Question

Industrial filling process. The characteristics of an industrialfilling process in which an expensive liquid is injectedinto a container were investigated in the Journal of QualityTechnology(July 1999). The quantity injected per containeris approximately normally distributed with mean 10

units and standard deviation .2 units. Each unit of fill costs\(20 per unit. If a container contains less than 10 units (i.e.,is underfilled), it must be reprocessed at a cost of \)10. A properly filled container sells for $230.

a. Find the probability that a container is underfilled. Notunderfilled.

b. A container is initially underfilled and must be reprocessed.Upon refilling, it contains 10.60 units. Howmuch profit will the company make on thiscontainer?

c. The operations manager adjusts the mean of the fillingprocess upward to 10.60 units in order to makethe probability of underfilling approximately zero.

Under these conditions, what is the expected profit percontainer?

Short answer

a.The underfilled probability is 0.5 and not underfilled probability is .5

b. The profit upon refilling the container is $8

c. The expected profit in each container is $18

Step-by-step solution

Given information

The quantity injected per container is approximately normally distributed with mean 10 units and standard deviation .2 units. Each unit of fill costs $20 per unit. If a container contains less than 10 units (i.e., is underfilled), it must be reprocessed at a cost of $10. A properly filled container sells for $230.

Finding the probability

Let x be the random variable follow normal distribution with mean 10 units and standard deviation 0.2 units.

$x~N(10,0.2)$

a.

The probability of underfill is given by,

role="math" localid="1660285516190" $$\begin{gathered} P(x<10)=P\left(\frac{x-10}{0.2}<\frac{10-10}{0.2}\right) \\ =P(z<0) \\ =0.50 \end{gathered}$$

The probability is 0.50

The probability of not underfill is given by,

$$\begin{gathered} P(x\ge 10)=P\left(\frac{x-10}{0.2}\ge \frac{10-10}{0.2}\right) \\ =1-P(z<0) \\ =1-0.50 \\ =0.50 \end{gathered}$$

Thus, the required probability is 0.50

Finding the profit

b.

The cost of each unit is $20

The reprocessed cost is $10

The selling cost is $230

The profit make the company upon refilling the container is,

$$\begin{gathered} profit=230-10-10.60\times 20 \\ =8 \end{gathered}$$

Thus, required the profit in refilling the container is 8$.

Finding expected profit

c.

The mean of filling process upward is 10.60 units. So,

$$\begin{gathered} Expectedprofit=230-10.60\times 20 \\ =18 \end{gathered}$$

Thus, the required he expected profit in each container is $18.