grâce à des inégalités de non-chevauchement

Formuler un problème d’ordonnancement juste-à-temps

grâce à des inégalités de non-chevauchement

Anne-Elisabeth FALQ

encadrée parPierre FouilhouxetSafia Kedad-Sidhoum

18 Novembre 2020, en ligne avec le Gscopde Grenoble

slides et tapuscrit disponibles sur

http://perso.eleves.ens-rennes.fr/~afalq494/recherche-these.html

Outline

1. Introduction

Scheduling around a common due date Known results about UCDDP and CDDP How to encode schedules?

2. A formulation for UCDDP using natural variables 3. How to manage this kind of formulations in practice 4. How to extend this formulation

5. Conclusion

1- Introduction • 1.1 Scheduling around a common due date

Scheduling around a common due-date on a single machine

An instance=• a set of tasks : J={1,2,3,4}

1 2 3 4

p₁=4 p₂=1 p₃=2 p₄=3

• common due date : d=10

• unit tardiness penalties : β₁=β₄=5, β₂=β₃=2

• unit earliness penalties : ∀j∈J,α_j = 2 A schedule:

p p p p p p p p p p p p p p p

0

|

d

1- Introduction • 1.1 Scheduling around a common due date

Scheduling around a common due-date on a single machine

An instance=• a set of tasks : J={1,2,3,4}

1 2 3 4

p₁=4 p₂=1 p₃=2 p₄=3

• common due date : d=10

• unit tardiness penalties : β₁=β₄=5, β₂=β₃=2

• unit earliness penalties : ∀j∈J,α_j = 2

A schedule:

p p p p p p p p p p p p p p p

0

|

d

1 / 25

1- Introduction • 1.1 Scheduling around a common due date

Scheduling around a common due-date on a single machine

An instance=• a set of tasks : J={1,2,3,4}

1 2 3 4

p₁=4 p₂=1 p₃=2 p₄=3

• common due date : d=10

• unit tardiness penalties : β₁=β₄=5, β₂=β₃=2

• unit earliness penalties : ∀j∈J,α_j = 2

A schedule:

p p p p p p p p p p p p p p p

0

|

d

3

1 4 2

1- Introduction • 1.1 Scheduling around a common due date

Scheduling around a common due-date on a single machine

An instance=• a set of tasks : J={1,2,3,4}

1 2 3 4

p₁=4 p₂=1 p₃=2 p₄=3

• common due date : d=10

• unit tardiness penalties : β₁=β₄=5, β₂=β₃=2

• unit earliness penalties : ∀j∈J,α_j = 2

A schedule:

p p p p p p p p p p p p p p p

0

|

d

3

1 4 2

4.5

1 / 25

1- Introduction • 1.1 Scheduling around a common due date

Scheduling around a common due-date on a single machine

An instance=• a set of tasks : J={1,2,3,4}

1 2 3 4

p1=4 p2=1 p3=2 p4=3

• common due date : d=10

• unit tardiness penalties : β1=β4=5, β2=β3=2

• unit earliness penalties : ∀j∈J,αj = 2

A schedule:

p p p p p p p p p p p p p p p

0

|

d

3

1 4 2

↓ 4,5×2=9

1- Introduction • 1.1 Scheduling around a common due date

Scheduling around a common due-date on a single machine

An instance=• a set of tasks : J={1,2,3,4}

1 2 3 4

p₁=4 p₂=1 p₃=2 p₄=3

• common due date : d=10

• unit tardiness penalties : β₁=β₄=5, β₂=β₃=2

• unit earliness penalties : ∀j∈J,α_j = 2

A schedule:

p p p p p p p p p p p p p p p

0

|

d

1 4 2 3

↓ 4×2=8

1 / 25

1- Introduction • 1.1 Scheduling around a common due date

Scheduling around a common due-date on a single machine

An instance=• a set of tasks : J={1,2,3,4}

1 2 3 4

p₁=4 p₂=1 p₃=2 p₄=3

• common due date : d=10

• unit tardiness penalties : β₁=β₄=5, β₂=β₃=2

• unit earliness penalties : ∀j∈J,α_j = 2

A schedule:

p p p p p p p p p p p p p p p

0

|

d

1 4 2 3

↓ 4×2=8

↓ 4

1- Introduction • 1.1 Scheduling around a common due date

Scheduling around a common due-date on a single machine

An instance=• a set of tasks : J={1,2,3,4}

1 2 3 4

p₁=4 p₂=1 p₃=2 p₄=3

• common due date : d=10

• unit tardiness penalties : β₁=β₄=5, β₂=β₃=2

• unit earliness penalties : ∀j∈J,α_j = 2

A schedule:

p p p p p p p p p p p p p p p

0

|

d

1 4 2 3

↓ 3

↓ 7

↓ 2.5

1 / 25

1- Introduction • 1.1 Scheduling around a common due date

Scheduling around a common due-date on a single machine

An instance=• a set of tasks : J={1,2,3,4}

1 2 3 4

p1=4 p2=1 p3=2 p4=3

• common due date : d=10

• unit tardiness penalties : β1=β4=5, β2=β3=2

• unit earliness penalties : ∀j∈J,αj = 2 A schedule:

p p p p p p p p p p p p p p p

0

|

d

1 4 2 3

↓ 3

↓ 7

↓ 2.5 3.5

↓ 3,5×2=7

1- Introduction • 1.1 Scheduling around a common due date

Scheduling around a common due-date on a single machine

An instance=• a set of tasks : J={1,2,3,4}

1 2 3 4

p₁=4 p₂=1 p₃=2 p₄=3

• common due date : d=10

• unit tardiness penalties : β₁=β₄=5, β₂=β₃=2

• unit earliness penalties : ∀j∈J,α_j = 2 A schedule:

p p p p p p p p p p p p p p p

0

|

d

4 2 3

↓ 3

↓ 7

↓ 2.5 1

↓

5 →17.5

1 / 25

1- Introduction • 1.1 Scheduling around a common due date

Scheduling around a common due-date on a single machine

An instance=• a set of tasks : J={1,2,3,4}

1 2 3 4

p1=4 p2=1 p3=2 p4=3

• common due date : d=10

• unit tardiness penalties : β1=β4=5, β2=β3=2

• unit earliness penalties : ∀j∈J,αj = 2 A schedule:

p p p p p p p p p p p p p p p

0

|

d

1 4 2 3

↓ 2

↓ 6

↓ 0

↓

6 →14F

1- Introduction • 1.1 Scheduling around a common due date

The Unrestrictive Common Due Date Problem (UCDDP)

An instance =

• a set of tasksJ

• their processing times (p_j)j∈J∈N^J

• anunrestrictivecommon due-dated>P

j∈J

pj

• their unit earliness penalties(α_j)j∈J

• their unit tardiness penalties(βj)j∈J

d

0

|

αj

1

βj

1

j j j⁰ j⁰ j⁰⁰

A solution schedule =a family of pairwise disjointprocessing intervals

The objective =minP

j∈J

αjEj+βjTj = minimize the sum ofearliness and tardiness penalties

2 / 25

1- Introduction • 1.1 Scheduling around a common due date

The Unrestrictive Common Due Date Problem (UCDDP)

An instance =

• a set of tasksJ

• their processing times (p_j)j∈J∈N^J

• anunrestrictivecommon due-dated>P

j∈J

pj

• their unit earliness penalties(α_j)j∈J

• their unit tardiness penalties(βj)j∈J

d

0

|

αj

1

βj

1

j j j⁰ j⁰ j⁰⁰

A solution schedule =a family of pairwise disjointprocessing intervals

The objective =minP

j∈J

αjEj+βjTj = minimize the sum ofearliness and tardiness penalties

1- Introduction • 1.1 Scheduling around a common due date

The Unrestrictive Common Due Date Problem (UCDDP)

An instance =

• a set of tasksJ

• their processing times (p_j)j∈J∈N^J

• anunrestrictivecommon due-dated>P

j∈J

pj

• their unit earliness penalties(α_j)j∈J

• their unit tardiness penalties(βj)j∈J

d

0

|

αj

1

βj

1

j j j⁰ j⁰ j⁰⁰

A solution schedule =a family of pairwise disjointprocessing intervals

The objective =minP

j∈J

αjEj+βjTj = minimize the sum ofearliness and tardiness penalties

2 / 25

1- Introduction • 1.1 Scheduling around a common due date

The Unrestrictive Common Due Date Problem (UCDDP)

An instance =

• a set of tasksJ

• their processing times (p_j)j∈J∈N^J

• anunrestrictivecommon due-dated>P

j∈J

pj

• their unit earliness penalties(α_j)j∈J

• their unit tardiness penalties(βj)j∈J

d

0

| αj

1

βj

1

j j j⁰ j⁰ j⁰⁰

A solution schedule =a family of pairwise disjointprocessing intervals

The objective =minP

j∈J

αjEj+βjTj = minimize the sum ofearliness and tardiness penalties

1- Introduction • 1.1 Scheduling around a common due date

The Unrestrictive Common Due Date Problem (UCDDP)

An instance =

• a set of tasksJ

• their processing times (p_j)j∈J∈N^J

• anunrestrictivecommon due-dated>P

j∈J

pj

• their unit earliness penalties(α_j)j∈J

• their unit tardiness penalties(βj)j∈J

d

0

|

αj

1

βj

1

j j j⁰ j⁰ j⁰⁰

A solution schedule =a family of pairwise disjointprocessing intervals

The objective =minP

j∈J

αjEj+βjTj = minimize the sum ofearliness and tardiness penalties

2 / 25

1- Introduction • 1.1 Scheduling around a common due date

The Unrestrictive Common Due Date Problem (UCDDP)

An instance =

• a set of tasksJ

• their processing times (p_j)j∈J∈N^J

• anunrestrictivecommon due-dated>P

j∈J

pj

• their unit earliness penalties(α_j)j∈J

• their unit tardiness penalties(βj)j∈J

d

0

|

αj

1

βj

1

j

j j⁰ j⁰ j⁰⁰

A solution schedule =a family of pairwise disjointprocessing intervals

The objective =minP

j∈J

αjEj+βjTj = minimize the sum ofearliness and tardiness penalties

1- Introduction • 1.1 Scheduling around a common due date

The Unrestrictive Common Due Date Problem (UCDDP)

An instance =

• a set of tasksJ

• their processing times (p_j)j∈J∈N^J

• anunrestrictivecommon due-dated>P

j∈J

pj

• their unit earliness penalties(α_j)j∈J

• their unit tardiness penalties(βj)j∈J

d

0

|

αj

1

βj

1

j

j j⁰

j⁰ j⁰⁰

A solution schedule =a family of pairwise disjointprocessing intervals

The objective =minP

j∈J

αjEj+βjTj = minimize the sum ofearliness and tardiness penalties

2 / 25

1- Introduction • 1.1 Scheduling around a common due date

The Unrestrictive Common Due Date Problem (UCDDP)

An instance =

• a set of tasksJ

• their processing times (p_j)j∈J∈N^J

• anunrestrictivecommon due-dated>P

j∈J

pj

• their unit earliness penalties(α_j)j∈J

• their unit tardiness penalties(βj)j∈J

d

0

|

αj

1

βj

1

j

j

j⁰

j⁰ j⁰⁰

A solution schedule =a family of pairwise disjointprocessing intervals

The objective =minP

j∈J

αjEj+βjTj = minimize the sum ofearliness and tardiness penalties

1- Introduction • 1.1 Scheduling around a common due date

The Unrestrictive Common Due Date Problem (UCDDP)

An instance =

• a set of tasksJ

• their processing times (p_j)j∈J∈N^J

• anunrestrictivecommon due-dated>P

j∈J

pj

• their unit earliness penalties(α_j)j∈J

• their unit tardiness penalties(βj)j∈J

d

0

|

αj

1

βj

1

j

j

j⁰

j⁰ j⁰⁰

A solution schedule =a family of pairwise disjointprocessing intervals

The objective =minP

j∈J

αjEj+βjTj = minimize the sum ofearliness and tardiness penalties

2 / 25

1- Introduction • 1.1 Scheduling around a common due date

The Unrestrictive Common Due Date Problem (UCDDP)

An instance =

• a set of tasksJ

• their processing times (p_j)j∈J∈N^J

• anunrestrictivecommon due-dated>P

j∈J

pj

• their unit earliness penalties(α_j)j∈J

• their unit tardiness penalties(βj)j∈J

d

0

|

αj

1

βj

1

j

j

j⁰

j⁰ j⁰⁰

A solution schedule =a family of pairwise disjointprocessing intervals

The objective =minP

j∈J

α_jE_j+β_jT_j = minimize the sum ofearliness and tardiness penalties

1- Introduction • 1.1 Scheduling around a common due date

The Unrestrictive Common Due Date Problem (CDDP)

An instance =

• a set of tasksJ

• their processing times (pj)j∈J∈N^J

• a unrestrictive common due-date d

• their unit earliness penalties(α_j)j∈J

• their unit tardiness penalties(β_j)j∈J

d

0

|

j j⁰ j⁰⁰

A solution schedule =a family of pairwise disjointprocessing intervals

The objective =minP

j∈J

αjEj+βjTj = minimize the sum ofearliness and tardiness penalties

3 / 25

1- Introduction • 1.2 Known results about UCDDP and CDDP

What are dominance properties?

LetT be a solution subset of an arbitrary optimization problem.

• T is adominantset if it contains at least one optimal solution

• T is astrictly dominantset if it contains all the optimal solutions In both cases,

→ the searching space can be reduced toT

→ other solutions can be discarded

1- Introduction • 1.2 Known results about UCDDP and CDDP

What are dominance properties?

LetT be a solution subset of an arbitrary optimization problem.

• T is adominantset if it contains at least one optimal solution

• T is astrictly dominantset if it contains all the optimal solutions In both cases,

→ the searching space can be reduced toT

→ other solutions can be discarded

4 / 25

1- Introduction • 1.2 Known results about UCDDP and CDDP

What are dominance properties?

LetT be a solution subset of an arbitrary optimization problem.

• T is adominantset if it contains at least one optimal solution

• T is astrictly dominantset if it contains all the optimal solutions

In both cases,

→ the searching space can be reduced toT

→ other solutions can be discarded

1- Introduction • 1.2 Known results about UCDDP and CDDP

What are dominance properties?

LetT be a solution subset of an arbitrary optimization problem.

• T is adominantset if it contains at least one optimal solution

• T is astrictly dominantset if it contains all the optimal solutions In both cases,

→ the searching space can be reduced to T

→ other solutions can be discarded

4 / 25

1- Introduction • 1.2 Known results about UCDDP and CDDP

Dominance properties and complexity

unrestrictive cased>P

j∈Jpj

dominance properties

• without idle time

• one on time task

(+ "V-shaped")

complexity

∀j∈J, αj=βj=ω→P

∀j∈J, αj=βj→

weakly

NP-hard

Kanet, 1981, Naval Research Logistics Quaterly Hall and Posner, 1991, Operations research

Hoogeveen and van de Velde, 1991, European Journal of Operational research Sourd, 2009, Informs Journal on Computing

1- Introduction • 1.2 Known results about UCDDP and CDDP

Dominance properties and complexity

unrestrictive cased>P

j∈Jpj

dominance

properties • without idle time

• one on time task

(+ "V-shaped")

complexity

∀j∈J, αj=βj=ω→P

∀j∈J, αj=βj→

weakly

NP-hard

Kanet, 1981, Naval Research Logistics Quaterly Hall and Posner, 1991, Operations research

Hoogeveen and van de Velde, 1991, European Journal of Operational research Sourd, 2009, Informs Journal on Computing

5 / 25

1- Introduction • 1.2 Known results about UCDDP and CDDP

Dominance properties and complexity

unrestrictive cased>P

j∈Jpj

dominance

properties • without idle time

• one on time task

(+ "V-shaped")

complexity

∀j∈J, αj=βj=ω→P

∀j∈J, αj=βj→

weakly

NP-hard

Kanet, 1981, Naval Research Logistics Quaterly Hall and Posner, 1991, Operations research

Hoogeveen and van de Velde, 1991, European Journal of Operational research Sourd, 2009, Informs Journal on Computing

1- Introduction • 1.2 Known results about UCDDP and CDDP

Dominance properties and complexity

unrestrictive cased>P

j∈Jpj

dominance

properties • without idle time

• one on time task

(+ "V-shaped")

complexity ^{∀j∈J, α}j=βj=ω→P

∀j∈J, αj=βj→

weakly

NP-hard

Kanet, 1981, Naval Research Logistics Quaterly

Hall and Posner, 1991, Operations research

Hoogeveen and van de Velde, 1991, European Journal of Operational research Sourd, 2009, Informs Journal on Computing

5 / 25

1- Introduction • 1.2 Known results about UCDDP and CDDP

Dominance properties and complexity

unrestrictive cased>P

j∈Jpj

dominance

properties • without idle time

• one on time task

(+ "V-shaped")

complexity ^{∀j∈J, α}j=βj=ω→P

∀j∈J, αj=βj→

weakly

NP-hard

Kanet, 1981, Naval Research Logistics Quaterly Hall and Posner, 1991, Operations research

Hoogeveen and van de Velde, 1991, European Journal of Operational research Sourd, 2009, Informs Journal on Computing

1- Introduction • 1.2 Known results about UCDDP and CDDP

Dominance properties and complexity

unrestrictive cased>P

j∈Jpj

dominance

properties • without idle time

• one on time task (+ "V-shaped") complexity ^{∀j∈J, α}j=βj=ω→P

∀j∈J, αj=βj→weakly NP-hard

Kanet, 1981, Naval Research Logistics Quaterly Hall and Posner, 1991, Operations research

Hoogeveen and van de Velde, 1991, European Journal of Operational research Sourd, 2009, Informs Journal on Computing

5 / 25

1- Introduction • 1.2 Known results about UCDDP and CDDP

Dominance properties and complexity

unrestrictive case d>P

j∈Jpj general case dominance

properties • without idle time

• without idle time

• one on time task

• one on time task or beginning at 0

(+ "V-shaped")

V-shaped

complexity ^{∀j∈J, α}j=βj=ω→P

∀j∈J, αj=βj=ω→NP-hard

∀j∈J, αj=βj→weakly NP-hard

∀j∈J, αj=βj→weakly NP-hard arbitraryαj, βj→Branch-and-Bound can solve up to 1000-task instances

Kanet, 1981, Naval Research Logistics Quaterly Hall and Posner, 1991, Operations research

Hoogeveen and van de Velde, 1991, European Journal of Operational research Sourd, 2009, Informs Journal on Computing

1- Introduction • 1.2 Known results about UCDDP and CDDP

Dominance properties and complexity

unrestrictive case d>P

j∈Jpj general case dominance

properties • without idle time • without idle time

• one on time task • one on time task orbeginning at 0 (+ "V-shaped") V-shaped

complexity ^{∀j∈J, α}j=βj=ω→P

∀j∈J, αj=βj=ω→NP-hard

∀j∈J, αj=βj→weakly NP-hard

∀j∈J, αj=βj→weakly NP-hard arbitraryαj, βj→Branch-and-Bound can solve up to 1000-task instances

Kanet, 1981, Naval Research Logistics Quaterly Hall and Posner, 1991, Operations research

Hoogeveen and van de Velde, 1991, European Journal of Operational research

Sourd, 2009, Informs Journal on Computing

5 / 25

1- Introduction • 1.2 Known results about UCDDP and CDDP

Dominance properties and complexity

unrestrictive case d>P

j∈Jpj general case dominance

properties • without idle time • without idle time

• one on time task • one on time task orbeginning at 0 (+ "V-shaped") V-shaped

complexity ^{∀j∈J, α}j=βj=ω→P ∀j∈J, αj=βj=ω→NP-hard

∀j∈J, αj=βj→weakly NP-hard

∀j∈J, αj=βj→weakly NP-hard arbitraryαj, βj→Branch-and-Bound can solve up to 1000-task instances

Kanet, 1981, Naval Research Logistics Quaterly Hall and Posner, 1991, Operations research

Hoogeveen and van de Velde, 1991, European Journal of Operational research

Sourd, 2009, Informs Journal on Computing

1- Introduction • 1.2 Known results about UCDDP and CDDP

Dominance properties and complexity

unrestrictive case d>P

j∈Jpj general case dominance

properties • without idle time • without idle time

• one on time task • one on time task orbeginning at 0 (+ "V-shaped") V-shaped

complexity ^{∀j∈J, α}j=βj=ω→P ∀j∈J, αj=βj=ω→NP-hard

∀j∈J, αj=βj→weakly NP-hard ∀j∈J, αj=βj→weakly NP-hard

arbitraryαj, βj→Branch-and-Bound can solve up to 1000-task instances

Kanet, 1981, Naval Research Logistics Quaterly Hall and Posner, 1991, Operations research

Hoogeveen and van de Velde, 1991, European Journal of Operational research

Sourd, 2009, Informs Journal on Computing

5 / 25

1- Introduction • 1.2 Known results about UCDDP and CDDP

Dominance properties and complexity

unrestrictive case d>P

j∈Jpj general case dominance

properties • without idle time • without idle time

• one on time task • one on time task orbeginning at 0 (+ "V-shaped") V-shaped

complexity ^{∀j∈J, α}j=βj=ω→P ∀j∈J, αj=βj=ω→NP-hard

∀j∈J, αj=βj→weakly NP-hard ∀j∈J, αj=βj→weakly NP-hard arbitraryαj, βj→Branch-and-Bound can solve up to 1000-task instances

Kanet, 1981, Naval Research Logistics Quaterly Hall and Posner, 1991, Operations research

Hoogeveen and van de Velde, 1991, European Journal of Operational research

Sourd, 2009, Informs Journal on Computing ^{5 / 25}

1- Introduction • 1.2 Known results about UCDDP and CDDP

Dominance properties and complexity

unrestrictive cased>P

jp_j general case dominance

properties • without idle time • without idle time

• one on time task • one on time task or beginning at 0 (+ "V-shaped")

V-shaped

complexity ^{∀j∈J, α}j=βj=ω→P ∀j∈J, αj=βj=ω→NP-hard

∀j∈J, αj=βj→weakly NP-hard ∀j∈J, αj=βj→weakly NP-hard

arbitrary αj, βj →NP-hard ^{arbitraryαj, βj→}Branch-and-Bound can solve up to 1000-task instances

Kanet, 1981, Naval Research Logistics Quaterly Hall and Posner, 1991, Operations research

Hoogeveen and van de Velde, 1991, European Journal of Operational research

Sourd, 2009, Informs Journal on Computing 5 / 25

1- Introduction • 1.3 How to encode schedules?

Time-indexed variables

d

0

|

First let us discretize the time horizon

asT=[d−p(J),d+p(J)]

.

Variables: ∀i∈J, ∀t∈ T: x_i,t =

1 ifi completes att

0 otherwise Objective function: P

j∈J

P

t∈T

c

x

1- Introduction • 1.3 How to encode schedules?

Time-indexed variables

p p p p p p p p p p p p p p p p p p p p p p p

d

0

|

First let us discretize the time horizon

asT=[d−p(J),d+p(J)]

.

Variables: ∀i∈J, ∀t∈ T: x_i,t =

1 ifi completes att 0 otherwise

Objective function: P

j∈J

P

t∈T

c

x

6 / 25

1- Introduction • 1.3 How to encode schedules?

Time-indexed variables

p(J) p(J)

p p p p p p p p p p p p p p p p p p p p p p p

d

0

|

First let us discretize the time horizon

asT=[d−p(J),d+p(J)]

.

Variables: ∀i∈J, ∀t∈ T: x_i,t =

1 ifi completes att 0 otherwise

Objective function: P

j∈J

P

t∈T

c

x

1- Introduction • 1.3 How to encode schedules?

Time-indexed variables

p(J) p(J)

p p p p p p p p p p p p p p p p p p p p p p p

d

0

|

First let us discretize the time horizon asT=[d−p(J),d+p(J)].

Variables: ∀i∈J, ∀t∈ T: x_i,t =

1 ifi completes att 0 otherwise

Objective function: P

j∈J

P

t∈T

c

x

6 / 25

1- Introduction • 1.3 How to encode schedules?

Time-indexed variables

p p p p p p p p p p p p p p p p p p p p p p p

d

0

| i

x_i,tt=1 First let us discretize the time horizon asT=[d−p(J),d+p(J)].

Variables: ∀i∈J,∀t∈ T: x_i,t =

1 ifi completes att 0 otherwise

Objective function: P

j∈J

P

t∈T

c

x

1- Introduction • 1.3 How to encode schedules?

Time-indexed variables

p p p p p p p p p p p p p p p p p p p p p p p

d

0

| i

x_i,tt=1 t⁰ x_i,t⁰=0 t⁰⁰

x_i,t⁰⁰=0

First let us discretize the time horizon asT=[d−p(J),d+p(J)].

Variables: ∀i∈J,∀t∈ T: x_i,t =

1 ifi completes att 0 otherwise

Objective function: P

j∈J

P

t∈T

c

x

6 / 25

1- Introduction • 1.3 How to encode schedules?

Time-indexed variables

p p p p p p p p p p p p p p p p p p p p p p p

d

0

| i

x_i,tt=1 t⁰ x_i,t⁰=0 t⁰⁰

x_i,t⁰⁰=0

First let us discretize the time horizon asT=[d−p(J),d+p(J)].

Variables: ∀i∈J,∀t∈ T: x_i,t =

1 ifi completes att 0 otherwise

Objective function: P

j∈J

P

t∈T

c

x

1- Introduction • 1.3 How to encode schedules?

Time-indexed variables

p p p p p p p p p p p p p p p p p p p p p p p

d

0

| i

xi,tt=1 t⁰ xi,t⁰=0 t⁰⁰

xi,t⁰⁰=0

First let us discretize the time horizon asT=[d−p(J),d+p(J)].

Variables: ∀i∈J,∀t∈ T: xi,t =

1 ifi completes att 0 otherwise

Objective function: P

j∈J

P

t∈T

c

x

Constraints: ^{• ∀i∈J,}P

t∈T

xi,t= 1 taski is placed

• ∀t∈ T,P

i∈J

P

s∈T s∈[t,t+p_i[

xi,s61 at most 1 task is in progress att

• ∀i∈J,∀t∈ T,xi,t∈Z integrity constraint

6 / 25

1- Introduction • 1.3 How to encode schedules?

Time-indexed variables

p p p p p p p p p p p p p p p p p p p p p p p

d

0

| i

xi,tt=1 t⁰ xi,t⁰=0 t⁰⁰

xi,t⁰⁰=0

First let us discretize the time horizon asT=[d−p(J),d+p(J)].

Variables: ∀i∈J,∀t∈ T: xi,t =1 ifi completes att 0 otherwise

Objective function: P

j∈J

P

t∈T

c

x

+ easy to formulate as a MIP + good relaxation value

− 2n p(J) binary variables= a pseudo polynomial number

− n+n p(J) inequalities= a pseudo polynomial number

1- Introduction • 1.3 How to encode schedules?

Completion time variables

d 0

Variables: ∀j∈J,C_j∈R+is the time when task j completes

Objective function: P

j∈J

α

[d −C

]

+ β

[C

−d]

7 / 25

1- Introduction • 1.3 How to encode schedules?

Completion time variables

d

0 j

Cj

Variables: ∀j∈J,C_j∈R+is the time when task j completes

Objective function: P

j∈J

α

[d −C

]

+ β

[C

−d]

1- Introduction • 1.3 How to encode schedules?

Completion time variables

d

0 j

Cj

i Ci

Variables: ∀j∈J,C_j∈R+is the time when task j completes

Objective function: P

j∈J

α

[d −C

]

+ β

[C

−d]

7 / 25

1- Introduction • 1.3 How to encode schedules?

Completion time variables

d

0 j

Cj

i Ci

d−Cj

Variables: ∀j∈J,C_j∈R+is the time when task j completes Objective function: P

j∈J

α

[d −C

]

+ β

[C

−d]

1- Introduction • 1.3 How to encode schedules?

Completion time variables

d

0 j

Cj

i Ci

d−Cj Ci−d

Variables: ∀j∈J,C_j∈R+is the time when task j completes Objective function: P

j∈J

α

[d −C

]

+ β

[C

−d]

7 / 25

1- Introduction • 1.3 How to encode schedules?

Completion time variables

d

0 j

Cj

i Ci

d−Cj Ci−d

Variables: ∀j∈J,C_j∈R+is the time when task j completes Objective function: P

j∈J

α

[d −C

]

+ β

[C

−d]

1- Introduction • 1.3 How to encode schedules?

Completion time variables

d

0 j

Cj

i Ci

d−Cj Ci−d

Variables: ∀j∈J,C_j∈R+is the time when task j completes Objective function: P

j∈J

α

[d −C

]

+ β

[C

−d]

7 / 25

1- Introduction • 1.3 How to encode schedules?

Completion time variables

d

0 j

Cj

i Ci

d−Cj Ci−d Cj

Ci

Variables: ∀j∈J,C_j∈R+is the time when task j completes Objective function: P

j∈J

α

[d −C

]

+ β

[C

−d]

1- Introduction • 1.3 How to encode schedules?

Completion time variables

d

0 j

Cj

i Ci

d−Cj Ci−d

Variables: ∀j∈J,C_j∈R+is the time when task j completes Objective function: P

j∈J

α

[d −C

]

+ β

[C

−d]

7 / 25

1- Introduction • 1.3 How to encode schedules?

Earliness–Tardiness variables

d

0

| j

Cj

i Ci

ej ti

Variables: ∀j∈J,ej=[d−Cj]⁺andtj=[Cj−d]⁺

Objective function: P

j∈J

α

e

+ β

t

←linear function of variablese andt

On the first example:

0

| p p p p p p p p p p p p p p p

1 3 2 4

d

1- Introduction • 1.3 How to encode schedules?

Earliness–Tardiness variables

d

0

| j

Cj

i Ci

ej ti

Variables: ∀j∈J,ej=[d−Cj]⁺andtj=[Cj−d]⁺ Objective function: P

j∈J

α

e

+ β

t

←linear function of variablese andt

On the first example:

0

| p p p p p p p p p p p p p p p

1 3 2 4

d

8 / 25

1- Introduction • 1.3 How to encode schedules?

Earliness–Tardiness variables

d

0

| j

Cj

i Ci

ej ti

Variables: ∀j∈J,ej=[d−Cj]⁺andtj=[Cj−d]⁺ Objective function: P

j∈J

α

e

+ β

t

On the first example:

0

| p p p p p p p p p p p p p p p

1 3 2 4

d

1- Introduction • 1.3 How to encode schedules?

Earliness–Tardiness variables

d

0

| j

Cj

i Ci

ej ti

Variables: ∀j∈J,ej=[d−Cj]⁺andtj=[Cj−d]⁺ Objective function: P

j∈J

α

e

+ β

t

0

| p p p p p p p p p p p p p p p

1 3 2 4

C1= 8

C3= 10

C2= 11 C4= 15 d

8 / 25

1- Introduction • 1.3 How to encode schedules?

Earliness–Tardiness variables

d

0

| j

Cj

i Ci

ej ti

Variables: ∀j∈J,ej=[d−Cj]⁺andtj=[Cj−d]⁺ Objective function: P

j∈J

α

e

+ β

t

0

| p p p p p p p p p p p p p p p

1 3 2 4

C3= 10

C2= 11 C4= 15 d−2

d

1- Introduction • 1.3 How to encode schedules?

Earliness–Tardiness variables

d

0

| j

Cj

i Ci

ej ti

Variables: ∀j∈J,e_j=[d−C_j]⁺andt_j=[C_j−d]⁺ Objective function: P

j∈J

α

e

+ β

t

0

| p p p p p p p p p p p p p p p

1

1 3 2 4

C3= 10

C2= 11 C4= 15 d−2

e1↓= 2 t1= 0

d

8 / 25

1- Introduction • 1.3 How to encode schedules?

Earliness–Tardiness variables

d

0

| j

Cj

i Ci

ej ti

Variables: ∀j∈J,e_j=[d−C_j]⁺andt_j=[C_j−d]⁺ Objective function: P

j∈J

α

e

+ β

t

0

| p p p p p p p p p p p p p p p

1

1 3 2 4

C2= 11 C4= 15

d−2 d

e1↓= 2 t1= 0

d

1- Introduction • 1.3 How to encode schedules?

Earliness–Tardiness variables

d

0

| j

Cj

i Ci

ej ti

Variables: ∀j∈J,e_j=[d−C_j]⁺andt_j=[C_j−d]⁺ Objective function: P

j∈J

α

e

+ β

t

0

| p p p p p p p p p p p p p p p

1

1 33 2 4

C2= 11 C4= 15

d−2 d

e1↓= 2 t1= 0

e3= 0↓ t1= 0 d

8 / 25

1- Introduction • 1.3 How to encode schedules?

Earliness–Tardiness variables

d

0

| j

Cj

i Ci

ej ti

Variables: ∀j∈J,e_j=[d−C_j]⁺andt_j=[C_j−d]⁺ Objective function: P

j∈J

α

e

+ β

t

0

| p p p p p p p p p p p p p p p

1

1 33 2 4

C4= 15

d−2 d

d+1 e1↓= 2

t1= 0

e3= 0↓ t1= 0 d

1- Introduction • 1.3 How to encode schedules?

Earliness–Tardiness variables

d

0

| j

Cj

i Ci

ej ti

Variables: ∀j∈J,e_j=[d−C_j]⁺andt_j=[C_j−d]⁺ Objective function: P

j∈J

α

e

+ β

t

0

| p p p p p p p p p p p p p p p

1

1 33 22 4

C4= 15

d−2 d

d+1 e1↓= 2

t1= 0

e3= 0↓ t1= 0

e2↓= 0 t1= 1 d

8 / 25

1- Introduction • 1.3 How to encode schedules?

Earliness–Tardiness variables

d

0

| j

Cj

i Ci

ej ti

Variables: ∀j∈J,e_j=[d−C_j]⁺andt_j=[C_j−d]⁺ Objective function: P

j∈J

α

e

+ β

t

0

| p p p p p p p p p p p p p p p

1

1 33 22 4

d−2 d

d+1 d+4

e1↓= 2 t1= 0

e3= 0↓ t1= 0

e2↓= 0 t1= 1 d