Keeping capacity for a higher-paying later request is not justified by the higher price alone. The useful comparison is the value of one more protected unit: how often would later demand actually use that particular unit, and what certain early revenue would be given up?
In a simple two-period model, let later demand be the nonnegative integer D, and let each later sale bring H. The expected value of the yth protected unit is H × P(D ≥ y). If an early sale would certainly bring L under the same revenue definition, protecting that extra unit changes expected revenue by H × P(D ≥ y) − L.
This guide derives that expression and checks every candidate policy in an original four-unit example. All services, demand probabilities and revenue values are fictional teaching assumptions. The model supplies no pricing recommendation, forecast for an actual business or guarantee of financial results. It deliberately excludes cancellations, overbooking and connected dates so that the marginal-capacity question can be understood clearly.
The Yield Management subject guide introduces expiring capacity, uncertainty and service promises. This branch develops its capacity-protection question into a complete finite calculation, then changes the assumptions to show which conclusions survive.
eduKateSG · Yield Management
Find your next learning step
Choose the route that matches the question you are trying to answer, or follow the chapters in order. Each worked example states its assumptions before asking you to apply the idea in a changed setting.
ROUTE 1 · CHAPTERS 1–4
Set up the service
Define the capacity unit, two sales periods and supplied demand.
ROUTE 3 · CHAPTERS 9–12
Test alternative assumptions
Examine discrete tails, ties, equal means and cost boundaries.
ROUTE 4 · CHAPTERS 13–16
Review the evidence
Separate feasibility, censored demand, choice and realised results.
ROUTE 5 · CHAPTERS 17–20
Practise and connect
Record assumptions, solve a new example and follow related lessons.
Chapter contents
Set up the service · Chapters 1–4
1. Define the unit and the period before valuing it
2. Make the arrival and sales assumptions visible
Value capacity at the margin · Chapters 5–8
5. Derive the tail-sum identity from unit indicators
6. Subtract the certain early sale at the margin
Test alternative assumptions · Chapters 9–12
9. Prevent the off-by-one error in a discrete demand tail
11. Compare two demand models with the same mean
12. Change the cost boundary before calling the result profit
Review the evidence · Chapters 13–16
13. Separate a capacity constraint from an allocation preference
14. Ask whether the demand data were censored by capacity
Practise and connect · Chapters 17–20
17. Prepare an evidence card for the allocation calculation
18. Practise a new three-unit service
CHAPTER 1 OF 20 · Set up the service
1. Define the unit and the period before valuing it
Our fictional service has four interchangeable places for one specified session. A place in a different session cannot repair a shortage in this one. The four units are already assumed deliverable; we are not estimating physical readiness or staff availability from a booking screen.
One accepted request consumes one unit. Every accepted customer is served under the model. There is no policy of accepting more commitments than capacity. This makes capacity protection different from overbooking, even though both subjects use probabilities in wider revenue-management discussions.
Let y be the number of units initially protected for later demand. It must be an integer from zero through four. The remaining 4 − y units are sold early under the conditions supplied in the next chapter. A protection level is an allocation choice, not a change to the offered price.
The distinction matters for interpretation. A policy can leave a unit unsold without having physically lost it; its sales opportunity expires when the session occurs. Another policy can fill every place and earn less revenue under the stated objective. Occupancy, revenue and profit are different outcomes.
Write the unit and date together on the worksheet. Then state which choices are allowed. If a learner quietly adds a second session, changes the price or permits extra commitments, the calculation no longer answers the original model. Richer decisions can be studied, but they need their own explicit problem.
CHAPTER 2 OF 20 · Set up the service
2. Make the arrival and sales assumptions visible
The teaching model has two non-overlapping periods. Early requests arrive first and are sufficient to fill every unit released to them. Each accepted early unit earns L = 45 revenue units. These sales are certain under the model and do not cancel.
Later requests then arrive with uncertain demand D. Each fulfilled later unit earns H = 100. No new early customers arrive in the later period, unused protected capacity has no salvage value, and the later-demand distribution is stipulated not to change with the protection choice.
Those assumptions are strong. They create a clean comparison between a certain early sale and an uncertain later opportunity. If early demand were insufficient, rejecting one early request would not always have the same opportunity cost. If customers switched between offers, the later distribution could depend on which early offers were available.
Our model treats each unit as interchangeable and every sale within a period as having the same revenue. It also ignores additional costs until we introduce a separate changed-boundary example. The initial objective is expected revenue, not full profit, customer welfare or every operational goal.
The purpose of this contract is not to claim that real services are this simple. It lets the learner identify why the next formula is valid. When a condition changes later, we can point to the exact step it affects instead of attaching a vague warning to an otherwise unexplained recommendation.
The original teaching distribution assigns probability 0.10 to no later requests, 0.20 to one, 0.30 to two, 0.25 to three and 0.15 to four. The probabilities sum to one. These are supplied model values, not estimated frequencies from an operating service.
| Later demand D | Supplied probability | Contribution to E[D] |
|---|---|---|
| 0 | 0.10 | 0 |
| 1 | 0.20 | 0.20 |
| 2 | 0.30 | 0.60 |
| 3 | 0.25 | 0.75 |
| 4 | 0.15 | 0.60 |
Mean later demand is 2.15 units. That average is useful but does not determine every unit’s chance of being sold. To value the first protected unit, we need P(D ≥ 1) = 0.90. For the second, P(D ≥ 2) = 0.70. The corresponding probabilities for the third and fourth are 0.40 and 0.15.
These are tail probabilities. Each asks whether enough later requests arrive to reach a particular unit. The first unit is used in every positive-demand case; the fourth only in the four-request case.
Check that the tails never increase with y. If a table says the fourth unit is more likely to be needed than the third under this single-resource model, the event definitions or probability calculations are inconsistent. A simple ordering check can catch an error before it propagates into an apparently sophisticated allocation rule.
CHAPTER 4 OF 20 · Set up the service
4. Explain what one protected unit actually sells
If y units are protected, the number sold later is min(D, y). Later sales cannot exceed either demand or protected capacity. For y = 2, the outcomes as D runs from zero to four are zero, one, two, two and two.
Using the supplied probabilities, expected later sales are 0 × 0.10 + 1 × 0.20 + 2 × 0.30 + 2 × 0.25 + 2 × 0.15 = 1.60. Protecting two units does not guarantee two later sales, and mean demand 2.15 does not make the expected fulfilled amount equal to 2.15.
Expected later revenue is therefore 100 × 1.60 = 160. The two unprotected units sell early for ninety, giving total expected revenue 250 under this policy.
The min operation is not a minor detail. Replacing it with D would count requests that cannot be served within the protected inventory. Replacing it with y would assume every protected unit sells regardless of demand. Both substitutions can make protection look more valuable than the stated model supports.
A learner should first enumerate the five outcomes by hand. That makes the capacity limit visible before expectation notation compresses it. The formula is useful because it summarises the complete outcome account; it should not replace understanding of which requests are fulfilled in each case.
CHAPTER 5 OF 20 · Value capacity at the margin
5. Derive the tail-sum identity from unit indicators
Write later sales as a sum of indicators: min(D, y) equals one if demand reaches the first unit, plus one if it reaches the second, and so on through the yth. In symbols, min(D, y) = Σ from k = 1 to y of 1{D ≥ k}.
Take expectations. Each indicator’s expectation is its event probability, so E[min(D, y)] = Σ from k = 1 to y of P(D ≥ k). For y = 2, that is 0.90 + 0.70 = 1.60, matching the full enumeration.
The identity does not require the indicator events to be independent. They are nested: reaching the fourth unit implies reaching the first three. Linearity of expectation still allows their probabilities to be added. Claiming independence here would add an unnecessary and false assumption.
For y = 3, expected later sales become 0.90 + 0.70 + 0.40 = 2.00. For y = 4, they become 2.15, matching mean demand because this particular distribution never exceeds four.
This derivation gives the marginal value directly. Increasing y from y − 1 to y adds the indicator for D ≥ y. The additional protected unit sells exactly when demand reaches that level. Its expected revenue is therefore H times that tail probability, rather than H times the mean demand or H times the probability of any positive demand.
CHAPTER 6 OF 20 · Value capacity at the margin
6. Subtract the certain early sale at the margin
Protecting one more unit gives up one certain early sale in our abundant-early-demand model. Its revenue cost is L. The added later value is H × P(D ≥ y). Thus the change from protection y − 1 to y is H × P(D ≥ y) − L.
For the four units, the later marginal values are ninety, seventy, forty and fifteen. Subtracting L = 45 gives expected-revenue changes of plus forty-five, plus twenty-five, minus five and minus thirty.
The first two additions improve the objective. The third and fourth reduce it. Because the tail probabilities decrease, these marginal changes cannot rise as y increases under this model. That structure makes the local comparison consistent with the complete finite-policy calculation.
This is the expected marginal-capacity reasoning associated with the classical two-class setting described in INFORMS' revenue-management history. The historical reference provides context; the distribution and calculations here are original teaching examples.
Do not read the rule as always wait for the higher offer. The relevant later value is probability weighted. Nor does it say always accept certain revenue. A sufficiently likely later use can have greater expected value. The rule’s usefulness comes from the exact comparison and the conditions supporting it, not from favouring certainty or high prices as general principles.
CHAPTER 7 OF 20 · Value capacity at the margin
7. Verify the optimum by checking every policy
There are only five protection levels, so calculate them all. Total expected revenue is R(y) = 45(4 − y) + 100E[min(D, y)]. This includes certain early revenue and the expected later amount under the supplied distribution.
| Protected units y | Early revenue | Expected later sales | Expected total revenue |
|---|---|---|---|
| 0 | 180 | 0 | 180 |
| 1 | 135 | 0.90 | 225 |
| 2 | 90 | 1.60 | 250 |
| 3 | 45 | 2.00 | 245 |
| 4 | 0 | 2.15 | 215 |
The maximum is 250 at y = 2. That agrees with the marginal calculation. The difference between adjacent totals is exactly the corresponding marginal change: forty-five, twenty-five, minus five and minus thirty.
This second route is an important check. If the marginal argument selected two but the enumerated table selected another policy, investigate the assumptions, tail definitions or arithmetic before trusting either result.
The conclusion is limited and concrete: among these five permitted policies, under this distribution and revenue-only objective, protecting two units maximises expected revenue. It is not a guarantee of the largest realised revenue in every demand outcome. It also does not settle a real service’s policy, because the model has deliberately left several consequential conditions outside its scope.
CHAPTER 8 OF 20 · Value capacity at the margin
8. Read the same comparison as an acceptance decision
Suppose the early period is still open and x units remain available. Accepting one early request earns forty-five and reduces the capacity left for later demand from x to x − 1. The expected later revenue lost is 100 × P(D ≥ x).
With four units left, that loss is fifteen, so the supplied expected-revenue comparison favours accepting the early request. With three left, the loss is forty, still below forty-five. With two left, the loss is seventy, greater than the early revenue. The model therefore stops early acceptance with two units remaining.
This state-by-state account gives the same protection level as the full-policy table. It also clarifies which tail probability belongs to the current decision. The marginal unit at stake is the last available later unit, not always the first one.
The state x must represent genuine remaining capacity after existing commitments. A total physical capacity of four is not the correct x if earlier accepted bookings already consume two. A current booking record matters before the mathematical comparison begins.
This is still a fixed two-period model. It does not describe repeated forecasting updates, mixed arrivals or changing prices. Those can alter the decision problem. Use the state version to understand the capacity opportunity cost in the supplied setting, rather than present a short formula as an algorithm for every real booking stream.
CHAPTER 9 OF 20 · Test alternative assumptions
9. Prevent the off-by-one error in a discrete demand tail
The yth protected unit sells when D is at least y, not only when D exceeds y. The difference matters because integer demand can put substantial probability exactly at the boundary.
For the second unit, P(D ≥ 2) is 0.70. P(D > 2) is only 0.40. Using the latter would omit the demand-two outcome, even though that outcome uses the second unit. The resulting expected value would be forty instead of seventy.
That error would reverse the comparison with the early revenue forty-five. It would incorrectly discourage protection of the second unit. The mistake is not a subtle optimisation issue; it is the wrong event for the marginal sale.
A reliable repair is to return to a specific outcome. If D = 2 and two units are protected, is the second sold? Yes. Therefore the equality case belongs in the tail. Derive the inequality from the event rather than choose whichever symbol looks familiar.
In a continuous model, a single exact demand value may have zero probability, making strict and non-strict thresholds equivalent for that distribution. That fact should not be carried into a discrete worksheet without checking. Model type matters. A carefully chosen boundary test can reveal a wrong rule even when most of the arithmetic and terminology appear correct.
Change only early revenue from forty-five to forty, leaving later revenue and demand unchanged. The marginal changes become plus fifty, plus thirty, zero and minus twenty-five. Protecting a third unit now leaves expected revenue unchanged.
The full totals are 160, 210, 240, 240 and 215 for y from zero through four. Both y = 2 and y = 3 maximise the stated objective. The expected-revenue model alone does not select one uniquely.
A separate, explicitly stated preference could distinguish them. A decision maker might consider another outcome or an agreed policy. Those considerations must be added to the account; they should not be disguised as a strict conclusion from equal expected revenues.
This is especially important in code. Replacing a strict comparison with a non-strict one can choose a different member of an optimal tie set. That may be acceptable if the tie rule is deliberate and documented. It is misleading to report that one policy had a larger expected value when the model says they were equal.
For a learner, ask what changed mathematically. The distribution did not change. The objective’s opportunity cost per protected unit did. The third unit’s later marginal value is forty, exactly matching the early sale given up. Recognising that equality is part of understanding the rule, not an awkward exception to be rounded away.
CHAPTER 11 OF 20 · Test alternative assumptions
11. Compare two demand models with the same mean
Consider two new teaching distributions, both with mean later demand two. In model A, D always equals two. In model B, D equals zero or four with probability one-half each. Keep capacity four, early revenue forty-five and later revenue one hundred.
Model A’s tails are one, one, zero and zero. Protecting the first two units is valuable; protecting more gives up early sales without adding later sales. Its expected-revenue totals for y = 0 through four are 180, 235, 290, 245 and 200. The optimum is two.
Model B’s four tails all equal one-half. Every additional protected unit has later expected revenue fifty, just above the early forty-five. Its totals are 180, 185, 190, 195 and 200. The optimum is four.
Mean demand is identical, yet the selected protection level differs. The marginal units depend on tail probabilities, not just the average number of requests. Replacing either distribution with its mean would remove precisely the information the allocation calculation needs.
This is not evidence that more variable demand always implies more protection. It is a specific counterexample to a mean-only rule. Different distributions, revenues and constraints can produce other results. A good explanation identifies which tail changed and how it affected each unit’s marginal value, rather than treat variability as a universal instruction to reserve more or less capacity.
CHAPTER 12 OF 20 · Test alternative assumptions
12. Change the cost boundary before calling the result profit
The first table maximises revenue because that is the objective supplied. If the decision concerns contribution after specified variable costs, use compatible contribution values for both periods. Do not subtract a cost from one offer while leaving the other on a gross-revenue basis.
For a separate invented example, let early revenue forty-five carry included variable cost five, giving contribution forty. Let later revenue one hundred carry included variable cost seventy, giving contribution thirty. Under these figures, even the first later unit’s expected contribution is only thirty times 0.90, or twenty-seven.
That is below the certain early contribution forty. All later marginal contributions are smaller still, so the same simple model would protect zero units when maximising this defined contribution. The gross-revenue optimum of two is not the answer to the changed objective.
This does not prove that later customers actually cost more in a real service. The costs were supplied specifically to expose the boundary issue. A full profit account may include other costs and effects that this contribution exercise omits.
The repair is to name what each amount includes and retain the same definition throughout the comparison. “Expected revenue” and “expected contribution under the listed costs” are useful, different statements. Calling both profit would make it harder for another reader to reconstruct why the model’s preferred allocation changed.
CHAPTER 13 OF 20 · Review the evidence
13. Separate a capacity constraint from an allocation preference
Capacity protection operates within the deliverable inventory. If one of the four units becomes unavailable, the feasible protection values change to zero through three. The same supplied demand tails can be used only if the demand model remains appropriate under the changed situation.
With capacity three and the original forty-five and one hundred revenue values, the marginal protection values remain ninety, seventy and forty. The model still protects two units, leaving only one early sale. The corresponding expected revenue is forty-five plus one hundred times 1.60, or 205.
That is not the old optimum 250 with one arbitrary deduction. The number of certain early units has changed, and the feasible policy set is different. Recalculate from the actual inventory definition rather than preserve a booking limit because it looked successful before the capacity update.
For connected dates or non-interchangeable units, a single scalar capacity can be insufficient. A two-night booking may need availability on both nights and perhaps a consistent room. The existing Checking Capacity Across Every Night of a Booking owns that feasibility problem.
The present branch assumes one period and interchangeable units so it can isolate uncertain opportunity value. A commercially attractive expected-value calculation does not make an infeasible request deliverable. Keep feasibility and preference as separate checks, with the record appropriate to each.
CHAPTER 14 OF 20 · Review the evidence
14. Ask whether the demand data were censored by capacity
The supplied D distribution describes later requests, not merely later sales. If a service had only one unit available, observing one sale would not establish that later demand was exactly one. Additional requests might have been refused or never observed.
A fictional record with repeated sell-outs can therefore understate demand if it is treated as an unconstrained request history. The protection calculation depends particularly on the upper tails, which are precisely the part a binding capacity limit can hide.
This does not justify inventing missing requests. It identifies a data limitation and the need for an appropriate estimation method or additional evidence. The root’s distinction between bookings and demand remains essential before probabilities are inserted into the marginal rule.
A useful classroom comparison labels a count as enquiries, accepted reservations, fulfilled units or requests under a specified observation process. Those counts can differ for valid reasons. A probability table built from the wrong event can be internally consistent and still answer the wrong question.
For an actual study, the observation process and model estimation need suitable expert treatment. Our worksheet begins with a complete stipulated demand distribution so that the optimisation can be checked exactly. It does not demonstrate that a real operator can recover an uncensored demand distribution from a short sales spreadsheet, or that a clean-looking histogram resolves every missing-demand problem.
CHAPTER 15 OF 20 · Review the evidence
15. Recognise when customer choice changes the model
We assumed later demand was unaffected by the early allocation. Real customers may have alternatives, change when they book or select another offer if one is unavailable. If those behaviours matter, a fixed exogenous D distribution may not describe every protection policy.
For a fictional counterexample, imagine that closing an early offer causes some customers to wait and request the later offer. Later demand now depends on the earlier availability choice. Comparing policies with one unchanged distribution would omit that relationship.
Another change is interleaved arrivals: early- and late-value requests appear in a mixed stream rather than separate periods. Information can update as time passes, and the remaining horizon matters. A static protection level might remain a useful model component, but its optimality does not follow automatically from our two-period derivation.
The historical Littlewood paper publication record provides an onward route into the origins of booking-control models. Its publisher’s accessible abstract establishes the historical context; it is not a current operating prescription. More elaborate contemporary systems require their own specified objectives, demand models and validation.
A learner should be able to say exactly which assumption changed. “Reality is more complicated” is true but unhelpful. “The distribution of later requests now depends on the early offer” identifies the missing relationship and explains why a probability table can no longer be reused unchanged across every candidate policy.
CHAPTER 16 OF 20 · Review the evidence
16. Compare expected performance with realised outcomes
Under the original optimum y = 2, the early sales earn ninety. If later demand is zero, realised total revenue is ninety, below the 180 obtained by filling all four units early. If later demand is four, two later units sell and total revenue is 290, above 180.
Neither realised outcome alone proves that the original expected-value calculation was right or wrong. The model compared repeated outcomes using the supplied probabilities. It did not promise to beat every alternative in every possible case.
Expected unused protected units are E[(2 − D)⁺], where the positive part means the larger of the expression and zero. Here the amount is two with probability 0.10, one with probability 0.20 and zero otherwise, giving 0.40 units on average.
Expected occupied units are two certain early sales plus 1.60 expected later sales, or 3.60 out of four. The corresponding ratio of expected occupied to fixed available capacity is ninety per cent. Full early occupancy is one hundred per cent but has lower expected revenue in this model.
These are different outcome measures. A service may have objectives that make unused capacity particularly costly or require a separate constraint. Add such conditions explicitly. Do not pretend the revenue maximum also maximises occupancy, removes disappointment or settles every purpose for which the session exists.
CHAPTER 17 OF 20 · Practise and connect
17. Prepare an evidence card for the allocation calculation
An allocation result should travel with enough information to reproduce it. State the service unit, period, actual capacity, allowed protection values, arrival order, early-demand assumption, revenue or contribution definition, later-demand distribution and treatment of unused inventory.
Then show either the full policy table or the marginal values, preferably both for a small example. Record ties and the rule for choosing among them. State which assumptions were supplied rather than estimated from evidence.
A reader should be able to ask what would change the answer. A higher early contribution raises the opportunity cost of protection. A change in upper-tail demand alters the later marginal values. A capacity update changes the feasible state. A customer-choice effect can require a different model rather than a new number in the old table.
This record is a learning and review tool, not an automatic authorisation to alter real offers. Actual commitments and the service purpose remain relevant. A mathematically optimal allocation under an inappropriate objective is not made appropriate by accurate arithmetic.
The practical standard is a conclusion narrow enough to inspect: “Among these stated policies, this one has the largest expected revenue under this distribution.” That sentence supports useful discussion. “The algorithm says this is best” conceals the objective, the assumptions and the other outcomes a responsible decision may need to consider.
Independent task A has capacity three, certain early revenue thirty per released unit, later revenue eighty, and later demand probabilities 0.20, 0.30, 0.30 and 0.20 for zero through three requests. Find the demand tails, marginal values and preferred protection level.
The tails are 0.80, 0.50 and 0.20. Later marginal values are sixty-four, forty and sixteen. After subtracting thirty, the changes are plus thirty-four, plus ten and minus fourteen. Protecting two units therefore maximises the stipulated expected-revenue objective.
Check every total. With y = 0, revenue is ninety. With one protected unit, it is sixty plus sixty-four, or 124. With two, it is thirty plus eighty times 1.30, or 134. With three, it is eighty times 1.50, or 120.
Task B changes early revenue to forty. The two middle totals become 144 and 144, so protection one and two tie. Explain why a program might return either if its documented tie rule differs.
Task C retains the original thirty but says early demand may fail to fill released inventory. Can the same certain early term thirty times capacity released still be used? Not without the additional demand account. Identify the missing relationship before calculating. That final task checks the assumption behind the opportunity cost, rather than only the arithmetic of the demand tails.
CHAPTER 19 OF 20 · Practise and connect
19. Questions about protecting capacity under uncertainty
Should every unit be protected when the later price is higher?
No. The relevant later value is conditional on enough demand reaching that unit. In the four-unit example, the fourth unit has later expected revenue fifteen even though the later selling revenue is one hundred.
Does mean demand determine the protection level?
No. The two mean-two distributions produce different optimal levels. The marginal comparison needs the relevant demand tails under the stated model.
Does the expected-revenue optimum guarantee the best realised result?
No. Different policies win in different demand outcomes. The objective averages those outcomes using the supplied probabilities. A separate risk or service objective requires its own explicit treatment.
Is capacity protection overbooking?
Not here. The model never accepts commitments beyond deliverable capacity. It decides how much capacity to release early versus retain for later requests. The existing airline overbooking guide studies another mechanism.
Can this calculation be copied into a real pricing system?
The worksheet does not establish that a real service satisfies its assumptions or objective. Inventory, customer terms, demand estimation, costs and operating constraints need the appropriate evidence and review. A correct model calculation is one part of a decision, not a complete service policy.
CHAPTER 20 OF 20 · Practise and connect
20. Continue through capacity, demand and service evidence
For the broader subject, return to Yield Management and Economics. For a different capacity question, read Checking Capacity Across Every Night of a Booking, which examines dated feasibility rather than uncertain marginal opportunity.
For uncertain arrivals beyond accepted capacity, use Airline Overbooking, Probability and Capacity Risk. Keep that guide’s show-up and shortage problem distinct from the two-period protection problem here. For records and distributions, follow Data and Probability.
Primary source routes reviewed on 22 October 2026 are INFORMS' account of revenue-management development and Ken Littlewood's 2005 publication record for work presented in 1972. The latter has an accessible publisher abstract, with full text subject to its access arrangements. The finite distributions, tail-sum derivation, policy tables, cost example and practice cases here are original teaching constructions.
Continue through Learning That Transfers into Life and How X Works. A useful final explanation values one marginal unit, checks the full table and names an assumption that could change the answer. That keeps the mathematics connected to a defined decision instead of letting a higher price answer a probability question by itself.