Deceptively simple??

[pawlowski6132]

OK, I'be been pounding my head against the wall all morning to trying to figure this out.

This is a real world problem.

1. I have an equation such that X divided by Y = Z.
2. Then, X and Y change at the same time.
3. The difference in Z between Original and Revised in this example is 18,800.
4. I'd like to explain or "break down" the 18,800 in terms of X and Y. "I.e, Of the 18,800, ____ was due to change in X and ____ was due to change in Y."
5. I thought I could understand the impact from each variable by changing one at a time and then add their impact but, it's not equal to changing both at once.
6. How can I quantify the "contribution" that X and Y made discreetly to 18,800??????

 

[tkhunny]

You must also define the change due to the interaction of x and y.

 

[pawlowski6132]

Hmm. Sorry, I'm not sure how to do that. Can you work it out here and provide a final solution?

In the mean time, I got a little further using a different method:

1. Change in X at original Y

Change in X. Original Y
12,360 at .3	41,200
15,000 at .3	50,000
Diff	8,800

2. Change in Y at New X

Change in Y New X
15,000 / .3	50,000
15,000 / .25	60,000
Diff	10,000

Diff Sum = 18,800 =

But why does this work?

Why can I use the

Change in X and ORIGINAL Y
+
Change in Y with REVISED Y

 

[Dr.Peterson]

5. I thought I could understand the impact from each variable by changing one at a time and then add their impact but, it's not equal to changing both at once.
6. How can I quantify the "contribution" that X and Y made discreetly to 18,800??????

The quick answer is, you can't. Since they interact in a nonlinear manner, there is no way to disentangle them so as to add two distinct effects together in a consistent way. There may be ways to approximate such a breakdown, but it can't work the way you want it to.

 

[JeffM]

[MATH]\dfrac{x + \Delta x}{y + \Delta y} = \dfrac{y * \dfrac{x}{y} + y * \dfrac{\Delta x}{y}}{y + \Delta y} = \dfrac{y * \dfrac{x}{y} + \Delta y * \dfrac{x}{y} - \Delta y * \dfrac{x}{y} + \Delta x * \dfrac{y}{y}}{y + \Delta y} =[/MATH]
[MATH]\dfrac{\dfrac{x}{y} * (1 + \Delta y) + \dfrac{y \Delta x - x \Delta y}{y}}{y + \Delta y} = \dfrac{x}{y} + \dfrac{y \Delta x - x \Delta y}{y(y + \Delta y)}.[/MATH]
[MATH]\therefore \left | \ \dfrac{x + \Delta x}{y + \Delta y} - \dfrac{x}{y} \ \right | = \left | \ \dfrac{y \Delta x - x \Delta y}{y(y + \Delta y)}\ \right | \approx \\ \left | \ \dfrac{y \Delta x - x \Delta y}{y^2} \ \right | \text { if } \left | \ \dfrac{\Delta y}{y} \ \right | \text { is small}.[/MATH]For example, 909/990 - 900/1000 is about 0.01818

(1000 * 9 - 900(-10))(1,000,000) = (9000 + 9000)/1,000,000 = 18/1000 = 0.018.

The relative change is easier to remember

[MATH]\dfrac{\dfrac{x + \Delta x}{y + \Delta y}}{\dfrac{x}{y}} - 1 = \dfrac{y}{x} * \left ( \dfrac{x}{y} + \dfrac{y \Delta x - x \Delta y}{y(y + \Delta y)} \right ) - 1 = \dfrac{y \Delta x - x \Delta y}{x(y + \Delta y)} \approx \\ \dfrac{\Delta x}{x} - \dfrac{\Delta y}{y} \text { if } \left | \ \dfrac{\Delta y}{y} \ \right | \text { is small}.[/MATH]In other words, there is a simple approximation to the change if changes are relatively small.

 

[tkhunny]

I might be tempted just to subtract.

Due to X: 50
Due to Y: 75
Total: 120
Due to Interaction: 120 - 50 - 75 = -5

Depends on what you want to know and why. This simplistic method WILL tell you what it is, but that may not be what you need.

You may also want to add variation due to other factors, such as x^2. You can make it as complicated as you like. Just don't over-do it. At some point, the additional amount of clarification will not be worth the effort.

In the example:
Due to X: 50
Due to Y: 75
Total: 120
Due to Interaction: 120 - 50 - 75 = -5
It may be sufficient to know that 5/120 = 4.17%, so, in some way, you have managed to explain 95.83% of the variation by examining the two pieces separately.

One more time: Depends on what you want to know and why.

 

[pawlowski6132]

Thank you Dr. Peterson and thank you Jeff and tkhunny.

Let me digest your replies.

 

[pawlowski6132]

"Depends on what you want to know and why"

Let me put this in the context of the actual real-world situation. Word problems, I think, allow us additional context to understand and explain:

We manage a factory that processes parts. (what the process is isn't relevant)
We plan to process them at the rate of .3 parts per hour. So, it takes us 3 hours (approx) to process one part.
We are planning to have to process 12,360 parts.
At the planned productivity rate of .3 parts per hour, we should plan to pay for 41,200 of labor hour

What actually happened was, we were shipped 15,00 parts to process.
In addition, because we had new hires, our productivity rate slowed and we were only able to process .25 parts per hour
This resulted in having to pay for 60,000 hours of labor
18,800 additional hours of unplanned labor.

Fill in the blanks as explanation to management of why our actual labor exceeded planned:


"Of the 18,800 in unplanned labor hours, ___________hrs were incurred due to the increased volume of parts to process and _________hrs were incurred because we processed all the parts at a slower rate."*


*Assume this as a constant or average

 

[pawlowski6132]

“One more time: Depends on what you want to know and why.”

I think I’m getting the picture now but, just to make sure, I’ll put this in word problem with the real world context.

• We process car parts in a factory
• For October, we were planning to process 12,360 parts (what the process is should be irrelevant)
• Our expected productivity was to process .3 parts every hour of labor
• At that rate, we expected to incur cost for 41,200 hours of labor
• After October ended, we actually processed 15,000 parts.
• Further we processed those parts at the rate of .25 parts every hour.
• We incurred 60,000 hours of labor.
• An additional 18,800 hours of labor above what we originally planned at the two key assumptions
• I’d like to explain this variance in hours to management like this (please help me fill in the blanks):

Of the 18,800 extra hours we incurred ______hrs were due to increased volume of parts and ________hrs were due to an overall reduced productivity.

I’m reading above that it’s not possible to derive a formula that consistently and accurately works.

That seems hard to believe but, I’m starting to understand.

 

[JeffM]

“One more time: Depends on what you want to know and why.”

I think I’m getting the picture now but, just to make sure, I’ll put this in word problem with the real world context.

• We process car parts in a factory
• For October, we were planning to process 12,360 parts (what the process is should be irrelevant)
• Our expected productivity was to process .3 parts every hour of labor
• At that rate, we expected to incur cost for 41,200 hours of labor
• After October ended, we actually processed 15,000 parts.
• Further we processed those parts at the rate of .25 parts every hour.
• We incurred 60,000 hours of labor.
• An additional 18,800 hours of labor above what we originally planned at the two key assumptions
• I’d like to explain this variance in hours to management like this (please help me fill in the blanks):

Of the 18,800 extra hours we incurred ______hrs were due to increased volume of parts and ________hrs were due to an overall reduced productivity.

I’m reading above that it’s not possible to derive a formula that consistently and accurately works.

That seems hard to believe but, I’m starting to understand.

I am going to take your example.

[MATH]\text {Estimated productivity} = 0.3 \text { units per hour}.[/MATH]
[MATH]\text {Estimated units produced} = 12,360 \text { units}.[/MATH]
[MATH]\text {Estimated hours} = \dfrac{12360}{0.3} = 41200 \text { hours}.[/MATH]
[MATH]\text {Actual units produced} = 15,000 \text { units}.[/MATH]
[MATH]\text {Actual hours} = 60,000 \text { hours}.[/MATH]
[MATH]\text {Actual productivity} = 0.25 \text { units per hour}.[/MATH]
Suppose we had hit the productivity goal but had had the increased volume.

[MATH]\dfrac{15000 \text { units}}{0.3 \text { units per hour}} = 50,000 \text { hours}.[/MATH]
So 50000 - 41200 = 8800.

Suppose we had just had the estimated volume but suffered the reduction in productivity.

[MATH]\dfrac{12360 \text { units}}{0.25 \text { units per hour}} = 49440 \text { hours}[/MATH]
So 49440 - 41200 = 8,240.

But we had a total of 18800, and 8800 + 8240 = 17,040. What accounts for the missing 1,760? The interaction of increased volume and reduced productivity.

Of the 18,800 extra hours we incurred, 8800 were incurred due solely to increased volume, 8240 were incurred due solely to reduced productivity, and 1760 were incurred because volume increased and productivity decreased simultaneously.

 

[pawlowski6132]

Thanx Jeff. That's exactly as far as I got except, I didn't reason away the 1760 like you did. That's while we always have an "OTHER" bucket on our variance reports.

What are your thoughts on this? This was another approach I took to explain the 18,800:

Impact from Volume	Hours
12,360 at .3	41,200
15,000 at .3	50,000
Diff	8,800
Impact from Productivity	Hours
15,000 @ FCST RATE	50,000
15,000 @ Actual Rate	60,000
Diff	10,000
Total Diff	18,800

 

[JeffM]

The problem is that it is not something "other." It is the combined effect of volume increases and productivity decreases interacting. Personally, I like my numbers to make as much sense as possible. I think it is not a matter of math, but of wording. Here is an alternate wording.

"Of the 18800 extra hours incurred, 8800 would have been incurred even if productivity had not decreased, 8240 would have been incurred even if volume had not increased. The remaining 1760 hours were incurred due to lower productivity on increased volume.

It's not that the math is hard. It is making sure that people grasp that some things are due to multiple causes. This three-way breakdown let's them see how much can be explained by single causes and how much must be explained by the interaction of multiple causes.

 

[pawlowski6132]

That makes sense Jeff. I like how you worded the 1760. Also, I totally agree with your overall message. I was 1/2 joking about the Other category. We have that to contain the explanations we haven't gotten to yet. Also, our leaders insist on asking "5 Why" and we are always in a better spot when we can quantify our answers.