1. A company that makes shopping carts for supermarkets and other stores recently purchased some new equipment that reduces the labor content of the jobs needed to produce the shopping carts. Prior to buying the new equipment, the company used five workers, who produced an average of 80 carts per hour. Workers receive $10 per hour, and machine cost was $40 per hour. With the new equipment, it was possible to transfer one of the workers to another department, and equipment cost increased by $10 per hour while output increased by four carts per hour.

a. Compute labor productivity under each system. Use carts per worker per hour as the measure of labor productivity.

b. Compute the multifactor productivity under each system. Use carts per dollar cost (labor plus equipment) as the measure.

c. Comment on the changes in productivity according to the two measures, and on which one you believe is the more pertinent for this situation.

a. Before: 80 5 = 16 carts per worker per hour.

After: 84 4 = 21 carts per worker per hour.

b. Before: $10 x 5 = $50 + $40 = $90; hence 80 ÷ $90 = .89 carts/$1.

After: $10 x 4 = $40 + $50 = $90; hence 84 ÷ $90 = .93 carts/$1.

c. Labor productivity increased by 31% (5/16).

Multifactor productivity increased by 4.5% (.04/.89).

2. An electrical contractor’s records during the last five weeks indicate the number of job requests:

Predict the number of requests for week 6 using each of these methods:

a. Naive.b. A four-period moving average.c. Exponential smoothing with α = .30. Use 20 for week 2 forecast.

Exponential smoothing

a 22

b.

22+18+21+224

=20 .75

c. F3 = 20 + .30(22 – 20) = 20.6

F4 = 20.6 + .30(18 – 20.6) = 19.82

F5 = 19.82 + .30(21 – 19.82) = 20.17

F6 = 20.17 + .30(22 – 20.17) = 20.72

3. Two independent methods of forecasting based on judgment and experience have been prepared each month for the past 10 months. The forecasts and actual sales are as follows:

a) Compute the MSE and MAD for each forecast. Does either forecast seem superior? Explain.

b) Compute MAPE for each forecast.

28. a. Forecast #1 Forecast #2

MonthA

Sales F Forecast(A–F) Error Error2 | e | Forecast Error Error2 | e |

1 770 771 –1 1 1 769 1 1 1

2 789 785 4 16 4 787 2 4 2

3 794 790 4 16 4 792 2 4 2

4 780 784 –4 16 4 798 –18 324 18

5 768 770 –2 4 2 774 –6 36 6

6 772 768 4 16 4 770 2 4 2

7 760 761 –1 1 1 759 1 1 1

8 775 771 4 16 4 775 0 0 0

9 786 784 2 4 2 788 –2 4 2

10 790 788 2 4 2 788 2 4 2

12 94 28 –16 382 36

b. Period Absolute Percentage Error (F1) Absolute Percentage Error (F2)

1 1 / 770 = .00130 1 / 770 = .0013

2 4 / 789 = .00507 2 / 789 = .00253

3 4 / 794 = .00504 2 / 794 = .00252

4 4 / 780 = .00513 18 / 780 = .02308

5 2 / 768 = .00260 6 / 768 = .00781

6 4 / 772 = .00518 2 / 772 = .00259

7 1 / 760 = .00132 1 / 760 = .00132

8 4 / 775 = .00516 0 / 775 = 0

9 2 / 786 = .00254 2 / 786 = .00254

10 2 / 790 = .00253 2 / 790 = .00253

Total .003587 .04622

MAPE F1 = .03587 / 10 = .003587

MAPE F2 = .04622 /10 = .004622

Since .003587 < .004622, choose forecasting method 1.

MSE =(A – F) 2

MAD =| e |

n – 1 N

Forecast #1: 94/9 = 10.44 28/10 = 2.8

Forecast #2 382/9 = 42.44 36/10 = 3.6

4. Freight car loadings over a 12-year period at a busy port are as follows:

a) Determine a linear trend line for freight car loadings. b) Use the trend equation to predict loadings for weeks 20 and 21.

a. t Y t*Y t2

1 220 220 1

2 245 490 4

3 280 840 9

4 275 1,100 16

5 300 1,500 25

6 310 1,860 36

7 350 2,450 49

8 360 2,880 64

9 400 3,600 81

10 380 3,800 100

11 420 4,620 121

12 450 5,400 144

13 460 5,980 169

14 475 6,650 196

15 500 7,500 225

16 510 8,160 256

17 525 8,925 289

18 541 9,738 324

171 7001 75,713 2109

∑ t i=171 ∑ Y i=7001

∑ t i2=2109 ∑ t i Y i=75 ,713

b=(n)(∑ t i Y i )−(∑ t i )(∑ Y i)

(n)(∑ t i2 )−(∑ ti )

2

b=(18 )(75 ,713)−(171)(7001 )(18 )(2109 )−(171 )2

=165 ,6638721

=19

a=(700118 )−19(171

18 )=388. 944−19( 9. 5 )=208 . 444

Y =208 .444+19 t

b. F = 208.444 + (19)(20) = 588.444

F = 208.444 + (19)(21) = 607.444

The forecasted demand for week 20 and 21 is 588.444 and 607.444 respectively.

5. The following table shows data on the average number of customers processed by several bank service units each day. The hourly wage rate is $25, the overhead rate is 1.0 times labor cost, and material cost is $5 per customer.

a. Compute the labor productivity and the multifactor productivity for each unit.

(1) (2) (3) (4) (5) (6) (7) (8) (9)

Customers Labor Material Overhead Total LP MFPUnit Employees processed Cost@$25 Cost@$5 @1 Cost (3) (2) (3) (7)

A 4 36 100 180 100 380 9 .095

B 5 40 125 200 125 450 8 .089

C 8 60 200 300 200 700 7.5 .086

D 3 20 75 100 75 250 6.7 .080