ISBN: 3540282300
TITLE: Multicriteria Scheduling
AUTHOR: T'kindt/Billaut
TOC: 

1. Introduction to scheduling 5
1.1 Definition 5
1.2 Some areas of application 6
1.2.1 Problems related to production 6
1.2.2 Other problems 7
1.3 Shop environments 7
1.3.1 Scheduling problems without assignment 8
1.3.2 Scheduling and assignment problems with stages 8
1.3.3 General scheduling and assignment problems 9
1.4 Constraints 9
1.5 Optimality criteria 12
1.5.1 Minimisation of a maximum function: "minimax" criteria 13
1.5.2 Minimisation of a sum function: "minimum" criteria 13
1.6 Typologies and notation of problems 14
1.6.1 Typologies of problems 14
1.6.2 Notation of problems 16
1.7 Project scheduling problems 17
1.8 Some fundamental notions 18
1.9 Basic scheduling algorithms 21
1.9.1 Scheduling rules 21
1.9.2 Some classical scheduling algorithms 22
2. Complexity of problems and algorithms 29
2.1 Complexity of algorithms 29
2.2 Complexity of problems 32
2.2.1 The complexity of decision problems 33
2.2.2 The complexity of optimisation problems 38
2.2.3 The complexity of counting and enumeration problems 40
2.3 Application to scheduling 48
3. Multicriteria optimisation theory 53
3.1 MCDA and MCDM: the context 53
3.1.1 MultiCriteria Decision Making 54
3.1.2 MultiCriteria Decision Aid 54
3.2 Presentation of multicriteria optimisation theory 55
3.3 Definition of optimality 57
3.4 Geometric interpretation using dominance cones 60
3.5 Classes of resolution methods 62
3.6 Determination of Pareto optima 64
3.6.1 Determination by convex combination of criteria 64
3.6.2 Determination by parametric analysis 70
3.6.3 Determination by means of the c-constraint approach 72
3.6.4 Use of the Tchebycheff metric 76
3.6.5 Use of the weighted Tchebycheff metric 79
3.6.6 Use of the augmented weighted Tchebycheff metric 81
3.6.7 Determination by the goal-attainment approach 86
3.6.8 Other methods for determining Pareto optima 91
3.7 Multicriteria Linear Programming (MLP) 92
3.7.1 Initial results 93
3.7.2 Application of the previous results 93
3.8 Multicriteria Mixed Integer Programming (MMIP) 94
3.8.1 Initial results 94
3.8.2 Application of the previous results 95
3.8.3 Some classical algorithms 97
3.9 The complexity of multicriteria problems 100
3.9.1 Complexity results related to the solutions 100
3.9.2 Complexity results related to objective functions 101
3.9.3 Summary 106
3.10 Interactive methods 107
3.11 Goal programming 108
3.11.1 Archimedian goal programming 110
3.11.2 Lexicographical goal programming 111
3.11.3 Interactive goal programming 111
3.11.4 Reference goal programming 112
3.11.5 Multicriteria goal programming 112
4. An approach to multicriteria scheduling problems 113
4.1 Justification of the study 113
4.1.1 Motivations 113
4.1.2 Some examples 114
4.2 Presentation of the approach 118
4.2.1 Definitions 118
4.2.2 Notation of multicriteria scheduling problems 121
4.3 Classes of resolution methods 122
4.4 Application of the process - an example 123
4.5 Some complexity results for multicriteria scheduling problems 124
5. Just-in-Time scheduling problems 135
5.1 Presentation of Just-in-Time (JiT) scheduling problems 135
5.2 Typology of JiT scheduling problems 136
5.2.1 Definition of the due dates 136
5.2.2 Definition of the JiT criteria 137
5.3 A new approach for JiT scheduling 139
5.3.1 Modelling of production costs in JiT scheduling for shop problems 141
5.3.2 Links with objective functions of classic JiT scheduling 145
5.4 Optimal timing problems 147
5.4.1 The $1\vertd_i,seq\vertF_\ell(\bar T1\alpha, \bar E^\beta)$ problem 147
5.4.2 The $P_\infty\vert prec, f_i convex\vert\Sigma_i f_i$ problem 149
5.4.3 The $1\vert f_i piecewise linear\vert F_\ell(\Sigma_i f_i, \Sigma_j I_j)$ problem 153
5.5 Polynomially solvable problems 153
5.5.1 The $1\vert d_i = d \ge\Sigma p_i 1\vert F_\ell(\bar E, \bar T)$ problem 153
5.5.2 The $1\vert d_i_= d unknown, nmit\vert F_\ell(\bar E, \bar T, d)$ problem 155
5.5.3 The $1\vert pi_ \subseteq [\underline p_i; \bar p_1i]\cap \mathbb N, d_i = d non restrictive\vert F_\ell (\bar E, \bar T, \bar {CC})$ problem 157
5.5.4 The $P\vert d_i = d non restrictive,nmit\vert F_\ell(\bar E,\bar T)$ problem 157
5.5.5 The $P\vert d_i = d unknown,nmit \vert F_\ell,(\bar E,\bar T)$ problem 159
5.5.6 The P\vert d_i = d unknown, p_i = p, nmit \vert F_\ell (\bar E, \bar T, d)$ problem 165
5.5.7 The R\vert p_{i,j} \in [\underline p_{i,j}; \bar p_{i,j}], d_i = d unknown\vert F_\ell(\bar T, \bar E,\bar{CC}^w)$ problem 169
5.5.8 Other problems 170
5.6 $\mathcal NP$-hard problems 173
5.6.1 The $1\vert d_i, nmit\vert F_\ell(\bar E^\alpha, \bar T^\beta)$ problem 173
5.6.2 The $F\vert prmu, d_i, nmit \vert F_\ell (\bar E^w, \bar T^w)$ problem 176
5.6.3 The $P\vert d_i = d non restrictive, nmit \vert f_max (\bar E^w, \bar T^w)$ problem 178
5.6.4 Other problems 182
5.7 Open problems 188
5.7.1 The $Q\vert d_i = d unknown, nmit\vert F_\ell(\bar E,\bar T)$ problem 188
5.7.2 Other problems 189
6. Robustness considerations 193
6.1 Introduction to flexibility and robustness in scheduling 193
6.2 Approaches that introduce sequential flexibility 195
6.2.1 Groups of permutable operations 195
6.2.2 Partial order between operations 197
6.2.3 Interval structures 199
6.3 Single machine problems 201
6.3.1 Stability vs makespan 201
6.3.2 Robust evaluation vs distance to a baseline solution 202
6.4 Flowshop and jobshop problems 203
6.4.1 Average makespan of a neighbourhood 203
6.4.2 Sensitivity of operations vs makespan 203
6.5 Resource Constrained Project Scheduling Problems (RCPSP) 204
6.5.1 Quality in project scheduling vs makespan 204
6.5.2 Stability vs makespan 205
7. Single machine problems 207
7.1 Polynomially solvable problems 207
7.1.1 Some $l\vert d_i\vert \bar C, f _max$ problems 207
7.1.2 The $1\vert s_i, pmtn, nmit\vert F_\ell(\bar C, P_max)$ problem 215
7.1.3 The $1\vert p_i \in [\underline p_i; \bar p_i], d_i\vert F_\ell(T_{max}, \bar {CC)^w)$ problem 216
7.1.4 The $1\vert p_i \in [p_i; \bar p_i], d_i\vert F_\ell(\bar C, \bar {CC)^w)$ problem 219
7.1.5 Other problems 219
7.2 $mathcal NP$-hard problems 222
7.2.1 The $1\vert d_i\vert \bar T, \bar C$ problem 222
7.2.2 The $l\vert r_i, p_i \in [\underline p_i; \bar p_i] \cap \mathbb N \vert F_\ell (C_{max}, \bar {CC}^w)$ problem 223
7.2.3 The $l\vert r_i, p_i \in [\underline p_i; \bar p_i] \cap \mathbb N \vert F_\ell (\bar U^w, \bar {CC}^w)$ problem 225
7.2.4 Other problems 226
7.3 Open problems 230
7.3.1 The $l\vert d_i\vert \bar U, T_{max}$ problem 230
7.3.2 Other problems 234
B. Shop problems 235
8.1 Two-machine flowshop problems 235
8.1.1 The $F2\vert prmu\vert Lex(C_{max}, \bar C)$ problem 235
8.1.2 The $F2\vert prmu\vert F_\ell(C_{max}, \bar C)$ problem 250
8.1.3 The $F2\vert prmu, r_i\vert F_\ell(C_{max}, \bar C)$ problem 256
8.1.4 The $F2\vert prmu\vert \in(\bar C/C_{max})$ problem 256
8.1.5 The $F2\vert prmu, d_i \vert # (C_{max}, T_{max})$ problem 262
8.1.6 The $F2\vert prmu, d_i \vert # (C_{max}, \underline U)$ problem 265
8.1.7 The $F2\vert prmu, d_i \vert # (C_{max}, T)$ problem 267
8.2 m-machine flowshop problems 270
8.2.1 The $F\vert prmu\vert Lex (C_{max}, \bar C)$ problem 270
8.2.2 The $F\vert prmu\vert # (C_{max}, \bar C)$ problem 272
8.2.3 The $F\vert prmu, d_i \vert \in (C_{max}, T_{max})$ problem 277
8.2.4 The $F\vert \underline p_{i,j} \in [\underline p_{i,j}; \bar p_{i,j}]prmu \vert F_\ell (C_{max}, \bar{CC^w)$ problem 280
8.2.5 The $ F\vert p_{i,j}=p_\in [\underline p_i;\overline p_i},prmu\vert #(C_{max},\bar{CC}^w)$ problem 281
8.3 Jobshop and Openshop problems 284
8.3.1 Jobshop problems 284
8.3.2 The $O2\vert\vert Lex(C_{max},\bar C)$ problem 284
8.3.3 The $O3\vert\vert Lex(C_{max},\bar C)$ problem 286
9. Parallel machines problems 287
9.1 Problems with identical parallel machines 287
9.1.1 The $P2\vert pmtn,d_i\vert \epsilon(L_{max},C_{max})$ problem 287
9.1.2 The $P3\vert pmtn,d_i\vert \epsilon (L_{max},C_{max})$ problem 290
9.1.3 The $P2\vert d_i\vert Lex(T_{max},\bar U)$ problem 293
9.1.4 The $P\vert d_i\vert # (\bar C, \bar U)$ problem 295
9.1.5 The $P\vert pmtn\vert Lex(\bar C,C_{max})$ problem 296
9.2 Problems with uniform parallel machines 297
9.2.1 The $Q\vert p_i = p\vert \epsilon (f_{max}/g_{max})$ problem 297
9.2.2 The $Q\vert p_i = p\vert \epsilon (\bar g/f_{max})$ problem 302
9.2.3 The $Q\vert pmtn\vert \epsilon (\bar C/C_{max})$ problem 303
9.3 Problems with unrelated parallel_ machines 310
9.3.1 The $R\vert p_{i,j} \in [\underline p_{i,j}, \bar p_{i,j}] F_\ell (\bar C, \bar {CC}^w)$ problem 310
9.3.2 The $R\vert pmtn\vert \epsilon (F_\ell(I_{max}, \bar M)/C_{max})$ problem 310
10. Shop problems with assignment 315
10.1 A hybrid flowshop problem with three stages 315
10.2 Hybrid flowshop problems with k stages 316
10.2.1 The $HFk, (PM^{(\ell)})^k_{\ell=1}\vert\vert F_\ell(C_{max}, \bar C)$ problem 316
10.2.2 The $HFk, (PM^{(\ell)})^k_{\ell=1}\vert\vert \epsilon(\bar C, C_{max})$ problem 318
10.2.3 The $HFk, (PM^{(\ell)})^k_{\ell=1}\vertr^{(1)}_i,d^{(k)}_i\vert \epsilon(C_{max}, T_{max})$ problem 318
A. Notations 323
A.1 Notation of data and variables 323
A.2 Usual notation of single criterion scheduling problems 323
B. Synthesis an multicriteria scheduling problems 329
B.1 Single machine Just-in-Time scheduling problems 329
B.2 Single machine problems 330
B.3 Shop problems 333
B.4 Parallel machines scheduling problems 333
B.5 Shop scheduling problems with assignment 334
References 335
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