Mixed integer programming formulations for single machine scheduling problems

In this paper, the computational performance of four different mixed integer programming (MIP) formulations for various single machine scheduling problems is studied. Based on the computational results, we discuss which MIP formulation might work best for these problems. The results also reveal that for certain problems a less frequently used MIP formulation is computationally more efficient in practice than commonly used MIP formulations. We further present two sets of inequalities that can be used to improve the formulation with assignment and positional date variables.     

Approaches to modeling train scheduling problems as job-shop problems with blocking constraints

The problem of scheduling a set of trains traveling through a given railway network consisting of single tracks, sidings and stations is considered. For every train a fixed route and travel times, an earliest departure time at the origin and a desired arrival time at the destination are given. A feasible schedule has to be determined which minimizes total tardiness of all trains at their destinations. This train scheduling problem is modeled as a job-shop scheduling problem with blocking constraints, where jobs represent trains and machines constitute tracks or track sections. Four MIP formulations without time-indexed variables are developed based on two different transformation approaches of parallel tracks and two different types of decision variables leading to job-shop scheduling problems with or without routing flexibility. A computational study is made on hard instances with up to 20 jobs and 11 machines to compare the MIP models in terms of total tardiness values, formulation size and computation time.     

Single machine scheduling with interfering job sets

We consider two single machine bicriteria scheduling problems in which jobs belong to either of two different disjoint sets, each set having its own performance measure. The problem has been referred to as interfering job sets in the scheduling literature and also been called multi-agent scheduling where each agent's objective function is to be minimized. In the first problem (P1) we look at minimizing total completion time and number of tardy jobs for the two sets of jobs and present a forward SPT-EDD heuristic that attempts to generate the set of non-dominated solutions. The complexity of this specific problem is NP-hard; however some pseudo-polynomial algorithms have been suggested by earlier researchers and they have been used to compare the results from the proposed heuristic. In the second problem (P2) we look at minimizing total weighted completion time and maximum lateness. This is an established NP-hard problem for which we propose a forward WSPT-EDD heuristic that attempts to generate the set of supported points and compare our solution quality with MIP formulations. For both of these problems, we assume that all jobs are available at time zero and the jobs are not allowed to be preempted.     

HEART: Unrelated parallel machines problem with precedence constraints for task scheduling in cloud computing using heuristic and meta-heuristic algorithms

Cloud computing is becoming a profitable technology because of it offers cost-effective IT solutions globally. A well-designed task scheduling algorithm ensures the optimal utilization of clouds resources and reducing execution time dynamically. This research article deals with the task scheduling of inter-dependent subtasks on unrelated parallel computing machines in a cloud computing environment. This article considers two variants of the problem-based on two different objective function values. The first variant considers the minimization of the total completion time objective function while the second variant considers the minimization of the makespan objective function. Heuristic and meta-heuristic (HEART) based algorithms are proposed to solve the task scheduling problems. These algorithms utilize the property of list scheduling algorithm of unrelated parallel machine scheduling problem. A mixed integer linear programming (MILP) formulation has been provided for the two variants of the problem. The optimal solution is obtained by solving MILP formulation using A Mathematical Programming Language (AMPL) software. Extensive numerical experiments have been performed to evaluate the performance of proposed algorithms. The solutions obtained by the proposed algorithms are found to out-perform the existing algorithms. The proposed algorithms can be used by cloud computing service providers (CCSPs) for enhancing their resources utilization to reduce their operating cost.     

0-1 Quadratic programming approach for optimum solutions of two scheduling problems

Two scheduling problems are considered: (1) scheduling n jobs non-preemptively on a single machine to minimize total weighted earliness and tardiness (WET); (2) scheduling n jobs non-preemptively on two parallel identical processors to minimize weighted mean flow time. In the second problem, a pre-ordering of the jobs is assumed that must be satisfied for any set of jobs scheduled on each specific machine. Both problems are known to be NP-complete. A 0-1 quadratic assignment formulation of the problems is presented. An equivalent 0-1 mixed integer linear programming approach for the problems are considered and a numerical example is given. The formulations presented enable one to use optimal and heuristic available algorithms of 0-1 quadratic assignment for the problems considered here.     

Scheduling two-stage hybrid flow shops with parallel batch, release time, and machine eligibility constraints

This paper investigates a difficult scheduling problem on a specialized two-stage hybrid flow shop with multiple processors that appears in semiconductor manufacturing industry, where the first and second stages process serial jobs and parallel batches, respectively. The objective is to seek job-machine, job-batch, and batch-machine assignments such that makespan is minimized, while considering parallel batch, release time, and machine eligibility constraints. We first propose a mixed integer programming (MIP) formulation for this problem, then gives a heuristic approach for solving larger problems. In order to handle real world large-scale scheduling problems, we propose an efficient dispatching rule called BFIFO that assigns jobs or batches to machines based on first-in-first-out principle, and then give several reoptimization techniques using MIP and local search heuristics involving interchange, translocation and transposition among assigned jobs. Computational experiments indicate our proposed re-optimization techniques are efficient. In particular, our approaches can produce good solutions for scheduling up to 160 jobs on 40 machines at both stages within 10?min.     

Lot sizing and furnace scheduling in small foundries

A lot sizing and scheduling problem prevalent in small market-driven foundries is studied. There are two related decision levels: (1) the furnace scheduling of metal alloy production, and (2) moulding machine planning which specifies the type and size of production lots. A mixed integer programming (MIP) formulation of the problem is proposed, but is impractical to solve in reasonable computing time for non-small instances. As a result, a faster relax-and-fix (RF) approach is developed that can also be used on a rolling horizon basis where only immediate-term schedules are implemented. As well as a MIP method to solve the basic RF approach, three variants of a local search method are also developed and tested using instances based on the literature. Finally, foundry-based tests with a real-order book resulted in a very substantial reduction of delivery delays and finished inventory, better use of capacity, and much faster schedule definition compared to the foundry's own practice.     

单机调度问题对偶集结迭代算法

具有到达时间约束、目标为最小化加权完工时间之和的单机调度问题是一个典型的NP-hard问题,采用时间下标建模的线性规划松弛方法可提供一个很强的下界,但优化求解存在维数困难.为此,本文提出了一种对偶集结优化策略,通过选择一个衰减集结矩阵集结对偶乘子变量,利用对偶理论获得模型的约束集结,从而降低计算复杂度.同时分析了集结模型的结构特性,并提出一种迭代算法来改善下界.仿真结果表明对偶集结迭代算法能够减少计算时间,同时改善下界性能,适用于大规模调度问题.     

Online over time processing of combinatorial problems

In an online environment, jobs arrive over time and there is no information in advance about how many jobs are going to be processed and what their processing times are going to be. In this paper, we study the online scheduling of Boolean Satisfiability (SAT) and Mixed Integer Programming (MIP) instances that are well-known NP-complete problems. Typical online machine scheduling approaches assume that jobs are completed at some point in order to minimize functions related to completion time (e.g., makespan, minimum lateness, total weighted tardiness, etc). In this work, we formalize and present an online over time problem where arriving instances are subject to waiting time constraints. We propose computational approaches that combine the use of machine learning, MIP, and instance interruption heuristics. Unlike other approaches, we attempt to maximize the number of solved instances using single and multiple machine configurations. Our empirical evaluation with well-known SAT and MIP instances, suggest that our interruption heuristics can improve generic ordering policies to solve up to 21.6x and 12.2x more SAT and MIP instances. Additionally, our hybrid approach observed up to 90% of solved instances with respect to a semi clairvoyant policy (SCP).     

An exact extended formulation for the unrelated parallel machine total weighted completion time problem

The plethora of research on \(\mathcal {NP}\)-hard parallel machine scheduling problems is focused on heuristics due to the theoretically and practically challenging nature of these problems. Only a handful of exact approaches are available in the literature, and most of these suffer from scalability issues. Moreover, the majority of the papers on the subject are restricted to the identical parallel machine scheduling environment. In this context, the main contribution of this work is to recognize and prove that a particular preemptive relaxation for the problem of minimizing the total weighted completion time (TWCT) on a set of unrelated parallel machines naturally admits a non-preemptive optimal solution and gives rise to an exact mixed integer linear programming formulation of the problem. Furthermore, we exploit the structural properties of TWCT and attain a very fast and scalable exact Benders decomposition-based algorithm for solving this formulation. Computationally, our approach holds great promise and may even be embedded into iterative algorithms for more complex shop scheduling problems as instances with up to 1000 jobs and 8 machines are solved to optimality within a few seconds.     

The complexity of customer order scheduling problems on parallel machines

This paper considers scheduling problems where a set of jobs (customer order) is shipped at the same time. The objective function is associated with the completion time of the orders. While a machine can process only one job at a time, multiple machines can process simultaneously jobs in an order. We first introduce this relatively new class of the customer order scheduling problems on parallel machines. Then, we establish the complexity of several problems with different types of objectives, job restrictions, and machine environments. For some tractable cases, we propose optimal solution procedures.     

MP or not MP: that is the question

It is well known that in the twentieth century, mathematical programming (MP) modeling and particularly linear programming (LP) modeling, even though strongly applied to combinatorial optimization, were not too successful when directed to scheduling problems. The purpose of this paper is to show that the field of successful applications of LP/MP modeling is still growing and includes also scheduling topics. We first focus on single machine scheduling. We consider a single machine scheduling model where a quadratic programming (QP) formulation handled by means of a QP solver is shown to be competitive with the state of the art approaches. Also, we discuss a single machine bicriterion scheduling problem and show that a standard LP formulation based on positional completion times performs reasonably well when handled by means of a LP solver. Then, we show how LP can be used to tighten bounds for approximation results in sequencing problems. Finally, we show how to enhance the complexity bounds of branch-and-reduce exact exponential algorithms by means of the so-called measure-and-conquer paradigm requiring always the solution of a specific MP model.     

Application of neural networks to heuristic scheduling algorithms

This paper considers the use of artificial neural networks (ANNs) to model six different heuristic algorithms applied to the n job, m machine real flowshop scheduling problem with the objective of minimizing makespan. The objective is to obtain six ANN models to be used for the prediction of the completion times for each job processed on each machine and to introduce the fuzziness of scheduling information into flowshop scheduling. Fuzzy membership functions are generated for completion, job waiting and machine idle times. Different methods are proposed to obtain the fuzzy parameters. To model the functional relation between the input and output variables, multilayered feedforward networks (MFNs) trained with error backpropagation learning rule are used. The trained network is able to apply the learnt relationship to new problems. In this paper, an implementation alternative to the existing heuristic algorithms is provided. Once the network is trained adequately, it can provide an outcome (solution) faster than conventional iterative methods by its generalizing property. The results obtained from the study can be extended to solve the scheduling problems in the area of manufacturing.     

多机Flow Shop加权完成时间调度问题的渐近最优算法研究

最近Chou、Queyranne和Simchi—Levi，Liu分别证明了恒速平行机调度问题和Flow shop调度问题的基于有效作业加权最短处理时间的启发式算法是渐近最优的。本文使用分组机器模型的方法证明：即使对于多机Flow shop加权完成时间调度问题，基于有效作业加权最短处理时间的启发式算法也是渐近最优的。关键词调度，多机Flow shop调度，启发式算法，渐近最优分析     

Unrelated parallel machine scheduling with dedicated machines and common deadline

This paper addresses a scheduling problem in the manufacturing of Polyvinyl Chloride pipes. There are two main attributes of PVC pipes: diameter and color. Each attribute has a corresponding attribute setup time and usually has several different levels. Each extruder produces different PVC pipe products based on the diameters as large, middle and small. The alternatives exist between these extruders, where the large and the middle type extruders can be used to produce the PVC pipes with the other diameters; the small type extruders can be used to produce the PVC pipes with middle diameters but cannot produce those with large diameters. The processing times are longer in all of the alternatives among different types of extruders. The objective is to minimize the total completion time for the unrelated parallel machine problem.Three dedicated machine heuristics are proposed herein for the problem and have been evaluated by comparing with the current scheduling method used in the case plant. The computational results show that the proposed constructive heuristics outperform the current scheduling method with significant improvements and can be used to solve large-size problems in reasonable computational times.     

An integer programming formulation for the project scheduling problem with irregular time-cost tradeoffs

Four integer programming formulations are studied for the irregular costs project scheduling problem with time/cost trade-offs (PSIC). Three formulations using standard assignment type variables are tested against a more novel integer programming formulation. Empirical tests show that in many instances the new formulation performs best and can solve problems with up to 90 activities in a reasonable amount of time. This is explained by a reduced number of binary variables, a tighter linear programming (LP) relaxation, and the sparsity and embedded network structure of the constraint matrix of the new formulation.     

Formal languages for integer programming modeling of shift scheduling problems

This paper approaches the problem of modeling optimization problems containing substructures involving constraints on sequences of decision variables. Such constraints can be very complex to express with Mixed Integer Programming (MIP). We suggest an approach inspired by global constraints used in Constraint Programming (CP) to exploit formal languages for the modeling of such substructures with MIP. More precisely, we first suggest a way to use automata, as the CP regular constraint does, to express allowed patterns for the values taken by the constrained sequence of variables. Secondly, we present how context-free grammars can contribute to formulate constraints on sequences of variables in a MIP model. Experimental results on both approaches show that they facilitate the modeling, but also give models easier to solve by MIP solvers compared to compact assignment MIP formulations.     

A branch-and-cut algorithm for a production scheduling problem with sequence-dependent and time-dependent setup times

This paper introduces and compares three different formulations of a production scheduling problem with sequence-dependent and time-dependent setup times on a single machine. The setup is divided into two parts: one that can be performed at any time and another one that is restricted to be performed outside of a given time interval. As a result, the setup time between two jobs is a function of the completion time of the first job. The problem can be formulated as a time-dependent traveling salesman problem, where the travel time between two nodes is a function of the departure time from the first node. We show that the resulting formulation can be strengthened to provide better linear programming relaxation lower bounds. We also introduce several families of valid inequalities which are used within a branch-and-cut algorithm. Computational experiments show that this algorithm can solve some instances with up to 50 jobs within reasonable computing times.     

Scheduling jobs on a single batch processing machine with incompatible job families and weighted number of tardy jobs objective

In this paper, we minimize the weighted and unweighted number of tardy jobs on a single batch processing machine with incompatible job families. We propose two different mixed integer linear programming (MILP) formulations based on positional variables. The second formulation does not contain a big-M coefficient. Two iterative schemes are discussed that are able to provide tighter linear programming bounds by reducing the number of positional variables. Furthermore, we also suggest a random key genetic algorithm (RKGA) to solve this scheduling problem. Results of computational experiments are shown. The second MILP formulation is more efficient with respect to lower bounds, while the first formulation provides better upper bounds. The iterative scheme is effective for the weighted case. The RKGA is able to find high-quality solutions in a reasonable amount of time.     

A hybrid VNS approach for the short-term production planning and scheduling: A case study in the pulp and paper industry

Mathematical formulations for production planning are increasing complexity, in order to improve their realism. In short-term planning, the desirable level of detail is particularly high. Exact solvers fail to generate good quality solutions for those complex models on medium- and large-sized instances within feasible time. Motivated by a real-world case study in the pulp and paper industry, this paper provides an efficient solution method to tackle the short-term production planning and scheduling in an integrated mill. Decisions on the paper machine setup pattern and on the production rate of the pulp digester (which is constrained to a maximum variation) complicate the problem. The approach is built on top of a mixed integer programming (MIP) formulation derived from the multi-stage general lotsizing and scheduling problem. It combines a Variable Neighbourhood Search procedure which manages the setup-related variables, a specific heuristic to determine the digester's production speeds and an exact method to optimize the production and flow movement decisions. Different strategies are explored to speed-up the solution procedure and alternative variants of the algorithm are tested on instances based on real data from the case study. The algorithm is benchmarked against exact procedures.