Deck 23: Linear Programming

Question 1

Identify three examples of resources that are typically constrained in a linear programming problem.

Answer 1

The textbook gives machinery,labor,money,time,and raw materials.

Question 2

Which of the following is not a requirement of a linear programming problem?

A)an objective function,expressed in linear terms
B)constraints,expressed as linear equations or inequalities
C)an objective function to be maximized or minimized
D)alternative courses of action
E)one constraint or resource limit for each decision variable

Answer 2

one constraint or resource limit for each decision variable

Question 3

One requirement of a linear programming problem is that the objective function must be expressed as a linear equation.

Answer 3

True

Question 4

In which of the following has LP been applied successfully?

A)minimizing distance traveled by school buses carrying children
B)minimizing 911 response time for police patrols
C)minimizing labor costs for bank tellers while maintaining service levels
D)determining the distribution system for multiple warehouses to multiple destinations
E)all of the above

Question 5

The ________ is a mathematical expression in linear programming that maximizes or minimizes some quantity.

Question 6

A linear programming problem contains a restriction that reads "the quantity of S must be no less than one-fourth as large as T and U combined." Formulate this as a linear programming constraint.

A)S / (T + U)≥ 4
B)S - .25T -.25U ≥ 0
C)4S ≤ T + U
D)S ≥ 4T / 4U
E)S ≥ .25T + S ≥ .25U

Question 7

________ are restrictions that limit the degree to which a manager can pursue an objective.

Question 8

A firm makes two products,Y and Z.Each unit of Y costs $10 and sells for $40.Each unit of Z costs $5 and sells for $25.If the firm's goal were to maximize profit,what would be the appropriate objective function?

A)Maximize profit = $40Y = $25Z
B)Maximize profit = $40Y + $25Z
C)Maximize profit = $30Y + $20Z
D)Maximize profit = 0.25Y + 0.20Z
E)Maximize profit = $50(Y + Z)

Question 9

What are the requirements of all linear programming problems?

Question 10

Which of the following represents a valid constraint in linear programming?

A)2X ≥ 7XY
B)(2X)(7Y)≥ 500
C)2X + 7Y ≥100
D)2X2 + 7Y ≥ 50
E)All of the above are valid linear programming constraints.

Question 11

A linear programming problem contains a restriction that reads "the quantity of Q must be no larger than the sum of R,S,and T." Formulate this as a linear programming constraint.

A)Q + R + S + T ≤ 4
B)Q ≥ R + S + T
C)Q - R - S - T ≤ 0
D)Q / (R + S + T)≤ 0
E)Q ≤ R + Q ≤ S + Q ≤ T

Question 12

________ is a mathematical technique designed to help operations managers plan and make decisions necessary to allocate resources.

Question 13

A linear programming problem contains a restriction that reads "the quantity of X must be at least three times as large as the quantity of Y." Which of the following inequalities is the proper formulation of this constraint?

A)3X ≥ Y
B)X ≤ 3Y
C)X + Y ≥ 3
D)X - 3Y ≥ 0
E)3X ≤ Y

Question 14

In a linear programming formulation,a statement such as "maximize contribution" becomes a(n):

A)constraint.
B)slack variable.
C)objective function.
D)violation of linearity.
E)decision variable.

Question 15

Linear programming is an appropriate problem-solving technique for decisions that have no alternative courses of action.

Question 16

In linear programming,a statement such as "maximize contribution" becomes an objective function when the problem is formulated.

Question 17

The requirements of linear programming problems include an objective function,the presence of constraints,objective and constraints expressed in linear equalities or inequalities,and ________.

Question 18

Linear programming helps operations managers make decisions necessary to make effective use of resources such as machinery,labor,money,time,and raw materials.

Question 19

A common form of the product-mix linear programming problem seeks to find that combination of products and the quantity of each that maximizes profit in the presence of limited resources.

Question 20

What is linear programming?

Question 21

The region that satisfies all of the constraints in linear programming is called the region of optimality.

Question 22

The corner-point solution method requires:

A)identifying the corner of the feasible region that has the sharpest angle.
B)moving the iso-profit line to the highest level that still touches some part of the feasible region.
C)moving the iso-profit line to the lowest level that still touches some part of the feasible region.
D)finding the coordinates at each corner of the feasible solution space.
E)none of the above

Question 23

In linear programming,if there are three constraints,each representing a resource that can be used up,the optimal solution must use up all of each of the three resources.

Question 24

Which of the following combinations of constraints has no feasible region?

A)X + Y ≥ 15 and X - Y ≤ 10
B)X + Y ≥ 5 and X ≥ 10
C)X ≥ 10 and Y ≥ 20
D)X + Y ≥ 100 and X + Y ≤ 50
E)X ≤ -5

Question 25

In the graphical solution to a linear program,the region that satisfies the constraint 4X + 15Z ≥ 1000 includes the origin of the graph.

Question 26

A linear programming problem contains a restriction that reads "the quantity of Q must be at least as large as the sum of R,S,and T." Formulate this as a linear programming constraint.

Question 27

For the following constraints,which point is in the feasible region of this minimization problem?
(1)14x + 6y > 42
(2)x - y > 3

A)x = -1,y = 1
B)x = 0,y = 4
C)x = 2,y = 1
D)x = 5,y = 1
E)x = 2,y = 0

Question 28

Using the iso-profit line solution method to solve a maximization problem requires that we:

A)find the value of the objective function at the origin.
B)move the iso-profit line away from the origin until it barely touches some part of the feasible region.
C)move the iso-cost line to the lowest level that still touches some part of the feasible region.
D)test the objective function value of every corner point in the feasible region.
E)none of the above

Question 29

Which of the following sets of constraints results in an unbounded maximization problem?

A)X + Y ≥ 100 and X + Y ≤ 50
B)X + Y ≥ 15 and X - Y ≤ 10
C)X + Y ≤ 10 and X ≥ 5
D)X ≤ 10 and Y ≤ 20
E)All of the above have a bounded maximum.

Question 30

A linear programming problem contains a restriction that reads "the quantity of S must be no more than one-fourth as large as T and U combined." Formulate this as a linear programming constraint.

Question 31

The feasible region in the diagram below is consistent with which one of the following constraints?

A)8X1 + 4X2 ≥ 160
B)4X1 + 8X2 ≤ 160
C)8X1 - 4X2 ≤ 160
D)8X1 + 4X2 ≤ 160
E)4X1 - 8X2 ≤ 160

Question 32

The optimal solution to a linear programming problem lies within the feasible region.

Question 33

An iso-profit line:

A)can be used to help solve a profit maximizing linear programming problem.
B)is parallel to all other iso-profit lines in the same problem.
C)is a line with the same profit at all points.
D)all of the above
E)none of the above

Question 34

Solving a linear programming problem with the iso-profit line solution method requires that we move the iso-profit line to each corner of the feasible region until the optimum is identified.

Question 35

What is the region that satisfies all of the constraints in linear programming called?

A)area of optimal solutions
B)area of feasible solutions
C)profit maximization space
D)region of optimality
E)region of non-negativity

Question 36

In terms of linear programming,the fact that the solution is infeasible implies that the "profit" can increase without limit.

Question 37

A linear programming problem contains a restriction that reads "the quantity of X must be at least twice as large as the quantity of Y." Formulate this as a linear programming constraint.

Question 38

The feasible region in the diagram below is consistent with which one of the following constraints?

A)8X1 + 4X2 ≤ 160
B)8X1 + 4X2 ≥ 160
C)4X1 + 8X2 ≤ 160
D)8X1 - 4X2 ≤ 160
E)4X1 - 8X2 ≤ 160

Question 39

For the constraints given below,which point is in the feasible region of this maximization problem?
(1)14x + 6y ≤ 42
(2)x - y ≤ 3
(3)x,y ≥ 0

A)x = 2,y = 1
B)x = 1,y = 5
C)x = -1,y = 1
D)x = 4,y = 4
E)x = 2,y = 8

Question 40

For a linear programming problem with the constraints 2X + 4Y ≤ 100 and 1X + 8Y ≤ 100,two of its corner points are (0,0)and (0,25).

Question 41

A manager must decide on the mix of products to produce for the coming week.Product A requires three minutes per unit for molding,two minutes per unit for painting,and one minute for packing.Product B requires two minutes per unit for molding,four minutes for painting,and three minutes per unit for packing.There will be 600 minutes available for molding,600 minutes for painting,and 420 minutes for packing.Both products have contributions of $1.50 per unit.
a.Algebraically state the objective and constraints of this problem.
b.Plot the constraints on the grid below and identify the feasible region.

Question 42

The ________ is the set of all feasible combinations of the decision variables.

Question 43

A maximizing linear programming problem has two constraints: 2X + 4Y ≤ 100 and 3X + 10Y ≤ 210,in addition to constraints stating that both X and Y must be nonnegative.What are the corner points of the feasible region of this problem?

A)(0,0),(50,0),(0,21),and (20,15)
B)(0,0),(70,0),(25,0),and (15,20)
C)(20,15)
D)(0,0),(0,100),and (210,0)
E)(0,0),(0,25),(50,0),(0,21),and (70,0)

Question 44

What combination of x and y will yield the optimum for this problem?
Minimize $3x + $15y,subject to
(1)2x + 4y ≤ 12 and
(2)5x + 2y ≤ 10 and
(3)x,y ≥ 0.

A)x = 2,y = 0
B)x = 0,y = 3
C)x = 0,y = 0
D)x = 1,y = 5
E)x = 0,y = 5

Question 45

What combination of a and b will yield the optimum for this problem?
Maximize $6a + $15b,subject to
(1)4a + 2b ≤ 12 and
(2)5a + 2b ≤ 20 and
(3)x,y ≥ 0.

A)a = 0,b = 0
B)a = 3,b = 3
C)a = 0,b = 6
D)a = 6,b = 0
E)a = 0,b = 10

Question 46

Suppose that an iso-profit line is given to be X + Y = 10.Which of the following represents another iso-profit line for the same scenario?

A)X + Y = 15
B)X - Y = 10
C)Y - X = 10
D)2X + Y = 10
E)none of the above

Question 47

Suppose that the feasible region of a maximization LP problem has corners of (0,0),(5,0),and (0,5).How many possible combinations of X and Y will yield the maximum profit if profit is given to be 5X + 5Y?

A)0
B)1
C)2
D)5
E)Infinite

Question 48

What are corner points? What is their relevance to solving linear programming problems?

Question 49

What combination of x and y will yield the optimum for this problem?
Maximize $3x + $15y,subject to
(1)2x + 4y ≤ 12 and
(2)5x + 2y ≤ 10 and
(3)x,y ≥ 0.

A)x = 2,y = 0
B)x = 0,y = 3
C)x = 0,y = 0
D)x = 1,y = 5
E)x = 0,y = 5

Question 50

Which of the following correctly describes all iso-profit lines for an LP maximization problem?

A)They all pass through the origin
B)They are all parallel.
C)They all pass through the point of maximum profit.
D)Each line passes through at least 2 corners.
E)all of the above

Question 51

Two methods of solving linear programming problems by hand include the corner-point method and the ________.

Question 52

The objective of a linear programming problem is to maximize 1.50A + 1.50B,subject to 3A + 2B ≤ 600,2A + 4B ≤ 600,1A + 3B ≤ 420,and A,B ≥ 0.
a.Plot the constraints on the grid below
c.Identify the feasible region and its corner points.Show your work.
d.What is the optimal product mix for this problem?

Question 53

Consider the following constraints from a two-variable linear program:X ≥ 1;Y ≥ 1;X + Y ≤ 9. If these are the only constraints,which of the following points (X,Y)CANNOT be the optimal solution?

A)(1,1)
B)(1,8)
C)(8,1)
D)(4,4)
E)The question cannot be answered without knowing the objective function.

Question 54

A linear programming problem has three constraints,plus nonnegativity constraints on X and Y.The constraints are: 2X + 10Y ≤ 100; 4X + 6Y ≤ 120; 6X + 3Y ≤ 90. What is the largest quantity of X that can be made without violating any of these constraints?

A)50
B)30
C)20
D)15
E)10

Question 55

Suppose that the feasible region of a maximization LP problem has corners of (0,0),(10,0),(5,5),and (0,7).If profit is given to be $X + $2Y what is the maximum profit the company can earn?

A)$0
B)$10
C)$15
D)$14
E)$24

Question 56

Explain how to use the iso-profit line in a graphical solution to maximization problem.

Question 57

A linear programming problem has two constraints 2X + 4Y ≥ 100 and 1X + 8Y ≤ 100,plus nonnegativity constraints on X and Y.Which of the following statements about its feasible region is TRUE?

A)There are four corner points including (50,0)and (0,12.5).
B)The two corner points are (0,0)and (50,12.5).
C)The graphical origin (0,0)is in the feasible region.
D)The feasible region is triangular in shape,bounded by (50,0),(33.3333,8.3333),and (100,0).
E)The feasible region cannot be determined without knowing whether the problem is to be minimized or maximized.

Question 58

What is the feasible region in a linear programming problem?

Question 59

A linear programming problem has two constraints 2X + 4Y ≤ 100 and 1X + 8Y ≤ 100,plus nonnegativity constraints on X and Y.Which of the following statements about its feasible region is TRUE?

A)There are four corner points including (50,0)and (0,12.5).
B)The two corner points are (0,0)and (50,12.5).
C)The graphical origin (0,0)is not in the feasible region.
D)The feasible region includes all points that satisfy one constraint,the other,or both.
E)The feasible region cannot be determined without knowing whether the problem is to be minimized or maximized.

Question 60

A linear programming problem has three constraints,plus nonnegativity constraints on X and Y.The constraints are: 2X + 10Y ≤ 100; 4X + 6Y ≤ 120; 6X + 3Y ≥ 90. What is the largest quantity of X that can be made without violating any of these constraints?

A)50
B)30
C)20
D)15
E)10

Question 61

A difference between minimization and maximization problems is that:

A)minimization problems cannot be solved with the corner-point method.
B)maximization problems often have unbounded regions.
C)minimization problems often have unbounded regions.
D)minimization problems cannot have shadow prices.
E)minimization problems are more difficult to solve than maximization problems.

Question 62

The graphical method of solving linear programs can handle only maximization problems.

Question 63

Suppose that a constraint is given by X + Y ≤ 10.If another constraint is given to be 3X + 2Y ≥ 15,and if X and Y are restricted to be nonnegative,determine the corners of the feasible solution.If the profit from X is 5 and the profit from Y is 10,determine the combination of X and Y that will yield maximum profit.

Question 64

A manager must decide on the mix of products to produce for the coming week.Product A requires three minutes per unit for molding,two minutes per unit for painting,and one minute for packing.Product B requires two minutes per unit for molding,four minutes for painting,and three minutes per unit for packing.There will be 600 minutes available for molding,600 minutes for painting,and 420 minutes for packing.Both products have contributions of $1.50 per unit.Answer the following questions; base your work on the solution panel provided.
a.What combination of A and B will maximize contribution?
b.What is the maximum possible contribution?
c.Are any resources not fully used up? Explain.

Question 65

In linear programming,statements such as "the blend must consist of at least 10% of ingredient A,at least 30% of ingredient B,and no more than 50% of ingredient C" can be made into valid constraints even though the percentages do not add up to 100 percent.

Question 66

________ is an analysis that projects how much a solution might change if there were changes in the variables or input data.

Question 67

What is sensitivity analysis?

Question 68

A synonym for shadow price is ________.

Question 69

In sensitivity analysis,a zero shadow price (or dual value)for a resource ordinarily means that the resource has not been used up.

Question 70

The diet problem is known in agricultural applications as the:

A)fertilizer problem.
B)feed-mix problem.
C)crop-rotation problem.
D)egg-choice problem.
E)genetic-transformation problem.

Question 71

Which of the following is an algorithm for solving linear programming problems of all sizes?

A)duplex method
B)multiplex method
C)shadow price method
D)simplex method
E)decision tree method

Question 72

Two methods of conducting sensitivity analysis on solved linear programming problems are ________ and ________.

Question 73

Suppose that a constraint for assembly time has a shadow price of $50/hour for 15 hours in either direction and that all available assembly time is currently used (would require overtime to do more).If the salary of workers is $30 and they receive 50% extra pay for overtime what should management do?

Question 74

Constraints are needed to solve linear programming problems by hand,but not by computer.

Question 75

Suppose that the shadow price for assembly time is $5/hour.The allowable increase for the assembly time constraint is 40 hours,and the allowable decrease is 30 hours.If all assembly hours were used under the initial LP solution and workers normally make $4/hour but can work overtime for $6/hour,what should management do?

A)do not change available hours for assembly time
B)decrease available hours for assembly time by 30 hours
C)increase available hours for assembly time by 40 hours
D)decrease available hours for assembly time by 5 hours
E)increase available hours for assembly time by 5 hours

Question 76

Sensitivity analysis can be applied to linear programming solutions by either (1)trial and error or (2)the analytic postoptimality method.

Question 77

A shadow price (or dual value)reflects which of the following in a maximization problem?

A)the marginal gain in the objective realized by subtracting one unit of a resource
B)the market price that must be paid to obtain additional resources
C)the increase in profit that would accompany one added unit of a scarce resource
D)the reduction in cost that would accompany a one unit decrease in the resource
E)the profit contribution necessary for that item to be included in the optimal solution

Question 78

A craftsman builds two kinds of birdhouses,one for wrens (X1),and one for bluebirds (X2).Each wren birdhouse takes four hours of labor and four units of lumber.Each bluebird house requires two hours of labor and twelve units of lumber.The craftsman has available 60 hours of labor and 120 units of lumber.Wren houses profit $6 each and bluebird houses profit $15 each.
Use the software output that follows to interpret the problem solution.Include a statement of the solution quantities (how many of which product),a statement of the maximum profit achieved by your product mix,and a statement of "resources unused" and "shadow prices."

Question 79

What is the usefulness of a shadow price (or dual value)?

Question 80

What is the simplex method?