The demand for a special small engine over the next five quarters is $200,150,300,250,$

and $400$ units, respectively. The manufacturer supplying the engine has different

production capacities estimated at $180,230,430,300,$ and $300$ for the five quarters.

Back$-$ordering is not allowed, but the manufacturer may use overtime to fill the

immediate demand, if necessary. The overtime capacity for each period is half the

regular capacity. The production costs per unit for the five periods are $$100,$$$96,$$$116,$

$$102,$ and $$106,$ respectively. The overtime production cost per engine is $50\%$ higher

than the regular production cost. If an engine is produced now for use in later periods,

an additional storage cost of $$4$ per engine per period is incurred.

Use this information to perform the following:

a$)[20$pt$]$ Formulate a linear programming problem mathematically to find the

optimum number of engines to be produced during regular time and overtime of

each period. Report the resulting transportation tableau.

Hint: See that problem is not balanced and you need to add a dummy destination

$($demand$).$ You need to represent, regular production and overtime production in

each quarter as a separate source.

Hint: Define a large enough cost to prevent back$-$ordering, e$.$g$.,$ cost of producing

engines for $Q1$ using regular or overtime production in $Q2$ can be $$1000.$

Regular supplies for each quarter is given in the problem. Overtime supplies are:

Supply Overtime