Six Sigma Case Study (DMAIC)

In a paper and accompanying model, we show how this opportunity is formulated as a Six Sigma project and showcase how Crystal Ball Professional Edition is used within each of the DMAIC (Define, Measure, Analyse, Improve, Control) process phases to deliver responsiveness to the customer and financial results to the business. 

As with most business situations of this nature, we are seeking the best (or most profitable) tradeoff between conflicting actions or strategies. This example specifically balances responsiveness to customer needs (higher levels of inventory given uncertain demand) against efficiency for the business (lower cost of goods sold). Of particular interest in seeking this optimal balance is how we can improve the solution and account for the effects of uncertainty in the key process input variables (KPIV’S). This method (stochastic optimisation) enables us to specify objective functions (Y’s) and requirements (constraints on X’s) in terms of likelihood of occurrence (i.e., maximize profit and provide at least 90% certainty that our total inventory costs would not exceed $169,000).

Background:

Fortune 1000 organisations have realised immense success by implementing the Define, Measure, Analyse, Improve, Control (DMAIC) methods made famous by Six Sigma practictioners. These practitioners can maximise their project results (e.g., cost savings and defect reduction) by incorporating real variability and uncertainty in their processes and into their process models or spreadsheets. 

Simulation, optimisation, and forecasting can be used throughout the DMAIC process. In this case study, a company that sold perishable inventory used Crystal Ball Professional Edition software within the Six Sigma process to create and maintain an optimal balance between lost sales and wasted inventory. In the recent past, this company all too often turned away customers due to unanticipated demand and a lack of inventory. This problem was identified and handed off to a Six Sigma project team for resolution. 

Define:

The first step in the Six Sigma project was to clearly define the defect and our project objectives. A defect was defined as any instance in which we turn a customer away because we do not have the materials to complete their order. In the past few years, these instances seem to have been happening with increasing frequency. We tracked the number of orders declined for the past ten years, and we saw an annual increase in the number of customers turned away. As you can see in Table 1, last year we turned away a record 1631 orders due to lack of inventory.

Using the historical sales data and CB Predictor, we forecast the number of lost sales given no change in our current processes. CB Predictor calculated an expected (mean) value of 2113 lost sales for this year (Figure 1). The output was saved as a normal probability distribution that described the uncertainty of the forecast.

We next used the CB Predictor results and Crystal Ball to forecast our losses due to inadequate inventory at $100 per lost sale. Crystal Ball created the forecast using Monte Carlo simulation, which used the probability distribution in Figure 1 to randomly select different possible values for the total number of lost sales. The results are shown in Figure 2.

Based on our forecast, we expected to lose $211,000 this year due to lost sales, with a minimum loss of $166,000 and a maximum loss of $258,000. While we wanted to reduce the cost of lost sales, we needed to balance this with our need to control the cost of perishable materials discarded due to expiration. Our project goal was to reduce the defect by 75% without increasing our inventory costs (cost of orders, cost of discarded materials, and cost of lost sales), which were $169,000 last year. In order for this project to be deemed a success, we needed 408 or less lost sales for the year and total inventory costs of no more than $169,000.

Measure:

In the Measure phase, we needed to develop and validate a process model. We built a spreadsheet in Excel (Figure 3) to describe our inventory costs.

We included weekly demand and lead time data for product delivery for the past year. We used these data points to develop custom distributions to describe the uncertainty for both variables. The probability distributions are shown in Figures 4 and 5.

We then applied these custom distributions to each week in our analysis to forecast our total cost, which we calculated as the sum of waste costs, order costs, and lost sales costs for the year. For the order quantity, reorder point, and initial inventory level, we used the values that applied to our current operating discipline (250, 250, 250). Without making any changes, our model indicated that we would lose between 1200 and 3100 sales due to lack of inventory (Figure 6). The mean value of lost sales was near 2100, which provided some validation of the model, since CB Predictor forecast the average number of lost sales this year at 2113.

The increased number of lost sales would be a major contributor to the 93% likelihood that our total inventory costs would exceed last year’s value of $169,000 (Figure 7).

Analyse

With the Define and Measure phases completed, we needed to determine the major controllable causes of lost sales. The Crystal Ball Tornado Chart tool allows the user to compare the impact of variability of multiple inputs on the output of the model. In this case, we used the Tornado Chart tool to determine which variables had the greatest influence on the Total Lost Sales. We found that the Reorder Point, Order Quantity, and Initial Inventory were most influential on the number of lost sales (Figure 8). Fortunately, these variables were within our control.

Improve

In the Improve phase, we defined Reorder Point, Order Quantity, and Initial Inventory as decision variables, which are model variables that we can control. OptQuest, the optimiser program available in Crystal Ball Professional Edition, uses a collection of algorithims to manipulate the values of the decision variables in order to minimize or maximise the objective function.  

We used OptQuest to find the optimal values for each of the three decision variables in order to minimise our expected (mean) lost sales. We also included a requirement that the optimal solution would provide at least 90% certainty that our total inventory costs would not exceed $169,000. Figure 9 is the Forecast Chart for Total Lost Sales after changing the decision variables to their optimal values. By changing these variables to an order quantity of 300, a reorder point of 1400, and an initial inventory of 700, our model indicated that we could reduce the expected number of lost sales to less than 20, with a 100% chance of reaching our goal of 408 or less lost sales!

This solution would also reduce our likelihood of exceeding last year’s inventory costs from 93% to 7% (Figure 10).

Using this knowledge, we turned to the question of whether we might be able to reach our goal of 75% defect reduction and achieve even more substantial gains in terms of reduced costs. We re-ran OptQuest, this time with the objective of minimising the expected (mean) total cost subject to the following requirements: (1) 90% certainty that the number of lost sales would not exceed our target value of 408, and (2) 90% certainty that total costs would not exceed last year’s value of $169,000.  

This optimisation gave us a new solution of an order quantity 200, a reorder point 800, and an initial inventory 400 (Figure 11). With this new solution, the expected number of lost sales increased to 65 (Figure 12), but we maintained 100% certainty of reaching our goal of 408 or less lost sales (assuming, of course, that our assumptions were correct).

On the upside, this solution virtually eliminated the possibility of any increase in our total inventory costs. With this solution, the total expected inventory cost was only $17,000, a $162,000 reduction from last year’s costs (Figure 13).

We found this solution to be optimum for our business, so we implemented it.

Control

In the final phase of DMAIC, we needed to develop a plan to insure that our anticipated gains were achieved and maintained. We used Crystal Ball to forecast the 90th percentile of the total inventory costs for each week (Figure 14), and we compared our total inventory costs to the P90 values. In future weeks, if the actual costs exceed the P90 value from our forecast, we can incorporate the latest demand and delivery lead times into the assumptions and re-run the optimisation to determine if the optimum reorder point or order quantity have changed.

Conclusion:

The Importance of Using Stochastic Methods If we were not using Crystal Ball Professional Edition, we still could have completed this project to reduce lost sales and minimise our total inventory costs. By excluding the variability in demand and lead delivery time and using the average values of 149 units and 2 weeks, we would have come up with an optimal solution of order quantity 300, reorder point 400, and initial inventory 500.  

However, when we included the actual, known variability in the model, and the results of the two methods are compared, we find that 99.6% of the time the stochastic solution results in lower costs than the deterministic solution (Figure 15). The value in using the stochastic method is, on average, over $85,000 per year for this project (Figure 16). How much difference would it make in YOUR business?

For more information on how Crystal Ball is used for Six Sigma, visit the Crystal Ball Web site