بررسی دینامیک نوسانی یک سیستم موجودی بازده ممنوع

We present an analytical investigation of the intrinsic oscillations in a nonlinear inventory system where excessive inventory cannot be returned to the supplier. Mathematically this is captured by a non-negative constraint on the replenishment order. By studying the eigenvalues of the characteristic matrices of the system, the criteria for different types of dynamic behaviour (including convergence, periodicity, quasi-periodicity, chaos, and divergence) are derived. The upper and lower bounds of the order and inventory oscillations are found via a time-domain analysis. Our results are verified by bifurcation diagrams. We find that the closer the replenishment rule feedback parameters are to the convergence area, the milder the intrinsic oscillation of the system.

A supply chain's inventory control policy needs to attenuate fluctuations in demand, so as to maintain a smooth production rate in the face of both externally and internally generated disturbances. It should also maintain inventory levels around target safety stock levels. One of the most well-studied oscillation phenomena in supply chains is the so-called bullwhip effect. Since the pioneering work of Lee et al. (1997), much effort has been devoted to this problem. Many factors affecting the bullwhip effect have been investigated including: the impact of forecasting methods (Chen et al., 2000 and Dejonckheere et al., 2003); statistical modelling of demand processes (Aviv, 2003 and Gaalman, 2006); cooperative mechanisms such as information sharing (Lee et al., 2000 and Dejonckheere et al., 2004); and Vendor Managed Inventory (Disney and Towill, 2003a). The integration of control theory and system dynamics approaches provides a powerful approach for quantifying and mitigating such effect (Disney and Towill, 2003b). However, in most of the previous theoretical studies on the bullwhip effect, linear inventory system models were adopted. In linear systems dynamical oscillations can only be generated by external events (such as demand). This has greatly limited the applicability of published results and has made it impossible to explain and describe oscillations caused by internal factors. To maintain linearity of inventory system models, order rates are permitted to take negative values. This means that all participants in a supply chain are allowed to return excess product freely. Specifically, a negative order rate value leads to a decrease in the inventory level at the consuming echelon and an immediate increase in the inventory level at the supplying echelon. This assumption may be difficult to realize in reality but we do recognize that it exists in some supply chains. For example in the consumer electronics and book publishing supply chains it is accepted practice that retailers may return unwanted product to the manufacturer/publisher. In practice this may also mean that the excess inventory is not physically moved from one location to another but instead will be considered to be in the possession of the upstream supplier until being used as part of a future replenishment (Hosoda and Disney, 2009). It has also been demonstrated that nonlinear effects play an important role in inventory systems, sometimes even a dominant role (Nagatani and Helbing, 2004). When linearity assumptions are removed complex dynamic behaviours are revealed. The behaviour may even become chaotic or hyper-chaotic. More importantly, oscillations generated internally by the system itself, rather than by the external environment, may arise. Mosekilde and Larsen (1988) adapted the beer game model (Sterman, 1989) to include both forbidden returns and lost sales constraints. To make the chaotic phenomenon more obvious, a long lead-time was used. Mosekilde and Larsen (1988) found that the operating cost of this constrained system could be 500 times higher than its linear counterpart. Thomsen et al. (1992) concluded that economic and business systems do not necessarily operate close to their steady state. Hwarng and Xie (2008) investigated several system factors that affect chaotic behaviour and discovered a ‘chaos-amplification’ phenomenon between supply chain echelons. Wu and Zhang (2007), using a supply chain model with a constrained discount rate and exponential demand function, found that the attractors of the model in the phase space moves with the assumed initial states, rendering it impossible to provide guidelines for avoiding chaos by bifurcation analysis. Wang et al. (2005) used the Lyapunov exponent to identify chaotic demand in real supply chain data and proposed an algorithm to cope with it from a time series aspect. The piecewise linear modelling approach has also been shown to be effective for certain nonlinear supply chain problems as the piecewise linear function is able to approximate any nonlinear function to any required level of accuracy. Liu (2005) and Rodrigues and Boukas (2006) analyzed the stability of supply chain inventory systems with piecewise linear techniques. Laugesen and Mosekilde (2006) and Mosekilde and Laugesen (2007) studied border-collision bifurcations in piecewise linear supply chain systems. However, mathematical properties of such systems, such as local and global stability conditions and bifurcations, are still “very hard to investigate” and “notoriously challenging” (Sun, 2010). This paper is concerned with identifying the range of oscillations in a constrained supply chain that are generated by the system itself rather than by the external environment. For our analysis a unit step demand input will be adopted (unless otherwise stated) to emphasize that it is the system itself rather than the environment that is generating these dynamical effects. The pattern and amplitude of each kind of oscillation will be characterized. Section 2 models the constrained one echelon supply chain system piecewise-linearly. Section 3 investigates the type of oscillation pattern produced by the inventory system. Section 4 focuses on the upper and lower bounds of oscillation with respect to the order volume and inventory level. Concluding remarks are given in Section 5.

We have highlighted the range of dynamic behaviours that are present in a constrained inventory system with only one constraint, forbidden returns. The stable region of the forbidden return system is identical to that of the linear system, indicating that existing knowledge and methodologies for studying the asymptotic stability of linear inventory systems can be applied to the forbidden returns case. We have shown that a complex and diversified set of behaviours and patterns exist outside the convergence region. When operating outside the convergence region, the inventory system may not escape to infinity as would be inferred by a linear analysis; the risk of catastrophic divergence is reduced by the physical constraint. Divergence only appears when there is a positive feedback on inventory or strong positive feedback on WIP. In other cases, if there is an oscillation, the oscillation will be bounded. Under certain parameter settings, the oscillation will show a periodic pattern, most noticeably when WIP feedback gain is larger than the inventory feedback gain. It is reasonable to conjecture that when facing a stochastic demand, system behaviour will be more predictable when the parameters are in this region. Moreover, we found it interesting that even a simple deterministic model with a short, known and constant lead-time is sufficient to generate such complex phenomena. We wonder what sort of impact stochastic demand, longer or random lead-times and more constraints will have? Through a numerical experiment, we have shown the bullwhip effect in a nonlinear inventory system is largely affected by the intrinsic oscillations produced by the system. Therefore it is essential, as a first step, to study the deterministic oscillation amplitude of the inventory system with respect to both order and inventory. We have demonstrated how to determine the type and bound of each oscillation. It is found that whilst system stability is determined by the modulus of eigenvalues, the oscillation amplitude is governed by their real parts. The upper bound of the oscillations can be expressed piecewise by solutions of the dominating difference equation, and the amplitude of order and inventory oscillations will be smaller if control parameters are set closer to the stable region. For period-2 oscillation, the amplitude of both order and inventory is constant. In the region 1<αSL<αS the oscillation of order is constrained to below αS. As can be predicted, setting parameters close to the divergence boundary will create massive order quantities, but with very long intervals between orders. As would be expected there is also a tendency that large orders are followed by long intervals of zero orders. This approach may be suitable in situations when there is a significant order set-up cost and it may be useful to compare performance to the Economic Order Quantity model. The lead-time of the inventory system is limited to one period in this paper, where the conventional settings, 0<{αS,αSL}≤1, are guaranteed to be stable. However when the lead-time increases this is not the case. Settings with a large inventory gain and small WIP gain are the most vulnerable to a lead-time increase in terms of stability. Nonlinear oscillations in such scenarios can then be analyzed using a similar approach. Moreover, there is a possibility that supply chain participants will use an intuitive ordering policy rather than a carefully designed one. In this case decision makers are more likely to move the parameters out of the stability area. Therefore an expansion of the control plane provides a more thorough understanding of the dynamic nature of a constrained inventory system. Potential future research could involve several aspects. One is to incorporate other constraints, such as lost sales and forbidden outsourcing, into the inventory system model. Our own preliminary investigation has shown that when lost sales are solely present, the stable region will be sensitive to the initial values of the inventory and orders. Another idea for future work is to expand our study of the oscillation when the system is exposed to random external disturbances. The intrinsic oscillation would then be coupled with random noise. Yet another avenue of investigation could be to study how different types of oscillation evolve along a multi-echelon supply chain, i.e., the so-called chaos-amplification effect, as a nonlinear counterpart to the linear ‘variance amplification’ (or bullwhip) effect.