The Supply Chain system is a network of many separate entities like suppliers, manufacturers, distributors and retailers that are responsible for all the tasks starting from manufacturing process to the finished goods then to the distribution of finished products to the customers. In some cases, customers are people and mostly, customers are retailers.
Normally, supply chain involves forward flow of materials and backward flow of information. Recently enterprises shows great interest in efficiency of supply chain systems due to the short life cycle of products, especially food products or the products that are fragile in nature. The efficiency in supply chain management system can lower the production cost, inventory cost and transportation cost and improved customer service throughout all the steps that are involved in the chain.
In supply chain, as we move from a downstream level to an upstream level, the phenomenon that involves the amplification of demand variability is called 'Bullwhip Effect'. The four major causes of Bullwhip effect are:
Price fluctuations due to special offers.
In the competitive market dynamic characteristics offer a great advantage in the modeling of the supply chain systems. The combined effect of the dynamics and supply chain system has influenced both industry and academia. Recently alternative approaches are preferred for the modeling of dynamic supply chain models, such as
Continuous-time differential equation model,
Discrete-time difference model,
Discrete event model,
Classical operational research method.
In supply chain management systems to design, analyze and simulate, control theory provides the mathematical tools, which are based on the dynamic models. In order to find the solutions for the bullwhip phenomenon, control theory is mostly used.
For supply chain modeling, we always need a feedback loop to correct the errors in the system for the proper operation. As a matter of fact, closed-loop system modeling is used in simulation and modeling of supply chain systems.
CHAPTER : LITERATURE REVIEW
MATLAB (Matrix Laboratory), a product of Mathworks,Â is a scientific software package designed to provide integrated numeric computation and graphics visualization in high-level programming language. MATLAB program consists of standard and specialized toolboxes allowing users to take advantage of the matrix algorithm based on LINPACK1 and EISPACK2 projects. MATLAB offers interactive features allowing the users great flexibility in the manipulation of data and in the form of matrix arrays for computationÂ and visualization. MATLAB inputs can be entered at the "command line" or from "mfiles", which contains a programming-like set of instructions to be executed by MATLAB. In the aspect of programming, MATLAB works differently from FORTRAN, C, or Basic, e.g. no dimensioning required for matrix arrays and no object code file generated. MATLAB offers some standard toolboxes and many optional toolboxes (at extra cost, of course!) such as financial toolbox and statistics toolbox. Users may create their own toolboxes consisted of "mfiles" written for specific applications. The original version of MATLAB was written in FORTRAN but later was rewritten in C.Â You are encouraged to use MATLAB's on-line help files for functions and commands associated with available toolboxes. MATLAB consists of a collection of toolboxes. These toolboxes contain library files called M-Files, which are also functions or command names, executable from the Command window.
Simulink (Simulation and Link) is an extension of MATLAB by Mathworks Inc. It works with MATLAB to offer modeling, simulation, and analysis of dynamical systems under a graphical user interface (GUI) environment.Â The construction of a model is simplified with click-and-drag mouse operations. Simulink includes a comprehensive block library of toolboxes for both linear and nonlinear analyses. Models are hierarchical, which allow using both top-down and bottom-up approaches. As Simulink is an integral part of MATLAB, it is easy to switch back and forth during the analysis process and thus, the user may take full advantage of features offered in both environments.
To start a Simulink session, you'd need to bring up Matlab program first.
From Matlab command window, enter:
Alternately, you may click on the Simulink icon located on the toolbar as shown:
Simulink's library browser window like one shown below will pop up presenting the block set for model construction.
To see the content of the blockset, click on the "+" sign at the beginning of each toolbox.
To start a model click on the NEW FILE ICON as shown in the screenshot above. Alternately, you may use keystrokes CTRL+N.
A new window will appear on the screen. You will be constructing your model in this window. Also in this window the constructed model is simulated. A screenshot of a typical working (model) window is shown below:
CHAPTER 2: APVIOBPCS
Automatic Pipeline Inventory and Order Based Production Control System (APVIOBPCS) deals with the periodic review ordering systems i.e. the system states are reviewed and a decision is made on placing orders on the upstream supply pipelines at regular and equally spaced points in time. Here, we are concerned with a production and inventory control system that incorporates pipeline or Work In Progress (WIP) feedback. This type of ordering policy requires an estimate of the delivery lead-times before it can generate orders.
The table below shows the different variants of IOBPCS family which APVIOBPCS is a part of,
IOBPCS system variant
Order Based Production Control System (OBPCS)
Inventory Based Production Control System (IBPCS)
Inventory and Order Based Production Control System (IOBPCS)
Variable Inventory and Order Based Production Control System (VIOBPCS)
Pipeline, Inventory, and Order Based Production Control System (PIOBPCS)
Pipeline, Variable Inventory, and Order Based Production Control System (PVIOBPCS)
Automatic Pipeline, Inventory and Order Based Production Control System (APIOBPCS)
Automatic Pipeline, Variable Inventory and Order Based Production Control System (APVIOBPCS)
Table 1: Some common IOBPCS variant.
The IOBPCS family of decision support systems is shown in the above table. At regular intervals of time the available system state is monitored and used to compute the next set of orders. According toÂ Coyle (1977)Â such a system is frequently observed in action in many market sectors. Author has studied the expected behaviour via industrial dynamics simulation models.Â Towill (1982)Â then recast the problem into a control engineering format with emphasis on predicting dynamic recovery, inventory drift, and noise bandwidth which led to variance estimations. Limited optimization was thereby enabled within the constraints imposed by having only two adjustable parameters controlling a third order model. An important feature of this 1982 paper was established hardware system "best practice" thus identifying good, workable, yet conservative designs capable of transfer into the production control arena.
The model was extended by Edghill and Towill (1989), which also led to the extension of theoretical analysis, by allowing the target inventory to be a function of observed demand. This Variable Inventory OBPCS is representative of that particular industrial practice where it is necessary to update the inventory cover over time. Usually the moving target inventory position is estimated from the forecast demand multiplied by a cover factor. Cover factor is a function of pipeline lead time often with an additional safety factor built in. A paper byÂ John et al. (1994)Â demonstrated that the addition of a further feedback loop based on orders in our pipeline provided the missing third control variable. This Automatic Pipeline IOBPCS model was subsequently optimized in terms of dynamic performance through the use of genetic algorithms.Â Disney et al. (2000).
However a further important conclusion emerged fromÂ John et al. (1994)Â indicating that the inventory drift would occur if the pipeline lead timeÂ estimateÂ used as part of the control algorithm was different from the currentÂ actualÂ lead time. This was recognized by the requirement for updating lead-time estimates online thus enabling the Adjustable APIOBPCS. Finally however, if we additionally include the Variable inventory target within the APIOBPCS ordering system, then we also encompass the Order-Up-To (OUT) Model developed initially within the OR fraternity and described recently byÂ Looman et al. (2002). The equivalence of the OUT andÂ APVIOBPCSÂ models was subsequently established byÂ Dejonckheere et al. (2003).
Figure 1: IOBPCS family.
As the IOBPCS family has evolved, it has become obvious that we were simulating much production control practice in addition to the situations described byÂ Coyle (1977) which is also linked back toÂ Roberts (1981)Â and hence to JayÂ Forrester (1958). The ORATE equationsÂ seems to have been discovered and rediscovered many times by management consultants who provide appropriateÂ softwareÂ to realize the algorithm. This was at the aggregate level an application advocated byÂ Axsäter (1985). But writingÂ softwareÂ using a generic ORATE equationÂ is not merely a necessary condition for good system design. The functions inherent in the equationÂ may take a standard form, but their settings within an operating scenario are context specific. There is little evidence that these settings are properly tuned, with a substantial gap in the knowledge exploited by user and system designer. This is at variance with the recommendations made byÂ Feltner and Weiner (1985)Â and endorsed byÂ Solberg (1992)Â that users should intellectually own suchÂ software
In contrast an IOBPCS variant has been designed and implemented in-house successfully controlling the delivery of 6000 health care aid products from the active catalogue (Cheema et al., 1997). This product range was divided up on a Pareto Curve basis (Koch, 1997) with availability targets varying according to the product ranking. Each product therefore had its own control law that was operated automatically and in parallel with the requisite control laws for all other items. This application shows that IOBPCS works at the individual product level. It is also an exemplifier of the management rule of thumb of simplifying the operating scenario first so that simple models are justifiable. This in turn is in line with the empirical evidence produced byÂ New (1998). The concept was to use the ordering system to drive the factory MRPÂ softwareÂ that was responsible for solving detailed scheduling problems.
Because of the inevitable and considerable interference between product routings on the shop floor, the actual lead-time for each product varied from target. Hence these lead-times can be considered to vary both between products and within products. One proposal for compensating for lead-time variation was further investigated byÂ Cheema (1994), having been implemented in an ad hoc manner within the health care products company. This required the on-line monitoring and automatic updating of pipeline targets through an exponentially smoothed non-linear feedback loop. Company Annual Reports substantiated that customer service levels and stock levels were considerably improved when this so-called To Make ordering system was incorporated as the MRP driver but the costs of maintaining accurate records of actual lead-times must have been notable.
We have already mentioned the need to cope with lead-time variation as a core function of the production control system. Initial results obtained byÂ John et al. (1994)Â on the importance of matching pipeline targets to current lead-times were confirmed byÂ Cheema (1994).Â The figure 1 belowÂ indicates our understanding through some simulation results with lead-time problem. These lead-time effects exhibit a situation where either there is an unavoidable material shortage or queue for usage of a particular manufacturing facility, These event usually occur in health care products company. The inventory reverts to its target position (zero) only if the estimated lead time is equal to the actual lead time, or else there is a positive or a negative offset. Note thatÂ Figs. 2(f) and (g), with zero off-set, correspond to the health care products control system with the exponential smoothing constant in the lead-time feedback loop set equal to unity.
Figure 2: Inventory drift after a step change in demand due to lead-time variation (a=1,Â Ta=3,Â Ti=Tw=2).
In the above figure APVIOBPCS model is used to demonstrate the inventory drift caused by pipeline lead time variation. The available variables for dynamic control are the feedback gains (1/Ti andÂ 1/Tw), the inventory coverÂ a, and the exponential smoother (Î±=1/(1+Ta)) used in sales forecasting feed forward channel. This model is derived in discrete time, hence the use of theÂ z-transform. In order to obtain consistent results the model adheres strictly to the order-of-events sequence initially defined byÂ Vassian (1955).
9Normal z transform analysis given below can be used to prove the existence of inventory offset. The first step is to find the inventory transfer function. This is easily obtained using standard block diagram techniques such as those mentioned inÂ Nise (1995)Â and is given by
To which we may apply the final value theorem (Eq.Â (1)), whereÂ I(z)=1/(1-z-1), the unit step input andÂ F(z)Â is the inventory transfer function (Eq.Â (1)).
Manipulation then yields the following final value of inventory for a step change in demand:
As the required target inventory level isÂ a, we note from Eq.Â (3)Â that if the estimate of the lead-time is wrong then an error is produced, unless:
â€¢ the error between our perception of the lead-time and the actual lead-time is zero and
â€¢Â TwÂ is set to infinity (i.e. there is no WIP feedback). Hence we have the original Inventory and Order Based Production Control System (IOBPCS) model. Towill (1982)
Hence, it makes the dynamic response more difficult to shape to match a desired behaviour.
Assuming that we wish to maintain a finiteÂ TwÂ (and indeed if we use theÂ Deziel and Eilon (1967)Â rule ofÂ Tw=Ti then we have no further choice in the matter) we must either accurately and continuously update our lead-time estimates, or, alternatively, find a different solution to the problem. We also have to keep in mind that lead-time estimates may be complicated if there are complex, interacting product channels (Burbidge, 1990). Also, accurate estimates are sometimes difficult even within our own company. if the pipeline crosses a number of process boundaries this problem multiplies. Designing for simplified material flow is one possible answer (Childerhouse and Towill, 2003) but is outside the scope of this project. Another possibility is the best matching of the manufacturer and material supplier levels of flexibility (Garavelli, 2003). Also, improved schedulingÂ software are now availableÂ as reviewed byÂ Knolmayer et al. (2002).
Solution to the inventory drift problem:
Figure 3: Block diagram of APVIOBPCS.
As shown in the above figure 3. The integrator in the established APVIOPBCS policy model WIP feedback loop sums the difference between ORATE and COMRATE. However, the WIP level can also be estimated as the sum of the previousÂ TpÂ ORATE signals as shown below in Eq.Â (4). It can be appreciated that the reason why the Final Value of the inventory levels (of APVIOPBCS) experience an offset is because the desired WIP level is based on theÂ perceptionÂ of the production lead-timeÂ Â and the actual WIP is based on theÂ actualÂ production lead-time. Therefore as already shown by the Final Value Theorem in Eq.Â (3)Â if our perception of the production lead-time is wrong then there is a difference between the desired WIP and actual WIP in the steady state:
Hence, a new system is proposed entirely focused at avoiding this effect by replacing the actual WIP signal with a WIP signal that would have been generated with our previousÂ (rather thanÂ Tp) ORATE signals, as shown in Eq.Â (5). Of course ifÂ Â then the two systems are equivalent. However, if is greater or less than or vice versaÂ then the systems will produce a different dynamic response. Hence the stability will be affected, but provided the system is still stable, the steady state inventory offset is eliminated. Thus, there is a need to establish stability boundaries for such a system. This new model is called EPVIOBPCS (Estimated Pipeline Variable Inventory and Order Based Production Control System). This model is different from the adaptive lead-time APIOBPCS modification considered byÂ Evans et al. (1997)Â because we do not attempt to obtain real-time accurate estimates of the lead-time and exploit them via a non-linear feedback loop. Instead the new model requires the WIP element within our block diagram.
Pipeline feedback has a beneficial effect on shaping the dynamic response of the IOBPCS class of periodic review production control systems (John et al., 1994). However, unless the pipeline target use the current value of delivery lead time there will be a room for inventory drift to occur. This, in turn will result in either excess stock build up or decreasing of customer service level. A situation like this is worsened if the lead times vary during normal plant operation (Cheema, 1994). The proposed solution to this problem is to use a WIP estimator based on the expected value of lead time and to incorporate this within the pipeline control loop. Early simulation results show that the inventory offset is indeed eliminated. Furthermore the penalty of incurring the long tailed response characteristic of the Proportional plus Integral control solution proposed by Cheema et al. (1997) is avoided. The proposed solution also avoids dealing with noisy, stale, or biased information that may affect the alternative approach utilising non linear lead time feedback. (Cheema et al., 1997)
Initial investigation indicates that if the control parameters remain fixed at their nominal values then the real world case where lead times increase will erode stability margins. This may cause a system that is designed to operate near the boundary to become unstable. Hence, a z-transform model is used, transformed it into the o-plane and applied the Routh-Hurwitz method to determine the stability boundaries for a range of delivery lead times. We have illustrated the procedure for the case of a volatile but initially stable system, which is destabilised by the lead time change. However, we have also demonstrated how such behaviour can be easily avoided by selecting a conservative design with parameters set in such a way that it will provide well damped dynamic response which is least affected by lead time changes.
The IOBPCS range of models is easily accomplished through spreadsheet based decision support systems. These are currently being exploited in new applications. The long term aim is to provide comprehensive analyses that enable the best match by selecting the most appropriate model for a given scenario, together with recommendations for good parameter settings. A virtue of the IOBPCS family is that by selecting extreme values we may simulate the wide spectrum of delivery scenarios ranging between the two extreme cases from Level Scheduling to Pass-On-Orders. Hence when dealing with many parallel pipelines it is easy to accomplish the same IOBPCS structure but tune the parameters according to Pare to Curve product classification. This enables availability targets to be met without excessive stockholdings as judged according to whether products are A, B or C rated.
CHAPTER 3: CASE STUDY (APVIOBPCS MODEL USING PID CONTROL SYSTEM)
Inventory holding costs and supply according to demand are the main issues in Supply Chain Management. Analytical techniques like 'operational research methods' and so many other techniques have been used, but no one method is appropriate enough to cope with all the issues. However, 'lean production' has a significant beneficial effects on the Supply Chain systems.
Figure: Schematic of the APVIOBPCS with cost function implemented in Simulink
In this APVIOBPCS model of a factory and sales system we see that the distributor produces virtual sales orders assuming a typical pattern of behaviour. These orders are further modified by the factory using human experience of the learning curve over time (this has the effect of producing an exponential delay). This is added to a fraction of the inventory error, plus a fraction of the Work In Progress (WIP) error. This comprises the order rate, which then will after a delay, cause production to be completed. From this completion rate the virtual sales rate is subtracted and the excess accumulation of these products leads to the inventory.
In the early descriptions of System Dynamics, delays were assumed to be exponential in form, in control terms exponential lags. Hence the delays due to the actual production processes, for example, are described by a simple single time constant Tp. This is typically the modelling process used in Stella Â® or Vensim Â®.
An Inventory and Order Based Production Control System lies at the heart of many commercial and bespoke ordering systems based on periodic review of stock and production targets. This simple and elegant control system works well, even when dealing with scenarios in which there are many competing value streams. However, such "interferences" inevitably cause some uncertainty in pipeline delivery times. We show via linear z-transform analysis that the consequences may include the possibility of inventory drift and instability. In this paper we establish the stability boundaries for such systems, and demonstrate an innovative method of eliminating inventory drift due to lead-time effect. This new principle is confirmed by simulation results.
Figure: The domain verification of inventory drift problem
EXISTANCE OF INVENTORY DRIFT PROBLEM IN VARIOUS SYATEMS:
The following table highlights the demand process, inventory model considered, analysis technique, whether the inventory drift problem exists if demand is a unit step input with an incorrect perception of the production lead-time used in the inventory model and some brief
comments on each reference.
Now, in order to stabilize the system, there are some stability conditions that should be taken into consideration, so that the system remains stable and yield maximum profits. Following are the conditions for stability of the system,
Figure: Stability conditions for MP(AP)VIOBPCS
So, to summarize the discussion, the following conclusions are made. The following chart shows the inter-relation between the parameter values when dealing with different types of models.
CHAPTER 4: EFFECTS OF CHANGING PARAMETERS ON APVIOBPCS MODEL
Using the initial values to obtain results,
Using these values, following results are generated,
STEP1: Now, varying value of Pb=1:28, we get the following results,
Conclusion: Increase in the value of Pb decresases the yield of the system.
STEP 2: To avoid huge impact on yield, changing the value of Pb in fractions from 0.1 to 0.9,
CONLUSION: Changing the values of Pb in fractions, causes un-noticeable difference on the output of the system.
STEP 3: Changing the value of Production Cost Pr from 0 to 10, we get the following results,
CONCLSION: Decrease in the production cost Pr, increases the yield of the system.
STEP 4: For the values of Pr from 35 to 45, we get the following results,
CONCLUSION: Increase in the production prices, decrease in the yield at the output of the system.
STEP 5: Varying Interest Rate i from 1 to 5,
CONCLUSION: For increase in the interest rate i, we get massive losses at the output.
STEP 6: For changing the value of Tv to 0, we get the following results,
CONCLUSION: If Tv=0, DINV becomes 0. So, If there's no desired inventory, that affects the output negatively.
STEP 7: For varying the value of Tv from 0.8 to 5, we get the following results,
CONCLUSION: If we have more desired inventory, the better the yield of system.
STEP 8: Now, in order to optimize the output of the system, we changed some parameters at the same time,
i = 0.1000
We got the following output,
STEP 9: Now, finally, we planned to change the values of PID gains, so we implemented the following values,
We got the following results,
CONCLUSION: The values of PID gains that we used, doesn't have a noticeable affect on the output of the system.