Question

You work for an engineering firm, with expertise in Decision Analysis, tasked to provide a demonstration for your firm’s client, a major land development company in the Detroit area. They are considering (2) options for lakefront development of high-end communities: (50) single, 2-story units ($950,000 each) and (75) duplex 2-story units ($800,000 each). They want a demonstration of how Decision Analysis can help their decision-making in which option to select. From a preliminary standpoint, the logistics are: 1) Problem Statement; 2) Decision Alternatives; 3) Chance Events; 4) Consequences. To accomplish this, develop an Influence Diagram with nodes (decision, chance, and consequence), arcs, and a Payoff Table. Discuss the steps and outcomes in MS Word, with your own example not found in the text. Include your diagrams.

Answer 1

As the Cash inflow ( selling price) detail is not given in question

so, as per instruction in question above "Discuss the steps and outcomes in MS Word, with your own example not found in the text. Include your diagrams.". please find a detailed problem on decission tree and diagram formation below.

Decision Analysis Example
	Consider a company that wishes to choose between two option Buy or rent .
	The buying has been used before and has a 50-50 chance of yielding a net saving of
	either $4 million or $7 million over a three-year period. With B (bad) and G (good) denoting these two
	possible states of nature, we have:
		Possible
		States	Profit	Probability
		B	4,000,000	0.5
		G	7,000,000	0.5
		Expected Profit =	5500000
		Variance =	2.25E+12
		Risk (Std. Dev.) =	1500000
		Mean/SD =	3.666666667
	The second choice of rent depends on whether the state of the market is Low (measured by
	standard indices), Medium, or High. If the market is Low, the anticipated profit is $3 million;
	if it is Medium, $6 million; and if it is High, $12 million. The (subjective) probabilities for these
	possible states of nature are 0.2, 0.5, and 0.3, respectively. Thus,
		Possible
		States	Profit	Probability
		Low	3,000,000	0.2
		Medium	6,000,000	0.5
		High	12,000,000	0.3
		Expected Profit =	7200000
		Variance =	1.116E+13
		Risk (Std. Dev.) =	3340658.618
		Mean/SD =	2.155263624
	Hence, if our objective is to maximize expected payoff, then Investment 2 should be chosen.
	Note however that Investment 2 has a higher risk.
	Alternatively, we could consider the ratio of the expected profit and risk. Investment 1 turns out
	to be better with respect to this criterion.
	As usual, the choice depends on one's attitude towards risk.
	Question: Can we reduce the risk/variability of Investment 2?
Acquiring Information:
	The probabilities 0.2, 0.5, 0.3 for the state of the market are called prior probabilities.
	This probability distribution is an estimate that can potentially be improved if we acquire more
	information regarding the state of the market. With more accurate information,
	we should be able to improve the performance of Investment 2. However, acquiring additional
	information also means additional cost, and we need to achieve a balance.
	Suppose a consulting firm offers to give us their projection of the state of the market at a cost of
	$1,000,000 (to be paid at the end of three years). Of course, projections can never be 100% perfect.
	Based on the firm's historical track record, the following conditional-probability table is presented to us:
		True State	Projection
		of Market	Low	Medium	High
		Low	0.9	0.05	0.05
		Medium	0.05	0.8	0.15
		High	0.05	0.1	0.85
	Thus, for example,
		P(Projection=Low | State=Low) = 0.9
	Note that the row sums in the above table should all equal to 1; and that if the
	firm is capable of making perfect projections, then we would see a table like this:
		True State	Projection
		of Market	Low	Medium	High
		Low	1	0	0
		Medium	0	1	0
		High	0	0	1
	This of course is too good to be true. Nevertheless, this can serve as a benchmark.
	We now need to consider the following questions:
		What is the fair value of having the firm's projection?
		Should we pay $1,000,000 for the firm's projection?
Posterior Probabilities:
	To assess the value of having the firm's projection, we need to calculate the posterior
	probabilities for the state of the economy, for any given projection. The point is that
	with the firm's projection, our original set of prior probabilities should be revised/updated
	to reflect the new information. This is done via the Bayes' Law.
	Consider for example the conditional probability for the true state of the economy to
	be Low, given that the firm's projection is Low. This is computed as follows.
			P(Projection=Low) =	0.22
			P(State=Low | Projection=Low) =	P(State=Low and Projection=Low)/P(Projection=Low)
			=	0.81818182
	Continuing such calculations then yields the following tables:
		Projection	Probability
		Low	0.22
		Medium	0.44
		High	0.34
		Given	True State
		Projection	Low	Medium	High
		Low	0.818181818	0.11363636	0.06818182
		Medium	0.022727273	0.90909091	0.06818182
		High	0.029411765	0.22058824	0.75
	Different rows in the last table give the posterior probabilities for the true state of the economy
	to be Low, Medium, or High, when the given projection is Low, Medium, or High.
	Note that the row sums should all equal to 1.
Decision Tree:
	Our problem can now be visualized in the following decision tree.

	Since the expected payoff with the firm's projection is given by 7,540,000, the maximum amount we
	are willing to pay for the firm's projection should be the difference between 7,540,000 and 7,200,000.
	In other words,
				Expected Value of Having the Firm's Projection =	340000
	Since this amount is less than what the firm is asking, we should not hire the consulting firm.
Expected Value of Perfect Information (EVPI):
	This is defined as the difference between the expected payoff when we have a perfect consultant and
	the expected payoff when we do not use a consultant.
	A similar analysis yields the following tree: