Lagrangian Relaxation Model Research Articles

Linearization Method for Large-Scale Hydro-Thermal Security-Constrained Unit Commitment

Security-constrained unit commitment (SCUC) is one of the most fundamental optimization problems in power systems. The objective of SCUC is to minimize the operating cost while respecting both system-wide and generator-specific constraints. It leads to a large-scale and mixed-integer programming (MIP) model with a large number of binary decision variables which is difficult to solve. This paper, based on the convex hull theory of single-unit, proposes a linearization method for the hydro-thermal SCUC problem with decoupled thermal units and variable-head hydro units. Then, the strategy of embedding two types of convex hulls in a multi-unit commitment and the heuristic method of constructing a feasible solution are designed, by which the multi-UC is approximated from large-scale mixed-integer programming to linear programming that can be solved in polynomial time. Finally, we theoretically prove that the optimal solution of the proposed LP model is always better than that of the Lagrangian Relaxation model. Numerical experiments on several large-scale test systems demonstrate the effectiveness and efficiency of the proposed method. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper proposes a linear programming model for the SCUC problem by lifting up to a higher-dimensional space. It realizes an important innovation in reducing the computational complexity of SCUC from the perspective of linearization. The proposed method can be well applied to large-scale long-term unit commitment problems. To better use this method, the following two properties should be highlighted: 1) the error of the proposed method is less than the Lagrangian relaxation method and decreases with the increasing system scales and 2) the computational efficiency of the proposed method is 10-100 times faster than that of the MIP model. We have tested many practical power systems and find that the error of the proposed LP model is usually very small compared with the precise MIP while the computational performance is significantly improved. In some practical cases, the decision makers usually do not want to find the precise optimal solution while only an approximation under a fast speed, because the boundary condition is imprecise. The proposed method is useful. Besides, for the cases that need the precise optimal solution, the proposed method can provide a high-quality initial solution for the MIP model to accelerate the convergence.

Hybridization in meta-heuristic techniques for green layout design in the tyre manufacturing industry: An optimal analysis

To stop or lessen factory pollution, many academics in the field of study have produced a vast array of strategies to achieve optimization goals. But, it is unable to follow a uniform solution for all production facilities. In this research work, green layout design (GLD) is proposed to minimize total material handling cost (TMHC) and CO2 emissions. The green facility layout refers to the facility layout with minimum CO2 emissions and to eliminate unnecessary energy dissipation from material handling equipments &amp; production machineries, as well as plant re-layout in-term on material handling cost. In this study context, a Lagrangian relaxation model has been used at the allocation level. It computes a suboptimal solution to the dual problem in order to obtain a feasible solution. This enhanced algorithm method iteratively tries to minimize the gap between the dual and feasible solutions on minimal of TMHC and CO2 emission rate. It has been experimentally simulated using the MATLAB (Matrix Laboratory) application and upgraded adaptive salp swarm optimization (ASSO), dragon fly optimization (DFO) and Hybrid (combination) of ASSO-DFO evolutionary optimization techniques. The primary focus of this implementation is to control material handling between departments or cells, production flow between departments, energy consumption (electrical and gas consumption for production), and manpower utilization within and across departments. This research has chosen an existing and acceptable designs from targeted tyre manufacturing company in South India. The hybrid ASSO-DFO implementation TMHC differs from ABC by 60%, DFO by 65%, 58% with simulated annealing (SA), and ASSO by 48%. Emission rate decreased by 26% ABC, 21% DFO, and 24% SA &amp; 18% with ASSO. An average of 50% decrease in TMHC and 25% reduction of CO2 emission rate in GLD have been achieved.

Optimization Techniques for Green Layout Design in Manufacturing Industries: A Meta-Heuristic Analysis

Many research papers and much legislation has been published in recent years to control or reduce factory pollution. However, only a few articles have discussed pollution from manufacturing facilities, specifically shop floors, even though this a specific single objective problem. In this research framework, a new variant technique of the jelly fish concept adaptive salp swarm optimization (ASSO) with a familiar Lagrangian relaxation model for lowering Total Material Handling Costs (TMHC) and carbon dioxide (CO2) emissions is presented. Using the Mat Lab software and the improved ASSO, the dragon fly optimization (DFO) algorithm technique, experimental simulations of the existing and recognized design of the studied industry were performed. The simulation results were validated and compared to those of other optimization techniques such as ant bee colony (ABC), simulated annealing (SA), and genetic algorithm (GA). It was determined that the proposed methodology, ASSO, was the most efficient, resulting in 40 % reduction compared to ABC, 38 % DFO, 50 % SA, and 40 % GA in the lowest TMHC, as well as an average 20 % reduction of emission rate in green layout design. These techniques could be combined into a hybrid format for further reduction of the emission rate up to 80 %.

Energy-efficient power allocation for IoT devices in CR-NOMA networks

Non-orthogonal multiple access is a promising technique to meet the harsh requirements for the internet of things devices in cognitive radio networks. To improve the energy efficiency (EE) of the unlicensed secondary users (SU), a power allocation (PA) algorithm with polynomial complexity is investigated. We first establish the feasible range of power consumption ratio using Karush-Kuhn-Tucker optimality conditions to support each SU's minimum quality of service and the effectiveness of successive interference cancellation. Then, we formulate the EE optimization problem considering the total transmit power requirements which leads to a non-convex fractional programming problem. To efficiently solve the problem, we divide it into an inner-layer and outer-layer optimization sub-problems. The inner-layer optimization which is formulated to maximize the sub-carrier PA coefficients can be transformed into the difference of convex programming by using the first-order Taylor expansion. Based on the solution of the inner-layer optimization sub-problem, the concave-convex fractional programming problem of the outer-layer optimization sub-problem may be converted into the Lagrangian relaxation model employing the Dinkelbach algorithm. Simulation results demonstrate that the proposed algorithm has a faster convergence speed than the simulated annealing algorithm, while the average system EE loss is only less than 2%.

A joint order acceptance and scheduling problem with earliness and tardiness penalties considering overtime

This paper considers a joint order acceptance and scheduling problem under a general scenario. A manufacturer receives multiple orders with a given revenue, processing time, release date, due date, deadline, and earliness and tardiness penalties. The manufacturer can be seen as a single-machine system. Due to limited capacity, the manufacturer cannot process every order and needs to determine the optimal set of accepted orders and corresponding production schedule such that the total profit is maximized. The manufacturer can extend its capacity with overtime by paying an additional cost. A time-indexed formulation is presented to model the problem. Two exact algorithms are proposed. The first algorithm, denoted by DPIA-GR, is a dynamic programming (DP)-based algorithm that starts by solving a relaxed version of the original model and successively recovers the relaxed constraint until an optimal solution to the original problem is achieved. The second algorithm, denoted by DPIA-LRGR, improves DPIA-GR by incorporating Lagrangian relaxation (LR). The subgradient method is employed to find the optimal Lagrangian multipliers. The relaxed model in DPIA-GR and the LR model in DPIA-LRGR can be represented using a weighted di-graph. Both algorithms are equivalent to finding the longest path in the graph and applying a graph reduction strategy to prevent unnecessary computational time and memory usage. A genetic algorithm (GA) is also proposed to solve large-scale versions of the problem. Numerical experiments show that both DPIA-GR and DPIA-LRGR solve the problem efficiently and outperform CPLEX and GA, but DPIA-LRGR offers better performance.

Lagrangian relaxation for the directional sensor coverage problem with continuous orientation

We present a mixed integer nonlinear programming formulation of the Directional Sensors Continuous Coverage Problem (DSCCP), where a given set of targets on a plane are to be covered by a set of sensors whose locations are known. Sensors are supposed to be directional, that is characterized by a discrete set of possible radii and aperture angles. The orientation (which is a continuous variable), the radius and the aperture angle of each sensor are to be decided, taking into account possibility of keeping one or more sensors switched off. The objective is to minimize cost coverage. We incorporate into the objective function penalty cost for possibly uncovered targets.We prove NP-hardness of DSCCP and introduce a Lagrangian relaxation model. We design a dual ascent procedure based on acting on a single multiplier at a time accompanied by a heuristics to find a feasible solution at each ascent iteration. We report also the results of the method on several test problems.

A combination of Lagrangian relaxation and column generation for order batching in steelmaking and continuous-casting production

This article considers the order batching problem in steelmaking and continuous-casting production. The problem is to jointly specify the slabs needed to satisfy each customer order and group all the slabs of different customer orders into production batches. A novel mixed integer programming model is formulated for the problem. Through relaxing the order assignment con- straints, a Lagrangian relaxation model is then obtained. By exploiting the relationship between Lagrangian relaxation and column generation, we develop a combined algorithm that contains nested double loops. At the inner loop, the subgradient method is applied for approximating the Lagrangian dual problem and pricing out columns of the master problem corresponding to the linear dual form of the Lagrangian dual problem. At the outer loop, column generation is employed to solve the master problem exactly and adjust Lagrangian multipliers. Computational experiments are carried out using real data collected from a large steel company, as well as on large-scaled problem instances randomly generated. The results demonstrate that the combined algorithm can obtain tighter lower bound and higher quality solution within an acceptable computation time as compared to the conventional Lagrangian relaxation algorithm. © 2011 Wiley Periodicals, Inc. Naval Research Logistics 58: 370-388, 2011

Optimization of multi-plants and SCM strategies with Lagrangian relaxation model

In this paper develop a optimization of multi-plants and Supply Chain Management strategies with Lagrangian relaxation model. Optimal product reliability with minimum total cost is derived using relaxation model analysis in a Supply Chain Management multisite collaboration system. This research also considers supply policyto construct the multiechelon system dynamic simulation model of supply system, and the simulation results which probes into the variations of each products reliability and mission capabilities

An Efficient Procedure for Designing Single Allocation Hub and Spoke Systems

Given the widespread use of the hub and spoke network architecture and its growing importance to competitiveness in logistics, communication, and mass transportation, there has been considerable interest by practitioners and researchers alike in finding efficient methods for designing such networks. This paper provides a method that delivers both high quality solutions and firm measures of that quality, and allows problems to be solved in reasonable time on a desktop computer. The approach begins with a previously proposed tight linear programming formulation and uses subgradient optimization on a lagrangian relaxation of the model. However, to dramatically improve the performance of this approach, we augment a subproblem of the lagrangian relaxation model with a cut constraint. In computational experiments on eighty-four standard test problems, average gaps are 0.048%. Maximum gaps are under 1% while average solution times on a Pentium-166 are under five minutes.