Combinatorial Optimization at Work 2020 Traffic Optimization Part I: Paths & Lagrange Relaxation Part II: Vehicles & Crewsfdafdsafdsfl Part III: Pollsters & Vehiclesfdsafdkfll Zuse Institute Berlin, 22.09.2020 Ralf Borndörfer Güvenç Şahin Luis Torres 1 Şahin | Traffic Optimization I | CO@Work 2020
Traffic of the Future information, control reliable, no pollution comfortable intermodal fast, efficient ▪ Needs data and communication to assess and control system status ▪ Needs mathematics to find smart solutions Şahin | Traffic Optimization I | CO@W 2020 2
Planning Problems in Public Transit Service Design Operational Planning Operations Control Passenger Information Şahin | Traffic Optimization I | CO@W 2020 3
Vehicle Rotation Planning is a Crucial Part of the Production Slide of DB Planning Process ◼ Which connections correspond best to demand? ◼ Market research ◼ Demand forecast ◼ What capacity is needed? Customer ◼ Passenger counting ◼ What kind of product should be offered? – … Marketing ◼ How can customer-friendly time tables be realised ◼ How can conflicts with other trains be avoided? Train Path Planning ◼ How can vehicles be employed efficiently? ◼ How can plans be made maintenance friendly and Vehicle Rotation Planning robust? ◼ How can maintenance capacity be used most Maintenance Planning efficiently ◼ How can the availability of vehicles be maximised? Train Preparation Planning ◼ How can vehicles be stabled, shunted, catered, cleaned etc. between maintenance and commercial Staff Planning service? ◼ Attractive timetable ◼ How can shifts be planned staff friendly? ◼ Reliability Customer ◼ How can high productivity of personnel be ◼ High product quality obtained? DB Bahn Fernverkehr AG | Fahrplan und Zugfahrt | 4th of July 2013 4
Slide of IVU Şahin | Traffic Optimization I | CO@W 2020 5
Workflow Oriented and Integrated Optimization: How fast business processes can follow IT? Slide of LSB Control Planning Crew Management Process Pairing Crew Hotel, DH, PickUp, Pairing & Roster Crew Tracking Generation Assignment Administration Maintenance Integrated Optimization Integrated Recovery of the Ressource of Aircraft and Crew Aircraft and Crew Maintenance Tail Maintenance Flight Movement Hub Planning Assignment Control Dispatch Control Control Operations Management Process Control Planning Şahin | Traffic Optimization I | CO@W 2020 29 Sep 2005 Optimization in NetLine/Crew Chart 6 Chart 6
Planning Problems in Public Transit Service Design Operational Planning Operations Control Passenger Information Şahin | Traffic Optimization I | CO@W 2020 7
The Shortest Path Problem 306 nodes, 445 edges Alexanderplatz ZIB Şahin | Traffic Optimization I | CO@W 2020 8
Shortest Path Problem 1 Def. (Single Source Shortest (st-)Path Problem (ShPP)): Let 𝐻 = (𝑊, 𝐹, 𝑑) be a (un)directed graph on 𝑜 nodes with edge weights 𝐹 , 𝑡, 𝑢 ∈ 𝑊 two nodes, 𝑄 𝐸 = 𝑄 𝑡𝑢 the set of all st-paths in D. 𝑑 ∈ ℝ ≥0 𝑡𝑢 shortet (st)-path problem (ShPP) min 𝑑(𝑄) , 𝑄 ∈ 𝑄 𝑡𝑢 a) ShPP conservative ∶⟺ 𝑑 𝐷 ≥ 0 ∀ (di)cycles 𝐷 ⊆ 𝐹 ShPP metric ∶⟺ 𝑑 𝑣𝑤 + 𝑑 𝑤𝑥 ≥ 𝑑 𝑣𝑥 ∀𝑣𝑤, 𝑤𝑥, 𝑣𝑥 ∈ 𝐹 ( Δ -inequality) b) c) ShPP Euclidean ∶⟺ 𝑑 𝑣𝑤 = 2 ∀𝑣𝑤 ∈ 𝐹 ⊆ ℝ 2 𝑣 − 𝑤 2 Obs. (Simplicity): A conservative ShPP has a simple optimal solution (no node repetitions). 𝑥 cycle 𝐷 𝑤 𝑣 Δ -inequality simple/non-simple path Şahin | Traffic Optimization I | CO@W 2020 9
Dijkstra's Algorithm 3 Alg. (Dijkstra's Algorithm, Dijkstra [1959]): Input: 𝐻 = 𝑊, 𝐹, 𝑑 , 𝑡, 𝑢 ∈ 𝑊, 𝑑 ∈ ℝ ≥0 Output: 𝑄 ∈ Argmin 𝐹 𝑑 𝑄 𝑄∈𝑄 𝑡𝑢 Data Structures: 𝑒 ∈ ℝ ∪ +∞ 𝑊 , pred ∈ 𝑊 ∪ nil 𝑊 , 𝑆 ⊆ 𝑊 reached nodes 1. 𝑒[𝑤] ← +∞, pred 𝑤 ← nil ∀𝑤 ∈ 𝑊, 𝑒 𝑡 ← 0, 𝑆 ← 𝑡 2. while 𝑆 ≠ ∅ do 𝑃(𝑜 × log 𝑜) 3. 𝑣 ← argmin 𝑤∈𝑆 𝑒 𝑤 , 𝑆 ← 𝑆 ∖ {𝑣} 4. forall 𝑣𝑤 ∈ 𝐹 do 𝑃 𝑛 5. if 𝑒 𝑣 + 𝑑 𝑣𝑤 < 𝑒[𝑤] then (amortized) 6. 𝑒 𝑤 ← 𝑒 𝑣 + 𝑑 𝑣𝑤 , 𝑆 ← 𝑆 ∪ 𝑤 , pred 𝑤 ← 𝑣 7. endif 8. endwhile 9. output (𝑢, pred 𝑢 , pred 2 𝑢 , … , 𝑡 or nil) 4 Prop. (Correctness and Run Time of Dijkstra's Algorithm): Alg. 3 is correct and runs in 𝑃(𝑜 log 𝑜 + 𝑛). Şahin | Traffic Optimization I | CO@W 2020 10
Shortest Path Problem 306 nodes, 445 edges 80 nodes, 122 edges Şahin | Traffic Optimization I | CO@W 2020 11
Dijkstra's Algorithm Şahin | Traffic Optimization I | CO@W 2020 12
IP Formulation of the ShPP 5 Def. (IP Formulation of the ShPP): Let 𝐸 = (𝑊, 𝐵, 𝑑) be a 𝐵 , 𝑡, 𝑢 ∈ 𝑊 two nodes. directed graph, with arc weights 𝑑 ∈ ℝ ≥0 min 𝑑 𝑈 𝑦 ShPP objective 𝑦 𝜀 + 𝑤 − 𝑦 𝜀 − 𝑤 i = 0 ∀𝑤 ≠ 𝑡, 𝑢 flow conservation 𝑦 𝜀 + 𝑡 ii = 1 flow constraint iii 0 ≤ 𝑦 ≤ 1 bounds iv 𝑦 integer integrality 𝑄 𝑇ℎ𝑄𝑄 ≔ conv {𝜓 𝑄 : 𝑄 ∈ 𝑄 a) st-path polytope 𝑡𝑢 } 𝑇ℎ𝑄𝑄 ≔ conv {𝑦 ∈ ℝ 𝐹 : (ShPP) i − iv } b) ShPP polytope 𝑄 𝐽 𝑇ℎ𝑄𝑄 ≔ conv {𝑦 ∈ ℝ 𝐹 : ShPP c) ShPP LP-relaxation 𝑄 i − iii } 𝑀𝑄 𝑇ℎ𝑄𝑄 is in general not true, but 6 Prop. (Path Polytopes): 𝑄 𝑇ℎ𝑄𝑄 = 𝑄 𝐽 𝑇ℎ𝑄𝑄 = 𝑄 𝑇ℎ𝑄𝑄 𝑑 𝑈 𝑦 contains a path for conservative 𝑑 . Proof: 𝑇ℎ𝑄𝑄 ; Argmin 𝑄 𝐽 𝑀𝑄 𝑦∈𝑄 𝑀𝑄 𝑇ℎ𝑄𝑄 allows subtours, 𝑄 𝑇ℎ𝑄𝑄 describes a flow. 𝑄 𝐽 𝑀𝑄 Şahin | Traffic Optimization I | CO@W 2020 13
Dijkstra's Algorithm – A* – Superoptimal Wind TXL-DXB Şahin | Traffic Optimization I | CO@W 2020 14
Flight Planning ▪ 3D, fuel, time-dependent (wind) ▪ Cost, temporary flight restrictions Şahin | Traffic Optimization I | CO@W 2020 15
Taipei – New York Boeing B777-300ER, 25 April 2017, great circle distance 12.565 km The green trajectory takes better advantage of the strong jet stream (~300 km/h). ▪ It is worth to take a long detour. ▪ Min Cost Track Besides saving fuel and time, the new route saves Min Fuel Track ▪ overflight fees by avoiding the expensive airspaces of Canada and Japan. using using old new heuristic dynamic GAIN search search space space reduction reduction distance 13.385 14.635 -1250 flown (km) flight time 14:40 13:55 0:45 (hours) 5665 fuel burn 95.524 89.859 = 17,8 t (kg) CO 2 overflight 2291 1139 1152 fees (USD) total cost 76.453 71.118 5335 (USD) * * based on: fuel price 500 USD / ton, flight time costs: 1400 USD / hour Şahin | Traffic Optimization I | CO@W 2020 16
Constrained Shortest Path Problem 5 Def. (IP Formulation of the Constrained ShPP (CSP)): Let 𝐵 , 𝑡, 𝑢 ∈ 𝑊 𝐸 = (𝑊, 𝐵, 𝑑) be a directed graph, with arc weights 𝑑 ∈ ℝ ≥0 two nodes, 𝐵𝑦 ≤ 𝑐 some linear constraints. min 𝑑 𝑈 𝑦 CSP objective 𝑦 𝜀 + 𝑤 − 𝑦 𝜀 − 𝑤 i = 0 ∀𝑤 ≠ 𝑡, 𝑢 flow conservation 𝑦 𝜀 + 𝑡 ii = 1 flow constraint iii 0 ≤ 𝑦 ≤ 1 bounds iv 𝐵𝑦 ≤ 𝑐 path constraints v 𝑦 integer integrality 𝑄 𝐷𝑇𝑄 ≔ conv {𝜓 𝑄 : 𝑄 ∈ 𝑄 a) const. st-path polytope 𝑡𝑢 , 𝐵𝜓 𝑄 ≤ 𝑐 } 𝐷𝑇𝑄 ≔ conv {𝑦 ∈ ℝ 𝐹 : (ShPP) i − v } b) CSP polytope 𝑄 𝐽 𝐷𝑇𝑄 ≔ conv {𝑦 ∈ ℝ 𝐹 : ShPP c) CSP LP-relaxation 𝑄 i − i𝑤 } 𝑀𝑄 6 Obs. (CSP): 𝑄 𝐷𝑇ℎ𝑄 ⊆ 𝑄 𝐷𝑇𝑄 ⊆ 𝑄 𝐷𝑇𝑄 ; equality does in general not 𝐽 𝑀𝑄 hold. The CSP is NP-hard. Şahin | Traffic Optimization I | CO@W 2020 17
IP Formulation of the Constrained ShPP 5 Def. (IP Formulation of the Constrained ShPP (CSP)): Let 𝐵 , 𝑡, 𝑢 ∈ 𝑊 𝐸 = (𝑊, 𝐵, 𝑑) be a directed graph, with arc weights 𝑑 ∈ ℝ ≥0 two nodes, 𝐵𝑦 ≤ 𝑐 some linear constraints. min 𝑑 𝑈 𝑦 CSP objective 𝑦 𝜀 + 𝑤 − 𝑦 𝜀 − 𝑤 i = 0 ∀𝑤 ≠ 𝑡, 𝑢 flow conservation 𝑦 𝜀 + 𝑡 ii = 1 flow constraint iii 0 ≤ 𝑦 ≤ 1 bounds iv 𝐵𝑦 ≤ 𝑐 path constraints v 𝑦 integer integrality 6 Obs. (CSP): 𝑄 𝐷𝑇ℎ𝑄 ⊆ 𝑄 𝐷𝑇𝑄 ⊆ 𝑄 𝐷𝑇𝑄 ; equality does in general not 𝐽 𝑀𝑄 hold. The CSP is NP-hard. Proof: Solves knapsack problem min 𝑑 𝑈 𝑦 , 𝑏 𝑈 𝑦 ≤ 𝑐, 𝑦 ∈ 0,1 𝑜 : 𝑑 𝑣𝑤 = 0, 𝑏 𝑣𝑤 = 0 … 𝑜 0 1 2 𝑑 𝑣𝑤 = 𝑑 𝑗 , 𝑏 𝑣𝑤 = 𝑏 𝑗 Şahin | Traffic Optimization I | CO@W 2020 18
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