Single Machine Batch Scheduling Problem Research Articles

Large-scale energy-conscious bi-objective single-machine batch scheduling under time-of-use electricity tariffs via effective iterative heuristics

Time-of-use (TOU) electricity pricing policy is widely encountered in the world, which provides new opportunities for power-intensive enterprises to save their energy cost. A good trade-off between the total electricity cost and production efficiency is desired by decision makers. This work addresses an energy-conscious bi-objective single-machine batch scheduling problem under TOU electricity tariffs, in which electricity price varies with time. The objective of the problem is to simultaneously minimize total electricity cost and makespan. Due to its strong NP-hard nature, two fast new ϵ-constraint-based constructive heuristic algorithms are developed to solve it. The core idea is to transform the bi-objective problem into a series of single-objective problems that are fast and heuristically solved to obtain an approximate Pareto front. Especially, for each transformed single-objective problem, two novel constructive heuristic algorithms are proposed by solving a series of multiple knapsack problems and 0–1 knapsack problems, respectively. Computational results on 145 benchmark and 80 newly generated larger-scale instances show that the proposed algorithms are quite efficient and are able to find high-quality Pareto solutions for large-scale problems with up to 200 batches.

Single-machine batch scheduling problem with job rejection and resource dependent processing times

This paper addresses single-machine batch scheduling with job rejection and convex resource allocation. A job is either rejected, in which case a rejection penalty will be incurred, or accepted and processed on the machine. The accepted jobs are combined to form batches containing contiguously scheduled jobs. For each batch, a batch-dependent machine setup time, which is a function of the number of batches processed previously, is required before the first job in the batch is processed. Both the setup times and job processing times are controllable by allocating a continuously divisible nonrenewable resource to the jobs. The objective is to determine an accepted job sequence, a rejected job set, a partition of the accepted job sequence into batches, and resource allocation that jointly minimize a cost function based on the total delivery dates of the accepted jobs, and the job holding, resource consumption, and rejection penalties. An dynamic programming solution algorithm with running time O (n6) is developed for the problem. It is also shown that the case of the problem with a common setup time can be solved in O (n5) time.

Single-machine batch scheduling to minimize the total setup cost in the presence of deadlines

We address the single-machine batch scheduling problem with the objective of minimizing the total setup cost. This problem arises when there are n jobs that are partitioned into F families and when setup operations are required whenever the machine switches from processing a job of one family to processing a job of another family. We assume that setups do not require time but are associated with a fixed cost which is identical for all setup operations. Each job has a processing time and an associated deadline. The objective is to schedule all jobs such that they are on time with respect to their deadlines and the total setup cost is minimized. We show that the decision version of this problem is NP-complete in the strong sense. Furthermore, we present properties of optimal solutions and an $$O(n\log n+nF)$$ algorithm that approximates the cost of an optimal schedule by a factor of F. The algorithm is analyzed in computational tests.

Optimal single-machine batch scheduling for the manufacture, transportation and JIT assembly of precast construction with changeover costs within due dates

The manufacture, transportation and on-site assembly sectors of precast construction projects are often considered separately and managed by rule of thumb, causing an inefficient use of resources and postponed delivery. This study views these sectors as a whole from the perspective of a single machine batch-scheduling problem. A dynamic programming algorithm, which aims to search for solutions that entail maximum production efficiency, was developed accordingly with the constraints of changeover costs and production deadlines. We tested the method’s ability by processing as many products as possible simultaneously using real data collected from a precast factory in a simulation and compared the effect with a previous study. We found that our method possesses great potential to improve the efficiency of precast production.

Bi-criteria single-machine batch scheduling with machine on/off switching under time-of-use tariffs

The industrial sector is the largest consumer of the world’s total energy and most of its consumption is in form of electricity. In recent years, to strengthen the peak load regulation capability, time-of-use (TOU) pricing has been implemented in many countries to encourage consumers to shift their use from peak to mid- and off-peak periods such that their energy bills can be reduced. In this paper, we study a new single-machine batch scheduling problem with machine on/off switching under TOU tariffs, which aims to simultaneously minimize total electricity cost and makespan. For the problem, we first develop a bi-objective mixed-integer linear programming model. Based on optimal batch rule analysis, an improved model is further provided which greatly reduces Pareto optimal solution search space. To efficiently solve large-size problems, we propose a heuristic based ε-constraint method. The results from extensive computational experiments confirm the effectiveness and efficiency of the proposed model and the algorithm.

Bi-objective optimization of single-machine batch scheduling under time-of-use electricity prices

Time-of-use (TOU) electricity pricing has been a common practice to enhance the peak load regulation capability of power grid. Meanwhile, it provides a good opportunity for industries to reduce energy costs, especially for energy-intensive ones, where batch scheduling is often involved. Majority of batch scheduling problems have been proved to be NP-hard, even for most single-machine environments. Optimizing batch scheduling under TOU policy in these industries will be of great significance. Single-machine batch scheduling is an important basis for more complicated shop scheduling problems. This paper investigates a bi-objective single-machine batch scheduling problem under TOU policy: the first objective is to minimize the makespan and the second is to minimize the total electricity costs. The considered problem is first formulated as a bi-objective mix-integer linear programming (MILP) model and is demonstrated to be NP-hard. Subsequently, the MILP is simplified by analyzing properties and search space for a Pareto optimal solution is greatly reduced. Then, an exact e -constraint method is adapted to obtain its Pareto front, which is accelerated due to these properties. Finally, a preferable solution is recommended for decision makers via a fuzzy-logic-based approach. Computational results on randomly generated instances show the effectiveness of the proposed approach.

Bi-objective optimization of a single machine batch scheduling problem with energy cost consideration

With the increasing energy price, the rapid growth of electricity demand and severe challenges for sustainable development, energy-efficient scheduling is becoming more and more important for power-intensive manufacturing industry, especially for batch production companies. This paper investigates a bi-objective single machine batch scheduling problem with non-identical job sizes, the time-of-use (TOU) electricity prices, and different energy consumption rates of the machine. The first objective is to minimize the makespan and the second is to minimize the total energy costs, by considering both the machine utilization and the economic cost. The problem is formulated as an integer programming model. Then, an exact ε-constraint method is adapted to obtain the exact Pareto front. To deal with large scale problems, based on decomposition ideas, two heuristic methods are developed to obtain approximate Pareto fronts. Computational experiments on randomly generated instances show the effectiveness of the methods. A study case of a real-world glass manufacturing company is also conducted to show that the proposed methods are promising for practical usages.

Bi-criteria sequencing of courses and formation of classes for a bottleneck classroom

In this paper, the problem of class formation and sequencing for multiple courses subject to a bottleneck classroom with an ordered bi-criteria objective is studied. The problem can be modelled as a single-machine batch scheduling problem with incompatible job families and parallel job processing in batches, where the batch size is family-dependent. For the minimisation of the number of tardy jobs, the strong NP-hardness is proven. For the performance measure of the maximum cost, we consider single criterion and bi-criteria cases. We present an O(n2logn) algorithm, n is the number of jobs, for both cases. An Integer Programming model as well as Simulated Annealing and Genetic Algorithm matheuristics to solve a fairly general case of the bi-criteria problem is presented and computationally tested.

Single machine batch scheduling problem to minimize makespan with controllable setup and jobs processing times

This paper concerns with a single-machine scheduling problem underbatch availability in which both the setup of each batch and the processing times of jobs are controllable by allocating a resource. The completion time ofa job in a batch is that of the last job in the batch. Two batchscheduling problems are investigated. The objective is todetermine the job sequence, the partition of the job sequence intobatches and the resource allocation scheme to minimize makespan,subject to the total amount of resource is bounded by a givenvalue $U$ in the first problem; while in the second problem is tominimize a total cost of makespan and resource without resourcelimitation, respectively. We show that the problems underlying canbe solved in polynomial time and present optimal algorithms.

A column generation approach for scheduling a batch processing machine with makespan objective

In this paper, the single machine batch scheduling problem with non-identical job processing times, non-identical job sizes, and a fixed machine size capacity is considered with the objective of minimising makespan. The problem is solved using a column generation approach in which the original problem is decomposed into a restricted master problem and a subproblem. The restricted master problem is then linearly relaxed and iteratively solved to optimality by utilising improving columns generated by solving the subproblem. Finally, the restricted master problem is resolved as an integer programme to obtain a final feasible solution. The results are compared to previous results obtained using simulated annealing, a genetic algorithm, and a commercial solver. The solution quality and run time compare very favourably with these other approaches, producing at least an equivalent solution in all of the test cases and generating a better solution in 65% of the cases that contained 50 or more jobs.

Batch scheduling with a rate-modifying maintenance activity to minimize total flowtime

We study a single machine batch scheduling problem with unit time jobs and an optional maintenance activity. The maintenance activity is assumed to be rate modifying, i.e. the processing times of the jobs processed after the maintenance are reduced. The objective function is minimum total flowtime. We focus first on the relaxed version of the problem, where batch sizes are not forced to be integers. For a given number of jobs, setup time, duration of the maintenance activity, and a rate-modifying factor, we show that the optimal solution has a unique property: the batch sizes of the jobs scheduled prior to the maintenance, and after it, form two decreasing arithmetic sequences. Based on this property, we introduce an optimal algorithm which is polynomial in the number of jobs. We propose a simple rounding procedure that guarantees an integer solution. Our numerical tests indicate that this procedure leads to very close-to-optimal schedules.

A new mathematical model for single machine batch scheduling problem for minimizing maximum lateness with deteriorating jobs

Article history: Received 10 March 2010 Received in revised form July, 25, 2011 Accepted 26 July 2011 Available online 27 July 2011 This paper presents a mathematical model for the problem of minimizing the maximum lateness on a single machine when the deteriorated jobs are delivered to each customer in various size batches. In reality, this issue may happen within a supply chain in which delivering goods to customers entails cost. Under such situation, keeping completed jobs to deliver in batches may result in reducing delivery costs. In literature review of batch scheduling, minimizing the maximum lateness is known as NP-Hard problem; therefore the present issue aiming at minimizing the costs of delivering, in addition to the aforementioned objective function, remains an NP-Hard problem. In order to solve the proposed model, a Simulation annealing meta-heuristic is used, where the parameters are calibrated by Taguchi approach and the results are compared to the global optimal values generated by Lingo 10 software. Furthermore, in order to check the efficiency of proposed method to solve larger scales of problem, a lower bound is generated. The results are also analyzed based on the effective factors of the problem. Computational study validates the efficiency and the accuracy of the presented model. © 2012 Growing Science Ltd. All rights reserved

Single machine batch scheduling problem with fuzzy batch size

In a batch scheduling problem, jobs are grouped (group is called batch) and scheduled in batches, and a setup time is incurred when starting a new batch. Processing times are assumed to be identical for all jobs. Setup times are assumed to be identical for all batches. Though all batch sizes cannot exceed a common upper bound, the upper bound is flexible and satisfaction degree with respect to the upper limit to be maximized is given. Also the other two objectives, i.e., the maximum completion time and the flow-time are to be minimized. Usually there exists no solution optimizing three objectives at a time. Therefore we define non-dominated solutions consisting of batch size, batch number and allocation of jobs to batches. First we propose an efficient algorithm for a sub-problem with fixed upper limit of batch size, fixed batch number based on a Lagrange relaxation procedure. Based on the properties of non-dominated solutions clarified in this paper, we propose an efficient algorithm to find some non-dominated solutions. Finally we summarize the results in this paper and discuss further research problems.

Scheduling a single machine with multiple job processing ability to minimize makespan

This paper investigates a new problem, called single machine scheduling with multiple job processing ability, which is derived from the production of the continuous walking beaming reheating furnace in iron and steel industry. In this problem, there is no batch and the jobs enter and leave the machine one by one and continuously, which is different from general single machine batch scheduling problem where the jobs in a batch share the same start and departure time. Therefore, the start time and the departure time of a job depend on not only the job sequence but also the machine capacity. This problem is also different from the single semi-continuous batching machine scheduling recently studied in the literature, where the jobs are processed in batch mode and a new batch cannot be started for processing until the processing of the previous batch is completed though jobs in the same batch enter and leave the machine one by one. The objective of this problem is to minimize the makespan. We formulate this problem as a mixed integer linear programming model and propose a particle swarm optimization (PSO) algorithm for this problem. Computational results on randomly generated instances show that the proposed PSO algorithm is effective.

Model Penjadwalan Batch Multi Item dengan Dependent Processing Time

This paper investigates a development of single machine batch scheduling for multi items with dependent processing time. The batch scheduling problem is to determine simultaneously number of batch (N), which item and its size allocated for each batch, and processing sequences of resulting batches. We use total actual flow time as the objective of schedule performance. The multi item batch scheduling problem could be formulated into a biner-integer nonlinear programming model because the number of batch should be in integer value, the allocation of items to resulting batch need binary values, and also there are some non-linearity on objective function and constraint due to the dependent processing time. By applying relaxation on the decision variable of number of batch (N) as parameter, a heuristic procedure could be applied to find solution of the single machine batch scheduling problem for multi items.

A single machine batch scheduling problem with bounded batch size

We address a single-machine batch scheduling problem to minimize total flow time. Processing times are assumed to be identical for all jobs. Setup times are assumed to be identical for all batches. As in many practical situations, batch sizes may be bounded. In the first setting studied in this paper, all batch sizes cannot exceed a common upper bound. In the second setting, all batch sizes share a common lower bound. An optimal solution consists of the number of batches and their (integer) size. We introduce an efficient solution for both problems.

Single machine batch scheduling to minimize total completion time and resource consumption costs

In this paper we study the single-machine batch scheduling problem under batch availability, where both setup and job processing times are controllable by allocating a continuously divisible nonrenewable resource. Under batch availability a set of jobs is processed contiguously and completed together, when the processing of the last job in the batch is finished. We present polynomial time algorithms to find the job sequence, the partition of the job sequence into batches and the resource allocation, which minimize the total completion time or the total production cost (inventory plus resource costs).

Single machine batch scheduling problem with family setup times and release dates to minimize makespan

In this paper we consider the single machine batch scheduling problem with family setup times and release dates to minimize makespan. We show that this problem is strongly NP-hard, and give an $$O(n (\frac{n}{m}+1)^m)$$ time dynamic programming algorithm and an $$O(mk^{k}P^{2k-1})$$ time dynamic programming algorithm for the problem, where n is the number of jobs, m is the number of families, k is the number of distinct release dates and P is the sum of the setup times of all the families and the processing times of all the jobs. We further give a heuristic with a performance ratio 2. We also give a polynomial-time approximation scheme (PTAS) for the problem.

Single machine scheduling with batch-dependent setup times

We address a single-machine batch scheduling problem. The setup times (incurred whenever starting a new batch) are assumed to be a function of the number of batches processed previously, i.e., batch-dependent. The objective is minimum total flow-time. We focus on the case of identical processing time jobs. Given the number of jobs and the setup times, we have to determine the optimal number of batches and their (integer) size. An efficient ( O ( n ) ) solution procedure is introduced.

Single machine batch scheduling with jointly compressible setup and processing times

A single machine batch scheduling problem is addressed. A batch is a set of jobs processed contiguously and completed together when the processing of all jobs in the batch is finished. Processing of a batch requires a machine setup time common for all batches. Two external resources can be used to linearly compress setup and job processing times. The setup times are jointly compressible by one resource and the job processing times are jointly compressible by another resource, i.e., the amount of resource consumption for setup time compression is the same for all setups and the amount of resource consumption for job processing time compression is the same for all jobs. Polynomial time algorithms are presented to find an optimal batch sequence and optimal amounts of resource consumption such that either total job completion time is minimized, subject to an upper bound on total weighted resource consumption, or total weighted resource consumption is minimized, subject to an upper bound on total job completion time. The algorithms are based on results from linear programming and from batch scheduling with fixed setup and processing times.