PROBLEM 2 (partially solved) Consider the following classical inventory problem.

Question

PROBLEM 2 (partially solved)

Consider the following classical inventory problem. A retailer must decide the quantity Q to order periodically to minimize expected annual cost. The retailer faces demand of D units per year and every order that is placed incurs an order processing fee of K dollars and a unit purchase cost of c dollars. It is easily seen that the average inventory is Q/2 on which an inventory carrying cost of h dollars/unit/year is charged. The total annual cost function, T(Q) may be expressed as:

T(Q) = K·D/Q +h·Q/2 + c·D

a) Write an expression for the number of orders placed per year?
b Write an expression for the time between orders?
c) Demonstrate that T(Q) is a convex function.

Take the second partial derivative (using the chain rule) with respect to Q. Then set that equal to zero.
T'(Q)=(-KD)/(Q^2)+(h/2)+0
T''(Q)=(2KD)/(Q^3)
0=(2KD)/(Q^3)

Now multiply each side by 1/2KD to get
Q^3=0
so Q=0
Because the second derivative is not negative or positive, the function is a convex function.
d) Determine a formula for the optimal order quantity that minimizes costs.
e) What would be the formula for the optimal order quantity if the term h·Q/2 in T(Q) were replaced by h·Qm/2, where m is a positive integer.

Explanation / Answer

(c)Take the second partial derivative (using the chain rule) with respect to Q. Then set that equal to zero. T'(Q)=(-KD)/(Q^2)+(h/2)+0 T''(Q)=(2KD)/(Q^3) 0=(2KD)/(Q^3) Now multiply each side by 1/2KD to get Q^3=0 so Q=0 Because the second derivative is not negative or positive, the function is a convex function.

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