A Preliminary Study on Preference Elicitation in DCOPs for - - PowerPoint PPT Presentation

A Preliminary Study on Preference Elicitation in DCOPs for Scheduling Devices in Smart Buildings

Atena M. Tabakhi Ferdinando Fioretto William Yeoh New Mexico State University

July 9, 2016

2 Fig.1 Fig.2

Home Automation

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The culture of impatience

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Fig.3 Fig.4

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Fig.5

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4 Fig.6

Network of smart buildings

Outline

• Background (DCOPs)
• Smart Building Device Scheduling (SBDS)
• Preference Elicitation in DCOPs
• Preliminary Results
• Conclusions and Future work

5

Background SBDS Preference Elicitation Results Conclusions

Distributed Constraint Optimization

6

Constraint graph fab fbc fac xc xb xa Constraint (cost table)

<X, D, F, A, α>:

• X: Set of variables.
• D: Set of finite domains for each variable.
• F: Set of constraints between variables.
• A: Set of agents, controlling the variables in X.
• α: Mapping of variables to agents.

xa xb cost 3 1 ∞ 1 2 1 1 5 Background SBDS Preference Elicitation Results Conclusions

Distributed Constraint Optimization

7

Background SBDS Preference Elicitation Results Conclusions

<X, D, F, A, α>:

• X: Set of variables.
• D: Set of finite domains for each variable.
• F: Set of constraints between variables.
• A: Set of agents, controlling the variables in X.
• α: Mapping of variables to agents.
• GOAL: Find a cost minimal assignment.

x⇤ = arg max

x

F(x) = arg max

x

X

f2F

f(x|scope(f))

F

min min

DCOP: Assumptions

• Agents coordinate an assignment for

their variables.

• Agents operate distributedly.

Communication:

• By exchanging messages.
• Restricted to agent’s local neighbors.

Knowledge:

• Restricted to agent’s sub-problem.
• Privacy preserving.

8

fab fbc fac xc xb xa xd fbd

Background SBDS Preference Elicitation Results Conclusions

Smart Building Device Scheduling (SBDS)

A SBDS problem is composed of:

• H: A neighborhood of smart buildings.
• Zi : A set of smart electric devices within each building hi.
• H: A time horizon for the device scheduling.

9

Background SBDS Preference Elicitation Results Conclusions

exact 2 Zi. The ener

problem uildings hi and whose

Smart Building Device Scheduling (SBDS)

A SBDS problem is composed of:

• H: A neighborhood of smart buildings.
• Zi : A set of smart electric devices within each building hi.
• H: A time horizon for the device scheduling.

10

Background SBDS Preference Elicitation Results Conclusions

exact 2 Zi. The ener

problem uildings hi and whose

Smart Building Device Scheduling (SBDS)

A SBDS problem is composed of:

• H: A neighborhood of smart buildings.
• Zi : A set of smart electric devices within each building hi.
• H: A time horizon for the device scheduling.
• θ : A pricing function expressing cost per kWh of energy consumed.

11

Background SBDS Preference Elicitation Results Conclusions

Smart Building Device Scheduling (SBDS)

A SBDS problem is composed of:

• H: A neighborhood of smart buildings.
• Zi : A set of smart electric devices within each building hi.
• H: A time horizon for the device scheduling.
• θ : A pricing function expressing cost per kWh of energy consumed.

12

Background SBDS Preference Elicitation Results Conclusions

Preferences: users express their discomfort for scheduling a device at a given time.

Smart Building Device Scheduling (SBDS)

• SBDS objective:

13

Background SBDS Preference Elicitation Results Conclusions

into a single one through the use of a weighted minimize X

t2T

X

hi2H

↵c · Ct

i + ↵u · U t i

monetary cost of hischedule at time t discomfort for the hi schedule at time t subject to:

1 ≤ szj ≤ T − zj ∀hi ∈ H, zj ∈ Zi X

t2T

t

zj = zj

∀hi ∈ H, zj ∈ Zi X

hi2H

P t

i ≤ `t

∀t ∈ T

Smart Building Device Scheduling (SBDS)

• SBDS objective:

14

Background SBDS Preference Elicitation Results Conclusions

into a single one through the use of a weighted minimize X

t2T

X

hi2H

↵c · Ct

i + ↵u · U t i

1 ≤ szj ≤ T − zj ∀hi ∈ H, zj ∈ Zi X

t2T

t

zj = zj

∀hi ∈ H, zj ∈ Zi X

hi2H

P t

i ≤ `t

∀t ∈ T

Device scheduling feasibility Maximum total load limit start time duration hi load at time t subject to:

DCOP mapping

SBDS

• A building hi ϵ H.
• Start time szj for a device zj

(in building hi)

• Schedule costs for a device zj
• Schedule preferences for zj
• Device scheduling feasibility
• Maximum total power limit

DCOP

• Agent ai ϵ A
• Variable xi ϵ X (controlled by ai)

with domain Di = {1,..,H}

• Local soft constraint
• Local soft constraint
• Local hard constraint
• Global hard constraint

15

Background SBDS Preference Elicitation Results Conclusions

Preference Elicitation in DCOPs

• DCOP assumption: Cost tables are known a priori.
• Unrealistic assumption in SBDS!
• User populated cost tables expressing preferences for each

device schedule.

• Lots of devices!

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Background SBDS Preference Elicitation Results Conclusions

Preference Elicitation in DCOPs

How to effectively elicit user’s preferences asking a few questions?

• Ask for the user’s input.
• Use historical data.

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Background SBDS Preference Elicitation Results Conclusions

Preference Elicitation in DCOPs

If we could ask only k questions, which k cost tables should be asked for user elicitation?

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Background SBDS Preference Elicitation Results Conclusions

Preference Elicitation in DCOPs

• Uncertain DCOP:

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describe our proposed techniques. Let ˆ P = hX, D, ˆ F, A, ↵i straints ˆ may have inaccurate straints F may ha ˆ F = Fr [ Fu whose cost tables

revealed cost tables (by the user) uncertain cost tables (estimated from historical data)

where:

xa xb cost N(μ0, σ20) 1 N(μ1, σ21) 1 N(μ2, σ22) 1 1 N(μ3, σ23) Random variables xa xb cost 3 1 1 1 2 1 1 5 Scalars Background SBDS Preference Elicitation Results Conclusions

Preference Elicitation in DCOPs

• Oracle DCOP:
• = accurate cost tables.
• Costs are sampled from the corresponding distributions of

the uncertain tables.

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cost N(μ0, σ20) N(μ1, σ21) N(μ2, σ22) N(μ3, σ23) Random variables xa xb cost 3 1 1 1 2 1 1 5 samples

let P = hX, D, F, A, ↵i constraints have accurate cost let P constraints F

• nly on

Background SBDS Preference Elicitation Results Conclusions

Preference Elicitation in DCOPs

• Given an oracle DCOP P and a value construct an

uncertain DCOP that reveals exactly k constraints per agent, minimizing the error:

21

is formalized alue k 2 N, constraints

• racle DCOP

DCOP ˆ P )

|F | · | | ✏ ˆ

P = E

⇥ |F ˆ

P(ˆ

x⇤) FP(x⇤)| ⇤

• racle DCOP

DCOP ˆ P )

• ptimal solution for a realization of
• ptimal solution for the oracle DCOP P

Background SBDS Preference Elicitation Results Conclusions

Preference Elicitation in DCOPs

• Given an oracle DCOP P and a value construct an

uncertain DCOP that reveals exactly k cost tables per agent, minimizing the error:

• Challenge: there are possible uncertain DCOPs.
• Solving each DCOP is NP-hard.
• We propose 5 heuristics to construct an uncertain DCOP.

22

is formalized alue k 2 N, constraints

• racle DCOP

DCOP ˆ P )

|F | · | | ✏ ˆ

P = E

⇥ |F ˆ

P(ˆ

x⇤) FP(x⇤)| ⇤

• racle DCOP

DCOP ˆ P )

• ptimal solution for a realization of
• ptimal solution for the oracle DCOP P

possible numbers is |F|

k·|A|

• .

, the preference

Background SBDS Preference Elicitation Results Conclusions

Preference Elicitation Heuristics

Goal: Elicit the first k cost tables, according to an ordering . Heuristics to enforce an ordering over cost tables:

• ≥[AE] Average of the expected costs of the uncertain constraints.
• ≥[AV] Average of the variance of the uncertain constraints.
• ≥[VE] Variance of the expected costs of the uncertain constraints.
• ≥[VV] Variance of the variance of the uncertain constraints.
• ≥[SD] Second-order stochastic dominance: Takes into account the

notion of the risk.

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relation ⌫.

Background SBDS Preference Elicitation Results Conclusions

Evaluation

• Analysis of at the increasing of the cost tables to elicit (k).
• 50 realizations of the uncertain DCOP.
• Results are averaged over 50 oracle DCOPs.

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|F ✏ ˆ

P

20 40 60 80 k (%) Normalized Error 0.0 0.1 0.2 0.3 0.4

RN SD VV VE AE AV

random ordering

Settings:

• |H| = 10
• |Z_i|= 10
• H = 12 (step = 30 min)
• Preferences sampled

from sampled in [1, 100] sampled in

ution N(ˆ µ, ˆ σ2),

√

with ˆ µ randomly N in [1,

√ˆ µ 2 ].

and ˆ σ2 in

Background SBDS Preference Elicitation Results Conclusions

Evaluation

• Analysis of at the increasing of the cost tables to elicit (k).
• 50 realizations of the uncertain DCOP.
• Results are averaged over 50 oracle DCOPs.

25

|F ✏ ˆ

P

random ordering

Settings:

• |H| = 10
• |Z_i|= 10
• H = 12 (step = 30 min)
• Preferences sampled

from sampled in [1, 100] sampled in

ution N(ˆ µ, ˆ σ2),

√

with ˆ µ randomly N in [1,

√ˆ µ 2 ].

and ˆ σ2 in

Background SBDS Preference Elicitation Results Conclusions

Evaluation

• Analysis of at the increasing of the cost tables to elicit (k).
• 50 realizations of the uncertain DCOP.
• Results are averaged over 50 oracle DCOPs.

26

|F ✏ ˆ

P

20 40 60 80 k (%) Normalized Error 0.0 0.1 0.2 0.3 0.4

RN SD VV VE AE AV

random ordering Background SBDS Preference Elicitation Results Conclusions

Main Results:

• 1. The error decreases

as the k increases.

• 2. AE and AV
• utperform other

heuristics.

Conclusions and Future Work

• We studied the effect of preference elicitation in scheduling

smart appliances within a network of interconnected buildings.

• We propose the SBDS problem and cast it as a DCOP.
• We propose 5 heuristics to select a subset of cost tables to elicit.
• Preliminary results:
• Our best heuristics are more accurate than a baseline method.
• Future work:
• More extensive analysis of our methods.
• More realistic setting for the SBDS agents.

27

Background SBDS Preference Elicitation Results Conclusions

References:

• Fig. 1: http://goo.gl/5znqip
• Fig. 2: goo.gl/dqwUz2
• Fig. 3: goo.gl/daWjOP
• Fig. 4: goo.gl/CeWvSn
• Fig. 5: goo.gl/afyUXV
• Fig. 6: goo.gl/WFzMhv

Thank You!

A Preliminary Study on Preference Elicitation in DCOPs for Scheduling Devices in Smart Buildings

Atena M. Tabakhi · Ferdinando Fioretto· William Yeoh

Preference Elicitation Heuristics (extra)

• ≥[SD] Second-order stochastic dominance:

29

Background SBDS Results Conclusions

Preference Elicitation

X

m=1

(fi(m) fj(m)) 0

fi ≥[SD] fj iff: where fi(m) is the expected cost of the m-th value assignment for the variables in the scope of fi.

X  |Σfi

x |,

alue assignment