Mat 540 Assign 3

Topics: Inch, Imperial units, English-language films Pages: 5 (1377 words) Published: September 7, 2013
A: Formulation of the LP Model

X1(Pizza), X2(hotdogs), X3(barbecue sandwiches)

Constraints:

Cost:

Maximum fund available for the purchase = $1500

Cost per pizza slice = $6 (get 8 slices) =6/8 = $0.75

Cost for a hotdog = $.45

Cost for a barbecue sandwich = $.90

Constraint: 0.75X1 + 0.45X2+ 0.90(X3) ≤ 1500

Oven space:

Space available = 3 x 4 x 16 = 192 sq. feet = 192 x 12 x 12 =27648 sq. inches

The oven will be refilled before half time- 27648 x 2 = 55296

Space required for pizza = 14 x 14 = 196 sq. inches

Space required for pizza slice = 196/ 8 = 24.50 sq. inches

Space required for a hotdog=16

Space required for a barbecue sandwich = 25

Constraint: 24.50 (X1) + 16 (X2) + 25 (X3) ≤ 55296

Constraint:

Julia can sell at least as many slices of pizza(X1) as hot dogs(x2) and barbecue sandwiches (X3) combined

Constraint: X1 ≥ X2 + X3 = X1 - X2 - X3 ≥ 0

Julia can sell at least twice as many hot dogs as barbecue sandwiches

X2/X3 ≥ 2 = X2 ≥2 X3 =X2 - 2 X3 ≥ 0

X1, X2, X3 >= 0 (Non negativity constraint)

Objective Function (Maximize Profit):

Profit =Sell- Cost

Profit function: Z = 0.75 X1 + 1.05 X2 + 1.35 X3

LPP Model:

Maximize Z = 0.75 X1 + 1.05 X2 + 1.35 X3

Subject to 24.5 X1 + 16 X2 + 25 X3 ≤ 55296

0.75 X1 + 0.45 X2 + 0.90 X3 ≤ 1500

X1 - X2 - X3 ≥ 0

X2 - 2 X3 ≥ 0

X1≥ 0, X2≥ 0 and X3 ≥0

Solve the LPM

Based on the excel solution the optimum solution:

Pizza (X1) = 1250; Hotdogs(X2) = 1250 and Barbecue sandwiches (X3) = 0

Maximum value of Z = $2250

Julia should stock 1250 slices of pizza, 1250 hot dogs and no barbecue sandwiches.

Maximum Profit = $2250.

|Maximum Profit |$ 2,250.00 | |Booth Rent per game |$ (1,000.00) | |Warming Oven 600 for total of 6 home |$ (100.00) | |games 600/6 =100

(A) Formulate and solve an L.P. model for this case.
There are three products or variables in this problem that we must consider for purchase. X1 = number of pizza slices Julia should purchase
X2 = number of hotdogs Julia should purchase
X3 = number of barbecue sandwiches Julia should purchase.
The reason why Julia is having a booth is to make some money. She wants to maximize her profit that she can get from selling the hotdogs, pizza, and barbeque sandwiches. The first thing to do is find the profit that Julia will make per Item. To find that per Item price, the cost of the Item will be subtracted from the selling price. Pizza: Julia can buy a pizza that contains 8 slices for 6$. That means each slice of pizza will cost her $0.75. She plans to sell each piece for 1.50. $1.50-$0.75= $0.75 profit

Hot dog:
$1.50 - $0.45 = $1.05 profit
Barbecue Sandwiches:
$2.25 - $0.90 = $1.35 profit
The objective function can now be written since we have found the potential profit of each food item. The objective of this function is to maximize Z(profit). Z= $0.75x1 + $1.05x2 + $1.35X3
The Budget is one thing that has to be taken into consideration. Julia has $1,500 on hand to purchase and prepare food for the first home game. A constraint must be formed for the budget. Since the cost of each item and money available is know. It is easy to form the budget constraint. This shows the total cost to purchase each item $0.75X1 + $0.45X2 + $0.90X3. Since the idea is to remain within the $1,500 budget this is the Budget Constraint: Budget Constraint $0.75X1 + $0.45X2 + $0.90X3 ≤ 1500

Space is the next constraint that must be formed. Julia is going to rent an oven that has 16 shelves, each shelve is worth 3feet by 4 feet. To calculate the space of the shelves: 3x4x16=192 square feet. This must be broken down in inches since the numbers of inches were given per item. There are 12 inches in each foot, it can be determined that 192x12x12=27,648 square inches. The oven will be filled and...
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