Industry: Manufacturing Product: ModelRisk Application: Optimizing assembly line performance under uncertainty
A consumer-electronics assembly plant signed off on its FY budget with a planned Overall Equipment Effectiveness of 92% — the number that fell out of multiplying availability (96%), performance (97%), and quality (99%) at their long-run averages. Six months in, realised OEE was running at 71%, and the plant manager could not explain to the executive committee where the missing 21 points had gone. The arithmetic of multiplying three averages had hidden the arithmetic of multiplying three distributions, and that is the difference between a number that looks plausible on a slide and a number that survives contact with a real production week.
The plant rebuilt its operations model in ModelRisk: a discrete-event Monte Carlo that simulates eight workstations in series, the inter-station buffers between them, the failures that take stations offline, and the defects that force rework loops. The deliverable was no longer a single planning OEE — it was a full distribution of weekly units produced, with explicit probabilities attached to budget, stretch and downside outcomes. The chart below is the result of 100,000 simulated weeks, and it tells the whole story in one shape.
Mean weekly output is 6,780 units, with realised OEE averaging 71% — exactly the gap the plant manager was seeing. The deterministic plan of 8,720 units (OEE 92%) sits at the 97th percentile of the simulated distribution: a number the line reaches in about three weeks out of a hundred, not as a baseline. The crucial insight is what the planning arithmetic destroyed — the spread. The naive way to model a week is to treat ten shifts as ten independent draws and add them up; do that and the shift-to-shift noise averages out (the central limit theorem at work), collapsing the weekly figure into a deceptively tight band. But real plant variation is dominated by week-level common cause — a chronic SMT fault, a marginal solder-paste lot, a thinly staffed week — that hits every shift in the week together and refuses to average away. Model that correctly and the distribution opens up: P10 = 5,337, P90 = 8,164.
The line runs at a nominal cycle time of 38 seconds with eight stations. Three sources of variability dominate, and each demands the right distribution:
Station uptime. Time between failures on each station was fitted with a Weibull distribution (shape β = 1.6, characteristic life η ≈ 480 hours for the most failure-prone SMT placer). The wear-out shape parameter matters: β > 1 is a station whose hazard grows with time-since-last-PM, and a flat exponential model — common in plant-engineering spreadsheets — systematically under-predicts the cluster of failures that drive a bad shift. Mean time to repair was fitted to a LogNormal with median 42 minutes and an upper tail to 4 hours when the maintenance crew is double-booked across the plant.
Defect counts per shift. The legacy model used Poisson with λ = 14 defects per shift. A goodness-of-fit on 18 months of operator log data rejected Poisson (variance/mean ratio = 3.1, not 1.0). The team replaced it with a Negative Binomial (mean 14, dispersion k = 7), which captures the over-dispersed reality of bad days clustering — a supplier batch with marginal solder paste produces a shift with 40+ defects, not 14.
Operator performance ratio. Bounded on (0, 1), modelled as Beta(α = 28, β = 5) with mean 0.85. The previous spreadsheet used Normal(0.85, 0.05), which silently drew performance values above 100% on roughly 1 % of iterations — physically impossible and an artefact that inflated planned throughput.
Running the deterministic version (point averages on every input) gives 8,720 units per week and OEE = 92%. Running the Monte Carlo with 100,000 simulated weeks tells a different story.
Mean weekly output is 6,780 units, P10 is 5,337, P50 is 6,826, and P90 is 8,164. The probability of hitting the budget commitment of 8,000 units/week is just 13%. The "missing" 21 points of OEE were never missing; they were the gap between point-arithmetic and the convolution of skewed, correlated input distributions — bounded availability and performance, over-dispersed defects, and the week-level common cause that makes a bad week bad across all ten shifts at once.
Sensitivity analysis ranks the drivers of P10 weekly output (the number that matters for service-level commitments to customers).
The SMT placer's Weibull η ranks first by a clear margin, followed by the supplier defect dispersion parameter. This reordered the maintenance capex priorities: a $180k retrofit of the SMT spindle was promoted above two other projects that had ranked higher on the deterministic ROI sheet but moved the throughput tail only weakly.
The original layout used a uniform 12-unit buffer between every pair of stations. The simulation reranked this. With the SMT placer identified as the dominant failure mode, the team tested moving WIP capacity to the downstream buffer immediately after that station — a classic Theory-of-Constraints move, but here justified probabilistically.
Reallocating 24 units of buffer capacity to the SMT-downstream slot (and removing 12 from each of two low-failure stations) lifts mean output from 6,780 to 7,188 units/week and lifts P10 from 5,337 to 5,989 — a 12 % improvement at the tail that protects the customer-commitment number, at zero capex.
The deterministic OEE of 92% was an arithmetical mirage — the product of three optimistic averages, multiplied as if independence and symmetry held. The Monte Carlo simulation in ModelRisk did not change a single physical input on the line; it just stopped lying about what those inputs actually produce in series.