# Decision Tables

## Decision Tables

Let's see how the example just discussed would look in a decision table. Tables are alternatives to diagrams and often work better than diagrams because they are much easier to work with if the number of summing nodes is more than a couple. Table 4-2 shows the decision we have just worked on in tabular format. Tables, like their diagram counterpart, handle discrete inputs very well, and because tables can easily be computed on spreadsheets or from databases, tables are excellent tools for decision analysis. Maintenance of information is much easier on a spreadsheet or in a database compared to a graphical depiction. We are all familiar with the calculation power of spreadsheets: a single number entry can be recalculated automatically throughout the entire spreadsheet. It is harder to do that with a graphic, which requires less common tools than spreadsheets if a graphic of even small complexity is to be maintained.

Table 4-2: Project Decision Example

Alternative ID

Description

Probability of Schedule Delay

Face Value of Delay, D, @ \$10,000 per Day

Expected Value of Delay

Acquisition Cost

Expected Value of Alternative

A

MAKE

0.6 Yes

0.4 No

\$200,000 Yes

\$0 No

\$120,000

\$125,000

\$245,000

B

0.2 Yes

0.8 No

\$200,000 Yes

\$0 No

\$40,000

\$200,000

\$240,000

For the remaining examples in this book, we will rely more on decision tables. We will use decision trees only when the tree form more easily conveys the concepts we are discussing.

## Decisions with Conditions

It would be nice if, but it is rare that, project managers can make decisions without consideration for other activities or constraints going on in the project. Other activities that bear on the decision complicate matters somewhat on the decision tree. Such activities "condition" the decision. Generally, conditions are of two types: independent conditions that establish prerequisites but do not in and of themselves affect performance thereafter, and dependent conditions that do affect performance. An example of an independent condition is that of a sponsor deciding whether or not to exercise a scope option, and then a make-buy decision by the project team that is conditioned only on the sponsor's decision to buy or not.

On the other hand, dependent conditions affect performance or value. That is to say, the performance in a project work package is conditioned on the first decision made. Using the foregoing example, if the sponsor's decision in some way affected the expected value of the maker or provider decision, then the project decision is dependent on the sponsor's decision.

### Decisions with Independent Conditions

Let us first discuss the project situation of a decision to be made conditioned on the prior or prerequisite activity of some other work package or some other external event. However, let us say that the alternatives we are deciding between per se are not affected by the prerequisite; just our ability to make decisions about the alternatives is affected.

For illustrating decision making in the context of independent conditions, we will continue with the decision scenario we have been developing in this chapter. The matter before the project team for decision is whether or not to make or buy an item on the WBS. Now, let us impose a condition that is independent of the performance of the in-house manufacturing or of the performance of the vendor if selected:

• The make or buy decision is conditioned on whether or not the project sponsor exercises an option to have the item included in the project deliverables. That is to say, the WBS item in question is optional with the sponsor. It is up to the sponsor to say whether or not it is to be delivered.

• The sponsor's decision is a random variable, S, with values 1 or 0. S = 1 means the item will be included in the project deliverables; S = 0 means it will not be included. Once this prerequisite is satisfied, then the project team can make the decision about make or buy.

• The performance, D, of the subsequent make or buy is independent of the sponsor's decision unless the sponsor decides not to exercise the option for the item. In that case, there would be no subsequent make or buy.

• Our account manager dealing with the sponsor estimates that there is a 75% chance the sponsor will decide in favor of exercising the option for the item. Probability of S = 1 is 0.75. In the "1-p" space there is a 25% chance the sponsor will not exercise the option for the item: probability of S = 0 is 0.25.

Under these new circumstances, what is the decision of the project team, what is the expected value of the item, and what are the downside considerations? We proceed as follows: We must add columns to the decision table to take into account the preconditioning of the sponsor's decision, S, about adding the item to the project deliverables. We then recalculate the outcomes taking special care to account for all events in the "1-p" spaces. Table 4-3 illustrates the results.

Table 4-3: Decision with Independent Conditions

Alternative ID

Description

Probability of 20-Day Schedule Overrun

Face Value of Delay, D, @ \$10,000 per Day

EV of Delay, D (p of Option * p of Overrun * \$Face Value * Sponsor's Decision Value)

EV of Acquisition Cost (p of Option * \$Face Value * D)

Expected Value of Alternative

A

MAKE

0.75 Yes

0.75 Yes

1

1

0.6 Yes

0.4 No

\$200,000 Yes

\$0 No

\$90,000 = 0.75 * 0.6 * \$200,000

\$93,750

\$183,750

A

MAKE

0.25 No

0.25 No

0

0

0.6 Yes

0.4 No

\$200,000 Yes

\$0 No

\$0

\$0

\$0

B

0.75 Yes

0.75 Yes

1

1

0.2 Yes

0.8 No

\$200,000 Yes

\$0 No

\$30,000 = 0.75 * 0.2 * \$200,000

\$150,000

\$180,000

B

0.25 No

0.25 No

0

0

0.2 Yes

0.8 No

\$200,000 Yes

\$0 No

\$0

\$0

\$0

Following the mathematics through the table row by row, you can see that the probability of the decision by the sponsor weights the probability of a subsequent delay and weights the acquisition cost. By this we mean that there could only be performance if the sponsor decides favorably to go forward and include the item in the project deliverables. Overall, the probability of delay is the probability that the sponsor makes the decision favorably times the probability of delay given that the decision is favorable. Certainly if the sponsor decides unfavorably, S = 0, so that there is to be no item in the project deliverables, and there is no chance for a delay.

Summing the buy and the make from Table 4-3:

 Expected value (buy) = \$0 + \$180,000 = \$180,000 Expected value (make) = \$0 + \$183,750 = \$183,750

Under the conditions of the scenario in Tables 4-2 and 4-3, we see that the decision is not changed: it is still "buy." We further see that the expected value of the final decision, \$180,000, has a higher unbudgeted downside risk compared to the decision tree without conditions:

Downside risk (make) \$183,750 - (\$125,000 acquisition + \$200,000 delay) -\$141,250

• Upside (make) = \$125,000, the acquisition cost without delay

Downside risk (buy) \$180,000 - (\$200,000 acquisition + \$200,000 delay) -\$220,000

• Upside (buy) = \$200,000, the acquisition cost without delay

The decision tree or table provides the project manager with the expected value of the decision-making process. As we know from previous discussion, expected value is the best single-number representation of a range of uncertainty. Any single instance of the project could fall anywhere in the range. Understanding the range is the purpose of the upside and downside analysis.

Furthermore, the acquisition cost of either alternative (\$125,000 for "make" or \$200,000 for "buy") has been transformed from deterministic to random by the dependency acquired from the effect of the sponsor's decision. In other words, the sponsor's decision to acquire the item is the random variable, S, with discrete density function S0 = 0, p = 0.25 and S1 = 1, p = 0.75 becomes the density of the acquisition cost, AC:

 AC0 = 0, p = 0.25, do not acquire the item AC1 = 1 * make or buy cost, p = 0.75, acquire the item

Therefore, for decision-making purposes the decision maker would look first to the expected values, weighing first the most advantageous expected value. Then the decision maker would look to the risks and opportunities, downside and upside, and weigh those values in terms of possible effects on the business. The decision policy elements for both risk consideration and expected value are considered jointly in the decision process.

Figure 4-8 shows the "make" part of the scenario we have been discussing. It is evident that decision tree charts grow unwieldy in the face of conditions. Thus, the project team should understand and use tables to simplify matters, especially since tables lend themselves to setup, computation, and maintenance in spreadsheets.

Figure 4-8: Decision Tree with Independent Conditions.

### Bayes' Theorem

Another degree of complication is introduced when the conditions of performance leading to a decision are interdependent. In this scenario, the probability structure becomes more difficult to manage without a good understanding of dependent probabilities. Thus, we introduce "Bayes' Theorem," named after the English mathematician Thomas Bayes who published his theory, Essay Towards Solving a Problem in the Doctrine of Chances, in 1764. Although Bayes' Theorem can be presented in a couple of forms, it is conveniently shown as follows for project management purposes:

p(A | B) = p(A and B)/p(B)

in which p(A | B) is read as "probability of A given B." Rearranging terms, Bayes' Theorem is also in the following two forms: p(A and B) = p(A | B) * p(B) and p(B) = p(A and B)/p(A | B).

If A and B are independent, then:

p(A | B) = p(A) * p(B)/p(B) = p(A)

because

p(A and B) = p(A) * p(B)

Let's try Bayes' Theorem in natural language with the project examples we have studied so far:

• "The probability of a 20-day delay, D, given a "make" decision, M = 1, is 0.6." In equation form: p(D | M1) = 0.6. If we were interested in solving for p(M1), we would have to estimate p(D and M1).

• "The probability of a 20-day delay, D, given a sponsor's decision, S, to include the item in the project deliverables" is p(D) since D and S are independent.

• "The probability of a 'make' decision, M, given a sponsor's decision, S = 1, to include the item in the project deliverables" is 1.33 * p(M1 and S1). We have no other information about p(M1 | S1) unless we independently measure or estimate p(M1 and S1).

### Decision Trees with Dependent Conditions

With Bayes' Theorem in hand, we can proceed to project decisions that are interdependent. Let's continue with our project for which there is an item on the WBS that may or may not be included by the sponsor's decision in the final project scope and for which there is a make or buy decision for satisfying the acquisition to be made by the project team. However, let's change the situation and recognize that a late 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 delayed beyond a point that the delay affects subsequent make or buy performance or value. In this example, we will say that our confidence in an on-time sponsor's 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 other words, we need to solve for:

• 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 carefully, you will see that there are actually six probabilities since "late or on time" is a shorthand notation for two distinctly different probabilities.

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)

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 "make late (or on time) and late SD" and "buy late (or on time) and late SD." We have already said that we have pretty high confidence that SD will be on time, so looking at a joint probability involving "SD late" will be a pretty small space. We do know one thing that is very useful: all the joint probabilities involving "SD 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. "Make on time" and "Make late" have probabilities of 0.4 and 0.6, respectively. If these were independent of "SD late," then the joint probabilities would multiply out to the multiples of the probabilities:

Make on time and SD late = 0.4 * 0.3 = 0.12

and

Make late and SD late = 0.6 * 0.3 = 0.18

However, in our situation "SD late" conditions performance, so the probabilities are not independent. Intuitively, the joint probability of being on time should be more pessimistic (smaller) since the likelihood of the joint event is more pessimistic than each event acting independently. In that case, the joint probability of being late is more optimistic (more likely to happen):

p(Make on time and SD late) p(Make on time) * p(SD late),

Estimate: p(Make on time and SD late) = 0.1

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 were independent gives:

p(Buy late and SD late) = 0.2 * 0.3 = 0.06

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 late),

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 resolve. The probability p(Make or Buy performance given SD late) is given in Table 4-5 and Table 4-6:

 p(Make performance given SD late) = p(Make performance and SD late)/p(SD late) p(Make 20 days late given SD late) = 0.2/0.3 = 0.67 p(Make 0 days late given SD late) = 0.1/0.3 = 0.33, and p(Buy performance given SD late) = p(Buy performance and SD late)/p(SD late), p(Buy 20 days late given SD late) = 0.07/0.3 = 0.23 p(Buy 0 days late given SD late) = 0.23/0.3 = 0.77

Table 4-5: Buy Calculations with Dependent Conditions

Probability of Situation Occurring

20 days and late SD decision

0.07

0 days and late SD decision

0.23

Total LATE SD decision

0.3

20 days and on-time SD decision [*]

0.14

0 days and on-time SD decision[*]

0.56

Total ON-TIME SD decision

0.7

Total SD decision

1.0

20 days given SD late = (20 days and SD late)/SD late

0.23

0 days given SD late = (0 days and SD late)/SD late

0.77

Total given SD late

1.0

20 days given SD on time = 20 days

0.2

0 days given SD on time = 0 days

0.8

Total given SD on time

1.0

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

Table 4-6: Make Calculations with Dependent Conditions

Project Situation: MAKE

Probability of Situation Occurring

20 days and late SD decision

0.2

0 days and late SD decision

0.1

Total LATE SD decision

0.3

20 days and on-time SD decision [*]

0.42

0 days and on-time SD decision[*]

0.28

Total ON-TIME SD decision

0.7

Total SD decision

1.0

20 days given SD late = (20 days and SD late)/SD late

0.67

0 days given SD late = (0 days and SD late)/SD late

0.33

Total given SD late

1.0

20 days given SD on time = 20 days

0.6

0 days given SD on time = 0 days

0.4

Total given SD on time

1.0

[*]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 has increased from 0.2 with no dependent conditions to 0.23 with dependent conditions. Correspondingly, the on-time prediction dropped from 0.8 to 0.77. For a make decision, the 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 the illustration of this project scenario. Take care when looking at this table. The acquisition costs of the make, \$125,000, or of the buy, \$200,000, are not affected by the late or on-time decision, SD, of the sponsor. Acquisition costs are only affected by the sponsor's decision, S, to have the item in the WBS or not. The value of the delay, if any, is taken care of with the value of the timeliness of the decision, SD, times the probability of the decision itself, S.

Table 4-7: Decision with Dependent Conditions

Alternative ID

Description

Probability of Sponsor Decision, SD, Late

Probability of 20-Day Schedule Delay

Face Value of Delay, D, @ \$10,000 per Day

EV of Delay, D (p of SD * p of Delay * \$Face Value)

EV of Acquisition Cost (p of Option * \$Face Value * D)

Expected Value of Alternative

A

MAKE

0.3 Late

0.3 Late

0.67 Yes

0.33 No

\$200,000 Yes

\$0 No

\$40,200 \$0

= 0.75 * (\$40,200

0.75 Yes

= \$125,000 * 0.75

+ \$84,000) + \$93,750

A

MAKE

0.7 On time

0.7 On time

0.6 Yes

0.3 No

\$200,000 Yes

\$0 No

\$84,000 \$0

= \$93,750

= \$186,900

A

MAKE

0.25 No

0.25 No

0.6 Yes

0.4 No

\$200,000 Yes

\$0 No

\$0

\$0

\$0

B

0.3 Late

0.3 Late

0.23 Yes

0.77 No

\$200,000 Yes

\$0 No

\$13,800

\$0

= 0.75 * (\$13,800

0.75 Yes

= \$200,000 * 0.75

+ \$28,000) + \$150,000

B

0.7 On time

0.7 On time

0.2 Yes

0.8 No

\$200,000 Yes

\$0 No

\$28,000 \$0

= \$150,000

= \$181,350

B

0.25 No

0.25 No

0.2 Yes

0.8 No

\$200,000 Yes

\$0 No

\$0

\$0

\$0

Note further that the decision to make or buy is not changed by the effect of a late decision of the sponsor. "Buy" comes out more advantageous in the face of a dependent condition with the sponsor's decision. The upside and downside of a decision in favor of "Buy" 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