Lagrangian Relaxation Technique Research Articles

A Reputation-based Distributed District Scheduling Algorithm for Smart Grids

In this paper we develop and test a distributed algorithm providing Energy Consumption Schedules (ECS) in smart grids for a residential district. The goal is to achieve a given aggregate load profile. The NP-hard constrained optimization problem reduces to a distributed unconstrained formulation by means of Lagrangian Relaxation technique, and a meta-heuristic algorithm based on a Quantum inspired Particle Swarm with Levy flights. A centralized iterative reputation-reward mechanism is proposed for end-users to cooperate to avoid power peaks and reduce global overload, based on random distributions simulating human behaviors and penalties on the effective ECS differing from the suggested ECS. Numerical results show the protocols effectiveness.

Using Lagrangian Relaxation to Solve Ready Mixed Concrete Dispatching Problems

The logistics and planning problem of delivering ready mixed concrete (RMC) to a set of demand customers from multiple depots is addressed. The RMC dispatching problem (RMCDP) is closely related to the vehicle routing problem, with the difference that a truck may visit demand nodes in the RMCDP more than once. This class of routing problems can be represented by using mixed-integer programming (MIP) and is known to be NP-hard. Solving RMC delivery problems is often achieved through heuristics and metaheuristic-based methods as exact solution approaches are often unable to find optimal solutions efficiently, in particular when multiple depots are represented in the model. Although a variety of methods are available to solve MIP models, in this paper an attempt is made to solve the RMCDP by using a Lagrangian relaxation technique. Namely, a solution algorithm based on Lagrangian relaxation is derived to reduce the complexity of the initial MIP model and show that the proposed relaxation is able to provide promising computation results on a realistic data set representative of an active RMCDP in the region of Adelaide, Australia.

Energy-efficient design of heterogeneous cellular networks from deployment to operation

The ever-increasing traffic demand has motivated mobile operators to explore how they can boost their network capacity with a minimal increase in their capital and operating expenditures. In order to tackle this problem, we investigate the energy-efficient design of heterogeneous cellular network (or simply HetNet), especially with a focus on deployment and operation strategies. We first formulate a general problem pertaining to minimizing the total energy consumption cost while satisfying the requirement of area spectral efficiency (ASE). We decompose this problem into a deployment problem at peak time and an operation problem at off-peak time. Under practical assumptions made from an observation on various topologies including an acquired real base-station deployment dataset, we demonstrate the submodularity of ASE function with respect to micro base-station deployment. Subsequently, we propose a greedy algorithm that is shown to be a constant-factor approximation to the optimal deployment. Although the greedy algorithm can be applied as an offline centralized solution for the operation problem, we further propose two online distributed algorithms with low complexity and signaling overhead using Lagrangian relaxation technique. Extensive simulations show that the proposed algorithms can significantly reduce the energy consumption with minimal deployment of micro base-stations.

The ship routing and freight assignment problem for daily frequency operation of maritime liner shipping

To cope with excess capacity and improve service quality, maritime international liner carriers have recently adopted a new operational model known as daily frequency. In this new model, carriers provide daily pickup and delivery service to customers at major ports along the Pacific Rim. We investigate the ship routing and freight assignment problem for daily frequency operation of liner shipping. A solution procedure that incorporates a Lagrangian relaxation technique and local search was proposed. The numerical results show that Shanghai, Hong Kong and Singapore are ports that are ideal for carriers in establishing daily frequency operations along the Pacific Rim.

Efficient Solutions Framework for Optimal Multitask Resource Assignments for Data Fusion in Wireless Sensor Networks

Motivated by the need to judiciously allocate scarce sensing resources to attain the highest benefit for the applications that sensor networks serve, in this article we develop a flexible solutions methodology for maximizing the overall reward attained, subject to constraints on the resource demands under fairly general reward or demand functions. We map a broad class of related problems for data fusion in wireless sensor networks into an integer programming problem and provide an iterative Lagrangian relaxation technique to solve it. Each iteration step involves solving for a maximum-weight independent set of an appropriately constructed graph, which, in many cases, can be obtained in polynomial time. We apply our methodology to the problem of tracking targets moving over a period of time through a nonhomogeneous, energy-constrained sensor field. With rewards represented by the quality of information attained in tracking, we study its trade-offs and relationship with energy consumption and periodic measurement taking. We finally illustrate other applications of our framework in sensor networks.

An A* Label-setting Algorithm for Multimodal Resource Constrained Shortest Path Problem

Multimodal resource constrained shortest path problem consists of finding a least travel time path with multiple resources and mode chain constraints on a multimodal transportation network. Existing solution approaches based on Lagrangian relaxation technique cannot take into account the sequence constraint of used modes in a multimodal path. In this study, we propose a fast A*-based label setting algorithm to solve efficiently dynamic multimodal resource constrained shortest path problem. A multilevel graph is proposed for modeling multimodal network provided with time-dependent travel time functions and timetable information. The mode chain constraint of a multimodal path is efficiently examined in a labeled graph by applying non-deterministic finite automata method. A speed-up technique based on the A*-algorithm and Access-Node routing is proposed to effectively reduce search space in the resource constrained route searching process. The computational results show that the proposed algorithm can efficiently find constrained shortest path for realistic multimodal network.

Lagrangian Relaxation for Stochastic Disassembly Line Balancing Problem

This study deals with the problem of profit oriented disassembly line balancing considering partial disassembly, presence of hazardous parts and uncertainty of task times. The objective of this paper is to design a serial line that obtains the maximum profit under uncertainty. Tasks of the best selected disassembly alternative are to be assigned to a sequence of workstations while respecting precedence and cycle time constraints. The line profit is computed as the difference between the positive revenue generated by the retrieved parts of the End of Life (EOL) product and the line operation cost. The latter includes the workstation operation costs and additional costs for handling hazardous parts. Task times are assumed to be random variables with known probability distributions. An AND/OR graph is used to model the disassembly alternatives and the precedence relationships among tasks and subassemblies. To cope with uncertainties, a solution method based on Lagrangian relaxation and Monte Carlo sampling technique is developed. To show the relevance and applicability of the proposed method, it is evaluated on a set of problem instances from the literature.

Lattice BLEU oracles in machine translation

The search space of Phrase-Based Statistical Machine Translation (PBSMT) systems can be represented as a directed acyclic graph (lattice). By exploring this search space, it is possible to analyze and understand the failures of PBSMT systems. Indeed, useful diagnoses can be obtained by computing the so-called oracle hypotheses, which are hypotheses in the search space that have the highest quality score. For standard SMT metrics, this problem is, however, NP-hard and can only be solved approximately. In this work, we present two new methods for efficiently computing oracles on lattices: the first one is based on a linear approximation of the corpus bleu score and is solved using generic shortest distance algorithms; the second one relies on an Integer Linear Programming (ILP) formulation of the oracle decoding that incorporates count clipping constraints. It can either be solved directly using a standard ILP solver or using Lagrangian relaxation techniques. These new decoders are evaluated and compared with several alternatives from the literature for three language pairs, using lattices produced by two PBSMT systems.

Optimal track class assignment for a railroad network

The track class and speed for North American railroads are governed by regulations from the Federal Railroad Administration. Better quality of track permits greater operational speeds and lower transportation cost but also requires greater maintenance expenditure. In this study, we considered the trade-off between maintenance and transportation costs, and developed an optimization framework determining the optimal assignment for track class based upon the characteristics of the track, traffic demand, and maintenance budget. We also presented a solution algorithm by using Lagrangian Relaxation techniques to improve the solution efficiency. This tool can help railroads maintain tracks at their most appropriate level so as to provide reliable and cost-effective services.

Low Power State Assignment Algorithm for FSMs Considering Peak Current Optimization

Finite state machine (FSM) plays a vital role in the sequential logic design. In an FSM, the high peak current which is drawn by state transitions can result in large voltage drop and electromigration which significantly affect circuit reliability. Several published papers show that the peak current can be reduced by post-optimization schemes or Boolean satisfiability (SAT)-based formulations. However, those methods of reducing the peak current either increase the overall power dissipation or are not efficient. This paper has proposed a low power state assignment algorithm with upper bound peak current constraints. First the peak current constraints are weighted into the objective function by Lagrangian relaxation technique with Lagrangian multipliers to penalize the violation. Second, Lagrangian sub-problems are solved by a genetic algorithm with Lagrangian multipliers updated by the subgradient optimization method. Finally, a heuristic algorithm determines the upper bound of the peak current, and achieves optimization between peak current and switching power. Experimental results of International Workshop on Logic and Synthesis (IWLS) 1993 benchmark suites show that the proposed method can achieve up to 45.27% reduction of peak current, 6.31% reduction of switching power, and significant reduction of run time compared with previously published results.

Lagrangian relaxation and cut generation for sequence-dependent setup time flowshop scheduling problems to minimise the total weighted tardiness

The Lagrangian relaxation and cut generation technique is applied to solve sequence-dependent setup time flowshop scheduling problems to minimise the total weighted tardiness. The original problem is decomposed into individual job-level subproblems that can be effectively solved by dynamic programming. Two types of additional constraints for the violation of sequence-dependent setup time constraints are imposed on the decomposed subproblems in order to improve the lower bound. The decomposed subproblem with the additional setup time constraints on any subset of jobs is also effectively solved by a novel dynamic programming. Computational results show that the lower bound derived by the proposed method is much better than those of CPLEX and branch and bound for problem instances with 50 jobs and five stages with less computational effort.

An inventory–routing problem with the objective of travel time minimization

In this paper, we consider an inventory–routing problem (IRP) in a large petroleum and petrochemical enterprise group. Compared to many other IRPs, the problem in this paper includes some special aspects due to the operational constraints, such as hours-of-service regulations of the company and the industry. Also, in some cases, it is more important to avoid stock out for any station, rather than purely focusing on transportation cost minimization. The objective is to minimize the maximum of the route travel time, which is not addressed in the literature so far. We present a tabu search algorithm to tackle the problem, which builds in an efficient and effective procedure to improve the search quality in each iteration. Moreover, lower bounds of reasonable sized problems, which are intractable in the formulated mathematical model by existing optimization software, are obtained via Lagrangian relaxation technique. Computational results indicate that the lower bounds are tight and the tabu search is capable of providing near optimal, close-to-lower-bound solutions in a computational time effective manner.

Extension of dynamic programming models for management optimisation from single structure to multi-structures level

The aim of this article is the management optimisation (inspection, maintenance and rehabilitation (IM&R)) of a group of structures. It is supposed that the optimisation is constrained by limited available budget at the beginning of each time period during the entire time horizon thus creating an economical dependence between the decisions related to each of the structures. A Lagrangian relaxation technique is used for the extension of existing dynamic programming methods from single structure to multi-structures level. The methodology is illustrated by using a Generalised Partially Observable Markov Decision Process having a decision tree composed of a sequence of two decisions at the beginning of each time period, namely an inspection decision followed by a maintenance action decision. A numerical example concerning the optimisation of IM&R of 16 different bridges is presented.

A Finite Capacity of Production Planning Approach for Industrial Manufacturing

This paper discusses a production planning approach that takes the finite capacity of the manufacturing system into account. The approach is based on a reduced level of detail modeling of the process flows. The paper considers explicitly only bottleneck steps. A mixed integer programming (MIP) formulation is suggested to determine completion time targets for the bottleneck steps of the lots. A Lagrange relaxation technique is applied to solve the MIP approximately. Intermediate capacity management is related to rough-cut capacity planning, linked to the master schedule, and capacity requirements planning, and linked to the material requirements plan.

Energy efficient k-barrier coverage in limited mobile wireless sensor networks

Energy cost and reliability are two main concerns in barrier coverage for wireless sensor networks. In this paper, we take the energy cost and reliability as objectives respectively to study two problems of k-barrier coverage: the minimum energy cost k-barrier coverage problem in static wireless sensor networks and the maximum k-barrier coverage problem in limited mobile wireless sensor networks. For the minimum energy cost k-barrier coverage problem, all sensors are stationary, and each sensor has l+1 sensing power levels in the network, the objective of the problem is to find a sensing level assignment to form k-barrier coverage such that the total power consumed by the k-barrier is minimized. We firstly transform it into a minimum cost flow problem with side constraints and use Lagrangian relaxation technique to solve the minimum cost flow problem. Then, we also propose a heuristic algorithm. For the maximum k-barrier coverage problem, each sensor can move within the limited range, the objective of the problem is to form more barriers while some sensors can move within limited range. We formulate the problem into an integer linear programming (ILP), then propose two heuristic algorithms based on the linear programming (LP) relaxation. The simulation results demonstrate our algorithms are efficient.

Virtual enterprises partner selection based on reverse auctions

Partner selection is a key issue in the development of an effective coalition formation mechanism for virtual enterprises (VE). Combinatorial reverse auction can be applied by a firm to select the best partners to minimize the cost in forming VE. The objectives of this paper are to propose architecture for selecting partners based on combinatorial reverse auction mechanism to minimize the cost of VE, develop algorithms to find a near-optimal solution efficiently, and implement a prototype system based on the proposed algorithms. We formulate the partner selection problem based on combinatorial reverse auctions and apply Lagrangian relaxation technique to solve the problem. Our partner selection solution algorithms include an algorithm for solving bidders’ subproblems by exploiting their problem structures, a subgradient algorithm for solving the dual problem, and a heuristic algorithm for finding a near-optimal solution. In addition to theoretical development, we also implement a prototype system based on the proposed algorithms and web services technologies to verify the effectiveness of our methodology.

Implementation of scheduling algorithm for optimisation by Lagrangian decomposition model

This paper mainly deals with the design and analysis of Lagrangian model using cyclic scheduling algorithm. An integer programming (IP) model is formulated and then decomposed by using Lagrangian relaxation technique. Two approaches were implemented, the first approach is based on the relaxation of the preference constraints and the second approach is based on the relaxation of the demand constraints. Since proper bounds were not obtained by relaxing the preference constraints, a solution methodology is developed based on the relaxation of demand constraints. Many researchers have discussed the different type of methods, such as bundle method, heuristic and variable fixing. A duality gap exists between the lower bound and the objective function value provided by an IP feasibility heuristic. A variable fixing scheme is introduced for the convergence criteria. Based on the numerical calculations and graphical representations, the minimum cost is obtained to cover the uncovered shifts.

Approximation Approach to Problem of Large-Scale Pavement Maintenance and Rehabilitation

This study focuses on large-scale pavement maintenance and rehabilitation planning. Pavement agencies usually deal with large and diverse networks of thousands of pavement sections. The complexity of pavement maintenance and rehabilitation planning increases exponentially with the size of the network. Exact optimization methods designed for small-scale problems thus suffer from dimensionality, which significantly compromises the ability to solve such problems in a reasonable time period. Consequently, such problems should be approached from the technique of approximation. A method with the Lagrangian relaxation technique and a branch-and-cut algorithm is proposed for use as a decomposition scheme for pavement maintenance and rehabilitation planning problems when the size of the problem poses a major hurdle for its solution. The proposed approach first decomposes the network (original problem) into individual sections (subproblems) and then solves the subproblems separately. As shown in the case study, the proposed approach is able to solve problems with a practical size efficiently. More specifically, the results show that the proposed method can solve problems with a network of 1,000 sections within a reasonable time period and can yield a solution close to the optimal one.

A heuristic algorithm based on Lagrangian relaxation for the closest string problem

The closest string problem that arises in both computational biology and coding theory is to find a string minimizing the maximum Hamming distance from a given set of strings. This study proposes an efficient heuristic algorithm for this NP-hard problem. The key idea is to apply the Lagrangian relaxation technique to the problem formulated as a mixed-integer programming problem. This enables us to decompose the problem into trivial subproblems corresponding to each position of the strings. Furthermore, a feasible solution can be easily obtained from a solution of the relaxation. Based on this, a heuristic algorithm is constructed by combining a Lagrangian multiplier adjustment procedure and a tabu search. Computational experiments will show that the proposed algorithm can find good approximate solutions very fast.

Simultaneous Technology Mapping and Placement for Delay Minimization

Technology mapping and placement have a significant impact on delays in standard cell-based very large scale integrated circuits. Traditionally, these steps are applied separately to optimize the delays, possibly since efficient algorithms that allow the simultaneous exploration of the mapping and placement solution spaces are unknown. In this paper, we present an exact polynomial time algorithm for delay-optimal placement of a tree and extend the same to simultaneous technology mapping and placement for the optimal delay in the tree. We extend the algorithm by employing Lagrangian relaxation technique, which assesses the timing criticality of paths beyond a tree, to optimize the delays in directed acyclic graphs. Experimental results on benchmark circuits in a 70 nm technology show that our algorithms improve timing significantly with remarkably less runtimes compared to a competitive approach of iterative conventional timing-driven mapping and multilevel placement.