IE 661
Scheduling Theory
Chapter 2
Rakesh Nagi
Department of Industrial Engineering
University at Buffalo (SUNY)
University at Buffalo (SUNY)
Department of Industrial Engineering
Chapter 2: Deterministic Models
Preliminaries
z
z
z
z
z
Processing time pij
Release date rj
Due date dj
Weight wj
Notation
¾α | β | γ
¾ α Machine environment
¾ β Processing characteristics and constraints
¾ γ Objective
University at Buffalo (SUNY)
Department of Industrial Engineering
Chapter 2: Deterministic Models
Preliminaries
z
α Machine environment
¾ Single machine (1)
¾ Identical machines in parallel (Pm)
¾ Machines in parallel with different speeds (Qm)
¾ Unrelated machines in parallel (Rm)
¾ Flow shop (Fm) (m machines in series)
¾ Flexible flow shop (FFc) (c stages with poss.
Identical machines)
¾ Job shop (Jm) (recrc for recirculation in β field)
¾ Flexible job shop (FJc)
¾ Open shop (Om) (scheduler can determine route)
University at Buffalo (SUNY)
Department of Industrial Engineering
Chapter 2: Deterministic Models
Preliminaries
z
β Processing characteristics and constraints
¾
¾
¾
¾
¾
¾
¾
¾
¾
¾
¾
Release dates (rj)
Sequence dependent setup times (sjk)
Machine specific sequence dependent setup times (sijk)
Premeptions (prmp)
Precedence constraints (prec)
Breakdowns (brkdwn)
Machine eligibility restrictions (Mj)
Permutation (prmu)
Blocking (block)
No-wait (nwt)
Recirculation (recrc)
University at Buffalo (SUNY)
Department of Industrial Engineering
Chapter 2: Deterministic Models
Preliminaries
z
γ Objective
¾ Makespan (Cmax)
¾ Max lateness (Lmax); Lj = Cj - dj
∑w C
¾ Total weighted completion time (
j
)
j
¾ Discounted total weighted completion time (
∑w T )
¾ Weighted number of tardy jobs ( ∑ w U
¾ Total weighted tardiness (
j
z
j
− rC j
))
j
j
z
∑ w (1 − e
j
)
Examples
Fm | pij = pj | ∑ w j C j
University at Buffalo (SUNY)
Department of Industrial Engineering
Chapter 2: Deterministic Models
Preliminaries
z
Classes of Schedules
¾ Nondelay Schedule: A feasible schedule is called
non-delay if no machine is kept idle while an
operation is waiting for processing (i.e., it prohibits
unforced idleness).
¾ A scheduling anomaly: Consider a P2 | prec | Cmax
with the following processing times
j 1
pj 8
2
7
University at Buffalo (SUNY)
3
7
4
2
5
3
6
2
7
2
8
8
9
8
1
0
1
5
Department of Industrial Engineering
Chapter 2: Deterministic Models
Preliminaries
University at Buffalo (SUNY)
Department of Industrial Engineering
Chapter 2: Deterministic Models
Preliminaries
z
Classes of Schedules
¾ Active Schedule: A feasible schedule is called active if it is
not possible to construct another schedule by changing the
order of processing on the machines and having at least one
operation finishing earlier and no operation finishing later.
¾ Semi-active Schedule: A feasible schedule is called semiactive if no operation can be completed earlier without
changing the order of processing on any one of the
machines.
Optimal
schedule
X
Nondelay
Active
Semi-Active
All schedules
University at Buffalo (SUNY)
Department of Industrial Engineering
Chap 2
z
Complexity
Hierarchy
¾ 1 || Σ Cj α
(reduces to)
1 || Σ wj Cj
¾ 1 || Σ wj Cj α
Pm || Σ wj Cj α
Qm |prec| Σ wj
Cj
University at Buffalo (SUNY)
Department of Industrial Engineering
Chapter 2: Deterministic Models
Preliminaries
z
Complexity hierarchy for Makespan (Fig. 2.8)
and Maximum Lateness problems (Fig. 2.9)
University at Buffalo (SUNY)
Department of Industrial Engineering