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Weekly UpdatesThus, for any particular day, taking a combination of reservations, a 90-percent customer-service level is achieved by way of the following inventory-depletion formula:
(.967) [X.sub.1] + (.972) [X.sub.2] = 786
where,
[X.sub.1] = booking level for expected arrivals
[X.sub.2] = booking level for stayovers
Thus, the following combinations can be authorized: (.967) 420 + (.972) 391 = 786 and (.967) 330 + (.972) 480 = 786.
In the end, we do nor have fixed cut-off points for reservations, but a dynamic series of combinations of expected arrivals and stayovers. It has been demonstrated that as long as the distributions of no-shows and early departures are statistically independent, reservations can be cur off when the sum of the optimally weighted reservations approaches 786. (19) Computer technicians can easily program these dynamic real-time decision rules into the hotel's reservation system.
Returning to the 150 luxury and handicap-accessible rooms, as we said, hotels normally do not overbook them. Stayovers can be a factor for those rooms, however. If the average unexpected stayover is six, then the hotel has a working inventory of only 150 - 6 = 144 special rooms. Given a 100-percent customer-service level for those rooms, the working inventory of 144 serves as the maximum combination of new arrivals and stayovers booked. However, if the 90-percent customer-service level were applied, then the following will hold:
Assume that the weighted average of the no-show and early departure rate is 4 percent, then,
p = .04
q = .96
[sigma] = [[square root of (pq/n)] = [[square root of ((.04) (.96)/144)] = .016
z = 1.28
The lower limit of the proportion of no-shows and early departures is therefore .04 - (1.28) (.016) = .02. This means that the upper limit of those showing up or staying is .98. Thus we can be 90-percent confident that no more than (.98) 146, or 143 protected rooms will be filled during peak demand periods. This, in turn, means that we can add 146 - 143 = 3 rooms to our working inventory of 786 standard rooms while maintaining the 90-percent customer-service level. This increases the maximum cut-off points for expected arrivals and stayovers, and affects the inventory-depletion ratios, all of which can be recomputed as we showed above.
Setting Booking Levels
In setting optimum booking levels, hotels need to determine only two factors: z (the customer-service level) and p (the probability of an empty room arising from a no-show or early departure). The z figure is much more difficult to determine, because it is derived by trading off the opportunity cost of empty rooms with the adverse consequences of oversales. The second can easily be computed from historical records. We assume a normal (bell-shaped) distribution of the proportion of no-shows and early departures, based on the properties of the Central Limit Theorem, while the magnitude of the standard deviation of proportion of no-shows and early departures can be approximated by binomial theory. Both are implicit in the optimizing formula that we explained above:
{1 - [p - z[square root of (pq/n)]]} X = C
which, through legitimate transformations, can be operationalized to the quadratic form:
[(1 - p).sup.2] [X.sup.2] - [2C(1 - p) + [z.sup.2]pq] [X.sup.2] + C.sup.2] = 0
We present the above overbooking model in its irreducible simplest form, so that rooms managers without any sophisticated knowledge of statistics or operations research can use it to augment their personal judgment regarding rooms-inventory management. (20) It is by no means a comprehensive model, because (as noted earlier) it's impossible to take into account every conceivable factor or variable that affects room occupancy. For instance, block reservations may have a distinctly different no-show rate than individual reservations do, and some groups are more reliable in showing up than are others.
Moreover, each of the hotel managers we interviewed knew which days of the week have the highest no-show rate, and they also mentioned that they could always expect a considerable number of early departures on a conference's penultimate day. Scheduled and unscheduled maintenance and repairs can also affect room capacity. Suffice it to say that operating with an overbooking model necessitates management intervention for extraordinary circumstances, such as when an entire city or district is sold out, making walks impossible.
The other purpose of advancing our simplified overbooking model is to allow rooms managers of large hotels currently using sophisticated computer programs to have some understanding of the sort of factors that allow the computer to make continuous real-time adjustments to inventory. We have already discussed many of those factors, including the number of expected arrivals and stayovers, historical rates of no-shows and early departures, the room capacity of the hotel and the average unexpected stayover rate, the effects of group reservations and provisions for reducing the size of a block over time as per sales agreement, the possibility of upgrades, management's input on the opportunity cost of empty rooms and loss of goodwill attendant to oversales, and complicated decision rules on displacement analysis associated with accepting multiple-night reservations at the expense of possibly walking an individual guest. Furthermore, our inventory rules suggest that all hotel overbooking models have algorithms that allow for continuous real-time adjustments to inventory as new reservations are accepted, cancellations are received, and early departures occur.
Note further that the customer-service level specified in our model applies only to peak periods. When total reservations (expected arrivals plus booked stayovers) do not deplete the revised working inventory of 789 standard rooms, oversales will not occur. Thus, on an annual basis, the customer-service level will be much higher. For our example, if the hotel overbooks to the maximum level permitted, it will have to walk guests 4 out of 40 times during peak periods. This means that it will walk guests only four days in a year. How many guests may get walked is explained next.
A 90-percent customer-service level does not mean that every guest will encounter a 10-percent chance of being walked. It merely means that during peak-demand periods, oversales will occur an average of one day out of ten. The actual number of walks involved would be relatively small. Exhibit 4 indicates that of the 13,759 guests expected to arrive, 69 were walked, representing a ratio of only 0.5 percent, or five out of 1,000, even though walks occurred 50 percent of the time during peak periods. The aggressive overbooking led to an average occupancy of 799 out of 800 standard rooms sold during peak periods, and washes (empty rooms) occurred on only nine out of 40 peak nights. A wash is not particularly bad as long as there are not too many empty rooms. Indeed, one hotel manager whom we interviewed considered nights with a few empty rooms to be sold out for all practical purposes.
Another point is that the instruction against exceeding the combined authorized booking levels by way of the inventory-depletion model applies only to the day of arrival. Like the airlines, hotels recognize that early reservations tend to be tentative, and may therefore overbook aggressively in advance of the arrival date to compensate for anticipated late cancellations and group attrition. Rooms managers are usually experienced enough to ensure that the authorized booking levels are seldom exceeded on the day of arrival.
One final observation is that hotels have done a much better job of managing the no-show problem than airlines. In particular, requiring guests to guarantee their reservations by credit card has reduced the hotel no-show rate from between 10 percent and 15 percent to about 4 percent. (21) All six rooms managers whom we interviewed volunteered the observation that since hotels started to accept only reservations guaranteed by credit cards, the no-show rate has plummeted. (22) This complements the fact that penalties on early departures have had a similar effect on the rate of unexpected departures. The airlines, on the other hand, have a no-show rate of 15 percent or more. (23) Additionally, the hotel industry has done a better job than the airlines of sharing information on no-shows, early departures, and group cancellations.
Our Recommendations
Whenever we asked our rooms-manager respondents about overbooking, we noticed that all of them indicated that the level of overbooking depends on the number of expected arrivals. While that is an important factor, we draw rooms managers' attention to the equal importance of early departures and stayovers, because they have the same effect on empty rooms as new guests who do or do not show up. We also suggest that managers take into consideration the working inventory of rooms when setting overbooking levels, rather than use the physical inventory as the benchmark.
We make the following additional recommendations. First, a hotel that wants to invoke early departure penalties must give guests prior notice of that policy when it confirms a reservation and again upon check-in. One must remain flexible, though, because early departure penalties can become a service issue. That is, if a guest complains about an early departure penalty (given that it is not always levied), the matter is better dropped--even if the guest has initialed a document confirming a check-out date. Half of the hotels that we surveyed attempt to levy early departure penalties, but all drop the fee if the guest becomes irate. Ironically, meeting planners favor the use of early departure penalties, because those fees prevent conventioneers from leaving early (thus invoking attrition penalties), and make it easier to plan for final banquets. Indeed, we have found that some meeting planners have started imposing their own no-show and early departure penalties on their meeting participants. Discussing early departure penalties at check-in also compels guests to announce their intention to shorten their stay (if that is the case), thus allowing other guests to gain access to rooms that would have been booked but left empty.
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