The output of Step 2 – Optimization is a three-part solution with an optimum offset, S, expressed in the context of a sentence dynamically concatenated from the execution,

such as, ‘‘CVR  elapsed time was aligned to FDR   elapsed

X i jaFDRs{alignCVRs jzjaFDRe{alignCVRej

ð5Þ

time by means of a linear programming model。 8 transmissions from the CVR were aligned with 8  identical

where alignCVRs and alignCVRe are defined in the constraints,

alignCVRs~SzCVRs ð6Þ

alignCVRe~SzCVRe ð7Þ

and the solution is subject to the  constraints

alignCVRs§aFDRs ð8Þ

alignCVRsƒaFDRs ð9Þ

alignCVRe§rFDRe ð10Þ

alignCVReƒaFDRe ð11Þ

where S is the CVR offset being  sought。

Equations 5 through 7 create a set of linear programming equations, similar in form to a linear regression (Anderson et al。, 2011)。 The addition of equations 8 through 11 provide the constraints needed to optimize the CVR/FDR alignment model consistent with Figure 4。

The answer produced in Step 2-Optimization, while an optimum solution, will have slack in the solution consistent with the nature of the FDR sampling rate and optimization models (Anderson et al。, 2011)。 While sensitivity analysis may offer insight into the slack in the constraints, a more user-friendly, automated approach can be had by running a relaxed solution two additional times。 The first time, Equation 5 is modified to

transmissions from the FDR。 The resulting 17-decision variable, 48-constraint linear programming model was solved resulting in an offset of 4000。042 seconds    ¡

0。117 seconds’’。 In the exemplar sentence, all numbers are dynamically calculated and replaced in the equation。 The output also contains a listing of the aligned events, as shown in Listing 4。 Additionally, a number of charts are produced, including a display of slack in the solution, an example is shown in Figure  6。

Figure 5。 Top time scale shows solution pushed towards the left and bottom time scale shows solution pushed towards the  right。

Figure 6。 CVR alignment solution slack per  event。

Listing 4

FEASIBLE STATUS: 6

AB_S54000。042 ,with Start/End Relaxations: 0。000/ 0。000

s cvr 4999。325 s fdr 4999。001 (delta s) +0。324 e cvr

5000。396 e fdr 5000。999 (delta e)  +0。603

s cvr 5049。285 s fdr 5049。001 (delta s) +0。284 e cvr

5055。529 e fdr 5055。999 (delta e)  +0。470

s cvr 5699。479 s fdr 5699。001 (delta s) +0。477 e cvr 5720。714 e fdr 5720。999 (delta e)  +0。285

s cvr 6009。092 s fdr 6009。001 (delta s) +0。091 e cvr

6012。431 e fdr 6012。999 (delta e)  +0。568

s cvr 6054。059 s fdr 6054。001 (delta s) +0。058 e cvr

6056。420 e fdr 6056。999 (delta e)  +0。579

s cvr 6559。482 s fdr 6559。001 (delta s) +0。481 e cvr 6567。667 e fdr 6567。999 (delta e)  +0。332

s cvr 6999。941 s fdr 6999。001 (delta s) +0。940 e cvr 7001。818 e fdr 7001。999 (delta e)  +0。181

s cvr 7004。577 s fdr 7004。001 (delta s) +0。576 e cvr

7007。316 e fdr 7007。999 (delta e)  +0。683

The test cases run in Step 1 – Matching were used as inputs to Step 2 – Optimization; Table 2 summarizes the test cases。 Cases 1 and 2 are perfect alignment cases and produce an offset with no slack。 Case 3 has random perturbations that were purposively contrived to create an infeasible solution, by adding e to both CVRs and CVRe。 As expected, Case 3 was infeasible。 While Case 4 is problematic, in that Step 1 found a match even though  the