Application Area 5: Multi-objective Optimization

Systems are required to satisfy multiple objectives. These problems require a slightly different approach.

Multi-objective optimization may be stated as follows:
Minimize f =   [ ψ 01 ( x , b ) , ψ 02 ( x , b ) , , ψ 0 n ( x , b ) ] T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qiqrFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabeqadiWa ceGabeqabeqadeqadeaakeaaieGaqaaaaaaaaaWdbiaa=zgacqGH9a qpcaWFGcWaamWaa8aabaWdbiaa=H8apaWaaSbaaSqaa8qacaaIWaGa aGymaaWdaeqaaOWdbmaabmaapaqaa8qacaWF4bGaaiilaiaa=jgaai aawIcacaGLPaaacaGGSaGaa8hOaiaa=H8apaWaaSbaaSqaa8qacaaI WaGaaGOmaaWdaeqaaOWdbmaabmaapaqaa8qacaWF4bGaaiilaiaa=j gaaiaawIcacaGLPaaacaGGSaGaa8hOaiabgAci8kaacYcacaWFGcGa a8hYd8aadaWgaaWcbaWdbiaaicdacaqGUbaapaqabaGcpeWaaeWaa8 aabaWdbiaa=HhacaGGSaGaa8NyaaGaayjkaiaawMcaaaGaay5waiaa w2faa8aadaahaaWcbeqaa8qacaWFubaaaaaa@55D3@ (objective function)
Subject to ψ i ( x , b )   0 ,   i = 1 ,   ,   p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qiqrFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabeqadiWa ceGabeqabeqadeqadeaakeaaieGaqaaaaaaaaaWdbiaa=H8apaWaaS baaSqaa8qacaWFPbaapaqabaGcpeWaaeWaa8aabaWdbiaa=HhacaGG SaGaa8NyaaGaayjkaiaawMcaaiaabckacqGHLjYScaaIWaGaaiilai aabckacaWFPbGaeyypa0JaaGymaiaacYcacaqGGcGaeyOjGWRaaiil aiaabckacaWFWbaaaa@4745@ (inequality constraints)
  ψ i ( x , b ) = 0 ,   i = p + 1 ,   ,   m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qiqrFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabeqadiWa ceGabeqabeqadeqadeaakeaaieGaqaaaaaaaaaWdbiaa=H8apaWaaS baaSqaa8qacaWFPbaapaqabaGcpeWaaeWaa8aabaWdbiaa=HhacaGG SaGaa8NyaaGaayjkaiaawMcaaiabg2da9iaaicdacaGGSaGaaeiOai aa=LgacqGH9aqpcaWFWbGaey4kaSIaaGymaiaacYcacaqGGcGaeyOj GWRaaiilaiaabckacaWFTbaaaa@4732@

b L   b     b U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qiqrFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabeqadiWa ceGabeqabeqadeqadeaakeaaieGaqaaaaaaaaaWdbiaa=jgapaWaaS baaSqaa8qacaWFmbaapaqabaGcpeGaaeiOaiabgsMiJkaa=jgacaqG GcGaeyizImQaaeiOaiaa=jgapaWaaSbaaSqaa8qacaWFvbaapaqaba aaaa@3E0E@

(equality constraints)

(design limits)

The functions ψ ( x , b ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qiqrFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabeqadiWa ceGabeqabeqadeqadeaakeaaieGaqaaaaaaaaaWdbiaa=H8adaqada WdaeaapeGaa8hEaiaacYcacaWFIbaacaGLOaGaayzkaaaaaa@37A1@ in the cost and constraint functions are assumed to have the form:

ψ ( x , b ) =   ψ 0 ( x , b ) + T 0 T f L   ( x , b )   d t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qiqrFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeGabeqadiWa ceGabeqabeqadeqadmaakeaaieGaqaaaaaaaaaWdbiaa=H8adaqada WdaeaapeGaa8hEaiaacYcacaWFIbaacaGLOaGaayzkaaGaeyypa0Ja a8hOaiaa=H8apaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeWaaeWaa8 aabaWdbiaa=HhacaGGSaGaa8NyaaGaayjkaiaawMcaaiabgUcaRiaa =bkadaGfWbqabSWdaeaapeGaa8hva8aadaWgaaadbaWdbiaaicdaa8 aabeaaaSqaa8qacaWFubWdamaaBaaameaapeGaa8NzaaWdaeqaaaqd baWdbiabgUIiYdaakiaa=XeacaWFGcWaaeWaa8aabaWdbiaa=Hhaca GGSaGaa8NyaaGaayjkaiaawMcaaiaa=bkacaWFKbGaa8hDaaaa@5260@

In MotionSolve, a multi-objective problem is transformed to a single-objective problem by redefining the cost function as follows:
Minimize: f =   j = 0 n w j ψ 0 j ( x , b ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qiqrFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeGabeqadiWa ceGabeqabeqadeqadmaakeaaieGaqaaaaaaaaaWdbiaa=zgacqGH9a qpcaWFGcWaaybCaeqal8aabaWdbiaa=PgacqGH9aqpcaaIWaaapaqa a8qacaWFUbaan8aabaWdbiabggHiLdaakiaa=DhapaWaaSbaaSqaa8 qacaWFQbaapaqabaGcpeGaa8hYd8aadaWgaaWcbaWdbiaaicdaieaa caGFQbaapaqabaGcpeWaaeWaa8aabaWdbiaa=HhacaGGSaGaa8Nyaa GaayjkaiaawMcaaaaa@4576@ (objective function)
Subject to: ψ i ( x , b ) 0 , i = 1 , , p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qiqrFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeGabeqadiWa ceGabeqabeqadeqadmaakeaaieGaqaaaaaaaaaWdbiaa=H8apaWaaS baaSqaa8qacaWFPbaapaqabaGcpeWaaeWaa8aabaWdbiaa=HhacaGG SaGaa8NyaaGaayjkaiaawMcaaGqaaiaa+bkacqGHLjYScaaIWaGaai ilaiaa+bkacaWFPbGaeyypa0JaaGymaiaacYcacaGFGcGaeyOjGWRa aiilaiaa+bkacaWFWbaaaa@4746@

ψ i ( x , b ) = 0 , i = p + 1 , , m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qiqrFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeGabeqadiWa ceGabeqabeqadeqadmaakeaaieGaqaaaaaaaaaWdbiaa=H8apaWaaS baaSqaa8qacaWFPbaapaqabaGcpeWaaeWaa8aabaWdbiaa=HhacaGG SaGaa8NyaaGaayjkaiaawMcaaiabg2da9iaaicdacaGGSaacbaGaa4 hOaiaa=LgacqGH9aqpcaWFWbGaey4kaSIaaGymaiaacYcacaGFGcGa eyOjGWRaaiilaiaa+bkacaWFTbaaaa@4736@

b L   b     b U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qiqrFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeGabeqadiWa ceGabeqabeqadeqadmaakeaaieGaqaaaaaaaaaWdbiaa=jgapaWaaS baaSqaa8qacaWFmbaapaqabaacbaGcpeGaa4hOaiabgsMiJkaa=jga caGFGcGaeyizImQaa4hOaiaa=jgapaWaaSbaaSqaa8qacaWFvbaapa qabaaaaa@3E12@

(inequality constraints)

(equality constraints)

(design limits)

w j > 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qiqrFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeGabeqadiWa ceGabeqabeqadeqadmaakeaaieGaqaaaaaaaaaWdbiaa=DhapaWaaS baaSqaa8qacaWFQbaapaqabaGcpeGaeyOpa4JaaGimaaaa@363E@ are a set f weights that reflect the relative importance of the various objectives.

Example 1

Design Goal
Considering a kinematic SLA suspension, the design goal is to modify hard points in a suspension to obtain desired ride behavior.


Figure 1.
The ride behavior is characterized by six objectives:
  • Ride Steer @ Design = -4°/m
  • Ride Steer @ Full Jounce = -4°/m
  • Ride Camber @ Design = +5°/m
  • Ride Camber @ Full Jounce = +5°/m
  • Ride Caster @ Design = -8°/m
  • Ride Caster @ Full Jounce = -8°/m

Ride Steer =   s t e e r A n g l e     r i d e H e i g h t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qiqrFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeGabeqadiWa ceGabeqabeqadeqadmaakeaaqaaaaaaaaaWdbmaalaaapaqaa8qacq GHciITieaacaWFGcacbiGaa43Caiaa+rhacaGFLbGaa4xzaiaa+jha caGFbbGaa4NBaiaa+DgacaGFSbGaa4xzaaWdaeaapeGaa8hOaiabgk Gi2kaa=bkacaGFYbGaa4xAaiaa+rgacaGFLbGaa4hsaiaa+vgacaGF PbGaa43zaiaa+HgacaGF0baaaaaa@4AB2@

Ride Camber =   s t e e r A n g l e     r i d e H e i g h t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qiqrFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeGabeqadiWa ceGabeqabeqadeqadmaakeaaqaaaaaaaaaWdbmaalaaapaqaa8qacq GHciITieaacaWFGcacbiGaa43Caiaa+rhacaGFLbGaa4xzaiaa+jha caGFbbGaa4NBaiaa+DgacaGFSbGaa4xzaaWdaeaapeGaa8hOaiabgk Gi2kaa=bkacaGFYbGaa4xAaiaa+rgacaGFLbGaa4hsaiaa+vgacaGF PbGaa43zaiaa+HgacaGF0baaaaaa@4AB2@

Ride Caster =   c a s t e r A n g l e   r i d e H e i g h t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qiqrFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeGabeqadiWa ceGabeqabeqadeqadmaakeaaqaaaaaaaaaWdbmaalaaapaqaa8qacq GHciITieaacaWFGcacbiGaa43yaiaa+fgacaGFZbGaa4hDaiaa+vga caGFYbGaa4xqaiaa+5gacaGFNbGaa4hBaiaa+vgaa8aabaWdbiabgk Gi2kaa=bkacaGFYbGaa4xAaiaa+rgacaGFLbGaa4hsaiaa+vgacaGF PbGaa43zaiaa+HgacaGF0baaaaaa@4A70@

Design Variables
Various hard point locations in the suspension are available as design variables. For this problem, there are 26 design variables.
Upper Control Arm Inboard hard-points (Fore/Aft) XYZ (6 DV)
Lower Control Arm Inboard hard-points (Fore/Aft) XYZ (6 DV)
Upper Ball Joint hard-point XYZ (3 DV)
Lower Ball Joint hard-point XYZ (3 DV)
Tie rod Inner & Outer XYZ (6 DV)
Spindle Alignment Point Location XZ (3 DV)
Results
For this example, the optimization method worked quite well.
Response Target Actual
Ride Steer @ Design -4 °/m -4.107 °/m
Ride Steer @ Full Jounce -4 °/m -3.898 °/m
Ride Camber @ Design +5 °/m +4.937 °/m
Ride Camber @ Full Jounce +5 °/m +5.024 °/m
Ride Caster @ Design -8 °/m -8.402 °/m
Ride Caster @ Full Jounce -8 °/m -7.594 °/m

Example 2

This is another suspension example for a bus. The task at hand is to design the steering system to meet required metrics. In a properly design steering system:
  • The left and right wheels have zero steer when the steering wheel angle is zero.
  • The steering angle difference between the left and right wheels should be zero at maximum steer and minimum steer.
  • Steering ratio refers to the ratio between the turn of the steering wheel and the turn of the wheels. The steering ratio at zero steering wheel angle is required to be 9.27.


Figure 2.
The problem has four objectives:
  • Left Wheel Steer Angle @ zero Steering Wheel Angle =0
  • Right Steer Angle @ zero Steering Wheel Angle =0
  • Steer Angle Difference between left and right wheels = 0
  • Steering Ratio @ Zero Steering Wheel Angle = 9.27
Design Variables
  • Drag Link Ball Position X-Y-Z Coordinates
  • Pitman Arm Ball Position X-Y-Z Coordinates
Results
For this example, the optimization method worked quite well. The results obtained are shown below:
Response Target Actual
Left Steer @ Zero SWA +0.001°
Right Steer @ Zero SWA +0.000°
Steering difference (L vs. R) -0.003°
Steering ratio @ Zero SWA 9.27 9.271