An Algorithm for Approximating High-Order Pareto Frontier in Multiobjective Integer Linear Problems

An algorithm for approximating the Pareto frontier (PF) in multiobjective integer linear problems is presented. As known, the feasible set in the objective space (FSOS) for an integer linear programming problem is not convex and only a few points of its Pareto frontier (so called supported points) can be found when constructing its convex hull. To get these points at the fisrt iteration of our algorithm we construct the convex Edgeworth-Pareto hull (CEPH) of the FSOC, i.e. a broader convex set that has the same Pareto frontier. The algorithm for constructing CEPH is more simple then one for constructing convex hull. It gives the result in the form of both a set of vertices and a system of linear inequalities. Using this information we can specify an unexplored region in the objective space which contains all not found yet Pareto frontier’s points. This region is restricted by the CEPH’s Pareto frontier and the Pareto frontier of the union of cones dominated by corresponding CEPH’s vertices. It can be easily vizualized on computer display. At each iteration, one unexplored subregion is selected to construct its partial CEPH. To formulate this subproblem, vertices and hyperplanes of CEPHs calculated at the previous iterations are used. As a result, new PF’s points for the initial problem can be found and the unexplored area is reduced. The proposed algorithm can be apllied for approximating Pareto frontier in the criteria space till 4-5 dimensions. It can be useful for Decision Support Systems in such areas as forest planning and management, water resources management and et

Um algoritmo para aproximação da fronteira de Pareto em problemas de programação inteira multiobjectivo

Neste artigo descreve-se um novo algoritmo iterativo que aproxima a fronteira de Pareto para os problemas MOILP (programação multiobjectivo linear com variáveis inteiras). O problema original decompõe-se numa sequência de sub-problemas. É dado um exemplo ilustrativo para o caso de dois critérios

CLUSTERING OF TERRITORIAL AREAS: A MULTI-CRITERIA DISTRICTING PROBLEM

Endogenous resources, economic profile and socio-economic issues are the criteria that define the development level and the identity features of a territorial unit. The territorial units that organize the country, in political and administrative terms – parishes and counties –, have a hierarquical structure, which initially reflected the organization of productive activities as well as the tradition State organization. The success of development policies addressed to territorial agglomerates depends on its homogeneity and of their territorial units. Facing to this the clustering of territorial areas can be stated as a districting multi-criteria problem. Thus, this paper aims to propose a framework for obtaining homogenous territorial clusters based on a Pareto frontier that includes multicriteria related to the territorial endogenous resources, economic profile and sociocultural features. This framework is developed in two phases. First, the criteria correlated with the development at the territory unit level are determined through statistical and econometric methods. Then, a multi-criteria approach is developed to allocate each territory unit to an agglomerate of territory according to the Pareto frontier established. The framework is applied to the context of a set of parishes and counties of the Alentejo Central region, southern Portugal. Results are presented and discussed in the scope of a regional strategy of development

Multi-objective Finite-Domain Constraint-Based Forest Management

This paper describes an implementation of a Constraint Programming approach to the problem of multi-criteria forest management optimization. The goal is to decide when to harvest each forest unit while striving to optimize several criteria under spatial restrictions. With a large number of management units, the optimization problem becomes computationally intractable. We propose an approach for deriving a set of efficient solutions for the entire region. The proposed methodology was tested for Vale do Sousa region in the North of Portugal

An Ecological-Economic Approach to Assess Impacts of the Expansion of Eucalyptus Plantations in Agroforest Landscapes of Northern Ethiopia

The conversion of fertile croplands to Eucalyptus woodlots in Ethiopian highlands, due to its business attractiveness to smallholders, raises concerns related to food production, water resources, carbon and other ecosystem services. This study was therefore designed to examine land allocation and plantation management decisions. Our emphasis was on the analysis of tradeoffs between the economic gains obtained from harvesting Eucalyptus timber and food production, carbon and water use. For that purpose, we considered a 1987 ha agroforest landscape in the Amhara region, Northern Ethiopia. With a planning horizon covering nine one-year periods, we developed and used nine Model I single objective linear programming (LP) models, and analyzed tradeoffs between objectives (e.g., land expectation value (LEV), Carbon, volume of ending inventory (VolEI), crop production and water use) using an LP-based Pareto frontier approach. The study revealed that the objective of maximizing the total economic gain from the sale of Eucalyptus wood poles favored a complete conversion of the available cropland into Eucalyptus woodlots. To meet the minimum annual crop production/consumption/requirements of households in the study area, the land under Eucalyptus should be limited to 1772 ha, with a sequestration potential of 1.5 to 1.57 × 107 kg yr−1 of carbon in the aboveground biomass. However, this land cover limit should be decreased to 921 ha so as to limit the total annual water use (for biomass production) below the amount available from rainfall (11,000 m3 ha−1 yr−1). Moreover, the study highlighted that maximizing the harvested wood volume or LEV would come at the cost of a decreased aboveground carbon stock and volume of ending inventory and higher total water use. It also provided alternative optimal Pareto-front points, among which decision makers will be able to select their preferred targets. The current study also showed the potential for the application of Pareto frontier approaches to support the development of effective ecological/economic management strategies and the design of land use policies in an Ethiopian context

A Web-Based Approach for Visualizing Interactive Decision Maps

This research expands the applicability of the Feasible Goals (FGoal) Pareto frontier multiple criteria method to display the Edgeworth–Pareto hull using interactive decision maps (IDMs). Emphasis is placed upon the development of a communication architecture to display the Pareto frontiers, which includes a client device, a web server, and a dedicated computation server implemented with sockets. A standalone application on the latter processes client-server requests and responses to display updated information on the client. Specifically, the dedicated computation server is responsible for calculating the information needed to generate the Edgeworth–Pareto hull. This is delivered to the web server to generate the IDM to be displayed on the client device. The key innovation of this work is a tool that is developed to aid decision-makers with a network-based computational architecture that includes a computational server constantly in communication with a web server for fast responses to client requests to represent IDMs. Results show that this innovation avoids time-consuming communication, and this approach to represent IDMs on the web facilitates collaboration among decision-makers because they can analyze several complex problems in different browser windows and decide which problem and solution better correspond to their aims

On the determination of space-time slip distribution with a regularized constraints inversion algorithm

The description of the seismic source from a set of their effects recorded at the surface constitutes a scientific problem, with solution reached through the methodological proposal of the Discrete Inverse Theory. In the case of finite sources it consists to determinate the evolution of vector slip field over the fault area: a 3D solution (time and two slip components to the points on the fault). The resolution of this problem, based on the discretization of the representation theorem, [1] [2], has been tried since to about 40 years. Currently there are several non-linear inversion formulations that produces numerically reasonable solutions and physically consistent. In this work we suggest an inversion method based on dual simplex algorithm, for reconstruct the kinematics rupture image of large earthquakes through space-time seismic slip distribution on finite faults planes. In its general setting, the method produces results from a near field strong ground motion waveforms, but can also be used with teleseismic waveforms as well as with geodesic data (static case). The solution of an auxiliary linear programming problem is an essential part of the developed method. To test this algorithm and examine its stability, non-uniqueness and robustness we applied it to a set of synthetic data obtained from the solution of a forward problem considering a particular slip distribution and near field Green functions, calculated by a finite differences method with a 3D structure model. The likeness between the inversion and the known solutions supports the credibility to use this method with data of real earthquakes

Clustering of Territorial Areas: A Multi-Criteria Districting Problem

Endogenous resources, economic profile and socio-economic issues are the criteria that define the level of development and identifying features of a territorial unit. The territorial units that organize the country, in political and administrative terms ? parishes and counties ?, have a hierarchical structure, which initially reflected the organization of productive activities as well as the traditional State organization. The success of development policies addressed to territorial agglomerates depends on the homogeneity of their territorial units. In this context, the clustering of territorial areas can be stated as a multi-criteria districting problem. Thus, this paper aims to propose a framework for obtaining homogenous territorial clusters based on a Pareto frontier that includes multiple criteria related to territories' endogenous resources, economic profile and socio-cultural features. This framework is developed in two phases. First, the criteria correlated with development at the territorial unit level are determined through statistical and econometric methods. Then, a multi-criteria approach is developed to allocate each territorial unit to a territorial agglomerate, according to the Pareto frontier established. The framework is applied to a set of parishes and counties of the Central Alentejo region in southern Portugal. Results are presented and discussed in the scope of a regional development strategy. The results of multiple linear regression analysis show us the most important variables in explaining the differences in development in the area considered. We conclude, as expected, that the more elderly the population or the higher the school drop-out rate, the lower the area's development. On the other hand, the greater the active population or the rate of employment in tertiary social activities, the greater is the development. In the 2nd part of the analysis, we started from the current situation in terms of administrative organization of parishes. The results of the Max-p-model show that tests to increase the homogeneity between parishes, using the variables of population size and area, it is possible to reduce the disparity between parishes, reducing the number of units. The simulations show that the number of parishes may be lower if the variable of analysis is population size. This result takes into account the wide disparity of the population in current parishes, as well as the small number of inhabitants in most places

Addressing Wildfire Risk in Forest Management Planning with Multiple Criteria Decision Making Methods

Wildfires impact the outcomes of forest management plans. Addressing that impact is thus critical for effective forest ecosystem management planning. This paper presents research on the use of multiple criteria decision making (MCDM) methods that integrate wildfire risk in planning contexts characterized by multiple objectives. Specifically, an a posteriori preference modeling approach is developed that adds wildfire criteria to a set of objectives representing ecosystem services supply values. Wildfire risk criteria are derived from stand-level wildfire occurrence and damage models as well as from the characteristics of neighboring stands that may impact wildfire probability and spread. A forested landscape classified into 1976 stands is used for testing purposes. The management planning criteria include the carbon stock, harvest volumes for three forest species, the volume of the ending inventory, and resistance to wildfire risk indicators. Results show the potential of multiple criteria decision making methods to provide information about trade-offs between wildfire risk and the supply of provisioning (timber) as well as regulatory (carbon) ecosystem services. This information may contribute to the effectiveness of forest ecosystem management planning.European Union’s Horizon 2020 research and innovation programme; Portuguese Science Foundation

Determinação da Distribuição Espaço Temporal de Deslizamentos Sísmicos Mediante Algoritmo de Inversão Linear Dísicamente Constrangido

In this work we suggest an algorithm for space-time seismic slip distribution on finite faults for large earthquakes, from sets of teleseismic and strong ground motion waveforms. The solution of an auxiliary linear programming problem is an essential part of the developed method. To test the algorithm and examine the stability, non-uniqueness and robustness we applied it to a set of synthetic data obtained from the solution of a forward problem based on the representation theorem applied to slip distribution schemes as real models. Were used green functions to near field stations, calculated by a finite differences method and 3D structure model. The likeness between the inversion and the known solution support the credibility to use this method with data of real earthquakes