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Putting all of these things together, the ideal scenario becomes: a customer orders an item or items (pull), the factory has the resources to produce the order (level loading), processes are configured to create the items ordered (flexible process), the order is produced and delivered to the customer exactly when he needs it .

Using Empirical Model Building to Optimize Empirical model building is a statistical approach for determining optimal process or design settings. It uses a series of experimental designs to reduce the total possible pro­cess or product design space to hone in on the optimal settings with regard to one or more requirements.

If you are new to design of experiments and empirical model building, a metaphor may prove helpful. Imagine that you suddenly wake up in a strange wilderness. You don' t know where you are, but you'd like to climb to the top of the nearest hill to see if there are any signs of civilization. What would you do?

A first step might be to take a good look around you. Is there anything you should know before starting out? You would probably pay particular attention to things that might be dangerous. If you are in a jungle these might be dangerous animals, quick­sand, and other things to avoid. You'd also look for things that could be used for basic survival, such as food, shelter, and clothing. You may wish to establish a "base camp" where you can be ensured that all the basic necessities are available; a safe place to return to if things get a bit too exciting. In empirical modeling we also need to begin by becoming oriented with the way things are before we proceed to change them. We will call this knowledge discovery activity Phase O.

Now that you have a feel for your current situation and you feel confident that you know something about where you are, you may begin planning your trip to the highest hill. Before starting out you will probably try to determine what you will need to make the trip . You are only interested in things that are truly important. However, since you are new to jungle travel, you decide to make a few short trips to be sure that you have what you need. For your first trip you pack up every conceivable item and set out. In all likelihood you will discover that you have more than you need. Those things that are not important you will leave at your camp. As part of your short excursions you also learn something about the local terrain close to your camp; not much, of course, but enough to identify which direction is uphill. This phase is equivalent to a screening experiment, which we call Phase I.

You now feel that you are ready to begin your journey. You take only those things you will need and head out into the jungle in the uphill direction. From time to time you stop to get your bearings and to be sure that you are still moving in the right direction. We call this hill-climbing steepest ascent, or Phase II.

At some point you notice that you are no longer moving uphill. You realize that this doesn' t mean that you are at the highest point in your area of the jungle, only that you are no longer moving in the right direction. You decide to stop and make camp. The next morning you begin to explore the local area more carefully, making a few short excursions from your camp. The jungle is dense and you learn that the terrain in the immediate vicinity is irregular, sometimes steep, sometimes less steep. This is in con­trast to the smooth and consistent uphill slope you were on during your ascent. We call this phase of your journey the factorial experiment, or Phase III.

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Now you decide that a more organized approach will be needed to locate the nearby peak. You break out the heavy artillery, the CPS you've been carrying since the beginning! (one of those cheap ones that don't have built-in maps). You take several altitude read­ings from near your camp, and others at a carefully measured distance on all major compass headings. Each time you carefully record the altitude on a hand-drawn map. You use the map to draw contour lines of equal altitude and eventually a picture emerges that clearly shows the location of the top of the hilL This is the composite design phase, which we call Phase IV.

At last you reach the top of the hilL You climb to the top of a tree and are rewarded with a spectacular view, the best for miles around. You decide that you love the view so much, you will build your home on this hill and live there permanently. You make your home sturdy and strong, able to withstand the ravages of wind and weather that are sure to come to your little comer of the jungle. In other words, your home design is robust, or impervious to changes in its environment. We call the activity of building products and processes that are insensitive to changes in their operating parameters robust product and process design, which is Phase V of the journey.

Now that this little tale has been told, let's go on to the real thing, improving your products, processes, and services.

Phase 0: Getting Your Bearings

"Where Are We Anyway?" Before any experimentation can begin the team should get an idea of what the major problems are, important measures of performance, costs, time and other resources available for experimentation, etc. Methods and techniques for conducting Phase 0 research are described in Chap. 10.

The central premise of the approach described in this section is that learning is, by its very nature, a sequential process. The experimenter, be it an individual or a team, begins with relatively little specific knowledge and proceeds to gain knowledge by conducting experiments on the process. As new knowledge is acquired, the learner is better able to determine which step is most appropriate to take next. In other words, experimentation always involves guesswork; but guesses become more educated as experimental data become available for analysis.

This approach is in contrast to the classical approach where an effort is made to answer all conceivably relevant questions in one large experiment. The classical approach to experimentation was developed primarily for agricultural experiments. Six Sigma applications are unlike agricultural applications in many ways, especially in that results become available quickly. The approach described here takes advantage of this to accelerate and direct learning.

We will use an example from electronic manufacturing. At the outset, a team of personnel involved in a soldering process received a mission from another team that had been evaluating problems for the factory as a whole. The factory team had learned that a leading reason for customer returns was solder problems. Another team discov­ered that the solder area spent more resources in terms of floor space than other areas; a major usage of floor space was for the storage of defective circuit boards and the repair of solder defects. Thus, the solder process improvement team was formed and asked to find ways to eliminate solder defects if possible, or to at least reduce them by a factor of 10. Team members included a Six Sigma technical leader, a process engineer, an inspector, a production operator, and a product engineer.

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The team spent several meetings reviewing Pareto charts and problem reports. It also performed a process audit which uncovered several obvious problems. When the problems were repaired the team conducted a process capability study, which revealed a number of special causes of variation, which were investigated and corrected. Over a 4-month period, this preliminary work resulted in a 50% reduction in the number of solder defects, from about 160 defects per standard unit to the 70 to 80 defect range. The productivity of the solder area nearly doubled as a result of these efforts. While impres­sive, the results were still well short of the 10xminimum improvement the team was asked to deliver.

Phase I: The Screening Experiment

"What's Important Here?" At this point the process was stable and the team was ready to move from the process control stage to the process improvement stage. This involved conducting designed experiments to measure important effects. The solder team decided to list as many items as possible that might be causing solder problems. Since many variables had already been studied as part of the Phase 0 work, the list was not unreasonably long. The team looked at ways to control the variables listed and was able to develop meth­ods for eliminating the effects of many variables on their list. The remaining list included the following factors:

Variable Low Level (-) High Level (+)

A: Prebaking of boards in an oven No Yes

B: Preheat time 10 s 20 s

C: Preheat temperature 150°F 200°F

D: Distance from preheat element to board 25 cm 50 cm surface

E: Line speed 3 fpm 5 fpm

F: Solder temperature 495°F 505°F

G: Circuit density Low High

H: Was the board in a fixture? No Yes

This information was used to create an experimental design using a statistical soft­ware package. There are many packages on the market that perform similar analyses to the one shown here.

Since this is only to be a screening experiment, the team was not interested in obtaining estimates of factor interactions. The focus was to identify important main effects. The software allows selection from among several designs. The Black Belt decided upon the design which would estimate the main effects with the smallest number of test units. This design involved testing 16 units. The data matrix pro­duced by the computer is shown in Table 11.3. The run order has been randomized by the computer. If the experiment cannot be conducted in that particular order, the computer software would allow the data to be run in blocks and it would adjust the analysis accordingly. The program also tells us that the design is of resolution IV,

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Run A B C D E F G H Response

1 + - - - - + + + 65

2 + - + + - + - - 85

3 + + - - + + - - 58

4 - + - - + - + + 57

5 - - - - - - - - 63

6 + + + + + + + + 75

7 - + - + - + + - 77

8 - + + - - + - + 60

9 + - + - + - - + 67

10 + + + - - - + - 56

11 - - + - + + + - 63

12 - - - + + + - + 81

13 + + - + - - - + 73

14 + - - + + - + - 87

15 - + + + + - - - 75

16 - - + + - - + + 84

TABLE 11.3 Screening Experiment Layout. Data matrix (Randomized)

which means that main effects are not confounded with each other or any two-factor interactions.

In Table 11.3 the" _If indicates that the variable is run at its low level, while a "+" sign indicates that it is to be run at its high level. For example, the unit for run #16 was processed as follows:

• Prebaking = No

• Preheat time = 10 sec

• Preheat temperature = 200°F

• Distance from preheat element to board surface = 50 cm

• Line speed = 3 £pm

• Solder temperature = 495°F

• Circuit density = High

• Fixture used = Yes

• Defects per standard unit = 84

Experimental data were collected using the randomized run order recommended by the software. The "response" column are data that were recorded in terms of defective solder joints per "standard unit," where a standard unit represented a

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Estimated effects and coefficients for response (coded units)

Term Effect Coef StDev coef T P

Constant 70.375 0.6597 106.67 0.000

A -0.750 -0.375 0.6597 -0.57 0.588

B 8.000 4.000 0.6597 6.06 0.001

C -0.500 -0.250 0.6597 -0.38 0.716

D -18.500 -9.250 0.6597 -14.02 0.000

E 0.000 0.000 0.6597 0.00 1.000

F -0.250 -0.125 0.6597 -0.19 0.855

G -0.250 -0.125 0.6597 -0.19 0.855

H 0.250 0.125 0.6597 0.19 0.855

ANOVA for defects (coded units)

Source of variation df Seq.SS Adj. SS Adj. MS F P-value

Main effects 8 1629.00 1629.00 203.625 29.24 0.000

Residual error 7 48.75 48.75 6.964

Total 15 1677.75

TABLE 11.4 Results of Experimental Data Analysis. Fractional Factorial Fit

circuit board with a median number of solder joints. * The results are shown in Table 11.4.

A model that fits the data well would produce residuals that fall along a straight line. The Black Belt concluded that the fit of the model was adequate.

The analysis indicates that factors B (preheat time) and D (distance from preheat element to board surface) produce significant effects. Figure 11.4 shows a normal prob­ability plot of the experimental effects. This figure plots the coefficients column from Table 11.4 on a normal probability scale. If the factor's effect was due to chance varia­tion it would plot close to the line representing normal variation. In Fig. 11.5 the effects of Band D are shown to be further from the line than can be accounted for by random variation.

The effects of the significant factors are graphed in response units in Fig. 11.5. Since the response is a defect count, the graph indicates that the low level of factor

D gives better results, while the high level of factor B gives the better results. This can also be seen by examination of the coefficients for the variables. When D is low the

*Technically, a Poisson model would be the correct choice here. However, use of a normal model, which the analysis assumes, is reasonably accurate for defect counts of this magnitude. The team also evaluated the variance, more specifically, the log of the variance. The variances at each factor combination did not differ significantly and are not shown here.

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1.5

1.0

~ 0.5 () C/)

Cii 0.0 E (; z -0.5

-1 .0

-1 .5

-1

Normal probability plot of the residuals (response is defects)

o Standardized residual

FIGURE 11.4 Residuals from experimental model.

1.5

1.0

OJ 0.5 (; () C/)

Cii 0.0 E (;

-0.5 z

-1 .0

-1 .5

-15

Normal probability plot of the standardized effects (Response is defects, alpha = 0.10)

-8

-10 -5 o 5

Standardized effect

FIGURE 11.5 Significant factor effects.

2

average defect rate is 18.5 defects per unit better than when D is high; when B is high the average defect rate is 8 defects per unit better than when B is low.

The team met to discuss these results. They decided to set all factors that were not found to be statistically significant to the levels that cost the least to operate, and factors B and D at their midpoints. The process would be monitored at these settings for a while to determine that the results were similar to what the team expected based on the experi­mental analysis. While this was done, another series of experiments would be planned to further explore the significant effects uncovered by the screening experiment.