A bakery deciding tomorrow’s fresh batch faces a different problem from a workshop replenishing a durable component throughout the year. Both hold inventory, but the cost of leftovers, the timing of replenishment and the uncertainty in demand differ. The newsvendor problem and economic order quantity make those differences explicit. A third model, Kingman’s queueing approximation, helps explain why filling every moment of capacity can create long waits even when average demand looks manageable.
The newsvendor balances too much against too little
The newsvendor chooses a quantity before demand is known for a selling period. Unsold units have a salvage value, while an unmet sale loses a contribution margin. Ordering more reduces shortage exposure but increases leftover exposure. The best quantity under the stated objective depends on both the demand distribution and this asymmetry of costs.
Play the newsvendor inventory game by choosing a quantity, observing a demand draw and comparing realised profit with expected profit. MIT’s supply-chain planning materials give the broader inventory context. A lucky high-demand round does not prove that a large order was the best decision in advance. Compare the same policy across the full specified distribution, not only the outcome it happened to produce.
Worked example: choose a quantity with salvage value
The game uses invented demand values 20, 40, 60, 80 and 100 with probabilities 0.1, 0.2, 0.4, 0.2 and 0.1. Each unit costs 6, sells for 10 and salvages for 2.
- Calculate the cost of being one unit short as 10 − 6 = 4 and the cost of one leftover unit as 6 − 2 = 4. The critical cumulative probability is 4 / (4 + 4) = 0.5.
- Accumulate demand probabilities: the distribution reaches 0.3 at forty units and 0.7 at sixty. The smallest quantity reaching the one-half threshold is sixty. This selects a policy before observing the round’s demand.
- At quantity sixty, profits across the five demand outcomes are −80, 80, 240, 240 and 240. Weight them by their probabilities to obtain expected profit 176. Compare another quantity using the same demand distribution rather than comparing two different lucky draws.
Economic order quantity balances ordering against holding
Economic order quantity, or EOQ, assumes a steady demand rate, a fixed cost per order, a holding cost per unit per year and instantaneous replenishment under the simplified model. Larger batches mean fewer orders but more average stock. With no shortages and the other assumptions fixed, the relevant annual cost is ordering cost plus holding cost: DS/Q + HQ/2.
In the EOQ simulation, annual demand is 1,200 units, ordering cost is 40 and holding cost is 2. A batch of 100 has relevant annual cost 580. The square-root optimum is about 219 units, with cost about 438. These values omit purchase expenditure that does not change with the batch size. Quantity discounts, spoilage, uncertain lead time and capacity limits can change the appropriate model.
Queueing adds a capacity question
A supply system also needs time to process orders or serve customers. In Kingman’s approximation, mean waiting time grows with variability and with utilisation relative to spare capacity. The model describes a stable single-server queue approximately; it is not a universal exact formula for a warehouse or an entire production network.
Kingman’s original paper supplies the heavy-traffic research context. The board’s mean service time of two minutes, utilisation 0.7 and arrival and service coefficients of variation equal to one give an approximate queue wait of 4.67 minutes. Reducing both variability measures to 0.5 gives about 1.17 minutes while mean service and utilisation remain fixed. Changing averages alone would miss that distinction.
Go a little deeper: Optimise the policy before judging one outcome
Expected profit averages possible outcomes under a stated probability model. Realised profit is one draw from that model. A policy can be reasonable before a draw and still produce a loss. Conversely, a poor policy can get lucky. Evaluate whether the probabilities, costs and objective were appropriate, then use repeated outcomes to learn where those assumptions need revision.
Choose a model around the decision
Answer these before using an inventory formula. Each question changes either the feasible actions or the cost of making the wrong commitment.
- Is this a one-period commitment or a repeating replenishment decision? State when an order is placed and when demand becomes known.
- What happens to leftovers and unmet demand? Distinguish salvage, disposal, lost sales, backorders and reputational effects rather than using one generic shortage penalty.
- Are demand, delivery and processing times predictable enough for the selected model? Use observations to investigate variability instead of assuming it away.
- Compare policies under the same scenarios and objective. Check constraints such as storage, cash, minimum batches and service commitments before using an unconstrained optimum.
Extend the exercise without making it opaque
If your next question is which available task to serve first, try shortest-job-first scheduling. It uses jobs available at time zero with known durations, so a reduced mean wait has a clear interpretation. Real scheduling may add unknown durations, priorities and arrivals that require a different rule.
If the question is how production experience affects unit cost, try Wright’s law. Its learning curve is a selected scenario rather than a forecast of guaranteed improvement. Keeping these questions separate lets you combine inventory, capacity and learning carefully without hiding uncertainty behind a single impressive-looking dashboard.
Try the ideas for yourself.
These are teaching models. Follow the assumptions in each experiment; the results are not real-world forecasts.
Sources & further reading
Current-event context was checked on October 28, 2026. Follow the original source for newer updates. Worked scenarios are illustrative unless explicitly identified as reported data.
- MIT OpenCourseWare — Supply chain planning ↗
University teaching materials on inventory and supply-chain planning.
- Kingman — The single server queue in heavy traffic ↗
Primary research behind the single-server heavy-traffic approximation.