Example 12.2

Operations Models

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Background Information Suppose that you are in the business of drilling a tiny

hole in the exact center of a square piece of wood.

In the past, the holes you have drilled were, on average, in the center of the wood, and the x- and y-coordinates each had a standard deviation of 0.1 inch.

Also, the drilling process has been stable – that is, the holes average being in the center of the square, and the deviations from the center of the square follow a normal distribution with mean 0 and standard deviation 0.1 inch.

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Background Information -- continued This mean, for example, that the x-coordinate is

within 0.1 inch 95% of the holes, and the x-coordinate is with 0.3 inch of the center for 99.7% of the holes.

This describes the inherent variability in the drilling process.

Without changing the hole-drilling process, you must live with this amount of variation.

Now suppose that you drill a hole and its x- and y- coordinates are x=0.1 and y=0.

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Background Information -- continued A natural reaction is to reduce the x-setting of the drill

by 0.1 to correct for the fact that the x-coordinate was too high.

Then if the next hole has coordinates x = -0.2 and y = 0.1, you might try to increase the x-coordinate by 0.2 and decrease the y-coordinate by 0.1.

Deming’s funnel experiment shows that this method of continually readjusting a stable process – he calls it “tampering” – will actually increase the variability of the coordinates of the position where the hole is drilled. In other words, tampering will generally make a process worse!

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Background Information -- continued To illustrate the effects of tampering, Deming placed

a funnel above a target on the floor and dropped small balls through the funnel in an attempt to hit the target.

As he demonstrated, many balls did not hit the target. His goal, therefore, was to make the balls fall as close to the target as possible.

Deming proposed four rules for adjusting the positioning of the funnel.

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Deming’s Rules

1. Never move the funnel

2. After each ball is dropped, move the funnel – relative to its previous position – to compensate for any error. To illustrate, suppose the target has coordinates (0,0) and the funnel begins directly over the target. If the ball lands at (0.5,.1) on the first drop, we compensate by repositioning the funnel at (0-0.5,0-0.1) = (-0.5,-1). If the second drop has coordinates (1,-2), we then reposition the funnel at (-0.5 – 1, -0.1 – (-2)) = (-1.5, 1.9).

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Deming’s Rules -- continued

3. Move the funnel – relative to its original position at (0,0) – to compensate for any error. For example, if the ball lands at (0.5,0.1) on the first drop, we compensate by repositioning the funnel at (0-0.5, 0-0.1) = (-0.5, -1). If the second drop has coordinates (1, -2), we then reposition the funnel at (0, -1, 0 – (-2)) = (-1,2).

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Deming’s Rules -- continued4. Always reposition the funnel directly over the last

drop. Therefore, if the first ball lands at (0.5,1), we position the funnel, (0.5, 1). If the second drop has coordinates (1, 2), we position the funnel at (1, 2). This rule might be followed, for example, by an automobile manufacturer’s painting department. With each new batch of paint, they attempt to match the color of the previous batch – regardless of whether the previous color was “correct”.

Do you believe any of the latter three rules will outperform rule 1, the “leave it alone” rule? If so, read on – you might be surprised.

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Solution To see how these rules work, we assume that the x-

coordinate on each drop is normally distributed with mean equal to the x-coordinate of the funnel position and standard deviation of 1.

A similar statement holds for the y-coordinate. Also, we assume that the x- and y- coordinates are selected independently of one another. These assumptions describe the inherent variability in the process of dropping the balls.

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Solution -- continued

To see how the rules work, let F0, X0, F1 be respectively, the x-coordinates of the funnel position on the previous drop, the outcome of the previous drop, and the repositioned funnel position for the next drop.

Then rule 1 never repositions, so that F1 = F0. Rule 2 repositions relative to the previous funnel position, so that F1 = F0 – X0. Rule 3 repositions relative to the original position (at 0), so that F1 = 0 – X0 = -X0. Finally rule 4 repositions at the previous drop, so that F1 = X0. Similar questions hold for the y-coordinate.

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Solution -- continued

For the simulation model, we simulate 50 consecutive drops of the ball from each of the four rules.

Our single output measure is the (straight-line) distance of the final drop from the target.

A rule is presumably a good one if the mean distance is small and the standard deviation of this distance is also small.

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FUNNEL.XLS Given the repositioning equations for the rules, the

simulation model is straightforward.

In fact, we use a RISKSIMTABLE function to test all four rules simultaneously.

The spreadsheet model appears on the next slide.

This file contains the model.

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Developing the Simulation Model The model can be developed with the following steps.

– Rule. Enter the formula =RISKSIMTABLE({1,2,3,4}) in cell B3 to indicate that we want to simulate all four rules. Note that if individual values are listed in RISKSIMTABLE, they must be enclosed in curly brackets. No curly brackets should be used if the list is referenced by a range.

– Position funnel. Enter 0 in cells B7 and C7 to indicate that the original funnel position is above the target at (0,0). Then implement the positioning equations by entering the formula =IF(Rule=1,B7,IF(Rule=2,B7-D7,IF(Rule=3,-D7,D7))) in cell B8 and copying it to the range B8:C56. Note how this formula references the location of the previous drop. The IF function captures the logic for all four rules.

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Developing the Simulation Model -- continued

– Simulate drops Simulate the positions of the drops by entering the formula =RISKNORMAL(B7,1) in cell D7 and copying it to the range D7:E56. This says that the ball’s drop position is normally distributed with mean equal to the funnel’s position and standard deviation 1.

– Distance. Calculate the final distance from the target in cell C58 with the formula =RISKOUTPUT( ) +SQRT(SUMSQ(D56:E56)). Here we have used SUMSQ function to get the sum of squares for the distance formula. We have also indicated that this is an output cell for @Risk.

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Using @RISK

We set the number of iterations to 1000 and the number of simulations to 4.

Selected summary measures for the final distance from the target for all four rules appears in the table shown here.

We also show histograms of this distance for rule, 1, 2, 3 on the next three slides.

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Using @RISK -- continued These results prove Deming’s point about tampering.

Rule 2 might not appear to be too much worse than rule 1, but its mean distance and standard deviation of distances are both about 40% higher than rule 1.

Rules 3 and 4 are disastrous. Their mean distances are more than seven times higher than for rule 1, and their standard deviations are also much higher.

The moral of the story, as Deming preached, is that you should not tamper with a stable process. If the process is not behaving as desired, then fundamental changes to the process are required, not a lot of tinkering.