A company receives an order for specially designed product. The cost of producin

Question

A company receives an order for specially designed product. The cost of producing each item is $40,000. The customer agrees to pay $350,000 for five good products, $420,000 for six good products, and $490,000 for seven good products. Other than the five, six or seven good ones all other remaining good or bad products, must be destroyed. The company agrees to pay the customer $200,000 if at least five good products are not produced. Each product is produced independently; the probability of a having good product is 0.85. Determine the number of products to produce, as well as the probability of losing money on the transaction.

Explanation / Answer

We know that the probability of producing a good product is 0.85, so let p=0.85. And we know that each product is produced independently, thus we can assume that they follow a binomial distribution.

Now we will find the probability distribution of the production schedule. This table shows the probability of producing good products among the total number of products produced:

Now we need to find out the expected profit in each of these combinations. For example, if we produce 5 products, how much profit does the company get if out of 5, 3 are good ones, or four are good ones etc.

We know that if the company produces less than 5 good products the company pays $200,000. And also we know that the cost of producing a particular good is $40,000, and the company gets paid $350,000 for 5 good products, $420,000 for 6 good products, $490,000 for 7 good products. So for example, the cost of producing 5 products is 40,000*5 = 200,000

Now if the number of good products is less than 5 then the company incurs a loss of -200,000 and an additional -200,000 that he promises to pay the customer, so a total of -400,000. Suppose all the 5 products are good, then the profit is 350,000-200,000 (Revenue - Cost) = 150,000

So we generate the following table:

Here notice that the profits beyond producing 7 goods does not increase, because there is no additional revenue produced from it.

For a given amount of good produced the expected profit is given by multiplying the net profit by the probability of its occurence and summing over all possible outcomes (here they refer to the number of good products in a batch). So we get the following table.

So for example the expected profit of producing 5 products is

150000*0.4437 - 400000*0.3915 - 400000*0.1382 - 400000*0.0244 - 400000*0.0022 - 400000*0.0001 = -155962.08

So, we see that the maximum expected profit is when the company produces 8 products. Hence it should produce 8 products.

The probability of losing money is the probability of the net profit being negative when 8 products are being produced. From the second table we see that the net income is negative if less than 5 good products are being produced. Therefore the probability of losing money is the sum of the probabilities of producing less than 5 good products in a batch of 8. So it is 0+0+0.0002+0.0026+0.0185 = 0.0214.

Hence there is only a 2% chance of losing money.

Total number of goods produced # of good products 5 6 7 8 9 10 11 12 0 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1 0.0022 0.0004 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 2 0.0244 0.0055 0.0012 0.0002 0.0000 0.0000 0.0000 0.0000 3 0.1382 0.0415 0.0109 0.0026 0.0006 0.0001 0.0000 0.0000 4 0.3915 0.1762 0.0617 0.0185 0.0050 0.0012 0.0003 0.0001 5 0.4437 0.3993 0.2097 0.0839 0.0283 0.0085 0.0023 0.0006 6 0.0000 0.3771 0.3960 0.2376 0.1069 0.0401 0.0132 0.0040 7 0.0000 0.0000 0.3206 0.3847 0.2597 0.1298 0.0536 0.0193 8 0.0000 0.0000 0.0000 0.2725 0.3679 0.2759 0.1517 0.0683 9 0.0000 0.0000 0.0000 0.0000 0.2316 0.3474 0.2866 0.1720 10 0.0000 0.0000 0.0000 0.0000 0.0000 0.1969 0.3248 0.2924 11 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1673 0.3012 12 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1422

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.