Distribution System Restoration via Subgradient-Based Lagrangian Relaxation

A Lagrangian relaxation (LR) approach was presented which is with machine capacity relaxation and operation precedence relaxation for solving a flexible job shop (FJS) scheduling problem from the steelmaking-refining-continuous casting process. Unlike the full optimization of LR problems in traditional LR approaches, the machine capacity relaxation is optimized asymptotically, while the precedence relaxation is optimized approximately due to the NP-hard nature of its LR problem. Because the standard subgradient algorithm (SSA) cannot solve the Lagrangian dual (LD) problem within the partial optimization of LR problem, an effective deflected-conditional approximate subgradient level algorithm (DCASLA) was developed, named as Lagrangian relaxation level approach. The efficiency of the DCASLA is enhanced by a deflected-conditional epsilon-subgradient to weaken the possible zigzagging phenomena. Computational results and comparisons show that the proposed methods improve significantly the efficiency of the LR approach and the DCASLA adopting capacity relaxation strategy performs best among eight methods in terms of solution quality and running time.

Convoy movement problem (CMP) is an important optimisation problem, especially occurring in military logistics, which focuses on routing convoys concurrently in a network. The CMP is subject to constraints for preventing certain interactions between convoys such that the objectives related to travel times of them are minimised. We propose a Lagrangian relaxation (LR) approach to efficiently obtain a lower bound on the optimal solution. Our LR approach does not have integrality property, and hence the lower bound obtained is always tigher than the lower bound obtained using the LP relaxation. As CMP is an NP-complete problem, lower bounds are useful to evaluate the solutions obtained by a heuristic. From our computational experimentation, it is found that the proposed LR approach is superior to the existing LR approach.

An alternative framework to Lagrangian relaxation approach for job shop scheduling

This paper addresses the short-term hydrothermal coordination problem. This problem is large-scale, combinatorial and nonlinear. It is usually solved using a Lagrangian relaxation approach. In the framework of the Lagrangian relaxation, this paper provides a novel, nonoscillating and efficient multiplier updating procedure. This procedure advantageously compares with previously reported procedures such us subgradient and bundle methods. A realistic large-scale case study is used to illustrate the behavior of the proposed procedure.

The prize collecting traveling salesman problem (PCTSP) is a generalization of the traveling salesman problem (TSP) in which a tour starting at a root node must visit a subset of nodes to collect a prescribed amount of total prize minimizing the summation of travel costs and penalty costs associated with nonvisited nodes. We propose new formulations and compare them with adaptations to the PCTSP of strong formulations from the literature for the asymmetric TSP. One of the formulations proposed in this paper provided the tighter linear relaxation bounds on computational experiments based on benchmark instances for the asymmetric TSP. We then develop a Lagrangian relaxation approach embedding heuristic schemes to obtain both lower and upper bounds. The proposed heuristic has shown to be successful in obtaining optimal or near‐optimal solutions with information from the Lagrangian subproblems. Our Lagrangian approach can thereupon provide small optimality gaps, being an efficient alternative to bound optimal values of large instances.

A Lagrangian relaxation approach to combinatorial exchange in freight logistics

This research addresses scheduling problem of n jobs on a Hybrid Flow Shop (HFS) with unrelated parallel machines in each stage. A monolithic mixed integer linear programming (MILP) model is presented to minimize the maximum completion time (makespan). As the research problem is shown to be strongly NP-hard, a Lagrangian relaxation (LR) algorithm is developed to handle the HFS scheduling problem. We design two approaches, simplification of subproblems and dominance rules, to solve the subproblems which are generated in each iteration. For evaluation purposes, numerical experiments with small and large size problems are randomly generated with up to 50 jobs and four stages. The experimental results show that the Lagrangian relaxation approaches outperform the MILP model with respect to CPU time. Furthermore, from the results, it can be conclude that the simplification of subproblems shows slightly better solutions in comparison with dominance rules to solve the subproblems.

Distributed Lagrangian Relaxation Protocol (DisLRP) has been proposed to solve a distributed combinatorial maximization problem called the Generalized Mutual Assignment Problem (GMAP). In DisLRP, when updating Lagrange multipliers (prices) of goods, the agents basically control their step length, which determines the degree of update, by a static rule. A merit of this updating rule is that since it is static, it is easy to implement even without a central control. Furthermore, if we choose this static rule appropriately, we have observed empirically that DisLRP converges to a state providing a good upper bound. However, it must be difficult to devise such a good static rule for updating step length since it naturally depends on problem instances to be solved. On the other hand, in a centralized context, the Lagrangian relaxation approach has conventionally computed step length by exploiting the least upper bound obtained during the search and a lower bound obtained through preprocessing. In this paper, we achieve this approach in a distributed environment where no central control exists and name the resultant protocol Adaptive DisLRP (ADisLRP). The key ideas of this new protocol are to 1) compute global information with a spanning tree, 2) update step length simultaneously with a synchronization protocol, and 3) estimate lower bounds during the search. We also show the robustness of ADisLRP through experiments where we compared ADisLRP with the previous protocols on the critically hard benchmark instances.

This paper proposes an enhanced differential evolution (EDE) approach to solve the problem of service restoration in distribution systems. Service restoration of distribution systems is to restore power to the blacked out but unfaulted area. Basically, it is a combinatorial optimization problem that often has a great number of candidate solutions to be evaluated by the system operators. To tackle the problem of service restoration, an EDE method which combines variable scaling DE and ant colony algorithm is employed to solve the multi-objective optimization problem. The proposed method was tested on a typical distribution system of the Taiwan Power Company (TPC) and compared with the heuristic-based fuzzy algorithm and classical DE method. Results show that the proposed method was superior to that of the existing methods in terms of convergence time and obtained restoration plans.

The new energy resources in distribution systems have led researchers to consider the operation of such elements during service restoration. Hence, the current state-of-the-art regarding the restoration problem considers the operation of distributed resources in both island and grid-connected modes. Few articles also address the system's restoration capacity changes throughout the time needed to repair the network after an outage, given the variations in the system's demand and availability of energy sources. However, the solutions proposed in the literature converge in proposing a single topology that is feasible throughout the entire repairing time. In this article, we propose addressing the mutable nature of the system's restoration capacity by splitting the repairing time into several time intervals and associating each of them with a specific configuration that minimizes the out-of-service load. Energy storage systems, intermittent and dispatchable generators, and voltage-control devices are also considered. The problem is formulated as a mixed-integer linear programming (MILP). The model and its scalability are tested using the 53-, 217-, and 404-node systems, considering different fault scenarios and reliability indices. The results show that the nonsupplied energy during the repairing time decreases when dynamic changes in the system topology are considered. Furthermore, the effects of storage systems operation and their integration with dynamic topology changes are highlighted.

The authors present a new method for determining the unit-commitment schedule of a power system which operates under fuel utilisation constraints. The proposed method employs the Lagrangian relaxation approach to determine a feasible suboptimal schedule. Then, linear programming is applied to improve the feasible solution, as well as seeking the optimal economic dispatch of the committed generators. The algorithm is implemented on a VAX11/780 machine to solve the unit-commitment problem of a system which consists of 26 thermal units and 6 fuel-constrained units. Numerical results presented in this paper describe the usefulness and practicality of the proposed method.

This paper studies a real-world hybrid flow shop problem arising from the steelmaking continuous casting process, which is the bottleneck of the iron and steel production. There are a variety of features to be taken into account, in particular the batch constraints and the variable processing times in the last stages. Based on a time-index formulation and machine capacity relaxation, three Lagrangian relaxation (LR) approaches are presented for addressing this scheduling problem. The three LR approaches decompose the relaxed problem into job-level problems, batch-level problems and machine-level problems, respectively. These subproblems are solved based on polynomial dynamic programming algorithms. The corresponding Lagrangian dual (LD) problems are solved by an efficient subgradient algorithm with global convergence. Computational results and comparisons demonstrate that the approach adopting job-level decomposition is most efficient among three approaches, whereas the approach adopting batch-level decomposition is most effective.

PurposeThis study aims to design a supply chain network in an uncertain environment while exists two options for distribution of the perishable product and production lot-sizing is concerned.Design/methodology/approachOwing to the complexity of the mathematical model, a solution approach based on a Lagrangian relaxation (LR) heuristic is developed which provides good-quality upper and lower bounds.FindingsThe model output is discussed through various examples. The introduction of some enhancements and using some heuristics results in better outputs in the solution procedure.Practical implicationsThis paper covers the modeling of some real-world problems in which demand is uncertain and managers face making some concurrent decisions related to supply chain management, transportation and logistics and inventory control issues. Furthermore, considering the perishability of product in modeling makes the problem more practically significant as these days there are many supply chains handling dairy and other fresh products.Originality/valueConsidering uncertainty, production, transshipment and perishable product in the inventory-routing problem makes a new variant that has not yet been studied. The proposed novel solution is based on the LR approach that is enhanced by some heuristics and some valid inequalities that make it different from the current version of the LR used by other studies.

This paper describes a new approximate maximum-likelihood (ML) MIMO detection approach by studying a Lagrangian dual relaxation (LDR) of ML. Unlike many existing relaxed ML methods, the proposed LDR employs a discrete domain for the problem formulation. We find that the proposed LDR exhibits an intriguing relationship to the lattice decoders (LDs) and the lattice reduction aided (LRA) detectors, both of which have caught much attention recently. Specifically, regularization in LDs, which was proposed to mitigate out-of-bounds symbol effects, can alternatively be interpreted as a way to constrain the symbol decision within bounds in a Lagrangian sense. We handle the LDR problem by using a projected subgradient method. The resultant method may physically be viewed as an adaptive regularization control in which a sequence of LDs are involved. Based on this newly developed insight, we propose two additional iterative LDR-based detectors using LRA decision-feedback (DF) and "lazy" DF. By simulation results, we show that the LDR LRA-DF and lazy-DF detectors yield better symbol error rate performance than the MMSE-regularized LRA-DF and DF detectors, respectively, where the SNR gaps can be more than 3dB.

Job shop scheduling problems with weighted quadratic tardiness costs are formulated as a class of transition firing problems of timed Petri nets, and a sequential Lagrangian relaxation approach is proposed to solve the scheduling problems. It is shown that the approach can not only avoid the solution oscillation in the Lagrangian relaxation approach proposed by Hoitomt and Luh, but also generate good near-optimal schedules. >