Bottleneck combinatorial optimization problems with uncertain costs and the OWA criterion

In this paper a class of bottleneck combinatorial optimization problems with uncertain costs is discussed. The uncertainty is modeled by specifying a discrete scenario set containing a finite number of cost vectors, called scenarios. In order to choose a solution the Ordered Weighted Averaging aggregation operator (OWA for short) is applied. The OWA operator generalizes traditional criteria in decision making under uncertainty such as the maximum, minimum, average, median, or Hurwicz criterion. New complexity and approximation results in this area are provided. These results are general and remain valid for many problems, in particular for a wide class of network problems.     

Ranking robustness and its application to evacuation planning

We present a new approach to handle uncertain combinatorial optimization problems that uses solution ranking procedures to determine the degree of robustness of a solution. Unlike classic concepts for robust optimization, our approach is not purely based on absolute quantitative performance, but also includes qualitative aspects that are of major importance for the decision maker.We discuss the two variants, solution ranking and objective ranking robustness, in more detail, presenting problem complexities and solution approaches. Using an uncertain shortest path problem as a computational example, the potential of our approach is demonstrated in the context of evacuation planning due to river flooding.     

Proportion-based robust optimization and team orienteering problem with interval data

In this paper, a proportion-based robust optimization approach is developed to deal with uncertain combinatorial optimization problems. This approach assumes that a certain proportion of uncertain coefficients in each solution are allowed to change and optimizes a deterministic model so as to achieve a trade-off between optimality and feasibility when the coefficients change. We apply this approach on team orienteering problem with interval data (TOPID), a variant of vehicle routing problem, which has not yet been studied before. A branch and price algorithm is proposed to solve the robust counterpart by using two novel dominance relations. Finally, numerical study is performed. The results show the usefulness of the proposed robust optimization approach and the effectiveness of our algorithm.     

Decomposition and local search based methods for the traveling umpire problem

The Traveling Umpire Problem (TUP) is a challenging combinatorial optimization problem based on scheduling umpires for Major League Baseball. The TUP aims at assigning umpire crews to the games of a fixed tournament, minimizing the travel distance of the umpires. The present paper introduces two complementary heuristic solution approaches for the TUP. A new method called enhanced iterative deepening search with leaf node improvements (IDLI) generates schedules in several stages by subsequently considering parts of the problem. The second approach is a custom iterated local search algorithm (ILS) with a step counting hill climbing acceptance criterion. IDLI generates new best solutions for many small and medium sized benchmark instances. ILS produces significant improvements for the largest benchmark instances. In addition, the article introduces a new decomposition methodology for generating lower bounds, which improves all known lower bounds for the benchmark instances.     

Generating hard instances for robust combinatorial optimization

While research in robust optimization has attracted considerable interest over the last decades, its algorithmic development has been hindered by several factors. One of them is a missing set of benchmark instances that make algorithm performance better comparable, and makes reproducing instances unnecessary. Such a benchmark set should contain hard instances in particular, but so far, the standard approach to produce instances has been to sample values randomly from a uniform distribution.In this paper we introduce a new method to produce hard instances for min-max combinatorial optimization problems, which is based on an optimization model itself. Our approach does not make any assumptions on the problem structure and can thus be applied to any combinatorial problem. Using the Selection and Traveling Salesman problems as examples, we show that it is possible to produce instances which are up to 500 times harder to solve for a mixed-integer programming solver than the current state-of-the-art instances.     

NP完全问题多项式时间算法研究

本文从代数及组合两个方面论证了NP完全问题存在多项式时间算法 .以往利用线性规划 (LP)技术来分析NP完全问题中的TSP问题 ,因其存在子环游问题 ,从而使问题得不到有效解决 .文中发展一分层网络 ,在求解TSP问题时 ,存在另一类(不完全 )子环游问题 .但两模型允许解集的交集避免了两类子环游基本可行解 ,从而使TSP问题可利用LP技术多项式时间内得以解决 ,同时给出了求哈密尔顿回路的多项式标记证明方法 ,开创了NPC问题研究的新局面 .     

A decomposition approach for optimal control problems with integer innerpoint state constraints

A large class of optimal control problems for hybrid dynamic systems can be formulated as mixed‐integer optimal control problems (MIOCPs). A decomposition approach is suggested to solve a special subclass of MIOCPs with mixed integer inner point state constraints. It is the intrinsic combinatorial complexity of the discrete variables in addition to the high nonlinearity of the continuous optimal control problem that forms the challenges in the theoretical and numerical solution of MIOCPs. During the solution procedure the problem is decomposed at the inner time points into a multiphase problem with mixed integer boundary constraints and phase transitions at unknown switching points. Due to a discretization of the state space at the switching points the problem can be decoupled into a family of continuous optimal control problems (OCPs) and a problem similar to the asymmetric group traveling salesman problem (AGTSP). The OCPs are transcribed by direct collocation to large‐scale nonlinear programming problems, which are solved efficiently by an advanced SQP method. The results are used as weights for the edges of the graph of the corresponding TSP‐like problem, which is solved by a Branch‐and‐Cut‐and‐Price (BCP) algorithm. The proposed approach is applied to a hybrid optimal control benchmark problem for a motorized traveling salesman. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A fuzzy-stochastic OWA model for robust multi-criteria decision making

All realistic Multi-Criteria Decision Making (MCDM) problems face various kinds of uncertainty. Since the evaluations of alternatives with respect to the criteria are uncertain they will be assumed to have stochastic nature. To obtain the uncertain optimism degree of the decision maker fuzzy linguistic quantifiers will be used. Then a new approach for fuzzy-stochastic modeling of MCDM problems will be introduced by merging the stochastic and fuzzy approaches into the OWA operator. The results of the new approach, entitled FSOWA, give the expected value and the variance of the combined goodness measure for each alternative. Robust decision depends on the combined goodness measures of alternatives and also on the variations of these measures under uncertainty. In order to combine these two characteristics a composite goodness measure will be defined. The theoretical results will be illustrated in a watershed management problem. By using this measure will give more sensitive decisions to the stakeholders whose optimism degrees are different than that of the decision maker. FSOWA can be used for robust decision making on the competitive alternatives under uncertainty.     

Possibilistic bottleneck combinatorial optimization problems with ill-known weights

In this paper a general bottleneck combinatorial optimization problem with uncertain element weights modeled by fuzzy intervals is considered. A possibilistic formalization of the problem and solution concepts in this setting, which lead to compute robust solutions under fuzzy weights, are given. Some algorithms for finding a solution according to the introduced concepts and evaluating optimality of solutions and elements are provided. These algorithms are polynomial for bottleneck combinatorial optimization problems with uncertain element weights, if their deterministic counterparts are polynomially solvable.     

Approximate solutions to convex optimization problems under stochastic uncertainty

In this paper, we survey two standard philosophies for finding minimizing solutions of convex objective functions affected by uncertainty. In a first approach, the solution should minimize the expected value of the objective w.r.t. uncertainty (average approach), while in a second one it should minimize the worst-case objective (worst-case, or min-max approach). Both approaches are however numerically hard to solve exactly, for general dependence of the cost function on the uncertain data. Here, a brief account is given on two techniques based on uncertainty randomization that permit to solve efficiently some suitable probabilistic relaxation of the indicated problems, with full generality with respect to the way in which the uncertainty enters the problem data. A specific application to uncertain least-squares problems is examined. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numreical solutoin of a class of parabolic partial differential equation arising in optimal control problems with uncertainty

In this paper the optimal control of uncertain parabolic systems of partial differential equations is investigated. In order to search for controllers that are insensitive to uncertainties in these systems, an iterative optimization procedure is proposed. This procedure involves the solution of a set of operator valued parabolic partial differential equations. The existence and uniqueness of solutions to these operator equations is proved, and a stable numerical algorithm to approximate the uncertain optimal control problem is proposed. The viability of the proposed algorithm is demonstrated by applying it to the control of parabolic systems having two different types of uncertainty.     

The biobjective travelling purchaser problem

The purpose of this article is to present and solve the Biobjective Travelling Purchaser Problem, which consists in determining a route through a subset of markets in order to collect a set of products, minimizing the travel distance and the purchasing cost simultaneously. The most convenient purchase of the product in the visited markets is easily computed once the route has been determined. Therefore, this problem contains a finite set of solutions (one for each route) and the problem belongs to the field of the Biobjective Combinatorial Optimization. It is here formulated as a Biobjective Mixed Integer Linear Programming model with an exponential number of valid inequalities, and this model is used within a cutting plane algorithm to generate the set of all supported and non-supported efficient points in the objective space. A variant of the algorithm computes only supported efficient points. For each efficient point in the objective space exactly one Pareto optimal solution in the decision space is computed by solving a single-objective problem. Each of these single-objective problems, in turn, is solved by a specific branch-and-cut approach. A heuristic improvement based on saving previously generated cuts in a common cut-pool structure has also been developed with the aim of speeding up the algorithm performance. Results based on benchmark instances from literature show that the common cut-pool heuristic is very useful, and that the proposed algorithm manages to solve instances containing up to 100 markets and 200 different products. The general procedure can be extended to address other biobjective combinatorial optimization problems whenever a branch-and-cut algorithm is available to solve a single-objective linear combination of these criteria.     

A rolling horizon approach for disruption management of railway rolling stock

This paper deals with real-time disruption management of rolling stock in passenger railway transportation. We describe a generic framework for dealing with disruptions of railway rolling stock schedules. The framework is presented as an online combinatorial decision problem, where the uncertainty of a disruption is modeled by a sequence of information updates. To decompose the problem and to reduce the computation time, we propose a rolling horizon approach: rolling stock decisions are only considered if they are within a certain time horizon from the time of rescheduling. The schedules are then revised as time progresses and new information becomes available. We extend an existing model for rolling stock scheduling to the specific requirements of the real-time situation, and we apply it in the rolling horizon framework. We perform computational tests on instances constructed from real-life cases of Netherlands Railways (NS), the main operator of passenger trains in the Netherlands. We explore the consequences of different settings of the approach for the trade-off between solution quality and computation time.     

Two-stage network constrained robust unit commitment problem

For a current deregulated power system, a large amount of operating reserve is often required to maintain the reliability of the power system using traditional approaches. In this paper, we propose a two-stage robust optimization model to address the network constrained unit commitment problem under uncertainty. In our approach, uncertain problem parameters are assumed to be within a given uncertainty set. We study cases with and without transmission capacity and ramp-rate limits (The latter case was described in Zhang and Guan (2009), for which the analysis part is included in Section 3 in this paper). We also analyze solution schemes to solve each problem that include an exact solution approach and an efficient heuristic approach that provides tight lower and upper bounds for the general network constrained robust unit commitment problem. The final computational experiments on an IEEE 118-bus system verify the effectiveness of our approaches, as compared to the nominal model without considering the uncertainty.     

A new population seeding technique for permutation-coded Genetic Algorithm: Service transfer approach

Genetic Algorithm (GA) is a popular heuristic method for dealing complex problems with very large search space. Among various phases of GA, the initial phase of population seeding plays an important role in deciding the span of GA to achieve the best fit w.r.t. the time. In other words, the quality of individual solutions generated in the initial population phase plays a critical role in determining the quality of final optimal solution. The traditional GA with random population seeding technique is quite simple and of course efficient to some extent; however, the population may contain poor quality individuals which take long time to converge with optimal solution. On the other hand, the hybrid population seeding techniques which have the benefit of good quality individuals and fast convergence lacks in terms of randomness, individual diversity and ability to converge with global optimal solution. This motivates to design a population seeding technique with multifaceted features of randomness, individual diversity and good quality. In this paper, an efficient Ordered Distance Vector (ODV) based population seeding technique has been proposed for permutation-coded GA using an elitist service transfer approach. One of the famous combinatorial hard problems of Traveling Salesman Problem (TSP) is being chosen as the testbed and the experiments are performed on different sized benchmark TSP instances obtained from standard TSPLIB [54]. The experimental results advocate that the proposed technique outperforms the existing popular initialization methods in terms of convergence rate, error rate and convergence time.     

Revising the OWA operator for multi criteria decision making problems under uncertainty

Multi criteria decision making (MCDM) problems are usually under uncertainty. One of these uncertain parameters is the decision maker (DM)’s degree of optimism, which has an important effect on the results. Fuzzy linguistic quantifiers are used to obtain the assessments of this parameter from DM and then, because of its uncertainty it is assumed to have stochastic nature. A new approach, entitled FSROWA, is introduced to combine the Fuzzy and Stochastic features into a Revised OWA operator.     

On inverse traveling salesman problems

The inverse traveling salesman problem belongs to the class of ??inverse combinatorial optimization?? problems. In an inverse combinatorial optimization problem, we are given a feasible solution for an instance of a particular combinatorial optimization problem, and the task is to adjust the instance parameters as little as possible so that the given solution becomes optimal in the new instance. In this paper, we consider a variant of the inverse traveling salesman problem, denoted by ITSP _W,A , by taking into account a set W of admissible weight systems and a specific algorithm. We are given an edge-weighted complete graph (an instance of TSP), a Hamiltonian tour (a feasible solution of TSP) and a specific algorithm solving TSP. Then, ITSP _W,A , is the problem to find a new weight system in W which minimizes the difference from the original weight system so that the given tour can be selected by the algorithm as a solution. We consider the cases ${W \in \{\mathbb{R}^{+m}, \{1, 2\}^m , \Delta\}}$ where ?? denotes the set of edge weight systems satisfying the triangular inequality and m is the number of edges. As for algorithms, we consider a local search algorithm 2-opt, a greedy algorithm closest neighbor and any optimal algorithm. We devise both complexity and approximation results. We also deal with the inverse traveling salesman problem on a line for which we modify the positions of vertices instead of edge weights. We handle the cases ${W \in \{\mathbb{R}^{+n}, \mathbb{N}^n\}}$ where n is the number of vertices.     

Heuristics and exact methods for number partitioning

Number partitioning is a classical NP-hard combinatorial optimization problem, whose solution is challenging for both exact and approximative methods. This work presents a new algorithm for number partitioning, based on ideas drawn from tree search, breadth first search, and beam search. A new set of benchmark instances for this problem is also proposed. The behavior of the new method on this and other testbeds is analyzed and compared to other well known heuristics and exact algorithms.     

Solving multiobjective vehicle routing problem with stochastic demand via evolutionary computation

This paper considers the routing of vehicles with limited capacity from a central depot to a set of geographically dispersed customers where actual demand is revealed only when the vehicle arrives at the customer. The solution to this vehicle routing problem with stochastic demand (VRPSD) involves the optimization of complete routing schedules with minimum travel distance, driver remuneration, and number of vehicles, subject to a number of constraints such as time windows and vehicle capacity. To solve such a multiobjective and multi-modal combinatorial optimization problem, this paper presents a multiobjective evolutionary algorithm that incorporates two VRPSD-specific heuristics for local exploitation and a route simulation method to evaluate the fitness of solutions. A new way of assessing the quality of solutions to the VRPSD on top of comparing their expected costs is also proposed. It is shown that the algorithm is capable of finding useful tradeoff solutions for the VRPSD and the solutions are robust to the stochastic nature of the problem. The developed algorithm is further validated on a few VRPSD instances adapted from Solomon’s vehicle routing problem with time windows (VRPTW) benchmark problems.