Deck 8: Linear Optimization Models

Question 1

A _____ refers to a constraint that holds as equality at the optimal solution.

A) dummy variable
B) first class constraint
C) slack variable
D) binding constraint

Answer 1

binding constraint

Question 2

The assumption that is necessary for a linear programming model to be appropriate and that ensures that the value of the objective function and the total resources used can be found by summing the objective function contribution and the resources used for all decision variables is known as _____.

A) proportionality
B) negativity
C) additivity
D) divisibility

Answer 2

additivity

Question 3

_____ is the process of translating a verbal statement of a problem into a mathematical statement.

A) Problem-solving approach
B) Data preparation
C) Data structuring
D) Problem formulation

Answer 3

Problem formulation

Question 4

In the case of a linear model with two decision variables, if the constraints are in the form of inequalities they are visually represented by regions called as:

A) half spaces.
B) curves.
C) 2-spaces.
D) regions of linear intersection.

Question 5

A(n) _____ refers to a set of points that yield a fixed value of the objective function.

A) objective function coefficient
B) infeasible solution
C) objective function contour
D) feasible region

Question 6

The nonnegativity constraints create a feasible region that is:

A) unbound by the horizontal axis only.
B) an area with no point satisfying all the constraints.
C) symmetric about the vertical axis around the origin.
D) bound by the horizontal and vertical axes.

Question 7

A mathematical function in which each variable appears in a separate term and is raised to the first power is known as a _____.

A) power function
B) linear function
C) what-if function
D) nonlinear function

Question 8

Nonnegativity constraints ensure that

A) the problem modeling includes only nonnegative values in the constraints.
B) the solution to the problem will contain only nonnegative values for the decision variables.
C) the objective function of the problem always returns maximum quantities.
D) there are no inequalities in the constraints.

Question 9

The _____ assumption necessary for a linear programming model to be appropriate, means that the contribution to the objective function and the amount of resources used in each constraint are in accordance to the value of each decision variable.

A) proportionality
B) divisibility
C) additivity
D) negativity

Question 10

Which algorithm, developed by George Dantzig, is effective at investigating extreme points in an intelligent way to find the optimal solution to even very large linear programs?

A) The ellipsoidal algorithm
B) The complex algorithm
C) The trial-and-error algorithm
D) The simplex algorithm

Question 11

A(n) _____ solution satisfies all the constraint expressions simultaneously.

A) feasible
B) objective
C) infeasible
D) extreme

Question 12

The intersections of half spaces represent_____.

A) objective functions
B) feasible solutions
C) decision variables
D) 2-spaces

Question 13

In problem formulation, the:

A) objective is expressed in terms of the decision variables.
B) constraints are expressed in terms of the obtained objective function coefficients.
C) nonnegativity constraints are always ignored.
D) optimal solution is decided upon.

Question 14

A controllable input for a linear programming model is known as a _____.

A) parameter
B) decision variable
C) dummy variable
D) constraint

Question 15

In a linear programming model, the _____ assumption plus the nonnegativity constraints mean that decision variables can take on any value greater than or equal to zero.

A) proportionality
B) divisibility
C) additivity
D) negativity

Question 16

The points where constraints intersect on the boundary of the feasible region are termed as _____.

A) feasible points
B) extreme vertices
C) extreme points
D) feasible edges

Question 17

When formulating a constraint, care must be taken to ensure that:

A) all the objective function coefficients are included.
B) there are no inequalities in the mathematical expression.
C) the decision variables are set at either maximum or minimum values.
D) the units of measurement on both sides of the constraint match.

Question 18

Constraints are:

A) quantities to be maximized in a linear programming model.
B) quantities to be minimized in a linear programming model.
C) restrictions that limit the settings of the decision variables.
D) input variables that can be controlled during optimization.

Question 19

_____ is an Excel tool that utilizes Dantzig's simplex algorithm to solve linear programs by systematically finding which set of constraints form the optimal extreme point of the feasible region.

A) Data Analysis
B) Goal Seeker
C) Excel Solver
D) Watch Window

Question 20

The term _____ refers to the expression that defines the quantity to be maximized or minimized in a linear programming model.

A) objective function
B) problem formulation
C) decision variable
D) association rule

Question 21

The study of how changes in the input parameters of a linear programming problem affect the optimal solution is known as_____.

A) regression analysis
B) cluster analysis
C) optimality analysis
D) sensitivity analysis

Question 22

Reference - 8.2: Use the information given below to answer questions
A canned food manufacturer has its manufacturing plants in three locations across a state. Their product has to be transported to three central distribution centers, which in turn disperse the goods to seventy-two stores across the state.
Reference - 8.2: Which of the following is most likely to be the objective function in this scenario?

A) Increasing the number of goods manufactured at the plant
B) Decreasing the cost of their raw material sourcing
C) Minimizing the cost of shipping goods from the plant to the store
D) Minimizing the quantity of goods distributed across the stores

Question 23

Which of the following error messages is displayed in Excel Solver when attempting to solve an unbounded problem?

A) Solver could not find a feasible solution.
B) Solver cannot improve the current solution. All constraints are satisfied.
C) Solver could not find a bounded solution.
D) Objective Cell values do not converge.

Question 24

The _____ for a decision variable is the shadow price of the nonnegativity constraint for that variable.

A) range of optimality
B) slack value
C) reduced cost
D) range of feasibility

Question 25

_____ is the situation in which no solution to the linear programming problem satisfies all the constraints.

A) Unboundedness
B) Divisibility
C) Infeasibility
D) Optimality

Question 26

The reduced cost indicates the change in the optimal objective function value that results from changing the right-hand side of the nonnegativity constraint from:

A) 1 to 0.
B) 0 to 1.
C) 1 to ∞.
D) ∞ to 1.

Question 27

In linear programming models of real problems, the occurrence of an unbounded solution means that the:

A) resultant values of the decision variables have no bounds.
B) mathematical models sufficiently represent the real-world problems.
C) problem formulation is improper.
D) constraints have been excessively used in modeling.

Question 28

A scenario in which the optimal objective function contour line coincides with one of the binding constraint lines on the boundary of the feasible region leads to _____ solutions.

A) infeasible
B) alternative optimal
C) binding
D) unique optimal

Question 29

Problems with infeasible solutions arise in practice because:

A) management doesn't specify enough restrictions.
B) too many restrictions have been placed on the problem.
C) of errors in objective function formulation.
D) there are too few decision variables.

Question 30

The situation in which the value of the solution may be made infinitely large in a maximization linear programming problem or infinitely small in a minimization problem without violating any of the constraints is known as _____.

A) infeasibility
B) unbounded
C) infiniteness
D) semi-optimality

Question 31

The slack value for binding constraints is:

A) always a positive integer.
B) zero.
C) a negative integer.
D) equal to the sum of the optimal points in the solution.

Question 32

A variable subtracted from the left-hand side of a greater-than-or-equal to constraint to convert the constraint into an equality is known as a(n) _____.

A) surplus variable
B) slack variable
C) unbounded variable
D) binding constraint

Question 33

Reference - 8.1: Use the information given below to answer questions
Rob is a financial manager with Sharez, an investment advisory company. He must select specific investments-for example, stocks and bonds-from a variety of investment alternatives.
Reference - 8.1. Which of the following statements is most likely to be the objective function in this scenario?

A) Minimization of the number of stocks held
B) Maximization of expected return
C) Minimization of tax dues
D) Maximization of investment risk

Question 34

The shadow price of nonbinding constraints

A) will always be zero.
B) always take positive values.
C) can never be equal to zero.
D) is no longer valid if the right-hand side of the constraint remains the same.

Question 35

The change in the optimal objective function value per unit increase in the right-hand side of a constraint is given by the _____.

A) objective function coefficient
B) shadow price
C) restrictive cost
D) right-hand side allowable increase

Question 36

Reference - 8.2: Use the information given below to answer questions
A canned food manufacturer has its manufacturing plants in three locations across a state. Their product has to be transported to three central distribution centers, which in turn disperse the goods to seventy-two stores across the state.
Reference - 8.2: Which of the following visualization tools could help understand this problem better?

A) A time-series plot
B) A scatter chart
C) A network graph
D) A contour plot

Question 37

Geometrically, binding constraints intersect to form the _____.

A) subspace
B) optimal point
C) decision cell
D) zero slack

Question 38

The _____ value for each less-than-or-equal-to constraint indicates the difference between the left-hand and right-hand values for a constraint.

A) objective function coefficient
B) slack
C) unbounded
D) surplus

Question 39

The _____ Report generated by Excel Solver gives information on the objective function value when variables are set to their limits.

A) Answer
B) Sensitivity
C) Classical
D) Limits

Question 40

Reference - 8.1: Use the information given below to answer questions
Rob is a financial manager with Sharez, an investment advisory company. He must select specific investments-for example, stocks and bonds-from a variety of investment alternatives.
Reference - 8.1. Restrictions on the type of permissible investments would be a _____ in this case.

A) feasible solution
B) surplus variable
C) slack variable
D) constraint

Question 41

A soft drink manufacturing company has 3 factories set up one in each of the three cities - Orland, Tampa, and Port St. Lucie and it supplies the produced soft drink bottles to 3 warehouses located in the city of Miami. The associated per-unit transportation cost between the factories and the warehouses is provided in the table below:

The factory at Orlando has a capacity of 14,000 units.
The factory at Tampa has a capacity of 25,000 units.
The factory at Port St. Lucie has a capacity of 23,000 units.
The requirements of the warehouses are:

Determine how much of the company's production should be shipped from each factory to each warehouse in order to minimize the total transportation cost?

Question 42

Three plants P1, P2, and P3 of a gas corporation supply gasoline to three of their distributors located in the city at three different locations A, B, and C. The plants' daily capacities are 4500, 3000, and 5000, gallons respectively, while the distributors' daily requirements are 5500, 2500, and 4200 gallons. The per-gallon transportation costs (in $) are provided in the table below:

Because of a failure of expected supply earlier, the distributors - A, B, and C this time have decided to charge a penalty of $0.45, $0.55, and $0.5 per gallon, respectively, to avoid any further delays.
Now, determine the optimum supply of gasoline to the distributors, in order to minimize the total transportation cost as well as the charges payable as penalty.

Question 43

A manager of a quality testing team wanted to test different lots of products using three resources, R1, R2, and R3. Each lot can be tested for quality using any one of the three procedures, P1, P2, or P3. The product once tested will be sent for packaging. The profit contribution per lot for each of these procedures varies and they are $4, $5, and $8, respectively. Also, resource A requires 2 hours, 3 hours, and 4 hours to test a lot using the procedure P1, P2, and P3, respectively. Resource B requires 3 hours, 2 hours, and 3 hours using the procedure P1, P2, and P3, respectively. Lastly, resource C requires 2 hours, 3 hours, and 4 hours using the procedure P1, P2, and P3, respectively. The available times for these three resources are 80 hours, 90 hours, and 65 hours, respectively. Formulate and solve a linear program and solve for the optimal solution for the above scenario by maximizing the profit.a. What will be the change in total profit, if machine M3 is given an extra hour of production time?

Question 44

Hire-a-Car System rents three types of cars at two different locations. The profit made per day for each car type and company at the two locations is listed below:

The management forecasts the demand per day by car type. A linear programming model developed to maximize profit is used to determine how many reservations to accept for each type of car. The demand forecast for a particular day is 125 rentals for Type 1 cars, 55 rentals for Type 2 cars, and 40 rentals for Type 3 cars. The company has 100 cars in location A and 120 cars in location B.
Use linear programming to determine how many reservations to accept for each car type and how the reservations should be allocated to the different locations. Is the demand for any car type not satisfied? Explain.

Question 45

Robin Tires, Inc. makes two types of tires, for SUV's and Hatchbacks. The firm has 500 hours of production time, 250 hours of packaging, and 150 hours available for shipping. The production time required per tire type is given in the following table:

Assuming that the company is interested in maximizing the total profit contribution, answer the following:a. What is the linear programming model for this problem? b. Develop a spreadsheet model and find the optimal solution using Excel Solver. How many tires of each model should Robin manufacture? c. What is the total profit contribution Robin can earn with the optimal production quantities?

Question 46

Three plants P1, P2, and P3 of a gas corporation supply gasoline to three of their distributors in the city located at A, B, and C locations. The plants' daily capacities are 4500, 3000, and 5000, gallons respectively, while the distributors' daily requirements are 5500, 2500, and 4200 gallons. The per-gallon transportation costs (in $) are provided in the table below:

Because of a failure of expected supply earlier, the distributors this time have decided to charge a penalty of $0.45, $0.55, and $0.5 per gallon, respectively for the locations A, B, and C to avoid any further delays.
Find an alternative optimal solution for this transportation problem? Hint: Use the procedure described in section 8.7.

Question 47

Gatson manufacturing company produces 2 types of tires: Economy tire; Premium tire. The manufacturing time and the profit contribution per tire are given in the following table:

Answer the following assuming that the company is interested in maximizing the total profit contribution. a. What is the linear programming model for this problem? b. Develop a spreadsheet model and find the optimal solution using Excel Solver. How many tires of each model should Gatson manufacture? c. What is the total profit contribution Gatson can earn with the optimal production quantities?

Question 48

Sunseel Industries produces two types of raw materials A and B, with a production cost of $4 and $8 per unit, respectively. The combined production of the raw materials A and B must be at least 700 units per month. At least 400 units of the raw material B and not more than 1200 units of the raw material A must be produced per month. The processing time for the raw material A is observed to be 5 hours and for B, it is found to be 4 hours. A total of 3000 such hours are available per month. How much of each raw material should be produced in order to minimize the cost. Develop a linear program that Sunseel Industries can use to determine how many units of each raw material to produce to minimize the total cost.

Question 49

Robin Tires, Inc. makes two types of tires, for SUV's and Hatchbacks. The firm has 500 hours of production time, 250 hours of packaging, and 150 hours available for shipping. The production time required per tire type is given in the following table:

Assuming that the company is interested in maximizing the total profit contribution, find the optimal solution using Excel Solver and answer the following:a. How many hours of production time will be scheduled in each department? b. What is the slack time in each department? c. If one more hour is available for packaging, what is the change in profit? d. What is the change in profit if one more hour is available for shipping?

Question 50

Ethan Steel, Inc. has two factories that manufacture steel components for four different rail projects located at four different sites. The demand for the steel components for the four projects, Project A, Project B, Project C, and Project D, are 3220, 3675, 4125, and 2975, respectively. The shipping details are as below:
Production Details:

Shipping Details (with per-unit shipping cost):

What is the optimal (cost minimizing) distribution plan for this transportation problem?

Question 51

The supervisor of a production company is trying to determine the number of two assembling parts, Part X and Part Y to be produced per day in three different sections of the plant. The time required for the production along with the profit contribution for each part are given in the following table:

Each part made (X and Y) must be processed in each of the three sections. No more than 60 units of Part X can be produced, but up to 70 units of Part Y can be produced per day. The company already has orders for 30 units of Part Y that must be satisfied. a. Develop a linear programming model and solve the model to determine the optimal production quantities of parts X and Y. b. If more time could be made available in Section 2, how much worth would it be?

Question 52

A Cake & Pastry shop makes 3 types of cakes which require three significant ingredients, given the combination of other ingredients vary. The data for the amount of these ingredients needed to make the cakes are provided in the table below:

Develop and solve a linear programming model to maximize the profit. What is the optimal solution for this problem?

Question 53

A soft drink manufacturing company has 3 factories set up one in each of the three cities - Orland, Tampa, and Port St. Lucie and it supplies the produced soft drink bottles to 3 warehouses located in the city of Miami. The associated per-unit transportation cost table is provided below:

The factory at Orlando has a capacity of 15,000 units.
The factory at Tampa has a capacity of 18,000 units.
The factory at Port St. Lucie has a capacity of 8,000 units.
The requirements of the warehouses are:
a. Determine how much of the company's production should be shipped from each factory to each warehouse in order to minimize the total transportation cost? b. Find an alternative optimal solution for this transportation problem? Hint: Use the procedure described in section 8.7.

Question 54

Jackson just obtained $240,000 by selling mutual funds and is now looking for other investment opportunities for these funds. His financial consultant recommends that all new investments be made in the stocks of industries such as like Agriculture, Healthcare, Banking, Manufacturing, and Real Estate. The projected annual rates of returns for the investments are as follows:

His consultant has set constraints on the investments based on the calculated risks involved with the industries:
1) Neither Agriculture nor Manufacturing industry should receive more than $100,000.
2) Neither Healthcare nor Banking should receive more than $50,000.
3) The amount invested in Manufacturing industry should not be more than 45 percent of the sum of the investment in Banking and Healthcare sectors.
4) The amount invested in Real Estate should be at least 20 percent of the sum of the investment in Banking and Healthcare sectors.
What portfolio recommendations-investments and amounts-should be made for the available $240,000?

Question 55

Northwest California Ventures Ltd. has decided to provide capital in five market areas for the start-ups. The investment consultant for the venture capital company has projected an annual rate of return based on the market risk, the product, and the size of the market.

The maximum capital provided will be $5 million.
The consultant has imposed conditions on allotment of capital based on the risk involved in the market.
• The capital provided to retail should be at most 40 percent of the total capital.
• The capital for education should be 26 percent of the total of other four markets (Electronics, Software, Logistics, and Retail)
• Logistics should be at least 15 percent of the total capital.
• The capital allocated for Software plus Logistics should be no more than the capital allotted for Electronics.
• The capital allocated for Logistics plus Education should not be greater than that allocated to Retail.
Calculate the expected annual rate of return based on the allocation of capital to each market area to maximize the return on capital provided. Also, show the allocation of capital for each market area.

Question 56

Clever Sporting Equipment, Inc. makes two types of balls: Soccer balls and Cork balls. The making of each soccer ball and cork ball requires 3 hours and 4 hours of production time, respectively. For the next month, the total production hours of 500 are available. Also, it is given that the combined production quantity for these two balls must be at least 150 units. The objective for this linear programming model is to fulfil the given production requirements at a minimum cost for the total production. The production cost for each Soccer ball is $9 and each Cork ball is $7. Formulate and solve for the recommended production quantities.

Question 57

A beverage cans manufacturer makes 3 types of soft drink cans needed for the beverage producers to fill soft drinks of three different volumes. The maximum availability of the machines' time allotted per day is 90 hours and the supply of metal is limited to 120 kg per day. The following table provides the details of the input needed to manufacture one batch of 100 cans.

Formulate and solve for the recommended production quantities for all the three different types cans by maximizing the profit.

Question 58

Two mining fields, field A and field B of a coal mining company produce Lignite and Bituminous coal. The operating cost per day for field A and field B are $55,000 and $45,000, respectively. The recent records at the company indicate that the mining field A can produce 250 tons of Lignite along with 300 tons of Bituminous coal per day whereas the mining field B can produce 200 tons of Lignite along with 450 tons of Bituminous coal per day. The demand for Lignite is expected to be 120,000 tons and for Bituminous coal, it is expected to be 170,000 tons. The expected demand should be met. To minimize the operating costs of the mining fields, how many days does the company need to operate each of these fields?

Question 59

Michael has decided to invest $40,000 in three types of funds. Fund A has projected an annual return of 8 percent, Fund B has projected an annual return of 10 percent, and Fund C has projected an annual return of 9 percent. He has decided to invest no more than 30 percent of the total amount in Fund B and no more than 40 percent of the total amount in Fund C. a. Formulate a linear programming model that can be used to determine the amount of investments Michael should allocate to each type of fund to maximize the total annual return. b. How much should be allocated to each type of fund? What is the total annual return?

Question 60

Ethan Steel, Inc. has two factories that manufacture steel components for four different rail projects located at four different sites. The demand for the steel components for the four projects, Project A, Project B, Project C, and Project D, are 3220, 3675, 4125, and 2975, respectively. The shipping details are as below:
Production details:

Shipping Details (with per-unit shipping cost):

Find an alternative optimal solution for this transportation problem? Hint: Use the procedure described in section 8.7.