Quick answer: an ATM is a small physical inventory system whose stock happens to be money. Banks and ATM operators forecast withdrawals at each machine, estimate forecast uncertainty, choose how much cash to load, decide when the next replenishment should occur, and coordinate replenishment logistics across a network. Too little cash creates stockouts and customer failure. Too much cash leaves value idle inside machines and increases handling, insurance and opportunity cost. The mathematical problem is therefore a joint forecast-and-inventory optimisation: how much cash should be positioned at each ATM so that service remains reliable without carrying more physical inventory than needed?

An ATM can contain perfectly good money in the wrong amount, at the wrong machine, on the wrong day.

Safety boundary: this article discusses forecasting and operations-research concepts at an abstract educational level. It does not disclose or recommend real cash-in-transit routes, security schedules, vault procedures or other operational details that could endanger staff or cash infrastructure.

Why this belongs in mathematics

ATM cash management combines time-series forecasting, seasonality, probability, inventory control, service-level design, clustering and vehicle-routing optimisation. It also gives a useful counterexample to the idea that modern banking is only digital. The Bank of England notes that wholesale cash distribution supplies banknotes and coin to ATMs, branches and retailers, and that most cash enters circulation in the UK through ATMs. See Wholesale Cash Supervision.

1. The ATM inventory equation

Let C_t be usable cash in an ATM at the start of day t, R_t the cash replenished during the day, W_t customer withdrawals and D_t eligible deposits/recycled cash where the machine supports them.

A simplified balance is:

C_t+1 = C_t + R_t + D_t − W_t.

The ATM fails its cash-withdrawal function if usable note inventory reaches the relevant operational minimum before replenishment arrives. The model therefore needs to predict W across the replenishment horizon, not merely today’s withdrawal total.

2. Cash demand has strong calendar structure

ATM withdrawals are rarely identically distributed from day to day. Demand can depend on:

day of week;
month-end and salary dates;
public holidays;
festive periods;
local events;
tourist seasons;
weather;
nearby branch or ATM closures;
the location’s customer population.

A downtown office ATM and a weekend shopping-centre ATM can have almost opposite weekly patterns. One global model can therefore blur useful local structure.

Research in the European Journal of Operational Research found that clustering ATMs with similar day-of-week withdrawal patterns before forecasting improved cash-demand predictions relative to treating the whole ATM population as one undifferentiated set. See Cash demand forecasting in ATMs by clustering and neural networks.

3. Start with simple forecasting baselines

Before using a neural network, an operator should know whether it can beat simple baselines such as:

same day last week;
rolling weekday average;
exponential smoothing;
seasonal ARIMA/ETS;
median of recent comparable days.

A complex model earns its place only if it reduces economically meaningful error after accounting for instability, maintenance and explainability.

4. Machine learning can capture nonlinear demand

ATM demand has been studied using neural networks, random forests, support-vector regression, LSTMs, GRUs and hybrid deep-learning models. A 2023 study using data from a large Indian commercial bank found that day-of-week information improved forecasting across methods and that LSTM performed strongly in the tested data. See ATM cash demand forecasting in an Indian bank with chaos and hybrid deep learning networks.

That does not mean “LSTM is always best.” Model ranking can change by ATM, sample size, regime and forecast horizon. The correct validation question is out-of-sample service and cost performance on the operator’s own network.

5. Forecast error must be translated into service risk

Suppose predicted three-day withdrawals are S$120,000 but forecast uncertainty is large. Loading exactly S$120,000 gives no protection against positive forecast error.

A simple inventory rule is:

Target cash = expected withdrawals over lead time + safety stock.

If demand over the replenishment horizon is approximately normal with mean μ and standard deviation σ, a simplified safety-stock rule can be:

Target = μ + zσ

where z reflects the chosen service level. This is only a teaching approximation. Real withdrawal distributions can be skewed, seasonal and heavy-tailed, and machine capacity can impose a hard upper bound.

6. Service level and cost pull in opposite directions

Loading more cash reduces the probability of stockout but increases idle inventory and handling cost. A simplified objective is:

Total expected cost = cash holding/opportunity cost + replenishment cost + expected stockout/service cost.

A model that minimises only idle cash will under-stock machines. A model that minimises only stockouts may load every ATM close to capacity. Neither solves the actual optimisation problem.

The operations-research literature on ATM forecasting explicitly identifies this trade-off: excessive cash creates opportunity and logistics cost, while shortages damage service. See also Optimization of ATM cash replenishment with group-demand forecasts.

7. A miniature replenishment example

Imagine an ATM has S$40,000 usable cash at Monday close. Forecast withdrawals are:

Tuesday: S$18,000;
Wednesday: S$22,000;
Thursday: S$30,000.

Total expected demand before a Thursday-night replenishment is S$70,000. The machine therefore needs at least S$30,000 additional expected inventory before accounting for safety stock.

If forecast-error analysis suggests an additional S$15,000 buffer for the chosen service level, the target replenishment becomes roughly S$45,000—subject to machine capacity, denomination needs and operational constraints.

If a holiday suddenly raises Thursday demand to S$55,000, the model’s calendar feature or exception process needs to catch it before the ATM reaches zero.

8. Forecast accuracy is not enough: under-forecast and over-forecast have different costs

A forecast that misses by S$20,000 high leaves idle cash. A forecast that misses by S$20,000 low can produce an ATM stockout. Equal absolute error can therefore create unequal economic loss.

This suggests using cost-sensitive validation in addition to MAE, RMSE, MAPE or SMAPE. The model should track:

stockout frequency;
cash idle-days;
emergency refill frequency;
forecast bias;
service failures during peak dates;
cost per successful withdrawal/service period.

A statistically superior forecast can be operationally inferior if its rare under-forecasts occur exactly on high-demand weekends and holidays.

9. Replenishment frequency is a decision variable

Replenishing every day reduces the inventory horizon but increases visit cost. Replenishing weekly reduces visits but requires more cash and increases exposure to forecast error.

The optimisation can therefore jointly choose:

next refill date;
refill amount;
acceptable stockout probability;
which ATMs can be grouped into a replenishment cycle;
which machines require exception visits.

Research integrating group-demand forecasts with replenishment decisions found that grouping geographically close ATMs and optimising replenishment intervals can reduce overall replenishment and inventory cost in example datasets.

10. Routing enters after the cash decision

Once replenishment quantities and deadlines are known, logistics becomes a vehicle-routing problem. At an abstract level, the optimiser can minimise travel and service cost subject to:

vehicle and operational capacity;
ATM service windows;
depot origin/return constraints;
required replenishment deadlines;
maximum route duration;
regulatory, safety and operational constraints that remain outside the public model.

Academic work has formulated ATM replenishment as variants of periodic or multiobjective vehicle-routing problems. The useful educational point is not the real route. It is that the forecasting problem and the routing problem are coupled: a poor forecast creates unnecessary emergency routing, while a route schedule can constrain how much forecast horizon each ATM must carry.

11. Clustering can help both forecasting and logistics

ATMs can be clustered in two different spaces:

demand-pattern clustering — machines with similar weekday/seasonal withdrawal behaviour;
geographic/logistics clustering — machines that are operationally sensible to service together.

Those clusters need not be identical. Two nearby ATMs can have radically different demand because one serves commuters and the other nightlife. Two distant ATMs can have similar seasonality but belong to different replenishment routes.

A strong system therefore distinguishes the feature space used for forecasting from the feature space used for routing.

12. Denomination mix creates another inventory constraint

An ATM can contain enough total value but run out of a denomination needed to complete ordinary withdrawal combinations. Total cash value is therefore not always sufficient to represent usable inventory.

A simplified inventory state can be a vector:

C = [number of note type 1, note type 2, …].

The withdrawal algorithm and local demand pattern then determine how quickly each cassette depletes. This is a multi-item inventory problem rather than one scalar cash balance.

13. Cash-recycling machines change net demand

Some machines can accept deposits and, under appropriate controls, recirculate fit cash. In that case gross withdrawals overstate external replenishment demand. The inventory equation needs both withdrawals and usable incoming deposits.

The broader cash cycle described by the Bank of England includes banknotes returning through branches, retailers and cash centres, being authenticated and recirculated to meet future demand. Physical cash is a circulation system, not a one-way supply chain.

14. Concept drift: declining cash use does not mean every ATM declines smoothly

Long-run cash use can decline while individual locations experience temporary increases because nearby branches close, population shifts, electronic payments fail or local businesses remain cash-heavy.

The ECB’s cash strategy explicitly treats access to cash as a continuing public requirement, and national authorities monitor ATM and branch access even as payment behaviour changes. The UK’s FCA access-to-cash regime likewise requires designated firms to assess and fill significant local gaps. See The Eurosystem Cash Strategy and Helping People Access Cash.

A bank should therefore not extrapolate a national cash-decline trend mechanically into every ATM forecast.

15. Electronic-payment outages are a stress scenario for physical cash

If card or electronic-payment systems suffer a material outage, local cash withdrawal demand can rise abruptly. The ECB notes that national central banks keep sufficient cash reserves to accommodate sudden and unforeseeable surges in banknote demand, including situations such as temporary electronic-payment outages.

ATM operations should therefore distinguish normal-demand forecasting from stress-demand scenarios. A weekly time-series model trained on ordinary behaviour is not automatically a crisis cash-demand model.

16. Creative-work lens: The Martian and the resupply window

The Martian supplies a useful operations-research intuition: finite inventory, uncertain future consumption and a resupply date interact. Having enough total food somewhere on Earth does not help a person on Mars before the next launch arrives. In the same way, cash in a central vault does not prevent an ATM stockout if replenishment lead time exceeds remaining inventory.

The creative work makes lead time and safety stock memorable. Real cash distribution remains governed by evidence, operational controls and security requirements.

17. The ATM cash-management pipeline

Reconcile ATM cash inventory and cassette state.
Clean historical withdrawal/deposit data.
Add calendar, holiday and local-event features.
Segment or cluster ATMs where demand patterns differ.
Run simple forecasting baselines.
Evaluate statistical and machine-learning challengers.
Forecast demand across the replenishment lead time.
Estimate forecast uncertainty and safety stock.
Choose service-level and cost objectives.
Determine refill amount and next replenishment window.
Optimise abstract logistics routing subject to operational constraints.
Monitor actual withdrawals against forecast in real time.
Trigger exceptions when stockout risk rises.
Backtest stockouts, idle cash and emergency visits.

18. Failure modes

Average-day forecasting. Weekends, paydays and holidays are compressed into one mean.
Accuracy-only optimisation. Lower RMSE is celebrated even though stockouts rise.
One-network model. Every ATM inherits the same seasonal pattern.
National-trend extrapolation. Declining cash use is assumed to reduce every local ATM equally.
Zero-safety-stock logic. Point forecasts are loaded with no buffer for forecast error.
Routing separation. Forecasts ignore that replenishment schedules constrain lead time.
Total-value blindness. Machine has enough value but wrong usable denomination mix.
Normal-regime assumption. Temporary electronic-payment disruption creates demand outside the training distribution.

19. Diagnostics and falsifiers

Which ATMs show persistent under-forecast bias?
How does forecast error change on salary days and holidays?
What proportion of idle cash is safety stock versus systematic over-forecast?
Which machine has the highest stockout probability before its next planned visit?
Does a complex model beat the seasonal-naive baseline after costs?
Which demand clusters remain stable over time?
How many emergency refills are caused by forecast error versus route disruption?
What happens to cash demand during a payment-system outage scenario?

Suppose someone claims, “The forecast is accurate because its average error is only 5%.” A falsifier is evidence that the model’s small average hides repeated severe under-forecasts on high-demand weekends that cause stockouts. Forecast accuracy must be judged against the operational loss function, not one summary statistic.

20. Verification and update triggers

backtest forecast error by ATM and calendar regime;
track stockouts and excess-cash days separately;
compare model performance with simple baselines;
re-cluster after location or customer-pattern changes;
update after nearby ATM or branch openings/closures;
stress public holidays and electronic-payment outages;
reconcile loaded cash, dispensed cash and returned inventory;
keep logistics and security data protected even while optimisation logic is audited.

Connections across the finance-and-banking algorithms lane

Intraday liquidity forecasting — another timing-sensitive inventory problem, but in settlement money rather than ATM notes.
Payment-system routing — the digital rail whose outages can alter physical-cash demand.
Operational risk and resilience — ATM availability depends on technology, cash logistics and third-party continuity.
A Good Mathematics Check Should Be Able to Disagree With the Working — the forecast baseline must remain capable of beating the sophisticated model.

Research anchors

The deeper lesson

ATM cash forecasting turns money back into physics. Notes occupy cassettes. Demand occurs at a place and time. Replenishment has lead time and cost. Safety stock protects against uncertainty. A route determines when new inventory can arrive. The strongest system therefore does not ask only “How much cash will people withdraw?” It asks “How uncertain is that demand, what service level matters, what is the cost of being wrong in either direction, and can the right inventory reach the right machine before its deadline?”

Educational note: This article explains public forecasting and operations-research concepts. It is not a security plan, cash-in-transit procedure, ATM servicing instruction or recommendation for any specific cash network.

Small Group Tutorials

How Banks Forecast ATM Cash Demand and Replenishment: Seasonality, Time-Series Models, Service Levels, Inventory Costs and Routing Optimisation