Zombie Optimization or How I Learned to Love...
Transcript of Zombie Optimization or How I Learned to Love...
![Page 1: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/1.jpg)
University of TorontoMechanical & Industrial Engineering
Zombie Optimizationor
How I Learned to Love Decomposition
J. Christopher BeckDepartment of Mechanical & Industrial Engineering
University of TorontoCanada
CPAIOR 2013 Master ClassMay 18, 2013
in collaboration with Mohammad Fazel-Zarandi, Stefan Heinz, Wen-Yang Ku,Daria Terekhov, Jens Schulz, Tony Tran, Peter Zhang
![Page 2: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/2.jpg)
University of TorontoMechanical & Industrial Engineering
2
![Page 3: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/3.jpg)
University of TorontoMechanical & Industrial Engineering
Disclaimer #1
• There is really nothing(more) about zombies in this talk– that was just to get
you in the room today
3
![Page 4: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/4.jpg)
University of TorontoMechanical & Industrial Engineering
Disclaimer #1
• There is really nothing(more) about zombies in this talk– that was just to get
you in the room today
3
![Page 5: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/5.jpg)
University of TorontoMechanical & Industrial Engineering
Disclaimer #2
• There is not reallymuch aboutcomputationalsustainability, either– well, there is some– I will try, with varying degrees of success, to
provide examples of decomposition in problems related to computational sustainability
4
![Page 6: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/6.jpg)
University of TorontoMechanical & Industrial Engineering
The Plan
• Decomposition &Modeling
• Logic-based Benders Decomposition(LBBD)
• Applying LBBD to Problems Somewhat Related to Computational Sustainability
• Beyond Decomposition
5
![Page 7: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/7.jpg)
University of TorontoMechanical & Industrial Engineering
The Plan
• Decomposition &Modeling
• Logic-based Benders Decomposition(LBBD)
• Applying LBBD to Problems Somewhat Related to Computational Sustainability
• Beyond Decomposition
6
wherein, I try to convince you that decomposition is central to applying
optimization to real problems
![Page 8: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/8.jpg)
University of TorontoMechanical & Industrial Engineering
Wind Farm Design7
![Page 9: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/9.jpg)
University of TorontoMechanical & Industrial Engineering
Wind Farm Design
• You want to build acommercial wind farm– what turbines do you buy? how many?– where do you build it? what do you build (e.g.,
turbine foundations, turbine layout, roads, electrical connections, energy storage)?
– how do you build it (construction planning)?– how do you operate it?
8
[Zhang 2013] MASc Thesis, University of Toronto.
![Page 10: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/10.jpg)
University of TorontoMechanical & Industrial Engineering
Wind Farm Design
• You want to build acommercial wind farm– what turbines do you buy? how many?– where do you build it? what do you build (e.g.,
turbine foundations, turbine layout, roads, electrical connections, energy storage)?
– how do you build it (construction planning)?– how do you operate it?
8
[Zhang 2013] MASc Thesis, University of Toronto.
Somehow you need to decide how to solve all these inter-related problems.
![Page 11: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/11.jpg)
University of TorontoMechanical & Industrial Engineering
Wind Farm Design
• You want to build acommercial wind farm– what turbines do you buy? how many?– where do you build it? what do you build (e.g.,
turbine foundations, turbine layout, roads, electrical connections, energy storage)?
– how do you build it (construction planning)?– how do you operate it?
8
[Zhang 2013] MASc Thesis, University of Toronto.
Somehow you need to decide how to solve all these inter-related problems.
The only reasonable way forward (as our scientific/engineering methodology has it) is to identify sub-problems we can solve
(more or less) independently.
![Page 12: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/12.jpg)
University of TorontoMechanical & Industrial Engineering
Problem 1: Turbine Placement
• Objective: maximize energy production or profit
• Constraints:– location: min. separation, land topology,
existing infrastructure– limit of input power to grid– turbine specifications
• Decisions:– turbine types, number, placement
9
Thanks to Peter Zhang.
![Page 13: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/13.jpg)
University of TorontoMechanical & Industrial Engineering
Turbine Placement Challenges10
Thanks to Peter Zhang.
![Page 14: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/14.jpg)
University of TorontoMechanical & Industrial Engineering
Turbine Placement Challenges10
Thanks to Peter Zhang.
![Page 15: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/15.jpg)
University of TorontoMechanical & Industrial Engineering
Problem 2: Infrastructure Layout
• Design the supporting structure– turbine foundations, electrical network, road
network, control, monitoring and data gathering
– reliability, maintenance, life time (stochastic!)• Power loss via transmission scales with
length– the turbine placement and electrical network
are interdependent
11
Thanks to Peter Zhang.
![Page 16: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/16.jpg)
University of TorontoMechanical & Industrial Engineering
Problem 3: Wind Energy Storage
• Smooth supply variations by storing energy (e.g., battery)– how big should the battery be?
• Depends on how it is used– connection with unit commitment problem– economic connection with turbine placement
12
Thanks to Peter Zhang.
![Page 17: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/17.jpg)
University of TorontoMechanical & Industrial Engineering
So What is My Point?13
• Standard approach: decomposition– focus on something we can solve
• Maybe particularly dangerous in computational sustainability– law of unintended consequences
• But the whole problem is just too big!
![Page 18: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/18.jpg)
University of TorontoMechanical & Industrial Engineering
Decomposition
• Hierarchical (the standard way)– overall problem is split into sub-problems
solved one at at time or independently• e.g., infrastructure layout after turbine placement
– no feedback• Integrated
– decisions really depend on each other but problem too big to solve in one model
– decomposition with feedback
14
![Page 19: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/19.jpg)
University of TorontoMechanical & Industrial Engineering
Decomposition
• Hierarchical (the standard way)– overall problem is split into sub-problems
solved one at at time or independently• e.g., infrastructure layout after turbine placement
– no feedback• Integrated
– decisions really depend on each other but problem too big to solve in one model
– decomposition with feedback
14
![Page 20: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/20.jpg)
University of TorontoMechanical & Industrial Engineering
The Plan
• Decomposition &Modeling
• Logic-based Benders Decomposition(LBBD)
• Applying LBBD to Problems Somewhat Related to Computational Sustainability
• Beyond Decomposition
15
wherein the basic idea is introduced
![Page 21: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/21.jpg)
University of TorontoMechanical & Industrial Engineering
Logic-BasedBendersDecomposition
16
![Page 22: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/22.jpg)
University of TorontoMechanical & Industrial Engineering
Logic-BasedBendersDecomposition
16
![Page 23: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/23.jpg)
University of TorontoMechanical & Industrial Engineering
Logic-BasedBendersDecomposition
16
![Page 24: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/24.jpg)
University of TorontoMechanical & Industrial Engineering
17
Resource Allocation & Scheduling
Assign jobs
[Hooker 2005] Constraints, 10, 385-401, 2005.
![Page 25: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/25.jpg)
University of TorontoMechanical & Industrial Engineering
17
Resource Allocation & Scheduling
Assign jobs
[Hooker 2005] Constraints, 10, 385-401, 2005.
![Page 26: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/26.jpg)
University of TorontoMechanical & Industrial Engineering
17
Resource Allocation & Scheduling
Assign jobs
Schedule Schedule Schedule
[Hooker 2005] Constraints, 10, 385-401, 2005.
![Page 27: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/27.jpg)
University of TorontoMechanical & Industrial Engineering
Problem Details
• Each job, j, has:– release date, Rj (earliest start time)– deadline, Dj (latest end time)– processing time, pjk, on resource k– resource requirement, rjk, for resource k– cost, cjk, to use resource k
• Goal: assign and schedule jobs to minimize total assignment cost while satisfying time windows and resource capacity
18
![Page 28: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/28.jpg)
University of TorontoMechanical & Industrial Engineering
CP Model19
[Heinz & B. 2012] CPAIOR, 211-227, 2012.
![Page 29: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/29.jpg)
University of TorontoMechanical & Industrial Engineering
CP Model19
xij = 1 if job j is assigned to resource i
[Heinz & B. 2012] CPAIOR, 211-227, 2012.
![Page 30: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/30.jpg)
University of TorontoMechanical & Industrial Engineering
CP Model19
xij = 1 if job j is assigned to resource i
all jobs assigned to one resource
[Heinz & B. 2012] CPAIOR, 211-227, 2012.
![Page 31: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/31.jpg)
University of TorontoMechanical & Industrial Engineering
CP Model19
xij = 1 if job j is assigned to resource i
all jobs assigned to one resource
resource capacity
[Heinz & B. 2012] CPAIOR, 211-227, 2012.
![Page 32: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/32.jpg)
University of TorontoMechanical & Industrial Engineering
CP Model19
xij = 1 if job j is assigned to resource i
all jobs assigned to one resource
resource capacity
time windows
[Heinz & B. 2012] CPAIOR, 211-227, 2012.
![Page 33: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/33.jpg)
University of TorontoMechanical & Industrial Engineering
CP Model19
xij = 1 if job j is assigned to resource i
all jobs assigned to one resource
resource capacity
time windows
Tends not to work too well(if goal is finding and proving optimality).
Why?
[Heinz & B. 2012] CPAIOR, 211-227, 2012.
![Page 34: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/34.jpg)
University of TorontoMechanical & Industrial Engineering
LBBD20
Global Model
[Hooker & Ottosson 2003] Mathematical Programming, 96, 33-60, 2003.
![Page 35: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/35.jpg)
University of TorontoMechanical & Industrial Engineering
LBBD20
Master Problem
Subproblem 1 Subproblem n. . .
[Hooker & Ottosson 2003] Mathematical Programming, 96, 33-60, 2003.
![Page 36: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/36.jpg)
University of TorontoMechanical & Industrial Engineering
LBBD20
Master Problem
Subproblem 1 Subproblem n. . .
Solution Solution
[Hooker & Ottosson 2003] Mathematical Programming, 96, 33-60, 2003.
![Page 37: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/37.jpg)
University of TorontoMechanical & Industrial Engineering
LBBD20
Master Problem
Subproblem 1 Subproblem n. . .
Solution SolutionCut Cut
[Hooker & Ottosson 2003] Mathematical Programming, 96, 33-60, 2003.
![Page 38: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/38.jpg)
University of TorontoMechanical & Industrial Engineering
LBBD
• Partition problem into– Master problem with decision variables, y– Sub-problem(s) with decision variables, x– When the y’s are fixed (to say, ŷ), sub-
problems are formed• MP & SP do not have to be any particular
form (e.g., IP/LP, IP/CP)• Each sub-problem is an inference dual
– What is the max. LB that can be inferred assuming y = ŷ?
21
![Page 39: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/39.jpg)
University of TorontoMechanical & Industrial Engineering
Making LBBD Work
• Sub-problem relaxation– MP solving needs to have some guidance or
else it just enumerates all MP solutions• Strong & cheap cuts
– Cuts should remove more than just the current MP solution
22
![Page 40: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/40.jpg)
University of TorontoMechanical & Industrial Engineering
Making LBBD Work
• Sub-problem relaxation– MP solving needs to have some guidance or
else it just enumerates all MP solutions• Strong & cheap cuts
– Cuts should remove more than just the current MP solution
22
Questions?Questions?
![Page 41: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/41.jpg)
University of TorontoMechanical & Industrial Engineering
23
Resource Allocation & Scheduling
Assign jobs
Schedule Schedule Schedule
![Page 42: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/42.jpg)
University of TorontoMechanical & Industrial Engineering
LBBD Master (MIP)24
![Page 43: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/43.jpg)
University of TorontoMechanical & Industrial Engineering
LBBD Master (MIP)24
Minimize resource assignment cost
![Page 44: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/44.jpg)
University of TorontoMechanical & Industrial Engineering
LBBD Master (MIP)24
Minimize resource assignment cost
Each activity is assigned to one resource
![Page 45: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/45.jpg)
University of TorontoMechanical & Industrial Engineering
LBBD Master (MIP)24
Minimize resource assignment cost
Sub-problem relaxation(Can we do better?)
Each activity is assigned to one resource
![Page 46: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/46.jpg)
University of TorontoMechanical & Industrial Engineering
LBBD Master (MIP)24
Minimize resource assignment cost
Sub-problem relaxation(Can we do better?)
Benders cut(Can we do better?)
Each activity is assigned to one resource
![Page 47: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/47.jpg)
University of TorontoMechanical & Industrial Engineering
Sub-problem Relaxation25
![Page 48: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/48.jpg)
University of TorontoMechanical & Industrial Engineering
Sub-problem Relaxation25
C
![Page 49: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/49.jpg)
University of TorontoMechanical & Industrial Engineering
Sub-problem Relaxation25
C
est lft
![Page 50: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/50.jpg)
University of TorontoMechanical & Industrial Engineering
Sub-problem Relaxation25
C
est lft
![Page 51: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/51.jpg)
University of TorontoMechanical & Industrial Engineering
Sub-problem Relaxation25
C
est lft
![Page 52: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/52.jpg)
University of TorontoMechanical & Industrial Engineering
Benders Cut
• Do not allow same assignment of activities (or a superset) to be assigned to the same resource
• Gets inserted into the master problem!
26
![Page 53: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/53.jpg)
University of TorontoMechanical & Industrial Engineering
Benders Cut
• Do not allow same assignment of activities (or a superset) to be assigned to the same resource
• Gets inserted into the master problem!
26
Counter for the iterations
![Page 54: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/54.jpg)
University of TorontoMechanical & Industrial Engineering
Benders Cut
• Do not allow same assignment of activities (or a superset) to be assigned to the same resource
• Gets inserted into the master problem!
26
Counter for the iterations
The set of jobs assigned to resource k in iteration h.
![Page 55: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/55.jpg)
University of TorontoMechanical & Industrial Engineering
LBBD Subproblem (CP)
• Single-machine, feasibility problem
27
![Page 56: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/56.jpg)
University of TorontoMechanical & Industrial Engineering
28
![Page 57: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/57.jpg)
University of TorontoMechanical & Industrial Engineering
A Tighter Relaxation29
C
est lft
![Page 58: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/58.jpg)
University of TorontoMechanical & Industrial Engineering
A Tighter Relaxation30
C
est lft
![Page 59: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/59.jpg)
University of TorontoMechanical & Industrial Engineering
A Tighter Relaxation30
C
est lft
![Page 60: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/60.jpg)
University of TorontoMechanical & Industrial Engineering
A Tighter Relaxation30
C
est lft
![Page 61: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/61.jpg)
University of TorontoMechanical & Industrial Engineering
A Tighter Relaxation30
C
est lft
![Page 62: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/62.jpg)
University of TorontoMechanical & Industrial Engineering
A Tighter Relaxation30
C
est lft
![Page 63: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/63.jpg)
University of TorontoMechanical & Industrial Engineering
A Tighter Relaxation31
C
est lft
![Page 64: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/64.jpg)
University of TorontoMechanical & Industrial Engineering
A Tighter Relaxation31
C
est lft
![Page 65: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/65.jpg)
University of TorontoMechanical & Industrial Engineering
A Tighter Relaxation31
C
est lft
![Page 66: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/66.jpg)
University of TorontoMechanical & Industrial Engineering
A Tighter Relaxation31
C
est lft
![Page 67: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/67.jpg)
University of TorontoMechanical & Industrial Engineering
A Tighter Relaxation32
[Hooker 2007] Integrated Methods for Optimization, 2007.
![Page 68: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/68.jpg)
University of TorontoMechanical & Industrial Engineering
A Tighter Relaxation32
[Hooker 2007] Integrated Methods for Optimization, 2007.
“Single” relaxation
![Page 69: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/69.jpg)
University of TorontoMechanical & Industrial Engineering
A Tighter Relaxation32
[Hooker 2007] Integrated Methods for Optimization, 2007.
“Single” relaxation
“Interval” relaxation
![Page 70: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/70.jpg)
University of TorontoMechanical & Industrial Engineering
A Stronger Benders Cut?
33
![Page 71: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/71.jpg)
University of TorontoMechanical & Industrial Engineering
A Stronger Benders Cut?
• Repeatedly resolve infeasible sub-problem, removing activities to identify a minimal infeasible subset of Jhk
33
![Page 72: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/72.jpg)
University of TorontoMechanical & Industrial Engineering
Results?
• Well it is a bit controversial– LBBD best for finding and proving optimality– MIP best for finding high-quality feasible
solutions– CIP competitive– CP good for finding high-quality feasible, bad
for proving optimality
34
![Page 73: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/73.jpg)
University of TorontoMechanical & Industrial Engineering
Results?
• Well it is a bit controversial– LBBD best for finding and proving optimality– MIP best for finding high-quality feasible
solutions– CIP competitive– CP good for finding high-quality feasible, bad
for proving optimality
34
![Page 74: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/74.jpg)
University of TorontoMechanical & Industrial Engineering
Results?
• Well it is a bit controversial– LBBD best for finding and proving optimality– MIP best for finding high-quality feasible
solutions– CIP competitive– CP good for finding high-quality feasible, bad
for proving optimality
34
![Page 75: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/75.jpg)
University of TorontoMechanical & Industrial Engineering
Parallel Machine Scheduling
1
2
M
Machines
Time
.
.
.
[Tran & B. 2012] ECAI, 774-779, 2012.
![Page 76: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/76.jpg)
University of TorontoMechanical & Industrial Engineering
Parallel Machine Scheduling
1
2
M
1 2 3 N
Jobs
Machines
Time
…
.
.
.
[Tran & B. 2012] ECAI, 774-779, 2012.
![Page 77: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/77.jpg)
University of TorontoMechanical & Industrial Engineering
Parallel Machine Scheduling
1
2
M
2 3 N
Jobs
Machines
Time
1
…
.
.
.
[Tran & B. 2012] ECAI, 774-779, 2012.
![Page 78: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/78.jpg)
University of TorontoMechanical & Industrial Engineering
Parallel Machine Scheduling
1
2
M
2 3 N
Jobs
Machines
Time
1
…
.
.
.
[Tran & B. 2012] ECAI, 774-779, 2012.
![Page 79: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/79.jpg)
University of TorontoMechanical & Industrial Engineering
Parallel Machine Scheduling
1
2
M
3 N
Jobs
Machines
Time
1 2
…
.
.
.
[Tran & B. 2012] ECAI, 774-779, 2012.
![Page 80: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/80.jpg)
University of TorontoMechanical & Industrial Engineering
Parallel Machine Scheduling
1
2
M
3 N
Jobs
Machines
Time
2 1
…
.
.
.
[Tran & B. 2012] ECAI, 774-779, 2012.
![Page 81: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/81.jpg)
University of TorontoMechanical & Industrial Engineering
Parallel Machine Scheduling
1
2
M
Machines
Time
2 1
3
N
.
.
.
[Tran & B. 2012] ECAI, 774-779, 2012.
![Page 82: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/82.jpg)
University of TorontoMechanical & Industrial Engineering
Parallel Machine Scheduling
1
2
M
Machines
Time
2 1
3
N
Makespan
.
.
.
[Tran & B. 2012] ECAI, 774-779, 2012.
![Page 83: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/83.jpg)
University of TorontoMechanical & Industrial Engineering
Parallel Machine Scheduling
1
2
M
Machines
Time
2 1
3
N
Makespan
.
.
.
[Tran & B. 2012] ECAI, 774-779, 2012.
Parallel machine scheduling with sequence and
machine dependent set-ups
![Page 84: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/84.jpg)
University of TorontoMechanical & Industrial Engineering
36
![Page 85: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/85.jpg)
University of TorontoMechanical & Industrial Engineering
36
xijk = 1 if k is processed directly after j on machine i
![Page 86: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/86.jpg)
University of TorontoMechanical & Industrial Engineering
36
xijk = 1 if k is processed directly after j on machine i
each job preceded and succeeded by at most one other job
![Page 87: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/87.jpg)
University of TorontoMechanical & Industrial Engineering
36
xijk = 1 if k is processed directly after j on machine i
each job preceded and succeeded by at most one other job
sets completion time of jobs based on sequence
![Page 88: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/88.jpg)
University of TorontoMechanical & Industrial Engineering
36
xijk = 1 if k is processed directly after j on machine i
each job preceded and succeeded by at most one other job
sets completion time of jobs based on sequence
only one job first on each machine
![Page 89: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/89.jpg)
University of TorontoMechanical & Industrial Engineering
Your Turn
• Develop an LBBD model– master problem?– sub-problem?– sub-problem
relaxation?– cut?
37
![Page 90: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/90.jpg)
University of TorontoMechanical & Industrial Engineering
Your Turn
• Develop an LBBD model– master problem?– sub-problem?– sub-problem
relaxation?– cut?
37
Remember: jobs needs to be assigned to machines and the jobs on a machine need to be
sequenced.
![Page 91: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/91.jpg)
University of TorontoMechanical & Industrial Engineering
Your Turn
• Develop an LBBD model– master problem?– sub-problem?– sub-problem
relaxation?– cut?
37
assign jobs to machines
Remember: jobs needs to be assigned to machines and the jobs on a machine need to be
sequenced.
![Page 92: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/92.jpg)
University of TorontoMechanical & Industrial Engineering
Your Turn
• Develop an LBBD model– master problem?– sub-problem?– sub-problem
relaxation?– cut?
37
assign jobs to machines
sequence each machine
Remember: jobs needs to be assigned to machines and the jobs on a machine need to be
sequenced.
![Page 93: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/93.jpg)
University of TorontoMechanical & Industrial Engineering
Your Turn
• Develop an LBBD model– master problem?– sub-problem?– sub-problem
relaxation?– cut?
37
assign jobs to machines
sequence each machine TSP
Remember: jobs needs to be assigned to machines and the jobs on a machine need to be
sequenced.
![Page 94: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/94.jpg)
University of TorontoMechanical & Industrial Engineering
Master Problem38
xij = 1 iff job j is assigned to machine iyijk = 1 iff job j is immediately before job k
on machine i
![Page 95: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/95.jpg)
University of TorontoMechanical & Industrial Engineering
Master Problem38
xij = 1 iff job j is assigned to machine iyijk = 1 iff job j is immediately before job k
on machine i
Sub-problem relaxation
![Page 96: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/96.jpg)
University of TorontoMechanical & Industrial Engineering
Master Problem38
xij = 1 iff job j is assigned to machine iyijk = 1 iff job j is immediately before job k
on machine i
Sub-problem relaxation
Generated by sub-problem
![Page 97: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/97.jpg)
University of TorontoMechanical & Industrial Engineering
Sub-problem
• Assymetric TSP– nodes = jobs– distance = set-up time
39
![Page 98: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/98.jpg)
University of TorontoMechanical & Industrial Engineering
Cut40
![Page 99: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/99.jpg)
University of TorontoMechanical & Industrial Engineering
Cut40
Optimal makespan on machine i in iteration h
![Page 100: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/100.jpg)
University of TorontoMechanical & Industrial Engineering
Cut40
Optimal makespan on machine i in iteration h
![Page 101: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/101.jpg)
University of TorontoMechanical & Industrial Engineering
Cut40
Optimal makespan on machine i in iteration h
Lower bound on job j’scontribution to the makespan
on machine i in iteration h
![Page 102: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/102.jpg)
University of TorontoMechanical & Industrial Engineering
Stopping Conditions
• All SPs find schedule with makespan ≤ makespan of MP, or
• MP finds solution with makespan equal to best feasible solution found so far– each iteration provides a feasible (but not
necessarily improving) solution
41
![Page 103: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/103.jpg)
University of TorontoMechanical & Industrial Engineering
Results42
0.01
0.1
1
10
100
1000
10000
100000
2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5
10 20 30 40 50 60
Run
time
(s)
# of Machines and Jobs
MIP
Benders
[Tran & B. 2012] ECAI, 774-779, 2012.
![Page 104: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/104.jpg)
University of TorontoMechanical & Industrial Engineering
43
![Page 105: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/105.jpg)
University of TorontoMechanical & Industrial Engineering
The Plan
• Decomposition &Modeling
• Logic-based Benders Decomposition(LBBD)
• Applying LBBD to Problems Somewhat Related to Computational Sustainability
• Beyond Decomposition
44
wherein we try to get back to the topic of the Master Class
![Page 106: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/106.jpg)
University of TorontoMechanical & Industrial Engineering
Problem 1: Turbine Placement
• Objective: maximize energy production or profit
• Constraints:– location: min. separation, land topology,
existing infrastructure– limit of input power to grid– turbine specifications
• Decisions:– turbine types, number, placement
45
Thanks to Peter Zhang.
![Page 107: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/107.jpg)
University of TorontoMechanical & Industrial Engineering
Turbine Placement Challenges46
Thanks to Peter Zhang.
![Page 108: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/108.jpg)
University of TorontoMechanical & Industrial Engineering
Turbine Placement Challenges46
Thanks to Peter Zhang.
Idea: use decomposition to separate linear from non-linear
parts of the problem
![Page 109: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/109.jpg)
University of TorontoMechanical & Industrial Engineering
Turbine Placement Challenges46
Thanks to Peter Zhang.
Idea: use decomposition to separate linear from non-linear
parts of the problem
![Page 110: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/110.jpg)
University of TorontoMechanical & Industrial Engineering
A Location-Allocation ProblemPotential facilities
110
![Page 111: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/111.jpg)
University of TorontoMechanical & Industrial Engineering
A Location-Allocation ProblemPotential facilities
Customers
110
![Page 112: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/112.jpg)
University of TorontoMechanical & Industrial Engineering
A Location-Allocation ProblemPotential facilities
Customers
110
![Page 113: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/113.jpg)
University of TorontoMechanical & Industrial Engineering
A Location-Allocation ProblemPotential facilities
Customers
110
![Page 114: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/114.jpg)
University of TorontoMechanical & Industrial Engineering
A Location-Allocation Problem
12
3
3
21
2
1
Potential facilities
Customers
110
![Page 115: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/115.jpg)
University of TorontoMechanical & Industrial Engineering
A Location-Allocation Problem
12
3
3
21
2
1
Potential facilities
Customers
110
![Page 116: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/116.jpg)
University of TorontoMechanical & Industrial Engineering
A Location-Allocation Problem
12
3
3
21
2
1
Potential facilities
Customers
110
![Page 117: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/117.jpg)
University of TorontoMechanical & Industrial Engineering
Computational Sustainability?
• Originally the facilities were to be recycling centres in the city of Tehran
111
![Page 118: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/118.jpg)
University of TorontoMechanical & Industrial Engineering
A Mixed Integer Model 112
:0:1
jpif site j is open
otherwise
:0:1
ijkxif client i is served by the kth vehicle of site j
otherwise
:0:1
jkzif a kth vehicle of site j
otherwise
[Alberada-Sambola et al. 2009], Computers & OR, 36(2): 597-611, 2009.
![Page 119: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/119.jpg)
University of TorontoMechanical & Industrial Engineering
113
[Alberada-Sambola et al. 2009], Computers & OR, 36(2): 597-611, 2009.
![Page 120: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/120.jpg)
University of TorontoMechanical & Industrial Engineering
113Fixed facility cost
[Alberada-Sambola et al. 2009], Computers & OR, 36(2): 597-611, 2009.
![Page 121: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/121.jpg)
University of TorontoMechanical & Industrial Engineering
113Fixed facility cost
Vehicle cost
[Alberada-Sambola et al. 2009], Computers & OR, 36(2): 597-611, 2009.
![Page 122: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/122.jpg)
University of TorontoMechanical & Industrial Engineering
113Fixed facility cost
Vehicle cost
Assignment cost
[Alberada-Sambola et al. 2009], Computers & OR, 36(2): 597-611, 2009.
![Page 123: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/123.jpg)
University of TorontoMechanical & Industrial Engineering
113Fixed facility cost
Vehicle cost
Assignment cost
Each client is served by one truck at one site
[Alberada-Sambola et al. 2009], Computers & OR, 36(2): 597-611, 2009.
![Page 124: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/124.jpg)
University of TorontoMechanical & Industrial Engineering
113Fixed facility cost
Vehicle cost
Assignment cost
Each client is served by one truck at one site
Distance Constraint
[Alberada-Sambola et al. 2009], Computers & OR, 36(2): 597-611, 2009.
![Page 125: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/125.jpg)
University of TorontoMechanical & Industrial Engineering
113Fixed facility cost
Vehicle cost
Assignment cost
Each client is served by one truck at one site
Distance Constraint
Capacity Constraint
[Alberada-Sambola et al. 2009], Computers & OR, 36(2): 597-611, 2009.
![Page 126: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/126.jpg)
University of TorontoMechanical & Industrial Engineering
113Fixed facility cost
Vehicle cost
Assignment cost
Each client is served by one truck at one site
Distance Constraint
Capacity Constraint
A client must be served by an open site and an allocated vehicle
[Alberada-Sambola et al. 2009], Computers & OR, 36(2): 597-611, 2009.
![Page 127: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/127.jpg)
University of TorontoMechanical & Industrial Engineering
113Fixed facility cost
Vehicle cost
Assignment cost
Each client is served by one truck at one site
Distance Constraint
Capacity Constraint
A client must be served by an open site and an allocated vehicle
Symmetry Constraint
[Alberada-Sambola et al. 2009], Computers & OR, 36(2): 597-611, 2009.
![Page 128: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/128.jpg)
University of TorontoMechanical & Industrial Engineering
113Fixed facility cost
Vehicle cost
Assignment cost
Each client is served by one truck at one site
Distance Constraint
Capacity Constraint
A client must be served by an open site and an allocated vehicle
Symmetry Constraint
Problem: The model doesn’t scale
40 clients, 20 possible locations: 75% of problems unsolved in 48
hours
[CP is even worse]
Problem: The model doesn’t scale
40 clients, 20 possible locations: 75% of problems unsolved in 48
hours
[CP is even worse]
[Alberada-Sambola et al. 2009], Computers & OR, 36(2): 597-611, 2009.
![Page 129: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/129.jpg)
University of TorontoMechanical & Industrial Engineering
Your Turn
• Develop an LBBD model– master problem?– sub-problem?– sub-problem
relaxation?– cut?
114
![Page 130: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/130.jpg)
University of TorontoMechanical & Industrial Engineering
Decisions to make
• Which facilities to open• Which customers to assign to which open
facilities• How many vehicles at each facility• Which customers to assign to which trucks
115
![Page 131: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/131.jpg)
University of TorontoMechanical & Industrial Engineering
Logic-Based Benders Decomposition (LBBD)
116
Capacity and Distance Constrained Plant Location
Problem
[Fazel-Zarandi & B 2012] IJOC, 24, 399-415, 2012.
![Page 132: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/132.jpg)
University of TorontoMechanical & Industrial Engineering
Logic-Based Benders Decomposition (LBBD)
116
Location-Allocation Master Problem
Truck AssignmentSubproblem 1
(TASP 1)
Truck AssignmentSubproblem n
(TASP n). . .
[Fazel-Zarandi & B 2012] IJOC, 24, 399-415, 2012.
![Page 133: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/133.jpg)
University of TorontoMechanical & Industrial Engineering
Logic-Based Benders Decomposition (LBBD)
116
Location-Allocation Master Problem
Truck AssignmentSubproblem 1
(TASP 1)
Truck AssignmentSubproblem n
(TASP n). . .
Solution Solution
[Fazel-Zarandi & B 2012] IJOC, 24, 399-415, 2012.
![Page 134: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/134.jpg)
University of TorontoMechanical & Industrial Engineering
Logic-Based Benders Decomposition (LBBD)
116
Location-Allocation Master Problem
Truck AssignmentSubproblem 1
(TASP 1)
Truck AssignmentSubproblem n
(TASP n). . .
Solution SolutionCut Cut
[Fazel-Zarandi & B 2012] IJOC, 24, 399-415, 2012.
![Page 135: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/135.jpg)
University of TorontoMechanical & Industrial Engineering
Can We Do Better?
• Why do I have to make the truck assignment right away?– introduces a lot of symmetry– delay detailed truck assignment until we have
a facility and customer assignment that looks good
• Triple index (xijk) is ugly
117
![Page 136: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/136.jpg)
University of TorontoMechanical & Industrial Engineering
Change the Model118
if site j is open
otherwise
if client i is served by site j
otherwise
:jnumVeh The number of vehicles assigned to facility j
:0:1
ijx
:0:1
jp
![Page 137: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/137.jpg)
University of TorontoMechanical & Industrial Engineering
119
![Page 138: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/138.jpg)
University of TorontoMechanical & Industrial Engineering
119
Each client is served by one facility
![Page 139: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/139.jpg)
University of TorontoMechanical & Industrial Engineering
119
Distance Constraints
Each client is served by one facility
![Page 140: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/140.jpg)
University of TorontoMechanical & Industrial Engineering
119
Distance Constraints
Each client is served by one facility
Capacity Constraint
![Page 141: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/141.jpg)
University of TorontoMechanical & Industrial Engineering
119
Distance Constraints
Each client is served by one facility
Capacity Constraint
![Page 142: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/142.jpg)
University of TorontoMechanical & Industrial Engineering
119
Distance Constraints
Each client is served by one facility
Capacity Constraint
We’ll talk about these later…
![Page 143: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/143.jpg)
University of TorontoMechanical & Industrial Engineering
Solving this model, we get:
• The open facilities (pj)• The assignment of customers to facilities
(xij)• The number of trucks at each facility
(numVehj)
120
![Page 144: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/144.jpg)
University of TorontoMechanical & Industrial Engineering
Solving this model, we get:
• The open facilities (pj)• The assignment of customers to facilities
(xij)• The number of trucks at each facility
(numVehj)
120
So we’re done, right?
![Page 145: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/145.jpg)
University of TorontoMechanical & Industrial Engineering
A Problem
• The customers assigned to a facility might not fit in the trucks we have allocated to that facility
121
![Page 146: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/146.jpg)
University of TorontoMechanical & Industrial Engineering
Truck Assignment Subproblem(TASP)• Given: Assigned clients & number of
vehicles at each open facility• Goal: Assign clients to vehicles such that
the vehicle distance constraints are satisfied
122
![Page 147: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/147.jpg)
University of TorontoMechanical & Industrial Engineering
Truck Assignment Subproblem(TASP)• Given: Assigned clients & number of
vehicles at each open facility• Goal: Assign clients to vehicles such that
the vehicle distance constraints are satisfied
122
TASP = bin packing[distance = “capacity”]
![Page 148: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/148.jpg)
University of TorontoMechanical & Industrial Engineering
TASP123
Bin packing(CP)
Send cut
Start
numVeh[j],I[j]
First Fit Decreasing
END
numFFD > numVeh
NO
numBin>numVeh
YES
YESNO
![Page 149: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/149.jpg)
University of TorontoMechanical & Industrial Engineering
Bin Packing Using CP124
:.tS ijj distItvehicleDispack ,,
jjj numVehFFDackingnumVehBinPnumVeh
min jackingnumVehBinP
![Page 150: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/150.jpg)
University of TorontoMechanical & Industrial Engineering
What about the cut?125
2
43
1
75 km
75 km75 km
75 km
Truck 1 Truck 2 Truck 3Truck distance capacity (l)= 100 km
100 km
![Page 151: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/151.jpg)
University of TorontoMechanical & Industrial Engineering
What about the cut?125
2
43
1
75 km
75 km75 km
75 km
Truck 1 Truck 2 Truck 3Truck distance capacity (l)= 100 km
1100 km
![Page 152: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/152.jpg)
University of TorontoMechanical & Industrial Engineering
What about the cut?125
2
43
1
75 km
75 km75 km
75 km
Truck 1 Truck 2 Truck 3Truck distance capacity (l)= 100 km
1
2
100 km
![Page 153: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/153.jpg)
University of TorontoMechanical & Industrial Engineering
What about the cut?125
2
43
1
75 km
75 km75 km
75 km
Truck 1 Truck 2 Truck 3Truck distance capacity (l)= 100 km
1 2100 km
![Page 154: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/154.jpg)
University of TorontoMechanical & Industrial Engineering
What about the cut?125
2
43
1
75 km
75 km75 km
75 km
Truck 1 Truck 2 Truck 3Truck distance capacity (l)= 100 km
1 2
3
100 km
![Page 155: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/155.jpg)
University of TorontoMechanical & Industrial Engineering
What about the cut?125
2
43
1
75 km
75 km75 km
75 km
Truck 1 Truck 2 Truck 3Truck distance capacity (l)= 100 km
1 2 3100 km
![Page 156: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/156.jpg)
University of TorontoMechanical & Industrial Engineering
What about the cut?125
2
43
1
75 km
75 km75 km
75 km
Truck 1 Truck 2 Truck 3Truck distance capacity (l)= 100 km
1 2 3
4
100 km
![Page 157: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/157.jpg)
University of TorontoMechanical & Industrial Engineering
What about the cut?125
2
43
1
75 km
75 km75 km
75 km
Truck 1 Truck 2 Truck 3 Truck 4Truck distance capacity (l)= 100 km
1 2 3 4100 km
![Page 158: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/158.jpg)
University of TorontoMechanical & Industrial Engineering
What about the cut?125
2
43
1
75 km
75 km75 km
75 km
Truck 1 Truck 2 Truck 3 Truck 4Truck distance capacity (l)= 100 km
1 2 3 4100 km
hIi
ijjhj xnumVehnumVeh 1*
![Page 159: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/159.jpg)
University of TorontoMechanical & Industrial Engineering
What about the cut?125
2
43
1
75 km
75 km75 km
75 km
Truck 1 Truck 2 Truck 3 Truck 4Truck distance capacity (l)= 100 km
1 2 3 4100 km
hIi
ijjhj xnumVehnumVeh 1*
4 trucks
![Page 160: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/160.jpg)
University of TorontoMechanical & Industrial Engineering
What about the cut?125
2
43
1
75 km
75 km75 km
75 km
Truck 1 Truck 2 Truck 3 Truck 4Truck distance capacity (l)= 100 km
1 2 3 4100 km
hIi
ijjhj xnumVehnumVeh 1*
4 trucks 1 truck
![Page 161: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/161.jpg)
University of TorontoMechanical & Industrial Engineering
Cuts
• Constraints added to the MP each time one of the sub-problems is not able to find a feasible solution
126
hIi
ijjhj xnumVehnumVeh 1*hJj
![Page 162: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/162.jpg)
University of TorontoMechanical & Industrial Engineering
Cuts
• Constraints added to the MP each time one of the sub-problems is not able to find a feasible solution
126
# vehicles at site jassigned in iteration h
hIi
ijjhj xnumVehnumVeh 1*hJj
![Page 163: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/163.jpg)
University of TorontoMechanical & Industrial Engineering
Cuts
• Constraints added to the MP each time one of the sub-problems is not able to find a feasible solution
126
# vehicles at site jassigned in iteration h
Max. decrease in the # vehicles needed given
reassigned clients
hIi
ijjhj xnumVehnumVeh 1*hJj
![Page 164: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/164.jpg)
University of TorontoMechanical & Industrial Engineering
127
Distance Constraints
Each client is served by one facility
Capacity Constraint
![Page 165: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/165.jpg)
University of TorontoMechanical & Industrial Engineering
127
Sub-problem Relaxation
Distance Constraints
Each client is served by one facility
Capacity Constraint
![Page 166: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/166.jpg)
University of TorontoMechanical & Industrial Engineering
127
Sub-problem Relaxation
Distance Constraints
Each client is served by one facility
Capacity Constraint
Benders cuts
![Page 167: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/167.jpg)
University of TorontoMechanical & Industrial Engineering
127
Sub-problem Relaxation
Distance Constraints
Each client is served by one facility
Capacity Constraint
Benders cuts
Location-Allocation Master Problem (LAMP)
![Page 168: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/168.jpg)
University of TorontoMechanical & Industrial Engineering
LBBD vs IP LBBD > 300 times faster than IP
128
IP CPU Time
LBB
D C
PU
Tim
e
[Fazel-Zarandi & B 2012] IJOC, 24, 399-415, 2012.
![Page 169: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/169.jpg)
University of TorontoMechanical & Industrial Engineering
129
![Page 170: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/170.jpg)
University of TorontoMechanical & Industrial Engineering
The Plan
• Decomposition &Modeling
• Logic-based Benders Decomposition(LBBD)
• Applying LBBD to Problems Somewhat Related to Computational Sustainability
• Beyond Decomposition
130
Does anyone notice any inconsistencies in the story I
am telling you so far?
![Page 171: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/171.jpg)
University of TorontoMechanical & Industrial Engineering
Decomposition
• Hierarchical (the standard way)– overall problem is split into sub-problems
solved one at at time or independently• e.g., infrastructure layout after turbine placement
– no feedback• Integrated
– decisions really depend on each other but problem too big to solve in one model
– decomposition with feedback
131
![Page 172: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/172.jpg)
University of TorontoMechanical & Industrial Engineering
Decomposition
• Hierarchical (the standard way)– overall problem is split into sub-problems
solved one at at time or independently• e.g., infrastructure layout after turbine placement
– no feedback• Integrated
– decisions really depend on each other but problem too big to solve in one model
– decomposition with feedback
131
![Page 173: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/173.jpg)
University of TorontoMechanical & Industrial Engineering
A Weakness in My Story
• Motivation was about taking really big complex problems and decomposing
• But all my examples have really been “small” problems (the type we normally solve in CP/AI/OR)– e.g., all the aspects of building a wind farm
not just turbine placement
132
![Page 174: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/174.jpg)
University of TorontoMechanical & Industrial Engineering
A Challenge
• Rather than decomposing what we already see as a single problem, can we unify what we think of as separate problems?
133
![Page 175: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/175.jpg)
University of TorontoMechanical & Industrial Engineering
Directions
• Integrating maintenanceplanning and production scheduling – long-term stochastic
reasoning combined with short-term combinatorial reasoning [Aramon Bajestani, forthcoming] PhD dissertation, University of Toronto
134
![Page 176: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/176.jpg)
University of TorontoMechanical & Industrial Engineering
135
Master Problem
. . .
Cut Cut
Subproblem 1 Subproblem n
Solution Solution
![Page 177: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/177.jpg)
University of TorontoMechanical & Industrial Engineering
136
Core Representation
. . .
Cut Cut
Subproblem 1 Subproblem n
Solution Solution
![Page 178: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/178.jpg)
University of TorontoMechanical & Industrial Engineering
136
Core Representation
. . .
Cut Cut
Subproblem 1 Subproblem n
Solution Solution
Reduced expressivity & a fast solver
![Page 179: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/179.jpg)
University of TorontoMechanical & Industrial Engineering
137
Core Representation
. . .
Cut Cut
“extra stuff” “extra stuff”
Solution Solution
Reduced expressivity & a fast solver
![Page 180: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/180.jpg)
University of TorontoMechanical & Industrial Engineering
137
Core Representation
. . .
Cut Cut
“extra stuff” “extra stuff”
Solution Solution
Reduced expressivity & a fast solver
Not [easily or efficiently] representable in core
![Page 181: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/181.jpg)
University of TorontoMechanical & Industrial Engineering
138
Core Representation
. . .
Cut Cut
“extra stuff” “extra stuff”
[Partial] Solution [Partial] Solution
Reduced expressivity & a fast solver
Not [easily or efficiently] representable in core
![Page 182: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/182.jpg)
University of TorontoMechanical & Industrial Engineering
139
Core Representation[SAT]
. . .
Cut Cut
“extra stuff”[T-solver 1]
“extra stuff”[T-solver n]
[Partial] Solution [Partial] Solution
Reduced expressivity & a fast solver
Not [easily or efficiently] representable in core
![Page 183: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/183.jpg)
University of TorontoMechanical & Industrial Engineering
139
Core Representation[SAT]
. . .
Cut Cut
“extra stuff”[T-solver 1]
“extra stuff”[T-solver n]
[Partial] Solution [Partial] Solution
Reduced expressivity & a fast solver
Not [easily or efficiently] representable in core
SAT Modulo Theory (SMT)SAT Modulo Theory (SMT)
![Page 184: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/184.jpg)
University of TorontoMechanical & Industrial Engineering
140
Core Representation[LP or MIP]
. . .
Cut Cut
“extra stuff”[cut generator]
“extra stuff”[cut generator]
[Partial] Solution [Partial] Solution
Reduced expressivity & a fast solver
Not [easily or efficiently] representable in core
![Page 185: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/185.jpg)
University of TorontoMechanical & Industrial Engineering
140
Core Representation[LP or MIP]
. . .
Cut Cut
“extra stuff”[cut generator]
“extra stuff”[cut generator]
[Partial] Solution [Partial] Solution
Reduced expressivity & a fast solver
Not [easily or efficiently] representable in core
Branch-and-cutBranch-and-cut
![Page 186: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/186.jpg)
University of TorontoMechanical & Industrial Engineering
141
Core Representation[logical state representation]
. . .
Cut Cut
“extra stuff”[LP]
“extra stuff”[temporal]
[Partial] Solution [Partial] Solution
Reduced expressivity & a fast solver
Not [easily or efficiently] representable in core
[Gregory et al. 2012] ICAPS, 65-73, 2012.
![Page 187: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/187.jpg)
University of TorontoMechanical & Industrial Engineering
141
Core Representation[logical state representation]
. . .
Cut Cut
“extra stuff”[LP]
“extra stuff”[temporal]
[Partial] Solution [Partial] Solution
Reduced expressivity & a fast solver
Not [easily or efficiently] representable in core
(AI) Planning Modulo Theory(AI) Planning Modulo Theory
[Gregory et al. 2012] ICAPS, 65-73, 2012.
![Page 188: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/188.jpg)
University of TorontoMechanical & Industrial Engineering
142
Core Representation[domain store + branching heuristics]
. . .
Cut Cut
“extra stuff”[global constraint 1]
“extra stuff”[global constraint n]
[Partial] Solution [Partial] Solution
Reduced expressivity & a fast solver
Not [easily or efficiently] representable in core
![Page 189: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/189.jpg)
University of TorontoMechanical & Industrial Engineering
142
Core Representation[domain store + branching heuristics]
. . .
Cut Cut
“extra stuff”[global constraint 1]
“extra stuff”[global constraint n]
[Partial] Solution [Partial] Solution
Reduced expressivity & a fast solver
Not [easily or efficiently] representable in core
CPCP
![Page 190: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/190.jpg)
University of TorontoMechanical & Industrial Engineering
Domain consistencyThesis
• CP itself can be seen as an instance of this decomposition pattern
• But a sub-problem “solver” (i.e. a constraint) has been almost always consistency enforcement
• It is time to move beyond this narrow view of a constraint and really exploit the choice of a rich constraint representation
143
![Page 191: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/191.jpg)
University of TorontoMechanical & Industrial Engineering
Things a Constraint Can Do
• Automatically detect independent sub-problems and solve them [Heinz, Ku, & B. 2013] CPAIOR, 12-27, 2013.
• Automated remodeling via dual presolving[Heinz, Schulz, & B. 2013] Constraints, 18, 166-201, 2013.
• Provide heuristic information (solution counting)[Pesant et al. 2012] JAIR, 43, 173-210, 2012.
• Generate clauses/explanations[Schutt et al. 2011] Constraints, 16, 173-194, 2011.
144
![Page 192: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/192.jpg)
University of TorontoMechanical & Industrial Engineering
Take Home Message I
• Decomposition (LBBD) is avaluable approach to solving hard combinatorial optimization problems– But it is non-trivial to use– Sub-problem relaxation and cuts critical
• Can it be used to integrate related problems currently solved separately?
145
![Page 193: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/193.jpg)
University of TorontoMechanical & Industrial Engineering
Take Home Message II
• LBBD is a pattern of delayedconstraint posting that can be seen in a number of techniques: SMT, B&Cut, and PMT– thinking of global constraints as such a sub-
problem (and more than just an inference mechanism) is a promising direction
146
![Page 194: Zombie Optimization or How I Learned to Love Decompositioncomputational-sustainability.cis.cornell.edu/... · University of Toronto Mechanical & Industrial Engineering Zombie Optimization](https://reader035.fdocuments.us/reader035/viewer/2022081613/5fb7f7b73cf11b2116504a04/html5/thumbnails/194.jpg)
University of TorontoMechanical & Industrial Engineering
No zombies were optimized in the making of this presentation
147