A Canadian Regional Jet (CRJ) is a 50-seater aircraft. We consider the case where a CRJ travels from Dorval airport in Montreal to airport B (outside Canada), and back to Dorval once a day. On board, three different types of wine are served: white, red and sparkling wine.
The CRJ has a maximal storage capacity for bottles of wine, which is distributed amongst the three types of wine. Each flight has a random number of passengers, that we shall model according to the empirical distribution from past data. From experience, it is known that passengers choose one type of wine and order only this type of wine. The amount of wine that passengers demand is random and bounded by the maximum allowed by AirLine regulations.
The inventory level of each type of wine is the amount of wine bottles in stock on-board the plain. Each time that we order wine we must pay a fixed amount for the truck to provision the plain. The inventory policy is as follows: we must choose a reorder level for each type of wine. If, upon inspection of the CRJ at Dorval one or more inventory levels has fallen below the reorder level, then we order wine bottles to fill up all capacity, even for those wines whose inventory levels are still above the reorder level.
In Flight, passengers will create the total demand of wine for each type and the inventory levels will go down accordingly. When a supply of wine is exhausted, no more bottles of this type will be available until the next flight departing from Montreal. The Airline makes a profit of $2.00 per 125 ml, regardless of type, but there is a penalty cost of $1.00 for each non-satisfied demand.
|X passengers||Probability of X passengers|
|0 < X<=10||0.005|
|10 < X<=20||0.015|
|20 < X<=30||0.020|
|30 < X<=40||0.035|
|40 < X<=50||0.025|
The above table shows a typical specification of a mixture of probability distributions and corresponds to the mathematical expression (1). It models cases where there are five scenarios or "types of flights". The probability that the flight is a "popular" one is, in this case 0.25: a quarter of flights have between 40 and 50 passengers. The actual number, given the type of day, has a uniform distribution.
Passengers are classified according to their preference for a single type of wine. Each passenger belongs to a class with a certain probability, independently of the rest of the passengers in this or other flights.
Wine is provided to passengers in 125ml mini-bottles. Once the class of the passenger is known, it has been determined by previous data that he or she will have between 0 and 3 mini-bottles per flight with the given probabilities:
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