Introduction to Management Science, 10e (Taylor) Chapter 3 Linear Programming: Computer Solution and...

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Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Introduction to Management Science, 10e (Taylor)

Chapter 3 Linear Programming: Computer Solution and Sensitivity Analysis

1) The reduced cost (shadow price) for a positive decision variable is 0.

Answer: TRUE

Diff: 2 Page Ref: 90

Main Heading: Computer Solution

Key words: shadow price, sensitivity analysis

2) When the right-hand sides of 2 constraints are both increased by 1 unit, the value of the

objective function will be adjusted by the sum of the constraints' prices.

Answer: FALSE



Key words: sensitive analysis, right-hand-side

3) When a linear programming problem is solved using a computer package decision variables

will always be integer and therefore decision variable values never need to be rounded.

Answer: FALSE

Diff: 1


Key words: computer solution

4) Sensitivity ranges can be computed only for the right hand sides of constraints.

Answer: FALSE




5) Sensitivity analysis determines how a change in a parameter affects the optimal solution.

Answer: TRUE


Main Heading: Sensitivity Analysis

Key words: sensitivity analysis

6) The sensitivity range for an objective function coefficient is the range of values over which

the current optimal solution point (product mix) will remain optimal.

Answer: TRUE



Key words: sensitivity analysis, objective coefficients

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7) The sensitivity range for an objective function coefficient is the range of values over which

the profit does not change.

Answer: FALSE




8) The sensitivity range for a constraint quantity value is the range over which the shadow price

is valid.

Answer: TRUE




9) If we change the constraint quantity to a value outside the sensitivity range for that constraint

quantity, the shadow price will change.

Answer: TRUE



Key words: sensitivity analysis, shadow price

10) The sensitivity range for a constraint quantity value is the range over which the optimal

values of the decision variables do not change.

Answer: FALSE




11) Linear programming problems are restricted to decisions in a single time period.

Answer: FALSE



Key words: linear programming problems, formulation

12) A maximization problem may be characterized by all greater than or equal to constraints.

Answer: FALSE



Key words: linear programming, formulation

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13) A change in the value of an objective function coefficient will always change the value of the

optimal solution.

Answer: FALSE




14) The terms reduced cost, shadow price, and dual price all mean the same thing.

Answer: TRUE




15) Sensitivity analysis can be used to determine the effect on the solution for changing several

parameters at once.

Answer: FALSE




16) For a profit maximization problem, if the allowable increase for a coefficient in the objective

function is infinite, then profits are unbounded.

Answer: FALSE




17) The reduced cost (shadow price) for a positive decision variable is __________.

Answer: zero




18) The sensitivity range for a __________ is the range of values over which the quantity values

can change without changing the shadow price

Answer: constraint quantity



Key words: computer solution, right-hand-side value

19) __________ is the analysis of the effect of parameter changes on the optimal solution.

Answer: Sensitivity analysis




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20) The sensitivity range for a constraint quantity value is also the range over which the

__________ is valid.

Answer: shadow price




21) The sensitivity range for an __________ coefficient is the range of values over which the

current optimal solution point (product mix) will remain optimal.

Answer: objective function




Consider the following linear program, which maximizes profit for two products, regular (R),

and super (S):

MAX 50R + 75S

s.t.

1.2R + 1.6 S ≤ 600 assembly (hours)

0.8R + 0.5 S ≤ 300 paint (hours)

.16R + 0.4 S ≤ 100 inspection (hours)

Sensitivity Report:

Final Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease

$B$7 Regular = 291.67 0.00 50 70 20

$C$7 Super = 133.33 0.00 75 50 43.75

Final Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease

$E$3 Assembly (hr/unit) 563.33 0.00 600 1E+30 36.67

$E$4 Paint (hr/unit) 300.00 33.33 300 39.29 175

$E$5 Inspect (hr/unit) 100.00 145.83 100 12.94 40

22) The optimal number of regular products to produce is __________, and the optimal number

of super products to produce is __________, for total profits of __________.

Answer: 291.67, 133.33, $24,583




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23) If the company wanted to increase the available hours for one of their constraints (assembly,

painting, or inspection ) by 2 hours, they should increase __________.

Answer: Inspection



Key words: computer solution, sensitivity analysis

24) The profit on the super product could increase by __________ without affecting the product

mix.

Answer: $50.




25) If downtime reduced the available capacity for painting by 40 hours (from 300 to 260 hours),

profits would be reduced by __________.

Answer: $1,333




26) A change in the market has increased the profit on the super product by $5. Total profit will

increase by __________.

Answer: $667




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Tracksaws, Inc. makes tractors and lawn mowers. The firm makes a profit of $30 on each tractor

and $30 on each lawn mower, and they sell all they can produce. The time requirements in the

machine shop, fabrication, and tractor assembly are given in the table.

Formulation:

Let x = number of tractors produced per period

y = number of lawn mowers produced per period

MAX 30x + 30y

subject to 2 x + y ≤ 60

2 x + 3y ≤ 120

x ≤ 45

The graphical solution is shown below.

27) How many tractors and saws should be produced to maximize profit, and how much profit

will they make?

Answer: 15 tractors and 30 saws for $1,350 in profit


Main Heading: Graphical Solutions of Linear Programming Models

Key words: graphical solution, simultaneous solution

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28) Determine the sensitivity range for the profit for tractors.

Answer: 20 ≤ x ≤ 60


Main Heading: Sensitivity Analysis and Graphical Solution

Key words: graphical solution, sensitivity analysis

29) What is the shadow price for assembly?

Answer: 0




30) What is the shadow price for fabrication?

Answer: $7.50



Key words: sensitivity analysis, right-hand-side value

31) What is the maximum amount a manager would be willing to pay for one additional hour of

machining time?

Answer: $1357.50 - $1350 = $7.50




32) A breakdown in fabrication causes the available hours to drop from 120 to 90 hours. How

will this impact the optimal number of tractors and mowers produced?

Answer: x = 22.5, y = 15, Z = 1125, so profits will fall by $1350 - $1125 = $225. Students

may also answer the question by determining the sensitivity range, which is from 60 to 180 hours,

resulting in a profit change of 30x$7.5 = $225.




33) What is the range for the shadow price for assembly?

Answer: allowable decrease = 45 - 15 = 30, and allowable increase is ∞.




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The production manager for the Whoppy soft drink company is considering the production of 2

kinds of soft drinks: regular (R) and diet (D). The company operates one "8 hour" shift per day.

Therefore, the production time is 480 minutes per day. During the production process, one of the

main ingredients, syrup is limited to maximum production capacity of 675 gallons per day.

Production of a regular case requires 2 minutes and 5 gallons of syrup, while production of a diet

case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and

profits for diet soft drink are $2.00 per case.

The formulation for this problem is given below.

MAX Z = $3R + $2D

s.t.

2R + 4D ≤ 480

5R + 3D ≤ 675

The sensitivity report is given below

Adjustable Cells

Final Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease

$B$6

Regular = 90.00 0.00 3 0.33 2

$C$6

Diet = 75.00 0.00 2 4 0.2

Constraints

Final Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease

$E$3

Production

(minutes) 480.00 0.07 480 420 210

$E$4 Syrup (gallons) 675.00 0.57 675 525 315

34) What is the optimal daily profit?

Answer: $420


Main Heading: Sensitivity Analysis and Computer Solution


35) How many cases of regular and how many cases of diet soft drink should Whoppy produce

to maximize daily profit?

Answer: 90 cases of regular and 75 cases of diet




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36) What is the sensitivity range for the per case profit of a diet soft drink?

Answer: 1.8 ≤ c2 ≤ 6.0




37) What is the sensitivity range of the production time?

Answer: 270 ≤ b1 ≤ 900




38) if the company decides to increase the amount of syrup it uses during production of these

soft drinks to 990 lbs. will the current product mix change? If show what is the impact on profit?

Answer: Yes., Increase in profit = 0.57(990 - 675) = $180




Mallory furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big

shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300

and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this

week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is

$300 and for each medium shelf is $150. Graphically solve this problem and answer the

following questions.

39) What is the optimal product mix and maximum profit?

Answer: 150 big shelves and no medium shelves. Profit = $45,000



Key words: formulation, objective function

40) Determine the sensitivity range for the profit on the big shelf.

Answer: the range is from $250 to infinity




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41) If the Mallory Furniture is able to increase the profit per medium shelf to $200, would the

company purchase medium shelves. If so, what would be the new product mix and the total

profit?

Answer: yes, Big = 90, Medium = 100 Z = $47,000




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The linear programming problem whose output follows is used to determine how many bottles of

fire red nail polish (x1), bright red nail polish (x2), basil green nail polish(x3), and basic pink

nail polish(x4) a beauty salon should stock. The objective function measures profit; it is assumed

that every piece stocked will be sold. Constraint 1 measures display space in units, constraint 2

measures time to set up the display in minutes. Note that green nail polish does not require any

time to prepare its display. Constraints 3 and 4 are marketing restrictions. Constraint 3 indicates

that the maximum demand for fire red and green polish is 25 bottles, while constraint 4 specifies

that the minimum demand for bright red, green and pink nail polish bottles combined is at least

50 bottles.

MAX 100x1 + 120x2 + 150x3 + 125x4

Subject to 1. x1 + 2x2 + 2x3 + 2x4 ≤108

2. 3x1 + 5x2 + x4 ≤ 120

3. x1 + x3 ≤ 25

4. x2 + x3 + x4 ≥ 50

x1, x2 , x3, x4 ≥ 0

Optimal Solution:

Objective Function Value = 7475.000

Objective Coefficient Ranges

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Right Hand Side Ranges

42) How much space will be left unused? How many minutes of idle time remaining for setting

up the display?

Answer: 0, 63




43) a) To what value can the per bottle profit on fire red nail polish drop before the

solution (product mix) would change?

b) By how much can the per bottle profit on green basil nail polish increase before

the solution (product mix) would change?

Answer: a) 87.5, b) 12




44) a) By how much can the amount of space decrease before there is a change in the

profit?

b) By how much can the amount of space decrease before there is a change in the product mix?

c) By how much can the amount of time available to setup the display can increase before the


d) What is the lowest value for the amount of time available to setup the display before the


Answer: a) 0 b) 8 c) 0 d) 57




45) You are offered the chance to obtain more space. The offer is for 15 units and the total price

is $1500. What should you do? Why?

Answer: reject the offer, (11 x 15) = 1125 < 1500




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46) Max Z = 5x1 + 3x2

Subject to: 6x1 + 2x2 ≤ 18

15x1 + 20x2 ≤ 60

x1 + x2 ≥ 0

Determine the sensitivity range for each constraint.

Answer: constraint 1: 6 - 24

constraint 2: 45 - 180



Key words: sensitivity analysis, sensitivity range for right hand sides

47) Max Z = 5x1 + 3x2


15x1 + 20x2 ≤ 60

x1 + x2 ≥ 0

Determine the sensitivity range for each objective function coefficient.

Answer: x1: 2.25 - 9.0 and, x2: 1.667 - 6.667



Key words: sensitivity analysis/range for objective function coefficients

48) Max Z = 3x1 + 3x2


25x1 + 50x2 ≤ 200

x1 , x2 ≥ 0

Determine the sensitivity range for each objective function coefficient.

Answer: x1: 1.5 - 7.5 and, x2: 1.2 - 6.0




49) For a maximization problem, assume that a constraint is binding. If the original amount of a

resource is 4 lbs., and the range of feasibility (sensitivity range) for this constraint is from 3 lbs.

to 6 lbs., increasing the amount of this resource by 1 lb. will result in the:

A) same product mix, different total profit

B) different product mix, same total profit as before

C) same product mix, same total profit

D) different product mix, different total profit

Answer: A




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50) A plant manager is attempting to determine the production schedule of various products to

maximize profit. Assume that a machine hour constraint is binding. If the original amount of

machine hours available is 200 minutes., and the range of feasibility is from 130 minutes to 340

minutes, providing two additional machine hours will result in:

A) the same product mix, different total profit

B) a different product mix, same total profit as before

C) the same product mix, same total profit

D) a different product mix, different total profit

Answer: D




The production manager for Beer etc. produces 2 kinds of beer: light (L) and dark (D). Two

resources used to produce beer are malt and wheat. He can obtain at most 4800 oz of malt per

week and at most 3200 oz of wheat per week respectively. Each bottle of light beer requires 12

oz of malt and 4 oz of wheat, while a bottle of dark beer uses 8 oz of malt and 8 oz of wheat.

Profits for light beer are $2 per bottle, and profits for dark beer are $1 per bottle.

51) If the production manager decides to produce of 0 bottles of light beer and 400 bottles of

dark beer, it will result in slack of

A) malt only

B) wheat only

C) both malt and wheat

D) neither malt nor wheat

Answer: A



Key words: graphical solution, constraints

52) Which of the following is not a feasible solution?

A) 0 L and 0 D

B) 0 L and 400 D

C) 200 L and 300 D

D) 400 L and 400 D

Answer: D



Key words: computer solution, infeasibility

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53) What is the optimal weekly profit?

A) $1000

B) $900

C) $800

D) $700

E) $600

Answer: C




Mallory Furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big

shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300

and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this

week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is

$300 and for each medium shelf is $150.

54) Which of the following is not a feasible purchase combination?

A) 0 big shelves and 200 medium shelves

B) 0 big shelves and 0 medium shelves

C) 150 big shelves and 0 medium shelves

D) 100 big shelves and 100 medium shelves

Answer: D



Key words: computer solution, infeasibility

55) If the Mallory Furniture company decides to purchase 150 big shelves and no medium

shelves, which of the two resources will be left over?

A) investment money only

B) storage space only

C) investment money and storage space

D) neither investment money nor storage space

Answer: B



Key words: computer solution, slack

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The production manager for the Whoppy soft drink company is considering the production of 2

kinds of soft drinks: regular and diet. The company operates one "8 hour" shift per day.

Therefore, the production time is 480 minutes per day. During the production process, one of the

main ingredients, syrup is limited to maximum production capacity of 675 gallons per day.

Production of a regular case requires 2 minutes and 5 gallons of syrup, while production of a diet

case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and

profits for diet soft drink are $2.00 per case.

56) Which of the following is not a feasible production combination?

A) 90 R and 75 D

B) 135 R and 0 D

C) 0 R and 120 D

D) 75 R and 90 D

E) 50 R and 50 D

Answer: D



Key words: graphical solution, feasibility

57) For the production combination of 135 regular cases and 0 diet cases, which resource is

completely used up (at capacity)?

A) only time

B) only syrup

C) time and syrup

D) neither time nor syrup

Answer: B



Key words: graphical solution, slack variables

58) The sensitivity range for the profit on a regular case of soda is

A) $2 to $3

B) $2 to $4

C) $1 to $3

D) $1 to $3.33

Answer: D




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59) Which of the following could not be a linear programming problem constraint?

A) A + B ≤ -3

B) A - B ≤ -3

C) A - B ≤ 3

D) A + B ≥ -3

E) -A + B ≤ -3

Answer: A



Key words: graphical solution, feasibility

60) Use the constraints given below and determine which of the following points is feasible.

(1) 14x + 6y ≤ 42

(2) x - y ≤ 3

A) x = 1; y = 5

B) x = 2; y = 2

C) x = 2; y = 8

D) x = 2; y = 4

E) x = 3; y = 0.5

Answer: B



Key words: constraints

61) For the constraints given below, which point is in the feasible region of this minimization

problem?

(1) 14x + 6y ≤ 42

(2) x + 3y ≥ 6

A) x = 0; y = 4

B) x = 2; y = 5

C) x = 1; y = 2

D) x = 2; y = 1

E) x = 2; y = 3

Answer: C



Key words: constraints

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62) What combination of x and y is a feasible solution that minimizes the value of the objective

function ?

Min Z = 3x + 15y

(1) 2x + 4y ≥ 12

(2) 5x + 2y ≥10

A) x = 0; y = 3

B) x = 0; y = 5

C) x = 5; y = 0

D) x = 6; y = 0

E) x = 4; y = 1

Answer: D



Key words: feasibility, constraints

63) A shadow price reflects which of the following in a maximization problem?

A) the marginal gain in the objective that would be realized by adding 1 unit of a resource

B) the marginal gain in the objective that would be realized by subtracting 1 unit of a resource

C) the marginal cost of adding additional resources

D) the marginal gain of selling one more unit

Answer: A



Key words: shadow price, sensitivity analysis

64) Given the following linear programming problem:

Max Z = 15x + 20 y

s.t.

8x + 5y ≤ 40

4x + y ≥ 4

What would be the values of x and y that will maximize revenue?

A) x = 5; y = 0

B) x = 0; y = 8

C) x = 0; y = 1

D) x = 1; y = 0

E) x = 3; y = 4

Answer: B



Key words: formulation, solution

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65) Given the following linear program that maximizes revenue:

Max Z = 15x + 20 y

s.t.

8x + 5y ≤ 40

4x + y ≥ 4

What is the maximum revenue at the optimal solution?

A) $120

B) $160

C) $185

D) $200

Answer: B




Given the following linear programming problem that minimizes cost.

Min Z = 2x + 8y

Subject to (1) 8x + 4y ≥ 64

(2) 2x + 4y ≥ 32

(3) y ≥ 2

66) Determine the optimum values for x and y.

A) x = 2; y = 6

B) x = 6; y = 2

C) x = 12; y = 2

D) x = 2; y = 2

E) x = 6; y = 5

Answer: C




67) At the optimal solution the minimum cost is:

A) $30

B) $40

C) $50

D) $52

E) $53.33

Answer: B




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68) What is the sensitivity range for the cost of x?

A) 0 to 2

B) 4 to 6

C) 2 to 4

D) 0 to 4

Answer: D




69) What is the sensitivity range for the third constraint, y ≥ 2?

A) 0 to 4

B) 2 to 5.33

C) 0 to 5.33

D) 4 to 6.33

Answer: C



Key words: sensitivity analysis, quantity ranges for constraints

70) For a maximization problem, the shadow price measures the __________ in the value of the

optimal solution, per unit increase for a given __________.

A) decrease, resource

B) increase, parameter

C) improvement, resource

D) change, objective function coefficient

E) decrease, parameter

Answer: C




71) Sensitivity analysis is the analysis of the effect of __________ changes on the __________.

A) price, company

B) cost, production

C) parameter, optimal solution

D) none of the above

Answer: C




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72) For a linear programming problem, assume that a given resource has not been fully used.

We can conclude that the shadow price associated with that constraint:

A) will have a positive value

B) will have a negative value

C) will have a value of zero

D) could have a positive, negative or a value of zero. (no sign restrictions)

Answer: C




73) For a resource constraint, either its slack value must be __________ or its shadow price must

be __________.

A) negative, negative

B) negative, zero

C) zero, zero

D) zero, negative

Answer: C




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Aunt Anastasia operates a small business: she produces seasonal ceramic objects to sell to

tourists. For the spring, she is planning to make baskets, eggs, and rabbits. Based on your

discussion with your aunt you construct the following table.

Your aunt also has committed to make 25 rabbits for a charitable organization. Based on the

information in the table, you formulate the problem as a linear program.

B = number of baskets produced

E = number of eggs produced

R = number of rabbits produced

MAX 2.5B + 1.5E + 2R

s.t.

0.5 B + 0.333E + 0.25R ≤ 20

B + E + R ≤ 50

0.25B + 0.333E + 0.75R ≤ 80

R ≥ 25

The Excel solution and the answer and sensitivity report are shown below.

The Answer Report:

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The Sensitivity Report:

74) Which additional resources would you recommend that Aunt Anastasia try to obtain?

A) mix/mold

B) kiln

C) paint and seal

D) demand

E) Cannot tell from the information provided

Answer: B




75) Suppose the charitable organization contacted Aunt Anastasia and told her that they had

overestimated the amount of rabbits they needed. Instead of 25 rabbits, they need 35. How would

this affect Aunt Anastasia's profits?

A) Profits would increase by $5.

B) Profits would decrease by $5

C) Profits would increase by $2.50

D) Profits would decrease by $2.50

E) Cannot tell from the information provided.

Answer: B




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76) Aunt Anastasia feels that her prices are too low, particularly for her eggs. How much would

her profit have to increase on the eggs before it is profitable for her to make and sell eggs?

A) $0.50

B) $1.00

C) $1.50

D) $2.50

E) None of the above

Answer: B




77) Aunt Anastasia's available hours for paint and seal have fallen from 80 hours to 60 hours

because of other commitments. How will this affect her profits?

A) Profits will decrease by $30.

B) Profits will increase by $30.

C) Profits will decrease by $20.

D) Profits will increase by $20.

E) Profits will not change.

Answer: E




78) Aunt Anastasia can obtain an additional 10 hours of kiln capacity free of charge from a

friend. If she did this, how would her profits be affected?

A) Profit would increase by $25.

B) Profits would decrease by $25.

C) Profits would increase by $6.25.

D) Profits would decrease by $6.25


Answer: C




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79) Aunt Anastasia is planning for next spring, and she is considering making only 2 products.

Based on the results from the linear program, which two products would you recommend that

she make?

A) baskets and eggs

B) baskets and rabbits

C) eggs and rabbits

D) She should continue to make all 3.


Answer: B




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Billy's Blues sells 3 types of T-shirts: Astro, Bling, and Curious. Manufacturing Astros

requires 2 minutes of machine time, 20 minutes of labor, and costs $10. Brand Bling requires

2..5 minutes of machine time, 30 minutes of labor, and costs $14 to produce. Brand Curious

requires 3 minutes of machine time, 45 minutes of labor, and costs $18 to produce. There are

300 machining hours available per week, 3,750 labor hours, and he has a budget of $3,000.

Brand Astro sells for $15, Brand Bling for $18, and Brand Curious for $25.

The LP formulation that maximizes week profit shown below.

MAX 15A +18B + 25 C

s.t.

2A + 2.5B + 3C ≤ 300

20A + 30B + 45C ≤ 3,750

10A + 14B + 18C ≤ 3,000

The solution from QM for Windows is show below.

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80) If Billy could acquire more of any resource, which would it be?

A) machining time

B) labor time

C) money

D) buyers

Answer: A


Main Heading: Linear Programming Models


81) If one of Billy's machines breaks down, it usually results in about 6 hours of downtime.

When this happens, Billy's profits are reduced by

A) $15

B) 18

C) $25

D) $35

Answer: D


Main Heading: Linear Programming Models


82) Billy's accountant made an error, and the budget has been reduced from $3000 to $2500.

Billy's profit will go down by

A) $0

B) $625

C) $1350

D) $1650

Answer: A


Main Heading: Irregular Types of Linear Programming Problems


83) Billy has decided that he can raise the price on the Curious t-shirt by 10% without losing

sales. If he raises the price, his profits will

A) increase by 10%

B) decrease by 10%

C) increase by $2.50

D) increase by $125

Answer: D


Main Heading: Irregular Types of Linear Programming Problems


Introduction to Management Science, 10e (Taylor) Chapter 3 Linear Programming: Computer Solution and...

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