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Predictive Analytics – Strategy @ Risk

Strategy @ Risk

Series: Predictive Analytics

Forecasting sales and forecasting uncertainty
This entry is part 1 of 4 in the series Predictive Analytics

Introduction

There are a large number of methods used for forecasting ranging from judgmental (expert forecasting etc.) thru expert systems and time series to causal methods (regression analysis etc.).

Most are used to give single point forecast or at most single point forecasts for a limited number of scenarios. We will in the following take a look at the un-usefulness of such single point forecasts.

As example we will use a simple forecast ‘model’ for net sales for a large multinational company. It turns out that there is a good linear relation between the company’s yearly net sales in million euro and growth rates (%) in world GDP:

with a correlation coefficient R= 0.995. The relation thus accounts for almost 99% of the variation in the sales data. The observed data is given as green dots in the graph below, and the regression as the green line. The ‘model’ explains expected sales as constant equal 1638M and with 53M in increased or decreased sales per percent increase or decrease in world GDP:

The International Monetary Fund (IMF) that kindly provided the historical GDP growth rates also gives forecasts for expected future change in the World GDP growth rate (WEO, April 2012) – for the next five years. When we put these forecasts into the ‘model’ we ends up with forecasts for net sales for 2012 to 2016 as depicted by the yellow dots in the graph above.

So mission accomplished! … Or is it really?

We know that the probability for getting a single-point forecast right is zero even when assuming that the forecast of the GDP growth rate is correct – so the forecasts we so far have will certainly be wrong, but how wrong?

“Some even persist in using forecasts that are manifestly unreliable, an attitude encountered by the future Nobel laureate Kenneth Arrow when he was a young statistician during the Second World War. When Arrow discovered that month-long weather forecasts used by the army were worthless, he warned his superiors against using them. He was rebuffed. “The Commanding General is well aware the forecasts are no good,” he was told. “However, he needs them for planning purposes.” (Gardner & Tetlock, 2011)

Maybe we should take a closer look at possible forecast errors, input data and the final forecast.

The prediction band

Given the regression we can calculate a forecast band for future observations of sales given forecasts of the future GDP growth rate. That is the region where we with a certain probability will expect new values of net sales to fall. In the graph below the green area give the 95% forecast band:

Since the variance of the predictions increases the further new forecasts for the GDP growth rate lies from the mean of the sample values (used to compute the regression), the band will widen as we move to either side of this mean. The band will also widen with decreasing correlation (R) and sample size (the number of observations the regression is based on).

So even if the fit to the data is good, our regression is based on a very small sample giving plenty of room for prediction errors. In fact a 95% confidence interval for 2012, with an expected GDP growth rate of 3.5%, is net sales 1824M plus/minus 82M. Even so the interval is still only approx. 9% of the expected value.

Now we have shown that the model gives good forecasts, calculated the confidence interval(s) and shown that the expected relative error(s) with high probability will be small!

So the mission is finally accomplished! … Or is it really?

The forecasts we have made is based on forecasts of future world GDP growth rates, but how certain are they?

The GDP forecasts

Forecasting the future growth in GDP for any country is at best difficult and much more so for the GDP growth for the entire world. The IMF has therefore supplied the baseline forecasts with a fan chart (( The Inflation Report Projections: Understanding the Fan Chart By Erik Britton, Paul Fisher and John Whitley, BoE Quarterly Bulletin, February 1998, pages 30-37.)) picturing the uncertainty in their estimates.

This fan chart ((Figure 1.12. from:, World Economic Outlook (April 2012), International Monetary Fund, Isbn 9781616352462)) shows as blue colored bands the uncertainty around the WEO baseline forecast with 50, 70, and 90 percent confidence intervals ((As shown, the 70 percent confidence interval includes the 50 percent interval, and the 90 percent confidence interval includes the 50 and 70 percent intervals. See Appendix 1.2 in the April 2009 World Economic Outlook for details.)) :

There is also another band on the chart, implied but un-seen, indicating a 10% chance of something “unpredictable”. The fan chart thus covers only 90% of the IMF’s estimates of the future probable growth rates.

The table below shows the actual figures for the forecasted GDP growth (%) and the limits of the confidence intervals:

Lower

Baseline

Upper

90%

70%

50%

50%

70%

90%

2012

2.5

2.9

3.1

.5

3.8

4.0

4.3

2013

2.1

2.8

3.3

4.1

4.8

5.2

5.9

The IMF has the following comments to the figures:

“Risks around the WEO projections have diminished, consistent with market indicators, but they remain large and tilted to the downside. The various indicators do not point in a consistent direction. Inflation and oil price indicators suggest downside risks to growth. The term spread and S&P 500 options prices, however, point to upside risks.”

Our approximation of the distribution that can have produced the fan chart for 2012 as given in the World Economic Outlook for April 2012 is shown below:

This distribution has: mean 3.43%, standard deviation 0.54, minimum 1.22 and maximum 4.70 – it is skewed with a left tail. The distribution thus also encompasses the implied but un-seen band in the chart.

Now we are ready for serious forecasting!

The final sales forecasts

By employing the same technique that we used to calculate the forecast band we can by Monte Carlo simulation compute the 2012 distribution of net sales forecasts, given the distribution of GDP growth rates and by using the expected variance for the differences between forecasts using the regression and new observations. The figure below describes the forecast process:

We however are not only using the 90% interval for The GDP growth rate or the 95% forecast band, but the full range of the distributions. The final forecasts of net sales are given as a histogram in the graph below:

This distribution of forecasted net sales has: mean sales 1820M, standard deviation 81, minimum sales 1590M and maximum sales 2055M – and it is slightly skewed with a left tail.

So what added information have we got from the added effort?

Well, we now know that there is only a 20% probability for net sales to be lower than 1755 or above 1890. The interval from 1755M to 1890M in net sales will then with 60% probability contain the actual sales in 2012 – see graph below giving the cumulative sales distribution:

We also know that we with 90% probability will see actual net sales in 2012 between 1720M and 1955M.But most important is that we have visualized the uncertainty in the sales forecasts and that contingency planning for both low and high sales should be performed.

An uncertain past

The Bank of England’s fan chart from 2008 showed a wide range of possible futures, but it also showed the uncertainty about where we were then – see that the black line showing National Statistics data for the past has probability bands around it:

This indicates that the values for past GDP growth rates are uncertain (stochastic) or contains measurement errors. This of course also holds for the IMF historic growth rates, but they are not supplying this type of information.

If the growth rates can be considered stochastic the results above will still hold, if the conditional distribution for net sales given the GDP growth rate still fulfills the standard assumptions for using regression methods. If not other methods of estimation must be considered.

Black Swans

But all this uncertainty was still not enough to contain what was to become reality – shown by the red line in the graph above.

How wrong can we be? Often more wrong than we like to think. This is good – as in useful – to know.

“As Donald Rumsfeld once said: it’s not only what we don’t know – the known unknowns – it’s what we don’t know we don’t know.”

While statistic methods may lead us to a reasonably understanding of some phenomenon that does not always translate into an accurate practical prediction capability. When that is the case, we find ourselves talking about risk, the likelihood that some unfavorable or favorable event will take place. Risk assessment is then necessitated and we are left only with probabilities.

A final word

Sales forecast models are an integrated part of our enterprise simulation models – as parts of the models predictive analytics. Predictive analytics can be described as statistic modeling enabling the prediction of future events or results ((in this case the probability distribution of future net sales)) , using present and past information and data.

In today’s fast moving and highly uncertain markets, forecasting have become the single most important element of the management process. The ability to quickly and accurately detect changes in key external and internal variables and adjust tactics accordingly can make all the difference between success and failure:
1. Forecasts must integrate both external and internal drivers of business and the financial results.
2. Absolute forecast accuracy (i.e. small confidence intervals) is less important than the insight about how current decisions and likely future events will interact to form the result.
3. Detail does not equal accuracy with respect to forecasts.
4. The forecast is often less important than the assumptions and variables that underpin it – those are the things that should be traced to provide advance warning.
5. Never relay on single point or scenario forecasting.
The forecasts are usually done in three stages, first by forecasting the market for that particular product(s), then the firm’s market share(s) ending up with a sales forecast. If the firm has activities in different geographic markets then the exercise has to be repeated in each market, having in mind the correlation between markets:
1. All uncertainty about the different market sizes, market shares and their correlation will finally end up contributing to the uncertainty in the forecast for the firm’s total sales.
2. This uncertainty combined with the uncertainty from other forecasted variables like interest rates, exchange rates, taxes etc. will eventually be manifested in the probability distribution for the firm’s equity value.
The ‘model’ we have been using in the example have never been tested out of sample. Its usefulness as a forecast model is therefore still debatable.

References

Gardner, D & Tetlock, P., (2011), Overcoming Our Aversion to Acknowledging Our Ignorance, http://www.cato-unbound.org/2011/07/11/dan-gardner-and-philip-tetlock/overcoming-our-aversion-to-acknowledging-our-ignorance/

World Economic Outlook Database, April 2012 Edition; http://www.imf.org/external/pubs/ft/weo/2012/01/weodata/index.aspx

Endnotes
27/04/2012
Inventory Management: Is profit maximization right for you?
This entry is part 2 of 4 in the series Predictive Analytics

Introduction

In the following we will exemplify how sales forecasts can be used to set inventory levels in single or in multilevel warehousing. By inventory we will mean a stock or store of goods; finished goods, raw materials, purchased parts, and retail items. Since the problem discussed is the same for both production and inventory, the two terms will be used interchangeably.

Good inventory management is essential to the successful operation for most organizations both because of the amount of money the inventory represents and the impact that inventories have on the daily operations.

An inventory can have many purposes among them the ability:
1. to support independence of operations,
2. to meet both anticipated and variation in demand,
3. to decouple components of production and allow flexibility in production scheduling and
4. to hedge against price increases, or to take advantage of quantity discounts.
The many advantages of stock keeping must however be weighted against the costs of keeping the inventory. This can best be described as the “too much/too little problem”; order too much and inventory is left over or order too little and sales are lost.

This can be as a single-period (a onetime purchasing decision) or a multi-period problem, involving a single warehouse or multilevel warehousing geographically dispersed. The task can then be to minimize the organizations total cost, maximize the level of customer service, minimize ‘loss’ or maximize profit etc.

Whatever the purpose, the calculation will have to be based on knowledge of the sales distribution. In addition, sales will usually have a seasonal variance creating a balance act between production, logistic and warehousing costs. In the example given below the sales forecasts will have to be viewed as a periodic forecast (month, quarter, etc.).

We have intentionally selected a ‘simple problem’ to highlight the optimization process and the properties of the optimal solution. The last is seldom described in the standard texts.

The News-vendor problem

The news-vendor is facing a onetime purchasing decision; to maximize expected profit so that the expected loss on the Qth unit equals the expected gain on the Qth unit:

I. , where

Co = The cost of ordering one more unit than what would have been ordered if demand had been known – or the increase in profit enjoyed by having ordered one fewer unit,

Cu = The cost of ordering one fewer unit than what would have been ordered if demand had been known – or the increase in profit enjoyed by having ordered one more unit, and

F(Q) = Demand Probability for q<= Q. By rearranging terms in the above equation we find:

II.

This ratio is often called the critical ratio (CR). The usual way of solving this is to assume that the demand is normal distributed giving Q as:

III. , where: , is normal distributed with zero mean and variance equal one.

Demand unfortunately, rarely haves a normal distribution and to make things worse we usually don’t know the exact distribution at all. We can only ‘find’ it by Monte Carlo simulation and thus have to find the Q satisfying the equation (I) by numerical methods.

For the news-vendor the inventory level should be set to maximize profit given the sales distribution. This implies that the cost of lost sales will have to be weighed against the cost of adding more to the stock.

If we for the moment assume that all these costs can be regarded as fixed and independent of the inventory level, then the product markup (% of cost) will determine the optimal inventory level:

IV.

In the example given here the critical ratio is approx. 0.8. The question then is if the inventory levels indicated by that critical ratio always will be the best for the organization.

Expected demand

The following graph indicates the news-vendors demand distribution. Expected demand is 2096 units ((Median demand is 1819 units and the demand lies most typically in the range of 1500 to 2000 units)), but the distribution is heavily skewed to the right ((The demand distribution has a skewness of 0.78., with a coefficient of variation of 0.45, a lower quartile of 1432 units and an upper quartile of 2720 units.)) so there is a possibility of demand exceeding the expected demand:

By setting the product markup – in the example below it is set to 300% – we can calculate profit and loss based on the demand forecast.

Profit and Loss (of opportunity)

In the following we have calculated profit and loss as:

Profit = sales less production costs of both sold and unsold items
Loss = value of lost sales (stock-out) and the cost of having produced and stocked more than can be expected to be sold

The figure below indicates what will happen as we change the inventory level. We can see as we successively move to higher levels (from left to right on the x-axis) that expected profit (blue line) will increase to a point of maximum ¤4963 at a level of 2729 units:

At that point we can expect to have some excess stock and in some cases also lost sales. But regardless, it is at this point that expected profit is maximized, so this gives the optimal stock level.

Since we include both costs of sold and unsold items, the point giving expected maximum profit will be below the point minimizing expected loss –¤1460 at a production level of 2910 units.

Given the optimal inventory level (2729 units) we find the actual sales frequency distribution as shown in the graph below. At this level we expect an average sale of 1920 units – ranging from 262 to 2729 units ((Having a lower quartile of 1430 units and an upper quartile of 2714 units.)).

The graph shows that the distribution possesses two different modes ((The most common value in a set of observations.)) or two local maxima. This bimodality is created by the fact that the demand distribution is heavily skewed to the right so that demand exceeding 2729 units will imply 2729 units sold with the rest as lost sales.

This bimodality will of course be reflected in the distribution of realized profits. Have in mind that the line (blue) giving maximum profit is an average of all realized profits during the Monte Carlo simulation given the demand distribution and the selected inventory level. We can therefore expect realized profit both below and above this average (¤4963) – as shown in the frequency graph below:

Expected (average) profit is ¤4963, with a minimum of –¤1681 and a maximum of ¤8186, the range of realized profits is therefore very large ((Having a lower quartile of ¤2991 and an upper quartile of ¤8129.)) ¤9867.

So even if we maximize profit we can expect a large variation in realized profits, there is no way that the original uncertainty in the demand distribution can be reduced or removed.

Risk and Reward

Increased profit comes at a price: increased risk. The graph below describes the situation; the blue curve shows how expected profit increases with the production or inventory (service) level. The spread between the green and red curves indicates the band where actual profit with 80% probability will fall. As is clear from the graph, this band increases in width as we move to the right indicating an increased upside (area up to the green line) but also an increased probability for a substantial downside (area down to the red line:

For some companies – depending on the shape of the demand distribution – other concerns than profit maximization might therefore be of more importance – like predictability of results (profit). The act of setting inventory or production levels should accordingly be viewed as an element for the boards risk assessments.

On the other hand will the uncertainty band around loss as the service level increases decrease. This of course lies in the fact that loss due to lost sales diminishes as the service level increases and the that the high markup easily covers the cost of over-production.

Thus a strategy of ‘loss’ minimization will falsely give a sense of ‘risk minimization’, while it in reality increases the uncertainty of future realized profit.

Product markup

The optimal stock or production level will be a function of the product markup. A high markup will give room for a higher level of unsold items while a low level will necessitate a focus on cost reduction and the acceptance of stock-out:

The relation between markup (%) and the production level is quadratic ((Markup (%) = 757.5 – 0.78*production level + 0.00023*production level²)) implying that markup will have to be increasingly higher, the further out on the right tail we fix the production level.

The Optimal inventory (production) level

If we put it all together we get the chart below. In this the green curve is the accumulated sales giving the probability of the level of sales and the brown curve the optimal stock or production level given the markup.

The optimal stock level is then found by drawing a line from the right markup axis (right y-axis) to the curve (red) for optimal stock level, and down to the x-axis giving the stock level. By continuing the line from the markup axis to the probability axis (left y-axis) we find the probability level for stock-out (1-the cumulative probability) and the probability for having a stock level in excess of demand:

By using the sales distribution we can find the optimal stock/production level given the markup and this would not have been possible with single point sales forecasts – that could have ended up almost anywhere on the curve for forecasted sales.

Even if a single point forecast managed to find expected sales – as mean, mode or median – it would have given wrong answers about the optimal stock/production level, since the shape of the sales distribution would have been unknown.

In this case with the sales distribution having a right tail the level would have been to low – or with low markup, to high. With a left skewed sales distribution the result would have been the other way around: The level would have been too high and with low markup probably too low.

In the case of multilevel warehousing, the above analyses have to be performed on all levels and solved as a simultaneous system.

The state of affairs at the point of maximum

To have the full picture of the state of affairs at the point of maximum we have to take a look at what we can expect of over- and under-production. At the level giving maximum expected profit we will on

average have an underproduction of 168 units, ranging from zero to nearly 3000 ((Having a coefficient of variation of almost 250%)). On the face of it this could easily be interpreted as having set the level to low, but as we shall see that is not the case.

Since we have a high markup, lost sales will weigh heavily in the profit maximization and as a result of this we can expect to have unsold items in our stock at the end of the period. On average we will have a little over 800 units left in stock, ranging from zero to nearly 2500. The lower quartile is 14 units and the upper is 1300 units so in 75% of the cases we will have an overproduction of less than 1300 units. However in 25% of the cases the overproduction will be in the range from 1300 to 2500 units.

Even with the possibility of ending up at the end of the period with a large number of unsold units, the strategy of profit maximization will on average give the highest profit. However, as we have seen, with a very high level of uncertainty about the actual profit being realized.

Now, since a lower inventory level in this case only will reduce profit by a small amount but lower the confidence limit by a substantial amount, other strategies giving more predictability for the actual result should be considered.
22/08/2012
Inventory management – Some effects of risk pooling

This entry is part 3 of 4 in the series Predictive Analytics

Introduction

The newsvendor described in the previous post has decided to branch out having news boys placed at strategic corners in the neighborhood. He will first consider three locations, but have six in his sights.

The question to be pondered is how many of the newspaper he should order for these three locations and the possible effects on profit and risk (Eppen, 1979) and (Chang & Lin, 1991).

He assumes that the demand distribution he experienced at the first location also will apply for the two others and that all locations (point of sales) can be served from a centralized inventory. For the sake of simplicity he further assumes that all points of sales can be restocked instantly (i.e. zero lead time) at zero cost, if necessary or advantageous by shipment from one of the other locations and that the demand at the different locations will be uncorrelated. The individual point of sales will initially have a working stock, but will have no need of safety stock.

In short is this equivalent to having one inventory serve newspaper sales generated by three (or six) copies of the original demand distribution:

The aggregated demand distribution for the three locations is still positively skewed (0.32) but much less than the original (0.78) and has a lower coefficient of variation – 27% – against 45% for the original ((The quartile variation has been reduced by 37%.)):

The demand variability has thus been substantially reduced by this risk pooling ((We distinguish between ten main types of risk pooling that may reduce total demand and/or lead time variability (uncertainty): capacity pooling, central ordering, component commonality, inventory pooling, order splitting, postponement, product pooling, product substitution, transshipments, and virtual pooling. (Oeser, 2011))) and the question now is how this will influence the vendor’s profit.

Profit and Inventory level with Risk Pooling

As in the previous post we have calculated profit and loss as:

Profit = sales less production costs of both sold and unsold items
Loss = value of lost sales (stock-out) and the cost of having produced and stocked more than can be expected to be sold

The figure below indicates what will happen as we change the inventory level. We can see as we successively move to higher levels (from left to right on the x-axis) that expected profit (blue line) will increase to a point of maximum – ¤16541 at a level of 7149 units:

Compared to the point of maximum profit for a single warehouse (profit ¤4963 at a level of 2729 units, see previous post), have this risk pooling increased the vendors profit by 11.1% while reducing his inventory by 12.7%. Centralization of the three inventories has thus been a successful operational hedge ((Risk pooling can be considered as a form of operational hedging. Operational hedging is risk mitigation using operational instruments.)) for our newsvendor by mitigating some, but not all, of the demand uncertainty.

Since this risk mitigation was a success the newsvendor wants to calculate the possible benefits from serving six newsboys at different locations from the same inventory.

Under the same assumptions, it turns out that this gives an even better result, with an increase in profit of almost 16% and at the same time reducing the inventory by 15%:

The inventory ‘centralization’ has then both increased profit and reduced inventory level compared to a strategy with inventories held at each location.

Centralizing inventory (inventory pooling) in a two-echelon supply chain may thus reduce costs and increase profits for the newsvendor carrying the inventory, but the individual newsboys may lose profits due to the pooling. On the other hand, the newsvendor will certainly lose profit if he allows the newsboys to decide the level of their own inventory and the centralized inventory.

One of the reasons behind this conflict of interests is that each of the newsvendor and newsboys will benefit one-sidedly from shifting the demand risk to another party even though the performance may suffer as a result (Kemahloğlu-Ziya, 2004) and (Anupindi and Bassok 1999).

In real life, the actual risk pooling effects would depend on the correlations between each locations demand. A positive correlation would reduce the effect while a negative correlation would increase the effects. If all locations were perfectly correlated (positive) the effect would be zero and a correlation coefficient of minus one would maximize the effects.

The third effect

The third direct effect of risk pooling is the reduced variability of expected profit. If we plot the profit variability, measured by its coefficient of variation (( The coefficient of variation is defined as the ratio of the standard deviation to the mean – also known as unitized risk.)) (CV) for the three sets of strategies discussed above; one single inventory (warehouse), three single inventories versus all three inventories centralized and six single inventories versus all six centralized.

The graph below depicts the situation. The three curves show the CV for corporate profit given the three alternatives and the vertical lines the point of profit for each alternative.

The angle of inclination for each curve shows the profits sensitivity for changes in the inventory level and the location each strategies impact on the predictability of realized profit.

A single warehouse strategy (blue) gives clearly a much less ability to predict future profit than the ‘six centralized warehouse’ (purple) while the ‘three centralized warehouse’ (green) fall somewhere in between:

So in addition to reduced costs and increased profits centralization, also gives a more predictable result, and lower sensitivity to inventory level and hence a greater leeway in the practical application of different policies for inventory planning.

Summary

We have thus shown through Monte-Carlo simulations, that the benefits of pooling will increase with the number of locations and that the benefits of risk pooling can be calculated without knowing the closed form ((In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a finite number of certain “well-known” functions.)) of the demand distribution.

Since we do not need the closed form of the demand distribution, we are not limited to low demand variability or the possibility of negative demand (Normal distributions etc.). Expanding the scope of analysis to include stochastic supply, supply disruptions, information sharing, localization of inventory etc. is natural extensions of this method ((We will return to some of these issues in later posts.)).

This opens for use of robust and efficient methods and techniques for solving problems in inventory management unrestricted by the form of the demand distribution and best of all, the results given as graphs will be more easily communicated to all parties than pure mathematical descriptions of the solutions.

References

Anupindi, R. & Bassok, Y. (1999). Centralization of stocks: Retailers vs. manufacturer. Management Science 45(2), 178-191. doi: 10.1287/mnsc.45.2.178, accessed 09/12/2012.

Chang, Pao-Long & Lin, C.-T. (1991). Centralized Effect on Expected Costs in a Multi-Location Newsboy Problem. Journal of the Operational Research Society of Japan, 34(1), 87–92.

Eppen,G.D. (1979). Effects of centralization on expected costs in a multi-location newsboy problem. Management Science, 25(5), 498–501.

Kemahlioğlu-Ziya, E. (2004). Formal methods of value sharing in supply chains. PhD thesis, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, July 2004. http://smartech.gatech.edu/bitstream/1853/4965/1/kemahlioglu ziya_eda_200407_phd.pdf, accessed 09/12/2012.

OESER, G. (2011). Methods of Risk Pooling in Business Logistics and Their Application. Europa-Universität Viadrina Frankfurt (Oder). URL: http://opus.kobv.de/euv/volltexte/2011/45, accessed 09/12/2012.

Endnotes

13/09/2012
Inventory management – Stochastic supply
This entry is part 4 of 4 in the series Predictive Analytics

Introduction

We will now return to the newsvendor who was facing a onetime purchasing decision; where to set the inventory level to maximize expected profit – given his knowledge of the demand distribution. It turned out that even if we did not know the closed form (( In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a finite number of certain “well-known” functions.)) of the demand distribution, we could find the inventory level that maximized profit and how this affected the vendor’s risk – assuming that his supply with certainty could be fixed to that level. But what if that is not the case? What if the supply his supply is uncertain? Can we still optimize his inventory level?

We will look at to slightly different cases:
1. one where supply is uniformly distributed, with actual delivery from 80% to 100% of his ordered volume and
2. the other where the supply have a triangular distribution, with actual delivery from 80% to 105% of his ordered volume, but with most likely delivery at 100%.
The demand distribution is as shown below (as before):

Maximizing profit – uniformly distributed supply

The figure below indicates what happens as we change the inventory level – given fixed supply (blue line). We can see as we successively move to higher inventory levels (from left to right on the x-axis) that expected profit will increase to a point of maximum.

If we let the actual delivery follow the uniform distribution described above, and successively changes the order point expected profit will follow the red line in the graph below. We can see that the new order point is to the right and further out on the inventory axis (order point). The vendor is forced to order more newspapers to ‘outweigh’ the supply uncertainty:

At the point of maximum profit the actual deliveries spans from 2300 to 2900 units with a mean close to the inventory level giving maximum profit for the fixed supply case:

The realized profits are as shown in the frequency graph below:

Average profit has to some extent been reduced compared with the non-stochastic supply case, but more important is the increase in profit variability. Measured by the quartile variation, this variability has increased by almost 13%, and this is mainly caused by an increased negative skewness – the down side has been raised.

Maximizing profit – triangular distributed supply

Again we compare the expected profit with delivery following the triangular distribution as described above (red line) with the expected profit created by known and fixed supply (blue line). We can see as we successively move to higher inventory levels (from left to right on the x-axis) that expected profits will increase to a point of maximum. However the order point for the stochastic supply is to the right and further out on the inventory axis than for the non-stochastic case:

The uncertain supply again forces the vendor to order more newspapers to ‘outweigh’ the supply uncertainty:

At the point of maximum profit the actual deliveries spans from 2250 to 2900 units with a mean again close to the inventory level giving maximum profit for the fixed supply case ((This is not necessarily true for other combinations of demand and supply distributions.)) .

The realized profits are as shown in the frequency graph below:

Average profit has somewhat been reduced compared with the non-stochastic supply case, but more important is the increase in profit variability. Measured by the quartile variation this variability has increased by 10%, and this is again mainly caused by an increased negative skewness – again have the down side been raised.

The introduction of uncertain supply has shown that profit can still be maximized however the profit will be reduced by increased costs both in lost sales and in excess inventory. But most important, profit variability will increase raising issues of possible other strategies.

Summary

We have shown through Monte-Carlo simulations, that the ‘order point’ when the actual delivered amount is uncertain can be calculated without knowing the closed form of the demand distribution. We actually do not need the closed form for the distribution describing delivery, only historic data for the supplier’s performance (reliability).

Since we do not need the closed form of the demand distribution or supply, we are not limited to such distributions, but can use historic data to describe the uncertainty as frequency distributions. Expanding the scope of analysis to include supply disruptions, localization of inventory etc. is thus a natural extension of this method.

This opens for use of robust and efficient methods and techniques for solving problems in inventory management unrestricted by the form of the demand distribution and best of all, the results given as graphs will be more easily communicated to all parties than pure mathematical descriptions of the solutions.

Average profit has to some extent been reduced compared with the non-stochastic supply case, but more important is the increase in profit variability. Measured by the quartile variation, this variability has increased by almost 13%, and this is mainly caused by an increased negative skewness – the down side has been raised.
23/09/2012