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
Home Automation Fig.1 Fig.2 2
The culture of impatience ? ? ? Fig.3 Fig.5 Fig.4 3
Network of smart buildings Fig.6 4
SBDS Preference Elicitation Results Conclusions Background Outline • Background (DCOPs) • Smart Building Device Scheduling (SBDS) • Preference Elicitation in DCOPs • Preliminary Results • Conclusions and Future work 5
SBDS Preference Elicitation Results Conclusions Background Distributed Constraint Optimization < 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. x a x a x b cost 0 0 3 f ab f ac 0 1 ∞ 1 0 2 1 1 5 x b x c f bc Constraint graph Constraint (cost table) 6
SBDS Preference Elicitation Results Conclusions Background Distributed Constraint Optimization < 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 min F ( x ) x min X = arg max f ( x | scope ( f ) ) x F f 2 F 7
SBDS Preference Elicitation Results Conclusions Background DCOP: Assumptions • Agents coordinate an assignment for their variables. x a • Agents operate distributedly. f ac f ab Communication: • By exchanging messages. x b x c f bc • Restricted to agent’s local neighbors. Knowledge: f bd • Restricted to agent’s sub-problem. x d • Privacy preserving. 8
SBDS Preference Elicitation Results Conclusions Background Smart Building Device Scheduling (SBDS) A SBDS problem is composed of: H : A neighborhood of smart buildings. • Z i : A set of smart electric devices within each building h i . • H: A time horizon for the device scheduling. • exact 2 Z i . The ener problem uildings h i and whose 9
SBDS Preference Elicitation Results Conclusions Background Smart Building Device Scheduling (SBDS) A SBDS problem is composed of: H : A neighborhood of smart buildings. • Z i : A set of smart electric devices within each building h i . • H: A time horizon for the device scheduling. • exact 2 Z i . The ener problem uildings h i and whose 10
SBDS Preference Elicitation Results Conclusions Background Smart Building Device Scheduling (SBDS) A SBDS problem is composed of: H : A neighborhood of smart buildings. • Z i : A set of smart electric devices within each building h i . • H: A time horizon for the device scheduling. • θ : A pricing function expressing cost per kWh of energy consumed. • 11
SBDS Preference Elicitation Results Conclusions Background Smart Building Device Scheduling (SBDS) A SBDS problem is composed of: H : A neighborhood of smart buildings. • Z i : A set of smart electric devices within each building h i . • H: A time horizon for the device scheduling. • θ : A pricing function expressing cost per kWh of energy consumed. • Preferences: users express their discomfort for scheduling a device at a given time. 12
SBDS Preference Elicitation Results Conclusions Background Smart Building Device Scheduling (SBDS) • SBDS objective: monetary cost of h i schedule at time t into a single one through the use of a weighted X X ↵ c · C t i + ↵ u · U t minimize i t 2 T h i 2 H discomfort for the h i schedule at time t subject to: 1 ≤ s z j ≤ T − � z j ∀ h i ∈ H , z j ∈ Z i X � t z j = � z j ∀ h i ∈ H , z j ∈ Z i t 2 T X P t i ≤ ` t ∀ t ∈ T h i 2 H 13
SBDS Preference Elicitation Results Conclusions Background Smart Building Device Scheduling (SBDS) • SBDS objective: into a single one through the use of a weighted X X ↵ c · C t i + ↵ u · U t minimize i t 2 T h i 2 H subject to: duration start time 1 ≤ s z j ≤ T − � z j ∀ h i ∈ H , z j ∈ Z i Device scheduling feasibility X � t z j = � z j ∀ h i ∈ H , z j ∈ Z i t 2 T Maximum total X P t i ≤ ` t ∀ t ∈ T load limit h i 2 H h i load at time t 14
SBDS Preference Elicitation Results Conclusions Background DCOP mapping SBDS DCOP • A building h i ϵ H. • Agent a i ϵ A • Variable x i ϵ X (controlled by a i ) • Start time s zj for a device z j (in building h i ) with domain D i = {1,..,H} • Schedule costs for a device z j • Local soft constraint • Schedule preferences for z j • Local soft constraint • Device scheduling feasibility • Local hard constraint • Maximum total power limit • Global hard constraint 15
SBDS Preference Elicitation Results Conclusions Background 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! • 16
SBDS Preference Elicitation Results Conclusions Background Preference Elicitation in DCOPs How to effectively elicit user’s preferences asking a few questions? Ask for the user’s input. • Use historical data. • 17
SBDS Preference Elicitation Results Conclusions Background Preference Elicitation in DCOPs If we could ask only k questions, which k cost tables should be asked for user elicitation? 18
SBDS Preference Elicitation Results Conclusions Background Preference Elicitation in DCOPs describe our proposed techniques. • Uncertain DCOP: Let ˆ P = h X , D , ˆ F , A , ↵ i straints ˆ may have inaccurate revealed cost tables (by the user) straints F may ha ˆ where: F = F r [ F u whose cost tables uncertain cost tables (estimated from historical data) x a x b cost x a x b cost N ( μ 0 , σ 2 0 ) 0 0 3 0 0 N ( μ 1 , σ 2 1 ) 0 1 1 0 1 N ( μ 2 , σ 2 2 ) 1 0 2 1 0 N ( μ 3 , σ 2 3 ) 1 1 5 1 1 Scalars Random variables 19
SBDS Preference Elicitation Results Conclusions Background Preference Elicitation in DCOPs • Oracle DCOP: let P = h X , D , F , A , ↵ i let P constraints have accurate cost = accurate cost tables. • constraints F only on • Costs are sampled from the corresponding distributions of the uncertain tables. x a x b cost cost N ( μ 0 , σ 2 0 ) 0 0 3 N ( μ 1 , σ 2 1 ) 0 1 1 N ( μ 2 , σ 2 2 ) 1 0 2 N ( μ 3 , σ 2 3 ) 1 1 5 samples Random variables 20
SBDS Preference Elicitation Results Conclusions Background Preference Elicitation in DCOPs is formalized • Given an oracle DCOP P and a value construct an alue k 2 N , oracle DCOP uncertain DCOP DCOP ˆ that reveals exactly k constraints per agent, constraints P minimizing the error: ) optimal solution for the oracle DCOP P |F | · | | ⇥ ⇤ x ⇤ ) � F P ( x ⇤ ) | | F ˆ P = E P (ˆ ✏ ˆ oracle DCOP optimal solution for a realization of DCOP ˆ P ) 21
SBDS Preference Elicitation Results Conclusions Background Preference Elicitation in DCOPs is formalized • Given an oracle DCOP P and a value construct an alue k 2 N , oracle DCOP uncertain DCOP DCOP ˆ that reveals exactly k cost tables per agent, constraints P minimizing the error: ) optimal solution for the oracle DCOP P |F | · | | ⇥ ⇤ x ⇤ ) � F P ( x ⇤ ) | | F ˆ P = E P (ˆ ✏ ˆ oracle DCOP optimal solution for a realization of DCOP ˆ P ) possible numbers � |F| • Challenge: there are possible uncertain DCOPs. � is . k ·| A | • Solving each DCOP is NP-hard. , the preference • We propose 5 heuristics to construct an uncertain DCOP. 22
SBDS Preference Elicitation Results Conclusions Background Preference Elicitation Heuristics Goal: Elicit the first k cost tables, according to an ordering . relation ⌫ � . 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. 23
SBDS Preference Elicitation Results Conclusions Background Evaluation |F Analysis of at the increasing of the cost tables to elicit (k). • ✏ ˆ P 50 realizations of the uncertain DCOP. • Results are averaged over 50 oracle DCOPs. • Settings: 0.4 RN - | H | = 10 random ordering SD - | Z _i|= 10 VV VE 0.3 - H = 12 (step = 30 min) AE Normalized Error - Preferences sampled AV from σ 2 ) , ution N (ˆ µ, ˆ 0.2 √ sampled in [1, 100] with ˆ N µ randomly √ ˆ σ 2 in sampled in µ and ˆ in [1 , 2 ] . 0.1 0.0 20 40 60 80 k (%) 24
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