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MAT540 Problems hw4

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					Problem
4-12

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.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.

Problem
4-20

Anna Broderick is the dietitian 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 1,500 and 2,000 calories
*At least 5 mg of iron
*At least 20 but no more than 60 g of fat
*At least 30 g of protein
*At least 40 g of carbohydrates
*No more than 30 mg of cholesterol

She selects the menu from the seven basic food items, as follows, with the nutritional
contribution per pound and the cost as given:
_______________________________________________________________________
                  Calories Iron       Protein Carbohydrates Fat         Cholesterol
                  (per lb) (mg/lb) (g/lb)          (g/lb)      (g/lb)      (mg/lb) $/lb
Chicken           520      4.4         17           0            30         180       0.80
Fish              500      3.3         85           0            5          90         3.70
Ground beef        860      0.3        82             0            75         350     2.30
Dried beans        600      3.4        10             30           3          0       0.90
Lettuce            50       0.5        6              0            0          0       0.75
Potatoes           460      2.2        10              70           0         0       0.40
Milk (2%)          240       0.2       16              22          10         20      0.83

The dietitian wants to select a menu to meet the nutritional guidelines while minimizing
the total cost per serving.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.
c. If a serving of each of the food items (other than milk) were limited to no more than a
    half pound, what effect would this have on the solution?


Problem
4-24

Brooks City has three consolidated high schools, each with a capacity of 1,200 students.
The school board has partitioned the city into five busing districts – north, south, east,
and central – each with different high school student populations. The three schools are
located in the central, west, and south districts. Some students must be bused outside their
districts, and the school board wants to minimize the total bus distance traveled by these
students. The average distances from each district to the three schools and the total
student population in each district are as follows:
____________________________________________________
                             Distance (miles)
                     --------------------------------------
                Central      West          South            Student
District       School        School       School            Population
____________________________________________________
North             8            11            14               700
South            12            9             ---              300
East              9            16             10              900
West             8             ---             9              600
Central          ---           8              12              500
____________________________________________________

The school board wants to determine the number of students to bus from each district to
each school to minimize the total busing miles traveled.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.
Problem
4-42

A finery blends four petroleum components into three grades of gasoline – regular,
premium, and diesel. The maximum quantities available of each component and the cost
per barrel are as follows:
_________________________________________________
                    Maximum Barrels
Component              Available/Day         Cost/Barrel
_________________________________________________
    1                   5,000                  $9
    2                   2,400                   7
    3                   4,000                   12
    4                   1,500                    6
_________________________________________________

To ensure that each gasoline grade retains certain essential characteristics, the refinery
has put limits on the percentages of the components in each blend. The limits as well as
the selling prices for the various grades are as follows:

_____________________________________________________________
Grade           Component Specification       Selling price/Barrel
_____________________________________________________________
Regular         Not less than 40% of 1            $12
                Not more than 20% of 2
                Not less than 30% of 3
Premium         Not less than 40% of 3              18
Diesel          Not more than 50% of 2              10
                Not less than 10% of 1
______________________________________________________________

The refinery wants to produce at least 3,000 barrels of each grade of gasoline.
Management wishes to determine the optimal mix of the four components that will
maximize profit.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.

				
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