# Formulating and Solving a Capacity Allocation Problem of a Plane

The passenger transport industry faces the problem of determining how to efficiently allocate transportation capacity when offering different prices or fees to their customers for a specific route. This is why they must consider sales revenues associated with each type of rate, estimated customer demand for such fees and the capacity of the transport in terms of the number of seats simultaneously. The next problem considers the formulation and computational problem of the capacity of allocation an airplane for an airline company. The complexity of the problem and the level of detail of the information is simplified for academic purposes.

## Capacity Allocation Problem

Let us consider an airline that flies the Santiago (Chile) route to Bogota (Colombia) with a layover in Lima (Peru). This route uses an airplane with a capacity of 200 passengers. The sales department has estimated market prices (in dollars) for combinations of source/destination of 3 types of rates currently offered by the company: “Fee Y” (first class), “Fee B” (standard) and “Fee C” (tourist).

Additionally and according to historical information about this route, the airline has estimated the maximum number of passages seats that will be in demand for each combination of rate on a flight segment. For example the maximum demand expected for the Santiago (SCL) to Bogota (BOG) in Fee B is 35 tickets.

With this information the airline wants to determine how to allocate the capacity of the aircraft in order to provide a certain number of tickets for each type of fee per flight segment. To do this we define the following linear programming model:

Decision Variables:

Xij: Number of tickets offered in the Fee i, for the Origin-Destination segment j

Where i=1,2,3 represent the various types of fees (Y, B and C, respectively) and j=1,2,3 combinations of source-destination (SCL-LIM, LIM-BOG and SCL-BOG, respectively).

Parameters:

Using a notation with parameters, we can represent an optimization model in a compact way.

Pij: Price in US\$ (Dollars) for the ticket fee i from Origin-Destination j segment

Dij: Maximum demand of tickets fee i on the Origin-Destination j segment

Objective Function:

Constraints:

We must offer for each fee in the source-destination combinations a number of tickets that does not exceed the market demand.

For each flight segment you must respect the total capacity of 200 passenger per aircraft.

When the plane takes off from Santiago to Lima, it takes passengers for both Lima and Bogota. Therefore independently of what fee each of these passengers paid (hence the sum in fees) may not exceed the total capacity of the aircraft. This is guaranteed by the first constraint of capacity. The second capacity restriction is for the stretch from Lima to Bogota, where passengers coming from Santiago are also considered.

Finally the non-negativity conditions are defined.

By using Solver to solve the above problem with the following optimal solution which determines how many tickets the airline should offer for each combination of rate and source destination is reached.

The optimal value of the problem of total income (in dollars) associated with the proposed optimal solution is US\$ 158,340.

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