Branch&Bound MIP

8/10/2019 Branch&Bound MIP

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Contents college 3 en 4

Book: Appendix A.1, A.3, A.4, 3.4, 3.5,4.1, 4.2, 4.4, 4.6 (not: 3.6 - 3.8, 4.2

- 4.3)

Extra literature on resource constrained

project scheduling (will be handed out)


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Planning and schedulingoptimization techniques

Dispatching Rules

Composite Dispatching Rules

Adaptive search Dynamic Programming

(Integer) Linear Programming

Cutting plane methods

Branch and Bound

Beam Search


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Linear programming (LP) model

LP:

Matrix form:

)n,,1j(0x

bxa...xaxa

bxa...xaxa

bxa...xaxa

:tosubject

xc...xcxcmin

j

mnmn22m11m

2nn2222121

1nn1212111

nn2211

0x

bAx

xcmin T

objective function

constraints

variable

restrictions

where:x, c: n-vectorA: m,n-matrixb: m-vector


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Linear programming example

)2,1j(0x

12x4x3

4xx2:tosubject

xxmax

j

21

21

21

0

0

x

x

12

4

x

x

43

12

:tosubjectx

x

1

1max

2

1

2

1

2

1

T

or:


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Linear programming example:graphical solution (2D)

1 2 3 4 5 6

x1

1

2

3

4

5

6x2

)objective(

6xx 21

4xx2 21

12x4x3 21

)2,1j(0x

12x4x3

4xx2

:tosubject

xxmax

j

21

21

21

0

solution

space


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Linear programming (cont.)

Solution techniques: (dual) simplex method

interior point methods (e.g. Karmarkar algorithm)

Commercial solvers, for example: CPLEX (ILOG)

XPRESS-MP (Dash optimization)

OSL (IBM) Modeling software, for example:

AIMMS

AMPL


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Integer programming (IP) models

Integer variable restriction

IP: integer variables only

MIP: part integer, part non-integer variables

BIP: binary (0-1) variables

General IP-formulation:

Complex solution space

)Zx(integerx

bAx

xcmin T


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Integer programming example:graphical solution (2D)

1 2 3 4 5 6

x1

1

2

3

4

5

6x2

)objective(

6xx 21

)2,1j(Zx

12x4x3

4xx2:tosubject

xxmax

j

21

21

21

02 optimal solutions!


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Total unimodularity property forinteger programming models

Suppose that all coefficients are integer in the model:

i.e.

Example: transportation problem

if A has the total unimodularity property(i.e. every square submatrix has determinant 0,1,-1)

there is an optimal integer solution x*

& the simplex method will find such a solution

0x

bAx

xcmin T

j,i,b,a iij


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Integer programming tricks

PROBLEM: x = 0 or x k

use binary indicator variable y=

restrictions:

kfor x,

0for x,

1

0

0,1yykx

x)onupperboundanis(MyMx


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PROBLEM: fixed costs: if xi>0 then costs C(x

i)

use indicator variable yi=

restrictions :

Integer programming tricks (2)

0x

bAx

)x(Cminimize

0.for x

0,for x

xck

0)x(C

:where

i

i

iii

i

0for x,

0for x,

1

0

i

i

0,1y

xcyk)x(C

yMx

i

iiiii

ii

)i(


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Hard vs. soft restrictions

hard restriction: must hold, otherwise unfeasibility

for example:

soft restriction: may be violated, with a penalty

for example:

(Integer) programming tricks (3)

0Y,0x

Y5xx

100Yxcminimize

21

T

5xx 21


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Conjunctive/disjunctive programming

- conjunctive set of constraints: must all be satisfied

- disjunctive set of constraints: at least one must be satisfied

example (Appendix A.4):

Integer programming tricks (5)

jkj

kjk

jjj

pxx

or

pxx

xwmin

}1,0{y

)y1(Mpxx

yMpxx

xwmin

2jkj

1kjk

jjj


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IP examplenonpreemptive single machine, total weighted

completion time (App. A.3)

function)(objectivextwimizemin

jjoboftimecompletionxt

jjoboftimecompletionxt

n

1j

1-Cmax

0t

jtj

1-Cmax

0tjt

jt

1xif jt

otherwise0andt,timeatjjobif,1xjt completes

objective function: minimize weighted completion time:

model definition:


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IP example (cont.)

Restriction: all jobs must be completed once:1x

1-Cmax

0tjt

Restriction: only one job per time t:

t)timeperjoboneexactly:on(restricti1x

t)duringprocessinisjjob(if1x

n

1j

pt

tsjs

pt

tsjs

j

j

if job j is in process during t, it must be completedsomewhere during [t,t+pj]


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IP example (cont.)

Complete IP-model:

1)-Cmax,0,tn,,1,j(for1,0x

1)-Cmax,0,t(for1x

n),1,j(for1x

:tosubject

xtwinimizem

jt

n

1j

pt

tsjs

1-Cmax

0tjt

n

1j

1-Cmax

0tjtj

j

nCmax

integer

variables


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IP example (cont.)Additional restriction: precedence constraints

Model definition: SUCC(j) = successors of job j

job j must be completed before all jobs in SUCC(j):

n),1,jSUCC(j),k(fortxtx

kjoboftimestartptx

jjoboftimecompletiontx

1maxC

0t

kt

1maxC

0t

jt

k

1maxC

0tkt

1maxC

0tjt


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Integer programmingsolution techniques

Heuristic vs. explicit approach: trade-off between solution quality and computation time

trade-off between implementation effort/costs and yield

(i.e. profits gained from solution quality improvement)

Heuristic methods; for example: local search (e.g. simulated annealing, tabu search, k-opt)

(composite) dispatching rules (e.g. EDD, SPT, MS)

adaptive search

rounding fractional solutions

beam search


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Explicit methods; 3 categories:

1. dynamic programming

2. cutting plane (polyhedral) methods3. branch and bound

or: hybrid methods (combination of the above)

Commercial IP solvers usually use acombination of heuristics and 2, 3

Integer programmingsolution techniques (cont.)


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Dynamic programming

Problem divided into stages

Each stage can have various states

A recursive objective function is used toiterate through all states and all stages(forwards or backwards)

))}x,i(i(F)x,i(c{min)i(F tt1t1ttttx

ttt

T),0,(tx t

ti

(constant)c)i(F 000


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Cutting plane methods

STEP 0: Create a relaxation of the problem byomitting restrictions

(e.g. the integrality restrictions)

STEP 1: Solve the current problem

STEP 2: If solution is infeasible then generate

a restriction that cuts of the solution,

and add it to the problem STEP 1

Otherwise: DONE


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Branch and bound

Enumeration in a search tree

each node is a partial solution, i.e. a part of

the solution space

...

...

root node

child nodes

child nodes

Level 0

Level 1

Level 2


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Branch and bound example 1

Disjunctive programming (appendix A.4):disjunctive set of constraints: at least one must be satisfied

xj = completion time of job j

restriction: )Ik,j(pxxorpxxeither jkjkjk

solve LP withoutdisjunctive restrictions

(= LP relaxation)

if disjunct. restr.violated for j & k

Level 0

Level 1

...

kjk pxx jkj pxx


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Branch and bound (cont.)

Upper bound: e.g. a feasible solution

Lower bound:

e.g. a solution to an easier problem

Node elimination (fathom/discard nodes):

when lower bound >= upper bound


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Branch and bound (cont.)

Branching strategy:

how to partition solution space

Node selection strategy: sequence of exploring nodes:

depth first (tries to obtain a solution fast)

breadth/best bound first (tries to find the best solution)

which nodes to explore (filter and beam width)

filter width: #nodes selected for thorough evaluation

beam width: #nodes that are branched on ( filter width)

Beam search


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Branch and bound example 2 Single machine, maximum lateness, release

and due dates

lower bound: EDD + preemption

Jobs 1 2 3 4

p(j) 4 2 6 5

r(j) 0 1 3 5

d(j) 8 12 11 10

(?,?,?,?)

(1,?,?,?) (4,?,?,?)(2,?,?,?)

Level 0

Level 1(3,?,?,?)


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Branch and bound example 2 Lower bound for: (1,?,?,?)

Lower bound: Lmax = max(0,17-12,15-11,0)=5

Jobs 1 2 3 4

p(j) 4 2 6 5r(j) 0 1 3 5

d(j) 8 12 11 10

t

r(2) r(3) r(4)

d(4)


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(?,?,?,?)

(1,?,?,?) (4,?,?,?)(2,?,?,?)

Level 0

Level 1(3,?,?,?)

Branch and bound example 2 (cont.)

LB=5 LB=7*=UB

(1,2,?,?) (1,3,?,?) Jobs 1 2 3 4

p(j) 4 2 6 5r(j) 0 1 3 5

d(j) 8 12 11 10

infeasible:

(1,3,4,3,2)

LB=6*=UB

(1,2,4,3)

LB=5*=UB(1,3,4,2)

DONE

(1,2,4,3) (1,3,4,2)


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Branch and bound example 3

1 2 3 4 5 6

x1

1

2

3

4

5

6x2

)objective(

6xx 21

0

4.2x

,8.0x

2

1

0x1 1x1

3x

,0x

2

1

2x

,1x

2

1

2x2 3x2

2x

,1x

2

1

3x

,0x

2

1

obj: 3 obj: 3

obj: 3 obj: 3

LP solution:

B h l 1


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Beam search example 1single-machine, total weighted tardiness

Upper bound: ATC rule (apparent tardiness cost):schedule 1 job at a timeevery time a machine comes available, determineranking of jobs:

pK)0,tpdmax(

j

j

j

jj

ep

w)t(I

MS rule

WSPTrule

look-ahead parameter:

K = 4.5 + R (R 0.5)

K = 6 - 2R (R 0.5)

= due date range factor maxminmax C/ddR


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Beam search example 1 (cont.)single-machine, total weighted tardiness

(?,?,?,?)

(1,?,?,?) (4,?,?,?)(2,?,?,?) (3,?,?,?)

1,2,3)(j

p

we

p

w)t(I

j

jpK

)0,tpdmax(

j

j

j

jj

Upper bound by ATC rule: )3,2,1j(0)0,tpdmax( jj

Jobs 1 2 3 4

p(j) 10 10 13 4

d(j) 4 2 1 12w(j) 14 12 1 12

w(j)/p(j) 1.4 1.2 0.1 3


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(?,?,?,?)

(1,?,?,?) (4,?,?,?)(2,?,?,?) (3,?,?,?)

Jobs 1 2 3 4

p(j) 10 10 13 4

d(j) 4 2 1 12

w(j) 14 12 1 12

w(j)/p(j) 1.4 1.2 0.1 3

Jobs C(j) d(j) T(j) w(j)*T(j)

1 10 4 6 84

2 24 2 22 264

3 37 1 36 36

4 14 12 2 24

Upper boundby ATC rule:

Total = 408


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(?,?,?,?)

(1,?,?,?) (4,?,?,?)(2,?,?,?) (3,?,?,?)

UB=408 UB=436 UB=814 UB=440

discardedexploredfurther

(beam width = 2)

4 nodesanalyzed(filter width=4)

B h l 1 ( t )


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Beam search example 1 (cont.)

(?,?,?,?)

(1,?,?,?) (4,?,?,?)(2,?,?,?) (3,?,?,?)

UB=408 436 814 440

(1,2,?,?) (1,3,?,?) (1,4,?,?)

UB=480 706 408

(1,4,2,3) (1,4,3,2)

UB=408 554

b t l ti

(2,1,?,?) (2,3,?,?) (2,4,?,?)

436

(2,4,1,3) (2,4,3,1)

436 608

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Branch&Bound MIP