8 Chapter 2 Linear Programming: Problem Formulation

 

SOLUTIONS TO END­OF­CHAPTER PROBLEMS:

 

Chapter 2 Linear Programming: Problem Formulation

 

 

 

 

2.1 Let X

1

= number of newspaper ads

X

2

= number of radio ads

 

MAX: 180X

1

+ 700X

2

                           

Subject to: 50X

1

+ 500X

2

< 2000 (Cost, $)

X

1

­ 3X

2

> 0 (Ratio:newspaper:radio)

(X

1

, X

2

) > 0 (Nonnegativity)

 

2.2 Let X

1

= number of regular cheeseburgers

X

2

= number of double cheeseburgers

 

MAX: 2.50X

1

+ 3.0X

2

 

  

Subject to: 4X

1

+ 8X

2

< 320 (Hamburger, ozs.)

1X

1

+ 2X

2

< 90 (Cheese, ozs.)

1X

1

+ 1X

2

< 120 (Buns)

(X

1

, X

2

) > 0 (Nonnegativity)

 

2.3 Let X

1

= pounds of soybeans

X

2

= pounds of wheat

X

3

= pounds of oats

X

4

= pounds of corn

X

5

= pounds of mineral supplement

 

MIN: 0.15X

1

+ 0.25X

2

+ 0.10X

3

+ 0.20X

4

+ 0.18X

5

 

   

Subj.to: 0.6X

1

­ 0.4X

2

­ 0.4X

3

+ 0.6X

4

­ 0.4X

5

> 0 (40% Corn & Soy)

­ 0.2X

1

+ 0.8X

2

­ 0.2X

3

­ 0.2X

4

­ 0.2X

5

< 0 (20% Wheat)

­ 0.3X

1

­ 0.3X

2

+ 0.7X

3

­ 0.3X

4

­ 0.3X

5

> 0 (30% Oats)

­ 0.1X

1

­ 0.1X

2

­ 0.1X

3

­ 0.1X

4

+ 0.9X

5

> 0 (10% Min Sup)

X

1

+ X

2

+ X

3

 

+ X

4

+ X

5

= 1000 (Bag weight)

(X

1

, X

2

, X

3

, X

4

, X

5

) > 0 (Nonnegativity)

 

  

Because the weight of each bag is known, an alternative

formulation can be used:

 

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Solutions Manual and Workbook

9

 

MIN: 0.15X

1

+ 0.25X

2

+ 0.10X

3

+ 0.20X

4

+ 0.18X

5

 

 

Subject to: X

1

+ X

4

> 400 (40% Corn & Soy)

X

2

< 200

(20% Wheat)

X

3

> 300

(30% Oats)

X

5

> 100

(10% Min Sup)

X

1

+ X

2

+ X

3

+ X

4

+ X

5

= 1000 (Bag Weight)

(X

1

, X

2

, X

3

, X

4

, X

5

) > 0 (Nonnegativity)

 

2.4 Let X

1

= number of G21 gears produced

X

2

= number of G22 gears produced

X

3

= number of G23 gears produced

X

4

= number of G24 gears produced

 

MAX: 251.55

1

X

1

+ 340.32X

2

+ 528.11X

3

+ 553.13X

4

       

  

Subject to: X

1

< 50 (Max. G21 Dem.)

X

2

< 50 (Max. G22 Dem.)

X

3

< 75 (Max. G23 Dem.)

X

4

< 25 (Max. G24 Dem.)

4X

1

+ 6X

2

+ 4X

3

+ 4.5X

4

< 800 (Forge Cap.)

2X

1

+ 3X

2

+ 5X

3

+ 5.5X

4

< 700 (Lathe Cap.)

3X

1

+ 3.5X

2

+ 5X

3

+ 7X

4

< 900 (Cutting Cap.)

2X

1

+ 3.5X

2

+ 4X

3

+ 8X

4

< 800 (Polishing Cap.)

(X

1

, X

2

, X

3

, X

4

) > 0 (Nonnegativity)

 

1

Profit for each gear type was determined in the following

manner:

 

G21 Gear Sales Price: $432.75

Less:

Forge Labor (4 * 5.5) $22.00

Lathe Labor (2 * 6.3) $12.60

Cutting Labor (3 * 10.5) $31.50

Polishing Labor (2 * 9.5) $19.00

Materials $54.00

Overhead $42.10 181.20

Gross Profit $ 251.55

 

 

2.5 Let X

1

= number of pints of blueberries

X

2

= number of jars of blueberry spread

X

3

= advertising dollars spent on blueberries

X

4

= advertising dollars spent on blueberry spread

 

MAX: 2.65X

1

+ 3.20X

2

­ 1X

3

­ 1X

4

 

 

Subject to: X

3

+ X

4

 

≥

500 (Min. Adver.)

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10 Chapter 2 Linear Programming: Problem Formulation

 

1.1X

1

+ 2.85X

2

+ X

3

+ X

4

< 4000 (Cost, $)

0.6X

1

­ 0.4X

2

          

> 0 (40% Fresh)

(X

1

, X

2

, X

3

, X

4

) > 0

(Nonnegativity)

 

2.6 Let X

1

= number of baseballs

X

2

= number of softballs

 

MAX: 6X

1

+ 9X

2

 

 

Subject to: 1.0X

1

+ 1.5X

2

< 200 (Leather)

4.0X

1

+ 5.0X

2

< 900 (Stitching)

550X

1

+1000X

2

 

<

100000 (Winding)

(X

1

, X

2

) > 0 (Nonnegativity)

 

2.7 Let X

1

= number of mixers

X

2

= number of coffee makers

X

3

= number of can openers

 

MAX: 26X

1

+ 28X

2

+ 15X

3

 

 

Subject to:19X

1

+ 22X

2

+ 10X

3

< 4000 (Prod. budget,$)

4X

1

+ 5X

2

+ 4X

3

 

<

1000 (Labor hours)

X

1

              

<

400 (Dem., mixers)

X

2

               

<

200 (Dem. Coff. makers)

X

3

              

< 170 (Dem. can open.)

(X

1

, X

2

, X

3

) > 0 (Nonnegativity)

 

2.8 Let X

1

= pounds of F1 produced at Nacogdoches

X

2

= pounds of F2 produced at Nacogdoches

X

3

= pounds of F1 produced at Lufkin

X

4

= pounds of F2 produced at Lufkin

 

MAX: 7X

1

+ 4.5X

2

+ 6X

3

+ 2.5X

4

 

 

Subject to: 1X

1

+ 1X

2

+ 2X

3

+ 3X

4

< 65000 (Prod. Bud. $)

X

1

+ X

2

< 3000 (Cap. Nacog.)

X

3

+ X

4

< 2000 (Cap. Luf.)

X

1

+ X

3

< 4000 (Max. Dem. F1)

X

2

+ X

4

< 3000 (Max. Dem. F2)

(X

1

, X

2

, X

3

, X

4

) > 0 (Nonnegativity)

 

 

2.9 Let X

1

= dollar amount invested in Stock A

X

2

= dollar amount invested in Stock B

X

3

= dollar amount invested in bonds

 

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Solutions Manual and Workbook

11

 

MAX: 0.06X

1

+ 0.09X

2

+ 0.10X

3

 

 

Subject to: X

1

+ X

2

+ X

3

= 1000 (Tot. Invest.)

X

3

= 300 (30% inv.bonds)

­ 0.4X

1

+ 0.6X

2

­ 0.4X

3

< 0 (40% inv.Stock B)

X

2

> 200 (Min. inv.Stock B)

1X

1

­ 1.5X

2

> 0 (Ratio B:A)

(X

1

, X

2

, X

3

) > 0 (Nonnegativity)

 

2.10 Let X

1

= number of newspaper ads

X

2

= number of Local Merchant ads

 

MAX: 1200X

1

+ 500X

2

 

 

Subject to: 100X

1

+ 50.50X

2

< 1000 (Tot. Adver. Bud. $)

X

1

< 30 (No.News.ads)

X

2

< 5 (No. L.M. ads)

X

1

­ 3X

2

> 0 (Ratio,News:L.M.,3:1)

(X

1

, X

2

) > 0 (Nonnegativity)

 

2.11 Let X

1

= mls. of orange drink mix

X

2

= mls. of 7­UP

X

3

= mls. of Triple Sec

 

MIN: 0.000658X1 + 0.000495X2 + 0.00899X3

 

Subject to: X

1

+ X

2

+ X

3

= 1000 (Amt = 1000 ml)

­ 0.7X

1

+ 0.3X

2

­ 0.7X

3

> 0 (7­UP > 70% mix)

­ 0.8X

1

+ 0.2X

2

­ 0.8X

3

< 0 (7­UP < 80% mix)

0.8X

1

­ 0.2X

2

­ 0.2X

3

> 0 (O.J mix > 20%)

0.7X

1

­ 0.3X

2

­ 0.3X

3

< 0 (O.J. mix < 30%)

­0.143X

1

­ 0.143X

2

 

+ 0.857X

3

= 0(Triple Sec)

(X

1

, X

2

, X

3

) > 0 (Nonnegativity)

 

2.12 Let X

1

= number of units of D1

X

2

= number of units of D2

X

3

= number of units of D3

X

4

= number of units of D4

 

MAX: 5.0X

1

+ 7.5X

2

+ 10.50X

3

+ 15.0X

4

 

Subject to: 0.5X

1

+ 0.5X

2

+ 1X

3

+ 1.1X

4

< 100 (Plastic)

5X

1

+ 5X

2

+ 7X

3

+ 7X

4

< 500 (Paint)

12X

1

+ 13X

2

+ 13X

3

+ 14X

4

< 600 (Nylon)

1X

1

+ 1.5X

2

+ 2X

3

+ 3X

4

< 200 (Cloth)

1X

1

+ 0.5X

2

+1.5X

3

+ 1.8X

4

< 300 (Labor)

(X

1

, X

2

, X

3

, X

4

) > 0 (Nonnegativity)

 

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12 Chapter 2 Linear Programming: Problem Formulation

 

2.13 Let X

1

= ozs. of beef

X

2

= ozs. of pork

X

3

= ozs. of chicken

X

4

= ozs. of filler

 

MIN: 0.10625X

1

+ 0.0625X

2

+ 0.059375X

3

+ 0.01875X

4

 

 

Subject to: X

1

+ X

2

+ X

3

+ X

4

= 16 (Tot.Wt.=16 ozs)

X

2

< 1.6 (Amt.pork, 10%)

X

3

        

≤

4 (Amt.chick. 30%)

X

1

> 4.8 (Amt.beef, 30%)

5X

1

+ 3X

2

+ 4X

3

+ 1X

4

> 10 (Min. mgs.Pro.)

3X

1

+ 5X

2

+ 2X

3

+ 3X

4

< 4 (Max. mgs.Fat)

3X

1

+10X

2

+ 4X

3

+ 3X

4

< 8 (Max. mgs.Water)

(X

1

, X

2

, X

3

, X

4

) > 0

(Nonnegativity)

 

2.14 Let X

1

= amount invested in gold

X

2

= amount invested in bonds

X

3

= amount invested in savings accounts

X

4

= amount invested in common stock

 

MAXIMIZE: 0.06X

1

+ 0.09X

2

+ 0.04X

3

+ 0.15X

4

 

 

Subject to: X

1

+ X

2

+ X

3

+ X

4

= 10000 (Tot. Invest., $)

X

2

+ X

3

< 2000 (Sav. & bonds: 20%)

X

1

> 500 (Gold, 5%)

X

4

< 6000 (C. Stk. 60%)

(X

1

, X

2

, X

3

, X

4

) > 0 (Nonnegativity)