1

Review problem #1Text problem 9 - 6, 7

• Weekly sales (in hundreds) of PERT shampoo at the SaveMor drug chain for the past 16 weeks are as follows:

Management wishes to forecast the annual demand for PERT shampoo in order to determine an optimal order policy.

• Graph this time series.

Week Sales1 652 613 684 675 746 827 758 639 7210 7811 7312 7513 6814 6415 7316 79

Sales

0

50

100

1 3 5 7 9 11 13 15

Sales

A stationary model seems to properly describe this time series.

Go to file ch 9-6.xlsand create the graph.

2

• Verify statistically that a stationary model is appropriate for forecasting this time series.

ANOVAdf SS MS F Significance F

Regression 1 71.11838 71.11838 2.008106 0.178329Residual 14 495.8191 35.41565Total 15 566.9375

H0: 1= 0H1: 1

Do not reject the null hypothesis. There is insufficient evidence for the existence of a trend at (even) 15% significance level.

3

• Compare the accuracy of the following stationary models:– 3, and 6 period moving average– 3 period weighted moving average with

optimal weights vs. Exponential smoothing with optimal alpha

4

Comparing moving average models

5

Obtaining the optimal weights 3 - period moving average

Don’t forget to select “Assume non-negative”in “Options”

The sum of the weights appears incell c9.

6

The optimal alpha Exponential smoothing model

Don’t forget to select “Assume non-negative”in “Options”

7

Review problem #3

A firm that manages and maintains several apartment house complexes has experienced the following expenses over the last five years. Quarter : t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Yt 27 39 41 50 44 47 58 63 73 92 94 102 123 119 128 147 155 158 164 187

 (a) Plot this time series using an Excel graphical tool (use an x-y scatter plot)

(b) Run a linear regression model and compare its performance to the best Holt’s model you can obtain, when optimizing alpha and gamma. Use the MAD criterion to compare.

(c) Repeat part (b), this time optimize alpha , gamma, the initial forecast and the initial trend. Which model do you select now based on the MAD?

8

Review problem 3 - Solution

0

50

100

150

200

0 5 10 15 20 25

• The plot indicates the presence of a linear trend.

• Thus, a linear regression model or the Holt’s model should perform well.

9

Review problem 3 - Solution

• Running linear regression and checking the p-value of the t-test for b1confirms the observation. The time series has a linear trend.

The p-value is very small, implying that there is sufficient evidence to support the hypothesis that b10

10

Review problem 3 – SolutionThe Linear regression Forecasting model

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Review problem 3 – Solution Optimizing alpha and gamma For Holt’s

Notice: Only alpha and gamma are optimized

Don’t forget to select “Assume

non-negative”in “Options”, since alpha and gamma are both

Alpha and gamma are both 1

12

Review problem 3 - Solution Optimizing alpha, gamma, L0, T0 for Holt’s

Notice: Both alpha, gamma,L0 and T0 are optimized

Don’t select “Assume non -negative” in “Options”, since L0 and T0 might be negative. But then, add the non-negativity constraints of alpha and gamma

13

Review problem 3 – Solution

• The Holt’s model with all the parameters optimized had the best performance over the past 20 periods based on MAD.– Holt’s (all parameters optimized) MAD = 5.39.

– Holt’s (only and are optimized) MAD = 6.89

– Linear regression MAD = 7.17

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Review problem 4

• Analyze the following time series. Compare the following four forecasting techniques, and determine which one you would use to make a forecast. (i) The exponential smoothing(ii) The weighted moving average(iii) The Holt’s method(v) The linear regression model.

• Write down each optimized model.• Perform a forecast by hand for period 61 and 62 using

the template results.• Show how you would calculate MAD for the first 4

periods of each technique.

15

Review problem 5

• Sport King is a chain of 12 sporting goods shops in South Carolina. Total chain demand for the Whamco flying disk averages 250 units a week at a cost. While the disks come in various colors, the firm’s policy is to determine the appropriate total order quantity using the EOQ formula and divide this total among the various colors in proportion to historic demand.

• Sports King buys the Whamco flying disks for $2.50 each, and sell them for $3.50 each. Sport King uses an annual inventory holding cost rate of 28%, and the cost to place an order is $40. Lead-time for delivery is two weeks, and Sport King desires a safety stock of 50 units. 

16

Problem 5 - continued

• Determine:– The optimal order quantity of disks.

– The number of calendar days between orders (cycle time).

– The reorder point for the disks.

– The total annual inventory cost (holding, ordering, procurement) of this policy.

• Solution• Annual demand = 250*52 = 13000 disks

• Unit holding cost = .28(2.5) = $.7 per unit per year

– The optimal order quantity

89.12187.

)40)(000,13(2

C

DC2*Q

h

o

17

– The cycle time T = Q*/D • T = Q*/D = 1218.89/13000 = .09376 years = (.09376)(52) =

4.875 weeks = 4.875*7 = 34.129 calendar days.

– The reorder point R = inventory position = Inventory at hand + Outstanding orders.

• There are no outstanding orders in our problem:

• Reorder point = Inventory at hand = (Lead time in weeks)(demand per week) = 2*250 = 500

4.875 weeks

2 weeks

Reorder point

Problem 5 - continued

18

– Total annual inventory costs =

23.388,33$CDSS2

QC

QD

C*)Q(TC*

h*o

Ann. Ordering cost Ann. Holding cost Procurement cost

Problem 5 - continued

19

– Whamco offers its customers the following all-units price discount schedule:

Number Price

Ordered per Unit1–499$ 2.50

500–999 $2.25

1000–1999 $2.10

2000–4999 $1.90

5000–9999 $1,80

10000 and above $1.75

Problem 5 - continued

20

– Assuming that holding costs are discounted, determine the following:

a. The optimal order quantity of Whamco disks.

b. The reorder point for the disks.

c. The number of calendar days between orders (cycle time).

d. The total annual inventory cost (holding, ordering, procurement) for this policy.

Problem 5 - continued

21

Review Problem 6

• The Compland Computer Store sells the HewPacX laser jet printer. Demand has been averaging 24 units per six-day week. The printers cost Compland $1095 each, and Compland uses a 20% annual inventory holding cost rate. The cost of placing an order with HewPac is $90, and the lead time is eight working days.

• If Compland runs out of the HewPacX printer, it gives its customers a loaner printer until the next shipment of HewPacX printers arrive. Each loaner costs Compland $1.50 per calendar day.

• Determine:– The optimal order quantity of HewPacX printers.– The reorder point for these printers.– The proportion of customers who will get loaner printers.– The total annual inventory cost (holding, ordering, shortage, and procure-

ment) of this policy.

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• SolutionD = 24*52 = 1248 per year. Ch = IC = .20(1095) = 219 per unit per year.

Co = 90. Cs = 1.50(365) = 547.5 per unit per year. Cb = 0 (not included)

– The optimal order quantity =

– The reorder point R = LD-S*. The optimal backorder is

Then R = 8(24/6) – 10.82 = 21.18

89.37

2*

2

sh

b

s

sh

h

O

CC

DC

C

CC

C

DCQ

8210.CCDCCQ

*Ssh

bh*

Problem 6 - continued

23

– Proportion of customers on backorder (assuming each customer orders one unit) = S*/Q* = 28.6%

– Total inventory costs =

8.487,372,12

)(2

)(),(

22

sboh CQ

SSCC

Q

DC

Q

SQSQTC

27.07

-10.82

27.89

-10.82

R = 21

8 daysQ =

37.

89

CycleLead time

Problem 6 - continued

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Review Problem 7

• Return to the Sports King disk sale problem.– Management has decided to consider uncertainty in the

demand pattern. A statistical analysis revealed that the weekly demand is normally distributed with:

• mean = 250 disks per week

• Standard deviation = 80 disks per week.

• Lead time is 2 weeks

• Required cycle service level is 92%.

– Find • The order quantity, the reorder point, the safety stock

• The total inventory costs

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– Calculating the reorder point: R = 500 + 1.405(113.13) = 658.94

– The safety stock = R – L = 1.45(113.13) = 164.03 disks.

• Solution– The order quantity does not change because we assume an EOQ

model, as before.– The reorder point depends on the service level.

P(DL > R) = .08 P(Z > Z.08) = .08 but

Z.08 = 1.405 so (R – )/ or R = L + Z.08L.– Before making the calculation, we need to adjust the mean and

standard deviation of the demand during the lead time.

L = 2(250) = 500; L2 = (2)(80)2 = 12800 L = 113.13

2 weeks variance per week

Problem 7 - continued