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Answered: - The following problem was solved in Excel and the following


The following problem was solved in Excel and the following spreadsheet was generated. Answer the following questions based on the excel spreadsheet. (16 pts.)

Note: the following model refers to a problem in which a Christmas Tree company is trying to decide how many Douglas Firs and how many Fraser Firs to plant in order to maximize their revenue. Each Douglas Fir sells for $40 and each Fraser Fir sells for $50. The problem also has restrictions on the minimum number of both Douglas and Fraser Firs to plant (minimum 200 each), limited planting ground space (each Douglas Fir requires 10 sq. ft and each Fraser Fir requires 12 sq. ft; only 10,000 sq. ft. available), and also limitations on the amount they can spend on shipping (they do not want to spend more than $4,000 on shipping; each Douglas Fir costs $6 to ship and each Fraser Fir costs $8 to ship).

Here is the model. Do not change it or try to solve. It has all been done for you. Simply answer the questions that follow.

Let D = number of Douglas Firs planted

?????? F = number of Fraser Firs planted

Max Revenue ($) 40D + 50F

(1)???? Minimum Douglas Firs Required

(2)???? Minimum Fraser Firs Required

(3)????? Planting Ground Space (Sq. Ft.)

(4)????? Shipping Cost Maximum ($)


s.t.s.t.??

State the complete solution to the problem in sentence form (optimal solution point and OFV).

Write the complete formula exactly as it would appear in each of the following cells (use cell references, symbols, etc. as you would in excel):

i.)B11

???????????????????????????????????????????????????????????????????????????????????????????????

ii.)B14

???????

???????????????????????????????????????????????????????????????

iii.)D17

c.)Which constraints are the binding constraints? How do you know this?

d.) i.) Which constraints are non-binding constraints?

ii.)Decide if there is Slack or Surplus for each non-binding constraint (this question refers to the definition of Slack and/or Surplus).

iii.)Calculate the amount of Slack or Surplus for each non-binding constraint (this question refers to the mathematical number/amount).


Name:

 

1.) A local restaurant owner is considering designing and selling logo t-shirts to his customers. The restaurant owner

 

believes he can sell the t-shirts for $20 each. He has done some research into local printing and design companies

 

and decides on a company that has the following costs: fixed cost of $3,000 to produce/design the shirts, plus $6 per

 

shirt. (16 pts.)

 

Let x = number of t-shirts sold.

 

a.) Develop a model to represent the total profit for selling x t-shirts.

 


 

b.)Calculate the profit if 100 t-shirts are sold. Is this a gain or loss?

 


 

c.)How many t-shirts will the restaurant have to sell to break-even? (Round to the nearest whole number.)

 


 

d.)What is your recommendation to the restaurant if the owner believes he can sell approximately ten t-shirts a

 

week? Should he go through with the project? Why or why not?

 


 

2.)A linear programming problem has the following constraint set:

 


 

x 2y

 

x

 

x y

 

x , y

 


 


 


 


 


 


 

10

 

5

 

6

 

0

 


 

(1)

 

(2)

 

(3)

 


 

Sketch the feasible region on the grid below:

 

Label each inequality, axis, and intercept point. Label the test point.

 


 


 

Show with arrows or shading the direction of each inequality.

 


 


 


 

Clearly outline the feasible region and shade it in.

 


 

Do not attempt to solve the problem. Do not find the intersection points of the lines. Show all work for

 

finding intercept points and determining shading direction using test points. (16 pts.)

 


 

x

 


 

3.)Solve the following linear programming problem using either the sliding lines method (i.e. graphical procedure

 

method) or the corner point method. Show all work- including work for finding intersection points. If you use

 

sliding lines, also graph your sample objective function line. The constraints have already been graphed for you and

 

the feasible region shaded and outlined. State the complete solution (optimal solution point and OFV) in sentence

 

form. (16 pts.)

 

Maximize:

 


 

4x + 3y

 


 

Subject to:

 

2x + y ? 12

 

x-y?6

 

2x + 3y ? 24

 

x, y ? 0

 


 

(1)

 


 

(1)

 

(2)

 

(3)

 


 

y

 


 


 


 


 


 


 


 

(3)

 


 

(2)

 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 

x

 


 

4.)Formulate the mathematical model (Define Variables, Objective Function, Constraints) for the following linear

 

programming problem. (Don?t forget units and labels!) DO NOT SOLVE THE PROBLEM. (16 pts.)

 

Holiday Tricks Costume Company is getting ready for Halloween by manufacturing two types of costume

 

accessories. The two types of accessories are a Dracula?s cape and a Witch?s broom. To produce one of

 

Dracula?s capes, it requires ? hour of production time, ? hour of packaging and inspection time, and 2 yards of

 

polyester. To produce one Witch?s broom it requires ? hour of production time, ? hour of packaging and

 

inspection time, and 3.5 yards of polyester. Each week, Holiday Tricks Costume Company has at most 61 hours

 

of production time, at most 9 hours of packaging and inspection time, and only 1500 yards of polyester

 

available to use producing these two types of costume accessories. Also, Holidays Tricks Costume Company is

 

expecting Dracula?s cape to be their biggest seller and thus want to produce at least 100 capes. One Dracula?s

 

cape produces a profit of $7 and one Witch?s broom produces a profit of $5. How many of each type of

 

accessory should Holiday Tricks Costume Company produce in order to maximize their profits each week?

 


 

5.) The following problem was solved in Excel and the following spreadsheet was generated. Answer the following

 

questions based on the excel spreadsheet. (16 pts.)

 

Note: the following model refers to a problem in which a Christmas Tree company is trying to decide how many

 

Douglas Firs and how many Fraser Firs to plant in order to maximize their revenue. Each Douglas Fir sells for

 

$40 and each Fraser Fir sells for $50. The problem also has restrictions on the minimum number of both

 

Douglas and Fraser Firs to plant (minimum 200 each), limited planting ground space (each Douglas Fir requires

 

10 sq. ft and each Fraser Fir requires 12 sq. ft; only 10,000 sq. ft. available), and also limitations on the amount

 

they can spend on shipping (they do not want to spend more than $4,000 on shipping; each Douglas Fir costs $6

 

to ship and each Fraser Fir costs $8 to ship).

 

Here is the model. Do not change it or try to solve. It has all been done for you. Simply answer the questions

 

that follow.

 

Let D = number of Douglas Firs planted

 

F = number of Fraser Firs planted

 


 

Max Revenue ($) 40D + 50F

 

s.t.

 

s.t.

 


 

D

 

10 D

 

6D

 

D

 


 

F

 

12 F

 

8F

 

,

 

F

 


 

200

 

200

 

10000

 

4000

 


 

0

 


 

(1)

 


 

Minimum Douglas Firs Required

 


 

(2)

 


 

Minimum Fraser Firs Required

 


 

(3)

 


 

Planting Ground Space (Sq. Ft.)

 


 

(4)

 


 

Shipping Cost Maximum ($)

 


 

a.) State the complete solution to the problem in sentence form (optimal solution point and OFV).

 


 

b.) Write the complete formula exactly as it would appear in each of the following cells (use cell references,

 

symbols, etc. as you would in excel):

 

i.)B11

 


 

ii.)B14

 


 

iii.)D17

 


 

c.)Which constraints are the binding constraints? How do you know this?

 


 

d.) i.) Which constraints are non-binding constraints?

 


 

ii.)Decide if there is Slack or Surplus for each non-binding constraint (this question refers to the definition

 

of Slack and/or Surplus).

 


 

iii.)Calculate the amount of Slack or Surplus for each non-binding constraint (this question refers to the

 

mathematical number/amount).

 


 

6.) Background: Cinnamon Rainbow Surf Shop in New Hampshire is already preparing for the upcoming Spring

 

Break and next summer season by developing a new production schedule for their three most popular surf boards:

 

the Lightning, the Wave Rider, and the Glory. Each of the three types of surf boards require time in the sanding

 

department, the finishing department, and the design (i.e. emblem, picture, colors, etc. of the board) department.

 

Cinnamon Rainbow expects to have 150 hours of sanding time, 100 hours of finishing time, and 200 hours of design

 

time available next production period. Specifically, the Lightning requires 3 hours in the sanding department, 3

 

hours in the finishing department, and 3 hours in the design department, the Wave Rider requires 2 hours in the

 

sanding department, 1 hour in the finishing department, and 3 hours in the design department, and the Glory requires

 

90 minutes in the sanding department, 75 minutes in the finishing department, and 2 hours in the design department.

 

Additionally, information from previous years? sales indicates that Cinnamon Rainbow typically sells 40 Glory surf

 

boards during one production cycle. Thus, Cinnamon Rainbow wants to guarantee that they produce at least 40

 

Glory surf boards. If the profit on each Lightning board is $150, on each Wave Rider is $125, and on each Glory

 

board is $75, develop a model that will determine Cinnamon Rainbow?s optimal production schedule while

 

maximizing profit.

 

The model follows:

 

Let L = the number of Lightening Boards Produced

 

W = the number of Wave Rider Boards Produced

 

G = the number of Glory Boards Produced

 

Maximize Profit ($) 150L + 125W + 75G

 

s.t

 


 

3L 2W

 

3L 1W

 

3L 3W

 

L

 


 

,

 


 

W

 


 

1.5G

 

1.25G

 

2G

 

G

 

,

 

G

 


 


 


 


 


 


 


 

150

 

100

 

200

 

40

 

0

 


 

Sanding Department (Hrs)

 

Finishing Department (Hrs)

 

Design Department (Hrs)

 

Glory Board Minimum

 


 

The Cinnamon Rainbow Surf Shop problem has been solved for you. Below is the sensitivity report that Solver

 

generated. Use the Sensitivity Report to answer the questions or state that the answer needs to be resolved and why

 

(in range, out of range, binding, etc.). Assume each part is the only change. (20 pts)

 

Sensitivity Report:

 


 

a.) State the complete solution to the problem (optimal solution point and OFV) in the CONTEXT of the problem.

 


 

b.) What is the range of optimality of CL, the coefficient associated with the Lightning variable?

 


 

c.) What will be the new optimal solution point and OFV if the profit on Lightning surf boards increases to $200 per

 

surf board?

 


 

d.) What is the range of feasibility for the Glory Surfboards Constraint?

 


 

e.) Suppose that Cinnamon Rainbow Surf Shop is required to produce 50 Glory surf boards. What will be the

 

optimal solution point and OFV?

 


 

f.) What is the range of feasibility for the Sanding Department Constraint?

 


 

g.) What will be the optimal solution point and OFV if the Sanding Department has an additional 2 hours of time

 

available?

 


 

 


Solution details:

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