Question: The production of three goods requires using two machines. Machine 1 can be utilized for b1 hours, while machine 2 can be utilized for b2 hours. The time spent for the production of one unit of each good is given by the following table:
The profits per unit produced of the three goods are 6, 3, and 4, respectively.
(a) Write down the linear programming problem this leads to.
(b) Show that the dual is
Solve this problem geometrically for b1 = b2 = 100.
(c) Solve the problem in (a) when b1 = b2 = 100.
(d) If machine 1 increases its capacity to 101, while b2 = 100, what is the new maximal profit?
(e) The maximum value of the profit in problem (a) is a function F of b1 and b2. What is the degree of homogeneity of the function F?