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This paper presents a comparative study between optimization-based and market-based approaches used for solving the Multirobot task allocation (MRTA) problem that arises in the context of multirobot systems (MRS). The two proposed approaches are used to find the optimal allocation of a number of heterogeneous robots to a number of heterogeneous tasks. The two approaches were extensively tested over a number of test scenarios in order to test their capability of handling complex heavily constrained MRS applications that include extended number of tasks and robots. Finally, a comparative study is implemented between the two approaches and the results show that the optimization-based approach outperforms the market-based approach in terms of optimal allocation and computational time.

In the last few years, the field of research in mobile robotics has encountered a significant shift as the researchers in this field have recently started focusing on MRS rather than single-robot systems. This increased interest in the community of mobile robotics research towards MRS comes from the significant advantages and higher potential provided by MRS than single-robot systems. The advantages of a robot team are many; some examples of these advantages include, but are not limited to, resolving task complexity, increased system reliability, increased system performance, and finally easier and simpler design [

One of the main areas of research in this field is the task allocation problem in MRS, where the mapping of robots to tasks is done in order to increase the overall performance of the system. The task allocation problem is a major issue in MRS as it focuses on the proper utilization of the available resource. In MRS, the available resources are the robots which are used to solve a problem or to perform a certain task. Thus, in order to increase the performance of the system, one must efficiently utilizes the available robots in order to solve the required tasks. Since the decision of which robot will do which task has a significant effect on the performance of the system, the allocation of the tasks to the proper robots strongly affects the performance of the system [

The first objective of this paper is to present a framework that is efficiently capable of modeling any instance of the MRTA problem and to provide a suitable solution to the given problem. The framework will generate a solution in which the given robots are efficiently allocated to the given tasks in such a way to maximize the overall performance and minimize the total cost of the allocation. The framework will take into consideration the real-world constraints of the system, the requirements of the tasks, and the capabilities of the robots. The second objective of this paper is to conduct a comparative study between two main solving approaches for the MRTA problem which are the optimization-based approach and the market-based approach.

The remainder of this paper is organized as follows. Section

Over the last few years, several MRS problems were addressed, and since the MRS have a higher potential of solving complex and sophisticated problems, the complexity of the addressed problem increased with time. Therefore, the complexity of the MRS had to increase to accommodate the complexity of the addressed problems. As more research was done on MRS, the researchers have encountered the question “Which robot will execute which task?” In order to answer this question, more focus was directed towards the task allocation problem in MRS.

The MRTA problem is a problem that arises in MRS where a number of robots are working together in the aim of achieving a common goal or task which is subdivided into a number of subtasks. The problem can be formulated as follows.

Given a set of available robots,

Given a set of available tasks,

Allocation of the available tasks to the robots occurs,

The output set

This allocation

Multiple Traveling Salesman Problem (mTSP) can be considered as the generalization of the original TSP which is used as a platform for the study of general methods that can be applied to a wide range of discrete optimization problems. In TSP

In the literature, various researchers have used the mTSP as a solution model for the MRTA due to the strong analogy between the two problems. In [

In the mTSP, a number of nodes

Since the proposed approach mainly aims to solve the task allocation problem of MRS in real-world applications, through the different phases of the development of the introduced approach, the central target was to introduce a generic approach that is capable of solving various MRTA problems of different features and challenges. This target had to be taken into consideration during the formulation of the problem and therefore the use of the mTSP formulation previously presented was not suitable enough to solve the task allocation problem in real-world MRS applications. Therefore, the previously presented formulation had to be extended and adapted in order to be used as a formulation for the MRTA problem. In order to appropriately adapt the mTSP formulation to be used as a formulation for the MRTA problem, one must properly categorize the MRTA problem in interest.

In this work, the solving approaches intends to solve MRTA problems that include heterogeneous single task robots, heterogeneous single-robot tasks, and instantaneous as well as time extended task assignment. After the categorization of the MRTA problem in interest, the mTSP formulation must be adapted to be used for solving the MRTA problems. This adaption is done through extending the mTSP formulation and the addition of extra features to the forming ingredients of the mTSP. Figure

Extending mTSP formulation for MRTA.

Since most real MRS applications require heterogeneous robots of different capabilities, it was a must to consider the heterogeneity of the robot in the proposed approach. Four main features of the robot were considered and thus were added to the traveling salesman in the implementation phase. The four features are as follows:

velocity of the robot,

robot capabilities,

energy level of the robot,

aging factor (efficiency).

In the same manner, the mTSP formulation for solving the MRTA problem needed to be adapted to handle the heterogeneity of the tasks and therefore it was a must to add extra features to the cities. The added features to the cities are as follows:

task requirements,

minimum time required to finish the task.

Although the MRTA problem is formulated as an instance of the mTSP, the same objective function of the mTSP previously explained in (

There are three main variations of the MRTA problem objective function than the mTSP objective function. The three proposed variations are:

a multiobjective function instead of a single-objective function.

the variable to be minimized is the time rather than the distance.

minimizing the time of the maximum subtour (MinMax) rather than minimizing the total time.

Then, for

Optimization is the branch of applied mathematics focusing on solving a certain problem in the aim of finding the optimum solution for this problem out of a set of available solutions. In other words, optimization techniques are applied in order to maximize the profit (maximization problem) or reduce the damage (minimization problem) of a certain problem. The set of available solutions is restricted by a set of constraints, and the optimum solution is chosen within these constrained solutions according to a certain criteria. These criteria define the objective function of the problem, where the objective function is a mathematical expression combining some variables in order to describe the goal of the system [

By reviewing the literature, it was found that different optimization approaches have been used in order to solve the general task allocation problem and was also used in order to solve the MRTA problem. In [

Another different optimization approaches were also used for solving the task allocation problem. For example, population-based approaches such as the genetic algorithm was used in [

The task allocation problem was also solved using hybrid optimization approaches such as the tabu search with random search method in [

The following subsections present a trajectory-based metaheuristic approach and a population-based metaheuristic approach to solve the MRTA problem.

The first introduced algorithm is the SA algorithm which is a metaheuristic algorithm of the trajectory-based approaches family. The trajectory-based family of metaheuristic algorithms is the family of algorithms that uses a single solution throughout the algorithm in order to find the optimal solution. The neighboring operator used in the proposed algorithm is randomly chosen at each iteration for the sake of diversity from one of the following operators:

swapping,

deletion and insertion,

inversion,

scrambling.

Algorithm

Distances between tasks

(1)

temperature

Geometric coefficient

Neighbor allocation

cost

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generateNeighborSolution (

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The second introduced algorithm is the GA which is an evolutionary algorithm of the population-based family of metaheuristic algorithms. Population-based algorithms is the family of algorithms that iteratively transforms a population of solutions throughout the algorithm in order to generate a new population of solutions in the aim of finding the optimal solution.

The mutation operators used in the proposed algorithms are

swapping,

deletion and insertion,

inversion,

scrambling,

partially mapped crossover,

order crossover.

Algorithm

(1)

Number of iterations

allocation

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generateValidSolution (

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Throughout time, humans have dealt with coordination and allocation problems for thousands of years with sophisticated market economies in which the individual pursuit of profit leads to the redistribution of resources and an efficient production of output. The principles of a market economy can be applied to multi-robot coordination [

In general, auctions are scalable, computationally cheap and have reduced communication requirements. They can be performed centrally, by an auctioneer or by the robots themselves, in a distributed way. Therefore, market-based coordination approaches have been studied in countless multi-agent systems [

For the market-based approach, the proposed algorithm was designed to follow the sealed-bid closed-cry auctions and the selected auction design is contract net protocol (CNP). The algorithm was tested over several sets of experiments and the results were not promising enough and thus it was suggested to add enhancements for the market-based approach. The first attempted enhancement is the relinquishing process enhancement, where the robot relinquishes a random task from its assigned tasks to explore more solutions. The second enhancement was to apply an optimization technique such as simulated annealing (SA) to optimize the final subtour of the allocation as single traveling salesman problem [

The first enhancement process is presented in Algorithm

(1) Number of robots

(2) Sub-tours list

(3) Relinquished task

(4)

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The final market-based approach with both enhancements, the relinquishing and the optimization enhancements, is presented in Algorithm

(1) Best cost

(2) Available tasks

(3) Bidding list

(4) Minimum bid

(5) Current allocation

(6) Current cost

(7) User define percentage

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The following subsections present the experiment setup, the emulation metrics, and a comparative study between the two proposed approaches.

All experiments are conducted on a Microsoft Windows operating system on a device whose specifications are presented in Table

System specifications.

Processor | 1.70 GHz |

Installed memory (RAM) | 6.00 GB |

System type | 64-bit operating system |

The proposed algorithms, both the market-based and the metaheuristic-based, are tested both qualitatively and quantitatively. A number of test scenarios that include different instances of the MRTA problem were proposed. The different proposed scenarios were used to test the algorithms qualitatively. However, in order to be able to quantitatively test the quality of the proposed algorithms, two evaluation metrics were proposed. Therefore, each algorithm was used to solve the MRTA problem in each qualitative test scenario and then the results and the quality of the solution provided were evaluated through the two quantitative evaluation metrics. The two quantitative metrics used to test both algorithms are as follows:

the best allocation found in terms of the objective function [

computational time taken to best allocation [

The first evaluation metric is the cost of the best allocation found which is the best solution in terms of the objective function previously discussed in (

Since one of the main sources of complexity of MRS applications is the size of the robot team as well as the number of tasks to be executed, it was essential to qualitatively test the proposed algorithms for their capability of handling complex MRTA problems of large number of tasks and robots. Analysis of the scalability of the proposed algorithms is done through applying each algorithm over a number of scenarios at which the number of tasks and the number of used robots increase over each scenario. It must be mentioned that in these scenarios there are no constraints applied on the domain of the problems being solved. The main reason of eliminating the constraints is to eliminate any other source of complexity to the problem rather than the increased number of robots and tasks, where eliminating other sources of complexity to the problem enables the adequate testing of only the scalability of the proposed algorithms with no other factors affecting the problem.

In the scalability test three scenarios were proposed which are as follows:

small-scale problem, five tasks, and three robots,

medium-scale problem, fifteen tasks, and five robots,

large-scale problem, fifty tasks, and fifteen robots.

Another significant cause of complexity of the newly tackled MRS applications is that the problems are highly constrained. The constraints of MRS applications mainly arise because of two main reasons. The first source of constraints is the introduction of the heterogeneous robots in the multirobot team as well as the heterogeneity of the tasks being executed. The second source of constraints is the amount of energy level of the used robots and the time requirements of the tasks to be executed. Therefore, it was a must to qualitatively test the capability of the proposed algorithms in this paper to solve constrained instances of the MRTA problem.

In the constraints test three scenarios were proposed which are as follows:

capabilities matching scenario,

time matching scenario,

heavily constrained scenario.

In this subsection, a comparative study is implemented in order to test the performance of the three proposed algorithms in this paper on different test scenarios. The performance of each algorithm is illustrated through plotting the values of the two evaluation metrics of each algorithm for each test scenario, where the

Figure

Small-scale MRTA scenario comparative study.

The three approaches were capable of converging to the same allocation cost in the small-scale scenario. GA was the fastest to reach the best allocation followed by SA and then the market-based approach.

Figure

Medium-scale MRTA scenario comparative study.

The performance of SA and GA in Figure

Figure

Large-scale MRTA scenario comparative study.

The results in Figure

Figure

Capabilities matching MRTA scenario comparative study.

The performance curves of SA and GA in Figure

Figure

Figure

Time matching MRTA scenario comparative study.

Figure

Heavily constrained MRTA scenario comparative study.

The reported results in Figure

This paper presented a comparative study between two well-known approaches, metaheuristic-based and market-based approaches, that are used extensively to solve the MRTA problem. The main intention in this paper was to propose generic approaches for solving the MRTA problem. The developed approaches were responsible for providing a solution that is not only a feasible solution but also an optimized one which enabled the appropriate use of the available resources and thus increasing the overall system performance and decreasing the costs. Another objective was to propose an approach that is capable of handling real-world application constraints such as time constraints, robot capabilities, and task requirements matching constraints. The proposed approaches were able to handle robotics applications where the number of robots and number of tasks are overextended. The analysis of the experimental and the comparative study that was conducted between the metaheuristic-based and the market-based approaches in this paper results showed that metaheuristic approach outperformed the market-based approach in a number of aspects such as the optimality of the found allocations as well as the total time taken to reach the optimal allocation. The results also demonstrated the better performance of the metaheuristic approaches relative to the market-based approach in the scalability scenarios while both approaches provided nearly similar results in the constraints handling scenarios.

In order to sum up the results of this research work, Table

MRTA approaches applicability results.

Algorithm | Market-based | SA | GA |
---|---|---|---|

Small scale | ∗ | ∗ | ∗ |

Medium scale | ∗ | (∗∗∗) | ∗∗ |

Large scale | ∗ | (∗∗∗) | ∗∗ |

Capabilities matching | (∗∗) | ∗ | ∗ |

Time matching | ∗ | ∗ | ∗ |

Heavily constraints | ∗ | ∗ |

As a future work, more aspects of MRTA will be investigated such as the reallocation capability of the approach to handle the robot failure and the communication burden required to execute the task allocation. The proposed MRTA approaches will be also tested using Khepera III real robots in a newly built tested arena for MRS simulations and experiments in the robotics and autonomous systems (RAS) laboratory.