Q$1.$ A mini$-$MP$3$ player manufacturing company has to manufacture two different models of its products, A$101$ and B$101,$ during next two months. The management wants to develop a production schedule indicating how many mini $mp3$ players of each model should be produced each month. Current demand forecasts call for $150\text{'}$A$101\text{'}$ and $125\text{'}$B$101\text{'}$ models to be shipped during the first month and $200\text{'}$A$101\text{'}$ and $150\text{'}$B$101\text{'}$ models to be shipped during the second month. Additional data are shown as follows.

$\backslash$table$[[$Model$,\backslash$table$[[$Production$],[$Cost$]],$Labor Requirement $($Hours$),\backslash$table$[[$Current$],[$Inventory$]]],[$Assembly$,20,],[$A$101,$$$120,2,1\mathrm{.}5,30],[$B$101,$$$90,1\mathrm{.}6,1,]]$

Last month company used a total of $1000$ hours of labor. The company's labor relations policy does not allow the combined total hours $($manufacturing plus assembly$)$ to increase or decrease by more than $100$ hours from one month to another month. In addition, the company charges monthly inventory at the rate of $2\%$ of the production cost based on the inventory levels at the end of the month. The company would like to have at least $25$ units of each model in inventory at the end of two months.

Establish a production schedule that minimizes production and inventory costs and satisfy all the constraints. Formulate the complete LP model for the same.

$[9]$