Risk-Based Analysis of Manufacturing Lead Time in Production Lines

Abstract

The paper proposes a method for assessing production risks related to potential exceedances of the agreed production lead time for batches of details in small and medium-sized enterprises. The study focuses on a linear production system composed of sequential technological operations, analyzed within the broader context of production and logistics processes. A stochastic model of the production flow has been developed, using dimensionless parameters to describe the state and trajectory of a product in a multidimensional technological space. The internal and external risk factors that affect the duration of operations are taken into account, including equipment failures, delays in material deliveries and labor availability. Analytical expressions enabling the quantitative assessment of the risk of production deadline violations and the resulting losses have been derived. The proposed method was validated on a production line for manufacturing wooden single-leaf windows. The results indicate that the presence of inter-operational reserves significantly reduces the probability of exceeding production deadlines and enhances the stability of the production process under stochastic disturbances. The use of inter-operational buffers in most cases ensured a reduction in the processing time of experimental batches of products by 18–25% and simultaneously led to a reduction in the level of production risk by several times, which confirms the effectiveness of the proposed approach and its practical significance for increasing the sustainability of production systems.

1. Introduction

Modern manufacturing enterprises operate in conditions of fierce competition and increasing demands for efficiency and reliability. One of the key problems in such conditions is managing risks associated with delays in order fulfillment and interruptions in technological operations, which also significantly affect internal logistics processes and material flow stability. To improve the sustainability of production processes and logistics efficiency, more and more companies, both large and small, are seeking to implement systematic approaches to assessing and minimizing production- and logistics-related risks [1,2,3]. A distinctive feature of production lines oriented toward serial and mass production of products is the presence of inter-operational reserves—time and material buffers formed between technological operations to ensure uninterrupted and synchronized operation of the production system [4,5]. Accurate risk assessment has a critical role in maintaining production flow and operational efficiency. Inaccurate or simplistic estimations may lead to underestimated safety buffers or excessive reserves, both of which negatively impact productivity and costs [6,7].

Manufacturing systems, even in their basic linear configurations, are complex dynamic structures with many interconnected processes and possible delays in the processing of products at process operations along the process route. Equipment failures, variability in raw material supplies, human factors and external conditions can cause stochastic deviations that spread throughout the entire technological route [8]. At the same time, a stop or failure in one operation does not necessarily lead to a complete stop of the production line but can cause an accumulation of inter-operational backlogs and uneven loading of equipment, which in turn can generate disruptions in the internal logistics flow [9]. A special feature of production lines oriented towards serial and mass production of products is the presence of inter-operational reserves—time and material buffers formed between technological operations in order to ensure uninterrupted and synchronized operation of the production system [4,10]. Inter-operational backlogs play a key role in maintaining the production rate when deviations and failures occur while also supporting the stability and continuity of material flows and logistics processes. In risk analysis, particular attention is paid to the probability of exceeding the agreed production deadlines, which directly affects the fulfillment of contractual obligations, logistics, and service levels, and can lead to significant financial losses. Methods for quantitative assessment of such risks are being developed but remain difficult to implement in practical application due to the complexity of the relationships and heterogeneity of factors [11]. Traditional approaches often rely on the assumption that risk (R) and reliability (N) are related through the simplistic formula R + N = 1. While useful in basic systems, such models do not capture the stochastic and interdependent nature of modern manufacturing environments [12,13]. Therefore, more sophisticated models that incorporate time-dependent behaviors and multi-state processes are necessary. Several attempts have been made to enhance the classical risk formulation by introducing dynamic factors, reliability distributions, or process interdependencies [14,15,16]. However, most of these models are either too complex for real-time applications or lack integration with internal logistics structures, which are vital for modern production lines [17].

This paper proposes a method for stochastic modeling of a production line that takes into account the probabilistic characteristics of process operations and the influence of production risks with special consideration for the impact on production logistics and material flow continuity. As a demonstration example, a line for the production of wooden single-leaf windows is considered, where the sequence of technological operations is strictly regulated. Using the subject-technological description of the system [18], random delays, changes in equipment states and variations in the flow of parts are displayed in the production line model. The proposed method goes beyond traditional queueing network models and PERT/CPM-based approaches. Unlike queueing models, which analyze system performance but often overlook the stabilizing effect of buffers, and unlike PERT/CPM methods, which focus on critical paths and schedules without fully accounting for stochastic variability, the proposed method simultaneously considers the process structure and statistical parameters of operations, including the probability and duration of process interruptions. This allows for a comprehensive, logistics-oriented evaluation of operational stability, identification of bottlenecks, and quantitative assessment of production losses.

The aim of the paper is to develop an analytical tool for assessing the risk of violation of production deadlines for a batch of products while taking into account the internal logistics structure of the production process, as well as to provide a quantitative assessment of potential production losses. The presented results are useful for developing practical solutions for risk management adapted to the specifics of discrete manufacturing systems. This enables a more realistic and logistics-oriented assessment of buffer systems and their effectiveness in reducing production risks.

2. Theoretical Background

2.1. Problem Statement and Assumptions

FMEA analysis (Failure Mode and Effects Analysis) is one of the most commonly used methods for risk assessment in products and manufacturing processes. Recommended by quality standards such as ISO 9001, ISO/TS 16949, and AS/EN 9100, it enables early detection and elimination of potential errors, failures, and their consequences. In industry, FMEA is used, among others, for the following [19,20]:

• Identifying and eliminating errors and their causes;
• Assessing risk and planning preventive actions;
• Detecting process weaknesses;
• Improving quality, reliability, and safety;
• Documenting and controlling production processes.

The analysis is carried out by an interdisciplinary team of experts [21], who evaluate three key parameters on a 1–10 scale:

• S (Severity)—the seriousness of the effect of the failure;
• O (Occurrence)—the likelihood of the failure occurring;
• D (Detection)—the likelihood of detecting the failure before it reaches the next process step.

Based on these values, the Risk Priority Number (RPN) is calculated as the product of the three parameters. Failure modes with higher RPN values are considered more critical and require more urgent corrective actions.

Traditional FMEA in production system risk assessment faces several limitations. It struggles to capture both the influence and interdependencies of severity (S), occurrence (O), and detection (D), while relying solely on these three factors, mostly from a process safety perspective. The resulting RPN is an abstract measure, not linked to cost, time, or performance, and cannot be directly associated with actual system losses. Risk evaluation is typically performed by engineers in isolation, overlooking system-level dependencies and interactions. Moreover, FMEA is time-consuming, prone to subjective ranking errors, and often carried out merely to meet customer requirements rather than to effectively reduce risk—making it inflexible and difficult for stakeholders to interpret [22,23,24,25,26,27,28]

In contrast, the proposed risk assessment method addresses these shortcomings by incorporating both the process flow structure and the statistical parameters of operations. Risk evaluation includes not only the probability that a process operation will remain in a given state (e.g., stoppage due to machine failure or reconfiguration), but also the distribution of time during which the product stays in that state, characterized by mathematical expectation and standard deviation. This enables a more comprehensive and logistics-oriented evaluation of the system’s operational stability. The key advantage of the new method lies in its ability to integrate process dynamics with statistical measures, providing a more realistic and decision-supportive framework for assessing production system risks.

The production process is a controlled transformation of initial blanks into finished products through the execution of a sequence of technological operations. A technological operation is understood as a completed part of a technological process, performed at one workstation, which can be divided into the following elements: transition, auxiliary, installation, fastening, working and auxiliary stroke, acceptance, adjustment, and sub-adjustment [29]. As a result of performing a technological operation, a continuous change in the state of the product occurs as a result of the transfer of technological resources to the product. Technological resources are understood as time, materials, energy, labor costs of the worker. Each operation is characterized by the consumption of a certain amount of resources and leads to a change in the state of the object being processed. These changes can be interpreted as the movement of points in a multidimensional space, in which the coordinates reflect the quantitative indicators of the resources expended. The sequence of such points forms an individual trajectory of product processing, called a technological trajectory [18]. In real production conditions, the process of transforming the resource intensity of a part is random. It is influenced by a wide range of factors: from equipment operating parameters to personnel qualifications. This means that the actual processing path deviates from the standard one specified by the technological regulations [18,30]. These deviations are interpreted as a manifestation of production risk which also impacts logistics coordination and work-in-progress inventory levels.

Factors influencing the stability of a technological process are usually classified as internal and external [31,32]. The first group includes parameters directly related to the functioning of production equipment, the quality of operations and the organization of work [16,33]. The second one includes supply conditions, availability of labor, volatility of prices for raw materials and components, and logistical failures [34,35]. Each of these factors can change the duration or resource load of technological operations, and in some cases lead to equipment downtime and the formation of inter-operational delays—which directly affect material flow timing and logistics buffers.

The state of the entire production system at any given moment in time can be viewed as a superposition of the states of many simultaneously processed products. Averaging these individual states yields generalized macro-parameters—indicators of load, productivity, efficiency, and stability and logistical performance. Their analysis allows us to quantitatively assess the impact of risks and formulate strategies for managing production sustainability and logistics flow robustness [7,12].

In this study, an approach based on a stochastic description of the transfer of technological resources within a one-dimensional coordinate space is applied. A generalized time reflecting the total duration of the product’s stay in the production process was selected as one coordinate axis. This choice simplifies the construction of a mathematical model, but retains the ability to take into account the random nature of deviations from standard parameters and allows the introduction of a macro description of the production line, applicable for the analysis and assessment of production, as well as logistical and financial losses. The model serves as a basis for quantitative assessment of the risk of non-fulfillment of the production plan within a given time window, which has direct implications for logistics scheduling and supply chain reliability. Particular attention within the framework of the proposed method is paid to taking into account inter-operational reserves—as one of the main instruments for risk compensation in serial and mass production of products—buffers that are essential for synchronizing production and logistics timing.

In this paper, a linear production line is considered. The technological route for manufacturing a product consists of $M$ sequentially arranged technological operations. As the product moves along the technological route, the initial workpiece turns into a finished product, accumulating technological resources during the execution of each technological operation [18,36].

2.2. Dimensionless Description of the Technological Process

For the technological process of manufacturing a product, the consumption rates of materials and standard processing time ${\eta }_{meanm}$ for the $m$-th technological operation are determined without taking into account the non-production downtime of equipment. The state of a detail at an arbitrary moment in time $t$ in the production process is expressed through dimensionless parameters:

$$\tau ={\displaystyle \frac{t}{t_{d\psi }}},{\tau }_{\psi }={\displaystyle \frac{t_{\psi }}{t_{d\psi }}},t_{\psi }={\displaystyle \sum _{k=1}^m\frac{{\eta }_{meank}}{t_{d\psi }}},{\tau }_{\psi }\in \left[0,1\right],{\vartheta }_{meanm}={\displaystyle \frac{{\eta }_{meanm}}{t_{d\psi }}},m=1..M,$$

(1)

where $t_{d\psi }$ is the average total standard operating time required to transform materials into a finished product; $t_{\psi }$ is the average time during which the product was subject to technological processing at the time $t$. If, at a given moment time $t$, technological processing of the $m$-th technological operation is completed for a part, then this time will be determined accordingly as the sum of the standard processing times from the first to the $m$-th technological operation $t_{\psi }$.

To model the change in the part’s state during the production process, the technological dimensionless coordinate space $\left(\tau ,{\tau }_{\psi },\vartheta \right)$ is used [18]. At any given moment in time ${\tau }_i$, the state of a product that is in the production process is determined by a point $D_i\left({\tau }_i,{\tau }_{\psi i},{\vartheta }_i\right)$ in this dimensionless coordinate space of states, $\left(i=0,\dots ,I\right)$. As the part continuously moves along the technological route, the sequence of such points forms a trajectory in the dimensionless coordinate space. This trajectory clearly characterizes the features of the technological processing of an individual part, from the original workpiece $D_0\left({\tau }_0,{\tau }_{\psi 0},{\vartheta }_0\right)$ to the final product $D_I\left({\tau }_I,{\tau }_{\psi I},{\vartheta }_I\right)$ (Figure 1).

The number of the technological operation at which the part is located in the technological processing is identified by the inequality

$${\displaystyle \sum _{k=1}^{m-1}{\vartheta }_{meank}}<{\tau }_{\psi }\le {\displaystyle \sum _{k=1}^m{\vartheta }_{meank}},{\tau }_{\psi }={\displaystyle \sum _{k=1}^{m-1}{\vartheta }_{meank}}+{\displaystyle \frac{\tau -{\tau }_m}{{\vartheta }_m}}{\vartheta }_{meanm},m=1\dots M,$$

(2)

where $\tau ={\tau }_m$ is the moment in time when the processing of the part at the $\left(m-1\right)$-th technological operation was completed and the part was transferred for technological processing to the $m$-th technological operation; ${\vartheta }_m=\left({\tau }_{m+1}-{\tau }_m\right)$ is the time of processing a part at the $m$-th technological operation. This is the time during which the part was directly in the processing process. Waiting time in inter-operational backlogs, equipment downtime and other non-production time losses are not taken into account.

3. Materials and Methods

3.1. Stochastic Modeling of Operation Durations

The execution time of the $m$-th technological operation is a random variable $\vartheta$, the value of which is determined by a large number of technological factors associated both directly with the technological parameters of the functioning of the equipment (workstation), and with the features of the execution of the technological operation, which include the heterogeneity of the material used when performing the operation or, for example, the different skill levels of the worker performing the technological operation. As a result of processing a part in the $m$-th technological operation, the part has a probability $r_z$ of being in one of the $z$-th states, the duration of which is characterized by the mathematical expectation ${\vartheta }_{meanzm}$ and the standard deviation ${\vartheta }_{stdzm}$ (

Table 1. Statistical characteristics of factors determining the technological operation [11].

Name Factor (the Operation State)	Probability	Statistical Characteristics of Event Duration (Time)
		Average	Standard Deviation
characteristics of workstation	$r_{0m}$	${\vartheta }_{mean0m}$	${\vartheta }_{std0m}$
additional adjustments and settings	$r_{1m}$	${\vartheta }_{mean1m}$	${\vartheta }_{std1m}$
breakdowns	$r_{2m}$	${\vartheta }_{mean2m}$	${\vartheta }_{std2m}$
delays or shortages in the delivery of materials	$r_{3m}$	${\vartheta }_{mean3m}$	${\vartheta }_{std3m}$
poor quality of materials	$r_{4m}$	${\vartheta }_{mean4m}$	${\vartheta }_{std4m}$
differences in technological and real times	$r_{5m}$	${\vartheta }_{mean5m}$	${\vartheta }_{std5m}$
absence	$r_{6m}$	${\vartheta }_{mean6m}$	${\vartheta }_{std6m}$

). Each state is determined by a certain technological factor. The factors defined in [11] were chosen as the main factors determining the time of execution of technological operations.

The average time to complete a process operation can be determined as follows:

$${\vartheta }_{meanm}={\displaystyle \underset{0}{\overset{\infty }{\int }}{\vartheta }_m{f}_m({\vartheta }_m)}d{\vartheta }_m={\displaystyle \sum _{z=1}^Z{r}_{z,m}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\vartheta }_{zm}{f}_{zm}({\vartheta }_{zm})}d{\vartheta }_{zm}},$$

(3)

where $f_m({\vartheta }_m)$, $f_{zm}({\vartheta }_{zm})$ are the distribution functions of random variables ${\vartheta }_m$, ${\vartheta }_{zm}$.

At a given moment in time $\tau$, the elementary volume $d\tau d{\tau }_{\psi }d\vartheta$ of the technological dimensionless coordinate space $\left(\tau ,{\tau }_{\psi },\vartheta \right)$ contains $dN=\varphi \left(\tau ,{\tau }_{\psi },\vartheta \right)d\tau d{\tau }_{\psi }d\vartheta$ details, the values of the parameters of which in the time interval $\left[\tau ;\tau +d\tau \right]$ are located at the location of the technological route $\left[{\tau }_{\psi };{\tau }_{\psi }+d{\tau }_{\psi }\right]$ with the time of technological processing limited by the limits $\left[\vartheta ;\vartheta +d\vartheta \right]$. The function $\varphi \left(\tau ,{\tau }_{\psi },\vartheta \right)$ determines the density of details in a unit volume of the dimensionless technological coordinate space of states $\left(\tau ,{\tau }_{\psi },\vartheta \right)$. As a result of technological processing of details, their condition changes. Thus, as a result of the change in state, some of the details leave the elementary volume $d\tau d{\tau }_{\psi }d\vartheta$ during the time interval $\left[\tau ;\tau +d\tau \right]$. On the other hand, this volume is filled with new details, the values of the parameters of which change over a period of time $\left[\tau ;\tau +d\tau \right]$ and fall into the elementary volume $d\tau d{\tau }_{\psi }d\vartheta$. This results in the density $\varphi \left(\tau ,{\tau }_{\psi },\vartheta \right)$ of details changing over time $\tau$. This change can be represented as follows:

$${\displaystyle \frac{d\varphi \left(\tau ,{\tau }_{\psi },\vartheta \right)}{d\tau }}={\displaystyle \frac{\mathsf{\partial }\varphi \left(\tau ,{\tau }_{\psi },\vartheta \right)}{\mathsf{\partial }\tau }}+{\displaystyle \frac{\mathsf{\partial }\varphi \left(\tau ,{\tau }_{\psi },\vartheta \right)}{\mathsf{\partial }{\tau }_{\psi }}}{\displaystyle \frac{d{\tau }_{\psi }}{d\tau }}+{\displaystyle \frac{\mathsf{\partial }\varphi \left(\tau ,{\tau }_{\psi },\vartheta \right)}{\mathsf{\partial }\vartheta }}{\displaystyle \frac{d\vartheta }{d\tau }}=J\left(\tau ,{\tau }_{\psi },\vartheta \right).$$

(4)

We will consider an established production process. For the established production process of manufacturing products, $J\left(\tau ,{\tau }_{\psi },\vartheta \right)=0$ [29]. Equation (4) is integrated over the interval of variation of the random variable $\vartheta$:

$${\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{\mathsf{\partial }\varphi \left(\tau ,{\tau }_{\psi },\vartheta \right)}{\mathsf{\partial }\tau }d\vartheta }+{\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{\mathsf{\partial }\varphi \left(\tau ,{\tau }_{\psi },\vartheta \right)}{\mathsf{\partial }{\tau }_{\psi }}\frac{d{\tau }_{\psi }}{d\tau }d\vartheta }+{\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{\mathsf{\partial }\varphi \left(\tau ,{\tau }_{\psi },\vartheta \right)}{\mathsf{\partial }\vartheta }\frac{d\vartheta }{d\tau }d\vartheta }=0.$$

(5)

The value $N^{-1}\varphi \left(\tau ,{\tau }_{\psi },\vartheta \right)d\tau d{\tau }_{\psi }d\vartheta$ can be interpreted as the probability of finding a part in a volume element $d\tau d{\tau }_{\psi }d\vartheta$. By analogy with the moments for the distribution density of a random variable, moments for the function $\varphi \left(\tau ,{\tau }_{\psi },\vartheta \right)$ are introduced:

$${\Theta }_{\mathrm{k}}\left(\tau ,{\tau }_{\psi }\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}\varphi \left(\tau ,{\tau }_{\psi },\vartheta \right){\vartheta }^{k}d\vartheta }.$$

(6)

which have the following physical meaning: ${\Theta }_0\left(\tau ,{\tau }_{\psi }\right)$ is the density of details at a point in time $\tau$ at a point in the process route, determined by the coordinate ${\tau }_{\psi }$; ${\Theta }_1\left(\tau ,{\tau }_{\psi }\right)$ is the flow of details at a point in time $\tau$ at a point in the process route determined by the coordinate ${\tau }_{\psi }$. Using the notation (6), Equation (5) takes the form

$${\displaystyle \frac{\mathsf{\partial }{\Theta }_0\left(\tau ,{\tau }_{\psi }\right)}{\mathsf{\partial }\tau }}+{\displaystyle \frac{\mathsf{\partial }{\Theta }_1\left(\tau ,{\tau }_{\psi }\right)}{\mathsf{\partial }{\tau }_{\psi }}}=0,$$

(7)

$${\Theta }_0(0,{\tau }_{\psi })=\psi \left({\tau }_{\psi }\right),{\Theta }_1(\tau ,0)=\gamma \left(\tau \right),{\Theta }_1(\tau ,{\tau }_{\psi })={\Theta }_{1\psi }(\tau ,{\tau }_{\psi }).$$

(8)

This is the equation of the production line model in a two-moment description using the zero moment ${\Theta }_0\left(\tau ,{\tau }_{\psi }\right)$ and the first moment ${\Theta }_1\left(\tau ,{\tau }_{\psi }\right)$ of the distribution function of details over states $\varphi \left(\tau ,{\tau }_{\psi },\vartheta \right)$ in the dimensionless technological coordinate space of states $\left(\tau ,{\tau }_{\psi },\vartheta \right)$ [13]. The equation is supplemented with initial and boundary conditions (8). The solution of the system of Equations (7) and (8) makes it possible to determine at an arbitrary moment in time $\tau$ the number of details at an arbitrary point ${\tau }_{\psi }$ in the technological route. The functions ${\Theta }_0\left(\tau ,{\tau }_{\psi }\right)$, ${\Theta }_1\left(\tau ,{\tau }_{\psi }\right)$ represent the linear distribution density of parts and the rate of processing of parts in a technological operation along the technological route of manufacturing a product (WIP: work-in-progress) at a point in time $\tau$ at a point ${\tau }_{\psi }\in \left[0,1\right]$ in the technological route in the dimensionless technological coordinate space of states $\left(\tau ,{\tau }_{\psi },\vartheta \right)$. The function $\psi \left({\tau }_{\psi }\right)$ determines the distribution of details ${\Theta }_0\left(\tau ,{\tau }_{\psi }\right)$ among technological operations at the initial moment of time $\tau =0$. The function $\gamma \left(\tau \right)$ sets the incoming intensity of details in the form of material or semi-finished products at the input of the technological route (to the first technological operation). Functions ${\Theta }_{1\psi }(\tau ,{\tau }_{\psi })$ and $\gamma \left(\tau \right)$ are given functions that are determined by the planned values of the enterprise’s production program. In accordance with the production program of the enterprise, production tasks are determined for the implementation of the production plan for each workplace. By integrating Equation (7) within the $m$-th technological operation, a system of equations is obtained that determines the number of parts in inter-operational backlogs ${\omega }_m\left(\tau \right)=\omega \left(\tau ,{\tau }_{\psi m}\right)$ with known values ${\Theta }_{1m}(\tau )={\Theta }_1(\tau ,{\tau }_{\psi m})$ for each $m$-th technological operation:

$${\displaystyle \frac{\mathsf{\partial }{\omega }_m\left(\tau \right)}{\mathsf{\partial }\tau }}+{\Theta }_{1m}(\tau )-{\Theta }_{1(m-1)}(\tau )=0,{\omega }_m\left(\tau \right)={\displaystyle \underset{{\tau }_{\psi m-1}}{\overset{{\tau }_{\psi m}}{\int }}{\Theta }_0(\tau ,{\tau }_{\psi })}d{\tau }_{\psi },m=1\dots M$$

(9)

For a synchronized production line ${\Theta }_{1m}(\tau )\cong {\Theta }_{1(m-1)}$, the number of details in the inter-operational backlog remains constant, ${\omega }_m\left(\tau \right)\cong const$. Ensuring synchronization of production operations is a primary problem of production planning. Planning the synchronized operation of the production line allows production losses associated with the downtime of process equipment to be avoided. For each $m$-th technological operation, the probability of being in a non-working state is determined. The probability of such a state is denoted as the risk $R_m\left(\tau \right)$ that the $m$-th technological operation is in a state of production downtime due to a number of production factors. The probability that a process operation is in a working condition is determined by the following formula [11,37]:

$$R_m\left(t\right)+N_m\left(t\right)=1, m=1\dots M$$

(10)

Then, for the $m$-th technological operation, the number of processed details per unit of time, taking into account losses associated with production risks, is determined by the expression

$${\Theta }_{1\mathrm{R}m}(\tau )=N_m\left(\tau \right){\Theta }_{1m}(\tau ),\hspace{1em}m=1\dots M$$

(11)

Thus, in accordance with Expression (11), the $m$-th technological operation per unit of time $\Delta \tau$ receives details ${\Theta }_{1m}(\tau )\Delta \tau$ for technological processing, and during this same time, it gives out a quantity of parts equal to ${\Theta }_{1\mathrm{R}m}(\tau )\Delta \tau$. The amount of losses in the technological processing of details per unit of time associated with the presence of production risks of the transition of the technological operation to a non-working state is determined by the difference $\Delta {\Theta }_{1\mathrm{R}m}(\tau )=\left({\Theta }_{1m}(\tau )-{\Theta }_{1\mathrm{R}m}(\tau )\right)$. The number of details received during a period of time $\Delta \tau$ at the input $m$-th technological operation is related to the number of details that have undergone technological processing during a period of time $\Delta t$ at the $\left(m-1\right)$-th technological operation:

$${\Theta }_{1m}(\tau )\Delta \tau ={\Theta }_{1\mathrm{R}(m-1)}(\tau )\Delta \tau ,m=1\dots M$$

(12)

Taking into account the relation (11), the last equality can be rewritten as

$${\Theta }_{1m}(\tau )={\Theta }_{1\mathrm{R}(m-1)}(\tau )=N_{(m-1)}\left(\tau \right){\Theta }_{1(m-1)}(\tau )=N_{(m-1)}(\tau )N_{(m-2)}(\tau ){\Theta }_{1(m-2)}(\tau )=\dots .$$

(13)

Then the production output of details per unit of time from the production line in the form of a finished product is equal to the flow of details from the last $M$-th technological operation:

$${\Theta }_{1\mathrm{M}}(\tau )={\Theta }_{11}(\tau ){\displaystyle \prod _{m=1}^M{N}_m}(\tau ),$$

(14)

$$R(\tau )=1-{\displaystyle \prod _{m=1}^M{N}_m}(\tau ),N(\tau )={\displaystyle \prod _{m=1}^M{N}_m(\tau )}.$$

(15)

Equation (14) describes the general approach to risk assessment in accordance with the concept (10) [11,12]. This approach to calculation of the risk $R(\tau )$ does not take into account the constraint (9), which links the input and output flow of details from the process operation. The features of using this concept are considered through the example of a linear production line.

3.2. Modeling Inter-Operational Buffers

Let a stationary rate of processing of details along a technological route ${\Theta }_{1\psi }\left(\tau ,{\tau }_{\psi }\right)={\Theta }_{1\psi d}\left({\tau }_{\psi }\right)$ be given, which determines the functioning of technological equipment with constant productivity, the statistical characteristics of which do not change over time: ${\Theta }_{1m}={\Theta }_1\left({\tau }_{\psi m}\right)$, ${\Theta }_{1m\psi d}={\Theta }_{1\psi d}\left({\tau }_{\psi m}\right)$. Then the values of the flow of products coming from the $m$-th technological operation at an arbitrary moment in time satisfy the inequality ${\Theta }_{1m}\left(\tau \right)\le {\Theta }_{1m\psi d}$ and are determined by solving the system of Equation (9). In the absence of downtime of process equipment in the limiting case $R_m\left(\tau \right)=0$ for the steady-state operating mode of a production line without inter-operational backlogs, the solution to the system of Equation (9) takes the form

$$\begin{array}{c}\left\{\begin{array}{l}{\Theta }_{1k}=\min \left({\Theta }_{11\psi d},\dots ,{\Theta }_{1m\psi d}\right),\\ {\Theta }_{1m}={\Theta }_{1k},\end{array}\right.,\left\{\begin{array}{c}{\omega }_m=1,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}m\le k,\\ {\omega }_m={\displaystyle \frac{{\Theta }_{1m}}{{\Theta }_{1m\psi d}}}={\displaystyle \frac{{\Theta }_{1k}}{{\Theta }_{1m\psi d}}},m>k,\end{array}\right.\\ m=1\dots M.\end{array}$$

(16)

The steady state is understood as the time interval from the moment the first detail leaves the production line until the moment the last detail leaves the production line. The rate of movement of products along the technological route is determined by the rate of processing of products at the $k$-th technological operation, which corresponds to the minimum rate of processing of details in comparison with other technological operations. This technological operation is a bottleneck in the technological route. For steady-state conditions, the rate of output of the finished details is ${\Theta }_{1M}={\Theta }_{1k}={\Theta }_{1\min \psi d}$ by the production time of a batch of $N_{batch}$ details is

$${\tau }_d=\max ({\tau }_{\psi })+{\displaystyle \frac{N_{batch}}{{\Theta }_{1M}}}=1+{\displaystyle \frac{N_{batch}-1}{\min \left({\Theta }_{11\psi d},\dots ,{\Theta }_{1M\psi d}\right)}}=1+{\displaystyle \frac{N_{batch}-1}{{\Theta }_{1\min \psi d}}},m=1\dots M,{\tau }_{\psi }\in \left[0,1\right].$$

(17)

The case when the risk of downtime of process equipment is different from zero is now considered. For periods of time when there is no downtime associated with the presence of factors of production risk $R_m\left(\tau \right)$ and the production line operates in a steady state after a production downtime associated with a production risk $R_m\left(\tau \right)$, the rate of processing of details ${\Theta }_{1m}$ and the average number of details ${\omega }_m$ within a technological operation are determined by Equation (16). Due to the occurrence of downtime of technological equipment associated with production factors, the output flow of finished details from the last technological operation will be equal to ${\Theta }_{1M}={\Theta }_{1k}N(\tau )$, where the value $N(\tau )$ is determined by Expression (15). The production time of a batch of details $N_{batch}$ will increase with the downtime of the production line due to factors associated with production risks $R_m\left(\tau \right)$, and accordingly, the rate of processing of a batch of products will decrease proportionally. Then Equation (17) for determining the production time of a batch of details will take the form

$${\tau }_{dR}=1+{\displaystyle \frac{N_{batch}-1}{N(\tau )\min \left({\Theta }_{11\psi d},\dots ,{\Theta }_{1M\psi d}\right)}}=1+{\displaystyle \frac{N_{batch}-1}{N(\tau ){\Theta }_{1\min \psi d}}},m=1\dots M.$$

(18)

Then, during the time ${\tau }_d$, when there are production risks $R_m\left(\tau \right)$ for a batch of size $N_{batch}$, the manufactured part of the batch is determined by the ratio

$${\displaystyle \frac{{\tau }_d}{{\tau }_{dR}}}={\displaystyle \frac{1+\frac{N_{batch}-1}{{\Theta }_{1\min \psi d}}}{1+\frac{N_{batch}-1}{N(\tau ){\Theta }_{1\min \psi d}}}},m=1\dots M.$$

(19)

The average rate of processing of products for a technological operation can be estimated by the following value: ${\Theta }_{1\mathrm{m}\psi d}~1/\Delta {\tau }_{\psi m}=1/\left({\tau }_{\psi m}-{\tau }_{\psi m-1}\right)~M$. Then, to quantify the portion of a batch of details manufactured during time ${\tau }_d$, the following expression can be used:

$${\displaystyle \frac{{\tau }_d}{{\tau }_{dR}}}\approx {\displaystyle \frac{1+\frac{N_{batch}-1}{M}}{1+\frac{N_{batch}-1}{N(\tau )M}}}={\displaystyle \frac{M+N_{batch}-1}{N(\tau )M+N_{batch}-1}}N(\tau )={\displaystyle \frac{1}{1-R(\tau )\alpha }}N(\tau ).$$

(20)

Taking into account that the parameter $R(\tau )\alpha$ is small, the following expression is obtained:

$${\displaystyle \frac{{\tau }_d}{{\tau }_{dR}}}\approx N(\tau )+N(\tau )R(\tau )\alpha =N(\tau )+N(\tau )\alpha -N^2(\tau )\alpha +\dots .,\alpha ={\displaystyle \frac{M}{M+N_{batch}-1}}={\displaystyle \frac{1}{1+\frac{N_{batch}-1}{M}}}.$$

(21)

When $N_{batch}>>M$, it follows that $\alpha \to 0$, and the share of manufactured products is determined as

$${\displaystyle \frac{{\tau }_d}{{\tau }_{dR}}}=N(\tau ),{\displaystyle \frac{{\tau }_{dR}-{\tau }_d}{{\tau }_{dR}}}=1-N(\tau )=R(\tau ).$$

(22)

Thus, for production lines without inter-operational backlogs, Formula (15) can be used with a sufficient degree of accuracy when the number of details in a batch is much greater than the number of technological operations in the technological route. The effects associated with the initial movement of a batch of parts along the production line do not have a significant impact on the methodology for calculating production losses associated with production risks. For small batches of details, when $N_{batch}\approx M$, the calculation of production losses using the methodology on the basis of which Expression (15) is written requires clarification.

3.3. Risk Estimation

In the absence of downtime of process equipment in the limiting case $R_m\left(\tau \right)=0$ for the steady-state operating mode of the production line at time $\tau$ during the $m$-th operation of the unlimited inter-operational backlogs, the solution to the system of Equation (9) takes the form

$$\left\{\begin{array}{l}{\omega }_m>1\\ {\Theta }_{1m}={\Theta }_{1m\psi d},\end{array}\right.,m=1\dots M.$$

(23)

At the $m$-th technological operation $m\le k$, ${\Theta }_{1k}=\min \left({\Theta }_{11\psi d},\dots ,{\Theta }_{1m\psi d}\right)$ inter-operational backlogs will accumulate provided that the rate of processing of products at the $\left(m-1\right)$-th technological operation is greater than the rate of processing of products at the $m$-th technological operation, ${\Theta }_{1\left(m-1\right)\psi d}>{\Theta }_{1m\psi d}$. For technological operations $m>k$, the state of inter-operational backlogs is determined by Equation (16), as in the case of a production line in the absence of buffers for inter-operational backlogs.

In this case, if downtime occurs at the $i$-th technological operation due to a production risk $R_m\left(\tau \right)>0$ at the technological operations $m<i$, technological processing of products will take place until the inter-operational buffers overflow. When $i>k$, the buffers are filled. In this regard, if a downtime occurs at the $i$-th technological operation due to a production risk $R_m\left(\tau \right)>0$, the production line will not stop until the inter-operational buffers overflow at the technological operations $m<i$, and at operations $m>i$, details will be in the inter-operational backlog. If the downtime ${\tau }_{Ri}$ due to production risk $R_i\left(\tau \right)$ is less than the time ${\tau }_{\omega i}$, required to process inter-operational backlogs for $m>i$ operations r to overfill inter-operational buffers at $m<i$ operations, then the production line will not experience this downtime and the overall rate of product output will not decrease. If ${\tau }_{Ri}>{\tau }_{\omega i}$, then the downtime of the technological line for a given episode will be reduced by the value ${\tau }_{\omega i}$ and will, accordingly, be determined by the value $\left({\tau }_{Ri}-{\tau }_{\omega i}\right)$. If the inter-operational backlog of technological operations contains a characteristic value $〈{\omega }_m〉$ of inter-operational backlogs, at a characteristic rate of processing of products in a technological operation ${\Theta }_{1\mathrm{m}\psi d}~1/\Delta {\tau }_{\psi m}=1/\left({\tau }_{\psi m}-{\tau }_{\psi m-1}\right)~M$, then in the time interval associated with the downtime of technological equipment due to production risk, there will be a flow of details during the time ${\tau }_{\omega i}$:

$${\tau }_{\omega i}={\displaystyle \frac{〈{\omega }_i〉}{{\Theta }_{1i\psi d}}}\approx 〈{\omega }_i〉M.$$

(24)

Then, when downtime of process equipment occurs due to production risks $R_m\left(\tau \right)$, the output flow of finished products from the last process operation will be equal to

$${\Theta }_{1M}={\Theta }_{1k}N(\tau )+{\displaystyle \frac{{\tau }_{\omega i}}{{\tau }_{Ri}}}{\Theta }_{1k}R(\tau )={\Theta }_{1k}\left(N(\tau )+{\displaystyle \frac{{\tau }_{\omega i}}{{\tau }_{Ri}}}R(\tau )\right),{\tau }_{\omega i}\le {\tau }_{Ri}.$$

(25)

The material flow is represented by a combination of two material flows. The first term represents the flow of material when all elements of the production line are functioning. The probability of the state is $N(\tau )$. The second term represents the average flow of material for the case when one of the elements of the production line is idle, but the output of finished details from the last technological operation continues due to the presence of inter-operational backlogs. The probability of the state is $R(\tau )$. An important element in Equation (25) is the characteristic average downtime ${\tau }_{Ri}$ due to production risk. For such a calculation, it is necessary to know not only the probability of the production line being in the state $N(\tau )$ or $R(\tau )$, but also the mathematical expectations of the time it will be in this state. Indeed, if the value of the inter-operational backlog is such that ${\tau }_{\omega i}={\tau }_{Ri}$, then the output flow of details acquires the value

$${\Theta }_{1M}={\Theta }_{1k}\left(N(\tau )+{\displaystyle \frac{{\tau }_{\omega i}}{{\tau }_{Ri}}}R(\tau )\right)={\Theta }_{1k}\left(N(\tau )+R(\tau )\right)={\Theta }_{1k},{\tau }_{\omega i}={\tau }_{Ri},$$

(26)

which is equal to the value of the output flow of details in the absence of production risks, $R_m\left(\tau \right)=0$. In addition, the condition ${\tau }_{\omega i}={\tau }_{Ri}$ allows us to calculate the minimum size of inter-operational backlogs, at which the value of the output flow of parts from the production line is not associated with the existence of production risks:

$$〈{\omega }_i〉={\tau }_{Ri}{\Theta }_{1i\psi d}.$$

(27)

This equation determines the production time of a batch of $N_{batch}$ details:

$${\tau }_{dR}=1+{\displaystyle \frac{N_{batch}-1}{{\Theta }_{1k}\left(N(\tau )+\frac{{\tau }_{\omega k}}{{\tau }_{Rk}}R(\tau )\right)}},m=1\dots M.$$

(28)

Then, during the time ${\tau }_d$, when there are production risks $R_m\left(\tau \right)$ for a batch of size $N_{batch}$, a portion of the batch determined by the ratio can be manufactured:

$${\displaystyle \frac{{\tau }_d}{{\tau }_{dR}}}\approx {\displaystyle \frac{1+\frac{N_{batch}-1}{M}}{1+\frac{N_{batch}-1}{M\beta }}}={\displaystyle \frac{M+N_{batch}-1}{\beta M+N_{batch}-1}}\beta ,\beta =\left(N(\tau )+{\displaystyle \frac{{\tau }_{\omega k}}{{\tau }_{Rk}}}R(\tau )\right).$$

(29)

In the absence of inter-operational backlogs, ${\tau }_{\omega k}\approx 0$, ${\tau }_{\omega i}<<{\tau }_{Ri}$, and the last expression takes the form of (20)

$${\displaystyle \frac{{\tau }_d}{{\tau }_{dR}}}\approx {\displaystyle \frac{M+N_{batch}-1}{\beta M+N_{batch}-1}}\beta \approx {\displaystyle \frac{M+N_{batch}-1}{\beta M+N_{batch}-1}}N(\tau ),\beta =N(\tau ),{\tau }_{\omega i}<<{\tau }_{Ri}.$$

(30)

For another limiting case ${\tau }_{\omega i}\ge {\tau }_{Ri}$, the following expression is obtained:

$${\displaystyle \frac{{\tau }_d}{{\tau }_{dR}}}\approx {\displaystyle \frac{M+N_{batch}-1}{\beta M+N_{batch}-1}}\beta =1,\beta =1,{\tau }_{\omega i}\ge {\tau }_{Ri}.$$

(31)

Thus, the presence of inter-operational reserves has a significant impact on ensuring the smooth functioning of the production line in the event of production risks.

4. Results

To demonstrate the application of the proposed stochastic model of the production line, the process of manufacturing a single-leaf wooden window in an industrial production environment is considered. The case study examines a linear production line, where a detail sequentially passes through a series of technological operations before turning into a finished product. The technological route consists of M consecutive operations, each of which is performed at a dedicated workstation. For each operation, standard processing times were defined in accordance with the technical documentation of the enterprise. These values do not include non-production downtime and are used as input parameters for the model.

Table 2. List of technological operations and workstations in the production of single-leaf windows: mean processing time (min).

m	Name of the Technological Operation	$\mathbf{Mean}\mathbf{Processing}\mathbf{Time},{\mathit{\eta }}_{\mathit{m}\mathit{e}\mathit{a}\mathit{n}\mathbf{z},\mathit{m}}$
		z = 0	z = 1	z = 2	z = 3	z = 4	z = 5	z = 6
E1	Cutting (WP: saw)	70	250	210	420	630	70	420
E2	Straightening (WP: PM)	60	120	120	360	540	60	360
E3	Planning (WP: PM—thickener)	70	120	140	420	630	70	420
E4	Profiling + cutting out (WP: bottom spindle milling machine)	163	340	640	700	1000	160	700
E5	Grinding (WP: wide belt grinder)	70	200	100	720	1000	70	720
E6	Assembling, drilling, removing finishing grinding (WP: worktable)	140	360	700	680	950	140	680
E7	Impregnation and painting (WP: industrial painting workshop)	120	130	500	600	900	120	600
	Total	693

WP—workplace; PM—planning machine;.

summarizes the technological key parameters for each technological operation, including mean processing time, standard deviation of processing time and identified main risk factors. The production line operates in a serial-batch mode, where batches of details are processed in sequence. Inter-operational buffers between operations are used to compensate for variability in operation times. The capacity of these buffers and their utilization rate are important factors in ensuring an uninterrupted flow of details. The enterprise aims to meet daily production targets while minimizing downtime and production losses. Variability in processing times arises from technological and risk factors (Table 1). These factors influence the probability of downtime at each operation and are incorporated into the stochastic model. The production line operates in steady-state conditions once the first part reaches the last workstation. Inter-operational buffer capacities remain constant during the observation period. The model parameters (mean, standard deviation, probability of downtime) are estimated from recorded production data over a representative period of time. The technological process of producing a batch of 60 single-leaf windows, including seven consecutive processing operations presented in Table 2, is analyzed [11]. The average time during which a detail is in the $z$-th state during processing at the $m$-th technological operation is designated as ${\eta }_{mean\mathrm{k},m}$. The values of the standard deviation of the execution time of a technological operation are determined on the basis of statistical processing of data and are equal to ${\eta }_{std0,m}=0.2{\eta }_{mean0,m}$ for $z=0$ and ${\eta }_{std\mathrm{z},m}=0.1{\eta }_{mean\mathrm{z},m}$ for $z>0$. In order to simplify the analysis, we will assume that the distribution function $f_{zm}({\vartheta }_{zm})$ (3) of random variables corresponds to the normal distribution law.

During the process of performing a technological operation, the equipment may be in one of the states (see Table 1) with the probability $r_z$ indicated in

Table 3. The state probability, $r_z$.

m	Name of the Technological Operation	$\mathbf{State}\mathbf{Probability},{\mathit{r}}_{\mathit{z}}$
		z = 0	z = 1	z = 2	z = 3	z = 4	z = 5	z = 6
E1	Cutting	0.834	0.036	0.020	0.024	0.072	0.006	0.008
E2	Straightening	0.963	0.002	0.002			0.009	0.024
E3	Planning	0.964	0.002	0.004			0.006	0.024
E4	Profiling + cutting out	0.896	0.048	0.025			0.015	0.016
E5	Grinding	0.903	0.025	0.042			0.006	0.024
E6	Assembling, drilling, removing finishing grinding	0.956	0.012	0.010			0.006	0.016
E7	Impregnation and painting	0.880	0.003	0.009	0.012	0.036	0.012	0.048

. These values were obtained as a result of processing experimental data in [11]. The total time that a part spends in technological processing consists of the time of the technological processing itself ${\eta }_{0,m}$ and the time required to solve production problems by taking the equipment out of the $z$-th state. At this time, the object is at the level of technological processing and does not allow other objects to start this technological processing operation. All other objects are awaiting their processing time in this technological operation, being in the inter-operational backlog.

Using the introduced dimensionless parameters (1), the parameters of technological operations can be presented in dimensionless form (

Table 4. Average dimensionless processing time.

m	Name of the Technological Operation	$\mathbf{Average}\mathbf{Processing}\mathbf{Time},{\mathit{\vartheta }}_{\mathit{m}\mathit{e}\mathit{a}\mathit{n}\mathbf{z},\mathit{m}}$
		z = 0	z = 1	z = 2	z = 3	z = 4	z = 5	z = 6
E1	Cutting	0.101	0.361	0.303	0.606	0.909	0.101	0.606
E2	Straightening	0.087	0.173	0.173	0.519	0.779	0.086	0.519
E3	Planning	0.101	0.173	0.202	0.606	0.909	0.101	0.606
E4	Profiling + cutting out	0.235	0.490	0.924	1.010	1.443	0.231	1.010
E5	Grinding	0.101	0.288	0.144	1.038	1.443	0.101	1.038
E6	Assembling, drilling, removing finishing grinding	0.202	0.519	1.010	0.981	1.371	0.202	0.981
E7	Impregnation and painting	0.173	0.188	0.721	0.865	1.298	0.173	0.866
	Total:	1.000

). In the limiting case, when production risk $r_z=0$ for factors $z>0$ and the process of performing technological operations is deterministic, the time required to produce a serial batch of 60 windows can be calculated using the formula

$$m_{{\tau }_{batch}}={\displaystyle \sum _{m=1}^M{\vartheta }_{mean\mathrm{m}}}+(N-1)\max ({\vartheta }_{mean})\approx 14.87$$

(32)

This formula assumes that there are no inter-operational reserves for the technological operations. This is a limiting case of the technological process. The value of time required for the production of a batch of 60 windows will be used in this work for a comparative analysis of different technological routes and modes of operation of the technological equipment. Note that the obtained value (32) corresponds to the operating time interval of a manufacturing enterprise during one calendar month $21.5\cdot 8\cdot 60/693\approx 14.89$, during which a batch of 60 windows is manufactured on a production line that does not provide for technological reserves.

If the distribution density function $f_{k,m}\left({\vartheta }_{k,m}\right)$ of a random variable ${\vartheta }_{k,m}$ is known for each technological operation, the distribution density $f_m\left({\vartheta }_m\right)$ of the random variable ${\vartheta }_m$ can be calculated according to Formula (3). Figure 2 shows the calculated distribution densities $f_m\left({\vartheta }_m\right)$ of the random variable ${\vartheta }_m$ for the $m$-th technological operation. The multimodal distribution density function $f_m\left({\vartheta }_m\right)$ of a random variable ${\vartheta }_m$ reflects the presence of a risk $r_z$ of occurrence of the $z$-th state for the $m$-th technological operation. The prevailing mode corresponds to the state $z=0$ with the probability of occurrence $r_0$, in which the product is directly processed during the technological operation.

Technological trajectories (implementation of a stochastic process for manufacturing a batch of products), formed using the distribution densities $f_m\left({\vartheta }_m\right)$ of the processing time of a part at the $m$-th technological operation for the limiting case of a production line for which no inter-operational backlogs are provided, are presented in Figure 3. Figure 3a illustrates the process paths of the first five details (details 1–5) in a production batch, while subfigure Figure 3b illustrates the process paths of the last five details (details 55–60). This difference highlights how stochastic variability and risk-related delays affect parts differently depending on their position in the production sequence. The solution to the problem regarding technological trajectories allows us to estimate the manufacturing time of a batch of details, which is correspondingly equal to the time interval between the start of processing the first detail in the first operation and the end of processing the last detail in the last operation.

Before moving on to assessing the risk of the production system exceeding the agreed production time for a batch of details, the effect of the size of the inter-operational backlog on the production time for a batch of 60 single-leaf windows for a deterministic model of the production line in the absence of production risks was considered: $r_z=0$ for factors $z>0$. The process trajectories of a batch of 60 wooden single-leaf windows for production line variants that provide different sizes ${\omega }_{\max m}$ of the inter-operational buffers are shown in Figure 4. For clarity of demonstration, the graphs contain technological trajectories, the product number of which is a multiple of five. The size ${\omega }_{\max m}$ of the inter-operational buffer is understood as the maximum number of details that are undergoing or have undergone processing at the $m$-th technological operation and are awaiting the start of processing at the $(m+1)$-th technological operation. In the qualitative analysis, it is assumed that the size ${\omega }_{\max m}$ of the inter-operational buffer is the same for each technological operation. Figure 4a shows process trajectories where no more than one detail is allowed to be present in a process operation. This option corresponds to Equation (15) for calculating the production risk $R(\tau )$ of the production system exceeding the agreed production time of a batch of details. When the size of the inter-operational buffer increases, the process trajectories are deformed in the interval from the first to the third process operation. This is explained by the fact that the fourth operation has the maximum processing time for the detail ${\vartheta }_{mean0,4}=0.235$ (Table 4). The fourth operation is the bottleneck of the production line and sets the overall pace of production of single-leaf wooden windows. Increasing the size ${\omega }_{\max m}$ of the inter-operation buffer results in more parts being shifted to the fourth process operation.

It is worth paying attention to the fact that the technological trajectories for ${\omega }_{\max m}=32$ and ${\omega }_{\max m}=64$ are the same. The explanation for this is quite simple. The number of details in the inter-operational buffer ${\omega }_3\left(\tau \right)$ before the fourth technological operation is determined by solving the system of Equation (9). From the graphical analysis of Figure 4f, it follows that in the third technological operation for cases ${\omega }_{\mathit{max}m}=32$ and ${\omega }_{\max m}=64$, the limitation, defined by ${\omega }_{\max m}$, for ${\omega }_m\le {\omega }_{\mathit{max}m}=32$ is not achieved for two variants. The next step is to analyze the process trajectories for the stochastic model of the production line taking into account production risks $r_z$. The process trajectories of a batch of 60 single-leaf windows for a stochastic model of a production line in the presence of production risks $r_z$ are shown in Figure 5. The process trajectories for the stochastic model are qualitatively consistent with the picture observed for the deterministic production line model. The fourth operation, as in the previous case, is a bottleneck in the technological route of manufacturing a batch of 60 wooden single-leaf windows. As the size ${\omega }_{\max m}$ of the inter-operational buffer increases, the process trajectories shift to the process operation that is identified as the bottleneck in the process route. Since process trajectories are implementations of a stochastic process of processing a batch of details taking into account the risk of the production system exceeding the agreed time for manufacturing a batch of details, the bottleneck in the process route can migrate from operation to operation depending on the probability of occurrence of a production risk and the duration of the time interval for eliminating the consequences associated with this risk. Obviously, the minimum production time for a batch of 60 single-leaf wooden windows will be in the case where, with equal processing time for the part in the technological operation, the bottleneck corresponds to the last technological operation.

In this case, the technological trajectories of a batch of details are gradually deformed towards the last production operation, creating a high level of inter-operational backlogs at these technological operations. This leads to the fact that the risk of the production system exceeding the agreed time for the production of a batch of details is reduced for two reasons, the first of which is the high level of inter-operational backlogs, and the second is that products accumulate at the last technological operations and the contribution of the occurrence of conditions associated with the presence of risks during the first technological operations is significantly reduced.

5. Discussion

We take into consideration the distribution function $F_b({\tau }_b)$, which determines the probability that the production time ${\tau }_{batch}$ of a batch of 60 wooden single-leaf windows will not exceed the time ${\tau }_b$. Figure 6a shows the curves for the distribution function $F_b({\tau }_b)$ for different values of the buffer size for inter-operational backlogs: ${\omega }_{\max m}=1$ (red curve), ${\omega }_{\max m}=2$ (blue curve), ${\omega }_{\max m}=4$ (green curve), ${\omega }_{\max m}=64$ (black curve). Increasing the size of the buffer for inter-operational backlogs leads to a decrease in the batch time of details.

Then the probability that the production time ${\tau }_{batch}$ of a batch of details will exceed the time ${\tau }_b$ can be determined as follows: $R_b({\tau }_b)=1-F_b({\tau }_b)$. The function $R_b({\tau }_b)$ represents the risk of the production system exceeding the agreed time for manufacturing a batch of products, that is, the probability that a batch of products cannot be manufactured within the specified time ${\tau }_b$. Figure 6b shows the risk function $R_b({\tau }_b)$ of exceeding the manufacturing time ${\tau }_b$ of a batch of 60 single-leaf wooden windows for different values of the buffer size for inter-operational backlogs. Figure 6b shows the curves $R_b({\tau }_b)$ corresponding to the results ${\omega }_{\max m}=1$ (red curve), ${\omega }_{\max m}=2$ (blue curve), ${\omega }_{\max m}=4$ (green curve), ${\omega }_{\max m}=64$ (black curve). The first curve on the right (shown in red) is for a production line that does not provide inter-operational backlogs, ${\omega }_{\max m}=1$. To build this curve, the calculation method for $R(\tau )$, corresponding to Formulas (15) and (21), was used. For example, with a probability of 0.75, it can be stated that the production time ${\tau }_{batch}=24$ for a batch of 60 wooden single-leaf windows will exceed the value for a production line that does not provide inter-operational backlogs (${\omega }_{\max m}=1$, red curve). For the case when the value is ${\omega }_{\max m}=2$, the probability is already ~0.35 (Figure 6a, blue curve). A further increase in the value of ${\omega }_{\max m}$ reduces the probability to ~0.15 (the black curve corresponds to ${\omega }_{\max m}=64$; Figure 6b).

Thus, the use of inter-operational buffers allows us to significantly reduce the probability of occurrence of production risks in accordance with Formula (27). Depending on the buffer configuration and the specifics of the process operations, the total processing time of the pilot batches was reduced by approximately 18–25% compared to the baseline without a buffer. At the same time, the estimated probability of failure to meet production deadlines was reduced several times, indicating a significant reduction in operational risks. This result is achieved due to the ability of interoperational buffers to compensate for random fluctuations in the duration of operations and localize their impact, preventing the cascading spread of failures along the process route. The results obtained confirm not only the theoretical but also the practical significance of the proposed approach to increasing the stability of discrete production systems. Each moment in time ${\tau }_b$ corresponds to a quantitative assessment of work in progress (WIP) within the production system $N_L$. Production losses can be characterized as the difference between the potential volume $N_{\det }$ of production for the reporting period ${\tau }_{\det }$ and the actual volume, $N_L=N_{\det }\left(1-{\tau }_{\det }/{\tau }_b\right)$, taking into account the absence of risks. A visual representation of losses due to production risks is shown in Figure 7. This graph represents the transformation of $F_b({\tau }_b)$, $R_b({\tau }_b)$ into $F_L(N_L)$, $R_L(N_L)$ by moving from variable ${\tau }_b$ to variable $N_L$. The transformation simplifies the interpretation of the obtained results, allowing the manufacturer to estimate losses in natural units, as well as the probability of these losses.

The quantitative and qualitative analysis performed demonstrates the significant influence of inter-operational reserves on the volume of losses arising from production risks. The losses calculated in accordance with Formula (15) are $N_L~24$ units with a probability of ~0.65, while with the same probability, the estimate of production losses can be reduced to $N_L~17$ units in the presence of inter-operational buffers of the required capacity.

The proposed method, although demonstrated using a wooden single-leaf window production line, has a general structure that makes it applicable to a wide variety of manufacturing systems characterized by sequential operations. It can be used in assembly lines in the automotive or electronics industries, where inter-operational buffers help mitigate disruptions caused by component delays, tooling changes, or rework. Similarly, the method is suitable for flexible manufacturing systems, where production times are highly variable due to customization and frequent equipment reconfigurations, as well as in metalworking and machining operations, where tool wear and operator variability significantly impact lead times. The use of dimensionless parameters in the model enables its application across production systems with varying scales, process structures, and technological settings.

The proposed method of analyzing production risks is not limited to strictly sequential technological operations. The use of an example with a linear production system in the article is due to the desire to simplify the mathematical transformations associated with parallel–sequential execution of technological operations, and thereby clearly demonstrate the key features of the developed approach. In this case, two parallel technological operations can be formally represented as one generalized operation, for which the statistical characteristics of the processing time are determined through the convolution of the distributions of random variables describing the execution time of each of the parallel operations. Issues of expanding the model to cases of flexible routing and parallel processing are planned to be considered in subsequent studies.

6. Conclusions

This paper is devoted to the justification of the use of the risk assessment method based on the assumption that risk (R), understood as the probability of occurrence of production losses, is synonymous with unreliability (Z) (15) [6] for production systems engaged in serial or mass production of products. The conditions for applying this approach have been defined. It is shown that calculation methods (15) and (21) assume that production lines do not provide buffers for inter-operational backlogs. A model of a production line based on the subject’s technological description of the production system was constructed and used to justify the applicability conditions of the proposed risk assessment concept, which assumes that risk (R) is equivalent to unreliability (Z) (15) [6] in production systems with serial or mass production of products. A new concept of the risk assessment method has been developed, based on the technological model of the production line. Analytical expressions for calculating production losses in serial and mass production conditions of a production line have been obtained. The risk assessment includes not only the probability that a process operation will remain in this state (for example, a stop due to a breakdown or reconfiguration of equipment), but also the distribution of the time the product will be in this state, described by the mathematical expectation and standard deviation. This approach allows for a more comprehensive and logistics-oriented evaluation of the operational stability of the production system. The key advantage of the proposed risk assessment method is its ability to simultaneously account for both the structure of the process route and the statistical parameters of operations. Importantly, the method demonstrates that when using the traditional risk formula, the result R is not zero; however, with the new formula, the risk R can approach zero as inter-operational reserves increase, which provides a more realistic tool for planning production buffers in logistics-intensive environments.

From a production management standpoint, the proposed model constitutes an analytically grounded instrument for supporting decision-making in the domains of production planning, scheduling, and capacity management within discrete manufacturing systems. By providing a quantitative framework for evaluating the effects of stochastic perturbations—such as variable operation durations and equipment downtimes—on lead time distribution and throughput performance, the model enables identification of bottleneck operations and estimation of the probability of schedule deviations. This facilitates evidence-based decisions regarding the configuration of inter-operational buffers, optimization of resource utilization, and adjustment of production schedules under uncertainty. Furthermore, the model functions as a decision-support tool for evaluating trade-offs between buffer sizing and operational reliability, thereby contributing to the formulation of robust production control strategies. In doing so, it establishes a methodological link between stochastic process modeling and practical applications in industrial operations management. The obtained results show that the presence of inter-operational buffers not only reduces the probability of violation of production batch deadlines but also stabilizes the production flow under stochastic disturbances. From the point of view of production risk management, this means that the size of the buffers should be considered as a control parameter that allows the level of production risks to be reduced to the required value. In practice, the proposed approach can be used to identify critical operations, assess possible production losses and make informed decisions on resource allocation, planning and configuration of production lines.

The vector of further research includes the following areas: (1) analysis of the risk of exceeding the established time for manufacturing a batch of details on a production line with sequential- and parallel-type technological routes; (2) development of a risk assessment model for a line that simultaneously processes several batches of details; (3) simulation modeling of batch processing in the presence of a stochastic incoming flow of orders; (4) analysis of the relationship between the size of the production batch and the parameters of production risk in order to identify optimal modes of organizing production; (5) comparative evaluation of the proposed approach with other established methods of production risk assessment, with an emphasis on validation using experimental data.

Although the model was validated on a specific example of wooden window manufacturing, its general structure based on dimensionless variables and stochastic characteristics of operation times makes it applicable to a wide range of manufacturing environments. Future research will focus on empirical validation in other industries, such as automotive, electronics, and mechanical engineering, using real production data. These comparative studies should assess the robustness of the model, provide empirical calibration, and contribute to the development of industry-specific guidelines for manufacturing risk management and buffer stock optimization.