Columbia IEOR 3402- 5.12 The buyer for Needless Markup, a famous “high end” department store, mus

Columbia IEOR 3402- 5.12 The buyer for Needless Markup, a famous “high end” department store, mus

5.12 The buyer for Needless Markup, a famous “high end” department store, must decide on the quantity of a high-priced women’s handbag to produce in Italy for the following Christmas season. The unit cost of the handbag to the store is $28.50 and the handbag will sell for $150.00. Any handbags not sold by the end of the season are purchased by a discount firm for $20.00. In addition, the store accountants estimate that there is a cost of $0.40 for each dollar tied up in inventory, as this dollar invested elsewhere could have yielded a gross profit. Assume that this cost is attached to unsold bags only.
a) Suppose that the sales of the bags are equally likely to be anywhere from 50 to 250 handbags during the season. Based on this, how many bags should the buyer purchase? (Hint: This means that the correct distribution of demand is uniform. You may solve this problem assuming either a discrete or a continuous uniform distribution.)

b) A detailed analysis of past data shows that the number of bags sold is better described by a normal distribution, with mean 150 and standard deviation 20. Now what is the optimal number of bags to be purchased?

c) The expected demand was the same in parts (a) and (b), but the optimal order quantities should have been different. What accounted for this difference?

5.13 An automotive warehouse stocks a variety of parts that are sold at neighborhood stores. One particular part, a popular brand of oil filter, is purchased by the warehouse for $1.5 each. It is estimated that the cost of order processing and receipt is $100 per order. The company uses an inventory carrying charge based on a 28 percent annual interest rate. The monthly demand for the filter follows a normal distribution with mean 280 and standard deviation 77. Order lead time is assumed to be four months. Assume that if a filter is demanded when the warehouse is out of stock, then the demand is back-ordered, and the cost assessed for each back-ordered demand is $12.80. Determine the following quantities:

a. The optimal values of the order quantity and the reorder level.

b. The average annual cost of holding, setup, and stock-out associated with this item assuming that an optimal policy is used.

c. Evalaute the cost pf uncertainity for this process. That is, compare the average annual cost you obtained in part (b) with the average annual cost that would be incurred if the lead time demand had zero variance.

5.39 The home appliance department of a large department store is using a lot size-reorder point system to control the replenishment of a particular model of FM table radio. The store sells an average of 10 radios each week. Weekly demand follows a normal distribution with variance 26. The store pays $20 for each radio, which it sells for $75. Fixed cost of replenishment amounts to $28. The accounting department recommends a 20 percent interest rate for the cost of capital. Storage costs amount to 3 percent and breakage to 2 percent of the value of each item. If a customer demands the radio when it is out of stock, the customer will generally go elsewhere. Loss-of-goodwill costs are estimated to be about $25 per radio. Replenishment lead-time is three months. Consider

a.	If lot sizes are based on the EOQ formula, what reorder level should be used for the radios?

b.	Find the optimal values of (Q,R).

c.	Compare the average annual costs of holding, ordering, and stock-out for the policies that you found in parts (a) and (b).

d.	Re-solve the problem usinf Equations (1) and (2) rather than (1) and (2). What is the effect of including lost sales explicitly?

5.40. Re-solve the problem faced by the department store mentioned in problem 39, replacing the stock-out with a 96 percent Type 1 service level.

Assignment 5

IEOR 3402: Assignment 5
• Hand in solutions to the starred problems only.

• You may work individually, or as a group of 2 students. (If you work as a group, please

submit one assignment per group.)

1. Nahmias, 5.12. You may assume fractional number of handbags. For part (b), use the

standard normal CDF table.

2. Page 245 of Nahmias suggests that when units are ordered in discrete quantities and demand

is continuous, then Q is rounded to the closest integer. Is it correct? If so, provide a

convincing argument. Otherwise, give a counter example.

3. * Page 245 of Nahmias claims that when the demand is assumed to be discrete, the “optimal

solution procedure is to locate the critical ratio between two values of F(Q) and choose the

Q corresponding to the higher value.”

Suppose, in the newsvendor model, that demand D is discrete in the sense that there exists

a sequence of real numbers, {x1, x2, · · · , }, representing all potential demand quantities, and

0 ? x1 ? x2 ? · · ·. Let pi be the probability of [D = xi

]. Furthermore, suppose that all of

these quantities are admissible order-up-to stocking levels. Prove the smallest y such that

F(y) ?


cu + co

is an optimal stock level.

4. * Christine and Tim are planning for a wedding reception at Logan Ridge, an upstate winery

overlooking one of the Finger Lakes. Since the invited guests have not replied on time, they

are uncertain about the actual number of guests that will be attending the reception. To the

best of the couple’s knowledge, the number of guests will be normally distributed with the

mean 170 and the standard deviation 20. A manager at Logan Ridge requests the couple for

an estimate for the number of guests in order to prepare food. He guarantees that he will

prepare food for 10 servings more than the estimate, and will charge based on the higher of

the estimate and the actual number of guests. In an unfortunate case that a guest cannot

be served, the goodwill cost will be 10 times the per-person charge. Christine and Tim need

to provide an estimate for the manager at Logan Ridge.

(a) Write down an appropriate objective function for the decision that Christine and Tim

need to make.

(b) What is the optimal solution? Use Excel if necessary.

5. * Consider the lot size reorder point (Q, R) system discussed in class.

(a) For computing the holding cost, the average inventory level is approximated by R ?

?? + Q/2. Why is it an approximation, not an exact value?

(b) For computing the penalty cost, we used the expected number n(R) of shortages that

occur in one cycle, and multiplied this quantity by a per-unit penalty cost p. How

is it different from computing the penalty cost for the “EOQ Model with Planned



6. * Let ?(·) be the standard normal probability density function. Show

?(z) = Z ?


t?(t)dt .

7. Nahmias, 5.13. You may assume demand is normally distributed. If necessary, use the

standard normal CDF table.

8. Nahmias, 5.39.

9. Nahmias, 5.40.

10. Waterloo Warehouse Limited (WWL) acts as a distributor for a product manufactured by

Norwich Company. WWL uses an (R, Q) type of control system. Norwich has a particular

way of delivering the quantity Q: They deliver Q/2 at the end of the lead time of one month,

and the other Q/2 at the end of two months. Assume that this business has the following


K = 100

R = 40

? = 360 units/year

c = $1.00

I = 25% per year.

Furthermore, the standard deviation of forecast errors over one month is 10 units.

(a) What is the numerical value of the safety stock? Explain your answer.

(b) Suppose that a replenishment is triggered at t = 0, and no order is outstanding at this

time. Describe in words what conditions must hold in order for there to be no stockout

up to time t = 2 months.

(c) Suppose that Norwich offered WWL the option to receive the whole quantity Q after

a lead time of one month, but at a higher unit price. Outline the steps of an analysis

to decide whether or not the option should be accepted.