Network planning is an essential phase in successfully operating state-of-the-art telecommunication systems. It helps carriers increase revenues by deploying the right technologies in a cost effective manner. More importantly, through the network planning phase, carriers determine the capital needed to build the network as well as the competitive pricing for the offered services. Through this phase, radio tower locations are selected from a pool of candidate locations so as to maximize the net revenue acquired from servicing a number of subscribers. In the Universal Mobile Telecommunication System (UMTS) which is based on the Wideband Code Division Multiple Access scheme (W-CDMA), the coverage area of each tower, called a cell, is not only affected by the signal's attenuation but is also affected by the assignment of the users to the towers. As the number of users in the system increases, interference levels increase and cell sizes decrease. This complicates the network planning problem since the capacity and coverage problems cannot be solved separately. To identify the optimal base station locations, traffic intensity and potential locations are determined in advance, then locations of base stations are chosen so as to satisfy minimum geographical coverage and minimum quality of service levels imposed by licensing agencies. This is implemented through two types of power control mechanisms. The power based power control mechanism, which is often discussed in literature, controls the power of the transmitted signal so that the power at the receiver exceeds a given threshold. On the other hand, the signal-to-interference ratio (SIR) based power control mechanism controls the power of the transmitted signal so that the ratio of the power of the received signal over the power of the interfering signals exceeds a given threshold. Solving the SIR based UMTS/W-CDMA network planning problem helps network providers in designing efficient and cost effective network infrastructure. In contrast to the power based UMTS/W-CDMA network planning problem, the solution of the SIR based model results in higher profits. In SIR based models, the power of the transmitted signals is decreased which lowers the interference and therefore increases the capacity of the overall network. Even though the SIR based power control mechanism is more efficient than the power based power control mechanism, it has a more complex implementation which has gained less attention in the network planning literature. In this thesis, a non-linear mixed integer problem that models the SIR based power control system is presented. The non-linear constraints are reformulated using linear expressions and the problem is exactly solved using a Benders decomposition approach. To overcome the computational difficulties faced by Benders decomposition, two novel extensions are presented. The first extension uses the analytic center cutting plane method for the Benders master problem, in an attempt to reduce the number of times the integer Benders master problem is solved. Additionally, we describe a heuristic that uses the analytic center properties to find feasible solutions for mixed integer problems. The second extension introduces a combinatorial Benders decomposition algorithm. This algorithm may be used for solving mixed integer problems with binary variables. In contrast to the classical Benders decomposition algorithm where the master problem is a mixed integer problem and the subproblem is a linear problem, this algorithm decomposes the problem into a mixed integer master problem and a mixed integer subproblem. The subproblem is then decomposed using classical Benders decomposition, leading to a nested Benders algorithm. Valid cuts are generated at the classical Benders subproblem and are added to the combinatorial Benders master problem to enhance the performance of the algorithm. It was found that valid cuts generated using the analytic center cutting plane method reduce the number of times the integer Benders master problem is solved and therefore reduce the computational time. It was also found that the combinatorial Benders reduces the complexity of the integer master problem by reducing the number of integer variables in it. The valid cuts generated within the nested Benders algorithm proved to be beneficial in reducing the number of times the combinatorial Benders master problem is solved and in reducing the computational time that the overall algorithm takes. Over 110 instances of the UMTS/W-CDMA network planning problem ranging from 20 demand points and 10 base stations to 140 demand points and 30 base stations are solved to optimality.

Closed-loop supply chain (CLSC) management provides opportunity for cost savings through the integration of product recovery activities into traditional supply chains. Product recovery activities, such as remanufacturing, reclaim a portion of the previously added value in addition to the physical material. Our problem setting is motivated by the practice of an Original Equipment Manufacturer (OEM) in the automotive service parts industry, who operates a well established forward network. The OEM faces customer demand due to warranty and beyond warranty vehicle repairs. The warranty based demand induces part returns. We consider a case where the OEM has not yet established a product recovery network, but has a strategic commitment to implement remanufacturing strategy. In accomplishing this commitment, complications arise in the network design due to activities and material movement in both the forward and reverse networks, which are attributed to remanufacturing. Consequently, in implementing the remanufacturing strategy, the OEM should simultaneously consider both the forward and reverse flows for an optimal network design, instead of an independent and sequential modeling approach. In keeping with these motivations, and with the goal of implementing the remanufacturing strategy and transforming independent forward and reverse supply chains to CLSCs, we propose to investigate the following research questions: 1. How do the following transformation strategies leverage the CLSC?s overall cost performance? ? Extending the already existing forward channel to incorporate reverse channel activities. ? Designing an entire CLSC network. 2. How do the following network flow integration strategies influence the CLSC?s overall cost performance? ? Using distinct forward and reverse channel facilities to manage the corresponding flows. ? Using hybrid facilities to coordinate the flows. In researching the above questions, we address significant practical concerns in CLSC network design and provide cost measures for the above mentioned strategies. We also contribute to the current literature by investigating the optimal CLSC network design. More specifically, we propose three models and develop mathematical formulations and novel solution approaches that are based on decomposition techniques, heuristics, and meta-heuristic approaches to seek a solution that characterizes the configuration of the CLSC network, along with the coordinated forward and reverse flows.

No description available.

Generation and transmission investment planning in deregulated markets faces new challenges particularly as deregulation has introduced more uncertainty to the planning problem. Tradi- tional planning techniques and processes cannot be applied to the deregulated planning problem as generation investments are profit driven and competitive. Transmission investments must facilitate generation access rather than servicing generation choices. The new investment plan- ning environment requires the development of new planning techniques and processes that can remain flexible as uncertainty within the system is revealed. The optimisation technique of Stochastic Dual Dynamic Programming (SDDP) has been success- fully used to optimise continuous stochastic dynamic planning problems such as hydrothermal scheduling. SDDP is extended in this thesis to optimise the stochastic, dynamic, mixed integer power system investment planning problem. The extensions to SDDP allow for optimisation of large integer variables that represent generation and transmission investment options while still utilising the computational benefits of SDDP. The thesis also details the development of a math- ematical representation of a general power system investment planning problem and applies it to a case study involving investment in New Zealand’s HVDC link. The HVDC link optimisation problem is successfully solved using the extended SDDP algorithm and the output data of the optimisation can be used to better understand risk associated with capital investment in power systems. The extended SDDP algorithm offers a new planning and optimisation technique for deregulated power systems that provides a flexible optimal solution and informs the planner about investment risk associated with uncertainty in the power system.

Quantity Flexibility contract is an arrangement where parties agree upon a scheme of forming ranges on volumes for their future transactions. The contract is based on setting upper and lower limits on replenishment orders as simple multiples of point estimates updated, published and committed by the buyers. We introduce a manufacturer with a limited capacity / also capable of subcontracting, for deliveries with a known lead time. He offers a Quantity Flexibility (QF) contract to a buyer while he has an active contract with another buyer serving a market with known demand forecast distributions. Using two-stage stochastic programming we study the effects of flexibility multiples and the environmental factors on the buyers&amp / #8217 / incentives and manufacturer&amp / #8217 / s capacity planning. Finally, the motivations of the Supply Chain actors to behave independently or to be involved into the integrated iv supply chain where information asymmetry is removed are investigated. Our experiments underline the critical roles played by the forecast accuracy and information sharing.

A extração e processamento de minério resultam na geração de elevadas quantidades de resíduos, cuja disposição gera impacto e risco ambiental. O lançamento direto em reservatórios contidos por diques consiste na forma mais comum de disposição dos resíduos de mineração em superfície. Estes diques podem ser construídos por alteamentos sucessivos a montante (upstream), prática desaconselhada em alguns países. Apesar de se tratar de um processo de baixo custo, o tipo de aterro a montante é uma operação de risco, pois a montante as barragens são sensíveis à liquefação e também porque a estabilidade das barragens pode ser ameaçada pelo excesso de poro pressão gerado dentro do depósito durante a construção. Para atender às exigências de projetos de áreas de disposição de resíduos é, portanto, fundamental entender o comportamento destes materiais. Resíduos exibem considerável variabilidade em suas características físicas, químicas e mineralógicas, que conferem um comportamento distinto dos geomateriais usualmente encontrados em depósitos naturais. Entre os diferentes resíduos destacam-se ouro e bauxita cuja prática de exploração no Brasil é revestida de considerável interesse econômico e ambiental. Nesta tese estes materiais são analisados a partir de uma extensa campanha de investigação geotécnica baseada em um programa de ensaios de laboratório destinados à caracterização destes resíduos e à definição de parâmetros constitutivos. Para fins comparativos são realizados ensaios com um material siltoso, porém inerte. A condutividade hidraúlica e compressibilidade dos resíduos foram obtidas através de ensaios oedométricos executados com estágios de carga e gradientes hidráulicos constantes. Ensaios triaxiais foram realizados com altos níveis de tensões de confinamento e distintas trajetórias de tensões, para a definição da linha do estado crítico e a identificação do comportamento destes materiais através de ensaios com Bender Elements. A análise dos resultados obtidos permitiu comparações entre a linha de consolidação oedométrica (LCO), a linha de consolidação isotrópica (LCI) e a linha do estado crítico (LEC) que definiram a resposta de liquefação estática do material. No espaço e - log (p´) o aparecimento de um comportamento instável alterou significativamente a inclinação da LEC com relação a LCI e a LCO. A mesma mudança de inclinação é observada tanto no parâmetro de estado como no módulo cisalhante do solo quando plotados contra a tensão efetiva média. Com base nessas evidências, é defendido que o LEC de resíduos de mineração é altamente não-linear, e servem como indicativo de três respostas distintas do solo: uma resposta estável quando LCO, LCI e a LEC exibem paralelismo, um comportamento meta-estável em baixas tensões de confinamento levando à liquefação e quebra de grãos para altas tensões de confinamento da amostra. O avanço nas pesquisas relativas à disposição de resíduos provenientes de indústrias de mineração com o conhecimento dos fenômenos que se processa, poderá evitar a ocorrência de ruptura de barragens, que teria como conseqüência desastres ambientais de grande impacto. / Production and deposition of mining tailings is a problem confronting geotechnical engineers given the dimension of tailings deposits and environmental risks. The direct launching of tailings in reservoirs contained by dikes consists in a common type of waste disposal. These dikes can be constructed by successive upstream embankments, a relatively simple and economical process adopted in some countries. Although it is a low cost process, the upstream embankment is an operation of great risk, especially because tailings are susceptible to liquefaction and also because the stability of the dams can be threatened by excess of pore pressures generated inside the deposit during construction. To meet design requirements of disposal of residues it is therefore essential to understand the behavior of these materials. Tailings show a considerable variability in their physical, chemical and mineral characteristics that confer a distinct behavior to the geo-materials usually found in natural deposits. The present work gives emphasis to gold and bauxite tailings that encompass substantial economic and environmental interest in Brazil. In the thesis, an extensive laboratory geotechnical investigation program is carried out in an attempt to characterize these residues and to establish design constituent parameters. For comparative purposes an inert silty material is also tested. Compressibility and hydraulic conductivity were measured in oedometer tests performed at constant load and constant hydraulic gradient tests. Triaxial tests were carried out at high stress levels and different stress paths in order to establish the position and shape of the critical state line. In addition measured with Bender Elements are evaluated. The results between the oedometer consolidation (OCL), isotropic consolidation (ICL) and critical sate (CSL) lines allowed the static liquefaction response to be directly assessed. In a e-log (p’) space, the onset of a meta-stable behaviour significantly changes the slope of the CSL relative to both ICL and OCL. The same change in slope is observed on both the state parameter and the small strain shear modulus measured by bender elements when plotted against mean stress. Based on these evidences it is advocated that the CSL of the tailing is highly non-linear in nature, which lead to three distinct soil responses: a stable response when OCL, ICL and CSL exhibit similar slopes, a meta-stable beahaviour at lower stress leading to liquefaction and a crushable response for higher mean stresses. Advanced research on tailings from mining industries with due recognition of the phenomena of liquefaction may prevent dam failure, and consequently may prevent major environmental disasters.

A extração e processamento de minério resultam na geração de elevadas quantidades de resíduos, cuja disposição gera impacto e risco ambiental. O lançamento direto em reservatórios contidos por diques consiste na forma mais comum de disposição dos resíduos de mineração em superfície. Estes diques podem ser construídos por alteamentos sucessivos a montante (upstream), prática desaconselhada em alguns países. Apesar de se tratar de um processo de baixo custo, o tipo de aterro a montante é uma operação de risco, pois a montante as barragens são sensíveis à liquefação e também porque a estabilidade das barragens pode ser ameaçada pelo excesso de poro pressão gerado dentro do depósito durante a construção. Para atender às exigências de projetos de áreas de disposição de resíduos é, portanto, fundamental entender o comportamento destes materiais. Resíduos exibem considerável variabilidade em suas características físicas, químicas e mineralógicas, que conferem um comportamento distinto dos geomateriais usualmente encontrados em depósitos naturais. Entre os diferentes resíduos destacam-se ouro e bauxita cuja prática de exploração no Brasil é revestida de considerável interesse econômico e ambiental. Nesta tese estes materiais são analisados a partir de uma extensa campanha de investigação geotécnica baseada em um programa de ensaios de laboratório destinados à caracterização destes resíduos e à definição de parâmetros constitutivos. Para fins comparativos são realizados ensaios com um material siltoso, porém inerte. A condutividade hidraúlica e compressibilidade dos resíduos foram obtidas através de ensaios oedométricos executados com estágios de carga e gradientes hidráulicos constantes. Ensaios triaxiais foram realizados com altos níveis de tensões de confinamento e distintas trajetórias de tensões, para a definição da linha do estado crítico e a identificação do comportamento destes materiais através de ensaios com Bender Elements. A análise dos resultados obtidos permitiu comparações entre a linha de consolidação oedométrica (LCO), a linha de consolidação isotrópica (LCI) e a linha do estado crítico (LEC) que definiram a resposta de liquefação estática do material. No espaço e - log (p´) o aparecimento de um comportamento instável alterou significativamente a inclinação da LEC com relação a LCI e a LCO. A mesma mudança de inclinação é observada tanto no parâmetro de estado como no módulo cisalhante do solo quando plotados contra a tensão efetiva média. Com base nessas evidências, é defendido que o LEC de resíduos de mineração é altamente não-linear, e servem como indicativo de três respostas distintas do solo: uma resposta estável quando LCO, LCI e a LEC exibem paralelismo, um comportamento meta-estável em baixas tensões de confinamento levando à liquefação e quebra de grãos para altas tensões de confinamento da amostra. O avanço nas pesquisas relativas à disposição de resíduos provenientes de indústrias de mineração com o conhecimento dos fenômenos que se processa, poderá evitar a ocorrência de ruptura de barragens, que teria como conseqüência desastres ambientais de grande impacto. / Production and deposition of mining tailings is a problem confronting geotechnical engineers given the dimension of tailings deposits and environmental risks. The direct launching of tailings in reservoirs contained by dikes consists in a common type of waste disposal. These dikes can be constructed by successive upstream embankments, a relatively simple and economical process adopted in some countries. Although it is a low cost process, the upstream embankment is an operation of great risk, especially because tailings are susceptible to liquefaction and also because the stability of the dams can be threatened by excess of pore pressures generated inside the deposit during construction. To meet design requirements of disposal of residues it is therefore essential to understand the behavior of these materials. Tailings show a considerable variability in their physical, chemical and mineral characteristics that confer a distinct behavior to the geo-materials usually found in natural deposits. The present work gives emphasis to gold and bauxite tailings that encompass substantial economic and environmental interest in Brazil. In the thesis, an extensive laboratory geotechnical investigation program is carried out in an attempt to characterize these residues and to establish design constituent parameters. For comparative purposes an inert silty material is also tested. Compressibility and hydraulic conductivity were measured in oedometer tests performed at constant load and constant hydraulic gradient tests. Triaxial tests were carried out at high stress levels and different stress paths in order to establish the position and shape of the critical state line. In addition measured with Bender Elements are evaluated. The results between the oedometer consolidation (OCL), isotropic consolidation (ICL) and critical sate (CSL) lines allowed the static liquefaction response to be directly assessed. In a e-log (p’) space, the onset of a meta-stable behaviour significantly changes the slope of the CSL relative to both ICL and OCL. The same change in slope is observed on both the state parameter and the small strain shear modulus measured by bender elements when plotted against mean stress. Based on these evidences it is advocated that the CSL of the tailing is highly non-linear in nature, which lead to three distinct soil responses: a stable response when OCL, ICL and CSL exhibit similar slopes, a meta-stable beahaviour at lower stress leading to liquefaction and a crushable response for higher mean stresses. Advanced research on tailings from mining industries with due recognition of the phenomena of liquefaction may prevent dam failure, and consequently may prevent major environmental disasters.

A extração e processamento de minério resultam na geração de elevadas quantidades de resíduos, cuja disposição gera impacto e risco ambiental. O lançamento direto em reservatórios contidos por diques consiste na forma mais comum de disposição dos resíduos de mineração em superfície. Estes diques podem ser construídos por alteamentos sucessivos a montante (upstream), prática desaconselhada em alguns países. Apesar de se tratar de um processo de baixo custo, o tipo de aterro a montante é uma operação de risco, pois a montante as barragens são sensíveis à liquefação e também porque a estabilidade das barragens pode ser ameaçada pelo excesso de poro pressão gerado dentro do depósito durante a construção. Para atender às exigências de projetos de áreas de disposição de resíduos é, portanto, fundamental entender o comportamento destes materiais. Resíduos exibem considerável variabilidade em suas características físicas, químicas e mineralógicas, que conferem um comportamento distinto dos geomateriais usualmente encontrados em depósitos naturais. Entre os diferentes resíduos destacam-se ouro e bauxita cuja prática de exploração no Brasil é revestida de considerável interesse econômico e ambiental. Nesta tese estes materiais são analisados a partir de uma extensa campanha de investigação geotécnica baseada em um programa de ensaios de laboratório destinados à caracterização destes resíduos e à definição de parâmetros constitutivos. Para fins comparativos são realizados ensaios com um material siltoso, porém inerte. A condutividade hidraúlica e compressibilidade dos resíduos foram obtidas através de ensaios oedométricos executados com estágios de carga e gradientes hidráulicos constantes. Ensaios triaxiais foram realizados com altos níveis de tensões de confinamento e distintas trajetórias de tensões, para a definição da linha do estado crítico e a identificação do comportamento destes materiais através de ensaios com Bender Elements. A análise dos resultados obtidos permitiu comparações entre a linha de consolidação oedométrica (LCO), a linha de consolidação isotrópica (LCI) e a linha do estado crítico (LEC) que definiram a resposta de liquefação estática do material. No espaço e - log (p´) o aparecimento de um comportamento instável alterou significativamente a inclinação da LEC com relação a LCI e a LCO. A mesma mudança de inclinação é observada tanto no parâmetro de estado como no módulo cisalhante do solo quando plotados contra a tensão efetiva média. Com base nessas evidências, é defendido que o LEC de resíduos de mineração é altamente não-linear, e servem como indicativo de três respostas distintas do solo: uma resposta estável quando LCO, LCI e a LEC exibem paralelismo, um comportamento meta-estável em baixas tensões de confinamento levando à liquefação e quebra de grãos para altas tensões de confinamento da amostra. O avanço nas pesquisas relativas à disposição de resíduos provenientes de indústrias de mineração com o conhecimento dos fenômenos que se processa, poderá evitar a ocorrência de ruptura de barragens, que teria como conseqüência desastres ambientais de grande impacto. / Production and deposition of mining tailings is a problem confronting geotechnical engineers given the dimension of tailings deposits and environmental risks. The direct launching of tailings in reservoirs contained by dikes consists in a common type of waste disposal. These dikes can be constructed by successive upstream embankments, a relatively simple and economical process adopted in some countries. Although it is a low cost process, the upstream embankment is an operation of great risk, especially because tailings are susceptible to liquefaction and also because the stability of the dams can be threatened by excess of pore pressures generated inside the deposit during construction. To meet design requirements of disposal of residues it is therefore essential to understand the behavior of these materials. Tailings show a considerable variability in their physical, chemical and mineral characteristics that confer a distinct behavior to the geo-materials usually found in natural deposits. The present work gives emphasis to gold and bauxite tailings that encompass substantial economic and environmental interest in Brazil. In the thesis, an extensive laboratory geotechnical investigation program is carried out in an attempt to characterize these residues and to establish design constituent parameters. For comparative purposes an inert silty material is also tested. Compressibility and hydraulic conductivity were measured in oedometer tests performed at constant load and constant hydraulic gradient tests. Triaxial tests were carried out at high stress levels and different stress paths in order to establish the position and shape of the critical state line. In addition measured with Bender Elements are evaluated. The results between the oedometer consolidation (OCL), isotropic consolidation (ICL) and critical sate (CSL) lines allowed the static liquefaction response to be directly assessed. In a e-log (p’) space, the onset of a meta-stable behaviour significantly changes the slope of the CSL relative to both ICL and OCL. The same change in slope is observed on both the state parameter and the small strain shear modulus measured by bender elements when plotted against mean stress. Based on these evidences it is advocated that the CSL of the tailing is highly non-linear in nature, which lead to three distinct soil responses: a stable response when OCL, ICL and CSL exhibit similar slopes, a meta-stable beahaviour at lower stress leading to liquefaction and a crushable response for higher mean stresses. Advanced research on tailings from mining industries with due recognition of the phenomena of liquefaction may prevent dam failure, and consequently may prevent major environmental disasters.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-04-24T13:25:57Z No. of bitstreams: 1 eduardomartinsviana.pdf: 1013683 bytes, checksum: 5d87e6aef68844b7f67e1941a0337702 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-04-24T16:55:49Z (GMT) No. of bitstreams: 1 eduardomartinsviana.pdf: 1013683 bytes, checksum: 5d87e6aef68844b7f67e1941a0337702 (MD5) / Made available in DSpace on 2017-04-24T16:55:49Z (GMT). No. of bitstreams: 1 eduardomartinsviana.pdf: 1013683 bytes, checksum: 5d87e6aef68844b7f67e1941a0337702 (MD5) Previous issue date: 2008-08-13 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Este trabalho propõe uma metodologia para auxiliar os estudos relacionados à recomposição fluente de sistemas elétricos de potência. Nesta fase inicial de recomposição, o sistema de potência encontra-se estruturado em longos corredores de transmissão e a definição de uma configuração de reatores constitui uma das estratégias adotadas para a eliminação de sobretensões. A ferramenta proposta, baseada em um modelo de otimização, visa o estabelecimento de procedimentos de rede que conduza a uma configuração mínima de reatores conectados ao sistema e a máxima tomada de carga durante esta fase de recomposição. Os diversos cenários de recomposição fluente, correspondentes à energização de novos trechos de linhas de transmissão, são representados através de subproblemas. A técnica de decomposição de Benders associada à metodologia primal-dual de pontos interiores é utilizada para solução do problema de otimização não-linear proposto. São realizados estudos de caso em ramos de recomposição fluente do Sistema Elétrico Brasileiro interligado para verificar a eficácia da metodologia. Os resultados obtidos são comparados com outros métodos propostos na literatura. / This work proposes a methodology to help the studies related to electric power systems fluent restoration. At this early restoration stage, power system is structured in long transmission corridors and the reactors set definition is one of the adopted strategies for overvoltages disposal. The proposed tool, based on an optimization model, aims the network procedures establishment leading to a minimum reactors configuration connected to the system and maximum load pick-up during this restoration phase. The various fluent restoration scenarios, corresponding to new transmission lines sections inrush, are represented by sub problems. Benders decomposition technique associated to primal-dual interior points methodology is used to solve the proposed nonlinear optimization problem. Case studies are conducted in fluent restoration branches of Brazilian Electric interconnected System to verify the methodology effectiveness. The results are compared with other proposed methods in the literature.

Underground mining is a complex operation that requires careful planning. The short-term scheduling, which is the scheduling of the tasks involved in the excavation process, is an important part of the planning process. In this master thesis we propose a new method for short-term scheduling of a cut-and-fill mine operated by the mining company Boliden AB. We include a new aspect of the problem by incorporating a priority between the excavation locations of the mine. The priority feature allows the user to control the output of the scheduling and to direct resources to the locations where they are most needed according to the long-term plans. Our solution method consists of two components: a constructive heuristic method that construct a complete solution by solving partial scheduling problems containing subsets of tasks, and a logic-based Benders decomposition scheme for solving these partial problems. The computational performance of the proposed method is evaluated on industrially relevant largescale instances generated from data provided by Boliden. Comparisons are made with applying a constraint programming solver on the complete problem and with replacing the logic-based Benders scheme by applying a constraint programming solver on the partial scheduling problems, respectively. Results show that the heuristic method combined with the logic-based Benders decomposition scheme outperforms the other two methods on all instances.