# Graphical and Computer Methods

Quantitative Analysis for Management, 11e (Render)

Chapter 7   Linear Programming Models: Graphical and Computer Methods

1) Management resources that need control include machinery usage, labor volume, money spent, time used, warehouse space used, and material usage.

Diff: 1

Topic:  INTRODUCTION

2) In the term linear programming, the word programming comes from the phrase “computer programming.”

Diff: 2

Topic:  INTRODUCTION

3) One of the assumptions of LP is “simultaneity.”

Diff: 2

Topic:  REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM

4) Any linear programming problem can be solved using the graphical solution procedure.

Diff: 1

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

5) An LP formulation typically requires finding the maximum value of an objective while simultaneously maximizing usage of the resource constraints.

Diff: 2

Topic:  FORMULATING LP PROBLEMS

6) There are no limitations on the number of constraints or variables that can be graphed to solve an LP problem.

Diff: 1

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

7) Resource restrictions are called constraints.

Diff: 1

Topic:  REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM

8) One of the assumptions of LP is “proportionality.”

Diff: 2

Topic:  REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM

9) The set of solution points that satisfies all of a linear programming problem’s constraints simultaneously is defined as the feasible region in graphical linear programming.

Diff: 2

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

10) An objective function is necessary in a maximization problem but is not required in a minimization problem.

Diff: 1

Topic:  REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM

11) In some instances, an infeasible solution may be the optimum found by the corner point method.

Diff: 2

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

12) The rationality assumption implies that solutions need not be in whole numbers (integers).

Diff: 2

Topic:  REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM

13) The solution to a linear programming problem must always lie on a constraint.

Diff: 2

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

14) In a linear program, the constraints must be linear, but the objective function may be nonlinear.

Diff: 2

Topic:  REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM

15) Resource mix problems use LP to decide how much of each product to make, given a series of resource restrictions.

Diff: 2

Topic:  FORMULATING LP PROBLEMS

16) The existence of non-negativity constraints in a two-variable linear program implies that we are always working in the northwest quadrant of a graph.

Diff: 2

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

17) In linear programming terminology, “dual price” and “sensitivity price” are synonyms.

Diff: 2

Topic:  SENSITIVITY ANALYSIS

18) Any time that we have an isoprofit line that is parallel to a constraint, we have the possibility of multiple solutions.

Diff: 2

Topic:  FOUR SPECIAL CASES IN LP

19) If the isoprofit line is not parallel to a constraint, then the solution must be unique.

Diff: 2

Topic:  FOUR SPECIAL CASES IN LP

AACSB:  Reflective Thinking

20) When two or more constraints conflict with one another, we have a condition called unboundedness.

Diff: 2

Topic:  FOUR SPECIAL CASES IN LP

21) The addition of a redundant constraint lowers the isoprofit line.

Diff: 2

Topic:  FOUR SPECIAL CASES IN LP

22) Sensitivity analysis enables us to look at the effects of changing the coefficients in the objective function, one at a time.

Diff: 2

Topic:  SENSITIVITY ANALYSIS

23) A widely used mathematical programming technique designed to help managers and decision making relative to resource allocation is called ________.

1. A) linear programming
2. B) computer programming
3. C) constraint programming
4. D) goal programming
5. E) None of the above

Diff: 1

Topic:  INTRODUCTION

24) Typical resources of an organization include ________.

1. A) machinery usage
2. B) labor volume
3. C) warehouse space utilization
4. D) raw material usage
5. E) All of the above

Diff: 1

Topic:  INTRODUCTION

25) Which of the following is not a property of all linear programming problems?

1. A) the presence of restrictions
2. B) optimization of some objective
3. C) a computer program
4. D) alternate courses of action to choose from
5. E) usage of only linear equations and inequalities

Diff: 2

Topic:  REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM

26) A feasible solution to a linear programming problem

1. A) must be a corner point of the feasible region.
2. B) must satisfy all of the problem’s constraints simultaneously.
3. C) need not satisfy all of the constraints, only the non-negativity constraints.
4. D) must give the maximum possible profit.
5. E) must give the minimum possible cost.

Diff: 2

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

27) Infeasibility in a linear programming problem occurs when

1. A) there is an infinite solution.
2. B) a constraint is redundant.
3. C) more than one solution is optimal.
4. D) the feasible region is unbounded.
5. E) there is no solution that satisfies all the constraints given.

Diff: 2

Topic:  FOUR SPECIAL CASES IN LP

28) In a maximization problem, when one or more of the solution variables and the profit can be made infinitely large without violating any constraints, the linear program has

1. A) an infeasible solution.
2. B) an unbounded solution.
3. C) a redundant constraint.
4. D) alternate optimal solutions.
5. E) None of the above

Diff: 2

Topic:  FOUR SPECIAL CASES IN LP

29) Which of the following is not a part of every linear programming problem formulation?

1. A) an objective function
2. B) a set of constraints
3. C) non-negativity constraints
4. D) a redundant constraint
5. E) maximization or minimization of a linear function

Diff: 2

Topic:  REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM

30) When appropriate, the optimal solution to a maximization linear programming problem can be found by graphing the feasible region and

1. A) finding the profit at every corner point of the feasible region to see which one gives the highest value.
2. B) moving the isoprofit lines towards the origin in a parallel fashion until the last point in the feasible region is encountered.
3. C) locating the point that is highest on the graph.
4. D) None of the above
5. E) All of the above

Diff: 2

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

31) The mathematical theory behind linear programming states that an optimal solution to any problem will lie at a(n) ________ of the feasible region.

1. A) interior point or center
2. B) maximum point or minimum point
3. C) corner point or extreme point
4. D) interior point or extreme point
5. E) None of the above

Diff: 1

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

32) Which of the following is not a property of linear programs?

1. A) one objective function
2. B) at least two separate feasible regions
3. C) alternative courses of action
4. D) one or more constraints
5. E) objective function and constraints are linear

Diff: 1

Topic:  REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM

33) The corner point solution method

1. A) will always provide one, and only one, optimum.
2. B) will yield different results from the isoprofit line solution method.
3. C) requires that the profit from all corners of the feasible region be compared.
4. D) requires that all corners created by all constraints be compared.
5. E) will not provide a solution at an intersection or corner where a non-negativity constraint is involved.

Diff: 2

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

34) When a constraint line bounding a feasible region has the same slope as an isoprofit line,

1. A) there may be more than one optimum solution.
2. B) the problem involves redundancy.
3. C) an error has been made in the problem formulation.
4. D) a condition of infeasibility exists.
5. E) None of the above

Diff: 2

Topic:  FOUR SPECIAL CASES IN LP

35) The simultaneous equation method is

1. A) an alternative to the corner point method.
2. B) useful only in minimization methods.
3. C) an algebraic means for solving the intersection of two or more constraint equations.
4. D) useful only when more than two product variables exist in a product mix problem.
5. E) None of the above

Diff: 2

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

36) Consider the following linear programming problem:

The maximum possible value for the objective function is

1. A) 360.
2. B) 480.
3. C) 1520.
4. D) 1560.
5. E) None of the above

Diff: 3

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

37) Consider the following linear programming problem:

The feasible corner points are (48,84), (0,120), (0,0), (90,0).  What is the maximum possible value for the objective function?

1. A) 1032
2. B) 1200
3. C) 360
4. D) 1600
5. E) None of the above

Diff: 2

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

38) Consider the following linear programming problem:

Which of the following points (X,Y) is not a feasible corner point?

1. A) (0,60)
2. B) (105,0)
3. C) (120,0)
4. D) (100,10)
5. E) None of the above

Diff: 2

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

39) Consider the following linear programming problem:

Which of the following points (X,Y) is not feasible?

1. A) (50,40)
2. B) (20,50)
3. C) (60,30)
4. D) (90,10)
5. E) None of the above

Diff: 2

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

40) Two models of a product — Regular (X) and Deluxe (Y) — are produced by a company.  A linear programming model is used to determine the production schedule.  The formulation is as follows:

The optimal solution is X = 100, Y = 0.

How many units of the regular model would be produced based on this solution?

1. A) 0
2. B) 100
3. C) 50
4. D) 120
5. E) None of the above

Diff: 1

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

41) Two models of a product — Regular (X) and Deluxe (Y) — are produced by a company.  A linear programming model is used to determine the production schedule.  The formulation is as follows:

The optimal solution is X=100, Y=0.

Which of these constraints is redundant?

1. A) the first constraint
2. B) the second constraint
3. C) the third constraint
4. D) All of the above
5. E) None of the above

Diff: 3

Topic:  FOUR SPECIAL CASES IN LP

AACSB:  Analytic Skills

42) Consider the following linear programming problem:

What is the optimum solution to this problem (X,Y)?

1. A) (0,0)
2. B) (50,0)
3. C) (0,100)
4. D) (400,0)
5. E) None of the above

Diff: 3

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

43) Consider the following linear programming problem:

This is a special case of a linear programming problem in which

1. A) there is no feasible solution.
2. B) there is a redundant constraint.
3. C) there are multiple optimal solutions.
4. D) this cannot be solved graphically.
5. E) None of the above

Diff: 3

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

44) Consider the following linear programming problem:

This is a special case of a linear programming problem in which

1. A) there is no feasible solution.
2. B) there is a redundant constraint.
3. C) there are multiple optimal solutions.
4. D) this cannot be solved graphically.
5. E) None of the above

Diff: 3

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

45) Which of the following is not acceptable as a constraint in a linear programming problem (maximization)?

1. A) Constraint 1
2. B) Constraint 2
3. C) Constraint 3
4. D) Constraint 4
5. E) None of the above

Diff: 2

Topic:  REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM

46) If one changes the contribution rates in the objective function of an LP,

1. A) the feasible region will change.
2. B) the slope of the isoprofit or isocost line will change.
3. C) the optimal solution to the LP is sure to no longer be optimal.
4. D) All of the above
5. E) None of the above

Diff: 2

Topic:  SENSITIVITY ANALYSIS

47) Sensitivity analysis may also be called

1. A) postoptimality analysis.
2. B) parametric programming.
3. C) optimality analysis.
4. D) All of the above
5. E) None of the above

Diff: 2

Topic:  SENSITIVITY ANALYSIS

48) Sensitivity analyses are used to examine the effects of changes in

1. A) contribution rates for each variable.
2. B) technological coefficients.
3. C) available resources.
4. D) All of the above
5. E) None of the above

Diff: 2

Topic:  SENSITIVITY ANALYSIS

49) Which of the following is a basic assumption of linear programming?

1. A) The condition of uncertainty exists.
2. B) Independence exists for the activities.
3. C) Proportionality exists in the objective function and constraints.
4. D) Divisibility does not exist, allowing only integer solutions.
5. E) Solutions or variables may take values from -∞ to +∞.

Diff: 2

Topic:  REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM

50) The condition when there is no solution that satisfies all the constraints simultaneously is called

1. A) boundedness.
2. B) redundancy.
3. C) optimality.
4. D) dependency.
5. E) None of the above

Diff: 2

Topic:  FOUR SPECIAL CASES IN LP

51) If the addition of a constraint to a linear programming problem does not change the solution, the constraint is said to be

1. A) unbounded.
2. B) non-
3. C) infeasible.
4. D) redundant.
5. E) bounded.

Diff: 2

Topic:  FOUR SPECIAL CASES IN LP

52) Which of the following is not an assumption of LP?

1. A) simultaneity
2. B) certainty
3. C) proportionality
4. D) divisibility

Diff: 2

Topic:  REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM

53) The difference between the left-hand side and right-hand side of a less-than-or-equal-to constraint is referred to as

1. A) surplus.
2. B) constraint.
3. C) slack.
5. E) None of the above

Diff: 2

Topic:  LINEAR PROGRAMMING MODELS: GRAPHICAL AND COMPUTER METHODS

54) The difference between the left-hand side and right-hand side of a greater-than-or-equal-to constraint is referred to as

1. A) surplus.
2. B) constraint.
3. C) slack.
5. E) None of the above

Diff: 2

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

55) A constraint with zero slack or surplus is called a

1. A) nonbinding constraint.
2. B) resource constraint.
3. C) binding constraint.
4. D) nonlinear constraint.
5. E) linear constraint.

Diff: 1

Topic:  SOLVING FLAIR FURNITURE’S LP PROBLEM USING QM FOR WINDOWS AND EXCEL

56) A constraint with positive slack or surplus is called a

1. A) nonbinding constraint.
2. B) resource constraint.
3. C) binding constraint.
4. D) nonlinear constraint.
5. E) linear constraint.

Diff: 1

Topic:  SOLVING FLAIR FURNITURE’S LP PROBLEM USING QM FOR WINDOWS AND EXCEL

57) The increase in the objective function value that results from a one-unit increase in the right-hand side of that constraint is called

1. A) surplus.
3. C) slack.
4. D) dual price.
5. E) None of the above

Diff: 2

Topic:  SENSITIVITY ANALYSIS

58) A straight line representing all non-negative combinations of X1 and X2 for a particular profit level is called a(n)

1. A) constraint line.
2. B) objective line.
3. C) sensitivity line.
4. D) profit line.
5. E) isoprofit line.

Diff: 2

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

59) In order for a linear programming problem to have a unique solution, the solution must exist

1. A) at the intersection of the non-negativity constraints.
2. B) at the intersection of a non-negativity constraint and a resource constraint.
3. C) at the intersection of the objective function and a constraint.
4. D) at the intersection of two or more constraints.
5. E) None of the above

Diff: 2

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

60) In order for a linear programming problem to have multiple solutions, the solution must exist

1. A) at the intersection of the non-negativity constraints.
2. B) on a non-redundant constraint parallel to the objective function.
3. C) at the intersection of the objective function and a constraint.
4. D) at the intersection of three or more constraints.
5. E) None of the above

Diff: 2

Topic:  FOUR SPECIAL CASES IN LP

61) Consider the following linear programming problem:

The maximum possible value for the objective function is

1. A) 360.
2. B) 480.
3. C) 1520.
4. D) 1560.
5. E) None of the above

Diff: 3

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

62) Consider the following linear programming problem:

Which of the following points (X,Y) is feasible?

1. A) (10,120)
2. B) (120,10)
3. C) (30,100)
4. D) (60,90)
5. E) None of the above

Diff: 2

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

63) Consider the following linear programming problem:

Which of the following points (X,Y) is in the feasible region?

1. A) (30,60)
2. B) (105,5)
3. C) (0,210)
4. D) (100,10)
5. E) None of the above

Diff: 2

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

64) Consider the following linear programming problem:

Which of the following points (X,Y) is feasible?

1. A) (50,40)
2. B) (30,50)
3. C) (60,30)
4. D) (90,20)
5. E) None of the above

Diff: 2

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

65) Which of the following is not an assumption of LP?

1. A) certainty
2. B) proportionality
3. C) divisibility
4. D) multiplicativity

Diff: 2

Topic:  REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM

66) Consider the following linear programming problem:

This is a special case of a linear programming problem in which

1. A) there is no feasible solution.
2. B) there is a redundant constraint.
3. C) there are multiple optimal solutions.
4. D) this cannot be solved graphically.
5. E) None of the above

Diff: 3

Topic:  FOUR SPECIAL CASES IN LP

AACSB:  Analytic Skills

67) Which of the following functions is not linear?

1. A) 5X + 3Z
2. B) 3X + 4Y + Z – 3
3. C) 2X + 5YZ
4. D) Z
5. E) 2X – 5Y + 2Z

Diff: 1

Topic:  REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM

68) Which of the following is not one of the steps in formulating a linear program?

1. A) Graph the constraints to determine the feasible region.
2. B) Define the decision variables.
3. C) Use the decision variables to write mathematical expressions for the objective function and the constraints.
4. D) Identify the objective and the constraints.
5. E) Completely understand the managerial problem being faced.

Diff: 2

Topic:  FORMULATING LP PROBLEMS

69) Which of the following is not acceptable as a constraint in a linear programming problem (minimization)?

1. A) Constraint 1
2. B) Constraint 2
3. C) Constraint 3
4. D) Constraint 4
5. E) Constraint 5

Diff: 2

Topic:  REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM

70) What type of problems use LP to decide how much of each product to make, given a series of resource restrictions?

1. A) resource mix
2. B) resource restriction
3. C) product restriction
4. D) resource allocation
5. E) product mix

Diff: 2

Topic:  FORMULATING LP PROBLEMS

71) Consider the following linear programming problem:

This is a special case of a linear programming problem in which

1. A) there is no feasible solution.
2. B) there is a redundant constraint.
3. C) there are multiple optimal solutions.
4. D) this cannot be solved graphically.
5. E) None of the above

Diff: 3

Topic:  FOUR SPECIAL CASES IN LP

AACSB:  Analytic Skills

72) Consider the following constraints from a linear programming problem:

2X + Y ≤ 200

X + 2Y ≤ 200

X, Y ≥ 0

If these are the only constraints, which of the following points (X,Y) cannot be the optimal solution?

1. A) (0, 0)
2. B) (0, 200)
3. C) (0,100)
4. D) (100, 0)
5. E) (66.67, 66.67)

Diff: 3

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

73) Consider the following constraints from a linear programming problem:

2X + Y ≤ 200

X + 2Y ≤ 200

X, Y ≥ 0

If these are the only constraints, which of the following points (X,Y) cannot be the optimal solution?

1. A) (0, 0)
2. B) (0, 100)
3. C) (65, 65)
4. D) (100, 0)
5. E) (66.67, 66.67)

Diff: 3

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

74) A furniture company is producing two types of furniture. Product A requires 8 board feet of wood and 2 lbs of wicker. Product B requires 6 board feet of wood and 6 lbs of wicker. There are 2000 board feet of wood available for product and 1000 lbs of wicker. Product A earns a profit margin of \$30 a unit and Product B earns a profit margin of \$40 a unit. Formulate the problem as a linear program.

Diff: 2

Topic:  FORMULATING LP PROBLEMS

AACSB:  Analytic Skills

75) As a supervisor of a production department, you must decide the daily production totals of a certain product that has two models, the Deluxe and the Special.  The profit on the Deluxe model is \$12 per unit and the Special’s profit is \$10.  Each model goes through two phases in the production process, and there are only 100 hours available daily at the construction stage and only 80 hours available at the finishing and inspection stage.  Each Deluxe model requires 20 minutes of construction time and 10 minutes of finishing and inspection time.  Each Special model requires 15 minutes of construction time and 15 minutes of finishing and inspection time.  The company has also decided that the Special model must comprise at least 40 percent of the production total.

(a)           Formulate this as a linear programming problem.

(b)           Find the solution that gives the maximum profit.

(b)  Optimal solution:  X1 = 120, X2 = 240      Profit = \$3,840

Diff: 3

Topic:  FORMULATING LP PROBLEMS and GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

76) The Fido Dog Food Company wishes to introduce a new brand of dog biscuits (composed of chicken and liver-flavored biscuits) that meets certain nutritional requirements.  The liver-flavored biscuits contain 1 unit of nutrient A and 2 units of nutrient B, while the chicken-flavored ones contain 1 unit of nutrient A and 4 units of nutrient B.  According to federal requirements, there must be at least 40 units of nutrient A and 60 units of nutrient B in a package of the new biscuit mix.  In addition, the company has decided that there can be no more than 15 liver-flavored biscuits in a package.  If it costs 1 cent to make a liver-flavored biscuit and 2 cents to make a chicken-flavored one, what is the optimal product mix for a package of the biscuits in order to minimize the firm’s cost?

(a)           Formulate this as a linear programming problem.

(b)           Find the optimal solution for this problem graphically.

(c)           Are any constraints redundant?  If so, which one or ones?

(d)           What is the total cost of a package of dog biscuits using the optimal mix?

(b)   Corner points (0,40) and (15,25)

Optimal solution is (15,25) with cost of 65.

(c)   2X1 + 4X2 ≥ 60 is redundant.

(d)   minimum cost = 65 cents

Diff: 3

Topic:  FORMULATING LP PROBLEMS and GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

77) Consider the following linear program:

(a)           Solve the problem graphically.  Is there more than one optimal solution?  Explain.

(b)           Are there any redundant constraints?

(a)   Corner points (0,50), (0,200), (50,50), (75,75), (50,150)

Optimum solutions:  (75,75) and (50,150).  Both yield a profit of \$3,000.

(b)   The constraint X1 ≤ 100 is redundant since 3X1 + X2 ≤ 300 also means that X1 cannot exceed 100.

Diff: 3

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

78) Solve the following linear programming problem using the corner point method:

Answer:  Feasible corner points (X,Y): (0,3) (0,10) (2.4,8.8) (6.75,3)

Maximum profit 70.5 at (6.75,3).

Diff: 3

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

79) Solve the following linear programming problem using the corner point method:

Answer:  Feasible corner points (X,Y): (0,2) (0,10) (4,8) (10,2)

Maximum profit is 52 at (4,8).

Diff: 3

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

80) Billy Penny is trying to determine how many units of two types of lawn mowers to produce each day. One of these is the Standard model, while the other is the Deluxe model.  The profit per unit on the Standard model is \$60, while the profit per unit on the Deluxe model is \$40.  The Standard model requires 20 minutes of assembly time, while the Deluxe model requires 35 minutes of assembly time.  The Standard model requires 10 minutes of inspection time, while the Deluxe model requires 15 minutes of inspection time.  The company must fill an order for 6 Deluxe models.  There are 450 minutes of assembly time and 180 minutes of inspection time available each day.  How many units of each product should be manufactured to maximize profits?

Maximum profit is \$780 by producing 9 Standard and 6 Deluxe models.

Diff: 3

Topic:  FORMULATING LP PROBLEMS and GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

81) Two advertising media are being considered for promotion of a product.  Radio ads cost \$400 each, while newspaper ads cost \$600 each.  The total budget is \$7,200 per week.  The total number of ads should be at least 15, with at least 2 of each type.  Each newspaper ad reaches 6,000 people, while each radio ad reaches 2,000 people.  The company wishes to reach as many people as possible while meeting all the constraints stated.  How many ads of each type should be placed?

Feasible corner points (R,N): (9,6) (13,2) (15,2)

Diff: 3

Topic:  FORMULATING LP PROBLEMS and GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

82) Suppose a linear programming (minimization) problem has been solved and the optimal value of the objective function is \$300.  Suppose an additional constraint is added to this problem. Explain how this might affect each of the following:

(a)  the feasible region,

(b)  the optimal value of the objective function.

(a)  Adding a new constraint will reduce the size of the feasible region unless it is a redundant constraint.  It can never make the feasible region any larger. However, it could make the problem infeasible.

(b)  A new constraint can only reduce the size of the feasible region; therefore, the value of the objective function will either increase or remain the same.  If the original solution is still feasible, it will remain the optimal solution.

Diff: 2

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM and FOUR SPECIAL CASES IN LP

AACSB:  Analytic Skills

83) Upon retirement, Mr. Klaws started to make two types of children’s wooden toys in his shopWuns and Toos.  Wuns yield a variable profit of \$9 each and Toos have a contribution margin of \$8 each. Even though his electric saw overheats, he can make 7 Wuns or 14 Toos each day.  Since he doesn’t have equipment for drying the lacquer finish he puts on the toys, the drying operation limits him to 16 Wuns or 8 Toos per day.

(a)           Solve this problem using the corner point method.

(b)           For what profit ratios would the optimum solution remain the optimum solution?

Corner points (0,0), (7,0), (0,8), (4,6)

Optimum profit \$84 at (4,6).

Diff: 3

Topic:  FORMULATING LP PROBLEMS and GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

84) Susanna Nanna is the production manager for a furniture manufacturing company.  The company produces tables (X) and chairs (Y).  Each table generates a profit of \$80 and requires 3 hours of assembly time and 4 hours of finishing time.  Each chair generates \$50 of profit and requires 3 hours of assembly time and 2 hours of finishing time.  There are 360 hours of assembly time and 240 hours of finishing time available each month.  The following linear programming problem represents this situation.

Maximize            80X + 50Y

Subject to:            3X + 3Y ≤ 360

4X + 2Y ≤ 240

X, Y ≥ 0

The optimal solution is X = 0, and Y = 120.

(a)           What would the maximum possible profit be?

(b)           How many hours of assembly time would be used to maximize profit?

(c)           If a new constraint, 2X + 2Y ≤ 400, were added, what would happen to the maximum possible profit?

(a) 6000, (b) 360, (c) It would not change.

Diff: 3

Topic:  FORMULATING LP PROBLEMS and GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

85) As a supervisor of a production department, you must decide the daily production totals of a certain product that has two models, the Deluxe and the Special.  The profit on the Deluxe model is \$12 per unit, and the Special’s profit is \$10.  Each model goes through two phases in the production process, and there are only 100 hours available daily at the construction stage and only 80 hours available at the finishing and inspection stage.  Each Deluxe model requires 20 minutes of construction time and 10 minutes of finishing and inspection time.  Each Special model requires 15 minutes of construction time and 15 minutes of finishing and inspection time.  The company has also decided that the Special model must comprise at most 60 percent of the production total.

Formulate this as a linear programming problem.

Diff: 2

Topic:  FORMULATING LP PROBLEMS

AACSB:  Analytic Skills

86) Determine where the following two constraints intersect.

5X + 23Y ≤ 1000

10X + 26Y ≤ 1600

Diff: 1

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

87) Determine where the following two constraints intersect.

2X – 4Y = 800

−X + 6Y ≥ -200

Diff: 1

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

Diff: 2

Topic:  FORMULATING LP PROBLEMS

AACSB:  Analytic Skills

89) Suppose a linear programming (maximization) problem has been solved and the optimal value of the objective function is \$300.  Suppose a constraint is removed from this problem. Explain how this might affect each of the following:

(a)           the feasible region.

(b)           the optimal value of the objective function.

(a)           Removing a constraint may, if the constraint is not redundant, increase the size of the feasible region.  It can never make the feasible region any smaller.  If the constraint was active in the solution, removing it will also result in a new optimal solution. However, removing an essential constraint could cause the problem to become unbounded.

(b)           Removal of a constraint can only increase or leave the same the size of the feasible region; therefore, the value of the objective function will either increase or remain the same, assuming the problem has not become unbounded.

Diff: 2

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM and FOUR SPECIAL CASES IN LP

AACSB:  Analytic Skills

90) Consider the following constraints from a two-variable linear program.

(1) X ≥ 0

(2) Y ≥ 0

(3) X + Y ≤ 50

If the optimal corner point lies at the intersection of constraints (2) and (3), what is the optimal solution (X, Y)?

Answer:  Y = 0, so X + 0 = 50, or X = 50.  Thus the solution is (50, 0).

Diff: 1

Topic:  GRAPHICAL SOLUTION TO AN LP PROBLEM

AACSB:  Analytic Skills

91) Consider a product mix problem, where the decision involves determining the optimal production levels for products X and Y.  A unit of X requires 4 hours of labor in department 1 and 6 hours a labor in department 2.  A unit of Y requires 3 hours of labor in department 1 and 8 hours of labor in department 2.  Currently, 1000 hours of labor time are available in department 1, and 1200 hours of labor time are available in department 2.  Furthermore, 400 additional hours of cross-trained workers are available to assign to either department (or split between both).  Each unit of X sold returns a \$50 profit, while each unit of Y sold returns a \$60 profit.  All units produced can be sold.  Formulate this problem as a linear program.  (Hint: Consider introducing other decision variables in addition to the production amounts for X and Y.)

Maximize: 50X + 60Y

Diff: 3

Topic:  FORMULATING LP PROBLEMS

AACSB:  Analytic Skills

92) A plastic parts supplier produces two types of plastic parts used for electronics. Type 1 requires 30 minutes of labor and 45 minutes of machine time. Type 2 requires 60 minutes of machine hours and 75 minutes of labor. There are 600 hours available per week of labor and 800 machine hours available. The demand for custom molds and plastic parts are identical. Type 1 has a profit margin of \$25 a unit and Type 2 have a profit margin of \$45 a unit. The plastic parts supplier must choose the quantity of Product A and Product B to produce which maximizes profit.

(a)           Formulate this as a linear programming problem.

(b)           Find the solution that gives the maximum profit using either QM for Windows or Excel.

(b)           X1 = 914.28, X2 = 114.28

Diff: 2

Topic:  FORMULATING LP PROBLEMS and SOLVING FLAIR FURNITURE’S PROBLEM USING QM FOR WINDOWS and EXCEL

AACSB:  Analytic Skills

93) A company can decide how many additional labor hours to acquire for a given week. Subcontractor workers will only work a maximum of 20 hours a week. The company must produce at least 200 units of product A, 300 units of product B, and 400 units of product C. In 1 hour of work, worker 1 can produce 15 units of product A, 10 units of product B, and 30 units of product C. Worker 2 can produce 5 units of product A, 20 units of product B, and 35 units of product C. Worker 3 can produce 20 units of product A, 15 units of product B, and 25 units of product C. Worker 1 demands a salary of  \$50/hr, worker 2 demands a salary of \$40/hr, and worker 3 demands a salary of \$45/hr. The company must choose how many hours they should contract with each worker to meet their production requirements and minimize labor cost.

(a)           Formulate this as a linear programming problem.

(b)           Find the optimal solution.

(b) X1 = 0, X2 = 9.23, X3 = 7.69

Diff: 3

Topic:  FORMULATING LP PROBLEMS and SOLVING FLAIR FURNITURE’S PROBLEM USING QM FOR WINDOWS and EXCEL

AACSB:  Analytic Skills

94) Define dual price.

Answer:  The dual price for a constraint is the improvement in the objective function value that results from a one-unit increase in the right-hand side of the constraint.

Diff: 2

Topic:  SENSITIVITY ANALYSIS

95) One basic assumption of linear programming is proportionality.  Explain its need.

Answer:  Rates of consumption exist and are constant.  For example, if the production of 1 unit requires 4 units of a resource, then if 10 units are produced, 40 units of the resource are required. A change in the variable value results in a proportional change in the objective function value.

Diff: 2

Topic:  REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM

96) One basic assumption of linear programming is divisibility.  Explain its need.

Answer:  Solutions can have fractional values and need not be whole numbers.  If fractional values would not make sense, then integer programming would be required.

Diff: 2

Topic:  REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM

97) Define infeasibility with respect to an LP solution.

Answer:  This occurs when there is no solution that can satisfy all constraints simultaneously.

Diff: 2

Topic:  FOUR SPECIAL CASES IN LP

98) Define unboundedness with respect to an LP solution.

Answer:  This occurs when a linear program has no finite solution.  The result implies that the formulation is missing one or more crucial constraints.

Diff: 2

Topic:  FOUR SPECIAL CASES IN LP

99) Define alternate optimal solutions with respect to an LP solution.

Answer:  More than one optimal solution point exist because the objective function is parallel to a binding constraint.

Diff: 2

Topic:  FOUR SPECIAL CASES IN LP

100) How does the case of alternate optimal solutions, as a special case in linear programming, compare to the two other special cases of infeasibility and unboundedness?

Answer:  With multiple alternate solutions, any of those answers is correct.  In the other two cases, no single answer can be generated.  Alternate solutions can occur when a problem is correctly formulated whereas the other two cases most likely have an incorrect formulation.

Diff: 2

Topic:  FOUR SPECIAL CASES IN LP

AACSB:  Reflective Thinking