Scheduling and (Integer) Linear Programming master class/7... · Schedulingand(Integer)LinearProgramming ChristianArtigues LAAS-CNRS&UniversitédeToulouse,France [email protected]

Scheduling and (Integer) Linear Programming

Christian Artigues

LAAS - CNRS & Université de Toulouse, France

[email protected]

Master Class CPAIOR 2012 - Nantes

Christian Artigues Scheduling and (Integer) Linear Programming CPAIOR 2012, Nantes 1 / 78

Outline

1 Introduction

2 Polyhedral studies and cutting plane generation

3 (Mixed) integer programming for scheduling problems

4 Column generation

5 A few Computational results


Outline

1 Introduction


3 (Mixed) integer programming for scheduling problemsTime-indexed variablesTotal unimodularitySequencing and natural date variablesFlow (TSP) and natural date variablesPositional date and assignment variablesHybrid formulations

4 Column generation



A bit of history

Since the beginning, linear programming has been used to solvescheduling problems.

“The military refer to their various plans or proposedschedules of training, logistical supply and deployment ofcombat units as a program. When I first analyzed the AirForce planning problem and saw that it could be formulatedas a system of linear inequalities, I called my paperProgramming in a Linear Structure” (Georges Dantzig)

This presentation is a (non-exhaustive) survey of (integer) linearprogramming formulations and valid inequalities for schedulingproblems


A simple scheduling exampleOne machine scheduling with release dates and deadlines2 jobs J1 and J2 (p1 = 3, p2 = 2, r1 = 0, r2 = 1, d1 = 9, d2 = 7).1 machine. Objective function f (S) = C1 + C2 = S1 + S2 + p1 + p2.

minS1 + S2+p1 + p2S1 ≥ r1S2 ≥ r2

S1 + p1 ≤ d1S2 + p2 ≤ d2

S2 ≥ S1 + p1 ∨ S1 ≥ S2 + p2S1,S2 integer

0 1 2 3 4 5 6

J1 J2∑

Ci = 8 (optimal solution)


The scheduling polyhedron

Feasible set SThe feasible set is S the set of points S ∈ Rn that satisfy theconstraints

min S1 + S2+p1 + p2S1 ≥ r1S2 ≥ r2

S1 + p1 ≤ d1S2 + p2 ≤ d2

S2 ≥ S1 + p1 ∨ S1 ≥ S2 + p2S1, S2 integer

0 1 2 3 4 5 6

J1 J2∑

Ci = 8

S1

S2 S

conv(S)

Find a complete descriptionof conv(S) : hard in general



Convex hull conv(S)

The convex hull of S, i.e. the smallest convex set containing S :conv(S) =

{x ∈ Rn

∣∣∣∃λi ∈ Rn+, x =∑|S|

i=1 λiS i ,∑|S|

i=1 λi = 1}

min S1 + S2+p1 + p2S1 ≥ r1S2 ≥ r2

S1 + p1 ≤ d1S2 + p2 ≤ d2


0 1 2 3 4 5 6

J1 J2∑

Ci = 8

S1

S2 Sconv(S)




Optimization on S and conv(S)

Let f : Rn → R a linear function, minS∈S

f (S) = minS∈conv(S)

f (S)

min S1 + S2+p1 + p2S1 ≥ r1S2 ≥ r2

S1 + p1 ≤ d1S2 + p2 ≤ d2


0 1 2 3 4 5 6

J1 J2∑

Ci = 8

S1

S2 Sconv(S)



The scheduling polyhedronconv(S) is a polyhedronThere exists A ∈ Rm×n and b ∈ Rn with m finite such thatconv(S) = {x ∈ Rn|Ax ≥ b}. Hence min

S∈conv(S)f (S) is a LP.

min S1 + S2+p1 + p2S1 ≥ r1S2 ≥ r2

S1 + p1 ≤ d1S2 + p2 ≤ d2


0 1 2 3 4 5 6

J1 J2∑

Ci = 8

S1

S2 Sconv(S)



Definitions : valid inequalities, faces and facetsLet P denote a polyhedron.

A valid inequality αx ≥ β is such that ∀S ∈ P, S verifies theinequality (P is included in the halfspace induced by theinequality)A face is the intersection of P with the hyperplane{x ∈ Rn|αx = β}A vertex is a face of dimension 0A facet is a face of dimension dim(P)− 1

S1

S2

3S1 + 5S2 ≥ 7 : a valid inequality

P ∩ {S|S1 + 3S2 = 6} : a 0-dimensional face (3, 1)P ∩ {S|2S1 + 3S2 = 9} : a facet

A polyhedron isfully described bythe set of allfacet-inducingvalid inequalities




S1

S2

3S1 + 5S2 ≥ 7 : a valid inequality

P ∩ {S|S1 + 3S2 = 6} : a 0-dimensional face (3, 1)

P ∩ {S|2S1 + 3S2 = 9} : a facet





S1

S2

3S1 + 5S2 ≥ 7 : a valid inequalityP ∩ {S|S1 + 3S2 = 6} : a 0-dimensional face (3, 1)

P ∩ {S|2S1 + 3S2 = 9} : a facet





S1

S2

3S1 + 5S2 ≥ 7 : a valid inequalityP ∩ {S|S1 + 3S2 = 6} : a 0-dimensional face (3, 1)

P ∩ {S|2S1 + 3S2 = 9} : a facet



(Integer) linear programming-based scheduling

Given a scheduling problem and the logical description of S1 Perform a polyhedral study :

Find a complete description of conv(S) with a polynomialnumber of linear inequalities ? → The problem can be solved byLPFind a complete description of conv(S) and show it has asupermodular structure ? → The problem can be solved by agreedy algorithmFind a partial description of conv(S) ? → This gives useful validinequalities

2 Design a (mixed)-integer programming formulationMore polyhedral studies, solve with branch-and-cut,branch-and-price, branch-and-cut-and-price, heuristics, . . . (onlypartly addressed here)



























Outline

1 Introduction



4 Column generation



A scheduling problem that can be solved by LP

Project scheduling SPS

• Precedence constraints E .• lij : minimum distance between the startof job i (Si) and the start of job j (Sj).SPS = {S ∈ R+n|Sj − Si ≥ lij , ∀(i , j) ∈ E}

0

1

2

3

4

5

1

0

3

2

2

1

0

−1

find the shortest scheduleminSn+1

Sj − Si ≥ lij ∀(i , j) ∈ ES0 = 0Sj ≥ 0 ∀j ∈ J

max∑

(i ,j)∈Elijxij

∑i∈Γ−1(j)

xij =∑

i∈Γ(j)xji ∀j ∈ J \ {0}

∑i∈Γ(0)

x0i = 1

x0i ≥ 0(longest path model)


Supermodular polyhedron : definitions [Sch96]

Consider a set N = {1, . . . , n} and its power set 2N (set of allsubsets of N) and a supermodular set function f : 2N → R, i.e. a setfunction that verifies :{

f (∅) = 0f (A ∪ B) + f (A ∩ B) ≥ f (A) + f (B), ∀A,B ⊆ N

P(f )={x ∈ R|N||∑

i∈A xi≥ f (A), ∀A⊆N} supermodular polyhedron

There is an O(n log n) greedy algorithm to findx∗ = argminx∈P(f ) cx :

if ∃i ∈ N, ci < 0, the problem is unbounded.otherwise x∗ ∈ B(f ). Solve the problem with the followinggreedy algorithm{

x∗1 = f ({1})x∗j = f ({1, . . . , j})− f ({1, . . . , j − 1}), j = 2, . . . , n.


Supermodular polyhedron : definitions [Sch96]

Consider a set N = {1, . . . , n} and its power set 2N (set of allsubsets of N) and a supermodular set function f : 2N → R, i.e. a setfunction that verifies :{

f (∅) = 0f (A ∪ B) + f (A ∩ B) ≥ f (A) + f (B), ∀A,B ⊆ N

P(f )={x ∈ R|N||∑

i∈A xi≥ f (A), ∀A⊆N} supermodular polyhedron

There is an O(n log n) greedy algorithm to findx∗ = argminx∈P(f ) cx :

if ∃i ∈ N, ci < 0, the problem is unbounded.otherwise x∗ ∈ B(f ). Solve the problem with the followinggreedy algorithm{

x∗1 = f ({1})x∗j = f ({1, . . . , j})− f ({1, . . . , j − 1}), j = 2, . . . , n.


A scheduling supermodular polyhedron [Que93]

Single-machine scheduling SSM

Set of feasible schedules for a set of jobs J on a single machine :

SSM =

{C ∈ RJ

∣∣∣∣∣ Ci ≥ pi , ∀j ∈ J ,Ci ≥ Cj + pi ∨ Cj ≥ Ci + pj , ∀i , j ∈ J , i 6= j

}

Queyranne [Que93] showed thatconv(SSM) = {C ∈ RJ |

∑j∈A pjCj ≥ g(A), ∀A ⊆ J} where

g(A) = 12

((∑j∈A pj

)2+∑

j∈A p2j

).

These valid inequalities can be obtained by Smith’s rule (WSPT).Optimum for

∑i∈J wiCi is obtained by sorting jobs in non

decreasing wi/pi .For A = {i1, . . . , is} ⊆ J , set wi = pi for i ∈ A and wi = 0 fori ∈ J \ AOpt = pi1pi1 + pi2(pi1 + pi2) +. . .+ pis (pi1 + pi2 +. . .+ pis ) = g(A)





SSM =

{C ∈ RJ


}



g(A) = 12

((∑j∈A pj

)2+∑

j∈A p2j

).








SSM =

{C ∈ RJ


}



g(A) = 12

((∑j∈A pj

)2+∑

j∈A p2j

).








SSM =

{C ∈ RJ


}



g(A) = 12

((∑j∈A pj

)2+∑

j∈A p2j

).








SSM =

{C ∈ RJ


}



g(A) = 12

((∑j∈A pj

)2+∑

j∈A p2j

).








SSM =

{C ∈ RJ


}



g(A) = 12

((∑j∈A pj

)2+∑

j∈A p2j

).






Each inequality ∑j∈S pjCj ≥ g(S) defines a facet for conv(Q).conv(SSM) is a supermodular polyhedron and the greedyalgorithm for minimizing ∑i∈J wiCi coincides with the Smith’srule WSPTThis rediscovers the WSPT rule but also provides validinequalities for other (NP-hard) scheduling problems. AnO(n log n) separation algorithm is available to find an inequalityviolated by a given vector C .More scheduling supermodular polyhedra :

the convex hull of the feasible start time for unit jobs on parallelmachines with nonstationary speeds [QS95]the convex hull of mean busy time vectors of preemptiveschedules for jobs with release dates on a single machine[GQS+02]


















Valid inequalities for NP-hard scheduling problems

Single-machine scheduling with release dates SSMR

Set of feasible schedules for a set of jobs J on a single machine withrelease dates :SSMR =

{S ∈ RJ

∣∣∣∣∣ Si ≥ ri , ∀j ∈ J ,Si ≥ Sj + lij ∨ Sj ≥ Si + lji , ∀i , j ∈ J , i 6= j

}

The problem of minimizing∑

S∈S wiSi is NP-hard =⇒ no chanceto have a complete characterization of conv(SSMR)

However, Balas [Bal85] studied P = conv(S) and derivedfacet-defining inequalities. He showed that a facet-defining inequalityfor a subset of jobs K ⊆ J induces also a facet for J .

∀i ∈ J , Si ≥ ri induces a facet of P∀i , j ∈ J , i 6= j ,(lij + ri − rj)Si + (lji + rj − ri )Sj ≥ dijdji + Lidji + Ljdij induces afacet of P if and only if −dji < rj − ri < dij





{S ∈ RJ


}









{S ∈ RJ


}









{S ∈ RJ


}






Cutting plane generation for NP-hard schedulingproblems

From a partial descrition of conv(S) (relaxation), iteratively solve theseparation problem for families of valid inequalities.

S1

S2

3S1 + 5S2 ≥ 7S1 + 3S2 ≥ 62S1 + 3S2 ≥ 9

Optimum has been found after adding Balas inequality(lij + ri − rj)Si + (lji + rj − ri )Sj ≥ dijdji + Lidji + Ljdij

Separation is NP-hard in general =⇒ the optimum cannot be foundquickly by pure cutting plane generation.Christian Artigues Scheduling and (Integer) Linear Programming CPAIOR 2012, Nantes 15 / 78



S1

S2

3S1 + 5S2 ≥ 7

S1 + 3S2 ≥ 62S1 + 3S2 ≥ 9





S1

S2

3S1 + 5S2 ≥ 7

S1 + 3S2 ≥ 6

2S1 + 3S2 ≥ 9





S1

S2

3S1 + 5S2 ≥ 7S1 + 3S2 ≥ 6

2S1 + 3S2 ≥ 9





S1

S2

3S1 + 5S2 ≥ 7S1 + 3S2 ≥ 6

2S1 + 3S2 ≥ 9



Outline

1 Introduction



4 Column generation



Principle of (mixed)-integer programming

Design a good MIP formulation for the scheduling problemSolve by branch-and-bound

min S1 + S2+p1 + p2S1 ≥ r1S2 ≥ r2

S1 + p1 ≤ d1S2 + p2 ≤ d2

S2 − S1 + Mx ≥ p1S1 − S2 + M(1− x) ≥ p2

S1, S2 integerx ∈ {0, 1}

S1

S2 S

Remark : once x is fixed, extreme points are integral =⇒ no need for integer constraints on S




min S1 + S2+p1 + p2S1 ≥ r1S2 ≥ r2

S1 + p1 ≤ d1S2 + p2 ≤ d2

S2 − S1 + Mx ≥ p1S1 − S2 + M(1− x) ≥ p2


S1

S2 S





min S1 + S2+p1 + p2S1 ≥ r1S2 ≥ r2

S1 + p1 ≤ d1S2 + p2 ≤ d2

S2 − S1 + Mx ≥ p1S1 − S2 + M(1− x) ≥ p2


S1

S2 S





min S1 + S2+p1 + p2S1 ≥ r1S2 ≥ r2

S1 + p1 ≤ d1S2 + p2 ≤ d2

S2 − S1 + Mx ≥ p1S1 − S2 + M(1− x) ≥ p2


S1

S2 S

Left node x = 1





min S1 + S2+p1 + p2S1 ≥ r1S2 ≥ r2

S1 + p1 ≤ d1S2 + p2 ≤ d2

S2 − S1 + Mx ≥ p1S1 − S2 + M(1− x) ≥ p2


S1

S2 S

Right node x = 0

0 1 2 3 4 5 6J1 J2

∑Ci = 8



Case study : the resource-constrained projectscheduling problem

Resource-constrained project scheduling with irregularstarting time costs• n jobs (set J ) with integer start times Si ∈ T = {0, 1, . . . ,T}.• Precedence constraints E such that (i , j) ∈ E =⇒ Sj ≥ Si + lij .• m resources (set R). Constant availability Bk , k ∈ R.• For each job i ∈ J : duration pi and resource requirements bik , k ∈ R.• Resource constraints

∑i∈J |Si≤t≤Si +pi−1 bik ≤ Bk ,∀t ∈ T ,∀k ∈ R.

• Cost function wj : {1, . . . ,T} → R.• Find a schedule that minimizes

∑i∈J wj(Sj).

Remark 1 : |R| = 1, B1 = 1, and bi1 = 1,∀i ∈ J =⇒ 1-machine problem.Remark 2 : |R| = 1, B1 ≥ 2 and bi1 = 1, ∀i ∈ J =⇒ parallel machine problem.






∑i∈J wj(Sj).







∑i∈J wj(Sj).







∑i∈J wj(Sj).







∑i∈J wj(Sj).







∑i∈J wj(Sj).







∑i∈J wj(Sj).







∑i∈J wj(Sj).







∑i∈J wj(Sj).



Scheduling objectivesObjective min

∑i∈J wj(Sj) models most standard objectives

Makespan minmaxi∈J CiLet J = {1, . . . , n + 1} with n + 1 dummy end jobs.

wi (t) = 0, ∀i ∈ J \ {n + 1}, wn+1(t) = t

Maximum lateness minmaxi∈J Ci − diSame modeling by adding arc (i , n + 1) in E with li ,n+1 = pi − di

Earliness-tardiness costsmin

∑i∈J (αi max(0, di − Ci ) + βi max(0,Ci − di )

wi (t) = αi max(0, di − t − pi ) + βi max(0, t + pi − di )

(weighted completion time if, in addition, di = 0 ∧ αi = 0, ∀i ∈ J )Weighted number of late jobs

∑i∈J wiUi

wi (t) =

{0 if t + pi ≤ diwi otherwise





wi (t) = 0, ∀i ∈ J \ {n + 1}, wn+1(t) = t






∑i∈J wiUi

wi (t) =






wi (t) = 0, ∀i ∈ J \ {n + 1}, wn+1(t) = t






∑i∈J wiUi

wi (t) =






wi (t) = 0, ∀i ∈ J \ {n + 1}, wn+1(t) = t






∑i∈J wiUi

wi (t) =



Outline

1 Introduction



4 Column generation



MILP Formulations with time-indexed variables

time-indexed variables xit = 1⇔ Si = t ⇔ Si =∑T

t=0 txitSingle machine

min∑j∈J

∑t∈T

wj(t)xjt∑t∈T

xjt = 1 ∀j ∈ J

T∑t=0

txjt −T∑

t=0txit ≥ lij ∀(i , j) ∈ E

∑j∈J

t∑s=t−pj +1

xjs ≤ 1 ∀t ∈ T

xjt ∈ {0, 1} ∀j ∈ J ,∀t ∈ T


MILP Formulations with time-indexed variables

time-indexed variables xit = 1⇔ Si = t ⇔ Si =∑T

t=0 txitParallel machines

min∑j∈J

∑t∈T

wj(t)xjt∑t∈T

xjt = 1 ∀j ∈ J

T∑t=0

txjt −T∑


∑j∈J

t∑s=t−pj +1

xjs ≤ B ∀t ∈ T

xjt ∈ {0, 1} ∀j ∈ J ,∀t ∈ T


MILP Formulations with time-indexed variablestime-indexed variables xit = 1⇔ Si = t ⇔ Si =

∑Tt=0 txit

RCPSP

min∑j∈J

∑t∈T

wj(t)xjt∑t∈T

xjt = 1 ∀j ∈ J

T∑t=0

txjt −T∑


∑j∈J

t∑s=t−pj +1

bjkxjs ≤ Bk ∀t ∈ T ,∀k ∈ R

xjt ∈ {0, 1} ∀j ∈ J ,∀t ∈ T

nT variables, |E | precedence constraints, |R|T resource constraints.


MILP Formulations with time-indexed variablestime-indexed variables xit = 1⇔ Si = t ⇔ Si =

∑Tt=0 txit

RCPSP (disaggregated precedence constraints)

min∑j∈J

∑t∈T

wj(t)xjt∑t∈T

xjt = 1 ∀j ∈ J

T∑s=t

xis +t+lij−1∑

s=0xjs ≤ 1 ∀(i , j) ∈ E , ∀t ∈ T

∑j∈J

t∑s=t−pj +1


xjt ∈ {0, 1} ∀j ∈ J ,∀t ∈ T

nT variables, |E |T precedence constraints, |R|T resource constraints.


Outline

1 Introduction



4 Column generation



Total unimodularity

A matrix A is totally unimodular (TU) if and only if every squaresubmatrix has determinant 0, 1 or -1.if A is TU, minAx≥bcx has an integer optimal solution or there isno solutionA is TU if (sufficient condition) rows can be partitionned intotwo disjoint sets B and C such that

each column of A has at most two non-zero entries,each entry of A is 0, 1 or -1if two non-zeros in a column have opposite signs, they are in thesame subset of rows (both in B or both in C).if two non-zeros in a column have the same sign, there is one inB and the other one in C

A is also TU if its transpose is TU.


Total unimodularity and scheduling

Project scheduling with irregular starting time costsn jobs (set J ) with integer start times Si ∈ T = {0, 1, . . . ,T}. Precedenceconstraints E such that (i , j) ∈ E =⇒ Sj ≥ Si + lij . Cost functionwj : {1, . . . ,T} → R. Find a schedule that minimizes

∑j∈J wj(Sj).

time-indexed variable zit = 1⇔ Si ≤ t (zit =∑T

t=0 xit)

min∑j∈J

∑t∈T

(wj(t)− wj(t + 1)) zjt

zjT = 1 ∀j ∈ Jzjt − zj,t+1 ≤ 0 ∀j ∈ J , ∀t ∈ Tzj,t+lij − zit ≤ 0 ∀(i , j) ∈ E ,∀t ∈ T

zjt ∈ {0, 1} ∀j ∈ J ,∀t ∈ T

The matrix totally unimodular =⇒ no integer restriction needed !Christian Artigues Scheduling and (Integer) Linear Programming CPAIOR 2012, Nantes 24 / 78

Comparison between formulations

Other IP with time-indexed variables xit = 1⇔ Si = t (nT variables)

min∑j∈J

∑t∈T

wj (t)xjt∑t∈T

xjt = 1 ∀j ∈ J

T∑s=t

xis +

t+lij−1∑s=0

xjs ≤ 1 ∀(i , j) ∈ E , ∀t ∈ T

xjt ∈ {0, 1} ∀j ∈ J , ∀t ∈ T

Totally unimodular matrix, integer polyhedron

min∑j∈J

∑t∈T

wj (t)xjt∑t∈T

xjt = 1 ∀j ∈ J

T∑t=0

txjs −T∑

t=0

txis ≥ lij ∀(i , j) ∈ E

xjt ∈ {0, 1} ∀j ∈ J , ∀t ∈ T

Polyhedron is not integer !

For the RCPSP, the LP relaxation of the time-indexed model withdisagreggated precedence constraints is tighter.


MILP Formulations with time-indexed variables :facetsFacet-inducing inequalities for the single-machine polyhedron(without precedence constraints) [SW92]∑

t∈Txjt ≤ 1 ∀j ∈ J

∑j∈J

t∑s=t−pj +1

xjs ≤ 1 ∀t ∈ T

t+∆−1∑s=t−pj +1

xjs +∑i 6=j

t∑s=t−pi +∆

xis ≤ 1 ∀j ∈ J ,∀t ∈ T ,

∀∆ ∈ {2, . . . ,maxi 6=j

pi}

All facet-inducing inequalities with rhs = 1 [vHS99].Christian Artigues Scheduling and (Integer) Linear Programming CPAIOR 2012, Nantes 26 / 78

Sousa and Wolsey valid inequalities [SW92]

T∑u=1

xju2 =

12∑

i∈J

t∑s=t−pi +1

xis2 ≤

12

∑i∈J

t+∆−1∑s=t−pi +∆

xis2 ≤

12

bT∑

u=1

xju2 +

n∑i=1

t∑s=t−pi +1

xis2 +

∑i∈J

t+∆−1∑s=t−pi +∆

xis2 c ≤ b

32c

t+∆−1∑s=t−pj +1

xjs +∑i 6=j

t∑s=t−pi +∆

xis ≤ 1


Sousa and Wolsey valid inequalities [SW92]

For each time period t ∈ T , for each task j ∈ T and for each∆ ∈ {2, . . . ,maxi 6=j pi}

j

i ∈ J \ {j}≤ 1

t − pj + 1 t + ∆− 1

t − pi + ∆ t


van den Akker et al. valid ineqalities [vHS99]

j1

j2

i ∈ J\{j1, j2}

≤ 2

u1 − pj1 + 1 u1 + ∆1 − 1 u2 − pj1 − v u1 + ∆1 + z u2 − pj1 + 1 u2 + ∆2 − 1

u1 − pj2 + ∆1 u1 max{u2 − v, u1 + ∆1} − pj2 min{u1 + ∆1 + z, u2} u2 − pj2 + ∆2 u2

u1 − pi + ∆1 + z + 1 u1 u2 − pi + 1u1 + ∆1 − 1 u2 − pi + ∆2u2 − v − 1

L M U

With the same principle all facet-inducing inequalities with rhs = 2are derived.

In [WJNS02], relation with valid ineqalities for the node packingproblem is established.


Valid inequalities for the time-indexed RCPSP ILPHardin et al. [HNS08] extend the Sousa and Wolsey 1-machine cutsto the RCPSP.

Let F be a forbidden set (or cover), i.e. ∑j∈F bi > B.∑j∈F

∑ts=t−pj +1 xjs ≤ |F | − 1 is a valid inequality ∀t ∈ T .

Consider now a “special” task j ∈ F and an interval v ≥ 0∑i∈F\{j}

∑ts=t−pj +1+v xiq +

∑t+vs=t−pj +1 xjs ≤ |F | − 1 is a valid

inequality ∀t ∈ T .The inequality defines a facet for the polyhedron reduced to jobsin C if C is a minimal forbidden set, i.e. ∑C\{i} bi ≤ B,∀i ∈ C .Hardin et al. propose lifting procedures and conditions for theresulting inequalities to be facet-inducing for the completepolyhedron.










































Outline

1 Introduction



4 Column generation



Sequencing variables

Time-indexed models yield good LP relaxation but the number ofvariables can be huge =⇒ need to consider more compact models.

Binary variable yij = 1 if and only if Sj ≥ Si + pi

Disjunctive case : yij = 1− yji , ∀(i , j) ∈ D (half of the variablescan be dropped) and yij is the incidence vector of linearorderings :If there are no release dates and in the 1-machine caseCj =

∑i∈D\{j} yijpi + pj

Not possible to consider start dependent costs =⇒ objective∑i∈J wiCi






























Sequencing variables : simplest case

Simple case (one-machine, no release dates, no precedenceconstraints)

Consider yij for i < j =⇒ Cj =∑

i<j yijpi +∑

i>j(1− yji)pi + pj∑j wjCj =

∑1≤i<j≤n(wjpi −wipj)yij +

∑1≤i<j≤n wipj +

∑j∈J wjpj

The objective is easily optimized by LP :min∑1≤i<j≤n(wjpi − wipj)yij0 ≤ yij ≤ 1 ∀i ∈ J , ∀ ∈ J , j > i

It suffices to fix yij = 1 if wjpi − wipj ≤ 0 and to 0 otherwise=⇒ WSPT rule [QS94].





i<j yijpi +∑




∑j∈J wjpj







i<j yijpi +∑




∑j∈J wjpj







i<j yijpi +∑




∑j∈J wjpj




Sequencing variables : one machine, precedenceconstraints (1)

Set E of precedence constraints(MILP due to [Pot80])

min∑i ,j∈J piwjyij +∑

j∈J pjwj

yij + yji = 1 ∀i , j ∈ J , i 6= jyij + yjk − yik ≤ 1 ∀i , j , k ∈ J , i 6= j 6= k

yij = 1 ∀(i , j) ∈ Eyij = 0 ∀(j , i) ∈ Eyij ∈ {0, 1} ∀i , j ∈ J , i 6= j

(NP-hard)


Sequencing variables : one machine, precedenceconstraints (2)

A vertex cover LP-relaxation. Let i ↔j denote {(i , j), (j , i)} ∩ E = ∅

min∑i ,j∈J piwjyij +∑

j∈J pjwj

yij + yji ≥ 1 ∀i , j ∈ J , i 6= j , i ↔ jyik + ykj ≥ 1 ∀(i , j) ∈ E ,∀k ∈ J , i ↔ k , k ↔ jyil + ykj ≥ 1 ∀(i , j), (k , l) ∈ E , i ↔ l , j ↔ k

yij ≥ 0 ∀i , j ∈ J , i 6= j , i ↔ j

For Series-Parallel precedence constraints, the polyhedron is integraland yields a feasible solution [CS04]


Sequencing and natural date variables : onemachine, release datesJobs have release dates ri , assuming r1 ≤ r2 ≤ . . . ≤ rnIdle times =⇒ need to consider natural date variables Si

MILP by [NS92] (improving [DW90] inequalities) linearizing :Sj ≥ (ri + pi)yij +

∑k 6=i ,j

pkyikykj ∀i , j ∈ J , i 6= j

min∑j∈J wjSj

Sj ≥ (ri +pi)yij +∑

k<i ,k 6=j pk(yik−yjk)+∑

k>i ,k 6=j pkykj ∀i , j ∈J , i 6= jSj ≥ ri ∀i ∈ J

yij + yji = 1 ∀1 ≤ i < j ≤ nyij + yjk − yik ≤ 1 ∀i , j , k ∈ J , i 6= j 6= k

δjk ∈ {0, 1} ∀i , j ∈ J , i 6= j




∑k 6=i ,j


min∑j∈J wjSj





δjk ∈ {0, 1} ∀i , j ∈ J , i 6= j




∑k 6=i ,j


min∑j∈J wjSj





δjk ∈ {0, 1} ∀i , j ∈ J , i 6= j


Sequencing and natural date variables : the generaldisjunctive scheduling problem

min∑j∈J wjSj

Sj ≥ ri ∀i ∈ JSj − Si + Mij(1− yij) ≥ pi ∀(i , j) ∈ D

yij + yji = 1 ∀(i , j) ∈ Dyij + yjk − yik ≤ 1 ∀(i , j), (j , k), (k , i) ∈ D, i 6= j 6= k

Sj − Si ≥ lij ∀(i , j) ∈ Eyij ∈ {0, 1} ∀(i , j) ∈ D

with Mij an upper bound on Si + pi − Sj , e.g. di − rj in case ofdeadlines.(n(n − 1)/2 integer variables, n continuous variables)


Sequencing and natural date variables : RCPSPForbidden set F ∈ F : ∃k ∈ R, ∑j∈F bjk > Bk . min∑j∈J wjSjMIP issued from [AVT93]

Sj ≥ ri ∀i ∈ JSj − Si + Mij(1− yij) ≥ pi ∀(i , j) ∈ D∑

i ,j∈F ,i 6=jyij ≥ 1 ∀F ∈ F

yij + yji ≤ 1 ∀i , j ∈ Jyij + yji = 1 ∀(i , j) ∈ D

yij + yjk − yik ≤ 1 ∀(i , j), (j , k), (ki) ∈ D, i 6= j 6= kSj − Si ≥ lij ∀(i , j) ∈ E

yij ∈ {0, 1} ∀(i , j) ∈ D

Let SY the set of feasible (S, y) vectors.(n(n − 1) binary variables, n continuous variables, |F| ? ?)


Sequencing and natural date variables : compactmodel for the RCPSP

Replace exponential number of constraints∑i ,j∈F ,i 6=j yij ≥ 1 ∀F ∈ F

by resource flow networks (0 dummy source job and n + 1 dummysink job)

∑j∈J

f0jk =∑j∈J

fj,n+1,k = Bk ∀j ∈ J ,∀k ∈ R∑j 6=i

fijk =∑j 6=i

fjik = bik ∀i ∈ J ,∀k ∈ R

0 ≤ fijk ≤min(bik , bjk)yij ∀i , j ∈ J , i 6= j ,∀k ∈ R

A compact model is obtained A. et al. [AMR00].




by resource flow networks (0 dummy source job and n + 1 dummysink job)∑

j∈Jf0jk =

∑j∈J

fj,n+1,k = Bk ∀j ∈ J ,∀k ∈ R∑j 6=i

fijk =∑j 6=i







by resource flow networks (0 dummy source job and n + 1 dummysink job)∑

j∈Jf0jk =

∑j∈J

fj,n+1,k = Bk ∀j ∈ J ,∀k ∈ R∑j 6=i

fijk =∑j 6=i





(Facet-inducing) valid inequalities for the RCPSPpolyhedronPolyhedral study for the forbidden set-based formulation,Alvarez-Valdés et al. [AVT93]

let Q = conv({(S, y) ∈ Rn2|(S, y) ∈ SY}) (the RCPSPpolyhedron)The dimension of Q is dQ = n2 − |E ∗| − |D|.Example : yij + yji ≤ 1 (a) induces a facet of Q if(i , j) 6∈ |E ∗| ∪ D

(a) is a face of Q as (i) we can find a solution (S, y) withyij = 1, yji = 0 and (S, y) ∈ Q ∩ {(S, y)|yij + yji = 1} (possibleif the time horizon is sufficiently large) (ii) we can find solutions(S, y) with yij = yji = 0 with (S, y) 6∈ {(S, y)|yij + yji = 1}.(a) is of dimension dQ − 1 as by setting yij + yji = 1 weincrement |D|


















More valid inequalities for disjunctive schedulingand the RCPSP

Let C denote the set of cliques in D. Applegate and Cook[AC91] derived cuts for the job-shop problem from one-machinescheduling valid inequalities (especially [DW90] inequalities)such as

Sj ≥ rj +∑i∈C

yijpi ∀C ∈ C,∀j ∈ C

Sj ≥ rk +∑

i∈C\{j}yijpi −

∑i∈C

yik(rl − rj) ∀C ∈ C,∀j ∈ C ,

Alvarez-Valdés et al. [AVT93] also proposed such cuts for theRCPSP. Let Γ−1(i) (Γ(i)) be the set of ancestors (descendants)of i in E .Sj ≥ Si +pi +

∑k∈Γ(i)

pkykj +∑

k∈Γ−1(j)pkyik +

∑k∈D(i)∪D(j)

k 6∈Γ(i)∪Γ−1(j)

pk(yik +ykj−1)


More valid inequalities for disjunctive schedulingand the RCPSP

Let C denote the set of cliques in D. Applegate and Cook[AC91] derived cuts for the job-shop problem from one-machinescheduling valid inequalities (especially [DW90] inequalities)such as

Sj ≥ rj +∑i∈C

yijpi ∀C ∈ C,∀j ∈ C

Sj ≥ rk +∑

i∈C\{j}yijpi −

∑i∈C

yik(rl − rj) ∀C ∈ C,∀j ∈ C ,

Alvarez-Valdés et al. [AVT93] also proposed such cuts for theRCPSP. Let Γ−1(i) (Γ(i)) be the set of ancestors (descendants)of i in E .Sj ≥ Si +pi +

∑k∈Γ(i)

pkykj +∑

k∈Γ−1(j)pkyik +

∑k∈D(i)∪D(j)

k 6∈Γ(i)∪Γ−1(j)

pk(yik +ykj−1)


Constraint-propagation-based cutting planes forthe RCPSP

So far the studied valid inequalities ignore the existence of atight upper bound.Compute UB with an efficient heuristic and update distancematrix dij through constraint propagation (where dij is a lowerbound of Sj − Si).Cuts can be derived from such updates. d c

ij denotes the value ofdij if constraint c is satisfied. We consider constraints such ask||h, k ≺ h, k � h.

Example 1 : Sj − Si ≥ dk||hij + dk≺h

ij ykh + dk�hij yhk , for any 4 jobs

i , j , k, h.Example 2 : xij ≥ xhk , ∀i , j , h, i , dh≺k

ij ≥ pi

Demassey, A. and Michelon [DAM05]Christian Artigues Scheduling and (Integer) Linear Programming CPAIOR 2012, Nantes 42 / 78







ij ≥ pi








ij ≥ pi








ij ≥ pi








ij ≥ pi


Outline

1 Introduction



4 Column generation



Flow and natural date variablesFor one-machine and parallel machine problem, sequencing variablesyij may be replaced by flow variables fij ∈ {0, 1} stating that i is animmediate predecessor of j in the sequence.min∑i∈J wiSi∑

j∈Jf0j =

∑j∈J

fj,n+1 = m∑

j∈J\{i}fij =

∑j∈J\{i}

fji = 1 ∀i ∈ J

Si ≥ ri ∀i ∈ JSj − Si + Mij(1− fij) ≥ pi ∀i , j ∈ J , i 6= j

Sj − Si ≥ lij ∀(i , j) ∈ Efij ∈ {0, 1} ∀i , j ∈ J , i 6= j

When there are no relase dates, machine assignment and subtourelimitation constraints can be used to avoid big−M coefficients.


Flow and natural date variables (variant)Big-M constraints can be removed, by considering continuous variableSij ≥ 0 which equals 0 unless i is the immediate predecessor of j .

min∑i∈J

∑j∈J\{i}

wiSij

∑j∈J

f0j =∑j∈J

fj,n+1 = m

∑j∈J\{i}

fij =∑

j∈J\{i}fji = 1 ∀i ∈ J

ri fij ≤ Sij ≤ di fij ∀i , j ∈ J , i 6= j∑j∈J\{i}

Sij −∑

k∈J\{i}(Ski + pk fki ) ≥ 0 ∀i ∈ J

∑j∈J\{j}

Sjk −∑

k∈J\{i}Sik ≥ lij ∀(i , j) ∈ E

fij ∈ {0, 1},Sij ≥ 0 ∀i , j ∈ J , i 6= jChristian Artigues Scheduling and (Integer) Linear Programming CPAIOR 2012, Nantes 45 / 78

Outline

1 Introduction



4 Column generation



Positional date and assignment variables (1 mach)Compact formulation without big −M coefficients ? Lasserre andQueyranne [LQ92]• Time horizon is divided in a set E of n positions (or events).

• Positional dates te , fe ≥ 0, for each e ∈ E start, end of event e (n var.)• Assignment variable aie ∈ {0, 1} of job j to event j (n2 var.)

min∑

e∈EwiCi∑

e∈Eaje = 1 ∀j ∈ J∑

j∈Jaje = 1 ∀e ∈ E

te +∑

j∈Jpiaie = fe ∀e ∈ E

te −∑

j∈Jrjaje ≥ 0 ∀e ∈ E

te ≥ te−1 ∀e ∈ ECi + M(1− aie) ≥ fe ∀i ∈ J ,∈ ∀e ∈ E

te ≥ 0 ∀e ∈ Eaie ∈ {0, 1} ∀i ∈ J ,∈ ∀e ∈ E


Positional date and assignment variables (1 mach)Compact formulation without big −M coefficients ? Lasserre andQueyranne [LQ92]• Time horizon is divided in a set E of n positions (or events).• Positional dates te , fe ≥ 0, for each e ∈ E start, end of event e (n var.)

• Assignment variable aie ∈ {0, 1} of job j to event j (n2 var.)min

∑e∈E

wiCi∑e∈E

aje = 1 ∀j ∈ J∑j∈J

aje = 1 ∀e ∈ E

te +∑


te −∑



te ≥ 0 ∀e ∈ Eaie ∈ {0, 1} ∀i ∈ J ,∈ ∀e ∈ E


Positional date and assignment variables (1 mach)Compact formulation without big −M coefficients ? Lasserre andQueyranne [LQ92]• Time horizon is divided in a set E of n positions (or events).• Positional dates te , fe ≥ 0, for each e ∈ E start, end of event e (n var.)• Assignment variable aie ∈ {0, 1} of job j to event j (n2 var.)

min∑

e∈EwiCi∑

e∈Eaje = 1 ∀j ∈ J∑


te +∑


te −∑



te ≥ 0 ∀e ∈ Eaie ∈ {0, 1} ∀i ∈ J ,∈ ∀e ∈ E


Positional date and assignment variables (1 mach)Compact formulation without big −M coefficients ? Lasserre andQueyranne [LQ92]• Time horizon is divided in a set E of n positions (or events).• Positional dates te , fe ≥ 0, for each e ∈ E start, end of event e (n var.)• Assignment variable aie ∈ {0, 1} of job j to event j (n2 var.)

min∑

e∈EwiCi∑

e∈Eaje = 1 ∀j ∈ J∑


te +∑


te −∑



te ≥ 0 ∀e ∈ Eaie ∈ {0, 1} ∀i ∈ J ,∈ ∀e ∈ EChristian Artigues Scheduling and (Integer) Linear Programming CPAIOR 2012, Nantes 47 / 78

Positional date and assignment variables (RCPSP)How many events needed ?∀i ∈ J either Si = 0 or ∃j ∈ J , Si = Sj + pj =⇒ |E| ≤ n + 1Start/End Event-based formulation (SEE) Koné, A., Lopez, Mongeau[KALM11]

Variable xie ∈ {0, 1} : job i starts at event e.

Variable yie ∈ {0, 1} : job i ends at event e.

te time of event e

2n2 + 2n binary variables, (n + 1) continuous variablesOn/Off Event-based formulation (OOE) Koné, A., Lopez, Mongeau[KALM11]

Variable zie ∈ {0, 1} : zie is set to 1 if job i starts at event e or if itstill being processed immediately after event e

n2 binary variables, (n + 1) continuous variablesChristian Artigues Scheduling and (Integer) Linear Programming CPAIOR 2012, Nantes 48 / 78

Positional date and assignment variables (RCPSP)Example

e 0 1 2 3x6e 0 1 0 0y6e 0 0 1 0z6e 0 1 0 0x7e 0 1 0 0y7e 0 0 0 1z7e 0 1 1 0

t 2 3 4 5 6 7 8x6t 0 1 0 0 0 0 0x7t 0 1 0 0 0 0 0


Start/End Event-based formulation (SEE)

min tn

t0 = 0

tf ≥ te + pi xie − pi (1− yif ) ∀(e, f ) ∈ E2, f > e,∀i ∈ Jte+1 ≥ te ∀e ∈ E, e < n∑

e∈Exie = 1,

∑e∈E

yie = 1 ∀i ∈ J∑e

v=0yiv +

∑n

v=exiv ≤ 1 ∀i ∈ J , ∀e ∈ E∑n

e′=eyie′ +

∑e−1

e′=0xje′ ≤ 1 ∀(i , j) ∈ E , ∀e ∈ E

r0k =∑

i∈Abikxi0 ∀k ∈ R

rek = r(e−1)k +∑

i∈Jbikxie −

∑i∈J

bikyie ∀e ∈ E, e ≥ 1, k ∈ R

rek ≤ Bk ∀e ∈ E, k ∈ Rxie ∈ {0, 1}, yie ∈ {0, 1} ∀i ∈ J ∪ {0, n + 1}, ∀e ∈ Ete ≥ 0, rek ≥ 0 ∀e ∈ E, k ∈ R.


On/Off Event-based formulation (OOE)

minCmax

Cmax ≥ te + (zie − zi(e−1))pi ∀e ∈ E,∀i ∈ J

t0 = 0, te+1 ≥ te ∀e 6= n − 1 ∈ E

tf ≥ te + ((zi−e − zi(e−1))− (zif − zi(f−1))− 1)pi ∀(e, f , i) ∈ E2 × J , f > e 6= 0e−1∑e′=0

zie′ ≥ e(1− (zie − zi(e−1))),

n−1∑e′=e

zie′ ≥ e(1 + (zie − zi(e−1))) ∀e 6= 0 ∈ E∑e∈E

zie ≥ 1 ∀i ∈ J

zie +

e∑e′=0

zje′ ≤ 1 + (1− zie)e ∀e ∈ E, ∀(i , j) ∈ E

n−1∑i=0

bikzie ≤ Bk ∀e ∈ E, ∀k ∈ R

te ≥ 0 ∀e ∈ Ezie ∈ {0, 1} ∀i ∈ J , ∀e ∈ E


Outline

1 Introduction



4 Column generation



Arc-time indexed formulationsMixing flow-based and time-indexed formulations.

x tij = 1 if job j starts immediately at the end of job i at time t.

min∑i∈J

∑j∈J\{i}

∑t∈T

wj(t)x tij

∑j∈J\{i}

∑t∈T

x tij = 1 ∀i ∈ J

∑i∈J

x00i = m∑

j∈J\{i}x t

ji −∑

j∈J\{i}x t+pi

ij = 0 ∀i ∈ J ,∀t ∈ T

x tij ∈ {0, 1} ∀i ∈ J ,∀j ∈ J \ {i},∀t ∈ T

At least as strong LP relaxation as time-indexed one if variables x tii

are omitted except for i = 0 (dummy “depot” job) [PUPR10]


Arc-time indexed formulationsMixing flow-based and time-indexed formulations.x t

ij = 1 if job j starts immediately at the end of job i at time t.

min∑i∈J

∑j∈J\{i}

∑t∈T

wj(t)x tij

∑j∈J\{i}

∑t∈T

x tij = 1 ∀i ∈ J

∑i∈J

x00i = m∑

j∈J\{i}x t

ji −∑

j∈J\{i}x t+pi

ij = 0 ∀i ∈ J ,∀t ∈ T

x tij ∈ {0, 1} ∀i ∈ J ,∀j ∈ J \ {i},∀t ∈ T






min∑i∈J

∑j∈J\{i}

∑t∈T

wj(t)x tij

∑j∈J\{i}

∑t∈T

x tij = 1 ∀i ∈ J

∑i∈J

x00i = m∑

j∈J\{i}x t

ji −∑

j∈J\{i}x t+pi

ij = 0 ∀i ∈ J ,∀t ∈ T

x tij ∈ {0, 1} ∀i ∈ J ,∀j ∈ J \ {i},∀t ∈ T






min∑i∈J

∑j∈J\{i}

∑t∈T

wj(t)x tij

∑j∈J\{i}

∑t∈T

x tij = 1 ∀i ∈ J

∑i∈J

x00i = m∑

j∈J\{i}x t

ji −∑

j∈J\{i}x t+pi

ij = 0 ∀i ∈ J ,∀t ∈ T

x tij ∈ {0, 1} ∀i ∈ J ,∀j ∈ J \ {i},∀t ∈ T


are omitted except for i = 0 (dummy “depot” job) [PUPR10]Christian Artigues Scheduling and (Integer) Linear Programming CPAIOR 2012, Nantes 53 / 78

Outline

1 Introduction



4 Column generation



Column generation for time-indexed formulations

How to deal with large-horizons when using time-indexedformulations ? =⇒ Dantzig-Wolfe decomposition.

min∑j∈J

∑t∈T

wj(t)xjt (one-machine problem)∑t∈T

xjt = 1 ∀j ∈ J

∑j∈J

t∑s=t−pj +1

xjs ≤ 1 ∀t ∈ T

xjt ∈ {0, 1} ∀j ∈ J ,∀t ∈ T

Let PS = {x ∈ [0, 1]n+T |∑j∈J∑t

s=t−pj +1 xjs ≤ 1,∀t ∈ T }.PS is integral as the matrix is totally unimodular [vHS00]. Set ofpseudo-schedules




min∑j∈J

∑t∈T


xjt = 1 ∀j ∈ J

∑j∈J

t∑s=t−pj +1

xjs ≤ 1 ∀t ∈ T

xjt ∈ {0, 1} ∀j ∈ J ,∀t ∈ T

Let PS = {x ∈ [0, 1]n+T |∑j∈J∑t





min∑j∈J

∑t∈T


xjt = 1 ∀j ∈ J

∑j∈J

t∑s=t−pj +1

xjs ≤ 1 ∀t ∈ T

xjt ∈ {0, 1} ∀j ∈ J ,∀t ∈ T

Let PS = {x ∈ [0, 1]n+T |∑j∈J∑t

s=t−pj +1 xjs ≤ 1,∀t ∈ T }.

PS is integral as the matrix is totally unimodular [vHS00]. Set ofpseudo-schedules




min∑j∈J

∑t∈T


xjt = 1 ∀j ∈ J

∑j∈J

t∑s=t−pj +1

xjs ≤ 1 ∀t ∈ T

xjt ∈ {0, 1} ∀j ∈ J ,∀t ∈ T

Let PS = {x ∈ [0, 1]n+T |∑j∈J∑t



Column generation for time-indexed formulationsWe have x =

∑rq=1 λqaq, ∀x extreme point of conv({PS}) where∑r

q=1 λq = 1, λ ≥ 0 and aq is the qst point of PS.

Equivalent LP (of equal LP relaxation value)

minr∑

q=1

∑j∈J

∑t∈T

wj(t)aqjt

λq

r∑q=1

(∑t∈T

aqjt

)λq = 1 ∀j ∈ J

r∑q=1

λq = 1

0 ≤ λq ≤ 1 q = 1, . . . , |T |

Start with a restricted set of pseudo schedules and solve the LPrelaxation by column generation.




q=1 λq = 1, λ ≥ 0 and aq is the qst point of PS.Equivalent LP (of equal LP relaxation value)

minr∑

q=1

∑j∈J

∑t∈T

wj(t)aqjt

λq

r∑q=1

(∑t∈T

aqjt

)λq = 1 ∀j ∈ J

r∑q=1

λq = 1

0 ≤ λq ≤ 1 q = 1, . . . , |T |





q=1 λq = 1, λ ≥ 0 and aq is the qst point of PS.Equivalent LP (of equal LP relaxation value)

minr∑

q=1

∑j∈J

∑t∈T

wj(t)aqjt

λq

r∑q=1

(∑t∈T

aqjt

)λq = 1 ∀j ∈ J

r∑q=1

λq = 1

0 ≤ λq ≤ 1 q = 1, . . . , |T |




minr∑

q=1(∑j∈J

∑t∈T

wj(t)aqjt)λq

r∑q=1

(∑t∈T

aqjt

)λq = 1 ∀j ∈ J

(πj)

r∑q=1

λq = 1

(γ)

0 ≤ λq ≤ 1 q = 1, . . . , |T |Reduced cost of a pseudo schedule

cq =∑j∈J

∑t∈T

wj(t)aqjt

−∑j∈J

πj(∑t∈T

aqjt)− γ

=∑j∈J

∑t∈T

(wj(t)− πj)aqjt − γ

Finding a negative reduced cost variable amounts to find a shortest pathin an acyclic graph with O(nT ) arcs.



minr∑

q=1(∑j∈J

∑t∈T

wj(t)aqjt)λq

r∑q=1

(∑t∈T

aqjt

)λq = 1 ∀j ∈ J (πj)

r∑q=1

λq = 1 (γ)


cq =∑j∈J

∑t∈T

wj(t)aqjt −

∑j∈J

πj(∑t∈T

aqjt)− γ

=∑j∈J

∑t∈T





minr∑

q=1(∑j∈J

∑t∈T

wj(t)aqjt)λq

r∑q=1

(∑t∈T

aqjt

)λq = 1 ∀j ∈ J (πj)

r∑q=1

λq = 1 (γ)


cq =∑j∈J

∑t∈T

wj(t)aqjt −

∑j∈J

πj(∑t∈T

aqjt)− γ

=∑j∈J

∑t∈T





minr∑

q=1(∑j∈J

∑t∈T

wj(t)aqjt)λq

r∑q=1

(∑t∈T

aqjt

)λq = 1 ∀j ∈ J (πj)

r∑q=1

λq = 1 (γ)


cq =∑j∈J

∑t∈T

wj(t)aqjt −

∑j∈J

πj(∑t∈T

aqjt)− γ

=∑j∈J

∑t∈T




Column generation for time-indexed RCPSP (1)How to strengthen the time-indexed formulation ? Mingozzi et al. [MM98]

min∑j∈J

∑t∈T

wj(t)xjt

∑t∈T

xjt = 1 ∀j ∈ J

∑j∈J

t∑s=t−pj +1


T∑t=0

txjs −T∑

t=0txis ≥ lij ∀(i , j) ∈ E

xjt ∈ {0, 1} ∀j ∈ J ,∀t ∈ T

Integer Dantzig-Wolfe decomposition of resource constraintsChristian Artigues Scheduling and (Integer) Linear Programming CPAIOR 2012, Nantes 58 / 78

Column generation for time-indexed RCPSP (2)Introduce additional variables

y ∈ {0, 1}n+T

∑

j∈J∑T

s=0 bjkyjs ≤ Bk , ∀t ∈ T ,∀k ∈ R∑Ts=0 yjs = pj , ∀j ∈ J

xjt ≥ yjt − yj,t−1, ∀j ∈ J

Perform an integer Danzig-Wolfe decomposition ofPS =

{y ∈ {0, 1}n+T

∣∣∣∑j∈J∑T

s=0 bjkyjs ≤ Bk ,∀t ∈ T ,∀k ∈ R}

Subproblem is decomposable for each t ∈ T and each t ∈ Rnyielding mT multidimentional knapsack problems.

Better LP relaxation than the time-indexed formulation, butpractically intractable.

Best known lower bounds for the RCPSP (before “SAT” results [Hor10])where obtained by computing relaxation of this formulation ( Bruckerand Knust [BK00], Baptiste, A., Demassey Michelon [DABM04], Baptiste andDemassey[BD04]) integrating CP-based filtering and cuts.Christian Artigues Scheduling and (Integer) Linear Programming CPAIOR 2012, Nantes 59 / 78


y ∈ {0, 1}n+T

∑

j∈J∑T




{y ∈ {0, 1}n+T

∣∣∣∑j∈J∑T






y ∈ {0, 1}n+T

∑

j∈J∑T




{y ∈ {0, 1}n+T

∣∣∣∑j∈J∑T






y ∈ {0, 1}n+T

∑

j∈J∑T




{y ∈ {0, 1}n+T

∣∣∣∑j∈J∑T






y ∈ {0, 1}n+T

∑

j∈J∑T




{y ∈ {0, 1}n+T

∣∣∣∑j∈J∑T





More on Column generation for time-indexedRCPSP

Can we combine strengthening the TI formulation and reductionof the number of variables ? (Cut and Column generationdirectly on the time-indexed formulation Sadykov andVanderbeck [SV11] ?)Other decomposition schemes : partition the job in m subsets sothat there is a feasible single machine schedule for each subset.van den Akker et al. [vHv99, vHv06, Mv07]


More on Column generation for time-indexedRCPSP

Can we combine strengthening the TI formulation and reductionof the number of variables ? (Cut and Column generationdirectly on the time-indexed formulation Sadykov andVanderbeck [SV11] ?)Other decomposition schemes : partition the job in m subsets sothat there is a feasible single machine schedule for each subset.van den Akker et al. [vHv99, vHv06, Mv07]


Outline

1 Introduction



4 Column generation



Machine scheduling results (1)

Efficiency of Sousa and Wolsey/van den Akker et al. cuts forlower bound computation(from van den Akker et al. [vHS99])

Columns 1(2) : average or maximal gap for inequalities with rhs 1(2)Columns n1(n2) : number of integer solutions found by cutting planegeneration with inequalities with rhs 1 (2)


Machine scheduling results (2)

Column-and-cut generation (Sadykov and Vanderbeck 2011)

[R] Time-indexed formulation[M] Dantzig-Wolfe decomposition of [vHS00]Column cpu : time to solve the LP relaxation


Machine scheduling results results (3)

Positional and sequencing based formulations cannot bediscarded but have to be combined with heuristics (Hoogeveenand van de Velde [Hv95], Danna et al. [DRL05])Comparison of formulations for parallel machine scheduling Unluand Mason [UM10]Recommendation is to use time-indexed models for smalldurations and flow-based model for other cases


RCPSP results : comparison of ILP formulations vsCP

(from Kone et al.[KALM11])

No formulationdominates the other,the accurateformulation has to bechosen depending oninstance charactristicsILP formulations arenot dominated by CP


RCPSP results : a cyclic example, MIP vs CPInstances n DSP hybrid/HD hybrid/GS CP ILP+ (P+

λ) CG (DW +

λ) λ0

λdsp λhyb CPUs λhyb CPUs λCP λILP+ CPUs λCG CPUsadpcm-st231.1 86 80 - - - - 80 - - 55 301 52adpcm-st231.2 142 139 - - - - 139 - - 82 305 82gsm-st231.1 30 30 29 2 28 2 28 28∗ 256 25 8 24gsm-st231.2 101 93 - - - - 93 - - 61 301 59gsm-st231.5 44 36 36 10 36 17 - 36∗ 3343 36 37 26gsm-st231.6 30 27 27 3 27 4 27 27∗ 7 27 3 17gsm-st231.7 44 41 41 13 41 17 41 41∗ 256 41 66 28gsm-st231.8 14 12 12 0.3 12 0.3 - 12∗ 0.6 12 <0.1 9gsm-st231.9 34 32 32 2 34 4 32 32∗ 62 31 12 28gsm-st231.10 10 8 8 0.2 8 0.1 8∗ 8∗ 0.2 8 <0.1 6gsm-st231.11 26 24 24 1 24 1 24 24∗ 5 24 1.5 20gsm-st231.12 15 13 13 0.3 13 0.4 13 13∗ 0.7 13 <0.1 10gsm-st231.13 46 43 43 168 42 440 43 41 11265 41 125 27gsm-st231.14 39 34 34 6 34 10 34 33∗ 3766 33 17 20gsm-st231.15 15 12 12 0.3 12 0.3 12 12∗ 0.9 12 <0.1 9gsm-st231.16 65 59 59 145 59 144 60 58 8656 48 300 38gsm-st231.17 38 33 33 202 - - 33 32 12786 33 19 23gsm-st231.18 214 194 - - - - 193 - - - - 120gsm-st231.19 19 15 15 0.4 15 0.6 15 15∗ 1.6 15 0.2 12gsm-st231.20 23 20 20 1 20 1.4 20 20∗ 17 20 0.9 13gsm-st231.21 33 30 30 6 30 6 31 30∗ 6105 29 7 20gsm-st231.22 31 29 29 3 29 4 29 29∗ 51 28 7 18gsm-st231.25 60 55 - - 55 75 57 55∗ 11589 48 300 37gsm-st231.29 44 42 42 13 42 15 42 42∗ 63 42 68 28gsm-st231.30 30 25 25 3 25 6 25 25∗ 14 25 6 16gsm-st231.31 44 39 39 13 39 17 39 39∗ 833 39 59 26gsm-st231.32 32 30 30 4 30 5 30 30∗ 11 30 5 21gsm-st231.33 59 52 - - - - 46 45 9697 46 300 33gsm-st231.34 10 8 7 <0.1 7 0.1 7∗ 7∗ 0.2 7 <0.1 6gsm-st231.35 18 16 14 0.4 14 0.5 14 14∗ 4 14 0.2 11gsm-st231.36 31 29 24 2 24 10 24 24∗ 321 24 4 18gsm-st231.39 26 23 21 1.5 21 3 21 21∗ 376 20 2 15gsm-st231.40 21 17 16 0.5 18 1.3 17 16∗ 28 16 0.6 12gsm-st231.41 60 50 - - 47 286 49 46 10069 46 300 34gsm-st231.42 23 19 18 0.7 19 4 18 18∗ 20 18 1 14gsm-st231.43 26 23 21 2 22 2 20 20∗ 101 20 2 15#best(opt) 23 25 25 27(2) 27(27)

From [AABH12], on a cyclic RCPSP with unit duration tasks, CP is able to find very good

solutions but ILP is better at proving optimality.Christian Artigues Scheduling and (Integer) Linear Programming CPAIOR 2012, Nantes 66 / 78

ConclusionProblems of increasing size are solved by MILP models with the helpof

Strong valid inequalitiesEfficient heuristicsBenefits from hybrid-methods (CP/MILP/LS)

LP Lower bounds are often still too slow : cut-and-column generationperspective.

“In retrospect it is interesting to note that the originalproblem that started my research is still outstanding–namely the problem of planning or scheduling dynamicallyover time, particularly planning dynamically underuncertainty. If such a problem could be successfully solved itcould (eventually through better planning) contribute to thewell-being and stability of the world.” (George Dantzig)Christian Artigues Scheduling and (Integer) Linear Programming CPAIOR 2012, Nantes 67 / 78

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References IV

E. Danna, E. Rothberg, and C. Le Pape, Exploring relaxationinduced neighborhoods to improve mip solutions, MathematicalProgramming 102 (2005), no. 1, 71–90.

M.E. Dyer and L.A. Wolsey, Formulating the single machineproblem with release dates as a mixed integer program, DiscreteApplied Mathematics 26 (1990), 255–270.

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J. R. Hardin, G. L. Nemhauser, and M. W. P. Savelsbergh,Strong valid inequalities for the resource-constrained schedulingproblem with uniform resource requirements, DiscreteOptimization 5 (2008), 19–35.Christian Artigues Scheduling and (Integer) Linear Programming CPAIOR 2012, Nantes 71 / 78

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J. M. van den Akker, C.P.M. Van Hoesel, and M.W.P.Savelsbergh, A polyhedral approach to single-machine schedulingproblems, Mathematical Programming 85 (1999), 541–572.

M. van den Akker, C. A. J. Hurkens, and M. W. P. Savelsbergh,Time-indexed formulations for machine scheduling problems :Column generation, INFORMS Journal on Computing 12 (2000),no. 2, 111–124.M. van den Akker, J. A. Hoogeveen, and S. L. van de Velde,Parallel machine scheduling by column generation, OperationsResearch 47 (1999), no. 6, 862–872.

M. van den Akker, J. A. Hoogeveen, and J. W. van Kempen,Parallel machine scheduling through column generation :Minimax objective functions, ESA, 2006, pp. 649–659.Christian Artigues Scheduling and (Integer) Linear Programming CPAIOR 2012, Nantes 77 / 78

References XI

H. Waterer, E.L. Johnson, P. Nobili, and M.W.P. Savelsbergh,The relation of time indexed formulations of single machinescheduling problems to the node packing problem, MathematicalProgramming (2002), no. 93, 477–494.


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