PROBLEM C: A SIMULATION PROBLEM

Use simulation techniques to solve the following problem on inventory management:

Consider the company in Task $3$ planning its inventory system over the next n months. The sizes of the demands at each point of time, D$($t$),$ are IID random variables $($independent of when the demands occur$),$ and distributed uniformly within the range $[0,100].$ At the beginning of each month, the company reviews the inventor level and decides how many items to order from its supplier. As before, if the company orders Z items, it incurs a cost of K$+$iZ$,$ where K$=35$ is the setup cost and i$=4$ is the incremental cost per item ordered. $($If Z$=0,$ no cost is incurred.$)$ When an order is placed, the time required for it to arrive $($lead time$)$ is $1$ month.

The company uses a stationary $($s$,$S$)$ policy to decide how much to order, i$.$e$.,$

Z$($t$)=\{($S$-$I$($t$)$ if I$($t$)=$s$)$

Where I$($t$)$ is the inventory level at the beginning of the month t$.$

When a demand occurs, it is satisfied immediately if the inventory level is at least as large as the demand. If the demand exceeds the inventory level, the excess of demand over supply is backlogged and satisfied by future deliveries. $($In this case, the new inventory level is equal to the old inventory level minus the demand size, resulting in a negative inventory level.$)$ When an order arrives, it is first used to eliminate as much of the backlog $($if any$)$ as possible; the remainder of the order $($if any$)$ is added to the inventory.

There are two types of costs, holding and shortage costs. Let I$($t$)$ be the inventory level at time t $[$note that I$($t$)$ can be positive, negative, or zero$],$ let I$^+($t$)=$max$\{$I$($t$),0\}$ be the number of items physically on hand in the inventory at time t $[$note that I$^+($t$)>=0],$ and let I$^-($t$)=$max$\{-$I$($t$),0\}$ be the backlog at time t $[$note that I$^-($t$)>=0].$ We assume the company incurs a holding cost of h$=2$ per item per month held in $($positive$)$ inventory. Similarly, suppose that the company incurs a backlog cost of p$=7$ per item per month in backlog.

Assume that the initial inventory level is I$(0)=60$ and that no order is outstanding.

Simulate the inventory system for n$=120$ months and use the average total cost per month $($which is the sum of the average ordering cost per month, the average holding cost per month, and the average shortage cost per month$)$ to compare the following nine inventory policies:

s $202020204040406060$

S $406080100608010080100$

Suppose an alternative system where the holding cost is h$^\text{'}=3,$ and backlog cost is p$^\text{'}=1.$ Compare the above nine policies in this case.