In this section, a planning model is proposed for the UAV monitoring problem. This model determines the flight paths of the UAV as well as the surveillance height, taking into account the weather conditions. Each surveillance area has a specific surveillance requirement level that must be met by drones. Since wind speed and direction affect the speed of UAVs and their monitoring time, the objective of the model is to minimize the total time required to complete all monitoring tasks. The mission planning problem is investigated to identify flight paths to monitor the covered areas on a specific network including nodes and flight altitudes. The movement of the UAV between the nodes will be in the form of perpendicular movements so that first the UAV moves horizontally and then the monitoring height is adjusted with the vertical movement of the UAV. Therefore, the distance between two nodes (i,r) and (j,s) is defined as Manhattan distance. The monitoring height of the UAV changes according to the required monitoring accuracy, so that a lower height covers more accuracy and a smaller area, and a higher height covers less accuracy and a wider area. Figures 1 and 2 shows the flight height and how to cover the monitoring areas.

The final speed of the UAV changes under the influence of the wind speed and direction, so that if the x-y coordinate axis is assumed to be a horizontal surface and the z-w coordinate axis is assumed to be a vertical surface (where w is in the direction of the wind), the final speed of the UAV is according to Eq. 1 for horizontal plane and according to Eq. 2 for vertical plane. According to Figure 3, the final speed of the UAV according to a fixed nominal speed, as well as hypothetical nodes, is provided for the example. The notations used in equations are explained in Section 2.2.

vf = √(vn × cosφ + vw ̃ × cosδ) + (vn × sinφ + vw ̃ × sinδ) , (1)

ijk k ij k ij

2 2 (2) vfrsk = √(vnk) − (vw ̃) , vw ̃ (3) φ = ωij − arcsin [ sin(δ − ωij)].

ij

vnk Finally, according to the above relationships, the travel time between different nodes at different heights is calculated as follows.

In this section, the notations to be used in this study are introduced as follows.

The main assumptions are as follows:

I. Each UAV tour must start and end at the same depot.

II. The distance between (i,r) and (j,s) is calculated as a Manhattan distance so that the drone first adjusts its height and then moves horizontally or vice versa.

III. Weather forecasting has already been done with sufficient accuracy and a certain tolerance.

Z1 = Min ∑ ∑ ∑ ∑ ∑ tirjsk Xirjsk + ∑ ∑ ∑ mir Yirk,

i r j s k i r k

s. t.

∑ ∑ ∑ cair Yirk ≥ 1, ∀a,

i−{0} r−{0} k

∑ Yirk ≤ 1, ∀i − {0}, r − {0},

∑ ∑ Xirjsk = Yirk , ∀i, r, k,

∑ ∑ X00jsk = 1 , ∀k,

(10)

∑ ∑ Xir00k = 1, ∀k, (11)

∑ ∑ Xirjsk = ∑ ∑ Xjsirk, ∀i, r, k,

j s j s

(12)

∑ ∑ ∑ ∑ tirjsk Xirjsk + ∑ ∑ mirYirk ≤ TMk, ∀k,

i r j s i−{0} r−{0}

U = 0, ∀k, (13) U ≥ U + 1 − |I| × |R| (1 − X ), ∀i, r, j, s, k; (i, r) ≠ (j, s), (14)

j,s,k irk irjsk

Ui,r,k ≤ |I| × |R| − 1, ∀i, r, k, (15)

Ui,r,k ≥ 0, ∀i, r, k, (16) Xirjsk ∈ {0,1}, ∀i, r, j, s, k, (17) Y ∈ {0,1}, ∀i − {0}, r − {0}, k. (18)

irk

The objective Function (5) minimizes the total time required to perform the monitoring operation. Constraint (6) ensures that each area is covered by at least one drone while meeting the surveillance accuracy requirements. Constraint (7) ensures that there will not be more than one UAV stop at the same node. Constraint (8) expresses the relationship between the decision variables Xirjsk and Yirk. Constraints (9) and (10) show that all tours of each drone start and end at node 0. Constraint (11) shows the conservation of current at different nodes. Constraint (12) specifies that the travel time and monitoring time of each drone cannot exceed its maximum allowed flight time. Constraints (13)-(15) are subtour elimination constraints. Constraints (16)-(18) define the types of variables.

The proposed model becomes non-fuzzy according to the mathematical programming methods and based on the credibility measure. To do this, we need both optimistic and pessimistic values as critical values. Ifξ is a fuzzy variable andαϵ (0,1], the optimistic and pessimistic values for the variableξ at theαlevel are defined as follows (see [17-18]).

ξ (α) = sup{r|Cr(ξ ≥ r) ≥ α}, (19)

sup

ξ (α) = inf{r|Cr(ξ ≤ r) ≥ α}. (20)

inf

In this paper, the variableξ is considered as a triangular fuzzy number, for example,ξ= (a1, a2, a3). And it is assumed that α≥0.5, so optimistic and pessimistic values are obtained as follows (see [20]).

ξ (α) = (2α − 1)a1 + (2 − 2α)a2, (21)

sup

ξ (α) = (2 − 2α)a2 + (2α − 1)a3. (22)

inf

Due to optimistic and pessimistic values, the problem constraints that include fuzzy parameters become non-fuzzy. If r is a definite value in the constraint, the following two equations are used to make the constraint non-fuzzy.

Cr(ξ ≥ r) ≥ α ⟺ r ≤ ξsup(α), (23) Cr(ξ ≤ r) ≥ α ⟺ r ≥ ξinf(α). (24)

In the proposed model, wind speed is considered fuzzy; therefore, Eqs. (1), (2), and (3) are affected by the existing uncertainty and are transformed into non-fuzzy according to Eqs. (19) to (24).

vfijk

(25)

≤ √(vn × cosφ + [(2α − 1)vw + (2 − 2α)vw ] × cosδ) + (vn × sinφ + [(2α − 1)vw + (2 − 2α)vw ] × sinδ) ,

k ij 1 2 k ij 1 2

vfijk

(26)

≥ √(vn × cosφ + [(2 − 2α)vw + (2α − 1)vw ] × cosδ) + (vn × sinφ + [(2 − 2α)vw + (2α − 1)vw ] × sinδ) ,

k ij 2 3 k ij 2 3

√( )2 [( ) ( ) ]2 (27)

vfrsk ≤ vnk − 2 − 2α vw2 + 2α − 1 vw3 , √( )2 [( ) ( ) ]2 (28)

vfrsk ≥ vnk − 2α − 1 vw1 + 2 − 2α vw2 ,

[(2 − 2α)vw2 + (2α − 1)vw3] (29)

φij ≤ ωij − arcsin [ sin(δ − ωij)], vnk

[(2α − 1)vw1 + (2 − 2α)vw2] (30)

φij ≥ ωij − arcsin [ sin(δ − ωij)].

vnk

To evaluate the presented model, a numerical example has been studied. For this purpose, a specific and divided area is considered, as shown in Fig. 5, which requires aerial monitoring operations. The indicated nodes indicate the points of monitoring by different drones. Accordingly, according to the degree of importance of imaging each area (according to Table 1), 3 drones at 3 different heights can perform monitoring. The flight speed of the drones is assumed to be the same (6 m⁄s), but the maximum flight duration is 600, 400, and 350 seconds, respectively. Also, wind speed is a fuzzy number (2m⁄s, 3m⁄s, 4m⁄s) and α=0.6. The height that can be covered by each area indicates the sensitivity with which each area is imaged. Height 1 is the lowest possible height and has the highest accuracy, and height 3 is the highest drone imaging level, which has the lowest accuracy. For example, area 4 can be covered from all three heights, but area 1 will only be covered from height 1 due to the sensitivity required for imaging.

By solving the mathematical model provided by the GAMS software, the results of the drones' routing for imaging the specified areas are obtained. Fig. 5 shows the drones' stopping points and the imaging range of each drone. According to the results obtained, drone 1, moving from point 0 at the height of level 1, is stationed at point 1 and then goes to point 2 at the same level. Also, drone 3, moving from the starting point, first goes to the height of level 2 at point 5 and then goes to point 7 at the same level. The return of both drones will be to the initial origin. According to Fig. 5, areas 1 and 2 are monitored by UAV 1 with greater sensitivity, and areas 3 to 8 are imaged by UAV 3 with less sensitivity.