# Math

A mathematical model of the problem.

## Variables

• An integer variable $\text{plane}_i$ per plane i indicates its landing time.

$$\forall i \in [1,10],\, \text{plane}_i = [\![LT_{i,0},LT_{i,2}]\!]$$

• An integer variable $\text{earliness}_j$ per plane i indicates how early a plane lands.

$$\forall i \in [1,10],\, \text{earliness}_i = [\![0,LT_{i,1} - LT_{i,0}]\!]$$

• An integer variable $\text{tardiness}_j$ per plane i indicates how late a plane lands.

$$\forall i \in [1,10],\, \text{tardiness}_i = [\![0,LT_{i,2} - LT_{i,1}]\!]$$

• An integer variable $tot_{dev}$ totals all costs:

$$tot_{dev} = [\![0, {+\infty})$$

## Constraints

• One plane per runway at a time:

$$\forall i,j \in [1,10]^2,\, i\ne j, \text{plane}_{i} \ne \text{plane}_{j}$$

We saw this type of constraint before, it is an alldifferent constraint.

• the earliness of a plane i:

$$\forall i \in [1,10],\, \text{earliness}_i = max(0,-\text{plane}_i + \text{LT}_{i,1})$$

When the plane is on time or late, its earliness is equal to 0. Otherwise, a positive value is computed that corresponds to the difference between the real landing time and the target landing time.

• the tardiness of a plane i:

$$\forall i \in [1,10],\, \text{tardiness}_i = max(0,\text{plane}_i - \text{LT}_{i,1})$$

When the plane is on time or early, its tardiness is equal to 0. Otherwise, a positive value is computed that corresponds to the difference between the real landing time and the target landing time.

• the separation time required between two planes has to be satisfied:

$$\forall i,j \in [1,10]^2,\, i\ne j, (\text{plane}_{i} + \text{ST}_{i,j} \le \text{plane}_{j}) \oplus (\text{plane}_{j} + \text{ST}_{j,i} \le \text{plane}_{i})$$

If plane i lands before plane j then plane j lands at least ST[i][j] after plane i. Otherwise, plane j lands first and plane i waits at least ST[j][i] after it to land. The two conditions cannot hold together because of the XOR logical operator. In this case a simple inequality constraint fits the need.

• the deviation cost has then to be maintained:

$$tot_{dev} = \sum_{i = 1}^{10} \text{PC}_{i,0} \cdot \text{earliness}_i + \text{PC}_{i,1} \cdot \text{tardiness}_i$$

## Objective

The objective is to find a solution that minimizes tot_dev.