Fuzzy Reasoning and Optimization Based on a Generalized Bayesian Network

Bayesian networks have been widely used as the knowledge bases with uncertainty. However, in most literatures, the uncertainty measure in Bayesian networks are limited in probability distributions and crisp variables, which restricts the practical usefulness of Bayesian networks when incomplete knowledge or linguistic vagueness is involved in the reasoning system. This study intends to develop a generalized Bayesian network in which the fuzzy variables, crisp variables, fuzzy parameters and crisp parameters can be considered. Based on the generalized Bayesian network, the fuzzy reasoning model for prediction, diagnosis, and optimization, can be designed. This study also develops the algorithms for fuzzy reasoning and optimization. The proposed network model will be applied to a two-echelon supply chain. (Generalized Bayesian Networks; Fuzzy Reasoning; Optimization; Supply Chain Management) 1 Research Background and Objectives Bayesian networks (Pearl 1988, Castillo et al. 1996, Castillo et al. 1997, Pearl 2000) are directed acyclic graphs (DAG) in which the nodes represent the variables, the arcs represent the direct causal influences between the linked variables, and the strengths of these influences are expressed by conditional probabilities. The semantics of Bayesian networks demands a clear correspondence between the topology of a DAG and the dependency relationships portrayed by it. They are widely used in knowledge representation and reasoning tools for various domains under uncertainty (Tatman and Shachter 1990, Dagum et al. 1992, Kao et al. 2000, Galán et al. 2002). Several methods have been developed for solving abductive or diagnostic reasoning problems in Bayesian networks. Exact methods exploit the independence structure contained in the network to efficiently propagate uncertainty (Pearl 1988, Castillo et al. 1996, Castillo et al. 1997). Meanwhile, stochastic simulation methods provide an alternative approach suitable for highly connected networks, in which exact algorithms can be inefficient (Pearl 1988, Castillo et al. 1997). Recently, search-based approximate algorithms, which search for high probability configurations through a space of possible values, have emerged as a new alternative (Pool 1993). On the other hand, two key approaches have been proposed for symbolic inference in Bayesian networks, namely: the symbolic probabilistic inference algorithm (SPI) and symbolic calculations based on slight modifications of standard numerical propagation algorithms (Shacher et al. 1990, Castillo et al. 1996, Castillo et al. 1997). The methods in the literatures have several limitations for reasoning from a Bayesian network: 1. All network nodes or domain variables must be crisp. 2. All parameters, including costs and utilities, of the network models are usually assumed crisp. 3. Different reasoning tasks, such as prediction, diagnosis and decision-making, cannot be done in a complete model. This study intends to develop a generalized Bayesian network in which crisp nodes (variables), fuzzy nodes, and fuzzy parameters are included. Based on the generalized Bayesian network model, fuzzy reasoning techniques are designed to answer different queries from the network. The decision makers can make prognosis as well as diagnosis from the generalized Bayesian network with the fuzzy reasoning methods. Furthermore, alternative actions in response to the (potential) problems can be evaluated and selected based on the diagnostic report. 2 Problem and model development This section introduces the generalized Bayesian network, the problem formulation, and the algorithm for fuzzy reasoning and optimization. 2.1 Generalized Bayesian networks Generally, a Bayesian network is defined as (1). BN= (V, L, P) (1) In (1), V denotes the set of nodes (vertices), L denotes the set of links (arcs), and P denotes the probability model describing the network, where V V L × ⊂ (2) In most literatures, V, and P are assumed crisp. If a crisp set is regarded as one special subset of fuzzy sets, then the definition of a Bayesian network can be extended into a generalized Bayesian network as follow. ) ~ , ~ , ~ ( P L V GBN = (3) In (3), the set of nodes, probability distributions, and consequently the links, are no more limited to crisp sets, which allow greater modeling flexibilities of Bayesian networks. Furthermore, the composition of the node set V~ can be expressed as (4). } ~ , ~ , ~ { ~ U R D V V V V = (4) where D V ~ denotes the decision nodes, R V ~ represents the random nodes which is defined as the nodes in a conventional Bayesian network, U V ~ denotes the utility nodes which stand for the objectives to be optimized. By (4), an influence diagram is included in the definition. 2.2 Problem formulation We first hypothesize a case of the two-echelon automotive supply chain (Naim et al. 2002). Case 1: After a field survey on the automotive supply chains, the engine assemblers and their suppliers can catch the whole picture of the supply chain performance. One main outputs of the field research is the cause-and-effect diagram shown in Figure 1 (Naim et al. 2002). In Figure 1, there are two levels of factors: the upper level of the customers (the engine assemblers, achromatic) and the lower level of the suppliers (in color). The arrows in the diagram represent the causal links between the keys of the two-echelon supply chain. The detailed description of Figure 1 is given in Table 1. Because there is a feedback loop in Figure 1, the two-echelon supply chain is a dynamic network.