$40$

DISCRETE$-$EVENTSYSTEM SIMULATION

Example $2\mathrm{.}4$: Simulation of an Order$-$Up$-$To$-$LeveI Inventory System Consider a situation in which a company sells refrigerators. The system they use for maintaining inventory is to review the situation after a fixed number of days $($say $N)$ and make a decision about what is to be done. The policy is to order up to a level $($the order up to level$-$say, $M),$ using the following relationship:

Order quantity $=(Order-up-to$ level$)-(Ending$ inventory$)+(Shortage$ quantity$)$

Let's say that the order$-$up$-$to level $(M)$ is $11$ and the ending inventory is three. Further, let's say that the review period $(N)$ is five days. Thus, on the fifth day of the cycle, $8$ refrigerators will be ordered from the supplier. If there is a shortage of two refrigerators on the fifth day, then $13$ refrigerators will be ordered. $($There can't be both ending inventory and shortages at the same time.$)$ If there were a shortage of three refrigerators, the first three received would be provided to the customers when the order arrived. That's called "making up backorders." The lost sales case occurs when customer demand is lost if the inventory is not available.

The number of refrigerators ordered each day is randomly distributed as shown in Table $2\mathrm{.}19.$ Another source of randomness is the number of days after the order is placed with the supplierbef ore arrival, or lead time. The distribution of lead time is shown in Table $2\mathrm{.}20.$ Assume that the orders are placed at the end of the day. If the lead time is zero, the order from the supplier will arrive the next morning, and the refrigerators will be available for distribution that next day. If the lead time is one day, the order from the supplier arrives the second morning after, and will be available for distribution that day.

The simulation has been started with the inventory level at $3$ refrigerators and an order for $8$ refrigerators to arrive in $2$ days' time. The simulation table is shown in Table $2\mathrm{.}21.$

Following the simulation table for several selected days indicates how the process operates. The order for $8$ refrigerators is available on the morning of the third day of the first cycle, raising the inventory level from zero refrigerators to $8$ refrigerators. Demands during the remainder of the first cycle reduced the ending inventory level to $2$ refrigerators on the fifth day. Thus, an order for $9$ refrigerators was placed. The lead time for this order was $2$ days. The order for $9$ refrigerators was added to inventory on the morning of day $3$ of cycle $2.$

Table $2\mathrm{.}19$ Random Digit Assignments for Daily Demand

$\backslash$table$[[$Demand$,$Probability,$\backslash$table$[[$Cumulative$],[$Probability$]],\backslash$table$[[$Random Digit$],[$Assignment$]]],[0,0\mathrm{.}10,0\mathrm{.}10,01-10$