effective solution approaches for solving stochastic and integer problems michel gendreau cirrelt...
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Effective solution approaches for solving stochastic and integer problems
Michel GendreauCIRRELT and MAGIÉcole Polytechnique de Montréal
SESO 2015 International Thematic WeekENSTA and ENPC Paris, June 22-26, 2015
Effective solution approaches for stochastic and integer problems
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Outline
1. Introduction
2. An exact method for stochastic, fixed cost, capacitated multi-commodity network design problems
3. A heuristic approach for stochastic, fixed cost, capacitated multi-commodity network design problems
4. Problems with integer 2nd stage
5. Conclusion and perspectives
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Acknowledgements
Walter ReiCIRRELT and ESG UQÀM
Section 3 is also coauthored with Teodor G. Crainic, Xiaorui Fu, and Stein W. Wallace
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Introduction
Effective solution approaches for stochastic and integer
problems 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!
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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!
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problems 7
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).
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An exact method for stochastic, fixed cost, capacitated multi-commodity network design problems
Effective solution approaches for stochastic and integer problems
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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)
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Decision variables
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The model
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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.
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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
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Stochastic demand
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Formulation
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Formulation (cont’d)
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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)
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Final model
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Important observations
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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...
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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.
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The L-Shape algorithm1. Initialize counters for the number of feasibility and
optimality cuts, and the number of iterations.2. Solve the relaxed master problem
Minimize (First-stage objective) + θ s.t. First-stage constraints, if any.
Feasibility cutsOptimality cuts (which involve θ)
3. 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
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The L-Shape algorithm
4. Recover the simplex multipliers corresponding to the optimal solutions of all the current scenario subproblems solved in Step 3.
5. 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.
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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.
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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.
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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.
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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!
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A heuristic approach for the fixed cost, capacitated multi-commodity network design problem
Effective solution approaches for stochastic and integer problems
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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.
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Original model
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Slight model modification
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Scenario decomposition
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Scenario decomposition
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Scenario decomposition
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Scenario decomposition
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Scenario decomposition
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Scenario decomposition
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General solution approach
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General solution approach
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General solution approach
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General solution approach
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Computational results
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Computational results
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Computational results
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Computational results
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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.
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Problems with integer 2nd stage
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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).
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A stochastic VRP formulation
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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.
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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.
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Conclusions and perspectives
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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.
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