Role of Uncertainty-Based Models in Decision Making and Design

Singiresu S. Rao
Professor
School of Mechanical Engineering
Purdue University
West Lafayette, IN 47907-1288

Most design activities involve decision making in terms of selecting the concept, configuration, materials, geometry, and conditions of operation. The information and data necessary for decision making is known with different degrees of confidence at different stages of design. For example, at the preliminary or conceptual design stage, very little information is known about the system. However, as we progress towards the final design, more and more data will be known about the system and its behavior. Thus the ability to handle different types of uncertainty in decision making becomes extremely important. The probability theory, fuzzy theory, Dempster-Shafer theory, and a combination of fuzzy and random theories can be used for handling the various types of uncertainties encountered in the design process.

1. Probability theory:

In many design problems, the component geometry (due to machine limitations and tolerances), material strength (due to variations in manufacturing processes and chemical composition of materials) and loads (due to component wearout, unbalances and uncertain external effects) are to be treated as random variables with known mean and variability characteristics. The resulting design procedure is known as reliability-based design. The reliability-based design is recognized as a more rational procedure compared to the traditional factor of safety-based design methods.

2. Fuzzy theory:

In certain situations, it may not be possible to describe the uncertainty present using probability distributions. For example, design information such as, "The system is to be designed such that the natural frequency is substantially away from the forcing frequency" and "The fiber content of the composite beam is very low", cannot be described using probability principles. In such cases, the designer can specify relative preferences in terms of membership functions and use fuzzy arithmetic and fuzzy calculus methods in decision making.

3. Dempster-Shafer theory:

The Dempster-Shafer theory can be used to represent situations in which different kinds of ignorance exist in our knowledge about a phenomenon or a system. The method uses belief functions which are based on a basic probability assignment, which is a measure of the belief committed exactly to a given hypothesis or a subset of the frame of descernment. The theory has been applied recently to several decision making problems such as the selection of bearings for a specific application and for mulpiple objective optimization of engineering systems. Since most practical design problems are solved iteratively, the computational information accumulated in each iteration can be used to develop an evidence-based model using Dempster-Shafer theory for solving design problems more efficiently.

4. Combined Uncertainty models:

Many realistic design situations involve different types of uncertainties, which can be classified as subjective, objective and hybrid type of uncertainties. For example, stochastic uncertainty is associated with the geometry or material properties of the system (based on experimental measurements). At the same time, design imprecision may be involved in the subjective selection of the parameter values among design alternatives based on personal knowledge and desirability. This requires the development of a combined uncertainty-based decision making model using probability and fuzzy theories.

References

1. S. S. Rao, "Reliability-Based Design," McGraw-Hill, New York, 1992.

2. S. S. Rao, "Multiobjective Optimization of Structural Design with Uncertain Parameters and Stochastic Processes," AIAA Journal, Vol. 22, pp. 1670-1678, 1984.

3. S. S. Rao and J. P. Sawyer, "A Fuzzy Finite Element Approach for the Analysis of Imprecisely-Defined Systems," AIAA Journal, Vol. 33, pp. 2364-2370, 1995.

4. A. C. Butler, F. Sadeghi, S. S. Rao and S. R. LeClair, "Computer-Aided Design/Engineering of Bearing Systems using the Dempster-Shafer Theory," Artificial Intelligence for Engineering Design, Analysis and Manufacturing, Vol. 9, 00. 1-11, 1995.

5. L. Chen and S. S. Rao, "Determination of Optimal Machining Conditions: A Coupled Unceretainty Model," ASME Journal of Manufacturing Science and Engineering, 1997 (in press).