There seems to be disagreement on when to use fuzzy and when to use probability. An example in manufacturing and production of machined parts may be helpful. In manufacturing and production we use statistical process control (SPC) to maintain processes so that parts are produced within specified tolerances and will perform within specifications. However, we know from experience that performance varies from within that tolerance range. Those parts that perform the best and have the greatest reliability are those parts whose production parameters most closely match the nominal dimensions in the design tolerances. As a consequence using SPC we attempt to reduce the process variance and keep the process means centered on their corresponding nominal tolerance values. Although often confused, the tolerance values and the process values are not the same thing. The process values are characterized by distribution parameters determined empirically through collecting event data on the process results. The engineers who designed the product set the tolerances. When we generate a plan for manufacturing a product we match the available processes to the tolerance. So a tolerance is not a probabilistic specification. The nominal dimension is the location where full compliance with the design requirements exists. The upper and lower tolerances are the upper and lower limits beyond which the performance is unacceptable (zero existence). Within these tolerance boundaries, acceptable design compliance exists. However, this acceptable design compliance is not uniform. The closer to the nominal dimension the closer to full design compliance. Thus, a gradient exists between the upper and lower tolerances and the nominal dimension. Conceptually, this matches the description of a fuzzy set and these gradients can be modeled with fuzzy sets. In addition, the actual performance of any manufactured part has the same characteristics. Those parts whose parameters are closest to the nominal dimension will perform the best, those closest to the upper or lower tolerance will perform the worst. Think of the "Monday/Friday car" that no one wants. They are all within tolerance.
In moving a part to production we match the tolerances to a process by aligning its nominal dimension to a process mean and the upper and lower tolerance to their respective locations on the distribution. The probability mass outside these limits determines the scrap rate. The probability mass within these limits determines the "good" parts. In accepting or rejecting a part we make no distinction between parts within these limits and statistically we treat them the same. However, we recognize they are not the same by constantly trying to reduce the process variance while keeping the process mean centered on the nominal value. This is to produce more parts with parameters close to the nominal dimension because they will more fully match the design intent and have better performance. Thus, the process is probabilistic and we control it using SPC but the behavior of the design and the intent of the designer more closely matches the description of a fuzzy set. With this in mind, we use fuzzy sets to model tolerances and assess designs, and probability to model processes and assess the outcome of production. Hope this adds some insight. Robert Young ========================================================= Dr. Robert E. Young Professor of Industrial Engineering Department of Industrial Engineering, Campus box 7906 North Carolina State University Raleigh, NC 27695-7906 USA personal site: http://www.ie.ncsu.edu/young research site: http://www.ie.ncsu.edu/wisdem Phone: [+1] 919/515-7201 FAX: [+1] 919/515-5281 email: [EMAIL PROTECTED] =========================================================
