Julia Robertson, a senior at Tech, explores various options to support her final year at school and weighs the option of hiring a food booth outside the institute for football games as she knows from her personal experience that during the games, people eat lot of food. Facts/Market survey: Before arriving at the final decision, she does some market survey and gathers the following information. Hiring cost of booth will be $1000 per game. She would be able to sell only food items from the booth and not the drinks.
So depending on the popularity of the food items, she decides to sell only slices of cheese pizza, hot dogs and barbeque sandwiches. As per the eligibility conditions All food items would be sold during the hour before the game starts and during the half time. She anticipates that it will not be possible for her to prepare the food herself while she will be selling it as such she had to prepare the food ahead of time and store it.
She considers the idea of leasing an oven for storing and warming the food and she considers filling the oven before the beginning of the game and again just before the half time. During her survey she finds out that oven would cost her $ 600 for the 6 game seasons and oven contains 16 shelves, each shelf of size 3 feet by 4 feet. Julia negotiates with a local pizza company to supply 14 inch cheese pizza twice each game meeting her sale requirements. She estimates that each pizza would cost her $6 containing 8 slices, and each hot dog would cost her $0.45 and each barbeque sandwich would cost her $0.90 if she makes the barbeque herself the night before.
On taking the measurement of food items, she finds that hot dog would require 16 square inch of space and barbeque sandwich would require 25 square inch of space. She decides the selling price of items as: Each slice of Pizza @ $1.50, Hot dog @ $1.5/piece & Barbeque sandwich @ $2.25/ piece. Total cash available with her for purchase of food items before the first game is $1500. To finance further games she will utilize the sale proceeds of first game. Also, from her survey, she anticipates that she would be able to sell at least as many slices of pizza as hot dogs and barbeque sandwiches combined and she would sell at least twice as many hot dogs as barbeque sandwiches.
She believes that she will sell everything she could stock. Submission/proposition: She thinks that if she could make a profit of $1000 in each game after paying all her expenses, then it is worth choosing the option. So final question in her mind is “Can this proposition result in +$1000 profit in each game.” As the moot question is the quantum of profit which can be earned through this food kiosk/booth, the aim is to maximize the profit within the given parameters. Therefore the objective is one of Maximization of the profit. But it will be constrained with the given limitations as found in the market survev. To evaluate these constraints and to put them in mathematical form, i reviewed the factual position and write them as under.
Julie considers that she will sell everything she can stock. Her capability to stock depends on two factors: Capacity of the warming oven which she intends to hire, as she has to stock the material before the game and before the interval break. Her financial capacity to purchase or self prepare the food items as she has limited resource in the beginning which is $1500. To put the objective and constrains in mathematical perspective or equations, let us use certain notations as follows: = X1 Let the No of Pizza sold Let No of Hot dogs sold = X2 Let No of Barbeque Sandwich sold = X3. To evaluate the storing capacity of Oven as mentioned in 1(1) Size of shelf = 3feet *4 feet Area of one shelf = 36 inch* 48 inch = 1728 sq inch Area of 16 shelves = 16*1728 27648 sq inch As oven is filled twice, one before start of game and other before half time, therefore total area for storage of food items available is double of 27648 sq inch. Total storing capacity 2* 27648 = 55296 sq. inch.
Julie considers buying 14 inch cheese Pizza from a local vendor. The area of cheese pizza would be equal to n r2 (where n= 22/7 and r=7 inch as diameter of pizza is 14 inch) Area of Pizza= 22/7 * (7)*(7) = 154 sq. inch But as in given problem, no mention has been made about the space requirement for storing a pizza in oven, it is assumed that one pizza will require a space of 14* 14 inch of area in the oven as the space available at each corner of the circular pizza will not be utilized for storing any item. So area requirement for storing one pizza = 14* 14 sq. inch = 196 sq. inch. Facts gathered by Julie are that space requirement. For each hot dog = 16 sq. inch For each barbeque sandwich. = 25 sq. in So total space requirement for storing the entire food material will be 196 X1 + 16 X2 + 25 X3. The total space available in oven is = 55296 sq. inch. Therefore the constraint for maximization of profit is as follows: Space constraint: 196 X1 + 16 X2 + 25 X3 < 55296 ——– (1st Constraint) 4).
Formulate the constraint for financial capacity as mentioned at 1(ii) above: The cost of 1 pizza for Julie $6 Total cost for X1 pizza would be = $6* X1 Cost of 1 Hot dog to Julie = $0.45 Cost of X2 Hot dogs $ 0.45 * X2 Similarly cost of X3 Barbeque sandwiches to Julie = $0.90 * X3 As she has only $1500 for purchasing food articles, and the cost of all the articles cannot exceed this amount of $1500, the constraint can be expressed as 6*X1 + 0.45 * X2 +0.9 * x3 = 1500 ——————(2nd Constraint) The other conditions as anticipated by Julie and mentioned in problem statement are: Julie expects to sell at least as many slices of pizza as hot dog and barbeque sandwiches combined. This is expressed mathematically as 8 *X1 2 X2+ X3 ————————— (3rd constraint) (This is because one pizza contains 8 slices) Julie anticipates that she would probably sell at least twice as many hot dogs as barbeque sandwiches. This is stated as X2 > 2×3 —( 4th Constraint)
