Download pdf - Bin$Packing$with$Linear$Usage$Costs$$€¦ · Outline$ 1. The$EnergeTIC$Project(Problem$Formulaon)$ 2. A$key$sub5problem$BPUC$(Bin$Packing$with$Usage$Costs)$ – Lower$bounds$based$on$LP$

Bin Packing with Linear Usage Costs An applica)on to Energy Management in Data

Centres

Hadrien Cambazard G-‐SCOP, Université de Grenoble

Deepak Mehta, Barry O’Sullivan, Helmut Simonis Cork Constraint ComputaAon Centre, Cork, Ireland

Outline 1.  The EnergeTIC Project (Problem FormulaAon)

2.  A key sub-‐problem BPUC (Bin Packing with Usage Costs) –  Lower bounds based on LP –  A Constraint Programming view-‐point

3.  Back to the applicaAon –  A Lower bound by column generaAon relying on BPUC –  Upper bound via Large Neighborhood Search

4.  Overview of experimental results

The EnergeTIC Project

•  Interest increased by the Roadef Challenge with google •  EnergeTIC is a project located in Grenoble •  Companies involved:

Schneider, Bull, Eolas Business & Decision, UXP

Design of energy efficient data centres

•  Permanent increase of energy price (50 % by 2020) •  Growing market for cloud compuAng


Problem formulaAon Assign virtual machines to servers

over mulAple Ame-‐periods

Energy models Characterize

each component

Demand predicAon Forecast the cpu needs

Energy indicators -‐ Usage -‐ Energy

ExploitaAon constraints

Solver Plan





each component




Solver Plan

EnergeTIC: Modeling Equipment

•  Characterize energy consumpAon of heterogeneous servers – Measures performed on Bull equipment –  Linear model of energy consumpAon

Fixed cost Usage cost





each component




Solver Plan

Problem formulaAon

…

1 migration

…

t-1 t t+1

STAND BY

STAND BY

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Uit

MiUmaxj

Mmaxj

•  A server has a CPU, MEMORY and CARDINALITY capaciAes •  A virtual machine has a CPU consumpAon changing over Ame and a

fixed MEMORY consumpAon •  The number of migraAons is limited from one period to the next •  ObjecAve : state (ON/STAND_BY) and load of servers

changes of state migraAons

Problem formulaAon

1 migration

… …

t-1 t t+1

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MiUmaxj

Mmaxj

A series of consecuAve “cost aware” mulAdimensional Bin-‐Packing problems (linked by a limited number of changes)

Problem formulaAon

1 migration

… …

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MiUmaxj

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Coupling constraints

Problem formulaAon

1 migration

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Coupling constraints

•  A set of items

•  A set of bins

Bin Packing with Usage Costs S = {w1, . . . , wn}

B = {{C1, f1, c1}, . . . , {Cm, fm, cm}}

w1 w2. . .

Cj

Fixed cost for opening a bin

Usage cost depending on the load

Cost

Load

Load

{Cj , fj , cj }

cjfj

•  Minimize

costj = fj + Loadjcj

Pm

j=1|Loadj>0 costj

Bin Packing with Usage Costs

3

9

{9,0,1} {3,0,2} {3,0,2} {3,0,2} {3,0,2}3

2

Items Bins

•  A set of items •  A set of bins •  Minimize the sum of the costs of the used bins

S = {w1, . . . , wn}B = {{C1, f1, c1}, . . . , {Cm, fm, cm}}

Bin Packing with Usage Costs •  A set of items •  A set of bins •  Minimize the sum of the costs of the used bins

S = {w1, . . . , wn}B = {{C1, f1, c1}, . . . , {Cm, fm, cm}}

3

2

3

9

{9,0,1} {3,0,2} {3,0,2} {3,0,2} {3,0,2}

(P1) (P2)

{9,0,1} {3,0,2} {3,0,2} {3,0,2} {3,0,2}

8x1 + 3x2 + 3x2 + 3x2 = 26 9x1 + 2x2 + 2x2 + 2x2 + 2x2 = 25

Lower bounds for BPUC

Bin j is open Item i is placed on bin j Load of bin j

“Standard” LP formulaCon:


Bin j is open Item i is placed on bin j Load of bin j

“Standard” LP formulaCon:

Linear relaxaCon easy to characterize: •  A unit of space on bin j will cost at least: •  Sort the bins by non-‐decreasing •  Fill the “cheapest” bins first

rj = fj/Cj + cjrj


rj = fj/Cj + cj

9

f1 =c1 =

9 5

r1 = 6

14 3

7

f3 =c3 =r3 = 5

1

5

f4 =c4 =10 r4 =10.2

3

f2 =c2 =r2 =

1 5 5.33

12

f5 =c5 =r5 =

12 10 11

Cj =

3

5


rj = fj/Cj + cj

9

f1 =c1 =

9 5

r1 = 6

14 3

7

f3 =c3 =r3 = 5

1

5

f4 =c4 =10 r4 =10.2

3

f2 =c2 =r2 =

1 5 5.33

12

f5 =c5 =r5 =

12 10 11

9

r1 = 6

7

r3 = 5

5

r4 =10.2

3

r2 =5.33

12

r5 = 11

3

5

3


rj = fj/Cj + cj

9

f1 =c1 =

9 5

r1 = 6

14 3

7

f3 =c3 =r3 = 5

1

5

f4 =c4 =10 r4 =10.2

3

f2 =c2 =r2 =

1 5 5.33

12

f5 =c5 =r5 =

12 10 11

Total Load = 18 Lower bound = 7x5 + 3x5.33 + 6x8 = 99

r1 = 6 r3 = 5 r4 =10.2 r2 =5.33 r5 = 11 O(mlog(m) + n)

7 8

3

5

3


Total Load = 18

Lower bound = 99

r1 = 6 r3 = 5 r4 =10.2 r2 =5.33 r5 = 11

OpAmal soluAon

Opt = 129

7 8

3

5

Lower bounds for BPUC based on LP

2 3 4 5 70

F

0 5 9 7 6 15 18 2400 5 12 7

S = {2, 2, 3, 5} B = {{3,1,2},{4,3,1},{7,3,3}}

z3: Arc-‐Flow Model of Carvalho

z1: Standard LP formulaAon z2: Gilmore and Gomory

i-‐th cuing pajern of bin j

C C z2 z3 z1

Solving BPUC exactly

Arc-‐Flow Model Standard LP formulaAon

Gilmore and Gomory

BPUC in the applicaAon domain goes with: 1.  Large scale 2.  Side constraints (ex: cardinality constraints, 2D, …)

MIP



Gilmore and Gomory


MIP

Not so easy to extend with side constraints



Gilmore and Gomory


MIP



Gilmore and Gomory

Bin-‐Packing global constraint +

Cost propagaAon using bound of LP

MIP CP


Cost propagaAon for BPUC

r1 = 6 r3 = 5 r4 =10.2 r2 =5.33 r5 = 11

7 3

8

Idea: Use LP bound to compute the minimum/maximum possible filling of each bin due to cost

LB = 99


r1 = 6 r3 = 5 r4 =10.2 r2 =5.33 r5 = 11

Suppose a UB=130, load on bin 3 (cheapest bin) must be >= 1 (if not LB = 132 > 130)

7 3

8 1


LB = 99


r1 = 6 r3 = 5 r4 =10.2 r2 =5.33 r5 = 11

7 3

8 1


1

UB = 130

We know now that 3 and 1 must be open so: •  Count their fixed costs in the bound •  Apply the same reasoning on restricted problem unAl a fixed point is reached


r1 = 6 r3 = 5 r4 =10.2 r2 =5.33 r5 = 11

7 3

8 3

0

3 3

5


UB = 130

9

Update upper bounds Update lower bounds

O(m)


r1 = 6 r3 = 5 r4 = 10.2 r2 = 5.33 r5 = 11

7

3

8 3

0

3 3

5

UB = 130 9

Update upper bounds Update lower bounds

•  Lower bound in and filtering in for each bin •  Upper/lower bounds can also be Aghtened by Dynamic Programming

(during search for CP, iniAally for MIP) •  On the example, the bound at the root node (with UB = 130):

–  LP Bound: 114.4 –  CP Bound: 119.66

•  UpdaAng load bounds triggers the filtering of the Bin-‐Packing global constraint commiing/forbidding items to bins

O(mlog(m) + n) O(m)

Results on BPUC

•  CP is good at handling large BPUC problems •  Need to implement branch and price for the cuing-‐

stock formulaAon to compare with CP

Lower bounds

Back to the applicaAon

•  Upper bounds: Large Neighborhood Search designed for the ROADEF Challenge 2012 (Mehta, O’Sullivan, Simonis)

•  Lower bounds: Column generaAon relying on BPUC

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Back to the applicaAon (Lower bound)

•  A column is a Bin-‐Packing (one Ame period) … •  The pricing problem can be well modeled with 3 global

constraints: –  BinPackingWithUsageCost (for cpu cost pertubated by dual variables) –  BinPackingWithUsageCost (for cardinality cost due to dual variables) –  BinPacking (for memory limitaAons) –  GlobalCardinality (for limits on the number of items on each bin)

…

1 migration

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t-1 t t+1

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Back to the applicaAon (Lower bound)

•  A column is a Bin-‐Packing (one Ame period) …

•  The pricing problem can be well modeled with 3 global constraints: –  BinPackingWithUsageCost (for cpu cost pertubated by dual variables) –  BinPackingWithUsageCost (for cardinality cost due to dual variables) –  BinPacking (for memory limitaAons) –  GlobalCardinality (for limits on the number of items on each bin)

•  Pricing is intractable but we are only looking for a lower bound:

–  Use a lower bound of the best reduced costs to get the dual bound of the column generaAon

–  Problem reduced to one Ame period are relaAvely easy in pracAce

Results on the applicaAon •  Benchmark: 74 instances with at most 242 virtual machines, 20 servers,

287 Ame-‐periods. Time-‐limit 600s. •  CG lower bound outperforms the LP and MIP lower bounds

(avg:0.3%, med:0.1%, max:7% of best known upper bounds) •  LNS scales very well in quality and size

(avg:0.5%, med:0%, max:4.5% of the best known lower bounds) •  Zoom on some instances:

Lower bound Upper bound

Conclusion •  Study of a key variant of Bin-‐Packing (BPUC):

–  Characterize LP bound + Cost based propagaAon –  Propose an extension of the Bin-‐Packing global constraint with a useful cost-‐model for this applicaAon domain

–  CP gives an effecAve exact solver for BPUC specially to handle large-‐scale problems and side constraints

•  Minimizing energy in data centres: –  Design a lower bound based on column generaAon that has some generality for the applicaAon domain

–  Assert the quality of LNS on real benchmark –  We are currently tesAng the scalability on problems where the bin-‐packings are 4 Ames larger:

1000 vms, 80 servers, 300 Ame-‐periods