Decision Analysis

[p.lucia.p]

Dear friends and students,
could you please help me with this math problem?

Milford Company of Chicago has requests to haul two shipments, one to St. Louis and one to Detroit. Because of a scheduling problem, Milford will be able to accept only one of these assignments. The St. Louis customer has guaranteed a return shipment, but the Detroit customer has not. Thus, if Milford accepts the Detroit shipment and cannot find a Detroit-Chicago return shipment, the truck will return to Chicago empty. The payoff table showing profit is as follows.

Shipment	Return Shipment from Detroit	No Return Shipment from Detroit
Shipment	s1	s2
St. Louis d1	2000	2000
Detroit d2	2500	1000

a) If the probability of a Detroit return shipment is 0,4, what should Milford do?
b) Use graphical sensitivity analysis to determine the values of the probability of state of nature s1 for which d1 has the largest expected value.
c) What is the expected value of perfect information that would tell Milford Trucking whether Detroit has a return shipment?

Milford can phone a Detroit truck dispatch center and determine whether the general Detroit shipping activity is busy (I1) or slow (I2). If the report is busy, the chances of obtaining a return shipment will increase. Suppose that the conditional probabilities are
P(I1/s1) = 0,6
P(I1/s2) = 0,3
P(I2/s1) = 0,4
P(I2/s2) = 0,7
d) What should Milford do?
e) If the general Detroit shipping activity is busy (I1), what is the probability that Milford will obtain a return shipment if it makes the trip to Detroit?
f) What is the efficiency of the phone information?

My answers are:
a) accept the St. Louis shipment
b) for p =< 2/3
c) EVPI = 200
d) accept the St. Louis shipment
e) 0,5714
f) Efficiency = 0

Are these solutions correct?

THANK YOU very much!

 

[JeffM]

Dear friends and students,
could you please help me with this math problem?

Milford Company of Chicago has requests to haul two shipments, one to St. Louis and one to Detroit. Because of a scheduling problem, Milford will be able to accept only one of these assignments. The St. Louis customer has guaranteed a return shipment, but the Detroit customer has not. Thus, if Milford accepts the Detroit shipment and cannot find a Detroit-Chicago return shipment, the truck will return to Chicago empty. The payoff table showing profit is as follows.

Shipment	Return Shipment from Detroit	No Return Shipment from Detroit
Shipment	s1	s2
St. Louis d1	2000	2000
Detroit d2	2500	1000

a) If the probability of a Detroit return shipment is 0,4, what should Milford do?
b) Use graphical sensitivity analysis to determine the values of the probability of state of nature s1 for which d1 has the largest expected value.
c) What is the expected value of perfect information that would tell Milford Trucking whether Detroit has a return shipment?

Milford can phone a Detroit truck dispatch center and determine whether the general Detroit shipping activity is busy (I1) or slow (I2). If the report is busy, the chances of obtaining a return shipment will increase. Suppose that the conditional probabilities are
P(I1/s1) = 0,6
P(I1/s2) = 0,3
P(I2/s1) = 0,4
P(I2/s2) = 0,7
d) What should Milford do?
e) If the general Detroit shipping activity is busy (I1), what is the probability that Milford will obtain a return shipment if it makes the trip to Detroit?
f) What is the efficiency of the phone information?

My answers are:
a) accept the St. Louis shipment
b) for p =< 2/3
c) EVPI = 200
d) accept the St. Louis shipment
e) 0,5714
f) Efficiency = 0

Are these solutions correct?

THANK YOU very much!

Your answer to (a) is correct.

\(\displaystyle expected\ value\ of\ SL = 2000 * 1.0 = 2000.\)

\(\displaystyle expected\ value\ of\ Detroit = 2500 * 0.4 + 1000 * (1.0 - 0.4) = 1000 + 600 = 1600.\ And\ 1600 < 2000.\)

I don't understand your answer to (b). The value of d1 is not sensitive to the state of nature. Did you state the problem correctly?

Your answer to c is correct. With perfect information

\(\displaystyle expected\ value = 0.4 * 2500 + (1 - 0.4) * 2000 = 1000 + 1200 = 2200.\ And\ 2200 - 2000 = 200.\)

Before I check your remaining answers, I want to know whether you copied the problem correctly. The conditional probabilities specify what the information reported will be given the state of nature, correct? Did you convert those to conditional probabilities specifying what the state of nature will be given the information reported?

 

[JeffM]

Thank You very much!
I am grateful to You for Your help!

b)
P (s1) = p
P (s2) = 1-p

EV(d1) = P(s1)*2000 + P(s2)*2000 = 2000p + (1-p)*2000 = 2000
EV(d2) = P(s1)*2500 + P(s2)*1000 = 2500p + (1-p)*1000 = 1500p + 1000

value of p for which the EV of d1 and d2 are equal:
2000 = 1500p + 1000
p = 2/3
for p =< 2/3 has d1 the largest EV

e)

	I1	I2
s1	P(I1/s1) = 0,6	P(I2/s1) = 0,4
s2	P(I1/s2) = 0,3	P(I2/s2) = 0,7

	P(s)	P(I1/s)	P(I1∩s)	P(s/I1)
s1	0,4	0,6	0,4*0,6 = 0,24	0,24/0,42 = 0,5714
s2	0,6	0,3	0,6*0,3 = 0,18	0,18/0,42 = 0,4286

P(I1) = 0,42

	P(s)	P(I2/s)	P(I2∩s)	P(s/I2)
s1	0,4	0,4	0,4*0,4 = 0,16	0,16/0,58 = 0,2759
s2	0,6	0,7	0,6*0,7 = 0,42	0,42/0,58 = 0,7241

P(I2) = 0,58

f)
EVSI = 2000 - 2000 = 0
Eff = EVSI/EVPI * 100% = 0%

I still have no clue what you are doing with question b. Please give it EXACTLY as it was stated in your book or handout.

I have not checked your arithmetic for the later questions, but the logic looks good to me.

 

[p.lucia.p]

Thank You,
I copied this problem exactly from my book.

The question b) is referred to the table below:

http://mcu.edu.tw/~ychen/op_mgm/notes/decision.html

Could You, please, explain to me, why is the efficiency of the information 0%?

 

[JeffM]

Thank You,
I copied this problem exactly from my book.

The question b) is referred to the table below:

http://mcu.edu.tw/~ychen/op_mgm/notes/decision.html

Could You, please, explain to me, why is the efficiency of the information 0%?

First of all, I do not understand the definition of "uncertainty" that you are being asked to use in that attachment. My understanding of the concept of "uncertainty" is that it is not possible to determine the probability of the states of nature. I think that my definition of "uncertainty" goes back to Frank Knight, who introduced the concept into economics about a century back. But I was a student a long time ago and may not understand modern vocabulary.

Second, the attachment does not help me at all to understand problem (b). As I read that problem, it asks what is the probability of s₁ that gives the maximum expected value to decision d₁. But that value is the same no matter what the probability of s₁ so the question seems meaningless to me. This gives me even less confidence that I am up to date.

Third, based on what I learned, information has zero value when it will not affect the decision being made. So, assuming your arithmetic was done correctly, the information learned from calling the dispatch center will not change the decision that would be made in the absence of that information, so THIS information has no value for THIS decision.

Given that my definition of "uncertainty" differs from what you are being taught and question (b) makes no sense to me, I think it would be best for you to talk with your teacher. I may be leading you astray, and that is certainly not my intention.

 

[p.lucia.p]

Ok, I understand.
Thank You very much once again!