Robust network optimization under polyhedral demand uncertainty is NP-hard

AbstractMinimum cost network design/dimensioning problems where feasibility has to be ensured w.r.t. a given (possibly infinite) set of scenarios of requirements form an important subclass of robust LP problems with right-hand side uncertainty. Such problems arise in many practical contexts such as Telecommunications, logistic networks, power distribution networks, etc. Though some evidence of the computational difficulty of such problems can be found in the literature, no formal NP-hardness proof was available up to now. In the present paper, this pending complexity issue is settled for all robust network optimization problems featuring polyhedral demand uncertainty, both for the single-commodity and multicommodity case, even if the corresponding deterministic versions are polynomially solvable as regular (continuous) linear programs. A new family of polynomially solvable instances is also discussed

Optimization with interval data: new problems, algorithms, and applications.

The parameters of real-world optimization problems are often uncertain due to the failure of exact estimation of data entries. Throughout the years, several approaches have been developed to cope with uncertainty in the input parameters of optimization problems, such as robust optimization, stochastic optimization, fuzzy programming, parametric programming, and interval optimization. Each of these approaches tackles the uncertainty in the input data with different assumptions on the source of uncertainty and imposes different requirements on the returned solutions. In this dissertation, the approach we take is that of interval optimization, and more specifically, interval linear programming. The two main problems we consider in this context are, considering all realizations of the interval data, the problems of finding the range of the optimal values and determining the set of all possible optimal solutions. While the theoretical aspects of these problems are well-studied, the algorithmic aspects and the engineering implications have not been explored. In this dissertation, we partially fill these voids. In the first part of the dissertation, we present and test three heuristics to find bounds on the worst optimal value of the equality-constrained interval linear program, which is known to be an intractable problem. In the second part of the dissertation, motivated by a real-case problem in the healthcare context, we define and analyze a new problem, the outcome range problem, in interval linear programming. The solution to the problem would help decision-makers quantify unintended/further consequences of optimal decisions made under uncertainty. Basically, the problem finds the range of an extra function of interest (different from the objective function) over all possible optimal solutions of an interval linear program. We analyze the problem from the theoretical and algorithmic perspectives. We evaluate the performance of our algorithms on simulated problem instances and on a real-world healthcare application. In the third part of the dissertation, we extend our analysis of the outcome range problem, exploring different theoretical properties and designing several new solution algorithms. We also test our solution approaches on different datasets, highlighting the strengths and weaknesses of each approach. Finally, in the last part of the dissertation, we discuss an application of interval optimization in the sensor location problem in the traffic management context. Particularly, we propose an optimization approach to handle the measurement errors in the full link flow observability problem. We show the applicability of our approach on several real-world traffic networks

OSPF routing with optimal oblivious performance ratio under polyhedral demand uncertainty

We study the best OSPF style routing problem in telecommunication networks, where weight management is employed to get a routing configuration with the minimum oblivious ratio. We consider polyhedral demand uncertainty: the set of traffic matrices is a polyhedron defined by a set of linear constraints, and a routing is sought with a fair performance for any feasible traffic matrix in the polyhedron. The problem accurately reflects real world networks, where demands can only be estimated, and models one of the main traffic forwarding technologies, Open Shortest Path First (OSPF) routing with equal load sharing. This is an NP-hard problem as it generalizes the problem with a fixed demand matrix, which is also NP-hard. We prove that the optimal oblivious routing under polyhedral traffic uncertainty on a non-OSPF network can be obtained in polynomial time through Linear Programming. Then we consider the OSPF routing with equal load sharing under polyhedral traffic uncertainty, and present a compact mixed-integer linear programming formulation with flow variables. We propose an alternative formulation and a branch-and-price algorithm. Finally, we report and discuss test results for several network instances. © 2009 Springer Science+Business Media, LLC

The Chinese Postman Problem with Load-Dependent Costs

[EN] We introduce an interesting variant of the well-known Chinese postman problem (CPP). While in the CPP the cost of traversing an edge is a constant (equal to its length), in the variant we present here the cost of traversing an edge depends on its length and on the weight of the vehicle at the moment it is traversed. This problem is inspired by the perspective of minimizing pollution in transportation, since the amount of pollution emitted by a vehicle not only depends on the travel distance but also on its load, among other factors. We define the problem, study its computational complexity, provide two mathematical programming formulations, and propose two metaheuristics for its solution. Extensive computational experiments reveal the extraordinary difficulty of this problem.The work by Angel Corberan, Isaac Plana, and Jose M. Sanchis was supported by the Spanish Ministerio de Economia y Competitividad and Fondo Europeo de Desarrollo Regional (FEDER) through [project MTM2015-68097-P] (MINECO/FEDER) and by the Generalitat Valenciana [project GVPROMETEO2013-049]. Gilbert Laporte was supported by the Canadian Natural Sciences and Engineering Research Council under [Grant 2015-06189].Corberán, Á.; Erdogan, G.; Laporte, G.; Plana, I.; Sanchís Llopis, JM. (2018). The Chinese Postman Problem with Load-Dependent Costs. Transportation Science. 52(2):370-385. https://doi.org/10.1287/trsc.2017.0774S37038552

A mathematical programming approach to stochastic and dynamic optimization problems

Includes bibliographical references (p. 46-50).Supported by a Presidential Young Investigator Award. DDM-9158118 Supported by matching funds from Draper Laboratory.Dimitris Bertsimas

A mathematical programming approach to stochastic and dynamic optimization problems

Includes bibliographical references (p. 46-50).Supported by a Presidential Young Investigator Award. DDM-9158118 Supported by matching funds from Draper Laboratory.Dimitris Bertsimas

Minimum cost b-matching problems with neighborhoods

In this paper, we deal with minimum cost b-matching problems on graphs where the nodes are assumed to belong to non-necessarily convex regions called neighborhoods, and the costs are given by the distances between points of the neighborhoods. The goal in the proposed problems is twofold: (i) finding a b-matching in the graph and (ii) determining a point in each neighborhood to be the connection point among the edges defining the b-matching. Different variants of the minimum cost b-matching problem are considered depending on the criteria to match neighborhoods: perfect, maximum cardinality, maximal and the a-b-matching problems. The theoretical complexity of solving each one of these problems is analyzed. Different mixed integer non-linear programming formulations are proposed for each one of the considered problems and then reformulated as Second Order Cone formulations. An extensive computational experience shows the efficiency of the proposed formulations to solve the problems under study

Safety Aware Vehicle Routing Algorithm, A Weighted Sum Approach

Driving is an essential part of work life for many people. Although driving can be enjoyable and pleasant, it can also be stressful and dangerous. Many people around the world are killed or seriously injured while driving. According to the World Health Organization (WHO), about 1.25 million people die each year as a result of road traffic crashes. Road traffic injuries are also the leading cause of death among young people. To prevent traffic injuries, governments must address road safety issues, an endeavor that requires involvement from multiple sectors (transport, police, health, education). Effective intervention should include designing safer infrastructure and incorporating road safety features into land-use and transport planning. The aim of this research is to design an algorithm to help drivers find the safest path between two locations. Such an algorithm can be used to find the safest path for a school bus travelling between bus stops, a heavy truck carrying inflammable materials, poison gas, or explosive cargo, or any driver who wants to avoid roads with higher numbers of accidents. In these applications, a path is safe if the danger factor on either side of the path is no more than a given upper bound. Since travel time is another important consideration for all drivers, the suggested algorithm utilizes traffic data to consider travel time when searching for the safest route. The key achievements of the work presented in this thesis are summarized as follows. Defining the Safest and Quickest Path Problem (SQPP), in which the goal is to find a short and low-risk path between two locations in a road network at a given point of time. Current methods for representing road networks, travel times and safety level were investigated. Two approaches to defining road safety level were identified, and some methods in each approach were presented. An intensive review of traffic routing algorithms was conducted to identify the most well-known algorithms. An empirical study was also conducted to evaluate the performance of some routing algorithms, using metrics such as scalability and computation time. This research approaches the SQPP problem as a bi-objective Shortest Path Problem (SPP), for which the proposed Safety Aware Algorithm (SAA) aims to output one quickest and safest route. The experiments using this algorithm demonstrate its efficacy and practical applicability