Robust Discrete Optimization Under Discrete and Interval Uncertainty: A Survey

Kasperski, Adam; Zieliński, Paweł

doi:10.1007/978-3-319-33121-8_6

Cited by 76 publications

(99 citation statements)

References 74 publications

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“…Although weight values were varied as much as possible, SQI variation was slight. These results demonstrated the statistical robustness of the index [51]. Soil quality index values in the agricultural Valley of Culiacan range from 0.54 to 0.76.…”

Section: Simulation and Sensitivity Analysis Resultsmentioning

confidence: 60%

Robust Soil Quality Index for Tropical Soils Influenced by Agricultural Activities

Rangel‐Peraza¹,

Padilla-Gasca²,

López-Corrales³

et al. 2017

JACEN

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The knowledge of the soil quality plays a vital role in the agricultural sector. Despite its importance, there is scarce scientific information concerning this regard. The objective of this research is to develop a methodology to identify and select the most appropriate indicators of Soil Quality Index (SQI) in a region with high agricultural activity. For its conformation, a descriptive statistical analysis and a Pearson correlation matrix were performed and the indicators that showed greater variation were identified using a Principal Components Analysis (PCA). A sensitivity analysis was carried out and the most sensible soil indicators of SQI were identified. This statistical procedure was also used to specify the weights of the indicators in SQI. The variables resulting from the multiparametric statistical analysis were pH, organic matter, sodium, calcium, iron, zinc, cation exchange capacity and electrical conductivity. The robustness of the SQI obtained in this study was demonstrated through simulations carried out by the numerical optimization through simplex method. The Soil Quality Index range obtained (0.54 -0.75) locates Culiacan Valley soils as moderate/high quality.

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Section: Simulation and Sensitivity Analysis Resultsmentioning

confidence: 60%

Robust Soil Quality Index for Tropical Soils Influenced by Agricultural Activities

Rangel‐Peraza¹,

Padilla-Gasca²,

López-Corrales³

et al. 2017

JACEN

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“…Assume w.l.o.g that C 1 ≤ C 2 ≤ · · · ≤ C n . Consider the following LP relaxation of (23)a: Let (x x x * , y y y * , u u u * ) be an optimal solution to (24). We first note that given y y y * , the optimal values of x x x * can be obtained in the following greedy way.…”

Section: Robust Two-stage Selection Problemmentioning

confidence: 99%

“…Single-stage robust combinatorial optimization problems, under various uncertainty sets, have been extensively discussed over the last decade. Survey of the results in this area can be found in [2,24,20,10]. For these problems a complete solution must be determined before the true scenario is revealed.…”

Section: Introductionmentioning

confidence: 99%

Two-stage combinatorial optimization problems under risk

Goerigk

Kasperski

Zieliński

2020

Theoretical Computer Science

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In this paper a class of robust two-stage combinatorial optimization problems is discussed. It is assumed that the uncertain second stage costs are specified in the form of a convex uncertainty set, in particular polyhedral or ellipsoidal ones. It is shown that the robust two-stage versions of basic network and selection problems are NP-hard, even in a very restrictive cases. Some exact and approximation algorithms for the general problem are constructed. Polynomial and approximation algorithms for the robust two-stage versions of basic problems, such as the selection and shortest path problems, are also provided.Typically, the first stage costs are known while the second stage costs can only be predicted to belong to an uncertainty set U . First such models were discussed in [16,18,26,23], where the robust two-stage spanning tree and perfect matching problems were considered. In these papers, the uncertainty set U contains K explicitly listed scenarios. Several negative and positive complexity results for this uncertainty representation were established. Some of them have been recently extended in [19], where also the robust two-stage shortest path problem has been investigated. In [25] and [13] the robust two-stage selection problem has been explored. The problem is NP-hard for discrete uncertainty representation but it is polynomially solvable under a special case of polyhedral uncertainty set, called continuous budgeted uncertainty (see [13]).Robust two-stage problems belong to the class of three-level, min-max-min optimization problems. In mathematical programming, this approach is also called adjustable robustness (see, e.g. [5,31]). Namely, some variables must be determined before the realization of the uncertain parameters, while the other part are variables that can be chosen after the realization. Several such models have been recently considered in combinatorial optimization, which can be represented as a 0-1 programming problem. Among them there is the robust two-stage problem discussed in this paper, but also the robust recoverable models [11,12] and the k-adaptability approach [9]. In general, problems of this type can be hard to solve exactly. A standard approach is to apply row and column generation techniques, which consists in solving a sequence of MIP formulations (see, e.g., [32]). However, this method can be inefficient for larger problems, especially when the underlying deterministic problem is already NP-hard. Therefore, some faster approximation algorithms can be useful in this case.In this paper we consider the class of robust two-stage combinatorial problems under convex uncertainty, i.e. when the uncertainty set U is convex. Important special cases are polyhedral and ellipsoidal uncertainty, which are widely used in single-stage robust optimization. Notice that in the problems discussed in [16,18,26,23], U contains a fixed number of scenarios, so it is not a convex set. The problem formulation and description of the uncertainty sets are provided in Section 2. The complexity status of ba...

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“…Roy [22] presents different definitions of robustness and discusses how these variants can be used in the wide area of operations research. Kasperski and Zieliński [11] review recent results on robust discrete optimization.Despite of the modeling approach or the robustness criterion, the RSP is known to be NP-hard [1,8,19,25]. Murthy and Her [18] proposed one of the first approaches to solve the RSP.…”

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confidence: 99%

An exact method for a class of robust shortest path problems with scenarios

Duque

Medaglia

2019

Networks

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In this variant of the robust shortest path problem, the cost of traversing an arc is given by a discrete set of scenarios. The problem is then to find a (robust) path that takes into account the information arising from the multiple cost realizations of the possible scenarios. To account for a robust path, we adopt the bw‐robustness criterion, which ameliorates the dramatic role played by worst‐case approaches. Under this criterion, the parameter b represents a desirable upper bound for the cost that the decision maker wants for most of the scenarios; while parameter w strictly bounds the cost and represents a value that the decision maker is not willing to exceed in any scenario. To solve the problem, we extend the pulse algorithm, a general‐purpose solution strategy that has been used on shortest path problems with side constraints. The proposed algorithm compares favorably against an integer programming approach both in terms of speed and scalability on networks with up to 39 377 nodes and 192 094 arcs.

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Robust Discrete Optimization Under Discrete and Interval Uncertainty: A Survey

Cited by 76 publications

References 74 publications

Robust Soil Quality Index for Tropical Soils Influenced by Agricultural Activities

Robust Soil Quality Index for Tropical Soils Influenced by Agricultural Activities

Two-stage combinatorial optimization problems under risk

An exact method for a class of robust shortest path problems with scenarios

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