Problem 3-2

National Scan, Inc., sells radio frequency inventory tags. Monthly sales for a seven-month period were as follows:

    

 MonthSales(000)Units

  Feb. 15

  Mar. 20

  Apr. 14

  May. 24

  Jun. 18

  Jul. 20

  Aug. 30

   

b. Forecast September sales volume using each of the following:

    

(1) A linear trend equation.(Round your intermediate calculations and final answer to 2 decimal places.)

   

 Yt 27.14  thousands  

    

(2) A five-month moving average. (Round your answer to 2 decimal places.)

    

  Moving average 21.20  thousands  

    

(3) Exponential smoothing with a smoothing constant equal to .40, assuming a March forecast of 17(000). (Round your intermediate calculations and final answer to 2 decimal places.)

    

  Forecast 23.61  thousands  

   

(4) The naive approach.

    

   Naive approach30  \ thousands  

    

(5) A weighted average using .60 for August, .10 for July, and .30 for Jun e. (Round your answer to 2 decimal places.)

     

  Weighted average

25.40  thousands  

rev: 03_15_2012, 05_23_2013_QC_30974

Explanation:

b.(1)

t y    ty

1 15 15       

2 20 40       

3 14 42       

4 24 96       

5 18 90       

6 20 120       

7 30 210       

28 141 613       

with n = 7, Σt = 28, Σt2 = 140

    

b =

nΣty − ΣtΣy

=

7(613) − 28(141)

= 1.75

nΣt2 − (Σt)2 7(140) − 28(28)

a =

Σy − bΣt

=

141 − 1.75(28)

= 13.14

n 7

  For Sept., t = 8, and Yt = 13.14 + 1.75(8) = 27.14 (000)

(2)

MA5=

14 + 24 + 18 + 20 + 30

= 21.20

5

(3)

  Month Forecast=   F( old ) +.40[Actual − F(Old)]

  April   18.20    =   17.00 +.40 [20 − 17.00]    

  May   16.52 =   18.20 +.40 [14 − 18.20]    

  June   19.51 =   16.52 +.40 [24 − 16.52]    

  July   18.91 =   19.51 +.40 [18 − 19.51]    

  August   19.35 =   18.91 +.40 [20 − 18.91]    

  September  23.61 =   19.35 +.40 [30 − 19.35]    

(5)

.60(30) + .10(20) + .30(18) = 25.40

Problem 3-7

Freight car loadings over an 18-week period at a busy port are as follows:

    

Week Number Week Number Week Number

1 220 7 350 13 461

2 245 8 360 14 475

3 277 9 404 15 502

4 275 10 380 16 510

5 340 11 441 17 543

6 310 12 450 18 541

   

a. Determine a linear trend line for expected freight car loadings. (Round your intermediate calculations and final answers to 3 decimal places.)

   

   = 212.732  + 19.034  t

     

b. Use the trend equation to predict expected loadings for weeks 22 and 23. (Round your intermediate calculations and final answers to 3 decimal places.)

   

 The forecasted demand for week 22 and 23 is 631.481  and 650.515  respectively.  

    

c. The manager intends to install new equipment when the volume exceeds 829 loadings per week. Assuming the current trend continues, the loading volume will reach that level in approximately how many more weeks? (Round your intermediate calculations to 3 decimal places and final answer to the nearest whole number.)

    

  It would take approximately 15  more weeks.

rev: 02_25_2012, 03_15_2012, 01_31_2013_QC_25862, 09_05_2013_QC_34423

Explanation:

a.

t y t*y t2

1   220   220   1  

2   245   490   4  

3   277   831   9  

4   275   1,100   16  

5   340   1,700   25  

6   310   1,860   36  

7   350   2,450   49  

8   360   2,880   64  

9   404   3,636   81  

10   380   3,800   100  

11   441   4,851   121  

12   450   5,400   144  

13   461   5,993   169  

14   475   6,650   196  

15   502   7,530   225  

16   510   8,160   256  

17   543   9,231   289  

18   541   9,738   324  

171   7,084   76,520   2109  

  Σti = 171   Σyi = 7,084

  Σti2 = 2109   Σtiyi = 76,520

b =

(n)(Σtiyi) − (Σti)(Σyi)

(n)(Σti2) − (Σti)2

b =

(18)(76,520) − (171)(7,084)

=

165,996

= 19.034

(18)(2109) − (171)2 8,721

a =

Σy − bΣt

or

n

a =

7,084 − 19.034(171)

   

18

a =

3,829.186= 212.733

 

18

b.

F = 212.733 + (19.034)(22) = 631.481

F = 212.733 + (19.034)(23) = 650.515

The forecasted demand for week 22 and 23 is 631.481 and 650.515 respectively.

c.

829 − 541

= 15.13  Weeks

19.034

It would take approximately 15 more weeks. Since we have just completed week 18, the loading volume is expected to reach 829 by week 33 (18 + 15).

Problem 3-13

The manager of a fashionable restaurant open Wednesday through Saturday says that the restaurant does about 35 percent of its business on Friday night, 35 percent on Saturday night, and 18 percent on Thursday night. What seasonal relatives would describe this situation?(Round your answers to 2 decimal places.)

    

   

  Wednesday 0.48   

  Thursday 0.72   

  Friday 1.40   

  Saturday 1.40   

 Explanation:

     

  Wednesday=.12 × 4 =    .48  

  Thursday =.18 × 4 =    .72  

  Friday =.35 × 4 =  1.40  

  Saturday =.35 × 4 =  1.40  

rev: 03_15_2012

Problem 3-20

An analyst must decide between two different forecasting techniques for weekly sales of roller blades: a linear trend equation and the naive approach. The linear trend equation is Ft = 124 + 2.1t, and it was developed using data from periods 1 through 10. Based on data for periods 11 through 20 as shown in the table, which of these two methods has the greater accuracy if MAD and MSE are used? (Round your answers to 2 decimal places.)

      

t Units Sold

11 144

12 146

13 152

14 142

15 152

16 149

17 152

18 154

19 157

20 164

       

    

  MAD (Naive) 5.11  

  MAD (Linear)

5.47   

  MSE (Naive) 22.97   

  MSE (Linear)40.83   

     

  Naive method  provides forecasts with less average error and less average squared error.

rev: 03_15_2012, 01_04_2013