Question

Kilmade Co., a brand manufacturer of relay switches for vehicles, is conducting an end-of-year analysis on its production of the K7 Relay Switch, the output good. In economics, it is common to estimate cost functions for a product to help with forecasting production quotas. Since you have been appointed director of the team for this analysis, you are charged with the following:

i) Determine the optimal lifetime (as measured by months of operation) that the K7 Relay Switches should be calibrated to, such that that production costs will be minimized. The company will use this level to establish a warranty period.

For simplicity, assume that costs are function of months of operation for relay switches. The cost function takes on the following form:

TC = X2 – 1,270X + 430

ii) Based on what you find, determine the warranty period so that no more than 28% of the product line are covered under the warranty program. Assume your optimal lifetime value represents the mean of a distribution that is distributed normally, with an assumed value for the standard deviation of 40 months.

Answer 1

The optimal lifetime of the K7 Relay Switches is determined by finding the vertex of the quadratic cost function, and the warranty period is established by using the normal distribution with the optimal lifetime as the mean and considering the 28% upper tail of the distribution.

Step-by-step explanation:

The cost function provided is TC = X² – 1,270X + 430, which is a quadratic function, and its graph is a parabola. To minimize the production costs and determine the optimal lifetime of the K7 Relay Switches, the first step is to find the vertex of the parabola, as it represents the minimum point on a graph of this kind. The vertex form of a parabola is given by TC = a(X - h)² + k, where (h, k) are the coordinates of the vertex. In this case, h can be found using the formula -b/(2a), where a is the coefficient of X², and b is the coefficient of X. Thus, h = 1,270 / 2, giving us the optimal lifetime in months.

Next, to establish a warranty period in which no more than 28% of the product line are covered, we need to use the normal distribution curve. With the optimal lifetime as the mean, and a standard deviation of 40 months, we can refer to a z-table to find the value that corresponds to 28% in the upper tail. As we want only 28% of the switches to be in warranty, we look for the z-score corresponding to 72% (as normal distribution is symmetrical), which is going to be the cutoff point for our warranty period.