Elizabeth Burke wants to develop a model to more effectively plan production for the next year. Currently, PLE has a planned capacity of producing $9,100$ mowers each month, which is approximately the average monthly demand over the previous year. However, looking at the unit sales figures for the previous year, she observed that the demand for mowers has a seasonal fluctuation, so with this "level" production strategy, there is overproduction in some months, resulting in excess inventory buildup, and underproduction in others, which may result in lost sales during peak demand periods.

Ms$.$ Burke explained that she could change the production rate by using planned overtime or undertime $($producing more or less than the average monthly demand$),$ but this incurs additional costs, although it may offset the cost of lost sales or of maintaining excess inventory. Consequently, she believes that the company can save a significant amount of money by optimizing the production plan.

Ms$.$ Burke saw a presentation at a conference about a

$D_t=$ decrease in production rate from period $t-1$ to period $t$

Material balance constraint:

$x_t+I_{t-1}-I_t+L_t=$ demand $in$ month $t$

Overtime$/$undertime constraint:

$O_t-U_t=x_t-$ normal production capacity

Production rate$-$change constraint:

$x_t-x_{t-1}=R_t-D_t$

Ms$.$ Burke also provided the following data and estimates for the next year: unit production cost $=$$$70\mathrm{.}00$; inventory$-$holding cost $=$$$1\mathrm{.}40$ per unit per month; lost sales cost $=$$$200$ per unit; overtime cost $=$$$6\mathrm{.}50$ per unit; undertime cost $=$$$3\mathrm{.}00$ per unit; and production$-$rate$-$change cost $=$$$5\mathrm{.}00$ per unit, which applies to any increase or decrease in the production rate from the previous month. Initially, $900$ units are expected to be in inventory at the beginning of January, and the production rate for the past December was $9,100$ units. She