Demand for rug-cleaning machines at Clyde’s U-Rent-It is shown in the following

Question

Demand for rug-cleaning machines at Clyde’s U-Rent-It is shown in the following table. Machines are rented by the day only. Profit on the rug cleaners is $19 per day. Clyde has 4 rug-cleaning machines.

a. Assuming that Clyde’s stocking decision is optimal, what is the implied range of excess cost per machine?

b. Suppose now that the $19 mentioned as profit is instead the excess cost per day for each machine and that the shortage cost is unknown. Assuming that the optimal number of machines is four, what is the implied range of shortage cost per machine?

DemandFrequency .30 . 20 .20 .15 .10 .05 1.00

Explanation / Answer

The question states that there will be a profit of $19 is a machine is rented out. This is also called cost of shortage, or Cs. What is not mentioned directly is the excess cost, or Ce. We need to compute this. We are told that 4 machines is optimal. If we look at the cumulative probabilities of demand, we get the following chart:

A) If 4 is the optimal number of machines, then the service level would be between 0.85 and 0.95.

We compute the service level using the formula: Service Level (SL) = Cs/(Cs+Ce)

Let us now set SL to the lower limit of 0.85. We know that Cs = $19.

We get $0.85 = $19/($19+Ce)

or ($19+Ce) = $19/0.85 = 22.353

Or Ce=22.353-19=$3.353

How about setting the service level to the maximum?

We get $0.95 = $19/($19+Ce)

or ($19+Ce) = $19/0.95 = 20

Or Ce = 1.

Thus, the implied range of excess cost per machine (Ce) is $1.00 <=Ce <=3.353

B) In this part, we are told that $19 is now Ce. Cs is now unknown. 4 is still the optimal number of machines. Thus, the service levels again vary between 0.85 and 0.95

Setting the service level to the lower bound of 0.85, we get

Cs/(Cs+19) = 0.85

Cs/0.85 = Cs+19

Cs ( 1/0.85 - 1) = 19

Cs/(0.176) = 19

Or CS = 19 * .176= 3.35

How about the upper limit?

Cs/(Cs+19) = 0.95

Cs/0.95 = Cs+19

Cs ( 1/0.95 - 1) = 19

Or Cs = 1

Thus, the Cost of Excess, Cs also ranges between $1.00 and $3.55.

Demand Probability Cumulative Probability 0 .3 .3 1 0.2 0.5 2 0.2 0.7 3 0.15 0.85 4 0.10 0.95 5 0.05 1.00

Related Questions

Navigate

• Browse (All)
• Browse D
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.