Modelos e algoritmos para variações do problema de balanceamento de linhas de produção...

(1)

(2)

(3)

Ficha catalográfica elaborada pela Biblioteca Prof. Achille Bassi e Seção Técnica de Informática, ICMC/USP,

com os dados fornecidos pelo(a) autor(a)

A658m Araújo, Felipe Francisco Bezerra Modelos e algoritmos para variações do problema de balanceamento de linhas de produção e designação de trabalhadores / Felipe Francisco Bezerra Araújo; orientador Alysson Machado Costa. -- São Carlos, 2016.

90 p.

Tese (Doutorado - Programa de Pós-Graduação em Ciências de Computação e Matemática Computacional) Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, 2016.

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

i; j s N S

pi i

Dj j

C

xsi 1

(19)

M in C X s2 S sxsi X s2 S

sxsj 8j 2 N; 8i 2 Dj

X

s2 S

xsi = 1 8i 2 N

X

i 2 N

(20)

W1 W2 W3 W4

W1

W2

W3

W4

W1 W3 W2 W4

W2

(21)

W4

W2 W1

W3

W2 W4

(22)

i; j w s N W S

pwi i w

Dj j

Iw w

C

xswi 1

0

ysw 1 w s

0

M in C

X

w2 W

X

s2 S

(23)

X

s2 S

ysw = 1 8w 2 W

X

w2 W

ysw = 1 8s 2 S

X w2 W X s2 S sxswi X w2 W X s2 S

sxswj 8j 2 N; 8i 2 Dj

X

w2 W

X

i 2 N

pwixswi C 8s 2 S

X

i 2 N

xswi jN jysw 8w 2 W; 8s 2 S

(24)

t1; t2; :::; ti; :::; tn n

(25)

1

T = t11 +

1

t2 + ::: +

1

ti + ::: +

1 tn >

1 ti

1

(26)

(27)

(28)

(29)

(30)

T n

(31)

W4 W2 W3

W1

W2 W3

1

17+141 = 7; 6s

W4

W2 W4

(32)

n

T1; T2; :::; Tn T

1

T =

P n i = 1T1i

j Tj Ti 8i 2 [1::n]

Tj Tj=T > 1

j dTj=Te 1 T

dTj=Te 1

A B T d 20 1 1 10 +201

e 1 = d20( 1

10+201)e

1 = 2 t = 6; 7s

(33)

t = 10s A t = 13; 3s

t = 20s

zsk

k KM ax

ti s 1 i s

0

zsk 1 s k

0

M in C

X

s2 S

sti s

X

s2 S

stj s 8i; j 2 N ji 2 Dj

X

w2 W

(34)

xswi ti s 8s 2 S; 8w 2 W; 8i 2 N

KXM ax

k= 0

zsk = 1 8s 2 S

X

w2 W

ysw = KXM ax

k= 0

kzsk 8s 2 S

X

s2 S

ysw = 1 8w 2 W

X

s2 S

ti s = 1 8i 2 N

xswi ysw 8s 2 S; 8w 2 W; 8i 2 N

xswi = 0 8w 2 W; 8s 2 S; 8i 2 Iw

X

w2 W jysw= 1

1 P

i 2 N pwixswi

1

C 8s 2 Sjzs0= 0

(35)

k

zs0= 1

F

fsw

w s w s ysw = 1

fsw = P _{i 2 N} _p1_{w i}_x_{sw i} P i 2 N pwifswxswi = 1 fsw = 0

vswi fsw i

w s 0 vswi = fswxswi

M Mw

w

M = _P jW j

i 2 N pw i Mw = _min_{i 2 N}1 _p_{w i}

M

i

pwi Mw

w

(36)

(3:2) (3:10)

X

w2 W

fsw F M zs0 8s 2 S

X

i 2 N

pwivswi = ysw 8w 2 W; 8s 2 S

fsw Mwysw 8w 2 W; 8s 2 S

vswi fsw Mw(1 xswi) 8s 2 S; 8w 2 W; 8i 2 N

vswi jN jxswi 8s 2 S; 8w 2 W; 8i 2 N

vswi

fsw = 0 w

s

vswi vswi = fsw xswi = 1

(37)

c KM ax

ci

i

ti i

ci ti

ci

ti

jj > i 2 N j ci

(38)

c KM ax

n = 0 t(Sn) = 0

Uw W

U N

k KM ax

W0 _U

w k

T0 W

W0

k W0

T0

W Uw = UwnW0 U = UnTW0 n = n + 1

Uw 6= ;

U = ;

c

M axPWS

ci + P _{j > i 2 N} minw2 W(twj)

M axPWS+

ci + P j > i 2 N maxw2 W(twj)

M axPWD

ci + P _{j > i 2 N} minw2 W0(t_wj) W0

M axPWD+

(39)

n W0

n

jW0_{j = 1} _n n

1;2

jW0_{j = 1} _n n

1;4

jW0_{j = 1} _n n

1;6

jW0_{j = 1} _n n

1;8

jW0_{j = 1} _n n

2

jW0_{j = 1} _n n

3

n jW0_j

P

i 2 N 00mi nw 2 W 00tw i

(40)

LC UC c = LC

c

c < = UC c = c + passo

LC = maxf maxi 2 N ti ; dP i 2 N ti =jWjeg

maxi 2 N ti

(41)

(42)

s v vulti mo = v smelhor = s

svi zi nho smelhor

v

svi zi nho smelhor

smelhor = svi zi nho vulti mo= v

svi zi nho s

v vulti mo= v

smelhor

(43)

(44)

KM ax = 2

(45)

(46)

(47)

KM ax = 2

1

(48)

(49)

i; j w s k N W S

pwi i w

Dj j

Iw w

KM ax

xswi k 1

0

yswk 1 w s

0

zk 1 k 0

M ax KXM ax

p= 1 Fk X w2 W X s2 S

xswi k = zk 8i 2 N; 8k 2 [1; KM ax]

X

s2 S KXM ax

k= 1

yswp = 1 8w 2 W

X

w2 W

(50)

X

w2 W

X

s2 S

sxswi p=

X

w2 W

X

s2 S

sxswj k 8i; j 2 N ji 2 Dj; 8k 2 [1; KM ax]

X

i 2 N

xswi k M yswk 8w 2 W; 8s 2 S; 8k 2 [1; KM ax]

xswi k = 0 8w 2 W; 8s 2 S; 8i 2 Iw; 8k 2 [1; KM ax]

X

w2 W

fswk Fp M (1

X

w2 W

yswk) 8s 2 S; 8k 2 [1; KM ax]

X

i 2 N

pwivswi k = yswk 8w 2 W; 8s 2 S; 8k 2 [1; KM ax]

fswk M yswk 8w 2 W; 8s 2 S; 8k 2 [1; KM ax]

vswi k fswk M (1 xswi k) 8s 2 S; 8w 2 W; 8i 2 N; 8k 2 [1; KM ax]

(51)

vswi k

fswk = 0

w s k

vswi k

(52)

W = f w1; w2; w3g

f w1g f w2g f w3g f w1; w2g f w1; w3g f w2; w3g f w1; w2; w3g

f f w1g; f w2g; f w3gg f f w1g; f w2; w3gg

f f w1; w2g; f w3gg f f w1; w3g; f w2gg f f w1; w2; w3gg

f w2g

f w1; w3g

(53)

1

40+181 12; 41s

1

(54)

(55)

(56)

KM ax

i < j i < k

(57)

xi i

xi = 0; 8i 2 Iw;

xi + xj 1 + xk; 8i; j ; k 2 N nIwji < k; k < j :

i j k

X

i

xi

xi = 1 i

x1; x2; x3; x4 x1 < x2 < x3 x4

x2

(58)

0 1 3 6 2 4 5 7 8 w w

w 2 W Tw = ;

I Iw

s i

I IwnI g

s 6 j 2 Tw

Tw = Tw[ s

Tw

w I I

(59)

Iw

w1 w2

f 1; 2; 4; 5; 8g w1 f 2; 3; 4; 5; 6; 7; 8g w2

0

I = ; w Iw = f 5; 7g

Iw f ; ; f 5g; f 7g; f 5; 7gg I = ; s = f 0; 1; 2; 3; 4; 6g

I = f 5g s = f 6g 5

I = f 5; 7g I = f 7g s = f 1; 2; 3; 4; 8g

(60)

w

w Tw

awti i

t w ywt w

t

jTwj X

t= 1

ywt 1; 8w 2 W

X

w2 W jTwj X

t= 1

ai wtywt 1; 8i 2 N:

Wi

(61)

X

w2 W jTwj X

t= 1

ywt

P jTwj

t= 1 ywt = 1

P jT_{w 1}j t= 1 yw1t+

P jT_{w 2}j t= 1 yw2t+ ::: + P jTw nj

t= 1 ywnt < = n 1 w1; w2; :::; wn

Tw

Wr = ; Lc= ; Lw = ;

i = 0

9w 2 W Lw; s 2 Twji 2 s

Lc= Lc[ f sg

Lw = Lw[ f wg

((N Lc) = ; )

Wr = Wr [ f Lwg

Lw

Lc

i 2 (N Lc)

set list 6= ;

Lw

Lc

Wr

f w1; w2; :::wng

(62)

i = 0

w = w1 s = Sw1;1 = f 0; 1; 3; 6; 7; 8g i 2 s Lw = f w1g Lc =

f Sw1;1g w = w2 w2

s = Sw2;1 = f 1; 2; 3; 4; 6g 2 2 s Lw = f w1; w2g

Lc = f Sw1;1; Sw2;1g 5

w2 Lw Sw2;1 Lc

2 s = Sw2;2 = f 1; 2; 3; 4; 8g Lw = f w1; w2g Lc= f Sw1;1; Sw2;2g

5 w2

Sw2;2 w = w2

w1 Sw1;1

0 s = Sw2;1 = f 0; 1; 2; 3; 4; 6g

Lw = f w2g Lc = f Sw2;1g i = 5

s = Sw1;2 = f 5; 6; 7; 8g Lw = f w2; w1g Lc = f Sw2;1; Sw1;2g

(63)

KM ax = 1

WR = f W1; W2; :::WjWRjg Cr

Wr qwr w

Wr er Wr

WR

M ax X

r 2 WR 1 Crer

jWXRj

r = 1

qwrer = 1; 8w 2 W:

KM ax = 2 Wi 2 WR

Wj 2 WRjWj = W Wi Ci Cj

Wi Wj

C = 1

1

C 1+C 21 =

C1 C2 C1+ C2

KM ax 3

(64)

Wi

Wr

solucoes = ;

Wi Wr

[ Wi = W

W1\ W2= ; 8W1; W22 Wi

solucoes = solucoes [ f Wig

solucoes

k

KM ax = 2 KM ax = 3

KM ax = 2

(65)

KM ax = 2

KM ax = 3

(66)

(67)

(68)

w i; j s r k W N S R

Dj j

Iw w

KM ax

pwi i w

F

zwi w i

ti sr i s

r

xswi r i w

s r

yswr w w

r

askr s r k

Fr r

fswr w s r

vswi r

(69)

M ax X

w2 W

X

i 2 N

zwi

s:a: X

s2 S

sti sr

X

s2 S

stj sr 8i; j 2 N ji 2 Dj; 8r 2 R

X

w2 W

xswi r k(ti sr + askr 1) 8s 2 S; 8k 2 [0; KM ax]; 8i 2 N; 8r 2 R

xswi r ti sr 8s 2 S; 8w 2 W; 8i 2 N; 8r 2 R

KXM ax

k= 0

askr = 1 8s 2 S; 8r 2 R

X

w2 W

yswr = KXM ax

k= 0

kaskr 8s 2 S; 8r 2 R

X

s2 S

yswr = 1 8w 2 W; 8r 2 R

X

s2 S

ti sr = 1 8i 2 N; 8r 2 R

xswi r yswr 8s 2 S; 8w 2 W; 8i 2 N; 8r 2 R

xswi r = 0 8w 2 W; 8s 2 S; 8i 2 Iw; 8r 2 R

X

w2 W

fswr Fr M as0r 8s 2 S; 8r 2 R

X

i 2 N

pwivswi r = yswr 8w 2 W; 8s 2 S; 8r 2 R

fswr M yswr 8w 2 W; 8s 2 S; 8r 2 R

vswi r fswr M (1 xswi r) 8s 2 S; 8w 2 W; 8i 2 N; 8r 2 R

vswi r M xswi r 8s 2 S; 8w 2 W; 8i 2 N; 8r 2 R

zwi

X

r 2R

xswi r 8s 2 S; 8w 2 W; 8i 2 N

X

r 2 R

Fr F

(70)

k

vswi r

vswi r = fswr xswi r = 1

vswi r = 0

(71)

(72)

c i

dwi w

c = c (1 + dwi)

(73)

R

Fr r

ar wi i w

r F

xr r

zwi i w

M ax X

w2 W

X

i 2 N

zwi

s:a: X

r 2 R

Frxr F

zwi

X

r 2 R

ar wixr 8w 2 W; 8i 2 N

X

r 2 R

(74)

i r

i

i r Fi + Fr (jPj 1) F

Fi F Fr (jPj 1)

pool

(75)

xswi lr w i

s l r

yswlr w s

l r

fswlr w s l r

vswi lr

Flr l r

zwi i w

M ax X

w2 W

X

i 2 N

zwi

s:a: X

w2 W

X

s2 S

xswi lr = 1 8i 2 N; 8l 2 [1; LM ax]; 8r 2 R

X

s2 S

yswlr = 1 8w 2 W; 8l 2 [1; LM ax]; 8r 2 R

X

w2 W

yswlr 1 8s 2 S; 8l 2 [1; LM ax]; 8r 2 R

X

w2 W

X

s2 S

sxswilr =

X

w2 W

X

s2 S

sxswj lr 8i; j 2 N ji 2 Dj; 8l 2 [1; LM ax];

8r 2 R X

i 2 N

xswi lr M yswlp 8w 2 W; 8s 2 S; 8l 2 [1; LM ax];

8r 2 R

xswi lr = 0 8w 2 W; 8s 2 S; 8i 2 Iw;

8l 2 [1; LM ax]; 8r 2 R

X

w2 W

fswlr F M (1

X

w2 W

yswlr) 8s 2 S; 8l 2 [1; LM ax]; 8r 2 R

X

i 2 N

ewivswi lr = yswr 8w 2 W; 8s 2 S; 8l 2 [1; LM ax];

8r 2 R

fswlr M yswlr 8w 2 W; 8s 2 S; 8l 2 [1; LM ax];

(76)

vswi lr fswlr M (1 xswi lr) 8s 2 S; 8w 2 W; 8i 2 N;

8l 2 [1; LM ax]; 8r 2 R

vswi lr M xswi lr 8s 2 S; 8w 2 W; 8i 2 N;

8l 2 [1; LM ax]; 8r 2 R

zwi

X

r 2 R

xswi lr 8s 2 S; 8w 2 W; 8i 2 N;

8l 2 [1; LM ax]

X

l2 [1;LM ax] X

r 2R

(77)

c k

n = 0 t(Sn) = 0

Uw = W

U = N

w 2 W

Tw w

k w Tw

Uw = Uwnw U = UnTw n = n + 1

U = ;

(78)

2 n=(jWj (1 + jWj))

R

Fr r

ar wi i w r

br w w r

F

xr p r p

zwi i w

M ax X

w2 W

X

i 2 N

zwi

s:a: X

p2 P

X

r 2 R

Frxr p F

zwi

X

p2 P

X

r 2 R

ar wixr p 8w 2 W; 8i 2 N

X

r 2R

(79)

ti slr i s

l r

xswi lr i w

s l r

asklr s l r k

Flr l r

fswlr w s l

r vswi lr

M ax X

w2 W

X

i 2 N

(80)

s:a: X

s2 S

sti slr

X

s2 S

stj sr 8i; j 2 N ji 2 Dj; 8l 2 [1; LM ax];

8r 2 R X

w2 W

xswi lr k(ti slr + asklr 1) 8s 2 S; 8k 2 [0; KM ax]; 8i 2 N;

8l 2 [1; LM ax]; 8r 2 R

xswi lr ti slr 8s 2 S; 8w 2 W; 8i 2 N;

8l 2 [1; LM ax]; 8r 2 R KXM ax

k= 0

asklr = 1 8s 2 S; 8l 2 [1; LM ax]; 8r 2 R

X

w2 W

yswlr = KXM ax

k= 0

kasklr 8s 2 S; 8l 2 [1; LM ax]; 8r 2 R

LXM ax

l= 1

X

s2 S

yswlr = 1 8w 2 W; 8r 2 R

X

s2 S

ti slr = 1 8i 2 N; 8l 2 [1; LM ax]; 8r 2 R

xswi lr yswlr 8s 2 S; 8w 2 W; 8i 2 N;

8l 2 [1; LM ax]; 8r 2 R

xswi lr = 0 8w 2 W; 8s 2 S; 8i 2 Iw;

8l 2 [1; LM ax]; 8r 2 R

X

w2 W

fswlr Flr M as0lr 8s 2 S; 8l 2 [1; LM ax]; 8r 2 R

X

i 2 N

ewivswi lr = yswlr 8w 2 W; 8s 2 S; 8l 2 [1; LM ax];

8r 2 R

fswlr M yswlr 8w 2 W; 8s 2 S; 8l 2 [1; LM ax];

8r 2 R

vswi lr fswlr M (1 xswi lr) 8s 2 S; 8w 2 W; 8i 2 N;

8l 2 [1; LM ax]; 8r 2 R

vswi lr M xswi lr 8s 2 S; 8w 2 W; 8i 2 N;

8l 2 [1; LM ax]; 8r 2 R

zwi

X

r 2 R

xswi lr 8s 2 S; 8w 2 W; 8i 2 N;

8l 2 [1; LM ax] LXM ax

l= 1

X

r 2 R

(81)

z zmi n zmax

z = zmi n Q = 0% z = zmax Q = 100%

Q

Q = z z_z mi n

max zmi n

(82)

zmi n = jN j

jPj jWj

jWjjN j zmax = min(jPj; jWj)jN j

Q = _{min(jPj; jWj)jN j jN j =}z jN j _{(min(jPj; jWj) 1)jN j}z jN j

Q > 100%

(83)

F

(84)

(85)

(86)

(87)

(88)

(89)

(90)

(91)

(92)

(93)

(94)

(95)

KM ax = 2

KM ax = 3

(96)

KM ax = 2

(97)

KM ax = 3

(98)

(99)

KM ax = 2

(100)

KM ax = 3