Lagrangian Relaxation Method Research Articles

Sequential recovery of disrupted facilities

For a bipartite graph with disrupted facilities, we confront the key concern that disrupted facilities can only be recovered one after one due to resource shortage. The objective of this paper is to design a sequential recovery schedule such that the summation of waiting time and transportation cost of all customers to regain service is minimized. For such a sequential disruption recovery problem, we consider two situations: sequential disruption recovery for uncapacitated facilities (SDRUF) and sequential disruption recovery for capacitated facilities (SDRCF). We model the corresponding optimization problems as integer linear programs, and propose two heuristic algorithms for each situation, based on both Lagrangian relaxation and greedy method, to generate feasible solutions. In the end, computational experiments are carried out to implement the proposed heuristic algorithms. The computational results show that the two heuristic algorithms are efficient and effective for solving the SDRUF and the SDRCF problems, and act as good alternatives for each other based on problem size.

Sustainable, resilient and responsive mixed supply chain network design under hybrid uncertainty with considering COVID-19 pandemic disruption.

The occurrence of the COVID-19 pandemic is a disruption that has adversely affected many supply chains (SCs) around the world and further proved the necessity of combination and interaction of resilience and sustainability. In This paper, a multi-objective mixed-integer linear programming model is developed for responsive, resilient and sustainable mixed open and closed-loop supply chain network design (SCND) problem. The uncertainty of the problem is handled with a hybrid robust-stochastic optimization approach. A Lagrangian relaxation (LR) method and a constructive heuristic (CH) algorithm are developed for overcoming problem complexity and solving large-scale instances. In order to assess the performance of the mathematical model and solution methods, some test instances are generated. The computations showed that the model and the solution methods are efficient and can obtain high-quality solutions in suitable CPU times. Other analyses and computations are done based on a real case study in the tire industry. The results demonstrate that resilient strategies are so effective and can improve economic, environmental and social dimensions substantially. Research findings suggest that the proposed model can be used as an efficient tool for designing sustainable and resilient SCs and the related decision-makings. Also, our findings prove that resilience is necessary for continued SC sustainability. It is concluded that using proposed resilience strategies simultaneously brings the best outcome for SC objectives. Based on the sensitivity analyses, the responsiveness level significantly affects SC objectives, and managers should consider the trade-off between responsiveness and their objectives.

Joint Dynamical VNF Placement and SFC Routing in NFV-Enabled SDNs

Due to that Service Function Chain (SFC) permits the forwarding of flows along a predetermined sequence chain of Virtual Network Functions (VNFs), it has become a common service in Network Function Virtualization (NFV)-enabled Software Defined Networks (SDNs). Generally, since there are multiple same VNF-instances in NFV-Enabled SDNs, this brings a great challenge for selecting or placing the required VNF-instances to satisfy the routing of SFC Request flows (SRs). In this paper, we study the routing problem for SRs by jointly considering dynamical VNF placement and multiple Resources and Quality of Service (QoS) constraints in NFV-Enabled SDNs. Specifically, we first define two optimization problems: one is the Dynamical VNF Placement and Routing Problem for SRs (DVPRP) and the other is the Delay, packet Loss and Jitter Aware Dynamical VNF Placement and Routing Problem for SRs (DLJA-DVPRP). We then formulate the two problems as Integer Linear Programming (ILP) problems. Next, we creatively devise an auxiliary edge-weight graph and propose two efficient algorithms to solve the problems with the aim of minimizing the resource consumption costs as well as ensuring multiple QoS constraints. Especially, we utilize the shortest path algorithm based on Lagrange relaxation method to solve the DLJA-DVPRP with multiple QoS constraints. Compared with existing algorithms, simulation results demonstrate our proposed algorithms have better performance in terms of throughput, traffic acceptance rate and load balance.

SoC threshold optimization for battery storage in frequency regulation considering uncertainty of SoC measurement and automatic generation control fatigue loss of thermal power system

The min/max state of charge (SoC) thresholds of battery storage (BS), which challenge the economics of frequency regulation (FR), have a certain degree of uncertainty and need to be artificially re-calibrated after a period of operation time. To tackle this challenge, a SoC threshold optimization method is proposed for joint FR by thermal power (TP) and battery storage (BS) by comprehensively considering the automatic generation control (AGC) fatigue loss cost of TP and the aging degradation cost of BS. The proposed method consists of three parts: 1) the AGC fatigue loss cost of TP is computed by considering motion loss cost (e.g., steam turbine rotor, oil servo-motor, steam gate valve stem) and static loss cost (e.g., steam drum, cylinder); 2) the aging degradation cost of BS is formulated by considering also the uncertainty of SoC; 3) the min/max SoC operation thresholds of BS are optimized through the surrogate Lagrangian relaxation (SLR) method by comprehensive considering of the AGC fatigue loss and aging degradation cost of FR. Results of extensive simulation tests have clarified and verified the economic effectiveness of proposed min/max SoC threshold optimization method for the joint frequency regulation by thermal power and battery storage.

An Integer Mathematical Programming Model for Production Scheduling Problem in Open-Pit Mines under Grade Uncertainty and Solving Using the Firefly Algorithm

Long-term production scheduling in open-pit mines is a crucial issue in mining planning and determines the distribution of cash flow throughout the life of the mine. The purpose of the planning is to maximize the net present value by taking into account all operational constraints such as slope, mixing of different grades, mineral production, and extraction capacity. The uncertainties associated with model data play an important role in optimizing long-term production plans. Among the uncertainties, grade uncertainty plays a major role. In this paper, hybrid models are presented by the Lagrangian relaxation (LR) method, augmented Lagrangian relaxation (ALR) method, and firefly algorithm (FA) to solve the long-term production scheduling problem of open-pit mines with the assumption of deterministic and also considering the grade uncertainty. The firefly algorithm is used to update the Lagrange multipliers. The newly proposed approaches are based on optimizing Lagrangian multipliers and comparing them with the results of combined Lagrangian relaxation method and augmented Lagrangian relaxation with the Genetic Algorithm (GA), and the traditional sub-gradient (SG) method. For solving and validating the obtained model, Chadarmelo iron ore mine is considered as a suitable case study. The results of the case study show that the combined strategy (ALR-FA) can provide a near-optimal solution over other methods such that, over a given period, the net present value using the proposed hybrid approach is 20.11% higher than the traditional method is available. Also, the CPU speed of the proposed model is 4.7% more than the other methods.

Optimal data placement strategy considering capacity limitation and load balancing in geographically distributed cloud

At present, the amount of data from users is increasing exponentially, and most of the data is stored in data centers distributed in different geographic locations. The cost of transferring large amounts of data across geographically distributed data centers can become prohibitive. Therefore, to shorten the data transmission time and reduce the cost of data transmission bandwidth and maintain the load balance of the geographically distributed cloud system, an optimal data placement strategy considering capacity limitation and load balancing in a geographically distributed cloud is proposed. Firstly, the capacity limitation, load balancing, and bandwidth cost of each cloud data center in the geographically distributed cloud are considered, and the data placement problem in the geographically distributed cloud is mathematically modeled. Secondly, the Floyd algorithm is used to model the cost of data transmission bandwidth and find the minimum transmission bandwidth cost. Finally, the Lagrangian relaxation method is used to obtain the optimal data placement scheme for the transmission time. To show the performance advantages of the proposed algorithm, comparative experiments are carried out. When the bandwidth is 15 Mbps, in terms of the Load Balancing Degree (LBD), the proposed algorithm is 40.3% higher than the Hash on average, 35.6% higher than the Closest on average, and 25.7% higher than the CRANE on average. Moreover, the experimental results show that the proposed algorithm can reduce the data transmission cost, and improve the load balancing in a geographically distributed cloud system.

Agent based simulation of centralized electricity transaction market using bi-level and Q-learning algorithm approach

The market participants face risks in market price fluctuations and uncertainties in the demand behaviour, regardless of the series of restructuring reforms of China power industry. To address this gap, the strategic pricing and bidding behaviour of generation and electricity selling enterprises in double-sided auctions of centralized electricity transaction market with elastic demand is examined. The market is modelled with two-level (Bi-level) mathematical optimization problem and Q-Learning algorithm. First, the upper level solves the profit maximization of the individual market players. Second, the lower level represents the market clearing at uniform transaction price by a market auctioneer using Lagrange relaxation method. Agent learning approach of Q-Learning algorithm is used to solve the two level mathematical problem. Both generation and electricity selling enterprises are modelled as Q-Learning agents with incomplete information about their counterparts. The simulation results show significant reduction in energy transaction price with the participation of flexible consumers in electricity market auctions, hence boosting the consumer savings. This study offers positive effects in reducing market prices in double-sided auction of electricity markets with elastic consumers using Q-learning algorithm as compared to a market with inelastic consumers.

Modeling and Optimization Sustainable Forest Supply Chain Considering Discount in Transportation System and Supplier Selection under Uncertainty

In recent decades, the forest industry has been growingly expanded due to economic conditions, climate changes, environmental and energy policies, and intense demand changes. Thus, appropriate planning is required to improve this industry. To achieve economic, social and environmental goals, a supply chain network is designed based on a multi-period and multi-product Mixed-Integer Non-Linear Programming (MINLP) model in which the objective is to maximize the profit, minimize detrimental environmental effects, improve social effects, and minimize the number of lost demands. In addition, to improve forest industry planning, strategic and tactical decisions have been implemented throughout the supply chain for all facilities, suppliers and machinery. These decisions significantly help to improve processes and product flows and to meet customers’ needs. In addition, because of the presence of uncertainty in some parameters, the proposed model was formulated and optimized under uncertainty using the hybrid robust possibilistic programming (HRPP-II) approach. The ε-constraint technique was used to solve the multi-objective model, and the Lagrangian relaxation (LR) method was utilized to solve the model of more complex dimensions. A case study in Northern Iran was conducted to assess the efficiency of the suggested approach. Finally, a sensitivity analysis was performed to determine the impact of important parameters on objective functions. The results of this study show that increasing the working hours of machines instead of increasing their number, increasing the capacity of some facilities instead of establishing new facilities and expanding the transport fleet has a significant impact on achieving predetermined goals.

Data Center Network Design for Internet‐Related Services and Cloud Computing

Data center networks provide the physical infrastructure that hosts Internet‐related services and cloud computing. Designing data center networks properly is imperative for Internet‐related service and cloud computing providers to gain competitive edges through cost efficiency and service quality. In this study, we formulate a mathematical programming model to address the data center network design problem, in which the objective is to minimize total operating cost and the service delay penalty by optimizing data center location, footprint allocation, and resource provisioning decisions, while incorporating essential features, such as latency, power, multiple resources, configuration limits, and interdependent footprints. We employ a queueing model to approximate the service latency and provide tractable reformulations. To enhance computational efficiency for large‐scale problems, we further develop Lagrangian relaxation methods and generate strengthening cuts by exploiting the structural properties of the problem. Our numerical studies demonstrate that the proposed model, which jointly optimizes location, allocation, and resource provisioning, can achieve significant cost reductions and improvements in service quality compared with a hierarchical approach that optimizes these decisions sequentially. Moreover, our proposed solution methods outperform state‐of‐the‐art commercial software in terms of computational efficiency. Based on real‐world datasets, the proposed model selects data centers that have been chosen by major cloud computing infrastructure providers. We also draw managerial insights that can be used as design guidelines in practice.

Incorporating sales and marketing considerations into a competitive multi-echelon distribution network design problem with pricing strategy in a stochastic environment

This paper presents a multi-echelon distribution network design problem with pricing strategy in a stochastic environment, where location, inventory, and pricing decisions in retail and wholesale channels are made simultaneously. The considered network is comprised of a central warehouse, a set of distribution centers, and a set of retailers and wholesalers. The objective of the problem is to maximize the supply chain profit. The decisions include the location of distribution centers and the allocation of retailers and wholesalers to them, order-size for each distribution center and product price at retail and wholesale channels for different payment conditions. A mixed-integer nonlinear mathematical model was formulated and solved using the Lagrangian relaxation method and a Genetic algorithm. Computational results indicate that Lagrangian relaxation algorithm has good performance in terms of objective function value and runtime, even in large sized problems.

Developing a two-stage model for a sustainable closed-loop supply chain with pricing and advertising decisions

Closed-Loop Supply Chain (CLSC) has become a critical problem due to its effects on various factors including economic motivations, environmental concerns, and social impacts. Moreover, there are coordination tools, such as pricing and advertising, which impact its performance. In this paper, we offer a two-stage approach to model and solve a sustainable CLSC, taking into account pricing, green quality, and advertising. In the first stage, optimal decisions on pricing, greening, and advertising are made, while in the second stage, a fuzzy multi-objective Mixed Integer Linear Programming (MILP) model is used to maximize the total profit, reduce CO2 emissions, and improve social impacts. Suitable solution methods are introduced according to the scale of the problem. For small-scale instances, an augmented ϵ-constraint method is used to solve the problem. For large-scale instances, approximations are required, and a Lagrangian relaxation algorithm solves the problem in polynomial time. The performance of the proposed model is evaluated through various numerical examples. The results illustrate the applicability and efficiency of the model, while confirming significant improvements in sustainable objectives under optimal pricing, green quality, and advertising. Besides, the proposed Lagrangian relaxation method significantly reduces the computational time for large-scale instances, with only a 2.308% deviation from the optimal results.

Improve communication network issues for multi UAV

In this paper, we address the problems of power efficiency, real-time routing, and rate allocation in flying ad-hoc networks (FANETs). The study designed a cross-layer optimization framework with delay constraints to solve the proposed problems and then used Lagrangian relaxation and dual decomposition methods to decompose the joint optimization problem into several lower complexity sub-problems. Moreover, Case model 3 is employed that allows each relay node to complete the optimization of different sub-problems through local information. The results show that the proposed algorithm can effectively increase the network throughput and reduce the packet-timeout ratio and power efficiency.

RETRACTED ARTICLE: Closed-loop supply chain network design with sustainability and resiliency criteria.

Today, the research on the closed-loop supply chain network design with sustainability and resiliency criteria is a very active research topic. This paper provides a new closed-loop supply chain under uncertainty with the use of resiliency, sustainability, and reliability dimensions among the first studies. To model this problem, a two-stage stochastic programming approach is used. To create robust solutions against uncertainty, a conditional value at risk criterion is contributed. The proposed model aims to minimize the total cost, environmental pollution, and energy consumption while maximizing the job opportunities as the social factor. In addition to the sustainability goals, the energy consumption is considered to be the last objective to be minimized. To show the applicability of the proposed model, an automobile assembler industry is applied. To solve the model, the Lp-metric method is employed to transform this multi-objective model into a single objective one. Since this closed-loop supply chain model is complex and NP-hard, a Lagrangian relaxation method with fix-and-optimize heuristic is employed to find the upper and lower bounds for the model via different random test problems. With an extensive analysis, the proposed model shows an improvement to the total cost, CO2 emissions, job opportunities and energy consumption.

Cooperative optimization of cutting parameters, process routes, and scheduling considering carbon emissions with analytic target cascading

Cutting parameters, process routes, and scheduling significantly affect carbon emissions and makespan of workshops. However, most current researches carried out cutting parameters, process routes, and scheduling decision independently and ignored their comprehensive effects on carbon emissions. To bridge the gap, this paper is aimed at solving cooperative optimization problem of cutting parameters, process routes, and scheduling (CCPJS) employing analytical target cascading (ATC). Firstly, a CCPJS-ATC model is presented and elements of the model are formulated, where carbon emissions and makespan are as optimization objectives. Secondly, a novel-augmented Lagrangian relaxation method is designed to solve the CCPJS-ATC model in a distributed manner. Finally, a case study is employed to verify the feasibility and effectiveness of the proposed method. It shows that the proposed method could achieve more carbon emissions and solving time savings compared with the traditional all-in-one approaches.

A new model for operating room scheduling with elective patient strategy

Hospitals are dealing with the daunting task of scheduling patients in operating rooms under a limited budget, time, and staff. This article finds the optimal schedule of surgeries by minimizing operating rooms’ idle times while maximizing the number of scheduled surgeries during the most effective and desirable time windows. Surgeries during ideal time windows are encouraged by assigning bonus weights in the objective function. Stated and implied benefits of this strategy include mitigating financial loss, complications, and death rate due to a reduction in surgery delays. We introduce a binary programming model for scheduling operating rooms and a mixed integer binary program for planning and scheduling both operating and recovery rooms for elected patients under deterministic conditions. We apply an open scheduling strategy for assigning operating rooms to surgeons and a Lagrangian relaxation method for finding promising solutions. We move hard constraints to the objective to reduce the complexity of the proposed NP-hard model. We incorporate a sub-gradient method that selects the best penalty vector. Finally, we benchmark the results to evaluate the efficiency of the proposed solutions.

A Novel Robust Model for Health Care Facilities Location-Allocation Considering Pre Disaster and Post Disaster Characteristics

Health care centers are one of the most important municipal facilities that are directly involved in providing personal and social health. In this paper, a new Robust Mathematical Modeling is simultaneously provided to locate and efficiently allocate healthcare facilities, including different service levels of medical care, concerning the characteristics in normal and disaster situations. Furthermore, to help victims and prevent overcrowding in hospitals and medical centers during disasters, the establishment of temporary and outpatient centers is allowed for emergency provision of basic services. Also, the possibility to send equipment and medical teams to these centers is considered in the proposed model. Since the considered problem is included as NP-Complete problems, to solve the problem, two metaheuristic algorithms, harmony search algorithm and hybrid Tabu search combined with variable neighborhood search algorithm, and a lower bound based on Lagrangian relaxation method are presented. Finally, to examine the proposed algorithms, a number of sample problems are randomly generated in small and large sizes and the results of exact solutions and the lower bound resulting from the Lagrangian relaxation method are evaluated and compared with the results of the meta-heuristic algorithms. The results show the good performance of the proposed algorithms.

A dual approach to multi-dimensional assignment problems

In this paper, we extend the purely dual formulation that we recently proposed for the three-dimensional assignment problems to solve the more general multidimensional assignment problem. The convex dual representation is derived and its relationship to the Lagrangian relaxation method that is usually used to solve multidimensional assignment problems is investigated. Also, we discuss the condition under which the duality gap is zero. It is also pointed out that the process of Lagrangian relaxation is essentially equivalent to one of relaxing the binary constraint condition, thus necessitating the auction search operation to recover the binary constraint. Furthermore, a numerical algorithm based on the dual formulation along with a local search strategy is presented. The simulation results show that the proposed algorithm outperforms the traditional Lagrangian relaxation approach in terms of both accuracy and computational efficiency.

A multi-block ADMM based approach for distribution market clearing with distribution locational marginal price

This work investigates a distribution locational marginal price (DLMP) scheme for promoting the market penetration of small-scale prosumers/consumers connected at distribution level and boosting their potential of demand response. In the distribution market, the distribution market operator (DMO) negotiates with each participant a node-specific DLMP and the respective demand profile of customer, so as to maximize the social welfare while respecting the operational constraints of the distribution system and load devices. First, different customers at demand side are modeled, including prosumers with photovoltaic (PV) generation, distributed storages and electric vehicles. All of them communicate with DMO and realize automated demand response by intelligent trading agent. Then, the distribution market clearing model for a massive number of customers in an ex-ante trade scheme is formulated as a market-based multi-period alternating current optimal power flow (ACOPF). As this clearing model is separable, the augmented Lagrangian relaxation (ALR) method is adopted to solve the market clearing model in a parallel manner. In the ALR, a second-order penalty term at each node is introduced to ensure the convergence, whereas it is not separable for customers. Hence, there are a multi-block demand variable coupled within the second-order penalty term at each node. A standard alternating direction method of multipliers (ADMM) algorithm cannot converge in this situation. To tackle this multi-block problem, we exploit the multi-block ADMM and distributed Dauglas-Rarchford Splitting method (d-DRSM), both of which proximally regularize the customer-specific subproblem to ensure global convergence. Furthermore, the prediction-correction mechanism in d-DRSM algorithm can achieve better rate of convergence in comparison with the multi-block ADMM. Finally, case studies are conducted and numerical results have shown the effectiveness of the proposed methodology, and the performance of different algorithms are analyzed for improving the clearing method’s efficiency.

Equitable routing of rail hazardous materials shipments using CVaR methodology

The low probability – high consequence nature of hazardous materials (hazmat) incidents dictate a risk-averse route planning approach. However, preparing routing plans for multiple hazmat shipments between various origin–destination pairs also raises the question of risk-equity, and not just minimization of hazmat risk. Hence, the objective is to plan an equitable routing plan for different rail hazmat shipments while not only ensuring the safety of citizens but also precluding certain population zones from being subjected to intolerable levels of hazmat risk. To this end, we propose an analytical framework that makes use of a conditional value-at-risk (CVaR) measure of risk to generate minimum risk shipment routes while promoting risk-equity in both the arcs and the yards of the railroad network. While the commercial solver, CPLEX, lacks the ability to generate integer solutions for even small problem instances, a Lagrangian relaxation method aimed at being maximized using the Subgradient optimization algorithm is applied to provide a lower bound. The proposed framework is finally used to study several problem instances using the realistic infrastructure of a railroad operator, and to conclude that risk-equity can be achieved by re-routing, and that the design of train services along with the trade-off between yard-risk and arc-risk determine the number of re-routing and the optimal value of CVaR.

Lagrangian Relaxation Method in Approximating the Pareto Front of Multiobjective Optimization Problems

In this paper, we propose that the Lagrangian relaxation approach can be used to approximate the Pareto front of the multiobjective optimization problems. We introduce Lagrangian relaxation approach to solve scalarized subproblems. The scalarization is a technique employed to transform multiple objectives optimization problems into single-objective optimization problems so that existing optimization techniques are used to solve the problems. The relaxation approach exploits transformation and creates a Lagrangian problem in which some of the constraints are replaced from the original problem to make the problem easier to solve. The method is very effective when the problem is large scale and difficult to solve; this means if the problem has nonconvex and nonsmooth structure, then our proposed method efficiently solves the problem. We succeed in establishing proper Karush Kuhn-Tucker type necessary conditions for our proposed approach. We establish the relation between our proposed approach and the well-known existing approach weighted-sum scalarization methods. We conduct extensive numerical experiments and demonstrated the advantages of the proposed method of adopting a test problem. GANIT J. Bangladesh Math. Soc. 40.2 (2020) 126-133