# Fair Task Allocation When Cost of Task Is Multidimensional
## 2 Problem Definition
### UAV Agricultural Plant Protection System  
This section presents details of the UAV task allocation problem in agricultural plant protection environment. Compared to traditional task allocation problem, UAV task allocation in agricultural plant protection environment is a more complicated problem. UAVs considered in this work are special identical agents for agricultural plant protection, which can handle agricultural plant protection tasks. Tasks have cost to UAVs. The integral factors of the problem are defined as follows. 1. Tasks are indivisible, a task is allocated to one UAV. 2. Task produces 3 dimensions of cost to UAVs: time cost, energy cost, and UAV service life cost. The total cost is calculated based on these 3 costs. 3. The relationship between different dimensions of cost is non-deterministic, but it is in a range (i.e., energy cost by the UAV for one hour of flight is not deterministic because the real-time environment is complex, but its lower and upper bounds can be determined). 4. As the status of UAVs is different, a task has a different cost to different UAVs in a single dimension (e.g., a task may has different time cost for two UAVs because the positions of the two UAVs are different and time spent flying is counted in time cost). 5. Due to the inconsistent state of UAVs, different UAVs have different weights over tasks (e.g., a UAV that is about to run out of energy has a higher weight on energy cost, and an aging UAV has a higher weight on service life cost). 6. Due to the complexity of the actual situation: the cost of tasks to UAVs is unknown. 7. UAVs'preference over the tasks in single dimension can be known (e.g., UAV is farther away from task 2 than task 1, then in the time cost dimension: \(task1\preceq task2\)). Based on these, we will formally present our problem model: the setting and meaning of the parameters can be found in Table 2, according to condition 1 the cost of a task \(j\) to UAV \(i\) can be defined as Equation (5). \[C_{i,j}=w_{i}(t)\cdot c_{i,j}(t)+w_{i}(e)\cdot c_{i,j}(e)+w_{i}(l)\cdot c_{i,j}(l) \tag{5}\]  
**Example 5**. : _Assuming that there are 2 UAVs and 2 tasks. UAV \(1\)'s weights are (0.2, 0.2, 0.6), each dimension corresponds to the weight of time cost, the weight of energy cost, and the weight of UAV service life cost, respectively. Similarly, UAV 2's weights are (0.3, 0.2, 0.5). The cost of agents by tasks can be seen in Tables 3 and 4._  
_Then, the cost of UAV \(1\) by task \(1\) is \(0.2\times 1+0.2\times 4+0.6\times 7=5.2\) The cost of UAV \(1\) by task \(2\) is \(0.2\times 3+0.2\times 15+0.6\times 12=10.8\) The cost of UAV \(2\) by task \(1\) is \(0.3\times 2+0.2\times 6+0.5\times 10=6.8\) The cost of UAV \(2\) by task \(2\) is \(0.3\times 1+0.2\times 8+0.5\times 4=3.9\)_ \begin{table}
\begin{tabular}{l l l l} \hline \hline  & **Time Cost** & **Energy Cost** & **Service Life Cost** \\ \hline UAV 1 & 3 & 15 & 12 \\ \hline UAV 2 & 1 & 8 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Cost of UAVs by Task 2. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter List** & **Description** \\ \hline \(U_{i}\) for \(i=1,2...n\) & The UAV set \(N\) cotains \(n\) UAVs \\ \hline \(T[j]\) for \(j=1,2...m\) & The task set \(M\) cotains \(m\) tasks \\ \hline \(c_{i,j}(t)\) & \(T[j]\)’s time cost to \(U_{i}\) \\ \hline \(c_{i,j}(e)\) & \(T[j]\)’s energy cost to \(U_{i}\) \\ \hline \(c_{i,j}(l)\) & \(T[j]\)’s service life cost to \(U_{i}\) \\ \hline \(k_{t,e},K_{t,e}\) & The lower and upper bounds of the relationship between time cost and energy cost \\ \hline \(k_{t,l},K_{t,j}\) & The lower and upper bounds of the relationship between time cost and service life cost \\ \hline \(w_{i}(t)\) & Weight of \(U_{i}\)’s time cost \\ \hline \(w_{i}(e)\) & Weight of \(U_{i}\)’s energy cost \\ \hline \(w_{i}(l)\) & Weight of \(U_{i}\)’s service life cost \\ \hline Matrix\(X:N\times M\rightarrow\{0,1\}\) & If task \(j\) is assigned to the UAV \(i\) \\  & then \(x_{i,j}=1\); otherwise \(x_{i,j}=0\) \\ \hline \hline \end{tabular}
\end{table}