Question

1. The postal service uses mail processing plants to sort letters. Once a letter is received, it must be sorted into bins by zip code. When a letter arrives, a machine reads the address of the letter and sorts the letter into two bins, "local" and "nonlocal." However, the process of reading the letter is complex, so the machine sometimes cannot read the address. In this case, the machine directs the letter to a third bin, the "manual sort" bin, where a plant operator then manually sorts the letter The postal service would like to know how many letters are not initially processed correctly. The plant considers a correctly processed letter to be a letter that is directed to the correct bin by either a machine or a plant operator. Thus, a letter that is sorted into the "manual sort" bin and then into the correct bin is initially processed correctly. However, the plant operator and the machine can sometimes sort the letter to the wrong bin. In addition, the plant operator has several other responsibilities and can forget to sort the letters. Any letter in the "manual sort" bin at the end of the day is considered an incorrectly processed letter Engineering found the likelihood of occurrence for each of the events: Letter address is readable by the machine: 0.850 Letter is sorted by the machine correctly:0.967 Letter in "manual sort" bin is not sorted in time: 0.038 Plant operator incorrectly sorts the letter: 0.192 . . a. What is the probability that a letter is correctly sorted? b. If the plant processes 200,000 letters a year, how many letters are wrongly sorted each year due to omission errors? c. The plant determines that each incorrectly processed letter costs the postal service an additional $0.42 in costs. If the plant buys another address reading machine, the total reliability of reading the letter address of the two machines together is 0.981. The

Answer 1

Let us define the following events as

R - Letter address is readable by the machine.

S - Letter is sorted by machine correctly

M - Letter in manual sort bin not sorted in time

O - Plant operator incorrectly sorts the letter

Then, P(R) = 0.85, P(S) = 0.967, P(M) = 0.038, P(O) = 0.192

a.

Probability that the letter is correctly sorted = P( Letter address is readable by the machine) * P(Letter is sorted by machine correctly) + P( Letter address is not readable by the machine) * P( Letter in manual sort bin sorted in time) * P(Plant operator correctly sorts the letter)

= P(R) * P(S) + (1 - P(R)) * (1 - P(M)) * (1 - P(O))

= 0.85 * 0.967 + (1 - 0.85) * (1 - 0.038) * (1 - 0.192)

= 0.9385444

b.

Probability that the letters are wrongly sorted due to omission errors = P( Letter address is not readable by the machine) * P( Letter in manual sort bin not sorted in time)

= (1 - 0.85) * 0.038 = 0.0057

Number of letters wrongly sorted each year due to omission errors = 200,000 * 0.0057 = 1140

c.

If we have only one machine, as in part (a), probability of incorrectly processed letter = 1 - Probability that the letter is correctly sorted = 1 - 0.9385444 = 0.0614556

For 200,000 letters each year, additional cost due to incorrectly sort = 200,000 * 0.0614556 * 0.42 = $5162.27

If we have two machines, the P( Letter address is readable by the machine) = 0.981

Now, Probability that the letter is correctly sorted =

= P(R) * P(S) + (1 - P(R)) * (1 - P(M)) * (1 - P(O))

= 0.981 * 0.967 + (1 - 0.981) * (1 - 0.038) * (1 - 0.192)

= 0.9633956

If we have only two machines, probability of incorrectly processed letter = 1 - Probability that the letter is correctly sorted = 1 - 0.9633956 = 0.0366044

For 200,000 letters each year, additional cost due to incorrectly sort = 200,000 * 0.0366044 * 0.42 = $3074.77

Total cost of using two machines = 3074.77 + 1500 = 4574.77

Since the additional cost of using two machines is less than the additional cost due to incorrectly sort by one machine, the plant should add the new machine.