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Question
(a) A firm produces four products: P, Q, R, and S. Each unit of product P requires two hours of milling, one hour of assembly, and $10 worth of in-process inventory. Each unit of product Q requires one hour of milling, three hours of assembly, and $5 worth of in-process inventory. Each unit of R requires 2.5 hours of milling, 2.5 hours of assembly, and $2 worth of in-process inventory. Finally, each unit of product S requires five hours of milling, no assembly, and $12 of in-process inventory.
The firm has 120 thousand hours of milling time and 160 thousand hours of assembly time available. In addition, not more than $1 million may be tied up in in-process inventory.
Each unit of product P returns a profit of $40; each unit of Q returns a profit of $24; each unit of product R returns a profit of $36; and each unit of product S returns a profit of $23. Not more than 20 000 units of product P can be sold; not more than 16 000 units of product R can be sold; and any number of units of products P and S may be sold. However, at least 10 000 units of product S must be produced and sold to satisfy a contract requirement.
The objective of the firm is to maximize the profit resulting from the sale of the four products. Formulate the above as a linear programming problem.
(b) Use the simplex method to maximize
z = x1 + x2
subject to
x1 + 5x2 ≤ 5
2x1 + x2 ≤ 4
with
x1, x2 ≥ 0
Using only the digits 1-6 how many five digit numbers can be formed? How many of these have at least a 5? How many of them have either no 5 or no 6?
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