## Decision Trees with Dependent Conditions

With Bayes' Theorem in hand, we can proceed to project decisions that are interdependent. Let' is an item on the WBS that may or may not be included by the sponsor's decision in the final pro buy decision for satisfying the acquisition to be made by the project team. However, let's change decision by the sponsor affects the subsequent performance of either make or buy:

■ Let SD be the random variable that represents a sponsor's decision that may or may not be affects subsequent make or buy performance or value. In this example, we will say that our decision is 70%, 0.7. In that event, using our "1-p" analysis, we have p(SD late) = 0.3, 30%.

■ If SD is on time, then the situation reverts to a case of independent conditions.

The problem at hand is to determine what is p(Make or Buy performance given SD late). In othe p(Make performance given SD late) = p(Make performance and SD late/p(SD late)

and p(Buy performance given SD late) = p(Buy performance and SD late)/p(SD late) where performance can be on time or late.

Table 4-4 arrays the scenarios that fall out of the situation in this project. Looking at this table cs six probabilities since "late or on time" is a shorthand notation for two distinctly different probabili

Table 4-4: Dependent Scenarios

Project Situation: MAKE

MAKE 1:

p[MAKE late (or on time) given SD late] = p[MAKE late (or on time) AND SD late]/p(SD late) MAKE 2:

p[MAKE late (or on time) given SD on time] = p[MAKE late (or on time)]_

p[BUY late (or on time) given SD late] = p[BUY late (or on time) AND SD late]/p(SD late) BUY 2:

p[BUY late (or on time) given SD on time] = p[BUY late (or on time)]_

Now we come to a vexing problem: to make progress we must estimate the joint probabilities of "buy late (or on time) and late SD." We have already said that we have pretty high confidence th probability involving "SD late" will be a pretty small space. We do know one thing that is very use late" have to fit in the space of 30% confidence that SD will be late:

p(Make late and SD late) + p(Make on time and SD late) = p(SD late) = 0.3, or p(Buy late and SD late) + p(Buy on time and SD late) = p(SD late) = 0.3

We now must do some estimating based on reasoning about the project situation as we know it. probabilities of 0.4 and 0.6, respectively. If these were independent of "SD late," then the joint pr multiples of the probabilities:

However, in our situation "SD late" conditions performance, so the probabilities are not independ on time should be more pessimistic (smaller) since the likelihood of the joint event is more pessi independently. In that case, the joint probability of being late is more optimistic (more likely to ha p(Make on time and SD late) = p(Make on time) * p(SD l.

and then:

Estimate: p(Make late and SD late) = 0.2 = 0.3 - 0.1 We can make similar estimates for the "buy" situation. Multiplying probabilities as though they w>

and p(Buy on time and SD late) = 0.8 * 0.3 = 0.24 Following the same reasoning about pessimism as we did in the "make" case:

p(Buy on time and SD late) = p(Buy on time) * p(SD lat Estimate: p(Buy on time and SD late) = 0.23

and then:

Estimate: p(Buy late and SD late) = 0.07 = 0.3 - 0.23

We now apply Bayes' Theorem to our project situation and calculate the question we started to r performance given SD late) is given in Table 4-5 and Table 4-6:

p(Make performance given SD late) p(Make 20 days late given SD late) p(Make 0 days late given SD late) p(Buy performance given SD late) p(Buy 20 days late given SD late) p(Buy 0 days late given SD late)

p(Make performance and SD late)/p(SD late) 0.2/0.3 = 0.67 0.1/0.3 = 0.33, and p(Buy performance and SD late)/p(SD late), 0.07/0.3 = 0.23 0.23/0.3 = 0.77

 Project Situation: BUY Probab 20 days and late SD decision 0 days and late SD decision Total LATE SD decision 20 days and on-time SD decision [*] 0 days and on-time SD decision!*] Total ON-TIME SD decision Total SD decision 20 days given SD late = (20 days and SD late)/SD late 0 days given SD late = (0 days and SD late)/SD late Total given SD late 20 days given SD on time = 20 days 0 days given SD on time = 0 days Total given SD on time [*]These events are independent so the joint probabilities are the product of the probabilities.
 Project Situation: MAKE Probab 20 days and late SD decision 0 days and late SD decision Total LATE SD decision 20 days and on-time SD decision [*] 0 days and on-time SD decisional Total ON-TIME SD decision Total SD decision 20 days given SD late = (20 days and SD late)/SD late 0 days given SD late = (0 days and SD late)/SD late Total given SD late

20 days given SD on time = 20 days

Total given SD on time__

[*]These events are independent so the joint probabilities are the product of the probabilities.

Notice the impact on the potential for being late with a buy. The probability of a 20-day delay ha: conditions to 0.23 with dependent conditions. Correspondingly, the on-time prediction dropped fi probability of delay went from 0.6 to 0.67.

Let us now compute the dollar value of the outcomes of the decision tables. Table 4-7 provides Take care when looking at this table. The acquisition costs of the make, \$125,000, or of the buy on-time decision, SD, of the sponsor. Acquisition costs are only affected by the sponsor's decisic The value of the delay, if any, is taken care of with the value of the timeliness of the decision, SD S.

 Alternative ID Description Probability Sponsor Exercises Option Probability of Sponsor Decision, SD, Late Probability of 20-Day Schedule Delay Face Value of Delay, D, @ \$10,000 per Day Va A MAKE 0.3 Late 0.3 Late \$0 No \$4 \$0 0.75 Yes A MAKE 0.7 On time 0.7 On time \$0 No \$8 \$0 A MAKE 0.25 No 0.25 No \$0 No \$0 B BUY 0.3 Late 0.3 Late \$0 No \$1 \$0 0.75 Yes B BUY 0.7 On time 0.7 On time \$0 No \$2 \$0 B BUY 0.25 No 0.25 No \$0 No \$0

Note further that the decision to make or buy is not changed by the effect of a late decision of th advantageous in the face of a dependent condition with the sponsor's decision. The upside and ■ are:

Upside of an on-time sponsor's decision is the "Buy" acquisition cost = \$200,000 Downside of late sponsor's decision = \$181,350 - (\$200,000 + \$200,000) = -\$218,650

Team LiB