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Weekly UpdatesUsing historical and accounting records, plus ones own judgment, one can assign explicit values to the following relevant variables: opportunity cost of empty rooms, prevalence of walk-ins as a last-minute substitute for no-shows and early departures, value of customer goodwill, the availability of nearby hotels of comparable standard, the cost of walking displaced guests, and legal penalties of walking a conventioneer protected by a no-walk agreement. Suppose that the overall cost of an empty room less variable cost is high and that walk-in traffic is insignificant. Also suppose that there are excellent prospects for room upgrades because the luxury and handicap-accessible rooms are seldom sold out, and there are many comparable hotels in the vicinity. Additionally, there are reciprocal-walk arrangements with comparable hotels offering discounts (usually around 33 percent). While the presence of such arrangements suggests that the hotels maintain a high overbooking rate and a low customer-service level, most of the hotel's guests are repeat business travelers and many of its group-sales agreements include no-walk provisions with heavy penalties for oversales. Moreover, suppose that employee morale has been undermined by having to deal with unhappy displaced guests. Those factors argue for a low overbooking rate and a high customer-service target. Given all available data and judgments, suppose that the hotel considers a 90-percent customer-service level to be best. Thus, guests are walked 10 percent of the time during peak periods when the hotel overbooks.
Hotels can define their customer-service level in this fashion because they have advance warning of being overbooked. Unlike airlines, which cannot know that they are oversold until a short time before the plane takes off, hotels are aware of their oversale situations by checkout time (normally 11:00 AM). Thus, hotels can make advance arrangements for alternative accommodation, determine which guests to protect, and alert the front desk to the likelihood of walking guests--all before the guests present themselves. Our rooms managers informed us that walking guests is one of the most stressful parts of their duties exactly because they cannot conduct an impromptu auction and ask for volunteers to be walked. Thus, the issue is not the number of walks so much as it is the fact that the front desk must gear up for managing any walks at all.
For our purposes, the customer-service level is defined as the percentage of peak days during which the hotel walks at least one guest due to overbooking. Determining the appropriate customer-service level is an extremely challenging task. To set customer-service levels, one has to balance the cost of empty rooms--which is relatively easy to calculate--with the consequences of an oversale--which is difficult to quantify, given the loss of customer goodwill and adverse effect on employee morale.
Airlines routinely set customer-service levels. For instance, on the Singapore-to-Jakarta route, Singapore Airlines and Garuda Airlines could reduce their usual customer-service levels because they operated 32 round trips per day between them and could make a revenue-pooling arrangement work. Moreover, since their flights were closely scheduled, bumping an oversold passenger would have meant a 30-minute travel delay for that individual--usually not a hardship. (14) In analogous fashion, many large downtown hotels ate located near one another. Thus, a displaced guest would for the most part not be terribly inconvenienced, and a reduced customer-service level could be acceptable.
Having explained the factors underlying a determination of the customer-service level, we calibrate the model, using our customer-service level of 90 percent on peak days. As shown in Exhibit 4, of the 40 days when the hotel had overbooked during peak periods, there were 20 days when at least one guest had to be walked. In other words, the hotel had an implicit 50-percent customer-service level during peak periods, far less than the optimal level of 90 percent. The model can help us determine the authorized booking level that will permit the desired level of service.
To begin building the model, each reservation can be treated as a Bernoulli problem: the guest either shows up or is a no-show, and will either stay over as expected or depart early. (15) This problem is addressed by calculating the standard error of proportions, as follows:
[sigma] = [square root of (pq/n)]
where:
[sigma] = standard error of proportion of no-shows or early departure
p = probability of a no-show or early departure
q = probability of a show up or stayover
n = number of expected arrivals or stayovers
Next, the Central Limit Theorem will guarantee that the sampling distribution of the proportion of no-shows or early departures is approximately normal (i.e., a bell-shaped curve) if the sample size (the average number of guests with reservations) is sufficiently large. (16)
Given the mean proportion, standard error, and the assumed normality of the distribution of no-shows or early departures, a one-sided confidence interval reflecting the desired customer-service level can be constructed to give us the lower limit of the normal distribution, since we need to protect only against the lower proportion of no-shows or early departures to prevent oversales.
Mathematically, this is expressed as:
{1 - [p - z [square root of (pq/n)]]} X = C
where:
X = authorized booking level
C = working inventory of rooms, which is the number of rooms less the average number of unexpected stayovers
z = standard normal deviate corresponding to the desired customer-service level
Note that we have one equation with only one unknown, where p, q, z, and C are all known constants and X (the authorized booking level) is the only unknown variable. Our goal is to create a single factor to express X (allowing us to plug in known values to derive a value for the authorized booking level, X). By a process of legitimate mathematical transformations, we can obtain the following cubic equation, which has three values for X:
[(1 - p).sup.2] [X.sup.3] - [2C(1 - p) + [z.sup.2]pq] [X.sup.2] + [C.sup.2] X = 0
Factoring out X, it is clear that one of the three roots of the cubic equation is X = 0, allowing us to work with the following equation:
X[[(1 - p).sup.2] [X.sup.2] - [2C (1 - p) + [z.sup.2]pq)] X + [C.sup.2]] = 0
Dividing throughout by X, we reduce the cubic equation to the following quadratic form, which will have two possible values for X:
[(1 - p).sup.2] [X.sup.2] - [2C (1 - p) + [z.sup.2]pq)] X + [C.sup.2]] = 0
Restating the above equation in terms of X, the solution to the general quadratic equation is:
X = -b[+ or -][square root of ([b.sup.2] - 4ac)]/2a
where:
a = [(1 - p).sup.2]
b = - [2C(1 - p) + [z.sup.2]pq]
c = [C.sup.2]
Since the discriminant [b.sup.2] - 4ac must always be positive (as is clear from inspection), the quadratic equation will have two real and distinct roots, [r.sub.1] and [r.sub.2]. However, only one of the roots will satisfy the original equation. The solution will therefore be unique, and there will be only one authorized booking level. (We will clarify this calculation by calibrating the model with data from Exhibit 4.) Note that the following equality must always hold:
stayovers + unexpected stayovers - early departures + expected arrivals - no-shows + walk-ins - upgrades = rooms occupied + walks.
Also note that when overbooking leads to oversales, no walk-ins are accepted, and upgrades may be made.
The estimated no-show probability for the expected arrivals and the early departure probability for stayovers can be computed by totaling the relevant columns in Exhibit 4 for division:
For expected arrivals, p = 583/13,759 = .042
For stayovers, p = 698/19,025 = .037
The computed estimates of the probabilities are based on large total samples and can therefore be taken to be nearly equal to the true pro Initially confining our analysis to expected arrivals, we note that the probability of no-shows (p) equals .042, and that the standard normal deviate for a 90-percent customer-service level is 1.28. Even though the number of standard rooms is 800, the average number of unexpected stayovers is 14.35 (see Exhibit 4), giving us a working inventory of only 786 standard rooms. (18) We now have all the values that can be substituted into the previously derived optimizing equation:
[(1 - .042).sup.2] [X.sup.2] - [2 (786) (1-.042) + [1.28.sup.2] (.042) (.958)] X+[786.sup.2] = 0
Using the formula to solve the general quadratic equation, the authorized booking level (X) is 813. Similarly, since the probability of early departures (p) equals .037, we have the following equation for the authorized booking level (X):
[(1-.037).sup.2] [X.sup.2] - [2 (786) (1-.037) + [1.28.sup.2]
(.037) (.963)] X+[(786).sup.2] = 0
Similarly solving, X = 809
This means that the hotel can have 813 expected arrivals or 809 stayovers booked on any particular day, each with different statistically independent probabilities of materializing. Clearly, both of those eventualities cannot occur at once. That is, the hotel cannot accommodate all guests if it has 813 expected arrivals and has 809 stayovers booked. Since the individual authorized booking levels were computed on a working capacity of 786 standard rooms, each would be counted as follows in terms of realizable demand:
expected arrivals: 786 / 813 = .967
stayovers: 786 / 809 = .972
This means that whenever a new arrival is accepted, the hotel must remove .967 rooms from its working inventory of 786 standard rooms, and when it has one stayover booked, it must remove .972 rooms from that working inventory. Conversely, a cancellation or no-show will release .967 rooms for the day of arrival and .972 rooms for all subsequent days on a multiple night booking. We stop raking reservations when the working rooms inventory reaches zero.
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