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### Example 2: Balancing a Matrix Using Gravity ModelYou are given the travel times between zones, trips originating at each zone (zone1 =15, zone 2=15) trips destined for each zone (zone 1=10, zone 2 = 20) and asked to use the classic gravity modelTravel Time OD Matrix (Cij{\displaystyle C_{ij}})Impedance Matrix (f(Cij){\displaystyle f(C_{ij})})Balancing Iteration 0 (Set-up)Balancing Iteration 1 (Tij,iteration1=TO,iTD,jf(Cij){\displaystyle T_{ij,iteration1}=T_{O,i}T_{D,j}f(C_{ij})})Row Total TO,i′{\displaystyle T'_{O,i}}Normalizing Factor NO,i=TO,i/TO,i′{\displaystyle N_{O,i}=T_{O,i}/T'_{O,i}}Balancing Iteration 2 (Tij,iteration2=Tij,iteration1∗NO,i,iteration1{\displaystyle T_{ij,iteration2}=T_{ij,iteration1}*N_{O,i,iteration1}})Normalizing Factor ND,j=TD,j/TD,j′{\displaystyle N_{D,j}=T_{D,j}/T'_{D,j}}Balancing Iteration 3 (Tij,iteration3=Tij,iteration2∗ND,j,iteration2{\displaystyle T_{ij,iteration3}=T_{ij,iteration2}*N_{D,j,iteration2}})Normalizing Factor = ND,j=TD,j/TD,j′{\displaystyle N_{D,j}=T_{D,j}/T'_{D,j}}Balancing Iteration 4 (Tij,iteration4=Tij,iteration3∗NO,i,iteration3{\displaystyle T_{ij,iteration4}=T_{ij,iteration3}*N_{O,i,iteration3}})Balancing Iteration 16 (Tij,iteration16=Tij,iteration15∗ND,i,iteration5{\displaystyle T_{ij,iteration16}=T_{ij,iteration15}*N_{D,i,iteration5}})So while the matrix is not strictly balanced, it is very close, well within a 1% threshold, after 16 iterations. The threshold refers to the proximity of the normalizing factor to 1.0.## Additional Questions## Variables- Trips arriving at destination- Effective Trips arriving at destination, computed as a result for calibration to the next iteration- Total number of trips between origin- Calibration parameter for Origins- Calibration parameter for Destinations- Cost function between origin## Further reading