Demand Forecasting Problems

Production & Operation Management

Forecasting AssignmentForecasting AssignmentForecasting AssignmentForecasting Assignment (No: 7)(No: 7)(No: 7)(No: 7) PGDM 2009PGDM 2009PGDM 2009PGDM 2009----2011, Sec: A, 2011, Sec: A, 2011, Sec: A, 2011, Sec: A, Group No: 5Group No: 5Group No: 5Group No: 5 Submitted To: Mr. N Raju GundalaSubmitted To: Mr. N Raju GundalaSubmitted To: Mr. N Raju GundalaSubmitted To: Mr. N Raju Gundala

Group Members Name Roll Number

Amit Shankar ChoudharyAmit Shankar ChoudharyAmit Shankar ChoudharyAmit Shankar Choudhary 05050505

Arup ChakrabortyArup ChakrabortyArup ChakrabortyArup Chakraborty 12121212 Ashvani KumariAshvani KumariAshvani KumariAshvani Kumari 14141414

Bala KishoreBala KishoreBala KishoreBala Kishore SwamiSwamiSwamiSwami 15151515 Bornali DeyBornali DeyBornali DeyBornali Dey 17171717 Mini DingraMini DingraMini DingraMini Dingra 41414141

1. The number of cans of soft drinks sold in a machine each week is recorded below. Develop forecasts using a three period moving average. 338, 219, 278, 265, 314, 323, 299, 259, 287, 302 Let Xt denote the number of cans of soft drinks sold in a machine of the tth week. Now the 3 month moving average of the t+1th week is given by

Xt + Xt-1 + Xt-2 Ft= 3 , t=3, 4, & 5 Now the three periods moving average forecast is given in the following table

Three-period moving average

Week (t) Xt Forecast

(Ft)

1 338

2 219

3 278

4 265 278.33

5 314 254.00

6 323 285.67

7 299 300.67

8 259 312.00

9 287 293.67

10 302 281.67

2. Use a four period moving average to forecast attendance at baseball games. Historical records show 5346, 7812, 6513, 5783, 5982, 6519, 6283, 5577, 6712, and 7345

Let Xt denote the number of viewers of baseball game of the tth period. Now the 4 period mov-ing average of the t+1th period is given by

Xt + Xt-1 + Xt-2 + Xt-3 Ft= 4 , t= 4, 5, 6 & 7 Now the four periods moving average forecast is given in the following table

Four period moving average

Period

(t) Xt

Forecast

(Ft)

1 5346

2 7812

3 6513

4 5783

5 5982 6363.5

6 6519 6522.5

7 6219 6199.25

8 6283 6125.75

9 5577 6250.75

10 6712 6149.5

11 7345 6197.75

3. A hospital records the number of floral deliveries its patients receive each day. For a two week period, the records show 15, 27, 26, 24, 18, 21, 26, 19, 15, 28, 25, 26, 17, 23 Use exponential smoothing with a smoothing constant of .4 to forecast the number of deliveries.

The formula is Ft=α*At-1+ (1- α) Ft-1 Here, α= 0.4

A=actual data F=forecasted data t=period number

α 0.4

Week Day Delivery Ft by exponential smoothing

1

1 15

2 27 15

3 26 19.8

4 24 22.28

5 18 22.968

6 21 20.9808

7 26 20.98848

2

8 19 22.993088

9 15 21.3958528

10 28 18.83751168

11 25 22.50250701

12 26 23.5015042

13 17 24.50090252

14 23 21.50054151

4. The number of girls who attend a summer basketball camp has been recorded for the seven years the camp has been offered. Use exponential smoothing with a smoothing constant of .8 to forecast attendance for the eighth year. 47, 68, 65, 92, 98, 121, 146 A

Ft=Ft-1+α (At-1-Ft-1) Here; α= 0.8

α 0.8

Year Attendance Ft by exponential smoothing

1 47

2 68 47

3 65 63.8

4 92 64.76

5 98 86.552

6 121 95.7104

7 146 115.94208

8 139.988416

5. The number of pizzas ordered on Friday evenings between 5:30 and 6:30 at a pizza delivery location for the last 10 weeks is shown below. Use exponential smoothing with smoothing constants of .2 and .8 to forecast a value for week 11. Compare your forecasts using MSE. Which smoothing constant would you prefer? 58, 46, 55, 39, 42, 63, 54, 55, 61, 52 Ft=Ft-1+α (At-1-Ft-1)

α 0.2 MSE 84.12344725

Weeks No of Piz-

za Ft by exponential smoothing Error Squared

Error

1 58

2 46 58 -12 144

3 55 55.6 -0.6 0.36

4 39 55.48 -16.48 271.5904

5 42 52.184 -10.184 103.713856

6 63 50.1472 12.8528 165.1944678

7 54 52.71776 1.28224 1.644139418

8 55 52.974208 2.025792 4.103833227

9 61 53.3793664 7.620634 58.07405647

10 52 54.90349312 -2.90349 8.430272298

11 54.3227945

α 0.8 MSE 107.1703721

Weeks No of Piz-

za Forecast using exponential smoothing Error Squared Er-

ror

1 58

2 46 58 -12 144

3 55 48.4 6.6 43.56

4 39 53.68 -14.68 215.5024

5 42 41.936 0.064 0.004096

6 63 41.9872 21.0128 441.5377638

7 54 58.79744 -4.79744 23.01543055

8 55 54.959488 0.040512 0.001641222

9 61 54.9918976 6.008102 36.09729445

10 52 59.79837952 -7.79838 60.81472314

11 53.5596759

It is clear by MSE that if α value increases the error is also increase.

6. A trend line for the attendance at a restaurant's Sunday brunch is given by Number = 264 + .72(t) How many guests would you expect in week 20? We have to put the value 20 in the place of‘t’ and we get, Number=264+ (0.72*20) we get

Number= 278.4 nos. Or 278 nos.

7. The number of new contributors to a public radio station's annual fund drive over the last ten years is 63, 58, 61, 72, 98, 103, 121, 147, 163, 198 Develop a trend equation for this information, and use it to predict next year's number of new contributors.

Let y denote the number of new contributors to a public radio station’s annual fund drive and x

denote the year. Assume that x and y is linearly related. Let y xα β= + be the suggested linear

relationship.

By the method of least squares the estimates of α and β are given by,

2 2

( )( )ˆ

( )

i i i i

i i

n x y x y

n x xβ

−=

−

∑ ∑ ∑∑ ∑

And

ˆˆ i i

y x

n

βα

−=∑ ∑

x Y Xy x2 y2

1 63 63 1 3969

2 58 116 4 3364

3 61 183 9 3721

4 72 288 16 5184

5 98 490 25 9604

6 103 618 36 10609

7 121 847 49 14641

8 147 1176 64 21609

9 163 1467 81 26569

10 198 1980 100 39204

55 1084 7228 385 138474

From the given data we have

n =10, i

x∑ =55, i

y∑ =1084, i i

x y∑ =7228, 2

ix∑ =385 and 2

iy∑

=138474

Thus

2

10*7228 55*1084ˆ10*385 (55)

β−

=− = 15.3455 and

1084 15.3455*55ˆ

10α

−=

= 24

Thus the trend equation is

y = 24 + 15.3455 x The next year’s number of new contributors can be obtained by substituting x = 11 in the re-gression equation y = 24 + 15.3455 x and is given by,

y = 24 + 15.3455*11 = 192.8 = 193

8. The average SAT verbal score for students from one high school over the last ten exams is 508, 490, 502, 505, 493, 506, 492, 490, 503, 501 Do the scores support an increasing or a decreasing trend?

Exam SAT Verbal

Score

1 508

2 490

3 502

4 505

5 493

6 506

7 492

8 490

9 503

10 501

It is a downward trend, as per the above graph shows.

y = -0.3515x + 500.93R² = 0.0231

485

490

495

500

505

510

0 2 4 6 8 10 12

SAT Verbal Score

Exams

SAT Verbal Score

SAT Verbal Score

9. Use the following to forecast a value for period 14, a second quarter. T = 16.32 - .18(t) C2 = .91 S2 = .75 t=0.91*0.75 = 0.6825 Putting the value of t in above equation we get T= 16.32- 0.18*0.6825 T= 16.32-0.12285 T= 16.19715

10. The number of properties newly listed with a real estate agency in each quarter over the last four years is given. Calculate the seasonal index values.

Year Quarter 1 2 3 4

1 73 81 76 77 2 89 87 85 92 3 123 115 108 131 4 92 93 87 101

Quarter 1 Average Index 2 Average Index 3 Average Index 4 Average Index

1 73 94.25 0.7745 81 94 0.8617 76 89 0.8539 73 114.25 0.6389

2 89 94.25 0.9443 87 94 0.9255 85 89 0.9551 120 114.25 1.0503

3 123 94.25 1.3050 115 94 1.2234 108 89 1.2135 176 114.25 1.5405

4 92 94.25 0.9761 93 94 0.9894 87 89 0.9775 88 114.25 0.7702

TOTAL 285 376 356 457

11. Quarterly billing for water usage is shown below.

Year Quarter 1 2 3 4 Winter 64 66 68 73 Spring 103 103 104 120

Summer 152 160 162 176 Fall 73 72 78 88

a. Find the seasonal index for each quarter. b. De-seasonalize the data. c. Find the trend line. d. Assume there is no cyclical component and

forecast the summer billing for year 5.

a.

Quarter 1 Average Index 2 Average Index 3 Average Index 4 Average Index

Winter 64 98 0.653061 66 100.25 0.658354 68 103 0.660194 73 114.25 0.63895

Spring 103 98 1.05102 103 100.25 1.027431 104 103 1.009709 120 114.25 1.050328

Summer 152 98 1.55102 160 100.25 1.59601 162 103 1.572816 176 114.25 1.540481

Fall 73 98 0.744898 72 100.25 0.0288 78 103 0.757282 88 114.25 0.770241

TOTAL 392 401 412 457

b.

(1) (2) (3) (4) (5) (6) '3/5 (7) 1*1 (8) 1*6

Period

(x)

Quarter Actual Billing

(y)

Average of the same quarters for each

year

Seasonal Factor Deseasonalized Billing

(yb)

Square of

Period

x*yb

1 Winter 64 67.75 0.000751734 85136.53137 1 85136.531

2 Spring 103 107.75 1.037304452 99.29582367 4 198.59165

3 Summer 152 162.5 1.564380265 97.16307692 9 291.48923

4 Fall 73 77.75 0.748495788 97.52893891 16 390.11576

5 Winter 66 0.000751734 87797.04797 25 438985.24

6 Spring 103 1.037304452 99.29582367 36 595.77494

7 Summer 160 1.564380265 102.2769231 49 715.93846

8 Fall 72 0.748495788 96.19292605 64 769.54341

9 Winter 68 0.000751734 90457.56458 81 814118.08

10 Spring 104 1.037304452 100.2598608 100 1002.5986

11 Summer 162 1.564380265 103.5553846 121 1139.1092

12 Fall 78 0.748495788 104.2090032 144 1250.508

13 Winter 73 0.000751734 97108.85609 169 1262415.1

14 Spring 120 1.037304452 115.6844548 196 1619.5824

15 Summer 176 1.564380265 112.5046154 225 1687.5692

16 Fall 88 0.748495788 117.5691318 256 1881.1061

136 1662

c. Avg Of X = 136/16= 8.5, ∑xyb – n*avg(x)avg(yb) B= = -1805.25 ∑x2 – n*avg(x)2 A= Avg(yb) – B*avg(x) = 37953.73 Therefore Y=A+Bx Y = 37953.73+(-1805.25)x

12. A customer comment phone line is staffed from 8:00 a.m. to 4:30 p.m. five days a week. Records are available that show the number of calls received every day for the last five weeks.

Week Day Number Week Day Number

1 M 28 4 M 35 T 12 T 17

W 16 W 16

TH 15 TH 20

F 23 F 29 2 M 29 5 M 37 T 10 T 19

W 14 W 18

TH 14 TH 21

F 26 F 28 3 M 32 T 15 W 15 TH 18 F 27

a. Use this information to calculate a seasonal index. b. De-seasonalize the data. c. Find the trend line. d. Assume there is no cyclical component and forecast

the calls for week 6.

SlopeSlopeSlopeSlope InterceptInterceptInterceptIntercept

0.28 17.67

Week n Day Number Trend Seasonal Index Deseasonal Data

1 1 M 28 17.9538 1.51 18.57

2 T 12 18.2376 0.68 17.56

3 W 16 18.5214 0.74 21.63

4 TH 15 18.8052 0.82 18.20

5 F 23 19.089 1.25 18.47

2 6 M 29 19.3728 1.51 19.24

7 T 10 19.6566 0.68 14.63

8 W 14 19.9404 0.74 18.93

9 TH 14 20.2242 0.82 16.99

10 F 26 20.508 1.25 20.88

3 11 M 32 20.7918 1.51 21.23

12 T 15 21.0756 0.68 21.95

13 W 15 21.3594 0.74 20.28

14 TH 18 21.6432 0.82 21.85

15 F 27 21.927 1.25 21.68

4 16 M 35 22.2108 1.51 23.22

17 T 17 22.4946 0.68 24.87

18 W 16 22.7784 0.74 21.63

19 TH 20 23.0622 0.82 24.27

20 F 29 23.346 1.25 23.29

5 21 M 37 23.6298 1.51 24.54

22 T 19 23.9136 0.68 27.80

23 W 18 24.1974 0.74 24.33

24 TH 21 24.4812 0.82 25.49

25 F 28 24.765 1.25 22.48

6 26 M 25.0488 1.51

27 T 25.3326 0.68

28 W 25.6164 0.74

29 TH 25.9002 0.82

30 F 26.184 1.25

14. A 24-hour coffee/donut shop makes donuts every eight hours. The manager must forecast donut demand so that the bakers have the fresh ingredients they need. Listed below is the actual number of glazed donuts (in dozens) sold in each of the preceding 13 eight-hour shifts.

Date Shift Demand(dozens) June 3 Day 59

Evening 47

Night 35 June 4 Day 64

Evening 43 Night 39

June 5 Day 62 Evening 46 Night 42

June 6 Day 64 Evening 50 Night 40

June 7 Day 69

Forecast the demand for glazed donuts for the three shifts of June 8 and the three shifts of June 9.

Slope Intercept

0.4945 47.308

Date n Shift Demand(dozens) Trend

3-Jun 1 Day 59 47.80

2 Evening 47 48.30

3 Night 35 48.79

4-Jun 4 Day 64 49.29

5 Evening 43 49.78

6 Night 39 50.28

5-Jun 7 Day 62 50.77

8 Evening 46 51.26

9 Night 42 51.76

6-Jun 10 Day 64 52.25

11 Evening 50 52.75

12 Night 40 53.24

7-Jun 13 Day 69 53.74

14 Evening

54.23

15 Night 54.73

8-Jun 16 Day 55.22

17 Evening 55.71

18 Night

56.21

9-Jun 19 Day

56.70

20 Evening 57.20

21 Night 57.69

15. In order to forecast the attendance at an annual tennis tournament, a model has been developed which uses attendance from the previous year and the amount spent for advertising this year. From the years shown in the table, forecast the attendance for years 2-5 and calculate the forecast error.

Year

Attendance

Advertising Expenditure

Forecast

Error

1 8363 750 2 9426 1250 3 9318 3200 4 10206 4500 5 11018 5600

The multiple regression model is Attendance = 6738 + .23($) + .25 (Attlag) Attlag is last year’s actual attendance. So, if we take the 1st year attendance and put in to the equation we get, Attendance = 6738 + 0.23*1250 + 0.25*8363 like wise we get from excel sheet.

Advertising

Year Attendance Expenditure Forecast Error

1 8363 750

2 9426 1250 9116.25 309.75

3 9318 3200 9830.5 -512.5

4 10206 4500 10102.5 103.5

5 11018 5600 10577.5 440.5