Minmax regret combinatorial optimization problems: an Algorithmic Perspective

Candia-Vejar, A (reprint author), Univ Talca, Modeling & Ind Management Dept, Curico, Chile.Uncertainty in optimization is not a new ingredient. Diverse models considering uncertainty have been developed over the last 40 years. In our paper we essentially discuss a particular uncertainty model associated with combinatorial optimization problems, developed in the 90's and broadly studied in the past years. This approach named minmax regret (in particular our emphasis is on the robust deviation criteria) is different from the classical approach for handling uncertainty, stochastic approach, where uncertainty is modeled by assumed probability distributions over the space of all possible scenarios and the objective is to find a solution with good probabilistic performance. In the minmax regret (MMR) approach, the set of all possible scenarios is described deterministically, and the search is for a solution that performs reasonably well for all scenarios, i.e., that has the best worst-case performance. In this paper we discuss the computational complexity of some classic combinatorial optimization problems using MMR. approach, analyze the design of several algorithms for these problems, suggest the study of some specific research problems in this attractive area, and also discuss some applications using this model

New results on minimax regret single facility ordered median location problems on networks

We consider the single facility ordered median location problem with uncertainty in the parameters (weights) defining the objective function. We study two cases. In the first case the uncertain weights belong to a region with a finite number of extreme points, and in the second case they must also satisfy some order constraints and belong to some box, (convex case). To deal with the uncertainty we apply the minimax regret approach, providing strongly polynomial time algorithms to solve these problems

Robust mean absolute deviation problems on networks with linear vertex weights

This article deals with incorporating the mean absolute deviation objective function in several robust single facility location models on networks with dynamic evolution of node weights, which are modeled by means of linear functions of a parameter. Specifically, we have considered two robustness criteria applied to the mean absolute deviation problem: the MinMax criterion, and the MinMax regret criterion. For solving the corresponding optimization problems, exact algorithms have been proposed and their complexities have been also analyzed.Ministerio de Ciencia e Innovación MTM2007-67433-C02-(01,02)Ministerio de Ciencia e Innovación MTM2009-14243Ministerio de Ciencia e Innovación MTM2010-19576-C02-(01,02)Ministerio de Ciencia e Innovación DE2009-0057Junta de Andalucía P09-TEP-5022Junta de Andalucía FQM-584

A review of network location theory and models

Cataloged from PDF version of article.In this study, we review the existing literature on network location problems. The study has a broad scope that includes problems featuring desirable and undesirable facilities, point facilities and extensive facilities, monopolistic and competitive markets, and single or multiple objectives. Deterministic and stochastic models as well as robust models are covered. Demand data aggregation is also discussed. More than 500 papers in this area are reviewed and critical issues, research directions, and problem extensions are emphasized.Erdoğan, Damla SelinM.S

On the complexity of the upgrading version of the Maximal Covering Location Problem

In this article, we study the complexity of the upgrading version of the maximal covering location problem with edge length modifications on networks. This problem is NP-hard on general networks. However, in some particular cases, we prove that this problem is solvable in polynomial time. The cases of star and path networks combined with different assumptions for the model parameters are analysed. In particular, we obtain that the problem on star networks is solvable in (Formula presented.) time for uniform weights and NP-hard for non-uniform weights. On paths, the single facility problem is solvable in (Formula presented.) time, while the (Formula presented.) -facility problem is NP-hard even with uniform costs and upper bounds (maximal upgrading per edge), as well as, integer parameter values. Furthermore, a pseudo-polynomial algorithm is developed for the single facility problem on trees with integer parameters.</p

Mathematical models for the design and planning of transportation on demand in urban logistics networks

Falta palabras claveThe freight-transport industry has made enormous progress in the development and application of logistics techniques that has transformed its operation, giving raise to impressive productivity gains and improved responsiveness to its consumers. While the separation of passenger and freight traffic is a relatively new concept in historic terms, recent approaches point out that most freight-logistics techniques are transferable to the passenger-transport industry. In this sense, passenger logistics can be understood as the application of logistics techniques in urban contexts to the passenger-transport industry. The design of an urban logistic network integrates decisions about the emplacement, number and capacities of the facilities that will be located, the flows between them, demand patterns and cost structures that will validate the profitability of the process. This strategic decision settles conditions and constraints of latter tactical and operative decisions. In addition, different criteria are involved during the whole process so, in general terms, it is essential an exhaustive analysis, from the mathematical point of view, of the decision problem. The optimization models resulting from this analysis require techniques and mathematical algorithms in constant development and evolution. Such methods demand more and more a higher number of interrelated elements due to the increase of scale used in the current logistics and transportation problems. This PhD dissertation explores different topics related to Mathematical models for the design and planning of transportation on demand in urban logistics networks. The contributions are divided into six main chapters since and, in addition, Chapter 0 offers a basic background for the contents that are presented in the remaining six chapters. Chapter 1 deals with the Transit Network Timetabling and Scheduling Problem (TNTSP) in a public transit line. The TNTSP aims at determining optimal timetables for each line in a transit network by establishing departure and arrival times of each vehicle at each station. We assume that customers know departure times of line runs offered by the system. However, each user, traveling later of before their desired travel time, will give rise to an inconvenience cost, or a penalty cost if that user cannot be served according to the scheduled timetable. The provided formulation allocates each user to the best possible timetable considering capacity constraints. The problem is formulated using a p-median based approach and solved using a clustering technique. Computational results that show useful applications of this methodology are also included. Chapter 2 deals with the TNTSP in a public transit network integrating in the model the passengers' routings. The current models for planning timetables and vehicle schedules use the knowledge of passengers' routings from the results of a previous phase. However, the actual route a passenger will take strongly depends on the timetable, which is not yet known a priori. The provided formulation guarantees that each user is allocated to the best possible timetable ensuring capacity constraints. Chapter 3 deals with the rescheduling problem in a transit line that has suffered a eet size reduction. We present different modelling possibilities depending on the assumptions that need to be included in the modelization and we show that the problem can be solved rapidly by using a constrained maxcost- ow problem whose coe_cient matrix we prove is totally unimodular. We test our results in a testbed of random instances outperforming previous results in the literature. An experimental study, based on a line segment of the Madrid Regional Railway network, shows that the proposed approach provides optimal reassignment decisions within computation times compatible with real-time use. In Chapter 4 we discuss the multi-criteria p-facility median location problem on networks with positive and negative weights. We assume that the demand is located at the nodes and can be different for each criterion under consideration. The goal is to obtain the set of Pareto-optimal locations in the graph and the corresponding set of non-dominated objective values. To that end, we first characterize the linearity domains of the distance functions on the graph and compute the image of each linearity domain in the objective space. The lower envelope of a transformation of all these images then gives us the set of all non-dominated points in the objective space and its preimage corresponds to the set of all Pareto-optimal solutions on the graph. For the bicriteria 2-facility case we present a low order polynomial time algorithm. Also for the general case we propose an efficient algorithm, which is polynomial if the number of facilities and criteria is fixed. In Chapter 5, Ordered Weighted Average optimization problems are studied from a modeling point of view. Alternative integer programming formulations for such problems are presented and their respective domains studied and compared. In addition, their associated polyhedra are studied and some families of facets and new families of valid inequalities presented. The proposed formulations are particularized for two well-known combinatorial optimization problems, namely, shortest path and minimum cost perfect matching, and the results of computational experiments presented and analyzed. These results indicate that the new formulations reinforced with appropriate constraints can be effective for efficiently solving medium to large size instances. In Chapter 6, the multiobjective Minimum cost Spanning Tree Problem (MST) is studied from a modeling point of view. In particular, we use the ordered median objective function as an averaging operator to aggregate the vector of objective values of feasible solutions. This leads to the Ordered Weighted Average Spanning Tree Problem (OWASTP), which we study in this work. To solve the problem, we propose different integer programming formulations based in the most relevant MST formulations and in a new one. We analyze several enhancements for these formulations and we test their performance over a testbed of random instances. Finally we show that an appropriate choice will allow us to solve larger instances with more objectives than those previously solved in the literature.Premio Extraordinario de Doctorado U

Ordered Weighted Average optimization in Multiobjective Spanning Tree Problem

Rework adversely impacts the performance of building projects. In this study, data were analyzed from 788 construction incidents in 40 Spanish building projects to determine the effects of project and managerial characteristics on rework costs. Finally, regression analysis was used to understand the relationships among contributing factors and to develop a model for rework prediction. Interestingly, the rework prediction model showed that only the original contract value (OCV) and the project location in relation to the company’s headquarters contributed to the regression model. Project type, type of organization, type of contract, and original contract duration (OCD), which represents the magnitude and complexity of a project, were represented by the OCV. This model for rework prediction based on original project conditions enables strategies to be put in place prior to the start of construction, to minimize uncertainties, to reduce impacts on project cost and schedule, and, thus, to improve productivity.Peer ReviewedPostprint (author's final draft