In this section we present an MISOCP model for solving the CVTSP, which is an ex- tension of the continuous model presented in [103]. The novel feature of the model is that the target visiting sequence is a decision to be taken, which is expressed by assign- ment variables. In addition, decision variables representing target points coordinates are introduced. The input parameters of the problem are:
n number of target points
qi set of target points coordinates in R2 i = 1, . . . , n
qmin vector of the minimum of the target point coordinates
qmax vector of the maximum of the target point coordinates
Vv vehicle speed
Vc carrier speed
a vehicle autonomy (in time units)
po coordinates in R2 of the starting point of the trajectory
pf coordinates in R2 of the ending point of the trajectory.
Qi coordinates in R2 of the i-th target point to be visited i = 1, . . . , n
wij binary variable taking value 1 if target point j is
visited in position i (i.e., Qi = qj) i, j = 1, . . . , n
pto,i coordinates in R2 of the take-off point for the visit of Qi i = 1, . . . , n
pl,i coordinates in R2 of the landing point after the visit of Qi i = 1, . . . , n
tto,li,1 time taken by the vehicle to reach Qi from pto,i i = 1, . . . , n
tto,li,2 time taken by the vehicle to reach pl,i from Qi i = 1, . . . , n
tto,li time taken by the carrier to reach pl,i from pto,i i = 1, . . . , n
tl,to1 time taken by the carrier to reach pto,1 from po
tl,toi time taken by the carrier to reach pto,i from pl,i−1 i = 2, . . . , n
tl,ton+1 time taken by the carrier to reach pf from pl,n
A formulation for the CVTSP is then given by the following model.
zCV T SP = min n X i=1 tto,li + n+1 X i=1 tl,toi (2.1) s.t.
kQi− pto,ik ≤ Vvtto,li,1 ∀ i = 1, . . . , n (2.2) kQi− pl,ik ≤ Vvtto,li,2 ∀ i = 1, . . . , n (2.3) kpto,i− pl,ik ≤ Vctto,li ∀ i = 1, . . . , n (2.4)
kpo− pto,1k ≤ Vctl,to1 (2.5)
kpl,i−1− pto,ik ≤ Vctl,toi ∀ i = 2, . . . , n (2.6)
kpf − pl,nk ≤ Vctl,ton+1 (2.7)
tto,li,1 + tto,li,2 ≤ tto,li ∀ i = 1, . . . , n (2.8)
Qi = n X j=1 wi,jqj ∀ i = 1, . . . , n (2.9) n X j=1 wi,j = 1 ∀ i = 1, . . . , n (2.10) n X i=1 wi,j = 1 ∀ j = 1, . . . , n (2.11) tto,li,1 ≥ 0 ∀ i = 1, . . . , n (2.12) tto,li,2 ≥ 0 ∀ i = 1, . . . , n (2.13) 0 ≤ tto,li ≤ a ∀ i = 1, . . . , n (2.14) tl,toi ≥ 0 ∀ i = 1, . . . , n + 1 (2.15) qmin≤ Qi≤ qmax ∀ i = 1, . . . , n (2.16) wij ∈ {0, 1} ∀ i, j = 1, . . . , n. (2.17)
The objective function (2.1) is the mission completion time, namely the time required by the carrier to travel between consecutive take-off and landing points until it reaches the final destination.
Constraints (2.2) and (2.3) define the time spent by vehicle to perform a take-off/landing process. More precisely, the time required to get to the target location from the take- off position is computed in (2.2), while (2.3) express the time taken to return to the carrier deck after leaving the target position.
Similarly, constraints (2.5)-(2.7) model the time the carrier requires to travel between consecutive take-off and landing positions.
Inequalities (2.8) express the synchronization between the carrier and the vehicle tra- jectories at landing points.
The sequencing (or assignment) variables are subject to constraints (2.9)-(2.11), which impose that target position variables assume all target point coordinates with no rep- etitions.
The time variables are required to be non-negative in bounds (2.12)-(2.15); in partic- ular, bound (2.14) takes into account the limited stand-alone autonomy of the vehicle. Target points variables can be safely limited as bound (2.16) prescribe: although re- dundant, these conditions may help optimization solvers to handle the model. Finally, bounds (2.7) express the requirement for assignment variables to be binary.
Model (2.1)-(2.17) is an MISOCP. Indeed, it is constituted by a linear objective func- tion (2.1), second-order conic constraints (2.2)-(2.7) and linear constraints (2.8)-(2.11). Since both the number of variables and constraints is polynomial in n, as a first at- tempt we used state-of-the-art optimization solvers as an exact solution strategy for the model.
The results presented in Section 2.4 were obtained by using solvers with algorithms tailored for conic problems. We also performed a preliminary testing by using global optimization solvers, such as COUENNE ([30]), ANTIGONE (Misener and Floudas [192]) and BARON ([220]). Such solvers do not identify the conic structure in con- straints (2.2)-(2.7); therefore they treat them as general non-convex constraints and they perform a term-by-term convexification. Hence, the global solvers prove to be ex- tremely slow in finding provably optimal solutions, even for very small-sized instances with five target points.