Assignment 1 2019 PDF

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Assignment 1...

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INSE 6400: Principles of Systems Engineering

Concordia Institute of Information System Engineering (CIISE), Concordia University, QC, Canada

Assignment 1 Due Date: July. 23th

Summer 2019

Farnoosh Naderkhani Office: S-EV 7-647 Email: [email protected]

Problem 1 Betsy Pitzer makes decisions according to Bayes’ decision rule. For her current problem, Betsy has constructed the following payoff table (in units of dollars):

Alternative A1 A2 A3 Prior probability

S1 50 0 20 0.5

State of Nature S2 100 10 40 0.3

S3 -100 -10 -40 0.2

a. Which alternative should Betsy choose? b. Find expected value for perfect information (EVPI). c. What is the most that Betsy should consider paying to obtain more information about which state of nature will occur? Problem 2 Consider the decision analysis problem having the following payoff table:

Alternative A1 A2 A3 Prior probability

S1 -100 -10 10 0.2

State of Nature S2 10 20 10 0.3

S3 100 50 60 0.5

a) Based on three decision criteria (i.e., maximin payoff criterion, maximum likelihood criterion, and Bayes’s decision rule), which alternative should be chosen? What is the resulting expected payoff, respectively? b) You are offered the opportunity to obtain information which will tell you with certainty whether the first state of nature S1 will occur. What is the maximum amount you should pay for the information? Assuming you will obtain the information, how should this information be used to choose an alternative? What is the resulting expected payoff (excluding the payment)? (note: use Bayes’s decision rule ) c) Now suppose that the opportunity is offered to provide information which will tell you with certainty which state of nature will occur (perfect information). What is the maximum amount you should pay for the information? Assuming you will obtain the information, how should this information be used to choose alternative? What is the resulting expected payoff (excluding the payment)? (note: use Bayes’s decision rule )

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Problem 3 Company A is considering whether to invest a system development project. The profit of the project depends on the number of investors (no including Company A). The expected profit for having 2 or more investors is $20,000, for having 1 investor $10,000, and for having no investor -$15,000. The number of investors is approximated by a Poisson distribution with mean  = 0.8. a) Develop the payoff table for Company A to make the decision. b) Determine the optimal decision under Bayes’ decision rule. c) Company A can ask a consultant for estimating the number of investors. The table below shows the accuracy (or quality) of the estimation by the consultant. Determine the posterior probabilities. d) Company A decides not to invest only if the consultant predicts for no investor. Determine the maximum amount Company A should pay for the consultant’s information.

Estimation by consultant No investor 1 investor 2 or more invest

0 50% 40% 10%

( hint: Poisson Distribution : P(X=x) =

Actual number of investor(s) 1 40% 50% 10%

2 or more 20% 40% 40%

𝑒 −𝜆 𝜆𝑥 ) 𝑥!

Problem 4 Consider the inventory example introduced in the lecture, but the following change in the ordering policy. If the number of cameras on hand at the end of each week is 0 or 1, two additional cameras will be ordered. Otherwise, no ordering will take place. Assume that the storage costs are the same as given in the lecture. a) Find the steady-state probabilities of the state of this Markov chain. b) Find the long-run expected average storage cost per week.

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Problem 5 Consider the following blood inventory problem in a hospital. There is a need for a rare blood type, namely, type AB, Rh-negative blood. The demand D (in pints) over any 3-day period is given by P{D=0} = 0.4, P{D=1} = 0.3, P{D=2} = 0.2 and P{D=3} = 0.1. Suppose that there are 3 days between deliveries. The hospital proposes a policy of receiving 1 pint at each delivery and using the oldest blood first. If more blood is required than is on hand, an expensive emergency delivery is made. Blood is discarded if it is still on the shelf after 21 days. Denote the state of the system as the number of pints on hand just after a delivery. Thus, because of the discarding policy, the largest possible state is 7. a) Find the (one-step) transition matrix of this Markov chain. b) Find the steady-state probabilities of the state of the Markov chain.

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