Cycle or Chase — fixed-cycle scheduling against reactive replanning

Cycle or Chase

Fixed-cycle scheduling against reactive replanning

A published timetable is useful even to the passenger who misses their train, because the next departure is already known. Fixed-cycle production scheduling rests on the same property. The model below runs one production line against 100 different SKU — stock-keeping unitOne specific sellable item. A 500 ml bottle & a 1 litre bottle of the same drink count as two, because they are stocked, counted & ordered separately.products, split the way nearly every real range is: a classic Pareto profile, or power lawThe familiar pattern in which a small share of items accounts for most of the total, often summarised as 80/20. It governs city sizes, book sales, word frequencies & very nearly every product range. The practical consequence here is that a plant cannot sensibly treat all its products the same way, because they are nowhere near the same size.Pareto profile, the same power-law concentration that turns up in city sizes, book sales & word frequencies. Twenty A, B & C items — ABC classificationSorting a range into bands by sales volume so that each band can be planned differently. A items are the fast movers, B the middle group, C the slow-moving tail. It is the standard operational answer to a Pareto profile: make the A items often & the C items rarely.A items carry 70% of the volume, thirty B items carry 20%, & a tail of fifty C items shares the remaining 10%. Under a Product wheelA fixed, repeating production sequence. The line works through the same order of products over & over, so every item has a known next slot. A bus timetable for a factory.product wheel the line follows a fixed repeating sequence, so every item has a known next slot: A items run every two weeks, B items every four & C items every eight, which is a two-week base cycle & an EPEI — every product every intervalHow long the line takes to work through its whole range & return to the start. An EPEI of eight weeks means every product is made at least once every eight weeks. A shorter EPEI means smaller batches & less stock, at the cost of more changeovers.EPEI of eight weeks. Under reactive replanning that sequence is torn up & rebuilt each week around whichever item is closest to running out. Both plants receive the same demand, the same hours & the same disruption. What separates them is not the depth of the service loss but the behaviour of the recovery.

A simulator by Alvin J Lin  ·  alvinlinjr@gmail.com  ·  open source under the MIT License

Start here if this is not your field

Five things make the whole model make sense, & none of them need any background in manufacturing.

One line, a hundred products

A single production line has to make 100 different SKU — stock-keeping unitOne specific sellable item. A 500 ml bottle & a 1 litre bottle of the same drink count as two, because they are stocked, counted & ordered separately.products. It has 132 working hours a week & needs about 100 of them just to make what customers buy. The margin is thin, which is the whole reason the planning rule matters.

Switching costs real time

Whenever the line stops making one product & starts another, it has to be cleared, cleaned & reset. That is a Changeover (setup)Everything needed to stop making one product & start the next: clear down, clean, swap tooling, reset, run a first-off check. Nothing sellable comes off the line while it happens.changeover, it takes about 45 minutes, & nothing sellable comes off the line while it happens. Twenty of them in a week is fifteen hours gone. A plant that switches constantly can run out of time to make anything at all.

A, B & C items

Twenty of the hundred products, the A, B & C items — ABC classificationSorting a range into bands by sales volume so that each band can be planned differently. A items are the fast movers, B the middle group, C the slow-moving tail. It is the standard operational answer to a Pareto profile: make the A items often & the C items rarely.A items, account for 70% of everything sold. Thirty B items account for 20%. The remaining fifty C items share the last 10%. That lopsidedness is the classic Pareto profile, or power lawThe familiar pattern in which a small share of items accounts for most of the total, often summarised as 80/20. It governs city sizes, book sales, word frequencies & very nearly every product range. The practical consequence here is that a plant cannot sensibly treat all its products the same way, because they are nowhere near the same size.Pareto pattern, & it is why a plant sorts its range into bands & plans each band differently: make the A items often, the C items rarely.

Share of sales against share of products, from the 100 products in the model. The straight line is what it would look like if everything sold equally.

Two ways to decide what to make

The first plant follows a fixed repeating schedule: the same products, in the same order, whatever else is happening that week. The second rebuilds its plan every week around whichever product is closest to running out. One is a timetable. The other is a queue with a lot of people pushing in.

What to actually watch

Both plants get hit equally hard when the disruption lands, & neither escapes it. The interesting part is afterwards: how fast service climbs back, & whether it ever fully does. In the first chart, follow the blue line against the orange one.

You do not need to read any further to use this. Press Play weeks below to watch the two boards fill up, or press Supplier shortage & look at the first chart. Any underlined word has a plain definition attached, so hover over it, or tap it on a phone. There is a full glossary further down, & an appendix at the foot setting out the mathematics for anyone who would rather check the reasoning than take it on trust.

Departures

The same week on both plants. One is working through a sequence published cycles in advance, so its runs can be on time. The other is working a hot list drawn up that morning: nothing on it was scheduled, & the amber jobs are already late enough to need an ExpeditePushing a job to the front of the queue because somebody has escalated. It costs everything behind it, & usually costs extra changeover time because the materials were never staged.expedite.

Scheduled serviceFixed product wheel WK 00
Unscheduled serviceReactive replanning WK 00
Week 0

Introduce a disruption

Both plants get exactly the same customer orders, the same working hours & the same disruption. The only difference between them is how they decide what to make, so anything that happens next is caused by that decision alone.

Service recovery profile

Case fill rateOf all the units customers asked for, the share shipped on time. It counts volume, so it is dominated by the fast movers & can look healthy while the tail is failing.Case fill rate by week: the share of what customers asked for that was actually handed over on time. The shaded band marks the disruption.

Product wheel Reactive replanning Recovery: CFR sustained above 97% for four weeks

Setup hours per week
Under a wheel this quantity is determined by the structure of the schedule rather than by events, which is why the blue line is close to flat & can be budgeted a year in advance. Reactive replanning sets up more often, out of sequence, & frequently without materials staged, so its setup load rises at precisely the point where throughput matters most. Appendix §2 gives the identity.
Backlog carried forward
Unfilled demand awaiting supply. The behaviour worth noting falls after the disruption window closes: the reactive plant frequently records its trough there, once the accumulated backlog arrives & effectively every setup becomes an expedite.

Cover position, week 0

One tile per product, ordered by sales volume, so the A items fill the top rows & the C tail the bottom ones. Colour shows CoverHow long the stock on hand would last at normal sales. Two weeks of cover means you run out in a fortnight if nothing more is made.cover, meaning how long the stock would last. Use the week slider above to move through time.

Product wheel
Reactive replanning
Backorder outstanding Under 1 week cover 1–2 weeks 2–4 weeks 4 weeks or more Above its own cycle requirement

Interpretation

Reading the run

    The shortage itself is not preventable. What is settled in advance is whether, on the far side of it, a customer is waiting for a scheduled departure or waiting for somebody to return their call.

    Scope of the model

    Glossary

    Every term used anywhere on this page, in plain language. Nothing here assumes prior knowledge.

    SKU stock-keeping unitOne specific sellable item. A 500 ml bottle & a 1 litre bottle of the same drink count as two, because they are stocked, counted & ordered separately.
    Pareto profile or power lawThe familiar pattern in which a small share of items accounts for most of the total, often summarised as 80/20. It governs city sizes, book sales, word frequencies & very nearly every product range. The consequence here is that a plant cannot sensibly treat all its products alike, because they are nowhere near the same size.
    A, B & C items ABC classificationSorting a range into bands by sales volume so each band can be planned differently. A items are the fast movers, B the middle group, C the slow-moving tail. It is the standard operational answer to a Pareto profile: make the A items often & the C items rarely.
    Changeover (setup)Everything needed to stop making one product & start the next: clear down, clean, swap tooling, reset, run a first-off check. Nothing sellable comes off the line while it happens.
    Product wheelA fixed, repeating production sequence. The line works through the same order of products over & over, so every item has a known next slot. A bus timetable for a factory.
    EPEI every product every intervalHow long the line takes to work through its whole range & return to the start. An EPEI of eight weeks means every product is made at least once every eight weeks. A shorter EPEI means smaller batches & less stock, at the cost of more changeovers.
    Level loadingArranging the schedule so each week carries roughly the same workload, rather than alternating heavy & light weeks.
    CoverHow long the stock on hand would last at normal sales. Two weeks of cover means you run out in a fortnight if nothing more is made.
    Cycle stockThe stock you must hold simply because you make things in batches rather than continuously. Shop fortnightly & you need a fortnight of food in the house; shop weekly & you need half as much.
    Safety stockExtra stock held on top of the normal batch, to absorb the weeks when demand comes in higher than expected.
    Case fill rateOf all the units customers asked for, the share shipped on time. It counts volume, so it is dominated by the fast movers & can look healthy while the tail is failing.
    Line fill, or OTIF (on time in full)Of all the order lines customers placed, the share delivered complete. Every product counts equally. A customer ordering ten items & receiving nine has had a line fail, even though 90% of the units arrived.
    BacklogOrders taken but not yet shipped, carried forward week to week until there is stock to fill them.
    ExpeditePushing a job to the front of the queue because somebody has escalated. It costs everything behind it, & usually costs extra changeover time because the materials were never staged.
    UtilisationThe share of the line’s available hours already committed. The higher it is, the less there is spare to recover with when something goes wrong.
    Schedule attainmentThe share of the published plan actually executed as published. The standard measure of whether a plant does what it said it would.
    Coefficient of variationVariability measured relative to the average. A CV of 0.6 means demand swings by roughly 60% of its own average from week to week, which is typical of slow movers.
    Lead timeThe time between deciding to make something & having it available to sell. One week throughout this model.
    ELSP economic lot scheduling problemThe classic operations research problem of deciding how often to make each product on a shared machine, trading changeover cost against the cost of holding stock.

    Appendix — the arithmetic underneath

    Nothing in the simulation depends on a hidden preference for cycles. The behaviour follows from four standard results in production scheduling: the capacity identity, the feasibility bound on cycle length, the economic lot scheduling frequency rule, & the variance of a stochastic replenishment interval. Each is set out below with the figures for the configuration you currently have loaded, so the theory & the simulation can be checked against one another.

    If you do not read equations, you will not miss anything. Every section states its result in words before it states it in symbols, & each one closes with a plain-terms panel. Reading only those panels gives the complete argument. The symbols are there so the reasoning can be checked rather than believed.

    Dtotal demand, units per week
    rrun rate, units per hour
    Havailable line hours per week
    splanned setup time, hours
    kunplanned setup multiplier
    NcSKUs in class c
    Tcreplenishment interval, weeks
    μimean weekly demand, SKU i
    σiits standard deviation
    Lproduction lead time, 1 week
    Λsetup load, hours per week
    ρutilisation of available hours
    Btbacklog at week t
    Qitrun quantity, SKU i week t

    §1 The demand process

    Weekly demand is drawn independently for each SKU from a normal distribution truncated at zero, with the Coefficient of variationVariability measured relative to the average. A CV of 0.6 means demand swings by roughly 60% of its own average from week to week, which is typical of slow movers.coefficient of variation rising as volume falls. C items additionally have an 18% chance of no demand at all in a given week, which reproduces the intermittency typical of a long tail.

    dit = max(0, μi(1 + cvi·Zit)),   Zit ~ N(0,1) iid(1)

    Volume is allocated in two layers of concentration: between the A, B & C bands by the imposed 70/20/10 split, & within each band by lognormal weights, so the largest A item is several times the smallest. Strictly this yields a Pareto-like concentration curve rather than a true power law, since the band shares are set rather than emergent. It reproduces the shape of a real range without claiming to be drawn from a fitted distribution. Both policies are driven by the same realised series from a fixed seed. Items are then classified into the A, B & C bands by realised volume rank, as a real ABC review does, so no B item outsells an A item & every interval matches the volume the item actually carries. Opening stock is staggered by each item’s position in the cycle so the run begins in steady state, & both policies are handed the identical opening position.

    In plain terms

    Each SKU has its own steady average, & the smaller the SKU the more erratically it actually arrives. That asymmetry matters later: the tail is where variability is highest & where a reactive rule does most of its damage.

    §2 The capacity identity

    In any week, the hours consumed by the items scheduled must fit inside the hours available. This is the only hard constraint in the model.

    ∑i∈St ( si + Qit/r )  ≤  Ht(2)

    For a wheel, membership of the scheduled set St is fixed by the schedule rather than by circumstances, so the setup load is a constant determined entirely by the frequency ladder:

    Λwheel = ∑c Nc·s / Tc(3)

    Which divides the week's hours into three parts, on your current settings:

    In plain terms

    Only the first line of that ledger makes product. The second is the toll paid for variety, & the third is what remains to absorb anything unexpected. A wheel fixes the toll in advance, which means the slack is a budgeted quantity that can be relied upon when something goes wrong. Reactive replanning leaves the toll to be determined by events, & events tend to raise it at the worst possible moment.

    §3 The shortest cycle you can actually keep

    Suppose every one of the N items ran on a single common interval T. Over one revolution the line must absorb N setups plus the production itself, which gives a lower bound on T:

    N·s + (D/r)·T  ≤  H·T   ⇒   T  ≥  N·s / (H − D/r)  =  Tmin(4)

    Writing ρrun = D/(rH) for the share of hours spent producing, the same bound becomes:

    Tmin = N·s / ( H·(1 − ρrun) )(5)
    In plain terms

    As the line fills up, the shortest schedule you are able to hold gets longer, & it lengthens hyperbolically rather than proportionally. This is the scheduling counterpart of the familiar queueing result that delay grows with ρ/(1−ρ): high utilisation does not simply make you late, it removes your ability to keep any short cycle at all. It is also the reason “run everything more often” is not available as a remedy, & why the frequency ladder in §4 is a necessity rather than a refinement.

    §4 Why A items run more often

    The independent solution to the ELSP — economic lot scheduling problemThe classic operations research 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, & yields an interval inversely proportional to the square root of demand:

    Ti* = √( 2s / (h·μi) )   ⇒   Ti ∝ μi−1/2   ⇒   Tc/TA = √(μA/μc)(6)
    In plain terms

    The habit of doubling the interval at each step down the ABC ladder turns out to be very close to the cost-minimising allocation for a 70/20/10 volume split. That is worth knowing in an argument, because it means the structure of the wheel can be defended on cost grounds & not only on the grounds of discipline. It also explains why running the tail as often as the head is not a conservative choice but an expensive one: the setups consumed would come directly out of the hours available to the items carrying the volume.

    §5 What the cycle costs in inventory

    Stock built to cover an interval is drawn down over it, so average Cycle stockThe stock you must hold simply because you make things in batches rather than continuously. Shop fortnightly & you need a fortnight of food in the house; shop weekly & you need half as much.cycle stock is half the run quantity. Aggregated across the range:

    Icycle = ∑i μiTi/2 = (D/2)·∑c wcTc     cover = Icycle/D + SS(7)
    In plain terms

    This is the weekly shop. If you go to the supermarket once a fortnight you need a fortnight of food in the house; go weekly & you need half as much on the shelf, but you make twice as many trips. The wheel’s higher stock is not an artefact of how the model was built, it is that same arithmetic, & it is the honest price of the schedule. Any case made for cycles should put this figure on the table rather than leave it to be found.

    §6 The term that separates the two policies

    Under periodic review with a known interval R & lead time L, safety stock covers demand variability over the protection interval in the standard way:

    SS = z·σd·√(R + L)(8)

    If the interval is itself a random variable, the variance of demand over the protection interval acquires a second component, & the requirement becomes:

    SS = z·√( (R̄+L)·σd2  +  μd2·σR2 )(9)
    In plain terms

    This is the formal content of trusting a timetable. Under a wheel σR is approximately zero by construction: the next slot is published, so the second term vanishes & safety stock has only demand variability to cover. Under reactive replanning, R for any given item is decided by its position in a queue driven by every other item’s shortfall. σR is therefore both large & outside the planner’s control, & no amount of safety stock discipline removes the penalty, because a service factor cannot be set against a variance nobody can observe. The value of a cycle is not that the interval is short. It is that the interval has almost no variance.

    §7 Why the failure is not proportional

    Utilisation of available hours includes the setup load, so a policy that sets up more often is running a busier plant on the same equipment:

    ρ = ( D/r + Λ ) / H(10)

    Kingman’s approximation for waiting time in a single-server queue then indicates how delay responds, with ca & cs the coefficients of variation of arrivals & service:

    W  ≈  ( (ca2 + cs2) / 2 ) · ( ρ / (1 − ρ) ) · τ(11)
    In plain terms

    Reactive replanning moves both factors the wrong way at once. It raises ρ by consuming more hours in setup, & it raises cs2 by making setup times bimodal, since planned work takes s & expedited work takes ks. Because the second factor behaves as ρ/(1−ρ), the consequence is not proportional to the cause. That is what you are seeing if you drag the available hours down through the region around 130: the inputs change smoothly & the output changes character.

    §8 Exogenous against endogenous setup load

    Backlog evolves as the difference between demand & throughput, & throughput is whatever the hours left after setup will produce:

    Bt+1 = Bt + Dt − Yt,     Yt = r·(Ht − Λt)(12)

    Under reactive replanning the number of expedited setups et increases with the backlog, & each carries the multiplier k, so throughput becomes a decreasing function of arrears:

    ∂Yt/∂Bt = −r·s·(k − 1)·∂et/∂Bt  <  0(13)

    Under a wheel the setup load is set by the schedule & does not respond to the backlog at all:

    ∂Λt/∂Bt = 0   ⇒   ∂Yt/∂Bt = 0(14)
    In plain terms

    These two lines are the whole argument. Under reactive replanning, falling behind reduces the rate at which you are able to catch up, which is a reinforcing loop; whenever its gain exceeds one the backlog diverges until something else stops it, & the sawtooth in the service chart is what that looks like from outside. Under a wheel, throughput in a bad week is the same as throughput in a good one, so arrears are cleared at a constant rate & the recovery is monotone. The railway comparison is exact at this point: a delayed service does not make the following service slower.

    §9 Two service measures that disagree

    Case fill rateOf all the units customers asked for, the share shipped on time. It counts volume, so it is dominated by the fast movers & can look healthy while the tail is failing.Case fill rate weights each product by its volume. Line fill, or OTIF (on time in full)Of all the order lines customers placed, the share delivered complete. Every product 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 product equally, which is why the two can disagree sharply.

    CFRt = ∑i min(xit, dit) / ∑i dit      LFRt = |{ i : xit ≥ dit }| / |{ i : dit > 0 }|(15)
    In plain terms

    The tail carries half the line-fill weight & a tenth of the case fill weight. A plant can therefore report a respectable case fill rate while a customer ordering across the range finds a much smaller proportion of their lines complete. When a reactive rule sheds C items under load, CFR barely registers it & OTIF falls sharply, which is why any comparison of the two policies conducted on volume alone understates the difference by construction.

    §10 What the model leaves out

    Several of these omissions work in the wheel’s favour & several against it. They are listed together so the balance can be judged rather than asserted.

    • A single line with no parallel resources & no routing choice, so no load balancing between assets.
    • Setup time independent of sequence. Real wheels are sequenced to minimise changeover, following grade, colour or allergen order, so a genuine wheel would gain more than it does here. This omission understates the wheel’s advantage.
    • A deterministic one-week production lead time, with no inbound supply variability outside the shortage scenario.
    • All unmet demand is backordered & none is lost, & there is no substitution between items. Lost sales would penalise both policies, the reactive one more.
    • Demand is independent across SKUs & across weeks: no seasonality, promotions, trend or forecast bias, & therefore no externally generated bullwhip.
    • Capacity is a hard constraint, with no overtime, additional shifts or subcontract available. In practice a firefighting plant buys its way out with premium hours, which moves the cost from service onto the P&L rather than removing it.
    • Perfect inventory accuracy & no yield loss outside the quality-hold scenario.
    • The wheel is assumed to be adhered to. Sustaining adherence against commercial pressure is the genuinely difficult part of running one, & this model assumes it away. Schedule attainmentThe share of the published plan actually executed as published. The standard measure of whether a plant does what it said it would.Schedule attainment on the departure board shows what the wheel achieves when material allows, not how hard it was to defend.

    Of the parameters, the two carrying the most weight are the minimum economic run length, which governs how much capacity the reactive rule wastes on small items, & the unplanned setup multiplier k. Both are measurable in a real plant. Anyone intending to use this to make a case should set them from their own changeover studies first.

    Created by

    Alvin J Lin

    alvinlinjr@gmail.com

    Licence

    Open source under the MIT License. You are free to use, copy, modify, merge, publish, distribute, sublicense & sell this work, including commercially, provided the copyright notice & the licence text travel with it. It is provided without warranty of any kind.

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    Reuse & adaptation

    The whole simulator is a single self-contained HTML file with no dependencies, so it can be saved, edited & rehosted freely. Every model assumption sits in one configuration block near the top of the script, & the appendix recomputes itself from whatever you change. If you adapt it with your own plant data, the attribution above is all that is asked.