Linear Programming

Linear Programming Models: Graphical and Computer Methods l CHAPTER 7

TRUE/FALSE 7.1

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

7.2

In the term linear programming, the word programming comes from the phrase computer programming. ANSWER: FALSE

7.3

Linear programming has few applications in the real world due to the assumption of certainty in the data and relationships of a problem. ANSWER: FALSE

7.4

Any linear programming problem can be solved using the graphical solution procedure. ANSWER: FALSE

7.5

Linear programming is designed to allow some constraints to be maximized. ANSWER: FALSE

7.6

A typical LP involves maximizing an objective function while simultaneously optimizing resource constraint usage. ANSWER: TRUE

7.7

Resource restrictions are called constraints. ANSWER: TRUE

7.8

Industrial applications of linear programming might involve several thousand variables and constraints. ANSWER: TRUE

7.9

An important assumption in linear programming is to allow the existence of negative decision variables. ANSWER: FALSE

7.10

7.11

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. ANSWER: TRUE An objective function is necessary in a maximization problem but is not required in a minimization problem. ANSWER: FALSE

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7.12

In some instances, an infeasible solution may be the optimum found by the corner-point method. ANSWER: FALSE

7.13

The analytic postoptimality method attempts to determine a range of changes in problem parameters that will not affect the optimal solution or change the variables in the basis. ANSWER: TRUE

7.14

The solution to a linear programming problem must always lie on a constraint. ANSWER: TRUE

7.15

In a linear program, the constraints must be linear, but the objective function may be nonlinear. ANSWER: FALSE

7.16

Early applications of linear programming were primarily industrial in nature, later the technique was adopted by the military for scheduling and resource management. ANSWER: FALSE

7.17

One can employ the same algorithm to solve both maximization and minimization problems. ANSWER: TRUE

7.18

One converts a minimization problem to a maximization problem by reversing the direction of all constraints. ANSWER: FALSE

7.19

The graphical method of solution illustrates that the only restriction on a solution is that the solution must lie along a constraint. ANSWER: FALSE

7.20

Anytime we have an iso-profit line which is parallel to a constraint, we have the possibility of multiple solutions. ANSWER: TRUE

7.21

If the iso-profit line is not parallel to a constraint, then the solution must be unique. ANSWER: TRUE

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7.22

The iso-profit solution method and the corner-point solution method always give the same result. ANSWER: TRUE

7.23

When two or more constraints conflict with one another, we have a condition called unboundedness. ANSWER: FALSE

7.24

The addition of a redundant constraint lowers the iso-profit line. ANSWER: FALSE

7.25

Sensitivity analysis enables us to look only at the effects of changing the coefficients in the objective function. ANSWER: FALSE

*7.26

All linear programming problems require that we maximize some quantity. ANSWER: FALSE

*7.27

If we do not have multiple constraints, we do not have a linear programming problem. ANSWER: FALSE

*7.28

Inequality constraints are mathematically easier to handle than equality constraints. ANSWER: TRUE

*7.29

Every solution to a linear programming problem lies at a “corner point.” ANSWER: FALSE

*7.30

A linear programming problem can have, at most one, solution. ANSWER: FALSE

*7.31

A linear programming approach can be used to solve any problem for which the objective is to maximize some quantity. ANSWER: FALSE

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MULTIPLE CHOICE 7.32

Typical management resources include (a) (b) (c) (d) (e)

machinery usage. labor volume. warehouse space utilization. raw material usage. all of the above

ANSWER: e 7.33

Which of the following is not a property of all linear programming problems? (a) (b) (c) (d) (e)

the presence of restrictions optimization of some objective a computer program alternate courses of action to choose from usage of only linear equations and inequalities

ANSWER: c 7.34

Which of the following is not a basic assumption of linear programming? (a) (b) (c) (d) (e)

The condition of certainty exists. Proportionality exists in the objective function and constraints. Additivity exists for the activities. Divisibility exists, allowing non-integer solutions. Solutions or variables may take values from   to + .

ANSWER: e 7.35

A feasible solution to a linear programming problem (a) (b) (c) (d)

must satisfy all of the problem's constraints simultaneously. need not satisfy all of the constraints, only the non-negativity constraints. must be a corner point of the feasible region. must give the maximum possible profit.

ANSWER: a

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7.36

An optimal solution to a linear program (a) (b) (c) (d) (e)

will always lie at an extreme point of the feasible region. could be any point in the feasible region of the problem. will always be unique (only one optimal solution possible for any one problem). will always include at least some of each product or variable. must always be in whole numbers (integers).

ANSWER: a 7.37

Infeasibility in a linear programming problem occurs when (a) (b) (c) (d) (e)

there is an infinite solution. a constraint is redundant. more than one solution is optimal. the feasible region is unbounded. there is no solution that satisfies all the constraints given.

ANSWER: e 7.38

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

an infeasible solution. an unbounded solution. a redundant constraint. alternate optimal solutions. none of the above

ANSWER: b 7.39

Which of the following is not a part of every linear programming problem formulation? (a) (b) (c) (d) (e)

an objective function a set of constraints non-negativity constraints a redundant constraint maximization or minimization of a linear function

ANSWER: d

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7.40

The optimal solution to a maximization linear programming problem can be found by graphing the feasible region and (a) finding the profit at every corner point of the feasible region to see which one gives the highest value. (b) moving the iso-profit lines towards the origin in a parallel fashion until the last point in the feasible region is encountered. (c) locating the point which is highest on the graph. (d) none of the above (e) all of the above ANSWER: a

7.41

Which of the following is not true about product mix linear programming problems? (a) (b) (c) (d) (e)

Two or more products are produced. Limited resources are involved. They always have integer (whole number) solutions. The feasible region cannot include negative areas. none of the above

ANSWER: c 7.42

The graphical solution to a linear programming problem (a) (b) (c) (d) (e)

includes the corner-point method and the iso-profit line solution method. is useful for four or fewer decision variables. is inappropriate for more than two constraints. is the most difficult approach, but is useful as a learning tool. can only be used if no inequalities exist.

ANSWER: a 7.43

Which of the following about the feasible region is false? (a) (b) (c) (d) (e)

It is only found in product mix problems. It is also called the area of feasible solutions. It is the area satisfying all of the problem's resource restrictions. All possible solutions to the problem lie in this region. all of the above

ANSWER: a

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7.44

The corner-point solution method: (a) (b) (c) (d) (e)

will yield different results from the iso-profit line solution method. requires that the profit from all corners of the feasible region be compared. will provide one, and only one, optimum. requires that all corners created by all constraints be compared. will not provide a solution at an intersection or corner where a non-negativity constraint is involved.

ANSWER: b 7.45

When a constraint line bounding a feasible region has the same slope as an iso-profit line, (a) (b) (c) (d) (e)

there may be more than one optimum solution. the problem involves redundancy. an error has been made in the problem formulation. a condition of infeasibility exists. none of the above

ANSWER: a 7.46

The simultaneous equation method is (a) (b) (c) (d) (e)

an alternative to the corner-point method. useful only in minimization methods. an algebraic means for solving the intersection of two constraint equations. useful only when more than two product variables exist in a product mix problem. none of the above

ANSWER: c 7.47

Consider the following linear programming problem: Maximize Subject to:

12X + 10Y 4X + 3Y  480 2X + 3Y  360 all variables  0

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

(0,120) (120,0) (180,0) (60,80) none of the above

ANSWER: c

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7.48

Consider the following linear programming problem: Maximize Subject to:

12X + 10Y 4X + 3Y  480 2X + 3Y  360 all variables  0

The maximum possible value for the objective function is (a) (b) (c) (d) (e)

360. 480. 1520. 1560. none of the above

ANSWER: c 7.49

Consider the following linear programming problem: Maximize Subject to:

12X + 10Y 4X + 3Y  480 2X + 3Y  360 all variables  0

Which of the following points (X,Y) is not feasible? (a) (b) (c) (d) (e)

(0,120) (100,10) (20,90) (60,90) none of the above

ANSWER: d 7.50

Consider the following linear programming problem: Maximize Subject to:

4X + 10Y 3X + 4Y  480 4X + 2Y  360 all variables  0

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Linear Programming Models: Graphical and Computer Methods l CHAPTER 7

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

1032 1200 360 1600 none of the above

ANSWER: b 7.51

Consider the following linear programming problem: Maximize Subject to:

5X + 6Y 4X + 2Y  420 1X + 2Y  120 all variables  0

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

(0,60) (105,0) (120,0) (100,10) none of the above

ANSWER: c 7.52

Consider the following linear programming problem: Maximize Subject to:

5X + 6Y 4X + 2Y  420 1X + 2Y  120 all variables  0

The maximum possible value for the objective function is (a) (b) (c) (d) (e)

640. 360. 525. 560. none of the above

ANSWER: d

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7.53

Consider the following linear programming problem: Maximize Subject to:

5X + 6Y 4X + 2Y  420 1X + 2Y  120 all variables  0

Which of the following points (X,Y) is not feasible? (a) (b) (c) (d) (e)

(50,40) (20,50) (60,30) (90,10) none of the above

ANSWER: a 7.54

Consider the following linear programming problem: Maximize Subject to:

20X + 8Y 4X + 2Y  360 1X + 2Y  200 all variables  0

The optimum solution occurs at the point (X,Y) (a) (b) (c) (d) (e)

(100,0). (90,0). (80,20). (0,100). none of the above

ANSWER: b 7.55

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: Maximize profit = 50X + 60 Y Subject to: 8X + 10Y  800 (labor hours) X + Y  120 (total units demanded) 4X + 5Y  500 (raw materials) all variables  0 The optimal solution is X = 100 Y = 0.

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How many units of the regular model would be produced based on this solution? (a) (b) (c) (d) (e)

0 100 50 120 none of the above

ANSWER: b 7.56

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: Maximize profit = 50X + 60 Y Subject to: 8X + 10Y  800 (labor hours) X + Y  120 (total units demanded) 4X + 5Y  500 (raw materials) all variables  0 The optimal solution is X = 100 Y = 0. How many units of the raw materials would be used to produce this number of units? (a) (b) (c) (d) (e)

400 200 500 120 none of the above

ANSWER: a 7.57

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: Maximize profit = 50X + 60 Y Subject to: 8X + 10Y  800 (labor hours) X + Y  120 (total units demanded) 4X + 5Y  500 (raw materials) X, Y  0 The optimal solution is X=100, Y=0.

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Which of these constraints is redundant? (a) (b) (c) (d) (e)

the first constraint the second constraint the third constraint all of the above none of the above

ANSWER: a 7.58

Consider the following linear programming problem. Minimize Subject to

20X + 30Y 2X + 4Y  800 6X + 3Y  300 X, Y  0

The optimum solution to this problem occurs at the point (X,Y) (a) (b) (c) (d) (e)

(0,0). (50,0). (0,100). (400,0). none of the above

ANSWER: b 7.59

Consider the following linear programming problem. Maximize Subject to:

20X + 30Y X + Y  80 6X + 12Y  600 X, Y  0

This is a special case of a linear programming problem in which (a) (b) (c) (d) (e)

there is no feasible solution. there is a redundant constraint. there are multiple optimal solutions. this cannot be solved graphically. none of the above

ANSWER: c

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7.60

Consider the following linear programming problem. Maximize Subject to

20X + 30Y X + Y  80 8X + 9Y  600 3X + 2Y  400 X, Y  0

This is a special case of a linear programming problem in which (a) (b) (c) (d) (e)

there is no feasible solution. there is a redundant constraint. there are multiple optimal solutions. this cannot be solved graphically. none of the above

ANSWER: a 7.61

Adding a constraint to a linear programming (maximization) problem may result in (a) (b) (c) (d) (e)

a decrease in the value of the objective function. an increase in the value of the objective function. no change to the objective function. either (c) or (a) depending on the constraint. either (c) or (b) depending on the constraint.

ANSWER: d 7.62

Deleting a constraint from a linear programming (maximization) problem may result in (a) (b) (c) (d) (e)

a decrease in the value of the objective function. an increase in the value of the objective function. no change to the objective function. either (c) or (a) depending on the constraint. either (c) or (b) depending on the constraint.

ANSWER: e

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7.63

Which of the following is not acceptable as a constraint in a linear programming problem (maximization)? Constraint 1 Constraint 2 Constraint 3 Constraint 4 (a) (b) (c) (d) (e)

X + XY + Y  12 X  2Y  20 X + 3Y = 48 X + Y + Z  150

Constraint 1 Constraint 2 Constraint 3 Constraint 4 none of the above

ANSWER: a 7.64

If two corner points tie for the best value of the objective function, then (a) (b) (c) (d) (e)

the solution is infeasible. there are an infinite number of optimal solutions. the problem is unbounded. the problem is degenerate. none of the above

ANSWER: b 7.65

If one changes the contribution rates in the objective function of an LP, (a) (b) (c) (d) (e)

the feasible region will change. the slope of the iso-profit or iso-cost line will change. the optimal solution to the LP will no longer be optimal. all of the above none of the above

ANSWER: b 7.66

Changes in the technological coefficients of an LP problem (a) (b) (c) (d) (e)

often reflect changes in the state of technology. have no effect on the objective function of the linear program. can produce a significant change in the shape of the feasible solution region. all of the above none of the above

ANSWER: d

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7.67

Sensitivity analysis may also be called (a) (b) (c) (d) (e)

postoptimality analysis. parametric programming. optimality analysis. all of the above none of the above

ANSWER: d 7.68

Sensitivity analyses are used to examine the effects of changes in (a) (b) (c) (d) (e)

contribution rates for each variable. technological coefficients. available resources. all of the above none of the above

ANSWER: d 7.69

Which of the following is a property of all linear programming problems? (a) (b) (c) (d) (e)

alternate courses of action to choose from minimization of some objective a computer program usage of graphs in the solution usage of linear and nonlinear equations and inequalities

ANSWER: a 7.70

Which of the following is a basic assumption of linear programming? (a) (b) (c) (d) (e)

The condition of uncertainty exists. Proportionality exists in the objective function and constraints. Independence exists for the activities. Divisibility exists, allowing only integer solutions. Solutions or variables may take values from   to + .

ANSWER: b 7.71

A point which satisfies all of a problem's constraints simultaneously is a(n) (a) (b) (c) (d) (e)

feasible point. corner point. intersection of the profit line and a constraint. intersection of two or more constraints. none of the above

ANSWER: a 7.72

The condition when there is no solution which satisfies all the constraints is called: (a) boundedness

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(b) (c) (d) (e)

redundancy optimality dependency none of the above

ANSWER: e 7.73

In a minimization problem, when one or more of the solution variables and the cost can be made infinitely large without violating any constraints, then the linear program has (a) (b) (c) (d) (e)

an infeasible solution. an unbounded solution. a redundant constraint. alternate optimal solutions. none of the above

ANSWER: e 7.74

If the addition of a constraint to a linear programming problem does not change the solution, the constraint is said to be (a) (b) (c) (d) (e)

unbounded. non-negative. infeasible. redundant. bounded.

ANSWER: d 7.75

The following is not true about product mix linear programming problems: (a) (b) (c) (d) (e)

Two or more products are produced. Individual resources are used in only a single product. They never have integer (whole number) solutions. Cost is always to be minimized. none of the above

ANSWER: b

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7.76

The graphical solution to a linear programming problem (a) (b) (c) (d) (e)

is a useful tool for solving practical problems. is useful for four or less decision variables. is useful primarily in helping one understand the linear programming solution process. is the most difficult approach. can only be used in a maximization problem.

ANSWER: c 7.77

In order for a linear programming problem to have a unique solution, the solution must exist (a) (b) (c) (d) (e)

at the intersection of the non-negativity constraints. at the intersection of a non-negativity constraint and a resource constraint. at the intersection of the objective function and a constraint. at the intersection of two or more constraints. none of the above

ANSWER: d 7.78

In order for a linear programming problem to have a multiple solutions, the solution must exist (a) (b) (c) (d) (e)

at the intersection of the non-negativity constraints. on a constraint parallel to the objective function. at the intersection of the objective function and a constraint. at the intersection of three or more constraints. none of the above

ANSWER: b 7.79

Consider the following linear programming problem: Maximize Subject to:

12X + 10Y 4X + 3Y  480 2X + 3Y  360 all variables  0

Which of the following points (X,Y) could be a feasible corner point? (a) (b) (c) (d) (e)

(40,48) (120,0) (180,120) (30,36) none of the above

ANSWER: b

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7.80

Consider the following linear programming problem: Maximize Subject to:

12X + 10Y 2X + 3Y  480 4X + 3Y  360 all variables  0

The maximum possible value for the objective function is (a) (b) (c) (d) (e)

360. 480. 1520. 1560. none of the above

ANSWER: e 7.81

Consider the following linear programming problem. Maximize Subject to:

12X + 10Y 4X + 3Y  480 2X + 3Y  360 all variables  0

Which of the following points (X,Y) is feasible? (a) (b) (c) (d) (e)

(10,120) (120,10) (30,100) (60,90) none of the above

ANSWER: c 7.82

Consider the following linear programming problem. Maximize Subject to:

6X +8Y 3X + 4Y  480 4X + 2Y  360 all variables  0

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

540 1200 360 960 none of the above

ANSWER: d 7.83

Consider the following linear programming problem. Maximize Subject to:

5X + 6Y 4X + 2Y  420 1X + 2Y  120 all variables  0

Which of the following points (X,Y) is in the feasible region? (a) (b) (c) (d) (e)

(30,60) (105,5) (0,210) (100,10) none of the above

ANSWER: d 7.84

Consider the following linear programming problem. Maximize Subject to:

6X +5Y 4X + 2Y  420 1X + 2Y  120 all variables  0

The maximum possible value for the objective function is (a) (b) (c) (d) (e)

530. 360. 525. 560. none of the above

ANSWER: a

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7.85

Consider the following linear programming problem. Maximize Subject to:

5X + 6Y 4X + 2Y  420 1X + 2Y  120 all variables  0

Which of the following points (X,Y) is feasible? (a) (b) (c) (d) (e)

(50,40) (30,50) (60,30) (90,20) none of the above

ANSWER: e 7.86

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: Maximize profit = 50X + 60 Y Subject to: 8X +10Y  800 (labor hours) X + Y  120 (total units demanded) 4X + 5Y  500 (raw materials) all variables  0 The optimal solution is X = 100 Y = 0. How many units of the Deluxe model would be produced based on this solution? (a) (b) (c) (d) (e)

0 100 50 120 none of the above

ANSWER: a 7.87

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: Maximize profit = 50X + 60 Y Subject to: 8X + 10Y  800 (labor hours) X +Y  120 (total units demanded) 4X + 5Y  500 (raw materials) all variables  0 The optimal solution is X = 100 Y = 0. How many units of the labor hours would be used to produce this number of units?

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Linear Programming Models: Graphical and Computer Methods l CHAPTER 7

(a) (b) (c) (d) (e)

400 200 500 120 none of the above

ANSWER: e 7.88

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: Maximize profit = 50X + 60 Y Subject to: 8X + 10Y  500 (labor hours) X + Y  120 (total units demanded) 4X + 5Y  800 (raw materials) X, Y  0 Which of the constraints is active in determining the solution? (a) (b) (c) (d) (d)

the first constraint the second constraint the third constraint constraints (a) and (b) none of the above

ANSWER: a 7.89

Consider the following linear programming problem. Maximize Subject to:

18X + 36Y X + Y  80 6X + 12Y  600 X, Y  0

This is a special case of a linear programming problem in which (a) (b) (c) (d) (e)

there is no feasible solution. there is a redundant constraint. there are multiple optimal solutions. this cannot be solved graphically. none of the above

ANSWER: c

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7.90

Consider the following linear programming problem. Maximize Subject to:

20X + 30Y X + Y  80 12X + 12Y  600 3X + 2Y 400 X, Y  0

This is a special case of a linear programming problem in which (a) (b) (c) (d) (e)

there is no feasible solution. there is a redundant constraint. there are multiple optimal solutions. this cannot be solved graphically. none of the above

ANSWER: b 7.91

Removing a constraint from a linear programming (maximization) problem may result in (a) (b) (c) (d) (e)

a decrease in the value of the objective function. an increase in the value of the objective function. either an increase or decrease in the value of the objective function. no change in the value of the objective function. either (b) or (d)

ANSWER: e 7.92

Adding a constraint to a linear programming (maximization) problem may result in (a) (b) (c) (d) (e)

a decrease in the value of the objective function. an increase in the value of the objective function. either an increase or decrease in the value of the objective function. no change in the value of the objective function. either a decrease or no change in the value of the objective function.

ANSWER: e

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7.93

Which of the following is not acceptable as a constraint in a linear programming problem (minimization)? Constraint 1 Constraint 2 Constraint 3 Constraint 4 Constraint 5 (a) (b) (c) (d) (e)

X + Y  12 X - 2Y  20 X + 3Y = 48 X + Y + Z  150 2X - 3Y + Z > 75

Constraint 1 Constraint 2 Constraint 3 Constraint 4 Constraint 5

ANSWER: e 7.94

Changes in the contribution rates in the objective function of an LP may represent (a) (b) (c) (d) (e)

changes in the technology used to produce the good. changes in the price for which the product can be sold. changes in government rules and regulations. changes in the raw materials used. none of the above

ANSWER: b 7.95

Changes in the technological coefficients of an LP problem may represent (a) (b) (c) (d) (e)

changes in the price for which the product can be sold. changes in the value of the resources used. changes in the amount of resources used for a product. changes in the degree to which a resource contributes to the cost of a product. none of the above

ANSWER: c *7.96

Consider the following linear programming problem. Maximize Subject to

10X + 30Y X + 2Y  80 8X + 16Y  640 4X + 2Y  100 X, Y  0

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Linear Programming Models: Graphical and Computer Methods l CHAPTER 7

This is a special case of a linear programming problem in which (a) (b) (c) (d) (e)

there is no feasible solution. there is a redundant constraint. there are multiple optimal solutions. this cannot be solved graphically. none of the above

ANSWER: b *7.97

Which of the following is not acceptable as a constraint in a linear programming problem (maximization)? Constraint 1 Constraint 2 Constraint 3 Constraint 4 (a) (b) (c) (d) (e)

X + Y  12 X  2Y  20 X + 3Y = 48 X2 + Y + Z  150

Constraint 1 Constraint 2 Constraint 3 Constraint 4 none of the above

ANSWER: d *7.98

Consider the following linear programming problem. Maximize Subject to:

12X + 10Y 4X + 3Y  480 2X + 3Y  360 all variables  0

Which of the following points (X,Y) could be a feasible corner point? (a) (b) (c) (d) (e)

(40,48) (120,0) (180,120) (30,36) none of the above

ANSWER: b

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*7.99

Consider the following linear programming problem. Maximize Subject to:

12X + 10Y 2X + 3Y  480 4X + 3Y  360 all variables  0

The maximum possible value for the objective function is (a) (b) (c) (d) (e)

360. 480. 1520. 1560. none of the above

ANSWER: e 7.100

Consider the following linear programming problem. Maximize Subject to:

12X + 10Y 4X + 3Y  480 2X + 3Y  360 all variables  0

Which of the following points (X,Y) is feasible? (a) (b) (c) (d) (e)

(10,120) (120,10) (30,100) (60,90) none of the above

ANSWER: c *7.101 Consider the following linear programming problem. Maximize Subject to:

5X + 6Y 4X + 2Y  420 1X + 2Y  120 all variables  0

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Which of the following points (X,Y) is not in the feasible region? (a) (b) (c) (d) (e)

(30,30) (60,40) (100,5) (20,40) none of the above

ANSWER: b *7.102 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: Minimize cost Subject to:

= 60X + 50 Y 8X + 10Y  800 (labor hours) X + Y  120 (total units demanded) 4X + 5Y  500 (raw materials) all variables  0

How many units of the Deluxe model would be produced based on this solution? (a) (b) (c) (d) (e)

0 100 50 120 none of the above

ANSWER: a *7.103 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: Maximize profit = 60X + 50 Y Subject to: 6X +10Y  500 (labor hours) X + Y  120 (total units demanded) 6X + 5Y  800 (raw materials) X, Y  0 Which of the constraints is active in determining the solution? (a) (b) (c) (d) (d)

the first constraint the first and third constraints the third constraint the second constraint none of the above

ANSWER: a

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*7.104 Consider the following linear programming problem. Maximize Subject to:

15X + 36Y X + Y  80 27.5X + 55Y  1200 X, Y  0

This is a special case of a linear programming problem in which (a) (b) (c) (d) (e)

there is no feasible solution. there is a redundant constraint. there are multiple optimal solutions. this cannot be solved graphically. none of the above

ANSWER: e

PROBLEMS 7.105

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 man-hours available daily at the construction stage and only 80 man-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. ANSWER: (a) Let X1 = number of deluxe models produced X2 = number of special models produced Maximize

12X1 + 10X2

Subject to:

1/3X1 + 1/4X2  100 1/6X1 + 1/4X2  80 0.4X1 + 0.6X2  0 X1, X2  0

(b) Optimal solution: X1 = 0, X2 = 400 7.106

Profit = $4,000

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

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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 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) (b) (c) (d)

Formulate this as a linear programming problem. Solve this problem graphically, giving the optimal values of all variables. Are any constraints redundant? If so, which one or ones? What is the total cost of a package of dog biscuits using the optimal mix?

ANSWER: (a) Let X1 = number of liver flavored biscuits in a package X2 = number of chicken flavored biscuits in a package Minimize Subject to:

X1 + 2X2 X1 + X2  40 2X1 + 4X2  60 X1  15 X1, X2  0

(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 7.107

Consider the following linear program. Maximize Subject to:

30X1 + 10X2 3X1 + X2  300 X1 + X2  200 X1  100 X2  50 X1  X2  0 X1, X2  0

(a) Solve the problem graphically. Is there more than one optimal solution? Explain. (b) Are there any redundant constraints?

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ANSWER: (a) Corner points (0,50), (0,200), (50,50), (75,75), (50,150) Optimum solutions: (75,75) and (50,150). Both yield profit of $3,000. (b) The constraint X1  100 is redundant since 3X1 + X2  300 also means that X1 cannot exceed 100. 7.108

The No-Glare Company is making two types of antique-style lamps, type #1 and type #2. There is enough skilled labor to make either 1,000 type #1 or 2,000 type #2 lamps per day. There are only 6,000 inserts available per day, of which the type #1 lamp requires 3 and the type #2 lamp requires 4. Besides these shared constraints, there are only enough fancy switches to make 1,400 of the type #2 lamps per day. Marginal profit (contribution) is $3 per type #1 lamp and $4 per type #2 lamp. Let X1 = the hundreds of type #1 lamps per day, etc. (a) Identify each corner point bounding the feasible region and find the total variable profit at each point. (b) How many type #1 and type #2 lamps should be produced? What is the maximum possible profit? ANSWER: (a) Corner points X1 X2 0 0 0 1,400 133.33 1,400 400 1,200 1,000 0

Profit($) 0 5,600 6,000 * 6,000 * 3,000

b) * Produce 133.33 type #1 lamps and 1,400 type #2 lamps for a profit of $6,000 or produce 400 type #1 lamps and 1,200 type #2 lamps for a profit of $6,000. 7.109

Solve the following linear programming problem using the corner point method. Maximize Subject to:

10X + 1Y 4X + 3Y  2X + 4Y  Y X, Y 

36 40 3 0

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).

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7.110

Solve the following linear programming problem using the corner point method. Maximize Subject to:

3 X + 5Y 4X + 4Y  48 1X + 2Y  20 Y 2 X, Y  0

ANSWER: Feasible corner points (X,Y): (0,2) (0,10) (4,8) (10,2) Maximum profit is 52 at (4,8). 7.111

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? ANSWER: Let X = number of standard model to produce Y = number of deluxe model to produce Maximize Subject to:

60X + 40Y 20X + 35Y  450 10X + 15Y  180 X6 X, Y  0

Maximum profit is $780 by producing 9 standard and 8 deluxe models. 7.112

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?

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ANSWER: Let R = number of radio ads placed N = number of newspaper ads placed Maximize Subject to:

2000R + 6000N R + N  15 400R + 600N  7200 R2 N2 R, N  0

Feasible corner points (R,N): (2,2) (9,6) (13,2) (2,10.67) Maximum exposure 68,020 with 2 radio and 10.67 newspaper ads. 7.113

Suppose a linear programming (maximization) 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 ANSWER: (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. (b) A new constraint can only reduce the size of the feasible region; therefore, the value of the objective function will either decrease or remain the same. If the original solution is still feasible, it will remain the optimal solution.

7.114

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 apiece. 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?

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ANSWER: Let X1 = numbers of wuns/day X2 = number of toos/day Maximize Subject to:

9X1 + 8X2 2X1 + 1X2  14 1X1 + 2X2  16 X1, X2  0

Corner points (0,0), (7,0), (0,8), (4,6) Optimum profit $84 at (4,6). 7.115

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 Subject to:

80X + 50Y 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? ANSWER: (a) 6000, (b) 360, (c) it would not change 7.116

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 man-hours available daily at the construction stage and only 80 man-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.

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ANSWER: Let X1 = number of deluxe models produced X2 = number of special models produced

7.117

Maximize

12X1 + 10X2

Subject to:

1/3X1 + 1/4X2  100 1/6X1 + 1/4X2  80 1.5X1 + X2  0 X1, X2  0

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 twice as many units of nutrient A as of nutrient B in a package of the new mix. In addition, the company has decided that there can be no more than 15 liver flavored biscuits, and at least 10 chicken 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) Are any constraints impossible to achieve? If so which one(s)? ANSWER: (a) Let X1 = number of liver flavored biscuits in a package X2 = number of chicken flavored biscuits in a package Minimize Subject to:

X1 + 2X2 3X1 + 7X2 0 X1  15 X2  X1, X2  0

Ratio of A to B Maximum liver Minimum chicken Non-negativity

(b) The constraint, 3X1 + 7X2 0, is impossible to achieve. 7.118

The No-Glare Company is making two types of antique-style lamps, type #1 and type #2. There is enough skilled labor to make either 1,000 type #1 or 2,000 type #2 lamps per day. There are only 6,000 inserts available per day, of which the type #1 requires 3 and the type #2 requires 4. Besides these shared constraints, there are only enough fancy switches to make 1,400 of the type #2 lamps per day. Management would like to make at least 10 percent more type #2 lamps than type #1 lamps; however, they do not believe that they can sell more than 25 percent more type #2 lamps than type #1 lamps. Marginal profit (contribution) is $3 per type #1 lamp and $4 per type #2 lamp. (a) Formulate this as a linear program. (b) What constraint may be unrealistic? ANSWER:

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(a) Let X1 = the hundreds of type #1 lamps per day, etc. Maximize: 300X1 + 400 X2 Subject to: 0.10X1 + 0.05X2 < 1 3X1 + 4X2  60 X2  14 1.1X1 – X2  0 1.25X1 – X2 0 X1 , X2 0

Labor Inserts Fancy switches Minimum Type #2 to Type #1 ratio Maximum Type #2 to Type #1 ratio

(b) The labor constraint may be unrealistic because it assumes a continuous tradeoff between labor required for the Type #1 and Type #2 lamps. 7.119

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, and there should be no more than 19 ads in total. The company does not want the number of newspaper ads to exceed the number of radio ads by more than 25 percent. Each newspaper ad reaches 6,000 people, 50 percent of whom will respond; while each radio ad reaches 2,000 people, 20 percent of whom will respond. The company wishes to reach as many respondents as possible while meeting all the constraints stated. Develop the appropriate LP model for determining the number of ads of each type that should be placed? ANSWER: Let R = number of radio ads placed N = number of newspaper ads placed Maximize: or Maximize: Subject to:

0.20*2000R + 0.50*6000N 500R + 3000N R + N  15 R + N  19 400R + 600N  7200 1R - N  R2 N2 R, N  0

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7.120

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 ANSWER: (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. (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.

*7.121 Suppose a linear programming (maximization) 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 ANSWER: (a) Adding a new  constraint will either, leave the feasible region as it was, or make it smaller. (b) A new constraint can only reduce the size of the feasible region. Therefore, the value of the objective function will either stay the same or be lowered. *7.122 The 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 2 units of nutrient A and 1 unit of nutrient B, while the chicken flavored ones contain 3 units of nutrient A and 4 units of nutrient B. According to federal requirements, there must be a ratio of 3 units of A to 2 of B in the new mix. In addition, the company has decided that there can be no more than 10 liver flavored biscuits, and that there must be least 10 chicken flavored biscuits in a package. If it costs 3 cents 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) What is the solution?

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ANSWER: (a) Let X1 = number of liver flavored biscuits in a package X2 = number of chicken flavored biscuits in a package Minimize Subject to:

3X1 + 2X2 X1 - 6X2 0 X1  10 X2 0 X1, X2  0

Ratio of A to B Maximum liver Minimum chicken Non-negativity

(b) X1 = 12, X2 = 10: Liver: 12 biscuits

Chicken: 10 biscuits

*7.123 The No-Glare Company is making two types of automobile headlights, type #1 and type #2. There is enough skilled labor to make either 2,000 type #1 or 4,000 type #2 lamps per day. There are only 12,000 inserts available per day, of which the type #1 lamp requires 6 and the type #2 lamp requires 8. Besides these shared constraints, there are only enough fancy switches to make 2,800 of the type #2 lamps per day. Marginal profit (contribution) is $4 per type #1 lamp and $6 per type #2 lamp. Let X1 = the hundreds of type #1 lamps per day, etc. (a) Identify each corner point bounding the feasible region and find the total variable profit at each point. (b) How many type #1 and type #2 lamps should be produced? What is the maximum possible profit? ANSWER: (a) Corner points X1 X2 0 0 0 15 20 0

Profit($) 0 9,000 * 8,000

b) * Produce 0 type #1 lamps and 1500 type #2 lamps for a profit of $9,000 *7.124 Billy Penny is trying to determine how many units of two types of lawnmowers 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 and the deluxe model, 30 minutes of assembly time. The standard model requires 15 minutes of inspection time; the deluxe model, 30 minutes. The company must fill an order for 12 deluxe models. There are 525 minutes of assembly time and 220 minutes of inspection time available each day. How many units of each product should be manufactured to maximize profits?

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ANSWER: Let X1 = number of standard model to produce X2 = number of deluxe model to produce Maximize Subject to:

60X + 40Y 20X + 30Y  525 15X + 30Y < 220 X  12 X, Y  0

Maximum profit is $870 by producing 6.5 standard and 12 deluxe models.

SHORT ANSWER/ESSAY 7.125

List at least three typical management resources that warrant control. ANSWER: machinery usage, labor volume, dollars spent, time used, warehouse space usage, raw material usage

7.126

The basic assumption of linear programming is certainty. Explain its need. ANSWER: Objective function rates and resource consumption are known and do not change during the analyzed time period.

7.127

One basic assumption of linear programming is proportionality. Explain its need. ANSWER: Rates of consumption exist, e.g., 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.

7.128

One basic assumption of linear programming is additivity. Explain its need. ANSWER: The total of all activities equals the sum of individual activities.

7.129

One basic assumption of linear programming is divisibility. Explain its need. ANSWER: Solutions need not be whole numbers.

7.130

One basic assumption of linear programming is non-negativity. Explain its need. ANSWER: Only solution values of zero or positive values are allowed.

7.131

Define infeasibility with respect to an LP solution. ANSWER: When there is no solution that can satisfy all constraints simultaneously.

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7.132

Define unboundedness with respect to an LP solution. ANSWER: a solution variable that is allowed to increase without limit while satisfying all constraints

7.133

Define redundancy with respect to an LP solution. ANSWER: the presence of one or more constraints that have no effect on the feasible solution area

7.134

Define alternate optimal solution with respect to an LP solution. ANSWER: More than one optimal solution point exists.

*7.135 Explain, briefly, the difference between an unbounded solution and a bounded solution. ANSWER: In a bounded solution, one or more of the constraints restricts the solution. In an unbounded solution, the solution is not restricted. *7.136 Mathematically, what are the requirements for multiple solutions? ANSWER: For multiple solutions to occur, the objective function must be parallel to an active constraint.

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