Deck 22: Linear Programming

Question 1

The region that satisfies the constraint 4X + 15Z ≥ 1000 includes the origin of the graph.

Answer 1

False

Question 2

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

Answer 2

True

Question 3

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

Answer 3

False

Question 4

Sensitivity analysis of linear programming solutions can use trial and error or the analytic postoptimality method.

Question 5

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 6

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 7

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

Question 8

Which of the following represents valid constraints in linear programming?

A)2X ≥ 7X × Y
B)2X × 7Y ≥ 500
C)2X + 7Y ≥ 100
D)2X2 + 7Y ≥ 50
E)2X × 7Y ≥ 650

Question 9

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 10

The optimal solution of a linear programming problem that consists of two variables and six constraints will probably not satisfy all six constraints as equalities.

Question 11

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 labour 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 12

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 13

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

Question 14

The graphical method of solving linear programming can handle only maximizing problems.

Question 15

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

Question 16

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

Question 17

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

Question 18

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

Question 19

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

Question 20

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%.

Question 21

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 22

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 23

For the two constraints given below,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 24

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)a,b ≥ 0.

A)a = 0,b = 0
B)a = 3,b = 3
C)a = 0,b = 6
D)a = 6,b = 0
E)cannot solve without values for a and b

Question 25

The region which satisfies all of the constraints in graphical linear programming is called the

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

Question 26

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 27

Which of the following sets of constraints results in an unbounded maximizing 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 28

In linear programming,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 29

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 30

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)none of the above

Question 31

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 32

If cars (C sell for $500 profit and trucks (T)sell for $300 profit which of the following represents the objective function?

A)Maximize = 500C + 300T
B)Minimize = 500C + 300T
C)Maximize = 500C - 300T
D)Minimize = 300T- 500C
E)Minimize = 500T - 300C

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

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 = 4

Question 35

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 36

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.The corner points of the feasible region of this problem are

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)none of the above.

Question 37

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

A)The points (100,0)and (0,25)both lie outside the feasible region.
B)The two corner points are (33-1/3,8-1/3)and (50,0).
C)The graphical origin (0,0)is not in the feasible region.
D)The feasible region is a straight line segment,not an area.
E)All of the above are true.

Question 38

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), (33-1/3,8-1/3),and (100,0).
E)The feasible region cannot be determined without knowing whether the problem is to be minimized or maximized.

Question 39

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

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

Question 40

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)All of the above have a feasible region.

Question 41

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 constraint ready for use in problem solving software.

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

Question 42

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,the appropriate objective function would be

A)maximize $40Y = $25Z.
B)maximize $40Y + $25Z.
C)maximize $30Y + $20Z.
D)maximize 0.25Y + 0.20Z.
E)maximize $20Y + $50Z.

Question 43

Sensitivity analysis helps to

A)see the value of increased scarce resources.
B)determine even better solutions.
C)see the impact of parameter changes.
D)A and C.
E)A,B,and C.

Question 44

Consider the following constraints from a two-variable linear program. (1)X ≥ 1
(2)Y ≥ 1
(3)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 45

The 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)None of the above are true.

Question 46

A maximizing linear programming problem with constraints C1,C2,and C3 has been solved.The dual values associated with the problem are C1 = $2,C2 = $0.50,and C3 = $0.Which statement below is false?

A)One more unit of the resource in C1 would add $2 to the objective function value.
B)One more unit of the resource in C2 would add one more unit each of X and Y.
C)The resource in C3 has not been used up.
D)The resources in C1 and in C2,but not in C3,are scarce.
E)The resource in C2 must be used up first.

Question 47

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 cost level that must be achieved for profitability

Question 48

________ is a mathematical technique designed to help operations managers plan and make decisions relative to the trade-offs necessary to allocate resources.

Question 49

Suppose that 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 50

Suppose that 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)$12

Question 51

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)They all maximize the profit lines.

Question 52

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)2Y - X = 15

Question 53

In sensitivity analysis,a zero shadow price (or dual value)for a resource ordinarily means that

A)the resource is scarce.
B)the resource constraint was redundant.
C)the resource has not been used up.
D)something is wrong with the problem formulation.
E)the resource is very costly.

Question 54

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 constraint ready for use in problem solving software.

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 + S + T)≤ 4

Question 55

Suppose that the shadow price for assembly time is $5/hour.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
C)increase available hours for assembly time
D)not enough information
E)Either A or C will result in larger profits than B.

Question 56

Suppose that an iso-profit line is given to be X + Y = 15.What would be the profit made from producing 20X and 10Y?

A)15
B)30
C)0
D)20X and 10Y is not a feasible solution.
E)Unable to determine.

Question 57

A linear programming problem has three constraints: 2X + 10Y ≤ 1004X + 6Y ≤ 1206X + 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 58

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 59

A linear programming problem has three constraints: 2X + 10Y ≤ 1004X + 6Y ≤ 1206X + 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 60

A linear programming maximization problem has been solved.In the optimal solution,two resources are scarce.If an added amount could be found for only one of these resources,how would the optimal solution be changed?

A)The shadow price of the added resource will rise.
B)The solution stays the same;the extra resource can't be used without more of the other scarce resource.
C)The extra resource will cause the value of the objective to fall.
D)The optimal mix will be rearranged to use the added resource,and the value of the objective function will rise.
E)The extra resource will cause the value of the objective to rise.

Question 61

What is sensitivity analysis?

Question 62

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

Question 63

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 constraint ready for use in problem solving software.

Question 64

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

Question 65

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

Question 66

A synonym for shadow price is ________.

Question 67

What is linear programming?

Question 68

What is the simplex method?

Question 69

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 constraint ready for use in problem solving software.

Question 70

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

Question 71

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

Question 72

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

Question 73

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

Question 74

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 75

What are the requirements of all linear programming problems?

Question 76

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

Question 77

In a linear programming problem,what is the relationship between the constraints and the feasible region? Explain with reference to a problem with two variables.

Question 78

What is the feasible region in a linear programming problem?

Question 79

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

Question 80

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