Enginola, Inc., assembles amplifiers on a two$-$stage production line. The first stage

makes a chassis and the second stage does the custom assembly. The chassis stage

consists of $20$ parallel stations, each staffed by an operator; the amplifier stage consists of $15$ parallel stations, each also staffed by a single operator. Because all chassis are identical, the time for an operator to build one is almost constant, at $15$ minutes. But, because there are many different amplifiers assembled from the standard chassis, the time for an operator to assemble an amplifier is highly variable, with a mean of $20$ minutes $($more specifically, here it is assumed that at each of the $15$ stations of the amplifier stage, the next amplifier to be built by the station operator can belong to any of the available types$).$ From the above description it is evident that the chassis stage has more capacity than the amplifier stage. This excess capacity is utilized by having the chassis operators perform additional tasks, whenever they are not needed to build chassis. Also, Enginola has implemented a KANBAN system to ensure that the inventory of completed chassis waiting at the amplifier stage does not become excessive. This system makes use of paper cards, which are attached to the finished chassis. Whenever an amplifier operator takes a chassis out of stock, s$/$he removes the card and hands it upstream to the chassis stage. The card is given to a chassis operator as a signal to build another chassis. When the operator completes the chassis, s$/$he attaches the card to it and delivers the chassis to the stockpoint at the amplifier stage. Since chassis operators are not allowed to build chassis without a card, and there are only m cards in the system, the total amount of chassis inventory at the amplifier stage can never exceed m$.$ Assuming that the $15$ stations at the amplifier stage operate independently from each

other, and that the time to build an amplifier at that stage can be approximated by an

exponential distribution $($this assumption is justified on the basis of the high variety of

the amplifiers that are built at that stage$),$ determine a value for the number of cards m so that the probability that an amplifier operator will find a chassis in stock when s$/$he needs one is at least $99$ percent.

Hints: Refer the above operation a stochastic inventory model, by looking at the KANBAN mechanism described above as an order generating mechanism. What is the replenishment lead time in this inventory model? To answer the problem question, you will need to characterize the demand during a replenishment lead time interval. Remember the connection between the exponential and the Poisson distribution, and also some key properties of the Poisson distribution