Multi-machine Scheduling with Setup Times

We present and solve a joint production and qualification planning problem for a manufacturing environment with high process flexibility. Various factors contribute to the complexity of the problem, one of which is product‐machine mapping: Each machine may be qualified to produce multiple products and each product can be produced on multiple machines. To meet a build‐plan, the factory needs to not only determine a multi‐machine multi‐product production schedule that accounts for sequence‐dependent setup time, but also a qualification schedule which prescribes whether and when a machine should undergo a qualification process such that it is ready to produce a product. We consider processing characteristics including sequence‐dependent setups, job splitting, and machine eligibility in addition to qualification. We formulate the mathematical model as an MILP problem that minimizes the total weighted delay. In this study, we describe two heuristic solution approaches developed for this complex decision setting and the application at Intel. We compare our approach with Intel's current approach which is a spreadsheet‐based manual approach that relies on the experience of the factory planner. The results indicate that our approach, which we call the GS approach, dominates in terms of minimizing the delay. Our approach performs well when capacity is tight and additional qualifications are considered while the Intel approach may perform better on reducing the setup and qualification time in certain problem instances, particularly those with loose capacity. As a result, an integrated approach which selects the better solution from both approaches is proposed, which shows significant reduction over the current approach in both the weighted delay and the setup and qualification time.

The three-machine flowshop scheduling problem to minimize total completion time is studied where setup times are treated as separate from processing times. Setup and processing times of all jobs on all machines are unknown variables before the actual occurrence of these times. The lower and upper bounds for setup and processing times of each job on each machine is the only information that is available. In such a scheduling environment, there may not exist a unique schedule that remains optimal for all possible realizations of setup and processing times. Therefore, it is desired to obtain a set of dominating schedules (which dominate all other schedules) if possible. The objective for such a scheduling environment is to reduce the size of dominating schedule set. We obtain global and local dominance relations for a three-machine flowshop scheduling problem. Furthermore, we illustrate the use of dominance relations by numerical examples and conduct computational experiments on randomly generated problems to measure the effectiveness of the developed dominance relations. The computational experiments show that the developed dominance relations are quite helpful in reducing the size of dominating schedules.

We present a methodology for automatically generating an online scheduling process for an arbitrary objective with the help of evolution strategies. The scheduling problem comprises independent parallel jobs and multiple identical machines and occurs in many real massively parallel processing systems. The system owner defines the objective that may consider job waiting times and priorities of user groups. Our scheduling process is a variant of the simple and commonly used greedy scheduling algorithm in combination with a repeated sorting of the waiting queue. This sorting uses a criterion whose parameters are evolutionary optimized. We evaluate our new scheduling process with real workload data and compare it to the best offline solutions and to the online results of the standard EASY backfill algorithm. To this end, we partition the user of the workloads into groups and select an exemplary objective that prioritizes some of those groups over others

In this paper, we investigate the problem of online task scheduling of jobs such as MapReduce jobs, Monte Carlo simulations and generating search index from web documents, on cloud computing infrastructures. We consider the virtualized cloud computing setup comprising machines that host multiple identical virtual machines (VMs) under pay-as-you-go charging, and that booting a VM requires a constant setup time. The cost of job computation depends on the number of VMs activated, and the VMs can be activated and shutdown on demand. We propose a new bi-objective algorithm to minimize the maximum task delay, and the total cost of the computation. We study both the clairvoyant case, where the duration of each task is known upon its arrival, and the more realistic non-clairvoyant case.

Many firms have to deal with the problems of scheduling and transportation allocation. The problems of assembly scheduling mainly focus on how to arrange orders in proper sequence on the assembly line with the purpose of minimizing the maximum completion time before they are flown to their destinations. Transportation allocation problems arise in how to assign processed orders to transport modes in order to minimize penalties such as earliness and tardiness. The two problems are usually separately discussed due to their complexity. This paper simultaneously deals with these two problems for firms with multiple identical parallel machines. We formulate this problem as a mixed integer programming model. The problem belongs to the class of NP-complete combinatorial optimization problems. This paper develops a hybrid genetic algorithm to obtain a compromised solution within a reasonable CPU time. We evaluate the performance of the presented heuristic with the well-known GAMS/CPLEX software. The presented approach is shown to perform well compared with well-known commercial software.

This paper deals with the capacitated single-stage production lot sizing and scheduling problem with multiple items, setup time, stochastic demand and unrelated parallel machines. A stochastic mixed-integer linear programming model is proposed to formulate the problem. Based on the uncertain constraints, the chance constrained programming approach is used to transform them into equivalent deterministic constraints and then obtain an optimal solution by deterministic mixed-integer linear programming model. Due to the complexity of problem, the mixed-integer programming (MIP) of the equivalence deterministic model of the capacitated single-stage production lot sizing problem with multiple items, setup time, stochastic demand and unrelated parallel machines is reformulated as a shortest path reformulation problem. The proposed algorithm is evaluated through a numerical example. Computational results show that the proposed method gives optimal or near optimal solution and have good-quality result for the test problem.

An exact algorithm for two-dimensional vector packing problem with volumetric weight and general costs

The vigorous development of electric vehicles (EVs) is an important means of reducing carbon emissions and mitigating environmental problems such as the greenhouse effect. Battery swapping stations (BSSs) can both provide battery swapping services for large-scale EVs and charge batteries centrally. As the supply of fully charged batteries in the BSS shrinks, it becomes necessary to schedule the charging of the depleted batteries rapidly that users have swapped for fully-charged ones. The charging schedule for depleted batteries must be made without knowledge of future battery arrivals. In this context, this paper develops a mathematical model for online charging scheduling of BSSs, formulates the charging strategy as a two-dimensional rectangle packing problem, and quickly calculates the scheduling arrangement of batteries by partitioning the remaining available capacity of a BSS. Since there are limited battery types within the BSS which can provide battery replacement services, this paper supplements the proposed model with known battery types, which improves the utilization of the available capacity of BSSs. Finally, numerical results verify the effectiveness of the proposed model. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Electric vehicles (EVs) are becoming an alternative way to reduce carbon emissions in transportation systems. Herein, the optimal battery charging problem is the core problem when it comes to dispatching a huge number of EVs. Up to now, battery-swapping is widely used for EVs due to its simple, convenient way. Furthermore, a business model for the battery swapping stations (BSSs) is brought up, where EV users send their depleted batteries to the BSS and the BSS provides the users with a fully charged replacement battery from its warehouse, which only takes a few minutes. Since the maximum charging power of the BSS is limited by the capacity of the transformer connecting the BSS to the power grid, the BSS will adopt an optimal charging schedule that maximizes the charging benefit for large quantities of depleted batteries in the warehouse. However, the challenge is that the charging schedule for depleted batteries must be made without knowledge of future battery arrivals because the EV behaviors are difficult to predict. To address this problem, this paper developed an online charging scheduling algorithm, which formulates the charging strategy as a two-dimensional rectangle packing problem. The proposed method can provide battery replacement services in real-time and solve quickly without any information about incoming depleted EV batteries. The proposed model and method have been tested on the system with different numbers of batteries to show the effectiveness. Besides, the online two-dimensional rectangle packing problem can provide an online decision for BSSs.

Minimizing the sum of earliness/tardiness in multi-machine scheduling: a mixed integer programming approach

This paper considers two types of stochastic group scheduling problems, one involves a single machine, and the other involves an m -machine flow shop. In both problems, jobs are classified into groups. The jobs of the same group are to be processed one after another, and a sequence independent setup time is incurred between each two consecutively scheduled groups. The job processing times are independent random variables and group setup times are functions of the job processing times. The objective is to find both a group sequence and a job sequence that minimizes the expected total flow time. An optimal group sequence and an optimal job sequence are obtained by processing jobs in increasing stochastic order of their mean processing times and group setup times for a single machine problem. A precedence relationship is derived for the m -machine flow shop problem.

We consider the two-dimensional bin packing and strip packing problem, where a list of rectangles has to be packed into a minimal number of rectangular bins or a strip of minimal height, respectively. All packings have to be non-overlapping and orthogonal, i.e., axis-parallel. Our algorithm for strip packing has an absolute approximation ratio of 1.9396 and is the first algorithm to break the approximation ratio of 2 which was established more than a decade ago. Moreover, we present a polynomial time approximation scheme ($\mathcal{PTAS}$) for strip packing where rotations by 90 degrees are permitted and an algorithm for two-dimensional bin packing with an absolute worst-case ratio of 2, which is optimal provided $\mathcal{P} \not= \mathcal{NP}$.

SUMMARY A recent trend in the analysis of scheduling models integrates batching decisions with sequencing decisions. The interplay between batching and sequencing reects the realities of the small-volume, high-variety manufacturing environment and adds a new feature to traditional scheduling problems. Practical interest in this topic has given rise to new research eorts, and there has been a series of articles in the research literature surveying the rapidly developing state of knowledge. Exam- ples include Ghosh (1), Liaee and Emmons (2), Potts and Van Wassenhove (3), and Webster and Baker (4). This paper deals with an important theoretical and practical problem in this area. We examine the single-machine model with family (or group) set-up times and a criterion of minimizing total weighted job completion time (weighted owtime). We propose new lower bounds for this problem, and then turn our attention to renement of a previously proposed branch-and-bound algorithm. The benets of our renements are illustrated by computational experiments.

This paper presents the comparative use of Simulated Annealing (SA) and Genetic Algorithm (GA) in a scheduling problem of unrelated parallel machines with set-up times. The problem accounts for allotting batched work parts to the unrelated parallel machines, where each batch is composed of a fixed number of identical jobs. Processing time of each job of a batch is determined according to the machine group as well as the batch group to which the job belongs. Major or minor set-up times are required between two subsequent batches, depending on the batch sequence but yet independent of machines. The objective of the problem is to minimise the Total Weighted Tardiness (TWT) for the unrelated parallel machine scheduling. SA and GA algorithms are proposed to obtain near-optimal solutions of the problem. The performance of proposed heuristics is compared through computational experiments with real data from dicing operations of a compound semiconductor manufacturing facility.

Capacitated lot-sizing with minimum batch sizes and setup times

New and improved level heuristics for the rectangular strip packing and variable-sized bin packing problems