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.
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