Dynamic updating of safety time
Development and simulation of calculation models for PipeChain AB
Ellinor Berkenäs Måns Bynke
Faculty of Engineering at Lund University The Department of Industrial Management and Logistics
Division of Production Management
This article is a summary of a master thesis conducted autumn 2010 to spring 2011 at PipeChain AB in Lund. The thesis has involved the development of models for dynamic updating of safety time and examination of what improvement these new models could bring to the inventory control principles used in the PipeChain software, a software that uses a static and manually determined safety time at the moment. Our findings show that a dynamic safety time often brings an improvement compared to a static time when it comes to the achievement of the service level. The service level goal is reached without an unacceptable increase in inventory levels.
Background
PipeChain is a company that is developing and distributing a software solution, with the same name as the company, with the purpose of controlling a supply chain. The inventory control logics being used in PipeChain is founded on a time concept, where inventory levels are measured in days of supply instead of quantities. This means that instead of using a fixed safety stock for the pur-pose of covering uncertainties and fluctuation in demand, a safety time is used that corresponds to the number of days the inventory should last under the current demand.
Choosing a high value for the safety time means that bigger uncertainties and fluctuations in demand can be handled and customers can still be kept satisfied as there is stock on hand. At the same time, in case of steady demand, stock might be unnecessarily kept meaning that capital is tied up needlessly.
Even if analytical calculations for determination of a safety stock level has existed for a long time, fairly recent studies have shown that only a minority of Swedish companies use such calculations. In case there are calculations being made, these are often performed only once per year or even more seldom. It is not uncommon that as rough estimations as a fixed percentage of the total inventory volume or a fixed number of days of demand for all articles, are being used as the basis for determination of safety stock. The absence of analytical safety stock calculations is believed to be wide spread among PipeChain’s customers as well. If analytical calculations for safety stock or safety time are carried out continuously the expectation is that the goal service level, in this case defined as the desired percentage of demand that should be possible to satisfy directly from the shelf (fill rate), could be reached with higher accuracy. The achievement of the service level should also be compared to any changes in mean inventory levels.
For example, under the assumption of 98 percent goal service level, a lowering of the measured service level from 99 to 98 is only desirable if the mean inventory level is also lowered.
Objective
The objective of the study has been to develop a model for dynamic updating of safety time and test this model using simulations to examine the validity and to describe what improvement, if any, the new model brings.
PipeChain – the software
The purpose of PipeChain is to innovate the customers supply chain for improved profitability and a competitive edge. The inventory control logics being used is best defined as the theoretical method time phased order point and can be described as follows:
1. Start with the current inventory position Calculate using the forecasted
demand the time when the inventory position reaches zero. 2. Calculate the reception time: –
Count the safety time backwards from the time
3. Calculate the order time: – – Count the lead time backwards from
the reception time 4. Calculate the order quantity:
emand during – inventor the time – Inventory position should be high
enough to cover forecasted demand during the max time at the time of delivery
Apart from this, PipeChain also considers a range of practical circumstances as for example a minimum delivery quantity, the fact that deliveries only can be received on certain days, etc. In this study however, the delimitation has been made that no such restrictions should be considered.
Method
First, the so called original model that corresponds to the current logics of the inventory control in PipeChain has been developed. Then, from the basis of a literature review and with the help of consecutive testing using computer simulations,
four different models of dynamic safety time updating have been developed.
The performance of the dynamic models has been compared to the original model using simulations. The results have been subject to statistical hypothesis testing where output in the form of deviation from the goal service level, the mean and the standard deviation of the service level and the mean inventory level have been compared between the different models.
Input to the simulations has been both theoretical demand generated from statistical distributions and demand from one of PipeChain’s customers. The results from using the theoretical demand have been the basis for general conclusions while the results from the real demand illustrates the theoretical results. The length of the simulations has been 25 years for the theoretical demand and almost four years for the real demand.
The theoretical demand has been generated from a compound distribution consisting of a Poisson distribution with six different arrival rates (50, 10, 3, 1/2, 1/10 and 1/40 days) and a uniform distribution which models the size of each order as a random number between one and ten. This has given six different cases of stationary demand that has been generated 30 times with different pseudo-random numbers to obtain 30 theoretically generated articles. The generated demand has been combined with different lead times and max times to obtain 120 different cases of what is believed to cover a wide range of realistic situations with both low and high intensity demand. The lead time is the time from when the goods are ordered from the supplier until they are received by the customer and has been chosen as 2, 5, 10, 20 and 40 days. The max time corresponds to the order quantity and is the number of days of supply for which the inventory should last when the delivery is received. The max time has been chosen as 5, 10, 20 and 60 days.
To also cover systematic changes in demand, a case where dynamic safety time is believed to be of high importance, three different cases of changing arrival rates have been constructed to form the non-stationary cases short yearly, long yearly and
The original model has been used in conjunction with a static safety time that has been chosen according to:
Empirical studies have shown that this is a reasonable value for a static safety time that has been chosen without analytical calculations. The model is called original(static). To also have an optimal static safety time model to compare with, repeated 25 years-simulations have been carried out to find the most optimal static time when it comes to the fulfillment of goal service level for a certain case. This is a theoretically valid model only, as such a perfect static time cannot be found under realistic circumstances where demand is not known in advance. This model is called original(optimal). The forecast accuracy is of great importance to the performance of the inventory control. PipeChain has the support for using advanced forecasts but as these forecasts have not been reconstructable, simple exponential smoothing (as described by Axsäter, 2006) has been used in this study with one exception. This exception was a special test case with the purpose of examining the general effect of a better forecast. In this case a forecast, generated according to the same principles as the demand, was used.
Dynamic models for the calculation
of safety time
From the basis of the literature review, four different models of dynamic safety time calculation have been developed. These are traditional
algorithm, traditional algorithm with adjustments, traditional algorithm with adjustments and time based deviation and simulation.
Traditional algorithm
When simulations are run, every day starts with the calculation of toda ’s forecast value. In this model, the safety time is updated directly thereafter, i.e. every day. The calculation of the safety time has been carried out with the traditional algorithm (see for example Axsäter, 2006) which is based on the order quantity , the standard deviation of the forecast error during the lead time and the
desired fill rate. The standard deviation is based on 80 values of forecast errors per day and is adjusted for the lead time. Since the real order quantities are changing according to the forecasted demand
during the max time, the order quantity to be used in the algorithm is calculated as an average of the ten latest order quantities to get a smoothed quantity not affected by temporary deviations. When the safety stock has been computed it is transferred to a safety time by accumulating forecast values until the sum reaches the computed safety stock. The number of days needed to reach this level is the safety time to be used.
Traditional algorithm with adjustments
This model is based on the same calculations as the
traditional algorithm but is supplemented with
empirically developed adjustments, which aim for a raise in the performance, particularly for cases of systematic changes in demand. There are two methods of adjustment complementing each other. The first adjustment method is that a dynamic goal service level is used. This dynamic goal service level is computed by first measuring the achieved service level during the last 60 days. This values is then exponentially smoothed using a smoothing constant of 0.2. The difference between this service level and the goal service level is then computed. The dynamic goal service is the original goal service level minus the difference. This results in a dynamic goal service level that can be either higher or lower than the original level. The limitation that the dynamic goal service level is never allowed to fall below the original service level minus 2 percent units is also introduced. The second adjustment is based on the performance of the most recent week. An accumulated difference (acc. diff) between the received and the desired service level is computed on a daily basis. Also, an accumulated change (acc. change) of service level is computed, i.e. the change in achieved service level on a daily basis during the last week. Both measures are computed in percent units. Depending on the size of acc. diff. and acc. change an adjustment of the new safety time might take place and a smoothing constant will be determined. In the final step the safety time is determined from those values according to the following exponential smoothing:
to e used Current new computed For an overview of the exact boundaries for adjustments based on acc diff and acc change, please refer to the thesis.
Traditional algorithm with time based deviation
In this model, computations are carried out in the same way as in traditional algorithm with
adjustments with one dissimilarity. The difference
is that is computed as a time based deviation (as
proposed by Krupp 1997). However, in contrast to Krupp’s time based MAD-value (Mean Absolute Deviation, see e.g. Axsäter, 2006), we instead use a time-based standard deviation.. When the forecast is based on simple exponential smoothing the outcome from using this model is virtually the same as that from the traditional algorithm with
adjustments. Hence, this technique is only tested
when a more sophisticated forecast with unique day-by-day values is used.
Simulations of a suitable safety time
This model is based on the simulation principle as described by e.g. Zizka (2005) and Køhler and de Kok (2002). Hence, this model is not connected to the previously described. Just as in the other models however, a goal service level is used. With certain optimization intervals a simulation over historical data is run. In this study, 30 days has been used as the optimization interval. The purpose of the simulation over historical data is to find the safety time that ought to have been used to reach the dynamic goal service level (same principle for the dynamic goal service level as described under
traditional algorithm with adjustments). By running
the simulation over historical data several times, all possible safety times can be tested until the safety time which gives an achieved service level as close as possible to the dynamic goal service level has been found. This time is set as the safety time to be used for the coming 30 days. Hence, the assumption that the safety time that was optimal during the last 30 days is optimal also for the coming 30 days is made. In a case where several different safety times results in the same service level, the lowest safety time that satisfies the dynamic goal service level is chosen.
Results and analysis
The results show that all the dynamic models (traditional algorithm, traditional algorithm with
adjustments, and simulation) outperforms
original(static) when it comes to the deviation from
the goal service level. This result holds for both high intensity (arrival rate 50, 10, 3) and low intensity (arrival rate 1/2, 1/10, 1/40) demand. A
representative example of results, in this case for high intensity demand with a goal service level of 95 percent, is shown in the table below.
Model
The deviation from the goal service level The standard deviation of the service level Original (optimal) 0,119 1,769 Original (static) 1,196 0,857 Traditional algorithm 0,354 1,658 Traditional algorithm with adjust-ments 0,252 1,716 Simulation 0,234 1,852 The inventory levels for the same case:
Model Total mean inventory Original (optimal) 085 809 Original (static) 112 945 Traditional algorithm 086 142 Traditional algorithm with adjustments 087 380 Simulation 086 737
All dynamic models have a lower deviation from the goal service level than original(static). They also have a lot lower total mean inventory level.
Simulation is the dynamic model with the lowest
goal service level deviation, at the same time it does not have the lowest total mean inventory level and actually the highest standard deviation. This pattern is often recognized also in the other cases.
An important result is that the dynamic models never or seldom can outperform original(optimal), implying that a very sophistically set static safety time is better than a dynamically updated one. The authors’ belief is however, that such a perfect static time hardly ever can be set.
The results from systematic changes in demand shows that traditional algorithm with adjustments and simulation outperform original(static) whereas they cannot surpass original(optimal). An important result from systematic changes is that traditional
This can be explained by the fact that the underlying formula is based on an assumption about normal distributed demand that does not hold. When it comes to the tests with real demand from a company that uses PipeChain, results confirm the positive impact of dynamic safety times which has been shown for the theoretically generated articles. A test with traditional algorithm with adjustments
and time based deviation was conducted using a
better forecast. Our findings show that this method is not better, but worse, than traditional algorithm
with adjustments. A possible explanation for this is
that the systematic changes in demand have been seasonal variations instead of distinct trends which was used by Krupp. Another possibility is that the method needs an even more sophisticated forecast than the one being used in this study to outperform the other methods.
A potential problem that has been identified in this study is that the dynamic models tend to trigger orders far more often than the static model. This has been found to be an outcome of the fact that the safety time is dynamically adjusted to, or close to, the max time. A solution to this would therefore be to prevent the safety time of reaching such a high level or even better, also adjusting the max time under certain circumstances.
Conclusions
The general conclusion is that using a dynamic safety time is a realistic and suitable approach to the problem of setting a safety time that could be chosen for both stationary and non-stationary demand. It is not easy to point out the best model as it depends on if a received service level as close as possible to the goal service level, a small standard deviation of the service level or a small mean inventory level is the most prioritized need. If a general conclusion should be made, the recommendation would be to use the model
traditional algorithm with adjustments if it is not
known whether demand is stationary or non-stationary and traditional algorithm if demand is known to be stationary. Simulation generally gives the smallest deviation from the goal service level but often has a higher standard deviation of the service level and a higher mean inventory level. Dynamic models seem to trigger more orders than a static model. If this is a problem further
development is needed where also an, at least to some extent, dynamic max time is considered.
Further research
One topic for further research is the evaluation of a dynamic max time. An idea is to adjust the max time upwards when the dynamic models repeatedly choose a safety time near the max time and adjust the max time downwards if they repeatedly choose a safety time of zero and the service level is still above the goal.
It would also be favorable to test the models under more realistic assumptions, for example with restricted delivery days or predetermined batch sizes.
The parameters that have been chosen in the models in this study are based on empirical testing. They are not optimized and further studies could be used to find the optimal values and rules for parameters like acc. diff. and acc. change. Also, simulation could probably be improved vastly by carrying out optimization simulations more often but letting the simulations still incorporate the same number of days back in time.
Dynamic safety stock calculation is a quite new scientific field and last but not least it should therefore be pointed out that continuous testing and development should be made to improve safety stock calculation and simplify the process of inventory control.
References
Axsäter, S. Inventory Control. New York: Springer, 2006.
Krupp, J. Safety stock management. Production
and Inventory management journal, 3rd qtr, 1997,
s. 11-18.
Køhler Gudum, C. och de Kok, T. G. A Safety Stock
Adjustment Procedure to Enable Target Service Levels in Simulation of Generic Inventory Systems.
Copenhagen Business School, Department of Management Science and Statistics, Preprint No 1, 2002.
Zizka, M. The Analytic Approach vs. the
Simulation Approach to Determining Safety Stock.
Problems & Perspectives in Management, nr. 3,
