OM HW Solution (chapters 3-5)

3.4Use simple linear regression to forecast annual demand for the firm's RVs for next year.

INDUSTRY SALES
x / ANNUAL SALES
y / x2 / xy / y2
536 / 98 / 287,296 / 52,528 / 9,604
791 / 137 / 625,681 / 108,367 / 18,769
650 / 112 / 422,500 / 72,800 / 12,544
813 / 145 / 660,969 / 117,885 / 21,025
702 / 120 / 492,804 / 84,240 / 14,400
575 / 103 / 330,625 / 59,225 / 10,609
684 / 116 / 467,856 / 79,344 / 13,456
x=4,751 / y=831 / x2=3,287,731 / xy=574,389 / y2=100,407

x2y - xxy 3,287,731(831) - 4,751(574,389)

a = ────────── = ──────────────────────

nx2 - (x)2 7(3,287,731) - (4,751)2

3,182,322

a = ────────── = 7.198

442,116

nxy - xy 7(574,389) - 4,751(831)

b = ───────── = ─────────────────

nx2 - (x)2 442,116

4,020,723 – 3,948,081 72,642

b = ──────────────── = ─────── = .1643

442,116 442,116

Y = a + bX = 7.198 + .1643X

Y8 = 7.198 + .1643(725) = 7.198 + 119.118

Y8 = 126.316 or 126.3 RVs

3.9a.Develop moving average forecasts for the past 10 months (Months 15-24) for AP = 2, 4, 6, and 8 months.

Absolute / Absolute / Absolute / Absolute
Month / Lawsuits / AP = 2 / Error / AP = 4 / Error / AP = 6 / Error / AP = 8 / Error
1 / 16
2 / 25
3 / 16
4 / 24
5 / 38
6 / 46
7 / 54
8 / 52
9 / 51
10 / 56
11 / 67
12 / 45
13 / 53
14 / 61
15 / 55 / 57.0 / 2.0 / 56.50 / 1.50 / 55.500 / 0.500 / 54.875 / 0.125
16 / 69 / 58.0 / 11.0 / 53.50 / 15.50 / 56.167 / 12.833 / 55.000 / 14.000
17 / 63 / 62.0 / 1.0 / 59.50 / 3.50 / 58.333 / 4.667 / 57.125 / 5.875
18 / 57 / 66.0 / 9.0 / 62.00 / 5.00 / 57.667 / 0.667 / 58.625 / 1.625
19 / 48 / 60.0 / 12.0 / 61.00 / 13.00 / 59.667 / 11.667 / 58.750 / 10.750
20 / 55 / 52.5 / 2.5 / 59.25 / 4.25 / 58.833 / 3.833 / 56.375 / 1.375
21 / 61 / 51.5 / 9.5 / 55.75 / 5.25 / 57.833 / 3.167 / 57.625 / 3.375
22 / 51 / 58.0 / 7.0 / 55.25 / 4.25 / 58.833 / 7.833 / 58.625 / 7.625
23 / 56 / 56.0 / 0.0 / 53.75 / 2.25 / 55.833 / 0.167 / 57.375 / 1.375
24 / 53 / 53.5 / 0.5 / 55.75 / 2.75 / 54.667 / 1.667 / 57.500 / 4.500
Sum = / 54.5 / Sum = / 57.25 / Sum = / 47.000 / Sum = / 50.625
MAD = / 5.45 / MAD = / 5.73 / MAD = / 4.70 / MAD = / 5.06

b.Which AP results in the lowest mean absolute forecasting error? Which AP would you recommend? Why?

AP = 6 results in the lowest MAD, so this would be the recommended forecasting model since it demonstrated the best forecasting accuracy over the past 10 months.

c.Using your recommended AP, forecast the number of lawsuits expected for next month (Month 25).

Month 25: F25 = (48 + 55 + 61 + 51 + 56 + 53)/6 = 54.00 lawsuits

3.13a.Use exponential smoothing to forecast monthly plastic pellet prices. Compute what the forecasts would have been for all the months of historical data for  = 0.1,  = 0.3, and  = 0.5 if the assumed forecast for all ’s in the first month is $0.39.

Forecast / Absolute / Forecast / Absolute / Forecast / Absolute
Month / Plastic / alpha = .1 / Error / alpha = .3 / Error / alpha = .5 / Error
1 / 0.39 / 0.39 / 0.39 / 0.39
2 / 0.41 / 0.390 / 0.390 / 0.390
3 / 0.45 / 0.392 / 0.396 / 0.400
4 / 0.44 / 0.398 / 0.412 / 0.425
5 / 0.40 / 0.402 / 0.421 / 0.433
6 / 0.41 / 0.402 / 0.414 / 0.416
7 / 0.38 / 0.403 / 0.023 / 0.413 / 0.033 / 0.413 / 0.033
8 / 0.36 / 0.400 / 0.040 / 0.403 / 0.043 / 0.397 / 0.037
9 / 0.35 / 0.396 / 0.046 / 0.390 / 0.040 / 0.378 / 0.028
10 / 0.38 / 0.392 / 0.012 / 0.378 / 0.002 / 0.364 / 0.016
11 / 0.39 / 0.391 / 0.001 / 0.379 / 0.011 / 0.372 / 0.018
12 / 0.43 / 0.390 / 0.040 / 0.382 / 0.048 / 0.381 / 0.049
13 / 0.37 / 0.394 / 0.024 / 0.396 / 0.026 / 0.406 / 0.036
14 / 0.38 / 0.392 / 0.012 / 0.389 / 0.009 / 0.388 / 0.008
15 / 0.36 / 0.391 / 0.031 / 0.386 / 0.026 / 0.384 / 0.024
16 / 0.39 / 0.388 / 0.002 / 0.378 / 0.012 / 0.372 / 0.018
Sum = / 0.231 / Sum = / 0.250 / Sum = / 0.266
MAD = / 0.023 / MAD = / 0.025 / MAD = / 0.027

b.Which alpha () value results in the least mean absolute deviation for Months 7-16.

 = .1 results in the lowest MAD.

c.Use the best alpha from Part b to compute the forecasted plastic pellets price for Month 17.

F17 = .1(0.39) + .9(0.388) = 0.3882 dollars per pound

3.27Use seasonalized time series regression analysis to develop a forecast for next year's quarterly sales revenues for personal computers:

First, compute the seasonal indexes:

═════════════════════════════════════════════════════════════════

QUARTERLY SALES ($ MILLION)

──────────────────────────────────── ANNUAL

YEAR Q1 Q2 Q3 Q4 TOTAL

─────────────────────────────────────────────────────────────────

1 9.2 5.4 4.3 14.1 33.0

2 10.3 6.4 5.4 16.0 38.1

─────────────────────────────────────────────────────────────────

TOTALS 19.5 11.8 9.7 30.1 71.1

QUARTER AVERAGE 9.75 5.90 4.85 15.05 8.8875

─────────────────────────────────────────────────────────────────

SEASONAL INDEX (SI) 1.097 .664 .546 1.693

═════════════════════════════════════════════════════════════════

SIQ1 = 9.75/8.8875, SIQ2 = 5.9/8.8875, SIQ3 = 4.85/8.8875

Next, deseasonalize the data by dividing each observation by its SI:

═══════════════════════════════════════════════════════════

QUARTERLY SALES ($ MILLION)

────────────────────────────────────

YEAR Q1 Q2 Q3 Q4

───────────────────────────────────────────────────────────

1 8.39 8.13 7.88 8.33

2 9.39 9.64 9.89 9.45

═══════════════════════════════════════════════════════════

Next, perform time series regression on the deseasonalized data:

═══════════════════════════════════════════════════════════

YEAR QUARTER x y x2 xy

───────────────────────────────────────────────────────────

1118.39 1 8.39

228.13 416.26

337.88 923.64

448.331633.32

2159.392546.95

269.643657.84

379.894969.23

489.456475.60

─────────────────────────────────────────────────────────

Totals 36 71.10 204 331.23

x = 36, y = 71.10, x2 = 204, xy= 331.23, n = 8

x2y - xxy 204(71.10) - 36(331.23)

a = ─────────── = ─────────────────

nx2 - (x)2 8(204) - (36)2

14,504.40 - 11,924.28 2,580.12

a = ─────────────── = ────────── = 7.679

1,632 - 1,296 336

nxy - xy 8(331.23) - 36(71.10)

b = ────────── = ────────────────

nx2 - (x)2 336

2,649.84 - 2,559.6

b = ───────────── = .26857

336

Y8 = a + bX8 = 7.679 + .26857(X)

Next, compute the deseasonalized forecasts for periods 9 - 12:

Y9 = 7.679 + .26857(9) = 10.096

Y10 = 7.679 + .26857(10) = 10.365

Y11 = 7.679 + .26857(11) = 10.633

Y12 = 7.679 + .26857(12) = 10.902

Next, use the seasonal indexes to seasonalize the forecasts:

═════════════════════════════════════════════════════════

SEASONALIZED

DESEASONALIZED FORECASTS

QUARTER SI FORECASTS ($ MILLION)

(1) (2) (3) [COL 2 X COL 3] ─────────────────────────────────────────────────────────

Q1 1.097 10.096 11.08

Q2 .664 10.365 6.88

Q3 .546 10.633 5.81

Q4 1.693 10.902 18.46

Chapter 4

4.13a.Which alternative is best?

Which is best depends on the annual production quantity, which is not yet known.

b.At what annual production volume would they be indifferent between the two approaches?

TCL-T = TCH-TTC = FC + v(Q)

55,000 + 62.50(Q) = 87,000 + 48.25(Q)

14.25(Q) = 32,000

Q = 2,246 parts

c.For what range of annual production volumes would each approach be preferred?

Low-Tech is preferred for Q = 0 to 2,246 parts.

High-Tech is preferred for Q  2,246 parts.

d.What other considerations should be important in the decision?

Product quality, flexibility, forecasts of future annual volumes.

5.4a.Use a decision tree analysis and recommend a course of action for this new-product idea.

The company should lease the concept to company A. Notice, however, that other alternatives are very close in their payoffs.

b.If the company follows your recommendation, what returns should the company expect to receive?

If the firm's estimates are correct, it will receive either $2,800,000 or $2,200,000.

5.8a.What is the payback period for the construction cost of each location?

First, compute the annual profit for each alternative.

Singapore:

ProfitS = TRS – TCS = pS(Q) – [FCS + vS(Q)]

= 210(250,000) – [2,800,000 + 130(250,000)]

= 52,500,000 – 35,300,000 = 17,200,000

Taiwan:

ProfitT = TRT – TCT = pT(Q) – [FCT + vT(Q)]

= 210(250,000) – [1,600,000 + 155(250,000)]

= 52,500,000 – 40,350,000 = 12,150,000

Next, compute the payback period for each alternative.

PBS = (Initial cost)/(Annual profit)

= (68,000,000)/(17,200,000)

= 3.95 years

PBT = (Initial cost)/(Annual profit)

= (53,000,000)/(12,150,000)

= 4.36 years

b.What variable cost per product for Taiwan would make it equally attractive as Singapore?

Let ProfitT = ProfitS = 17,200,000 and solve for vT.

ProfitT = TRT – TCT = pT(Q) – [FCT + vT(Q)] = 17,200,000

= 210(250,000) – [1,600,000 + vT (250,000)] = 17,200,000

vT = [210(250,000) – 1,600,000 – 17,200,000]/250,000

= $134.80 per product

c.What other factors should be considered in the location decision?

Taxes, export tariffs and laws, labor laws, economic and political stability, availability of labor and raw materials, transportation and communications infrastructures, etc.