Parallel prerequisites can finish later than their average durations suggest

A launch that requires two approvals cannot proceed when the average approval is finished. It proceeds when both are finished. This distinction matters when a cash forecast replaces each uncertain task duration with its average and then treats the resulting schedule as an expected completion date.

The Government Accountability Office discusses this issue as merge bias: parallel paths meet at a milestone, and the latest required path controls its timing. Uncertainty on several paths can therefore change the milestone's timing even when each individual duration estimate looks reasonable.

Begin by establishing the actual dependency. Are both approvals mandatory? Can delivery begin in stages? Does one task start only after another finishes? A mathematically careful calculation applied to the wrong dependency structure can still put the receipt in the wrong week.

Suppose an invented service launch requires two independent reviews, A and B, that start together. Each review takes one week with 50% probability or three weeks with 50% probability. The launch occurs as soon as both reviews finish, and there are no other prerequisites.

Each review's expected duration is two weeks: half of one week plus half of three weeks. A schedule that inserts those two expected durations shows both reviews completing in week two and places the launch there.

But the actual modeled outcomes are different. Both reviews finish in week one in 25% of cases. In the other 75%, at least one takes three weeks. The launch therefore occurs in week one or week three, never in week two. Its expected timing is 0.25 times one plus 0.75 times three, or 2.5 weeks.

The 2.5-week result is the probability-weighted average of the launch dates. It is not a date on which the launch can occur in this two-outcome example. Week three is the most likely completion week, while only 25% of launches occur by the end of week two.

A decision requiring 80% confidence of completion by a deadline would therefore need week three under these assumptions. Calling week two the expected date because it contains both mean task durations would overstate the probability of meeting that commitment.

Report the measure the decision needs. Staffing may depend on the distribution of operating weeks, a customer promise may depend on the chance of meeting a date, and a cost estimate may use an expected duration. Those are related outputs, but none is a substitute for the others.

Assume the hypothetical contract pays $100,000 immediately at launch. The forecast distribution has a 25% chance of receiving the entire amount in week one and a 75% chance of receiving it in week three. Expected receipts are $25,000 in week one and $75,000 in week three.

Those expected weekly amounts do not describe partial customer payments. If payroll requires cash in week two, the business has either received $100,000 or received nothing in this model. Test the cash balance under the complete scenarios rather than assuming the $25,000 expected receipt is available.

Now suppose holding the launch team costs $15,000 for every week until launch, with no minimum commitment or partial-week complication. Expected holding cost is $15,000 times 2.5, or $37,500. Using the two-week average-duration schedule would show $30,000 and miss $7,500 of expected cost.

The four equally likely cases require independence. Separate rows in a schedule do not establish it. Both reviews might depend on the same missing customer records, the same specialist or the same access window. Those shared conditions can change their joint outcomes.

For a comparison, suppose the two reviews always finish together: both take one week half the time and both take three weeks half the time. Each individual review still has a two-week expected duration, but the launch's expected timing is now two weeks rather than 2.5.

That example does not make shared risks desirable. It shows why the milestone calculation needs the joint distribution, not just two separate averages. Another dependency structure could make long delays more severe or affect additional tasks outside these simplified paths. Model the cause and the relevant outcome together.

Suppose an optional service guarantees review A finishes in week one while review B retains its original one-or-three-week distribution. Launch then occurs in week one half the time and week three half the time, giving an expected launch at two weeks.

The expected improvement is half a week, not the full one-week reduction in A's individual mean. In scenarios where B takes three weeks, finishing A earlier does not move the launch. Under the assumed $15,000 weekly holding cost, the modeled expected saving is $7,500 before the acceleration fee.

Use that result only within the stated example. A penalty, a lost customer or a payment discount could make specific dates disproportionately valuable. Calculate those contractual effects in each scenario instead of multiplying every benefit by the average time saved.

Two activities cannot be treated as fully parallel if both need the same person at the same time and that person cannot serve them simultaneously. Add the resource constraint or sequence the work before assessing duration uncertainty. Otherwise the base schedule contains capacity that does not exist.

Match the duration units to the payment calendar as well. Five working days are not always one calendar week, and an approval received after an invoice cutoff can move collection by more than the physical delay. Model the actual handoff and billing rule where they affect liquidity.

Ask each owner for the conditions behind the short and long outcomes, then check which conditions are shared. For a small number of paths, a table of complete scenarios can make the issue visible without a complex simulation.

Keep the milestone rule, joint probabilities and cash triggers with the forecast. When a review finishes, replace its uncertainty with the observed result and update the remaining paths. The forecast becomes more useful as it narrows around actual progress, while preserving the difference between an average duration and a date the business can safely promise.