Branch-and-cut (and-price) for the chance constrained vehicle routing problem

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1 Branch-and-cut (and-price) for the chance constrained vehicle routing problem Ricardo Fukasawa Department of Combinatorics & Optimization University of Waterloo May 25th, 2016 ColGen 2016 joint work with Thai Dinh and James Luedtke (University of Wisconsin) Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 1 / 29

2 The deterministic vehicle routing problem G = (V,E) V = {0} V + Edge lengths l e, e E K vehicles, capacity b Find a set of K routes with minimum total length depot Client demands d i, i V + Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 2 / 29

3 The deterministic vehicle routing problem G = (V,E) V = {0} V + Edge lengths l e, e E K vehicles, capacity b Find a set of K routes with minimum total length depot Client demands d i, i V + Let S j be the set of clients served by route j. Then d(s j) b Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 2 / 29

4 The stochastic vehicle routing problem G = (V,E) V = {0} V + Edge lengths l e, e E K vehicles, capacity b Find a set of K routes with minimum total length depot Client demands d i, i V + Demands D i, i V +: random variables that only get realized after routes have been decided Let S j be the set of clients served by route j. Then d(s j) b Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 2 / 29

5 The chance-constrained vehicle routing problem G = (V,E) V = {0} V + Edge lengths l e, e E K vehicles, capacity b Find a set of K routes with minimum total length depot Client demands d i, i V + Demands D i, i V +: random variables that only get realized after routes have been decided Let S j be the set of clients served by route j. Then d(s j) b Then P{D(S j) b} 1 ǫ Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 2 / 29

6 Literature review Deterministic VRP State-of-the-art methods use branch-and-cut-and-price Citation: Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 3 / 29

7 Literature review Deterministic VRP State-of-the-art methods use branch-and-cut-and-price Citation: Do I need any? Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 3 / 29

8 Literature review Deterministic VRP State-of-the-art methods use branch-and-cut-and-price Citation: Do I need any? Stochastic VRP (2-stage) Heuristics: Stewart & Golden (1983), Dror & Trudeau (1986), Savelsbergh & Goetschalckx (1995), Novoa et al. (2006), Secomandi and Margot (2009),... Integer L-Shaped: Gendreau et al. (1994), Laporte et al. (2002),... Branch-and-cut: Laporte et al. (1989),... Branch-and-price: Christiansen et al. (2007) Branch-and-cut-and-price: Gauvin et al. (2014) Stochastic VRP (chance-constrained) Reduction to deterministic case: Stewart & Golden (1983) Branch-and-cut: Laporte et al. (1989) Branch-and-cut: Beraldi et al. (2015) Branch-and-cut for Robust VRP: Gounaris, Wiesemann, Floudas (2013) Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 3 / 29

9 State of CCVRP Distribution BC BP BCP Deterministic Independent Normal Correlated Normal - Stewart and Golden (83): Reduction to deterministic, only applies to some distributions e.g. Poisson, Binomial. Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 4 / 29

10 State of CCVRP Distribution BC BP BCP Deterministic Independent Normal Correlated Normal Goal - Stewart and Golden (83): Reduction to deterministic, only applies to some distributions e.g. Poisson, Binomial. Develop exact methods for chance-constrained SVRP with very few assumptions on the demand uncertainty. Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 4 / 29

11 State of CCVRP Distribution BC BP BCP Deterministic Independent Normal Correlated Normal Goal - Stewart and Golden (83): Reduction to deterministic, only applies to some distributions e.g. Poisson, Binomial. Develop exact methods for chance-constrained SVRP with very few assumptions on the demand uncertainty. Assumption: Quantile { Q p(s) := inf b : P{ } D i b } p i S can be computed for any S V + and any p [0,1]. Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 4 / 29

12 State of CCVRP Distribution BC BP BCP Deterministic Independent Normal Correlated Normal Computable Q p(s) Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 5 / 29

13 State of CCVRP Distribution BC BP BCP Deterministic Independent Normal Correlated Normal Computable Q p(s) Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 5 / 29

14 Edge formulation for deterministic VRP d i: deterministic demand at customer i V + r(s): number of trucks required to serve S V + x e: number of times a vehicle traverses edge e E min l x ex e e E s.t. x e = 2, i V + e δ({i}) e δ({0}) e δ(s) x e 1, x e = 2K x e 2r(S), S V + x e Z +, e E. e E \δ({0}) Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 6 / 29

15 Edge formulation for chance-constrained VRP Modified capacity inequalities e δ(s) x e 2r ǫ(s), S V + r ǫ(s): Minimum number of trucks required to serve customer set S, where probability of capacity violation is at most ǫ for each truck Requires solving stochastic bin-packing Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 7 / 29

16 Edge formulation for chance-constrained VRP Modified capacity inequalities e δ(s) x e 2r ǫ(s), S V + r ǫ(s): Minimum number of trucks required to serve customer set S, where probability of capacity violation is at most ǫ for each truck Requires solving stochastic bin-packing Challenge How to obtain valid lower bounds on r ǫ(s)? Laporte et al. (1989): If demands are independent normal, can use Q1 ǫ(s) where Q p(s) be pth quantile of the random variable i S Di, i.e. Q p(s) := inf { b : P{ i S Di b } p }. b Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 7 / 29

17 Edge formulation for chance-constrained VRP Modified capacity inequalities e δ(s) x e 2r ǫ(s), S V + r ǫ(s): Minimum number of trucks required to serve customer set S, where probability of capacity violation is at most ǫ for each truck Requires solving stochastic bin-packing Challenge How to obtain valid lower bounds on r ǫ(s)? Laporte et al. (1989): If demands are independent normal, can use Q1 ǫ(s) where Q p(s) be pth quantile of the random variable i S Di, i.e. Q p(s) := inf { b : P{ i S Di b } p }. Not valid in general. b Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 7 / 29

18 Bad example for Laporte et al. bound 3 2 Scenarios Clients Probability Table: Demands in each scenario 4 1 b = 2 ǫ = 0.1 Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 8 / 29

19 Bad example for Laporte et al. bound 3 2 Scenarios Clients Probability Table: Demands in each scenario 4 1 b = 2 ǫ = 0.1 Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 8 / 29

20 Bad example for Laporte et al. bound 3 2 Scenarios Clients Probability Table: Demands in each scenario 4 1 b = 2 ǫ = 0.1 Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 8 / 29

21 Bad example for Laporte et al. bound 3 2 Scenarios Clients Probability Table: Demands in each scenario 4 1 b = 2 ǫ = 0.1 Solution depicted is feasible Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 8 / 29

22 Bad example for Laporte et al. bound 3 2 Scenarios Clients Probability Table: Demands in each scenario 4 1 b = 2 ǫ = 0.1 Solution depicted is feasible However, for S = {1,2,3,4}, Q 0.9(S) = 5 Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 8 / 29

23 Bad example for Laporte et al. bound 3 2 Scenarios Clients Probability Table: Demands in each scenario 4 1 b = 2 ǫ = 0.1 Solution depicted is feasible However, for S = {1,2,3,4}, Q 0.9(S) = 5 Q1 ǫ (S) Thus using requires 3 vehicles to enter S = {1,2,3,4} b Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 8 / 29

24 Bounds on required trucks more generally Simple general bound { } 1 P D i b 1 ǫ k ǫ(s) = i S 2 otherwise x e 2k ǫ(s), S V + e δ(s) k ǫ(s) r ǫ(s) but sufficient to define a valid formulation Cheap to compute for a given set S (thus easy for x Z E +) Cuts may be weak Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 9 / 29

25 Improved general bound Lemma r ǫ(s) Q1 ǫrǫ(s)(s) b Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 10 / 29

26 Improved general bound Lemma r ǫ(s) Q1 ǫrǫ(s)(s) b But we don t know r ǫ(s)! Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 10 / 29

27 Improved general bound Lemma r ǫ(s) Q1 ǫrǫ(s)(s) b But we don t know r ǫ(s)! Lemma For any k 2, { } Q1 ǫ(k 1) (S) r ǫ(s) min k, b Proof: Either r ǫ(s) k or r ǫ(s) k 1. Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 10 / 29

28 Improved general bound (2) Use best k: { { } kǫ(s) Q1 ǫ(k 1) (S) = max min k, b } : k = 2,...,K. Always at least as good as first simple bound Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 11 / 29

29 Improved general bound (2) Use best k: { { } kǫ(s) Q1 ǫ(k 1) (S) = max min k, b } : k = 2,...,K. Always at least as good as first simple bound Improvements are possible for special cases: Independent and Correlated normal: Can use a stronger closed form formula (derived from robust CVRP). Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 11 / 29

30 State of CCVRP Distribution BC BP BCP Deterministic Independent Normal Correlated Normal Computable Q p(s) Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 12 / 29

31 State of CCVRP Distribution BC BP BCP Deterministic Independent Normal Correlated Normal Computable Q p(s) Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 12 / 29

32 Set partitioning formulation for DETERMINISTIC Sets : Ω: set of elementary routes satisfying capacity Parameters : a ir: number of times vertex i appears in route r Variables : λ r: (binary) whether to choose route r min λ s.t. c rλ r r Ω a ir λ r = 1, i V + r Ω λ r = K r Ω λ r {0,1}, r Ω Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 13 / 29

33 Set partitioning formulation for DETERMINISTIC Sets : Ω: set of elementary routes satisfying capacity Ω : set of non-elementary routes satisfying capacity. Parameters : a ir: number of times vertex i appears in route r Variables : λ r: (binary) whether to choose route r min λ s.t. c rλ r r Ω r Ω a ir λ r = 1, i V + r Ω λ r = K λ r {0,1}, r Ω Pseudo-polynomial pricing. Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 13 / 29

34 Set partitioning formulation for STOCHASTIC Sets : Ω s: set of elementary routes satisfying chance-constraint Parameters : a ir: number of times vertex i appears in route r Variables : λ r: (binary) whether to choose route r min λ s.t. c rλ r r Ω s r Ω s a ir λ r = 1, i V + r Ω s λ r = K λ r {0,1}, r Ω Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 13 / 29

35 Set partitioning formulation for STOCHASTIC Sets : Ω s: set of elementary routes satisfying chance-constraint Ω s : set of non-elementary routes satisfying chance-constraint. Parameters : a ir: number of times vertex i appears in route r Variables : λ r: (binary) whether to choose route r min λ s.t. c rλ r r Ω s a ir λ r = 1, i V + r Ω s λ r = K r Ω s λ r {0,1}, r Ω Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 13 / 29

36 Theorem Finding the least cost non-elementary route in a graph that respects the capacity chance constraint under the finite distribution model is strongly NP-hard. Theorem Finding the least cost non-elementary route in a graph that respects the capacity chance constraint under the independent normal distribution model is strongly NP-hard. Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 14 / 29

Theorem Finding the least cost non-elementary route in a graph that respects the capacity chance constraint under the finite distribution model is strongly NP-hard.

37 Theorem Finding the least cost non-elementary route in a graph that respects the capacity chance constraint under the finite distribution model is strongly NP-hard. Theorem Finding the least cost non-elementary route in a graph that respects the capacity chance constraint under the independent normal distribution model is strongly NP-hard. Proof Idea: Use chance-constraint to enforce elementarity. Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 14 / 29

38 State of CCVRP Distribution BC BP BCP Deterministic Independent Normal Hard Correlated Normal Hard Computable Q p(s) Hard Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 15 / 29

39 State of CCVRP Distribution BC BP BCP Deterministic Independent Normal Hard Correlated Normal Hard Computable Q p(s) Hard Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 15 / 29

40 State of CCVRP Distribution BC BP BCP Deterministic Independent Normal Hard Correlated Normal Hard Computable Q p(s) Hard Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 15 / 29

41 BCP idea Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 16 / 29

42 BCP idea Deterministic: Elementary (strongly NP-hard) Non-elementary (pseudo-polynomial) Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 16 / 29

43 BCP idea Deterministic: Elementary (strongly NP-hard) Non-elementary (pseudo-polynomial) Chance-constrained Elementary (strongly NP-hard) Non-elementary (strongly NP-hard) Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 16 / 29

44 BCP idea Deterministic: Elementary (strongly NP-hard) Non-elementary (pseudo-polynomial) Chance-constrained Elementary (strongly NP-hard) Non-elementary (strongly NP-hard) Relax chance-constraint Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 16 / 29

45 Relaxed pricing scheme Exact capacity chance constraint y i: binary indicator of whether or not node i is visited { F ǫ = y {0,1} V+ : P { D T y b } } 1 ǫ Idea Find w Z V+ + and τ Z + such that: Use R(w,τ) instead of F ǫ : F ǫ R(w,τ) := { } y Z V+ : w T y τ Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 17 / 29

46 Relaxed pricing scheme Exact capacity chance constraint y i: binary indicator of whether or not node i is visited { F ǫ = y {0,1} V+ : P { D T y b } } 1 ǫ Idea Find w Z V+ + and τ Z + such that: Use R(w,τ) instead of F ǫ : F ǫ R(w,τ) := { } y Z V+ : w T y τ Capacity cuts ensure only solutions to F ǫ will be picked Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 17 / 29

47 Generic relaxed pricing scheme (cont d) How to choose coefficients? Natural choice: w i = E[D i] Given w, optimize τ in preprocessing phase: τ = max {w T y : P { D T y b } } 1 ǫ,y {0,1} V+ Stochastic binary knapsack problem Joint normal random demands Binary second-order cone program Scenario model of random demands Structured binary integer program (Song et al., 2014) Any easily computable upper bound on the above maximum can be used. Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 18 / 29

48 Relaxed pricing with joint normal demands Idea With joint normal random demands, binary second-order cone program can be replaced with a semidefinite program With mean vector µ and covariance matrix Σ: { } P D T y b 1 ǫ µ T y +Φ 1 (1 ǫ) y T Σy b Get a lower bound on y T Σy in terms of µ T y Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 19 / 29

49 Relaxed pricing with joint normal demands Idea With joint normal random demands, binary second-order cone program can be replaced with a semidefinite program With mean vector µ and covariance matrix Σ: { } P D T y b 1 ǫ µ T y +Φ 1 (1 ǫ) y T Σy b Get a lower bound on y T Σy in terms of µ T y Find η such that η µ T y y T Σy for all y {0,1} V+ µ T y +Φ 1 (1 ǫ) η µ T y b (2) Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 19 / 29

50 Relaxed pricing with joint normal demands Idea With joint normal random demands, binary second-order cone program can be replaced with a semidefinite program With mean vector µ and covariance matrix Σ: { } P D T y b 1 ǫ µ T y +Φ 1 (1 ǫ) y T Σy b Get a lower bound on y T Σy in terms of µ T y Find η such that η µ T y y T Σy for all y {0,1} V+ η found by solving an SDP µ T y +Φ 1 (1 ǫ) η µ T y b (2) η =max η,p,q η (3a) s.t. µ iη p i i V + (3b) Σ = diag(p 1,...,p n)+q Q 0, (3c) (3d) Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 19 / 29

51 Relaxed pricing with joint normal demands Idea With joint normal random demands, binary second-order cone program can be replaced with a semidefinite program With mean vector µ and covariance matrix Σ: { } P D T y b 1 ǫ µ T y +Φ 1 (1 ǫ) y T Σy b Get a lower bound on y T Σy in terms of µ T y Find η such that η µ T y y T Σy for all y {0,1} V+ η found by solving an SDP µ T y +Φ 1 (1 ǫ) η µ T y b (2) η =max η,p,q η (3a) s.t. µ iη p i i V + (3b) Σ = diag(p 1,...,p n)+q Q 0, Solve RCSP using constraint (2) on resource µ T y. (3c) (3d) Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 19 / 29

52 Pricing with independent normal demands Pricing for independent normal with mean vector µ and variance vector σ 2 { } P D T y b 1 ǫ µ T y +Φ 1 (1 ǫ) yi 2 σi 2 b i V + Relax to: µ T y +Φ 1 (1 ǫ) y T σ 2 b Resources: µ T y and y T σ 2 Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 20 / 29

53 Computational tests overview Test instances Based on deterministic VRP instances 32 to 55 customers Two variance settings: low ( 10% of mean) and high ( 20% of mean) Three distribution assumptions: independent normal, joint normal, scenario Implementation details Cplex Implemented in BCP code based from F. et al. (2006) 7200 second time limit BC BC BC J BCP r BCP i k ǫ(s) k ǫ(s) k J ǫ(s) Rel. pricing Rel. pricing (for ind. normal) Table: Strategies used Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 21 / 29

54 20 15 BCP i BCP r BC J BC BC % 20% Figure: Summary of results for instances with independent normal distribution. Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 22 / 29

55 20 15 BCP r BC J BC BC % 20% Figure: Summary of results for instances with joint normal distribution. Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 23 / 29

56 20 15 BCP r BC BC % 20% Figure: Summary of results for instances with scenario distribution. Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 24 / 29

57 Concluding remarks Summary Chance-constrained formulation avoids difficulties in modeling recourse actions Proposed method can solve chance-constrained VRP with correlations Builds on successful approaches for solving deterministic VRP Can be extended to other variants Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 25 / 29

58 Concluding remarks Summary Chance-constrained formulation avoids difficulties in modeling recourse actions Proposed method can solve chance-constrained VRP with correlations Builds on successful approaches for solving deterministic VRP Can be extended to other variants Distribution BC BP BCP Deterministic Independent Normal Hard Correlated Normal Hard Computable Q p(s) Hard Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 25 / 29

59 Future work Future work Incorporate more advanced features of deterministic VRP into solution approach Seek improved pricing friendly relaxation of chance-constrained capacity constraint Other models of handling uncertainty How well can deterministic constraints approximate chance-constraints? Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 26 / 29

60 THANK YOU! Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 27 / 29

61 Comparing solutions Experiment: For an instance, obtain chance-constrained and recourse model solutions Evaluate each solution in both model metrics Four instances, size up to 22 nodes, all independent normal Max Violation Prob. % % Increase Var CC Sol Rec Sol Expected Cost Low % % % High % % % Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 28 / 29

62 Comparing solutions: Correlated demands Recourse solution: Ignore correlation Evaluate each solution in both model metrics using true distribution Max Violation Prob. % Increase in Var CC Sol Rec Sol Expected Cost Low % % % % High % % % % Dinh, Fukasawa, Luedtke BCP for Chance-constrained VRP 29 / 29

Exact algorithms for the chanceconstrained

Exact algorithms for the chanceconstrained vehicle routing problem Thai Dinh, Ricardo Fukasawa & James Luedtke Mathematical Programming A Publication of the Mathematical Optimization Society ISSN 0025-5610