5-11(a) Approximate cash receipts = Yield per acre 1,000 Price per bushel
ACT
EventCornTomatoesBeetsAsparagusCauliflower
Dry$20,000$20,000$22,500$30,000$20,000
Moderate 35,000 40,000 30,000 25,000 20,000
Damp 40,000 20,000 45,000 20,000 20,000
(b)Inadmissible acts: cauliflower (dominated by every other act). The last column in the payoff table may be eliminated.
CornTomatoesBeetsAsparagus
Column minimum$20,000$20,000$22,500$20,000
The maximin payoff act is to plant beets.
Payoff Probability
EventProbabilityCornTomatoesBeetsAsparagus
Dry .3$ 6,000$ 6,000$ 6,750$ 9,000
Moderate .5 17,500 20,000 15,000 12,500
Damp .2 8,000 4,000 9,000 4,000
Expected payoff:$31,500$30,000$30,750$25,500
The act having the maximum expected payoff, $31,500, is to choose corn.
5-22See the following decision tree. The optimal course of action is to first test market. Then if the
results are favorable, market; but if they are unfavorable, abandon.
6-1(a)$1,000(1/200) + $0(199/200) = $5
(b)Since her insurance payment provides a payoff of $10, Shirley maximizes expected payoff
($5.00) by not buying the insurance.
(c)Her certainty equivalent for the stolen car risk is $10. Since her net worth would change by
$9 by buying the insurance and since $9 > $10, Shirley would buy.
6-2(a)(b)(c)(d)
(1) Expected payoff$35,000$19,300$2,500$ 0
(2) Risk premium 4,000 1,000 200 20,000
(3) Certainty equivalent 31,000 18,300 2,30020,000
6-5(a) Certainty equivalent = $10
(b) Risk premium = $5 ($10) = $5
Exercises for Chp. 7
7-6(a) See the following.
(b)
= 0.20MSE = 954.4516
PeriodActualForecast Error
tYear Yt FtYt Ft
11990205.00
21991206.00205.00 1.00
31992223.00205.2017.80
41993231.00208.7622.24
51994234.00213.2120.79
61995241.00217.3723.63
71996267.00222.0944.91
81997268.00231.0736.93
91998277.00238.4638.54
101999290.00246.1743.83
112000254.93
(c) = 0.30 = 0.20MSE = 297.1306
PeriodActualTrendSlopeForecast Error
tYear Yt Tt bt FtYt Ft
11990205.00
21991206.00205.001.00
31992223.00211.102.02206.0011.01
41993231.00218.483.09213.1217.88
51994234.00225.303.84221.5812.42
61995241.00232.704.55229.1411.86
71996267.00246.176.33237.2529.75
81997268.00257.167.26252.5115.49
91998277.00268.198.02264.4212.58
101999290.00280.358.85276.2113.79
112000289.20
(d)The twoparameter exponential smoothing is better because (1) it reflects trend and (2) it has the smaller MSE.
7-7 (a) See the following figure.
(b) = 0.30 = 0.20 = 0.40MSE = 14.48276
PeriodActualTrendSlope SeasonalForecast Error
tQuarter Yt Tt bt St Ft Yt Ft
11995W 3.90
2S 6.10 3.902.201.56
3S 4.30 5.562.090.77
4F10.80 8.602.281.26
51996W 7.80 9.952.100.78
6S10.6010.471.781.3418.858.25
7S 6.9011.251.580.71 9.472.57
8F13.5012.211.461.2016.122.62
91997W12.9014.501.620.8310.70 2.20
10S15.2014.681.331.2221.666.46
11S10.3015.571.240.6911.361.06
12F18.7016.461.171.1720.111.41
131998W13.9017.391.130.8214.560.66
14S14.4016.500.721.0822.598.19
15S10.2016.490.580.6611.891.69
16F17.3016.370.441.1320.012.71
171999W13.5016.740.420.8113.710.21
18S18.2017.060.401.0818.550.35
19S14.2018.660.640.7011.55 2.65
20F20.7019.030.591.1121.741.04
(c) 2000W 15.93
S 21.73
S 14.58
F 23.75
7-11(a) See the following figure.
(b)
X Y XY X2
14 68 952 196
23105 2,415 529
9 40 360 81
17 79 1,343 289
10 81 810 100
22 95 2,090 484
5 31 155 25
12 72 864 144
6 45 270 36
16 93 1,488 256
13470910,7472,140
=X=Y=XY= X2
a = 70.9 3.619(13.4) = 22.405
c) extra question. Consider the excel report given below.
SUMMARY OUTPUTRegression Statistics
Multiple R / 0,895347
R Square / 0,801646
Adjusted R Square / 0,776852
Standard Error / 11,81164
Observations / 10
ANOVA
df / SS / MS / F / Significance F
Regression / 1 / 4510,781 / 4510,781 / 32,3319 / 0,000462
Residual / 8 / 1116,119 / 139,5149
Total / 9 / 5626,9
Coefficients / Standard Error / t Stat / P-value / Lower 95% / Upper 95% / Lower 95,0% / Upper 95,0%
Intercept / 22,40476 / 9,31077 / 2,406328 / 0,042752 / 0,934089 / 43,87544 / 0,934089 / 43,87544
X Variable 1 / 3,619048 / 0,636471 / 5,686115 / 0,000462 / 2,151343 / 5,086753 / 2,151343 / 5,086753
-What can you say about the performance of regression?
Answer: r-square is 0.80, close to 1, showing that there is a significant linear relation between X and Y. This regression seems to be appropriate.
-Is the trend significant?
In the ANOVA table F-significance is 0.000462, almost 0. Since it is less than 5%, at a 95% confidence level, we are quite sure that there is a significant trend.
7-17(a)
(1)(2)(3) (4)(5) (6)
Sum ofPercentage
12Month 12Month Moving of Moving
MonthCashTotal Totals AverageAverage
1995J 2.7
F 5.4
M 9.3
A 2.4
M 6.1
J 7.3 81.9
J 6.5 82.1 164.06.8 95.6
A 9.7 83.1 165.26.9140.6
S13.4 83.9 167.07.0191.4
O10.6 85.6 169.57.1149.3
N 5.1 87.3 172.97.2 70.8
D 3.4 87.4 174.77.3 46.6
1996J 2.9 86.4 173.87.2 40.3
F 6.4 86.3 172.77.2 88.9
M10.1 86.4 172.77.2140.3
A 4.1 86.5 172.97.2 56.9
M 7.8 86.2 1.72.77.2108.3
J 7.4 85.6 171.87.2102.8
J 5.5 86.2 171.87.2 76.4
A 9.6 87.1 173.37.2133.3
S13.5 88.3 175.47.3184.9
O10.7 88.0 176.37.3146.6
N 4.8 88.3 176.37.3 65.8
D 2.8 87.8 176.17.3 38.4
1997J 3.5 88.8 176.67.4 47.3
F 7.3 88.1 176.97.4 98.6
M11.3 88.9 177.07.4152.7
A 3.8 89.7 178.67.4 51.4
M 8.1 91.4 181.17.5108.0
J 6.9 92.5 183.97.7 89.6
J 6.5 91.5 184.07.7 84.4
A 8.9 92.3 183.87.7115.6
S14.3 88.9 181.27.6188.2
O11.5 90.3 179.27.5153.3
N 6.5 91.4 181.77.6 85.5
D 3.9 92.6 184.07.7 50.6
1998J 2.5 93.7 186.37.8 32.1
F 8.1 94.1 187.87.8103.8
M 7.9 95.6 189.77.9100.0
A 5.2 96.7 192.38.0 65.0
M 9.2 97.4 194.18.1113.6
J 8.1 97.6 195.08.1100.0
J 7.6 99.3 196.98.2 92.7
A 9.3100.8 200.18.3112.0
S15.8105.3 206.18.6183.7
O12.6106.3 211.68.8143.2
N 7.2105.8 212.18.8 81.8
D 4.1106.0 211.88.8 46.6
(1)(2)(3) (4)(5) (6)
Sum ofPercentage
12Month 12Month Moving of Moving
MonthCashTotal Totals AverageAverage
1999J 4.2105.0 211.08.8 47.7
F 9.6105.5 210.58.8109.1
M12.4106.0 211.58.8140.9
A 6.2106.8 212.88.9 69.7
M 8.7106.5 213.38.9 97.8
J 8.3108.5 215.09.0 92.2
J 6.6
A 9.8
S16.3
O13.4
N 6.9
D 6.1
(b)
Percentages of Moving Averages
Seasonal Cash
19951996199719981999Median IndexRequired
J 40.3 47.3 32.1 47.7 43.80 43.0$ 430,000
F 88.9 98.6103.8109.1101.20 99.4 994,000
M140.3152.7100.0140.9140.60138.21,382,000
A 56.9 51.4 65.0 69.7 60.95 59.9 599,000
M108.3108.0113.6 97.8108.15106.31,063,000
J102.8 89.6100.0 92.2 96.10 94.4 944,000
J 95.6 76.4 84.4 92.7 88.55 87.0 870,000
A140.6133.3115.6112.0124.45122.31,223,000
S191.4184.9188.2183.7186.55183.31,833,000
O149.3146.6153.3143.2147.95145.41,454,000
N 70.8 65.8 85.5 81.8 76.30 75.0 750,000
D 46.6 38.4 50.6 46.6 46.60 45.8 458,000
1,221.20 1,200.0
Seasonal index = (1 ,200/1,221.20) Median
92 Letting XA = number of model A's to be made
XB = number of model B's to be made
XC = number of model C's to be made
XD = number of model D's to be made
Maximize P = 2.00XA + 1.50XB + 2.00XC + 2.50XD
Subject to 5XA + 6XB + 6XC + 7XD 100(plastic)
10XA + 12XB + 15XC + 15XD 500(beads)
4XA + 5XB + 5XC + 0XD 600(nylon)
0XA + 2XB + 3XC + 4XD 200(teflon)
.5XA + .4XB + .5XC + .8XD 300(labor)
XA XBXC XD0 (mixture)
XB 50(quantity B)
XD 20 (quantity D)
where XA, XB, XC, XD 0
96(a) LettingXA = dollar investment in bond A
XB = dollar investment in bond B
XC = dollar investment in bond C
XD = dollar investment in bond D
XE = dollar investment in bond E
Maximize P = .08XA + .09XB + .09XC + .l0XD + .09XE
(b)(1) XA + XB + XC + XD + XE = 100,000
(2) XB + XD + XE 50,000
(3)XA + XC + XE 30,000
(4)XA + XE 25,000
(5) XA + XB XC + XD 0 or XB + XD XA + XC
(6) .06XA + .0225XB + .0225XC + .025XD .0675XE 0
or .08XA + .09XE ¼(.08XA + .09XB + .09XC + .10XD + .09XE)
97Letting XH = pounds of hog bellies
XP = pounds of pork
XT = pounds of tripe
XC = pounds of chicken
XB = pounds of beef
Minimize C = .30XH + .20XT + .70XB + .60XP + .45XC
Subject to XH + XT .10(restriction)
XC .25(chicken)
XB .30 (beef)
3XH + 5XT + 4XB + 3XP + 3XC 3(protein)
5XH + 3XT + 2XB + 4XP + 3XC 4(fat)
6XH + 4XT + 5XB + 9XP + 4XC 8(water)
XH + XT + XB + XP + XC= 1 (total weight)
where all Xs 0
915LettingXij = fraction of time that type i persons are assigned to job j (if fractional assignment is not possible, all Xij must be {0,1} binary variables)
i = C, T, or S
j = F, B, or R
MinimizeC = 20XCF + 25XCB + 35XCR + 25XTF + 20XTB + 30XTR + 30XSF + 25XSB + 25XSR
Subject to
(clerk assmnt.) XCF + XCB + XCR= 1
(typist assmnt.) XTF + XTB + XTR= 1
(steno. assmnt.) XSF + XSB + XSR= 1
(fil. assmnt.) XCF + XTF + XSF = 1
(book. assmnt.) XCB + XTB + XSB = 1
(rep. assmnt.) XCR + XTR + XSR = 1
where all Xs 0
918 Letting Xij = quantity shipped from plant i to warehouse j
i = A or B
j = C, D, or E
Minimize C = 11XAC + 12XAD + 13XAE + 13XBC + 12XBD + 13XBE
Subject to XAC + XAD + XAE = 1,000(A capacity)
XBC + XBD + XBE= 500(B capacity)
XAC + XBC= 500(C demand)
XAD + XBD= 500(D demand)
XAE + XBE= 500(E demand)
where all Xs 0
10-17
Adjustable CellsFinal / Reduced / Objective / Allowable / Allowable
Cell / Name / Value / Cost / Coefficient / Increase / Decrease
$C$9 / Xt / 9.142857143 / 0 / 20 / 30 / 5
$D$9 / Xc / 2.857142857 / 0 / 15 / 5 / 9
$E$9 / Xb / 0 / -5 / 15 / 5 / 1E+30
Constraints
Final / Shadow / Constraint / Allowable / Allowable
Cell / Name / Value / Price / R.H. Side / Increase / Decrease
$I$5 / wood / 100 / 0.714285714 / 100 / 20 / 64
$I$6 / labor / 60 / 2.571428571 / 60 / 106.6666667 / 10
(a)$ 5/7 = $ .714 for wood
$18/7 = $2.571 for labor
(b)You may ignore this question.
(c) The value of the resources used in making one desk is
30(5/7) + 10(18/7) = 330/7 = $47.14
Since this is smaller than the unit profit of $50, desks should be made.
10-36(a) The problem is formulated in the text. The solution is:
X1 = 300X6 = 0
X2 = 300X7 = 500
X3 = 300X8 = 300
X4 = X9 = 200
X5 = 0X10 = 0
P = 4,810
(b)stated in the output report (1) (800, 1,066.67)(2) (8, 100, )(3) (1,900, )(4) (887, )
(c)stated in the output report (1) (1.19, 2.01)(2) (2.49, 3.16)(3) (.94, 1.76)(4) (, 3.65)
stated in the output report (5) (, 3.45)(6) (, 3.29)(7) 2.40, 2.91)(8) (2.60, 3.16)
(9) (2.70, 3.71)(10) (, 3.29)
(d)UM = 3.85UGA = .183
UST = 0UP1 = .452
UMO = 0UP2 = .295
UC = .002UOL = .405
USA = 0
U1 = 0U6 = .44
U2 = 0U7 = 0
U3 = 0U8 = 0THIS BLUE PART IS NOT COVERED IN THE LECTURE, JUST SKIP!
U4 = 2.15U9 = 0
U5 = 1.40U10 = 1.34
(e) Um=3.85 is greatest so, the greatest increase in total profit would occur if the available gallons of milk could be increased.
10-31Primal
Letting Xi = number of ads to be placed in publication i
Maximize P =
1.5X1 + 1.4X2 + .7X3 + .8X4 + 1.1X5 + 1.5X6 + .3X7 + .9X8 + 2.1X9 + 1.9X10 + 1.4X11 + 1.6X12 + 2.0X13 + 1.6X14 + .9X15
Subject to
(budget) (UB)2.0X1 + 2.5X2 + 1.5X3 + 1.0X4 + 2.0X5 + 2.0X6 + .8X7 + 1.2X8 + 3.0X9 + 1.8X10 + 1.5X11 + 1.5X12 + 3.0X13 + 2.0X14 + .8X15 100,000
(age mix) (UM1) 1X1 + 1X2 1X3 1X4 + 1X5 + 0X6 1X7 + 0X8 1X9 1X10 + 1X11 + 1X12 + 1X13 + 0X14 1X15 0
(gen. mix) (UM2).5X1 .5X2 .5X3 .5X4 .5X5 + 1X6 .5X7 + 1X8 .5X9 .5X10 .5X11 .5X12 .5X13 + 1X14 .5X15 0
(teen lim.) (UT)2.0X1 + 2.5X2 + 0X3 + 0X4 + 2.0X5 + 0X6 + 0X7 + 0X8 + 0X9 + 0X10 + 1.5X11+ 1.5X12 + 3.0X13 + 0X14 + 0X15 40,000
(pub. 1) (UM1) 1X1 10
(pub. 2) (UM2) 1X2 2
(pub. 3) (UM3) 1X3 10
(pub. 4) (UM4) 1X4 3
(pub. 5) (UM5) 1X5 2
(pub. 6) (UM6) 1X6 5
(pub. 7) (UM7) 1X7 4
(pub. 8) (UM8) 1X8 6
(pub. 9) (UM9) 1X9 7
(pub. 10) (UM10) 1X10 5
(pub. 11) (UM11) 1X11 5
(pub. 12) (UM12) 1X12 4
(pub. 13) (UM13) 1X13 2
(pub. 14) (UM14)1X14 2
(pub. 15) (UM15) 1X15 3
where all X’s 0
Dual
LettingUB= marginal value per dollar of budget
UM1= marginal value for teenyouth mix
UM2= marginal value for general mix
UT= marginal value per dollar of teen limit
Ui = marginal value per ad limit for publication i
(a) dual formulation was not covered in the lecture, just skip.
(b)as stated in the solution
UB = 0U1 = 1.5U5 = 1.1U9 = 2.1U13 = 2
UM1= 0U2 = 1.4U6 = 1.5U10 = 1.9U14 =1.6
UM2 = 0U3 = .7U7 = .3U11 = 1.4U15 = .9
UT = 0U4 = .8U8 = .9U12 = 1.6
C=91
(c)$0
(d)$0
11-9(a) 5 ⅓ regulars, 6 ⅔ deluxes, P = 153 ⅓
(b) 0 regulars, 10 deluxes, R = 600
11-10Let XR, XD denote the number of regular and deluxe models, respectively. The subscripts R, B, P, and M relate to the (1) revenue, (2) budget, (3) profit, and (4) mixture goals.
(a)Minimize C = 0XR + 0XD + 0YR+ + 1YR + 1YB+ + 0YB + 0YP+ + 1YP + 0YM+ + 2YM
Subject to:
(labor): 5XR + 8XD 80
(frames): 1XR + 1XD 12
GI (revenue): 30XR + 60XD (YR+ YR) = 500
G2 (budget):20XR + 45XD (YB+ YB) = 400
G3 (profit):10XR + 15XD (YP+ YP) = 140
G4 (mixture: 1XR 1XD (YM+ YM) = 0
where all variables 0
(b) when XR = 4 and XD = 7,
YR+ YR = 30(4) + 60(7) 500 = 40, YR+ = 40 and YR = 0
YB+ YB = 20(4) + 45(7) 400 = 5, YB+ = 0 and YB = 5
YP+ YP = 10(4) + 15(7) 140 = + 5, YP+ = 5 and YP = 0
YM+ YM =1(4) 1(7) 0 = 3, YM+ = 0 and YM = 3
C = 0(4) + 0(7) + 0(40) + 1(0) + 1(0) + 0(5) + 0(5) + 1(0) + 0(0) + 2(3) = 6