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Copyright ., 2012.

. All rights reserved.

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ABSTRACT .......................................................................................................................... 13

1: ..................................................................................................... 14

1.1 ......................................................................................................................... 15 1.2 ......................................................................... 16 2 : ................................................... 18

2.1 ..................................................................................................................... 19 2.2 ........................................................................ 20 2.3 ............................................. 23 2.4 ........................................... 25

3 : ................................... 30

3.1 ........................................................................ 31 3.1.1 ............. 31 3.1.2 ................................. 33

3.1.2.1 - ...................................................... 34 3.1.2.2 ....................................................... 37

3.2 ..........................................................................39

3.2.1 ............................................................................................................... 39 3.2.2 I ......................................................................... 39

3.2.3 (a priori methods) ................................................................ 41 3.2.4

(interactive methods) ........................................................................... 44

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3.2.5

(a posteriori methods) ......................................................................... 49 3.3 .................................................................................................. 53

4 : ........................................................... 55

4.1 ..................................................... 56 4.2 .................................................... 59 4.3 .................................................................................................................. 61 4.4.

(MIXED MODEL ASSEMBLY LINES).............................................. 63

5 : GAMS........................................................................................ 68

5.1 GAMS ...................................................................................... 69 5.2 GAMS........................................................................... 70

5.2.1 ................................................................................................................. 72 5.2.2 ........................................................................................ 73 5.2.3 ........................................................................................................ 74 5.2.4 ........................................................................................................ ..... 75

5.2.5 ............................................................................................ 77 5.2.6 ............................................................................................... 79 5.2.7 .............. 79 5.2.8 ......................................................... 80 5.2.9 GAMS ............................................................................................. 81 5.2.10 .......................................................................................... 84

6 : ........................................................................................................ 85

6.1 ................................................................. 86 6.2 ............................................................................................ 87 6.3 ................................................................................................................ 88 6.4 ....................................................................................................................... 90 6.5 ........................................................................... 91

7 : ........................................................................ 94 7.1 H...................................................................................................................... 95 7.2 (LEXICOGRAPHC OPTIMISATION) ....................... 96

7.3 (GLOBAL CRITERION) ................................................ 101 7.4 (SATISFACTORY LEVELS) ................................................... 105

7.5 ME (GLOBAL CRITERION SATISFACTORY LEVELS)............................................................ 109

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7.6 (-CONSTRAINT METHOD)....................................... 113 7.7 ...................................................................................... 119

8 : - ............................................................ 121

( GAMS).............................................................................. 126

( C)....................................................................................... 136

................................................................................................................. 140

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2.1: 2.2: 2.3: 2.4:

2.5: (i) (Di) 2 3.1: 3.2: AUGMECON 4.1: 4.2: 6.1: 6.2: 20 7.1.: 7.1.: 1 4 7.2: 7.3: 7.4: ( ) 7.5: 7.6: ( ) 7.7: 7.8: ( ) 7.9: 7.10: ( ) 7.11: 7.12: 7.13: 7.14: 7.15: A 7.16: A 1 7.17: A 2 7.18: 7.19: (Satisfactory Levels) 7.20: 7.21: 7.22: 7.23: 7.24: A 7.25: A 1 7.26: A 2 7.27: 7.28: (Global Criterion) &

(Satisfactory Levels) 7.29: A 7.30: A 7.31: A 2 7.32: 7.33: case study 7.34: case study 7.35: case study 7.36: 7.37: 7.38: 7.39: A 7.40: A 1 7.41: A 2

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7.42: 7.43: ,

. %, , .

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2.1: 3.1: 5.1: GAMS 5.2: GAMS

5.3: , 5.4: 7.1: 7.2: K 7.3: 7.4: 7.5: 7.6: 7.7: 7.8: case study 7.9: 7.10: , Pareto front 7.11: , Pareto front 7.12: 7.13: ,

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, . 2012 2012.

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, , , . , .

2012

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. , , , .

, , . , , . , , , , , . . (ixed Model Assembly Lines) . , (a priori, interactive a posteriori) GAMS. , C. , .

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ABSTRAC

Production planning is a very important issue for today's industries because with proper and effective response can yield significant advantages and benefits to companies. Also due to the high complexity of, is a big challenge for research, which therefore lately has focused on one of the most important problems of the science of organization and management, the problem of scheduling tasks in the production process.

Furthermore, as we are trying to approach as more accurately as possible the growing complexity problems of the modern world, we notice the introduction of more than one criterion, in the relevant analyses and decisions we have to take. Within this context, the area of Multicriteria Analysis could not be found despite a massive bloom of and invited to standardize the way in which the individual decision maker is involved in making a decision. The existence of multiple criteria, makes the collection, processing and utilization of information quite different from what we are accustomed to single objective corresponding analyses, offering far more choices and dire prospects decide, but who should be able to do the right choices in tactics and methodology to follow.

This thesis is an intersection of these scientific areas and asked to show how multi-criteria decision analysis can address the challenges posed by the complexity inherent in modern production environments products. This happens in practice through the study of scheduling tasks in assembly lines mixed model (Mixed Model assemble lines) and the drafting of an effective plan of production through the use of different methods for Multicriteria Analysis.

Specifically, I studied and evaluated techniques from all the different classes of methods for Multicriteria Analysis (a priori, interactive and a posteriori) after modeling and solving the corresponding integer mathematical programming problems in an environment GAMS. To enable the processing of large volumes of data, I also developed a related application in a language C. In the end, I present a comparative presentation of all techniques and the different effects on the production plan which emerged after the choices one decides, according to the profile - preferences that we have in this thesis.

Keywords: production planning, scheduling work, mixed model assembly lines, operations research, multivariable analysis, integer mathematical programming, decision making

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1

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1.1

, . , . , ( - ), - , .

, () . , ( ) . ( ) . 50 , . , , , (Steuer, 1989). , , . , (Multiple Criteria Decision Making), 1970 1970 (eleny, 1982).

(Multiple Objective Mathematical Programming) , . , , .

. , .

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. ( ), ( ), , ( ). , a priori, a posteriori, . a priori , a posteriori . , a posteriori ( ) , . , .

, , , , , .

1.2

. , , . , , . . , . , . , . , , . .

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, . , . GAMS, . , , . , , , , . . , (Lexicographic Optimization), (Global Criterion), (Satisfactory Levels) (-constraint). . , GAMS C, .

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2

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2.1

() . (Decision Maker), () . (Multiple Criteria Decision Making, MCDM) (Operational Research) .

. , . , , - , , .

. , , .

, . . , , , .

, . , . .

,

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, . ( , , ), , 2 . .

2.2

.

Roy (1985), , . ( 2.1) .

: Roy (1985)

2.1:

, , .

V V

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1:

(alternatives or actions) . () . (continuous set) (discrete set).

, (decision problematic), , .

O Roy (1985) ( 2.2): 1. : (choice)

2. : (sorting)

, .

3. : (ranking) .

4. : (description) , .

, . , , .

: (2000) 2.2:

1. X3

2. X1

3. X5

4. X6

5. X4

6. X2

X1, X

2, X

3, X

4, X

5, X

6

X1, X

2, X

3, X

4, X

5, X

6

X3

1

X2, X

4, X

5

2

X1, X

3, X

6

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2:

. , .

. (Roy, 1985). :

) , ) (, 2008).

, . (consistent family of criteria).

, Roy (1985) 2.3. (2008), . , (dimension) , (preference scale) .

: Roy (1985)

2.3:

g

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3:

( , ), (global evaluation model). . :

) , ) , ) , .

:

1. . ( , ).

2. . .

4:

, .

, , , .

2.3

. , . ,

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. Roy (1985) , :

1. (unique synthesis criterion).

2. (outranking synthesis approach).

3. (interactive local judgment approach).

(2008), :

1. (compensatory models) .

2. (non compensatory models) , .

, , Roy (1985), :

1. : .

2. : .

3. : .

Pardalos et al. (1995) , , . , , .

1. (multiobjective mathematical programming). 2. (multiattribute utility theory). 3. (outranking relations). 4. - (preference disaggregation approach).

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2.4, , , , - , . , , . , .

: (2000)

2.4:

2.4

, , . , .

. , CAUSE (Criteria, Alternatives, Uncertainty, Stakeholders, Environment) .

-

-

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( 2.1), , , ( ) .

Ei gij, j, . .

: (2008) 2.1:

, , , , .

:

(efficient dominant) (non-efficient dominated).

(efficiency, dominance) Pareto , . Pareto Pareto .

2.5, , i .

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: (2008) 2.5: (i) (Di) 2

, , 1 2 , 3 2 1, 2, 1. , 4, 5 EN, E3 1, K2. , i, Pareto . i 1 2. . , Di, i, . 2.5 Di, 4 , , E2 Dk.

, . , , .

:

: , , . .

: (, .) ( ).

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.

, (decision makers) , , . (stakeholders) , , . , , .

, , . , .

. , , .

:

(intra-criterion preferences): . , , .

(inter-criterion preferences): . .

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, , , . wj, , .

. , (Direct Rating) (Point Allocation), , (Swing) (SMART). , , .

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3

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3.1

O (Mathematical Programming) . , . , .

, , , , . , , .

. , . , , , , . , , .

, .

3.1.1

, , . (Winston and Venkataramanan, 2003):

( ) (Linear Programming), (Non Linear Programming).

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. Simplex 60 , . . .

. . . ( ), . 95% 0 1. (Integer Programming). (Mixed Integer Programming). (branch and bound), .

. (Stochastic Programming) (Fuzzy Programming). .

, ( ), (Multiobjective Programming, Multicriteria Programming). (vector optimization) (scalar optimization) . 70

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, . ( ) (, ) , .

, . , , . , , , .

3.1.2

(Multiple Objective Mathematical Programming) : (Multiple Criteria Decision Making) . , . . , () .

, . . , , , , .

, . .

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3.1.2.1 BA -

, (ill structured problems), ( ) , .

, . , , , , . ( ) , . , , .

. , . 30 , Simplex. , , . ( 2000).

, , . .

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(efficient, non-dominated solution):

x ( Pareto , Pareto) x S x S fi(x) fi(x) i=1,2,,p fi(x) > fi(x) i.

Pareto . , Pareto , . Pareto z = (f1 (x), , fp (x)) Rp. x S fi(x) fi(x) i = 1, 2, , p x (dominates) x x x. To Pareto Pareto (Pareto set). , Pareto.

, , . .

(final or best compromise solution):

, . .

(efficient extreme solution):

() .

(criterion vector):

x = (x1, x2, ,xn) Rn Rp () z = (z1, z2, , zp) z1=c1x, z2=c2x . z .

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(ideal point):

( , ideal vector) Rp, . , g* gi. ( ). (utopia point). ( 3.1).

: (1998) 3.1:

( S )

(alternative optima):

z* x1*, x2*, xk* k

(payoff table):

, , (pay-offtable). : gi(x), i= 1, 2, , n .

, , , , , ( 3.1).

, ,

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. , - (gij) : gj gk (gijgik) i, . . ( 1998).

: (1998) 3.1:

3.1.2.2

(linear vector maximum problem). . . :

max {c1 x = z1 } max {c2 x = z2 } . . . . . (2.1) max {cp x = zp } s.t. x S = {x Rn Ax = b, x 0, b Rm }

S: n: m: p: ci: i zi: i : (m n) b : (m 1) ( ) x : (n 1)

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. ( A, b ci) . , ( ). ( ) S. , . S ( ) . : : , . . . : . . ( ), (vector optimization). : . ( ) . . : ( ) . . (.. ).

, Rn, Rp. . , Rp Rn. , p, , n , .

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3.2

3.2.1

, ( Pareto ) . (Hwang and Masud, 1979). :

1. ( a priori, .. , goal programming),

2. ( , interactive methods)

3. ( a posteriori, . , generation methods).

.

3.2.2 I

, , . Edgeworth 1881 Pareto 1906, (Stadler, 1988).

Pareto (ophelimity) . , von Neumann Morgenstern (game theory). 1951 Koopmans Pareto, . Kuhn Tucker .

,

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. (goal programming) Charnes Cooper 1957 1961. Miller Starr 1959 , .

1960 Kuhn-Tucker , L. Zadeh, (fuzzy sets). H. Raiffa 1969 (multiattribute utility theory). 1960 1970 Geoffrion 1968, Benayoun .. 1969 ( STEM) Philip (1972), Evans Steuer (1973) Zeleny (1974). 1972 .

1970 . . (White, 1990).

1980 1990 , , . . . 1987-1992 153 1216 . 1994 217 31 , .

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3.2.3 (a priori methods)

. :

(lexicographic optimization):

. . ( ). .

... . .

. , .

:

(Global criterion method) , , .

(value function):

u(x) = u [(g1(x), g2(x), ..., gn(x)]

:

[max] u(x) x S

,

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. (multi-attribute utility theory). , u(x) , g1, g2, ..., gn :

( ) igi(x)

p1, p2, ..., pn .

:

u(x) > u(y) x y

u(x) = u(y) x y (3.1)

pi, i=1, 2,..., n , x y :

x: g1 g2... gi... gn

y: g1- g2... gi+1... gn

, gi y x g1 y.

, x, g1 gi, x y( x y). 3.1:

u(x) = u(y)

p1g1+ p2g2 + ... + pigi + ...pngn= p1 (g1-) + p2g2+ ... + pi(gi+1) + ... + pngn

:

= pi/ p1, i

, p1= 1 , (pi= ), , gi. , (trade off) .

( ) (g1, gi), i=1, ..., n

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() , .

, , , . , , , - , , , .. , . ( 1998)

, . , .

:

O a priori , ( ) ( [0, 1] . . (. Steuer 1989).

(scaling), (scaling factor). , , , . , (Keeney and Raiffa, 1976).

-Goal Programming:

, a priori - . O Charnes Cooper 1957 1961 (Ignizio, 1982). .

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- . . (preemptive goal programming) ...

( ). , .

- , . , , . . , .

3.2.4 (interactive methods)

(interactive methods) . . . : . , , . .

. , ,

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() . , - - . , , , .

1970 . . , , (reduced feasible region), (reduced weighting vector space) (direction or line search methods) (Steuer 1989, pp 361). (reference point) Rp - (aspiration point) (reference point methods). .

STEM (Step Method):

Benayoun, de Montgolfier, Tergny Laritchev 1971 (Benayoun et al, 1971) . STEM - . , , , .

min-max , Tchebycheff. (payoff table) . , .

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. .

STEM, , . . - . , (Steuer 1989, . 365). , , , (Isserman and Steuer, 1987, Reeves and Reid, 1988).

GDF ( Geoffrion, Dyer, Feinberg):

1972 (Geoffrion et al, 1972) (line search) . . Frank-Wolfe (steepest descent).

. (local gradients) . , , . (trade offs) . (line search), . (grid search) : . . , .

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GDF . . , , , , .

Zionts-Wallenius:

1976 (Ziont and Wallenius, 1976) 1983 (Ziont and Wallenius, 1983). . i. . i. i . .

Zionts-Wallenius , . . .

Interval Criterion Weights:

Steuer 1977 (Steuer, 1977). . . . k- , (k). (k), . . , i , . , . .

. ,

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. , , , .

Interactive weighted Tchebycheff:

Steuer Choo 1983 (Steuer and Choo, 1983). Interactive weighted sums/filtering approach , , . Tchebycheff . .

(Satisfactory Levels) :

(Satisfactory levels) . O , . , . , , .

, :

1. Interactive weighted sums/filtering approach 2. Interactive Surrogate Worth Tradeoff 3. Achievement scalarizing function 4. VIG (Korhonen and Laakso, 1986) Pareto Race (Korhonen and

Wallenius, 1988) 5. TRIMAP

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3.2.5 (a posteriori methods)

a posteriori ( ), . , . (Evans, 1984) .

: , . , , , ( ). , , . : , ( ) , . ( ) . (Breslawski and Zionts, 1992). .

:

(weighting method). i , . ( ). i, .

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NISE (Non Inferion Set Estimation):

Cohon (1978) . . , . . . (maximum allowable error), ( ) . . o NISE.

Min-Max

min-max -. - p (Rp, p ). -, Rp, ( p ) - . - . ( 2000).

Lp (Family of Lp-metrics):

Rk , () Rk. Lp .

M (-constraint method)

(-constraint method). , . , , . ,

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(binding constraints). , . .

- . . , (Chankong and Haimes, 1983). . . , .

AUGMECON (Augmented -constraint method)

.

, p-1 . . . ( p ). . , , Pareto ( ) . Pareto . , Pareto .

( ).

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. .

.

, , : p-1 . i- () grid points. (+) grid points RHS () i- . (+)x(+) x x (+). . grid points , , . 3.2.

3.2: AUGMECON

AUGMECON 2

Augmecon2 Augmecon. Augmecon . k rk qk ,

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:

stepk= rk/qk

, sk.. k- :

fk(x) sk = ek

To ek :

ek = fmink + tstepk

fmink k- t gridpoints (t = 0...qk).

. p b :

b = int(sp/stepp)

int() , sp stepp -constraint p- .

sp stepp Pareto , . b.

, .

3.3

. , . 0 1 ( ) 0-1 .

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(Mixed Integer MOLP) 0-1 (Mixed 0-1 MOLP) .

. . , , , , , , . ( , , , .), .

(, , , ) :

-

. , , , . .

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4

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4.1

' ' .

, , . 4.1 . :

(flow-shop)

(job-shop)

(batch-shop)

(Projects)

: (2004)

4.1:

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, .

. . , . , , , .

. -. - .

, . , , . , . , , -. .

. , . , .

(off-line) . , . , , . .

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(on-line) . .

. , , . .

, , , , . . , , , . .

. . . . . , .

, .

, : , , . .

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, :

: (2004)

4.2:

4.2

. , , . . .

, , ( ), . , 14 , 21 28 , . .

, . , ,

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, , . , .

. , , , , , .

, , , . , , , , .

, , . , , . , , , .

. . , .

. , . , . , , . , , , , .

( ). . .

.

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. , . , , .

. , . , , . , , , , .

, . , , , . . , . , , , , .

4.3

, , :

.

, .

, . :

: , , .

: , .

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: , .

: .

. . , , , , (, , ) . . , , , . . . , . . , .

:

( / )

, &

&

,

,

. , .

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. , .

. . . - , , .

4.4 (MIXED-MODEL ASSEMBLY LINES)

, Just-In-Time (JIT),

, , . , "" . , (Monden, 1983, 1993). , , .

Just-In-Time (JIT) (Mixed Model Assembly Lines MMAL), , , , , (Monden, 1993). , ( Boysen, et al., 2009, ). H JIT. , .

(JIT MMALS-problem) , (

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- ), , . ( Miltenburg et ., 1990). , , heuristics metaheuristics ( , ) ( , .), ( Hyun et l., 1998, 1998 McMullen, 2000). , , , / ( Kubiak, 1993, ' Kindt Billaut, 2002 Boysen et al., 2009).

, , (Decision Maker - DM), () . , , , . (Multiple Objectives) , , , . H (MCDA) , , . T'kindt Billaut (2002).

, , - , , ( Figueira et al., 2005 Ehrgott et al., 2010 ). , , . , :

A priori (. , , ) Interactive (. , STEM, Zionts-Wallenius ) A posteriori (. , -constraint).

a priori, , . , ,

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. (interactive methods), , . , . . , Pareto, . Pareto non-dominated ( )

, , . , . , ( Miltenburg et ., 1990, ). , heuristics metaheuristics Pareto , . , , , (Miettinen, 1999, Ehrgott, 2000) .

JIT MMALS , (Boysen et al., 2009). (Monden, 1983): , , , , , . , . Monden Toyota , the Goal Chasing Method, (Monden, 1983).

Miltenburg et al. (1990) Monden . ,

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. Miltenburg Goldstein (1991) , . papers, .

Bard et al. (1994) trade-offs . - Tabu, branch and bound. Bautista et al.(1996) Zhu Ding (2000) . , .

Aigbedo and Monden (1997) , , , , - . Zeramdini et al.(2000) , . , . -, , .

Korkmazel Meral (2001) , . Kostaras et al. (2000), , . , .

O Hyun et al.(1998) MMAL , . :

. , . . ,

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, Pareto stratumniche cubicle. Tavakkoli-Moghaddam Rahimi-Vahed (2006) (MA), MMALS (Mansouri, 2005), .

McMullen (1998) , Miltenburg (1989), , Tabu Search (TS). McMullen Frazier (2000) TS. McMullen - Kohonen ( ) ant colony (McMullen, 2000), (McMullen, 2001a) (McMullen, 2001b). , SA TS .

(Rahimi-Vahed et al., 2007a) SFLA(shuffled frog-leaping algorithm) BO (bacteria optimization) (Rahimi-Vahed Mirzaei, 2007) MO-MMALS-, Hyun et al (1998). , . MOPSO (MO-Particle swarm optimization) (Rahimi-Vahed et al., 2007b) MO PSO TS (Mirghorbani et al., 2007). .

Rabbani et al. (2007) Javadi et al. (2008) - (GP), MMALS , Hyun et al. (1998), . GP, , , .

Mahdavi et al. (2009) , (FMOLP). , maxmin , . , , .

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5

GAMS

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5.1 GAMS

GAMS (General Algebraic Modeling System) , . , . . , GAMS, , , .

GAMS, . GAMS .

O GAMS , . (Rosenthal 2008).

GAMS :

. , ,

.

, , , , ,

, ,

, , ,

, ,

, ,

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, , . (Mc Carl 2008).

5.2 GAMS

GAMS, 5.1.

Inputs ( ):

Sets ()

Declaration - ( )

Assignment of members - ( )

Data (Parameters, Tables, Scalars) (, , )

Declaration - ( )

Assignment of values - ( )

Variables - ()

Declaration - ( )

Assignment of type - ( )

Assignment of bounds and/or initial values (optional) - ( )

Equations ()

Declaration - ( )

Definition - ( )

Model and Solve statements ( )

Display statement (optional) ( )

Outputs ( ):

Echo Print ( )

Symbol Reference Maps ( )

Equation Listings ( )

Status Reports ( )

Results ()

5.1. GAMS

[: Rosenthal 2008]

, GAMS:

GAMS GAMS. ( , ) .

GAMS . ,

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, .

; (semicolon). (compiler) GAMS , .

. . GAMS. , (*) GAMS. , , GAMS.

GAMS : (declaration) (assignment). . . , GAMS. GAMS, ,

31 .

GAMS. , * $ - .

(comment) , . GAMS, .

, , "*" . output. , GAMS. (blocks), GAMS . $ . $ontext $offtext. (Rosenthal 2008).

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5.2.1

(SETS) GAMS . S a, b c S = { a, b, c }. GAMS, , : set S / a, b, c / . set ( sets), S , a, b, c.

, GAMS :

SET Set Name Optional Explanatory Text

/ first Set Element Name Optional Explanatory Text

second Set Element Name Optional Explanatory Text

/;

(Set Name)

( ) GAMS, . , 31 . .

(Set Element Name) 10 . ( ). . , .

(Explanatory Text) 254 . , .

* . . , 1991 2000 : set t "time" /1991 * 2000 /;, 1991, 1992... , 2000. . , .

. : GAMS

73

5.2.2

GAMS , - , . , :

GAMS (Table). .

, :

Table Table Name (Set i, Set j ... ) Optional Explanatory Text

set_j_element_1 set_j_element_2

set_i_element_1 value_11 value_12

set_i_element_2 value_21 value_22;

(Table Name) ( ) GAMS. i, j.

(Scalars) GAMS . .

, GAMS :

Scalars

Scalar 1 Name Optional Explanatory Text / Numerical Value /

Scalar 2 Name Optional Explanatory Text / Numerical Value /

... ;

(Scalar Name)

( identifier). T (Numerical Value).

, GAMS (Parameter) :

. : GAMS

74

Parameters

Parameter Name (set dependency) Optional Explanatory Text

/ first Set Element Name Associated Value,

second Set Element Name Associated Value,

... /;

(Parameter Name) ( ). (data element), . / , . = ].

5.2.3

(Variables) . , . . , . .

, GAMS :

Variable type

first Variable Name (set dependency) Optional Explanatory Text

second Variable Name (set dependency) Optional Explanatory Text

;

(Variable type) . (Variable name) ( ) GAMS. : Free: , , -

+ Positive: , 0 +

. : GAMS

75

Negative: , - 0 Binary: , 0 1

Integer: , 0 100

.. GAMS . , , . , , . .

. GAMS, :

.lo .up .fx ,

. , .

.l . . .

.m .

.scale .

.prior .

5.2.4

(Equations), GAMS . , GAMS , .

, GAMS :

Equations

First Equation Name (set dependency) Optional Explanatory Text

Second Equation Name (set dependency) Optional Explanatory Text

;

. : GAMS

76

(Equation name) GAMS .

, :

Equation Name (set dependency) $Optional Logical Condition . .

Left Equation Terms Equation Type Right Equation Terms;

".." . (Left Term) , (Right Term) , .

(Equation Type) , : =E= (Equality): , =G= (Greater or equal):

, =L= (Lower or equal):

, =N= .

.

GAMS (Indexed Operations): sum : smin : smax :

. GAMS , Indexed Operations ( (Controlling Indices), Expression)

SUM (summation-

), . () sum (Controlling Indices) .

. : GAMS

77

, i xij ,

GAMS : sum(i, x(i,j)), , , ij cij*xij , GAMS : sum((i,j), c(i,j)*x(i,j)). smin smax ,

. smin smax sum.

card . , 1985 1995: set t time periods / 1985*1995 / card(t), 11, , 1985 1995. (c Carl, 2008).

5.2.5

(Model Statement) . GAMS :

Model Model Name Optional Explanatory Text / Model Contents / ;

(Model Name) GAMS, 10 . 80 .

(Model Contents) , all . , , .

. , . . :

. : GAMS

78

iterlim (iteration limit) , .

optcr (max relative MIP optimality gap) , . 0.1 (10%), , .

bratio (basis acceptance test) GAMS . 0.25 (25%), 1000 , GAMS 25%, 250 . 1 bratio , 0 GAMS. (Mc Carl, 2008).

GAMS . 2. . GAMS (.., ). , .

LP : Linear Programming - . ( ) .

NLP : Non-Linear Programming - . , , .

DNLP : Discontinuous Non-Linear Programming . - - ,

MIP : Mixed Integer Programming . , - .

RMIP : Relaxed Mixed Integer Programming - . .

MINLP : Mixed Integer Nonlinear Programming - . - - .

. : GAMS

79

RMINLP : Relaxed Mixed Integer Nonlinear Programming - . MINLP, .

MPEC : Mathematical Programs with Equilibrium Constraints - .

MCP : Mixed Complementarily Problem - M .

CNS : Constrained Nonlinear System - .

5.2. GAMS

[: Rosenthal 2008]

5.2.6

, (Solve Statement). GAMS (Solver) , . GAMS , . GAMS .

GAMS :

Solve Model Name Using Model Type Maximizing or Minimizing Equation Name;

, (Using) (Model Type), 2, .

5.2.7

GAMS , .txt. model_name.OUT. (Display). , :

FILE Local File Identifier / External File Location / ;

PUT Local File Identifier ;

PUT Item(s) ;

FILE , . External

File Location , ,

. : GAMS

80

PUT , . Local File Identifier, ( ), GAMS . Item(s) , , , , , .

PUT,

, , : PUT item 1: width: decimals, item 2: width: decimals, ... / ;

, .

, - (:), (width) , .

, - (:), (decimals), . (0), .

, : put CAPCHP.L:9:2, CAPABS.L:5:0 /;

:

9 ( )

CAPCHP.L, , .

5 ( ) CAPABS.L, .

Put close , GAMS. :

Put close Local File Identifier;

5.2.8

GAMS , . LOOP (), . GAMS :

. : GAMS

81

Loop ( (sets_to_vary),

Statements to execute

) ;

(sets to vary)

(indices) , (statement to execute) .

for :

for (scalar name = startvalue to endvalue, statements;

);

:

scalarname , ,

startvalue , endvalue , statements

.

5.2.9 GAMS

(Output) GAMS . GAMS . H , , (Compilation). :

(Echo Print) , , . .

(Error Report) . . GAMS

. : GAMS

82

. GAMS, , .

**** . , . . GAMS: , . , , , , .

(Symbol Reference Map) (, , , , , , ) , , .

, . $onsymxref .

3 , 4 :

GAMS

EQU Equation ()

MODEL Model ()

FILE Put file ( )

PARAM Parameter (, , )

SET Set ()

VAR Variable () 5.3. ,

[: Mc Carl, 2008]

. : GAMS

83

DECLARE

DEFINED ,

ASSIGNED

IMPL-ANS

CONTROL , , (sum, smax, smin).

REF ( )

5.4. [: Mc Carl, 2008]

(Equation Listing) . . , , . (Left Hand Side LHS), (Right Hand Side RHS) .

, . , . , , : Option limrow = r ; ( r, ).

(Column Listing). - . , . , , : Option limcol = c ; ( c, ).

. : GAMS

84

(- Minus Infinity), (0), (eps) (+ Plus Infinity) (Undefined), (Not Available) .

(Model Statistics) . , , , :

MODEL STATISTICS

BLOCKS OF EQUATIONS 48 SINGLE EQUATIONS 5,321

BLOCKS OF VARIABLES 40 SINGLE VARIABLES 5,567

NON ZERO ELEMENTS 18,898 DISCRETE VARIABLES 881

5.2.10

GAMS (Solve Summary), :

, , , ,

1.000 ,

1.000 (Solver Status),

(normal completion), (iteration interrupted) , , (non capable),

(Model Status),

(optimal), , (unbounded), (infeasible), , , , , (unknown), (non optimized), . (Mc Carl, 2008)

. : GAMS

85

6

. : GAMS

86

6.1

, , .

, . , first-in-first-out , , .

- - , .

( ), (Item Order, ). (, , , , .) (, : , ). , (IO) ( Short-term Production Planning - STP), , .

, , (s) , . A. (item) , , , , .

, :

: / . / , .

: ,

. : GAMS

87

( , , , , .) (, -, .) (ID .).

6.2

o A, , :

( ).

, , ,

, nlij, i l, j (nlij0)

: . , :

6.1.

6.2. 20

100

00

999

9

999

8

999

7

452

4

452

3

60

0

...

29

28

452

8

452

7

452

6

452

5

... ...

=10 =20

--- --- --- --- --- --- --- ---

--- --- --- --- --- --- --- ---

--- --- --- --- --- --- --- ---

20

...

2

1

4

3

=1

6

5

8

7

... ...

=2 ... ... ... ...

452

2

l

Set of Jobs JOs

Day x

Day x-1

Day x

Day x-1

Day 1

Day 1

Production

Production

Assembly Line 1

Assembly Line 2

. : GAMS

88

6.3

A, 3 :

: , .

/ : , , . , , . , , .

( ): , , .

A: : . [ ]

: . , yota Avensis Prius , Yaris Corolla (.. ABS, Air Conditioning, .). , , Yaris 1.3, Yaris 1.3 ABS. [ ]

: , . , , , 100 /, 100 /. . , , , . , o A, , . A, ( ). [ ]

. : GAMS

89

: , , (O). [ ] : ( ) +Dmax, . Dmax O. [ ]

: , , (.. , ). Toyota , 75 Air-conditions . [ ] : , . [ ]

: , . [ ] : / / . [/ ] :

o Air-conditions . [ ] o silver marketing. [/ ] o , , . . , 10 , 0 , , 20, [/ ]

. : GAMS

90

6.4

o A , , , :

(Sequence Length): , V. , 4 , , . , , , , . .

(JIT) : , . , ir-Conditions 200 , 2000 , 120 , 80 50 . , , , .

(Lateness): , . , . , , .

(Resources violation): , . . (distance ratio), . : ABS: 1/8 1 ABS 8 . , , (.. 2 ABS

. : GAMS

91

8 , ABS 1/8) , .

. , (.. ), . : , , ABS 1/8 1/4. : , 5 . . , .

o

A , , . :

V .

, o A, :

F = costsequence length+ costjust-in-time + costlateness + costutilization

, o A .

6.5

, . :

i: (i = 1, ..., n Families)

j: (i = 1, ..., n Days)

l: (l = 1, ..., n Lines)

dr: (dr = 1, ..., nDr)

m: (m = 1, ..., [dr])

k: (k = 1, ..., n Ctrt)

fam_linejl: 1 i l 0 .

expProdi: i.

. : GAMS

92

priorityi: .

late[prior]: .

demi: i.

cMinkjl: k j l.

cMaxkjl: k j l.

pMinjl: 1 j.

pMaxjl: 1 j.

minL: .

maxDiff: .

numeratordr, m / denominatordr, m: m- dr.

ec dr, l, j, m: m- dr, j l.

vcdr, m: m- dr

xik: 1 k, 0

, :

Xi,j,l : l j.

Zdr,l,j,m : m l j (Zdr, l, j, m = 1 m- dr j, l, 0)

Sj,l : l (Sj,l = 1, l j, 0)

X:

,

, : :

ij line(i)=line(j) place(i) place(j). (1) :

i, j, l: Xi,j,l = 0 if JOil = 0 (2) :

i, l and j > creationi + late[priorityi]: Xi,j,l = 0 (3) :

i : j l Xi,j,l = demi (4)

. : GAMS

93

, & : :

k, j, l: cMinkjl i/xik=1 Xi,j,l + j Sj,l cMinkjl cMaxkjl (5) :

j, l: pMinjl i/xik=1 Xi,j,l + j Sj,l pMinjl pMaxjl (6)

l, j minLength : Sj,l = 0, l: j Sj,l = 1 (7) :

dr, l and j:

m zdr,l,j,m pMaxjl (numeratordr,m/denominatordr,m) (idr) xi,j,l (8)

j: m zdr,l,j,m = 1 (9) X: 1-9. (Sequence Length):

Maximise f1(x) i j l Xi,j,l (Lateness):

Minimize f2(x) = i j l Xijl (j - expProdi)2

(JIT) :

Maximise f3(x)= l constrk j (50-j) i/k Xijl

(Resources violation):

Minimize f4(x) = dr,l,j,m (Zdr,l,j,m (ecdr,l,j,m + nvdr,l,d,m vcdr,m))

nvdr,l,j,m m- dr j 1.

. : GAMS

94

7

-

. : GAMS

95

7.1

, , , (Decision Maker, DM) (a priori methods), (progressive methods) (a posteriori methods) , .

, , . , ( ), () . - , / , , .

, , , . :

, ,

( , ).

, , . , :

a priori

(Lexicographic optimization)

O (Global criterion)

progressive

(Satisfactory Levels)

a posteriori

(-constraint)

O

. : GAMS

96

, :

850 6640

6.966 2685

2 4

15

25

4

7.1:

:

1. . ,

2. (scheduling), , (allocation).

7.2 (LEXICOGRAPHIC OPTIMISATION)

A, (pay-off table), . , 4 :

K

Sequence Length

Lateness

JIT

Resource Utilisation

7.2: K

. : GAMS

97

(DM) , o , , , . , , . , , , , . , (pay-off table):

7.1.:

, 1 4 :

7.1.: 1 4

1 2 3 4

Sequence Length 6928 4058 6897 5762

Lateness 129625 48106 246918 114800

JIT 357241 233713 377245 231884

Resource Utilisation 550 886 2239 190

0

50000

100000

150000

200000

250000

300000

350000

400000

Payoff Table

1 2 3 4

Sequence length 6928 4058 6897 5762

Resource utilisation 550 886 2239 190

010002000300040005000600070008000

Payofftable (2)

. : GAMS

98

, , , () :

7.2:

, , , :

7.3:

7.4: ( )

0,00%

20,00%

40,00%

60,00%

80,00%

100,00%

120,00%

C1 C2 C3 C4

Payoff Table (3)

ax C1

Min C2

Max C3

Min C4

Sequencelength

Lateness JITResourceutilisation

ax C1 Min C2 Max C3 Min C4

Sequence length 6928 4058 6897 5762

010002000300040005000600070008000

Sequence length (max )

ax C1 Min C2 Max C3 Min C4

Sequence length 100,00% 58,57% 99,55% 83,17%

0,00%

20,00%

40,00%

60,00%

80,00%

100,00%

120,00%

C1 Sequence length (max )

. : GAMS

99

7.5:

7.6: ( )

7.7:

Sequence length Lateness JITResourceutilisation

ax C1 Min C2 Max C3 Min C4

Lateness 129625 48106 246918 114800

0

50000

100000

150000

200000

250000

300000

Lateness (min )

ax C1 Min C2 Max C3 Min C4

Lateness 52,50% 19,48% 100,00% 46,49%

0,00%

20,00%

40,00%

60,00%

80,00%

100,00%

120,00%

C2 Lateness (min )

Sequence length Lateness JITResourceutilisation

ax C1 Min C2 Max C3 Min C4

JIT 357241 233713 377245 231884

050000

100000150000200000250000300000350000400000

Just in Time (max )

. : GAMS

100

7.8: ( )

7.9:

7.10: ( )

ax C1 Min C2 Max C3 Min C4

JIT 94,70% 61,95% 100,00% 61,47%

0,00%

20,00%

40,00%

60,00%

80,00%

100,00%

120,00%

C3 Just in Time (max )

Sequencelength

Lateness JITResourceutilisation

ax C1 Min C2 Max C3 Min C4

Resource utilisation 550 886 2239 190

0

500

1000

1500

2000

2500

Resource Utilisation (min)

ax C1 Min C2 Max C3 Min C4

Resource utilisation 24,56% 39,57% 100,00% 8,49%

0,00%

20,00%

40,00%

60,00%

80,00%

100,00%

120,00%

C4 Utilisation (min )

. : GAMS

101

7.3 (GLOBAL CRITERION)

, : pi , i = 1,2, ..., n trade-offs gi, i = 1,2, ..., n .

tradeoff sir

g

gi g

r

gr

g

gi. . s

ir

g

,

:

u(g) , tradeoffs :

, tradeoffs g

i g

r ( ),

i, .

, . , g

i

gr (pr=1).

: gr 1

gi?

: 1 gi m gr.

: (gi, gr) ; : , tradeoff . : (g

i, g

r)?

: .

1

(g)n

i i

i

u p g

g

1 2 1 2( , , , , , , , ) ~ ( , , 1, , , , )i r n i r ir ng g g g g g g g g s g

g

(g)

(g)i

ir

r

u

gs

u

g

. : GAMS

102

C1 (Sequence Length) , DM- :

1000 C2, 1

C1 1000 C3, 10

C1 100 C4, 10

C1

: F1 (x) = 1000*C1 1*C2 + C3*10 10*C4 (: 1000, -1, 10, -10) o , 6 , . :

Weight C1 Weight C2 Weight C3 Weight C4

Case Study 1 1000 -1 10 -10

Case Study 2 2000 -2 10 -10

Case Study 3 1000 -2 10 -20

Case Study 4 1000 -2 5 -20

Case Study 5 100 -1 10 -10

Case Study 6 100 -1 10 -50

7.3:

, , :

IDEAL

Case Study 1

Case Study 2

Case Study 3

Case Study 4

Case Study 5

Case Study 6

ax C1 Sequence

length 6928 6897 6928 6897 6897 6897 6897

Min C2 Lateness 48106 151408 147232 128797 123400 151618 149733

Max C3 JIT 377245 372443 367613 369235 367879 372463 371943

Min C4 Resource utilisation 190 427 398 304 357 425 255

7.4:

. : GAMS

103

, , :

7.11:

7.12:

7.13:

IDEALCase

Study 1Case

Study 2Case

Study 3Case

Study 4Case

Study 5Case

Study 6

Sequence length 6928 6897 6928 6897 6897 6897 6897

6880

6890

6900

6910

6920

6930

6940

C1 Sequence Length (max)

IDEALCase

Study 1Case

Study 2Case

Study 3Case

Study 4Case

Study 5Case

Study 6

Lateness 48106 151408 147232 128797 123400 151618 149733

0

20000

40000

60000

80000

100000

120000

140000

160000

C2 Lateness (min)

IDEALCase Study

1Case Study

2Case Study

3Case Study

4Case Study

5Case Study

6

JIT 377245 372443 367613 369235 367879 372463 371943

362000

364000

366000

368000

370000

372000

374000

376000

378000

C3 Just in Time (max)

. : GAMS

104

7.14:

(DM) , C1 C4. O , , case study 3.

, , case study :

7.15: A

7.16: A 1

IDEALCase

Study 1Case

Study 2Case

Study 3Case

Study 4Case

Study 5Case

Study 6

Resource utilisation 190 427 398 304 357 425 255

0

100

200

300

400

500

C4 Resource Utilisation (min)

0

100

200

300

400

500

600

700

800

900

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ITEMS (TOTAL)

0

100

200

300

400

500

600

700

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ITEMS (LINE 1)

. : GAMS

105

7.17: A 2

7.18:

7.4 (SATISFACTORY LEVELS)

(Satisfactory levels) . O , . , .

0

50

100

150

200

250

300

350

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ITEMS (LINE 2)

0,00%

20,00%

40,00%

60,00%

80,00%

100,00%

120,00%

1

16

31

46

61

76

91

10

6

12

1

13

6

15

1

16

6

18

1

19

6

21

1

22

6

24

1

25

6

27

1

28

6

30

1

31

6

33

1

34

6

36

1

37

6

39

1

DEMAND SATISFACTION

0,00%

20,00%

40,00%

60,00%

80,00%

100,00%

120,00%

1

16

31

46

61

76

91

10

6

12

1

13

6

15

1

16

6

18

1

19

6

21

1

22

6

24

1

25

6

27

1

28

6

30

1

31

6

33

1

34

6

36

1

37

6

39

1

. : GAMS

106

:

7.19: (Satisfactory Levels)

(DM) , , :

Sequence length Lateness JIT Resource utilisation

Cycle SL C1 SL C2 SL C3 SL C4 ax C1 Min C2 Max C3 Min C4

1 MAX 48106 377245 190 Not Feasible Not Feasible Not Feasible Not Feasible

2 MAX 70000 300000 220 Not Feasible Not Feasible Not Feasible Not Feasible

3 MAX 140000 300000 250 6928 139963 355705 250

4 MAX 130000 340000 250 6897 129981 340029 250

5 6800 MIN 360000 250 6800 114345 360000 250

6 6900 MIN 340000 250 6928 131648 353936 250

7 6800 140000 MAX 500 6897 139998 371130 500

8 6850 140000 MAX 350 6897 139959 370870 350

9 6850 130000 MAX 300 6897 129998 369445 300

10 6850 130000 MAX 250 6897 129984 367734 250

11 6800 140000 360000 MIN 6897 139997 360050 225

12 6850 130000 360000 MIN 6897 129998 360009 225

13 6850 100000 360000 MIN Not Feasible Not Feasible Not Feasible Not Feasible 7.5:

. : GAMS

107

:

7.20:

7.21:

7.22:

IDEAL 1 2 3 4 5 6 7 8 9 10

Sequence length 6928 6928 6897 6800 6928 6897 6897 6897 6897 6897 6897

6700

6750

6800

6850

6900

6950

C1 Sequence Length (max)

IDEAL 1 2 3 4 5 6 7 8 9 10

Lateness 48106 139963 129981 114345 131648 139998 139959 129998 129984 139997 129998

0

20000

40000

60000

80000

100000

120000

140000

160000

C2 Lateness (min)

IDEAL 1 2 3 4 5 6 7 8 9 10

JIT 377245 355705 340029 360000 353936 371130 370870 369445 367734 360050 360009

320000

330000

340000

350000

360000

370000

380000

390000

C3 Just in Time (max)

. : GAMS

108

7.23:

:

7.24: A

7.25: A 1

IDEAL 1 2 3 4 5 6 7 8 9 10

Resource utilisation 190 250 250 250 250 500 350 300 250 225 225

0

100

200

300

400

500

600

C4 Resource Utilisation (min)

0

100

200

300

400

500

600

700

800

900

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ITEMS (TOTAL)

0

100

200

300

400

500

600

700

800

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ITEMS (LINE 1)

. : GAMS

109

7.26: A 2

7.27:

7.5 O A (GLOBAL CRITERION SATISFACTORY LEVELS)

(Global Criterion) E (Satisfactorty Levels). , . , , .

0

50

100

150

200

250

300

350

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ITEMS (LINE 2)

0,00%

20,00%

40,00%

60,00%

80,00%

100,00%

120,00%

1

16

31

46

61

76

91

10

6

12

1

13

6

15

1

16

6

18

1

19

6

21

1

22

6

24

1

25

6

27

1

28

6

30

1

31

6

33

1

34

6

36

1

37

6

39

1

DEMAND SATISFACTION

0,00%

20,00%

40,00%

60,00%

80,00%

100,00%

120,00%

1

16

31

46

61

76

91

10

6

12

1

13

6

15

1

16

6

18

1

19

6

21

1

22

6

24

1

25

6

27

1

28

6

30

1

31

6

33

1

34

6

36

1

37

6

39

1

. : GAMS

110

, :

7.28: (Global Criterion) & (Satisfactory Levels)

Cycle SL C1 SL C2 SL C3 SL C4

1 6897 129998 360009 225 2 6928 139963 355705 250 3 6928 131648 353936 250

7.6:

Weight C1 Weight C2 Weight C3 Weight C4

Case Study A 2000 -2 10 -10

Case Study B 1000 -2 10 -20 Case Study C 100 -1 10 -50

7.7:

Sequence length

Lateness JIT Resource utilisation

Case study/Cycle ax C1 (X10) Min C2 Max C3 Min C4(X1000)

IDEAL 69280 48106 377245 190000

A1 68970 129992 362190 225000

A2 69280 139957 365094 250000

A3 Not Feasible Not Feasible Not Feasible Not Feasible

IDEAL 69280 48106 377245 190000

B1 68970 129992 362171 225000

B2 69280 139939 365097 250000

B3 Not Feasible Not Feasible Not Feasible Not Feasible

IDEAL 69280 48106 377245 190000

C1 68970 129983 362182 225000

C2 69280 139945 365111 250000

C3 Not Feasible Not Feasible Not Feasible Not Feasible

7.8: case study

. : GAMS

111

case study (2), , . :

7.29: A

7.30: A

7.31: A 2

0

100

200

300

400

500

600

700

800

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ITEMS (TOTAL)

0

100

200

300

400

500

600

700

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ITEMS (LINE 1)

0

20

40

60

80

100

120

140

160

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ITEMS (LINE 2)

. : GAMS

112

7.32:

7.33: case study

0,00%

20,00%

40,00%

60,00%

80,00%

100,00%

120,00%

1

16

31

46

61

76

91

10

6

12

1

13

6

15

1

16

6

18

1

19

6

21

1

22

6

24

1

25

6

27

1

28

6

30

1

31

6

33

1

34

6

36

1

37

6

39

1

DEMAND SATISFACTION

0,00%

20,00%

40,00%

60,00%

80,00%

100,00%

120,00%

1

16

31

46

61

76

91

10

6

12

1

13

6

15

1

16

6

18

1

19

6

21

1

22

6

24

1

25

6

27

1

28

6

30

1

31

6

33

1

34

6

36

1

37

6

39

1

ax C1 (X10) Min C2 Max C3 Min C4(X1000)

Sequence length Lateness JITResourceutilisation

IDEAL 69280 48106 377245 190000

A1 68970 129992 362190 225000

A2 69280 139957 365094 250000

0

50000

100000

150000

200000

250000

300000

350000

400000

Global Criterion - Satisfactory Level Case Study A

. : GAMS

113

7.34: case study

7.35: case study

7.6 EOO T (E-CONSTRAINT METHOD)

H , e-constraint , , . ,

ax C1 (X10) Min C2 Max C3 Min C4(X1000)

Sequence length Lateness JITResourceutilisation

IDEAL 69280 48106 377245 190000

B1 68970 129992 362171 225000

B2 69280 139939 365097 250000

050000

100000150000200000250000300000350000400000

Global Criterion - Satisfactory Level Case Study B

ax C1 (X10) Min C2 Max C3 Min C4(X1000)

Sequence length Lateness JITResourceutilisation

IDEAL 69280 48106 377245 190000

C1 68970 129983 362182 225000

C3 69280 139945 365111 250000

050000

100000150000200000250000300000350000400000

Global Criterion - Satisfactory Level Case Study C

. : GAMS

114

(. .. et al., 2007).

:

1. , .

2. , . , , .

3. ("" ) k k+1 . . "" . .

4. ( - ). . , .

5. , . .

6. , (trade-off) . ( 2000).

-constraint, , (Cohon, 1978 Chankong Haimes, 1983 Miettinen, 1998):

. : GAMS

115

:

A.

, B. , C. .

e-constraint (AUGMECON) ( 2009) .

i. , , (payoff table). .

ii. , - , Pareto ( Pareto ).

iii. , , . 2 .( ).

, e-constraint , ( , -constraint ( 1 - 8):

1.

( 7.10). 2. , e2 = z2

N = 89 3. , r1 = z2

N - z2I = 319 - 230 = 89

4. gridpoints 4 (g2), (r1), 10 .

5. r1 / (g2-1) =10 .

6. : Pareto. ( 7.11)

7. (DM) - : O Pareto front - (230 - 260),

8. , r2 = z2N - z2

I = 260 - 230 = 30

. : GAMS

116

9. gridpoints 4 (g2), (r2), 10 .

10. r2 / (g2-1) = 3 .

11. ( 7.12) 12. ( 7.13)

Lateness Resource utilisation

JIT Sequence length

Lateness 129625 319 353671 6928

Resource utilisation 145563 230 353134 6928

7.9:

7.36:

Lateness 145563 136969 131658 130627 130428 130168 129967 129827 129715 129625

Resource utilisation

230 240 250 260 270 280 290 300 310 319

7.10. , Pareto front

7.37:

230; 145563

319; 129625 128000130000132000134000136000138000140000142000144000146000148000

200 210 220 230 240 250 260 270 280 290 300 310 320 330

Payoff chart (Ideal and nadir points)

Point 2Point 1

120000

125000

130000

135000

140000

145000

150000

230 240 250 260 270 280 290 300 310 319

LATE

NES

S

RESOURCES UTILISATION

E-CONSTRAINT METHOD ( FIRST STAGE )

. : GAMS

117

7.11: , Pareto front

7.38:

Min C2 Min C4 Max C3 ax C1

Lateness Resource utilisation

JIT Sequence length

137035 236 354387 6928

7.12:

, , , :

7.39: A

120000

125000

130000

135000

140000

145000

150000

230 233 236 239 242 245 248 251 254 257 260

LATE

NES

S

RESOURCES UTILISATION

E-CONSTRAINT METHOD (SECOND STAGE)

0

100

200

300

400

500

600

700

800

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ITEMS (TOTAL)

Lateness 145563 142672 137035 136911 136734 132240 131986 131613 131470 131118 130977

Resource utilisation

230 233 236 239 242 245 248 251 254 257 260

. : GAMS

118

7.40: A 1

7.41: A 2

7.42:

0

100

200

300

400

500

600

700

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ITEMS (LINE 1)

0

50

100

150

200

250

300

350

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ITEMS (LINE 2)

0,00%

20,00%

40,00%

60,00%

80,00%

100,00%

120,00%

1

16

31

46

61

76

91

10

6

12

1

13

6

15

1

16

6

18

1

19

6

21

1

22

6

24

1

25

6

27

1

28

6

30

1

31

6

33

1

34

6

36

1

37

6

39

1DEMAND SATISFACTION

0,00%

20,00%

40,00%

60,00%

80,00%

100,00%

120,00%

1

16

31

46

61

76

91

10

6

12

1

13

6

15

1

16

6

18

1

19

6

21

1

22

6

24

1

25

6

27

1

28

6

30

1

31

6

33

1

34

6

36

1

37

6

39

1

. : GAMS

119

7.7

, , :

7.13: ,

7.43: , . % , , .

E-Constraint Global CriterionSatisfactory

LevelsGL with SL

ax C1 99,45% 99,01% 99,01% 99,45%

Min C2 284,86% 267,74% 270,23% 290,93%

Max C3 93,94% 97,88% 95,43% 96,78%

Min C4 124,21% 160,00% 118,42% 131,58%

0,00%

50,00%

100,00%

150,00%

200,00%

250,00%

300,00%

350,00%

ax C1 Min C2 Max C3 Min C4

Sequence length Lateness JIT Resource utilisation

E-Constraint 6928 137035 354387 236

Global Criterion 6897 128797 369235 304

Satisfactory Levels

6897 129998 360009 225

GL with SL 6928 139957 365094 250

. : GAMS

120

7.43: , . % , , .

E-Constraint Global CriterionSatisfactory

LevelsGL with SL

ax C1 99,45% 99,01% 99,01% 99,45%

Min C2 284,86% 267,74% 270,23% 290,93%

Max C3 93,94% 97,88% 95,43% 96,78%

Min C4 124,21% 160,00% 118,42% 131,58%

0,00%

50,00%

100,00%

150,00%

200,00%

250,00%

300,00%

350,00%

. : GAMS

121

8

-

. : GAMS

122

. , . , , , .

. , , , , .

E

GAMS. , (, , )

PRIORI

, , . , , , .

, , tradeoffs, , . , tradeoff 2 . , . , , .

. : GAMS

123

ME

, feedback. , , .

H Satisfactory levels . , , , . .

, Satisfactory levels, . , 2 . , (trade offs). A POSTERIORI

(a posteriori) a priori . , . , , .

, (a posteriori) . , , , . , . , . . .

. : GAMS

124

, -constraint , - . , , . -constraint . , -constraint , Pareto , .

, 2 , . , , , , .

, .

, , , , , .

, , , . , .

(gui), , , . , , .

, , , - . , - , , .

. : GAMS

125

, , , , trade off .

. : GAMS

126

K GAMS

. : GAMS

127

1. -CONSTRAINT ETHOD

Sets i / i1*i1342 /

j / j1*j12714 / ;

parameter

cons( i,j )