Production-Inventory System Controller Design and Supply Chain Dynamics

This paper deals with the modelling and control of aggregated production-inventory systems as described by differential equations. Hitherto, research in this area has been characterised by the approximation of production delays, by first order lags, rather than more realistic pure delays. We demonstrate the substantial qualitative differences between these two approaches and thus generate the motivation for the rest of the paper, which tackles pure delay systems. The application of some relatively new design methodologies for delay systems yields four design choices, which are tested for their performance over a range of criteria including stability robustness. The investigation is then extended to the model of a supply chain comprising many such production-inventory systems. The mechanism by which disturbances can be transmitted along the supply chain causing disruption and incurring costs to other supply chain echelons is elucidated. An heuristic feedback policy designed to adaptively tune the individual system designs in response to such disturbances is presented

Quantifying Bounded Rationality: Managerial Behaviour and the Smith Predictor

The concept of bounded rationality in decision making and research on its relegation to aggregate system dynamics is examined. By recasting one such example of a dynamic system, the Beer Game, as a Smith predictor control system is derived. A stability analysis is then employed to support the and qualify the assertion that the level of bounded rationality can adversely affect the aggregate dynamic behaviour of such supply chains. The analytical basis of these calculations enables the qualification of the potential cost improvements resulting from more desirable supply chain dynamics. This approach is designed to inform the strategic investment decision to purchase computational aids in order to overcome the level of bounded rationality in the system

The Optimal Control of Inventory Systems

The science of Operational Research has traditionally dominated in the field of inventory theory. Yet there have been sporadic attempts to apply methods from control theory in this domain. We review these efforts, concentrating particularly on optimal control approaches. We also enumerate the generic benefits and limitations of operational research methods. Through these efforts we judge whether the supremacy of operational research techniques in the field of inventory systems is warranted. We apply a novel optimal control algorithm to a differential equation model of an inventory system. This enables us to mimic the cost structures implied by quantity discounts and approximate capacity constraints. Some examples illustrate how optimal responses to instantaneous jumps in demand are generated and how these are affected by quantity discounts

The Stability of Supply Chains

A continuous time version of the well known beer game model is derived and its stability and robust stability properties are investigated. Novelty originates from the treatment of pure process delay rather than exponential lags and it is shown that this can lead to to diametrically different dynamics to the exponential lag case. The stability properties of the system are shown to support and quantify the qualitative empirical results of the beer game. Additional insight into the influence of certain model parameters is attained by the interpretation as the degree of mismatch in a Smith predictor regulator. The transient inability to supply all that is demanded is mimicked and shown to constitute an influential source of demand amplification. The analytical nature of these calculations engenders the capacity to improve supply chain dynamics through the synthesis and calibration of strategic supply chain trade off problems

Modelling the Dynamics of Supply Chains

Since the 1960's many academic and managerial researches have focused their attention on production-inventory-distribution systems. The application of diverse mathematical techniques from continuous differential equation systems to mathematical programming models have been attempted. Yet none have predominated over Operational Research techniques either in industry or the research literature. This aper aims to enumerate and appraise the various methodologies which have been applied to supply chain analysis over the last forty years. In particular we shall ask of each technique: To what extent does it reveal the dynamics of the process involved? These questions are important since only through knowledge of the dynamics can we gain a full appreciation and understanding of the factors which affect supply chain performance

The Effect of Batched Production on Demand Amplification

Demand amplification is the tendency of small fluctuations in demand at the retailer end of the supply chain to be amplified as they are communicated down the chain. A brief review of the literature on this phenomenon is presented, concentrating particularly on the causes propounded. A continuous-time differential equation model of a production-inventory system is then proposed. The application of a novel optimal control algorithm is applied in order to simulate the rational behaviour of inventory managers. This algorithm allows us to mimic the discontinuous cost structures implied by the advantages of batched production. By simulating the response of the system to small changes in demand, the relationship between batch size and the magnitude of demand amplification is investigated

Quantifying the Effects of Trust in Supply Chains During Promotional Periods

Analytical modelling of supply chains has tended to focus on the material flows whilst neglecting the study of information dissemination through the supply chain. An important factor influencing these flows is the level of trust between supply chain partners. The dimensions of such trust are examined here and a brief review of research aimed at evaluating its importance in this area is undertaken. An optimal control model is then constructed to calculate the cost implications of the prevailing level of trust in a multi-echelon supply chain faced with satisfying a promotion for the end product. The mechanism by which both low and high levels of trust distort demand information as it is transmitted along the supply chain is investigated. A sensitivity analysis reveals that these results are essentially unaffected by the cost structures assumed. Lastly, two methods designed to compensate for low levels of trust are appraised

A Lyapanov Function Method to Estimate the Stability Region of Nonlinear Systems Using the Lie Series

A method to estimate the region of asymptotic stability, about an equilibrium point of an autonomous nonlinear system is presented. The Lie series is used to approximate the solution trajectories of a general point in the stability region. The method is applied to several well-known examples of two and three dimensions