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

Question 1

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

Answer 1

True

Question 2

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

Answer 2

True

Question 3

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

Answer 3

True

Question 4

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

Question 5

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

Question 6

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

Question 7

There is no general method for solving all nonlinear problems.

Question 8

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

Question 9

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

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

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

Question 13

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

Question 14

Goal programming permits multiple objectives to be satisfied.

Question 15

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

Question 16

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

Question 17

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

Question 18

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

Question 19

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

Question 20

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

Question 21

The global optimum point may be superior to nearby points, but the local optimum point is the true optimal solution-n to a nonlinear programming problem.

Question 22

The constraint X₁ - X₂ ≤ 0 with 0 -1 integer programming allows for X₁ to be selected as part of the optimal solution only if X₂ is selected to be a part of the optimal solution, but not both.

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.

Question 24

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

Question 25

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

Question 26

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

Question 27

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.

Question 28

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)It maximizes instead of minimizing as in LP.

Question 29

The constraint X₁ - X₂ = 0 with 0 -1 integer programming allows for either both X₁ and X₂ to be selected to be a part of the optimal solution, or for neither to be selected.

Question 30

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

Question 31

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

Question 32

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.

Question 33

The constraint X₁ + X₂ + X₃+ X₄ ≤ 2 with 0 -1 integer programming allows at most two of the items X_1, X_2, X_3, and X₄ to be selected to be a part of the optimal solution.

Question 34

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

Question 35

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.

Question 36

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.

Question 37

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.

Question 38

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.

Question 39

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)deviation from certainty.

Question 40

Which of the following functions is nonlinear?

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

Question 41

Consider the following 0-1 integer programming problem: Minimize 20X + 36Y +24Z
Subject to: 2X + 4Y + 3Z ≥ 7
12X + 8Y + 10Z ≥ 25
X, Y, Z must be 0 or 1
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

Question 42

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

Question 43

The overall best solution in a nonlinear program is a

A)global optimum.
B)local optimum.
C)binary optimum.
D)nonlinear optimum.

Question 44

Which of the following is not considered nonlinear programming?

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

Question 45

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

Question 46

Consider the following 0-1 integer programming problem: Minimize 20X + 36Y + 24Z
Subject to: 2X + 4Y + 3Z ≥ 7
12X + 8Y + 10Z ≥ 25
X, Y, Z must be 0 or 1
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

Question 47

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

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

Question 48

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.

Question 49

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

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

Question 50

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: Where should Los Angeles receive their bicycles from?

A)Salt Lake City and Chicago
B)Salt Lake City and Dallas
C)Chicago and Dallas
D)Dallas

Question 51

The following Maximize: 7X₁ + 3X₂
Subject to: 5X₁ + 7X₂ ≤ 27
4X₁ + X₂ ≤ 14
3X₁ - 2X₂ ≤ 9
X₁, X₂ ≥ 0
X₁ integer
Represents a:

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

Question 52

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

Question 53

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 = 10000Y
D)X ≤ 10000Y

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.

Question 55

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

Question 56

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.

Question 57

Terms that are minimized in goal programming are called

A)deviational variables.
B)global variables.
C)decision variables.
D)minimization variables.

Question 58

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

Question 59

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

Question 60

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

Question 61

Johnny's apple shop sells homemade apple pies and freshly squeezed apple juice.Each apple pie requires 2 apples, and 1 apple yields 4 ounces of juice.Customer's use a self-service dispenser to pour apple juice in a container and are charged by the ounce at a rate of $0.50 per ounce.The contribution to profit of the apple pie, factoring in the apples and remaining ingredients are $2 per pie, and the contribution to profit of freshly squeezed apple juice is $0.20 per ounce.In a given day, there must be at least 100 ounces of apple juice produced and at least 10 apple pies.The company has a supply of 60 apples per day.What is the optimal solution? Apple pies must be produced in whole quantities, but any positive value is positive for juice production.

Question 62

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 63

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:
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.
Management has established a profit goal of $12,000 per day.
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 64

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 65

How many constraints are required to develop an integer solution to the haberdashery problem described in Table 10-9?

A)4
B)5
C)6
D)7

Question 66

A package express carrier is considering expanding the fleet of aircraft used to transport packages.Of primary importance is that there is a total of $350 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.Of secondary importance is that the company needs at least 10 new aircraft.It takes 150 hours per month to maintain the C1A, and 100 hours to maintain the C1B.The least level of importance is that there are a total of 1,200 hours of maintenance time available per month.
(a)First, formulate this as an integer programming problem to maximize the number of pounds that may be carried.
(b)Second, rework the problem differently than in part (a)to suppose the company decides that what is most important to them is that they keep the ratio of C1Bs to C1As in their fleet as close to 1.2 as possible to allow for flexibility in serving their routes.Formulate the goal programming representation of this problem, with the other three goals having priorities P2, P3, and P4, respectively.

Question 67

Johnny's apple shop sells homemade apple pies and freshly squeezed apple juice.Each apple pie requires 2 apples, and 1 apple yields 4 ounces of juice.Customers use a self-service dispenser to pour apple juice in a container and are charged by the ounce at a rate of $0.50 per ounce.The contribution to profit of the apple pie, factoring in the apples and remaining ingredients are $2 per pie, and the contribution to profit of freshly squeezed apple juice is $0.20 per ounce.In a given day, there must be at least 100 ounces of apple juice produced and at least 10 apple pies.The company has a supply of 60 apples per day.Formulate this problem as a mixed integer program.Apple pies must be produced in whole quantities, but any positive value is positive for juice production.

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

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.

Capacity 800 hr/day 1,500 comp/day
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:
Management had decided to severely limit overtime.
Management has established a profit goal of $15,000 per day.
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.
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 70

Define deviational variables.

Question 71

Define quadratic programming.

Question 72

Bastille College is going to purchase new computers for both faculty and staff.There are a total of 50 people who need new machines - 30 faculty and 20 staff.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 assigned to staff must be less than half the number of deluxe computers assigned to faculty.The College feels that it must purchase at least 5 deluxe computers for the faculty; if possible, it would like to purchase as many as 20 deluxe computers for the faculty.Staff members do feel somewhat "put upon" by having a limit placed upon the number of deluxe machines purchased for their use, so the College would like to purchase as many deluxe machines for the staff as possible (up to 10).The budget is $100,000.Develop a goal programming formulation of this problem that treats each of the requirements stated above as an equally weighted goal.

Question 73

Allied Manufacturing has three factories located in Dallas, Houston, and New Orleans.They each produce the same 281 products 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 in the table below:

There is no way to meet the demand for each warehouse.Therefore, the company has decided to set the following equally weighted goals: (1)each source should ship as much of its capacity as possible, (2)the number shipped to each destination should be as close to the demand as possible, (3)the capacity of New Orleans should be divided as evenly as possible between warehouses #1 and #2, and (4)the total cost should be less than $1,400.Formulate this as a goal program, which includes a strict requirement that capacities cannot be violated.

Question 74

How many constraints are required to develop an integer solution to the haberdashery problem described in Table 10-9?

A)4
B)5
C)6
D)7

Question 75

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 76

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 77

A bakery produces muffins and doughnuts.Let x₁ be the number of doughnuts produced and x₂ be the number of muffins produced.The profit function for the bakery is expressed by the following equation: profit = 4x₁ + 2x₂ + 0.3x₁² + 0.4x₂².The bakery has the capacity to produce 800 units of muffins and doughnuts combined and it takes 30 minutes to produce 100 muffins and 20 minutes to produce 100 doughnuts.There is a total of 4 hours available for baking time.There must be at least 200 units of muffins and at least 200 units of doughnuts produced.Formulate a nonlinear program representing the profit maximization problem for the bakery.

Question 78

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 79

State the advantage of goal programming over linear programming.

Question 80

A bakery produces muffins and doughnuts.Let x₁ be the number of doughnuts produced and x₂ be the number of muffins produced.The profit function for the bakery is expressed by the following equation: profit = 4x₁ + 2x₂ + 0.3x₁² + 0.4x₂².The bakery has the capacity to produce 800 units of muffins and doughnuts combined and it takes 30 minutes to produce 100 muffins and 20 minutes to produce 100 doughnuts.There is a total of 4 hours available for baking time.There must be at least 200 units of muffins and at least 200 units of doughnuts produced.How many doughnuts and muffins should the bakery produce in order to maximize profit?