Problem $3(35$ pts$)$

You're an intern at a cereal bar manufacturing company that produces three types of bars: bar

$1,$ bar $2,$ and bar $3.$ Each cereal bar is composed of varying proportions of four ingredients:

honey, peanuts, chocolate, and pistachios. Bar $1$ is a mix in weight of $88\%$ honey, $9\%$ peanuts,

and $3\%$ pistachios. Bar $2$ is a mix in weight of $92\%$ honey, $3\%$ chocolate, and $5\%$ pistachios.

Bar $3$ is a mix in weight of $87\%$ honey, $4\%$ peanuts, $6\%$ chocolate, and $3\%$ pistachios. The

manufacturing company sells bar $1$ for $$2$ apiece, bar $2$ for $$2\mathrm{.}50$ apiece, and bar $3$ for $$3\mathrm{.}00$

apiece. Each bar weighs about the same at $0\mathrm{.}01$ kg $.$ The maximum daily production is $200$ bars

$1,300$ bars $2,$ and $250$ bars $3.$ The suppliers can provide up to $5$ kg of honey, $2$ kg of peanuts,

$1$ kg of chocolate, and $1$ kg of pistachios each day, at a cost of $$100$ per kg of honey, $$120$ per kg

of peanuts, $$110$ per kg of chocolate, and $$200$ per kg of pistachios.

a$)$ Develop a linear program to determine the optimal daily production of the bars to maximize

profit. Write out the LP in full $($no i$\text{'}$s$,$ no j$\text{'}$s$,$ no sigmas$).(15$ pts$)$

b$)$ Solve the LP from part a using the Simplex Method. $(10$ pts$)$

c$)$ One day, the manufacturer mistakenly receives one extra kg of honey. To ensure the extra

honey does not go to waste, the manufacturer decides that at least $73\%$ of all production that

day must be of bar $1$ and bar $2.$ Given this new information, make any necessary changes to the

LP found in part a$.$ Be sure to also explain in words what changes were made. $(10$ pts $)$