Intermediate Due Dates Research Articles

Efficient formulation for transportation scheduling of single refinery multiproduct pipelines

Multiproduct pipeline transportation scheduling is a complex operations research problem that is characterized by the movement of the cargo rather than the carrier. Hence, it cannot be solved using vehicle routing methods. While most formulations for short-term scheduling adopt a continuous-time representation, they often lead to suboptimal solutions because of the dependence on the number of time slots in the grid, which is difficult to predict. Furthermore, some of these formulations have poor linear relaxations due to the presence of inefficient big-M constraints. In this paper, we develop a discrete-time mixed integer linear programming (MILP) model for the detailed scheduling of a straight pipeline with a single refinery and multiple depots. The proposed formulation rigorously detects interface material generated between adjacent products and considers planned shutdowns in pipeline segments due to maintenance operations, as well as local market demands occurring at multiple intermediate due dates. The main novelty is that continuous tasks can span multiple time slots to enforce minimum batch sizes on injection and delivery nodes, which allows for the model to generate better schedules than those obtained with previously proposed formulations. To ensure an efficient model by design, we rely on generalized disjunctive programming (GDP) and on the convex hull reformulation of disjunctions, which results in stronger and often more computationally efficient formulations. We present numerical results for a set of benchmark instances and show that the proposed model applies to large-scale industrial cases.

Single Reference Grid Continuous-Time Formulation for Batch Scheduling

Abstract Short term scheduling of multipurpose batch processes has received growing attention over past decades. It concerns the optimal allocation of a set of limited resources to tasks over time in order to enhance the revenue of plants. This paper addresses the short term scheduling of batch processes through a continuous-time mixed integer linear programming (MILP) formulation based on the state-task network (STN) representation that allows to consider multiple intermediate due dates for market requirements. The proposed formulation can be classified as a slot-based approach that views the time horizon as a set of ordered blocks of unknown and variable lengths. Compared to previous similar approaches, it is simpler and leads to a smaller mathematical model without decoupling tasks from units. A few benchmark problems are used to illustrate the computational advantages of the proposed optimization approach.

Mixed-integer linear programming model for tree-like pipeline scheduling problem with intermediate due dates on demands

Multi-product pipelines are a significant and extensive mean of transporting petroleum based products from refineries to distribution centers. Previous contributions on tree-like pipeline scheduling problem have considered a simple structure with a single refinery connected to a mainline and some secondary lines only emerged from the mainline. In practice, however, a tree-like pipeline may also have several branches on a secondary line resulting in a complex structure, the so called multi-level tree-like pipeline. This paper addresses the short-term scheduling of multi-level tree-like pipelines with multiple refineries through a continuous time mixed-integer linear programming (MILP) model that considers multiple intermediate due dates for product demands. The objective is to satisfy product demands on time at the minimum operational costs, such as pumping, interface and backorder costs. The proposed model performance’s is shown by solving four examples.

Hybrid Time Formulation for Diesel Blending and Distribution Scheduling

This work focuses on the scheduling of an in-line diesel blending and distribution subsystem of an oil refinery. The formulation is based on a hybrid time representation in which time points are equally distributed along the time horizon, within which time slots of variable length are postulated. The hybrid time representation takes advantage of the flexibility of the continuous time representation and enables handling of intermediate due dates with the use of fixed time points. Time variables are defined in terms of resources instead of transfer operations leading to a smaller size formulation. Results from a real-world case are used to validate the proposed formulation that takes into consideration capacities and operating rules while minimizing costs. Results of the multiperiod scheduling are also compared to a day-ahead production planning policy. Computational efficiency is achieved after adding valid inequalities and symmetry breaking constraints.

Heuristic rules for tardiness problem in flow shop with intermediate due dates

In this paper, the tardiness flow shop models with intermediate due date were considered. The flow shop models consist of a set of jobs each having a number of operations, while each operation is performed in a single machine. All the jobs are considered having the same unidirectional precedence order. In the tardiness flow shop models with intermediate due date, which we call the generalized tardiness flow shop models, there exist a due date associated with the completion of each operation, and we want to find a schedule which minimizes the total tardiness of the jobs. This is a more general version of tardiness flow shop in the sense that, by assigning a large value to each of the intermediate due dates, we can obtain the traditional flow shop models. Considering the generalized tardiness flow shop models as the NP-hard problems, a set heuristic sequencing rules for finding the best permutation schedule for such problems is proposed. We conducted an extensive computational experiment using randomly generated test problems for evaluating the efficiency of the proposed rules in obtaining a near-optimal solution. The efficiency of the rules was evaluated, and those rules with better solutions were designated and reported.

Multi-period scheduling of a multi-stage multi-product bio-pharmaceutical process

There have been several works in the literature for scheduling of multi-product continuous processes with significant attention laid on short-term scheduling. This work presents a continuous-time model for multi-period scheduling of a multi-stage multi-product process from bio-pharmaceutical industry. The overall model is a mixed-integer linear programming (MILP) formulation based on state-task-network (STN) representation of the process using unit-specific event-based continuous-time representation. The proposed model is an extension of model by Shaik and Floudas (2007, Industrial & Engineering Chemistry Research, 46, 1764) with several new constraints to deal with additional features such as unit and sequence dependent changeovers, multiple intermediate due dates, handling of shelf-life and waste disposal, and penalties on backlogs and late deliveries. Improved tightening and sequencing constraints have been presented. The validity of the proposed model has been illustrated through an example from the literature.

Production Planning and Scheduling of Parallel Continuous Processes with Product Families

The goal of this paper is the development of a mixed integer programming formulation for the production planning and scheduling of parallel (single-stage) continuous processes in the presence of sequence-dependent switchover times and costs for product families, and sequence-independent switchovers for products belonging to the same family. Our formulation combines a discrete-time planning (big-bucket) grid with a continuous-time treatment of the scheduling decisions within each period and across adjacent periods. At the production planning level, it handles product orders at intermediate due dates and accounts for holding and backlog costs. At the scheduling level, it accounts for equipment unit constraints, switchover times and costs, maintenance activities, and idle production periods. The proposed model was motivated from and implemented to a real-world production facility. It can effectively address industrial-scale planning−scheduling problems. Importantly, it yields solutions that are considerably better than the solutions currently obtained using commercial tools.

Optimal scheduling of continuous plants with energy constraints

This work addresses the scheduling of continuous single stage multiproduct plants with parallel units and shared storage tanks. Processing tasks are energy intensive and we consider time-dependent electricity pricing and availability together with multiple intermediate due dates, handled as hard constraints. A new discrete-time aggregate formulation is proposed to rapidly plan the production levels. It is combined with a continuous-time model for detailed scheduling as the essential part of a rolling-horizon algorithm. Their computational performance is compared to traditional discrete and continuous-time full-space formulations with all models relying on the Resource-Task Network (RTN) process representation. The results show that the new models and algorithm can generate global optimal schedules much more efficiently than their counterparts in problems involving unlimited power availability. Under restricted power, the aggregate model underestimates the electricity cost, which may cause the rolling-horizon approach to converge to a suboptimal solution, becoming the discrete-time model a better approach.

New Continuous-Time Scheduling Formulation for Continuous Plants under Variable Electricity Cost

This work addresses the scheduling of continuous plants subject to energy constraints related to time-dependent electricity pricing and availability. Discrete- and continuous-time formulations are presented that can address these issues together with multiple intermediate due dates. Both formulations rely on the resource−task network process representation. Their computational performance is compared for the objective of total electricity minimization with the results favoring the discrete-time model due to the more natural way of handling such a wide variety of discrete events. In particular, it can successfully handle problems of industrial size. Nevertheless, the new continuous-time model is a major breakthrough since it is the first model of its type that is able to effectively incorporate time-variable utility profiles. When compared to a simple manual scheduling procedure, the proposed scheduling approaches can lead to potential electricity savings around 20% by switching production from periods of ...

Multi-product batch scheduling with intermediate due dates using priced timed automata models

In this contribution, we discuss an extension of the earlier work on scheduling using reachability analysis of timed automata (TA) models, specifically addressing the problem of tardiness minimization. In the TA-based approach the resources, recipes and additional timing constraints are modeled independently as sets of priced timed automata. The sets of individual automata are synchronized by means of synchronization labels and are composed by parallel composition to form a global automaton. The global automaton has an initial location where no operations have been started and at least one target location where all operations that are required to produce the demanded quantities of end-products within the specified due dates have been finished. A cost-optimal symbolic reachability analysis is performed on the composed automaton to derive schedules with the objective of minimizing tardiness. The model formulation is extended to include release dates of the raw materials and due dates of the production orders. The meeting of due dates is modeled by causing additional costs (e.g. penalties for late delivery and storage costs for early production). The modeling approach and the performance of the approach are tested for two different case studies and the results are compared with that of a MILP formulation solved using the standard solver CPLEX. The numerical experiments demonstrate, that the TA-based approach is competitive compared to standard commercial solvers and good feasible solutions are obtained with considerably reduced computational effort.

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.

Production Scheduling of a Large-Scale Industrial Batch Plant. I. Short-Term and Medium-Term Scheduling

The medium-term production scheduling of a multipurpose, multiproduct industrial batch plant is modeled using a novel mathematical framework. The methodology consists of the decomposition of the whole scheduling period into successive short horizons of a few days. A decomposition model is implemented to determine each short horizon and the corresponding products to be included. Then, a novel continuous-time formulation for short-term scheduling of batch processes with multiple intermediate due dates is applied to each short horizon selected, leading to a large-scale mixed-integer linear programming (MILP) problem. The scheduling model includes over 80 pieces of equipment and can take into account the processing recipes of hundreds of different products. Several characteristics of the production plant are incorporated into the scheduling model and actual plant data are used to model all parameters. Two different modes of operation are presented along with representative production schedules to demonstrate ...

Enhanced Continuous-Time Unit-Specific Event-Based Formulation for Short-Term Scheduling of Multipurpose Batch Processes: Resource Constraints and Mixed Storage Policies

An enhanced continuous-time formulation is presented for the short-term scheduling of multipurpose batch plants with intermediate due dates. The proposed formulation is based on the continuous-time formulation introduced by Ierapetritou and Floudas (Ind. Eng. Chem. Res. 1998, 37, 4341 and 4360) and incorporates and elaborates on several features that include (i) various storage policies (UIS, FIS, NIS, and ZW), (ii) resource constraints, (iii) variable batch sizes and processing times, (iv) batch mixing and splitting, and (v) sequence-dependent changeover times. The key features of the proposed formulation include a continuous-time representation utilizing a necessary number of event points of unknown location corresponding to the activation of a task. Also, tasks are allowed to continue over several event points, enabling resource quantities to be correctly determined at the beginning of each resource utilization. Several examples are presented to illustrate the effectiveness of the proposed formulation, and comparisons with other approaches are provided.

Continuous-Time Optimization Approach for Medium-Range Production Scheduling of a Multiproduct Batch Plant

The medium-range production scheduling problem of a multiproduct batch plant is studied. The methodology consists of a decomposition of the whole scheduling period to successive short horizons. A mathematical model is proposed to determine each short horizon and the products to be included. Then a novel continuous-time formulation for short-term scheduling of batch processes with multiple intermediate due dates is applied to each time horizon selected, leading to a large-scale mixed-integer linear programming (MILP) problem. Special structures of the problem are further exploited to improve the computational performance. An integrated graphical user interface implementing the proposed optimization framework is presented. The effectiveness of the proposed approach is illustrated with a large-scale industrial case study that features the production of 35 different products according to a basic three-stage recipe and its variations by sharing 10 pieces of equipment.

Scheduling of Single-Stage and Continuous Processes on Parallel Lines with Intermediate Due Dates

This paper addresses scheduling problems in single-stage and continuous multiproduct processes on parallel lines with intermediate due dates and especially restrictions on minimum run lengths. Whereas previous approaches require predetermined time slots or a maximum number of time slots to which only one task should be assigned, the proposed formulation does not use time slots within a time period. Therefore, the size of a model and its computation time are significantly reduced, with restrictions on several realistic situations, such as minimum run lengths, idle times of processes, and tests/maintenance, that are addressed in previous literature. Our approach is illustrated with various examples presented in the literature.

Effective Continuous-Time Formulation for Short-Term Scheduling. 3. Multiple Intermediate Due Dates

The problem of short-term scheduling often involves the satisfaction of variable product demands at specific due dates within the time horizon under consideration. Ierapetritou and Floudas (Ind. Eng. Chem. Res. 1998, 37, 4341−4359, 4360−4374) presented a new continuous time formulation to effectively address the problem of short-term scheduling in batch, continuous, and mixed production facilities where product demands are specified at the end of the time horizon. The primary objective of this paper is to extend the continuous-time formulation so as to deal with intermediate due dates. The mathematical model is developed and the operation mode of the plant (batch or semicontinuous) is further exploited to result in the most efficient solution strategy. Several examples are provided to illustrate the capabilities of the proposed continuous-time formulations, and it is demonstrated that a variety of problems presented in the literature can be addressed efficiently.

Short-term scheduling of batch plants with multiple intermediate due dates

The problem of short-term scheduling often involves the satisfaction of variable product demands at specific due dates within the time horizon under consideration. Ierapetritou and Floudas, [2, 3] presented a novel continuous time formulation to effectively address the problem of short-term scheduling in batch, continuous and mixed production facilities where product demands are specified at the end of time horizon. The primary objective of this paper is to extend the continuous time formulation so as to deal with intermediate due dates. The mathematical model is developed and the features of the problem are further exploited to result in the most efficient solution of the problem. Two examples are provided to illustrate the capabilities of the proposed approach.