J Intell Manuf DOI 10.1007/s10845-015-1143-4 Probabilistic Boolean network modeling of an industrial machine Pedro J. Rivera Torres 1 · E. I. Serrano Mercado 2 · Luis Anido Rifón 1 Received: 18 June 2015 / Accepted: 18 August 2015 © Springer Science+Business Media New York 2015 Abstract Theoretical modeling of manufacturing processes assists the design of new systems for predic- tions of future behavior, identifies improvement areas, and evaluates changes to existing systems. A novel approach is proposed to model industrial machines using probabilistic Boolean networks (PBNs) to study the relationship between machine components, their reliability and function. Once a machine is modeled as a PBN, through identification of regulatory nodes, predictors and selection probabilities, sim- ulation and property verification are used to verify model correctness and behavior. Using real machine data, model parameters are estimated and a PBN is built to describe the machine, and formulate valid predictions about proba- bility of failure through time. Two models were established: one with non-deterministic inputs (proposed), another with components’ MTBFs inputs. Simulations were used to gen- erate data required to conduct inferential statistical tests to determine the level of correspondence between predic- tions and real machine data. An ANOVA test shows no difference between expected and observed values of the two models (p value = 0.208). A two-sample T test demon- strates the proposed model provides values closer to expected values; consequently, it can model observable phenomena (p value = 0.000). Simulations are used to generate data required to conduct inferential statistical tests to determine the level of correspondence between model prediction and B Pedro J. Rivera Torres privera@ece.uprm.edu 1 ETSET-Universidade de Vigo, Campus Universitario, s/n, 36310 Vigo, Spain 2 Polytechnic University of Puerto Rico, 377 Ponce De Leon Ave., San Juan, PR 00918, USA real machine data. This research demonstrates that using PBNs to model manufacturing systems provides a new mech- anism for the study and prediction of their future behavior at the design phase, assess future performance and identify areas to improve design reliability and system resilience. Keywords Bio-inspired modeling · Biological manufac- turing systems · Probabilistic Boolean networks Introduction Probabilistic Boolean networks (PBN) are mathematical con- structs that can be used to model Gene Regulatory Networks (GRN). GRNs are collections of DNA segments within a cell that interact indirectly with other segments and substances in a cell in order to govern the expression levels of genes. They can be used to better understand the general rules that govern gene regulation in genomic DNA. PBNs are transi- tion systems that satisfy the Markov Property (Markov 1954), such that the probability that the system will take a transi- tion from a given state to another depends exclusively on the current state, and is not dependent on the past history of the system. PBNs were proposed by Ilya Shmulevich in several publications (Shmulevich et al. 2002a, b; Shmulevich and Dougherty 2010) as an extension of Stuart Kauffman’s Boolean Network (BN) concept (Kauffman 1969a, b). This alternative to modeling GRNs combines the rule-based mod- eling of Kauffman’s BNs with uncertainty principles. PBNs consist of a group of constituent BNs that have assigned selection probabilities, where each Boolean Network can be considered a “context”. Data for the cells comes from dif- ferent sources; each source represents a context of the cell (Shmulevich and Dougherty 2010). In a given time t , a sys- tem can be governed by one of the constituent BNs, and at 123