Average flow constraints and stabilizability in uncertain production-distribution systems

Abstract

Consider multi-inventory systems with controlled flows and uncertain demands (disturbances) bounded within assigned compact sets. The system is modelled as a first order one integrating the discrepancy between controlled flows and demands at different sites/nodes. Thus, the buffer levels at the nodes represent the system state. Given a long-term average demand, we are interested in a control strategy that satisfies just one of the two requirements: i) meeting any possible demand at each time (worst case stability) or ii) achieving a pre-defined flow in the average (average flow constraints). Necessary and sufficient conditions * D. Bauso is with Dipartimento di Ingegneria Informatica, Universitàd iP a l e r m o ,P a l e r m o-I t a l ydario.bauso@unipa.it † F. Blanchini is with Dipartimento di Matematica e Informatica, Università di Udine, Udine -Italy blanchini@uniud.it ‡ R. Pesenti is with Dipartimento di Matematica Applicata, Università "Ca' Foscari" di Venezia -Italy pesenti@unive.it § Research supported by PRIN "Robustness optimization techniques for high performance control systems", and MURST-PRIN 2007ZMZK5T "Decisional model for the design and the management of logistics networks characterized by high interoperability and information integration".for the achievement of both goals have been proposed by the authors. In this paper we face the case in which these conditions are not satisfied. We show that if we ignore the requirement on worst case stability, we can find a control strategy driving the expected value of the state to zero. On the contrary, if we ignore the average flow constraints, we can find a control strategy that satisfies worst case stability while optimizing any linear cost on the average control. In the latter case we provide a tight bound for the cost.

Optimization of Long-Run Average-Flow Cost in Networks With Time-Varying Unknown Demand

“…4 Proof: We first prove (26) under the assumption that w(t) is known on-line. To do this, consider the following strategy u(t) = Φ(x(t), w(t)) = Φ 1 (w(t)) + Φ 2 (x(t)), for all t ≥ 0, where Φ 1 (w(t)) is any static strategy, for instance Φ PWA (w(t)), which solves problem (10), and Φ 2 (x) is a stabilizing term that we define later on.…”

“…To be more clear, in contrast with most existing literature, we are not considering the average cost of the flows (see for instance [26]) but the cost of the average flows. Clearly there is no distinction between the two concepts cost-of-theaverage and average-cost in the case of linear functionals (see, e.g., [4]). …”

Balancing time-varying demand-supply in distribution networks: An internal model approach

“…Cooperative control systems have been widely investigated in a variety of different contexts [16,12,15,3]. Less attention has been devoted to cooperative control in the framework of dynamical flow networks, with some interesting exceptions [9,6,8,7,17,5]. The aim of this paper is to study a class of cooperative control algorithms in the context of distribution networks under exogenous inputs.…”

“…The work [5] did not consider the presence of external inputs. A large amount of work on the topic of flow control in the presence of disturbances has been carried out in works such as [6,8,7] where the problem is cast in the robust control framework. The approach in our paper is based on the theory of output regulation and to the best of our knowledge this has not been considered before.…”

A fuzzy solution approach for a multi-objective integrated production-distribution model with multi products and multi periods under uncertainty

“…The first approach is widely investigated in the literature. In recent studies, the integration of these two main supply chain problems is developed (Bard & Nananukul, 2010;Armentano et al, 2011;Bauso et al, 2010;Bilge, 2010;Shiguemoto & Armentano, 2010). Fig.…”