**General Overview of the food booth model: This is a sample answer to this case study:**

Julia seeks to establish a food booth and is looking for advice on the viability of her business venture. In response to this, there is a need to carry out a sensitivity analysis of the available resources, the projected business activity levels in the area, and then compare with the profits that she is expecting from the business. The provided analysis will be done using POM-QM qualitative software. This assignment will use linear programming approach to be able to model the food booth data, but it will be in two steps. The first step is the model for the current game.

The decision Variables include:

X_{1}= Number of pizza that she buys at the first time.

X_{2}= Number of hot dog pieces she intends to sell at the first time.

X_{3}= Number of sandwich she intends to sell at the first time.

Objective Function: Maximizing the amount of profits that Julia can earn in one game.

Contribution from selling of food item = Selling price – cost of food preparation/buying.

Costs: cost of booth charges per game + oven charges per game:

Maximize (1.50-0.75)X_{1}+(1.50-0.45)X_{2}+(2.25-0.90)X_{3} – 1000-100

Thus, the objective function becomes

Maximize 0.75X_{1}+1.05X_{2}+1.35X_{3} – 1100

However, as the POM-QM software does not allow for constants in the objective function line, we will therefore use the objective function as:

Maximize 0.75X1+1.05X2+1.35X3 and later subtract the fixed costs from the contribution margin to realize the profits.

**Constants:**

Oven Space: This has 16 shelves with each shelf area being 3 ft by 4 ft.

The total oven space= 16*3*12*4*12=27648 in^{2 }but since the oven is used twice, the cumulative space used is =27648*2 = 55296 in^{2}

Space occupied by a single Pizza: take P*(14/2)^2 in^{2}/ 8 = 19.24 Approximately 20 in^{2 }. Therefore, the constraint becomes: 20X_{1}+16X_{2}+25X_{3}<=55296 for the total space occupied by the pizzas.

Cash available constraint: this is the amount of money that she will need to purchase or prepare the food items. We will take the assumption that the lease fee and costs can be made later during the day from the realized profits.

0.75X1+0.45X2+0.90X3<=1500

Constraint involving the number of pizzas sold versus the number of hot dogs and sandwiches.

X_{1}>X_{2}+X_{3} == -X1+X2+X3<=0

Constraint involving of hot dogs sold versus the number of sandwiches.

X2>2X3 == -X2+2X3<=0

The L.P Model for the first game therefore is;

Maximize 0.75X1+1.05X2+1.35X3

Subject to the following constraints:

20X_{1}+16X_{2}+25X_{3}<=55296

0.75X_{1}+0.45X_{2}+0.90X_{3}<=1500

-X_{1}+X_{2}+X_{3}<=0

-X_{2}+2X_{3}<=0

Xi>=0

On maximizing, Julia if Julia happens to sell 1250 pieces of pizza and 1250 pieces of hot dogs in the first game, she will realize a profit of $2250.

This model can further be extended to cover all the 6 games.

Number of variables will be extended to 18 since we have 3 for each game.

X_{4} – number of pizza pieces procured for selling for second game

X_{5 }– number of hot dog pieces procured for selling for second game

X_{6}– number of sandwich pieces procured for selling for second game respectively

X_{7}

X_{8}

X_{9……}

All other constraints will remain the same except for cash constraint as it will be modified each time a game ends. In all instances, the available money will be $1500 + {profits realized in previous game(s)}. We will also assume that number of ovens is limited to one. This implies that in all other games, she will sell 1536 pizza pieces and 1536 hot dogs implying that total gross revenue will be $16,074 from all the 6 games. Deducting costs ($6000+ $600) the profits = $ 9,474.

Average profit per game= $1579, although the profit is the first game = $1150. She can improve on this by borrowing money before starting the business and plan for 1536 pizzas and 1536 hot dogs. For these items she would need $1843.20 therefore, she should borrow $343.20 and with this her profit margin would rise to $9,903.

In the case Julia feels that she can’t be able to handle all the work, she can hire a friend of hers to help her for $100 per game. From earlier analysis, the new partner will help to create efficiency but will not help in improving the number of sales made per game. If Julia will spend less time in preparing the food items the better. Julia should go ahead and hire her friend, though if she can be able to negotiate for a smaller pay, she would still make good profits.

Other Quantitative and Qualitative factors

There are many factors that have been overlooked/ assumed but they still have an effect in our recommendation. Such are:

- All throughout the analysis, we have assumed that all food items that Julia produces will be sold out without any wastage. This is a very optimistic assumption and in real life situation it doesn’t occur. But even if we factor the wastage fraction, still the projected profits will be more than 50% of what Julia expected.

- Also, in the oven spacing, we didn’t factor in the wastage of space due to uneven shapes of food items in the oven. This can’t have much effect on the sales and the associated costs.

- There is a need for Julia to observe the cost of food preparation and procurement closely, if the cost is around what we analyzed above, she should venture into her project.

**Conclusions Of Julia Food Booth Case Study:**

By critical and sensitivity analysis of Julia’s Business project, She stands a chance to make adequate sales that will be able to support her business and also help her realize good profits from the venture. The projected amount of money that she will need to borrow from her friends is about $343.20. In the beginning, she will only need the available oven space.