Solving 0-1 Semidefinite Programs for Distributionally Robust - PowerPoint PPT Presentation

Solving 0-1 Semidefinite Programs for Distributionally Robust Allocation of Surgery Blocks Yiling Zhang 1 Joint work with Prof. Siqian Shen 1 , Prof. S. Ayca Erdogan 2 1 Industrial and Operations Engineering, University of Michigan 2 Industrial and Systems Engineering, San Jose State University Supporeted by NSF grant #1433066. 1/25

Outline Introduction DR Chance-Constrained Model Formulation Ambiguity Set 0-1 SDP Reformulation Solving Approaches Cutting-Plane Method 0-1 SOCP Approximation Computational Studies Setup Results Conclusion 2/25

Allocation of Surgery Blocks Operating rooms (ORs): ◮ 40% of a hospital’s total revenues; BUT, a similarly large proportion of its total expenses 1 ◮ Average OR runs at only 68% capacity 1 ◮ Uncertain service duration of surgical procedure 1 Healthcare Financial Management Association 2003 3/25

Allocation of Surgery Blocks Operating rooms (ORs): ◮ 40% of a hospital’s total revenues; BUT, a similarly large proportion of its total expenses 1 ◮ Average OR runs at only 68% capacity 1 ◮ Uncertain service duration of surgical procedure Works on allocation of surgery blocks: ◮ Blake and Donald (2002): MILP ◮ Denton, Miller, Balasubramanian, and Huschka (2010): two-stage stochastic integer program ◮ Shylo, Prokopyev, and Schaefer (2012): chance-constrained formulation ◮ Deng, Shen, and Denton (2016): distributionally robust formulation ◮ ... 1 Healthcare Financial Management Association 2003 3/25

Applications Applications with similar settings (bin packing structure): ◮ Cloud computing server planning: uncertain job hours requested ◮ Shen and Wang (2014) ◮ Machine scheduling: uncertain task duration ◮ Skutella and Uetz (2005) cloudcomputingcafe.com theideasmith.net 4/25

Stochastic OR Allocation Problem 𝑡 1 𝑈 1 𝑡 2 𝑈 2 𝑡 3 𝑈 3 𝑡 4 Surgeries ORs 5/25

Stochastic OR Allocation Problem 𝑡 1 𝑈 1 𝑡 2 𝑈 2 𝑡 3 𝑈 3 𝑡 4 Surgeries ORs (random service duration) 5/25

Stochastic OR Allocation Problem 𝑡 11 𝑡 21 𝑡 31 𝑈 1 𝑡 12 𝑨 1 =? 𝑡 22 𝑡 32 𝑈 2 𝑡 13 𝑨 2 =? 𝑡 23 𝑡 33 𝑈 3 𝑡 14 𝑨 3 =? 𝑡 24 𝑡 34 Surgeries ORs (random service duration) 5/25

Stochastic OR Allocation Problem 𝑡 11 𝑡 21 𝑡 31 𝑈 1 𝑡 12 𝑨 1 = 1 𝑡 22 𝑡 32 𝑈 2 𝑡 13 𝑨 2 = 1 𝑡 23 𝑡 33 𝑈 3 𝑡 14 𝑨 3 = 0 𝑡 24 𝑡 34 Surgeries ORs (random service duration) Decisions: o z i ∈ { 0 , 1 } : z i = 1 if we open OR i , and = 0 if not. 5/25

Stochastic OR Allocation Problem 𝑡 11 𝑡 21 𝑡 31 𝑈 1 𝑡 11 𝑡 13 𝑡 12 𝑡 12 𝑨 1 = 1 𝑡 22 𝑡 32 𝑈 2 𝑡 24 𝑡 13 𝑨 2 = 1 𝑡 23 𝑡 33 𝑈 3 𝑡 14 𝑨 3 = 0 𝑡 24 𝑡 34 Surgeries ORs (random service duration) Decisions: o z i ∈ { 0 , 1 } : z i = 1 if we open OR i , and = 0 if not. o y ij ∈ { 0 , 1 } : y ij = 1 if allocate surgery j to OR i 5/25

A Chance-Constrained Formulation Let s i = [ s ij , j ∈ J ] T , y i = [ y ij , j ∈ J ] T c y � c z � � min i z i + ij y ij z , y i ∈ I i ∈ I j ∈ J o Objective: Minimize the cost of opening ORs 6/25

A Chance-Constrained Formulation Let s i = [ s ij , j ∈ J ] T , y i = [ y ij , j ∈ J ] T c y � c z � � min i z i + ij y ij z , y i ∈ I i ∈ I j ∈ J s.t. y ij ≤ ρ ij z i ∀ i ∈ I , j ∈ J � y ij = 1 ∀ j ∈ J i ∈ I y ij , z i ∈ { 0 , 1 } ∀ i ∈ I , j ∈ J o Objective: Minimize the cost of opening ORs o Deterministic constraints: Feasible surgery allocation 6/25

A Chance-Constrained Formulation Let s i = [ s ij , j ∈ J ] T , y i = [ y ij , j ∈ J ] T c y � c z � � min i z i + ij y ij z , y i ∈ I i ∈ I j ∈ J s.t. y ij ≤ ρ ij z i ∀ i ∈ I , j ∈ J � y ij = 1 ∀ j ∈ J i ∈ I y ij , z i ∈ { 0 , 1 } ∀ i ∈ I , j ∈ J � � s T i y i ≤ T i ≥ 1 − α i , ∀ i ∈ I P f s o Objective: Minimize the cost of opening ORs o Deterministic constraints: Feasible surgery allocation o Chance constraint: “Total operating time ≤ time available in OR i ” at 1 − α i probability, given the distribution f s 6/25

Outline Introduction DR Chance-Constrained Model Formulation Ambiguity Set 0-1 SDP Reformulation Solving Approaches Cutting-Plane Method 0-1 SOCP Approximation Computational Studies Setup Results Conclusion 7/25

Distributionally Robust (DR) Model � � � c y c z min i z i + ij y ij (2) z , y i ∈ I i ∈ I j ∈ J s.t. y ij ≤ ρ ij z i ∀ i ∈ I , j ∈ J (3) � y ij = 1 ∀ j ∈ J (4) i ∈ I y ij , z i ∈ { 0 , 1 } ∀ i ∈ I , j ∈ J (5) � � s T inf i y i ≤ T i ≥ 1 − α i , ∀ i ∈ I (6) f s ∈D i P f ◮ (6): The worst-case probability given by any f s ∈ D i is guaranteed at least 1 − α i (a DR chance constraint). 8/25

Literature Review Distributionally robust optimization ◮ Scarf, Arrow, and Karlin (1958); Delage and Ye (2010); Bertsimas, Doan, Natarajan, and Teo (2010); Goh and Sim (2010), Wiesemann, Kuhn, and Sim (2014), Esfahani and Kuhn (2016)... Distributionally robust chance-constrained programming ◮ Zymler, Kuhn, and Rustem (2013); Jiang and Guan (2015) Jointly chance-constrained binary packing ◮ Song, Luedtke, and K¨ u¸ c¨ ukyavuz (2014) DR chance-constrained knapsack/bin packing ◮ Zhang, Denton, and Xie (2015): mean + variance ◮ Wagner (2008): mean + covariance ◮ Cheng, Delage, and Lisser (2014): mean + covariance 9/25

Outline Introduction DR Chance-Constrained Model Formulation Ambiguity Set 0-1 SDP Reformulation Solving Approaches Cutting-Plane Method 0-1 SOCP Approximation Computational Studies Setup Results Conclusion 10/25

Moment-based Ambiguity Set ◮ Ambiguity set (Delage and Ye, 2010): � � i f ( s i ) ds i = 1 � s i ∈ Ξ ∗ D i = D M i ( µ 0 i , Σ 0 i , γ 1 , γ 2 ) = f ( s i ) : ( E [ s i ] − µ 0 i ) T (Σ 0 i ) − 1 ( E [ s i ] − µ 0 i ) ≤ γ 1 E [( s i − µ 0 i )( s i − µ 0 i ) T ] � γ 2 Σ 0 i ∗ Ξ i = R | J | 11/25

Moment-based Ambiguity Set ◮ Ambiguity set (Delage and Ye, 2010): � � i f ( s i ) ds i = 1 � s i ∈ Ξ ∗ D i = D M i ( µ 0 i , Σ 0 i , γ 1 , γ 2 ) = f ( s i ) : ( E [ s i ] − µ 0 i ) T (Σ 0 i ) − 1 ( E [ s i ] − µ 0 i ) ≤ γ 1 E [( s i − µ 0 i )( s i − µ 0 i ) T ] � γ 2 Σ 0 i ∗ Ξ i = R | J | ◮ decrease γ 2 with fixed γ 1 ◮ decrease γ 1 with fixed γ 2 γ 2 = 5 γ 1 = 5 𝜹 𝟐 , 𝜹 𝟑 = (𝟐, 𝟐) 𝜹 𝟐 , 𝜹 𝟑 = (𝟐, 𝟔) 𝜹 𝟐 , 𝜹 𝟑 = (𝟐, 𝟑) 𝜹 𝟐 , 𝜹 𝟑 = (𝟑, 𝟔) 𝚻 𝟐/𝟑 𝚻 𝟐/𝟑 𝜹 𝟐 , 𝜹 𝟑 = (𝟐, 𝟔) 𝜹 𝟐 , 𝜹 𝟑 = (𝟔, 𝟔) 𝛎 𝛎 ∗ ( µ, Σ): True mean and covariance pair 11/25

Moment-based Ambiguity Set ◮ Ambiguity set (Delage and Ye, 2010): � � i f ( s i ) ds i = 1 � s i ∈ Ξ ∗ i ( µ 0 i , Σ 0 D i = D M i , γ 1 , γ 2 ) = f ( s i ) : ( E [ s i ] − µ 0 i ) T (Σ 0 i ) − 1 ( E [ s i ] − µ 0 i ) ≤ γ 1 E [( s i − µ 0 i )( s i − µ 0 i ) T ] � γ 2 Σ 0 i ∗ Ξ i = R | J | ◮ decrease γ 2 with fixed γ 1 ◮ decrease γ 1 with fixed γ 2 γ 2 = 2 γ 1 = 2 𝜹 𝟐 , 𝜹 𝟑 = (𝟐, 𝟐) 𝜹 𝟐 , 𝜹 𝟑 = (𝟐, 𝟔) 𝜹 𝟐 , 𝜹 𝟑 = (𝟑, 𝟔) 𝜹 𝟐 , 𝜹 𝟑 = (𝟐, 𝟑) 𝚻 𝟐/𝟑 𝚻 𝟐/𝟑 𝜹 𝟐 , 𝜹 𝟑 = (𝟔, 𝟔) 𝜹 𝟐 , 𝜹 𝟑 = (𝟐, 𝟔) 𝛎 𝛎 ∗ ( µ, Σ): True mean and covariance pair 11/25

Moment-based Ambiguity Set ◮ Ambiguity set (Delage and Ye, 2010): � � i f ( s i ) ds i = 1 � s i ∈ Ξ ∗ D i = D M i ( µ 0 i , Σ 0 i , γ 1 , γ 2 ) = f ( s i ) : ( E [ s i ] − µ 0 i ) T (Σ 0 i ) − 1 ( E [ s i ] − µ 0 i ) ≤ γ 1 E [( s i − µ 0 i )( s i − µ 0 i ) T ] � γ 2 Σ 0 i ∗ Ξ i = R | J | ◮ decrease γ 2 with fixed γ 1 ◮ decrease γ 1 with fixed γ 2 γ 2 = 1 γ 1 = 1 𝜹 𝟐 , 𝜹 𝟑 = (𝟐, 𝟐) 𝜹 𝟐 , 𝜹 𝟑 = (𝟐, 𝟔) 𝜹 𝟐 , 𝜹 𝟑 = (𝟐, 𝟑) 𝜹 𝟐 , 𝜹 𝟑 = (𝟑, 𝟔) 𝚻 𝟐/𝟑 𝚻 𝟐/𝟑 𝜹 𝟐 , 𝜹 𝟑 = (𝟔, 𝟔) 𝜹 𝟐 , 𝜹 𝟑 = (𝟐, 𝟔) 𝛎 𝛎 ∗ ( µ, Σ): True mean and covariance pair 11/25

Outline Introduction DR Chance-Constrained Model Formulation Ambiguity Set 0-1 SDP Reformulation Solving Approaches Cutting-Plane Method 0-1 SOCP Approximation Computational Studies Setup Results Conclusion 12/25

0-1 SDP Reformulation with D i = D M i Jiang and Guan (2015) show that ◮ By introducing the dual variables, the DR chance constraints (6) ⇔ SDP constraints (exact): γ 2 Σ 0 i · G i + 1 − r i + Σ 0 i · H i + γ 1 q i − α i λ i ≤ 0 (7a) � G i � 0 1 − p i � � 2 y i − � 0 (7b) − p T 1 2 y T λ i + y T i µ 0 1 − r i i − T i z i i i � G i � � H i � − p i p i ∈ S ( | J | +1) × ( | J | +1) ∈ S ( | J | +1) × ( | J | +1) , , − p T + p T + 1 − r i q i i i λ i ≥ 0 . (7c) 13/25