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Efficient Scheduling of Discrete Industrial Processes Through Continuous Modeling

Ruike Lyu, Xiangbo Su, Ershun Du, Hongye Guo, Qixin Chen, Chongqing Kang

arXiv:2608.26487v1eess.SY

TL;DR

Legacy RTN models accurately represent discrete industrial processes but use many binary variables and scale poorly with production targets. This paper proposes cRTN, a continuous-variable reformulation with carefully designed constraints that preserves discrete-process behavior. Compared with legacy RTN, cRTN maintains high accuracy, reduces solution time by more than 89.0%, and can reduce energy costs by up to 25% in certain scenarios.

  • Problem

    Legacy RTN models require many binary variables and batch-specific formulations, limiting computational scalability for discrete industrial-process scheduling.

  • Method

    cRTN decouples modeling from batches with continuous resource and task variables while using designed constraints and selected binaries to preserve discrete process characteristics.

  • Results

    More than 89.0% solution-time reduction is reported while maintaining high accuracy, and energy costs decrease by up to 25% under certain scenarios.

  • Takeaways & Limitations

    cRTN provides a unified framework for discrete industrial-process modeling and improves computational tractability for larger production targets and integrated demand-response applications.

Abstract

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The resource-task network (RTN) model has been widely applied to represent the technical constraints of complex industrial processes (IPs) such as steel-making, providing the basis for industrial demand response. However, the legacy RTN model contains numerous binary variables and applies different formulations for non-flexible and flexible processes, restricting its computational efficiency and applicability. To systematically improve the computational performance of IP models, we propose continuous RTN model (cRTN), a novel modeling approach that uses continuous variables to represent production tasks and progresses, which are then integrated into unified as well as computationally favorable formulations for the technical constraints in discrete IPs, including resource balance, task execution, waiting time limits, and production targets. Compared to the legacy models, cRTN features fewer binary variables, shorter solving time, and better scalability while maintaining the same accuracy. Numerical tests based on a steel plant demonstrate that cRTN is in typical cases 10 times faster than legacy models and remains tractable with increasing batch sizes, which in legacy models leads to larger problem scales and infeasible solving time. cRTN also achieves a reduction in energy costs by resolving the issue of rounding errors reported in legacy models.

I. INTRODUCTION

Industrial demand response needs scalable models for discrete industrial processes, but legacy RTN formulations become computationally burdensome as production targets grow. The paper proposes cRTN, a unified reformulation that preserves discrete-process constraints while improving computational performance.

  • Motivation: Renewable-dependent power systems need demand-side flexibility, and industrial loads are important resources because production processes consume most industrial energy.Industrial response can change consumption in response to electricity prices or grid instructions.
  • Process characteristics: Discrete industrial processes produce materials in batches, unlike continuous processes where production quantities vary continuously with processing time.In a steel plant, an electric arc furnace takes 80 minutes to melt a batch, with no molten steel available before completion.
  • Existing modeling: RTN represents manufacturing devices and products as resources and production or transportation operations as tasks, accurately expressing discrete-process constraints.It has been widely applied to production scheduling and industrial demand-response applications, including steel-making.
  • Existing modeling: Legacy RTN uses numerous binary variables and models each batch separately, causing poor computational performance and problem sizes that grow with production targets.These binaries represent device operating states and batch-based material flows, while the batch-by-batch formulation increases variables and constraints proportionally.
  • Proposed approach: cRTN uses continuous variables for task operating times and progress, with unified formulations for resource balance, task execution, waiting times, and production objectives.The reformulation handles ordinary and flexible production processes without affecting the original model’s representability.
  • Results: Numerical experiments report fewer binary variables, shorter solving time, better scalability, and lower energy costs while maintaining representability.The model is evaluated through numerical tests using a steel plant case and addresses previously reported rounding errors.

II. PROBLEM DESCRIPTION AND METHODOLOGY

Discrete industrial processes operate in batches, finite equipment states, and coupled stages with transfer and waiting limits. The cRTN approach preserves RTN’s graph structure while using continuous variables and status variables to improve modeling efficiency and enforce technical constraints.

  • Characteristics of discrete production processes: Discrete industrial processes use minimum processing units, so products become available only after complete batch operations.An EAF may process several tons as one batch, preventing partial output during the operation.
  • Characteristics of discrete production processes: Finite equipment operating states and restrictions on state switching require integer variables in conventional models.Discrete states can coexist with continuously adjustable operating time, energy consumption, and material production within a state.
  • Characteristics of discrete production processes: Coupled production stages require subsequent processing to follow completed prior stages, while transferred intermediate products may face waiting-time limits.Molten steel, for example, must be processed before cooling beyond an acceptable threshold.
  • Proposed methodology: cRTN introduces status variables for uninterruptible processes while replacing legacy discrete resource and task variables with batch-independent continuous variables.The approach targets strict technical-constraint satisfaction and improved computational efficiency.
  • Modeling objectives: The model must represent executable technical constraints and remain solvable within an acceptable time frame.These objectives concern both factory implementation of schedules and computational tractability in commercial solvers.
  • Proposed methodology: cRTN retains RTN’s graph-based representation while introducing continuous variables for resource quantities and task operations.It keeps binary variables only where essential to represent scheduling and material flow accurately.

III. THE LEGACY RTN MODEL

The legacy RTN models production devices and products as resources and production or transport operations as tasks on a discrete time horizon. Its batch-specific formulation expresses resource balances, task execution, waiting limits, delivery, energy cost, and operating modes, but can suffer from scaling and rounding issues.

  • Resource-task network representation: RTN represents equipment and products as resources, while production and transportation operations are modeled as tasks connecting resource conversions.A steel plant example contains melting, decarburization, refining, and casting stages plus transfer tasks.
  • Resource balance: The interaction matrix records resource consumption or generation at each time offset after a task starts.For steelmaking, molten steel is generated 80 minutes, or 16 five-minute slots, after melting begins.
  • Task execution: Legacy RTN binds tasks to individual batches, so model size increases with production targets and limits scalability.Different batches are represented by different process and transfer tasks.
  • Resource balance: Resource balance updates each resource from its previous value and task-induced changes at every time slot.Applied to equipment, the constraint limits a device to one task within a given period.
  • Resource balance: The equipment-use constraint can produce rounding errors, which are analyzed in the case study.The same formulation also represents energy consumption across time slots without direct inter-slot coupling.
  • Production targets and objective: The legacy formulation requires each batch to complete every task once, reach the final stage, and minimize production energy cost when that objective is selected.Multiple operating modes can be represented by modeling equipment states as different tasks.
  • Task execution: Transfer-task constraints enforce immediate execution, avoiding an additional waiting period before transfer.This reflects a common steel-manufacturing requirement and keeps the formulation simpler.
  • Waiting time limit: Intermediate products may wait between stages, but the combined transfer and waiting duration is bounded by a material-specific maximum.Such limits reflect requirements such as processing molten steel before it cools excessively.

C. Flexible operating mode

Flexible operating mode allows task processing time and power to vary within feasible bounds while preserving uninterruptible processing and required energy completion conditions.

  • Processing time: Flexible task processing time is bounded by minimum and maximum durations and linked to the start of the subsequent task.Transfer tasks are assumed to execute immediately, so the subsequent-task start time represents the current task’s end time.
  • Processing power: Flexible operation introduces a continuous power variable constrained by minimum and maximum processing powers.The power variable is defined over time during task operation.
  • Processing status: The processing-status variable identifies whether a task is active, with active tasks generally remaining uninterrupted until completion.Status is determined by the task start and subsequent-task times.
  • Energy conversion: Each flexible operating mode must satisfy the energy required to complete its task.Under the equal-energy assumption, required energy can be determined from nominal power multiplied by nominal processing time.

IV. CONTINUOUS RTN FORMULATION

cRTN decouples model size from production batches and replaces batch- and state-based decisions with continuous task-operation durations and progress variables. Unified constraints represent resource balances, task execution, production targets, and energy consumption while retaining discrete-process structure.

  • Model structure: cRTN models each product or process at a stage with one resource or task, so model scale does not increase with the production target.This decouples task and resource modeling from individual batches.
  • Continuous variables: cRTN replaces starting times, operating states, and per-slot power with operating time for each state within each time slot.The continuous formulation uses the demand-response time scale to improve computational performance while partly sacrificing accuracy.
  • Continuous variables: Resource quantities Rr,t are relaxed to continuous variables representing progress toward completion of the task generating resource r.Rr,t = 0 or 1 represents no progress or task completion, while intermediate values represent ongoing processing.
  • Continuous variables: Device operation is represented by duration in each state within a time slot rather than on/off-state variables.Figure 5 illustrates the replacement of discrete device states with state-specific operating durations.
  • Unified process modeling: Idle, minimum-speed, and maximum-speed states provide a common representation for ordinary and flexible production processes.The formulation uses k = 0 for idle and k = 1 and k = 2 for minimum and maximum processing speeds.
  • Resource representation: The task-duration variable Di,k,t records how long task i operates in state k during slot t, while G records per-unit-time resource changes.The association matrix links task states to generated or consumed resources.
  • Technical constraints: Continuous task variables make resource changes continuous, enabling a resource-balance constraint for the cRTN formulation.The model uses a ≤ relation to avoid conflicts with discrete time-slot constraints; energy-cost minimization enforces equality in the stated setting.
  • Technical constraints: Task execution constrains Di,k,t to nonnegative values no longer than a time interval, with total operating time filling that interval.Unlike legacy RTN task execution, the formulation allows execution duration to vary continuously through task start or end times within a slot.

B. Modeling Discrete IPs

cRTN uses continuous progress and task-duration variables while adding constraints that preserve the batchwise and sequential behavior of discrete industrial processes. Virtual transfer tasks and waiting-task formulations encode completion, interruption, transfer, and timing requirements.

  • Modeling discrete processes: cRTN is a reformulation rather than a relaxation, using continuous resources and task progress while preserving discrete industrial-process behavior.The model is designed to retain the representational capabilities of legacy RTN.
  • Uninterruptible tasks: Uninterruptible tasks cannot return to the idle state after their generated resource has positive progress.When Rri+,t > 0, Di,0,t = 0, so the task continues at a processing speed between the minimum and maximum states.
  • Batch-based processing: Virtual output and input tasks transfer a completed resource between process stages by setting the preceding resource to 0 and the subsequent resource to 1.This construction expresses batch processing and sequential relationships without complicating the model.
  • Batch-based processing: A subsequent task can consume resource r only after the preceding task has completed it, represented by Rr,t = 1.Output and input tasks enforce this completion condition between waiting and processing tasks.
  • Waiting time limits: Waiting tasks represent transfer and waiting progress, with generation rates 1/(wi + Wi) and 1/wi for the minimum and maximum rates.Here wi is transfer time and Wi is maximum waiting time; waiting tasks are also uninterruptible.
  • Waiting time limits: After input materials arrive, processing tasks must start immediately because waiting-task parameters already account for transfer and waiting time.This is represented by preventing the task from remaining in its idle state while its consumed resource is present.
  • Implementation boundary: Finite-horizon implementations require additional initial- and final-slot constraints, including preventing tasks longer than two slots from starting in the final slot.The section presents fundamental principles and illustrations rather than all finite-horizon implementation constraints.

V. AN ILLUSTRATIVE EXAMPLE

The illustrative example models an electric arc furnace in a secondary steelmaking plant, where scrap steel is heated and melted in batches. It compares RTN and cRTN scheduling for minimizing electricity costs under technical constraints and production targets.

  • Example setup: The example uses an electric arc furnace that processes scrap steel through batch heating and melting.It is drawn from a secondary steelmaking plant and uses a six-hour scheduling horizon with one-hour time slots.
  • Example setup: The scheduling objective is to minimize electricity costs while satisfying the furnace’s technical constraints and production targets.The example is designed to expose the conceptual and mathematical differences between RTN and cRTN.

A. Conventional RTN Modeling

The conventional RTN example models furnace processing, transfer, and waiting with batch-specific tasks and resources. Flexible-task power variables convert the formulation to MILP, while cRTN keeps model dimensions independent of batch count and uses a unified state-rate representation.

  • Conventional RTN Modeling: The RTN example includes processing, transferring, and waiting tasks, with transfer and waiting optionally combined into one waiting task.The model uses five resource types and four task types, with separate batch indexes for b1 and b2.
  • Scalability: The legacy RTN’s resources, tasks, and variables scale with the number of batches because processes and transfers are modeled separately for each batch.This batch binding limits the scalability of the same production-line model as production targets grow.
  • Conventional RTN Modeling: The RTN scheduling formulation is an integer programming problem whose objective calculates electricity procurement cost under time-of-use prices.The example denotes its scheduling outcome as the Basic RTN solution.
  • Flexible-task formulation: Flexible RTN tasks require additional power variables and constraints to represent adjustable power, producing a mixed-integer linear programming formulation.The stated adjustable range is from 1/2 to 4/3 of nominal value.
  • Continuous RTN example: The cRTN example uses four resource types and four task types, combining transfer and waiting into a simplified waiting task.Its task parameters represent minimum and maximum rates rather than batch-specific task instances.
  • Scalability: Unlike legacy RTN, cRTN keeps the number of resources, tasks, and corresponding variables independent of batch count.For flexible EAF operation, the association entries encode progress rates for idle, minimum-power, and maximum-power states.
  • Scalability: cRTN requires at most 4ST binary variables, independent of batches, whereas RTN requires 4bST binary variables scaling linearly with batch count.The comparison concerns binary-variable growth in the illustrated formulation.

C. Comparison of scheduling results

Across two electricity-price scenarios, cRTN reproduces RTN scheduling when their discrete timing aligns and avoids RTN’s rounding-error delay when it does not.

  • Scenario 1: In Scenario 1, RTN and cRTN produce identical schedules when four low-price slots match the time required for two batches at nominal power.This agreement supports cRTN’s preservation of discrete batch-processing behavior.
  • Scenario 2: In Scenario 2, RTN requires four slots because its one-task-per-resource-per-slot rule delays the second batch after the first finishes mid-slot.cRTN avoids this delay and can complete two batches within the lower-price period.
  • Method settings: The numerical tests use 5-minute time slots and convert steel-plant technical parameters into cRTN processing rates, including virtual tasks.For example, a waiting task with processing rate 1/3 requires three time slots in its specified operating state.

A. Task Description and Method Settings

The evaluation compares cRTN with legacy RTN across PJM price scenarios, focusing on energy costs, load-profile accuracy, resource-balance accuracy, and solution time.

  • Rounding error: 2.13% RMSE measures the relative difference between RTN and cRTN optimal load profiles across the 15 price scenarios.The difference is attributed to RTN rounding errors caused by discrete time-slot restrictions.
  • Rounding error: cRTN eliminates the rounding error and enables an additional approximately 0.5% reduction in energy costs.The impact is expected to increase with longer time slots.
  • Optimality gap: With stricter than 1e-2 optimality gaps, RTN solution time exceeds one hour, whereas cRTN remains comparatively stable across gap settings.For optimality gaps of 0.1 or smaller, the maximum resource-balance gap is less than 10^-6.

C. Model Scalability

cRTN scales better than legacy RTN because its formulation decouples modeling from production batches while retaining discrete-process constraints.

  • Model scalability: At 10 heats, cRTN remains solvable within an hour, whereas RTN exceeds two hours beyond 7 heats.RTN solution time increases close to exponentially as production targets grow.
  • Model scalability: At heat = 9, cRTN has 4197 variables, while RTN has over 20000 binary variables.The legacy model’s variable and constraint counts grow linearly with production objectives, contributing to exponential MILP solution-time growth.
  • Performance: cRTN reduces solution time by more than 89.0% while maintaining high accuracy relative to legacy RTN.The model uses continuous variables for resource quantities and task operations, with designed binary variables preserving discrete process states.
  • Energy costs: Resolving legacy RTN rounding errors allows cRTN to reduce energy costs by up to 25% under certain scenarios.The paper positions this as an energy-cost improvement in addition to computational gains.
  • Scope: The study evaluates one industrial facility, while aggregating multiple facilities and applying cRTN to real-time optimization remain future research directions.The authors identify flexibility aggregation and real-time optimization as potential applications rather than demonstrated results here.

APPENDIX A RESOURCE BALANCE IN STN AND CRTN MODELS

The appendix expresses resource evolution through task operating durations and production or consumption rates, yielding a unified balance formulation for STN and cRTN.

  • Resource balance: The STN resource balance tracks material states over time using the continuous duration of each task’s operation at each operating point.Production and consumption rates determine how material states evolve during each interval.
  • Unified formulation: The unified notation uses gr,i,k, positive for production and negative for consumption, to represent task effects on resource r.This compact formulation combines resource-production and resource-consumption terms.
  • cRTN relation to STN: cRTN adds constraints to the unified resource-balance formulation to preserve the discrete nature of production processes.The formulation is therefore presented as more general than STN, with continuous processes as a degenerate case of discrete processes.
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