No usable demand series existed, so one was reconstructed: twenty months of shift history, parsed out of the plant's weekly operating workbooks by a purpose-built reader. Shortage per shift was defined as the sum of four absence categories. Whether to include long-term absence was not assumed either way; both definitions were tested against payroll, and only the inclusive one reconciled.
The cost structure that emerged is the insight. Premium-rate coverage outweighs direct crew wages by roughly three to one. The baseline is not a staffing cost with some overtime attached, it is an overtime cost with some staffing attached.
The reframing: absence is modelled as exogenous demand, but it is partly a response to the coverage system itself. A missed shift reliably creates a premium-rate opening, which gives colleagues a standing financial interest in those openings existing. Sizing a crew against that demand risks sizing it against a problem the incentive structure helps produce.
Four structural options were priced against the same twenty months rather than argued in the abstract: status quo at full premium coverage, a permanent crew sized to peak, a permanent crew plus seasonal contract, and single-wave against phased hiring. Sizing to peak was rejected on the summer profile, since it leaves the crew idle most of the year and idle time recovers only a fraction of its wage. The split structure was retained because the demand series is bimodal.
The method rejection that mattered. The textbook newsvendor shortcut computes a critical ratio and reads the answer off the demand distribution. It was built first and discarded, because it assumes a single overstock-to-understock cost ratio and the real structure has several: one crew spans weekdays carrying different premium multipliers, so no single ratio exists. The replacement is direct minimisation across candidate crew sizes, defensible only because the cost function is piecewise-linear with monotone slopes and therefore convex, so a single scan provably returns the global optimum. The shortcut was not wrong so much as inapplicable.
An accounting option that moved the answer. Benefits could be loaded as one flat percentage or split. Splitting was chosen after testing both: the fixed tranche is a charge on base salary and does not move when someone works additional hours, while only the variable tranche rides every dollar paid. That makes a premium hour cost materially less than a flat load implies, shifting the hire-versus-cover trade in a direction a flat assumption would have hidden.
Cost for a crew carrying n relief against D missing:
Cost(n) = n × wage − salvage × MAX(0, n−D) + premium_rate × MAX(0, D−n)
Try it: where does the crew size land?
Illustrative model. Invented numbers, real structure.
With two premium rates the critical ratio no longer has a single ratio to use. It reads n = 8 where the true optimum is n = 10, so acting on it leaves money on the table every shift. That is why the model scans instead. The curve is flat around the optimum, which is why being one position out costs almost nothing while being five out costs a great deal.
The salvage term stops an idle employee being modelled as pure loss: idle relief perform non-conforming sorting at a measured rate, displacing contracted labour billed per unit. Carrying it explicitly shifts the optimum upward, so omitting it would have systematically under-hired.
Three modelling decisions, each with a reason. One-time costs stay out of the solver, because training and attrition move the payback date rather than the steady-state optimum; only the seasonal wage discount enters the decision, since a seasonal never reaches full rate. The permanent wage ramp applies to forecasts only, because after two years everyone is at full rate and letting the solver see a temporary discount would over-hire against a rate that expires. Training is modelled as paid-versus-useful rather than as a lump, so every recruit is paid while contributing no coverage and residual premium spend during ramp-up is captured; no separate departure cost is added, which would double-count wages already charged.
Validation. With the correct crew count the model's implied premium-shift count matched what payroll had actually paid, against an independent export rebuilt from a malformed single-column file. The policy was then priced across a held-out window excluded from fitting. Removing the three most extreme days in twenty months moves the recommendation by a single position, and an apparently sensible outlier trim was tested and shown to cost money, because at that threshold it removed ordinary busy days rather than anomalies.
Eight defects were caught before publication, three of which moved the headline materially: a silent parser fault returning zero vacation absence across hundreds of sheets, which understated mean shortage by nearly half with no error raised; two mixed accounting conventions producing a saving inflated by roughly half against the corrected figure; and a premium rate understated because two pay lines were read as alternatives rather than summed.
Approved for implementation in September 2026, and the highest-value project presented at the Q3 productivity summit. Year one delivers a reduction against baseline while still absorbing the training cost, and steady state is materially better again once the ramp is complete. The training investment pays back inside the first year, and premium-rate spend falls sharply.
The saving is concentrated rather than even: most of the first year's benefit lands in the summer quarter, and the opening quarter is nearly flat because it carries the first training wave. Both are surfaced rather than smoothed into an annual average, because a plan sold as a large annual number that loses money in its opening quarter will have lost the room by week four.
The second wave is a balancing order, not a commitment. Two live inputs size it: actual survivors from the first wave, and a re-solve against data generated after that wave is working. That matters because of the baseline insight. Between the waves the plant produces several months of absence data under materially reduced premium availability, and if absence falls the optimum falls with it. Three outcomes are defined in advance, including the one where the model turns out to have over-recommended.