MAN $341$

Fall $2024$

Reliability Example I

MANAGERIAL ISSUES:

Managers are concerned with the reliability of their products. An important design performance measure is the reliability of the product. A highly reliable product sells for higher prices in the marketplace and low reliability of a product results in failures. If a product fails when the customer is using it$,$ the company will incur some costs: Out of pocket replacement costs and intangible loss of goodwill costs. Thus, managers try to ensure high reliability in the products$$ designs.

One of the ways to signal that a product is high quality is to offer warranty contracts. They also help managers hedge against possible failures. A freely available warranty contract is more desirable by customers; but, sometimes, customers may be willing to pay some premium for the contracts, as well. The question here is how to price such warranty contracts? The premium should be at least sufficient to cover the expected total costs associated with reliability and contract structure.

The key factor in this computation is the probability that the product will function over some prescribed timeframe. This probability, in turn, is a function of the distribution of the $($random$)$ lifetime of a product. Designers may use empirical distributions $($actual lifetimes observed in experiments$)$ or theoretical distributions fitted to the observed data such as normal, exponential or Weibull distributions.

The example below illustrates the uses of the concept of reliability.

Tests conducted by the R&D team on a new product have determined that the product$$s operational lifetime follows a normal distribution with a mean of six years and a standard deviation of one$-$half year. The product costs $$120.$ Using this information, answer the following questions.

$1.$ What is the probability that a randomly selected product will be operational for at least five years? $($That is$,$ it will not fail before five years.$)$