Introduction to Design of Experiments

1. Introduction to Design of Experiments Professional engineers are very frequently responsible for devising experiments to answer practical problems. There are well-tried methods for planning experiments that will provide the maximum reliable information. Several different factors may be present and may affect the results of experiments that are not readily controlled. “Interfering factors” affect the results but are not of prime interest. Also, the different factors may likely interact with each other in the sense that a higher value of one factor makes the results either more or less sensitive to another factor.

2. Experimentation vs. Use of Routine Operating Data • Routine production data often provides useful clues to desirable changes in operation conditions. However, these clues are often ambiguous due to multiple governing factors changing simultaneously in an unplanned pattern. • For example: the composition of crude oil fed to a refinery changes with increases and decreases of rates of flow from individual wells. In this case, an appreciable amount of time is required before steady, reliable data are obtained, during which time, further changes may occur. • An alternative, is to use: Planned Experiments

3. Scale of Experimentation • Experiments should be done on as small a scale as will provide the necessary data. • Managers in charge of full-scale industrial production units are often reluctant to allow experimentation concerning operating conditions in their units due to possible negative affects on products. These managers may be more agreeable to experimentation if a technique such as EVOP is used. EVOP is a series of small planned changes in operating conditions known as evolutionary operation. The changes in each step are small enough that serious consequences are avoided and the results are evaluated after each step in order to decide the most logical next step. • Experimentation can also be conducted on a laboratory scale at a very modest cost, or through the use of a pilot plant which has similar characteristics to the full scale plant but is considerably smaller.

4. One-factor-at-a-time vs. Factorial Design • One-factor-at-a-time: set up standard operating conditions for all factors, then vary conditions from the standard set, one factor at a time  Poor choice due to interdependency of governing factors and lack of efficiency. If governing factors are independent, then One-factor-at-a-time would be a reasonable plan, but inefficient, since the result of changing two factors would simply be the sum of the effects of changing each factor separately. • Frequently, factors interact  changing factor A causes the process to be more or less sensitive to any change in factor B. For example, the effect of increasing the temperature may be greater at higher pressures.

5. One-factor-at-a-time vs. Factorial Design • If there was no interaction between the temperature and pressure, the simplest model of the relation would be: where K is a constant, P is pressure, T is temperature, is the error for test i, and Ri is the response for test i. • For the situation where interaction is present between the temperature and pressure, the simplest model would be: where the interaction between temperature and pressure is second-order because it involves two independent variables (temp and pressure). If it involved three independent variables it would be third order.

6. One-factor-at-a-time vs. Factorial Design • Second order interactions are very common. The possibility for interaction between factors should always be considered; however, One-factor-at-a-time design does not give any precise information about interactions, and the results of this plan of experimentation may be extremely misleading. In order to determine the effects of interaction between factors, the effects of increasing one variable at different values of a second variable must be analyzed. • A good alternative to the One-at-a-time design is the Factorial Design, where tests are conducted at as many as possible combinations of the operating factors. For example, tests might be conducted at three different values (levels) of the first factor and two different levels of the second factor. Then measurements at each of the three levels of the first factor would be conducted at each of the two levels of the second factor  6 different tests.

7. One-factor-at-a-time vs. Factorial Design • When tests at combinations of different levels of factors are conducted, suitable algebraic manipulation of the data can be used to separate the results of changes in the various factors from one another. Multilinear regression and analysis of variance, to be explored later in this course, are tools often used to analyze the data. • In the early stages of industrial experimentation it is often best to choose only two levels for each factor varied in the factorial design. Further experiments can then be designed logically on the basis of results from the early experiments. When a full factorial design is conducted, experiments are done at every possible combination of the chosen levels for each factor. In other words, a factorial design should be sequential or evolutionary in nature. • If a full factorial design is not feasible  fractional factorial design to be discussed later

8. Replication • Replication  conducting each trial at the same set of conditions multiple times • The mean of repeated results for the same conditions has a standard deviation that becomes smaller as the number of replications increases. Therefore, a larger number of replications contributes to more reliable results. Also, as the sample size increases, the distribution of sample means more closely approximates the normal distribution.

9. Bias Due to Interfering Factors • Interfering factors (or lurking variables) are often present in industrial experiments. These factors may cause systematic error, unless suitable precautions are taken, that will bias the results. • Examples: • Temperature of surrounding air affecting temperature of a measurement • Unknown trace contaminant in the feed to a CSTR • Wear on machine to be tested

10. Bias Due to Interfering Factors Preventing Bias by Randomization  Random choices of assignment of material to different experiments and the order in which experiments are done • Interfering factors are “averaged out” and the biases are minimized • Random choices can be made using tables of random numbers • Randomization should always be used (whenever possible) if interfering factors may be present • If randomization is not practical, then some alternate scheme may be required to minimize the effects of interfering factors

11. Bias Due to Interfering Factors Preventing Bias by Blocking  dividing the complete experiment into groups or blocks in which the interfering factors are more homogeneous • Blocking increases the precision of an experiment by reducing interfering factors • A paired t-test is an example of experimental design using blocking • Randomization can still be used within each block to protect from unknown sources of bias • Block design does not give as much information as complete factorial design, but generally requires fewer tests • Blocking should be used when it is desired to eliminate the distortion caused by an interfering variable, but knowledge of the effects of that variable is not desired

12. Fractional Factorial Designs • As the number of factors increases, so does the number of experiments necessary for a full factorial design • A complete factorial design for n factors at two levels requires 2n measurements • In many cases, through careful design, fewer experiments can be conducted with the less important information being neglected • Half-fraction design is more useful when the number of factors is larger. A half-fraction factorial design gives essentially the same information as the full factorial design if: • At least one factor has negligible effect on any of the results • Only main effects and second-order interactions are appreciable One of these two criteria are often met. Therefore, the half-fraction design can often be conducted as a preliminary test to determine if the remaining tests for the full factorial design need to be completed