Capacitated Machines Research Articles

Novel mathematical formulations for parallel-batching processing machine scheduling problems

We study mathematical formulations for batch-processing machine scheduling problems (BPMPs), which are the challenging issues in the machine scheduling literature where machines are capable of processing a batch of jobs simultaneously if jobs with non-identical sizes can be packed in a capacitated machine. In this paper, we tackle single- and parallel-machine BPMPs, and other interesting problem variants that aim at minimizing the makespan. We develop novel formulations along with valid inequalities and an algorithm framework that makes use of dual information and bounding techniques to achieve efficiency when instances are intractable. Extensive computational experiments on benchmark instances show that our approaches achieve state-of-the-art results and prove the optimality of intractable instances in the literature.

Modified differential evolution and heuristic algorithms for dump tippler machine allocation in a typical sugar mill in Thailand

This paper focuses on a computational tool for scheduling and sequencing sugarcane vehicles for dump tippler machines at the mill yard of a sugar mill, which can be represented as an NP-hard problem for parallel capacitated machines with machine restrictions, job grouping, and sequencing independent setup time. This research aims to determine the optimal sequencing of jobs, i.e. minimizing the makespan by considering machine restrictions, capacitated machines, group size, number of jobs, and the constraints of the sugar mill. In the considered problem machines alternatively operate, that distinguishes it from a general parallel machine problem. The mixed integer linear programing model is developed for solving the small-scale problem instances. Large-scale instances are handled by four heuristics, and four differential evolution (DE) metaheuristics. In order to improve the computational results, solution quality and computation time were considered. In addition, modified DE algorithms were used in encoding operation (initial solution), mutation and local search operation. The computational results revealed that the modified DE algorithms had higher relative improvement on the makespan. Furthermore, this decision-making support tool was implemented as a prototype in the sector of cane and sugar industry in Thailand and extended to other similar industries.

Integrating scheduling with optimal sublot for parallel machine with job splitting and dependent setup times

This paper addresses a novel problem of the parallel capacitated machines with job splitting and dependent setup times (PCMS), Pmc|split, pj, sjp|Cmax. A mixed integer programming (MIP) model is developed to find an optimal schedule with minimum makespan. Since the problem is NP-hard, metaheuristics are required to find a near optimal solution for larger, more practical problems. Therefore, this paper presents the first applications of two effective metaheuristics, Particle Swarm Optimization (PSO) and Differential Evolution (DE) with a solution representation scheme for solving the problem. To evaluate the metaheuristics’ performances, the lower bound is also firstly developed. Experimental results reveal that, in the small-size problems, there are no distinctive differences between the two algorithms’ performances, since both algorithms are able to find the optimal solutions. However, for medium to large size problems, the DE outperforms the PSO by providing significantly superior results of makespan for 22 out of 27 instances.

Double-row facility layout with replicate machines and split flows

This paper presents a new variant of the double row layout problem (DRLP) that features types of capacitated machines. This variant dramatically improves the fidelity with which the DRLP represents the complex production alternatives in many production environments, especially in the microcomputer industry. The problem consists of placing different classes of machines, each with multiple replicates, on either side of a central aisle. This is a common type of layout problem observed in many production and service environments. This work extends the DRLP literature by utilizing machine replicates in the layout. Multiple replicates of the same machine allow the product flow to split into multiple branches, each representing a parallel flow line. Flow splitting can improve machine utilization and reduces the number of machines needed. We also introduce machine sharing, which lets different products use the same machine. This eliminates the need for dedicated assignments of products to machines, which is a common assumption in the current literature. The consideration of re-entrant product flow, whereby each product can visit the same machine multiple times, represents another unique aspect of this research. The problem is formulated as a nonlinear mixed integer program. Due to the computational complexity of this problem a hierarchical optimization approach is developed and evaluated on test problems of various sizes. Computational results show that the proposed method produces consistent and high quality solutions across problem sizes.

Two-stage hybrid flow shop batching and lot streaming with variable sublots and sequence-dependent setups

A paint manufacturing firm's customers typically place orders for two or more products simultaneously. Each product belongs to a family that denotes batching compatibility during manufacturing. Further, products can be split into several sublots to allow overlapping production in a two-stage hybrid flow shop wherein various identical, capacitated machines operate in parallel at each stage. We present a mixed-integer linear program (MILP) for this integrated batching and lot streaming problem with variable sublots, incompatible job families, and sequence-dependent setup times. The model determines the number and size of sublots for each product and the production sequencing for each sublot such that the total weighted completion time is minimised. To promote practical implementation, we develop and evaluate heuristics to efficiently solve this problem.

Влияние частоты на амплитуды тока и момента синхронных машин

State of the art . The magnitude of winding EMF in a rotary electric machine depends on the rotational speed, which gives rise to an intuitive assumption that the current and torque amplitudes for an inductive synchronous electric machine (or voltage and torque amplitudes for a capacitive machine) will too. Research methods . This research uses mathematical modeling. Results . The following theorems hold true. Theorem 1. The current amplitude in the inductive load of an inductive synchronous electric machine does not depend on the rotational speed. Theorem 2. The torque amplitude for an inductive synchronous electric machine with an inductive load does not depend on the rotational speed. Theorem 3. The voltage amplitude for the capacitive load of a capacitive synchronous electric machine does not depend on the rotational speed. Theorem 4. The amplitude and torque for a capacitive synchronous electric machine with a capacitive load does not depend on the rotational speed. Conclusions . Contrary to a possible intuitive assumption, the current and torque amplitudes in an inductive synchronous electric machine with an inductive load do not depend on the rotational speed. With another load type, they do. For a capacitive synchronous electric machine with a capacitive load, the voltage and torque amplitude do not depend on the rotational speed.

Теоретически установленная независимость амплитуд тока и момента синхронной машины с индуктивной нагрузкой от частоты

The current and torque of an inductive synchronous electrical machine operating on an inductive load are considered. For an inductive synchronous machine operating on an inductive load, the following theorem is valid: the amplitude of the current through the inductive load of an inductive synchronous electrical machine does not depend on the rotation frequency. This theorem is proved by equating the Kirchhoff second law and the law of electromagnetic induction as applied to a rotating machine. The following theorem is also valid: the torque amplitude for an inductive synchronous electrical machine operating on an inductive load does not depend on the rotation frequency. This theorem is proved by analyzing the Ampere law as applied to a rotating machine. It has been established that with an active load, the current and torque amplitudes depend linearly on the machine rotation frequency. With the machine operating on a capacitive load, the current and torque amplitudes are quadratic functions of the rotation frequency. The dependence of the machine power on the nature of its load is established. For an inductive load, the reactive electrical power is a linear function of the frequency; for a resistive load, the machine power is a quadratic function of the frequency; and for a capacitive load, its power is a cubic function of the frequency. For a synchronous capacitive machine operating on a capacitive load, the following theorems are valid: The voltage amplitude across the capacitive load of a capacitive synchronous electrical machine does not depend on the rotation frequency. The torque amplitude for a capacitive synchronous electrical machine operating on a capacitive load does not depend on the rotation frequency.

Energy-aware lot sizing problem: Complexity analysis and exact algorithms

The single-item lot sizing problem under a periodic energy limitation is considered in this paper. Identical and parallel capacitated machines constitute the production system, each one consuming a certain amount of energy when being switched on, when reserved, and when producing. We consider a cost for starting-up the machines, a reservation cost to keep the machines ready for production, in addition to classical lot sizing costs, such as, joint setup cost, unit production cost and unit holding cost, all being time-dependent. Besides the classical lot sizing decisions of how much and in which periods to produce, we have to decide the number of machines to switch on and to switch off in each period. We show that this problem is NP-hard even under restricted conditions. In contrast, assuming stationary energy parameters, we propose two polynomial time dynamic programming algorithms to solve the problem to optimality. The first algorithm is proposed for the case with null setup cost, null reservation cost and null reservation energy consumption, and runs in O(M5T4) time, with M being the number of machines and T the number of periods. We show that we can extend this algorithm to solve the generalized version of the problem in time complexity O(M6T6).

Capacity acquisition for the single-item lot sizing problem under energy constraints

We study a single-item lot sizing problem integrated with some energy constraints, called energy-LSP in the rest of the paper. We consider a production system composed of identical and capacitated machines in parallel. One has to decide how many machines to start and how much to produce to serve the demand and/or to replenish the inventory. In addition to the production capacity limit in each period, induced by the machines on, we have a limit on the energy that can be consumed by the start-ups of the machines and the production of units. We first provide a mixed integer programming formulation for the problem in the general case. We then develop efficient polynomial time algorithms running in O(Tlog T), with T the length of the planning horizon, for several subproblems: (i) energy-LSP where all energy parameters are set to 0, (ii) energy-LSP where only the machine start-ups consume energy, (iii) energy-LSP where only the production of units consumes energy. We show that, in all these special cases, energy-LSP can be seen as an extension of the integrated capacity acquisition and lot sizing problem. This allows us, as by-product of our approach, to significantly improve the existing result proposed for the capacity acquisition problem without energy constraint nor subcontracting.

Strongly Fully Polynomial Time Approximation Scheme for the weighted completion time minimization problem on two-parallel capacitated machines

We consider the total weighted completion time minimization for the two-parallel capacitated machines scheduling problem. In this problem, one of the machines can process jobs until a certain time T1 after which it is no longer available. The other machine is continuously available for performing jobs at any time. We prove the existence of a strongly Fully Polynomial Time Approximation Scheme (FPTAS) for the studied problem, which extends the results for the unweighted version (see [I. Kacem, Y. Lanuel and M. Sahnoune, Strongly Fully Polynomial Time Approximation Scheme for the two-parallel capacitated machines scheduling problem, Int. J. Plann. Sched. 1 (2011) 32–41]). Our FPTAS is based on the simplification of a dynamic programming algorithm. Moreover, we present a set of numerical experiments and we discuss the results.

Strongly Fully Polynomial Time Approximation Scheme for the Weighted Completion Time Minimization Problem on Two-Parallel Capacitated Machines

We consider the total weighted completion time minimization for the two-parallel capacitated machines scheduling problem. In this problem, one of the machines can process jobs until a certain time T1 after which it is no longer available. The other machine is continuously available for performing jobs at any time. We prove the existence of a strongly Fully Polynomial Time Approximation Scheme (FPTAS) for the studied problem, which extends the results for the unweighted version (see Kacem, Lanuel and Sahnoune (2011)). Our FPTAS is based on the simplification of a dynamic programming algorithm.

A New Type of Capacitive Machine

The paper proposes a new type of the synchronous capacitive machine operated on a principle of the electric field effect. The proposed machine has smaller size and lighter weight than the standard electromagnetic synchronous machines with the same rated parameters. Another important advantage is a simple structure of the machine, which simplifies the production process and reduces the costs of the motor. The paper also presents extensive simulation results of the proposed capacitive machine. The simulation results show that the proposed machine is able to reach the same power output as the electromagnetic machines.

ONLINE AND SEMI-ONLINE SCHEDULING ON CAPACITATED TWO-PARALLEL MACHINES

In this paper, we investigate the capacitated two-parallel machines scheduling problem, where one machine is only available for a special period of time after which it can no longer process any job while the other machine is continuously available. Our objective is to minimize the completion time of the machine which is continuously available. The offline version of the problem is equivalent to the minimization version of the Subset-Sum problem. We first show the lower bound of the online version is infinite. We also consider the semi-online version with known the total job processing time in advance, for which both lower bound and semi-online algorithms are given.

Strongly Fully Polynomial Time Approximation Scheme for the two-parallel capacitated machines scheduling problem

We study the n–job two–parallel machines scheduling problem with the aim of minimising the total flow–time. In this problem, instead of allowing both machines to be continuously available as it is often assumed in the literature, we consider that one of the machines is available for a specified period of time after which it can no longer process any job. On the basis of the modification of an exact algorithm execution, we establish the existence of a strongly Fully Polynomial Time Approximation Scheme (FPTAS) for the above–mentioned problem.

Fully Polynomial Time Approximation Scheme for the Two-Parallel Capacitated Machines Scheduling Problem Under Unavailability Constraint

Decision Support Systems (DSS) ensure the computer-based support for the conscientious decision-making in solving problems that require a large amount of information processing and complex scenarios. DSS for Transportation (DSST) are intelligent systems that are used at operational and organizational management levels. Operating a DSST in a public transportation web-based monitoring system is presented in this paper.

Smooth Priorities for Multi-Product Inventory Control

Whenever dealing with periodic review multi-products inventory control for capacitated machines, one of the main issues that has to be addressed concerns the dynamic capacity allocation. That is, how to assign capacity to the several competing products that require more than it is available. One typical approach is to assign priorities to the products, according to some degree of relative importance. Whereas priority based capacity allocation is attractive, due to its simplicity and for the fact that it makes sense in a significant number of settings, we contend this to be too unfair to lower priority products, given their access to production tends to be highly variable and uncertain. Departing from what we call strict priorities, we propose a priority based mechanism that improves on this drawback, termed smooth priorities. This new policy for multi-product, limited capacity production systems with stochastic demand is studied. Theoretical comparisons are made to the common policies of strict priorities and linear scaling. An optimizer based on Infinitesimal Perturbation Analysis, IPA, simulation is devised and results of practical comparison between smooth and strict priorities are presented. The structure of the cost function with smooth priorities is studied through function plots obtained from simulation and numerical results show consistent better performances than those achieved under strict priorities.

Capacitated single machine scheduling and its on-line heuristics

The capacitated single machine scheduling problem arises from a manufacturing system. The machine works periodically and is capacitated in each period. This is a variant of the bin packing problem. We theoretically compare two types of production settings: in the first type, the machine processes each demand without splitting and in the second, splitting is allowed but a setup time occurs. From the worst case viewpoint, we show that the optimal schedule length of the production without splitting may be nearly twice as long as that of the production with splitting. The ratio of two is asymptotically tight. Then we adapt the well-known bin packing heuristics for the on-line version of the capacitated single machine scheduling problem and give their worst case analysis.

Capacitated single machine scheduling and its on-line heuristics

The capacitated single machine scheduling problem arises from a manufacturing system. The machine works periodically and is capacitated in each period. This is a variant of the bin packing problem. We theoretically compare two types of production settings: in the first type, the machine processes each demand without splitting and in the second, splitting is allowed but a setup time occurs. From the worst case viewpoint, we show that the optimal schedule length of the production without splitting may be nearly twice as long as that of the production with splitting. The ratio of two is asymptotically tight. Then we adapt the well-known bin packing heuristics for the on-line version of the capacitated single machine scheduling problem and give their worst case analysis.

On the resolution of the single product capacitated machine siting problem

The single product capacitated machine siting problem (SPCMSP) is an extension of the simple plant location problem, in which plant production depends on installing capacitated machines. In this paper we compare, both theoretically and computationally, three heuristic algorithms for the SPCMSP based upon Lagrangean relaxation and reduction tests of a mixed-integer formulation of the problem, which is NP-hard. We test the performance of the algorithms with examples involving up to 100 potential plants, 1000 customers and six potential machines per plant, which we obtain encouraging results.

Cyclical Schedules for Capacitated Lot Sizing with Dynamic Demands

Cyclical scheduling, where the time between production periods for each item is constant, offers simplicity and ease of control compared with noncyclical scheduling, where production periods are irregularly spaced. However, when demands are dynamic, flexibility in the spacing of production periods permits noncyclical scheduling to result in lower total costs. This study investigates the additional cost of cyclical scheduling. We focus upon the capacitated lot sizing problem (CLSP), which deals with planning production on a single, capacitated machine serving multiple items with dynamic demands. To establish cyclical schedules, a mathematical programming model is developed along with a heuristic solution technique and a Lagrangian-based lower bounding procedure. For problems solved in this study, differences between solution costs and lower bounds average less than 1%. Experiments are performed to evaluate the heuristic technique and to compare cyclical schedules with noncyclical ones. For comparison purposes, a noncyclical heuristic and problem sets from the literature are used. New problems based upon operating data from Ford Motor Company are also generated and solved. In this study, cyclical schedules average 4.4% higher in cost than noncyclical ones. Coefficient of demand variation most seriously influences differences in costs. Other significant factors include capacity utilization, setup time, time between orders, and number of items.