Scheduling for a processor sharing system with linear slowdown

We consider the problem of scheduling arrivals to a congestion system with a finite number of users having identical deterministic demand sizes. The congestion is of the processor sharing type in the sense that all users in the system at any given time are served simultaneously. However, in contrast to classical processor sharing congestion models, the processing slowdown is proportional to the number of users in the system at any time. That is, the rate of service experienced by all users is linearly decreasing with the number of users. For each user there is an ideal departure time (due date). A centralized scheduling goal is then to select arrival times so as to minimize the total penalty due to deviations from ideal times weighted with sojourn times. Each deviation penalty is assumed quadratic, or more generally convex. But due to the dynamics of the system, the scheduling objective function is non-convex. Specifically, the system objective function is a non-smooth piecewise convex function. Nevertheless, we are able to leverage the structure of the problem to derive an algorithm that finds the global optimum in a (large but) finite number of steps, each involving the solution of a constrained convex program. Further, we put forward several heuristics. The first is the traversal of neighbouring constrained convex programming problems, that is guaranteed to reach a local minimum of the centralized problem. This is a form of a “local search”, where we use the problem structure in a novel manner. The second is a one-coordinate “global search”, used in coordinate pivot iteration. We then merge these two heuristics into a unified “local–global” heuristic, and numerically illustrate the effectiveness of this heuristic.     

The resource constrained project scheduling problem with multiple crashable modes: A heuristic procedure

We present a heuristic procedure for a nonpreemptive resource constrained project scheduling problem in which the duration/cost of an activity is determined by the mode selection and the duration reduction (crashing) applied within the selected mode. This problem is a natural combination of the time/cost trade-off problem and the resource constrained project scheduling problem. The objective is to determine each activity's start (finish) time, mode and duration so that the total project cost is minimized. Total project cost is the sum of all activity costs and the penalty cost for completing the project beyond its due date. We introduce a multi-pass algorithm. We report computational results with a set of 100 test problems and demonstrate the efficacy of the proposed heuristic procedure.     

Job scheduling with forbidden setups and two objectives using genetic algorithms and penalties

One of the most important tasks in service and manufacturing systems is how to schedule arriving jobs such that some criteria will be satisfied. Up to now there have been defined a great variety of scheduling problems as well as corresponding models and solution approaches. Most models suffer from such more or less restrictive assumptions like single machine, unique processing times, zero set-up times or a single criterion. On the other hand some classical approaches like linear or dynamic programming are practicable for small-size problems only. Therefore over the past years we can observe an increasing application of heuristic search methods. But scheduling problems with multiple machines, forbidden setups and multiple objectives are scarcely considered. In our paper we apply a Genetic Algorithm to such a problem which was found at a continuous casting plant. Because of the forbidden setups the probability for a random generated schedule to be feasible is nearly zero. To resolve this problem we use three kinds of penalties, a global, a local and a combined approach. For performance investigations of these penalty types we applied our approaches to a real world test instance with 96 jobs, three machines and two objectives. We tested five different penalty levels with 51 independent runs to evaluate the impact of the penalties.     

A two pass heuristic algorithm for scheduling ‘blocked out’ units in continuous process industry

This paper addresses the problem of scheduling cascaded ‘blocked out’ continuous processing units separated by finite capacity storage tanks. The raw materials for the product lines arrive simultaneously on the input side of the first unit. But every unit can process only one product line at a time, thus giving rise to the possibility of spillage of raw material due to limited storage capacity. The need to process multiple product lines and the added constraint of multiple intermediate upliftment dates aggravate the problem. This problem is quite common in petrochemical industry. The paper provides a MINLP (Mixed Integer Non-Linear Programming) formulation of the problem. However, for any realistic scheduling horizon, the size of the problem is too large to be solved by standard packages. We have proposed a depth first branch and bound algorithm, guided by heuristics, to help planners in tackling the problem. The suggested algorithm could output near optimal solutions for scheduling horizons of 30 time periods when applied to real life situations involving 3 units and 3 product lines. Preliminary version of the paper appeared in the proceedings of MISTA, 2005.     

Simultaneously scheduling multiple turns for steel color-coating production

This paper investigates a large-scale scheduling problem in the iron and steel industry, called color-coating production scheduling (CCPS). The problem is to generate multiple production turns for the galvanized coils that dynamically arrive from upstream lines within a given scheduling horizon, and at the same time determine the sequence of these turns so that the productivity and product quality are maximized while the production cost and the number of generated turns are minimized. We formulate this problem as a mixed integer nonlinear program and propose a tabu search heuristic to obtain satisfactory solutions. Results on real production instances show that the presented model and heuristic are more effective and efficient with comparison to manual scheduling. A practical scheduling system for CCPS combining the model and heuristic has been developed and successfully implemented in a major iron and steel enterprise in China.     

A production scheduling heuristic for an electronics manufacturer with sequence-dependent setup costs

In this paper, we develop a three-step heuristic to address a production scheduling problem at a high volume assemble-to-order electronics manufacturer. The heuristic provides a solution for scheduling multiple product families on parallel, identical production lines so as to minimize setup costs. The heuristic involves assignment, sequencing, and time scheduling steps, with an optimization approach developed for each step. For the most complex step, the sequencing step, we develop a greedy randomized adaptive search procedure (GRASP). We compare the setup costs resulting from the use of our scheduling heuristic against a heuristic previously developed and implemented at the electronics manufacturer that assumes approximately equal, sequence-independent, setup costs. By explicitly considering the sequence-dependent setup costs and applying GRASP, our empirical results show a reduction in setups costs for an entire factory of 14–21% with a range of single production line reductions from 0% to 49%.     

Supply chain scheduling in a collaborative manufacturing mode: model construction and algorithm design

In collaborative manufacturing, the supply chain scheduling problem becomes more complex according to both multiple product demands and multiple production modes. Aiming to obtain a reasonable solution to this complexity, we analyze the characteristics of collaborative manufacturing and design some elements, including production parameters, order parameters, and network parameters. We propose four general types of collaborative manufacturing networks and then construct a supply chain scheduling model composed of the processing costs, inventory costs, and two penalty costs of the early completion costs and tardiness costs. In our model, by considering the urgency of different orders, we design a delivery time window based on the least production time and slack time. Additionally, due to the merit of continuously processing orders belonging to the same product type, we design a production cost function by using a piecewise function. To solve our model efficiently, we present a hybrid ant colony optimization (HACO) algorithm. More specifically, the Monte Carlo algorithm is incorporated into our HACO algorithm to improve the solution quality. We also design a moving window award mechanism and dynamic pheromone update strategy to improve the search efficiency and solution performance. Computational tests are conducted to evaluate the performance of the proposed method.

     

Learning and forgetting effects on a group scheduling problem

This paper proposes to investigate learning and forgetting effects on the problem of scheduling families of jobs on a single machine to minimize total completion time of jobs. A setup time is incurred whenever the single machine transfers job processing from a family to another family. To analyze the impact of learning and forgetting on this group scheduling problem, we structure three basic models and make some comparisons through computational experiments. The three models, including no forgetting, total forgetting and partial forgetting, assume that the processing time of a job is dependent on its position in a schedule. Some scheduling rules and a lower bound are derived in order to constitute our branch-and-bound algorithm for searching an optimal sequence. In addition, an efficient and simply-structured heuristic is also built to find a near-optimal schedule.     

Minimizing makespan in a two-machine flowshop scheduling with batching and release time

This paper studies two-machine flowshop scheduling with batching and release time, whose objective is to minimize the makespan. We formulate the scheduling problem as a mixed integer programming model and show that it is a strongly NP-hard problem. We derive a lower bound and develop dynamic programming-based heuristic algorithms to solve the scheduling problem. Computational experiments are carried out to evaluate the performance of the heuristic algorithms. The numerical results show that some of the heuristic algorithms can indeed find effective solutions for the scheduling problem.     

A bicriteria parallel machine scheduling with a learning effect of setup and removal times

In studies on scheduling problems, generally setup times and removal times of jobs have been neglected or by including those into processing times. However, in some production systems, setup times and removal times are very important such that they should be considered independent from processing times. Since, in general jobs are done according to automatic machine processes in production systems processing times do not differ according to process sequence. But, since human factor becomes influential when setup times and removal times are taken into consideration, setup times will be decreasing by repeating setup processes frequently. This fact is defined with learning effect in scheduling literature. In this study, a bicriteria m-identical parallel machines scheduling problem with a learning effect of setup times and removal times is considered. The objective function of the problem is minimization of the weighted sum of total completion time and total tardiness. A mathematical programming model is developed for the problem which belongs to NP-hard class. Results of computational tests show that the proposed model is effective in solving problems with up to 15 jobs and five machines. We also proposed three heuristic approaches for solving large jobs problems. According to the best of our knowledge, no work exists on the minimization of the weighted sum of total completion time and total tardiness with a learning effect of setup times and removal times.     

A mixed integer programming model for scheduling orders in a steel mill

The problem of scheduling orders at each facility of a large integrated steel mill is considered. Orders are received randomly, and delivery dates are established immediately. Each order is filled by converting raw materials into a finished saleable steel product by a fixed sequence of processes. The application of a deterministic mixed integer linear programming model to the order scheduling problem is given. One important criterion permitted by the model is to process the orders in a sequence which minimizes the total tardiness from promised delivery for all orders; alternative criteria are also possible. Most practical constraints which arise in steelmaking can be considered within the formulation. In particular, sequencing and resource availability constraints are handled easily. The order scheduling model given here contains many variables and constraints, resulting in computational difficulties. A decomposition algorithm is devised for solving the model. The algorithm is a special case of Benders partitioning. Computational experience is reported for a large-scale problem involving scheduling 102 orders through ten facilities over a six-week period. The exact solution to the large-scale problem is compared with schedules determined by several heuristic dispatching rules. The dispatching rules took into consideration such things as due date, processing time, and tardiness penalties. None of the dispatching rules found the optimal solution.     

Improved bid prices for choice-based network revenue management

One of the latest developments in network revenue management (RM) is the incorporation of customer purchase behavior via discrete choice models. Many authors presented control policies for the booking process that are expressed in terms of which combination of products to offer at a given point in time and given resource inventories. However, in many implemented RM systems—most notably in the hotel industry—bid price control is being used, and this entails the problem that the recommended combination of products as identified by these policies might not be representable through bid price control. If demand were independent from available product alternatives, an optimal choice of bid prices is to use the marginal value of capacity for each resource in the network. But under dependent demand, this is not necessarily the case. In fact, it seems that these bid prices are typically not restrictive enough and result in buy-down effects.We propose (1) a simple and fast heuristic that iteratively improves on an initial guess for the bid price vector; this first guess could be, for example, dynamic estimates of the marginal value of capacity. Moreover, (2) we demonstrate that using these dynamic marginal capacity values directly as bid prices can lead to significant revenue loss as compared to using our heuristic to improve them. Finally, (3) we investigate numerically how much revenue performance is lost due to the confinement to product combinations that can be represented by a bid price.The heuristic is not restricted to a particular choice model and can be combined with any method that provides us with estimates of the marginal values of capacity. In our numerical experiments, we test the heuristic on some popular networks examples taken from peer literature. We use a multinomial logit choice model which allows customers from different segments to have products in common that they consider to purchase. In most problem instances, our heuristic policy results in significant revenue gains over some currently available alternatives at low computational cost.     

Multi-mode project payment scheduling problems with bonus–penalty structure

This paper involves the multi-mode project payment scheduling problem with bonus–penalty structure where activities can be performed with several modes and a bonus–penalty structure exists at the deadline of the project. In the problem the decisions on when to schedule events and payments, the magnitude of each payment, and the performing mode of each activity need to be optimized. A two-module simulated annealing heuristic is proposed to solve the mixed integer non-linear programming models for the contractor and the client, and a satisfactory solution, which consists of payment event set, event schedule, and payment amount set, may be found through iterations between the heuristic’s two modules. The profits of the two parties of the contract are changed significantly by the bonus–penalty structure and the structure may be considered as a coordination mechanism essentially, which may enhance the flexibility of payment scheduling and be helpful for the two parties to get more profits from the project. Through solving and analyzing an instance the insight that the bonus–penalty structure may advance the project completion effectively and improve the profits of the two parties in the meantime can be obtained.     

A penalty-based heuristic algorithm for the permutation flowshop scheduling problem with sequence-dependent set-up times

This paper considers the permutation flowshop scheduling problem with sequence-dependent set-up times and develops a penalty-based heuristic algorithm to find an approximately minimum makespan schedule. The proposed algorithm determines the penalty in time associated with a particular sequence and selects the sequence with the minimum time penalty as the best heuristic solution. Computational results comparing the effectiveness and efficiency of the proposed penalty-based heuristic algorithm with an existing savings index heuristic algorithm are reported and discussed.     

Mathematical model for scheduling operations in cascaded continuous processing units

The scenario under consideration involves n cascaded continuous processing units responsible for processing m product lines. Each product line needs to be processed by all the units in the same sequence, and has dedicated finite capacity storage tanks before and after every processing unit. A unit can process only one product line at a time. Inputs for all the product lines arrive continuously and simultaneously on the input side of the first unit in the sequence. There are multiple intermediate due dates for the final products. An optimal schedule for the units calls for a trade-off among spillage costs, upliftment failure penalties and changeover costs. A mathematical model is developed for the purpose and the resulting MINLP is linearized using standard techniques. The MILP has been tested using GAMS for three units and three product lines as encountered in a refinery situation. The model could output optimal schedules for a ten day scheduling horizon within reasonable time.     

Safe scheduling: Setting due dates in single-machine problems

We consider single-machine stochastic scheduling models with due dates as decisions. In addition to showing how to satisfy given service-level requirements, we examine variations of a model in which the tightness of due-dates conflicts with the desire to minimize tardiness. We show that a general form of the trade-off includes the stochastic E/T model and gives rise to a challenging scheduling problem. We present heuristic solution methods based on static and dynamic sorting procedures. Our computational evidence identifies a static heuristic that routinely produces good solutions and a dynamic rule that is nearly always optimal. The dynamic sorting procedure is also asymptotically optimal, meaning that it can be recommended for problems of any size.     

Learning heuristics for basic block instruction scheduling

Instruction scheduling is an important step for improving the performance of object code produced by a compiler. A fundamental problem that arises in instruction scheduling is to find a minimum length schedule for a basic block—a straight-line sequence of code with a single entry point and a single exit point—subject to precedence, latency, and resource constraints. Solving the problem exactly is known to be difficult, and most compilers use a greedy list scheduling algorithm coupled with a heuristic. The heuristic is usually hand-crafted, a potentially time-consuming process. In contrast, we present a study on automatically learning good heuristics using techniques from machine learning. In our study, a recently proposed optimal basic block scheduler was used to generate the machine learning training data. A decision tree learning algorithm was then used to induce a simple heuristic from the training data. The automatically constructed decision tree heuristic was compared against a popular critical-path heuristic on the SPEC 2000 benchmarks. On this benchmark suite, the decision tree heuristic reduced the number of basic blocks that were not optimally scheduled by up to 55% compared to the critical-path heuristic, and gave improved performance guarantees in terms of the worst-case factor from optimality.     

Fast algorithms for scheduling with learning effects and time-dependent processing times on a single machine

We consider scheduling problems with learning/deterioration effects and time-dependent processing times on a single machine, with or without due date assignment considerations. By reducing them to a special assignment problem on product matrices, we solve all these problems in near-linear time. This improves the time complexity of previous algorithms for some scheduling problems and establishes the fast polynomial solvability for several other problems.     

Customer order scheduling to minimize the number of late jobs

In the order scheduling problem, every job (order) consists of several tasks (product items), each of which will be processed on a dedicated machine. The completion time of a job is defined as the time at which all its tasks are finished. Minimizing the number of late jobs was known to be strongly NP-hard. In this note, we show that no FPTAS exists for the two-machine, common due date case, unless P = NP. We design a heuristic algorithm and analyze its performance ratio for the unweighted case. An LP-based approximation algorithm is presented for the general multicover problem. The algorithm can be applied to the weighted version of the order scheduling problem with a common due date.     

Algorithm for minimizing weighted earliness penalty in single-machine problem

In this paper, the problem of minimizing the weighted earliness penalty in a single-machine scheduling problem is addressed. For this problem, every job is assumed to be available at time zero and must be completed before or on its deadline. No tardy job is allowed. Each job has its own earliness penalty and deadline. The paper identifies several local optimality conditions for sequencing of adjacent jobs. A heuristic algorithm is developed based on these local optimality conditions. Sample problems are solved and the solutions obtained from the heuristic are compared to solutions obtained from the heuristics developed by Chand and Schneeberger. Also, comparisons are performed between the solutions obtained from the heuristic and the optimal solutions obtained from a mathematical modeling approach for problems involving 10 and 15 jobs. The results show that the developed heuristic produces solutions close to optimal in small size problems, and it also outperforms the Chand and Schneeberger's method.