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2. An airline is planning their flights between different cities of Turkey. We focus on the...

2. An airline is planning their flights between different cities of Turkey. We focus on the flight they will have on August 1

2. An airline is planning their flights between different cities of Turkey. We focus on the flight they will have on August 1st of this year, that departs from Istanbul, stops in Ankara and continues to Antalya. The plan is to sell the tickets to three types of passengers: the ones traveling from Istanbul to Ankara (passenger type 1), the ones traveling from Ankara to Antalya (passenger type 2), and the passengers traveling from Istanbul to Antalya (passenger type 3). The aircraft can seat 30 passengers, and the airline offers three fair classes of economy (E), business (B) and first class (F). Ticket prices have been set and advertised as Pij, where i refers to the passenger type and represents the fair classes. Also, based on past experiences demand forecasters have determined the upper bounds on the number of potential customers of wij for all origin-destination/ fair class combinations. The aim is to find the number of tickets to sell to each of these 9 combinations with the maximum revenue. (a) Formulate this as an integer linear programming problem without overbooking or exceeding the forecasted maximum demand. (b) Consider the case where the airline needs to pay taxes for each flight route as follows: If the sold tickets are less than 16 then they would pay $1000 per pas- senger, else $2400 per passenger. How would you incorporate this requirement into your model?
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Answer #1

a) The Linear Program is as follows:

The objective function is to maximize the total profit

Total Profit = \sum_{i=1}^{3}\sum_{j=1}^{3}p_{ij}\times x_{ij}

Decision variables: x_{ij} (Number of passengers of each type in each fare class)

Constraints:

1) Number of people traveling from Istanbul to Ankara and Antalya across all the three fare classes should be less than or equal to 30

x_{1j} and x_{3j} represent the passengers traveling from Istanbul

\sum_{j=1}^{3}x_{1j} + \sum_{j=1}^{3}x_{3j} \leq 30

2) Number of people traveling from Istanbul to Antalya and Ankara to Antalya across all the three fare classes should be less than or equal to 30

x_{3j} represent the passengers traveling from Istanbul to Antalya and x_{2j} represent the passengers traveling from Ankara to Antalya

\sum_{j=1}^{3}x_{2j} + \sum_{j=1}^{3}x_{3j} \leq 30

3) The number of passengers of any type traveling in any fare class x_{ij} should be less than or equal to the upper bound uij

x_{ij} \leq u_{ij}

b) The cost of taxes are to be accounted

We introduce a variable y such that y has to take a binary value (i.e. 0 and 1)

Cost = 1000 * x_{ij} * y + 2400 * x_{ij} * (1-y)

x_{ij} >= 16 - M * y

Where M is a very large number (around 100000)

So, if x_{ij} is more than 16, y will be 0 and hence M * y will be 0 and in the cost function 1000 * x_{ij} * y term will be zero, thereby leaving only 2400 * x_{ij} * (1-y) or 2400 * x_{ij} in the cost function.

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