DUALITAS DAN ANALISIS SENSITIVITAS

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1 /5/008 DUALITAS DAN ANALISIS SENSITIVITAS Dr. Mohd Adul Mukhy, SE., MM. Prl Prole P ze z cx suject to Ax x 0 optu vlue s z* Dul Prole xze suject to D v π πa c optu vlue s v* Theore. (Strog Dulty) If oth P d D re fesle, the z* v*.

2 /5/008 Tel Prl Dul Ler Progrg PRIMAL Koefse NK DUAL koefse Koefse fugs tuju (ksss) NK C C C Koefse fugs tuju (ksss) 3 Tel Huug tr prl dul Prl (tu Dul) Bts I Dul (tu Prl) Vrel I Fugs tuju Nl k 4

3 /5/008 The cost flow prole d ts dul Mze (,j) A c j x j j x j k x k for ll Î N. d x j 0 for ll (,j) Î A. Prl Mze π Dul suject to π π c for ll (, j) A j j 5 MASALAH PRIMAL MASALAH DUAL MA : Z 3 5 S.T.: > 0 > 0 MIN : S.T.:

4 /5/008 PENELESAIAN PRIMAL : OBJECTIVE FUNCTION VALUE ) VARIABLE VALUE REDUCED COST ROW SLACK OR SURPLUS DUAL PRICES ) ) ) PENELESAIAN DUAL OBJECTIVE FUNCTION VALUE ) VARIABLE VALUE REDUCED COST ROW SLACK OR SURPLUS DUAL PRICES ) ) Kedl ktf 7 PENELESAIAN TABEL DENGAN PRIMAL vrel dsr Z Z NK Keterg

5 /5/008 vrel dsr Z Z NK Keterg /0 ~ / /5 6 vrel dsr Z Z NK Keterg / / /3 5 9 vrel dsr Z Z NK Keterg / / / /0 ~ /3 5 5/6 5/6 vrel dsr Z NK Keterg Z /6 / 7½ l optl /9 /3 6⅓ / /8 /6 5/6 0 5

6 /5/008 Apl ts : 3 5 druh ejd 3 6 l y k tetp ( ) 5 9 3, Z (3 x 0,56) Z, Z 8, x ,3 6,7 5 6, x 3 0,56 5,34 (5 x 5,34) 8,37 7,5 0,87 Apl ts 3 : druh ejd () Z (3 x ) (5 x 5) Z x Z 8 7,5 0,5 x 3 6

7 /5/008 Huug tr vrelvrel PrlDul dl Ler Progrg Vrel Prl Vrel Dul Vrel sl : Vrel Slck : Vrel surplus : Z j C j Vrel Asl : D,, j,, 3 PENIMPANGANPENIMPANGAN DARI BENTUK STANDAR j j Nl j Kovers Betuk Buk Stdr Mejd Betuk Stdr Dl Model Ler Progrg Betuk uk stdr : Mss Z j j j j tdk terts Betuk stdr : Mksss Z j j j j j j j j j ( ' ' ' ), ' 0, ' ' 0 j j j j 4 7

8 /5/ ,. 0; d c c c : / Mss 0 t s Mecr etuk dul dr sutu slh dul Peruh ke dl euk stdr 0 0,. 0; d c c c : / ) ( Mksss 0 t s 6 Dul dr dul terseut (prl) 0 0,. 0; d : / C C C (Z) Mss t s 0 0,. 0; d : / C C C Z Mss t s Peruh ke dl etuk stdr

9 /5/008 Dul dr sutu slh dul tdk l dlh slh prly. Bts yg egdug td pers () dperluk sepert lyky ts ertd ; tetp ts oegtf g dul vrel yg ersgkut hrus dhlgk (ytu vrel yg tdk terts ly). Meghlgk ts oegtf pd slh prl k egktk peruh ts pd slh dul ejd etuk pers () 7 Huug etuketuk Prl Dul Mslh Prl (Dul) Mx Z (tu 0 ) Bts etuk etuk Vrel j (tu j ) j 0 j 0 dhlgk Mslh Dul (Prl) M 0 (tu Z) Vrel j (tu j ) j 0 j 0 dhlgk Bts j etuk etuk 8 9

10 /5/008 PRIMAL PROBLEM: xze z 3x 4x 6x 3 8x 4 suject to x x x 3 x 4 x 3x 4x 3 5x 4 3 x, x, x 3, x 4 0 DUAL PROBLEM: ze y 3y Suject to y y 3 y 3y 4 y 4y 6 y 5y 8 Oservto. The costrt trx the prl s the trspose of the costrt trx the dul. Oservto. The RHS coeffcets the prl ecoe the cost coeffcets the dul. 9 PRIMAL PROBLEM: xze z 3x 4x 6x 3 8x 4 suject to x x x 3 x 4 x 3x 4x 3 5x 4 3 x, x, x 3, x 4 0 DUAL PROBLEM: ze y 3y Suject to y y 3 y 3y 4 y 4y 6 y 5y 8 Oservto 3. The cost coeffcets the prl ecoe the RHS coeffcets the dul. Oservto 4. The prl ( ths cse) s x prole wth equlty costrts d oegtve vrles The dul ( ths cse) s zto prole wth costrts d vrles ucostred sg. 0 0

11 /5/008 OBJECTIVE FUNCTION VALUE ) VARIABLE VALUE REDUCED COST ROW SLACK OR SURPLUS DUAL PRICES ) ) OBJECTIVE FUNCTION VALUE ) VARIABLE VALUE REDUCED COST ROW SLACK OR SURPLUS DUAL PRICES ) ) Alss Sestvts Kre terjd peruhperuh dl vrelvrel, pkh fugs tuju upu fugs kedl deg cr eftk kdhkdh prldul etode splek seksl ugk. Kre tujuy dlh peyeles optl, k lss dseut pul Post Optlty. Peruhperuh yg ugk terjd:. Keterts kpsts suer (fugs ts).. Koefsekoefse fugs tuju. 3. Koefsekoefse teks fugs ts 4. Peh vrelvrel ru. 5. Peh tsts ru

12 /5/008 OBJECTIVE FUNCTION VALUE ) VARIABLE VALUE REDUCED COST ROW SLACK OR SURPLUS DUAL PRICES ) ) ) SENSITIVIT ANALSIS OBJ COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE INFINIT RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE INFINIT OBJECTIVE FUNCTION VALUE ) VARIABLE VALUE REDUCED COST ROW SLACK OR SURPLUS DUAL PRICES ) ) SENSITIVIT ANALSIS? OBJ COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE INFINIT RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE INFINIT

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