Theoretical framework for formulating MIP scheduling models with multiple and non-uniform discrete-time grids

We present a framework for the formulation of MIP scheduling models based on multiple and nonuniform discrete time grids. In a previous work we showed that it is possible to use different (possibly non-uniform) time grids for each task, unit, and material. Here, we generalize these ideas to account for general resources, and a range of processing characteristics such as limited intermediate storage and changeovers. Each resource has its own grid based on resource consumption and availability allowing resource constraints to be modeled more accurately without increasing the number of binary variables. We develop algorithms to define the unit-, task-, material-, and resource-specific grids directly from problem data. Importantly, we prove that the multi-grid formulation is able to find a schedule with the same optimal objective as the discrete-time single-grid model with an arbitrarily fine grid. The proposed framework leads to the formulation of models with reduced number of binary variables and constraints, which are able to find good solutions faster than existing models.     

A novel precedence-based and heuristic approach for short-term scheduling of multipurpose batch plants

Many continuous-time formulations have been proposed during the last decades for short-term scheduling of multipurpose batch plants. Although these models establish advantages over discrete-time representations, they are still inefficient in solving moderate-size problems, such as maximization of profit in long horizon, and minimization of makespan. Unlike existing literature, this paper presents a new precedence-based mixed integer linear programming (MILP) formulation for short-term scheduling of multipurpose batch plants. In the new model, multipurpose batch plants are described with a modified state-task network (STN) approach, and binary variables express the assignments and sequences of batch processing and storing. To eliminate the drawback of precedence-based formulations which commonly include large numbers of batches, an iterative procedure is developed to determine the appropriate number of batch that leads to global optimal solution. Moreover, four heuristic rules are proposed to selectively prefix some binary variables to 0 or 1, thereby reducing the overall number of binary variables significantly. To evaluate model performance, our model and the best models reported in the literature (S&K model and I&F model) are utilized to solve several benchmark examples. The result comparison shows that our model is more effective to find better solution for complex problems when using heuristic rules. Note that our approach not only can handle unlimited intermediate storage efficiently as well as the I&F model, but also can solve scheduling problems in limited intermediate storage more quickly than the S&K model.     

A simpler better slot-based continuous-time formulation for short-term scheduling in multipurpose batch plants

Short-term scheduling of multipurpose batch plants is a challenging problem for which several formulations exist in the literature. In this paper, we present a new, simpler, more efficient, and potentially tighter, mixed integer linear programming (MILP) formulation using a continuous-time representation with synchronous slots and a novel idea of several balances (time, mass, resource, etc.). The model uses no big-M constraints, and is equally effective for both maximizing profit and minimizing makespan. Using extensive, rigorous numerical evaluations on a variety of test problems, we show that in contrast to the best model in the literature, our model does not decouple tasks and units, but still has fewer binary variables, constraints, and nonzeros, and is faster.     

基于特定单元事件点的改进间歇生产调度模型

建立有效的间歇调度模型一直是生产调度问题调度研究的热点,而连续时间模型是优化短期间歇生产调度问题的有效工具。基于特定单元事件点的概念,建立一种改进的间歇调度连续时间混合整数线性规划(MILP)模型。该调度模型引入了新变量,使模型处理物料在不同设备间的传输过程更加灵活。结果表明,提出的改进模型只需要较少的事件点,就可以快速有效处理无限中间存储(UIS)间歇调度问题。     

Multiple and nonuniform time grids in discrete-time MIP models for chemical production scheduling

The modeling of time plays a key role in the formulation of mixed-integer programming (MIP) models for scheduling, production planning, and operational supply chain planning problems. It affects the size of the model, the computational requirements, and the quality of the solution. While the development of smaller continuous-time scheduling models, based on multiple time grids, has received considerable attention, no truly different modeling methods are available for discrete-time models. In this paper, we challenge the long-standing belief that employing a discrete modeling of time requires a common uniform grid. First, we show that multiple grids can actually be employed in discrete-time models. Second, we show that not only unit-specific but also task-specific and material-specific grids can be generated. Third, we present methods to systematically formulate discrete-time multi-grid models that allow different tasks, units, or materials to have their own time grid. We present two different algorithms to find the grid. The first algorithm determines the largest grid spacing that will not eliminate the optimal solution. The second algorithm allows the user to adjust the level of approximation; more approximate grids may have worse solutions, but many fewer binary variables. Importantly, we show that the proposed models have exactly the same types of constraints as models relying on a single uniform grid, which means that the proposed models are tight and that known solution methods can be employed. The proposed methods lead to substantial reductions in the size of the formulations and thus the computational requirements. In addition, they can yield better solutions than formulations that use approximations. We show how to select the different time grids, state the formulation, and present computational results.     

A tighter continuous time formulation for the cyclic scheduling of a mixed plant

In this paper, based on the cyclic scheduling formulation of Schilling and Pantelides [Schilling, G., & Pantelides, C. (1999). Optimal periodic scheduling of multipurpose plants. Computers& Chemical Engineering, 23, 635–655], we propose a continuous time mixed integer linear programming (MILP) formulation for the cyclic scheduling of a mixed plant, i.e. a plant composed of batch and continuous tasks. The cycle duration is a variable of the model and the objective is to maximize productivity. By using strengthening techniques and the analysis of small polytopes related to the problem formulation, we strengthen the initial formulation by tightening some initial constraints and by adding valid inequalities. We show that this strengthened formulation is able to solve moderate size problems quicker than the initial one. However, for real size cases, it remains difficult to obtain the optimal solution of the scheduling problem quickly. Therefore, we introduce MILP-based heuristic methods in order to solve these larger instances, and show that they can provide good feasible solutions quickly.     

Scheduling of displacement batch digesters using discrete time formulation

This paper provides mathematical programming based optimization model and computational results for short-term scheduling of displacement batch digesters in a pulp industry. The scheduling problem involves development of an optimal solution that yields the best sequence of operations in each of the parallel batch digesters sharing common resources. The constraints are imposed on meeting the demand of pulp of different qualities within a specified time horizon. The problem comprises of both fixed-time and variable time durations of the tasks, different storage policies, zero-wait and finite wait times, and handling of shared resources. The scheduling problem is formulated using a state-task-network (STN) representation of production recipes, based on discrete time representation resulting in a mixed-integer linear programming (MILP) problem which is solved using GAMS software. The basic framework is adapted from the discrete-time model of Kondili et al. (Comput. Chem. Eng., 1993, 17, 211–227). Different case studies involving parallel digesters in multiple production lines are considered to demonstrate the effectiveness of the proposed formulation using two different objective functions.     

Optimal planning and campaign scheduling of biopharmaceutical processes using a continuous-time formulation

This work addresses the optimal planning and campaign scheduling of biopharmaceutical manufacturing processes, considering multiple operational characteristics, such as the campaign schedule of batch and/or continuous process steps, multiple intermediate deliveries, sequence dependent changeovers operations, product storage restricted to shelf-life limitations, and the track-control of the production/campaign lots due to regulatory policies. A new mixed integer linear programing (MILP) model, based on a Resource Task Network (RTN) continuous time single-grid formulation, is developed to comprise the integration of all these features. The performance of the model features is discussed with the resolution of a set of industrial problems with different data sets and process layouts, demonstrating the wide application of the proposed formulation. It is also performed a comparison with a related literature model, showing the advantages of the continuous-time approach and the generality of our model for the optimal production management of biopharmaceutical processes.     

Modeling and Optimization for Short-term Scheduling of Multipurpose Batch Plants

In the past two decades, short-term scheduling of multipurpose batch plants has received significant attention. Most scheduling problems are modeled using either state-task-network or resource-task-network (RTN) process representation. In this paper, an improved mixed integer linear programming model for short-term scheduling of multipurpose batch plants under maximization of profit is proposed based on RTN representation and unit-specific events. To solve the model, a hybrid algorithm based on line-up competition algorithm and linear programming is presented. The proposed model and hybrid algorithm are applied to two benchmark examples in literature. The simulation results show that the proposed model and hybrid algorithm are effective for short-term scheduling of multipurpose batch plants.     

Design,synthesis and scheduling of multipurpose batch plants via an effective continuous-time formulation

Design, synthesis and scheduling issues are considered simultaneously for multipurpose batch plants. An earlier proposed continuous-time formulation for scheduling is extended to incorporate design and synthesis. Processing recipes are represented by the State-Task Network (STN). The superstructure of all possible plant designs is constructed according to the potential availability of all processing/storage units. The proposed model takes into account the trade-offs between capital costs, revenues and operational flexibility. Computational studies are presented to illustrate the effectiveness of the proposed formulation. Both linear and nonlinear models are included, resulting in MILP and mixed-integer nonlinear programming (MINLP) problems, respectively. The MILP problems are solved using a branch and bound method. Globally optimal solutions are obtained for the nonconvex MINLP problems based on a key property that arises due to the special structure of the resulting models. Comparisons with earlier approaches are also presented.     

A rule-based genetic algorithm for the scheduling of single-stage multi-product batch plants with parallel units

This paper presents a heuristic rule-based genetic algorithm (GA) for large-size single-stage multi-product scheduling problems (SMSP) in batch plants with parallel units. SMSP have been widely studied by the researchers. Most of them used mixed-integer linear programming (MILP) formulation to solve the problems. With the problem size increasing, the computational effort of MILP increases greatly. Therefore, it is very difficult for MILP to obtain acceptable solutions to large-size problems within reasonable time. To solve large-size problems, the preferred method in industry is the use of scheduling rules. However, due to the constraints in SMSP, the simple rule-based method may not guarantee the feasibility and quality of the solution. In this study, a random search based on heuristic rules was proposed first. Through exploring a set of random solutions, better feasible solutions can be achieved. To improve the quality of the random solutions, a genetic algorithm-based on heuristic rules has been proposed. The heuristic rules play a very important role in cutting down the solution space and reducing the search time. Through comparative study, the proposed method demonstrates promising performance in solving large-size SMSP.     

Source-based discrete and continuous-time formulations for the crude oil pooling problem

The optimization of crude oil operations in refineries is a challenging scheduling problem due to the need to model tanks of varying composition with nonconvex bilinear terms, and complicating logistic constraints. Following recent work for multiperiod pooling problems of refined petroleum products, a source-based mixed-integer nonlinear programming formulation is proposed for discrete and continuous representations of time. Logistic constraints are modeled through Generalized Disjunctive Programming while a specialized algorithm featuring relaxations from multiparametric disaggregation handles the bilinear terms. Results over a set of test problems from the literature show that the discrete-time approach finds better solutions when minimizing cost (avoids source of bilinear terms). In contrast, solution quality is slightly better for the continuous-time formulation when maximizing gross margin. The results also show that the specialized global optimization algorithm can lead to lower optimality gaps for fixed CPU, but overall, the performance of commercial solvers BARON and GloMIQO are better.     

Continuous‐time scheduling formulation for straight pipelines

Pipelines represent the most cost‐effective way of transporting large quantities of refined petroleum products over large distances but can be challenging to operate. In this article, we propose a new mixed‐integer linear programming formulation for scheduling straight pipelines with multiple single and dual purpose nodes. The model allows for simultaneous injections and deliveries, and interacting pumping runs, in which a segment of the pipeline simultaneously receives material from its refinery and upstream segment. In contrast to previous batch centric models, it uses segment dependent coordinates. To make it efficient by design, we rely on generalized disjunctive programming and develop disjunctions for which the convex hull reformulation is simple (roughly the same number of variables and constraints as its big‐M counterpart). Through the solution of a set of test cases from the literature, we show a better utilization of the pipeline capacity that is translated into a lower makespan. © 2016 American Institute of Chemical Engineers AIChE J, 63: 1923–1936, 2017     

A novel approach to scheduling multipurpose batch plants using unit‐slots

Several models for scheduling multipurpose batch plants exist in the literature. The models using unit‐specific event points have shown better solution efficiency on various literature examples. This article presents a novel approach to scheduling multipurpose batch plants, which uses unit‐slots instead of process‐slots to manage shared resources such as material storage. We develop two slightly different models that are even more compact and simpler than that of Sundaramoorthy and Karimi, Chem Eng Sci. 2005;60:2679–2702. Although we focus on material as a shared resource, our multi‐grid approach rationalizes, generalizes, and improves the current multi‐grid approaches for scheduling with shared resources. Our models allow nonsimultaneous transfers of materials into and out of a batch. We show by an example that this flexibility can give better schedules than those from existing models in some cases. Furthermore, our approach uses fewer slots (event‐points) on some examples than even those required by the most recent unit‐specific event‐based model. Numerical evaluation using literature examples shows significant gains in solution efficiency from the use of unit‐slots except where the number of unit‐slots required for the optimal solution equals that of process slots. We also highlight the importance of constraint sequencing in GAMS implementation for evaluating mixed‐integer linear programming based scheduling models fairly. © 2009 American Institute of Chemical Engineers AIChE J, 2010     

Multiproduct batch plants design using linear process performance models

In this contribution, a novel linear generalized disjunctive programming (LGDP) model is developed for the design of multiproduct batch plants optimizing both process variables and the structure of the plant through the use of process performance models. These models describe unit operations using explicit expressions for the size and time factors as functions of the process variables with the highest impact. To attain a linear formulation, values of the process variables as well as unit sizes are selected from a set of meaningful discrete values provided by the designer. Regarding structural alternatives, both kinds of unit duplications in series and in parallel are considered in this approach. The inclusion of the duplication in series requires different detailed models that depend on the structure selected. Thus, in a new approach for the multiproduct batch plant design, a set of potential structural alternatives for the plant is defined. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

A novel MILP formulation for short-term scheduling of multi-stage multi-product batch plants with sequence-dependent constraints

This paper presents a continuous-time mixed-integer linear programming (MILP) model for short-term scheduling of multi-stage multi-product batch plants. The model determines the optimal sequencing and the allocation of customer orders to non-identical processing units by minimizing the earliness and tardiness of order completion. This is a highly combinatorial problem, especially when sequence-dependent relations are considered such as the setup time between consecutive orders. A common approach to this scheduling problem relies on the application of tetra-index binary variables, i.e. (order, order, stage, unit) to represent all the combinations of order sequences and assignments to units in the various stages. This generates a huge number of binary variables and, as a consequence, much time is required for solutions. This paper proposes a novel formulation that replaces the tetra-index binary variables by one set of tri-index binary variables (order, order, stage) without losing the model's generality. By the elimination of the unit index, the new formulation requires considerably fewer binary variables, thus significantly shortening the solution time.     

基于状态设备网络的改进间歇生产调度模型

建立有效的间歇生产调度模型一直是生产调度问题研究的热点,基于特定事件点的连续时间建模方法是优化短期间歇生产调度问题的有效工具。基于状态设备网络和特定事件点概念,建立非线性的连续时间间歇生产调度模型。为了解决非线性引起的求解困难,该模型使用替代方法线性化模型中的双线性项,替代法不仅将建立的混合整数非线性规划模型转化为混合整数线性规划模型,且由于其不包含大M松弛项,能使模型搜索空间更紧凑,模型求解效率更高。通过3个实例对比实验表明了基于状态设备网络描述的改进间歇生产调度模型搜索高效性。另外,模型中还给出了不同存储条件下,基于状态设备网络描述的间歇生产调度模型约束,扩展了模型适用性。     

A CP formulation for scheduling multiproduct multistage batch plants

The short-term scheduling of multiproduct multistage batch plants is tackled in this paper by means of a constraint programming (CP) methodology. This approach, consisting of both a model and a search strategy, easily handles different features found in industrial environments: finite unit ready times, dissimilar parallel equipment at each stage, sequence-dependent changeovers, topology constraints, forbidden job-equipment assignments, order release times, as well as renewable resources limitations. It can also address various interstage storage and operational policies: UIS, NIS/ZW, NIS/UW, and mixed ones. Besides, it introduces two simple and efficient search methodologies based on domain knowledge, whose great impact on the computational performance is shown. The approach was extensively tested by means of several examples having various difficulty degrees. It rendered good computational results for a variety of interstage storage policies and objective functions. Moreover, this work shows that the default depth-first search strategy does not perform well for scheduling problems.     

新的多产品间歇生产调度的MILP模型

提出了一种新的多产品厂间歇调度问题的连续时间混合整数线性规划 (MILP)模型,该模型的整数变量体系不依赖于时间块(或者事件点)的概念,并且利用了变量物理概念上的对称互补性,使得与传统的建模方法相比不仅整数变量的数目减少了一半以上,而且建模思想、建模理论都有了新的改进.通过对一个算例的考察证实了新模型可以快速地求得全局最优解.     

A novel adaptive surrogate modeling‐based algorithm for simultaneous optimization of sequential batch process scheduling and dynamic operations

A novel adaptive surrogate modeling‐based algorithm is proposed to solve the integrated scheduling and dynamic optimization problem for sequential batch processes. The integrated optimization problem is formulated as a large scale mixed‐integer nonlinear programming (MINLP) problem. To overcome the computational challenge of solving the integrated MINLP problem, an efficient solution algorithm based on the bilevel structure of the integrated problem is proposed. Because processing times and costs of each batch are the only linking variables between the scheduling and dynamic optimization problems, surrogate models based on piece‐wise linear functions are built for the dynamic optimization problems of each batch. These surrogate models are then updated adaptively, either by adding a new sampling point based on the solution of the previous iteration, or by doubling the upper bound of total processing time for the current surrogate model. The performance of the proposed method is demonstrated through the optimization of a multiproduct sequential batch process with seven units and up to five tasks. The results show that the proposed algorithm leads to a 31% higher profit than the sequential method. The proposed method also outperforms the full space simultaneous method by reducing the computational time by more than four orders of magnitude and returning a 9.59% higher profit. © 2015 American Institute of Chemical Engineers AIChE J, 61: 4191–4209, 2015