Use the provided excel file for answering the questions.

Case $1$: Tahina company

Tahina is a company that specializes in producing Tahini sauce. Their products are based on high$-$quality ground sesame paste mixed with water. To make the final product, they add lemon juice, garlic and fresh parsley. Their goal is to set prices such that their profit is maximized, while making sure they meet their customers$$ demand.

Their two main products are White tahini and Green tahini. Preparing $1$ gallon of White tahini sauce requires $0\mathrm{.}6$ gallons of sesame paste, $32$ garlic cloves, $2$ cups of fresh lemon juice and $16$ table spoons of fresh parsley. Preparing $1$ gallon of Green tahini sauce requires $0\mathrm{.}5$ gallon of sesame paste, $24$ garlic cloves, $2\mathrm{.}5$ cups of fresh lemon juice and $28$ table spoons of fresh parsley.

To maintain high$-$quality production, the company relies on daily supply of fresh ingredients, which contains $15$ gallons of sesame paste, $700$ garlic cloves, $50$ cups of fresh lemon juice and $500$ table spoons of fresh parsley. Production cannot exceed $28$ gallons a day.

Tahina company has a strong position in the market, but demand is sensitive to price. Tahina$$s marketing consultant has come up with the following demand curves:

White tahini Demand $($in gallons$)=20010*($White tahini price per gallon$)$

Green tahini Demand $($in gallons$)=30020*($Green tahini price per gallon$)$

The company$$s unit costs are $$1\mathrm{.}5$ per one gallon of White tahini and $$2$ per one gallon of Green tahini.

$1.$ What are the decision variables?

$2.$ What is the objective?

$3.$ Maximize $/$ minimize the objective?

$4.$ What are the constraints?

$5.$ Set the problem in spreadsheet $$Problem $1$ and solve it$.$ What is your solution?

$6.$ What type of problem do we have in case $1?$ a$.$ A linear programming problem

b$.$ A non$-$linear problem

c$.$ An integer programming problem

d$.$ A problem with a soft constraint Explain why.

$7.$ Assume the demand for white tahini has the formula:

White tahini Demand $($in gallons$)=$ X $$ Y $*($White tahini price per gallon$),$

but the parameters X and Y may change daily. What we do know is that X is drawn from a normal distribution with mean $200$ and standard deviation $10,$ and Y is drawn from a uniform distribution between $5$ and $15.$ Estimate the average X and Y parameters over the next $7$ days. Use your estimated parameters to predict the demand for white tahini, and solve the problem again using excel. What would be your new solution?

$8.$ What method did you use to estimate the demand? a$.$ Decision tree

b$.$ Optimization

c$.$ Monte Carlo simulation d$.$ Regression