Regression of Cost Dependent CERs

1. Regression of Cost Dependent CERs The Aerospace Corporation 15049 Conference Center Drive, CH1-410 Suite 600 Chantilly, VA 20151

2. Outline Introduction Traditional Regression and Risk Analysis Alternative Approach to Regression and Risk Analysis An Example Conclusions

3. Introduction There are two basic types of Cost Estimating Relationships (CERs). Design Parameter Dependent: Subsystem Hardware (HW) CERs which use weight (or other design parameter) as a base. Cost Dependent: Systems Engineering, Integration and Test, and Program Management (SEITPM) CERs which use estimated cost as a base. Throughout this paper, assume all CERs that do not use cost as a base are weight (mass)-based CERs.

4. Introduction CERs are Developed in Parallel. Subsystem HW costs are regressed against Subsystem actual weights to generate CERs. SEITPM actual costs are regressed against Spacecraft HW actual costs to generate CERs. But�CERs are Used in Series in Risk Analysis. Subsystem HW estimated costs are driven by subsystem estimated weights. SEITPM estimated costs are driven by HW cost estimates. This results in mishandling of errors. Problem: Cost dependent CERs, like SEITPM, are not correctly developed.

5. Traditional Regression When developing a CER for SEITPM, we usually regress actual SEITPM (SEITPM$Act) against actual spacecraft (SC$Act) cost to come up with an estimate of SEITPM as a function of actual SC cost. e.g� The result is a CER for SEITPM, in which uncertainty can be quantified by the standard error (SE) of the regression (minimum unbiased percentage error (MUPE)).

6. Traditional Regression However, when applying this CER to an estimate, we don�t know SC$Act, so we estimate it with SC$Est. But, SC$Est is the result of several summed regressions, relating subsystem cost (SS$) to weight (WT), and containing its own uncertainty! e.g�

7. Traditional Risk Analysis When we estimate SEITPM$ as a function of SC$Est, we fail to fully and accurately account for the uncertainty that is carried along with SC$Est, even through functional correlation. This is because the CER for SEITPM$ is a function of SC$Act, not SC$Est. There is a disconnect in logic here. Recall� The result is that we estimate SEITPM$Est using SC$Est, but we only account for the uncertainty related to SEITPM$Est as if its independent variable had no uncertainty. i.e., despite functional correlation, the uncertainty attached to SEITPM$Est(SC$Est) is different than that of SEITPM$Est(SC$Act)!

8. Mathematically Total estimate is a sum of the subsystem and SEITPM costs:* Subsystem estimates follow the form: SEITPM estimate follows the form:

9. Mathematically So, the SEITPM estimate is actually represented by: And the total estimate is represented by: But, since our regressions use actual costs as the independent variable in SEITPM CERs, then in practice:

10. Mathematically So, the unaccounted error in the uncertainty calculation is:

11. An Alternative Approach We suggest an alternative approach. Start with the desired end result and work backward to determine the required inputs. We want to be able to estimate SEITPM costs as a function of estimated spacecraft cost. SEITPM$Est (SC$Est) That means we should regress actual SEITPM against estimated spacecraft cost.

12. Mathematically Total estimate is a sum of the subsystem and SEITPM costs: Again, the SEITPM estimate is actually represented by:

13. Mathematically So, the total estimate is represented by: But, if our regressions use estimated cost as the independent variable in SEITPM, then: These two quantities are identical, therefore, the unaccounted error in the uncertainty calculation is ZERO.

14. An Example A fictitious set of data for seven programs Three subsystem weights and �actual costs� SEITPM �actual costs� CERs developed using the traditional method

15. An Example Now calculate SEITPM as a function of estimated SC costs (highlighted in yellow) The CER for SEITPM using this method is

16. An Example The CERs have different percentage errors The errors manifest themselves in Statistical Uncertainty Simulations

17. Example Using Weight How about SEITPM as a function of weight? MUPE larger than traditional method but less than proposed method But not all SS$ function of weight (Might be SW) Alternate method used because CERs are developed and used in parallel

18. Conclusions SEITPM CERs are usually derived from actual SC costs. But, since we use estimated HW costs in these CERs, we tend to misrepresent the total cost variance, leading to total cost distributions that are misleading. To remedy this problem, we should derive SEITPM CERs as a function of estimated SC costs. This adjusts the total cost variance, leading to more accurate total cost distributions. Now the SEITPM CER input data matches the type of data used to produce the CER. May result in larger or smaller total error