Comparing Markov Chains: Aggregation and Precedence Relations Applied to Sets of States, with Applications to Assemble-to-Order Systems

Ana Bušić (1, 2) , Ingrid M. H. Vliegen (3) , Alan Scheller-Wolf (4)

1 LINCS - Laboratory of Information, Network and Communication Sciences
2 DYOGENE - Dynamics of Geometric Networks
3 Department of Industrial Engineering and Innovation Sciences
4 Tepper School of Business

Solving Markov chains is, in general, difficult if the state space of the chain is very large (or infinite) and lacking a simple repeating structure. One alternative to solving such chains is to construct models that are simple to analyze and provide bounds for a reward function of interest. We present a new bounding method for Markov chains inspired by Markov reward theory: Our method constructs bounds by redirecting selected sets of transitions, facilitating an intuitive interpretation of the modifications of the original system. We show that our method is compatible with strong aggregation of Markov chains; thus we can obtain bounds for an initial chain by analyzing a much smaller chain. We illustrate our method by using it to prove monotonicity results and bounds for assemble-to-order systems.

Mots clés: Markov chains; precedence relations; bounding models; aggregation; supermodularity; assemble-to-order (ATO) systems

Domaines: Mathématique discrète [cs.DM]; Performance et fiabilité [cs.PF]; Probabilités [math.PR]

Ana Bušić, Ingrid M. H. Vliegen, Alan Scheller-Wolf. Comparing Markov Chains: Aggregation and Precedence Relations Applied to Sets of States, with Applications to Assemble-to-Order Systems. Mathematics of Operations Research, 2012, 37 (2), ⟨10.1287/moor.1110.0533⟩. ⟨hal-00361430v2⟩