MAT 540 Week 8 Chapter 4 Homework

Chapter
4




  1. Betty Malloy, owner of the Eagle Tavern in Pittsburgh,
    is preparing for Super Bowl Sunday, and she must determine how much beer
    to stock. Betty stocks three brands of beer- Yodel, Shotz, and Rainwater.
    The cost per gallon (to the tavern owner) of each brand is as follows:



 




Brand


Cost/Gallon




 


 




Yodel


$1.50




Shotz


0.90




Rainwater


0.50




 

The tavern has a budget of $2,000
for beer for Super Bowl Sunday. Betty sells Yodel at a rate of $3.00 per
gallon, Shotz at $2.50 per gallon, and Rainwater at $1.75 per gallon. Based on
past football games, Betty has determined the maximum customer demand to be 400
gallons of Yodel, 500 gallons of shotz, and 300 gallons of Rainwater. The
tavern has the capacity to stock 1,000 gallons of beer; Betty wants to stock up
completely. Betty wants to determine the number of gallons of each brand of
beer to order so as to maximize profit.

 




  1. Formulate a linear
    programming model for this problem.





 




    1. Solve the model by using the
      computer.





 


  1. As result of a recently passed bill, a congressman’s
    district has been allocated $3 million for programs and projects. It is up
    to the congressman to decide how to distribute the money. The congressman
    has decide to allocate the money to four ongoing programs because of their
    importance to his district- a job training program, a parks project, a
    sanitation project, and a mobile library. However, the congressman wants
    to distribute the money in a manner that will please the most voters, or,
    in other words, gain him the most votes in the upcoming election. His
    staff’s estimates of the number of votes gained per dollar spent for the
    various programs are as follows.



 




Program


Votes/Dollar




Job training


0.03




Parks


0.08




Sanitation


0.05




Mobile library


0.03




 

In order also to satisfy several
local influential citizens who financed his election, he is obligated to
observe the following guidelines:




      • None of the programs can
        receive more than 30% of the total allocation





 




      • The amount allocated to parks
        cannot exceed the total allocated to both the sanitation project and the
        mobile library.





 




      • The amount allocated to job
        training must at least equal the amount spent on the sanitation project.





Any money
not spent in the district will be returned to the government; therefore, the
congressman wants to spend it all. Thee congressman wants to know the amount to
allocate to each program to maximize his votes.

 




  1. Formulate a linear
    programming model for this problem.





 




    1. Solve the model by using the
      computer.





 


  1. Anna Broderick is the dietician for the State
    University football team, and she is attempting to determine a nutritious
    lunch menu for the team. She has set the following nutritional guidelines
    for each lunch serving:






      • Between 1,300 and 2,100
        calories

      • At least 4 mg of iron

      • At least 15 but no more than
        55g of fat

      • At least 30g of protein

      • At least 60g of carbohydrates

      • No more than 35 mg of
        cholesterol





She
selects the menu from seven basic food items, as follows, with the nutritional
contributions per pound and the cost as given:

 





Calories
(per lb.)


Iron
(mg/lb.)


Protein
(g/lb.)


Carbohydrates
(g/lb.)


Fat
(g/lb.)


Cholesterol
(mg/lb)


Cost
($/lb.)




Chicken


500


4.2


17


0


30


180


0.85




Fish


480


3.1


85


0


5


90


3.35




Ground beef


840


0.25


82


0


75


350


2.45




Dried beans


590


3.2


10


30


3


0


0.85




Lettuce


40


0.4


6


0


0


0


0.70




Potatoes


450


2.25


10


70


0


0


0.45




Milk (2%)


220


0.2


16


22


10


20


0.82




 

The dietician wants to select a menu
to meet the nutritional guidelines while minimizing the total cost per serving.


 



  1. Formulate a linear programming model for this problem
    and solve.




 



    1. If a serving of each of the food items (other than
      milk) was limited to no more than a half pound, what effect would this
      have on the solution?




 


  1. Dr. Maureen Becker, the head administrator at Jefferson
    County Regional Hospital, must determine a schedule for nurses to make
    sure there are enough of them on duty throughout the day. During the day,
    the demand for nurses varies. Maureen has broken the day in to twelve
    2hour periods. The slowest time of the day encompasses the three periods
    from 12:00 A.M. to 6:00 A.M., which beginning at midnight; require a
    minimum of 30, 20, and 40 nurses, respectively. The demand for nurses
    steadily increases during the next four daytime periods. Beginning with
    the 6:00 A.M.- 8:00 A.M. period, a minimum of 50, 60, 80, and 80 nurses
    are required for these four periods, respectively. After 2:00 P.M. the
    demand for nurses decreases during the afternoon and evening hours. For
    the five 2-hour periods beginning at 2:00 P.M. and ending midnight, 70,
    70, 60, 50, and 50 nurses are required, respectively. A nurse reports for
    duty at the beginning of one of the 2-hour periods and works 8 consecutive
    hours (which is required in the nurses’ contract). Dr. Becker wants to
    determine a nursing schedule that will meet the hospital’s minimum
    requirement throughout the day while using the minimum number of nurses.



 




  1. Formulate a linear
    programming model for this problem.





 




    1. Solve the model by using the
      computer.





 


  1. The production manager of Videotechnics Company is attempting
    to determine the upcoming 5-month production schedule for video recorders.
    Past production records indicate that 2,000 recorders can be produced per
    month. An additional 600 recorders can be produced monthly on an overtime
    basis. Unit cost is $10 for recorders produced during regular working
    hours and $15 for those produced on an overtime basis. Contracted sales
    per month are as follows:






Month


Contracted Sales (units)




1


1200




2


2100




3
4
5


2400
3000
4000




 

Inventory carrying costs are $2 per recorder
per month. The manager does not want any inventory carried over past the fifth
month. The manager wants to know the monthly production that will minimize
total production and inventory costs.

 



  1. Formulate a linear programming model for this problem.




 



    1. Solve the model by using the computer.


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