Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 2

Outline

I Vehicle routing problem;

I How interior point methods can help;

I Interior point branch-price-and-cut: central primal-dual solutions;

I Results for vehicle routing variants;

I Conclusion and future developments.

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 3

Vehicle routing problem

I One of the most studied combinatorial optimization problems;

I Theoretical reason: very difficult to solve by standard methods; tough

testbed for new algorithms;

I “the literature growth is almost perfectly exponential with a 6.09%

annual growth rate” (Eksioglu et al. 2009; Braekers et al., 2016)

I Practical reason: models important real-life situations, faced by many

companies around the world; impact our day-to-day lives;

I Airline companies; train and bus schedules; freight transportation ...

I “We want it at the best price and on time!”

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 3

Vehicle routing problem

I One of the most studied combinatorial optimization problems;

I Theoretical reason: very difficult to solve by standard methods; tough

testbed for new algorithms;

I “the literature growth is almost perfectly exponential with a 6.09%

annual growth rate” (Eksioglu et al. 2009; Braekers et al., 2016)

I Practical reason: models important real-life situations, faced by many

companies around the world; impact our day-to-day lives;

I Airline companies; train and bus schedules; freight transportation ...

I “We want it at the best price and on time!”

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 3

Vehicle routing problem

I One of the most studied combinatorial optimization problems;

I Theoretical reason: very difficult to solve by standard methods; tough

testbed for new algorithms;

I “the literature growth is almost perfectly exponential with a 6.09%

annual growth rate” (Eksioglu et al. 2009; Braekers et al., 2016)

I Practical reason: models important real-life situations, faced by many

companies around the world; impact our day-to-day lives;

I Airline companies; train and bus schedules; freight transportation ...

I “We want it at the best price and on time!”

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 4

VRP with time windows (VRPTW)

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 4

VRP with time windows (VRPTW)

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 4

VRP with time windows (VRPTW)

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 5

Vehicle routing problem

I To effectively solve vehicle routing problems, we need to rely on a variety

of formulations and state-of-the-art solution methods;

I Heuristic, exact and hybrid methods;

I Vehicle flow formulation (xkij) and set partitioning formulation (λp);

(completely different, but the same actually! – See: Munari, A generalized

formulation for vehicle routing problems, ArXiv, 2016)

I Branch-and-cut; column generation and branch-and-price;

I Auxiliary techniques: dynamic programming; implicit enumeration; and

many others to speed up the generation of routes and/or valid inequalities;

I Typical things that you need to solve large-scale problems.

5

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 6

Large-scale optimization problems

I A formulation that challenges state-of-the-art implementations;

I Special structure in the coefficient matrix, which allows a reformulation

(e.g. Dantzig-Wolfe decomposition, Lagrangian relaxation, Benders

decomposition, etc);

I Geoffrion (1970); Vanderbeck and Wolsey (2010);

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 7

Large-scale optimization problems

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 7

Large-scale optimization problems

Master problem

Subproblems

Decomposition

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 7

Large-scale optimization problems

Master problem

Subproblems

Column generation

Decomposition

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 7

Large-scale optimization problems

Master problem

Subproblems

Column generation

Decomposition

TOO MANY VARIABLES

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 8

Large-scale discrete optimization problems

ℤn

8

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 8

Large-scale discrete optimization problems

ℤn ℝnMaster

Subproblems

8

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 8

Large-scale discrete optimization problems

ℤn ℝnMaster

Subproblems

Column generation

8

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 8

Large-scale discrete optimization problems

ℤn ℝnMaster

Subproblems

Column generation

8

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 8

Large-scale discrete optimization problems

ℤn ℝnMaster

Subproblems

Column generation

Branch-and-price

8

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 9

Large-scale optimization problems

I In column generation and branch-and-price, we typically have to solve

hundreds of thousands of linear programming (LP) problems in sequence;

I This way, it is important to use a fast LP method;

I Are we really interested in an optimal solution of these problems?

I What informations would be relevant in this context?

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 9

Large-scale optimization problems

I In column generation and branch-and-price, we typically have to solve

hundreds of thousands of linear programming (LP) problems in sequence;

I This way, it is important to use a fast LP method;

I Are we really interested in an optimal solution of these problems?

I What informations would be relevant in this context?

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 10

Column generation

I We are interested in solving a linear programming problem with a huge

number of columns, called the Master Problem (MP):

z? := min∑j∈N

cjλj ,

s.t.∑j∈N

ajλj = b,

λj ≥ 0, ∀j ∈ N.

I N is too big;

I The columns (cj , aj) ∈ A are not known explicitly;

I We know how to generate them!

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 10

Column generation

I We are interested in solving a linear programming problem with a huge

number of columns, called the Master Problem (MP):

z? := min∑j∈N

cjλj ,

s.t.∑j∈N

ajλj = b,

λj ≥ 0, ∀j ∈ N.

I N is too big;

I The columns (cj , aj) ∈ A are not known explicitly;

I We know how to generate them!

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 11

Column generation

I Restricted Master Problem (RMP):

zRMP := min∑j∈N

cjλj ,

s.t.∑j∈N

ajλj = b, (u)

λj ≥ 0, ∀j ∈ N.I with N ⊂ N .

I Let u be a dual optimal solution of the RMP;

I Pricing subproblem (oracle):

zSP := min{0, cj − uT aj |(cj , aj) ∈ A}.

I (cj , aj) are the variables in the subproblem;

I If zSP < 0, then new columns are generated;

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 11

Column generation

I Restricted Master Problem (RMP):

zRMP := min∑j∈N

cjλj ,

s.t.∑j∈N

ajλj = b, (u)

λj ≥ 0, ∀j ∈ N.I with N ⊂ N .

I Let u be a dual optimal solution of the RMP;

I Pricing subproblem (oracle):

zSP := min{0, cj − uT aj |(cj , aj) ∈ A}.

I (cj , aj) are the variables in the subproblem;

I If zSP < 0, then new columns are generated;

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 12

Standard column generation

I Optimal solutions, typically obtained by the simplex method:

⇒ Extreme points of the RMP;

I Bang-bang: they oscillate too much between consecutive iterations;

⇒ uj+1 is typically far from uj ;

I Heading-in and tailing-off;

I Degeneracy;

I see Vanderbeck (2005); Lubbecke and Desrosiers (2005);

I Extreme points result in slow convergence of the method.

12

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 13

Oscillation in a VRP instance

‖uj − uj+1‖2, for each iteration j:������������� ���

�

���

���

���

���

�

���

���

� � � �� �� �� � �� � �� �� �� �� �� �� �� � � � �� �� �� ��

������������������������ ����������������

���������

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Munari, P.; Gondzio, J. Column generation and branch-and-price with interior point methods. Pro-ceeding Series of the Brazilian Society of Computational and Applied Mathematics, v. 3 (1), 2015.

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 13

Column generation variants

I Stabilization techniques: avoid extreme solutions!

⇒ use a point in the interior of the feasible set;

I Most of them: modify the master problem!

I Add variables, bounds, constraints, penalties, ...

⇒ The master problem may become more difficult to solve;

⇒ Some of them may be difficult to implement;

⇒ Several parameters to tune.

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 14

Column generation variants

I Stability center and/or safety region in the dual space;

RMP (dual)

(a)

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 14

Column generation variants

I Stability center and/or safety region in the dual space;

RMP (dual)

(a)

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 14

Column generation variants

I Stability center and/or safety region in the dual space;

RMP (dual)

(a)

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 14

Column generation variants

I Stability center and/or safety region in the dual space;

RMP (dual)

(a)

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 14

Column generation variants

I Stability center and/or safety region in the dual space;

RMP (dual)

(a)

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 15

Column generation variants

I Stabilization techniques: avoid extreme solutions!

⇒ use a point in the interior of the feasible set;

I Different strategies:

I dynamic boxes and penalties (Marsten et al., 1975, du Merle 1999;

Ben Amor et al. 2009);

I smoothing (Wentges, 1997; Neame, 1999; Pessoa, 2013);

I bundle and nonlinear penalties (Frangioni, 2002; Briant et al., 2004)

I interior points (Goffin and Vial, 2002; Rosseau et al., 2003);

I and many others;

15

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 16

Column generation variants

I Most of them: modify the master problem or require additional

control on the dual solutions;

I Add variables, bounds, constraints, penalties, ...

⇒ The master problem may become more difficult to solve;

⇒ Some of them may be difficult to implement;

⇒ Several parameters to tune.

I They all agree on one thing:

I “Column generation is more efficient when based on well-centered

interior points of the feasible set;”

I So, why not using a primal-dual interior point method?

I This is straightforward: does not require any changes in the RMP

nor additional control (naturally stable solutions).

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 16

Column generation variants

I Most of them: modify the master problem or require additional

control on the dual solutions;

I Add variables, bounds, constraints, penalties, ...

⇒ The master problem may become more difficult to solve;

⇒ Some of them may be difficult to implement;

⇒ Several parameters to tune.

I They all agree on one thing:

I “Column generation is more efficient when based on well-centered

interior points of the feasible set;”

I So, why not using a primal-dual interior point method?

I This is straightforward: does not require any changes in the RMP

nor additional control (naturally stable solutions).

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 16

Column generation variants

I Most of them: modify the master problem or require additional

control on the dual solutions;

I Add variables, bounds, constraints, penalties, ...

⇒ The master problem may become more difficult to solve;

⇒ Some of them may be difficult to implement;

⇒ Several parameters to tune.

I They all agree on one thing:

I “Column generation is more efficient when based on well-centered

interior points of the feasible set;”

I So, why not using a primal-dual interior point method?

I This is straightforward: does not require any changes in the RMP

nor additional control (naturally stable solutions).

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 17

Interior point method

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 17

Interior point method

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 17

Interior point method

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 18

Primal-dual column generation method (PDCGM)

I Primal-dual interior point method to get primal-dual solutions

(Gondzio and Sarkissian, 1996; Gondzio et al., 2013;

Munari and Gondzio, 2013; Gondzio et al., 2016);

I Suboptimal solution (λ, u) (ε-optimal solution): we stop the interior point

method with optimality tolerance ε.

I The distance to optimality ε is dynamically adjusted according to the

relative gap;ε = min{εmax, gap/D}

I gap = (UB− LB)/(1 + |UB|);

I D: degree of optimality (fixed, D > 1);

I We save time and stop with a well-centered dual solution!

17

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 18

Primal-dual column generation method (PDCGM)

I Primal-dual interior point method to get primal-dual solutions

(Gondzio and Sarkissian, 1996; Gondzio et al., 2013;

Munari and Gondzio, 2013; Gondzio et al., 2016);

I Suboptimal solution (λ, u) (ε-optimal solution): we stop the interior point

method with optimality tolerance ε.

I The distance to optimality ε is dynamically adjusted according to the

relative gap;ε = min{εmax, gap/D}

I gap = (UB− LB)/(1 + |UB|);

I D: degree of optimality (fixed, D > 1);

I We save time and stop with a well-centered dual solution!

17

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 18

Primal-dual column generation method (PDCGM)

I Primal-dual interior point method to get primal-dual solutions

(Gondzio and Sarkissian, 1996; Gondzio et al., 2013;

Munari and Gondzio, 2013; Gondzio et al., 2016);

I Suboptimal solution (λ, u) (ε-optimal solution): we stop the interior point

method with optimality tolerance ε.

I The distance to optimality ε is dynamically adjusted according to the

relative gap;ε = min{εmax, gap/D}

I gap = (UB− LB)/(1 + |UB|);

I D: degree of optimality (fixed, D > 1);

I We save time and stop with a well-centered dual solution!

17

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 18

Primal-dual column generation method (PDCGM)

I Primal-dual interior point method to get primal-dual solutions

(Gondzio and Sarkissian, 1996; Gondzio et al., 2013;

Munari and Gondzio, 2013; Gondzio et al., 2016);

I Suboptimal solution (λ, u) (ε-optimal solution): we stop the interior point

method with optimality tolerance ε.

I The distance to optimality ε is dynamically adjusted according to the

relative gap;ε = min{εmax, gap/D}

I gap = (UB− LB)/(1 + |UB|);

I D: degree of optimality (fixed, D > 1);

I We save time and stop with a well-centered dual solution!

17

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 18

Primal-dual column generation method (PDCGM)

I Primal-dual interior point method to get primal-dual solutions

(Gondzio and Sarkissian, 1996; Gondzio et al., 2013;

Munari and Gondzio, 2013; Gondzio et al., 2016);

I Suboptimal solution (λ, u) (ε-optimal solution): we stop the interior point

method with optimality tolerance ε.

I The distance to optimality ε is dynamically adjusted according to the

relative gap;ε = min{εmax, gap/D}

I gap = (UB− LB)/(1 + |UB|);

I D: degree of optimality (fixed, D > 1);

I We save time and stop with a well-centered dual solution!

17

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 19

Non-optimal solutions from interior point method

(b)

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Non-optimal solutions from interior point method

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Non-optimal solutions from interior point method

(c)

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PDCGM: Algorithm

1. Input: Initial RMP; parameters κ, εmax, D > 1, δ > 0, .

2. set LB = −∞, UB =∞, gap =∞, ε = 0.5;

3. while (gap > δ) do

4. find a well-centered ε-optimal solution (λ, u) of the RMP;

5. UB = min{UB, zRMP };

6. call the oracle with the query point u;

7. LB = max{LB, κzSP + bT u};

8. gap = (UB− LB)/(1 + |UB|);

9. ε = min{εmax, gap/D};

10. if (zSP < 0) then add the new columns into the RMP;

11. end(while)

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 20

PDCGM: Algorithm

1. Input: Initial RMP; parameters κ, εmax, D > 1, δ > 0, .

2. set LB = −∞, UB =∞, gap =∞, ε = 0.5;

3. while (gap > δ) do

4. find a well-centered ε-optimal solution (λ, u) of the RMP;

5. UB = min{UB, zRMP };

6. call the oracle with the query point u;

7. LB = max{LB, κzSP + bT u};

8. gap = (UB− LB)/(1 + |UB|);

9. ε = min{εmax, gap/D};

10. if (zSP < 0) then add the new columns into the RMP;

11. end(while)

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 20

PDCGM: Algorithm

1. Input: Initial RMP; parameters κ, εmax, D > 1, δ > 0, .

2. set LB = −∞, UB =∞, gap =∞, ε = 0.5;

3. while (gap > δ) do

4. find a well-centered ε-optimal solution (λ, u) of the RMP;

5. UB = min{UB, zRMP };

6. call the oracle with the query point u;

7. LB = max{LB, κzSP + bT u};

8. gap = (UB− LB)/(1 + |UB|);

9. ε = min{εmax, gap/D};

10. if (zSP < 0) then add the new columns into the RMP;

11. end(while)

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 21

PDCGM: well-centered solutions

I Primal-dual interior point method with symmetric neighborhood

(Gondzio, 2015):

I (λ, u) is well-centered in the feasible set:

γµ ≤ (cj − uT aj)λj ≤ (1/γ)µ, ∀j ∈ N,

for some γ ∈ (0.1, 1], where µ = (1/|N |)(cT − uTA)λ;

I Natural way of stabilizing dual solutions.

20

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 22

PDCGM: Convergence

TheoremLet z? be the optimal value of the MP. Given the optimality tolerance δ > 0,

the primal-dual column generation method converges in a finite number of

steps to a primal feasible solution λ of the MP with objective value z that

satisfies:

(z − z?) < δ(1 + |z|).

Gondzio, J.; Gonzalez-Brevis, P. and Munari, P. New developments in the Primal-Dual Column

Generation Technique, European Journal of Operational Research 224, pp. 41-51, 2013;

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 23

PDCGM: Algorithm

I Implementation in C, using interior point solver HOPDM;

I Publicly available code

http://www.maths.ed.ac.uk/~gondzio/software/pdcgm.html

I Source-code examples are provided for different applications:

I Cutting stock problem;I Vehicle routing problem;I Capacitated lot sizing problem with setup times;I Multiple kernel learning;I Two-stage stochastic programming;I Multicommodity network flow.

Gondzio, J.; Gonzalez-Brevis, P.; Munari, P. Large-Scale Optimization with the Primal-Dual Column

Generation Method. Mathematical Programming Computation, v. 8 (1), p. 47-82, 2016.

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Interior point branch-price-and-cut (IPBPC)

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Interior point branch-price-and-cut (IPBPC)

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Interior point branch-price-and-cut (IPBPC)

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Interior point branch-price-and-cut (IPBPC)

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Interior point branch-price-and-cut (IPBPC)

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Interior point branch-price-and-cut (IPBPC)

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Interior point branch-price-and-cut (IPBPC)

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Interior point branch-price-and-cut (IPBPC)

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Interior point branch-price-and-cut (IPBPC)

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 24

Interior point branch-price-and-cut (IPBPC)

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 25

Interior point branch-price-and-cut (IPBPC)

I The primal-dual interior point algorithm will be used to provide

well-centered, suboptimal solutions:

I Column generation;

I Valid inequalities;

I Branching (early termination; central branching).

I More stable primal and dual solutions;

I Deeper columns and cuts;

I Speed up solution times.

I Very few attempts in the literature (du Merle et al., 1999; Elhedhli

and Goffin, 2004, Munari and Gondzio, 2013);

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 25

Interior point branch-price-and-cut (IPBPC)

I The primal-dual interior point algorithm will be used to provide

well-centered, suboptimal solutions:

I Column generation;

I Valid inequalities;

I Branching (early termination; central branching).

I More stable primal and dual solutions;

I Deeper columns and cuts;

I Speed up solution times.

I Very few attempts in the literature (du Merle et al., 1999; Elhedhli

and Goffin, 2004, Munari and Gondzio, 2013);

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 25

Interior point branch-price-and-cut (IPBPC)

I The primal-dual interior point algorithm will be used to provide

well-centered, suboptimal solutions:

I Column generation;

I Valid inequalities;

I Branching (early termination; central branching).

I More stable primal and dual solutions;

I Deeper columns and cuts;

I Speed up solution times.

I Very few attempts in the literature (du Merle et al., 1999; Elhedhli

and Goffin, 2004, Munari and Gondzio, 2013);

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 25

Interior point branch-price-and-cut (IPBPC)

I The primal-dual interior point algorithm will be used to provide

well-centered, suboptimal solutions:

I Column generation;

I Valid inequalities;

I Branching (early termination; central branching).

I More stable primal and dual solutions;

I Deeper columns and cuts;

I Speed up solution times.

I Very few attempts in the literature (du Merle et al., 1999; Elhedhli

and Goffin, 2004, Munari and Gondzio, 2013);

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 25

Interior point branch-price-and-cut (IPBPC)

I The primal-dual interior point algorithm will be used to provide

well-centered, suboptimal solutions:

I Column generation;

I Valid inequalities;

I Branching (early termination; central branching).

I More stable primal and dual solutions;

I Deeper columns and cuts;

I Speed up solution times.

I Very few attempts in the literature (du Merle et al., 1999; Elhedhli

and Goffin, 2004, Munari and Gondzio, 2013);

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 26

Interior point branch-price-and-cut (IPBPC)

I Oracle: two types of subproblems;

ORACLE

Pricing subproblem

Separation subproblem

new column (s)

new cut (s)

primal and dual solutions

I We start calling the separation subproblem as soon as the gap falls

below a tolerance εc (= 0.1), at every Kc iterations;

I Early branching: stop CG with a loose tolerance εb (= 10−3) and branch!

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 26

Interior point branch-price-and-cut (IPBPC)

I Oracle: two types of subproblems;

ORACLE

Pricing subproblem

Separation subproblem

new column (s)

new cut (s)

primal and dual solutions

I We start calling the separation subproblem as soon as the gap falls

below a tolerance εc (= 0.1), at every Kc iterations;

I Early branching: stop CG with a loose tolerance εb (= 10−3) and branch!

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 27

Interior point branch-price-and-cut (IPBPC)

I Two-steps:

Preprocessing step

Quickly obtain a suboptimal

solution of the MP

Branch?

yes

Branch node

Step 1

I To quickly solve the master problem: looser optimality tolerance;

heuristics in the pricing subproblem.

25

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 27

Interior point branch-price-and-cut (IPBPC)

I Two-steps:

Preprocessing step

Quickly obtain a suboptimal

solution of the MP

Branch? no

yes

Branch node

Find an optimal solution of the MP

Step 1 Step 2

I To quickly solve the master problem: looser optimality tolerance;

heuristics in the pricing subproblem.

25

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 27

Interior point branch-price-and-cut (IPBPC)

I Two-steps:

Preprocessing step

Quickly obtain a suboptimal

solution of the MP

Branch? no

yes

Branch node

Find an optimal solution of the MP

Branch? noPrune node

yes

Step 1 Step 2

I To quickly solve the master problem: looser optimality tolerance;

heuristics in the pricing subproblem.

25

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 28

VRP with time windows (VRPTW)

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 29

VRP with time windows (VRPTW)

I Set partitioning formulation:

min∑r∈R

crλr

s.t.∑r∈R

ariλr = 1, i = 1, . . . , n,

(ui)

λr ∈ {0, 1}, r ∈ R.

I R: set of feasible routes;

I Routes are generated by solving a Resource Constrained Elementary

Shortest Path Problem (subproblem).

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 30

VRP with time windows (VRPTW)

I IPBPC implementation for the VRPTW;

I RMP: primal-dual interior point method (HOPDM);

I Subproblem: label-setting algorithm with improvements (Feillet et

al., 2004; Righini and Salani, 2008; Desaulniers et al. 2008);

I Valid inequalities: Subset row cuts (Jepsen et al., 2008);

I Solomon’s instances (standard benchmark for VRPTW);

I Comparison to a state-of-the-art simplex-based BPC by Desaulniers,

Lessard and Hadjar (2008).

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 30

VRP with time windows (VRPTW)

I IPBPC implementation for the VRPTW;

I RMP: primal-dual interior point method (HOPDM);

I Subproblem: label-setting algorithm with improvements (Feillet et

al., 2004; Righini and Salani, 2008; Desaulniers et al. 2008);

I Valid inequalities: Subset row cuts (Jepsen et al., 2008);

I Solomon’s instances (standard benchmark for VRPTW);

I Comparison to a state-of-the-art simplex-based BPC by Desaulniers,

Lessard and Hadjar (2008).

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 30

VRP with time windows (VRPTW)

I IPBPC implementation for the VRPTW;

I RMP: primal-dual interior point method (HOPDM);

I Subproblem: label-setting algorithm with improvements (Feillet et

al., 2004; Righini and Salani, 2008; Desaulniers et al. 2008);

I Valid inequalities: Subset row cuts (Jepsen et al., 2008);

I Solomon’s instances (standard benchmark for VRPTW);

I Comparison to a state-of-the-art simplex-based BPC by Desaulniers,

Lessard and Hadjar (2008).

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 31

Nodes_100

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RC108

R101

R102

R103

R104

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R106

R107

R108

R109

R110

R111

R112

0 20 40 60 80 100 120

Number of nodes

DLH08 IPBPC

Nodes

Inst

an

ce

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 32

Comparing to a simplex-based BPC

Number of nodes

DLH08 IPBPC Ratio

C1 9 9 1.00

RC1 104 78 1.33

R1 239 182 1.31

352 269 1.31

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 33

Cuts_100

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R111

R112

0 100 200 300 400 500 600 700 800

Number of valid inequalities

DLH08 IPBPC

Valid inequalities

Inst

an

ce

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 34

Comparing to a simplex-based BPC

Number of valid inequalities

DLH08 IPBPC Ratio

C1 0 0 1.00

RC1 2199 1191 1.85

R1 3391 2140 1.58

5590 3331 1.68

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 35CPUtime_100

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R112

0 2000 4000 6000 8000 10000 12000 14000 16000 18000

CPU time

DLH08 IPBPC

Seconds

Inst

an

ce

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 36

Comparing to a simplex-based BPC

CPU time (sec)

DLH08 IPBPC Ratio

C1 158 28 5.69

RC1 17198 3472 4.95

R1 27928 4621 6.04

45284 8121 5.58

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 37

IPBPC: impact of suboptimal solutionsAuthor's personal copy

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30

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 38

Interior point branch-price-and-cut (IPBPC)

I Oracle: two types of subproblems;

ORACLE

Pricing subproblem

Separation subproblem

new column (s)

new cut (s)

primal and dual solutions

I We start calling the separation subproblem as soon as the gap falls

below a tolerance εc (= 0.1), at every Kc iterations;

I Early branching: stop CG with a loose tolerance εb (= 10−3) and branch!

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 39

VRPTW with multiple deliverymen (VRPTWMD)

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 39

VRPTW with multiple deliverymen (VRPTWMD)

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 39

VRPTW with multiple deliverymen (VRPTWMD)

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 39

VRPTW with multiple deliverymen (VRPTWMD)

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 40

VRPTW with multiple deliverymen (VRPTWMD)

I Pureza, Morabito and Reimann (2012):

I Vehicle flow (compact) formulation and metaheuristics;

I Objective function:

ω1

L∑l=1

∑j∈C

xl0j + ω2

L∑l=1

∑j∈C

lxl0j + ω3

L∑l=1

∑i∈N

∑j∈N

dijxlij

(nb of vehicles) (nb of deliverymen) (distance)

I We propose a set partitioning formulation and an interior point

branch-price-and-cut method;

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 40

VRPTW with multiple deliverymen (VRPTWMD)

I Pureza, Morabito and Reimann (2012):

I Vehicle flow (compact) formulation and metaheuristics;

I Objective function:

ω1

L∑l=1

∑j∈C

xl0j + ω2

L∑l=1

∑j∈C

lxl0j + ω3

L∑l=1

∑i∈N

∑j∈N

dijxlij

(nb of vehicles) (nb of deliverymen) (distance)

I We propose a set partitioning formulation and an interior point

branch-price-and-cut method;

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 40

VRPTW with multiple deliverymen (VRPTWMD)

I Pureza, Morabito and Reimann (2012):

I Vehicle flow (compact) formulation and metaheuristics;

I Objective function:

ω1

L∑l=1

∑j∈C

xl0j + ω2

L∑l=1

∑j∈C

lxl0j + ω3

L∑l=1

∑i∈N

∑j∈N

dijxlij

(nb of vehicles) (nb of deliverymen) (distance)

I We propose a set partitioning formulation and an interior point

branch-price-and-cut method;

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 40

VRPTW with multiple deliverymen (VRPTWMD)

I Pureza, Morabito and Reimann (2012):

I Vehicle flow (compact) formulation and metaheuristics;

I Objective function:

ω1

L∑l=1

∑j∈C

xl0j + ω2

L∑l=1

∑j∈C

lxl0j + ω3

L∑l=1

∑i∈N

∑j∈N

dijxlij

(nb of vehicles) (nb of deliverymen) (distance)

I We propose a set partitioning formulation and an interior point

branch-price-and-cut method;

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 41

VRPTWMD: Set partitioning formulation

minL∑

l=1

∑p∈P l

clpλlp

s.t.L∑

l=1

∑p∈P l

alpiλlp = 1, i = 1, . . . , n,

(ui)

L∑l=1

∑p∈P l

lλlp ≤ D,

(σ)

λlp ∈ {0, 1}, l = 1, . . . , L, p ∈ P l.

I P l: set of all feasible routes in mode l, l = 1, . . . , L;

I λlp: 1, if the p-th route in mode l is chosen; 0, o.w.

35

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 42

VRPTWMD: Set partitioning formulation

I Columns: visited customers and mode (number of deliverymen)

alp =

0

1

1...

0

l

→ customer 1 is not visited

→ customer 2 is visited

→ route mode

I Cost of a route p in mode l:

clp = ω1

L∑l=1

∑j∈C

xlp0j + ω2

L∑l=1

∑j∈C

lxlp0j + ω3

L∑l=1

∑i∈N

∑j∈N

cijxlpij ,

where xlpij = 1 if and only if route p ∈ P l visits node i and goes directly

to node j.

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 43

VRPTWMD: Exact hybrid method

I Hybrid: math programming + heuristics (matheuristic);

I They have been proposed for many different types of problems, in special

for VRP variants (Archetti and Speranza, 2014);

I Combine the best of two worlds! (Metaheuristic helps BPC with UB)

I We propose combining the IPBPC (exact method) with two

metaheuristics:

I ILS: Iterated Local Search;

I LNS: Large Neighbourhood Search;

I These metaheuristics have been successfully used to find feasible solutions

of the VRPTWMD and many other variants∗.

∗Alvarez, A. and Munari, P. Metaheuristic approaches for the vehicle routing problem with time

windows and multiple deliverymen. Journal of Management & Production, v. 23, p. 279-293, 2016.

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 43

VRPTWMD: Exact hybrid method

I Hybrid: math programming + heuristics (matheuristic);

I They have been proposed for many different types of problems, in special

for VRP variants (Archetti and Speranza, 2014);

I Combine the best of two worlds! (Metaheuristic helps BPC with UB)

I We propose combining the IPBPC (exact method) with two

metaheuristics:

I ILS: Iterated Local Search;

I LNS: Large Neighbourhood Search;

I These metaheuristics have been successfully used to find feasible solutions

of the VRPTWMD and many other variants∗.

∗Alvarez, A. and Munari, P. Metaheuristic approaches for the vehicle routing problem with time

windows and multiple deliverymen. Journal of Management & Production, v. 23, p. 279-293, 2016.

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 44

VRPTW with multiple deliverymen (VRPTWMD). Exact hybrid method

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40

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 45

VRPTW with multiple deliverymen (VRPTWMD). Computational experiments

I Solomon’s instances (VRPTW): 100 customers;

I R1 (12), C1 (9), RC1 (8);

I R2 (11), C2 (8), RC2 (8); → larger capacities and time windows

I Service times (Pureza et al., 2012):

sli =min{2× di, T −max{wa

i , t0i} − ti,n+1}l

I ω1 = 1, ω2 = 0.1, ω3 = 0.0001

(vehicles, deliverymen, travel costs);

I Linux PC with Intel Core i7 3.1 GHz CPU, 16 GB of RAM;

I Time limit: 1 hour.

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 46

VRPTW with multiple deliverymen (VRPTWMD). Average results for each class

Hybrid IPBPC IPBPC/Hybrid

Class Objective Objective Ratio (%)

C1 11.0828 11.0828 0.00

R1 15.4842 15.9677 3.64

RC1 16.2522 16.0784 -1.15

C2 3.3588 3.3628 0.12

R2 3.8978 4.3083 11.52

RC2 4.6681 5.3600 15.08

Alvarez, A. and Munari, P. An exact hybrid method for the vehicle routing problem with time

windows and multiple deliverymen. Computers & Operations Research, 2017. (Accepted)

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 47

VRPTW with multiple deliverymen (VRPTWMD). Best results from the literature (from metaheuristics)

Best results IPBPC/Best Hybrid/Best

Class Objective nV nD Dist Ratio (%) Ratio (%)

C1 11.0839 10.00 10.00 838.80 -0.01 -0.01

R1 15.2567 12.60 31.70 1263.20 4.66 1.49

RC1 17.0796 13.40 35.30 1496.30 -5.86 -4.84

C2 3.3624 3.00 3.00 623.70 0.01 -0.11

R2 3.8947 3.00 7.90 1046.80 10.62 0.08

RC2 4.4951 3.40 9.70 1251.30 19.24 3.85

Alvarez, A. and Munari, P. An exact hybrid method for the vehicle routing problem with time

windows and multiple deliverymen. Computers & Operations Research, 2017. (Accepted)

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 48

VRPTW with multiple deliverymen (VRPTWMD). Hybrid method: source of best solutions

Source 600 sec 3600 sec

Best % Best %

MHinitial 34 60.71 34 60.71

MIPinitial 11 19.64 8 14.29

Integer RMP 0 0.00 0 0.00

MIP heur 4 7.14 2 3.57

MH polish 7 12.50 12 21.43

I MHinitial: Initial solution provided by metaheuristics, before starting the BPC;

I MIPinitial: MIP heuristic, using initial columns only (before starting the BPC);

I Integer RMP: Integer solution found from the linear relaxation of the RMP;

I MIP heur: MIP heuristic at the end of the node;

I MH polish: Metaheuristics after finding a new incumbent of the BPC.

43

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 49

Conclusions and future developments

I Similar to most combinatorial optimization problems, VRP requires

sophisticated solution methods to work well in practice;

I Interior point methods offer advantageous features when integrated to

these methods;

I Natural way of stabilizing column generation and improving cut

generation and branching;

I Reductions in iterations, nodes and CPU time;

I In addition, hybrid methods that combine exact and heuristic approaches

seem to be a good option in practice, to solve real-life problems;

I Wide range of applications may benefit from these tools.

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 50

VRP under uncertainty

I Uncertainty on demand, travel times, service times;

I Robust optimization and Stochastic programming;

I Very active area in the last few years!

(Oyola et al., 2016; Gendreau et al., 2016)

I How to effectively incorporate uncertainty to set partitioning formulations,

to be able to solve large-scale problems?

I Ongoing project in collaboration with Jacek Gondzio and Douglas Alem;

Solving challenging vehicle routing problems: you better follow the central pathPedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 51

Obrigado :)

Acknowledgments

http://www.dep.ufscar.br/docentes/munari

munari@dep.ufscar.br