1. Introduction
Multi-agent resource allocation is one of the major research topics in artificial intelligence [
1], and fair allocation algorithms for items have been considered. Minmax share (MMS) fairness was proposed by the authors of [
2] as a fairness concept for allocation of indivisible items. The goal is to allocate the items among the agents in a fair manner. The concept coincides with the standard proportionality fairness concept if the items to be allocated are divisible [
3,
4]. An agent’s MMS is the best if that agent can guarantee that they are allowed to partition the tasks, but then receives the least preferred bundle [
2]. If all agents have positive utilities for the items, the items are viewed as goods. On the other hand, if all agents have negative utilities (i.e., negative utilities can be expressed as cost) for the items, the items are viewed as tasks (i.e., some form of work, task is expressed as chore). However, in all the work above, utilities of goods or the cost of tasks limited in single dimension and the situation that cost is multidimensional has not been taken into account.
In reality, we will face the task allocation problem when the cost of the task is multidimensional, such as when allocating tasks of plant protection to Unmanned Aerial Vehicles (UAV) in the agricultural environment. With the accelerated development of industrialization and urbanization, the shortage of the main rural labor force has led to a sharp rise in the cost of agricultural labor. There are approximately 2 billion hectares of arable land in the world, dozens of major pests and diseases occur all year round, and a large number of agricultural plant protection operations are needed. According to statistics from the World Health Organization (WHO) in recent years, more than 3 million cases of pesticide poisoning occur every year around the world [
5]. With advancement in technologies, UAV can be useful in agricultural plant protection for crops to execute multiple tasks. To allocate tasks to UAVs in an agricultural plant protection environment, various situations need to be considered, such as reducing energy cost while minimizing overall task completion time. In order to find a method that can accomplishes fairness in allocation tasks to UAVs in the agricultural plant protection environment, evaluation of the cost of UAVs from multiple dimensions is needed. According to the authors of [
6] and combined with the actual situation, the cost of UAVs by tasks usually comes from three dimensions: time cost, energy cost, and UAV service life cost. Time cost mainly includes working time and flight time of UAVs arriving at tasks. Energy cost is UAV cost of electricity and fuel during flight and working. Because of the limited life of UAVs and their assembled lithium battery, the UAV service life cost includes the life loss of UAVs and lithium battery. When allocating tasks, the cost of all dimensions needs to be considered comprehensively, not just a single dimension. The Round-Robin algorithm is a classic task allocation algorithm, its process is that the agent chooses a task according to its own preferences every round until all tasks are allocated. Although there are some variations of this algorithm, such as Greedy Round-Robin or Double Round-Robin, none of the existing algorithms can solve the task allocation problem when the cost of task is multidimensional.
In this work, we take the agricultural plant protection environment as an example to study the task allocation problem when the cost of task is multidimensional. The fairness concept we use in this work is the intensively studied and well-established MMS fairness and all tasks are set as indivisible. It needs to be emphasized that this work only uses the agricultural plant protection environment as an example to conduct research, and the conclusions obtained can be applied to all task allocation issues. The main contributions of this paper are as follows.
Designing of a task allocation model for UAV applications in the agricultural plant protection environment.
Defining the Minmax share when the cost of tasks is multidimensional.
Improving the traditional Round-Robin algorithm to solve the task allocation problem when the cost of task is multidimensional.
Prove in the form of a theorem that the Minmax share approximation can be guaranteed.
Test and evaluate the performance of the proposed method using datasets generated based on real agricultural plant protection environment at the lab.
The remainder of the paper is structured as follows.
Section 2 explains the problem and detailed model of the task allocation system in agricultural plant protection environment.
Section 3 describes the key elements involved in the implementation of the algorithm we proposed and we analyze the properties of the algorithm in detail.
Section 4 discusses the experiments and results of the implemented methodology.
Section 5 presents the literature survey.
Section 6 concludes the findings of this research and proposes future work.
2. Problem Definition
Agricultural plant protection UAV task allocation can be formalized as a multi-agent task allocation problem. In this section, we will first introduce the multi-agent task allocation problem and MMS, which is an indicator to measure the fairness of task allocation. Then, according to the background of UAV agricultural plant protection, we define the allocation problem that the cost of tasks is multidimensional. In addition, we also define the relationship between the total cost and the cost of each dimension when the cost of tasks is multidimensional.
2.1. Fair Allocation Problem
For the fair allocation problem, N is a set of n agents and M is a set of m indivisible tasks. For any $i\in N$ and $j\in M$, ${C}_{i,j}$ represents the cost of task j to agent i. Next, we will illustrate the definition of preference.
Definition 1. Preference
Imagine that agent $i\in N$ and $j,k\in M$, then for i, $j\u2aafk$ if and only if ${C}_{i,j}\le {C}_{i,k}.$
Example 1. Assuming there are 2 agents and 2 tasks, the cost of agents by tasks can be seen in Table 1, then for agent 1, task 1 ⪯ task 2, and for agent 2, task 2 ⪯ task 1. Different agents may have different preferences for these tasks and these preferences are generally captured by cost functions ${C}_{i}:{2}^{M}\to \mathbb{R}$. As the goal of the problem is to fairly allocate all the tasks to these agents, the MMS fairness is defined in the following.
Definition 2. MMS fairness:
Imagine that agent i gets the opportunity to partition all the tasks into n bundles, but she is the last to choose a bundle. Then, her best strategy is to partition the tasks such that the biggest cost of a bundle is minmized. Let $\mathsf{\Pi}(M)$ denote the set of all possible n-partitionings of M. Then, the Minmax share (MMS) of agent i is defined as where ${X}_{j}$ is the task bundle that agent i partitions, and ${C}_{i}$ represents that the cost of agent i.
Example 2. (Continues from Example 1)
Assuming that agent 1 gets the opportunity to partition all tasks into 2 bundles and she is the last to choose, her best strategy is to partition the tasks into {task 1} and {task 2}, then the Minmax share of agent 1 is 3. Similarly, the Minmax share of agent 2 is 4.
Next, we will introduce the formal definition of MMS allocation.
Definition 3. MMS allocation:
Let $x={({x}_{i})}_{i\in N}$ be an allocation, where ${({x}_{i,j})}_{j\in M}$ and ${x}_{i,j}\in \{0,1\}$ indicates if agent i gets task j under allocation x. Then, x is a MMS allocation if and only if for all $i\in N$ Example 3. (Continues from Example 2) Assuming that the task 1 is allocated to agent 1 and task 2 is allocated to agent 2, then the coat of agent 1 is 2 and the coat of agent 2 is 1. All agents’ costs do not exceed their value of MMS, so this allocation is a MMS allocation.
The authors of [
7] have shown that a MMS allocation does not necessarily exist, and to make it more accurate to measure the fairness of allocation, the definition of approximated MMS allocation has been proposed.
Definition 4. $r\u2014$approximated MMS allocation:
Let $x={({x}_{i})}_{i\in N}$ be an allocation, where ${({x}_{i,j})}_{j\in M}$ and ${x}_{i,j}\in \{0,1\}$ indicates if agent i gets task j under allocation x. Then, x is a $r\u2014$approximated MMS allocation if and only if for all $i\in N$ In Equation (
3),
$r>1$, because when
$r\le 1$ an allocation will be called MMS-fairness, which has been shown in Definition 3. Note that a feasible allocation guarantees
${\sum}_{i\in N}{x}_{i,j}=1$ because a task can be assigned to only one agent when tasks are indivisible. It is assumed that all agents have underlying cardinal additive utilities over the tasks. Next, we will present the definition of the preference model that will be used in this paper.
Definition 5. Ordinal model:
Agents are only allowed to express their preference over tasks but the cost of the tasks to agents is unknown.
We have explained the additive valuations in
Section 2. Next, we will present the formal definition of the additive valuations for tasks.
Definition 6. Additive valuations:
Let N denote the set of agents and M denote the set of indivisible tasks, agents are said to have additive valuations when, for any subset of tasks $S\subseteq M$, the cost of agent i satisfies Example 4. (Continues from Example 1) Assuming that the task 1 and task 2 are allocated to agent 1, then according to Table 1, the cost of agent 1 is $2+3=5$. In the next section of this work, our research is based on an Ordinal model and additive valuations.
2.2. 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.
Tasks are indivisible, a task is allocated to one UAV.
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.
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).
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).
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).
Due to the complexity of the actual situation: the cost of tasks to UAVs is unknown.
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: $task1\u2aaftask2$).
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).
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 Table 3 and Table 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 $0.3\times 1+0.2\times 8+0.5\times 4=3.9$
As our research focuses on additive valuations, the total cost of UAV
i is calculated in Equation (
6), different types of cost have a relationship, which has been explained in condition 3, and are expressed as Equations (
7) and (
8).
According to condition 7, UAVs’ preferences over the tasks has been reported and the cost of tasks to UAVs is unknown. As there are 3 types of cost, according to Definition 1, UAVs have a preference for tasks based on each type of cost. Formally speaking, for all
$i\in N$, the UAV
i’s preference are shown in Equations (
9)–(
11).
Example 6. (Continues from Example 5) From Table 3 and Table 4, we can see that the preference of UAV 2 are as follows. In time cost, ${T}_{t}\left[2\right]\u2aaf{T}_{t}\left[1\right]$;
In energy cost, ${T}_{e}\left[1\right]\u2aaf{T}_{e}\left[2\right]$;
In service life cost, ${T}_{l}\left[2\right]\u2aaf{T}_{l}\left[1\right]$;
According to above conditions, our research is to find an allocation that satisfies $r\u2014$approximated MMS allocation. That is for all $i\in N:{C}_{i}\le r\xb7MM{S}_{i}$ by Definition 4.
5. Related Work
There are already some studies on the problems related to the use of UAVs in agricultural plant protection. The researchers in [
10] proposed a path planning algorithm with the goal of minimizing the total energy consumption of the work, realizing the full coverage path planning of the field and the optimization of the return point location. The researchers in [
11] proposed a path planning method based on Grid-GSA algorithm, which can plan a reasonable return point for the field with irregular boundaries, which makes the non-plant protection operation time the shortest. The researchers in [
6] fully consider the influence of the additional flight distance caused by the change of flight height, and studied a path planning method for plant protection UAVs that can be used for three-dimensional terrain. However, these works did not take into account the differences between plant protection tasks. In this work, we will study the problem of fair allocation when there are differences in the cost of plant protection tasks.
The fair allocation of indivisible items is a central problem in multi-agent systems [
3,
12]. Fair allocation has been extensively studied for allocation of divisible goods, commonly known as cake cutting [
13]. In order to identify fair allocations, one needs to formalize what fairness means. Multiple solution concepts, such as envy-freeness, proportionality, and equitability, have been proposed to formally capture fairness [
14,
15]. The authors of [
16] proposed the idea of envy-freeness up to one good (EF1); it has been further popularized in [
17], the researchers have shown that a natural modification of the Nash welfare maximizing rule satisfies EF1 and Pareto-optimal (PO), in addition they proposed envy freeness up to the least valued good (EFX), which is a stronger fairness concept. MMS fairness was proposed in [
2] as a fairness concept for allocation of indivisible items. The concept coincides with the standard proportionality fairness concept if the items to be allocated are divisible [
3,
4]. An agent’s MMS is the “most preferred bundle he could guarantee himself as a divider in divide-and-choose against adversarial opponents” [
2]. The authors of [
2] studied the conditions under which MMS allocations would exist.
Although MMS is a highly attractive fairness concept, the authors of [
3,
7] have shown that an MMS allocation of items does not exist in general. This fact initiated research on approximate MMS allocations of goods in which each agents gets some fraction of her MMS guarantee [
18]. There have been several works on algorithms that find an approximate MMS allocation [
19,
20,
21,
22,
23]. [
7] showed that although an MMS allocation of goods may not always exist, but there exists a polynomial-time algorithm that returns a
$\frac{2}{3}$-approximate MMS allocation. [
21] developing a simple and efficient algorithm that achieves the same approximation guarantee. [
23] proved that MMS allocations do not always exist but can be 2-approximated by a simple algorithm and also presented a PTAS for relaxation of MMS called optimal MMS.
Fair allocation of indivisible items problem can be divided into good allocation and task allocation (i.e., in some works, it is expressed as chore allocation). If all agents have positive utilities for the items, we view the items as goods. On the other hand, if all agents have negative utilities for the items, we can view the items as tasks [
18]. The authors of [
17] have shown that a natural modification of the Nash welfare maximizing rule satisfies EF1 and Pareto-optimal (PO) for the case of goods. Algorithms for computing approximate MMS allocations of goods are being used in practice for fair division in real-world problems [
24]. The authors of [
19,
20,
21] proposed an algorithm that can guarantee an approximate MMS allocation. The work mentioned above only focused on the case of goods. There are many settings in which agents may have negative utilities such as when tasks are allocated. The authors of [
1,
23] researched on fair allocation of of indivisible tasks and the authors of [
21] presented an improved approximation algorithm for MMS allocation of tasks. The authors of [
18,
23] showed that fair allocations of goods and tasks have some fundamental connections but differences as well. The problem of combinations of goods and tasks has received attention as well [
25].
Agents are said to have additive valuations when for any subset of goods
$S\subseteq M$, the valuation of agent
i satisfies
${v}_{i}(S):={\sum}_{j\in S}{v}_{i}(j)$. Agents are said to have submodular valuations when for any
$A\subseteq B\subset M$ and
$g\in M\setminus B$, the valuation of agent i satisfies
${v}_{i}(B\cup \left\{g\right\})-{v}_{i}(B)\le {v}_{i}(A\cup \left\{g\right\})-{v}_{i}(A)$. The authors of [
21] have shown that under additive valuations a
$\frac{2}{3}$-approximate MMS allocation can be achieved and under submodular valuations a
$\frac{1}{10}$-approximate MMS allocation can be guaranteed. The authors of [
1] consider MMS fairness in 3 preference models elicited from the agents: the cardinal model in which agents are allowed to report their utilities over the items, the ordinal model in which agents are only allowed to express their ordinal rankings over the items, and the public ranking model in which the ordinal preferences are publicly known. The authors of [
1] present an MMS approximation can be guaranteed while the algorithm is strategy proof in 3 preference models. However, in all the work above, utilities of goods or the cost of tasks limited in single dimension and the situation that cost comes from multiple dimensions has not been taken into account.
There are some optimization algorithms that can be used to solve multidimensional cost problem through multi-objective optimization models. The authors of [
26] used multi-objective Genetic Algorithms (GA) to solve multi-UAV mission planning. The authors of [
27] proposed a methodology with heuristic based on Earliest Available Time algorithm to solve the UAV scheduling problem in an indoor environment incorporated Particle Swarm Optimization (PSO) algorithm, with an objective of minimizing the makespan. However, as the space for the solutions of our problem increases exponentially with the number of tasks and these optimizations usually requires thousands of iterations to converge to the optimal solution when dealing with NP-Hard problems. Therefore, this type of algorithm takes a long time and is not suitable for embedding into agents with low hardware facilities.
In this work, we consider a model that has costs of tasks from multiple dimensions, and in each dimension, agents are only allowed to express their ordinal rankings over tasks. In addition, the cost of tasks under additive valuations. For this model, we proposed a task allocation algorithm that guarantees low time complexity while still achieving high fairness allocation.
6. Conclusions and Future Work
In this work, to solve indivisible tasks fair allocation problem when the cost of tasks is multidimensional, a task allocation problem model using agricultural plant protection environment as an example is established. We improve the traditional Round-Robin algorithm to solve this problem and get smallest approximate ratio of Min-Max share, which is a key indicators of fairness. We have proved with a theorem that the approximate ratio of Min-Max share of our proposed algorithm is no more than $2+\frac{m\xb7{\alpha}_{i}\xb7(1+n)-n}{{n}^{2}}$ and the time complexity of the algorithm is $O(mlogm)$. Experiments validate our analysis; we compare our proposed method with RSD and SequPick algorithm. Compared with RSD, our proposed algorithm achieves a lower approximation ratio of MMS when the running time is similar. For the comparison with the SequPick algorithm, although the running time of the proposed algorithm is slightly longer, we have a significantly lower approximation ratio of MMS and more stable performance.
In future work, we will improve our allocation algorithm to deal with more complex problems, such as consider the impact of task allocation order on cost. This will lead to a significant increase in the time complexity of the algorithm. In order to control the time complexity of the algorithm within an acceptable range, a specific branch and bound method can be applied.