Reverse an additional discount customer if $p>\frac{R_g-R_d}{R_g}$

Not all those who book will actually turn up and Bodily and Weatherford $(1995),$ using

decision trees, show that the form of the solution remains unchanged.

A simpler approach for deciding how many discounts to offer is given in Weatherford

and Bodily $(1992).$ Another discounted unit is offered provided that the probability of

overbooking, $Pr(ob)($i$.$e$.$ the probability that discounted demand plus full price

demand will exceed capacity$),$ is greater than the ratio of reduced price contribution $R_d$

to the full price contribution $R_f,$ i$.$e$.$

Allow discount $ifPr(ob)>\frac{R_d}{R_f}$

Thus the closer the discounted price is to the full price the more $q_d,($the quantity

discounted$)$ approaches the capacity of the facility $C.$

In the approaches outlined above it was assumed that there was no correlation

between the demand for full and discounted price accommodation. This is most

unlikely in practice. Brumelle et al$.(1990)$ attempt to overcome this problem by

advancing a rule for situations where there is a positive correlation between the number

of discounted customers $(q_d)$ and the number of full price paying customers $(q_f)$ and

where discounts are issued first and there is no overlap in the time period during which

discounts and full price are offered. Their rule is:

$100[\frac{q_f-(100-q)}{10}]=\frac{75}{100}\frac{q_f-(100-q)}{10}=(0\mathrm{.}75)=0\mathrm{.}5752q=100-5\mathrm{.}75=94\mathrm{.}25P_g(q^{**})$

Using the Brummell Model found on page $51.$

Hotel with $100$ rooms

Forecast demand varies $($see below PHI$)$

Full Price Roome is $$100\mathrm{.}00$

Using a discount of $25\%$ for early booking

Finish the table below to determine number $$ of discounted rooms

$\backslash$table$[[,,$PHI,S$.$D$.],[,,0\mathrm{.}1587,10],[,,0\mathrm{.}2119,10],[,,0\mathrm{.}3085,10],[,,0\mathrm{.}4207,10],[,,0\mathrm{.}5,10],[,,0\mathrm{.}6554,10],[,,0\mathrm{.}7257,10],[,,0\mathrm{.}8159,10],[,,0\mathrm{.}9032,10]]$