Scheduling a real-world hybrid flow shop with variable processing times using 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

We consider a car rental network revenue management (RM) problem, accounting for the key operational characteristics of car rental services such as the varying length of rentals and mobility of inventories, which imply the intertemporal and spatial correlations of rental demands for inventories across different locations and days. The problem is formulated as an infinite‐horizon cyclic stochastic dynamic program to account for the time‐varying and cyclic nature of car rental businesses. To tackle the curse of dimensionality , we propose a Lagrangian relaxation (LR) approach with product‐ and time‐dependent Lagrangian multipliers to decomposing the dynamic network problem into multiple single‐station single‐day subproblems. We show that the Lagrangian dual problem is a convex program and then develop a subgradient‐based algorithm to solve the dual problem and derive an LR‐based bid price policy. To improve the scalability of the LR approach, we further propose three simpler LR‐based bid price policy variants with either location‐dependent or leadtime‐dependent Lagrangian multipliers, or both. Our numerical study indicates that the LR‐based bid price policies can outperform some commonly used heuristics. Using a set of real‐world booking data, we provide a case study in which we empirically demonstrate the operational characteristics of car rental services, calibrate the arrival process of booking requests using a Poisson regression model, and demonstrate that the LR‐based bid price policies indeed outperform other heuristics consistently in both in‐sample and out‐of‐sample horizons.

Solving stochastic hydrothermal unit commitment with a new primal recovery technique based on Lagrangian solutions

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.

Single-machine due-window assignment scheduling with delivery times and variable processing times is investigated, where the variable processing time of a job means that the processing time is a function of its position in a sequence and its resource allocation. Currently, there are multiple optimization objectives for the due-window assignment problem, and there is a small amount of research on optimization problems where the window starting time, the rejected cost and the optimal scheduling are jointly required. The goal of this paper is to minimize the weighed sum of scheduling cost, resource consumption cost and outsourcing measure under the optional job outsourcing (rejection). Under two resource allocation models (i.e., linear and convex resource allocation models), the scheduling cost is the weighted sum of the number of early–tardy jobs, earliness–tardiness penalties and due-window starting time and size, where the weights are positional-dependent. The main contributions of this paper include the study and data simulation of single-machine scheduling with learning effects, delivery times and outsourcing cost. For the weighed sum of scheduling cost, resource consumption cost and outsourcing measure, we prove the polynomial solvability of the problem. Under the common and slack due-window assignments, through the theoretical analysis of the optimal solution, we reveal that four problems can be solved in O(n6) time, where n is the number of jobs.

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

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.

We present a methodological framework for solving large‐scale linear programming (LP) problems via a Lagrangian optimization approach. Beginning with a formulation of the Lagrangian dual (LD) problem, we relate this approach to standard LP theory in two steps: first, we demonstrate that the optimal objective function value obtained by solving the Lagrangian dual is optimal to the original linear program; and second, theorems and propositions that characterize the mathematical properties of the Lagrangian dual and establish the equivalence of this approach to classical LP duality are included. Recognizing that the Lagrangian dual formulation is a nondifferentiable optimization problem, a subgradient algorithm is prescribed for solving the LD problem, and related dual convergence results are presented. Furthermore, to recover the primal solution of the original linear program, an improved primal‐dual proximal point algorithm along with its convergence properties is also described. Computational results for a small numerical example serve to illustrate the theoretical and algorithmic concepts elucidated in this article, and finally, a brief extension of this approach for solving integer programs can also be found at the end of this article.

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.

Lagrangian relaxation with subgradient iterations is applied to the problem of optimal restoration of distribution systems after a blackout. The method breaks the restoration horizon into intervals and develops a restoration plan by finding the status of each of the feeders at each time interval. The Lagrangian relaxation approach allows a computationally efficient calculation of the timing and selection of feeders to be energized. The subgradient iterative approach relates to the selection of a vector of Lagrange multipliers in the optimization process. A restoration index obtained from the dual formulation of the problem is also shown. This index identifies the feeders that are closer to restoration. The proposed algorithm applies to radially configured distribution systems and is an operator-permissive, automated approach. Examples are shown including a 100-feeder restoration case.

In a typical circuit optimization flow, one essential decision is to select the implementation for each gate according to a cell library. An implementation implies specific gate size, threshold voltage, etc. The selection normally needs to handle multiple and often conflicting objectives. An effective approach for multi-objective optimization is Lagrangian relaxation (LR), which has been adopted in continuous gate sizing. When LR is applied to the gate implementation selection, the Lagrangian dual problem is no longer convex like in continuous gate sizing, and conventional sub-gradient method becomes inefficient. In this paper, we propose a projection-based descent method and a new technique of Lagrangian multiplier distribution for solving the Lagrangian dual problem in discrete space. Experimental results demonstrate that our approach leads to significantly better solution quality and faster convergence compared to the sub-gradient method.

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.