Suppose a startup company called $$Antivirus Mountaineers$$ has developed a test kit for a

dangerous virus. Their orders for the next $12$ months are as follows $($in million kits$)$: $200,300,$

$400,400400,500,400,300,300,200,200,200.$ The executives of the company need to develop

their production schedule for the upcoming year. Assume that the planning horizon is only for the

next $12$ months, but at the end of the $12$th month the company plans to have $50$ million kits in

inventory going into the next planning year. At the beginning of the first year they are starting

from zero test kits in inventory.

The company can produce more than the required number of kits in each month and hold the

surplus to the next month. However, costs of storage in a refrigerated unit are $$0\mathrm{.}1$ per kit per

month. The storage capacity each month is $60$ million kits.

Increasing production from one month to the next requires hiring new employees. This increases

production costs by $$0\mathrm{.}1$ per additional kit produced that month relative to the number of units

produced in the previous month. On the other hand, decreasing production from one month to the

next requires laying off employees. Laying off employees also incurs costs, which depend on the

magnitude of the change in production from one month to next. Specifically, each unit reduction

in production volume costs $$0\mathrm{.}05$ per kit for the month when the change in production took place.

Baseline production costs are $$0\mathrm{.}2$ per kit assuming no change in the number of produced kits

relative to the previous month. For example, if the company produced $10$ kits in one month and

$12$ kits in the next month, then in the second month cost of $10$ units is $$0\mathrm{.}2$ per unit and cost of $2$

additional units is $$0\mathrm{.}2+$$$0\mathrm{.}1=$$$0\mathrm{.}3$ per kit. Similarly, if production decreased from $10$ to $8$ then in

the second month $8$ units are baseline, costing $$0\mathrm{.}2$ per kit. However, the firm also incurs costs of

$2*$$$0\mathrm{.}05$ for production decrease.

a$)$ Develop a linear programming model to determine the optimal schedule of production for

the next $12$ months.