Continuous-time Mixed-integer Linear Programming Research Articles

A novel approach for steelmaking scheduling with self-generation under real-time and demand charge tariffs

It is difficult for energy-intensive enterprises equipped with self-generation to incentivize consumers to shift load to low-demand periods through time-based pricing. This paper addresses a scheduling problem of steel plant equipped with self-generation under real-time electricity prices. Considering the possibility of centralized load shifting resulting in excessive loads for some periods, we also consider the demand charge pricing. A continuous-time mixed-integer linear programming model based on event expression and generalized disjunctive programming is proposed. As industrial scale problems cannot be solved by some commercial solvers in an acceptable time, a SPEA2 algorithm is tailored. Effective encoding based on decomposition-and-offsetting mechanism and decoding based on backward-and-rules are used to generate the initial population and scheduling scheme, respectively. Meanwhile, crossover and mutation operators incorporating conflict resolution strategies can strike a balance between exploring and exploiting to produce high-quality solutions. Numerical experiments indicate that the proposed approach has superiority in performance and efficiency.

Continuous-time formulation and differential evolution algorithm for an integrated batching and scheduling problem in aluminium industry

This paper investigates an integrated batching and scheduling problem of electrolysis and caster in aluminium industry. The problem is to determine the assignment and scheduling of orders considering sequence-dependent setup times caused by technological and operational constraints of electrolysis cells, and determine the batching and scheduling of orders in the following casters. A novel unit-specific event-based continuous-time mixed integer linear programming model (MILP) is proposed to describe the problem. In this model, the event point is stage specific, and lower bounds are specified to tighten the model. A hybrid pointer-based differential evolution algorithm with new individual representation scheme is designed to solve the problem of industrial scale. An improved hybrid pointer-based mutation operator and a new point-cross crossover operator are proposed to enhance the performance of the algorithm. Computational experiments show that the proposed algorithm is more efficient when compared with CPLEX for medium and large size instances. Comparisons with the lower bound demonstrate that the algorithm is effective.

Two-stage parallel speed-scaling machine scheduling under time-of-use tariffs

As one of the demand-side programs, time-of-use (TOU) tariffs brings opportunities of maintaining power grid stability for electricity providers and chances of energy conservation for manufacturers, but it also brings challenge for enterprises to optimize scheduling schemes. This paper studies a two-stage parallel machine scheduling problem under TOU to minimize total electricity costs. The two-stage parallel machine system is composed of identical parallel speed-scaling machines at stage 1 and unrelated parallel machines at stage 2. The key issues lie in assigning a group of jobs to a set of parallel machines at each stage and choosing the appropriate processing speed for all jobs at stage 1, and then determining the interval of processing time for jobs on each selected machine. To solve this problem, a new continuous-time mixed-integer linear programming model is formulated. According to the characteristics of this model, a tabu search-greedy insertion hybrid (TS-GIH) algorithm is designed, which realizes job-machine assignment based on load balancing principle, job insertion with greedy mechanism as well as movement and speed adjustment strategies to find more suitable positions for jobs. The effectiveness of the proposed TS-GIH is demonstrated by comparing with CLPEX and improved genetic algorithm (IGA) through real-life and randomly generated instances. The results show that TS-GIH can realize the trade-off between computation time and solution quality. Compared with CLPEX, the computation time of TS-GIH is significantly less, and the solution quality is much better than IGA.

A holistic MILP model for scheduling and inventory management of a multiproduct oil distribution system

This paper addresses the optimal scheduling of an oil transportation system characterized by a straight multiproduct pipeline featuring multiple input and output nodes, where products are dispatched to local markets often by tanker trucks. We present a new continuous-time mixed integer linear programming (MILP) model that is designed based on real-world necessities and that requires significantly fewer binary variables than previous work. As main contributions, the model: i) can rigorously avoid forbidden product sequences in every pipeline segment; ii) considers filler batch constraints to avoid large contamination volumes; and iii) includes inventory management constraints in the different pipeline nodes. We first use an illustrative example before testing the approach with a new large-scale example problem and three real-world cases from the literature. Results show that the proposed model has a tight linear programming (LP) relaxation and is very efficient computationally. It is thus a significant contribution to the state-of-the-art.

An MILP formulation for optimizing detailed schedules of a multiproduct pipeline network

This paper addresses how to optimize detailed schedules of a multiproduct pipeline network. A continuous-time mixed integer linear programming (MILP) formulation is developed to seek the minimum makespan of transportation tasks. Operation constraints about sequence of products, size of single-product batches, flow rate of delivery and injection operations, pumping rate of pipeline segments, batch tracking, conditions of implementing injection or delivery operations, inventory management of stations and consumer markets’ demand are all satisfied. The MILP formulation is illustrated by three pipeline networks. Results show the proposed MILP model has a better performance than previous works.

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.

A MILP model based on flowrate database for detailed scheduling of a multi-product pipeline with multiple pump stations

Multi-product pipelines usually transport several products in batches to respective delivery stations. As for a multi-product pipeline with multiple pump stations, this paper develops a continuous-time mixed-integer linear programming (MILP) model based on flowrate database to optimize its detailed scheduling. In the proposed model, various unit pump cost and flowrate constraints, which strongly depend on pump operation schemes, are introduced for the economy and safety of solved scheduling plans. Moreover, this paper considers the actual field processing constraints which vary with batch interface migration and rarely considered in previous work. And a novel method of historical flowrate database preprocessing is presented to enhance solving efficiency. Finally, through comparing with three real-world cases solved by another two available models, the proposed one performs the best in scheduling optimization as well as substantial reduction of pump cost.

A TWO-STAGE GREEDY HEURISTIC FOR A FLOWSHOP SCHEDULING PROBLEM UNDER TIME-OF-USE ELECTRICITY TARIFFS

This paper studies a two-machine flowshop scheduling problem under time-dependent electricity tariffs, in which electricity prices may vary from time to time throughout the day. The main issue is to assign a set of jobs to available time slots with different electricity prices to minimise the total resource cost required for processing the jobs. The main contribution of this work is two-fold. First, a new continuous-time mixed-integer linear programming (MILP) model is proposed for the problem. Second, a two-stage greedy heuristic is developed. A computational experiment on randomly generated instances demonstrates that the greedy algorithm can improve the objective function by almost 40 percent. The algorithm can be applied by production managers to scheduling jobs in a flowshop under time-of-use (TOU) electricity tariffs to save electricity costs.

Improved mixed-integer linear programming model and heuristics for bi-objective single-machine batch scheduling with energy cost consideration

ABSTRACTThis article addresses bi-objective single-machine batch scheduling under time-of-use electricity prices to minimize the total energy cost and the makespan. The lower and upper bounds on the number of formed batches are first derived and a continuous-time mixed-integer linear programming model is proposed, which improves an existing discrete-time model in the literature. Two improved heuristics are proposed based on the improved model. Computational experiments demonstrate that the improved model and heuristics can run hundreds of times faster than the existing ones for large-size instances.

Optimal Scheduling of Multiproduct Pipelines in Networks with Reversible Flow

This paper presents a new continuous-time mixed-integer linear programming formulation for scheduling pipeline systems transporting liquid products. It is a generic formulation with respect to the structure of the network (nodes and connecting segments form the building blocks) and to the direction of flow that can change as many times as required. Nodes are connected to one or more pipeline segments and external interactions with refineries (at a constant rate) and local markets may be included for better inventory control. To ensure a computationally efficient formulation by design, constraints with binary and continuous variables were first written as disjunctions and then convex-hull reformulated. It is validated through the solution of six benchmark problems from the literature involving straight pipelines with reversible flow as well as tree-like and mesh structures with unidirectional flow. The schedules obtained by the optimization reflect, in most cases, a better utilization of the pipeline capacity.

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.

Energy-conscious unrelated parallel machine scheduling under time-of-use electricity tariffs

This paper investigates an energy-conscious unrelated parallel machine scheduling problem under time-of-use (TOU) electricity pricing scheme, in which the electricity price varies throughout a day. The problem lies in assigning a group of jobs to a set of unrelated parallel machines and then scheduling jobs on each separate machine so as to minimize the total electricity cost. We first build an improved continuous-time mixed-integer linear programming (MILP) model for the problem. To tackle large-size problems, we then propose a two-stage heuristic. Specifically, at the first stage, jobs are assigned to machines aiming at minimizing the total electricity cost under the preemptive circumstance. At the second stage, the jobs assigned to each machine are scheduled using an insertion heuristic. Computational results on a real-life instance for turning process and random test instances demonstrate that the proposed MILP approach is able to solve small-size problems while the two-stage heuristic is appropriate for large-size problems. The case study for turning process also reveals that the proposed optimization approaches can contribute to cleaner production.

MIP formulations for an application of project scheduling in human resource management

In the literature, various discrete-time and continuous-time mixed-integer linear programming (MIP) formulations for project scheduling problems have been proposed. The performance of these formulations has been analyzed based on generic test instances. The objective of this study is to analyze the performance of discrete-time and continuous-time MIP formulations for a real-life application of project scheduling in human resource management. We consider the problem of scheduling assessment centers. In an assessment center, candidates for job positions perform different tasks while being observed and evaluated by assessors. Because these assessors are highly qualified and expensive personnel, the duration of the assessment center should be minimized. Complex rules for assigning assessors to candidates distinguish this problem from other scheduling problems discussed in the literature. We develop two discrete-time and three continuous-time MIP formulations, and we present problem-specific lower bounds. In a comparative study, we analyze the performance of the five MIP formulations on four real-life instances and a set of 240 instances derived from real-life data. The results indicate that good or optimal solutions are obtained for all instances within short computational time. In particular, one of the real-life instances is solved to optimality. Surprisingly, the continuous-time formulations outperform the discrete-time formulations in terms of solution quality.

Implementation of an integrated production and electricity optimization system in melt shop

The production of stainless steel begins by melting of scrap in melt shop that is known as one of the most electricity intensive processes. Volatile and uncertain renewable energy sources cause high fluctuations in electricity prices and thus usually higher costs for the final user such as a melt shop. This paper presents an integrated production and electricity optimization system that can help the melt shop to adjust its production schedule to volatile electricity prices and thus lower its costs. An existing production scheduling optimization system based on a continuous-time mixed-integer linear programming model has been further developed to account for electricity costs as well. It deploys an intelligent heuristics which decomposes the overall optimization problem into several sub-problems of smaller size in order to achieve faster and more robust solution. The electricity-aware optimization system has been successfully tested and implemented in a melt shop where it managed to reduce electricity costs by around 3%. At the same time, the system has improved coordination between different production stages, and thus made the entire melt shop more flexible, agile and responsive to unexpected events. The system has also been recognized as a very useful tool for running various simulations, what-if analysis and business scenarios on the melt shop in order to identify bottlenecks and further increase its production rate.

Short-term operational planning of refined products pipelines

Fil: Cafaro, Diego Carlos. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Santa Fe. Instituto de Desarrollo Tecnologico para la Industria Quimica. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnologico para la Industria Quimica; Argentina

An efficient greedy insertion heuristic for energy-conscious single machine scheduling problem under time-of-use electricity tariffs

This paper addresses an energy-conscious single machine scheduling problem under time-of-use (TOU) or time-dependent electricity tariffs, in which electricity prices may vary from hour to hour throughout a day. The key issue is to assign a set of jobs to available time periods with different electricity prices so as to minimize the total electricity cost required for processing the jobs. The main contribution of this work is two-fold. First, a new continuous-time mixed-integer linear programming (MILP) model is proposed for the problem. Second, an efficient greedy insertion heuristic is developed. In the proposed heuristic, the jobs are inserted into the available time periods one after another in non-increasing order of their electricity consumption rates and each job is inserted into the time period(s) with minimum electricity cost. A real-life case study from a Chinese company reveals that the total electricity cost can be reduced by about 30% with the proposed algorithm. Computational experiment on randomly generated instances also demonstrates that our algorithm can yield high-quality solutions with low electricity costs within dozens of seconds for large-scale single machine scheduling problems with 5000 jobs. The algorithm can be applied by production managers to scheduling jobs on a single machine under TOU electricity tariffs to save electricity costs.

Short-term scheduling of multiple source pipelines with simultaneous injections and deliveries

This paper addresses the optimal scheduling of straight pipelines featuring multiple intermediate nodes acting as dual-purpose stations, with a continuous-time Mixed-Integer Linear Programming formulation partly derived from Generalized Disjunctive Programming. The new model allows for an intermediate station to act as an output and input terminal at the same time so as to reduce the number of segment switches between active and idle, and consequently decrease operating costs. Contrary to previous approaches, decisions related to batch sizing, batch sequencing and timing are determined in a single step. Several examples of growing complexity are solved to illustrate the effectiveness and computational advantage of the proposed model in both solution quality and CPU time.

A Novel Monolithic MILP Framework for Lot-Sizing and Scheduling of Multiproduct Treelike Pipeline Networks

Detailed scheduling of long-distance multiproduct pipelines has received growing attention in the past few years. It helps the planner to reduce the number of pump and segment switches between active and idle conditions to obtain savings on pump operating and maintenance costs. Most contributions on the detailed scheduling of multiproduct pipelines concern networks with a single straight line. Large-scale pipeline networks, however, usually have a treelike configuration, featuring a mainline and several secondary lines, transporting smaller volumes of refined petroleum products over shorter distances. This work addresses the scheduling of a multiproduct treelike pipeline through a continuous-time mixed integer linear programming (MILP) model that allows the execution of simultaneous deliveries from a unique refinery to multiple downstream terminals so as to get a substantial increase in transportation capacity. Contrary to previous contributions on treelike pipeline systems, the new model solves batch siz...

Improving the Mathematical Formulation for the Detailed Scheduling of Refined Products Pipelines by Accounting for Flow Rate Dependent Pumping Costs

This work presents a continuous-time mixed-integer linear programming (MILP) formulation to find the best detailed schedule for single-source multi-product pipelines, minimizing the total operating costs. By knowing the aggregate plan including pumping and delivery tasks, the best detailed solution tends to minimize pump stoppage/restart costs as well as pumping energy charges associated to the head loss inside the pipeline, which are strongly dependent on the flow rate at every pipeline segment.

An improvement-based MILP optimization approach to complex AWS scheduling

The automated wet-etch station (AWS) is one of the most critical stages of a modern semiconductor manufacturing system (SMS), which has to simultaneously deal with many complex constraints and limited resources. Due to its inherent complexity, industrial-sized automated wet-etch station scheduling problems are rarely solved through full rigorous mathematical formulations. Decomposition techniques based on heuristic, meta-heuristics and simulation-based methods have been traditionally reported in literature to provide feasible solutions with reasonable CPU times.This work introduces an improvement MILP-based decomposition strategy that combines the benefits of a rigorous continuous-time MILP (mixed integer linear programming) formulation with the flexibility of heuristic procedures. The schedule generated provides enhanced solutions over time to challenging real-world automated wet etch station scheduling problems with moderate computational cost. This methodology was able to provide more than a 7% of improvement in comparison with the best results reported in literature for the most complex problem instances analyzed.