Which Assumption Could Change Our Decision?

Botie’s candidate catches more faults than the old rule, but it requires more inspections. Our first calculation says it is worthwhile. That conclusion depends on how costly a missed fault is, and the team only has a rough estimate.

We can inspect that dependence directly. Keep the observed policy differences fixed, vary the uncertain cost, and find where the decision changes. Then we know which missing evidence deserves attention.

Change the cost per miss or the extra inspections. Positive savings favour the candidate under this cost model.

We already know how to calculate a policy’s loss. Subtract candidate loss from baseline loss to obtain savings. We can simplify that difference into avoided miss loss minus added inspection cost.

This arrangement shows exactly which assumption drives the recommendation. Keep the sign convention beside it: positive savings favour the candidate, negative savings favour the baseline.

Savings = avoided_misses × cost_per_miss
          − extra_inspections × cost_per_inspection
At miss cost 20: 2 × 20 − 5 × 4 = 20 units

The comparison covers these consequences only. Added upkeep would reduce savings further.

Use a low, central, and high estimate only if those labels mean something in the problem. You might have several documented operating scenarios instead. There is no requirement to use three.

Here we try miss costs of 5, 10, and 20 units because they show a change of sign. The list is an exploratory choice. It has no implied probability or confidence level.

A result that stays positive across a justified range is more robust to that input than one that changes sign after a small revision.

At break-even, savings are zero. If avoided misses is positive, divide the extra inspection cost by the avoided miss count to find the miss cost at that boundary.

If avoided misses is zero, that division is undefined. A candidate that adds inspection work while avoiding no misses cannot justify itself through this particular benefit. A different benefit would need to be stated.

0 = 2 × cost_per_miss − 5 × 4
Break-even cost_per_miss = (5 × 4) / 2 = 10 units

At exactly the boundary, these terms tie. A tie does not establish that the designs are equivalent in every other respect.

import numpy as np
miss_cost = np.array([4., 8., 12., 16.])
savings = 3*miss_cost - 6*5
print(savings.tolist())

[-18.0, -6.0, 6.0, 18.0]

For this linear comparison, one extra unit of miss cost changes savings by the number of avoided misses. That slope is a sensitivity. It tells us how strongly the output responds, but not how likely a large cost change is.

An input with a small sensitivity can still matter if its uncertainty range is large. Conversely, a sensitive input that is measured precisely may contribute little uncertainty. Examine both properties.

Do not call an arbitrary parameter sweep a statistical confidence interval. A probability statement needs an uncertainty model and supporting assumptions.

The number of faults, inspection duration, and server load may all increase during the same difficult operating conditions. Varying one input at a time can miss that combination. Describe scenarios that preserve plausible relationships.

For example, a busy shift might contain more pumps and longer inspections because access is harder. A design that passes isolated changes could fail their combination. On the other hand, combining every worst value may describe an impossible scenario.

Keep a reason for each scenario and identify which conditions the design promises to handle.

If the recommendation changes within a plausible miss-cost range, review the evidence for that cost before polishing a model score. If capacity fails under the busy-shift scenario, measure inspection duration there.

There may also be a reason to choose a design that is slightly worse under the central estimate but acceptable across more plausible conditions. Explain the trade to the person responsible for the decision.

Botie’s workshop leaves the uncertain inputs connected to the calculation. The conclusion can move when the assumptions move.