A UAV aims to reap the most profits from providing MEC to the GSAs, while a GSA aims to seek the service of a UAV to meet its performance requirements. This can be formulated as a multi-objective optimization problem as given in the following problem
P1. The variables to be optimized are as follows:
1
\(\Theta \triangleq \left\{ \theta _{n,m} \right\}\): The association between the UAVs and GSAs.
2
\({\textbf {A}}\triangleq \left\{ \alpha _{n, k_m}[t]\right\}\): The transmission time assigned by a UAV to the IoTDs in each time slot.
3
\({\textbf {B}}\triangleq {\left\{ \beta _{n, k_m}[t]\right\} }\): The executing time assigned by a UAV to the IoTDs in each time slot.
4
\(\textbf{F}\triangleq {\left\{ f_n[t]\right\} }\): The computing power set by a UAV in each time slot.
5
\(\textbf{J}\triangleq {\{f_{k_m}[t]\}}\): The computing power set by an IoTD in each time slot.
6
\({\textbf {Q}}\triangleq \left\{ Q_{n,m}\right\}\): The flight trajectory of a UAV.
$$\begin{aligned} \begin{array}{lll} ({\textbf {P1}}) &{} \max _{\Theta , {\textbf {A}}, {\textbf {B}}, \textbf{F}, \textbf{J}, {\textbf {Q}},T}{\sum _{n\in \mathcal {N}}{\sum _{m\in \mathcal {M}}{\theta _{n, m}W_{n, m}}}} &{}\quad (26\textrm{a}) \\ &{} \min _{\Theta , {\textbf {A}}, {\textbf {B}}, \textbf{F}, \textbf{J}, {\textbf {Q}}}{\sum _{n\in \mathcal {N},T}{\sum _{m\in \mathcal {M}}{\theta _{n, m}G_{n, m}}}}&{}\quad (26\textrm{b}) \\ \text {s.t. \ } &{} \sum _{m\in \mathcal {M}}{\theta _{n, m}=1}, \quad \forall\; n\in \mathcal {N}, &{}\quad (26.1) \\ &{} \sum _{n\in \mathcal {N}}{\theta _{n, m}=1}, \quad \forall\; m\in \mathcal {M}, &{}\quad (26.2) \\ &{} 0\le f_{k_m} [t]\le f_{k_m}^{{\max}}, \quad \forall\; t\in \mathcal {T}, &{}\quad (26.3) \\ &{} 0\le f_n[t]\le f_n^{{\max}}, \quad \forall\; t \in \mathcal {T}, &{}\quad (26.4) \\ &{} \sum _{k_m\in \mathcal {K}_m}{\alpha _{n, k_m} [t]} \le 1, \ 0\le \alpha _{n, k_m}[t]\le 1, \forall t\in \mathcal {T}, &{}\quad (26.5) \\ &{} \sum _{k_m\in \mathcal {K}_m}{\beta _{n, k_m} [t]}\le 1, \ 0\le \beta _{n, k_m} [t]\le 1, \forall t \in \mathcal {T}, &{}\quad (26.6) \\ &{} c_{n, k_m}[1]=0,\ \beta _{n, k_m}[1]=0, \ f_n[1]=0, \quad \forall\; k_m\in \mathcal {K}_m, &{}\quad (26.7) \\ &{} b_{n, k_m}[T]=0 \ {\text {and}}\ \alpha _{n, k_m}[T]=0, \quad \forall\; k_m\in \mathcal {K}_m, &{}\quad (26.8) \\ &{} \sum _{i=1}^{t-1}{b_{n, k_m} [i]}\ge \sum _{i=2}^{t}{c_{n, k_m} [i]}, 2\le t\le T,\ \forall k_m\in \mathcal {K}_m, &{}\quad (26.9) \\ &{} \sum _{t=2}^{T}{c_{n, k_m}[t]}+\sum _{t=1}^{T}{l_{k_m}[t]}\ge L_{k_m}, \quad \forall\; k_m\in \mathcal {K}_m, &{}\quad (26.10) \\ &{} q_{n,m}[T]=q_{n,m}[1], \quad \forall\; n\in \mathcal {N}, &{}\quad (26.11) \\ &{} \parallel v_n[t] \parallel \le v_n^{{\max}}, \quad \forall\; t \in \mathcal {T}, &{}\quad (26.12) \\ &{} \sum _{t\in \mathcal {T}}^{}\left( {E_{k_m}^{\text {C}}[t]+E_{n, k_m}^{\text {Tx}} [t]} \right) \le E_{k_m}^{{\max}}, \forall k_m\in \mathcal {K}_m, \quad \forall\; m\in \mathcal {M}, &{}\quad (26.13) \\ &{} \sum _{t \in \mathcal {T}}{E_n^{\text {F}}[t]}+\sum _{t\in \mathcal {T}}{\sum _{k_m\in \mathcal {K}_m}{E_{n, k_m}^{\text {C}}[t]}}\le E_{n}^{{\max}}, \forall n\in \mathcal {N}. &{}\quad (26.14) \\ \end{array} \end{aligned}$$
where constraints (26.1) and (26.2) (introduced in Sec. III-A) are the
association constraints for the UAVs and the GSAs, respectively, constraints (26.3 and (26.4) (introduced in Sec. III-B and Sec. III-C, respectively) are the
maximum computing power constraints on a UAV and an IoTD, respectively, constraints (26.5) and (26.6) (introduced in Sec. III-B) are the
time-integrity constraints for each of the UAVs, constraints (26.7), (26.8), and (26.9) (introduced in Sec. III-B) are the
information-causality constraints caused by the used of the assembly line working mode, constraint (26.10) (introduced in Sec. III-B) is the
task-integrity constraint for each of the IoTDs, constraint (26.11) indicates that at the end of a mission cycle a UAV must return to its starting point for charging and maintenance operations, constraint (26.12) (introduced in Sec. III-C) is the
maximum velocity constraint on each of the UAVs, and constraints (26.14) and (26.13) (introduced in Sec. IV-A and Sec. IV-B, respectively) are the
maximum available energy constraints on the UAVs and IoTDs, respectively.