1. You have been asked to perform an analysis to assist in deciding how much of two products to
produce to maximize the profit. The total profit associated with producing a units of product a
and b units product b is given as a single function that is given below:
ππ (ππ , ππ ) = 0.2ππ 2 β 1.1ππ + 12βππ β 2.5ππ
where
P is the total profit ($)
a is the amount of product a to produce
b is the amount of product b to produce (tons) The coefficients in front of each term (i.e., 0.2, 1.1, 12 and 2.5) convert the terms into USΒ dollars. Find the values of a and b that maximize C.
Note: You might not have done this in calculus, but it is not hard because it extends what youΒ learned for one variable. i) Recall, to find the maximum or minimum of a function, you take the derivative, set it equal to
zero, and solve for the independent variable. So if this problem had been P(a) and not P(a,b), you would have taken the derivative of P with respect to a (ππππ
ππππ), set this to zero, and solved for a. ii) To extend this to two variables, here a and b, take the derivate of P with respect to a
assuming b is constant and set it equal to zero. This is just like i) because you are assuming b is
a constant.iii) Then, take the derivate of C with respect to b assuming a is constant and set it equal to zero.
This is just like i) because you are assuming a is a constant.iv) Now, you have two equations β one from ii) and one from iii). Use algebra to solve them and
you have the optimal values.[For completeness, with one variable it was necessary to take the second derivate to see if the point you found was a maximum, minimum, or inflection point. This part of the two variable problem is complicated so you will need to believe me that the point you found yields maximum profit.]2. A ΒΌ inch titanium bolt is being consumed in a factory at a fairly steady rate of 60 per week.
The bolts cost $0.50 each and it costs the plant $12 to place an order with the supplier. Plant management has no idea of the real cost to hold inventory, so they are estimating the holding cost as the product of the annual interest rate multiplied by the cost of the item. Currently, the CFO requires all calculations to use an interest rate of 25%/year.a) What is the order quantity that minimizes the sum of the cost of ordering plus the cost of
holding inventory. Assume holding costs are based on average inventory over a cycle and the cost per unit per unit time is the interest rate * cost of the part. Also, assume instantaneous replenishment.b) If you order the quantity you computed in part a), what is the time between placing orders.
c) If you order the quantity computed in part a), what is the total yearly holding cost? What is the total annual ordering cost?
