Resource Constrained Shortest Paths with Side Constraints and Non - PowerPoint PPT Presentation

Resource Constrained Shortest Paths with Side Constraints and Non Linear Costs Stefano Gualandi stefano.gualandi@unipv.it http://www-dimat.unipv.it/~gualandi twitter: @famo2spaghi Thursday, September 19, 13

Outline 1. Introduction 2. Non Linear Costs 3. Iterated Preprocessing 4. Lower bounding via Lagrangian Relaxation 5. Computational Results Thursday, September 19, 13

Column Generation Overview What is F in Crew Scheduling problems? Thursday, September 19, 13

Resource Constrained Shortest Path The problem of putting together a set of pieces of work into a single duty, that is a column or variable of problem (LP-MP), is formalized as a Resource Constrained Shortest Path Problem Example 12 pieces of works, 3 depots Thursday, September 19, 13

Resource Constrained Shortest Path Let G = ( N , A ) be the compatibility graph, weighted, directed, and acyclic: N = P ∪ {{ s h , t h }| h ∈ D } a node for each PoW, and a pair of nodes for each depot A has an arc for each pair ( i , j ) of compatible PoW, and ( s h , i ) (pull-out) and ( i , t h ) (pull-in) ∀ h ∈ D and i ∈ P Thursday, September 19, 13

Resource Constrained Shortest Path ∀ ∈ ∈ each arc ( i , j ) has associated a set of resources r k ij , for each k ∈ K , e.g. working time , driving time, and break time (other resources may be used to model working regulation) Thursday, September 19, 13

Example of Crew Schedules (with reosurces) Resources: 1 spread time (red) 2 driving time (light blue), corresponds to PoW 3 out-of-service time (yellow) 4 long break (grey) 5 breaks (green), very important how they are located Thursday, September 19, 13

Non Linear Costs Thursday, September 19, 13

Non Linear Costs Thursday, September 19, 13

Non Linear Costs 800 600 f(t(P)) = step(t(P)) + (t(P))^2 400 200 0 0 60 120 180 240 300 360 420 480 540 600 660 720 Time Thursday, September 19, 13

Resource Constrained Shortest Paths (RCSP) 3 4 · Arc-flow IP formulation with non linear costs f h : 3 ÿ 4 ÿ ÿ ÿ r k min f h w e x e + e x e e ∈ A k ∈ K h ∈ H e ∈ A Y + 1 if i = s _ ] ÿ ÿ s.t. x e − x e = b i = − 1 if i = t ∀ i ∈ N _ 0 otherwise e ∈ δ + e ∈ δ − [ i i ÿ r k e x e ≤ U k ∀ k ∈ K e ∈ A + side non linear constraints x e ∈ { 0 , 1 } ∀ e ∈ A . Thursday, September 19, 13

Resource Constrained Shortest Paths (RCSP) w e , t e a b 5, 1 5, 1 5, 1 5, 1 s i t 5, 0 10, 0 5, 0 10, 0 c d Thursday, September 19, 13

Resource Constrained Shortest Paths (RCSP) We restrict to super additive functions : c ( P 1 ∪ P 2 ) ≥ c ( P 1 ) + c ( P 2 ) w e , t e a b 5, 1 5, 1 5, 1 5, 1 s i t 5, 0 10, 0 5, 0 10, 0 c d Thursday, September 19, 13

Resource Constrained Shortest Paths (RCSP) We restrict to super additive functions : c ( P 1 ∪ P 2 ) ≥ c ( P 1 ) + c ( P 2 ) " 2 Example: er c ( P ) = w ( P ) + f ! " !q = q e ∈ P w e + P e ∈ P t e : w e , t e a b 5, 1 5, 1 5, 1 5, 1 s i t 5, 0 10, 0 5, 0 10, 0 c d Thursday, September 19, 13

Resource Constrained Shortest Paths (RCSP) We restrict to super additive functions : c ( P 1 ∪ P 2 ) ≥ c ( P 1 ) + c ( P 2 ) " 2 Example: er c ( P ) = w ( P ) + f ! " !q = q e ∈ P w e + P e ∈ P t e : w e , t e a b 5, 1 5, 1 5, 1 5, 1 s i t 5, 0 10, 0 5, 0 10, 0 c d Optimal Path: P 2 = {s, c, i, b, t}, c(P 2 ) = 25 + 4 = 29 Thursday, September 19, 13

Resource Constrained Shortest Paths (RCSP) We restrict to super additive functions : c ( P 1 ∪ P 2 ) ≥ c ( P 1 ) + c ( P 2 ) " 2 Example: er c ( P ) = w ( P ) + f ! " !q = q e ∈ P w e + P e ∈ P t e : w e , t e 14 a b 5, 1 5, 1 5, 1 5, 1 s i t 5, 0 10, 0 5, 0 10, 0 15 c d Optimal Path: P 2 = {s, c, i, b, t}, c(P 2 ) = 25 + 4 = 29 Thursday, September 19, 13

Resource Constrained Shortest Paths (RCSP) We restrict to super additive functions : c ( P 1 ∪ P 2 ) ≥ c ( P 1 ) + c ( P 2 ) " 2 Example: er c ( P ) = w ( P ) + f ! " !q = q e ∈ P w e + P e ∈ P t e : w e , t e 14 14 a b 5, 1 5, 1 5, 1 5, 1 s i t 5, 0 10, 0 5, 0 10, 0 15 c d Optimal Path: P 2 = {s, c, i, b, t}, c(P 2 ) = 25 + 4 = 29 Thursday, September 19, 13

Resource Constrained Shortest Paths (RCSP) We restrict to super additive functions : c ( P 1 ∪ P 2 ) ≥ c ( P 1 ) + c ( P 2 ) " 2 Example: er c ( P ) = w ( P ) + f ! " !q = q e ∈ P w e + P e ∈ P t e : w e , t e 36 a b 5, 1 5, 1 5, 1 5, 1 s i t 5, 0 10, 0 5, 0 10, 0 c d Path: P 3 = {s, a, i, b, t}, c(P 2 ) = 20 + 16 = 36 Thursday, September 19, 13

Resource Constrained Shortest Paths (RCSP) We restrict to super additive functions : c ( P 1 ∪ P 2 ) ≥ c ( P 1 ) + c ( P 2 ) " 2 Example: er c ( P ) = w ( P ) + f ! " !q = q e ∈ P w e + P e ∈ P t e : w e , t e 14 a b 5, 1 5, 1 5, 1 5, 1 s i t 5, 0 10, 0 5, 0 10, 0 15 c d Bellmann’s optimality conditions do not hold! Thursday, September 19, 13

Outline 1. Introduction 2. Non Linear Costs 3. Iterated Preprocessing 4. Lower bounding via Lagrangian Relaxation 5. Computational Results Thursday, September 19, 13

Resource-based Preprocessing (Beasley and Christofides, 1989; Dumitrescu and Boland, 2003; Sellmann et al., 2007) ⌫  jt ) > U k then remove arc e = ( i , j ) if r k ( P ∗ si ) + r k e + r k ( P ∗ where P ∗ si and P ∗ jt are shortest ( k -th resource) paths. � � Resource consumption of each arc. Upper resource bound U = 7. Thursday, September 19, 13

Resource-based Preprocessing (Beasley and Christofides, 1989; Dumitrescu and Boland, 2003; Sellmann et al., 2007) ⌫  jt ) > U k then remove arc e = ( i , j ) if r k ( P ∗ si ) + r k e + r k ( P ∗ where P ∗ si and P ∗ jt are shortest ( k -th resource) paths. � � Resource consumption of each arc. Upper resource bound U = 7. r e 6 a b 2 1 2 s 3 1 t 2 4 c d 1 Thursday, September 19, 13

Resource-based Preprocessing (Beasley and Christofides, 1989; Dumitrescu and Boland, 2003; Sellmann et al., 2007) ⌫  jt ) > U k then remove arc e = ( i , j ) if r k ( P ∗ si ) + r k e + r k ( P ∗ where P ∗ si and P ∗ jt are shortest ( k -th resource) paths. � � Resource consumption of each arc. Upper resource bound U = 7. r e 6 a b 2 1 2 s 3 1 t 2 4 c d 1 Thursday, September 19, 13

Resource-based Preprocessing (Beasley and Christofides, 1989; Dumitrescu and Boland, 2003; Sellmann et al., 2007) ⌫  jt ) > U k then remove arc e = ( i , j ) if r k ( P ∗ si ) + r k e + r k ( P ∗ where P ∗ si and P ∗ jt are shortest ( k -th resource) paths. � � Resource consumption of each arc. Upper resource bound U = 7. r e 6 a b 2 1 2 s 3 1 t 2 4 c d 1 Thursday, September 19, 13

Resource-based Preprocessing (Beasley and Christofides, 1989; Dumitrescu and Boland, 2003; Sellmann et al., 2007) ⌫  jt ) > U k then remove arc e = ( i , j ) if r k ( P ∗ si ) + r k e + r k ( P ∗ where P ∗ si and P ∗ jt are shortest ( k -th resource) paths. � � Resource consumption of each arc. Upper resource bound U = 7. r e 6 a b 2 1 2 s 3 1 t 2 4 c d 1 Thursday, September 19, 13

Resource-based Preprocessing (Beasley and Christofides, 1989; Dumitrescu and Boland, 2003; Sellmann et al., 2007) ⌫  jt ) > U k then remove arc e = ( i , j ) if r k ( P ∗ si ) + r k e + r k ( P ∗ where P ∗ si and P ∗ jt are shortest ( k -th resource) paths. � � Resource consumption of each arc. Upper resource bound U = 7. r e 6 a b 2 1 2 s 3 1 t 2 4 c d 1 Thursday, September 19, 13

Cost-based Preprocessing UB=7 6 a b c e 2 1 2 s 3 1 t 2 4 c d 1 Thursday, September 19, 13

Cost-based Preprocessing UB=7 6 a b c e 2 1 2 s 3 1 t 2 4 c d 1 Thursday, September 19, 13

Cost-based Preprocessing ⌫  if LB ( c ( P ∗ → t )) ≥ UB then remove arc e e s − where P ∗ → t is a shortest path from s to t via arc e . e s − � � UB=7 6 a b c e 2 1 2 s 3 1 t 2 4 c d 1 Thursday, September 19, 13

Outline 1. Introduction 2. Non Linear Costs 3. Iterated Preprocessing 4. Lower bounding via Lagrangian Relaxation 5. Computational Results Thursday, September 19, 13

Resource Constrained Shortest Paths (RCSP) ! · " Arc-flow IP formulation with non linear costs f h : 3 ÿ 4 ÿ ÿ ÿ r k min w e x e + f h e x e e ∈ A k ∈ K h ∈ H e ∈ A Y + 1 if i = s _ ] ÿ ÿ s.t. x e − x e = b i = − 1 if i = t ∀ i ∈ N _ 0 otherwise e ∈ δ + e ∈ δ − [ i i ÿ r k e x e ≤ U k ∀ k ∈ K e ∈ A + side non linear constraints x e ∈ { 0 , 1 } ∀ e ∈ A . Thursday, September 19, 13

Resource Constrained Shortest Paths (RCSP) ! · " : Arc-flow IP formulation with non linear costs f 3 ÿ 4 ÿ r 1 min w e x e + f e x e e ∈ A e ∈ A Y + 1 if i = s _ ] ÿ ÿ s.t. x e − x e = b i = − 1 if i = t ∀ i ∈ N _ 0 otherwise e ∈ δ + e ∈ δ − [ i i ÿ r k e x e ≤ U k ∀ k ∈ K e ∈ A x e ∈ { 0 , 1 } ∀ e ∈ A . [G. Tsaggouris and C. Zaroliagis, ESA2004] Thursday, September 19, 13