A UAV location and routing problem with spatio-temporal synchronization constraints solved by ant colony optimization

Abstract

In this study, we introduce an optimization problem which attempts to optimize location and routing of a homogeneous unmanned aerial vehicle fleet. The problem also allocates the available capacity to the potential locations while it sustains the feasibility defined by synchronization constraints which include time windows at visited points, capacity monitoring in the stations and a limited number of multiple sorties. A mixed integer linear programming formulation for the problem is given and a heuristic method based on ant colony optimization approach is suggested. The suggested heuristic is compared to a commercial solver, a greedy heuristic and a simpler version of the suggested heuristic. We have observed that the suggested heuristic provides the best solutions, while the commercial solver is able to produce only poor solutions in longer time periods. The learning component, which is the main difference between the suggested heuristic and its simplified version, makes a significant change. The results of the experiments strongly suggest the usage of our metaheuristic method for the introduced problem.

Appendix

To define the feasible space and objective function of the discussed problem formally as an MILP formulation, the notation for indices, sets, parameters, variables are given below.

Sets:

u $\in$ U :

set of UAVs.

i, j$\in$I:

set of interest points.

i, j$\in$S:

set of stations.

k $\in$ K :

set of platforms (ships).

p $\in$ P :

set of periods (sorties).

t $\in$ T :

set of time.

Parameters:

p _i :

importance of interest point i.

d _ij :

expected time in flight between i and j.

t _i :

expected time on interest point or on station i (when refueling).

y _max :

maximum number of active stations allowed.

b _i :

beginning time for time window of interest point i.

e _i :

ending time for time window of interest point i.

t _max :

maximum time between first takeoff and last landing.

C _k :

capacity of platform k.

FT :

maximum flight time before refueling.

Variables:

X _ijup :

binary variable indicating if UAV u has a leg from point i to point j in period p, or not.

Y _i :

binary variable indicating if station i is activated, or not.

S _ki :

binary variable indicating if platform k is assigned to station i, or not.

PL _iu :

binary variable indicating if UAV u is assigned to station i, or not.

A _uip :

arrival time of UAV u to interest point or station i in period p.

AR _uipt :

binary variable indicating if UAV u arrives to station i in period p before time t, or not.

DE _uipt :

binary variable indicating if UAV u departures from station i in period p before time t, or not.

D _uip :

departure time of UAV u from station i in period p.

F _ijup :

a continuous variable used to prevent subtours.

Since the formulation of the problem takes several pages, for the ease of reading, we give the formulation along with explanations of the objective function and the constraints.

The objective function (A1) aims to maximize the total of importance values collected from interest points while it tries to avoid unnecessary assignments of ships to stations. ε represents a small positive real number.

$$ {\rm max} z = \mathop \sum \limits_{j \in I \cup S} \mathop \sum \limits_{i \in I} \mathop \sum \limits_{u \in U} \mathop \sum \limits_{p \in P} X_{jiup} p_{i} - \mathop \sum \limits_{k \in K} \mathop \sum \limits_{i \in S} S_{ki} *\varepsilon $$

(A1)

Constraint (A2) limits the number of stations that can be activated.

$$ \mathop \sum \limits_{i \in S} Y_{i} \le y_{max} $$

(A2)

Constraints (A3 and A4) force each UAV to start its route from only one station which is active, for each period.

$$ \mathop \sum \limits_{j \in I} \mathop \sum \limits_{u \in U} \mathop \sum \limits_{p \in P} X_{ijup} \le Y_{i} \left| I \right|\quad \forall i \in S $$

(A3)

$$ \mathop \sum \limits_{i \in S} \mathop \sum \limits_{j \in I} X_{ijup} \le 1\quad \forall u \in U,p \in P $$

(A4)

Constraint (A5) serves as flow conservation as in its widely known classical meaning.

$$ \mathop \sum \limits_{i \in I \cup S} X_{ijup} = \mathop \sum \limits_{i \in I \cup S} X_{jiup} \quad \forall j \in I, u \in U,p \in P $$

(A5)

Constraint (A6) limits the departures from interest points to one.

$$ \mathop \sum \limits_{i \in I \cup S} \mathop \sum \limits_{u \in U} \mathop \sum \limits_{p \in P} X_{jiup} \le 1\quad \forall j \in I $$

(A6)

Constraints (A7 and A8) prevent infeasible tours, where ε is a small positive real number.

$$ \mathop \sum \limits_{i \in I \cup S} \mathop \sum \limits_{u \in U} F_{jiup} - \mathop \sum \limits_{i \in I \cup S} \mathop \sum \limits_{u \in U} F_{ijup} \le \varepsilon * \mathop \sum \limits_{i \in I \cup S} \mathop \sum \limits_{u \in U} X_{ijup} \quad \forall j \in I,p \in P $$

(A7)

$$ F_{ijup} \le X_{ijup} \quad \forall i \in I \cup S, j \in I \cup S,u \in U,p \in P $$

(A8)

Constraint (A9) ensures that if the station is not active, UAV cannot land on that station at any period.

$$ \mathop \sum \limits_{j \in I} \mathop \sum \limits_{u \in U} \mathop \sum \limits_{p \in P} X_{jiup} \le Y_{i} \left| I \right|\quad \forall i \in S $$

(A9)

Constraint (A10) forces each UAV to land on at most one station at any period.

$$ \mathop \sum \limits_{i \in S} \mathop \sum \limits_{j \in I} X_{jiup} \le 1\quad \forall u \in U,p \in P $$

(A10)

Constraint (A11) ensures that if a UAV takes off in a period, it must land.

$$ \mathop \sum \limits_{i \in S} \mathop \sum \limits_{j \in I} X_{ijup} = \mathop \sum \limits_{i \in S} \mathop \sum \limits_{j \in I} X_{jiup} \quad \forall u \in U,p \in P $$

(A11)

Constraints (A12–A14) provide the satisfaction of time window restrictions, while Constraint (A15) ensures that all UAVs should return to a station before its fuel is exhausted, where arrival time to the station cannot be smaller than the sum of the arrival time to the last interest point, the elapsed time on the last interest point and the elapsed time in flight between the last interest point and the station. The constant M, used in Constraints (A12–A15), represents a positive real number greater than t_max.

$$ A_{ujp} \le \left( {1 - \mathop \sum \limits_{i \in I \cup S} X_{ijup} } \right) M + E_{j} \quad \forall j \in I, u \in U,p \in P $$

(A12)

$$ A_{ujp} \ge \left( {\mathop \sum \limits_{i \in I \cup S} X_{ijup} - 1} \right) M + B_{j} \quad \forall j \in I, u \in U,p \in P $$

(A13)

$$ A_{ujp} \ge A_{uip} + X_{ijup} t_{i} + d_{ij} + \left( {X_{ijup} - 1} \right) M\quad \forall i \in I \cup S,\forall j \in I, u \in U,p \in P $$

(A14)

$$ A_{ujp} + t_{i} + X_{jiup} d_{ij} + \left( {X_{jiup} - 1} \right) M \le A_{uip} \quad \forall i \in S, j \in I, u \in U,p \in P $$

(A15)

Constraint (A16) forces each platform to be assigned at most to one station.

$$ \mathop \sum \limits_{i \in S} S_{ki} \le 1\quad \forall k \in K $$

(A16)

Constraint (A17) ensures that if the station is not active, no platform can be assigned to that station.

$$ \mathop \sum \limits_{k \in K} S_{ki} \le Y_{i} \left| K \right|\quad \forall i \in S $$

(A17)

Constraint (A18) limits the departures from station to interest point.

$$ D_{uip} + X_{ijup} d_{ij} + \left( {X_{ijup} - 1} \right)t_{max} \le A_{ujp} \quad \forall i \in S, j \in I, u \in U,p \in P $$

(A18)

Constraint (A19) limits that the number of UAVs assigned to a station cannot exceed the total capacity of platforms which are assigned to that station.

$$ \mathop \sum \limits_{u \in U} PL_{iu} \le \mathop \sum \limits_{k \in K} S_{ki} C_{k} \quad \forall i \in S $$

(A19)

Constraint (A20) forces that each UAV can be assigned at most to one station.

$$ \mathop \sum \limits_{i \in S} PL_{iu} \le 1\quad \forall u \in U $$

(A20)

At the first period, constraint (A21) forces that each UAV can only depart from the station to which it is assigned. For successor periods, constraint (A22) forces that each UAV can only depart from the station which it arrives at the previous period.

$$ PL_{iu} \ge \mathop \sum \limits_{j \in I} X_{ijup,p = 1} \quad \forall i \in S, u \in U $$

(A21)

$$ \mathop \sum \limits_{j \in I} X_{{jiup^{{\prime }} ,p^{{\prime }} + 1 = p}} \ge \mathop \sum \limits_{j \in I} X_{ijup} \quad \forall i \in S, u \in U, p,p^{{\prime }} \in P,p \Leftrightarrow p \ge 2 $$

(A22)

Constraint (A23 and A24) declares that departure/arrival time from a station cannot be later than t_max and if there is no departure/arrival, the value of corresponding variables must be equal to zero.

$$ D_{uip} \le \mathop \sum \limits_{j \in I} X_{ijup} t_{max} \quad \forall i \in S, u \in U, p \in P $$

(A23)

$$ A_{uip} \le \mathop \sum \limits_{j \in I} X_{jiup} t_{max} \quad \forall i \in S, u \in U, p \in P $$

(A24)

Constraints (A25–A31) limit the capacity of each station at each time from the beginning to the end of operations.

$$ \mathop \sum \limits_{t \in T} DE_{uipt} \le \mathop \sum \limits_{j \in I} X_{ijup} t_{max} \quad \forall i \in S, u \in U, p \in P $$

(A25)

$$ \mathop \sum \limits_{t \in T} DE_{uipt} \le t_{max} - D_{uip} + 1\quad \forall i \in S, u \in U, p \in P $$

(A26)

$$ \mathop \sum \limits_{t \in T} AR_{uipt} \le \mathop \sum \limits_{j \in I} X_{jiup} t_{max} \quad \forall i \in S, u \in U, p \in P $$

(A27)

$$ \mathop \sum \limits_{t \in T} AR_{uipt} \le t_{max} - A_{uip} + 1\quad \forall i \in S, u \in U, p \in P $$

(A28)

$$ (t - D_{uip} )/t \le DE_{uipt} + \left( {1 - \mathop \sum \limits_{j \in I} X_{ijup} } \right)\quad \forall i \in S, u \in U,p \in P,t \in T $$

(A29)

$$ (t - A_{uip} ) /t \le AR_{uipt} + \left( {1 - \mathop \sum \limits_{j \in I} X_{jiup} } \right)\quad \forall i \in S, u \in U,p \in P,t \in T $$

(A30)

$$ \mathop \sum \limits_{u \in U} PL_{iu} - \mathop \sum \limits_{u \in U} \mathop \sum \limits_{p \in P} DE_{uipt} + \mathop \sum \limits_{u \in U} \mathop \sum \limits_{p \in P} AR_{uipt} \le \mathop \sum \limits_{k \in K} S_{ki} C_{k} \quad \forall i \in S,t \in T $$

(A31)

Constraint (A32) prevents departure/arrival from being at the same time.

$$ \mathop \sum \limits_{i \in S} A_{uip} + \left( {1 - \mathop \sum \limits_{i \in S} \mathop \sum \limits_{j \in I} X_{ijup} } \right)t_{max} - \mathop \sum \limits_{i \in S} D_{uip} \ge 2\quad \forall u \in U,p \in P $$

(A32)

Constraint (A33) forces that if there is a departure at the first period, it has to start at least at time 1.

$$ \mathop \sum \limits_{i \in S} \mathop \sum \limits_{j \in I} X_{ijup,p = 1} \le \mathop \sum \limits_{i \in S} D_{uip,p = 1} \quad \forall u \in U $$

(A33)

At the first period, constraints (A34 and A35) force that a UAV cannot depart before expected refuel time at the station.

$$ D_{uip,p = 1} + \left( {1 - \mathop \sum \limits_{j \in I} X_{ijup,p = 1} } \right)t_{max} \ge ti\quad \forall i \in S, u \in U $$

(A34)

$$\begin{aligned} &D_{uip} + \left( {1 - \sum \limits_{j \in I} X_{ijup} } \right)\left( {t_{max} + ti} \right) - A_{{uip^{{\prime }} :p^{{\prime }} + 1 = p}}\ge ti\\ &\qquad \forall i \in S, u \in U, p,p^{{\prime }} \in P,p \Leftrightarrow p \ge 2 \end{aligned}$$

(A35)

Constraint (A36) forces that, if a UAV does not depart from a station at all, it cannot be assigned to that station.

$$ PL_{iu} \le \mathop \sum \limits_{j \in I} \mathop \sum \limits_{p \in P} X_{ijup} \quad \forall i \in S, u \in U $$

(A36)

Constraint (A37) forces that, if there is no departure/arrival to a station in any period, that station cannot be active.

$$ Y_{i} \le \mathop \sum \limits_{j \in I} \mathop \sum \limits_{u \in U} \mathop \sum \limits_{p \in P} X_{ijup} + \mathop \sum \limits_{j \in I} \mathop \sum \limits_{u \in U} \mathop \sum \limits_{p \in P} X_{jiup} \quad \forall i \in S $$

(A37)

Constraint (A38) limits for each period that the time between departure and arrival of any UAV cannot be more than allowed flight time before refuel.

$$ \mathop \sum \limits_{i \in S} A_{uip} - \mathop \sum \limits_{i \in S} D_{uip} \le FT\quad \forall u \in U, p \in P $$

(A38)

Constraints (A39–A47) identify the domains for decision variables.

$$ X_{ijup} \in \left\{ {0,1} \right\}\quad \forall i \in I \cup S, j \in I \cup S,u \in U,p \in P $$

(A39)

$$ Y_{i} \in \left\{ {0,1} \right\}\quad \forall i \in S $$

(A40)

$$ F_{ijup} \ge 0\quad \forall i \in I \cup S, j \in I \cup S,u \in U,p \in P $$

(A41)

$$ A_{ujp} \ge 0\quad \forall j \in I, u \in U,p \in P $$

(A42)

$$ S_{ki} \in \left\{ {0,1} \right\}\quad \forall i \in S, k \in K $$

(A43)

$$ PL_{iu} \in \left\{ {0,1} \right\}\quad \forall i \in S, u \in U $$

(A44)

$$ AR_{uipt} \ge 0\quad \forall i \in S,u \in U,p \in P,t \in T $$

(A45)

$$ DE_{uipt} \ge 0\quad \forall i \in S,u \in U,p \in P,t \in T $$

(A46)

$$ D_{uip} \ge 0\quad \forall i \in S,u \in U,p \in P. $$

(A47)