The catapult or trebuchet model is a classic example used to teach Design of Experiments. It illustrates Monte Carlo simulation and tolerance analysis.
Suppose you are manufacturing catapults and customers demand the distance the catapult throws a standard ball is 25 meters, plus or minus 1 meter. There are many design specifications involved in producing your catapults, such as:
- Angle of Launch
- Mass of the Ball
- Distance Pulled
- Spring Constant
Each of the design factors contains an @RISK probability distribution to represent different possible values each factor could take. @RISK probability distributions can be entered directly as formulas or by using the Define Distribution icon on the @RISK toolbar. For example, a Uniform distribution represents the possible values for Distance Pulled.
The output is Distance Thrown, and contains a RiskSixSigma property function defining Lower Specification Limit, Upper Specification Limit, and Target for Distance Thrown. Like inputs, an @RISK output can be typed into the formula bar or defined via dialog box using the Add Output button on the @RISK toolbar.
Capability metrics Cpk, Cpk Upper, Cpk Lower, Sigma Level, and DPM are calculated for the catapult, enabling you to determine whether it is ready for production.
The resulting distribution of Distance Thrown shows that about 60% of the time the distance is outside of specification limits.
Sensitivity analysis identifies the most important design factors affecting Distance Thrown as the Distance Pulled, followed by the Mass of the Ball.
This model can help explore the theory of Taguchi or Robust Parameter Design. Taguchi theory states that there are two types of variables which define a system – those whose levels affect the process variation, and those whose levels do not. The idea behind Taguchi Design is to set variables of the first type at a level which minimizes total process variation. Variables which don’t affect process variation are used to control and/or adjust the process.
In the catapult model, you can adjust various design parameters – such as Pull Distance and Mass of Ball – to try to minimize the variation in the output Distance Thrown. Considering that 60% of the time the Distance Thrown is outside the specification limits of 24 to 26 meters, there is room for improvement.
Minimum Edition: @RISK Industrial. This model analyzes a metallic burst cup manufactured by welding a disk onto a ring. The model relates the weld strength to process and design factors, models the variation for each factor, and forecasts the product performance in relation to the engineering specifications. An engineer could attempt to reduce or better control the variation within the weld time and amplitude limits. Alternatively, RISKOptimizer can be used to find the optimal process and design targets to minimize the annual cost from defectives.
Suppose you are analyzing a metallic burst cup manufactured by welding a disk onto a ring (see below). The product functions as a seal and a safety device, so it must hold pressure in normal use, and it must separate if the internal pressure exceeds the safety limit.
The model relates the weld strength to process and design factors, models the variation for each factor, and forecasts the product performance in relation to the engineering specifications. Modeling a response based on multiple factors can often be accomplished by generating a statistically significant function through experimental design or multiple regression analysis.
In this example, @RISK simulates the variation using Normal distributions for each factor. @RISK distributions support cell referencing so that you can easily set-up a tabular model that can be updated throughout a product and process development lifecycle.
The uncertain factors are:
- Disk thickness
- Horn wall thickness
- Horn length
- Weld pressure
- Weld time
- Trigger point
Adding a distribution to each factor is as easy as clicking on the Define Distribution icon on the @RISK toolbar. From there you can select a Normal distribution and input its parameters or cell references, as shown below. You could also type the formula directly into Excel’s formula bar for each input. For example, the cell for Well Pressure contains the formula
The output is Weld Strength (N) in the Design & Process Performance section, and contains a RiskSixSigma property function that includes the Lower Specification Limit (LSL), Upper Specification Limit (USL), and Target value specified. As with defining input distributions, you can type the output formula directly in the output cell or use the Add Output dialog. The formula would be:
=RiskOutput("Weld Strength (N)",,,RiskSixSigma(D82,E82,105,0,1))+ [the mathematical calculation]
After you run the simulation, Six Sigma statistics were generated using @RISK Six Sigma functions for Cpk-Upper, Cpk-Lower, Cpk, and PPM Defects (or DPM). Standard @RISK statistics functions (like RiskMean) were also used.
The @RISK output distribution displays the expected performance based on the design and process input variation and shows LSL, USL, and Target value with markers. You can easily access the output statistics using the reporting features or through @RISK functions.
The @RISK Sensitivity Analysis clearly shows that the Weld Time and Amplitude parameters are driving the Weld Strength variation.
The next steps for this problem could include two options: The engineer can attempt to reduce or better control the variation within the Weld Time and Amplitude, or use RISKOptimizer to find the optimal process and design targets to maximize yield or reduce scrap cost.
This model demonstrates the use of RISKOptimizer in experimental design. RISKOptimizer combines Monte Carlo simulation with genetic algorithm-based optimization. Using these two techniques, RISKOptimizer is uniquely capable of solving complex optimization problems that involve uncertainty.
With RISKOptimizer, you can choose to maximize, minimize, or approach a target value for any given output in your model. RISKOptimizer tries many different combinations of controllable inputs that you specify in an effort to reach its goal. Each combination is called a “solution,” and the total group of solutions tried is called the “population.” “Mutation” refers to the process of randomly trying new solutions unrelated to previous trials. You can also set constraints that RISKOptimizer must abide by during the optimization.
For uncertain, uncontrollable factors in your model, you define @RISK probability distribution functions. For each trial combination of inputs, RISKOptimizer also runs a Monte Carlo simulation, sampling from those @RISK functions and recording the output for that particular trial. RISKOptimizer can run thousands of trials to get you the best possible answer. By accounting for uncertainty, RISKOptimizer is far more accurate than standard optimization programs.
In this example, as above, the part under investigation is a metallic burst cup manufactured by welding a disk onto a ring. The product functions as a seal and a safety device, so it must hold pressure in normal use, and it must separate if the internal pressure exceeds the safety limit.
The model relates the weld strength to process and design factors, models the variation for each factor, and forecasts the product performance. RISKOptimizer was used to search for the optimal combination of process settings and nominal design values to minimize scrap cost, called Annual Defect Cost in the model. This is the same as maximizing yield.
The process and design variables RISKOptimizer will adjust are:
- Disk thickness
- Horn wall thickness
- Horn length
- Weld pressure
- Weld time
- Trigger point
All in an effort to minimize the output Annual Defect Cost. Clicking on RISKOptimizer’s Model Definition icon lets you define which cells to adjust, what your output is, and what constraints to use. In addition to the inputs and outputs described above, we will also define a constraint where the Trigger Point must always be less than or equal to Weld Time.
When you click Start Optimization, the RISKOptimizer Progress window appears, showing you a summary status of the analysis.
After simulation and optimization, RISKOptimizer efficiently found a solution that reduced the Annual Defect Cost to under $8,000.
Using RISKOptimizer can save time and resources in a quality improvement and cost reduction effort. The next steps for this problem would be to validate the model and optimized solution through experimentation.
This simple DC circuit consists of two voltage sources - one independent and one dependent - and two resistors. The independent source specified by the design engineer has an operational power range of 5,550 W + 300 W. If the power draw on the independent voltage source is outside of the specification, the circuit will be defective. The design performance results clearly indicate that the design is not capable of performance with a percentage of the circuits failing on both the high and low end of the limits. The PNC values identify the Percent of Nonconforming units expected on the upper and lower ends of the specification.
The basic model logic follows:
The model calculates the standard deviation for each component based on known information and the following assumptions within this model: 1) The mean of the component values are centered within the tolerance limits. 2) The component values are normally distributed. Note that @RISK can be used to fit a probability distribution to a data set or to model other types of probability distributions, if needed.
A RiskSixSigma property function in the Output cell PowerDEP defines Upper Limit, Lower Limit, and Target that are used for Six Sigma results calculations. @RISK Six Sigma functions are used to calculate Cpk Lower, Cpk Upper, Cpk, Cp, DPM, PNC Upper and PNC Lower.
The @RISK Sensitivity Analysis identifies the input variables driving variation in the output. The sensitivity shows that the two voltage sources are the main contributors to the variation in power consumption. Armed this information, the engineering team can focus their improvement efforts on the voltage sources instead of the resistors.
The model can be used to test different components and tolerances, performances and yields can be compared, and the optimal solution can be selected to maximize yield and reduce cost.
This model represents the process flow of a company's internal sales quotation process. The process is taken from an actual company and has over 36 individual steps involving 10 individuals or departments. For example, when management saw from the simulation results that it took over 24 hours to complete 35 minutes of value-added work, they saw the need for process improvement.
Minimum Edition: @RISK Professional. This model is of a manufacturer that needs to reduce the number of defective units produced. It uses @RISK to pinpoint the manufacturing stage that is the worst offender. It also obtains key process capability metrics for each stage, as well as for the entire process, that will help to improve quality and reduce waste. Given historical data, it also uses @RISK's distribution fitting feature to define distribution functions describing the number of defective parts at each stage of the process.
This model is used to calculate the percentage of defective products. Each product component is nondefective if some property of its finished state lies within defined tolerance limits. The project component cells are designated as @RISK outputs with RiskSixSigma property functions defining LSL, USL, and Target value. In this way, you can see graphs of the components' quality and calculate Six Sigma statistics on each component. .
This is an extention of the DMAIC (Define Measure, Analyze, Improve, Control) failure model for use in quality control and planning. It includes the use of RiskTheo functions (in this case RiskTheoXtoP) for determining the failure rate without actually running a simulation. The model also illustrates @RISK outputs with RiskSixSigma property functions defining LSL, USL, and Target values for each component.