Exact Algorithms for the Chance-Constrained Vehicle Routing Problem - PowerPoint PPT Presentation

Exact Algorithms for the Chance-Constrained Vehicle Routing Problem Ricardo Fukasawa Department of Combinatorics & Optimization University of Waterloo January 7, 2016 Aussois 2016 joint work with Thai Dinh and James Luedtke (University of Wisconsin) Dinh, Fukasawa, Luedtke Chance-constrained VRP 1 / 28

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 Client demands d i , ∀ i ∈ V + depot Dinh, Fukasawa, Luedtke Chance-constrained VRP 2 / 28

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 Client demands d i , ∀ i ∈ V + depot Let S j be the set of clients served by route j . Then d ( S j ) ≤ b Dinh, Fukasawa, Luedtke Chance-constrained VRP 2 / 28

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 Client demands d i , ∀ i ∈ V + Demands D i , ∀ i ∈ V + : random variables that only get realized after routes have been decided depot Let S j be the set of clients served by route j . Then d ( S j ) ≤ b Dinh, Fukasawa, Luedtke Chance-constrained VRP 2 / 28

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 Client demands d i , ∀ i ∈ V + Demands D i , ∀ i ∈ V + : random variables that only get realized after routes have been decided depot Let S j be the set of clients served by route j . Then d ( S j ) ≤ b Question What to do when sum of loads exceed b ? Dinh, Fukasawa, Luedtke Chance-constrained VRP 2 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained) Recourse-based model (two-stage) Follow planned routes depot Dinh, Fukasawa, Luedtke Chance-constrained VRP 3 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained) Recourse-based model (two-stage) Follow planned routes If capacity is exceeded at a customer, determine alternative service plan depot Dinh, Fukasawa, Luedtke Chance-constrained VRP 3 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained) Recourse-based model (two-stage) Follow planned routes If capacity is exceeded at a customer, determine alternative service plan Common assumption: Make round-trip to depot and back (simpler) Minimize expected cost (including cost depot of recourse) Dinh, Fukasawa, Luedtke Chance-constrained VRP 3 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained) Recourse-based model (two-stage) Follow planned routes If capacity is exceeded at a customer, determine alternative service plan Common assumption: Make round-trip to depot and back (simpler) Minimize expected cost (including cost depot of recourse) Other recourse options Alternate recourse actions may be better (e.g., Novoa et al., 2006) Dinh, Fukasawa, Luedtke Chance-constrained VRP 3 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained) Recourse-based model (two-stage) Follow planned routes If capacity is exceeded at a customer, determine alternative service plan Common assumption: Make round-trip to depot and back (simpler) Minimize expected cost (including cost depot of recourse) Other recourse options Alternate recourse actions may be better (e.g., Novoa et al., 2006) Dinh, Fukasawa, Luedtke Chance-constrained VRP 3 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained) Recourse-based model (two-stage) Follow planned routes If capacity is exceeded at a customer, determine alternative service plan Common assumption: Make round-trip to depot and back (simpler) Minimize expected cost (including cost depot of recourse) Other recourse options Alternate recourse actions may be better (e.g., Novoa et al., 2006) but more expensive computationally Dinh, Fukasawa, Luedtke Chance-constrained VRP 3 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained) Chance-constrained model Limit the likelihood that capacity in each truck is exceeded to a specified threshold ǫ And ignore cases when capacity is exceeded! depot Dinh, Fukasawa, Luedtke Chance-constrained VRP 3 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained) Chance-constrained model Limit the likelihood that capacity in each truck is exceeded to a specified threshold ǫ And ignore cases when capacity is exceeded! Let S j be the set of clients served by route j . Then P { D ( S j ) ≤ b } ≥ 1 − ǫ depot Dinh, Fukasawa, Luedtke Chance-constrained VRP 3 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained) Chance-constrained model Limit the likelihood that capacity in each truck is exceeded to a specified threshold ǫ And ignore cases when capacity is exceeded! Let S j be the set of clients served by route j . Then P { D ( S j ) ≤ b } ≥ 1 − ǫ Advantages No need to model complicated depot recourse actions Customers more likely to receive “regular” service Possible driver efficiency Dinh, Fukasawa, Luedtke Chance-constrained VRP 3 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained) Chance-constrained model Limit the likelihood that capacity in each truck is exceeded to a specified threshold ǫ And ignore cases when capacity is exceeded! Let S j be the set of clients served by route j . Then P { D ( S j ) ≤ b } ≥ 1 − ǫ Advantages No need to model complicated depot recourse actions Customers more likely to receive “regular” service Possible driver efficiency Disadvantage: When you are the unlucky customer that is left stranded with probability ǫ Dinh, Fukasawa, Luedtke Chance-constrained VRP 3 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained) Chance-constrained model Limit the likelihood that capacity in each truck is exceeded to a specified threshold ǫ And ignore cases when capacity is exceeded! Let S j be the set of clients served by route j . Then P { D ( S j ) ≤ b } ≥ 1 − ǫ Advantages No need to model complicated depot recourse actions Customers more likely to receive “regular” service Possible driver efficiency Disadvantage: When you are the unlucky customer that is left stranded with probability ǫ e.g. Dey and F. (Sunday at train station) Dinh, Fukasawa, Luedtke Chance-constrained VRP 3 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained) Experiment: For an instance, take routing decisions based on chance-constrained ( ǫ = 5%) and recourse (2-stage) model Evaluate each solution by probability that a truck’s capacity is violated and by its expected 2-stage cost (2-stage solution is optimal) Four instances, size up to 22 nodes, all independent normal (low and high variance) Max Violation Prob. % % Increase Var CC Sol 2-stage Sol Expected Cost Low 1.7 50.0 2.3% 5.0 7.8 0.9% 2.4 2.4 0 3.1 6.4 0.6% High 4.0 8.3 3.4% 3.6 23.7 2.9% 1.0 1.0 0 0.7 16.9 0.3% Dinh, Fukasawa, Luedtke Chance-constrained VRP 4 / 28

Literature review Deterministic VRP State-of-the-art methods use branch-and-cut-and-price E.g., F. et al. (2006), Baldacci et al. (2008), Baldacci et al. (2011), Pecin et al. (2014) 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) Dinh, Fukasawa, Luedtke Chance-constrained VRP 5 / 28

Literature review Deterministic VRP State-of-the-art methods use branch-and-cut-and-price E.g., F. et al. (2006), Baldacci et al. (2008), Baldacci et al. (2011), Pecin et al. (2014) 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) Not aware of any exact SVRP method that has been tested for problems with correlation (most assume independent normal) Dinh, Fukasawa, Luedtke Chance-constrained VRP 5 / 28