A university orders all office supplies, including red grading pens, out of a catalog from the same supplier. Conservative estimates of the demand for these red pens are 12,000 pens per year. There is a $25 charge for placing an order and the university has a $10 annual cost for holding these pens. Prices for the pens are based on the quantity purchased, so if less than 100 are ordered, the unit price is $2.50, if 100 to 299 are ordered, the unit price is $2.40, if 300 to 499 are ordered, the unit price is $2.30, and if 500 or more are ordered, the unit price is $2.20. What order quantity will result in the lowest total annual cost to the university?
A)100
B)245
C)300
D)500

Question

Quizplus · Accepted Answer

To determine the order quantity that results in the lowest total annual cost, we need to consider the total cost for each quantity option and then select the one with the lowest total cost. For the lowest quantity option (less than 100), the total cost would be: Total cost = (12,000 pens / order quantity) x $2.50/unit + $25 + $10/year For the next quantity option (100-299), the total cost would be: Total cost = (12,000 pens / order quantity) x $2.40/unit + $25 + $10/year For the third quantity option (300-499), the total cost would be: Total cost = (12,000 pens / order quantity) x $2.30/unit + $25 + $10/year For the highest quantity option (500 or more), the total cost would be: Total cost = (12,000 pens / order quantity) x $2.20/unit + $25 + $10/year We can simplify the calculation by using algebra. Let x be the order quantity, then the total cost equation for each quantity option can be written as follows: Total cost = (12,000/x)P + 25 + 10 where P is the unit price based on the quantity purchased: P = $2.50 for x<100 P = $2.40 for 100<=x<300 P = $2.30 for 300<=x<500 P = $2.20 for x>=500 Now we can find the derivative of the total cost equation with respect to x and set it to zero to find the minimum: d(Total cost)/dx = -(12,000/x^2)P + (12,000/x) dP/dx = 0 Solving for x, we get: x = sqrt[(12,000 dP/dx) / P] Plugging in the values of dP/dx for each quantity option, we get: x = sqrt[(12,000 (-0.10)) / 2.50] = 245.98 for 100<=x<300 x = sqrt[(12,000 (-0.10)) / 2.40] = 257.57 for 300<=x<500 x = sqrt[(12,000 (-0.10)) / 2.20] = 306.18 for x>=500 Out of the three options, x=306 will result in the lowest total cost: Total cost = (12,000/306) x $2.20/unit + $25 + $10/year = $448.63 Therefore, the best choice is D, ordering 500 or more red grading pens.

A University Orders All Office Supplies, Including Red Grading Pens