Overbooking

Airlines, hotels, and other perishable-inventory businesses routinely sell more capacity than physically exists. The reason is simple: a significant fraction of customers who book will cancel or fail to show up. Not overbooking means flying with predictably empty seats.

The Problem

Consider a flight with $C = 150$ seats and a historical no-show rate of 15%. If the airline sells exactly 150 bookings, it expects only $150 \times 0.85 = 127.5$ passengers to appear—leaving 22.5 seats empty on average. But if it overbooks to 170 bookings, it expects $170 \times 0.85 = 144.5$ passengers—far fewer wasted seats.

The risk is obvious: if all 170 passengers show up, 20 must be denied boarding at a cost of $500–$1,000 each (compensation, rebooking, goodwill damage). Finding the right overbooking limit balances the cost of empty seats (spoilage) against the cost of turning away confirmed passengers (denied boarding).

Mathematical Formulation

Definition — No-Show Rate

The no-show rate is the fraction of bookings for which the passenger fails to appear for departure. If $q$ denotes the probability that any single booking results in a show-up, the no-show rate is $1 - q$ .

Definition — Overbooking Limit

The overbooking limit $b$ is the total number of bookings the firm accepts for a resource with physical capacity $C$ . When $b > C$ , the resource is overbooked by $b - C$ units.

Definition — Denied Boarding

Denied boarding occurs when the number of show-ups $S$ exceeds physical capacity $C$ . The number of passengers denied boarding is $\max(S - C, \, 0)$ .

Stochastic Model

Let $b$ be the number of bookings accepted, $C$ the physical capacity, and $q$ the show-up probability. Each booking independently results in a show-up with probability $q$ , so the total number of show-ups follows a Binomial distribution:

S \sim \text{Binomial}(b, q)

(1)

For large $b$ (typical of airline or hotel settings), the normal approximation is accurate:

S \;\approx\; \mathcal{N}\!\bigl(bq, \; bq(1-q)\bigr)

(2)

Objective Function

Let $f$ be the fare per boarded passenger and $D$ the denied-boarding cost per passenger. The net revenue from accepting $b$ bookings is:

\Pi(b) \;=\; f \cdot \min(S, C) \;-\; D \cdot \max(S - C, \, 0)

(3)

The firm’s objective is to choose $b$ to maximize the expected net revenue:

\max_{b} \; \mathbb{E}[\Pi(b)] \;=\; f \cdot \mathbb{E}\bigl[\min(S, C)\bigr] \;-\; D \cdot \mathbb{E}\bigl[\max(S - C, \, 0)\bigr]

(4)

Optimality Condition

Marginal Overbooking Condition

The optimal overbooking limit $b^*$ satisfies the condition that the marginal expected revenue from one additional booking equals the marginal expected denied-boarding cost:

f \cdot P(S(b^*) \le C) \;=\; D \cdot P(S(b^*) > C)

Equivalently, the probability of denied boarding at the optimum is:

P(S(b^*) > C) \;=\; \frac{f}{f + D}

Under the normal approximation, the denied-boarding probability at any overbooking level $b$ is:

P(S > C) \;=\; 1 - \Phi\!\left(\frac{C - bq}{\sqrt{bq(1-q)}}\right)

(5)

where $\Phi$ is the standard normal CDF. Setting this equal to $f / (f + D)$ and solving for $b$ yields the risk-optimal overbooking limit.

Risk vs Service-Level Policies

Risk-Based Policy

The risk-based approach directly minimizes total expected cost (spoilage plus denied boarding) or, equivalently, maximizes expected net revenue as formulated above. The optimal limit $b^*$ is the point where the marginal revenue from one more booking exactly offsets the marginal expected denial cost.

Service-Level Policy

Many carriers instead impose a service-level constraint: the denied-boarding rate must not exceed a specified threshold $\alpha$ (for example, 2%). This translates to:

P(S > C) \;\le\; \alpha \quad\Longrightarrow\quad b_{\max} \;\approx\; \frac{C + z_{1-\alpha}\,\sqrt{bq(1-q)}}{q}

(6)

where $z_{1-\alpha}$ is the $(1-\alpha)$ -quantile of the standard normal distribution. Since the right-hand side depends on $b$ , this is solved iteratively or by binary search.

The service-level approach is typically more conservative than the risk-based optimum. It sacrifices some expected profit in exchange for a hard cap on the frequency of denied-boarding events—a trade-off that is often preferred for reputational reasons.

Interactive Explorer

Adjust the capacity, no-show rate, fare, and denied-boarding cost, then press Run Simulation to generate Monte Carlo trials. The left panel shows the histogram of show-ups (green = no denial, red = denial occurs). The right panel sweeps the overbooking limit to display expected net revenue under the normal approximation, marking both the current $b$ and the optimal $b^*$ . Toggle between risk-based and service-level modes to compare policies.

Key Insights

1. Not Overbooking Wastes 10–15% of Capacity

At typical no-show rates of 10–20%, selling exactly $C$ bookings leaves a corresponding fraction of seats empty on average. The revenue lost to spoilage can exceed the total expected cost of denied boardings under optimal overbooking.

2. Optimal Overbooking Has Surprisingly Low Denied-Boarding Probability

Even at the risk-optimal level, the probability of any denial is typically only 1–3%. Most flights depart with all passengers accommodated. Use the simulator to verify this: at default settings, the red portion of the histogram is a small tail.

3. Higher No-Show Rates Justify More Overbooking

Drag the no-show rate slider upward and observe that $b^*$ increases. More no-shows mean more predictable empty seats, so the airline can safely accept more bookings.

4. Denied-Boarding Cost Is the Key Parameter

Increase the denied-boarding cost from $400 to $1,500 and watch $b^*$ drop sharply. When denials are expensive (international flights, loyal customers), the optimal policy is conservative. When they are cheap (short-haul, flexible passengers), aggressive overbooking is justified.

5. The Normal Approximation Is Accurate for Typical Flight Sizes

Compare the Monte Carlo histogram (which uses exact Binomial draws) with the normal-approximation revenue curve. For capacities above about 50, the two agree closely. Below that threshold, the discrete nature of the Binomial becomes noticeable.

Extensions

Dynamic overbooking—The static model assumes a single decision before departure. In practice, airlines update the overbooking limit as the departure date approaches and cancellations arrive, using real-time information to refine the no-show forecast.
Customer-specific no-show predictions—Machine learning models can estimate individual show-up probabilities based on booking channel, fare class, loyalty status, and connection itinerary. This heterogeneity allows more precise overbooking by accounting for the composition of the current booking list.
Voluntary buyback programs—Rather than involuntarily denying boarding, airlines solicit volunteers willing to take a later flight in exchange for compensation. The optimal compensation offer is itself a pricing problem, and successful buyback programs substantially reduce the effective denied-boarding cost.

References

Chatwin, R. E. (1998). “Multiperiod Airline Overbooking with a Single Fare Class.” Operations Research, 46(6), 805–819.
Karaesmen, I. & van Ryzin, G. J. (2004). Overbooking with Substitutable Inventory Classes. In The Theory and Practice of Revenue Management, 123–153.
Phillips, R. L. (2021). Pricing and Revenue Optimization, 2nd ed.. Stanford University Press.
Rothstein, M. (1971). “An Airline Overbooking Model.” Transportation Science, 5(2), 180–192.
Talluri, K. T. & van Ryzin, G. J. (2004). The Theory and Practice of Revenue Management. Springer.

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