scholarly journals An Improved Lagrangian Relaxation Algorithm for the Robust Generation Self-Scheduling Problem

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Ping Che ◽  
Zhenhao Tang ◽  
Hua Gong ◽  
Xiaoli Zhao
Keyword(s):  
Lagrangian Relaxation ◽  
Single Unit ◽  
Mixed Integer ◽  
Scheduling Problem ◽  
Two Stage ◽  
Splitting Technique ◽  
Relaxation Algorithm ◽  
Special Cases ◽  
Lagrangian Relaxation Algorithm ◽  
Relaxed Problem

The robust generation self-scheduling problem under electricity price uncertainty is usually solved by the commercial solver, which is limited in computation time and memory requirement. This paper proposes an improved Lagrangian relaxation algorithm for the robust generation self-scheduling problem where the quadratic fuel cost and the time-dependent exponential startup cost are considered. By using the optimal duality theory, the robust generation self-scheduling problem, which has a max-min structure, is reformulated as a minimization mixed integer nonlinear programming (MINLP) problem. Upon the reformulation, the Lagrangian relaxation algorithm is developed. To obtain a solvable relaxed problem, the variable splitting technique is introduced before the relaxation. The obtained relaxed problem is decomposed into a linear programming-type subproblem and multiple single-unit subproblems. Each single-unit subproblem is solved optimally by a two-stage backward dynamic programming procedure. The special cases of the problem are discussed and a two-stage algorithm is proposed. The proposed algorithms are tested on test cases of different sizes and the numerical results show that the algorithms can find near-optimal solutions in a reasonable time.

scholarly journals Lagrangian relaxation algorithm for the truck scheduling problem with products time window constraint in multi-door cross-dock

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Binghai Zhou ◽  
Yuanrui Lei ◽  
Shi Zong
Keyword(s):  
Lagrangian Relaxation ◽  
Time Window ◽  
Transportation Cost ◽  
Scheduling Problem ◽  
Relaxation Algorithm ◽  
Physical Conditions ◽  
Cross Docking ◽  
Truck Scheduling ◽  
Storage Area ◽  
Lagrangian Relaxation Algorithm

<p style='text-indent:20px;'>Cross-docking is a kind of process that products are unloaded in front of the inbound doors, consolidated based on the downstream demand, and then directly transferred to the outbound doors without a long storage process during the transportation. In this paper, a multi-door cross-dock truck scheduling problem is investigated in which the scheduling and sequencing assignment of trucks need to be considered, with the objectives of minimizing the inner transportation cost in the cross-dock and the total truck waiting cost. The major contribution of this paper is that a novel product-related time window constraint and the temporary storage area are firstly introduced to adapt to different physical conditions of goods considering real-world requirements. Then, a Lagrangian relaxation algorithm is proposed which aims to decompose the relaxed problem into several easy-to-be-solved sub-problems. Besides, a subgradient algorithm is used at each iteration to further deal with these sub-problems. Finally, theory analysis and simulation experiments of different problem scales are carried out during the comparison with a Greedy algorithm to evaluate the performance of the proposed algorithm. Results indicate that the Lagrangian relaxation algorithm is able to achieve more satisfactory near-optimal solutions within an acceptable time.</p>

Multi Project Scheduling and Material Planning Using Lagrangian Relaxation Algorithm

2021 ◽  
Vol 20 (4) ◽  
pp. 580-587
Author(s):  
Alim Al Ayub Ahmed ◽  
Ngakan Ketut Acwin Dwijendra ◽  
NareshBabu Bynagari ◽  
A.K. Modenov ◽  
M. Kavitha ◽  
...  
Keyword(s):  
Lagrangian Relaxation ◽  
Project Scheduling ◽  
Relaxation Algorithm ◽  
Lagrangian Relaxation Algorithm

scholarly journals Convex approximations for two-stage mixed-integer mean-risk recourse models with conditional value-at-risk

2019 ◽  
Vol 181 (2) ◽  
pp. 473-507 ◽  
Author(s):  
E. Ruben van Beesten ◽  
Ward Romeijnders
Keyword(s):  
At Risk ◽  
Error Bounds ◽  
Value At Risk ◽  
Risk Measure ◽  
Mixed Integer ◽  
Conditional Value At Risk ◽  
Two Stage ◽  
Special Cases ◽  
Integer Recourse ◽  
Mean Risk

Abstract In traditional two-stage mixed-integer recourse models, the expected value of the total costs is minimized. In order to address risk-averse attitudes of decision makers, we consider a weighted mean-risk objective instead. Conditional value-at-risk is used as our risk measure. Integrality conditions on decision variables make the model non-convex and hence, hard to solve. To tackle this problem, we derive convex approximation models and corresponding error bounds, that depend on the total variations of the density functions of the random right-hand side variables in the model. We show that the error bounds converge to zero if these total variations go to zero. In addition, for the special cases of totally unimodular and simple integer recourse models we derive sharper error bounds.

scholarly journals Solving the Short-Term Scheduling Problem of Hydrothermal Systems via Lagrangian Relaxation and Augmented Lagrangian

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Rafael N. Rodrigues ◽  
Edson L. da Silva ◽  
Erlon C. Finardi ◽  
Fabricio Y. K. Takigawa
Keyword(s):  
Lagrangian Relaxation ◽  
Large Scale ◽  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Real Life ◽  
Hydrothermal Systems ◽  
Bundle Method ◽  
Mixed Integer ◽  
Scheduling Problem ◽  
Short Term

This paper addresses the short-term scheduling problem of hydrothermal power systems, which results in a large-scale mixed-integer nonlinear programming problem. The objective consists in minimizing the operation cost over a two-day horizon with a one-hour time resolution. To solve this difficult problem, a Lagrangian Relaxation (LR) based on variable splitting is designed where the resulting dual problem is solved by a Bundle method. Given that the LR usually fails to find a feasible solution, we use an inexact Augmented Lagrangian method to improve the quality of the solution supplied by the LR. We assess our approach by using a real-life hydrothermal configuration extracted from the Brazilian power system, proving the conceptual and practical feasibility of the proposed algorithm. In summary, the main contributions of this paper are (i) a detailed and compatible modelling for this problem is presented; (ii) in order to solve efficiently the entire problem, a suitable decomposition strategy is presented. As a result of these contributions, the proposed model is able to find practical solutions with moderate computational burden, which is absolutely necessary in the modern power industry.

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