tempt $=13969268$cmid$=1882738$

Tuck Inc. is planning its aggregate production schedule for the next $3$ months. Units may be produced at regular time or overtime. The relevant costs and capacities are shown in the table. The demand for each month is also shown. There are three ways of meeting this demand: inventory, current production, and back$-$orders. Units produced in a particular month may be sold in that month or kept in inventory for sale in a later month. There is a $$1$ cost per unit for each month an item is held in inventory. Initially, there are $15$ units in inventory. Also, sales can be backordered at a cost of $$2$ per unit per month. Backorders represent production in future months to satisfy demand in past months, and hence incur an additional cost. Initially there were no backorders. There should be no inventory or backorders at end of month $3.$

$\backslash$table$[[,,$Capacity $($Units$),$Production Cost $($$$/$Unit$)],[$Month$,$Demand,Regular,Overtime,Regular,Overtime$],[1,60,70,20,20,23],[2,140,90,10,17,22],[3,80,100,20,17,22]]$

a$.$ Develop a linear program $($LP$)$ that when solved will yield the optimal production plan. Clearly define your decision variables and write down the mathematical formulation.

b$.$ Solve the LP on Excel and use the sensitivity analysis to answer the following questions $($explain your answers fully$)$:

i$.$ If the company managed to decrease the production cost using regular time to $17$$ per unit in month $1,$ what effect will this have on the solution?

ii$.$ If the company managed to decrease the production cost using overtime to $22$$ per unit in month $1,$ what effect will this have on the solution?

iii. Describe how an increase in demand of $2$ units in month $1(62$ units instead of $60$ units$)$ can affect the current solution?