CASE $15\mathrm{.}2$ Ebony Bath Soap

greater than $$100,000$ in each case?

Management of Ebony, a leading manufacturer of bath soap, is trying to control its inventory costs. The weekly cost of holding one unit of soap in inventory is $$30($one unit is $1000$ cases of soap$).$ The marketing department estimates that weekly demand averages $120$ units, with a standard deviation of $15$ units, and is reasonably well modeled by a normal distribution. If demand exceeds the amount of soap on hand, those sales are lost $-$ that is$,$ there is no backlogging of demand. The production department can produce at one of three levels: $110,120,$ or $130$ units per week. The cost of changing the production level from one week to the next is $$3000.$

Management would like to evaluate the following production policy. If the current inventory is less than $L=30$ units, they will produce $130$ units in the next week. If the current inventory is greater than $U=80$ units, they will produce $110$ units in the next week. Otherwise, Ebony will continue at the previous week's production level.

Ebony currently has $60$ units of inventory on hand. Last week's production level was $120.$

Questions

Develop a simulation model for $52$ weeks of operation at Ebony. Graph the inventory of soap over time. What is the total cost $($inventory cost plus production change cost$)$ for the $52$ weeks?

Run the simulation for $500$ iterations to estimate the average $52-$week cost with values of $U$ ranging from $30$ to $80$ in increments of $10.$ Keep $L=30$ throughout.

Report the sample mean and standard deviation of the $52$ week cost under each policy. Using the simulated results, is it possible to construct valid $90\%$ confidence intervals for the average $52-$week cost for each value of $U?$ In any case, graph the average $52-$week cost versus $U.$ What is the best value of $U$ for $L=30?$

What other production policies might be useful to investigate?