Towards a formal analysis of the multi-robot task allocation problem using set theory

Bulletin of Electrical Engineering and Informatics
Vol. 10, No. 2, April 2021, pp. 1092~1104
ISSN: 2302-9285, DOI: 10.11591/eei.v10i2.2395  1092
Journal homepage: http://beei.org
Towards a formal analysis of the multi-robot task allocation
problem using set theory
Farouq Zitouni1
, Ramdane Maamri2
, Saad Harous3
1
Department of Computer Science, Kasdi Merbah University-Ouargla, Algeria
1,2
LIRE Laboratory, Abdelhamid Mehri University-Constantine, Algeria
2
Department of Computer Science, Abdelhamid Mehri University-Constantine, Algeria
1,3
Department of Computer Science and Software Engineering UAE University-Abu Dhabi, United Arab Emirates
Article Info ABSTRACT
Article history:
Received Mar 17, 2020
Revised Dec 31, 2020
Accepted Feb 6, 2021
Nowadays, the multi-robot task allocation problem is one of the most
challenging problems in multi-robot systems. It concerns the optimal
assignment of a set of tasks to several robots while optimizing a given
criterion subject to some constraints. This problem is very complex,
particularly when handling large groups of robots and tasks. We propose a
formal analysis of the task allocation problem in a multi-robot system, based
on set theory concepts. We believe that this analysis will help researchers
understand the nature of the problem, its time complexity, and consequently
develop efficient solutions. Also, we used that formal analysis to formulate
two well-known taxonomies of multi-robot task allocation problems. Finally,
a generic solving scheme of multi-robot task allocation problems is proposed
and illustrated on assigning papers to reviewers within a journal.
Keywords:
Assigning papers to reviewers
Generic solving scheme
Multi-robot systems
Set theory
Task allocation problem
Time complexity
This is an open access article under the CC BY-SA license.
Corresponding Author:
Farouq Zitouni
Department of Computer Science,
Kasdi Merbah University-Ouargla, Algeria
Email: farouqzitouni@uaeu.ac.ae
1. INTRODUCTION
During the last decade, intensive research on multi-robot systems (MRS) has been conducted to
solve several real-world problems. Usually, MRS are fault-tolerant and can efficiently solve complex
problems [1-3]. The issue of task allocation is one of the main challenges in MRS. In such a situation, a
complex problem is usually divided into elementary and straightforward tasks, where each one is then
assigned to a robot or several robots [4]. Solving a multi-robot task allocation (MRTA) problem is to assign
tasks to robots while optimizing a given objective function under several constraints [5].
Several centralized (i.e., using a central agent) and decentralized [5-7] approaches have been
proposed to solve MRTA problems in the literature. Centralized systems reduce information duplication, and
the check of generated solutions’ quality is effortless [8]. However, maintaining connectivity between robots
and the central agent is difficult, especially in large real-world problems [5]. Also, a load of communications
on the central agent can sometimes exceed its tolerable threshold. Besides, centralized approaches are not
robust (i.e., if the central agent fails, the system fails too). Decentralized approaches generally overcome the
disadvantages of centralized ones, particularly robustness. However, no guarantee [2] is provided on the
quality of generated solutions. Currently, we mainly distinguish three taxonomies of MRTA problems, i.e.,
each one classifies MRTA instances according to some features and criteria [9-11]. Taxonomies [9, 10] are
certainly the most adopted. We discuss them in the next paragraphs.

Bulletin of Electr Eng & Inf ISSN: 2302-9285 
Towards a formal analysis of the multi-robot task allocation problem using set theory (Farouq Zitouni)
1093
The taxonomy proposed in work [9] counts eight classes: i) [single-task robots, single-robot tasks,
instantaneous assignment]; ii) [single-task robots, single-robot tasks, time-extended assignment];
iii) [single-task robots, multi-robot tasks, instantaneous assignment]; iv) [single-task robots, multi-robot
tasks, time-extended assignment]; v) [multi-task robots, single-robot tasks, instantaneous assignment]; vi)
[multi-task robots, single-robot tasks, time-extended assignment]; vii) [multi-task robots, multi-robot tasks,
instantaneous assignment], and viii) [multi-task robots, multi-robot tasks, time-extended assignment]. In the
rest of the paper, this taxonomy is called “taxonomy A” for simplicity’s sake.
− Single-task robots: This means that each robot can perform a single task at the same time.
− Multi-task robots: This means that some robots can perform multiple tasks simultaneously.
− Single-robot tasks: This means that each task requires exactly one robot for its accomplishment.
− Multi-robot tasks: This means that some tasks require the cooperation of several robots for their
accomplishment.
− Instantaneous assignment: This means that the available information only allows instantaneous allocations
of tasks to robots, i.e., we do not use time windows, precedence, or synchronization constraints.
− Time-extended assignment: This means that more information is available, i.e., we use time windows,
precedence, or synchronization constraints, which allows time-extended allocations of tasks to robots;
in other words, the use of planning is unavoidable.
The taxonomy proposed in work [10] distinguishes 21 classes, which are: no dependencies (1 or 2),
in-schedule dependencies (2, 5 or 6), cross-schedule dependencies (1, 2, 3, 4, 5, 6, 7 or 8), and complex
dependencies (1, 2, 3, 4, 5, 6, 7 or 8). Where the numbers between parentheses represent one of the eight
classes defined in work [9]. In this taxonomy, each robot 𝑟 is supposed to have a schedule of assigned tasks.
In the rest of the paper, this taxonomy is called “taxonomy B” for simplicity’s sake.
− No dependencies: When allocating a task 𝑡 to a robot 𝑟, we consider 𝑡 and 𝑟.
− In-schedule dependencies: When allocating a task 𝑡 to a robot 𝑟, we consider 𝑟 and its schedule.
− Cross-schedule dependencies: When allocating a task 𝑡 to a robot 𝑟, we consider 𝑟, its schedule, and
other robots’ schedules. It is worth pointing out that the robots’ schedules are static, i.e., schedules
cannot be changed during the allocation process.
− Complex dependencies: When allocating a task 𝑡 to a robot 𝑟, we considered 𝑟, its schedule, and other
robots’ schedules. It is worth pointing out that the schedules are dynamic, i.e., schedules can be
changed during the allocation process.
All MRTA problems are firmly NP-hard [11]. The computational time to generate optimal solutions
for such problems increases exponentially with the problem size (i.e., numbers of robots and tasks).
Accordingly, analytical methods are strongly discouraged [12], especially when the problem size is large. In
the following, we present some related work.
There exist various common points between the following problems: i) the multi-robot task
allocation, ii) the multiple traveling repairmen [13], and iii) the multiple traveling salesmen [14]. The last
problems are derived from the well-known traveling salesman problem [15], which has been demonstrated to
be NP-hard in combinatorial optimization [16]. Therefore, the multi-robot task allocation problem is also NP-
hard [5]. The goal of the multiple traveling repairman problem is to determine a tour for each repairer while
optimizing the average waiting time of customers. The aim of the multiple traveling salesmen problem is to
allow each salesperson to visit all the cities and then return to the first one while optimizing their traveling
distances. These criteria (i.e., time and distance) are usually the most adopted in MRTA problems. In the last
decades, several approaches have been proposed in the literature to solve the different MRTA problems. We
mainly distinguish between centralized and distributed methods [17-20].
In centralized approaches, an intermediate robot collects all the required information and allocates
tasks to robots. The global objective function is optimized based on complete data [21, 22]. The work
presented in [23] combines a map partitioning method with a Markovian Decision Process to assign a set of
exploration tasks to a fleet of robots. Centralized approaches have many advantages: the objective function is
easily optimized, the simplicity of implementation, and the running time is significantly reduced, to mention
but a few. Nevertheless, they have some drawbacks, e.g., i) the scalability in real-life applications is not
always reachable: i.e., guarantee that all robots are connected to the central station leads to a heavy traffic
burden on it [24]; ii) the coverage area is drastically restricted because the allocations are computed on the
central station: i.e., the computational demand on the central robot is too high; and iii) the centralized
approaches are not robust: i.e., single failure point. The work described in [25] proposed a methodology to
solve a scenario of fire extinguishment that combines quantum genetic algorithms and reinforcement learning.
In distributed approaches, each robot runs the allocation algorithm individually. Usually, distributed
methods comprise a consensual phase to guarantee robots’ convergence to consistent allocations, despite the
network topologies [26]. Consensual stages add supplementary computing charges to the allocation process.
Consequently, the makespan of these approaches is increased (a significant amount of time is taken by the

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Bulletin of Electr Eng & Inf, Vol. 10, No. 2, April 2021 : 1092 – 1104
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robots to converge to a consistent allocation) [27]. Auction-based approaches [28-31] are considered as
distributed methods. They are generally employed to solve problems where each robot can execute one task
at a time, each task is strictly assigned to one robot, and allocations are time-extended (i.e., ST-SRTA).
Auction-based algorithms are shown to produce sub-optimal solutions [9] efficiently. In auction-based
approaches, robots offer bids on tasks according to their respective contextual information. The robot with
the highest bid wins. The auctioneer might be one of the robots or a dedicated central station [32, 33]. In
many ST-SR-TA task allocation scenarios, tasks, and robots are tightly bound [34]. In single-round
combinatorial auctions [35], robots consider task bundles when calculating their bids, and the produced
allocations are near-optimal. However, the computing time of all possible task bundles is exponentially
increased with the number of tasks. In sequential single-item auctions [36], this shortcoming is overcome.
Robots place their bids only on unallocated tasks in each round until allocating all tasks. In sequential
simultaneous auctions [22], robots place their proposals on unallocated tasks in each round. Only one task is
assigned at a time to each robot. An unmissable shortcoming in auction-based algorithms is the network
topology, as the robots must place bids on tasks and transmit them to the auctioneer. The approaches
proposed in [37-40] presented some auction-based strategies to solve MRTA problems. They used several
metaheuristics, such as genetic algorithms, firefly algorithm, ant colony system, quantum genetic algorithms,
artificial bee colony optimization. In work [41], the authors examined the MRTA problem using auctions.
They analyzed the influence of lossy communications on solutions’ quality. In work [42], the authors
introduced a method to resolve the MRTA problem. They transformed the situation into a transportation
problem and proposed a collision-free task allocation scheme.
In work [43], the authors proposed an algorithm called the “consensus-based bundle algorithm”
(CBBA). They combined the consensus-based and the auction-based approaches to take their advantages. The
consensus phase is used to agree on the winning bid values. The algorithm proposed in work [34] is derived from
CBBA. It is shown to have exceptional performance compared to the CBBA. Other algorithms [44-48] extended
from CBBA have also been shown to produce better solutions when solving ST-SR-TA problems with
temporal constraints. This family of algorithms is proven to be very robust, despite the network topology [5].
However, they cannot handle the situations where schedules are dynamic. Besides, they produce sub-optimal
allocations due to greedy-based strategies in the task inclusion phase [5]. The work in [49] uses CBBA, and
allocations are computed using ant colony optimization. The work described in [50] uses the CBBA, and
assignments are computed combining ant colony optimization and bat algorithm.
Many algorithms are available in the literature to solve MRTA problems [51, 52]. The authors of the
work [53] solved the multi-objective coalition formation problem using the Pareto archived evolution
strategy algorithm. The proposed method is centralized and does not consider dynamic scenarios. The
approach described in [18] extended the work in [34] to handle online rescheduling. The algorithm is not
efficient if much information arrives frequently. The work in [5] presented a solution to the task allocation
problems in a distributed MRS, under critical time constraints. The authors of the work [54] proposed a multi-
objective optimization of multi-robot task allocation with precedence constraints. The work in [55] discussed a
new coordination control for multiple non-holonomic robots for target tracking with limited communications.
We give a mathematical formulation of the MRTA problem using set theory concepts. It allows a
formal comprehension of the problem and gives the required tools to develop efficient solutions. At the end
of the paper, we provide a generic framework for solving any MRTA problem. The work presented in this
paper has three significant contributions. First, we propose a formal analysis of MRTA problems based on set
theory concepts. It is not meant to be final or exhaustive, and indeed it has limitations. However, we believe
that will help researchers understanding the nature of the problem, its components, and its complexity.
Second, we formalize the taxonomies proposed in [9, 10], using our formal analysis. It will help recognize
the type of a given problem from its components’ properties (i.e., robots, tasks, allocations.). Third, we
propose a generic solving scheme (i.e., an algorithm) of MRTA problems. Each step of this scheme is explained,
and its time complexity is given. Also, we illustrate the proposed generic solving method in a challenging situation,
which is the assignment of papers to reviewers within a journal. In conclusion, the work presented in this paper is
not meant to be better than other literature formulations [9-11]; it just complements them.
The rest of the paper is organized as follows. Some standard definitions specific to set theory are
given in section 2. Section 3 shows the proposed formal analysis of task allocation in MRS. Sections 4, and 5 provide
the formulation of different MRTA classes of taxonomies A [9] and B [10], respectively. Section 6 discusses
the proposed generic solving scheme of MRTA problems and highlights its time complexity. Section 7
illustrates the proposed algorithm on a real-world problem, which is the assignment of papers to reviewers
within a journal. Finally, a conclusion and some future work are provided in section 8.

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2. BACKGROUND ON SET THEORY
In this section, we give some standard definitions specific to the set theory.
Definition 1 A set is a collection of objects (i.e., set elements). We write 𝑒 ∈ 𝐸 to denote that 𝑒 is an element
of 𝐸. For example, the set 𝐸 = {𝑒1, 𝑒2, 𝑒3} contains three elements. The symbol |𝐸| denotes the cardinality of
𝐸. The symbol ∅ represents the empty set [56]
Definition 2 A subset is a set whose elements are all members of another set. The term 𝐹 ⊆ 𝐸 is read: 𝐹 is a
subset of 𝐸. The term 𝐹 ⊂ 𝐸 is read: 𝐹 is a proper subset of 𝐸 [56].
Definition 3 Let 𝐸 be a set. The power set of 𝐸, denoted as ℘(𝐸), is the set that contains all possible subsets
of 𝐸. The cardinality of ℘(𝐸) is 2|𝐸|
[56]. The power set of 𝐸 is given by (1).
℘(𝐸) = {𝑋|𝑋 ⊆ 𝐸} (1)
Definition 4 A collection of subsets 𝐹 of a set 𝐸 is called a family of subsets of 𝐸. For example, let 𝐸 =
{𝑒1, 𝑒2, 𝑒3, 𝑒4, 𝑒5} be a set, an instance of a family of subsets of 𝐸 might be given by 𝐹 =
{{𝑒1, 𝑒2, 𝑒3}, {𝑒4, 𝑒5}, {𝑒1, 𝑒2, 𝑒4}}. It is worth pointing out that any family of subsets of 𝐸 is a subset of the
power set ℘(𝐸) if it has no repeated members [56]. The cardinality of a family of subsets 𝐹 is equal to the
number of subsets it contains. The terms “family of subsets 𝐹” and “family 𝐹” will be used interchangeably.
Example 1 Let 𝐸 = {𝑒1, 𝑒2, 𝑒3} be a set.
− The power set of 𝐸 is: ℘(𝐸) = {∅, {𝑒1}, {𝑒2}, {𝑒3}, {𝑒1, 𝑒2}, {𝑒1, 𝑒3}, {𝑒2, 𝑒3}, {𝑒1, 𝑒2, 𝑒3}}.
− Collections of subsets 𝐹1 = {{𝑒1, 𝑒2}, {𝑒2, 𝑒3}}, 𝐹2 = {{𝑒1}, {𝑒2}} and 𝐹3 = {{𝑒1, 𝑒2, 𝑒3}} are three families
of 𝐸. The cardinality of 𝐹2 is |𝐹2| = 2.
Definition 5 The Cartesian product 𝐸1 × … × 𝐸𝑛 of sets 𝐸1, … , 𝐸𝑛 is defined as the set of all ordered tuples
(𝑒1, … , 𝑒𝑛), with 𝑒1 ∈ 𝐸1, … , 𝑒𝑛 ∈ 𝐸𝑛 as follows [56].
𝐸1 × … × 𝐸𝑛 = {(𝑒1, … , 𝑒𝑛)|𝑒1 ∈ 𝐸1, … , 𝑒𝑛 ∈ 𝐸𝑛} (2)
Definition 6 Let 𝐸1, … , 𝐸𝑛 be 𝑛 sets. An n-ary relation on these sets is a subset of 𝐸1 × … × 𝐸𝑛. The sets
𝐸1, … , 𝐸𝑛 are called domains of the relation, and 𝑛 is its degree [56].
Definition 7 A 𝑘-permutation is an ordered arrangement of 𝑘 subsets from a family of 𝑛 subsets (𝑘 ≤ 𝑛). If a
given family of subsets 𝐹 has 𝑛 subsets, then the number of 𝑘-permutations generated from 𝐹, denoted as
𝑃(𝑛, 𝑘), is given by (3) [56].
𝑃(𝑛, 𝑘) =
𝑛!
(𝑛−𝑘)!
(3)
Example 2 Let 𝐹 = {{𝑒1}, {𝑒2}, {𝑒3}, {𝑒1, 𝑒2}} be a family. It is evident that 𝐹 has 24 different 4-permutations,
e.g., {{𝑒1}, {𝑒2}, {𝑒1, 𝑒2}, {𝑒3}} is one of them.
Definition 8 Let 𝐸′ be a subset of a set 𝐸 and 𝐹′ be a subset of a set 𝐹. According to work discussed in [10],
we can define a utility measure, denoted as 𝑈(𝐸′
, 𝐹′), for subsets 𝐸′ and 𝐹′, (4).
𝑈(𝐸′
, 𝐹′) = {
𝑉(𝐸′
, 𝐹′) , 𝑖𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 𝐸′
𝑎𝑛𝑑 𝐹′
𝑎𝑟𝑒 𝑟𝑒𝑙𝑎𝑡𝑒𝑑
∞ , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
(4)
The term 𝑉(𝐸′
, 𝐹′) is any function that returns a numerical value according to relationships between
elements of subsets 𝐸′ and 𝐹′. For example, if elements of 𝐸′ and 𝐹′ are points in the space, 𝑉(𝐸′
, 𝐹′) may
return the average of distances between points of 𝐸′ and points of 𝐹′. In other words, the function 𝑉(𝐸′
, 𝐹′) is
given (5).
𝑉(𝐸′
, 𝐹′) =
∑ ∑ ‖𝑒𝑖−𝑓𝑗‖
|𝐹′|
𝑗=1
|𝐸′|
𝑖=1
|𝐸′|.|𝐹′|
(5)
3. FORMAL ANALYSIS OF MRTA PROBLEMS
In this section, we give the proposed formal analysis of multi-robot task allocation problems.
Definition 9 A robot is an autonomous (logical or physical) entity that perceives its environment, makes
decisions, and performs actions [11]. The set 𝐴 = {𝑎1, … , 𝑎𝑛} designates a group of 𝑛 robots.
Definition 10 A task is an elementary and independent action accomplished by one or several robots [11].
The set 𝑇 = {𝑡1, … , 𝑡𝑚} designates a collection of 𝑚 tasks.

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Definition 11 A resource refers to any tool (logical or physical) that a task requires for its accomplishment.
For example, suppose we want to measure the ambient temperature of someplace (i.e., task). In that case, we
need a thermometer (i.e., resource). In MRTA scenarios, robots usually offer resources and tasks use them.
The set Ω = {𝜔1, … , 𝜔𝑘} designates a collection of 𝑘 resources.
Definition 12 Resources offered by robots is a ternary relation over sets 𝐴, Ω, and ℬ = {0,1}. It is a set of
ordered triples (𝑎, 𝜔, 𝑏) consisting of elements: 𝑎 ∈ 𝐴, 𝜔 ∈ Ω, and 𝑏 ∈ ℬ. It encodes the relation’s
information: robot 𝑎 has 𝑏 instances of resource 𝜔. This relation is denoted as 𝕆 (𝕆 ⊆ 𝐴 × Ω × ℬ).
Definition 13 Resources required by tasks is a ternary relation over sets 𝑇, Ω, and ℬ = {0,1}. It is a set of
ordered triples (𝑎, 𝜔, 𝑏) consisting of elements: 𝑡 ∈ 𝑇, 𝜔 ∈ Ω, and 𝑏 ∈ ℬ. It encodes the relation’s
information: task 𝑡 requires 𝑏 instances of resource 𝜔 for its accomplishment. This relation is denoted as ℝ
(ℝ ⊆ 𝑇 × Ω × ℬ).
Definition 14 Valid assignments over robots and tasks is a quinary relation over sets 𝐴, ℬ = {0,1}, Ω, 𝑇, and
ℬ = {0,1}. It is a set of ordered quintuples (𝑎, 𝑏, 𝜔, 𝑡, 𝑏′) consisting of elements: 𝑎 ∈ 𝐴, 𝑏 ∈ ℬ, 𝜔 ∈ Ω, 𝑡 ∈ 𝑇,
and 𝑏′ ∈ ℬ. It encodes the relation: robot 𝑎 has 𝑏 instances of resource 𝜔 and can be assigned to task 𝑡,
which needs 𝑏′ instances of resource 𝜔. This relation is denoted as 𝕍 (𝕍 ⊆ 𝐴 × ℬ × Ω × T × ℬ).
This relation is given by (6), where 𝑄𝑖(𝑎, 𝑏, 𝜔, 𝑡, 𝑏′) are conditions that determine allowed quintuples. For
example, suppose the distance separating a robot and a task is larger than a given threshold. In that case,
this robot cannot be assigned to this task even if it has the required resources. In other words, these
conditions translate the constraints of the considered problem.
𝕍 = {(𝑎, 𝑏, 𝜔, 𝑡, 𝑏′)|(𝑎, 𝜔, 𝑏) ∈ 𝕆, (𝑡, 𝜔, 𝑏′) ∈ ℝ, (𝑏, 𝑏′) ≥
(1,1), 𝑄1(𝑎, 𝑏, 𝜔, 𝑡, 𝑏′), 𝑄2(𝑎, 𝑏, 𝜔, 𝑡, 𝑏′), 𝑄3(𝑎, 𝑏, 𝜔, 𝑡, 𝑏′), … } (6)
NB: The set ℬ = {0,1} can be replaced by ℕ = {0,1,2,3, … }. However, for simplicity, we use the first set.
Definition 15 A coalition is a temporary group of robots working together on some task. Generally, when the
task is performed, the coalition is dissolved [11]. If we consider a set 𝐸, then one can enumerate all possible
coalitions using Definition 3. In this context, we should exclude the empty set ∅ since it is not a coalition.
Thus, for a given set 𝐸, we have 2|𝐸|
− 1 possible coalitions.
Definition 16 Let 𝐸 be a set. An overlapping family 𝐹 of 𝐸 is a family whose subsets are not duplicated and
not necessarily pairwise disjoint. An overlapping family 𝐹 should simultaneously verify the following
conditions:
a. ∅ ∉ 𝐹.
b. ∃𝐹1, 𝐹2 ∈ 𝐹|(𝐹1 ≠ 𝐹2) ⇒ (𝐹1 ∩ 𝐹2 ≠ ∅).
c. ⋃ 𝐹𝑖
|𝐹|
𝑖=1 = 𝐸.
Property 1 signifies that the empty subset is not considered (i.e., the empty coalition is not considered).
Property 2 implies that a given element can be a member of several subsets simultaneously (i.e., a given
robot can belong to several coalitions at the same time). Property 3 denotes that all members of set 𝐸 should
appear in subsets of family 𝐹 (i.e., all robots are allocated). The symbol Θ(𝐸) designates the collection of all
possible overlapping families of subsets of a set 𝐸, the symbol Θ𝑖 denotes the ith
family in the collection Θ(𝐸),
and the symbol Θ𝑖𝑗 expresses the jth
subset in the family Θ𝑖. The number of overlapping families of a set 𝐸 is
given by (7) [57].
|Θ(𝐸)| =
1
2
∑ (−1)𝑘 |𝐸|!
(|𝐸|−𝑘)!𝑘!
22|𝐸|−𝑘
|𝐸|
𝑘=0 (7)
Definition 17 Let 𝐸 be a set. A disjoint family 𝐹 of 𝐸 is a family whose subsets are not duplicated and
mandatory pairwise disjoint. A disjoint family 𝐹 should simultaneously verify the following conditions:
a. ∅ ∉ 𝐹.
b. ∀𝐹1, 𝐹2 ∈ 𝐹|(𝐹1 ≠ 𝐹2) ⇒ (𝐹1 ∩ 𝐹2 = ∅).
c. ⋃ 𝐹𝑖
|𝐹|
𝑖=1 = 𝐸.
Property 1 signifies that the empty subset is not considered (e.g., the empty coalition is not considered).
Property 2 implies that a given element cannot be a member of several subsets simultaneously (e.g., any task
is allocated one time). Property 3 denotes that all members of set 𝐸 should appear in subsets of family 𝐹
(e.g., all tasks are allocated). The symbol ∆(𝐸) designates the collection of all possible disjoint families of
subsets of a set 𝐸, the symbol ∆𝑖 denotes the ith
family in the collection ∆(𝐸), and the symbol ∆𝑖𝑗 expresses
the jth
subset in the family ∆𝑖. The number of disjoint families of a set 𝐸 is given by (8) [58].
|∆(𝐸)| = ∑ (
1
𝑘!
∑ (−1)𝑖 𝑘!
(𝑘−𝑖)!𝑖!
(𝑘 − 𝑖)|𝐸|
𝑘
𝑖=0 )
|𝐸|
𝑘=0 (8)

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Remark 1 If for each family ∆𝑖∈ ∆(𝐸) we consider its |∆𝑖|-permutations, we can define an augmented
collection ∆′(𝐸), where each family ∆𝑖∈ ∆(𝐸) is replaced by its |∆𝑖|-permutations. The symbol ∆𝑖′ will
denote the ith
family in the collection ∆′(𝐸) and the symbol ∆𝑖𝑗′ will express the jth
subset in the family ∆𝑖′.
The number of augmented disjoint families of a set 𝐸 is given by (9).
|∆′(𝐸)| = ∑ ((
1
𝑘!
∑ (−1)𝑖 𝑘!
(𝑘−𝑖)!𝑖!
(𝑘 − 𝑖)|𝐸|
𝑘
𝑖=0 ) 𝑘!)
|𝐸|
𝑘=0 (9)
Example 3 Let 𝐸 = {𝑒1, 𝑒2} be a set.
− The collection of all possible overlapping families is:
Θ(𝐸) = {{{𝑒1}, {𝑒2}}, {{𝑒1, 𝑒2}}, {{𝑒1}, {𝑒1, 𝑒2}}, {{𝑒2}, {𝑒1, 𝑒2}}, {{𝑒1}, {𝑒2}, {𝑒1, 𝑒2}}}.
− The collection of all possible disjoint families is:
∆(𝐸) = {{{𝑒1}, {𝑒2}}, {{𝑒1, 𝑒2}}}.
− The collection of all possible augmented disjoint families is:
∆′(𝐸) = {{{𝑒1}, {𝑒2}}, {{𝑒2}, {𝑒1}}, {{𝑒1, 𝑒2}}}.
Definition 18 Let 𝐴 = {𝑎1, … , 𝑎𝑛} be a set of robots and 𝑇 = {𝑡1, … , 𝑡𝑚} be a set of tasks. Let Θ(𝐴) =
{Θ1, … , Θ𝑝} be the collection of overlapping families of 𝐴 and ∆′(𝑇) = {∆1
′
, … , ∆𝑞′} be the augmented
collection of disjoint families of 𝑇. Terms Θ𝑖 and ∆𝑗′ are, respectively, called family of robots’ coalitions and
family of task’ coalitions.
Partial allocations are a binary relation over collections Θ(𝐴) and ∆′(𝑇). It is a set of ordered pairs
(Θ𝑖, ∆𝑗′) consisting of elements: Θ𝑖 ∈ Θ(𝐴) and ∆𝑗′ ∈ ∆′(𝑇). It is a subset of the Cartesian product Θ(𝐴) ×
∆′(𝑇). This relation is given by (10) and denoted as Γ.
Γ = {(Θ𝑖, ∆𝑗′)|Θ𝑖 ∈ Θ(𝐴), ∆𝑗′ ∈ ∆′(𝑇), |Θ𝑖| = |∆𝑗′|, 𝑄1(Θ𝑖), … , 𝑄1′(∆𝑗′), … , 𝑄1′′(Θ𝑖, ∆𝑗′), … } (10)
Each pair (Θ𝑖, ∆𝑗′) ∈ Γ is read as follows: each robots’ coalition Θ𝑖𝑘 ∈ Θ𝑖 is assigned to tasks’ coalition
∆𝑗𝑘′ ∈ ∆𝑗′. For example, let Θ𝑖 = {{𝑎1, 𝑎3}, {𝑎1, 𝑎2}} and ∆𝑗
′
= {{𝑡1, 𝑡2}, {𝑡3}}. We read: robots’ coalition
{𝑎1, 𝑎3} is assigned to tasks’ coalition {𝑡1, 𝑡2}, and robots’ coalition {𝑎1, 𝑎2} is assigned to tasks’ coalition
{𝑡3}. 𝑄1(Θ𝑖), … , 𝑄1′(∆𝑗′), … , 𝑄1′′(Θ𝑖, ∆𝑗′), … are conditions that determine allowed partial allocations.
Definition 19 A global allocation Γ′ (or merely an allocation) is a sub-relation of Γ whose pairs (Θ𝑖, ∆𝑖′)
should simultaneously verify the following conditions:
a. ⋃ ⋃ Θ𝑖𝑗 = 𝐴
|Θ𝑖|
𝑗=1
|Γ′|
𝑖=1 .
b. ⋃ ⋃ ∆𝑖𝑗′ = 𝑇
|∆𝑖′|
𝑗=1
|Γ′|
𝑖=1 .
c. ⋂ ⋂ ∆𝑖𝑗′
|∆𝑖′|
𝑗=1
|Γ′|
𝑖=1 = ∅.
d. ∀(Θ𝑖, ∆𝑖′) ∈ Γ′
∀Θ𝑖𝑗 ∈ Θ𝑖∀∆𝑖𝑗
′
∈ ∆𝑖
′
|{𝜔|𝜔 ∈ (Θ𝑖𝑗 × ℬ × Ω × ∆𝑖𝑗
′
× ℬ)} ∩ {𝜔|𝜔 ∈ (∆𝑖𝑗
′
× Ω × ℬ)} =
{𝜔|𝜔 ∈ (∆𝑖𝑗
′
× Ω × ℬ)}.
Relation (Θ𝑖𝑗 × ℬ × Ω × ∆𝑖𝑗
′
× ℬ) ⊆ 𝕍 represents valid assignments over robots in Θ𝑖𝑗 and tasks in ∆𝑖𝑗
′
.
Relation (∆𝑖𝑗
′
× Ω × ℬ) ⊆ ℝ represents the required resources by tasks in ∆𝑖𝑗
′
. Condition 1 denotes that all
robots are solicited to be assigned to tasks. Condition 2 means that all tasks should be assigned. Condition 3
signifies that a given task is exactly assigned to one robots’ coalition. Condition 4 states that each coalition
of robots should offer the required resources to the corresponding coalition of tasks. The optimal global
allocation is the one that optimizes the utility measure given by (11).
𝑈
̂ = ∑ ∑ 𝑈(Θ𝑖𝑗, ∆𝑖𝑗′)
|Θ𝑖|
𝑗=1
|Γ′|
𝑖=1 (11)
Definition 20 Let (𝐹1, 𝐹2) ∈ Γ be a partial allocation. Let 𝐹1𝑖 and 𝐹2𝑖 be the ith
robots’ and tasks’ coalitions
of families 𝐹1 and 𝐹2, respectively. Potential assignments are a sub-relation of the relation 𝕍 given by
Definition 14 (valid assignments) whose quintuples modeled by (12). This relation is denoted as ℙ.
ℙ = {(𝑎, 𝑏, 𝜔, 𝑡, 𝑏′)|(𝑎, 𝑏, 𝜔, 𝑡, 𝑏′) ∈ 𝕍, 𝑎 ∈ 𝐹1𝑖, 𝑡 ∈ 𝐹2𝑖} (12)

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Robots in 𝐹1𝑖 ar related to tasks in 𝐹2𝑖 (4), if and only if the relation ℙ is not empty.
Example 4 In this example, we consider 𝐴 = {𝑎1, 𝑎2}, 𝑇 = {𝑡1, 𝑡2, 𝑡3}, and Ω = {𝜔1, 𝜔2}. Relations 𝕆, ℝ,
and 𝕍 are given as follow:
− 𝕆 = {(𝑎1, 𝜔1, 1), (𝑎1, 𝜔2, 0), (𝑎2, 𝜔1, 1), (𝑎2, 𝜔2, 1)}.
− ℝ = {(𝑡1, 𝜔1, 1), (𝑡1, 𝜔2, 0), (𝑡2, 𝜔1, 1), (𝑡2, 𝜔2, 0), (𝑡3, 𝜔1, 0), (𝑡3, 𝜔2, 1)}.
− 𝕍 = {(𝑎1, 1, 𝜔1, 𝑡1, 1), (𝑎1, 1, 𝜔1, 𝑡2, 1), (𝑎2, 1, 𝜔1, 𝑡1, 1), (𝑎2, 1, 𝜔1, 𝑡2, 1), (𝑎2, 1, 𝜔2, 𝑡3, 1)}.
Let 𝐹1 = {{𝑎1, 𝑎2}, {𝑎2}} be a family of robots’ coalitions; and 𝐹2 = {{𝑡1}, {𝑡2, 𝑡3}} be a family of tasks’
coalitions. If we consider 𝐹12 = {𝑎2} and 𝐹22 = {𝑡2, 𝑡3}, then ℙ = {(𝑎2, 1, 𝜔1, 𝑡2, 1), (𝑎2, 1, 𝜔2, 𝑡3, 1)}.
Definition 21 Let Γ′
= {(𝐸1, 𝐹1), … , (𝐸𝑝, 𝐹𝑝)} be a global allocation. A family of potential assignments is the
set [ℙ] = {ℙ1, … , ℙ𝑝} of potential assignments given by Definition 20 (each term ℙ𝑖 ∈ [ℙ] is computed as in
Example 4).
The concept of a family of potential assignments allows us to build task schedules of robots. The use of this
concept will be detailed in Section 5.
4. FORMULATION OF TAXONOMY A
This section gives the formulation of taxonomy A [9], using our proposed formulation.
Definition 22 Let 𝕆 be the relation of resources offered by robots given in Definition 12. All robots of a
given problem are said to be Single-Task, if the following condition is satisfied.
∀𝑎 ∈ 𝐴∃! 𝜔 ∈ Ω|(𝑎, 𝜔, 1) ∈ 𝕆 (13)
Condition 13 signifies that each robot 𝑎 ∈ 𝐴 offers one and only one resource 𝜔 ∈ Ω for accomplishing tasks.
Definition 23 Let 𝕆 be the relation of resources offered by robots given in Definition 12. Some robots of a
given problem are said to be Multi-Task, if the following condition is satisfied.
∀𝑎 ∈ 𝐴∃𝜔1, … , 𝜔𝑝 ∈ Ω|(𝑎, 𝜔1, 1) ∈ 𝕆 ∧ … ∧ (𝑎, 𝜔𝑝, 1) ∈ 𝕆 (14)
Condition 14 signifies that some robots 𝑎 ∈ 𝐴 offer several resources 𝜔 ∈ Ω for accomplishing tasks.
Definition 24 Let ℝ be the relation of resources required by tasks given in Definition 13. All tasks of a given
problem are said to be Single-Robot, if the following condition is satisfied.
∀𝑡 ∈ 𝑇∃! 𝜔 ∈ Ω|(𝑡, 𝜔, 1) ∈ ℝ (15)
Condition 15 signifies that each task 𝑡 ∈ 𝑇 requires one and only one resource 𝜔 ∈ Ω for its
accomplishment.
Definition 25 Let ℝ be the relation of resources required by tasks given in Definition 13. Some tasks of a
given problem are said to be Multi-Robot, if the following condition is satisfied.
∀𝑡 ∈ 𝑇∃𝜔1, … , 𝜔𝑝 ∈ Ω|(𝑡, 𝜔1, 1) ∈ ℝ ∧ … ∧ (𝑡, 𝜔𝑝, 1) ∈ ℝ (16)
Condition 16 means that some tasks 𝑡 ∈ 𝑇 require several resources 𝜔 ∈ Ω for their accomplishment.
= {(𝐹1, 𝐹1′), … , (𝐹𝑝, 𝐹𝑝′)} be an allocation.
− If the execution of tasks in Γ′
are associated with any temporal model (e.g., time intervals [59], simple
temporal networks [60]). In this case, we say that the allocation of robots to tasks is performed in a
time-extended manner.
− Conversely, we say that the allocation of robots to tasks is performed in an instantaneous manner (i.e.,
no temporal model is provided, and tasks are allocated as they come).’
5. FORMULATION OF TAXONOMY B
In this section, we formulate the taxonomy described in [10], using the proposed analysis. It is worth
pointing out that each robot should have a schedule of tasks, which is a critical issue in this context. The
following example shows how robots’ schedules are built.
Example 5 In this example, we consider 𝐴 = {𝑎1, 𝑎2, 𝑎3}, 𝑇 = {𝑡1, 𝑡2, 𝑡3, 𝑡4}, and Ω = {𝜔1, 𝜔2}. Relations 𝕆,
ℝ, 𝕍, and Γ′ are given as follow:
− 𝕆 = {(a1, ω1, 1), (a1, ω2, 1), (a2, ω1, 1), (a2, ω2, 1), (a3, ω1, 1), (a3, ω2, 1)}.

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− ℝ = {
(t1, ω1, 1), (t1, ω2, 1), (t2, ω1, 1), (t2, ω2, 1),
(t3, ω1, 1), (t3, ω2, 1), (t4, ω1, 1), (t4, ω2, 1)
}.
− 𝕍 =
{
(a1, 1, ω1, t1, 1), (a1, 1, ω1, t2, 1), (a1, 1, ω1, t3, 1), (a1, 1, ω1, t4, 1),
(a1, 1, ω2, t1, 1), (a1, 1, ω2, t2, 1), (a1, 1, ω2, t3, 1), (a1, 1, ω2, t4, 1),
(a2, 1, ω1, t1, 1), (a2, 1, ω1, t2, 1), (a2, 1, ω1, t3, 1), (a2, 1, ω1, t4, 1),
(a2, 1, ω2, t1, 1), (a2, 1, ω2, t2, 1), (a2, 1, ω2, t3, 1), (a2, 1, ω2, t4, 1),
(a3, 1, ω1, t1, 1), (a3, 1, ω1, t2, 1), (a3, 1, ω1, t3, 1), (a3, 1, ω1, t4, 1),
(a3, 1, ω2, t1, 1), (a3, 1, ω2, t2, 1), (a3, 1, ω2, t3, 1), (a3, 1, ω2, t4, 1)}
.
− 𝛤′
= {({{𝑎1}}, {{𝑡1}}), ({{𝑎1, 𝑎2}, {𝑎1, 𝑎2, 𝑎3}}, {{𝑡2}, {𝑡3, 𝑡4}})}.
𝛤′
is a global allocation, which means that: robots’ coalition {𝑎1} is assigned to tasks’ coalition {𝑡1}, robots’
coalition {𝑎1, 𝑎2} is assigned to tasks’ coalition {𝑡2}, and robots’ coalition {𝑎1, 𝑎2, 𝑎3} is assigned to tasks’
coalition {𝑡3, 𝑡4}. The family of potential assignments corresponding to 𝛤′
is [ℙ] = {ℙ1, ℙ2, ℙ3}, where:
− Potential assignments ℙ1 over subsets {𝑎1} and {𝑡1} are:
ℙ1 = {(a1, 1, ω1, t1, 1), (a1, 1, ω2, t1, 1)}
Robots’ coalition {𝑎1} is assigned to tasks’ coalition {𝑡1}. We have only one partial schedule: i.e., robot a1
offers both resources ω1 and ω2 to task t1.
− Potential assignments ℙ2 over subsets {𝑎1, 𝑎2} and {𝑡2} are:
ℙ2 = {(a1, 1, ω1, 𝑡2, 1), (a1, 1, ω2, 𝑡2, 1), (a2, 1, ω1, 𝑡2, 1), (a2, 1, ω2, 𝑡2, 1)}
Robots’ coalition {𝑎1, 𝑎2} is assigned to tasks’ coalition {𝑡2}. We have four partial schedules: i.e., robot a1
offers both resources ω1 and ω2 to task 𝑡2; robot a2 offers both resources ω1 and ω2 to task 𝑡2; robot a1
offers resource ω1 to task 𝑡2 and robot a2 offers resource ω2 to task 𝑡2; or robot a1 offers resource ω2 to
task 𝑡2 and robot a2 offers resource ω1 to task 𝑡2.
− Potential assignments ℙ3 over subsets {𝑎1, 𝑎2, 𝑎3} and {𝑡3, 𝑡4} are:
ℙ3 = {
(a1, 1, ω1, t3, 1), (a1, 1, ω2, t3, 1), (a1, 1, ω1, t4, 1), (a1, 1, ω2, t4, 1),
(a2, 1, ω1, t3, 1), (a2, 1, ω2, t3, 1), (a2, 1, ω1, t4, 1), (a2, 1, ω2, t4, 1),
(a3, 1, ω1, t3, 1), (a3, 1, ω2, t3, 1), (a3, 1, ω1, t4, 1), (a3, 1, ω2, t4, 1)
}
Robots’ coalition {𝑎1, 𝑎2, 𝑎3} is assigned to tasks’ coalition {𝑡3, 𝑡4}. We have 32×2
= 81 partial schedules
(we only cite one instance): e.g., robot a1 offers resource ω1 to task t3, robot a1 offers resource ω2 to task
t3, robot a2 offers resource ω1 to task t4, and robot a3 offers resource ω2 to task t4.
Generally, if we have three sets 𝐴, 𝑇 and Ω of robots, tasks, and resources, respectively. In this case, we
should consider |𝐴|(|𝑇|×|Ω|)
different partial schedules of size |𝑇| × |Ω. In our example, we should consider
1 × 4 × 81 global schedules, and the schedule of each robot will contain its assigned tasks: e.g., if we take
the partial schedules: i) robot a1 offers both resources ω1 and ω2 to task t1; ii) robot a1 offers resource ω1
to task t2 and robot 𝑎2 offers resource ω2 to task t2; and iii) robot a1 offers resource ω1 to task t3, robot a1
offers resource ω2 to task t3, robot 𝑎2 offers resource ω1 to task t4 and robot 𝑎3 offers resource ω2 to task
t4. In this case, i) the schedule of robot a1 contains tasks {(𝑡1, 𝜔1), (𝑡1, 𝜔2), (𝑡2, 𝜔1), (𝑡3, 𝜔1), (𝑡3, 𝜔2)}; ii)
the schedule of robot 𝑎2 contains tasks {(𝑡2, 𝜔2), (𝑡4, 𝜔1)}; and iii) the schedule of robot 𝑎3 contains one
task {(𝑡4, 𝜔2)}. It is worth pointing out that the order of execution of tasks of each robot depends on how
tasks are allocated to robots, i.e., instantaneous or time-extended assignments.
= {(𝐸1, 𝐹1), … , (𝐸𝑝, 𝐹𝑝)} be an allocation and [ℙ] = {ℙ1, … , ℙ𝑝} be its corresponding
family of potential assignments. We consider robots 𝑎1, … , 𝑎𝑛. We showed, via Example 5, how robots’
schedules are built. Let 𝑆𝑎𝑖
(𝜏) be the schedule of robot 𝑎𝑖 at time 𝜏. We define the features of taxonomy B
[10] as follows.
− No dependencies:
As shown in (17) shows that a given robot’s utility measure for a specific task depends on them.
∀𝑡𝑗 ∈ 𝑆𝑎𝑖
(0): 𝑈({𝑎𝑖}, {𝑡𝑗}) = 𝑓(𝑎𝑖, 𝑡𝑗) (17)
− In-schedule dependencies:
As shown in (18) shows that a given robot’s utility measure for a specific task depends on its schedule.
(0): 𝑈({𝑎𝑖}, {𝑡𝑗}) = 𝑓 (𝑎𝑖, 𝑆𝑎1
(0)) (18)
− Cross-schedule dependencies:

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As shown in (19) shows that a given robot’s utility measure for a specific task depends on all robots’
schedules, including itself. However, the schedules are static.
(0): 𝑈({𝑎𝑖}, {𝑡𝑗}) = 𝑓 (𝑎𝑖, 𝑆𝑎1
(0), … , 𝑆𝑎𝑛
(0)) (19)
− Complex dependencies:
As shown in (20) shows that a given robot’s utility measure for a specific task depends on all robots’
schedules, including itself. However, the schedules are dynamic.
(0): 𝑈({𝑎𝑖}, {𝑡𝑗}) = 𝑓 (𝑎𝑖, 𝑆𝑎1
(𝜏), … , 𝑆𝑎𝑛
(𝜏)) (20)
6. SOLVING SCHEME OF MRTA PROBLEMS
This section overviews how the proposed formulation is used in a generic solving scheme and
explains its analytical time complexity. Algorithm 1 highlights the main steps in solving a MRTA problem.
− Line 1 computes the relation 𝕍, a subset of the Cartesian product of relations 𝕆 and ℝ. The relation 𝕍
contains all valid assignments between tasks and robots. The time complexity of this step is
𝒪(𝑚𝑎𝑥{|𝕆|, |ℝ|}2).
− Line 2 computes the collection Θ(𝐴) of overlapping families of subsets of the set 𝐴; i.e., a given robot
may belong to several coalitions, since it can perform several tasks (simultaneously or sequentially). The
time complexity of this step is 𝒪(|℘(𝐴)| × 2|℘(𝐴)|
).
− Line 3 computes the collection ∆(𝑇) of disjoint families of subsets of the set 𝑇; i.e., a given task must
belong to only one coalition, since it must be done once. The time complexity of this step is also
𝒪(|℘(𝑇)| × 2|℘(𝑇)|
).
− Line 4 computes the collection ∆′(𝑇) of all possible permutations of families in ∆(𝑇). The time
complexity of this step is 𝒪 ( max
∆𝑖∈∆(𝑇)
{|∆𝑖|!}).
− Line 5 computes the relation Γ, a subset of the Cartesian product Θ(𝐴) × ∆′(𝑇). This relation contains all
partial allocations. The time complexity of this step is 𝒪(max{|Θ(𝐴)|, |∆′(𝑇)|}2).
− Line 6 computes all global allocations. The time complexity of this step is 𝒪(|℘(Γ)| × 2|℘(Γ)|
).
− Finally, each global allocation is evaluated using the considered utility measure subject to some
constraints. The optimal assignment of tasks in 𝑇 to robots in 𝐴 is returned.
Agorithm 1. Generic solving of a MRTA problem
It is worth pointing out that MRTA problems are strongly NP-hard, i.e., shown through complexity
analysis. Therefore, exact methods are not highly recommended to handle large-scale MRTA problems.
Thus, the use of metaheuristics-based techniques is recommended [37, 49].

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7. ILLUSTRATIVE EXAMPLE
We give an illustrative example to show how Algorithm 1 is employed to solve a given MRTA
problem. We choose the problem of assigning papers to reviewers within a journal. We suppose that the
Bulletin of Electrical Engineering and Informatics has received 20 papers to consider for publication on a
given day. The editor-in-chief (EIC) has ten potential reviewers. After checking all the papers, the EIC
concluded that they are articulated around five topics. Hence, he should assign each paper to some reviewers
(e.g., two reviewers) while optimizing a given criterion under some constraints. This situation is indeed a
MRTA problem, where: (a) reviewers are the robots (Definition 9), i.e., |𝐴| = 10; (b) papers are the tasks
(Definition 10), i.e., |𝑇| = 20; and (c) topics are the resources (Definition 11), i.e., |Ω| = 5. Topics for which
reviewers are experts are summarized in Table 1 (Definition 12), i.e., relation 𝕆. Topics covered by papers
are summarized in Table 2 (Definition 13), i.e., relation ℝ.
Line 1 of Algorithm 1: We compute 𝕍, i.e., valid assignments over robots and tasks, using (6) and
Tables 1 and 2 (Definition 14). We suppose that no constraint is considered: i.e., {𝜀1, … , 𝜀𝑞} is empty (all
valid assignments are accepted). It is worth pointing out that in the worst case, we should have |𝐴| × |𝑇| ×
|Ω| valid assignments: i.e., |𝕍| = |𝐴| × |𝑇| × |Ω|. For our example, |𝕍| = 10 × 20 × 5 = 1000, which is a
large number to consider.
Line 2, 3, and 4 of Algorithm 1: We compute Θ(𝐴) (Definition 16), ∆(𝑇) (Definition 17), and ∆′(𝑇)
(Remark 1), respectively. Cardinalities of collections Θ(𝐴), ∆(𝑇), and ∆′(𝑇) are given as follows. There is a
tip to reduce the size of collections Θ(𝐴) and ∆′(𝑇). For instance, if the journal policy states that each paper
is assigned exactly to 2 reviewers; therefore, the EIC might consider: (a) families Θ𝑖 ∈ Θ(𝐴) for which
|Θ𝑖𝑗| = 2; and (b) families ∆𝑖′ ∈ ∆′(𝑇) for which |∆𝑖𝑗′| = 1.
|Θ(𝐴)| =
1
2
∑ (−1)𝑘 10!
(10−𝑘)!𝑘!
2210−𝑘
10
𝑘=0
|∆(𝑇)| = ∑ (
1
𝑘!
∑ (−1)𝑖 𝑘!
(𝑘−𝑖)!𝑖!
(𝑘 − 𝑖)20
𝑘
𝑖=0 )
20
𝑘=0
|∆′(𝑇)| = ∑ ((
1
𝑘!
∑ (−1)𝑖 𝑘!
(𝑘−𝑖)!𝑖!
(𝑘 − 𝑖)20
𝑘
𝑖=0 ) 𝑘!)
20
𝑘=0
Line 5 and 6 of Algorithm 1: We compute Γ = Θ(𝐴) × ∆′(𝑇), partial allocations, given by (10) and
℘(Γ), the powerset of relation Γ. Line 7 to 13 of Algorithm 1: For each subset in ℘(Γ), we check if it is a
global allocation (Definition 19). If this is the case, we evaluate the global allocations and return the optimal
one. For the current example, line 9 is ignored since assignments are instantaneous. For example, if we have:
Γ′
=
{
({a6, a7}, {t1}), ({a9, a7}, {t2}), ({a4, a10}, {t3}), ({a4, a1}, {t4}), ({a7, a10}, {t5}),
({a8, a7}, {t16}), ({a5, a10}, {t17}), ({a9, a10}, {t18}), ({a2, a6}, {t19}), ({a10, a9}, {t20})}
We evaluate Γ′
using the objective function given by (21). We sum portions of topics offered by two
reviewers compared to topics needed by a paper. The optimal global assignment is the one that maximizes
this sum. Of course, the EIC might use other criteria to evaluate a global allocation: e.g., impose a maximum
occurrence of a given reviewer.
𝑈
̂ = ∑ ∑ (
|{𝜔|𝜔∈(Θ𝑖𝑗×ℬ×Ω×∆𝑖𝑗
′
×ℬ)}|
|{𝜔|𝜔∈(∆𝑖𝑗
′ ×Ω×ℬ)}|
)
|Θ𝑖|
𝑗=1
|Γ′|
𝑖=1
(21)
Table 1. Topics of reviewers (relation 𝕆)
𝑎1 𝑎2 𝑎3 𝑎4 𝑎5 𝑎6 𝑎7 𝑎8 𝑎9 𝑎10
𝜔1 1 1 0 1 0 1 1 0 1 1
𝜔2 1 0 0 0 1 0 1 1 0 0
𝜔3 0 0 0 1 0 1 1 0 1 1
𝜔4 0 0 1 1 1 1 0 0 0 0
𝜔5 1 1 0 0 1 1 0 0 0 1

 ISSN: 2302-9285
1102
Table 2. Topics covered by papers (relation ℝ)
𝑡1 𝑡2 𝑡3 𝑡4 𝑡5 𝑡6 𝑡7 𝑡8 𝑡9 𝑡10 𝑡11 𝑡12 𝑡13 𝑡14 𝑡15 𝑡16 𝑡17 𝑡18 𝑡19 𝑡20
𝜔1 1 1 0 1 0 0 1 0 1 0 1 1 1 1 0 1 1 0 1 1
𝜔2 1 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 0 0 1 0
𝜔3 0 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 0 1 1
𝜔4 0 1 1 0 0 0 1 1 0 1 1 0 1 1 1 1 0 0 1 0
𝜔5 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0
8. CONCLUSION AND FUTURE WORK
This paper proposed a formal analysis of multi-robot task allocation problems using elementary set
theory concepts. We formulated two well-known taxonomies of MRTA problems (i.e., the taxonomy of
Gerkey and Matarić entitled “A formal analysis and taxonomy of task allocation in multi-robot systems”; and
the taxonomy of Korsah, Stentz, and Dias entitled “A comprehensive taxonomy for multi-robot task
allocation”), then we developed them using the proposed analysis. A generic solving scheme of MRTA
problems is presented, and its time complexity is discussed. Finally, the proposed generic solving method is
employed to solve a real-world problem: i.e., the assignment of papers to reviewers within a journal. In
conclusion, the work introduced in this paper is not intended to be better than the previous formulations; it
merely complements them. As future work, we plan to represent the taxonomy of Ernesto, Marie, Hakim, and
Maria entitled “A taxonomy for task allocation problems with temporal and ordering constraints” using our
framework.
ACKNOWLEDGEMENTS
This research work is supported by UAEU Grant: 31T102-UPAR-1-2017.
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BIOGRAPHIES OF AUTHORS
Farouq Zitouni obtained his PhD in Computer Science from Abdelhamid Mehri University,
Constantine, Algeria. He is currently a Professor at the department of computer science at Kasdi
Merbah University - Ouargla and member of LIRE Laboratory, Abdelhamid Mehri University-
Constantine, Algeria. He is working on the field of multi-robot task allocation and multi-robot
systems. His main research interests include optimization and metaheuristic algorithms. He
published some papers in international conferences and journals. Currently he is visiting the
UAE University.
Ramdane Maamri is an associate professor of computer science in University of Constantine 2,
Algeria. He received his master from University of Minnesota, USA, and his Ph.D in Computer
Science from University of Mentouri Constantine, Algeria. He has participated in many national
and international research projects and supervised a number of masters and Ph.D students. He is
currently the head of software technologies and information systems department. His research
interests include software engineering, verification and validation, web services, multi agent
systems and ambient systems.
Saad Harous obtained his PhD in computer science from Case Western Reserve University,
Cleveland, OH, USA in 1991. He has more than 30 years of experience in teaching and research
in three different countries: USA, Oman and UAE. He is currently a Professor at the College of
Information Technology, in the United Arab Emirates University. His teaching interests include
programming, data structures, design and analysis of algorithms, operating systems and
networks. His research interests include parallel and distributed computing, P2P delivery
architectures, wireless networks and the use of computers in education and processing Arabic
language. He has published more than 200 journal and conference papers and book chapters. He
is an IEEE senior member.

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