The Prominence of Stabilization Techniques in Column Generation: the case of Freight Transportation Ruslan Sadykov 1 , 2 Alexander A. Lazarev 3 Arthur Pessoa 4 Eduardo Uchoa 4 François Vanderbeck 2 , 1 1 2 3 4 Inria Bordeaux, University Institute of Univ. Federal Bordeaux, Control Sciences, Fluminense Talence, France Talence, France Moscow, Russia Niteroi, Brazil Odysseus 2015 Ajaccio, France, June 1 1 / 21
Contents Freight railcar routing application Column generation approach Stabilization Results and conclusions 2 / 21
The freight car routing application: overview transportation demands initial car distribution 3 / 21
Specificity of freight rail transportation in Russia The state company Independent freight car management companies ◮ Freight car blocking Transp. costs matrix ◮ Assignment of ◮ Freight train Transp. times matrix transportation scheduling demands to freight ◮ Locomotives cars car movements management ◮ Freight car routing ◮ Personnel management 4 / 21
Specificity of freight rail transportation in Russia The state company Independent freight car management companies ◮ Freight car blocking Transp. costs matrix ◮ Assignment of ◮ Freight train Transp. times matrix transportation scheduling demands to freight ◮ Locomotives cars car movements management ◮ Freight car routing ◮ Personnel management Distances are large, and average freight train speed is low ( ≈ 300 km/day): discretization in periods of 1 day is reasonable 4 / 21
The freight car routing application: input and output Input ◮ Railroad network (stations) ◮ Initial location of cars (sources) ◮ Transportation demands and associated profits ◮ Transportation times between stations ◮ Costs: transfer costs and standing (waiting) daily rates; 5 / 21
The freight car routing application: input and output Input ◮ Railroad network (stations) ◮ Initial location of cars (sources) ◮ Transportation demands and associated profits ◮ Transportation times between stations ◮ Costs: transfer costs and standing (waiting) daily rates; Output: operational plan ◮ A set of accepted demands and their execution dates ◮ Empty and loaded cars movements to meet the demands (car routing) Objective Maximize the total net profit 5 / 21
Similar applications in the literature [Fukasawa et al., 2002] ◮ Train schedule is known ◮ Cars should be assigned to trains to be transported ◮ Discretization by the moments of arrival and departure of trains. ◮ Smaller time horizon (7 days) 6 / 21
Similar applications in the literature [Fukasawa et al., 2002] ◮ Train schedule is known ◮ Cars should be assigned to trains to be transported ◮ Discretization by the moments of arrival and departure of trains. ◮ Smaller time horizon (7 days) Other works ◮ [Holmberg et al., 1998] ◮ [Löbel, 1998] ◮ [Lulli et al., 2011] ◮ [Caprara et al., 2011] 6 / 21
Commodity graph [Stratonnikov and Shiryaev, 2012] Commodity c ∈ C represents the flow (movements) of cars of type c . Graph G c = ( V c , A c ) for commodity c ∈ C : · · · · · · · · · · · · · · · waiting arc station 3 empty transfer arc · · · station 2 loaded transfer arc · · · station 1 · · · · · · · · · · · · time Each vertex v ∈ V c represent location of cars of type c on a certain station at a certain time standing at a certain rate g a — cost of arc a ∈ A c 7 / 21
Multi-commodity flow formulation Notations Q — set of demands, C q — set of car types compatible with demand q ∈ Q , n max — maximum number of cars to assign to demand q ∈ Q , q n c � v — number of cars of type c situated initially in vertex v ∈ V . Variables x c a ∈ Z + — flow size along arc a ∈ A c , c ∈ C 8 / 21
Multi-commodity flow formulation Notations Q — set of demands, C q — set of car types compatible with demand q ∈ Q , n max — maximum number of cars to assign to demand q ∈ Q , q n c � v — number of cars of type c situated initially in vertex v ∈ V . Variables x c a ∈ Z + — flow size along arc a ∈ A c , c ∈ C � � g a x c min a c ∈ C a ∈ A c � � x c a ≤ n max ∀ q ∈ Q q c ∈ C q a ∈ A cq � x c � x c n c a = � a − ∀ c ∈ C , v ∈ V c v a ∈ δ + ( v ) a ∈ δ − ( v ) x c a ∈ Z + ∀ c ∈ C , a ∈ V c 8 / 21
Contents Freight railcar routing application Column generation approach Stabilization Results and conclusions 9 / 21
Path reformulation ◮ S c — set of type c car “sources” = { v ∈ V c : � n v > 0 } ) ◮ P c s — set of paths (type c car routes) from source s ∈ S c Variables ◮ λ p — flow size along path p ∈ P c s , s ∈ S c , c ∈ C g path � � � min λ p p c ∈ C s ∈ S c p ∈ P s � � � λ p ≤ n max ∀ q ∈ Q q c ∈ C q s ∈ S c s : q ∈ Q path p ∈ P c p � n c λ p = � ∀ c ∈ C , s ∈ S c s p ∈ P c s ∀ c ∈ C , s ∈ S c , p ∈ P c λ p ∈ Z + s 10 / 21
Column generation for path reformulation ◮ Pricing problem decomposes to shortest path problems, one for each source ◮ slow: number of sources are thousands 11 / 21
Column generation for path reformulation ◮ Pricing problem decomposes to shortest path problems, one for each source ◮ slow: number of sources are thousands ◮ To accelerate, for each c ∈ C , we search for an in-tree of shortest paths to the terminal vertex from all s ∈ S c ◮ drawback: some demands are severely “overcovered”, bad convergence 11 / 21
Column generation for path reformulation ◮ Pricing problem decomposes to shortest path problems, one for each source ◮ slow: number of sources are thousands ◮ To accelerate, for each c ∈ C , we search for an in-tree of shortest paths to the terminal vertex from all s ∈ S c ◮ drawback: some demands are severely “overcovered”, bad convergence ◮ We developed an iterative procedure: repeat Find an in-tree T from all non-exhausted sources; foreach path p in T in the order of its reduced cost do Find n ′ — the maximum number of cars able to follow p ; if n ′ > 0 then Add λ p to the restricted master; Reduce by n ′ the number of cars in the source of p ; Reduce by n ′ the volume of all demands covered by p ; until iteration limit k or all demands are covered or all sources are exhausted ; 11 / 21
Diving Heuristic Master problem solution λ ∗ can be fractional, so we apply the diving heuristic [Joncour et al., 2010] ◮ use Depth-First Search ◮ at each node of the tree ◮ select least fractional column ¯ λ p : rounded value ⌈ ¯ λ p ⌋ > 0 ◮ add ⌈ ¯ λ p ⌋ to the partial solution ◮ update right-hand-side of the master constraints ◮ apply preprocessing, which may lead to a change in the pricing problem variables bounds ◮ solve the updated master LP 12 / 21
Contents Freight railcar routing application Column generation approach Stabilization Results and conclusions 13 / 21
Column generation in the dual space L ( π ) — Lagrangian dual function η ( π t , η t ) L ( π ) π Outer approximation 14 / 21
Column generation in the dual space L ( π ) — Lagrangian dual function η ( π t , η t ) L ( π ) π Outer approximation 14 / 21
Column generation in the dual space L ( π ) — Lagrangian dual function η η ( π t , η t ) π, ˆ (ˆ L ) L ( π ) L ( π ) π π Outer approximation Inner approximation 14 / 21
Dual Price Smoothing [Wentges, 1997] π t = α ˆ π + ( 1 − α ) π t ˜ ( π in , η in ) := (ˆ π, ˆ L ) ( π out , η out ) := ( π t , η t ) ( π sep , η sep ) := α ( π in , η in ) + ( 1 − α ) ( π out , η out ) OUT SEP OUT SEP IN IN 15 / 21
Auto-adaptative α -schedule [Pessoa et al., 2014] π out increase α L ( π ) = ˆ L decrease α π ˜ g (˜ π ) π in 16 / 21
Penalty functions [du Merle et al., 1999] π, ˆ (ˆ L ) π t = argmax � � L t ( π ) − ˆ S t ( π ) π ∈ R m + L ( π ) − ˆ S ( π − ˆ π ) ˆ S q ( π q − ˆ π q ) γ = 0 . 9 · n max ˆ π q π q ∆ Here: q ∈ Q | π 1 π 0 � q − ˆ q | ∆ = | Q | · κ 17 / 21
Contents Freight railcar routing application Column generation approach Stabilization Results and conclusions 18 / 21
Stabilization results Real-life instances : 40-140 days horizon, 1,025 stations, up to 5,300 demands, 11 car types, 12,651 cars, and 8,232 sources. Up to ≈ 230 thousands nodes and 7.5 millions arcs. 19 / 21
Stabilization results Real-life instances : 40-140 days horizon, 1,025 stations, up to 5,300 demands, 11 car types, 12,651 cars, and 8,232 sources. Up to ≈ 230 thousands nodes and 7.5 millions arcs. No stabilization 10h Smoothing stabilization Pen. func. stabilization ( κ = 0 . 05) solution time (log scale) Combined stabilization ( κ = 0 . 1) 1h 10m size CG Smooth Pen Comb 40 23 19 32 21 60 117 73 155 95 1m 80 570 234 178 114 100 2481 607 278 152 120 8947 1465 410 213 40 60 80 100 120 140 140 28884 3069 756 338 planning horizon length, days 19 / 21
Diving heuristic results All instances solved to optimality by the diving heuristic. (means Lagrangian bound is equal to the optimal solution value). 20 / 21
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