$$Good Intentions$($GI$)$ company produces two products, which it sells on both a cash and credit basis. Revenues from credit sales will not have been received but are included in determining profit earned during the current six$-$month period. Sales during the next six months can be made either from units produced during the next six months or from the beginning inventory. Relevant information about products one and two is as follows.

$$ During the next six months, at most $150$ units of product type $1$ can be sold on a cash basis, and at most $100$ units of product $1$ can be sold on a credit basis. It costs $35$ to produce each unit of product type $1,$ and each sells for $40.$ A credit sale of a unit of product $1$ yields $0\mathrm{.}50$ less profit than a cash sale $($because of delays in receiving payment$).$ Two hours of production time are needed to produce each unit of product $1.$ At the beginning of the six$-$month period, $60$ units of product $1$ are in the inventory.

$$ During the next six months, at most $175$ units of product type $2$ can be sold on a cash basis, and at most $250$ units of product type $2$ can be sold on a credit basis. It costs $45$ to produce each unit of product type $2,$ and each sells for $52\mathrm{.}5.$ A credit sale of a unit of product type $2$ yields $1$ less profit than a cash sale. Four hours of production time are needed to produce each unit of product type $2.$ At the beginning of the six$-$month period, $30$ units of product type $2$ are in the inventory.

$$ During the next six months, GI has $1000$ hours for production available. At the end of the next six months, GI incurs a $10\%$ holding cost on the value of ending inventory $($measured relative to production cost$).$ An opportunity cost of $5\%$ is also assessed against any cash on hand at the end of the six$-$month period.

$($a$)$ Formulate and solve a linear programming model that yields GI's maximum profit during the next six months. What is GI's ending inventory position? Assuming an initial cash balance of $0$ pounds, what is GI's ending cash balance?

$($b$)$ Since an ending inventory and cash position of $0$ is undesirable $($for ongoing operations$),$ GI is considering other options. At the beginning of the six$-$month period, GI can obtain a loan $($secured by ending inventory$)$ that incurs an interest cost equal to $5\%$ of the value of the loan. The maximum value of the loan is $75\%$ of the value of the ending inventory. The loan will be repaid one year from now. GI has the following goals $($listed in order of priority$)$:

Goal $1$: Make thGoal$-1$: Make the ending cash balance of $Gl$ come as close as possible to $75.$

Goal$-2$: Make profit come as close as possible to the profit level obtained in part $($b$).$

Goal$-3$: At any time, GI's current ratio is defined to be:

$($GI$\text{'}$s assets$)/($Gi$\text{'}$s liabilities$)$

Assuming initially that current liabilities equal $150,$ six months from now $*$ Gl$\text{'}$s

current ratio will equal:

$\frac{CB+*AR+*EI}{150+\text{S}\text{i}\text{z}\text{e}of*\text{l}\text{o}\text{a}\text{n}}$

where $CB$ is the ending cash balance, $AR$ is the value of accounts receivable, and $EI$ is

the value of the ending inventory. Six months from now, $\text{'}$Gl desires the current

ration to be as close as possible to $2.$

Use preemptive goal programming to determine Gl$\text{'}$s production and financial

strategy. $(9$ points$)$