1-16 $\text{BEP}=5$ units

1-18 $\text{BEP}=700$ units

1-20 $\text{BEP}=160$ per week; $\text{total} \text{revenue}=\text{\$}6,400$

1-22 $s=45$

1-24

(a) $\text{BEP}=25$ (b) $\text{BEP}=18.333$

2-14 0.30

2-16

(a) 0.10 (b) 0.04 (c) 0.25 (d) 0.40

2-18

(a) 0.20 (b) 0.09 (c) 0.31 (d) Dependent

2-20

(a) 0.3 (b) 0.3 (c) 0.8 (d) 0.49 (e) 0.24 (f) 0.27

2-22 0.719

2-24

(a) 0.08 (b) 0.84 (c) 0.44 (d) 0.92

2-26

(a) 0.995 (b) 0.885 (c) Assumed events are independent.

2-28 0.78

2-30 2.85

2-32

(a) 0.1172 (b) 0.0439 (c) 0.0098 (d) 0.0010 (e) 0.1719

2-34 0.328, 0.590

2-36 0.776

2-38

(a) 0.0548 (b) 0.6554 (c) 0.6554 (d) 0.2119

2-40 18,29.27

2-42

(a) 0.5 (b) 0.27425 (c) 48.2

2-44 0.7365

2-46 0.162

3-18 Maximin criterion; Texan

3-20

(a) Stock market (b) $21,500

3-22 (b) CD

3-24 (b) Medium

3-26 8 cases

3-30 (b) Very large

(c) Small (d) Very large (e) Very large

3-32 Minimax regret decision: option 2; minimum EOL decision: option 2

3-34 $\mathrm{-}$$0.526

3-36 Construct the clinic (EMV = 30,000).

3-40 Do not gather information; build quadplex.

3-42 0.533; 0.109

3-44 (c) Conduct the survey. If the survey is favorable, ­produce razor; if not favorable, don’t produce razor.

3-46

(a) 0.923, 0.077, 0.25, 0.75 (b) 0.949, 0.051, 0.341, 0.659

3-48 Do not use study. Risk avoiders.

3-50

(a) Broad (b) Expressway (c) Risk avoider

4-10

(b) SST = 29.5. SSE = 12. SSR = 17.5. $\widehat{Y}=6+0.1X$ (c) $\widehat{Y}=10$

4-12 (a) $\widehat{Y}=6+0.1X$

4-16

(a) $149,542 (b) The model predicts the average price for a house this size. (c) Age, number of bedrooms, lot size (d) 0.3969

4-18 For $X=1200,$ $\widehat{Y}=2.35$; for $X=2400,$ $\widehat{Y}=3.67$

4-22 The model with just age is best because it has the highest $r^2\left(0.70\right).$

4-24 $\widehat{Y}=91,446.49+29.86X_1+2,116.86X_2-1,504.77X_3=142,\mathrm{465.16.}$

4-26 The model with horsepower is $\widehat{Y}=53.87-0.269X_1; r^2=0.77$.

The model with weight is $\widehat{Y}=57.53-0.01X_2;$ $r^2=\mathrm{0.73.}$

4-30 $\widehat{Y}=77.190+0.047X$

4-32

(a) $\widehat{Y}=42.43+0.0004X$ (b) $\widehat{Y}=-31.54+0.0058X$

5-16 $\text{MAD}=6.48$ for 3-month moving average; $\text{MAD}=7.78$ for 4-month moving average

5-18 $Y=2.22+1.05X$

5-20 Forecast for year 12 is 11.789; $\text{MAD}=2.437$.

5-23 Forecasts for year 6 are 565.6 and 581.4.

5-25 Forecast for year 6 is 555.

5-27 $\text{MAD}=5.6$ for trend line; $\text{MAD}=74.56$ for expo­nential smoothing; $\text{MAD}=67$ for moving average

5-29 (b) $\text{MAD}=2.60; \text{RSFE}=5.11$ at week 10

5-34 $\text{MAD}=3.34$

5-38 $F_{11}=6.26; \text{MAD}=0.58 \text{for} \alpha =0.8$ is lowest.

5-40 270, 390, 189, 351

6-20

(a) 20,000 (b) 50 (c) 50

6-22 $45 more; $\text{ROP}=4,000$

6-24 8 million

6-26 28,284; 34,641; 40,000

6-28

(a) 10 (b) 324.92 (c) 6.5 days; 65 units (d) $\text{Maximum}=259.94$, $\text{average}=129.97$ (e) 7.694 runs; (f) $192.35; $37,384.71 (g) 5

6-30

(a) $Z=2.05$ (b) 3.075 (c) 23.075 (d) $4.61

6-32 Add 160 feet, $1,920

6-34 2,697

6-36 Take the discount. $\text{Cost}=\$49,912.50$.

6-38 $4.50; $6.00; $7.50

6-46 Item 4 $\text{EOQ}=45$

6-48 Order 200 units.

7-14 40 air conditioners, 60 fans, $\text{profit}=\$1,900$

7-16 175 radio ads, 10 TV ads

7-18 40 undergraduate, 20 graduate, $160,000

7-20 $20,000 petrochemical; $30,000 utility; $\text{return}=\$4,200; \text{risk}=6$

7-22 $X=18.75, Y=18.75, \text{profit}=\$150$

7-24 (1358.7, 1820.6), $3,179.30

7-26

(a) $\text{Profit}=\$2,375$ (b) 25 barrels pruned, 62.5 barrels regular (c) 25 acres pruned, 125 acres regular

7-28

(a) Yes (b) Doesn’t change

7-34

(a) 25 units product 1, 0 units product 2 (b) 25 units of resource 1 are used, slack = 20; 75 units of resource 2 are used, slack = 12; 50 units of resource 3 are used, slack = 0; constraint 3 is binding and others are not. (c) 0, 0, and 25 (d) Resource 3; up to $25 (dual price) (e) Total profit would decrease by 5 (value of reduced cost).

7-36 24 coconuts, 12 skins; $\text{profit}=5,040$ rupees

7-44 Use 7.5 lb of C-30, 15 lb of C-92, 0 lb of D-21, and 27.5 lb of E-11; $\text{cost}=\$3.35$ per lb

8-2 (b) $50,000 in LA bonds, $175,000 in Palmer Drugs, $25,000 in Happy Days

8-4 1.33 pounds of oat product per horse, 0 pounds of grain, 3.33 pounds of mineral product

8-6 6.875 TV ads; 10 radio ads; 9 billboard ads; 10 news­paper ads

8-8 Use 30 of the 5-month leases starting in March, 100 5-month leases starting in April, 170 5-month leases starting in May, 160 5-month leases starting in June, and 10 5-month leases starting in July.

8-10 Send 400 students from sector A to school B, 300 from A to school E, 500 from B to school B, 100 from C to school C, 800 from D to school C, and 400 from E to school E.

8-12 (b) 0.499 lb of ground meat, 0.173 lb of chicken, 0.105 lb of spinach, and 0.762 lb of potatoes; total $\text{cost}=\$1.75$

8-14 13.7 trainees begin in August, and 72.2 begin in October.

9-8 Des Moines to Albuquerque 200, Des Moines to Boston 50, Des Moines to Cleveland 50, Evansville to Boston 150, Ft. Lauderdale to Cleveland 250; $\text{cost}=\$3,200$

9-10 5 units from Pineville to 3, 30 units from Oak Ridge to 2, 10 units from Oakville to 3, 30 units from Mapletown to 1; $\text{cost}=\$230$; multiple optimal solutions

9-12 $\text{Total} \text{cost}=\$14,700$

9-18 New Orleans’ $\text{cost} \text{is} \$20,000;$ Houston’s is $19,500, so Houston should be selected.

9-20 Optimal cost using East St. Louis: $17,400; optimal cost using St. Louis: $17,250.

9-22 Job A12 to machine W, Job A15 to machine Z, Job B2 to machine Y, Job B9 to machine X; $\text{time}=50$ hours

9-24 4,580 miles

9-26 $\text{Total} \text{rating}=335$

9-28 $\text{Maximum} \text{total} \text{rating}=75.5$

9-30

(a) $2,591,200 (b) $2,640,400 (c) $2,610,100 and $2,572,100

9-34 200 on path 1–2–5–7–8, 200 on path 1–3–6–8, and 100 on path 1–4–8; = 500. This increases to 700 with the changes.

9-36 $\text{Total} \text{flow}=13$

9-38 Shortest path is 1–3–7–11–14–16. Total distance is 74.

9-40

(a) 1,200 miles (b) 1,000 miles

9-42 $\text{Total} \text{distance}=40$

10-10 (a) 2 prime-time ads per week, 4.25 off-peak ads per week, audience = 38,075 (b) 2 prime-time ads per week, 4 off-peak ads per week, $\text{audience}=36,800$ (c) 4 prime-time ads per week, 1 off-peak ad per week, $\text{audience}=37,900$

10-12 3 large posters and 4 small posters

10-16 (b) Build at Mt. Auburn, Mt. Adams, Norwood, Covington, and Eden Park.

10-18 (a) $X_1\ge X_2$ (b) $X_1+X_2+X_3=2$

10-20 (b) 0 TV, 0.73 radio, 0 billboard, and 88.86 newspaper ads

10-24 $X_1=15,$ $X_2=20$

10-28 (b) 18.3 XJ6 and 10.8 XJ8; $\text{revenue}=70,420$

10-30 0.333 in stock 1 and 0.667 in stock 2; $\text{variance}=0.102;$ $\text{return}=0.09$

11-22 (a) 0.50 (b) 0.50 (c) 0.97725 (d) 0.02275 (e) 43.84

11-24

(a) 0.0228 (b) 0.3085 (c) 0.8413 (e) 0.9772

11-28 14

11-32

(b) Critical path A–C takes 20 weeks; path B–D takes 18 weeks (c) 0.222 for A–C; 5 for B–D (d) 1.00 (e) 0.963 (f) Path B–D has more variability and has a higher probability of exceeding 22 weeks.

11-38 Yes. The cost to crash is $2,200.

12-10 Total costs for 1, 2, 3, and 4 clerks are $564, $428, $392, and $406, respectively.

12-12 (a) 4.167 cars (b) 0.4167 hours (c) 0.5 hours (d) 0.8333 (e) 0.1667

12-14 (a) 0.512, 0.410, 0.328 (b) 0.2 (c) 0.8 minutes (d) 3.2 (e) 4 (f) 0.429, 0.038 minutes, 0.15, 0.95

12-16 (a) 0.2687 hours (b) 3.2 (c) Yes; $\text{savings}=\$142.50$ per hour.

12-18 (a) 0.0397 hours (b) 0.9524 (c) 0.006 hours (d) 0.1524 (e) 0.4286 (f) 0.4 (g) 0.137

12-20 With one person, $L=3, W=1 \text{hour}, L_q=2.25,$ and $W_q=0.75$ hour. With two people, $L=0.6, W=0.2$ hour, $L_q=2.225,$ and $W_q=0.075$ hour.

12-22 No

12-24 (a) 0.1333 hours (b) 1.333 (c) 0.2 hour (d) 2 (e) 0.333

12-26 (a) 80 customers per day (b) 10.66 hours, $266.50 (c) 0.664 hours, $16.60 (d) 2 tellers, $208.60

12-30 (a) 0.576 (b) 1.24 (c) 0.344 (d) 0.217 hours (e) 0.467 hours

12-32 (a) 0.27 hours (b) 3.2 (c) 4 (d) 0.33 hours (e) 0.4096

13-14 No

13-16 Expected value is 6.35 (using formula). The average number is 7 in Problem 13-15.

13-18 (b) Average number $\text{delayed}=0.40$; average number of $\text{arrivals}=2.07$

13-26 (a) Cost/hour is generally more expensive replacing 1 pen each time. (b) Expected cost/hour with 1 pen $\text{policy}=\$1.38$ (or $58/breakdown); expected cost/hour with 4-pen $\text{policy}=\$1.12$ (or $132/breakdown)

14-8 (b) 90% (c) 30%

14-10 Next month, 4/15, 5/15, 6/15; three months, 0.1952, 0.3252, 0.4796

14-12 (a) 70% (b) 30% (c) 40%

14-14 25% for Battle’s; 18.75% for University; 26.25% for Bill’s; 30% for College

14-16 111 at Northside, 75 at West End, and 54 at Suburban

14-18 Horizon will have 104,000 customers, and Local will have 76,000.

14-20 New MFA = (5,645.16, 1,354.84)

14-22 50% Hicourt, 30% Printing House, and 20% Gandy

14-26 Store 1 will have one-third of the customers, and store 2 will have two-thirds.

15-8 45.034 to 46.966 for $\overline{x}$, 0 to 4.008 for R

15-10 16.814 to 17.187 for $\overline{x}$, 0.068 to 0.932 for R

15-12 2.236 to 3.728 for $\overline{x}$, 0 to 2.336 for R, in control

15-16 (a) 1011.8 for $\overline{x}$ and 96.3 for R (b) 956.23 to 1067.37 (c) Process is out of control.

15-18 $\text{LCL}=0, \text{UCL}=4$

M1-4 $\text{Sun}- 0.80$

M1-6 $\text{Lambda}=3.0445,$ $\text{CI}=0.0223\text{,} \text{RI}=0.58\text{,} \text{CR}=0.0384$

M1-8 Car 1, 0.4045

M1-10 University B has highest weighted average, 0.4995.

M2-6 1–2–6–7, with a total distance of 10 miles

M2-8 Shortest route is 1–2–6–7, with a total distance of 14 miles.

M2-10 Shortest route is 1–2–5–8–9, with a distance of 19 miles.

M2-12 4 units of item 1, 1 unit of item 2, and no units of items 3 and 4

M2-14 Ship 6 units of item 1, 1 unit of item 2, and 1 unit of item 3.

M2-16 The shortest route is 1–3–6–11–15–17–19–20.

M3-6 (a) $\text{OL}=\$8\left(20,000-X\right)$ for $X\le 20,000; \text{OL}=0$ otherwise (b) $0.5716 (c) $0.5716 (d) 99.99% (e) Print the book.

M3-8 (a) $\text{BEP}=1,500$ (b) $\text{Expected} \text{profit}=\$8,000$

M3-10 (a) $\text{OL}=\text{\$}10\left(30-X\right)$ for $X\le 30; \text{OL}=0$ otherwise (b) $\text{EOL}=\$59.34$ (c) $\text{EVPI}=\$59.34$

M3-12 (a) Use new process. New $\text{EMV}=\$283,000.$ (b) Increase selling price. New $\text{EMV}=\$296,000.$

M3-14 $\text{BEP}=4,955$

M3-16$\text{EVPI}=\$249.96$

M3-18$\text{EVPI}=\$51.24$

M4-8 Strategy for $X:X_2;$ strategy for $Y:Y_1;$ value of the $\text{game}=6$

M4-10 $X_1{=}^{35}{/}_{57}; X_2{=}^{22}{/}_{57}; Y_1{=}^{32}{/}_{57}; Y_2{=}^{25}{/}_{57};$ value of $\text{game}=66.70$

M4-12 (b) $\text{Q}=41/72,$ $1-Q=31/72,$ $P=55/72,$ $1-P=17/72$

M4-14 Value of $\text{game}=9.33$

M4-16 Saddle point exists. Shoe Town should invest $15,000 in advertising and Fancy Foot should invest $20,000 in advertising.

M4-18 Eliminate dominated strategy $X_2.$ Then $Y_3$ is dominated and may be eliminated. The value of the game is 6.

M4-20 Always play strategy $A_{14}.$ $3 million.

M5-8 $X{=}^{-3}{/}_2; Y{=}^1{/}_2; Z{=}^7{/}_2$

M5-16 $\left(\begin{array}{lll}-48/60\hfill & 6/60\hfill & 32/60\hfill \\ 6/60\hfill & -12/60\hfill & 6/60\hfill \\ 12/60\hfill & 6/60\hfill & -8/60\hfill \end{array}\right)\begin{array}{ccc}& & \\ & & \\ & & \end{array}$

M5-18 $0X_1+4X_2+3X_3=28; 1X_1+2X_2+2X_3=16$

M6-6 (a) $Y^n=12X-6$ (b) $Y^n=80X^3+12X$ (c) $Y^n=6/X^4$ (d) $Y^n=500/X^6$

M6-8 (a) $Y^n=30X^4-1$ (b) $Y^n=60X^2+24$ (c) $Y^n=24/X^5$ (d) $Y^n=250/X^6$

M6-10$X=5$ is point of inflection.

M6-12 $Q=2,400, TR=1,440,000$

M6-14 $P=5.48$

M7-18 (b) $14X_1+4X_2\le 3,360;10X_1+12X_2\le 9,600$ (d) $S_1=3,360,S_2=9,600$ (e) $X_2$ (f) $S_2$ (g) 800 units of $X_2$ (h) 1,200,000

M7-20 $X_1=2,$ $X_2=6,$ $S_1=0,$ $S_2=0,$ $P=\$36$

M7-22$X_1=14,$ $X_2=33,$ $C=\$221$

M7-24 Unbounded

M7-26 Degeneracy; $X_1=27,$ $X_2=5,$ $X_3=0,$ $P=\$177$

M7-28 (a) Min. $C=9X_1+15X_2$

(b) $X_1=0, X_2=20, C=\$300$

M7-30 8 coffee tables, 2 bookcases, $\text{profit}=96$

M7-34 (a) 7.5 to infinity (b) Negative infinity to $40 (c) $20 (d) $0

M7-36 (a) 18 Model 102, 4 Model H23 (b) $S_1=\text{slack time for soldering,}$ $S_2=\text{slack time for inspection}$ (c) Yes—shadow price is $4. (d) No—shadow price is less than $1.75.

M7-38 (a) Negative infinity to $6 for phosphate, $5 to infinity for potassium (b) Basis won’t change, but $X_1,$ $X_2,$ and $S_2$ will change.

M7-40 $\text{Max} P=50 U_1+4U_2$ $12U_1+1U_2\le 120$ $20U_1+3U_2\le 250$

M8-16 Total cost is $230.

M8-18 Total cost is $14,700.

M8-20 Unbalanced; total $\text{cost}=\$5,310$

M8-22 Total $\text{cost}=\$28,300$

M8-24 Job A12 to machine W, Job A15 to machine Z, Job B2 to machine Y, Job B9 to machine X; $\text{time}=50 \text{hours}$

M8-26 4,580 miles and 6,040 miles, respectively.

M8-28 $\text{Total}=86$

M8-32 $\text{Total} \text{cars} \text{per} \text{hour}=5$

M8-34 $\text{Total} \text{distance}=430$

M8-41 (a) The shortest path is 1–3–7–11–14–16 with a distance of 74. (b) The distance increases by 2.

M8-43 (a) 1,200 miles (b) 1,000 miles