Moment-based Distributionally Robust Server Allocation and - PowerPoint PPT Presentation

Moment-based Distributionally Robust Server Allocation and Scheduling Problems Yiling Zhang 1 , Siqian Shen 1 , Ayca Erdogan 2 1 : Dept. of IOE, University of Michigan 2 :Dept. of ISE, San Jos´ e State University Zhang, S., Erdogan INFORMS 2015 1/32

Outline Introduction Distributionally Robust Server Allocation Modeling Solution Algorithms Distributionally Robust Appt. Scheduling Modeling Computational Results Server Allocation Appointment Scheduling Conclusions Zhang, S., Erdogan INFORMS 2015 2/32

Two Common Problems in Service Operations P1: Server Allocation 8:00 12:00 √ √ x Surgeries Operating Rooms start) ≥ P2: Appointment Scheduling completion) ≥ Zhang, S., Erdogan INFORMS 2015 3/32

Generic Problem Settings Common issues: 1) service time uncertainty; 2) unknown distributions with limited data. Allocation phase: Given a set of servers and jobs: ◮ Decisions: Which servers to open and how to allocate jobs. ◮ Objective: Minimize the total operational cost. ◮ Constraint: Low overtime probability in each open server. Zhang, S., Erdogan INFORMS 2015 4/32

Generic Problem Settings Common issues: 1) service time uncertainty; 2) unknown distributions with limited data. Allocation phase: Given a set of servers and jobs: ◮ Decisions: Which servers to open and how to allocate jobs. ◮ Objective: Minimize the total operational cost. ◮ Constraint: Low overtime probability in each open server. Scheduling phase: Given appointments assigned to a server: ◮ Decisions: Arrival time of each appointment ◮ Objective: Minimize the total waiting (+ idleness) ◮ Constraint: Low overtime probability Zhang, S., Erdogan INFORMS 2015 4/32

Literature Review Allocation: ◮ Deterministic: Blake and Donald (2002), Jebali et al. (2006) ◮ Stochastic multi-OR allocation: Denton et al. (2010) ◮ Chance-constrained multi-OR allocation: Shylo et al. (2012) Scheduling: ◮ Under random service durations: Weiss (1990), Van den Bosch and Dietz (2000), Denton and Gupta (2003), Gupta and Denton (2008), Pinedo (2012), Erdogan and Denton (2013) ◮ Near-optimal scheduling policy: Mittal et al. (2014), Begen and Queyranne (2011), Begen et al. (2012), Ge et al. (2013) ◮ Simulation and queuing theories: Bailey (1952); Brahimi and Worthington (1991); Ho and Lau (1992); Rohleder and Klassen (2002); Hassin and Mendel (2008); Zeng et al. (2010) ◮ Distributionally Robust (DR) appointment scheduling: Mak et al. (2014) and Kong et al. (2014) Zhang, S., Erdogan INFORMS 2015 5/32

In this Talk... Under random service time, we consider ◮ Problem 1: Multiple Server Allocation; ◮ Problem 2: Single Server Appointment Scheduling We study their Distributionally Robust (DR) variants, and employ ◮ Moment ambiguity sets of the unknown distribution We reformulate the DR models as ◮ Allocation: 0-1 SDP (cross-moment), 0-1 SOCP (exact 1st & 2nd-moment matching), 0-1 SOCP (Gaussian Approximation) ◮ Scheduling: SDP (cross-moment ambiguity set) We optimize the 0-1 SDP via a cutting-plane algorithm, and directly compute the rest in off-the-shelf solvers. Zhang, S., Erdogan INFORMS 2015 6/32

Outline Introduction Distributionally Robust Server Allocation Modeling Solution Algorithms Distributionally Robust Appt. Scheduling Modeling Computational Results Server Allocation Appointment Scheduling Conclusions Zhang, S., Erdogan INFORMS 2015 7/32

Notation ◮ Set of Servers: I (operating cost τ i and time limit T i ) ◮ Set of Jobs: J ( ρ ij = 1 if job j can be operated on server i ) ◮ Random service durations: s = [ s ij , i ∈ I , j ∈ J ] T ◮ Decision Variable ◮ z i ∈ { 0 , 1 } : whether or not to operate server i , such that � 1 operate server i z i = 0 o.w. ◮ y ij ∈ { 0 , 1 } : whether to assign job j to server i , with � 1 allocate job j to server i y ij = 0 o.w. Zhang, S., Erdogan INFORMS 2015 8/32

0-1 Chance-Constrained Formulation Let α i be the risk tolerance of having overtime on server i , ∀ i ∈ I . � min τ i z i z , y i ∈ I s.t. y ij ≤ ρ ij z i , ∀ i ∈ I , j ∈ J � y ij = 1 , ∀ j ∈ J i ∈ I ( j )     � P s ij y ij ≤ T i  ≥ 1 − α i , ∀ i ∈ I  j ∈ J ( i ) y ij , z i ∈ { 0 , 1 } , ∀ i ∈ I , j ∈ J . A variant of chance-constrained binary packing (see, e.g., Song, Luedtke, and K¨ u¸ c¨ ukyavuz (2014)) Zhang, S., Erdogan INFORMS 2015 9/32

Moment-based Ambiguity Sets Consider s i = [ s ij , j ∈ J ] T as random service time of server i . Due to limited data, we may not know the exact distributions of s i , and thus �� � cannot accurately evaluate P j ∈ J ( i ) s ij y ij ≤ T i . Thus, we consider ◮ Cross-moment Ambiguity Set (Delage and Ye (2010)):  �  s i ∈ Ξ i f ( s i ) ds i = 1     D i M ( µ i 0 , Σ i ( E [ s i ] − µ i 0 ) T (Σ i 0 ) − 1 ( E [ s i ] − µ i 0 , γ 1 , γ 2 ) = f ( s i ) : 0 ) ≤ γ 1   E [( s i − µ i 0 )( s i − µ i 0 ) T ] � γ 2 Σ i   0 ◮ Special Case Ambiguity Set (Exact Mean and Covariance Matching): � � � s i ∈ Ξ i f ( s i ) ds i = 1 , E [ s i ] = µ i 0 D i C ( µ i 0 , Σ i 0 ) = f ( s i ) : E [( s i − µ i 0 )( s i − µ i 0 ) T ] = Σ i 0 Zhang, S., Erdogan INFORMS 2015 10/32

DR Chance Constraint ◮ A DR Allocation Model: Replace     � s ij y ij ≤ T i  ≥ 1 − α i , ∀ i ∈ I P  j ∈ J ( i ) with     � inf s ij y ij ≤ T i  ≥ 1 − α i , ∀ i ∈ I . f ( s i ) ∈D P  j ∈ J where D is either D i M or D i C . Zhang, S., Erdogan INFORMS 2015 11/32

Outline Introduction Distributionally Robust Server Allocation Modeling Solution Algorithms Distributionally Robust Appt. Scheduling Modeling Computational Results Server Allocation Appointment Scheduling Conclusions Zhang, S., Erdogan INFORMS 2015 12/32

Allocation ⇒ 0-1 SDP when D = D i M �� � To reformulate inf f ( s i ) ∈D P j ∈ J s ij y ij ≤ T i ≥ 1 − α i , define � H i p i � : dual of ( E [ s i ] − µ i 0 ) T (Σ i 0 ) − 1 ( E [ s i ] − µ i 0 ) ≤ γ 1 ◮ ( p i ) T q i 0 ) T ] � γ 2 Σ i ◮ G i : dual variables with E [( s i − µ i 0 )( s i − µ i 0 ◮ r i : dual variables with � s i ∈ Ξ i f ( s i ) ds i = 1. Following Jiang and Guan (2015), ◮ the DR chance constraint is equivalent to SDP constraints. ◮ the DR server allocation model then becomes a 0-1 SDP. Thus, we propose a cutting-plane algorithm that decomposes the 0-1 SDP into two stages. Zhang, S., Erdogan INFORMS 2015 13/32

Master Problem: 0-1 Integer Linear Program A Master Problem (MP) without enforced DR chance constraints: � min τ i z i z , y i ∈ I s.t. y ij ≤ ρ ij z i , ∀ i ∈ I , j ∈ J � y ij = 1 , ∀ j ∈ J i ∈ I ( j ) C i ( y i ) ≤ 0 , i ∈ I y ij , z i ∈ { 0 , 1 } , ∀ i ∈ I , j ∈ J , where C i ( y i ) ≤ 0 include linear cuts from solving server-based subproblems that evaluate whether y can satisfy the server-based DR chance constraints. Zhang, S., Erdogan INFORMS 2015 14/32

Subproblem Dual and Valid Cuts Given y from MP, we formulate a subproblem for each i ∈ I as the equivalent SDP of the DR chance constraint by letting D = D i M . Take the dual of the SDP subproblem (also an SDP): i d i + ( y T 0 − T i ) u i ≤ 0 y T i µ i SUB i ( y i )-Dual: max Q i , d i , u i � Q i � γ 2 Σ i d i � � 0 0 − � 0 ( d i ) T u i 0 1 � Q i d i � 0 � � 0 + � 0 ( d i ) T u i 0 − α i � Q i d i � ∈ S ( | J ( i ) | +1) × ( | J ( i ) | +1) . ( d i ) T u i + d i + ( y T Consider optimal ( ˜ i ˜ d i , ˜ u i ). If y T i µ i u i > 0, then 0 − T i )˜ generate a valid cut (linear in y i ). Zhang, S., Erdogan INFORMS 2015 15/32

A Cutting-Plane Approach 1. Initial MP without C i ( y i ) ≤ 0 , i ∈ I . 2. Iterate the following steps until no cuts are needed: i. Solve MP and obtain ( z , y ). If fail, claim infeasible, exit. ii. Otherwise, for i ∈ I do ◮ Solve SUB i ( y i )-Dual and obtain optimal dual ( Q i , d i , u i ). ◮ If (( d i ) T + d i ( µ i 0 ) T ) y i − u i T i > 0, generate a cut (( d i ) T + u i ( µ i 0 ) T ) y i − u i T i ≤ 0 into cut set C i ( y i ) ≤ 0 of MP. iii. If no cut generated from SUB i ( y i )-Dual for ∀ i ∈ I , then ( z , y ) is optimal; exit. Zhang, S., Erdogan INFORMS 2015 16/32

Allocation ⇒ 0-1 SOCP when D = D i C �� � We replace inf f ( s i ) ∈D P j ∈ J s ij y ij ≤ T i ≥ 1 − α i by an SOCP constraint given: Theorem (Wagner, 2008) Given the first and second order information µ i 0 and Σ i 0 of the service duration vector s i , given the ambiguity set D i C and probability α i , then an equivalent formulation for C P [ s T inf f ( s i ) ∈D i i y i ≤ T i ] ≥ 1 − α i is � � α i ( T i − ( µ i 0 ) T y i ) , ∀ i ∈ I . y T i Σ i 0 y i ≤ 1 − α i Alternatively, the DR allocation model is a 0-1 SOCP and is directly optimized by CVX 2.1 + Gurobi solver. Zhang, S., Erdogan INFORMS 2015 17/32

Outline Introduction Distributionally Robust Server Allocation Modeling Solution Algorithms Distributionally Robust Appt. Scheduling Modeling Computational Results Server Allocation Appointment Scheduling Conclusions Zhang, S., Erdogan INFORMS 2015 18/32