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Real world problems usually have to deal with some uncertainties. This is particularly true for the planning of services whose requests are unknown a priori.
Several approaches for solving stochastic problems are reported in the literature. Metaheuristics seem to be a powerful tool for computing good and robust solutions. However, the efficiency of...
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... main topologies for the design of a SONET network are available. The first topology consists in the assignment of each customer to exactly one ring by using one ADM and allowing connection between different rings through a unique federal ring composed by one DXC for each connected ring. The objective of this problem, say srap and depicted in Figure 1, is to minimize the number of ...
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Citations
... Another method for solving chance-constrained programs, suggested in Aringhieri (2004), combines a tabu search heuristic with simulation. The evaluation of the feasibility of a solution is realized using two different methods. ...
In this paper, we study an interpretation of the sample-based approach to chance-constrained programming problems grounded in statistical testing theory. On top of being simple and pragmatic, this approach is theoretically well founded, non parametric and leads to a general method for leveraging existing heuristic algorithms for the deterministic case to their chance-constrained counterparts. Throughout this paper, this algorithm design approach is illustrated on a real world graph partitioning problem which crops up in the field of compilation for parallel systems. Extensive computational results illustrate the practical relevance of the approach, as well as the robustness of the obtained solutions.
... Compute bounds and approximate solutions [128], [126], [137], Usually computationally demanding [80], [11], [51], [52], [110], [129], [102] (Meta)Heuristics Use of precedent techniques for computing distribution [153], [107], [12], [13], [3] ...
... Another method for solving CCPs, suggested in [3], combines a tabu search heuristic with simulation. The evaluation of the feasibility of a solution is realized using two dierent methods. ...
This PhD thesis is devoted to the study of combinatorial optimization problems related to massively parallel embedded architectures when taking into account uncertain data (e.g. execution time). Our focus is on chance constrained programs with the objective of finding the best solution which is feasible with a preset probability guarantee. A qualitative analysis of the uncertain data we have to treat (dependent random variables, multimodal, multidimensional, difficult to characterize through classical distributions) has lead us to design a non parametric method, the so-called "robust binomial approach", valid whatever the joint distribution and which is based on robust optimization and statistical hypothesis testing. We also propose a methodology for adapting approximate algorithms for solving stochastic problems by integrating the robust binomial approach when verifying for solution feasibility. The paractical relevance of our approach is validated through two problems arising in the compilation of dataflow application for manycore platforms. The first problem treats the stochastic partitioning of networks of processes on a fixed set of nodes, by taking into account the load of each node and the uncertainty affecting the weight of the processes. For finding stochastic solutions, a semi-greedy iterative algorithm has been proposed which allowed measuring the robustness and cost of the solutions with regard to those for the deterministic version of the problem. The second problem consists in studying the global placement and routing of dataflow applications on a clusterized architecture. The purpose being to place the processes on clusters such that it exists a feasible routing, a GRASP heuristic has been conceived first for the deterministic case and afterwards extended for the chance constrained variant of the problem.
... Allowing infeasible intermediate solutions during the search is sometimes beneficial in TS [22]. This was also found to be the case in Ref. 23, where a TS was designed for solving CCPs. Simulation was used to find approximate probabilities that each constraint was violated and two alternative move evaluation functions were tested: allowing infeasible solutions with a penalty gave better results than avoiding infeasible solutions altogether. ...
A large number of publications have been written about metaheuristics and their use in solving deterministic optimization problems. As metaheuristic techniques become more widespread in the production of decision support systems, it is clear that they also need to cope with problems, based on real-world applications, that contain noise and uncertainty. An increasing amount of work is being put into the study of metaheuristics for stochastic problems, and into how metaheuristics must be adapted to efficiently handle different sources of stochasticity. We give examples of how metaheuristics have been used to solve stochastic problems, thereby illustrating the additional considerations that are required in this context.
Keywords:
optimization;
uncertainty;
noise;
local search;
population search;
constructive search
... Consequently, there has been a lot of interest in formulations with discretely distributed random parameters, which are often created by sampling. Given problem SP with decision variables that are pure integer, Tayur et al.[32]solve the problem using an algebraic geometry approach and Aringhieri[3]applies a tabu search for finding feasible solutions. For problems with random T(x) and possibly continuous decision variables, a deterministic equivalent problem can be formulated as a 'big-M' mixed 0–1 integer program[20]. ...
We present a new method for solving stochastic programs with joint chance constraints with random technology matrices and discretely distributed random data. The problem can be reformulated as a large-scale mixed 0-1 integer program. We derive a new class of optimality cuts called IIS cuts and apply them to our problem. The cuts are based on irreducibly infeasible subsystems (IIS) of an LP defined by requiring that all scenarios be satisfied. We propose a method for improving the upper bound of the prob- lem when no cut can be found. We derive and implement a branch-and-cut algorithm based on IIS cuts, and refer to this algorithm as the IIS branch-and-cut algorithm. We report on computational results with several test instances from optimal vaccine allocation. The computational results are promising as the IIS branch-and-cut algorithm gives better results than a state-of-the-art commercial solver on one class of problems.
... Lutz et al. (1998), Finke et al. (2002, Dengiz and Alabas (2000) adopt a standard TS algorithm that uses an external simulator to estimate the solution cost. Aringhieri (2004) proposed an estimation-based TS for 2.2 Metaheuristics for stochastic combinatorial optimization two stochastic combinatorial optimization problems that arise in the design of telecommunication networks. Cheung et al. (2007) tackled a stochastic delivery problem using an estimation-based TS. ...
In stochastic combinatorial optimization, problem parameters are affected by uncertainty; however, probability distributions describing the uncertainty are known or can be estimated. Stochastic routing problems, a prominent class of stochastic combinatorial optimization problems, involve finding an efficient way to distribute or collect goods across a logistic network. In order to tackle these problems, I considered a typical setting in which the cost of each solution is a random variable, and the goal is to find the solution with the minimum expected cost. It has been shown that, for some problems and for known probability distributions, the expectation can be computed analytically. Unfortunately, this typically involves complex analytical developments and computationally expensive procedures. Moreover, computing the expectation through the analytical computation approach is a highly problem-specific issue and it requires a deep understanding of the underlying probabilistic model. An alternative approach is empirical estimation, which estimates the expectation through Monte Carlo simulation. The main advantage of the empirical estimation approach over the analytical computation one is generality: a sample estimate of the expected cost of a given solution can be obtained by simply averaging sample cost estimates over a number of realizations of the random variable.
... Among the second group of papers (adapting simulation to the different components of TS), there are Costa and Silver (1998) Aringhieri (2004), and Cheung et al. (2007). Costa and Silver (1998) describe a TS algorithm for a problem in the context of cause-effect analysis, where the true cause of an undesirable effect must be recognized and eliminated. ...
... An application where sampling is employed to save time with respect to using the exact objective function is the one by Aringhieri (2004), that applies TS to a Chance Constrained Program (Definition 4). Constraints are supposed to be linear functions, that is, in Definition 4 we pose H i (x,x) = P j a ij x jb i (x), with j = 1,2..., n and x 2 S & R n : Note that in this problem, only the vector b is assumed to be random. ...
Metaheuristics such as Ant Colony Optimization, Evolutionary Compu- tation, Simulated Annealing, Tabu Search and Stochastic Partitioning Methods are introduced, and their recent applications to a wide class of combinatorial optimization problems under uncertainty are reviewed. The flexibility of metaheuristics in being adapted to dierent modeling ap- proaches and problem formulations emerges clearly. This paper provides also a description and classification of the modeling approaches of opti- mization under uncertainty. Moreover, a formal description of the main formulations corresponding to more classical domains in the literature is provided. In this survey, the reader familiar to metaheuristics finds also pointers to classical algorithmic approaches to optimization under uncer- tainty, while the reader new to metaheuristics should find a good tutorial in those metaheuristics that are being applied to optimization under un- certainty.
... Lutz et al. (1998), Finke et al. (2002, Dengiz and Alabas (2000) adopt a standard TS algorithm that uses an external simulator to estimate the solution cost. Aringhieri (2004) proposed an estimation-based TS for 2.2 Metaheuristics for stochastic combinatorial optimization two stochastic combinatorial optimization problems that arise in the design of telecommunication networks. Cheung et al. (2007) tackled a stochastic delivery problem using an estimation-based TS. ...
We describe an optimization model for interdicting smuggling of illicit nuclear material. Modeled on a directed network, the interdictor's goal is to select radiation detector installation locations, subject to a budget constraint, to minimize a smuggler's evasion probability. A smuggler, characterized by a random origin-destination pair, traverses the network knowing the detector locations. An additional layer of uncertainty is captured when considering a set of budget scenarios in place of a fixed budget value. An optimal solution to this model yields a priority list of detector installation locations. After the budget is revealed, detectors are installed from highest to lowest priority until the budget is exhausted. This notion captures time-phased securing of a transportation network as resources become available. Modeled as bilevel stochastic mixed-integer programs, both the fixed and random budget models are computationally challenging. Hence, we develop a tabu search heuristic to approximately solve both problem types, and we present computational results to assess the performance of our algorithm.
The compilation process design for massively parallel multicore-embedded architectures requires solving a number of difficult optimization problems, nowadays solved mainly using deterministic approaches. However, one of the main characteristics of these systems is the presence of uncertain data, such as the execution times of the tasks. The authors consider that the embedded systems design is one of themajor domains forwhich applying optimization under uncertainty is legitimate and highly beneficial. This chapter introduces the most suitable techniques from the field of optimization under uncertainty for the design of compilation chains and for the resolution of the associated optimization problems.
