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Dynamic Stochastic Model for a Single Airport Ground Holding Problem Avijit Mukherjee Mark Hansen University of California at Berkeley 1 Literature Static Integer Programming Many papers on deterministic problem Stochastic problem studied in Richetta and Odoni (1993); Hoffman (1997); Ball et al.(2003) Equity issues addressed by Vossen et al. (2002) 2 Literature Dynamic Models: Richetta and Odoni (1994) Inability to revise previously assigned ground delays Objective function is to minimize expected delay cost No longer Linear Programming Problem if nonlinear measure of delay introduced Can handle specific type of scenario tree 3 Research Contributions Dynamic Stochastic Model for SAGHP Ability to revise ground delays of some flights (non-departed) Can handle any generalized scenario tree Alternative Objective Functions Expected Squared Deviation from RBS Allocation Multi-Criteria Optimization 4 Capacity Scenario Tree Scenario 1 (p=0.3) P(Scenario 2)=1 Scenario 2 (p=0.2) P(Scenario 2)=0.4 P(Scenario 2)=0.2 Scenario 3 (p=0.4) Scenario 4 (p=0.1) 1 2 3……. τ1 τ2 τ3…….. T 5 Decision Making Process ξ1 ξ2 ξ3 ξ4 1 τ1 df τ2 τ3 af T 6 Decision Variables X ⎧⎪1 if flight f is planned to arrive by time period = ⎨ t under scenario q; f ,t ⎪⎩0 otherwise q f ,t X 2 f ,t 1 X 1 f ,t =X 2 f ,t 0 X 1 df af 7 Model Formulation Objective Function Min. Expected Total Cost of Delay (sum of ground and airborne delays) Major Constraints Number of arrivals during any time interval (period) less than airport capacity Coupling Constraints: decisions cannot be based on a particular scenario until it is completely realized 8 Model Parameters and Input Data {1..T + 1} : set of time periods of uniform duration, T being the planning horizon Φ = {1..F } : Set of Flights Dep f ∈{1..T } : scheduled departure time period of flight f Arr f ∈ {1..T } : scheduled arrival time period of flight f λ : Cost ratio between airborne and ground delay 9 Θ : set of capacity scenarios Pq : Probabilit y of occurrence of scenario q ∈ Θ q M t : Airport arrival capacity at time period t under capacity scenario q q M T +1 is set to a high value for all q ∈ Θ 10 B = total number of branches of the scenario tree; B ≥ Θ Ν = number of scenarios represente d by ith branch; i ∈ {1..B} i The scenarios represented by branch i is given by set i i i i 1 k Νi k Ω = {S ,.., S ,.., S }, S ∈ Θ i The time periods corresponding to start and end nodes of a branch are given by oi and µ i ; i ∈ {1..B} 11 B = 7; Θ = {χ1 , χ 2 , χ 3 , χ 4 }; Θ = 4 χ1 i=4 i = 2; Ν = 2; Ω = {χ , χ }; 2 2 1 2 ο = τ ; µ = (τ − 1) 2 1 2 2 i =6 i =3 i = 1; Ν 1 = 4 ; Ω1 = { χ 1 , χ 2 , χ 3 , χ 4 }; 2 3……. i = 5 ; Ν 5 = 1; χ3 Ω 5 = { χ 2 }; i = 7 χο 5 ο1 = 1; µ1 = (τ 1 − 1) 1 χ2 = τ ;µ = Τ 2 5 4 τ1 τ2 τ3…….. T+1 12 Decision Variables X ⎧⎪1 if flight f is planned to arrive by the end of = ⎨ time period t under scenario q; f ,t q ∈ Θ, f ∈ Φ, ⎪⎩0 otherwise q t ∈ { Arrf ..T + 1} ⎧1 if flight f is released for departure by the end of ⎪ q q ∈ Θ, f ∈ Φ, Y f ,t = ⎨ time period t under scenario q; ⎪0 otherwise t ∈ {Dep f ..T + 1} ⎩ W tq = number of aircraft subject to airborne queuing delay at time t for one or more time periods, under scenario q 13 Objective Function T T +1 q q ⎧⎪⎡ ⎫ ⎤ q⎪ Min ∑ Pq × ⎨⎢ ∑ ∑ (t − Arr f )× ( X f ,t − X f ,t −1 )⎥ + λ × ∑W t ⎬ t =1 q∈{1..Q} ⎪⎩⎢⎣ f ∈{1.. F } t = Arr f ⎪⎭ ⎥⎦ Constraints Decision Variables Non Decreasing q q X f , t − X f , t −1 ≥ 0 ; ∀f ∈ Φ, q ∈ Θ, t ∈ { Arr f ..T + 1} Planned Departure Time of Flights ⎧ q q ⎪ Y f , t = ⎨ X f , t + Arr f − Dep f ; if ⎪1 otherwise ⎩ t + Arr f − Dep f ≤ T 14 Arrival Capacity ⎛ ⎞ W tq−1 − W tq + ∑ ⎜ X qf , t − X qf , t −1⎟ ≤ ⎠ f ∈Φ ⎝ M tq ; t ∈ {1..T + 1}, q ∈ Θ Feasibility Conditions W 0q = W Tq +1 = 0 X qf ,T +1 = 1 ∀f ∈ Φ, q ∈ Θ Coupling Constraints for Ground Holding Decision Variables Si i i Sk S1 Ni Y f , t = .... = Y f , t = ........ = Y f , t ; f ∈ Φ, t ∈ {1..T }; S i ∈ Ω i : Ν i ≥ 2 and ο i ≤ t ≤ µ i k 15 Static vs Dynamic Formulation Static Stochastic Model (Ball et al 2003, RichettaOdoni 1993) is a special case of dynamic model. The decisions are taken once at the beginning of day and not revised later. is same for all q ∈ Θ; i.e., superscript q in the decision variables can be dropped. q Therefore, X f ,t can be denoted as X f ,t ; ∀q ∈ Θ X q f ,t Similarly,Y f ,t = Y f ,t ; ∀q ∈ Θ q 16 Example 4 Scenarios, 4 Decision Stages 13 Time Periods 13 Flights Cost Ratio λ = 5 3 ξ3 ξ 4 2 Capacity ξ1 ξ 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 time period Probability Mass Function : P{ξ1} = 0.5; P{ξ 2 } = 0.3; P{ξ 3} = 0.1; P{ξ 4 } = 0.1 17 Scenario Tree ξ1 ξ2 ξ3 ξ4 1 2 3 4 5 6 7 8 9 10 11 12 13 18 Flight No. Dep. Arr. Decision Richetta-Odoni Stage Model MukherjeeHansen Model (Optimal Solution 1) ξ1 ξ 2 ξ3 ξ 4 ξ1 ξ 2 ξ3 MukherjeeHansen Model (Optimal Solution 2) ξ4 ξ1 ξ 2 ξ3 ξ4 2 6 7 1 3 3 3 3 1 2 5 5 1 2 4 4 3 2 8 1 0 0 0 0 1 1 1 1 1 1 1 1 7 5 9 1 0 0 0 0 0 0 0 0 0 0 0 0 8 7 9 2 0 3 3 3 0 1 2 2 0 1 3 3 12 9 11 3 0 0 1 1 0 0 1 1 0 0 1 1 19 Expected Cost of Delay 12 Expected Delay Cost 10 8 Airborne Delay Ground Delay 6 4 2 0 Richetta-Odoni Mukherjee-Hansen 20 Cumulative Arrivals at DFW July14_2003 cumulative number of arrivals 350 300 250 200 150 100 50 0 0:00 1:00 2:00 3:00 4:00 5:00 6:00 7:00 8:00 9:00 10:00 11:00 12:00 time of day 21 Case 1: Baseline 35 ξ1 ξ 2 ξ 3 ξ 4 ξ 5 ξ 6 15 0:00 8:00 8:30 9:00 9:30 10:00 10:30 12:00 Time of Day Probability Mass Function P{ξ1 } = 0.4; P{ξ 2 } = 0.2; P{ξ 3 } = 0.1; P{ξ 4 } = 0.1; P{ξ 5 } = 0.1; P{ξ 6 } = 0.1 Cost Ratio λ = 3 22 Scenario Tree for Baseline Case ξ1 ξ2 ξ3 ξ4 ξ6 ξ5 23 Other Cases Case 2: Change in Cost Ratio. λ = 25 Case 3: Change in PMF P{ξ1} = 0.1; P{ξ 2 } = 0.1; P{ξ3} = 0.1; P{ξ 4 } = 0.1; P{ξ5 } = 0.2; P{ξ 6 } = 0.4 Case 4: Early Branching. Scenarios are realized 30 minutes earlier 24 Results: Baseline Case Baseline Case Airborne Delay Ground Delay Expected Cost of Delay (Aircraft-Hr) 40 35 Mukherjee-Hansen Model Ground delays more severe Less airborne delays Total expected cost least 30 25 20 15 10 5 0 Ball et al. Ricehtta-Odoni Mukherjee-Hansen Delay reduction compared to Static Model 10% in MukherjeeHansen Model 2% in Richetta-Odoni 25 Planned Arrival Rates in Baseline Case Time Period Mukherjee-Hansen Model Richetta-Odoni Model ξ2 ξ3 ξ4 ξ5 ξ6 35 34 34 34 34 34 5 14 14 14 14 14 14 18 18 18 12 12 12 12 12 12 12 12 12 12 20 21 13 13 13 13 33 33 33 33 12 12 20 20 20 20 ξ5 ξ6 ξ1 ξ2 ξ3 ξ4 9:00AM9:15AM 33 25 25 25 25 25 9:15 AM-9:30 AM 16 5 5 5 5 9:30AM-9:45AM 12 26 18 9:45AM10:00AM 20 25 10:00AM10:15AM 12 12 ξ1 26 Cases 2 and 3 Expected Cost of Delay (Aircraft-Hr) 50 Case2 Case3 45 No airborne delays 40 35 Static model plans for worst scenario 30 25 20 15 10 5 0 Ball et al. Ricehtta-Odoni Mukherjee-Hansen Dynamic models adaptive to changing conditions Delays are higher in case 3 due to high probability of worse conditions 27 Case 4: 30 Minutes Early Information Expected Delay Cost (Aircraft-Hr) 40 Airborne Delay Ground Delay 35 30 Static model produces same delays as in baseline case 25 Value of early information 20 15 10 5 0 Ball et al. Ricehtta-Odoni Mukherjee-Hansen Mukherjee-Hansen Model Least delays Absorbs most of the delay by ground holding 28 Perfect Information Case ξ1 ξ2 ξ3 ξ4 ξ5 ξ6 0:00 12:15 29 Mukherjee-Hansen Richetta-Odoni Ball et al. Perfect Information 350 300 Percentage 250 200 150 100 50 0 0 1 2 Case # 3 4 30 Alternative Objective Functions Minimizing Expected Squared Deviation from RBS Allocation ⎧⎡ T +1 ⎪⎢ Min ∑ ∑ ∑ P{q} × ⎨⎢ q ∈ {1..Q} ⎪⎢ f ∈ {1..F } t = Arr f ⎩⎣ q ⎛ ⎜ t − RBS f ⎜ ⎝ ⎫ 2 ⎤ q q ⎞ T ⎪ ⎥+λ× q ⎟ × (X − X ) ∑W t ⎬ f ,t f ,t −1 ⎥ ⎟ t =1 ⎪ ⎠ ⎥⎦ ⎭ 31 Multi-Criteria Optimization ⎫ ⎧⎡ ⎤ q q T +1 ⎪ ⎪⎢ ⎥ ( ) ) + t − Arrf × ( X −X ∑ ⎪ ⎪⎢ ∑ f ,t f , t −1 ⎥ ⎪ ⎪⎢ f ∈ Φ t = Arr ⎥ f ⎦ ⎪ ⎪⎣ ⎡ ⎤ ⎪ 2 ⎪ q q q 1 T + ⎢ ⎥ ⎪ ⎪ ⎛ ⎞ )⎥ + ⎬ Min −X ∑ P{q} × ⎨weight × ⎢ ∑ ∑ ⎜ t − RBS ⎟ × ( X f f ,t f , t −1 q ∈ {1..Q} ⎪ ⎢ f ∈ Φ t = Arr f ⎝ ⎥ ⎪ ⎠ ⎣ ⎦ ⎪ ⎪ T ⎪ ⎪ q λ × ∑ Wt ⎪ ⎪ ⎪ ⎪ t =1 ⎪ ⎪ ⎭ ⎩ 32 Work in Progress Reformulating the model as a minimum cost network flow problem. Ability to handle time varying unconditional probabilities of the capacity scenarios 33 Acknowledgements Authors are thankful to Prof. Mike Ball for his thoughtful suggestions on this research. 34