# Linear Programming Break-Even Analysis, Sensitivity Analysis

Problem 1: Break-Even Analysis

A small organization builds TV satellite dishes. The investment in plant and equipment is $200,000. The variable cost per dish is $500. The price of the TV dish is $1000.

(Please show your work used to come up with this solution)

1. How many TV satellite dishes would be needed to be sold for the firm to break even?

2. What profit or loss can be anticipated with a demand of 500 dishes?

3. If the demand is 300 dishes, what is the minimum price that the firm must charge to breakeven?

Problem 2: Linear Programming Model Development

A manufacturer makes two products, doors and windows. Each must be processed through two work areas. Work area #1 has 60 hours of available production time. Work area #2 has 48 hours of available production time. Manufacturing of a door requires 4 hours in work area #1 and 2 hours in work area #2. Manufacturing of a window requires 2 hours in work area #1 and 4 hours in work area #2. Profit is $8 per door and $6 per window. As you respond to the following questions, please note that you are only setting the problem up. A solution to determine the number of doors and windows to be manufactured and the resulting profit is NOT necessary.

1. Define the decision variables that will tell how many units to build (doors and windows)

2. Develop an objective function that will maximize profits.

3. Develop production constraints for work areas #1 and #2.

Problem 3: Model Development, Sensitivity Analysis, Interpretation of Output

A large sporting goods store is placing an order for bicycles with its supplier. Four models can be ordered: the adult Open Trail, the adult Cityscape, the girl's Sea Sprite, and the boy's Trail Blazer. It is assumed that every bike ordered will be sold, and the profits for each, respectively, are $30, $25, $22, and $20. The linear programming model should maximize profit. There are several conditions that the store needs to be concerned about. One of these is the space to hold the inventory. The adult bikes need two feet each, but each children's bike needs only one foot. The store has 500 feet of space available for the bicycles. There are 1200 hours of assembly time available. The children's bikes need 4 hours each to assemble; the Open Trail needs 5 hours each; and the Cityscape needs 6 hours each. The store would like to place an order for 275 bicycles.

Requirements:

1. Formulate a model for the problem, state the objective function and the subject to constraints.

2. Refer to Appendix 1 to answer the following:

A. Based on the optimal solution, how many of each kind of bicycle should be ordered and what will the total profit be?

B. What would the profit be if the store had 100 more feet of storage space?

C. Over what range of assembly hours is the dual price available?

D. Which resource should the company work to increase? - inventory space or assembly time?

Appendix 1

Objective Function Value = 6850.000

Variable Value Reduced Costs

Open Trail 100.000 0.000

City Scape 0.000 13.000

Sea Sprite 175.000 0.000

Trail Blazer 0.000 2.000

Constraint Slack/Surplus Dual Price

1 (Space Available) 125.000 0.000

2 (Assembly Hours Available) 0.000 8.000

3 (Order Desired) 0.000 - 10.000

Right Hand Side Ranges:

Constraint Lower Limit Current Value Upper Limit

1 (Space Available) 375.000 500.000 No Upper Limit

2 (Assembly Hours Available) 1100.000 1200.000 1325.000

3 (Order Desired) 240.000 275.000 300.000

Problem 4: Transportation Problem

Chicago Transport is to move goods from three factories to three distribution centers. Information about the move is as follows:

Factory Units Supplied Distribution Center Units Demanded

A 200 X 100

B 100 Y 150

C 150 Z 200

Costs per unit to ship from each factory to each distribution center are:

Distribution Center

Factory X Y Z

A $3 $2 $5

B $9 $10 $7

C $5 $6 $4

Requirements:

1. Develop a Network representative of this problem.

2. Formulate the problem as a linear program showing the objective function and the constraints

3. Using Appendix 2, what is the total shipping cost of the optimal solution?

4. Based on the optimal solution of Appendix 2, how many units will be shipped from each factory to each distribution center? Note Variable X1 represents units shipped from Factory A to Distribution Center X; X2 - A to Y; X3 - A to Z; X4 - B to X; X5 - B to Y; X6 - B to Z; X7 - C to X; X8 - C to Y; and X9 - C to Z.

Appendix 2

Objective Function Value = 1800.000

Variable Value Reduced Costs

X1 50.000 0.000

X2 150.000 0.000

X3 0.000 3.000

X4 0.000 1.000

X5 0.000 3.000

X6 100.000 0.000

X7 50.000 0.000

X8 0.000 2.000

X9 100.000 0.000

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#### Solution Summary

This posting contains solution to following problems on Linear Programming Break-Even Analysis, Sensitivity Analysis and Transportation.

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