Model enhancement: Improving theoretical optimization with simulation

doi:10.1016/j.simpat.2007.02.003

Simulation Modelling Practice and Theory

Volume 15, Issue 6, July 2007, Pages 703-715

https://doi.org/10.1016/j.simpat.2007.02.003 Get rights and content

Abstract

Many optimization techniques are based on mathematical representations. In this case, important modeling simplifications need to be made. The solution thus provided, even if proven to be theoretically one of the best, might not be so good in practice. Simulation can be used to evaluate the actual performance of the solution. We propose here a coupling between optimization and simulation that tries to improve the solution provided by a mathematical model. This approach, named “model enhancement” here, still focuses on optimizing the theoretical objective function, contrary to the common optimization–simulation coupling that focuses on improving the objective function evaluated from simulation. We propose to illustrate this approach on a routing problem, and present numerical results on the quality of the solution and the efficiency of both coupling approaches.

Introduction

The goal of this paper is to discuss a way to improve the practical quality of a solution provided by an optimization process. There are several advanced optimization techniques (mixed integer programming: branch-and-bound, branch-and-cut, etc. [7], decomposition methods: Benders, Dantzig–Wolfe, etc. [5]) that can solve efficiently problems formalized with mathematical models. Major results have been stated to prove the optimality or the quality of the solution (approximation algorithms can ensure the solution found to be close to the optimal solution according to a given precision [4]), and to ensure the efficiency of the techniques (their complexity, their speed to converge to a solution, etc.).

However, these methods have significant drawbacks when looking for their practical implementation. First, they are not robust to changes in the structure of the problem: adding a new kind of constraints might make the problem unsolvable with the previous optimization technique (e.g. linear constraints, solved with the simplex method, that become non-linear). Secondly, and more of concern in this paper, major simplifications on the modeling of the problem often have to be considered. As a result, a solution that is optimal in theory may not be so good in practice.

Therefore, we propose an optimization–simulation coupling called model enhancement, that attempts to reinforce the mathematical model in order to make the solution more adapted to practice than a straightforward optimization. This approach is mildly inspired by decomposition methods used for exact resolution of optimization problems, like Benders’ decomposition or the column generation [5].

Section 2 presents the routing problem that is used as illustration all along the article. Section 3 recalls the common optimization–simulation coupling sketch, usually called simulation optimization, and formalizes the problem that we propose to discuss in this article. Section 4 explains what straightforward optimization implies on the formulation and the solutions of the problem. Section 5 finally presents the idea of model enhancement, its goal and sketch. Section 6 proposes numerical results for each optimization approach, and compares the quality of their solution and their computational efficiency.

Section snippets

Study of a routing problem

We propose to consider a public transportation company in an urban network, such as a bus company. Basically, this company needs to design low cost bus routes while satisfying potential customers.

Let us consider a directed graph G = (V, E) where V is the set of vertexes and E the set of edges. Each vertex represents a potential bus stop or a crossroad in the real-life network. Each edge represents a road between two stops or crossroads.

We assume the customer demands are known, i.e. there is a

Simulation optimization

An optimization problem can be expressed as finding the best solution x to a real problem (P_r), i.e. minimizing or maximizing a function f_r(x). A solution x fits problem (P_r) if it satisfies a set of constraints that defines the space C_r of feasible solutions.

In our example, x can represent the route of a bus in the city, C_r the constraints on this route (its length, the streets it can use, etc.), and f_r(x) represents the satisfaction of the customers using the route x. $(P_{r}) \{\begin{matrix} optimize & f_{r} (x) \\ subject \end{matrix}$

Straightforward optimization

Optimization techniques, independent of simulation, usually need a mathematical model (P_o). This representation is also an approximation of the real problem (P_r). $(P_{o}) \{\begin{matrix} optimize & f_{o} (x), f_{o} \sim f_{r} \\ subject to & x \in C_{o}, C_{o} \sim C_{r} \end{matrix}$ However, we can reasonably assume that the simulation optimization problem (P_s) is closer to the real problem than the straightforward optimization problem (P_o): in our example, the optimization model (P_o) will assume that there are constant waiting times at the bus stops, contrary to the

Model enhancement

From the previous assumptions, if Λ_o(x) is never empty, all solutions x in C_o are feasible solutions of problem (P_o). As C_o ⊇ C_s, any solution of the simulation optimization problem (P_s) is a feasible solution of (P_o). In particular, any optimal solution $x_{s}^{*}$ of (P_s) is a feasible solution of (P_o). The idea of model enhancement is to find a family of constraints Λ_o ∈ K_m such that the optimal solution $x_{o}^{*}$ of (P_o) is one of the optimal solutions $x_{s}^{*}$ of (P_s). We can state the model enhancement problem (

Numerical results

Simulation optimization provides good practical solutions, but needs a very long time to execute (cf. Fig. 12). At the opposite, straightforward optimization is very fast, but provides poor practical solutions. The numerical results presented here show that, with our example, model enhancement is a good compromise. The conditions of the experiment are now presented, and its numerical results are discussed.

Conclusion

Simulation optimization provides good practical solutions. However, it may take a long time to execute, with lots of simulation evaluations. At the opposite, for some problems, like the routing problem presented here, theoretical approaches can efficiently solve simplified formulations. But, usually, they find solutions that are not so good in practice.

In this article, we propose an approach called model enhancement that still focuses on the theoretical problem, and tries to improve its

Acknowledgements

We would like to thank Antoine Mahul, PhD, from Blaise Pascal University, Clermont–Ferrand, France, for his invaluable advice to formalize model enhancement. We are also very grateful to Mamadou K. Traoré, Assistant Professor at Blaise Pascal University, for his help to make the article more readable, and for his support.

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