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Tissue-engineered medical products (TEMPs) are gaining significant attention for their potential to address organ shortages and improve public health. However, as the field grows, optimizing production to achieve appropriate biofunctionality (e.g., high cell viability) remains a critical challenge. Each TEMP has its unique process plan, precedence requirements, and critical constraints on cell viability, which dictate its overall success. The waiting time spent by a work-in-progress TEMP is a significant factor in dictating viability of cells at each production stage and in the final product. In order to achieve a better control while manufacturing TEMPs and maintain the desired level of cell viability, we investigated the problem of assigning and scheduling of operations required by these products on different machines. This problem is akin to a flexible job-shop scheduling problem. We present a mixed-integer linear programming (MILP) model formulation for assigning and scheduling TEMP operations in a deterministic environment that ensures both the stage-wise sequencing, and also, machine-wise sequencing of the operations of an order, while ensuring the desired viability of cells by controlling the waiting time of each order. The outcome of the model gives insights into planning the capacity of different facilities as well as their effective utilization by appropriately assigning orders to these facilities. We also extend our methodology to a stochastic environment where the processing times follow different distributions.
Abstract
Tissue-engineered medical products (TEMPs) are gaining significant attention for their potential to address organ shortages and improve public health. However, as the field grows, optimizing production to achieve appropriate biofunctionality (e.g., high cell viability) remains a critical challenge. Each TEMP has its unique process plan, precedence requirements, and critical constraints on cell viability, which dictate its overall success. The waiting time spent by a work-in-progress TEMP is a significant factor in dictating viability of cells at each production stage and in the final product. In order to achieve a better control while manufacturing TEMPs and maintain the desired level of cell viability, we investigated the problem of assigning and scheduling of operations required by these products on different machines. This problem is akin to a flexible job-shop scheduling problem. We present a mixed-integer linear programming (MILP) model formulation for assigning and scheduling TEMP operations in a deterministic environment that ensures both the stage-wise sequencing, and also, machine-wise sequencing of the operations of an order, while ensuring the desired viability of cells by controlling the waiting time of each order. The outcome of the model gives insights into planning the capacity of different facilities as well as their effective utilization by appropriately assigning orders to these facilities. We also extend our methodology to a stochastic environment where the processing times follow different distributions.
Keywords
Biomanufacturing; MILP; Tissue Engineering; Production Optimization
(ProQuest: ... denotes formulae omitted.)
1. Introduction
The manufacturing of tissue-engineered medical products (TEMPs) is a highly complex and a multi-disciplinary process that integrates biology, materials science, and engineering. These products, which include engineered tissues, scaffolds, and cell-based therapies, are designed to restore, replace, or regenerate damaged tissues and organs. Duc to the intricate nature of biological systems and the stringent quality requirements for medical products, the production process involves numerous interdependent stages that perform the processes required for the final product.
For an industry involved in the production of a TEMP, appropriate planning and scheduling decisions are essential to maintain the quality and compliance of the product while earning the maximum cost-based benefits. Mathematical programming offers a robust framework to optimize and manage such complex processes by systematically addressing constraints and minimizing the cost incurred. To maintain the cell viability requirements, it is imperative to monitor and control the operations and the quality of the work-in-progress units. This can be achieved by incorporating appropriate process control strategies within the production environment.
This paper presents an integrated-framework-based methodology for tackling the requisite operational issues through process control and the application of mathematical programming to optimize the production process of TEMPs. By leveraging process control and optimization techniques, this work seeks to provide a comprehensive strategy to improve manufacturing efficiency while adhering to the strict requirements of the biomedical industry.
2. Manufacturing layout and challenges
Based on the relationship between the source of cellular raw material and the recipient of the final product, a TEMP can be broadly classified as Autologous and Allogenic [1]. An Autologous TEMP relies on extracting the cellular raw material from the patient and the final product is implanted to the same patient (patient-specific therapy). In case of an Allogenic product, the source of the raw material is a donor and the recipient is a patient (universal donor therapy). The production for an Autologous TEMP resembles a pull system, while that for an Allogenic TEMP can be viewed as a push system. Therefore, to attain better performance, the scaling-out approach is more suitable for the Autologous TEMP achieved by exploiting the benefits of parallel production. However, the Allogenic product is more amenable to the scaling-up approach, where the products are produced in large quantities. Despite these differences, the process flow for the production of both Autologous and Allogenic TEMPs can be categorized similarly as shown in Fig 1. However, additional complexity is caused by strict quality and other compliance rules, stochasticity of the process due to inherent nature of the cells, incorrect configuration of the environment variables of the bioreactors, and chances of contamination [2].
As more TEMP industries are moving towards the commercialization phase, the demand for good quality TEMP is going to grow as well [3]. To meet the expected increase in demand for the TEMPs in the future, it is necessary to implement appropriate practices to achieve industrial-scale production levels of good quality products. At that scale, production facilities would rely on multiple machines for each stage [4]. A subset of the available machines is capable of carrying out the operations that are required to produce different jobs. Hence, the production process flow of a TEMP is similar to a flexible job shop. The inherent unpredictable nature of the cells may alter the expected outcome of a selected treatment eventually leading to uncertain processing times. In extreme cases, this can lead to increased lead time, thereby, resulting in increased number of tardy jobs. Another extreme consequence of random processing time is decrease in conforming units since cells of other work-in-progress products may degrade due to waiting. Therefore, it is imperative to control the risk associated with random processing time while scheduling of jobs in the TEMP job-shop.
Cell viability is an important performance metric which can be monitored throughout the manufacturing stage using quality check strategies [5]. An appropriate cell viability measure can be relied upon to ensure the quality standards of the final product as cell viability directly correlates to the quality of the product. The viability of cells can be affected by the environment variables during the operations, and the time spent in or outside cry opreservation waiting for its next operation [1]. Control of the process parameters is essential for meeting the quality and density requirements of each sub-stage of the process. Failure to do so, may lead to contamination of the culture and the corresponding order would require rework from its initial production stage [6]. Therefore, to maintain quality and compliance requirements, process control-based methodologies are necessary as they assist in avoiding contamination and facilitate timely deliveries of good quality products.
3. Proposed Methodology
In view of the challenges encountered during the production of a TEMP as described in Section 2, in this section, we present an integrated framework for decision making during the planning and production phases of TEMP manufacturing. First, we present an architecture of this framework, followed by the description of a mathematical model to be used for providing a production schedule.
3.1 Architecture
The challenges faced during the planning and production phases of decision making stem from the complex nature of each process, strict requirements for the production environment, and quality of the product. To overcome these challenges, we propose an integrated framework that would assist in scheduling the jobs on different machines while regulating the process control parameters to maintain the desired objectives and prevent contamination through realtime control. Process control parameters are assumed to be monitored continuously throughout all processes during the production of TEMP products. A schematic representation of the proposed framework is depicted in Figure 2.
In the proposed framework, the decision making involved during the production of TEMP can be broadly categorized into two phases, namely, Planning Phase and Production Phase. The Planning Phase comprises two modules, namely, Meta-Model and Optimization Module. The Meta-Model Module provides information on the processing time distributions corresponding to different treatment methods. This information is fed to the Optimization Module to generate a schedule in accordance with the production requirements that meet the desired criteria. If the output schedule becomes unacceptable due to inherent uncertainty in the process (that may, for example, cause excessive completion time of the job(s)), then the information on alternate treatment strategies and their processing times are generated using a Meta-Model and supplied to the Optimization Module for generating a new schedule. This iteration is repeated until an acceptable schedule is obtained, which is then implemented on the factory floor. At an industrial-scale production level, the use of a mathematical model to generate optimal schedules is effective.
During the production phase, the operations are executed following the schedule suggested by the Optimization Module that relies on the data for various treatments supplied by a Meta-Model. Process parameters such as pH, DO, temperature, nutrients, among others, are continuously monitored throughout the operations. Some of these parameters can be regulated if they deviate from the permissible range for a selected treatment. We label these parameters as control parameters, and their regulation is necessary to maintain desired quality and facilitate requisite cell and tissue growth, and thereby, increasing cell viability and product quality. Based on the information recorded by monitoring the operations, the Process Control module would initiate appropriate control actions whenever the values of the process parameters deviate from their permissible range. In the case of contamination, machine breakdown, or large deviation of the actual processing time from its estimate, the Planning Phase is invoked to produce a new schedule for the in-process, and remaining jobs.
3.2 Optimization Module
An integral part of the proposed Optimization Module is to solve a mathematical model that outputs scheduling decisions based on the input provided by the designed Meta-Model. We present a novel model formulation of the processes involved in the production of a TEMP. For convenience, we use the terminology of scheduling problems and call a TEMP as a job and bio-reactors and others resources as machines. Assume a set of jobs J = {1,2,..., n^ are to be assigned to a set of machines M = {1,2, ....,&}. Each job j E J has unique processing requirements that are represented as set of operations = ^o^c^ where L7 represents the total number of operations needed for completion of job j E J. Each operation o E can be performed on machines M^° Ç M. The problem is to find the optimal permutation of the operations on the machine set M such that the production constraints are met while achieving the best possible value of the objective function. A facility producing a TEMP at industrial scale is assumed to receive orders that require cellular raw material from different donors. The jobs that correspond to different donors would require decontamination of the machine that processes these jobs, which indicates the time spent in setting up the machine for a subsequent job. The setup times may vary based on the degree of the decontamination required. This adds to the complexity of the model as the setup times in this case are sequence-dependent. Adding to this complexity is the consideration of non-negligible transfer times, which are to be controlled since higher transfer and waiting times would increase the chances of contamination of the culture. The notations for the sets, parameters and decision variables that we use in this paper are presented in Table 1.
Next, we present a Mixed Integer Linear Program (MILP) formulation for the TEMP production scheduling problem on hand.
... (1)
... (2)
... (3)
... (4)
... (5)
... (6)
... (7)
... (8)
... (9)
... (10)
... (11)
... (12)
... (13)
... (14)
... (15)
... (16)
Constraints (1) ensure that for every job j E J, each of its operations is assigned to exactly one machine, and the preceding and succeeding operations are assigned to other machines. Constraints (2) and (3) assign the operations of the last and first stages to the dummy start and end operations, respectively. Constraints (4) ensure that if operation o E О3 of job j E J is processed on a machine m EM, then the job also moves from machine m to another machine for operation о + 1. The structure of Constraints [(l)-(4)] can be viewed as s - t paths, where 5 and t are respectively, represented by the dummy start and end operations of job j E J\ {0,rc + 1}. The nodes on the path connecting 5 and t correspond to operations that are to be assigned to certain machines while maintaining the precedence requirement between different operations. Similarly, the precedence constraints of the operations of different jobs on each machine can also be viewed as 5 - t paths, where 5 and t correspond to dummy start and end jobs 0 and n + 1, respectively. The operation requirement for these jobs is a single dummy operation that is represented by 0° = (9и+1 = {1}. Constraints [(5)-(9)] enforce the precedence of different operations on each machine, and Constraints (8) and (9) maintain the relation between the machine-based and job-based s - t paths.
Constraints [(10)-(l 1)] enforce the precedence constraints between subsequent operations for each job, while constraints (12) ensure that there is no overlap between the processing of two operations on the same machine. Note that, U3is a sufficiently large positive number. Finally, Constraints (13) maintain the consistency between the assignment and time variables x and t, respectively. The remaining constraints present the domains of all the decision variables. An appropriate objective function can be used based on desired preference. This can pertain to, among others, minimization of makespan, number of tardy (late) jobs, and average tardindess. Based on the chosen objective, reformulation of the above model may be required by adding appropriate constraints and/or variables.
A restriction on wait time between two subsequent operations о - 1 and o^o E О3 \ {0} of job j E J\ {0,/i+ 1} can be enforced by evaluating the wait time w3,° using Equation (17). In this case, the time of travel from one machine to another is not included in the wait time between the corresponding operations.
... (17)
As mentioned earlier, due to the processing time for each operation being a random variable, there is a risk of the production schedule of TEMP not meeting due dates. To avert this risk, we propose minimizing the the ConditionalValue-at-Risk (CVaR) for this stochastic scheduling problem[7]. This would involve reformulating the problem as a two-stage stochastic program where the objective of the master problem would be to minimize ф = īļ + where q is the Value-At-Risk, a is a user-defined probability level, is the probability of occurrence of scenario s E S and ¡Lis represents the expected value of max(0,X - T|)+ with as the set of master problem variables. Simulationoptimization based techniques can be used that employ Monte Carlo simulation to generate scenarios for evaluating the objective function of the stochastic scheduling problem [8, 9].
4. Future work
Biomanufacturing relies on the ability to transition from small-scale laboratory production to efficient, reproducible, and cost-effective industrial-scale manufacturing. To achieve this, it is essential to implement the use of robust planning and production practices. In this paper, we have presented a methodology based on an integrated framework for decisions making that requires the development of three modules, namely, Optimization, Meta-Model and Process Control. We have also presented a mathematical model for the production environment of TEMP. It is interesting to note that the structure of the proposed formulation has characteristic of a Vehicle Routing problem (VRP). An effective and efficient solution methodology similar to that for the VRP can be employed for the solution of the proposed model formulation. Fast computation algorithm will ease the incorporation of stochasticity in processing times. Another important aspect for future exploration is the extension of the mathematical model to a dynamic production environment. Validating the proposed framework through experimental implementation and industry collaboration would be a crucial next step. Applying the framework to real-world manufacturing settings could provide insights into its scalability, feasibility, and potential modifications needed to align with regulatory and operational constraints. Further work could also focus on integrating supply chain and logistics considerations into the model by inclusion of supplier and order taking considerations. Given the complexity of sourcing biomaterials, ensuring cold-chain logistics for cell-based products, complying with stringent regulatory requirements, and including supply chain dynamics, an expanded model would provide a greater practical relevance.
References
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