1. Effective solution approaches for solving stochastic and integer problems
Michel Gendreau
CIRRELT and MAGI
École Polytechnique de Montréal
SESO 2015 International Thematic Week
ENSTA and ENPC  Paris, June 22-26, 2015

2. Effective solution approaches for stochastic and integer problems
Outline
Introduction
An exact method for stochastic, fixed cost, capacitated multi-commodity network design problems
A heuristic approach for stochastic, fixed cost, capacitated multi-commodity network design problems
Problems with integer 2nd stage
Conclusion and perspectives
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

3. Effective solution approaches for stochastic and integer problems
Acknowledgements
Walter Rei
CIRRELT and ESG UQÀM
Section 3 is also coauthored with Teodor G. Crainic, Xiaorui Fu, and Stein W. Wallace
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

4. Introduction

5. Stochastic and integer problems
Integer programming problems (pure or mixed) are among the most difficult optimization problems.
Stochastic optimization problems are also extremely difficult to deal with.
Problems that display both combinatorial and stochastic elements are thus among the most difficult!
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

6. Some important problems
In many different fields: network design problems
In logistics: stochastic vehicle routing and supply chain problems
In energy management: stochastic unit commitment problems
And many others!
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

7. Effective solution approaches for stochastic and integer problems
Modelling does matter
The intrinsic nature of a problem is important, but it is not the only thing!
Deciding which modelling framework is used to state the problem is critical.
For instance, if a problem can be formulated as a stochastic dynamic program with reasonable state and action spaces, it can be dealt with relatively easily.
In this talk, we focus on problems which can be attacked within the framework of stochastic programming with recourse (normally, two-stage).
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

8. An exact method for stochastic, fixed cost, capacitated multi-commodity network design problems

9. Effective solution approaches for stochastic and integer problems
The problem
Network design problems are encountered in a wide variety of settings.
Notation:
Set of nodes N
Set of arcs A = {(i, j), i ɛ N, j ɛ N} with
Fixed cost fij
Variable (unit) cost cij
Capacity uij
Set of commodities (O-D pairs) K : origin o(k), destination s(k), and volume of demand w(k)
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

10. Effective solution approaches for stochastic and integer problems
Decision variables
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Effective solution approaches for stochastic and integer problems

11. Effective solution approaches for stochastic and integer problems
The model
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Effective solution approaches for stochastic and integer problems

12. Stochastic variants of the CMND
Possible uncertain parameters:
Demand:
Volume of demand w(k)
Origin o(k) or destination s(k)
Costs
Arc failures
We should also specify objective and constraints.
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

13. Effective solution approaches for stochastic and integer problems
The main variant
We focus on the variant with stochastic demands:
Arcs must be selected before knowing demands
Commodity flows can be routed in an optimal fashion after demands have been observed.
We could allow some demands to be only partially satisfied at some cost (which can be quite high).
The objective is to minimize the expected total cost of the design:
Fixed costs + variable costs + penalties
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

14. Effective solution approaches for stochastic and integer problems
Stochastic demand
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Effective solution approaches for stochastic and integer problems

15. Effective solution approaches for stochastic and integer problems
Formulation
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Effective solution approaches for stochastic and integer problems

16. Effective solution approaches for stochastic and integer problems
Formulation (cont’d)
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

17. Effective solution approaches for stochastic and integer problems
Formulation (cont’d)
If we use continuous distributions for commodity demands, the problem will be very difficult to tackle.
One way of dealing with this stochastic complexity is either to:
Use directly a scenario-based description of uncertainty, i.e., define a finite set S of representative scenarios (this could be used, among other things, to account for correlations between demands).
Create a set of scenarios through Monte Carlo sampling (possibly coupled with techniques for enlarging the scenario set, such as sample average approximation)
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

18. Effective solution approaches for stochastic and integer problems
Final model
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Effective solution approaches for stochastic and integer problems

19. Important observations
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Effective solution approaches for stochastic and integer problems

20. Possible solution approaches
Brute force: if the model is not too large (small enough network and not too many scenarios), one could just dump the complete model in a commercial MIP solver (not very elegant, but easy to implement, if it works…).
Alternately, use the L-shape algorithm of Van Slyke and Wets (1969).
Or resort to heuristics...
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

21. Effective solution approaches for stochastic and integer problems
The L-Shape algorithm
This algorithm can be seen as a direct adaptation of Benders’ decomposition to the problem at hand:
The master problem corresponds to the first stage of the model with the yij design variables and cuts.
Subproblems are defined for each scenario and correspond to the second stage CMCMCFP for a given y vector and a given scenario s.
They generate feasibility and optimality cuts for the master problem.
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

22. Effective solution approaches for stochastic and integer problems
The L-Shape algorithm
Initialize counters for the number of feasibility and optimality cuts, and the number of iterations.
Solve the relaxed master problem
Minimize (First-stage objective) + θ
s.t. First-stage constraints, if any.
Feasibility cuts
Optimality cuts (which involve θ)
Given the current solution of the master problem (yv, θv), solve the |S| scenario subproblems As soon as an infeasible subproblem is encountered, create a feasibility cut, add it to the master problem and return to Step 2. Otherwise, go to Step 4
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

23. Effective solution approaches for stochastic and integer problems
The L-Shape algorithm
Recover the simplex multipliers corresponding to the optimal solutions of all the current scenario subproblems solved in Step 3.
Construct an optimality cut using these multipliers If the RHS of this cut, wv ≥ θv, the current solution is optimal. STOP Otherwise, add the optimality cut to the master problem and return to Step 2.
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

24. Notes about the L-Shape algorithm
In the L-Shape algorithm, the optimality cuts are used to construct iteratively a piecewise linear outer approximation of the recourse function (the expected cost associated with the flows given the current yv vector and expressed in terms of this vector).
When the subproblems are flow problems, feasibility cuts are capacity cuts, which force the addition of arcs with enough capacity to carry demand for each scenario.
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

25. Notes about the L-Shape algorithm
As in other applications of Benders’ decomposition, a naïve implementation will not perform well. One should first reinforce the master problem with constraints forcing the installation of some minimum capacity in the network (for instance by looking at the composite scenario with minimum demand for each commodity).
Standard acceleration procedures for Benders decomposition, such as the McDaniel-Devine (1977) procedure can definitely help.
More recent approaches, such as the one involving the addition of Local Branching cuts proposed by Rei et al. (2009), can significantly speed up convergence.
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

26. Some computational results
A fairly sophisticated implementation (Rei et al., 2009) was applied to a generalization of our stochastic network design problem, the Stochastic Integrated Logistics Network Design Problem, which also involves binary variables for location, product assignment, and sourcing decisions.
A benchmark of 72 instances with 20, 30, 40 scenarios was solved.
Out of these 72 instances, 22 easy instances were solved to optimality (< 1% gap) in 23 s on average, while 20 harder ones took on average of 163 s on a 2.4 GHz AMD Opteron 64-bit processor.
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

27. Some recent developments
Recent work by Crainic et al. (2014) shows that one can improve significantly the performance of this type of method by “keeping” some of the scenario subproblems in the master problem.
Obviously, choosing “well” the scenarios helps!
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

28. A heuristic approach for the fixed cost, capacitated multi-commodity network design problem

29. Effective solution approaches for stochastic and integer problems
A basic fact
While exact methods have made much progress in recent years, it is still difficult, if not impossible, to solve fairly large instances of hard combinatorial problems.
The situation gets even worse when dealing with stochastic variants of these problems.
In such cases, one must resort to heuristics.
Here, we want to tackle the same problem as before through an effective heuristic approach.
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

30. Effective solution approaches for stochastic and integer problems
Original model
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Effective solution approaches for stochastic and integer problems

31. Slight model modification
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Effective solution approaches for stochastic and integer problems

32. Scenario decomposition
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Effective solution approaches for stochastic and integer problems

33. Scenario decomposition
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

34. Scenario decomposition
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

35. Scenario decomposition
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

36. Scenario decomposition
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

37. Scenario decomposition
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

38. General solution approach
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Effective solution approaches for stochastic and integer problems

39. General solution approach
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Effective solution approaches for stochastic and integer problems

40. General solution approach
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

41. General solution approach
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

42. Computational results
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Effective solution approaches for stochastic and integer problems

43. Computational results
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

44. Computational results
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Effective solution approaches for stochastic and integer problems

45. Computational results
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

46. What about multi-stage problems ?!?
In theory, one could envision extending the method presented here to multi-stage problems, since (standard) Progressive Hedging is an effective method for (continuous) multi-stage stochastic programs.
However, it still remains to be done.
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

47. Problems with integer 2nd stage

48. What about integer 2nd stage ?!?
There are interesting problems that exhibit an integer 2nd stage (e.g., stochastic vehicle routing problems).
Because of the integer 2nd stage, the whole logic underlying Benders decomposition/L-shape method breaks down.
A possible answer is the so-called “Integer L-shape method” of Laporte and Louveaux, if one looks for an exact method.
There are other options, but they are usually restricted to fairly small instances (see the work of Rüdiger Schultz among others).
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

49. A stochastic VRP formulation
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

50. The Integer L-shaped method
The key element of the Integer L-shape method is the replacement of the feasibility and optimality cuts of the original L-shape algorithm by suitable equivalents.
In the Integer L-shape method, optimality cuts simply express the fact that we know the value of the recourse for the integer solutions that have been encountered. They are thus very weak.
Local branching cuts can also be added to extend the range of optimality cuts.
Usually, one needs to add lower bounding functionals derived from partial solutions to obtain good results on larger instances.
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

51. Effective solution approaches for stochastic and integer problems
Other options
Metaheuristics, especially those based on local search principles (such as tabu search) can be applied quite effectively to many problems.
However, the challenge is to find a way to compute effectively the recourse function.
Other exact approaches, such as column generation and branch-and-price can also be applied effectively, but this requires a lot (a lot!) of specialized, problem-dependent developments.
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems

52. Conclusions and perspectives

53. Conclusion and perspectives
Stochastic and integer problems represent an exciting area that is just emerging.
Up to now, not much has been done on many problems that look very interesting.
Decomposition techniques seem to be the key for effective solution approaches, except for rather small instances.
On the modelling side, correlation between uncertain parameters is possibly a major stumbling block in many application areas, but no one seems to work on ways to deal with it.
SESO June 22-26, 2015
Effective solution approaches for stochastic and integer problems