Deck 6: Integer, Goal, and Nonlinear Programming Models

Question 1

An investment strategy restricts the portfolio to a mix of two stocks A and B with the following price/share and annual returns:
$\begin{array}{ccc}\text { Stock } & \text { Price/Share } & \text { Annual Return } \\ \mathrm{A} & \$ 35 & 5 \% \\\mathrm{~B} & \$ 45 & 7 \%\end{array}$

Assume that the maximum amount available for investment is $100,000 with the following 2 prioritized investment goals:
Rank R1: Obtain an annual return of at least 6%.
Rank R2: Limit the investment in stock B to no more than 55% of the total investment.
Assume X? = dollar amount allocated to stock A,and X? = dollar amount allocated to stock B.
What is the objective function?

A)Min d?? + d??
B)Min R?(d??)+ R?(d??)
C)Min R?(d??)- R?(d??)
D)Min R?(d??)+ R?(d??)
E)Min R?(d??)+ R?(d??)

Answer 1

Min R?(d??)+ R?(d??)

Question 2

Consider the following constraint and its associated binary variables: XA + XB = 1.
This constraint is an example of an if-then constraint.

Answer 2

False

Question 3

Which of the following statements about nonlinear programming is False?

A)Either the objective function or at least one constraint is nonlinear.
B)The optimal solution must be a corner point solution.
C)There can be both local and global optimal solutions.
D)There is no precise way to know where to start the solution search process.

Answer 3

B

Question 4

The "bin" option is used in Solver to specify general integer variables.

Question 5

A company incurs a cost that is directly proportional to the magnitude of the decision variable.This is an example of a:

A)fixed charge
B)sunk cost
C)variable cost
D)marginal cost
E)inventory cost

Question 6

Consider the following constraint and its associated binary decision variables:
XA + XB + XC ≥ 2.
This constraint is an example of a mutually exclusive constraint.

Question 7

Which type of cost is independent of the volume of production?

A)inventory cost
B)marginal cost
C)sunk cost
D)variable cost
E)fixed charge

Question 8

If a company produces product A then it must also produce either product B or product C.Which of the following constraints enforces this condition?

A)XA - XB - XC ≥ 0
B)XA + XB + XC ≤ 2
C)XA + (XB - XC)≤ 0
D)XA + XB + XC ≥ 2
E)XA - XB - XC ≤ 0

Question 9

If a company produces Product A,then it must produce at least 200 units of Product A.Which of the following constraints model this condition?

A)X₁Y₁ ≤ 200
B)X₁ ≥ 200 + Y₁
C)X₁ ≤ 200Y₁
D)X₁ - 200Y₁ ≥ 0
E)X1 > 200

Question 10

Consider a model with 3 ranked goals.Solving this model requires us to solve:

A)a single LP problem
B)3 separate LP problems
C)2 separate LP problems
D)a single quadratic programming problem
E)3 quadratic programming problems

Question 11

These are problems in which some,but not all,the decision variables must have integer solutions:

A)mixed IP problems
B)pure IP problems
C)pure binary problems
D)mixed binary IP problems
E)goal programming problems

Question 12

Problems which can be stated as an assortment of desired objectives are known as:

A)quadratic programming
B)integer programming
C)goal programming
D)linear programming
E)binary programming

Question 13

Suppose that XA equals 10.What are the values for d₁⁺ and d₁⁻ in the following constraint?
XA + d₁⁻- d₁⁺ = 7

A)d₁⁻ = 0,d₁⁺ = 3
B)d₁⁻= 3,d₁⁺ = 0
C)d₁⁻ = 7,d₁⁺ = 0
D)d₁⁻ = 0,d₁⁺ = 7
E)d₁⁻ = 10,d₁⁺ = 3

Question 14

A company wants to select one product from a set of 3 possible products.Which of the following ensures that only one product will be selected?

A)XA + XB + XC = 1
B)XA + XB + XC ≤ 1
C)XA + XB + XC ≥ 1
D)XA + XB + XC ≥ 0
E)XA - XB - XC = 1

Question 15

Rounding off the solution to an LP relaxed problem may yield:

A)an infeasible solution
B)a non-optimal solution
C)a higher objective function value
D)an unbounded solution
E)A or B

Question 16

Nonlinear programming models can be approximated by linear or almost linear models through a procedure known as ________.

A)separable programming
B)goal programming
C)integer programming
D)quadratic programming
E)mixed integer programming

Question 17

The deviational variable di‾ indicates the amount by which a certain goal is:

A)overachieved
B)over attained
C)binding
D)underachieved
E)non-binding

Question 18

Consider the following constraint that needs to be expressed as a goal:
3X₁ + 4X₂ ≤ 11.
The correct format is:

A)3X₁ + 4X₂ - d₁⁻ + d₁⁺ = 11
B)3X₁ + 4X₂ + d₁⁻ - d₁⁺ = 11
C)3X₁ + 4X₂ + d₁⁻ - d₁⁺ ≥ 11
D)3X₁ + 4X₂ + d₁⁻ - d₁⁺ ≤ 11
E)3X₁ + 4X₂ + d₁⁻ - d₁⁺ ≠ 11

Question 19

These are problems in which all decision variables must have integer solutions:

A)mixed binary IP problems
B)pure binary IP problems
C)mixed IP problems
D)pure IP problems
E)goal programming problems

Question 20

A company wants to select 2 products from a set of four possible products.Which of the following constraints ensures that no more than 3 will be selected?

A)XA + XB + XC + XD = 3
B)XA + XB + XC + XD ≥ 0
C)XA + XB + XC + XD ≤ 3
D)XA + XB + XC + XD ≥ 3
E)XA + XB + XC + XD ≠ 3

Question 21

The optimal solution of to an NLP model must be a corner point of the feasible region.

Question 22

In mixed IP problems,some,but not all,of the decision variables have integer solutions.

Question 23

The deviational variable dᵢ⁻ typically denotes underachievement.

Question 24

In goal programming,the objective is typically to maximize the sum of the deviational variables.

Question 25

Use this information to answer the following questions.
A decision maker is provided with 5 different potential projects and must determine which projects to choose.The projects require different amounts of capital and different expected net present values (NPV)over the next three years.

Determine which set of projects should be selected in order to achieve the maximum net present value if the decision maker has $150,000 available for investment each year.

Question 26

An NLP model can have both local and global optimal solutions.

Question 27

The IP solution can sometimes produce a better objective function value than its LP relaxed problem.

Question 28

Use this information for the following questions.
Six products are currently waiting processing on a single machine.The machine is available today for a maximum of 25 hours.The processing time per hour for each product along with the profit contribution per product is show below.

Refer to the information above.Which products should be processed to maximize the total profit contribution?

Question 29

If overachievement is acceptable,the appropriate di⁺ variable can be dropped from the objective function.

Question 30

An integer programming problem assumes that its objective function and its constraints are linear.

Question 31

Use this information to answer the following questions.
A decision maker is provided with 5 different potential projects and must determine which projects to choose.The projects require different amounts of capital and different expected net present values (NPV)over the next three years.

Refer to the information above.Determine which set of projects should be selected in order to achieve the maximum net present value if the following two conditions must also be met:
a)If project 1 is selected,then project 2 must be selected,and vice versa.
b)Since projects 4 and 5 require outsourcing various operations,the decision maker wants at most one of these projects to be included in the solution and not both.

Question 32

There is no precise way to know where to start the solution search process for an NLP model.

Question 33

Consider the following objective function with prioritized goals: Min: R₁(d₁⁻)+ R₂(d₁⁻).
This implies that goal 2 is of higher rank than goal 1.

Question 34

The optimal solution to an IP model must be at a corner point of the feasible region.

Question 35

Rounding off the solution to an LP relaxation problem may sometimes yield an infeasible solution to an IP problem.

Question 36

Use this information for the following questions.
Six products are currently waiting processing on a single machine.The machine is available today for a maximum of 25 hours.The processing time per hour for each product along with the profit contribution per product is show below.

Refer to the information above.Assume that the following conditions must be met:
a)We want to ensure that exactly one of these products: A,B,or C must be processed.
b)We want to ensure that if product E is processed then product F must be processed.
Which products should be processed to maximize the total profit contribution?

Question 37

A goal programming problem assumes that its objective function and constraints are linear.

Question 38

When an objective function contains squared terms,and the problem's constraints are linear,it is referred to as a quadratic programming problem.

Question 39

Mr.Smith,an avid reader,must decide on which books to take with him during a long flight.Mr.Smith will use his carry-on bag,which can hold a maximum of 20 pounds.Mr.Smith has attached a satisfaction index for each book,based on a 5-point scale (1 = low satisfaction,5 =high satisfaction),as shown below.

Which books should Mr.Smith take with him on the flight to maximize his satisfaction index?

Question 40

Deviational variables assume the value zero if a goal is completely attained.

Question 41

Set up and solve the following nonlinear programming problem using Excel:

Question 42

Consider the following linear programming problem:

Reformulate and solve this problem as a goal programming problem if the following prioritized goals must be met.

Question 43

An investor is considering 7 different stocks: A,B,C,D,E,F,and G.The expected annual return for each stock is provided as follows:

The investor has imposed the following restrictions regarding the composition of the portfolio:
•the portfolio must include exactly one of the following stocks: either A or B.
•if stock B is selected,then stock F must also be selected.
•if stock D is selected,then stock E must be excluded.
Which stocks should be included in the portfolio to maximize annual return?

Question 44

A company currently has two factories: F1 and F2,and three retail outlets: R1,R2,and R3.The shipping costs per unit along with the monthly capacity and demand requirements are summarized below:

The firm has decided to build a new factory to expand its productive capacity.The two sites being considered are Philadelphia and Pittsburgh.The estimated shipping costs for the new factories along with their estimated fixed cost and production capacity are summarized below:

Which of the new locations will yield the lowest cost in combination with the existing factories and retail outlets?

Question 45

A marketing manager is considering the following advertising media to promote a new
product.

The marketing manager has established the following goals and weights:
Goal 1: Reach at least 500,000 individuals;Weight 40.
Goal 2: Limit the total spending to $150,000;Weight 60.
Formulate and solve this goal programming problem.

Question 46

A city is reviewing the location of its fire stations.The city is made up of a number of districts,as illustrated below.A fire station can be placed in any district and is able to handle the fires for both its neighborhood and any adjacent neighborhood.The objective is to minimize the number of fire stations used.

Question 47

Consider the following linear programming problem:

Reformulate and solve this problem as a goal programming problem if the following goals,with their associated weights,must be met.
Goal 1: Produce at least 15 units of product A;Weight 15
Goal 2: Produce at least 15 units of product B;Weight 15
Goal 3: Minimize overtime in assembly department;Weight 25
Goal 4: Minimize overtime in machine department;Weight 25
Goal 5: Achieve at least $100,000 in profit;Weight 20

Question 48

Consider the following linear programming problem:

Reformulate and solve this problem as a goal programming problem if the following goals,stated in no particular order,must be met.
Goal 1: Produce at least 15 units of each product.
Goal 2: Avoid overtime in the assembly and machine departments.
Goal 3: Achieve at least $100,000 in profit.

Question 49

Consider the following profit expected from each dollar (Xᵢ)spent in the following advertising media:

How should the advertising budget be allocated if the company wants to spend at least $1000 on each medium without exceeding its $10,000 budget?

Question 50

A developer wishes to expand an office complex and needs to determine how many small,medium,and large offices to include in the expansion.Each small office requires 400 square feet,each medium office requires 700 square feet,and each large office requires 1200 square feet.The current square footage available for expansion is 35,000 square feet.The developer expects to pay $10,000 for each small office,$20,000 for each medium office,and $35,000 for each large office.The developer wishes to limit his expenditures to $500,000.How many of each type of office should be built if the following goals,stated in no particular order,must be met?
Goal 1: The expansion should include approximately 6 small offices.
Goal 2: The expansion should include approximately 11 medium offices.
Goal 3: The expansion should include approximately 15 large offices.
Goal 4: The expansion should consist of approximately 35,000 square feet.
Goal 5: The expansion should cost approximately $500,000.

Question 51

Set up and solve the following nonlinear programming problem using Excel:

Question 52

Set up and solve the following nonlinear programming problem using Excel:

Question 53

A company produces four products each of which must undergo assembly,inspection,and packaging.The information below summarizes the hours required for each operation by each product,the total hourly capacity for each operation,and the fixed cost incurred by each product if it is produced.

Each unit of product 1 sold will contribute $60 to profit,and each unit of products 2,3,and 4 contributes $65,$70,and $55,respectively.What is the optimal product mix?

Question 54

A marketing manager is considering the following advertising media to promote a new
product.

The marketing manager has established the following goals,stated in no particular order,for the advertising campaign:
Goal 1: Reach at least 500,000 individuals.
Goal 2: Limit the total spending to $150,000.
Formulate and solve this goal programming problem.

Question 55

A company is considering producing 8 different products for the upcoming holiday season.The profit contribution per product is illustrated below:

The marketing manager has imposed the following restrictions on the production mix:
•Since products 1,2,and 3 are electronic gadgets,at least two of these products must be in the production mix.
•Since products 4 and 5 are kids-oriented,the production mix must include no more than one of these products.
•If product 7 is included in the mix,then product 8 must be included,and vice versa.
Which products should be included in the mix to maximize profit?