and would appreciate any help on how to formulate it.
The problem goes like this. Suppose a machine uses two procedures, C and D, for two types of produce, X and Y.
-C requires 6 units of raw material A and 4 units of raw material B, and creates 5 units of produce X and 2 units of material Y.
-D requires 3 units of raw material A and 5 units of raw material B, and creates 2 units of produce X and 4 units of material Y.
-There's 180 units of raw material A available.
-There's 200 units of raw material B available.
-For each round of procedures C, D the machine turns a profit of 2 units for C and 3 units for D.
I am to find how many cycles of C and D are required to maximize profit.
I cannot formulate this because I cannot understand how the procedures imply profit for produce X and Y. If there was a value for profit made by X,Y I guess I could work around it. But now I cannot see the values for X and Y come into play...
Thanks in advance for any help.
As the problem is currently stated, the products, X and Y, of the processes appear (as you have already observed) to be irrelevant, since the profit is described as depending only on the number of times the processes C and D are run. Assuming these numbers have to be integers, then the mathematical formulation of the problem would appear to me to be \begin{eqnarray} &\text{maximize}&\ \ \ \ \ 2c + 3d\\ &\text{subject to}&\cases{6c+3d\le 180\,,&\\ 4c+ 5d \le 200\,,\ \text{and}\\ c\ge 0,\ d\ge0\,,\ \text{integers,}} \end{eqnarray} where $\ c\ $ is the number of times process C is run, and $\ d\ $ is the number of times process D is run. If you ignore the integer restrictions on the variables, and solve this as a standard linear program, its solution is $\ c=0,\ d=40\ $, for a total profit of $120$ units. Since the values of $\ c\ $ and $\ d\ $ in this solution are, in fact, integers, it must also be the solution to the problem with those integer restrictions in force.