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.

 

 

 

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.

 

 

 

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.

 

 

 

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

 

 

 

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

 

 

 

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.

 

 

7) The sensitivity range for an objective function coefficient is the range of values over which the profit does not change.

 

 

 

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

 

 

 

9) If we change the constraint quantity to a value outside the sensitivity range for that constraint quantity, the shadow price will change.

 

 

 

10) The sensitivity range for a constraint quantity value is the range over which the optimal values of the decision variables do not change.

 

 

 

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

 

Answer:  FALSE

 

 

 

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

 

 

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

 

 

 

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

 

 

 

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

 

 

 

16) For a profit maximization problem, if the allowable increase for a coefficient in the objective function is infinite, then profits are unbounded.

 

 

 

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

 

 

 

18) The sensitivity range for a __________ is the range of values over which the quantity values can change without changing the shadow price

 

 

 

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

 

 

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

 

 

 

21) The sensitivity range for an __________ coefficient is the range of values over which the current optimal solution point (product mix) will remain optimal.

 

 

 

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 __________.

 

 

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 __________.

 

 

 

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

 

Key words:  computer solution

 

 

 

25) If downtime reduced the available capacity for painting by 40 hours (from 300 to 260 hours), profits would be reduced by __________.

 

 

 

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

 

 

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?

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

 

 

 

29) What is the shadow price for assembly?

 

 

 

30) What is the shadow price for fabrication?

 

 

 

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

 

 

 

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?

 

 

 

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

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?

 

 

 

35)  How many cases of regular and how many cases of diet soft drink should Whoppy produce to maximize daily profit?

 

 

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

 

 

 

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

 

 

 

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?

 

 

 

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?

 

Key words:  formulation, objective function

 

 

 

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

 

 

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?

 

 

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

 

 

 

Right Hand Side Ranges

 

 

 

 

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

 

 

 

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?

 

 

 

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 solution (product mix) would change?

 

d)   What is the lowest value for the amount of time available to setup the display before the solution (product mix) would change?

 

 

 

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?

 

 

46) Max Z = 5x1 + 3x2

 

Subject to: 6x1 + 2x2 ≤ 18

 

                  15x1 + 20x2 ≤ 60

 

                  x1 + x2 ≥ 0

 

Determine the sensitivity range for each constraint.

 

Main Heading:  Sensitivity Analysis and Computer Solution

 

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

 

 

 

47) Max Z = 5x1 + 3x2

 

Subject to:       6x1 + 2x2 ≤ 18

 

                        15×1 + 20×2 ≤ 60

 

                        x1 + x2 ≥ 0

 

Determine the sensitivity range for each objective function coefficient.

 

 

 

48) Max           Z = 3x1 + 3x2

 

Subject to:       10x1 + 4x2 ≤ 60

 

                        25x1 + 50x2 ≤ 200

 

                        x1 , x2 ≥ 0

 

Determine the sensitivity range for each objective function coefficient.

 

Main Heading:  Sensitivity Analysis and Computer Solution

 

Key words:  sensitivity analysis/range for objective function coefficients

 

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

 

 

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

 

 

 

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

 

 

 

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

53) What is the optimal weekly profit?

 

A) $1000

 

B) $900

 

C) $800

 

D) $700

 

E) $600

 

 

 

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

 

 

 

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

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                    

 

 

 

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

 

 

 

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

 

 

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

 

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

 

 

 

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

 

 

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

 

 

 

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

 

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

 

 

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

 

 

 

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

 

 

 

67) At the optimal solution the minimum cost is: 

 

A) $30

 

B) $40

 

C) $50

 

D) $52

 

E) $53.33

 

 

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

 

 

 

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

 

 

 

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

 

 

 

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

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)

 

 

 

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

 

 

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:

 

 

 

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

 

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.

 

 

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

 

 

 

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.

 

 

 

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

 

E) Cannot tell from the information provided.

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.

 

E) Cannot tell from the information provided.

 

 

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.

 

 

 

 

 

 

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

 

A) machining time

 

B) labor time

 

C) money

 

D) buyers

 

 

 

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

 

 

 

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

 

 

 

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