Deck 10: Integer Programming, Goal Programming, and Nonlinear Programming

Question 1

0-1 integer programming might be applicable to selecting the best gymnastics team to represent a country from among all identified teams.

Answer 1

True

Question 2

The following objective function is nonlinear: Max 5X - 8YZ.

Answer 2

True

Question 3

There is no general method for solving all nonlinear problems.

Answer 3

True

Question 4

The three types of integer programs are: pure integer programming, impure integer programming, and 0-1 integer programming.

Question 5

In goal programming, our goal is to drive the deviational variables in the objective function as close to zero as possible.

Question 6

If conditions require that all decision variables must have an integer solution, then the class of problem described is an integer programming problem.

Question 7

Goal programming permits multiple objectives to be satisfied.

Question 8

An integer programming solution can never produce a greater profit objective than the LP solution to the same problem.

Question 9

Quadratic programming contains squared terms in the constraints.

Question 10

In goal programming, if all the goals are achieved, then the value of the objective function will always be zero.

Question 11

When solving very large integer programming problems, we sometimes have to settle for a "good," not necessarily optimal, answer.

Question 12

The following objective function is nonlinear: Max 5X + (1/8)Y - Z.

Question 13

Requiring an integer solution to a linear programming problem decreases the size of the feasible region.

Question 14

The constraint X₁ + X₂ ≤ 1 with 0 -1 integer programming allows for either X₁ or X₂ to be a part of the optimal solution, but not both.

Question 15

In goal programming, the deviational variables have the same objective function coefficients as the surplus and slack variables in a normal linear program.

Question 16

A 0-1 programming representation could be used to assign sections of a course to specific classrooms.

Question 17

Unfortunately, goal programming, while able to handle multiple objectives, is unable to prioritize these objectives.

Question 18

Nonlinear programming is the case in which objectives and/or constraints are nonlinear.

Question 19

Unfortunately, multiple goals in goal programming are not able to be prioritized and solved.

Question 20

The transportation problem is a good example of a pure integer programming problem.

Question 21

Table 10-1
A company has decided to use 0-1 integer programming to help make some investment decisions. There are three possible investment alternatives from which to choose, but if it is decided that a particular alternative is to be selected, the entire cost of that alternative will be incurred (i.e., it is impossible to build one-half of a factory). The integer programming model is as follows: The optimal solution is X₁ = 0, X₂ = 1, X₃ = 1
According to Table 10-1, which presents an integer programming problem, the optimal solution is to select only two of the alternatives. Suppose you wished to add a constraint that stipulated that alternative 2 could only be selected if alternative 1 is also selected . How would this constraint be written?

A) X₁ = X₂
B) X₁ ≤ X₂
C) X₁ ≥ X₂
D) X₁ + X₂ = 2
E) None of the above

Question 22

Table 10-1
A company has decided to use 0-1 integer programming to help make some investment decisions. There are three possible investment alternatives from which to choose, but if it is decided that a particular alternative is to be selected, the entire cost of that alternative will be incurred (i.e., it is impossible to build one-half of a factory). The integer programming model is as follows: The optimal solution is X₁ = 0, X₂ = 1, X₃ = 1
In Table 10-1, which presents an integer programming problem, using the optimal solution means only two of the alternatives would be selected. How much slack is there in the third constraint?

A) 0
B) 3
C) 33
D) 36
E) None of the above

Question 23

An integer programming (maximization) problem was first solved as a linear programming problem, and the objective function value (profit) was $253.67. The two decision variables (X, Y) in the problem had values of X = 12.45 and Y = 32.75. If there is a single optimal solution, which of the following must be true for the optimal integer solution to this problem?

A) X = 12 Y = 32
B) X = 12 Y = 33
C) The objective function value must be less than $253.67.
D) The objective function value will be greater than $253.67.
E) None of the above

Question 24

Goal programming differs from linear programming in which of the following aspects?

A) It tries to maximize deviations between set goals and what can be achieved within the constraints.
B) It minimizes instead of maximizing as in LP.
C) It permits multiple goals to be combined into one objective function.
D) All of the above
E) None of the above

Question 25

In an integer programming problem, if it is desired to have variable X be exactly twice the value of variable Y, the constraint would be written:

A) 2X + Y = 0.
B) X + 2Y = 0.
C) 2X - Y = 0.
D) X - 2Y = 0.
E) None of the above

Question 26

A goal programming problem had two goals (with no priorities assigned). Goal number 1 was to achieve a profit of $2,400 and goal number 2 was to have no idle time for workers in the factory. The optimal solution to this problem resulted in a profit of $2,300 and no idle time. What was the value for the objective function for this goal programming problem?

A) 2300
B) 100
C) -100
D) 0
E) None of the above

Question 27

A type of integer programming is

A) pure.
B) mixed.
C) zero-one.
D) All of the above
E) None of the above

Question 28

Assignment problems solved previously by linear programming techniques are also examples of

A) pure-integer programming problems.
B) mixed-integer programming problems.
C) zero-one integer programming problems.
D) goal programming problems.
E) nonlinear programming problems.

Question 29

We do not have a general method for solving all types of ________ problems.

A) mixed-integer programming
B) 0-1 integer programming
C) goal programming
D) nonlinear programming
E) pure integer programming

Question 30

Table 10-1
A company has decided to use 0-1 integer programming to help make some investment decisions. There are three possible investment alternatives from which to choose, but if it is decided that a particular alternative is to be selected, the entire cost of that alternative will be incurred (i.e., it is impossible to build one-half of a factory). The integer programming model is as follows: The optimal solution is X₁ = 0, X₂ = 1, X₃ = 1
According to Table 10-1, which presents an integer programming problem, if the optimal solution is used, how much of the budget would be spent?

A) $32,000
B) $29,000
C) $61,000
D) $62,000
E) None of the above

Question 31

Table 10-2
According to Table 10-2, which presents a solution for an integer programming problem, at the optimal solution, how much slack exists in the third constraint?

A) 0
B) 9
C) 5
D) 6
E) -1

Question 32

Which of the following is a category of mathematical programming techniques that doesn't assume linearity in the objective function and/or constraints?

A) integer programs
B) goal programming problems
C) nonlinear programs
D) multiple objective programming problems
E) None of the above

Question 33

A goal programming problem had two goals (with no priorities assigned). Goal number 1 was to achieve a profit of $3,600 and goal number 2 was to have no wasted material. The optimal solution to this problem resulted in a profit of $3,300 and no wasted material. What was the value for the objective function for this goal programming problem?

A) 300
B) -300
C) 3300
D) 0
E) None of the above

Question 34

A model containing a linear objective function and linear constraints but requiring that one or more of the decision variables take on an integer value in the final solution is called ________

A) a goal programming problem.
B) an integer programming problem.
C) a nonlinear programming problem.
D) a multiple objective LP problem.
E) a branch-and-bound programming problem.

Question 35

Which of the following functions is nonlinear?

A) 4X + 2Y + 7Z
B) -4X + 2Y
C) 4X + (1/2)Y + 7Z
D) Z
E) 4X/Y + 7Z

Question 36

An integer programming (minimization) problem was first solved as a linear programming problem, and the objective function value (cost) was $253.67. The two decision variables (X, Y) in the problem had values of X = 12.45 and Y = 32.75. If there is a single optimal solution, which of the following must be true for the optimal integer solution to this problem?

A) X = 13 Y = 33
B) X = 12 Y = 32
C) The objective function value must be less than $253.67.
D) The objective function value will be greater than $253.67.
E) None of the above

Question 37

Table 10-1
A company has decided to use 0-1 integer programming to help make some investment decisions. There are three possible investment alternatives from which to choose, but if it is decided that a particular alternative is to be selected, the entire cost of that alternative will be incurred (i.e., it is impossible to build one-half of a factory). The integer programming model is as follows: The optimal solution is X₁ = 0, X₂ = 1, X₃ = 1
According to Table 10-1, which presents an integer programming problem, if the optimal solution is used, what would the value of the objective function be ________.

A) 21,000
B) 12,000
C) 16,000
D) 2
E) None of the above

Question 38

Goal programming is characterized by

A) all maximization problems.
B) setting of lower and upper bounds.
C) the deviation from a high-priority goal must be minimized before the next-highest-priority goal may be considered.
D) All of the above
E) None of the above

Question 39

A mathematical programming model that permits decision makers to set and prioritize multiple objective functions is called a

A) pure-integer programming problem.
B) mixed-integer programming problem.
C) zero-one integer programming problem.
D) goal programming problem.
E) nonlinear programming problem.

Question 40

In a goal programming problem with two goals at the same priority level, all the deviational variables are equal to zero in the optimal solution. This means

A) there is no feasible solution to the problem.
B) all goals are fully achieved.
C) nonlinear programming must be used to solve this.
D) this problem was an integer programming problem.
E) None of the above

Question 41

Which of the following statements is false concerning goal programming?

A) The objective function is the main difference between linear programming and goal programming.
B) The objective in goal programming is to minimize deviational variables.
C) Deviational variables are zero if a goal is completely obtained.
D) It is not possible for two goals to have equal priority.
E) The priorities of each goal are reflected in the objective function.

Question 42

Consider the following 0-1 integer programming problem: If we wish to add the constraint that no more than two of these variables must be positive, how would this be written?

A) 2X + 2Y + 2Z ≤ 3
B) X + Y + Z ≤ 2
C) X ≤ 2, and Y ≤ 2, and Z ≤ 2
D) X, Y, Z ≤ 2
E) None of the above

Question 43

An integer programming (maximization) problem was first solved as a linear programming problem, and the objective function value (profit) was $253.67. The two decision variables (X, Y) in the problem had values of X = 12.45 and Y = 32.75. Which of the following must be true for the optimal integer solution to this problem?

A) X = 12 Y = 32
B) X = 12 Y = 33
C) X = 12
D) Y = 32
E) None of the above

Question 44

The concept of "satisficing" is affiliated with which of the following?

A) mixed integer programming
B) integer programming
C) linear programming
D) nonlinear programming
E) goal programming

Question 45

The overall best solution in a nonlinear program is a ________.

A) global optimum
B) local optimum
C) binary optimum
D) nonlinear optimum
E) goal optimum

Question 46

A capital budgeting problem involving the selection of possible projects under budget constraints is solved by which of the following?

A) mixed-integer programming
B) 0-1 integer programming
C) goal programming
D) nonlinear programming
E) pure integer programming

Question 47

A transportation problem is an example of

A) a pure-integer programming problem.
B) a mixed-integer programming problem.
C) a zero-one integer programming problem.
D) a goal programming problem.
E) a nonlinear programming problem.

Question 48

The concept of a local optimum is affiliated with which of the following?

A) mixed integer programming
B) integer programming
C) linear programming
D) nonlinear programming
E) goal programming

Question 49

Table 10-4
A company has decided to use 0−1 integer programming to help make some investment decisions. There are three possible investment alternatives from which to choose, but if it is decided that a particular alternative is to be selected, the entire cost of that alternative will be incurred (i.e., it is impossible to build one-half of a factory). The integer programming model is as follows:
Table 10-4 presents an integer programming problem. What is the meaning of Constraint 2?

A) Both alternatives 1 and 2 must be selected.
B) If alternative 2 is selected, alternative 1 must also be selected.
C) Either alternative 1 or alternative 2 must be selected.
D) No more than one alternative may be selected.
E) None of the above

Question 50

A goal programming problem had two goals (with no priorities assigned). Goal number 1 was to achieve a cost of $3,600 and goal number 2 was to have no wasted material. The optimal solution to this problem resulted in a cost of $3,900 and no wasted material. What was the value for the objective function for this goal programming problem?

A) 300
B) -300
C) 3300
D) 0
E) None of the above

Question 51

Table 10-4
A company has decided to use 0−1 integer programming to help make some investment decisions. There are three possible investment alternatives from which to choose, but if it is decided that a particular alternative is to be selected, the entire cost of that alternative will be incurred (i.e., it is impossible to build one-half of a factory). The integer programming model is as follows:
Table 10-4 presents an integer programming problem. What is the meaning of Constraint 1?

A) If X₁ is selected, X₂ must also be selected.
B) No more than two alternatives may be selected.
C) At least two alternatives must be selected.
D) If X₂ is selected, X₁ must also be selected.
E) None of the above

Question 52

Which of the following is not considered nonlinear programming?

A) nonlinear objective and nonlinear constraints
B) linear objective with nonlinear constraints
C) nonlinear objective with linear constraints
D) binary decision variable with nonlinear constraints
E) integer decision variable with linear constraints

Question 53

A quadratic programming problem involves which of the following conditions?

A) squared terms in the objective function and linear constraints
B) linear objective function and squared terms in the constraints
C) squared terms in both the objective function and constraints
D) a strictly goal programming problem with squared terms in the objective function
E) None of the above

Question 54

If we wish to develop a stock portfolio wherein we maximize return and minimize risk, we would have to use

A) pure-integer programming.
B) goal programming.
C) zero-one integer programming.
D) mixed-integer programming.
E) nonlinear programming.

Question 55

Another name for a 0-1 variable is a ________ variable.

A) either-or
B) binary
C) yes-no
D) quadratic
E) on-off

Question 56

Consider the following 0 - 1 integer programming problem: If we wish to add the constraint that X must be positive, and that only Y or Z but not both can be positive, how would the additional constraint(s) be written?

A) X + Y + Z ≤ 3, Y + Z ≤ 1
B) X ≤ 1, Y + Z = 1
C) X ≤ 2, and Y ≤ 2, and Z ≤ 2
D) X = 1, Y + Z ≤ 1
E) None of the above

Question 57

Terms that are minimized in goal programming are called ________.

A) deviational variables
B) global variables
C) decision variables
D) minimization variables
E) None of the above

Question 58

A goal programming problem had two goals (with no priorities assigned). Goal number 1 was to achieve a cost of $2,400 and goal number 2 was to have no idle time for workers in the factory. The optimal solution to this problem resulted in a cost of $2,400 and no idle time. What was the value for the objective function for this goal programming problem?

A) 2300
B) 100
C) -100
D) 0
E) None of the above

Question 59

The following represents a:

A) goal programming problem.
B) mixed integer programming problem.
C) nonlinear programming problem.
D) 0-1 integer programming problem.
E) pure integer programming problem.

Question 60

As part of a larger problem, you are trying to determine whether or not to open a plant with a capacity of 10,000 units (using binary variable Y). You also define X as the number of units (if any) produced at that plant. How will you ensure that Y will equal 1 if the plant is open?

A) Y ≥ X
B) Y ≤ X
C) X + Y ≥ 2
D) X = 10000Y
E) X ≤ 10000Y

Question 61

Smalltime Investments Inc. is going to purchase new computers. There are ten employees, and the company would like one for each employee. The cost of the basic personal computer with monitor and disk drive is $2,000, while the deluxe version with VGA and advanced processor is $3,500. Due to internal politics, the number of deluxe computers should be less than half the number of regular computers, but at least three deluxe computers must be purchased. The budget is $27,000, although additional money could be used if it were deemed necessary. All of these are goals that the company has identified. Formulate this as a goal programming problem.

Question 62

The Elastic Firm has two products coming on the market: Zigs and Zags. To make a Zig, the firm needs 10 units of product A and 15 units of product B. To make a Zag, they need 20 units of product A and 15 units of product B. There are only 2,000 units of product A and 3,200 units of product B available to the firm. The profit on a Zig is $4 and on a Zag it is $6. Management objectives in order of their priority are:
(1) Produce exactly 50 Zigs.
(2) Achieve a target profit of at least $750.
(3) Use all of the product B available.
Formulate this as a goal programming problem.

Question 63

Table 10-6
Table 10-6 represents a solution for an integer programming problem. If this problem had been solved as a simple linear programming problem, what would you expect the value of the objective function to be?

A) less than 208
B) greater than 208
C) exactly 208
D) A or C
E) B or C

Question 64

Goal programming and linear programming differ in that

A) in LP, the objective function is maximized or minimized, while in goal programming, the deviation between goals and possible achievement is minimized.
B) slack variables are used in LP, while deviational variables are used in goal programming.
C) deviational variables have positive objective function coefficients in goal programming, but slack variables have 0 coefficients in LP.
D) All of the above
E) None of the above

Question 65

Table 10-5
Table 10-5 represents a solution to a goal programming problem. There are three goals (each represented by a constraint). Which of the goals is assigned the highest priority?

A) goal 1
B) goal 2
C) goal 3
D) goals 2 and 3
E) All goals have the same priority.

Question 66

A package express carrier is considering expanding the fleet of aircraft used to transport packages. There is a total of $220 million allocated for purchases. Two types of aircraft may be purchased - the C1A and the C1B. The C1A costs $25 million, while the C1B costs $18 million. The C1A can carry 60,000 pounds of packages, while the C1B can only carry 40,000 pounds of packages. The company needs at least eight new aircraft. In addition, the firm wishes to purchase at least twice as many C1Bs as C1As. Formulate this as an integer programming problem to maximize the number of pounds that may be carried.

Question 67

Classify the following problems as to whether they are pure-integer, mixed-integer, zero-one, goal, or nonlinear programming problems.
(a) Maximize Z = 5 X₁ + 6 X₁ X₂ + 2 X₂
Subject to: 3 X₁ + 2 X₂ ≥ 6
X₁ + X₂ ≤ 8
X₁, X₂ ≥ 0
(b) Minimize Z = 8 X₁ + 6 X₂
Subject to: 4 X₁ + 5 X₂ ≥ 10
X₁ + X₂ ≤ 3
X₁, X₂ ≥ 0
X₁, X₂ = 0 or 1
(c) Maximize Z = 10 X₁ + 5 X₂
Subject to: 8 X₁ + 10 X₂ = 10
4 X₁ + 6 X₂ ≥ 5
X₁, X₂ integer
(d) Minimize Z = 8 X₁² + 4 X₁ X₂ + 12 X₂²
Subject to: 6 X₁ + X₂ ≥ 50
X₁ + X₂ ≥ 40

Question 68

Smalltime Investments Inc. is going to purchase new computers for most of the employees. There are ten employees, and at least eight computers must be purchased. The cost of the basic personal computer with monitor and disk drive is $2,000, while the deluxe version with VGA and advanced processor is $3,500. Due to internal politics, the number of deluxe computers must be no more than half the number of regular computers, but at least three deluxe computers must be purchased. The budget is $27,000. Formulate this as an integer programming problem to maximize the number of computers purchased.

Question 69

A goal programming problem had two goals (with no priorities assigned). Goal number 1 was to achieve a cost of $3,600 and goal number 2 was to complete the task in 400 hours or fewer. The optimal solution to this problem resulted in a cost of $3,600 and a completion time of 420 hours. What was the value for the objective function for this goal programming problem?

A) 400
B) -400
C) 20
D) 0
E) None of the above

Question 70

Data Equipment Inc. produces two models of a retail price scanner, a sophisticated model that can be networked to a central processing unit and a stand-alone model for small retailers. The major limitations of the manufacturing of these two products are labor and material capacities. The following table summarizes the usages and capacities associated with each product. The typical LP formulation for this problem is:
Maximize $160 X₁ + $95 X₂
Subject to: 8 X₁ + 5 X₂ ≤ 800
20 X₁ + 7 X₂ ≤ 1500
X₁, X₂ ≥ 0
However, the management of DEI has prioritized several goals that are to be attained by manufacturing:
(1) Since the labor situation at the plant is uneasy (i.e., there are rumors that a local union is considering an organizing campaign), management wants to assure full employment of all its employees.
(2) Management has established a profit goal of $12,000 per day.
(3) Due to the high prices of components from nonroutine suppliers, management wants to minimize the purchase of additional materials.
Given the above additional information, set this up as a goal programming problem.

Question 71

A model containing a linear objective function and requiring that one or more of the decision variables take on an integer value in the final solution is called

A) an integer programming problem.
B) a goal programming problem.
C) a nonlinear programming problem.
D) a multiple objective LP problem.
E) insufficient information.

Question 72

Data Equipment Inc. produces two models of a retail price scanner, a sophisticated model that can be networked to a central processing unit and a stand-alone model for small retailers. The major limitations of the manufacturing of these two products are labor and material capacities. The following table summarizes the usages and capacities associated with each product. The typical LP formulation for this problem is:
Maximize P = $160 X₁ + $95 X₂
Subject to: 8 X₁ + 5 X₂ ≤ 800
20 X₁ + 7 X₂ ≤ 1500
X₁, X₂ ≥ 0
However, the management of DEI has prioritized several goals that are to be attained by manufacturing:
(1) Management had decided to severely limit overtime.
(2) Management has established a profit goal of $15,000 per day.
(3) Due to the difficulty of obtaining components from non-routine suppliers, management wants to end production with at least 50 units of each component remaining in stock.
(4) Management also believes that they should produce at least 30 units of the network model.
Given the above additional information, set this up as a goal programming problem.

Question 73

The Elastic Firm has two products coming on the market, Zigs and Zags. To make a Zig, the firm needs 10 units of product A and 15 units of product B. To make a Zag, they need 20 units of product A and 15 units of product B. There are only 2,000 units of product A and 3,000 units of product B available to the firm. The profit on a Zig is $4 and on a Zag it is $6. Management objectives in order of their priority are:
(1) Produce at least 40 Zags.
(2) Achieve a target profit of at least $750.
(3) Use all of the product A available.
(4) Use all of the product B available.
(5) Avoid the requirement for more product A.
Formulate this as a goal programming problem.

Question 74

Allied Manufacturing has three factories located in Dallas, Houston, and New Orleans. They each produce the same product and ship to three regional warehouses: #1, #2, and #3. The cost of shipping one unit of each product to each of the three destinations is given below. There is no way to meet the demand for each warehouse. Therefore, the company has decided to set the following goals: (1) the number shipped from each source should be as close to 100 units as possible (overtime may be used if necessary), (2) the number shipped to each destination should be as close to the demand as possible, (3) the total cost should be close to $1,400. Formulate this as a goal programming problem.

Question 75

Table 10-4
A company has decided to use 0−1 integer programming to help make some investment decisions. There are three possible investment alternatives from which to choose, but if it is decided that a particular alternative is to be selected, the entire cost of that alternative will be incurred (i.e., it is impossible to build one-half of a factory). The integer programming model is as follows:
Table 10-4 presents an integer programming problem. If the optimal solution is used, then only two of the alternatives would be selected. How much slack would there be in the third constraint?

A) 1000
B) 5000
C) 3300
D) 8000
E) None of the above

Question 76

Table 10-5
Table 10-5 represents a solution to a goal programming problem. There are three goals (each represented by a constraint). Which goals are only partly achieved?

A) number 1 only
B) number 1 and number 2
C) number 2 and number 3
D) number 1 and number 3
E) None of the above

Question 77

Table 10-4
A company has decided to use 0−1 integer programming to help make some investment decisions. There are three possible investment alternatives from which to choose, but if it is decided that a particular alternative is to be selected, the entire cost of that alternative will be incurred (i.e., it is impossible to build one-half of a factory). The integer programming model is as follows:
Table 10-4 presents an integer programming problem. Suppose you wish to add a constraint that stipulates that both alternative 2 and alternative 3 must be selected, or neither can be selected. How would this constraint be written?

A) X₂ = X₃
B) X₂ ≤ X₃
C) X₂ ≥ X₃
D) X₂ + X₃ = 1
E) None of the above

Question 78

Table 10-5
Table 10-5 represents a solution to a goal programming problem. There are three goals (each represented by a constraint). Goal number 3 represents a resource usage goal. How much of this resource would be used by this solution?

A) 50 units
B) 70 units
C) 2500 units
D) 240 units
E) None of the above

Question 79

Agile Bikes has manufacturing plants in Salt Lake City, Dallas, and Chicago. The Bikes are shipped to retail stores in Los Angeles, New York, Miami, and Seattle. Information on shipping costs, supply, and demand is given in the following table: What type of mathematical programming is required to solve this problem?

Question 80

Table 10-6
Table 10-6 represents a solution for an integer programming problem. If one uses the optimal solution presented, how much slack is there in the first equation?

A) 0 units
B) 5 units
C) 3 units
D) 2 units
E) None of the above