Analysis of contract net in multi-agent systems

doi:10.1016/j.automatica.2005.12.002

Automatica

Volume 42, Issue 5, May 2006, Pages 733-740

https://doi.org/10.1016/j.automatica.2005.12.002 Get rights and content

Abstract

Application of contract net protocol requires the development of a bid evaluation procedure specific to the problem. Care must be taken to apply contract net protocol to tasks that involve precedence constraints among different operations and heterogeneous resources. The lack of a process model in the original contract net protocol makes it difficult to determine the feasibility of the resulting contracts. We propose a model to facilitate the development of the bid evaluation procedure by extending our previous results to handle tasks with more complex process structure. We formulate an optimization problem to find a minimal cost feasible execution sequence for a task.

Introduction

In multi-agent system (MAS) (Ferber, 1999, Nilsson, 1998), a set of agents may form a coalition to execute a complex task based on contract net protocol (David & Smith, 1983; Smith, 1980). The set of agents that form a coalition to execute a task constitutes a contract net. Application of contract net protocol and MAS in manufacturing systems has been extensively studied (Baker, 1996, Brussel et al., 1998, Oliveira et al., 1997, Parunak, 1987, Zhang et al., 2003). Parunak applied MAS to model a factory using contract net protocol (Parunak, 1987). Baker proposed a market-driven contract net, where agents are connected over a network to represent the connection between the shop floor and the marketplace (Parunak, 1987). Oliveira et al. (1997) specified a coalition formation strategy based on the contract net protocol. Zhang proposed a price based negotiation mechanism to assign tasks in distributed manufacturing environment (Zhang et al., 2003).

In a contract net, an agent may play the role of a manager or a bidder. A bidder submits a bid to indicate the capabilities to execute the announced task. A manager may receive several bids and select one or more bids for the task using a task specific bid evaluation procedure. Despite the extension and/or modification of contract net in Fisher, Muller, Pischel, and Schier (1995), Takuya, Kazuo, and Yuichiro (1996), the original contract net and many of its later variants lacked a formal model to develop the bid evaluation procedure. Motivated by this drawback, Sandholm introduced a formal model (Sandholm, 1998) for contract net protocol that provably leads to desirable task allocation among agents based on marginal cost calculation and hill-climbing algorithm. However, the model is restricted to simple tasks that can be executed by a single agent. Modeling and analysis of contracts and the associated workflows have also been studied in Iwaihara, Jiang, and Kambayashi (2004) and Wil van der Aalst (2000). However, the issue to optimize contract awarding is not addressed in the above papers.

The goal of this paper is to develop an effective bid evaluation procedure for a manager to award contracts for complex tasks. This paper is differentiated from the results presented in Parunak (1987), Baker (1996), Oliveira et al. (1997), Zhang et al. (2003) as we model and analyze contract net based on Petri-net (PN) model. This paper also differs from the ones studied in Sandholm (1998), Hsieh, 2004a, Hsieh, 2004b as the task structure considered in this paper is much more complex and there are alternative ways to execute a given task, which poses an optimization issue in awarding the contracts. We adopt PN as the model. We use PN to model the interactions between managers’ workflow and the proposals submitted by the bidders and formulate a contract awarding problem to find a minimal cost feasible execution sequence for a task. We propose feasible conditions to award contracts and execute a given task. We test the feasible condition by exploiting the structure of the PN model.

The remainder of this paper is organized as follows. In Section 2, we propose the PN models for managers’ tasks and bidders’ proposals. In Section 3, we model collaboration of bidders and managers. We characterize the feasible condition to execute a task in Section 4. Section 5 presents the method to determine the minimal cost contracts. In Section 6, we propose a method to control the execution sequence. Section 7 concludes this paper.

Section snippets

An example

Example 1

Suppose a manager has been assigned a task. Fig. 1(a) shows a digraph representation of the operations in the task, where $p_{i}$ denotes a production state, $p_{0}$ is the initial state and a directed edge $p_{i} \to p_{k}$ denotes an operation. Each operation has its resource requirement. For each type of resources, there is a resource agent to allocate resources. Fig. 1(b) shows the set $R = {r_{1}, r_{2}, r_{3}, r_{4}, r_{5}, r_{6}, r_{7}}$ of resource types and the set $B = {b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6}, b_{7}}$ of bidders in the system.

The set of resources

Modeling collaboration of bidders and a manager

To evaluate the proposals submitted by the bidders, we construct a PN model to capture the interactions between the bidders’ proposals and the task. The PNs $G_{r}$ of the resources and the PN $G_{j}$ of a task are merged to form a PN $N_{j} = ∥_{r \in R} G_{r} ∥ G_{j}$ to model the interactions between resources and a task. The PN model in Fig. 5 is obtained by merging the PN models in Fig. 4 with the PN model in Fig. 2. Formally, $N_{j} (m_{j 0}, u_{j}) = (P_{j}, T_{j}, I_{j}, O_{j}, m_{j 0}, u_{j})$ , where $u_{j}$ is a commitment policy of $N_{j}$ defined as follows.

Definition 3.1

Feasible condition to execute a task

As $N_{j} = ∥_{r \in R} G_{r} ∥ G_{j}$ , and each $G_{j}$ is a SCSM, each feasible execution sequence in $S_{j} (m_{j 0})$ corresponds to a directed circuit in $G_{j}$ . Let $Γ_{j}$ denote the set of all directed circuits in $G_{j}$ . In Fig. 2, there are four circuits in $Γ_{j}$ . Let $t_{1} t_{2} t_{3} \dots t_{| γ |}$ denote the sequence of transitions in $γ \in Γ_{j}$ with $t_{1} = t_{j}^{r}, t_{1} = t_{j}^{f}$ . We proposed the concept of minimal resource requirements (MRR) to characterize $S_{j} (m_{j 0})$ .

Definition 4.1

$m_{γ}^{*}$ is the MRR of a circuit $γ$ with $m_{γ}^{*} (p) = \{\begin{matrix} R_{t_{1}} (r) \oplus R_{t_{2}} (r) \\ \oplus \dots \oplus R_{t_{| γ |}} (r) & if p = p_{r} for some r \in R, \\ 0 & otherwise, \end{matrix}$ where $γ \in Γ_{j}$ , $R_{t}$

Determination of minimal cost contracts

We optimize the contracts by finding the minimal cost circuit to execute a task. To find the optimal circuit, ${\bar{G}}_{j}$ is mapped to a digraph ${\bar{g}}_{j}$ with each place mapped to a node and each transition mapped to an arc. The arc associated with transition t is assigned a cost (weight) of $c (t)$ . Fig. 7(a) shows the digraph ${\bar{g}}_{j}$ corresponding to Fig. 6(b). Based on ${\bar{g}}_{j}$ , Dijkstra's shortest path algorithm (Dijkstra, 1959) can be applied to find the optimal directed path connecting the node $p_{j}^{r}$ to $p_{j}^{f}$ , where $p$

Control of execution sequence

To test whether ${\tilde{N}}_{j}$ can be kept live by applying Property 5.1, we have to test the coverability of $\tilde{m}_{j}^{*}$ as coverability analysis is computationally feasible only for small PNs (Murata, 1989). Instead, a more efficient and sufficient test procedure is developed here by exploiting the structure of ${\tilde{N}}_{j}$ . We check whether the set of resources that can return to idle state from a given marking $m$ dominates the MRR $\tilde{m}_{j}^{*}$ of ${\tilde{N}}_{j}$ . For each ${\tilde{G}}_{j}$ , we construct a PN ${MG}_{j} = ∥_{b \in B} ⋃_{k \in Ω_{b}^{d}} G_{b}^{k}∥ {\tilde{G}}_{j} = ({\tilde{P}}_{j}, {\tilde{T}}_{j}, {\tilde{I}}_{j}, {\tilde{O}}_{j}, {\tilde{m}}_{j}) .$

Conclusion

We formulate a contract awarding problem and develop an effective bid evaluation procedure for it based on a PN model, which captures the interactions between a manager task and bidders’ proposals. We find a minimal cost feasible execution sequence for a task by converting the problem to a shortest path problem and compute the token flow by exploiting the structure of the PN to test the feasible condition to execute a task.

Acknowledgement

This paper is supported in part by NSC under Grant NSC94-2416-H-324-014.

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Fu-Shiung Hsieh received the B.S. and M.S. degrees in control engineering from National Chiao-Tung University, Taiwan, Republic of China, in 1987 and 1989, respectively. He received the Ph.D. Degree from National Taiwan University in 1993. He served as research fellow in Industrial Technology Research Institute, Taiwan, from 1994 to 1999. He was an Assistant Professor and Associate Professor with The Overseas Chinese Institute of Technology from 1999 to July 2004 and August 2004 to July 2005, respectively. Since August 2005, he has been with Chaoyang University of Technology, where he is currently an Associate Professor at the Department of Computer Science and Information Engineering. He has served as a Guest Editor for Asian Journal of Control. He has also served as a referee for Automatica, Fuzzy sets and systems, IEEE Transactions on Systems, Man, & Cybernetics and IEEE Transactions on Robotics & Automation. He is listed in Who's Who in the World. His research interests are in the fields of multi-agent systems, electronic commerce, system theory, discrete event systems and manufacturing systems.

^☆: Some parts of this paper was presented in The 10th IFAC/IFORS/IMACS/IFIP Symposium on Large Scale Systems: Theory and Applications. This paper was recommended for publication in revised form under the direction of Editor Berç Rustem.

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Brief paperAnalysis of contract net in multi-agent systems☆

Abstract

Introduction

Section snippets

An example

Modeling collaboration of bidders and a manager

Feasible condition to execute a task

Determination of minimal cost contracts

Control of execution sequence

Conclusion

Acknowledgement

Automatica

Metaphor or realityA case study where agents bid with actual costs to schedule a factory

Reference architecture for holonic manufacturing systemsPROSA

Computers in Industry

Negotiation as a metaphor for distributed problem solving

Artificial Intelligence

A note on two problems in connection with graph

Numerische Mathematik

Multi-agent systems, an introduction to distributed artificial intelligence

Communication sequential processes

Communications of the ACM

Brief paper
Analysis of contract net in multi-agent systems☆