A corrugated container company is testing whether a computer decision model will improve the uptime of its box production line. Currently, knives used in the production process are checked manually and replaced when the operator believes the knives are dull. Knives are expensive, so operators are encouraged not to change the knives early. Unfortunately, if knives are left running for too long, the cuts are not made properly, which can jam the machines and require that the entire process be shut down for unscheduled maintenance. Shutting down the entire line is costly in terms of lost production and repair work, so the company would like to reduce the number of shutdowns that occur daily. Currently, the company experiences an average of 0.75 knife-related shutdowns per shift, exponentially distributed. In testing, the computer decision model reduced the frequency of knife-related shutdowns to an average of 0.20 per shift, exponentially distributed. The decision model is expensive, but the company will install it if it can help achieve the target of four consecutive shifts without a knife-related shutdown.
a. Under the current system, what is the probability that the plant would run four or more consecutive shifts without a knife-related shutdown?
b. Using the computer decision model, what is the probability that the plant could run four or more consecutive shifts without a knife-related shutdown? Has the decision model helped the company achieve its goal?
c. What would be the maximum average number of shutdowns allowed per day such that the probability of experiencing four or more consecutive shifts without a knife-related shutdown is greater than or equal to 0.70?