Cycle or Chase
Chasing the shortage feels like the responsible thing to do. But every time we tear up the schedule, we start a fire somewhere else - & next week we are putting that one out instead. That is how a plant ends up chasing its own tail for a month over a problem that lasted a week.
This runs one bottling line two ways. The first keeps to a fixed cycle, like a train timetable: when a run is missed, everyone still knows when the next one goes. The second chases whatever is closest to running out. Both get the same orders, the same hours & the same shortage.
The line makes 100 SKU - stock-keeping unitOne specific sellable item. A 750 ml bottle & a 1 litre bottle of the same liquid count as two: stocked, counted & ordered separately.SKUs on a classic Pareto profile, or power lawA small share of items carries most of the total - usually shorthanded as 80/20. It governs city sizes, book sales & very nearly every product range. The consequence: we cannot treat all SKUs alike, because they are nowhere near the same size.Pareto profile - 20 A, B & C items - ABC classificationSorting the range into bands by volume so each band is planned differently. A items are the runners, C the slow tail. The standard answer to a Pareto profile: make the A items often, the C items rarely.A items carrying 70% of volume, 30 B items carrying 20%, a tail of 50 C items sharing the last 10%. On the wheel, A items run every two weeks, B every four & C every eight, so the whole range is covered in eight weeks. Neither plant escapes the shortage. What differs is how long each one spends firefighting afterwards.
Before we start
Five grounding ideas to keep in mind.
One line, a hundred products
The line has 132 working hours a week & 100 SKU - stock-keeping unitOne specific sellable item. A 750 ml bottle & a 1 litre of the same liquid count as two: stocked, counted & ordered separately.SKUs to get through, & about 100 of those hours go on making what customers buy. That leaves a thin margin for everything else, which is why the planning rule matters.
Switching costs real time
Every switch from one product to the next means clear down, clean, reset. That is a Changeover, or setupClear down, clean, swap tooling, reset, first-off check. Nothing sellable comes off the line while it happens.changeover: 45 minutes, & nothing sellable comes off the line while it runs. Twenty a week is fifteen hours gone.
A, B & C items
Twenty products carry 70% of sales, thirty more carry 20%, & the remaining fifty share what is left. Not everything can be equally important - that is the Pareto profile, or power lawA small share of items carries most of the total - usually shorthanded as 80/20. It governs city sizes, book sales & very nearly every product range. The consequence: we cannot treat all SKUs alike, because they are nowhere near the same size.Pareto pattern, & it is why we make the A, B & C items - ABC classificationSorting the range into bands by volume so each band is planned differently. A items are the runners, C the slow tail. The standard answer to a Pareto profile: make the A items often, the C items rarely.A items often & the C items rarely.
Two ways to decide what to make
One plant holds a fixed sequence, whatever else is happening that week. The other rebuilds its plan around whatever is closest to running out. The second sounds more sensible, which is exactly what makes it a trap.
Where the difference shows
Both plants take the same hit, & no rule prevents it. The gap opens in the weeks after the shortage ends. That is usually where the orange line bottoms out, once all the pushed-back volume lands at once.
Press Supplier shortage, then Play weeks. Underlined words carry definitions; there is a glossary below, & an appendix with the mathematics.
Why a timetable still helps when the train is cancelled
Rows 1 to 5 are a railway with a timetable. Rows 6 on are the same railway without one - which is how most plants run a week.
Departure boards
Both boards show the same production week. One plant is working a sequence that was published cycles ahead; the other is working a hot list drawn up that morning, on which nothing was ever scheduled.
Break something
Both plants get identical orders, hours & disruption. The only difference is how they decide what to make next.
Both plants fall over. Only one gets back up
Case fill rateOf all the cases customers asked for, the share shipped on time. It counts volume, so it is dominated by the runners & can look healthy while the tail is failing.Case fill by week, meaning the share of orders we handed over on time. The shaded band is the disruption itself. The orange line does most of its damage after that band ends.
Where the stock actually sits, week 0
Each tile is one SKU, ordered by volume so the A items fill the top rows & the C tail the bottom. Colour shows CoverHow long the stock on hand would last at normal sales. Two weeks of cover means running out in a fortnight if nothing more is made.cover, which is how long that stock would last.
Reading the result
What just happened
We cannot stop the shortage. We can only decide whether it lands on a plant with a timetable, or a plant already chasing its tail.
Scope of the model
Glossary
Every term used on this page.
Appendix - the arithmetic underneath
This is the arithmetic behind the model. Four things drive everything on the page: how the hours have to add up, how short a cycle we can hold, how often each SKU should run, & what happens to stock when the gap between runs keeps moving.
Each section gives the result in words, then in symbols, then what it means on the line. Every number below comes from the settings you have loaded, so the maths & the simulation can be compared directly.
§1 Where the orders come from
Weekly demand per SKU is drawn from a normal distribution truncated at zero, with the Coefficient of variationVariability measured against the average. A CV of 0.6 means demand swings by roughly 60% of its own average, week to week - typical of slow movers.coefficient of variation rising as volume falls. C items also have an 18% chance of no demand in a week, which reproduces a real long tail.
dit = max(0, μi(1 + cvi·Zit)), Zit ~ N(0,1) iid(1)Volume is concentrated twice: across the bands by the 70/20/10 split, & within each band by lognormal weights. Strictly that is Pareto-like rather than a true power law, since the band shares are imposed. Items are then sorted into bands by actual volume, the way a real ABC review does it, & opening stock is staggered by position in the cycle so the run starts settled rather than all-full. Both rules start from the same position & see the same orders.
Each SKU has its own steady average, & the smaller the SKU the more erratically it actually arrives. The tail is where variability is highest, & where a reactive rule does most of its damage.
§2 Where the hours go
The hours consumed must fit inside the hours available. This is the only hard constraint.
∑i∈St ( si + Qit/r ) ≤ Ht(2)On a wheel, membership of St is
fixed by the schedule rather than by circumstances, so setup load is a constant set by the
frequency ladder:
Λwheel = ∑c Nc·s / Tc(3)Which splits the week into three parts:
Only the first line makes anything. The second is what variety costs us, & the third is what is left over to absorb surprises. A wheel fixes the middle line in advance, so we know how much spare time we have. Reactive lets events decide it, & events push it up at the worst possible moment.
§3 The shortest cycle we can actually keep
If all N items ran on one common interval T, the line must absorb N setups plus the production itself over each revolution:
N·s + (D/r)·T ≤ H·T ⇒ T ≥ N·s / (H − D/r) = Tmin(4)With ρrun = D/(rH) for the share of hours spent producing:
Tmin = N·s / ( H·(1 − ρrun) )(5)The busier the line, the longer the shortest schedule we can actually keep - & it stretches away fast, not gently. A full line does not just make us late; it takes away our ability to hold a short cycle at all. So “run everything more often” is not on the table, which is why the A/B/C ladder in §4 exists.
§4 Why A items run more often
The independent solution to the ELSP - economic lot scheduling problemThe classic problem of deciding how often to make each product on a shared machine, trading changeover cost against the cost of holding stock.economic lot scheduling problem trades setup cost against holding cost item by item, giving an interval inversely proportional to the square root of demand:
Ti* = √( 2s / (h·μi) ) ⇒ Ti ∝ μi−1/2 ⇒ Tc/TA = √(μA/μc)(6)Doubling the interval at each step down the ladder sits close to the cost-minimising allocation for a 70/20/10 split. So the wheel can be defended on cost, not only on discipline. Running the tail as often as the runners is not cautious but expensive: every changeover comes out of the hours the volume needs.
§5 The stock we build for ourselves
Stock built to cover an interval is drawn down over it, so average Cycle stockThe stock we hold purely because we make things in batches rather than continuously. Shop fortnightly & you need a fortnight of food in the house.cycle stock is half the run quantity:
Icycle = ∑i μiTi/2 = (D/2)·∑c wcTc cover = Icycle/D + SS(7)This is the weekly shop. Shop fortnightly & you keep a fortnight of food in the house; shop weekly & you keep half as much but make twice the trips. The wheel’s higher stock is that arithmetic, & it is the honest price of the schedule.
Cycle stock covers how often we choose to run. Safety stock covers what we cannot control. Plants that hold plenty of stock & still miss service have usually built the first & skimped on the second. A cycle does not carry more stock so much as make the amount predictable, which is what lets us size the safety stock instead of guessing at it.
§6 Why knowing the next slot is worth so much
With a known interval R & lead time L, safety stock covers the protection interval in the standard way:
SS = z·σd·√(R + L)(8)If the interval is itself random, the variance picks up a second component:
SS = z·√( (R̄+L)·σd2 + μd2·σR2 )(9)This is trusting a timetable, written as a formula. On a wheel σR is near zero, because the next slot is published: the second term disappears & safety stock only has to cover demand. Under re-planning, when a SKU runs next depends on every other SKU’s shortfall, so that term is large & nobody controls it. No amount of discipline buys it back, because we cannot size a buffer against something we cannot measure.
So what a cycle buys is not a short gap between runs but a gap that stays the same. Trains are still late sometimes. A timetable never promised otherwise: it promised you would know when the next one goes, & a wheel promises the same about the next production slot.
§7 Small cause, big mess
Utilisation includes setup load, so a rule that sets up more often runs a busier plant on the same equipment:
ρ = ( D/r + Λ ) / H(10)Kingman’s approximation then indicates how delay responds, with ca & cs the coefficients of variation of arrivals & service:
W ≈ ( (ca2 + cs2) / 2 ) · ( ρ / (1 − ρ) ) · τ(11)Reactive pushes both terms the wrong way at once. ρ goes up, because changeovers eat hours. Then cs2 goes up as well, because setups now come in two sizes: short when planned, long when rushed. With ρ/(1−ρ) in there, the damage grows far faster than the cause. Between about 140 hours & 125, the settings move smoothly & the result does not.
§8 Falling behind makes us slower
Backlog is the gap between demand & throughput, & throughput is whatever the hours left after setup will produce:
Bt+1 = Bt + Dt − Yt, Yt = r·(Ht − Λt)(12)Under re-planning the number of expedited setups et rises with the backlog, each carrying k, so throughput falls as arrears grow:
∂Yt/∂Bt = −r·s·(k − 1)·∂et/∂Bt < 0(13)On a wheel, setup load is set by the schedule & does not respond to the backlog at all:
∂Λt/∂Bt = 0 ⇒ ∂Yt/∂Bt = 0(14)Under re-planning, falling behind makes us slower at catching up, which makes us fall further behind. Once that loop is strong enough the backlog simply runs away, & the sawtooth in the service chart is what it looks like from outside. It is the same mechanism as the bullwhip: the upset takes far longer to settle than it lasted.
On a wheel, a bad week makes as much as a good one, so the backlog clears at a steady rate. A late train does not slow down the train behind it. A rush job does.
§9 Two ways of counting, two different answers
Case fill rateOf all the cases customers asked for, the share shipped on time. It counts volume, so it is dominated by the runners.Case fill weights each SKU by volume. Line fill, or OTIFOf all the order lines placed, the share delivered complete. Every SKU counts equally. A customer ordering ten items & receiving nine has had a line fail, even though 90% of the units arrived.Line fill weights every SKU equally, which is why the two can disagree sharply:
CFRt = ∑i min(xit, dit) / ∑i dit LFRt = |{ i : xit ≥ dit }| / |{ i : dit > 0 }|(15)The tail is half of every line we count, but only a tenth of every case. So we can report a healthy case fill while a customer who orders across the range finds half their lines short. When reactive drops C items, case fill barely notices & OTIF falls off. Judged on cases alone, the gap between the two rules is understated every time.
§10 What the model leaves out
Some of these work in the wheel’s favour & some against it, so they are listed together.
- One line, no parallel resources, so no load balancing.
- Setup time independent of sequence. Real wheels are sequenced to minimise changeover by grade, colour or allergen, so a genuine wheel would gain more than this one does. Understates the wheel.
- A flat one-week lead time, with no supply wobble outside the shortage itself.
- All unmet demand backordered, none lost, no substitution. Lost sales would penalise both, reactive more.
- Demand independent across SKUs & weeks: no seasonality, promotions or forecast bias, so no externally generated bullwhip.
- Capacity is hard: no overtime, extra shifts or subcontract. In practice a firefighting plant buys premium hours, which moves the cost onto the P&L rather than removing it.
- Perfect inventory accuracy, no yield loss outside the quality hold.
- The wheel is assumed to be adhered to. Sustaining that against commercial pressure is the genuinely hard part, & this model assumes it away. Flatters the wheel.
Two parameters carry most of the weight: the minimum economic run length, & the unplanned setup factor k. Both are measurable in a real plant, so set them from your own changeover studies before using this to make a case.