Introduction to Management Science Solutions

Posted on Oct 16, 2024 in Mathematics

Chapter 1

11) Demand and Total Revenue

A)

• D = 800 – 10(20) = 600
• D = 800 – 10(70) = 100

B)

• D = 800 – 10(26) = 540
• D = 800 – 10(27) = 260
• D = 800 – 10(68) = 120
• D = 800 – 10(69) = 110

C)

• Tr = d * p
• Tr = 800 – 10(p)

D)

• 800 – 10p^2
• 800(30) – 10(30)^2 = 15000
• 800(40) – 10(40)^2 = 16000
• 800(50) – 10(50)^2 = 15000

E)

• Tr/p = d * p/p
• D = tr/p
• D = 16000/40 = 400 Units

13) Micromedia Offers

A) 9600 + 120x

B) P = 600x – (9600 + 120x) = 480x – 9600

C) 480(30) – 9600 = 4800

D) P = 480x – 9600 = 0

X = 20 students

15) Preliminary Break-Even Analysis

tr = d * p

B = r – (fc + vc)

Tr = 300000x

A) 300000 – (4500000 + 150000x) = 30 boxes = b

B) 150000(50) – 4500000 = $3000000

50 Boxes

Chapter 2

2) Coffee Houses

Data not true to be equally likely. More preference for others.

16) Probability Calculations

A. 55 * .35 = .19

B. .55 + .35 – .19 = .71

C. 1 – .71 = .29

17) Conditional Probability

A) 196/989 = 19%

B. 177/196 = 90%

C. 177/989 = 17.8%

D. 177/759 = 23%

E. (759/989) + (196/989) – (177/989) = 78%

Chapter 4

4) Amy’s Leasing Decision

A) Lease with the lowest cost chance # of miles she’ll drive.

B)

• Hepburn Honda:
  • 36000 Miles (12000 Miles for 3 Years): 36($299) + $0.15(36000 – 36000) = $10764
  • 45000 Miles (15000 Miles for 3 Years): 36($299) + $0.15(45000 – 36000) = $12114
  • 54000 Miles (18000 Miles for 3 Years): 36($299) + $0.15(54000 – 36000) = $13464
• Midtown Motors:
  • 36000 Miles (12000 Miles for 3 Years): 36($310) + $0.20 * max(36000 – 450000) = $11160.00
  • 45000 Miles (15000 Miles for 3 Years): 36($310) + $0.20 * max(45000 – 450000) = $11160.00
  • 54000 Miles (18000 Miles for 3 Years): 36($310) + $0.20 * max(54000 – 450000) = $12960.00
• Hopkins Automotive Lease Option:
  • 36000 Miles (12000 Miles for 3 Years): 36($325) + $0.15 * max(36000 – 540000) = $11700
  • 45000 Miles (15000 Miles for 3 Years): 36($325) + $0.15 * max(45000 – 540000) = $11700
  • 54000 Miles (18000 Miles for 3 Years): 36($325) + $0.15 * max(54000 – 540000) = $11700

C)

• Optimistic: 10764 11160 11700
• Conservative: 13464 12960, 11700

D)

• Ev(Hepburn Honda) = 0.5($10764) + 0.4($12114) + 0.1($13464) = $11574
• Ev(Midtown Motors) = 0.5($11160) + 0.4($11160) + 0.1($12960) = $11340
• Ev(Hopkins Automotive) = 0.5($11700) + 0.4($11700) + 0.1($11700) = $11700

F)

• Ev(Hepburn Honda) = 0.3($10764) + 0.4($12114) + 0.3($13464) = $12114
• Ev(Midtown Motors) = 0.3($11160) + 0.4($11160) + 0.3($12960) = $11700
• Ev(Hopkins Automotive) = 0.3($11700) + 0.4($11700) + 0.3($11700) = $11700

13) Seneca Hill Winery

A.

• Decision: Type of grape to plant
• Chance: Level of demand
• Consequence: Low profit contribution
• Alternatives: Plant only ‘C’, only ‘R’, or both
• Chance Outcomes: (wW) (wS) (sW) & (sS)

C)

• Ev (plant Chardonnay) = 0.55(20) + 0.45(70) = 42.5
• Ev (plant Both Grapes) = 0.05(22) + 0.50(40) + 0.25(26) + 0.20(60) = 39.6
• Ev (plant Riesling) = 0.30(25) + 0.70(45) = 39.0
• Optimal Decision: Plant Chardonnay Only

D)

• Only Riesling is planted.
• Ev (plant Both Grapes) = 0.05(22) + 0.50(40) + 0.05(26) + 0.40(60) = 46.4
• Ev (plant Riesling) = 0.10(25) + 0.90(45) = 43.0
• Change: Plant Both Grapes

E)

• Only Ev to change is Plant Chardonnay
• Ev (plant Chardonnay) = 0.55(20) + 0.45(50) = 33.5
• Too low, best to plant both

15) Lake Placid Community Center

A)

• Decision: Maximize cash flow
• Ev (small) = 0.1(400) + 0.6(500) + 0.3(660) = 538
• Ev (medium) = 0.1(-250) + 0.6(650) + 0.3(800) = 605
• Ev (large) = 0.1(-400) + 0.6(580) + 0.3(990) = 605
• Build large or medium size

B)

• If demand is not large enough, best to build medium size. Smaller loss with medium size.

C)

• Evwpi = 0.1(400) + 0.6(650) + 0.3(990) = 727
• Evwopi = 605 & Evpi = 727 – 605 = 122000
• Should consider perfect info, worth 122k if researched further.

D)

• Ev (small) = 0.2(400) + 0.5(500) + 0.3(660) = 528
• Ev (medium) = 0.2(-250) + 0.5(650) + 0.3(800) = 515
• Ev (small) = 0.2(-400) + 0.5(580) + 0.3(990) = 507
• Build small size center

E)

• Ev (small) = 0.0(400) + 0.6(500) + 0.4(660) = 564
• Ev (medium) = 0.0(-250) + 0.6(650) + 0.4(800) = 710
• Ev (small) = 0.0(-400) + 0.6(580) + 0.4(990) = 744
• 744000 – 528000 = 216000
• 744000 – 605000 = 139000
• Mayor is risk averse. Option 2 is the best approach.

Chapter 6

7) Gasoline Sales

C)

• 3-week: 27.50
• 4-week: 9.648
• 5-week: 7.406
• Best option to use is week-5.

Chapter 7

23) Embassy Motor Company

Let E = # of units of the Ez-rider produced

Let L = # of units of the Lady-sport produced

A)

• Max 2400e + 1800l
• S.t.:
  • 6e + 3l ≤ 2100 (Engine Time)
  • L ≤ 280 (Lady-sport Maximum)
  • 2e + 2.5l ≤ 1000 (Assembly & Testing)
  • E, L ≥ 0

C) Manufacturing Time & Assembly & Testing

26) Sea Wharf Restaurant

Let N = Amount spent on newspaper advertising

Let R = Amount spent on radio advertising

A)

• Max 50n + 80r
• S.t.:
  • N + r = 1000 (Budget)
  • N ≥ 250 (Newspaper Minimum)
  • R ≥ 250 (Radio Minimum)
  • N – 2r ≥ 0 (News ≥ 2 * radio)
  • N, R ≥ 0

38) Applied Technologies

A)

Let S = Yards of the standard grade material per frame

Let P = Yards of the professional grade material per frame

• Min 7.50s + 9p
• S.t.:
  • 0.10s + 0.30p ≥ 6 (Carbon Fiber at least 20% of 30)
  • 0.06 + 0.12 ≤ 3 (Kevlar no more than 10% of 30 yards)
  • S + p = 30 (Total yards)
  • S, P ≥ 0

D)

• Doesn’t change. Value is reduced.
• 7.5(15) + 8(15) = $232.5

E)

• $7.4/yd
• Optimal solution: S = 10, P = 20
• Reduced: 7.50(10) + 7.40(20) = $223
• Lower, doesn’t change because of the requirement for a maximum percentage of Kevlar (10%).

Chapter 8

10) Innis Investments

A)

• S = 4000
• M = 10000
• Total Risk = 8(4000) + 3(10000) = 62000

B)

• Variable S Objective Coefficient Range:
  • 8.000 – 4.250 = 3.750
  • No upper limit
• Variable M:
  • No upper limit
  • 23.000 + 3.400 = 6.400

C) 5(4000) + 4(10000) = $60000

D) 60000/1200000 = 0.05 or 5%

E) 0.057 Risk Units

F) 0.057(100) = 5.7%

21) Roundtree Furniture

A)

Let S1 = Supersaver rentals allocated to Room Type I

Let S2 = Supersaver rentals allocated to Room Type II

Let D1 = Deluxe rentals allocated to Room Type I

Let D2 = Deluxe rentals allocated to Room Type II

Let B2 = Business rentals allocated to Room Type II

• Max 30s1 + 20s2 + 35d1 + 30d2 + 40b
• S.t.

Optimal Solution Is:

20 Supersaver rentals will have to be turned away if demands materialize as forecast.

B) Roundtree should accept 110 Supersaver reservations, 60 Deluxe reservations, and 50 Business reservations.

C) Yes, upgrade. Increase demand for Deluxe (60 – 26 = 34). Constraint 2 demand increase increases profit by $10. Added cost to breakfast: $5.

D) Convert to Type 1 room. Shadow price for Constraint 4: Increase profit by $30.

E) Yes, forecast for demand for each. Modify the right-hand side of the first 3 constraints and resolve.

30) Channel 10

A)

Let L = Minutes devoted to local news

Let N = Minutes devoted to national news

Let W = Minutes devoted to weather

Let S = Minutes devoted to sports

• Min 300l + 200n + 100w + 100s
• S.t.
  • l + n + w + s = 20 (Time Available)
  • L ≥ 3 (15% Local)
  • L + n ≥ 10 (50% Requirement)
  • W – s ≤ 0 (Weather – sports)
  • -l – n + s ≤ 0 (Sports Requirement)
  • W ≥ 4 (20% Weather)
  • L, N, W, S ≥ 0

Optimal Solution:

• L = 3
• N = 7
• W = 5
• S = 5
• Total Cost = $3300

B) Each additional minute increases cost by $100. Each reduction reduces cost by $100. Valid increase to 210 and decrease to 18 from the current level of 20.

C) Increase by 1, cost increases by $100.

D) Increase by $100.

E) Increase by 1 minute, no effect. Shadow price is 0.

Chapter 9

3) Employee Credit Union

Let A = Automobile

Let F = Furniture

Let S = Secured

Let SL = Signature

Let R = Risk

• Max 0.08a + 0.10f + 0.11s + 0.12sl + 0.09r
• S.t.
  • a + f + s + sl + r ≤ 2000000
  • R ≤ 600000
  • S + sl ≤ r
  • Sl ≤ 0.10(a + f + s + sl)
  • A + f + s + sl + r ≥ 0

Solution:

• A = 630000
• F = 170000
• S = 460000
• SL = 140000
• R = 600000
• Annual Return: $188800 (9.44%)

9) Epsilon Airlines

• Min m + t + w + r + f + a + s
• S.t.
  • r + f + a + s + m ≥ 75
  • F + a + s + m + t ≥ 50
  • A + s + m + t + w ≥ 45
  • S + m + t + w + r + f ≥ 90
  • T + w + r + f + a ≥ 75
  • W + r + f + a + s ≥ 45

Solution:

• M = 20
• T = 20
• W = 0
• R = 45
• F = 5
• A = 5
• S = 0
• Total # = 95
• Excess:
  • R = 25
  • S = 10
  • All others = 0

10) Shore Financial

A)

Let S = Stocks

Let B = Bonds

Let M = Mutual Funds

Let C = Cash

• Max 0.1s + 0.03b + 0.04m + 0.01c
• S.t.
  • s + b + m + c = 1 (Total Investment)
  • 0.8s + 0.2b + 0.3m + 0.0c ≤ 0.4 (Risk Constraint)
  • s ≥ 0.15 (Stocks Minimum)
  • b ≥ 0.10 (Bonds Minimum)
  • m ≥ 0.10 (Mutual Funds Minimum)
  • c ≥ 0.10 (Cash Minimum)

Allocation:

• Stocks: 40.9%
• Bonds: 14.5%
• Mutual Funds: 14.5%
• Cash: 30%
• Annual Return: 5.4%
• Total Risk = 0.409(0.8) + 0.145(0.2) + 0.145(0.3) + 0.300(0.0) = 0.4

B)

• Stocks: 0%
• Bonds: 36%
• Mutual Funds: 36%
• Cash: 28%
• Return: 2.53%
• Total Risk = 0.0(0.8) + 0.36(0.2) + 0.36(0.3) + 0.28(0.0) = 0.18

C)

• Stocks: 75%
• Bonds: 0%
• Mutual Funds: 15%
• Cash: 10%
• Return: 8.2%
• Total Risk = 0.75(0.8) + 0.0(0.2) + 0.15(0.3) + 0.10(0.0) = 0.65

D) Willing to take more risk. Increase investment in stocks to ≥ 75% or reduce cash requirement to at least 10%.

E) He can use this for new future investments to figure out maximum total risk and resolve.

Chapter 10

5) Premier Products

• Max 100ya + 125yb + 115yc + 100td + 120ra + 135rb + 115rc + 120rd + 155la + 150lb + 140lc + 130ld
• S.t.
• ya + yb + yc + yd …