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

Question 1

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

Answer 1

True

Question 2

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

Answer 2

False

Question 3

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

Answer 3

False

Question 4

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

Question 5

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

Question 6

There is no general method for solving all nonlinear problems.

Question 7

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

Question 8

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

Question 9

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

Question 10

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 11

Quadratic programming contains squared terms in the constraints.

Question 12

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

Question 13

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

Question 14

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

Question 15

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

Question 16

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

Question 17

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

Question 18

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

Question 19

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

Question 20

Goal programming permits multiple objectives to be satisfied.

Question 21

Table 10-2
Maximize Z = 34 X₁ + 43 X₂ + 29 X₃
Subject to: 5 X₁ + 4 X₂ + 7 X₃ ≤ 50
1 X₁ + 2 X₂ + 2 X₃ ≤ 16
3 X₁ + 4 X₂ + 1 X₃ ≤ 9
all X_i are integer and non-negative
Final Integer Solution: Z = 208
Decision
Variable Solution
X₁ 1
X₂ 0
X₃ 6
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 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:
Maximize 5000 X₁ + 7000X₂ + 9000X₃
Subject to: X₁ + X₂ + X₃ ≤ 2 (only 2 may be chosen)
25000X₁ + 32000X₂ + 29000X₃ ≤ 62,000 (budget limit)
16 X₁ + 14 X₂ + 19 X₃ ≤ 36 (resource limitation)
all variables = 0 or 1
where X₁ = 1 if alternative 1 is selected,0 otherwise
X₂ = 1 if alternative 2 is selected,0 otherwise
X₃ = 1 if alternative 3 is selected,0 otherwise
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 23

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:
Maximize 5000 X₁ + 7000X₂ + 9000X₃
Subject to: X₁ + X₂ + X₃ ≤ 2 (only 2 may be chosen)
25000X₁ + 32000X₂ + 29000X₃ ≤ 62,000 (budget limit)
16 X₁ + 14 X₂ + 19 X₃ ≤ 36 (resource limitation)
all variables = 0 or 1
where X₁ = 1 if alternative 1 is selected,0 otherwise
X₂ = 1 if alternative 2 is selected,0 otherwise
X₃ = 1 if alternative 3 is selected,0 otherwise
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 24

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 25

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:
Maximize 5000 X₁ + 7000X₂ + 9000X₃
Subject to: X₁ + X₂ + X₃ ≤ 2 (only 2 may be chosen)
25000X₁ + 32000X₂ + 29000X₃ ≤ 62,000 (budget limit)
16 X₁ + 14 X₂ + 19 X₃ ≤ 36 (resource limitation)
all variables = 0 or 1
where X₁ = 1 if alternative 1 is selected,0 otherwise
X₂ = 1 if alternative 2 is selected,0 otherwise
X₃ = 1 if alternative 3 is selected,0 otherwise
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 26

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 27

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 28

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 29

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 30

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 31

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 32

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 33

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 34

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 35

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 36

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 37

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 38

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:
Maximize 5000 X₁ + 7000X₂ + 9000X₃
Subject to: X₁ + X₂ + X₃ ≤ 2 (only 2 may be chosen)
25000X₁ + 32000X₂ + 29000X₃ ≤ 62,000 (budget limit)
16 X₁ + 14 X₂ + 19 X₃ ≤ 36 (resource limitation)
all variables = 0 or 1
where X₁ = 1 if alternative 1 is selected,0 otherwise
X₂ = 1 if alternative 2 is selected,0 otherwise
X₃ = 1 if alternative 3 is selected,0 otherwise
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 39

A type of integer programming is

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

Question 40

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 41

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 42

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 43

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 44

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 45

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 46

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 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

Another name for a 0-1 variable is a(n)________ variable.

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

Question 49

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 50

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 51

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 52

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 53

Table 10-3
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:
Maximize 5000 X₁ + 7000X₂ + 9000X₃
Subject to: X₁ + X₂ + X₃ ≤ 2 Constraint 1
-X₁ + X₂ ≤ 0 Constraint 2
25,000 X₁ + 32,000 X₂ + 29,000 X₃ ≤ 62,000 (budget limit)
16 X₁ + 14 X₂ + 19 X₃ ≤ 36 (resource limitation)
all variables = 0 or 1
where X₁ = 1 if alternative 1 is selected,0 otherwise
X₂ = 1 if alternative 2 is selected,0 otherwise
X₃ = 1 if alternative 3 is selected,0 otherwise
Solution x₁ = 1,x₂ = 0,x₃ = 1,objective value = 14,000.
Table 10-3 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 54

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 55

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 56

Table 10-3
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:
Maximize 5000 X₁ + 7000X₂ + 9000X₃
Subject to: X₁ + X₂ + X₃ ≤ 2 Constraint 1
-X₁ + X₂ ≤ 0 Constraint 2
25,000 X₁ + 32,000 X₂ + 29,000 X₃ ≤ 62,000 (budget limit)
16 X₁ + 14 X₂ + 19 X₃ ≤ 36 (resource limitation)
all variables = 0 or 1
where X₁ = 1 if alternative 1 is selected,0 otherwise
X₂ = 1 if alternative 2 is selected,0 otherwise
X₃ = 1 if alternative 3 is selected,0 otherwise
Solution x₁ = 1,x₂ = 0,x₃ = 1,objective value = 14,000.
Table 10-3 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 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

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 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

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 61

Table 10-5
Maximize Z = 34 X₁ + 43 X₂ + 29 X₃
Subject to: 5 X₁ + 4 X₂ + 7 X₃ ≤ 50
1 X₁ + 2 X₂ + 2 X₃ ≤ 16
3 X₁ + 4 X₂ + 1 X₃ ≤ 9
all X_i are integer and non-negative
Final Integer Solution: Z = 208
Decision
Variable Solution
X₁ 1
X₂ 0
X₃ 6
Table 10-5 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 62

Table 10-7
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 30 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.
Let X₁ = number of Zigs,X₂ = number of Zags.
d₁^- = underachievement of Zig goal
d₁⁺ = overachievement of Zig goal
d₂^- = underachievement of profit target
d₂⁺ = overachievement of profit target
d₃^- = unused product B
d₃⁺ = additional amount of product B needed
In the goal programming problem described in Table 10-7,what is the optimal solution?

A)X₁ = 75 + X₂ = 91.67
B)X₁ = 91.67 + X₂ = 50
C)X₁ = 50 + X₂ = 75
D)X₁ = 50 + X₂ = 91.67
E)X₁ = 75 + X₂ = 50

Question 63

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?

A)Linear programming
B)Integer programming
C)Mixed-integer programming
D)Zero-one integer programming
E)Nonlinear programming

Question 64

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 65

Table 10-3
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:
Maximize 5000 X₁ + 7000X₂ + 9000X₃
Subject to: X₁ + X₂ + X₃ ≤ 2 Constraint 1
-X₁ + X₂ ≤ 0 Constraint 2
25,000 X₁ + 32,000 X₂ + 29,000 X₃ ≤ 62,000 (budget limit)
16 X₁ + 14 X₂ + 19 X₃ ≤ 36 (resource limitation)
all variables = 0 or 1
where X₁ = 1 if alternative 1 is selected,0 otherwise
X₂ = 1 if alternative 2 is selected,0 otherwise
X₃ = 1 if alternative 3 is selected,0 otherwise
Solution x₁ = 1,x₂ = 0,x₃ = 1,objective value = 14,000.
Table 10-3 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 66

Table 10-7
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 30 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.
Let X₁ = number of Zigs,X₂ = number of Zags.
d₁^- = underachievement of Zig goal
d₁⁺ = overachievement of Zig goal
d₂^- = underachievement of profit target
d₂⁺ = overachievement of profit target
d₃^- = unused product B
d₃⁺ = additional amount of product B needed
In the goal programming problem described in Table 10-7,how is the goal of achieving a target profit of at least $750 expressed?

A)X₁ + X₂ + d₂^- = 750
B)X₁ + X₂ + d₂^- - d₂⁺ = 750
C)4X₁ +6 X₂ + d₂^- = 750
D)4X₁ + 6X₂ + d₂^- - d₂⁺ = 750
E)X₁ + X₂ - d₂⁺ = 750

Question 67

Table 10-4
Table 10-4 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 68

Table 10-4
Table 10-4 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 69

Table 10-3
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:
Maximize 5000 X₁ + 7000X₂ + 9000X₃
Subject to: X₁ + X₂ + X₃ ≤ 2 Constraint 1
-X₁ + X₂ ≤ 0 Constraint 2
25,000 X₁ + 32,000 X₂ + 29,000 X₃ ≤ 62,000 (budget limit)
16 X₁ + 14 X₂ + 19 X₃ ≤ 36 (resource limitation)
all variables = 0 or 1
where X₁ = 1 if alternative 1 is selected,0 otherwise
X₂ = 1 if alternative 2 is selected,0 otherwise
X₃ = 1 if alternative 3 is selected,0 otherwise
Solution x₁ = 1,x₂ = 0,x₃ = 1,objective value = 14,000.
Table 10-3 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 70

Table 10-6
The profit function for a set of two electronics products can be expressed by the following expression: X₁² -2X₁ -3X₂ + 2X₂² where X₁ = the number of product 1 units produced and X₂ = the number of product 2 units produced.At least 10 units of X₁ must be produced and at least 20 units of X₂ must be produced.No more than 50 units total can be produced.
What type of mathematical programming model is required for the problem described in Table 10-6?

A)An integer programming model
B)A goal programming model
C)A nonlinear programming model
D)A zero-one integer programming model
E)A mixed-integer programming model

Question 71

What is the optimal solution to the problem described in Table 10-6?

A)X₁ = 50;X₂ = 0
B)X₁ = 30;X₂ = 20
C)X₁ = 20;X₂ = 30
D)X₁ = 25;X₂ = 25
E)X₁ = 0;X₂ = 25

Question 72

Table 10-7
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 30 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.
Let X₁ = number of Zigs,X₂ = number of Zags.
d₁^- = underachievement of Zig goal
d₁⁺ = overachievement of Zig goal
d₂^- = underachievement of profit target
d₂⁺ = overachievement of profit target
d₃^- = unused product B
d₃⁺ = additional amount of product B needed
In the goal programming problem described in Table 10-7,what is the objective function?

A)Min: P₁d₁^- + P₁d₁⁺ + P₂d₂⁺ + P₂d₂^- + P₃d₃^- + P₃d₃⁺
B)Min: P₁d₁^- + P₁d₁⁺ + P₂d₂⁺ + P₃d₃⁺
C)Min: P₁d₁^- + P₁d₁⁺ + P₂d₂^- + P₃d₃^-
D)Min: P₁d₁^- + P₂d₂^- + P₃d₃^-
E)Min: P₁d₁⁺ + P₂d₂⁺ + P₃d₃⁺

Question 73

Table 10-7
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 30 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.
Let X₁ = number of Zigs,X₂ = number of Zags.
d₁^- = underachievement of Zig goal
d₁⁺ = overachievement of Zig goal
d₂^- = underachievement of profit target
d₂⁺ = overachievement of profit target
d₃^- = unused product B
d₃⁺ = additional amount of product B needed
In the goal programming problem described in Table 10-7,how is the goal of producing exactly 50 Zigs expressed?

A)X₁ + d₁⁺ = 50
B)X₁ - d₁⁺ = 50
C)X₁ + d₁^- = 50
D)X₁ - d₁^- = 50
E)X₁ + d₁^- - d₁⁺ = 50

Question 74

According to Table 10-6,how is the constraint expressed that no more than 50 units can be produced?

A)X₁ + X₂ ≥ 50
B)X₁ + X₂ ≤ 50
C)X₁² + X₂² ≥ 50
D)X₁² + X₂² ≤ 50
E)X₁ + X₂ = 50

Question 75

Table 10-4
Table 10-4 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 76

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 77

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 78

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 79

Table 10-5
Maximize Z = 34 X₁ + 43 X₂ + 29 X₃
Subject to: 5 X₁ + 4 X₂ + 7 X₃ ≤ 50
1 X₁ + 2 X₂ + 2 X₃ ≤ 16
3 X₁ + 4 X₂ + 1 X₃ ≤ 9
all X_i are integer and non-negative
Final Integer Solution: Z = 208
Decision
Variable Solution
X₁ 1
X₂ 0
X₃ 6
Table 10-5 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

Question 80

Table 10-7
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 30 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.
Let X₁ = number of Zigs,X₂ = number of Zags.
d₁^- = underachievement of Zig goal
d₁⁺ = overachievement of Zig goal
d₂^- = underachievement of profit target
d₂⁺ = overachievement of profit target
d₃^- = unused product B
d₃⁺ = additional amount of product B needed
In the goal programming problem described in Table 10-7,how is the goal of the use of all product B available expressed?

A)15X₁ + 30X₂ + d₃^- - d₃⁺ = 3200
B)X₁ + X₂ + d₃^- - d₃⁺ = 3200
C)X₁ + X₂ + d₃^- = 3200
D)15X₁ + 30X₂ + d₃^- = 3200
E)15X₁ + 3-X₂ - d₃⁺ = 3200