How Mathematics Improves The World | Keeping Supermarket Shelves Full Without Filling Warehouses With Waste
A supermarket shelf has two obvious ways to fail.
There is nothing left.
Or there is too much left.
The first failure is visible to the customer.
No milk.
No rice.
No eggs.
The second failure often happens behind the store.
Wilted vegetables.
Expired yoghurt.
Bread that will not sell tomorrow.
Frozen stock occupying expensive cold-storage space.
Both come from the same question:
How much should we order before we know exactly how much people will buy?
This is not merely a retail question.
Hospitals order medicines.
Factories order components.
Airlines hold spare parts.
Governments hold emergency stockpiles.
Every inventory system lives between scarcity and excess.
Mathematics gives that tension a language.
Quick Read
Inventory management decides how much stock to hold and when to replenish it. The problem exists because demand and supply are uncertain, while storage, spoilage and stockouts all have costs. For stable non-perishable products, classical models such as the economic order quantity balance ordering cost against holding cost. Under uncertainty, reorder points and safety stock protect against demand variation and delivery delays.
Perishable products create a sharper trade-off. The classical newsvendor problem asks how much of a single-period product to order before random demand is known. Order too little and profitable sales are lost. Order too much and leftover value is low or negative. The optimal quantity is a demand quantile determined by the ratio between underage and overage costs. Modern operations research extends this logic to many products, many stores, uncertain supply, dynamic pricing, substitution, lead times, delivery routing and multi-echelon networks.
MIT logistics teaching materials treat probabilistic demand and safety stock as core inventory concepts, while INFORMS describes the newsvendor problem as a fundamental building block of stochastic inventory theory for perishable products. In Singapore, NEA’s current food-waste strategy explicitly prioritises prevention and reduction at source, and its supermarket guidebooks encourage better ordering, stock management and redistribution of unsold edible food.
One-sentence answer: Mathematics improves the world by letting retailers translate uncertain future demand into explicit stocking decisions, balancing the cost of an empty shelf against the cost of excess inventory so customers are served with less capital, storage and food wasted.
Inventory Exists Because Supply and Demand Do Not Arrive Together
A customer wants milk now.
The dairy cannot produce and deliver that exact carton instantly.
So supermarkets hold inventory.
Inventory is time stored as product.
It allows something made yesterday to satisfy demand today.
That buffering is valuable.
It also costs money.
Products occupy shelves and warehouses.
Cash is tied up.
Cold rooms use electricity.
Insurance and handling costs accumulate.
Some goods become obsolete.
Food spoils.
The buffer protects service and creates holding cost simultaneously.
Three Costs Sit Under Almost Every Inventory Decision
Inventory models often separate several kinds of cost.
1. Ordering or setup cost
Each replenishment can create administrative, transport or production setup costs.
2. Holding cost
Each unit carried through time occupies space, ties up capital and may deteriorate.
3. Shortage cost
When stock runs out, a sale may be lost, delayed or substituted. Customer trust can fall and emergency replenishment may be expensive.
For perishables, add a fourth:
waste or markdown cost.
Inventory Mathematics is the art of making these costs disagree in an orderly way.
Economic Order Quantity: The First Beautiful Trade-Off
Suppose annual demand D is predictable.
Each order costs S.
Holding one unit for one year costs H.
Order a huge quantity.
Few orders, low ordering cost, high average stock.
Order tiny quantities frequently.
Low average stock, high ordering cost.
The classic economic order quantity is:
Q* = √(2DS/H)
The square root is the compromise point where marginal ordering and holding effects balance under the model.
This model is deliberately simple.
Real supermarkets face uncertain demand, quantity discounts, truck capacities, shelf life and coordinated deliveries.
But EOQ teaches the first principle:
the cheapest replenishment policy is not “order as much as possible” or “hold as little as possible”.
Cycle Stock: Inventory Created by Replenishment Batches
If a store orders Q units at a time and demand drains inventory steadily to zero, average cycle stock is approximately Q/2.
Larger batches increase average inventory.
Smaller batches reduce it.
But smaller batches require more frequent deliveries.
This is why transport economics and inventory economics cannot be separated.
A full truck may be cheap per carton and expensive in stock.
Reorder Point: Inventory Must Be Ordered Before It Runs Out
Suppose supplier lead time is five days.
Average daily demand is 100 units.
If demand were perfectly predictable, reorder when inventory position reaches:
5 × 100 = 500 units
The order travels while those 500 units are sold.
The replenishment arrives just as stock approaches zero.
That is the deterministic reorder point.
Reality introduces uncertainty.
Safety Stock: Inventory Held Because the Forecast Will Be Wrong
Maybe demand during lead time is 500 units on average and sometimes 620.
Maybe the supplier usually takes five days and occasionally seven.
If the store holds no buffer, normal random variation creates stockouts.
Safety stock is deliberate extra inventory held to absorb uncertainty.
In a simple normal-demand model with fixed lead time:
safety stock = z σLT
where σLT is standard deviation of demand during lead time and z is a service-factor chosen from the desired probability of not stocking out.
The reorder point becomes:
ROP = expected lead-time demand + safety stock
Buffer is uncertainty translated into units on a shelf.
Service Level: How Much Stockout Risk Is Acceptable?
A 99.9% service target sounds better than 95%.
It also costs more inventory.
The last fraction of service can require disproportionately large safety stock because it protects against increasingly rare demand spikes.
Different products deserve different targets.
Running out of table salt is inconvenient.
Running out of an essential medicine can be dangerous.
The service level is therefore a business and social decision translated into a probability.
Cycle Service Level and Fill Rate Are Different
A cycle service level asks:
What is the probability this replenishment cycle experiences no stockout?
Fill rate asks:
What fraction of total demand is filled immediately from stock?
A system can stock out in many cycles and still have high fill rate if each shortage is tiny.
Metrics change behaviour.
Choose the wrong service metric and the inventory policy optimises the wrong customer experience.
Forecasting: The Inventory Decision Begins With a Distribution, Not a Number
A forecast of 1,000 units is incomplete.
Is tomorrow’s demand almost certainly between 990 and 1,010?
Or could it be anywhere from 500 to 1,500?
The mean is the same.
The stocking decision is not.
Good demand forecasting estimates uncertainty as well as central tendency.
Inventory is chosen against a probability distribution.
A point forecast hides the risk the stock is supposed to absorb.
Seasonality: Saturday Is Not Monday
Grocery demand follows rhythms.
Day of week.
Payday.
School holidays.
Festivals.
Weather.
Promotions.
A forecast model may use seasonal indices, exponential smoothing, ARIMA-like time-series structures, causal regressors or machine learning.
The best method depends on data and product behaviour.
A frozen chicken and an umbrella have different seasonality.
Promotions: The Forecast Changes Because the Price Changes
Put detergent on 30% discount.
Demand rises.
How much?
Depends on price elasticity, display placement, advertising, competitor prices, stock availability and whether customers bring purchases forward from future weeks.
A promotion can create a demand spike and a post-promotion dip.
Inventory planning must forecast the entire response, not only the sale week.
Marketing changes the stochastic process.
The Newsvendor Problem: Buy Before You Know Demand
A newspaper seller chooses morning stock before knowing how many customers will arrive.
Unsold papers have little value tomorrow.
This gives the classic newsvendor model its name.
The same structure fits:
- fresh bread;
- flowers;
- seasonal fashion;
- airline seats;
- hotel rooms;
- vaccines with limited shelf life under some planning contexts;
- event merchandise.
One selling period.
Uncertain demand.
Overage and underage have different costs.
Underage Cost: What Does One Missing Unit Cost?
If the store orders one unit too few, what happens?
Lost gross margin.
Maybe customer substitution to another product.
Maybe a lost future customer.
Maybe an emergency replenishment.
The marginal underage cost Cu measures the cost of being one unit too low.
For a simple selling-price p and unit cost c with lost sales and no future effect:
Cu = p − c
Reality can be richer.
Overage Cost: What Does One Excess Unit Cost?
If one unit remains unsold, perhaps it can be sold tomorrow.
Or marked down.
Or donated.
Or discarded.
If salvage value is v, a simple overage cost is:
Co = c − v
Perishable food may have near-zero or negative end-of-life value once disposal and handling are included.
The overage cost can therefore be substantial even when purchase cost seems modest.
The Critical Fractile: Choose a Quantile, Not the Average
The newsvendor optimum satisfies:
F(Q*) = Cu / (Cu + Co)
where F is the cumulative distribution of demand.
If running out is much more expensive than leftover stock, the critical ratio is high and the store orders a high demand quantile.
If waste is very expensive and stockouts are tolerable, the ratio falls and the store orders more conservatively.
The optimal order is not necessarily expected demand.
It is the demand quantile matching the cost asymmetry.
This is one of operations research’s most elegant results.
A Bakery Example
A loaf costs $2 to produce.
It sells for $5.
Unsold end-of-day loaves can be sold to a recycler or other channel for $0.50 equivalent value.
Underage cost:
5 − 2 = $3
Overage cost:
2 − 0.5 = $1.50
Critical ratio:
3 / (3 + 1.5) = 0.667
The bakery should order around the 66.7th percentile of forecast demand under the simple model.
Not the mean.
Not the maximum ever observed.
The cost ratio chooses the percentile.
Perishability Turns Inventory Into an Age Distribution
Ten cartons of milk are not identical if some expire tomorrow and others next week.
The state of inventory includes age.
A simple stock count loses information.
Perishable-inventory models track inventory by remaining shelf life.
Each day:
- new stock arrives at full life;
- old stock ages;
- customer demand removes units;
- expired units become waste.
The optimisation decides how much fresh stock to add given what old stock already exists.
Time is part of inventory identity.
FIFO and FEFO: Which Unit Should Leave First?
First-in, first-out dispatches the oldest received inventory first.
First-expired, first-out dispatches the unit with earliest expiry first.
They are not always identical because different batches can have different shelf lives.
For perishables, FEFO often better aligns physical issue order with waste prevention.
But customers may choose fresher stock from the back of the shelf.
Real behaviour can defeat warehouse policy.
Retail inventory is a human system as well as a mathematical one.
Markdown Optimisation: Lower the Price Before the Food Becomes Waste
A yoghurt expires in two days.
Keep full price and perhaps no one buys it.
Discount 20% and demand may rise.
Discount too much and revenue is unnecessarily sacrificed.
Dynamic markdown optimisation estimates price elasticity and chooses timing and discount depth to maximise recovery while reducing waste.
The product has a deteriorating option value.
Today’s unsold unit can still be sold tomorrow.
After expiry, the option disappears.
Donation Changes Salvage Value
Unsold edible food does not necessarily have zero social value.
Donation programmes can redistribute safe excess food to people who need it.
NEA’s food-waste strategy explicitly places redistribution of unsold or excess food after prevention and reduction at source.
From an inventory model’s perspective, donation changes the terminal value and social objective of leftover stock.
But donation is not permission to over-order.
Prevention remains preferable because production, transport and refrigeration resources have already been consumed.
Singapore Food Waste: Inventory Error Has a National Cost
Singapore’s National Environment Agency reported that food waste accounted for about 11% of total waste generated in 2025, with 790,000 tonnes generated and an 18% recycling rate.
Not all of that comes from supermarkets.
But supermarkets sit at an important point between producers and households.
Order too much and perishables may become waste before consumers ever receive them.
Order too little and customers substitute, travel elsewhere or buy emergency imports.
Inventory quality is therefore part of food-system efficiency.
Shelf Availability: The Warehouse Can Be Full While the Shelf Is Empty
Inventory records show 20 units.
The customer sees zero.
Why?
Stock is in the back room.
Replenishment was delayed.
The item is misplaced.
The inventory record is wrong.
On-shelf availability is a separate operational metric from inventory quantity.
Computer vision, shelf sensors and labour scheduling can improve replenishment.
A perfect order policy is useless if stock never reaches the shelf.
Inventory Accuracy: The Database Can Have Stock the Store Does Not
The system says twelve cans.
Three were damaged.
Two were stolen.
One was scanned incorrectly at checkout.
Physical stock: six.
The replenishment system waits because it thinks twelve exist.
Phantom inventory creates stockouts even when the forecasting model is correct.
Cycle counting, RFID and computer vision reduce inventory-record error.
Again, optimisation depends on measurement.
Substitution: An Empty Shelf Does Not Mean Lost Demand
No Brand A orange juice.
Some customers buy Brand B.
Some buy apple juice.
Some leave without buying anything.
The demand for one SKU therefore depends on stock of neighbouring SKUs.
Substitution couples inventory decisions.
Stocking more of one product can protect service when another runs out.
But it can also cannibalise sales.
The aisle is a portfolio, not a set of independent products.
Lost Sales Hide True Demand
The store sells all 20 croissants by 10 a.m.
POS data says demand was 20.
Wrong.
Demand was at least 20.
Maybe another 15 customers arrived and found the shelf empty.
Sales are censored by stock availability.
Forecasting directly from sales can perpetuate understocking:
low stock → low observed sales → low forecast → low future stock
Demand estimation needs to account for stockout periods.
What was not sold can still be part of demand.
Lead Time: The Supplier Is Part of the Forecast
If supplier lead time is deterministic, safety stock protects mainly demand uncertainty.
If lead time also varies, uncertainty compounds.
A port delay.
Truck breakdown.
Customs inspection.
Factory shortage.
Lead-time distribution matters as much as demand distribution.
Safety stock should protect uncertainty during the replenishment exposure period.
Supplier Reliability: Cheap and Late Can Be Expensive
Supplier A charges $1.00 per unit and arrives reliably in two days.
Supplier B charges $0.96 and arrives in two to eight days.
B looks cheaper on purchase price.
Its variability requires more safety stock.
More inventory increases holding and spoilage.
The correct comparison uses total system cost, not unit price.
Mathematics reveals hidden costs of uncertainty.
Dual Sourcing: One Cheap Supplier and One Fast Supplier
A common risk strategy uses two suppliers.
One is cheap and slow.
One is expensive and fast.
Normal demand is supplied by the cheap source.
Unexpected spikes are covered by the fast source.
Operations-research work continues to study dual-sourcing policies under uncertainty; a 2026 Operations Research paper develops robust inventory rules for base and surge sourcing under uncertain demand.
The second supplier is a flexibility option.
The Bullwhip Effect: Small Customer Changes Become Large Supplier Swings
Customer demand rises 5%.
The supermarket fears shortage and orders 10% more.
The wholesaler sees the order spike and orders 20% more.
The factory sees the wholesale spike and schedules 30% more production.
Then demand normalises.
The chain is full of excess inventory.
This amplification is the bullwhip effect.
Causes include:
- forecast updating;
- batch ordering;
- price promotions;
- rationing games;
- long lead times.
Sharing point-of-sale demand and reducing lead times can dampen amplification.
Better information reduces unnecessary stock.
Multi-Echelon Inventory: The Same Can Is Counted in Several Places
Inventory can sit at:
- factory;
- national distribution centre;
- regional warehouse;
- store back room;
- store shelf;
- in transit.
Optimising each location independently creates duplication.
The warehouse holds safety stock.
Every store holds its own safety stock.
Total network buffer becomes excessive.
Multi-echelon models optimise inventory across levels jointly.
Where should uncertainty be buffered?
Central stock pools variability efficiently but may respond slowly.
Local stock responds quickly but duplicates buffer.
Network position matters.
Risk Pooling: Combine Uncertain Demand and Variability Often Falls
Two stores each have uncertain demand.
If their demand spikes are not perfectly correlated, a central warehouse can share inventory between them.
One store is busy while the other is quiet.
The combined demand is smoother relative to its mean.
This is risk pooling.
The square-root effect appears under common independent-demand assumptions: aggregated standard deviation grows with √n while mean grows with n.
Centralisation can therefore reduce total safety stock.
But it may increase delivery distance and response time.
Again, every efficiency creates another trade-off.
Lateral Transshipment: Move Stock Between Stores
Store A has ten excess units.
Store B is about to stock out.
Instead of waiting for the central warehouse, move stock from A to B.
Lateral transshipment reduces local shortages and waste.
But transport cost and handling matter.
The decision needs real-time inventory accuracy and demand forecasts.
A network with visibility can share stock dynamically.
Routing and Inventory Are One Problem
Deliver to Store A today or tomorrow?
If today, the truck route becomes longer.
If tomorrow, A needs more safety stock.
Inventory-routing problems jointly choose delivery quantities and vehicle routes under uncertain demand.
Operations Research continues to publish models in this area because logistics and inventory cannot always be optimised separately.
The shelf and the truck are one system.
ABC Analysis: Not Every Product Deserves Equal Attention
A supermarket carries thousands of SKUs.
Managing every product with equal analytical effort is wasteful.
ABC analysis ranks items by annual consumption value or other importance measures.
- A items: few items, large value.
- B items: intermediate.
- C items: many items, low individual value.
High-value or critical items receive tighter control and forecasting.
But value alone can be misleading.
A low-value baby formula SKU may be strategically important.
Classification should match the business objective.
XYZ Analysis: Demand Variability Changes the Policy
Products can also be classified by predictability.
- X: stable demand;
- Y: moderate variation or seasonality;
- Z: erratic demand.
An AX product—high value, stable demand—may deserve precise continuous replenishment.
A CZ product—low value, erratic demand—may be managed with a simpler policy or even ordered on demand depending on context.
Inventory policy should fit demand structure, not corporate habit.
Intermittent Demand: Some Products Sell Zero, Zero, Zero, Then Ten
Specialty products and spare parts often have intermittent demand.
Ordinary moving averages perform poorly because zeros dominate.
Croston-type methods model non-zero demand size and interval between demands separately.
The demand process has two questions:
- When will demand occur?
- How large will it be when it occurs?
One average cannot answer both.
Forecast Error Is More Important Than Forecast Accuracy Alone
A forecast can be unbiased and noisy.
Or precise and biased.
MIT supply-chain research on safety-stock policy notes that classic formulas often assume unbiased normally distributed forecast errors, while real organisational forecasts can be biased.
Systematic overforecast creates excess inventory.
Systematic underforecast creates chronic stockouts.
Safety stock should protect random error.
It should not be used to hide a biased planning process.
Sales and Operations Planning: Align the Forecast Before Buying Against It
Marketing expects a promotion.
Sales expects a new customer.
Operations knows the supplier is constrained.
Finance wants lower working capital.
If each department uses a different demand number, inventory chaos follows.
Sales and operations planning reconciles one cross-functional demand and supply plan.
The Mathematics is only as coherent as the organisation supplying its inputs.
Machine Learning: Better Forecasts, New Failure Modes
Machine-learning models can combine:
- historical sales;
- weather;
- promotions;
- holidays;
- local events;
- prices;
- web searches;
- store attributes.
They can forecast thousands of SKUs automatically.
But they can fail when behaviour changes.
Pandemic buying.
A competitor closes.
A new regulation changes packaging.
An unprecedented festival promotion.
The training distribution no longer matches reality.
Forecasting systems need anomaly detection and human override.
Prediction Intervals: One Number Should Not Drive the Order
A model predicts tomorrow demand 100.
90% interval: 98–102.
Very stable.
Another product also predicts 100.
90% interval: 20–220.
Wildly uncertain.
The order policy should react differently.
Probabilistic forecasting connects naturally to inventory optimisation because the decision needs the whole demand distribution.
Robust Optimisation: What If the Demand Distribution Is Wrong?
Classical newsvendor assumes the demand distribution is known.
In reality, the distribution is estimated from finite historical data.
Operations Research has developed robust and distributionally robust newsvendor models for exactly this ambiguity.
Instead of optimising against one fitted distribution, the policy protects against a set of plausible distributions consistent with known moments or data.
The solution may sacrifice some expected profit to reduce sensitivity to modelling error.
Robustness is insurance against being wrong about uncertainty itself.
Demand Shaping: Sometimes the Store Can Change the Demand
Inventory models often treat demand as external.
Retailers can influence it.
Price.
Promotion.
Product placement.
Loyalty rewards.
Substitution recommendations.
Price-setting newsvendor models jointly choose inventory and price under uncertain demand.
The decision is no longer “how much should we stock for demand?”
It becomes “what demand should we create, and how much stock should support it?”
Dynamic Pricing and Fairness
Discounting ageing food can reduce waste.
Dynamic prices can also feel unfair if different customers see different prices without transparency.
Mathematics can optimise revenue.
It does not decide what pricing practice society considers fair.
Retail optimisation needs governance just as traffic, electricity and spectrum optimisation do.
Waste Is an Objective, Not Merely a Cost
A profit-maximising model assigns waste a financial disposal and lost-cost value.
Society may care about more.
Water used to grow food.
Land.
Fertiliser.
Transport emissions.
Cold-chain energy.
Food security.
A multi-objective inventory model can include explicit waste penalties or carbon cost.
The numerical optimum changes when externalities are counted.
The Objective Function Is a Moral Boundary
If the objective minimises store cost, donating food may look worse than discarding it if donation requires labour.
If the objective includes social value and waste reduction, the ranking changes.
An optimiser faithfully serves the objective it is given.
That is precisely why objective design deserves scrutiny.
Mathematics does not turn values into truth.
It turns values into consequences.
Reinforcement Learning: Learn Ordering Policies Through Simulation
Complex multi-product inventory systems can be modelled as sequential decisions.
State:
- inventory by age;
- incoming orders;
- forecast;
- price;
- time to expiry.
Action:
- order quantity;
- markdown;
- transshipment.
Reward:
- sales margin;
- minus holding;
- minus waste;
- minus stockout penalties.
Reinforcement learning can search policies in simulation.
But the simulator becomes the teacher.
If simulated substitution, expiry or demand response is wrong, the learned policy exploits a fictional supermarket.
Policy validation must return to real operations.
A/B Testing: Does the New Inventory Policy Actually Help?
Deploy a new ordering model to half the stores.
Keep comparable stores on the old policy.
Measure:
- stockouts;
- waste;
- sales;
- gross margin;
- inventory days;
- customer substitution.
Randomisation helps separate policy effect from seasonal change.
Simulation proposed improvement.
Experiment checks whether reality agrees.
A Classroom Thought Experiment: How Many Sandwiches?
A school canteen sells sandwiches.
Tomorrow demand could be:
- 40 with probability 0.2;
- 60 with probability 0.5;
- 80 with probability 0.3.
Each sandwich costs $2 and sells for $5.
Leftovers have no value.
Should the canteen make 40, 60 or 80?
Students calculate expected profit for each quantity.
Then give leftovers a $1 salvage value.
The answer may change.
Now charge a $2 goodwill penalty for every customer turned away.
The answer changes again.
The probability distribution stayed the same.
The objective changed.
This is inventory optimisation in a classroom.
A Second Thought Experiment: Safety Stock With Dice
Daily demand equals the roll of two dice.
Lead time is three days.
Roll six dice to simulate three days of demand.
Repeat fifty times.
Plot the distribution of lead-time demand.
Choose a reorder point that covers 90% of trials.
Then 99%.
How much extra stock does the higher service target require?
Students see service level become inventory physically.
Primary Mathematics: Inventory Begins With “How Many?”
Primary students already learn:
- addition and subtraction;
- averages;
- fractions;
- graphs;
- money;
- time;
- rates;
- estimation.
A stock ledger is arithmetic.
A sales graph is data.
“How many should I buy?” is estimation under uncertainty.
The adult mathematics grows from ordinary counting with consequences.
Secondary Mathematics: Probability Enters the Warehouse
Secondary students add:
- probability distributions;
- standard deviation;
- normal quantiles;
- functions;
- optimisation;
- correlation;
- regression.
Safety stock becomes a quantile.
Forecasting becomes regression.
Risk pooling becomes variance arithmetic.
Markdown becomes price elasticity.
The supermarket becomes applied statistics.
Advanced Mathematics: Inventory as Stochastic Control
Modern inventory management draws on:
- probability;
- stochastic processes;
- dynamic programming;
- Markov decision processes;
- convex optimisation;
- integer programming;
- time-series forecasting;
- robust optimisation;
- game theory;
- machine learning.
Demand is random.
Inventory state evolves.
Orders arrive after delays.
Perishables age.
Today’s decision changes tomorrow’s options.
Inventory is a sequential decision problem under uncertainty.
Why This Improves the World
1. It keeps shelves available with less excess stock
Safety stock and reorder policies translate service targets into explicit inventory buffers.
2. It reduces food waste
Perishable models, markdowns and better forecasting reduce the number of products reaching expiry unsold.
3. It reduces capital tied up in warehouses
Replenishment models avoid holding inventory merely because uncertainty feels uncomfortable.
4. It makes supply reliability economically visible
Lead-time variability and stockout risk expose when a cheaper supplier creates higher total system cost.
5. It coordinates the whole supply chain
Multi-echelon models and demand sharing reduce duplicate buffers and bullwhip amplification.
6. It lets social objectives enter operational decisions
Waste, carbon, donation and service can be represented explicitly rather than disappearing behind purchase price.
What Mathematics Does Not Do
Mathematics does not make tomorrow’s demand certain.
It does not make a biased forecast unbiased.
It does not prevent supplier failure.
It does not guarantee inventory records match physical shelves.
It does not decide how society values waste versus low prices.
It does not make a profit-maximising policy socially optimal.
And no inventory model can eliminate a stockout when physical supply is fundamentally unavailable.
Frequently Asked Questions
What is safety stock?
Safety stock is additional inventory held above expected demand during replenishment lead time to protect against uncertainty in demand, supply or both.
What is a reorder point?
A reorder point is the inventory-position threshold that triggers replenishment. Under uncertainty it commonly equals expected demand during lead time plus safety stock.
What is the newsvendor problem?
The newsvendor problem models a one-period stocking decision under uncertain demand. The optimal order quantity balances the marginal cost of ordering too little against the marginal cost of ordering too much.
Why is the optimal newsvendor quantity not always average demand?
Because underage and overage costs are usually asymmetric. The optimal quantity is a demand quantile determined by the critical ratio Cu/(Cu + Co), not automatically the 50th percentile or mean.
Why can more safety stock be bad?
More safety stock reduces stockout risk but increases holding cost, capital use and—in perishables—waste. The correct level depends on service requirements and consequences of shortage.
How can supermarkets reduce food waste mathematically?
Useful methods include probabilistic demand forecasting, smaller replenishment batches, expiry-aware inventory, markdown optimisation, transshipment, improved inventory accuracy and measuring waste as an explicit planning objective. Operational changes must still meet food-safety requirements.
Sources and Further Reading
- MIT OpenCourseWare, Logistics Systems — Inventory Management, Probabilistic Demand and Safety Stock.
- INFORMS / Operations Research, literature on the newsvendor problem, including Petruzzi and Dada’s review of stochastic single-period inventory and pricing.
- MIT Center for Transportation and Logistics, Integrating Safety Stock Policies into Roche’s S&OP Process, on forecast bias and safety-stock assumptions.
- National Environment Agency Singapore, Food Waste Management, current national food-waste data and prevention hierarchy.
- National Environment Agency Singapore, Food Waste Management Strategies, including prevention at source and redistribution of unsold edible food.
Continue Through eduKateSG
Continue with How Mathematics Works. Compare this article with Keeping Electricity Flowing Through a Changing Grid: both systems hold reserve against uncertain demand. Also compare it with When the Right Match Can Save a Life, where optimisation again helps allocate scarce resources but cannot decide the human objective by itself.
Final Thought: The Best Shelf Is Not the Fullest Shelf
A perfectly full shelf looks safe.
Behind it may be a warehouse full of tomorrow’s waste.
An almost empty shelf looks efficient.
One unexpected delivery delay can turn it into a stockout.
Inventory lives between those pictures.
Forecast demand.
Measure uncertainty.
Choose a service level.
Count the cost of waste.
Order before the future arrives.
Then learn from what sold and what did not.
Mathematics improves the world here not by predicting every customer perfectly.
It improves the world by making uncertainty manageable enough that the shelf can stay useful without the warehouse becoming a monument to fear.