Lagrangian Relaxation Research Articles

Inventory Control for Omnichannel Retailing Between One Warehouse and Multiple Stores

In this article, we research a multiperiod two-echelon inventory control problem for omnichannel retailing [i.e., online channel, offline channel, and buy-online-and-pick-up-in-store (BOPS) channel]. Especially, we jointly consider the limited storage capacity of the stores and the delivery lead time of the warehouse. For the out of stock, the unsatisfied demands for multiple purchasing channels have different processes (i.e., lost sales or backlog), based on the different characteristics of multiple channels. First, we divide the multiperiod inventory control problem for omnichannel retailing into two subproblems (the inventory control problem for the stores and the inventory control problem for the warehouse). Then, to reduce the scale of the first subproblem and the computation effort, we propose a method based on Lagrangian relaxation and value iteration to solve the inventory control problem of stores. Further, based on the results of the inventory control problem of stores, we construct a delay-aware Markov decision process (DAMDP) for the inventory control problem of the warehouse and employ the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> -learning technique based on the constructed DAMDP to solve the second subproblem. Besides, we also extend our analysis to the case of the low-level backlog in buying-online channels (i.e., online and BOPS channels). Finally, some numerical experiments are conducted to illustrate the performance of the inventory control policy for omnichannel retailing and reveal some managerial insights.

Exploring Symmetry-Induced Divergence in Decentralized Electric Vehicle Scheduling

Due to the low computational efficiency and high communication pressure caused by centralized control, decentralized coordination of electric vehicles (EVs) has become a research hotspot in recent years. However, the integration of numerous EVs with homogeneous parameters brings unprecedented challenges for the optimal scheduling of the power system. To explore symmetry-induced divergence in decentralized electric vehicle scheduling, the established centralized optimization scheduling model of electric vehicles is decomposed into a distributed optimization model by Lagrange relaxation, achieving parallel control of each EV aggregator (EVA). To solving the oscillation caused by parameter symmetry in the process of optimization, we propose a convergence acceleration algorithm based on a perturbation function, which is a random distribution function used to break parameter symmetry within an acceptable calculation error range. On the premise of sensitivity analysis, a closed form of the perturbation limit is theoretically derived. Case studies based on different scales of users validate the effectiveness and computational efficiency of the proposed method.

Integrated optimization of planning and operation of a shared automated electric vehicle system considering the trip selection and opportunity cost

&lt;abstract&gt; &lt;p&gt;Shared autonomous electric vehicle systems (SAEVS) combine autonomous driving technology with shared electric vehicle services to provide advantages over traditional shared vehicle systems, including autonomous vehicle relocation and rapid response to user needs. In this study, we seek to enhance the operational efficiency and profitability of SAEVS by considering trip selection and the potential opportunity cost associated with unmet user demands. An integer linear programming (ILP) model is developed using a spatio-temporal state network to optimize the system design planning (e.g., charging facility, vehicle fleet sizing and distribution) and operational decisions (e.g., vehicle operational relocation and trip selection strategy). To handle the computational complexities of this model, we propose a Lagrangian relaxation (LR) algorithm. The performance of the LR algorithm is evaluated through a case study. The results, along with a parameter sensitivity analysis, reveal several key findings: (ⅰ) Allocating vehicles to stations with concentrated early peak demand, distributing charging facilities to stations with high total demand throughout the day and implementing vehicle relocation after the early demand peak can mitigate uneven vehicle distribution; (ⅱ) Implementing trip selection enhances SAEVS profitability; (ⅲ) Increasing opportunity cost meets user demands but at the expense of reduced profit; (ⅳ) It is recommended that SAEVS be equipped with charging facilities of suitable charging power based on operational conditions.&lt;/p&gt; &lt;/abstract&gt;

Probabilistic Electricity Price Forecast With Optimal Prediction Interval

The uncertainty of electricity prices poses a challenge to all market participants as their decisions highly depend on the accuracy of price forecasts. Prediction intervals become an efficient method to quantify the uncertainties that reside in electricity price forecasts. In this paper, we propose a probabilistic electricity price forecast with optimal prediction interval method that considers both reliability and sharpness requirements. Taking reliability and sharpness into account, we ensure the prediction interval has a narrow width without sacrificing reliability. In the proposed method, the quantile regression is utilized to estimate the upper and lower bounds of the prediction intervals to avoid electricity price distribution assumption. In addition, the extreme learning machine (ELM) method is embedded in the forecast method to capture the nonlinear relationship within price data, and its tuning parameters are optimized using the augmented Lagrangian relaxation method in this paper. The effectiveness of the proposed probabilistic forecast method is demonstrated using data from various electricity markets.

Lagrangian Relaxation Based Parallelized Quantum Annealing and Its Application in Network Function Virtualization

Lagrangian Relaxation Based Parallelized Quantum Annealing and Its Application in Network Function Virtualization

New ideas in Lagrangian relaxation for a scheduling problem with the weighted tardiness criterion

New ideas in Lagrangian relaxation for a scheduling problem with the weighted tardiness criterion

A two-point heuristic to calculate the stepsize in subgradient method with application to a network design problem

We introduce a heuristic rule for calculating the stepsize in the subgradient method for unconstrained convex nonsmooth optimization which, unlike the classic approach, is based on retaining some information from previous iteration. The rule is inspired by the well known two-point stepsize by Barzilai and Borwein (BB) [6] for smooth optimization and it coincides with (BB) in case the function to be minimised is convex quadratic.Under the use of appropriate safeguards we demonstrate that the method terminates at a point that satisfies an approximate optimality condition.The proposed approach is tested in the framework of Lagrangian relaxation for integer linear programming where the Lagrangian dual requires maximization of a concave and nonsmooth (piecewise affine) function. In particular we focus on the relaxation of the Minimum Spanning Tree problem with Conflicting Edge Pairs (MSTC). Comparison with classic subgradient method is presented. The results on some widely used academic test problems are provided too.

Synergistic Integration of Machine Learning and Mathematical Optimization for Unit Commitment

Unit Commitment (UC) is important for power system operations. With increasing challenges, e.g., growing intermittent renewables and intra-hour net load variability, traditional mathematical optimization could be time-consuming. Machine learning (ML) is a promising alternative. However, directly learning good solutions is difficult in view of the combinatorial nature of UC. This paper synergistically integrates ML within our recent decomposition and coordination method of Surrogate Lagrangian Relaxation to learn “good enough” subproblem solutions of deterministic UC. Compared to original UC, a subproblem is much easier to learn. Nevertheless, predicting good-enough subproblem solutions is still challenging because of the “jumps” of binary decisions and many types of constraints. To overcome these issues, subproblem dimensionality is reduced via aggregating multipliers. Multiplier distributions are novelly specified based on “jumps” for effective learning. Loss functions are innovatively designed to improve prediction qualities. Ordinal Optimization and branch-and-cut are used as backups for unfamiliar cases. Furthermore, online self-learning is seamlessly integrated with offline learning to exploit solutions from daily operations. Results on the IEEE 118-bus system and the Polish 2383-bus system demonstrate that continual learning keeps on improving the subproblem-solving process with near-optimality of the overall solutions maintained. Our method opens a new direction to solve complicated UC.

Multi-Agent Deep Reinforcement Learning for Spectral Efficiency Optimization in Vehicular Optical Camera Communications

In this paper, we propose a vehicular optical camera communication system that can meet low bit error rate (BER) and ultra-low latency constraints. First, we formulate a sum spectral efficiency optimization problem that aims at finding the speed of vehicles and the modulation order that maximizes the sum spectral efficiency subject to reliability and latency constraints. This problem is mixed-integer programming with nonlinear constraints, and even for a small set of modulation orders, is NP-hard. To overcome the entailed high computational and time complexity which prevents its solution with traditional methods, we first model the optimization problem as a partially observable Markov decision process. We then solve it using an independent Q-learning framework, where each vehicle acts as an independent agent. Since the state-action space is large we then adopt deep reinforcement learning (DRL) to solve it efficiently. As the problem is constrained, we employ the Lagrange relaxation approach prior to solving it using the DRL framework. Simulation results demonstrate that the proposed DRL-based optimization scheme can effectively learn how to maximize the sum spectral efficiency while satisfying the BER and ultra-low latency constraints. The evaluation further shows that our scheme can achieve superior performance compared to radio frequency-based vehicular communication systems and other vehicular OCC variants of our scheme.

Learning-Based Multi-UAV Assisted Data Acquisition and Computation for Information Freshness in WPT Enabled Space-Air-Ground PIoT

The explosion of information data in the power Internet of Things (PIoT) poses difficulties for data acquisition and computation of power devices located in remote areas with limited computing capability and energy. Therefore, we propose a multi-unmanned aerial vehicle (UAV) assisted wireless power transmission (WPT) enabled space-air-ground PIoT framework, in which the UAVs provide energy for data acquisition of power devices through WPT, and use mobile edge computing (MEC) technology to calculate the data. Finally, they act as relays to forward the data to the low earth orbit (LEO) satellite. For keeping the data fresh, we jointly optimize the number of UAV hovering positions, the hovering positions, the connection between UAVs and power devices, the energy transmission and data acquisition time of power devices, the computing resources, flight speed and trajectories of UAVs, aiming to minimizing the average age of information (AoI) of power devices. Through theoretical derivation the problem is decoupled into five independent subproblems, which are solved using improved K-means algorithm, Lagrangian dual decomposition method, interior point method, adaptive optimization, and Q-Learning algorithm, respectively. Extensive numerical simulations demonstrate the superior properties of our solution comparing to benchmark algorithms.

A Cooperative Dynamic Game via Hypergraph Distributed Optimization

We propose a novel cooperative dynamic game with the use of distributed optimization methods. Each coalition has an objective function and the goal of all the coalitions is to optimize the summation of their objectives given the interactions of their agents. The internal interactions are described by a multigraph but we will use directly a stochastic matrix to describe them while the external interactions are described by a hypergraph. The objective functions are quadratic and they encompass information for both types of interactions to highlight their importance in the outcome for the coalitions. The problem is formulated by means of distributed optimization where the actions of the agents at each iteration are the optimum solutions of the sequential optimization problem and the game is evolving by using a tilted matrix mechanism. We show that the optimization problem at each iteration can be solved in one step via a dual decomposition algorithm by exploiting the hypergraph structure of the optimization problem. For the complete description of the dynamic game we propose a switched dynamical system and we demonstrate its convergence numerically.

Distributed Model Predictive Control via Separable Optimization in Multiagent Networks

We present a distributed model predictive control method which enables a group of agents to compute their control inputs locally, while communicating with their neighbors over a communication network. While many distributed model predictive control methods require a central station for some coordination or computation of the optimization variables, our method does not require a central station, making our approach applicable to a variety of communication network topologies. With our method, each agent solves for its control inputs without solving for the control inputs of other agents, allowing for efficient optimization by each agent, unlike some other distributed methods. Further, our method attains linear convergence to the optimal control inputs in convex model predictive control problems, improving upon the sub-linear convergence rates provided by some other distributed methods such as dual decomposition methods. Moreover, our algorithm provides a closed-loop controller for convex model predictive control problems with affine constraints. We demonstrate our method in both convex and non-convex model predictive control problems in wireless transceiver alignment and satellite deployment, where we show robustness of our method to time delays.

Dual core-nilpotent decomposition and dual binary relation

In this paper, we present the definitions of appreciable index and dual index of a dual real square matrix. Then, we introduce the unique dual core-nilpotent (D-C-N) decomposition that exists universally for all dual real square matrices. By applying the decomposition, we get a characterization of the dual Drazin generalized inverse (D-Drazin), introduce D-Drazin, D-C-N, C-sharp C-N (C-C-N) and G-sharp C-N (G-C-N) binary relations of dual real matrices, and discuss relevant properties of these binary relations. We prove that D-Drazin binary relation is a pre-order, and D-C-N, C-C-N and G-C-N binary relations are partial orders. Furthermore, we discuss relationships among binary relations mentioned above.

Efficient asymmetric reposition strategy of rolling stock for urban rail transit systems

Balancing energy efficiency with high-quality service on urban rail lines with highly asymmetric passenger demand is a crucial yet challenging issue. This paper proposes an efficient asymmetric rolling stock reposition strategy to simultaneously optimize the train timetable, speed profiles and rolling stock circulation, given the passenger demand in the undersaturated direction and timetable in the oversaturated direction. By constructing space-time networks for passengers and rolling stocks, respectively, an integer linear programming model is formulated by considering the tradeoff between the passenger travel time and train energy cost. A rolling horizon scheme incorporating the Lagrangian relaxation decomposition technique is designed, which can be decomposed into two subproblems by relaxing the capacity constraints. The effectiveness of our proposed methodology is validated through numerical experiments on a test network and the Beijing Batong Line, demonstrating that the optimized train circulation plan significantly reduces energy consumption while incurring a minimal increase in travel time.

Designing a responsive-sustainable-resilient blood supply chain network considering congestion by linear regression method

Blood Components (BCs) are highly perishable. Perishability, the short lifespan of BCs, and unpredictability in demand volumes make managing the Blood Supply Chain (BSC) more complex. Therefore, it is essential to expand a proper blood network model to tackle uncertainty and minimize the time of delivering blood to patients. This study presents an innovative multi-objective multi-echelon multi-period mixed-integer non-linear model for efficient, responsive, and Green Blood Supply Chain (GBSC) networks considering resilience measures that are divided into three assessment metrics and congestion in Permanent Blood Centers (PBCs). The suggested model aims at significantly reducing prices, Waiting Time (WT) in the system, and environmental damages while simultaneously boosting the supply chain networks' level of resilience. Apart from that, a linear regression is proposed to forecast the demand for blood to decrease the possibility of a shortage of BCs supply in the Blood Supply Chain Network (BSCN). To meet the suggested model, firstly, LP-metric method is used for solving small-sized problem instances. However, this method was inefficient for solving large-sized instances. So, the Lagrangian Relaxation (LR) method is employed. Additionally, the presented decision model's efficacy is evaluated against a case study from real life. Moreover, a sensitivity analysis is undertaken on important problem parameters to offer insightful managerial information.

Resource management for sum-rate maximization in SCMA-assisted UAV system

This work presents a resource management framework for optimizing the sum-rate in a sparse code multiple access (SCMA)-assisted UAV downlink system. We formulate two optimization problems for maximizing the overall sum-rate: the first problem addresses UAV 3D deployment and trajectory optimization with energy constraints, while the second focuses on optimizing SCMA subcarrier and power allocation optimization, subject to factor graph matrix (FGM) constraints and a minimum user data rate. Since the optimization problems are non-convex, the complexity of finding the global optimal solutions is prohibitive. We propose a gradient ascent-based iterative algorithm to compute the optimal UAV 3D deployment and trajectory. Further, an effective channel state information-based algorithm is proposed for FGM assignment, followed by a Lagrange dual decomposition method to solve the power allocation problem efficiently. Our research findings demonstrate that the optimization of the UAV trajectory gives improved sum-rate within the specified energy budget. Further, employing CSI-based multiple subcarrier allocation and strategic power allocation can significantly improve system performance compared to the benchmark schemes.

An effective data placement strategy for IIoT applications

SummaryWith the rapid development of the Internet of things (IoT) and mobile communication technology, the amount of data related to industrial Internet of things (IIoT) applications has shown a trend of explosive growth, and hence edge‐cloud collaborative environment becomes one of the most popular paradigms to place the IIoT applications data. However, edge servers are often heterogeneous and capacity limited while having lower access delay, so there is a contradiction between capacity and latency while using edge storage. Additionally, when IIoT applications deployed crossing edge regions, the impact of data replication and data privacy should not be ignored. These factors often pose challenges to proposing an effective data placement strategy to take full advantage of edge storage. To address these challenges, an effective data placement strategy for IIoT applications is designed in this article. We first analyze the data access time and data placement cost in an edge‐cloud collaborative environment, with the consideration of data replication and data privacy. Then, we design a data placement strategy based on ‐constraint and Lagrangian relaxation, to reduce the data access time and meanwhile limit the data placement cost to an ideal level. As a result, our proposed data placement strategy can effectively reduce data access time and control data placement costs. Simulation and comparative analysis results have demonstrated the validity of our proposed strategy.

A sustainable battery scheduling and echelon utilization framework for electric bus network with photovoltaic charging infrastructure

Battery capacity degradation in battery electric buses (BEBs) poses a significant operational challenge for transit agencies. This study presents a sustainable battery scheduling and echelon utilization framework considering battery capacity fading and charging infrastructure integrated with solar photovoltaic (PV) and energy storage systems. The framework aims to minimize the sum of bus charging, battery replacement, and carbon emission costs during the BEB lifespan. We first present a power battery replacement problem for a single bus fleet, then extend it to a joint power battery replacement and fleet-depot matching problem. Finally, we propose a power battery replacement and fleet-depot problem by introducing solar PV and energy storage systems. Dynamic programming, Lagrange relaxation, and a two-step approach are adopted to solve the three problems. A case study involving six bus depots in Beijing demonstrates that optimal battery replacement schedules can significantly lower charging costs. Moreover, integrating solar PV and energy storage is shown to considerably reduce both charging costs and carbon emissions.

Multi-priority HVAC temperature control with resource constrained based on coordinated distributed zone MPC

The conventional HVAC system for multi-area buildings includes both centralized and in-situ equipment to collectively regulate indoor air temperature to meet the preferences of its occupants. However, due to the limited capacity of the total supply air mass rate in centralized equipment, maintaining comfortable indoor air temperature for all areas can face conflicts. The first conflict arises between satisfying diverse preferences and limited global resources, while the second conflict is between centralized air supply and distributed regulation. In this paper, a distributed zone model predictive control (DZMPC) with priority conflict resolution is proposed for the multi-area HVAC systems. This strategy is composed of a zone parameter optimization with dynamic priority as a coordinated upper layer and a lower DZMPC layer. In the upper layer, a zone parameter optimization with priority is presented to coordinate sub-controllers by adjusting variable references’ bounds for DZMPC, to solve the resource competition when it occurs. In the lower layer, model predictive control with zone control is formed to mitigate the tension of the shared resource. By integrating dual decomposition and Augmented Lagrange function to address global constraints, a distributed solvable control structure is formed. The resulting optimization can be solved using a distributed primal–dual algorithm. The effectiveness of the proposed scheme is proved by simulation examples.

A Two-Warehouse Lot Sizing Problem for Defective Items with a Completely Backlogged Shortage Under Limited Storage Capacity for Rented Warehouses

Determining an economic order quantity across inventory management is vital for any production or distribution company. There is an assumption in classical lot-sizing problems that the inventories are stored in a single warehouse with unlimited capacity. However, the mentioned assumption may be unrealistic, and in many markets, owned warehouses (OW) have limited capacity, therefore, in some cases, a rented warehouse (RW) is considered for storing goods to create more flexibility in terms of space capacity. In this study, we proposed a more realistic extension of a two-warehouse system, which intends to investigate an optimal EOQ considering limited storage for both OW and RW. This paper contributes to considering multiple RWs with limited capacity. For this purpose, a two-objective MINLP lot-sizing problem with a discreet period and finite time horizon is taken into account. The complexity of the model is to determine if there is a need to rent one or multiple warehouses regarding limited storage on OWs and RWs as well as budget restrictions to hire one or multiple warehouses. Also, all parameters are considered to be stochastic which means they follow a specific probability distribution. As the capacity of RWs and the total available budget to rent them are random parameters, the constraint on budget, as well as RW’s capacity in a period, can be formulated as chance constraints. Finally, four multi-objective decision-making methods namely Goal programming, Lp-Metric, Goal Attainment, and Augmented E-Constraint are utilized to solve the concerned problem. Then, the best method is chosen based on graphical, statistical, and TOPSIS analysis. Moreover, the Lagrangian Relaxation method is applied since solving this optimization model in large-size problems requires a considerable amount of time. In the end, the effect of changing the rates of three parameters on objective function values is evaluated through sensitivity analysis.