/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*.
/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
/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.: 8 3 5 6 5 30 > 0 > 0 MIN : 8 5 30 3 S.T.: 6 3 3 3 5 3 5 0 0 3 0 6 3
/5/008 PENELESAIAN PRIMAL : OBJECTIVE FUNCTION VALUE ) 7.5000000 VARIABLE VALUE REDUCED COST.833333.000000 5.000000.000000 ROW SLACK OR SURPLUS DUAL PRICES ) 6.333333.000000 3).000000.833333 4).000000.500000 PENELESAIAN DUAL OBJECTIVE FUNCTION VALUE ) 7.5000000 VARIABLE VALUE REDUCED COST.000000 6.333333.833333.000000 3.500000.000000 ROW SLACK OR SURPLUS DUAL PRICES ).000000.833333 3).000000 5.000000 Kedl ktf 7 PENELESAIAN TABEL DENGAN PRIMAL vrel dsr Z Z 3 4 5 NK Keterg 3 5 0 0 0 0 3 0 0 0 0 8 0 0 3 0 0 5 5 0 6 5 0 0 30 8 4
/5/008 vrel dsr Z Z 3 4 5 NK Keterg 3 5 0 0 0 0 3 0 0 0 0 8 8/0 ~ 0 0 3 0 0 5 5/3 5 5 0 6 5 0 0 30 30/5 6 vrel dsr Z Z 3 4 5 NK Keterg 3 0 0 5/3 0 5 3 0 0 0 0 8 0 0 0 /3 0 5 5 0 6 0 0 5/3 5 9 vrel dsr Z Z 3 4 5 NK Keterg 3 0 0 5/3 0 5 3 0 0 0 0 8 8/ 4 0 0 0 /3 0 5 5/0 ~ 5 0 6 0 0 5/3 5 5/6 5/6 vrel dsr Z 3 4 5 NK Keterg Z 0 0 0 5/6 / 7½ l optl 3 0 0 0 5/9 /3 6⅓ 0 0 0 /3 0 5 5 0 0 0 5/8 /6 5/6 0 5
/5/008 Apl ts : 3 5 druh ejd 3 6 l y k tetp 6 8 8 5 5 6( ) 5 9 3,3 5 5 3 Z (3 x 0,56) Z,67 5 5 Z 8,37 30 6 x 5 0 8 30 30 30 3,3 6,7 5 6,7 90 80 0 5 9 x 3 0,56 5,34 (5 x 5,34) 8,37 7,5 0,87 Apl ts 3 : 6 5 30 druh ejd 6 5 3 6 5 8 8 6() 5 6 5 5 5 3 5 3 3 8 8 3 5 5 5 Z (3 x ) (5 x 5) Z 3 5 8 5 x 5 93 75 8 5 Z 8 7,5 0,5 x 3 6
/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
/5/008 8 5 0 0,. 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
/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
/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
/5/008 OBJECTIVE FUNCTION VALUE ) 4.66666700 VARIABLE VALUE REDUCED COST.666667.000000.000000.666667 3.000000.333333 4.333333.000000 ROW SLACK OR SURPLUS DUAL PRICES ).000000.333333 3).000000.666667 OBJECTIVE FUNCTION VALUE ) 4.66666700 VARIABLE VALUE REDUCED COST.666667.000000.000000.666667 3.000000.333333 4.333333.000000 ROW SLACK OR SURPLUS DUAL PRICES ).000000.333333 3).000000.666667 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
/5/008 OBJECTIVE FUNCTION VALUE ) 7.5000000 VARIABLE VALUE REDUCED COST.833333.000000 5.000000.000000 ROW SLACK OR SURPLUS DUAL PRICES ) 6.333333.000000 3).000000.833333 4).000000.500000 SENSITIVIT ANALSIS OBJ COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE 3.000000 3.000000 3.000000 5.000000 INFINIT.500000 RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE 8.000000 INFINIT 6.333333 3 5.000000 3.000000.400000 4 30.000000 9.000000 5.000000 3 OBJECTIVE FUNCTION VALUE ) 7.5000000 VARIABLE VALUE REDUCED COST.000000 6.333333.833333.000000 3.500000.000000 ROW SLACK OR SURPLUS DUAL PRICES ).000000.833333 3).000000 5.000000 SENSITIVIT ANALSIS? OBJ COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE 8.000000 INFINIT 6.333333 5.000000 3.000000.400000 3 30.000000 9.000000 5.000000 RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE 3.000000 3.000000 3.000000 3 5.000000 INFINIT.500000 4