Ex$.3.$

From past data, the production manager of a factory knows that, by varying his production rate, he

incurs additional costs. He estimates that his cost per unit increases by $$0\mathrm{.}50$ when production is

increased from one month to the next. Similarly, reducing production increases costs by $$0\mathrm{.}25$ per

unit. A smooth production rate is obviously desirable.

Sales forecasts for the next twelve months are $($in thousands$)$:

June's production schedule already has been set at $4000$ units, and the July $1$ inventory level is

projected to be $2000$ units. Storage is available for only $10,000$ units at any one time. Ignoring

inventory costs, formulate a production schedule for the coming year that will minimize the cost of

changing production rates while meeting all sales demands. $($Hint: Express the change in production

from month $t$ to month $t+1$ in terms of nonnegative variables $x_t^{+}$and $x_t^{-}$as $x_t^{+}-x_t^{-}.$ Variable $x_t^{+}$

is the increase in production and $x_t^{-}$the decrease. It is possible for both $x_t^{+}$and $x_t^{-}$to be positive

in the optimal solution.$)$

Can I have the solution in writing please