The leadership team at AquaLux, a prominent bath soap manufacturer, is aiming to optimize their inventory expenses. The cost incurred weekly for holding a single unit of soap in stock amounts to $30($where one unit equates to $1000$ soap cases$).$ The marketing division has projected a weekly demand with an average of $120$ units and a standard deviation of $15$ units. This demand follows a relatively accurate normal distribution pattern. Sales exceeding the available soap quantity result in lost sales; in other words, there's no backlog for unmet demand.

The manufacturing department can operate at one of three production capacities: $110,120,$ or $130$ units per week. A production level shift incurs a $3000$ cost$.$ The management intends to assess the following production strategy: When the existing inventory falls below L $=30$ units, they will manufacture $130$ units in the subsequent week. If the current inventory surpasses U $=80$ units, they will produce $110$ units in the upcoming week. In all other cases, AquaLux will maintain the previous week's production level. At present, AquaLux maintains an inventory of $60$ units. The production level from the prior week was set at $120$ units.