# Linear programming model for revenue management application

Problem 30

Hanson Inn is a 96-room hotel located near the airport and convention center in Louisville, Kentucky. When a convention or a special event is in town, Hanson increases its normal room rates and takes reservations based on a revenue management system. The Classic Corvette Owners Association scheduled its annual convention in Louisville for the first weekend in June. Hanson Inn agreed to make at least 50% of its rooms available for convention attendees at a special convention rate in order to be listed as a recommended hotel for the convention. Although the majority of attendees at the annual meeting typically request a Friday and Saturday two-night package, some attendees may select a Friday night only or a Saturday night only reservation. Customers not attending the convention may also request a Friday and Saturday two-night package, or make a Friday night only or Saturday night only reservation. Thus, six types of reservations are possible: convention customers/two-night package; convention customers/Friday night only; convention customers/Saturday night only; regular customers/two-night package; regular customers/Friday night only; and regular customers/Saturday night only.

The cost for each type of reservation is shown here.

The anticipated demand for each type of reservation is as follows:

Hanson Inn would like to determine how many rooms to make available for each type of reservation in order to maximize total revenue.

A. Define the decision variables and state the objective function

CT = number of convention two-night rooms

CF = number of convention Friday only rooms

CS = number of convention Saturday only rooms

RT = number of regular two-night rooms

RF = number of regular Friday only rooms

RS = number of regular Saturday only room

___CT+___CF+___CS+____RT+____RF+____RS

B. Formulate a linear programming model for this revenue management application.

___CT+___CF+___CS+____RT+____RF+____RS

s.t.

1) ___CT___

2) ___CF___

3)___CS___

4) ___RT___

5)___RF___

6)___RS___

7)___CT+___CF___

8) ___CT+___CS___

9) ___CT+___CF+___RT+___RF

10) ___CT+___CS+___RT+___RS

C) What is the optimal allocation and the anticipated total revenue?

Variable Value

CT

CF

CS

RT

RF

RT

Total Revenue

d) Suppose that one week before the convention, the number of regular customers/Saturday night only rooms that were made available sell out. If another nonconvention customer calls and requests a Saturday only room, what is the value of accepting this additional reservation?

The shadow price for constraint 10 is ? and shows an added profit of ? if this additional reservation is accepted

https://brainmass.com/math/linear-programming/linear-programming-model-revenue-management-application-517125

#### Solution Preview

Problem 30

Hanson Inn is a 96-room hotel located near the airport and convention center in Louisville, Kentucky. When a convention or a special event is in town, Hanson increases its normal room rates and takes reservations based on a revenue management system. The Classic Corvette Owners Association scheduled its annual convention in Louisville for the first weekend in June. Hanson Inn agreed to make at least 50% of its rooms available for convention attendees at a special convention rate in order to be listed as a recommended hotel for the convention. Although the majority of attendees at the annual meeting typically request a Friday and Saturday two-night package, some ...

#### Solution Summary

The expert examines linear programming model for revenue management applications.

Introduction to Management Science: Quantitative Methods: Multiple choice questions

Question 1

In the scientific method, model parameters are:

a. always changing

b. constant during the process of solving a specific problem

c. defined as decision variables

d. found in the model solution process

Question 2

The components of break-even analysis are:

a. volume, cost and profit

b. fixed cost, variable cost and equipment structure

c. percentage of total revenue, percentage of total cost

d. total sales, total variable revenue

Question 3

A management science model:

a. is an abstract representation of an existing problem

b. appears most frequently as a mathematical relationship

c. can appear in the form of a graph or a chart

d. is accurately described by all of the above

Question 4

Management science is also referred to as:

a. management

b. quantitative analysis

c. qualitative analysis

d. computer science

Question 5

Management science can be described as:

a. strictly a science

b. both an art and a science

c. strictly an art

d. a deterministic technique

Question 6

Linear programming models exhibit certain common characteristics except:

a. decision variables for measuring the level of activity

b. linearity among some constraint relationships

c. an objective function to be maximized or minimized

d. a set of constraints

Question 7

For most graphs, the constraint equations which intersect to form a solution point must be solved simultaneously:

a. because the solution coordinates from the graph cannot be visually read with high precision

b. in order to confirm the mathematically determined coordinates

c. in order to determine all of the optimal point solution

d. because the slope b and the y-intercept a are not always integers

Question 8

The maximum number of constraints that could define the feasible solution space is:

a. 2

b. 3

c. 4

d. unlimited

Question 9

Which of the list of items below is not a component of a linear programming problem?

a. constraints

b. objective function

c. decision variables

d. a nonlinear residual

Question 10

The change in the value of the objective function per unit increase in the value of the right hand side is referred to as:

a. shadow price

b. quantity values

c. feasible range

d. optimal range

Question 11

In order to transform a ">=" constraint into an equality ("=") in a linear programming model,

a. add a slack variable

b. add a surplus variable

c. subtract a surplus variable

d. subtract a surplus variable and add a slack variable

Question 12

A decrease in fixed costs with everything else remaining constant

a. decreases the break-even point

b. increases the break-even point

c. keeps the break-even point same

d. increases the variable costs

Question 13

The term ____________ refers to testing how a problem solution reacts to changes in one or more of the model parameters.

a. priority recognition

b. decision analysis

c. analysis of variance

d. sensitivity analysis

Question 14

Which of the following could not be a linear programming problem constraint?

a. 1A + 2B

b. 1A + 2B = 3

c. 1A + 2B > 3

d. 1A + 2B < 3

Question 15

Non-negativity constraints restrict the decision variable to

a. 0

b. positive values

c. negative values

d. both a and b

Question 16

A graphical solution is generally limited to linear programming problems with

a. 1 decision variable

b. 2 decision variables

c. 3 decision variables

d. 4 decision variables

Question 17

The region which satisfies all of the constraints in a graphical linear programming problem is called the

a. region of optimality

b. feasible solution space

c. region of non-negativity

d. optimal solution space

Question 18

The optimal solution is the ___________ feasible solution.

a. best

b. only

c. worst

d. none of the above

Question 19

Multiple optimum solutions can occur when the objective function is _______ a constraint line.

a. unequal to

b. equal to

c. linear to

d. parallel to

Question 20

The optimal solution of a minimization problem is at the extreme point _________ the origin.

a. farthest from

b. closest to

c. exactly at

d. none of the above

Question 21

If the original amount of a resource is 25, and the range of feasibility for it can increase by 5, then the amount of the resource can increase to

a. 25

b. 30

c. 20

d. 125

Question 22

A shadow price reflects which of the following in a maximization problem?

a. the marginal gain in the objective that would be realized by adding 1 unit of a resource

b. the marginal gain in the objective that would be realized by subtracting 1 unit of a resource

c. the marginal cost of adding additional resources

d. none of the above

Question 23

The standard constraint form in a linear programming model requires that all decision variables be on the left side of the inequality (or equality) and numerical values on the right side.

a. True

b. False

Question 24

In financial management applications of linear programming in which funds are to be invested, the objective is to:

a. maximize risk

b. minimize return

c. maximize return

d. maximize cost

Question 25

A popular example of a linear programming model is the

a. product mix problem

b. diet problem

c. transportation problem

d. all of the above

Question 26

In an investment example of a linear programming problem, the decision variables are

a. the monetary amount invested in each investment alternative

b. the returns of the investment alternatives

c. the requirements for investing

d. none of the above

Question 27

A < constraint represents a

a. minimum requirement

b. maximum limit

c. either of the above

d. minimization constraint

Question 28

An optimal solution will always occur at

a. the intersection of two or more constraint lines

b. an extreme point

c. a corner point

d. any of the above

Question 29

The solution of a management science model provides a manager with useful information that can aid in the decision making process.

a. True

b. False

Question 30

Standard form requires that fractional relationships between variables in constraints be eliminated

a. True

b. False

Question 31

Which of the following is not a type of integer linear programming problem?

a. continuous

b. zero-one integer

c. mixed integer

d. pure integer

Question 32

A zero-one integer problem always finds an optimal integer solution.

a. True

b. False

Question 33

In a _____ integer model, some solution values for decision variables are integer and others can be non-integer.

a. total

b. 0 - 1

c. mixed

d. all of the above

Question 34

In a _____ integer model, the solution values of the decision variables are 0 or 1.

a. total

b. 0 - 1

c. mixed

d. all of the above

Question 35

A feasible solution to a maximization problem is ensured by rounding ________ non-integer solution values.

a. up and down

b. up

c. down

d. up or down