اثر فرکانس به روز رسانی اطلاعات در ثبات تولید سیستم های کنترل موجودی

Production ordering and inventory dynamics in a manufacturing system are analyzed using function transformation techniques (z-transform) and their conditions for stability are examined. A generic model which captures the mixing and variability in the production process is developed. Variation in the stability of the system operating under sufficient inventory coverage and under limited inventory coverage is highlighted. The effects of the frequency of information update on stability are then examined by relating the update frequency to the sampling interval of the underlying difference equations. System dynamics simulations are used to demonstrate the stable or unstable behaviour of the production–inventory system.

In current volatile markets, companies need to respond quickly to changes in customer demand and adapt quickly to changes in technology. This necessitates the modelling and analysis of the dynamic behaviour of companies, especially their production–inventory systems (Ortega and Lin, 2004). Analysis of production and inventory dynamics helps reveal periods of inventory build-ups, stock-outs, overtime production and production shutdown, which costs the companies in terms of profits, market position and customer satisfaction. Hence, it becomes critical to model and analyze the dynamics of production inventory systems. The dynamic behaviour or dynamic complexity is said to arise from the interaction between the various system components over time (Sterman, 2000). Dynamic systems, characterized by delays, feedbacks and nonlinearities, cannot be adequately captured using mathematical programming techniques such as linear/non-linear/stochastic programming. A natural choice to examine the production and inventory dynamics is the application of control theoretic techniques. This often involves capturing of the production–inventory system using feedback-based structures (Forrester, 1961; Towill, 1982; Axsäter, 1985; Edghill and Towill, 1990; Sterman, 2000) and analysis of the system through the application of control theoretic tools such as block diagrams, Bode plots, and functional transformations (Wikner et al., 1992; John et al., 1994; Grubbström and Wikner, 1996; Disney and Towill, 2002; Disney et al., 2004). In this research work, modified causal loop diagrams are used to capture (model) the production–inventory system, which is then analyzed by applying z-transformation technique (a type of function transformation technique). Function transformation technique maps the system from the time domain to the frequency domain; the advantages of which are summarized below (Disney and Towill, 2002): • Frequency response analysis has been found to be an efficient tool to examine the critical design parameters and identify ranges of parameter values that give good transient response performance (Ortega and Lin, 2004). •Standard techniques exists to analyze the system performance such as rise time, peak overshoots, and settling time, without recourse to simulation (Bissell, 1996), Frequency domain calculations can be computationally very simple (Bissell, 1996). Closed loop transfer functions of the system can be obtained that enables to gain insight into the stability of the system, Appropriate integration of transfer functions with simulation enables additional system analysis (Disney and Towill, 2002). • A number of techniques exists for transferring problems from one domain (Laplace, z, Fourier, w, frequency, etc) to another domain, to help gain insight from situations that have already been encountered and solved in other domains (Disney and Towill, 2002). •Transforms can be used to capture the stochastic properties by serving as moment generating functions (Grubbström, 1998). A comprehensive literature review on the use of control theoretic concepts for the dynamic analysis of production–inventory systems can be found in Ortega and Lin (2004) and Disney and Towill (2002). John et al. (1994) demonstrated the stabilizing effect of including a supply line (work-in-progress, WIP) component into an inventory and order based production control system (Towill, 1982), using block diagrams and Laplace transform. Towill et al. (1997) examined the critical design parameters within an adaptive model consisting of three feedback loops—inventory error loop, desired order in pipeline loop and the lead time loop, and highlighted how the total orders in the pipeline can be used for assessing the load of the internal manufacturing pipeline. Grubbström (1998) used Laplace transform, z-transform and Net Present Value on MRP systems and showed a three-fold use of transfer functions: (1) describes production, demand and inventory dynamics in a compact way, (2) captures stochastic properties by serving as moment generating functions, and (3) assesses the cash flows up capturing the net present value in the transfer functions. White (1999) has showed that simple inventory management systems are analogous to the proportional control in conventional control theory, and has demonstrated that the use proportional, integrative and derivative (PID) control algorithms can result in saving of up to 80%. Optimal control parameters for use in general production and inventory control systems have been found by Disney et al. (2000) using genetic algorithm. The performance measures characteristics considered by them include (1) inventory recovery to “shock” demands, (2) in-built filtering capability, (3) robustness to the production lead-time variations, (4) robustness to pipeline level information fidelity, and (5) systems selectivity. Dejonckheere et al. (2003) have employed filter theory to relate the dynamics of order replenishment to the production planning strategies ranging from lean systems to agile systems, highlighting the flexibility of their order replenishment policy. Lin et al. (2004) studied the stability and bullwhip effect assuming sufficient supply stock for a serial supply chain using z-transform analysis. They concluded that bullwhip effect will occur if the ordering policy includes forecasting; and developed control schemes to reduce the bullwhip effects. Disney et al. (2004) have studied a general production–inventory control system which is guaranteed to be stable through the use of Deziel–Eilon arbitrary setting (Deziel and Eilon, 1967). They have derived analytical expressions for the bullwhip and inventory variance produced by the control system, and highlighted the bullwhip boundary as a function of the inventory feedback gain. Using linear z-transform analysis, Disney and Towill (2005) have identified and proposed a method to eliminate the possibility of an inventory drift due to uncertain pipeline lead times. In this paper, a novel production ordering and inventory control system, which allows for the capture of mixing and variability in the production process and inventory management is presented using a causal loop diagram. The proposed system can be viewed as an extension of the automated pipeline inventory and order based production control system (APIOBPCS) family of models (Towill, 1982; John et al., 1994), with the key enhancements lying in the representation of the production process, information update frequency and shipment-backlog constraints. The difference equation models of the system are constructed and the generalized transfer function of the production release order is obtained using z-transform techniques for infinite inventory coverage and for limited inventory coverage. The boundary conditions for the system stability are computed based on Jury's Test (1964). Stability refers to the classical definition from the control system perspective, which indicates the ability of the system to return to the original state once the source of disturbance has been removed. Insights into the non-linear dynamic behaviour of the system are drawn from the mathematical models developed, the results of which are supported through simulation results. The effect of the frequency of information update on the stability of the system has been demonstrated by relating the update frequency to the sampling interval of the underlying difference equations. Hence, the existence of instability due to the improper selection of system parameters and the choice of sampling interval is confirmed. Finally, guidance for the selection of parameters to guarantee system stability is presented.

A general production ordering and inventory control system has been described and analyzed. Generalized stability conditions have been derived using z-transformation techniques with the system parameters including fractional adjustment of WIP, fractional adjustment of inventory, exponential smoothing constant for forecast, number of production stages (or order of production delay), production lead time and the sampling interval. The stability boundaries for system operating under inventory adequacy and inventory insufficiency have been established. Jury's Test has been employed to derive the stability conditions for the complex higher order polynomials. These results have been verified through the simulation of the dynamic behaviour of the original system. The effect of the frequency of information update on the stability of the system has also been analyzed. The frequency of information update has been mapped on to the sampling interval of the underlying difference equations. Results have revealed that aggressive ordering policies (higher values of the fractional adjustment rates for WIP and inventory) require a more frequent information update, i.e. lower sampling interval. Also, the stable Deziel–Eilon settings of the fractional adjustment rates have been found to be dependent on the sampling interval, stressing the need to select the appropriate control parameters also based on the sampling interval. Further investigations need to be carried out to study the effect of multiple parameters, multiple non-linear constraints and their interactions. The influence of the sampling interval on special ordering schemes and manufacturing policies also needs to be studied. The natural progression of this paper would involve the future extension of this work in two directions. One direction is a horizontal extension to supply chain settings, where the stability effects of information update frequency between members of the supply chain analyzed in relation to the strategy of the supply chain employed (viz. lean, agile, vendor managed inventory). The other direction is a vertical extension within an enterprise to analyze the dynamic interactions between hierarchical planning and scheduling systems resulting in stable or unstable behavioural patterns, especially in the presence of disturbances.