$($Road lighting$)$ Consider a road divided into m segments that is illuminated by n lamps. Let p$\_$i be the power of the i$-$th lamp. The illumination I$\_$j of the j$-$th segment is assumed to be $\_($i$=1)^$n$$a$\_$ij p$\_$i $,$ where a$\_$ij are known coefficients. Let I$\_$j$^*$ be the desired illumination of road j$.$ We are interested in choosing the lamp powers p$\_$i so that the illuminations I$\_$j are close to the desired illuminations I$\_$j$^*.$ Provide a reasonable linear programming formulation of this problem. Note that the wording of the problem is loose and there is more than one possible formulation.

$($Production and inventory planning$)$ A company must deliver d$\_$i units of its product at the end of the i$-$th month. Material produced during a month can be delivered either at the end of the same month or can be stored as inventory and delivered at the end of a subsequent month. However, there is a storage cost of h dollars per month for each unit of product held in inventory. The company begins the first month with zero inventory. If the company produces x$\_$i units in month i$,$ and x$\_($i$+1)$ units in month i$+1,$ it incurs a cost of s$|$x$\_($i$+1)-$x$\_$i $|$ dollars, reflecting the cost of switching to a new production level. Formulate a linear programming problem whose objective is to minimize the total cost of the production and inventory schedule over a period of $6$ months. Assume that the inventory left at the end of the last month has no value and does not incur any storage costs.