Finding minimum flow time cyclic schedules for non-identical, multistage jobs

Finding a minimum flow time cyclic schedule for a single, multistage job with a serial, re-entrant routing is known to be NP-hard. This paper addresses the problem of scheduling multiple, non-identical jobs in a cyclic fashion, where the job routings may be arbitrary partial orders as well as re-entrant. Given a fixed cycle length, our goal is to minimize a weighted sum of the job flow times. We present a general schedule construction algorithm for implementing a cyclic version of priority dispatch rules that accepts any user-defined tie-breaking function and naturally yields a feasible cyclic schedule. We also describe a pair of easily solvable subproblems that may be used to tighten existing cyclic schedules, as well as an iterative schedule improvement algorithm based on a technique called compression. A numerical study suggests that our schedule construction algorithm, called Cyclic PDR, outperforms its traditional noncyclic priority dispatch rule counterpart, as well as a previously proposed single-pass algorithm. The Cyclic PDR algorithm is shown to be particularly effective when used in conjunction with a least work remaining tie-breaking function. Taken together, our schedule construction and improvement techniques provide an effective solution approach for producing minimum flow time cyclic schedules.     

An optimization-based algorithm for job shop scheduling

Scheduling is a key factor for manufacturing productivity. Effective scheduling can improve on-time delivery, reduce inventory, cut lead times, and improve the utilization of bottleneck resources. Because of the combinatorial nature of scheduling problems, it is often difficult to find optimal schedules, especially within a limited amount of computation time. Production schedules therefore are usually generated by using heuristics in practice. However, it is very difficult to evaluate the quality of these schedules, and the consistency of performance may also be an issue. In this paper, near-optimal solution methodologies for job shop scheduling are examined. The problem is formulated as integer optimization with a “separable” structure. The requirement of on-time delivery and low work-in-process inventory is modelled as a goal to minimize a weighted part tardiness and earliness penalty function. Lagrangian relaxation is used to decompose the problem into individual part subproblems with intuitive appeal. By iteratively solving these subproblems and updating the Lagrangian multipliers at the high level, near-optimal schedules are obtained with a lower bound provided as a byproduct. This paper reviews a few selected methods for solving subproblems and for updating multipliers. Based on the insights obtained, a new algorithm is presented that combines backward dynamic programming for solving low level subproblems and interleaved conjugate gradient method for solving the high level problem. The new method significantly improves algorithm convergence and solution quality. Numerical testing shows that the method is practical for job shop scheduling in industries. This work was supported in part by the National Science Foundation under DMI-9500037, and the Advanced Technology Center for Precision Manufacturing, University of Connecticut.     

Task Scheduling Optimization in Cloud Computing by Rao Algorithm

Cloud computing is currently dominated within the space of high-performance distributed computing and it provides resource polling and on-demand services through the web. So, task scheduling problem becomes a very important analysis space within the field of a cloud computing environment as a result of user's services demand modification dynamically. The main purpose of task scheduling is to assign tasks to available processors to produce minimum schedule length without violating precedence restrictions. In heterogeneous multiprocessor systems, task assignments and schedules have a significant impact on system operation. Within the heuristic-based task scheduling algorithm, the different processes will lead to a different task execution time (makespan) on a heterogeneous computing system. Thus, a good scheduling algorithm should be able to set precedence efficiently for every subtask depending on the resources required to reduce (makespan). In this paper, we propose a new efficient task scheduling algorithm in cloud computing systems based on RAO algorithm to solve an important task and schedule a heterogeneous multiple processing problem. The basic idea of this process is to exploit the advantages of heuristic-based algorithms to reduce space search and time to get the best solution. We evaluate our algorithm's performance by applying it to three examples with a different number of tasks and processors. The experimental results show that the proposed approach significantly succeeded in finding the optimal solutions than others in terms of the time of task implementation.     

An approximate solution of machine scheduling problems by decomposition method

To obtain an approximate solution for a large-scale job-shop scheduling problem the decomposition method was investigated. This means that an original problem is decomposed into subproblems, which are solved separately, and then the solution of the original problem is composed from the subproblems' solutions. Different methods to decompose the problem were tested by computational experiments and evaluated from the viewpoint of the goodness of schedule and computation time.     

A new decomposition algorithm for a liquefied natural gas inventory routing problem

We consider an inventory routing problem (IRP) in the liquefied natural gas (LNG) supply chain, called the LNG-IRP. Here, an actor is responsible for the LNG production and inventory management at the liquefaction plants, the routing and scheduling of a heterogeneous fleet of LNG ships, as well as the inventories and sales at the regasification terminals. Furthermore, all ports have a limited number of berths available for loading and unloading. The LNG-IRP is more complicated than many other maritime inventory routing problems because a constant rate of the cargo evaporates in the tanks each day and is used as fuel during transportation. In addition, a variable number of tanks are unloaded at the regasification terminals. We introduce a new path flow formulation for this problem arising from a novel decomposition scheme based on parts of a ship schedule, called duties. A ship schedule for the entire planning horizon can be divided into duties consisting of a visit to a liquefaction plant, then one or two visits to a regasification terminal before ending in a liquefaction plant. The solution method suggested is based on a priori generation of duties, and the formulation is strengthened by valid inequalities. The same problem was previously solved by a branch-price-and-cut algorithm for a schedule-based formulation. Computational results show that the new formulation provides tighter bounds than the previous schedule-based formulation. Furthermore, on a set of 27 benchmark instances, the proposed algorithm clearly outperforms the previous branch-price-and-cut algorithm both with regard to computational time and the number of problems solved within a 10-h time limit.     

Integrated scheduling of a container handling system with simultaneous loading and discharging operations

The integrated scheduling of container handling systems aims to optimize the coordination and overall utilization of all handling equipment, so as to minimize the makespan of a given set of container tasks. A modified disjunctive graph is proposed and a mixed 0–1 programming model is formulated. A heuristic algorithm is presented, in which the original problem is divided into two subproblems. In the first subproblem, contiguous bay crane operations are applied to obtain a good quay crane schedule. In the second subproblem, proper internal truck and yard crane schedules are generated to match the given quay crane schedule. Furthermore, a genetic algorithm based on the heuristic algorithm is developed to search for better solutions. The computational results show that the proposed algorithm can efficiently find high-quality solutions. They also indicate the effectiveness of simultaneous loading and discharging operations compared with separate ones.     

A comparison of stochastic programming and bi-objective optimisation approaches to robust airline crew scheduling

A prominent problem in airline crew scheduling is the pairings or Tour-of-Duty planning problem. The objective is to determine a set of pairings (or Tours-of-Duty) for a crew group to minimise the planned cost of operating a schedule of flights. However, due to unforeseen events the performance in operation can differ considerably from planning, sometimes causing significant additional recovery costs. In recent years there has been a growing interest in robust crew scheduling. Here, the aim is to find solutions that are “cheap” in terms of planned cost as well as being robust, meaning that they are less likely to be disrupted in case of delays. Taking the stochastic nature of delays into account, Yen and Birge (Transp Sci 40:3–14, 2006) formulate the problem as a two-stage stochastic integer programme and develop an algorithm to solve this problem. Based on the contradictory nature of the goals, Ehrgott and Ryan (J Multi-Criteria Decis Anal 11:139–150, 2002) formulate a bi-objective set partitioning model and employ elastic constraint scalarisation to enable the solution by set partitioning algorithms commercially used in crew scheduling software. In this study, we compare the two solution approaches. We improve the algorithm of Yen and Birge (Transp Sci 40:3–14, 2006) and implement both methods with a commercial crew scheduling software. The results of both methods are compared with respect to characteristics of robust solutions, such as the number of aircraft changes for crew. We also conduct experiments to simulate the performance of the obtained solutions. All experiments are performed using actual schedule data from Air New Zealand.     

Scheduling railway traffic at a construction site

We consider the problem of rescheduling trains in the case where one track of a railway section consisting of two tracks in opposing directions is closed due to construction activities. After presenting an appropriate model for this situation we derive a polynomial algorithm for the subproblem of finding an optimal schedule with minimal latenesss if the subsequences of trains for both directions outside the construction site are fixed. Based on this algorithm we propose a local search procedure for the general problem of finding good schedules and report test results for some real world instances. Received: December 8, 1999 / Accepted: May 2, 2001     

Quay crane scheduling in container terminals

The problem of scheduling identical quay cranes moving along a common linear rail to handle containers for a ship is studied. The ship has a number of container-stacking compartments called bays, and only one quay crane can work on a bay at the same time. The objective of the scheduling problem is to find the work schedule for each quay crane which minimizes the ship’s stay time in port. Finding the optimal solution of the scheduling problem is computationally intractable and a heuristic is proposed to solve it. The heuristic first decomposes the difficult multi-crane scheduling problem into easier subproblems by partitioning the ship into a set of non-overlapping zones. The resulting subproblems for each possible partition are solved optimally by a simple rule. An effective algorithm for finding tight lower bounds is developed by modifying and enhancing an effective lower-bounding procedure proposed in the literature. Computational experiments were carried out to evaluate the performance of the heuristic on a set of test problems randomly generated based on typical terminal operations data. The computational results show that the heuristic can indeed find effective solutions for the scheduling problem, with the heuristic solutions on average 4.8% above their lower bounds.     

How to park freight trains on rail–rail transshipment yards: the train location problem

In modern rail–rail transshipment yards huge gantry cranes spanning all railway tracks allow for an efficient transshipment of containers between different freight trains. This way, multiple trains loaded with cargo for varying destinations can be consolidated to a reduced number of homogeneous trains, which is an essential requirement of hub-and-spoke railway systems. An important problem during the daily operations of such a transshipment yard is the train location problem, which assigns each train of a given pulse to a railway track (vertical position) and decides on each train’s parking position on the track (horizontal position), so that the distances of container movements are minimized and the overall workload is equally shared among cranes. For this problem a mathematical model is presented; different heuristic solution procedures are described and tested in a comprehensive computational study. The results show that our procedures allow for a remarkable reduction of train processing time compared with typical real-world train location policies.     

Steel-making process scheduling using Lagrangian relaxation

The steel-making process, including steel-making and continuous casting, is usually the bottleneck in iron and steel production. Effective scheduling of this process is thus critical to improve productivity of the entire production system. Unlike the production scheduling in the machinery industry, steel-making process scheduling is characterized by the following features: job grouping and precedence constraints, set-up and removal times on the machines, and high job waiting costs. These features add extra difficulties to the scheduling problem. The objective is to ensure continuity of the production process and just-in-time delivery of final products. In this paper, a novel integer programming formulation with a 'separable' structure is constructed considering all the above-mentioned features. A solution methodology is developed combining Lagrangian relaxation, dynamic programming and heuristics. After relaxing two sets of 'coupling constraints', the relaxed problem is decomposed into smaller subproblems, each involving one job only. These subproblems are solved efficiently by using dynamic programming at the low level while the Lagrangian multipliers are iteratively updated at the high level by using a subgradient method. At the termination of such iterations, a two-stage heuristic is then used to adjust subproblem solutions to obtain a feasible schedule. A numerical experiment demonstrates that the method generates high quality schedules in a timely fashion.     

A divide-and-conquer strategy with particle swarm optimization for the job shop scheduling problem

An optimization algorithm based on the ‘divide-and-conquer’ methodology is proposed for solving large job shop scheduling problems with the objective of minimizing total weighted tardiness. The algorithm adopts a non-iterative framework. It first searches for a promising decomposition policy for the operation set by using a simulated annealing procedure in which the solutions are evaluated with reference to the upper bound and the lower bound of the final objective value. Subproblems are then constructed according to the output decomposition policy and each subproblem is related to a subset of operations from the original operation set. Subsequently, all these subproblems are sequentially solved by a particle swarm optimization algorithm, which leads directly to a feasible solution to the original large-scale scheduling problem. Numerical computational experiments are carried out for both randomly generated test problems and the real-world production data from a large speed-reducer factory in China. Results show that the proposed algorithm can achieve satisfactory solution quality within reasonable computational time for large-scale job shop scheduling problems.     

Models and optimisation approaches for scheduling steelmaking–refining–continuous casting production under variable electricity price

This paper studies the steelmaking–refining–continuous casting (SRCC) scheduling problem with considering variable electricity price (SRCCSPVEP). SRCC is one of the critical production processes for steel manufacturing and energy intensive. Combining the technical rules used in iron-steel production practice, time-dependent electricity price is considered to reduce the electricity cost and the associate production cost. A decomposition approach is proposed for the SRCCSPVEP. Without considering the electrical factor, the first phase applies the mathematical programming method to determine the relative schedule plan for each cast. In the second phase, we formulate a scheduling problem of all casts subject to resource constraint and time-dependent electricity price. A heuristic algorithm combined with the constraint propagation is developed to solve this scheduling problem. To investigate and measure the performance of the proposed approach, numerous instances are randomly generated according to the collective data from a well-known iron-steel plant in China. Computational results demonstrate that our algorithm is very efficient and effective in providing high-quality scheduling plans, and the electricity cost can be reduced for the iron-steel plant.     

An optimisation approach for pre-runtime scheduling of tasks and communication in an integrated modular avionic system

In modern integrated modular avionic systems, applications share hardware resources on a common avionic platform. Such an architecture necessitates strict requirements on the spatial and temporal partitioning of the system to prevent fault propagation between different aircraft functions. One way to establish a temporal partitioning is through pre-runtime scheduling of the system, which involves creating a schedule for both tasks and a communication network. While avionic systems are growing more and more complex, so is the challenge of scheduling them. The scheduling of the system has an important role in the development of new avionic systems, since functionality is typically added to the system over a period of several years and a scheduling tool is used both to detect if the platform can host the new functionality and, if this is possible, to create a new schedule. For this reason an exact solution strategy for avionics scheduling is preferred over a heuristic one. In this paper we present a mathematical model for an industrially relevant avionic system and present a constraint generation procedure for the scheduling of such systems. We apply our optimisation approach to instances provided by our industrial partner. These instances are of relevance for the development of future avionic systems and contain up to 20,000 tasks to be scheduled. The computational results show that our optimisation approach can be used to create schedules for such instances within a reasonable time.     

Bicriteria scheduling in a two-machine permutation flowshop

In this paper we consider a production scheduling problem in a two-machine flowshop. The bicriteria objective is a linear combination or weighted sum of the makespan and total completion time. This problem is computationally hard because the special case concerning the minimization of the total completion time is already known to be strongly NP-hard. To find an optimal schedule, we deploy the Johnson algorithm and a lower bound scheme that was previously developed for total completion time scheduling. Computational experiments are presented to study the relative performance of different lower bounds. While the best known bound for the bicriteria problem can successfully solve test cases of 10 jobs within a time limit of 30?min, under the same setting our branch-and-bound algorithm solely equipped with the new scheme can produce optimal schedules for most instances with 30 or less jobs. The results demonstrate the convincing capability of the lower bound scheme in curtailing unnecessary branching during problem-solving sessions. The computational experience also suggests the practical significance and potential implications of this scheme.     

A discrete time reactive scheduling model for new order insertion in job shop,make-to-order industries

This research presents a new reactive scheduling methodology for job shop, make-to-order industries. An integer linear programming formulation previously developed by the authors to schedule these types of industries is extended to address the problem of inserting new orders in a predetermined schedule, which is important in order-driven industries. A reactive scheduling algorithm is introduced to iteratively update the schedules. Numerical results on realistic examples of job shops of different sizes illustrate the effectiveness of the approach. In each case, different alternatives for inserting a set of new orders in an initial schedule are optimally generated, enabling the user to choose the most convenient one. Solutions are characterised by measures of scheduling efficiency as well as stability measures that assess the impact of rescheduling operations in a previously defined scheduling solution.     

Service network design for freight transportation: a review

Freight transportation is a highly competitive market, where logistics service providers (LSP) or carriers have to offer their customers highly reliable and high-quality services at low prices. To meet this challenge, LSPs have to standardize and consolidate. They consolidate their freight in a network of hubs and terminals and build up regular services. The design of such services requires decisions about the frequency, mode, and route of the service, and the corresponding schedule and routing of the freight. In some cases, they also need to make decisions about the assignment of crew and vehicles as well as the repositioning of empty containers and vehicles. Recent publications show that realistic instances of such planning problems are difficult to solve. Nevertheless, some of these real-life problems are modeled and solved using mathematical programming techniques. In this paper, we review different problem formulations published in literature. We further analyze and compare the specific solution frameworks. Based on this we give recommendations for future research.     

Air cargo scheduling: integrated models and solution procedures

Designing a profitable flight schedule is a highly complex planning problem. Both passenger and cargo airlines usually follow a decomposition approach and break this problem into several subproblems which are then solved consecutively and iteratively using specific but isolated models. At cargo airlines, the four major interdependent decision problems are flight selection, fleet assignment, rotation planning, and cargo routing. In our research, we have developed a planning approach which differs from other OR-based planning approaches in two aspects. The approach is based on integrated models and it is based on the pragmatic planning paradigm to optimally modify an existing schedule. For this purpose, the planner has to identify mandatory and optional flights. Then the planning goal is to identify the best combination of optional flights to be included into the schedule. Our integrated planning models comprise several additional important planning aspects for cargo airlines such as available capacities on external flights (e.g. belly capacities from passenger flights or road-feeder services), cargo handling costs and constraints, and aircraft maintenance regulations. There are two main aspects which we present in this paper. First, we describe the planning problem and the specific planning paradigm, develop a set of complex mixed-integer programs representing the different subproblems, and finally present integrated problem formulations as well as several model extensions. Thereafter, we develop a branch and price and cut approach for solving the mathematical programs and present extensive computational results obtained for a set of generated yet highly practical problem instances for different types of carriers. The results show that our approach is able to find high quality solutions to problem instances of realistic size and complexity within reasonable time.     

A hybrid flowshop scheduling problem for a cold treating process in seamless steel tube production

Seamless steel tubes often have various categories and specifications, which further require complicated operations in production, especially in the cold treating process (CTP). This paper investigates the scheduling problem using the seamless tube plant of Baoshan Iron and Steel Complex as a study background. By considering the practical production constraints such as sequence-dependent setup times, maintenance schedule, intermediate material buffers, job-machine matches, we formulate the hybrid flowshop scheduling problem with a non-linear mixed integer programming model (NMIP). In addition, our model provides a flexibility to remove the permutation assumption, which is often a limitation in early studies. In order to obtain the solution of the above NMIP problem, a two-stage heuristic algorithm is proposed and it combines a modified genetic algorithm and a local search method. With real production instances, our computation experiments indicate that the proposed algorithm is efficient and it outperforms several other approaches. Industrial implementation also shows that such a scheduling tool brings a cost saving of more than 10% and it substantially reduces the computation time. Our study also illustrates the need of relaxing permutation assumption in such a scheduling problem with complicated operation sequences.     

A tabu search algorithm with solution space partition and repairing procedure for cyclic robotic cell scheduling problem

This paper proposes a tabu search (TS) algorithm to solve an NP-hard cyclic robotic scheduling problem. The objective is to find a cyclic robot schedule that maximises the throughput. We first formulate the problem as a linear program, provided that the robot move sequence is given, and reduce the problem to searching for an optimal robot move sequence. We find that the solution space can be divided into some specific subspaces by the maximal number of works-in-process. Then, we propose a TS algorithm to synchronously perform local searches in each subspace. To speed up our algorithm, dominated subspaces are eliminated by lower and upper bounds of the cycle time during the iterations. In the TS, a constructive heuristic is developed to generate initial solutions for each subspace and a repairing procedure is proposed to maintain the feasibility of the solutions generated in the initialisation stage and the neighbours search process. Computational comparison both on benchmark instances and randomly generated instances indicates that our algorithm is efficient for the cyclic robotic scheduling problem.