Simplex v3

Version PHP : 8.2.33
Moteur de resolution : Python (python3)

Entree du probleme

Fonction objectif (ex: max(4x1+3x2+6x3)) Contraintes (une par ligne)

Toutes les variables decisionnelles sont supposees non negatives (xi >= 0). Les contraintes explicites xi>=0 peuvent etre saisies mais ne sont pas affichees dans le tableau principal.

Noms de variables acceptes : identifiants en minuscules de la forme [a-z_][a-z0-9_]* (ex: x1, y2, abc). Un mot comme abc dans x1<=abc est interprete comme une variable. Les ecritures 0x1 ou 2x1 signifient 0*x1 / 2*x1 (pas un litteral hexadecimal).

Parametres

Afficher des fractions

Nombre de decimales (0 a 10)

Afficher les coefficients

Afficher les calculs dans les cellules

Afficher les rapports de pivot

Fonction objectif

Canonique : max(4x1+3x2+6x3)

Standard : max(0x1+0x2+0x3+0s1+0s2-a1-a2)

Standard (phase 2) : max(4x1+3x2+6x3)

Type : MAX

Methode : Deux phases

Contraintes

Canonique :

x1+x2+x3<=30
2x1+x2+3x3>=40
x1+2x2+x3=36

Standard :

x1+x2+x3+s1=30
2x1+x2+3x3-s2+a1=40
x1+2x2+x3+a2=36

Etape 0 - Phase 1

Entrante : x1 | Sortante : a1 | Pivot : L2, C1

Le plus petit rapport positif est : 20 sur la ligne a1 pour la colonne x1.

Base	x1 : 4	x2 : 3	x3 : 6	s1 : 0	s2 : 0	a1 : 0	a2 : 0	CST
s1 : 0	1	1	1	1	0	0	0	30(Rp : 30 / 1 = 30)
a1 : 0	2	1	3	0	-1	1	0	40(Rp : 40 / 2 = 20)
a2 : 0	1	2	1	0	0	0	1	36(Rp : 36 / 1 = 36)
Z	-3	-3	-4	0	1	0	0	-76

Etape 1 - Phase 1

Entrante : x2 | Sortante : a2 | Pivot : L3, C2

Le plus petit rapport positif est : 32/3 sur la ligne a2 pour la colonne x2.

Base	x1 : 4	x2 : 3	x3 : 6	s1 : 0	s2 : 0	a1 : 0	a2 : 0	CST
s1 : 0	(1)-((1)*(2)/(2)) = 0	(1)-((1)*(1)/(2)) = 0.5	(1)-((1)*(3)/(2)) = -0.5	(1)-((1)*(0)/(2)) = 1	(0)-((1)*(-1)/(2)) = 0.5	(0)-((1)*(1)/(2)) = -0.5	(0)-((1)*(0)/(2)) = 0	(30)-((1)*(40)/(2)) = 10(Rp : 10 / 0.5 = 20)
x1 : 4	(2)/(2) = 1	(1)/(2) = 0.5	(3)/(2) = 1.5	(0)/(2) = 0	(-1)/(2) = -0.5	(1)/(2) = 0.5	(0)/(2) = 0	(40)/(2) = 20(Rp : 20 / 0.5 = 40)
a2 : 0	(1)-((1)*(2)/(2)) = 0	(2)-((1)*(1)/(2)) = 1.5	(1)-((1)*(3)/(2)) = -0.5	(0)-((1)*(0)/(2)) = 0	(0)-((1)*(-1)/(2)) = 0.5	(0)-((1)*(1)/(2)) = -0.5	(1)-((1)*(0)/(2)) = 1	(36)-((1)*(40)/(2)) = 16(Rp : 16 / 1.5 = 10.67)
Z	((00)+(01)+(-1*0))-(0) = 0	((01/2)+(01/2)+(-1*3/2))-(0) = -1.5	((0-1/2)+(03/2)+(-1*-1/2))-(0) = 0.5	((01)+(00)+(-1*0))-(0) = 0	((01/2)+(0-1/2)+(-1*1/2))-(0) = -0.5	((0-1/2)+(01/2)+(-1*-1/2))-(-1) = 1.5	((00)+(00)+(-1*1))-(-1) = 0	(010)+(020)+(-1*16) = -16

Etape 2 - Phase 1

Entrante : - | Sortante : - | Pivot : -

Base	x1 : 4	x2 : 3	x3 : 6	s1 : 0	s2 : 0	a1 : 0	a2 : 0	CST
s1 : 0	(0)-((1/2)*(0)/(3/2)) = 0	(1/2)-((1/2)*(3/2)/(3/2)) = 0	(-1/2)-((1/2)*(-1/2)/(3/2)) = -0.33	(1)-((1/2)*(0)/(3/2)) = 1	(1/2)-((1/2)*(1/2)/(3/2)) = 0.33	(-1/2)-((1/2)*(-1/2)/(3/2)) = -0.33	(0)-((1/2)*(1)/(3/2)) = -0.33	(10)-((1/2)*(16)/(3/2)) = 4.67
x1 : 4	(1)-((1/2)*(0)/(3/2)) = 1	(1/2)-((1/2)*(3/2)/(3/2)) = 0	(3/2)-((1/2)*(-1/2)/(3/2)) = 1.67	(0)-((1/2)*(0)/(3/2)) = 0	(-1/2)-((1/2)*(1/2)/(3/2)) = -0.67	(1/2)-((1/2)*(-1/2)/(3/2)) = 0.67	(0)-((1/2)*(1)/(3/2)) = -0.33	(20)-((1/2)*(16)/(3/2)) = 14.67
x2 : 3	(0)/(3/2) = 0	(3/2)/(3/2) = 1	(-1/2)/(3/2) = -0.33	(0)/(3/2) = 0	(1/2)/(3/2) = 0.33	(-1/2)/(3/2) = -0.33	(1)/(3/2) = 0.67	(16)/(3/2) = 10.67
Z	((00)+(01)+(0*0))-(0) = 0	((00)+(00)+(0*1))-(0) = 0	((0-1/3)+(05/3)+(0*-1/3))-(0) = 0	((01)+(00)+(0*0))-(0) = 0	((01/3)+(0-2/3)+(0*1/3))-(0) = 0	((0-1/3)+(02/3)+(0*-1/3))-(-1) = 1	((0-1/3)+(0-1/3)+(0*2/3))-(-1) = 1	(014/3)+(044/3)+(0*32/3) = 0

Etape 3 - Phase 2

Entrante : x3 | Sortante : x1 | Pivot : L2, C3

Le plus petit rapport positif est : 44/5 sur la ligne x1 pour la colonne x3.

Base	x1 : 4	x2 : 3	x3 : 6	s1 : 0	s2 : 0	CST
s1 : 0	0	0	-0.33	1	0.33	4.67
x1 : 4	1	0	1.67	0	-0.67	14.67(Rp : 14.67 / 1.67 = 8.8)
x2 : 3	0	1	-0.33	0	0.33	10.67
Z	0	0	-0.33	0	-1.67	90.67

Etape 4 - Phase 2

Entrante : s2 | Sortante : s1 | Pivot : L1, C5

Le plus petit rapport positif est : 38 sur la ligne s1 pour la colonne s2.

Base	x1 : 4	x2 : 3	x3 : 6	s1 : 0	s2 : 0	CST
s1 : 0	(0)-((-1/3)*(1)/(5/3)) = 0.2	(0)-((-1/3)*(0)/(5/3)) = 0	(-1/3)-((-1/3)*(5/3)/(5/3)) = 0	(1)-((-1/3)*(0)/(5/3)) = 1	(1/3)-((-1/3)*(-2/3)/(5/3)) = 0.2	(14/3)-((-1/3)*(44/3)/(5/3)) = 7.6(Rp : 7.6 / 0.2 = 38)
x3 : 6	(1)/(5/3) = 0.6	(0)/(5/3) = 0	(5/3)/(5/3) = 1	(0)/(5/3) = 0	(-2/3)/(5/3) = -0.4	(44/3)/(5/3) = 8.8
x2 : 3	(0)-((-1/3)*(1)/(5/3)) = 0.2	(1)-((-1/3)*(0)/(5/3)) = 1	(-1/3)-((-1/3)*(5/3)/(5/3)) = 0	(0)-((-1/3)*(0)/(5/3)) = 0	(1/3)-((-1/3)*(-2/3)/(5/3)) = 0.2	(32/3)-((-1/3)*(44/3)/(5/3)) = 13.6(Rp : 13.6 / 0.2 = 68)
Z	((01/5)+(63/5)+(3*1/5))-(4) = 0.2	((00)+(60)+(3*1))-(3) = 0	((00)+(61)+(3*0))-(6) = 0	((01)+(60)+(3*0))-(0) = 0	((01/5)+(6-2/5)+(3*1/5))-(0) = -1.8	(038/5)+(644/5)+(3*68/5) = 93.6

Etape 5 - Phase 2

Entrante : - | Sortante : - | Pivot : -

Base	x1 : 4	x2 : 3	x3 : 6	s1 : 0	s2 : 0	CST
s2 : 0	(1/5)/(1/5) = 1	(0)/(1/5) = 0	(0)/(1/5) = 0	(1)/(1/5) = 5	(1/5)/(1/5) = 1	(38/5)/(1/5) = 38
x3 : 6	(3/5)-((-2/5)*(1/5)/(1/5)) = 1	(0)-((-2/5)*(0)/(1/5)) = 0	(1)-((-2/5)*(0)/(1/5)) = 1	(0)-((-2/5)*(1)/(1/5)) = 2	(-2/5)-((-2/5)*(1/5)/(1/5)) = 0	(44/5)-((-2/5)*(38/5)/(1/5)) = 24
x2 : 3	(1/5)-((1/5)*(1/5)/(1/5)) = 0	(1)-((1/5)*(0)/(1/5)) = 1	(0)-((1/5)*(0)/(1/5)) = 0	(0)-((1/5)*(1)/(1/5)) = -1	(1/5)-((1/5)*(1/5)/(1/5)) = 0	(68/5)-((1/5)*(38/5)/(1/5)) = 6
Z	((01)+(61)+(3*0))-(4) = 2	((00)+(60)+(3*1))-(3) = 0	((00)+(61)+(3*0))-(6) = 0	((05)+(62)+(3*-1))-(0) = 9	((01)+(60)+(3*0))-(0) = 0	(038)+(624)+(3*6) = 162

Solution

x1 = 0
x2 = 6
x3 = 24

Z = 162