Risk and Uncertainty – OT Course Notes

Dear John,

I would like to ask you about Example 1, Question c, Chapter 10 – Risk and Uncerntainty in OT Course Notes:

John has a factory capacity of 1,200 units per month.
Units cost him $6 each to make and his normal selling price is $11 each. However, the demand per month is
uncertain and is as follows:

Demand Probability
400 0.2
500 0.3
700 0.4
900 0.1
He has been approached by a customer who is prepared to contract to a fixed quantity per month at a price
of $9 per unit. The customer is prepared to sign a contract to purchase 300, 500, 700 or 800 units per month.
The company can vary production levels during the month up to the maximum capacity, but cannot carry
forward any unsold units in inventory.

(a) Calculate all possible profits that could result
(b) Determine for what quantity John should sign the contract, under each of the following criteria:
i) expected value
ii) maximin
iii) maximax
iv) minimax regret
(c) What is the most that John would be prepared to pay in order to obtain perfect knowledge as to
the level of demand?

Answer key:

(a)
Contract size\Demand 400u 500u 700u 900u
300u 2,900 3,400 4,400 5,400
500u 3,500 4,000 5,000 5,000
700u 4,100 4,600 4,600 4,600
800u 4,400 4,400 4,400 4,400

(b) (i) Expected value if contract size =
300 units = (0.2 ×2,900) + (0.3 × 3,400) + (0.4 × 4,400) + (0.1 × 5,400) = $3,900
500 units = (0.2 × 3,500) + (0.3 × 4,000) + (0.5 × 5,000) = $4,400
700 units = (0.2 × 4,100) + (0.8 × 4,600) = $4,500
900 units = $4,400
Sign contract for 700 units
(ii) maximin
Worst outcome from:
300 units = $2,900
500 units = $3,500
700 units = $4,100
800 units = $4,400
Sign contract for 800 units

(iii) Best outcome from

300 units = $5,400
500 units = $5,000
700 units = $4,600
800 units = $4,400
Sign contract for 300 units
(iv) Regret table:
Contract
size
Demand 400u 500u 700u 900u
300u 1,500 1,200 600 0
500u 900 600 0 400
700u 300 0 400 800
800u 0 200 600 1,000

Worst regret for

300 units = $1,500
500 units = $900
700 units = $800
800 units = $1,000
Sign contract for 700 units

(c) With perfect knowledge of the level of demand, the payoffs would be as follows:
Result of Decision Payoff
perf. know. Contract

400 800u 4,400
500 700u 4,600
700 500u 5,000
900 300u 5,400

The expected return with perfect knowledge =
(0.2 × 4,400) + (0.3 × 4,600) + (0.4 × 5,000) + (0.1 × 5,400) = $4,800

The expected return without perfect knowledge (from (b)(i) is $4,400
So the most to pay for perfect knowledge
= 4,800 – 4,400
= $400

My question is: in Section c, why’s EV without perfect information is 4,400 rather than 4,500 (the highest EV of all) ?

In addition, I also want to know whether the content “Value of information” is exempted from the syllabus?

I hope your reply soon.