Manufacturing unit production operation optimization method in order flow disturbance state

Abstract

According to the manufacturing unit production operation optimization method in the order flow disturbance state, the discrete state space model with the work-in-process inventory as the core variable is constructed, the internal disturbance type and the external disturbance type are distinguished, the dynamic order flow matrix reconstruction is adopted to restrain internal disturbance propagation, and therefore the production efficiency is improved. And external demand changes can be quickly responded through self-adaptive gain adjustment. According to the method, the calculation complexity is reduced by utilizing the decomposition type state equation, the global production efficiency is optimized by combining the load balancing strategy, the product fluctuation under disturbance is reduced by 30% in simulation, and the delivery punctuality rate is improved by 15%.

Description

Technical Field

[0001] The present invention belongs to the technical field of discrete manufacturing process control, and specifically relates to a dynamic optimization method for workshop production based on state space modeling, which is particularly suitable for the coordinated operation optimization of multiple manufacturing units under order flow disturbance conditions. Background Art

[0002] A discrete manufacturing plant is a system composed of multiple collaborative manufacturing units. The production process depends on the combination or configuration of different manufacturing units, and the logistics paths between units are constantly changing. Furthermore, the random occurrence of disturbances such as equipment failures and operator absences makes the production process complex and variable, making production control difficult to implement. Describing the state variables in a discrete plant system and interpreting the disturbances it experiences during operation are of great practical significance for the design, production control, and performance evaluation of discrete plant systems. In recent years, this has become a hot topic among scholars both domestically and internationally. Some have used state-space methods to study the production capacity control and work-in-process control of a manufacturing system composed of multiple workstations, and designed a closed-loop production control system. Others have also used state-space methods to analyze the predictive output and disturbance propagation problems of manufacturing networks.

[0003] A key characteristic of discrete manufacturing plants is that they are composed of multiple relatively simple individuals—manufacturing units. Each unit within the system has relatively independent dynamic characteristics, and there are interactions between the units. When a single unit makes a decision, it not only affects its own behavior but also the dynamic characteristics of other units, thereby affecting the overall characteristics of the system. Production control systems in discrete manufacturing plants are subject to disturbances such as urgent orders and fluctuating demand, which can be considered external characteristics of the system.

[0004] Classical control theory focuses on analyzing the relationship between a system's input and output, and can only reflect the system's external characteristics. Considering the order input flow (an abstraction of material flow) within a discrete shop floor system, analyzing the internal dynamic characteristics of a production control system requires selecting appropriate state variables to construct the system's state space. Work-in-process inventory is a key variable in describing the system's internal dynamic characteristics, and the amount of work-in-process directly or indirectly affects the system's damping ratio and natural frequency. Summary of the Invention

[0005] The purpose of the present invention is to provide a method for optimizing the production operation of a manufacturing unit under a state of order flow disturbance, so as to solve the problems raised in the above background technology:

[0006] (1) How to propose selecting work-in-process inventory as the state feedback variable of the discrete workshop production control system, establish the differential equations of the input, output and state variables of a single manufacturing unit in the discrete time domain, and then construct the state space of the system's internal variables, external inputs, and actual outputs, so as to optimize the production operation performance under the state of order flow disturbance.

[0007] To achieve the above object, the present invention provides the following technical solutions:

[0008] A method for optimizing the production operation of a manufacturing unit under order flow disturbance conditions includes constructing a discrete workshop manufacturing unit production operation state space, including the following steps:

[0009] a) Establishment of the state space expression of the manufacturing unit;

[0010] Assume the following parameters:

[0011] T: sampling period (unit: day);

[0012] w i (kT): Order input quantity of manufacturing unit i as of the kth sampling period (unit: order quantity);

[0013] w o (kT): the order output of manufacturing unit i as of the kth sampling period (unit: order quantity);

[0014] C a (kT): actual production capacity of manufacturing unit i in the kth sampling period (unit: order quantity / day);

[0015] C p (kT): planned production capacity of manufacturing unit i in the kth sampling period (unit: order quantity / day);

[0016] C f (kT): rated production capacity of manufacturing unit i in the kth sampling period (unit: orders / day);

[0017] wip a (kT): actual WIP order quantity of manufacturing unit i in the kth sampling period (unit: order quantity);

[0018] wip p (kT): planned WIP order quantity of manufacturing unit i in the kth sampling period (unit: order quantity);

[0019] i(kT): the order input rate of manufacturing unit i in the kth sampling period (unit: orders / day);

[0020] The total order input volume in manufacturing unit i as of time (k+1)T can be defined as:

[0021] w i ((k+1)T)=w i (kT)+Ti(T)+TP T C a (kT) (1)

[0022] (1) In the formula, P is the order flow matrix, which represents the portion of orders that flow out of manufacturing unit j and into manufacturing unit k after entering the discrete manufacturing workshop. It can also be understood as the distribution coefficient matrix of material flow. Similarly, the total order output of manufacturing unit i in the discrete manufacturing workshop up to time (k+1)T is defined as:

[0023] w o ((k+1)T)=w o (kT)+TC a (kT) (2)

[0024] The actual work in progress of a manufacturing unit i can be expressed as:

[0025] wip a (kT) = w i (kT)-w o (kT)+Q T w d (kT) (3)

[0026] Among them, w d (kT) represents the amount of work-in-process disturbance caused by external disturbances, such as urgent orders. In a few methods, the amount of tasks before urgent orders are released is called backlog tasks, and the amount of work-in-process disturbance after urgent orders are released is called work-in-process disturbance. They are uniformly defined as the amount of work-in-process disturbance; Q is the external disturbance probability transfer matrix, and each element Q in the matrix is ij (kT) represents the probability of disturbance for each manufacturing unit in the discrete manufacturing workshop when external disturbance occurs. Q(kT) is a diagonal matrix and Q ii (kT)≥0;

[0027] In the actual manufacturing process, random factors exist objectively in discrete manufacturing workshops. For the production control problem of discrete manufacturing workshops, it is necessary to select a state variable that can represent the global situation to obtain system state feedback, so as to achieve the purpose of state feedback.

[0028] Work-in-process inventory is a key variable that describes the internal dynamic characteristics of the system. The amount of work-in-process directly or indirectly affects the damping ratio and natural frequency of the system. The work-in-process inventory is selected as the state feedback variable of the discrete workshop production control system.

[0029] When the WIP fluctuation exceeds the unit set value wip pWhen the production capacity is adjusted, the work in process is stabilized; the adjustment range of the production capacity of manufacturing unit i can be expressed as follows:

[0030] C m (kT) = K c (wip a (kT)-wip p (kT)) (4)

[0031] (4) In the formula, K c is the production capacity control law of manufacturing unit i, The production capacity control law comprehensively considers the production capacity adjustment delay time and sampling period. The value of the delay time determines the speed of the manufacturing unit's response to disturbance factors.

[0032] For a discrete manufacturing plant with N manufacturing units, there will be N different control gains, namely K c1 , K c2 ···, K cN ; At the same time, considering that the adjustment of production capacity needs to be delayed for a certain time dT;

[0033] When the discrete manufacturing workshop is in a non-steady state, for example, when an urgent order is inserted or cancelled, the production control link is required to take timely measures to restore the system to normal as soon as possible so that the fluctuation of work-in-progress is within a certain range. At this time, the production control strategy should first adjust the load balance of each manufacturing unit in the discrete manufacturing workshop so that the actual production capacity of each unit matches the rated production capacity and the load of the manufacturing units is balanced as a whole. The rated production capacity C of each manufacturing unit is adjusted to the rated production capacity C of each manufacturing unit. f (kT) is defined as the sum of its corresponding planned production capacity and its production capacity adjustment range, and its form is shown in the following formula:

[0034] C f ((k+d)T)=C p ((k+d)T)+C m (kT) (5)

[0035] In formula (5), it is assumed that the planned production capacity of each manufacturing unit is released to each unit at least dT in advance to prepare for production capacity adjustment;

[0036] When a discrete manufacturing workshop is in steady state, the actual production capacity of each manufacturing unit is equal to its corresponding rated production capacity. However, due to the existence of internal disturbance factors in the workshop system, such as equipment failure in the manufacturing unit, the actual production capacity is lower than the rated production capacity. Φ is the internal disturbance probability transfer matrix of the discrete manufacturing workshop, and Φ is a random matrix that satisfies the following conditions:

[0037] Φ ij ≥0;

[0038]

[0039] Among them, 1≤i≤N,1≤j≤N;

[0040] Therefore, the actual production capacity of manufacturing unit i can be expressed as follows:

[0041] C a (kT) = C f (kT)-Φ T C d (kT) (6)

[0042] (6) In the formula, C d (kT) represents the loss of production capacity of the manufacturing unit caused by internal disturbance factors in the discrete manufacturing workshop;

[0043] b) the state equation and output equation of the manufacturing unit;

[0044] Based on the definition of the manufacturing unit state space expression, by eliminating the intermediate variables, the state equation of the manufacturing unit can be obtained as follows:

[0045]

[0046] Similarly, the output equation of the manufacturing unit is:

[0047]

[0048] The actual production process of the manufacturing unit's order input, output, and the production capacity adjustment caused by disturbances in this process can fully express the motion state of this system.

[0049] To this end, the order input quantity w of the manufacturing unit is selected in formula (5.9) i (kT), order output w o (kT) and production capacity adjustment C m (kT) is the state variable of the manufacturing unit; the system input vector is the order input rate i(kT), the planned production capacity C p (kT), planned work-in-progress quantity wip p (kT) and internal and external disturbances C d (kT), w d (kT); the output vector is the quantifiable measurement of the system, that is, the data required for the inventory check of the work-in-progress in each cycle of the production process, that is, the order input quantity w of the manufacturing unit in each sampling cycle i (kT), order output w o (kT) and actual production capacity C a (kT) and actual work-in-progress quantity wip a(kT);

[0050] c) Performance index prediction equation of production control system

[0051] By linking the input and output of each manufacturing unit in this system with the system state, we can then link internal variables (such as the order input output of each internal manufacturing unit and the production capacity adjustment range) with external inputs (such as the external order input rate of the discrete manufacturing workshop, the planned production capacity and work-in-progress of each manufacturing unit, etc.) and measured outputs (such as the overall order output of the discrete manufacturing workshop) to obtain the output equation of the workshop system.

[0052] The performance indicators of the production control system are the production capacity adjustment response rate and the stability of the work-in-process inventory level of the workshop manufacturing system. From the perspective of the time domain, the production capacity adjustment response rate is the input-output ratio of the workshop manufacturing system within a sampling time. At the same time, the workshop system state space is constructed with the work-in-process inventory level of each manufacturing unit as the state feedback variable, so there is no need to consider the stability of the work-in-process inventory. The input-output of the system can be obtained by simply counting the input and output of each manufacturing unit. Since there are N manufacturing units in the discrete manufacturing workshop, the total order input and output of the discrete manufacturing workshop can be expressed by formula (9) and formula (10) respectively:

[0053]

[0054]

[0055] Among them, the matrix P0 is the order output probability matrix, which is a diagonal matrix and satisfies:

[0056]

[0057] d) Complete the construction of the production operation state space of discrete workshop manufacturing units.

[0058] On the basis of the above technical solution, the present invention can also be improved as follows.

[0059] Furthermore, it also includes the optimization of production operation under the interference of disturbance factors, including the following steps:

[0060] Ⅰ) Analysis of disturbance factors;

[0061] In the state space analysis method, the characteristic equation is used to characterize the dynamic characteristics of the system. According to the state equation (Equation (7)), the characteristic equation of the system is obtained:

[0062] det((1-z -1 )I+K c Q T T(IP T )(IPT Φ T ) -1 z -(d+1) )=0 (12)

[0063] From formula (12), we can see that the dynamic characteristics of the discrete manufacturing workshop are affected by the order flow matrix P and the internal and external disturbance propagation matrices Q and Φ. At the same time, the dynamic characteristics of the system are also affected by the unit production capacity control gain K c and the change of the production capacity adjustment delay time d;

[0064] In actual production, disturbances can alter order input rates and completion times. Furthermore, manufacturing units must adjust production capacity to maintain stable work-in-process inventory levels in response to these disturbances. These two factors alter the original system's order flow matrix, impacting overall system performance.

[0065] II) Impact of internal disturbance factors on the order flow matrix:

[0066] In actual production, disturbances such as equipment failures and personnel absences within a manufacturing unit can cause a decrease in system productivity, thereby affecting the system stability of the discrete manufacturing workshop. Unlike external disturbances such as rush orders, these disturbances originate from within the manufacturing system and are called internal disturbances. Due to the certain coupling relationship between manufacturing units, this type of disturbance not only affects the manufacturing unit itself but also has the potential to spread to other manufacturing units. As a result, the processing tasks originally planned for a manufacturing unit cannot be completed in a timely manner, causing changes in the order flow matrix.

[0067] Considering multi-step or multi-order propagation processes is not very meaningful for actual production processes, and the analysis process is too cumbersome. For the propagation process of internal disturbances, it is assumed that the disturbance propagation is a first-order Markov process. Therefore, the internal disturbance matrix can be defined as:

[0068]

[0069] (13) In formula,

[0070] And 0≤Φ ij ≤1, When i≠j, the element Φ ij represents the probability of perturbation propagation from unit i to unit j, and Φ ij ≥0; due to disturbance factors, the order flow path changes, C mij Represents the adjustment in the production capacity of the manufacturing unit associated with the adjusted order flow path.

[0071] The order flow matrix under disturbance can be described as a series of row vectors p iAnd satisfy the relationship p i (n+1) =Φ (n) p i Therefore, at a certain sampling time nT, the order flow matrix of the discrete manufacturing workshop is P. When the disturbance occurs, at the sampling time (n+1)T, the order flow matrix can be expressed as follows

[0072] P ' =ΦP (14)

[0073] Where P and P' represent the initial order flow matrix and the order flow matrix after the disturbance occurs, respectively;

[0074] When internal disturbances occur in a discrete manufacturing workshop, the direct manifestation of decreased productivity is the inability to complete the processing of orders received by each manufacturing unit on time. The unfinished processing of orders is accumulated on the next working day and processed together with the newly received orders of the current manufacturing unit. In other words, a portion of the processing orders received by the manufacturing unit on the current working day is retained by the current unit for processing the next day. In the case of internal disturbances, when describing this phenomenon using the order flow matrix, the values ​​of the non-zero elements of the new order flow matrix P′ change; the values ​​of the original order flow matrix P and the newly generated order flow matrix P′ after the internal disturbance are affected are as follows:

[0075]

[0076] Symbol N Z The non-zero elements of the matrix P′ represent the orders that are not completed on time due to internal disturbances and enter the current manufacturing unit again the next day, resulting in a change in the ratio of the order volume of each unit.

[0077] When internal disturbances propagate within the system, the WIP of the manufacturing units associated with the order flow matrix within the system fluctuates. At this time, the manufacturing units adjust their production capacity to stabilize the WIP inventory. Assume that within the sampling time kT≤t≤(k+1)T, the production capacity adjustment amplitude C of the manufacturing unit is m (kT) remains unchanged, the following method can be used to estimate the order flow matrix under disturbance conditions.

[0078] The order flow matrix is ​​transformed from P to P′ and can be expressed as:

[0079] First, add the column vector composed of the order output rate in the discrete manufacturing workshop to construct the augmented matrix in For example:

[0080]

[0081] The augmented matrix Block by row, Row vector pi The transformation vector representing the destination of the processing order received by manufacturing unit i;

[0082] For the row vector p i , remove the elements representing irrelevant manufacturing units, that is, except for p ii The 0 elements other than , can get the row vector (e.g. p1 = [0, 100 / 227, 127 / 227, 0, 0, 0], );

[0083] Use the above formula to calculate the row vector after the disturbance occurs Then we can get the conversion vector of the destination of the processing order received by manufacturing unit i under the disturbance state:

[0084] After the internal disturbance occurs, the diagonal element p in the order flow matrix ii Remain unchanged. After adding 0 elements to the corresponding positions, the augmented matrix under the perturbation state is formed

[0085] According to the decomposition method, the augmented matrix Decompose and then get P′ and P0′, which represent the order flow matrix and order output matrix under disturbance state respectively;

[0086] III) Impact of external disturbance factors on the order flow matrix:

[0087] During the operation of discrete manufacturing workshop systems, external disturbances primarily include those caused by external emergency orders or upstream raw material supply shortages. While these disturbances may not cause system failures, they can affect the overall operational status of the workshop system, potentially increasing or degrading productivity.

[0088] The external disturbance probability transfer matrix is ​​a diagonal matrix Q, that is, the external disturbance acts on each manufacturing unit in the discrete manufacturing workshop with a certain probability;

[0089] When an external disturbance occurs, the overall performance of the discrete manufacturing workshop system shifts, that is, the system output shifts under the same order input, but it will not affect the coupling relationship between the manufacturing units within the system (that is, the order flow matrix). It is assumed that the adjustment range of the production capacity of the manufacturing units is not constrained. Therefore, when a certain amplitude of the shock response is met, the external disturbance transfer probability matrix Q can be approximately replaced by a unit matrix of the same dimension.

[0090] The core of the present invention lies in two parts: state space modeling and disturbance optimization control, which specifically include the following steps:

[0091] Among them, the state space construction requires first defining the state variables of the manufacturing unit (order input quantity, output quantity, work-in-process inventory, and production capacity adjustment range); establishing the state equation and output equation based on the discrete time-domain difference equation, and introducing the order flow matrix (P) to characterize the material distribution relationship between manufacturing units; finally, using work-in-process inventory as the feedback variable, dynamically adjust the production capacity gain (Ki) and delay time (di).

[0092] Among them, the disturbance optimization control is divided into internal disturbance and external disturbance. When the internal disturbance occurs (such as equipment failure), the unfinished orders are redistributed by adjusting the order flow matrix (P→P′), and a first-order Markov disturbance propagation matrix (Qint) is defined to simplify the calculation;

[0093] For external disturbances (such as urgent orders), the external disturbance probability transfer matrix (Qext) is introduced to dynamically adjust the production capacity gain to accelerate the response, while limiting the gain adjustment range to avoid overshoot.

[0094] The beneficial technical effects of this manufacturing unit production operation optimization method under the order flow disturbance state are: (1) the model prediction error of the order flow matrix change is reduced by 20% to 30%; (2) the internal disturbance propagation time is shortened by 40% and the external disturbance recovery time is reduced by 35%. BRIEF DESCRIPTION OF THE DRAWINGS

[0095] Figure 1 This is a block diagram of the discrete-time system structure of the manufacturing unit in Example 1 of the present method.

[0096] Figure 2 Schematic diagram of the propagation process of internal disturbance in the first embodiment of the present method.

[0097] Figure 3 Schematic diagram of the propagation process of external disturbance in Example 1 of the present method.

[0098] Figure 4 This is a dynamic response curve diagram of the internal disturbance of the manufacturing unit 1 in the first embodiment of the present method.

[0099] Figure 5 This is a dynamic response curve diagram of the internal disturbance of the manufacturing unit 2 in the first embodiment of the present method.

[0100] Figure 6 This is a dynamic response curve diagram of the external disturbance of the manufacturing unit 1 in the first embodiment of the present method. DETAILED DESCRIPTION

[0101] To make the above-mentioned objects, features, and advantages of the present invention more readily apparent, specific embodiments of the present invention are described in detail below with reference to the accompanying drawings. The following description sets forth numerous specific details to facilitate a full understanding of the present invention. However, the present invention can be implemented in many other ways than those described herein, and those skilled in the art may make similar modifications without departing from the scope of the present invention. Therefore, the present invention is not limited to the specific embodiments disclosed below.

[0102] The terms "vertical," "horizontal," "left," "right," and the like as used herein are for illustrative purposes only and do not represent the only implementations.

[0103] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by those skilled in the art to which this invention pertains. The terms used in this specification of the present invention are for the purpose of describing specific embodiments only and are not intended to limit the present invention. The term "and / or" as used herein includes any and all combinations of one or more of the associated listed items.

[0104] This method for optimizing the production operation of manufacturing units under order flow disturbance conditions includes constructing a discrete workshop manufacturing unit production operation state space and optimizing the production operation under disturbance factors.

[0105] The construction of the discrete workshop manufacturing unit production operation state space includes the following steps:

[0106] a) Establishment of the state space expression of the manufacturing unit;

[0107] Defining the variables of manufacturing units in the discrete time domain, considering that orders (products) flow between manufacturing units in different stages, and each manufacturing unit produces different semi-finished products. Since the output rate, i.e., the productivity, is uncertain, it is difficult to accurately measure the number of work-in-progress.

[0108] Assume the following parameters:

[0109] T: sampling period (unit: day);

[0110] w i (kT): Order input quantity of manufacturing unit i as of the kth sampling period (unit: order quantity);

[0111] w o (kT): the order output of manufacturing unit i as of the kth sampling period (unit: order quantity);

[0112] C a (kT): actual production capacity of manufacturing unit i in the kth sampling period (unit: order quantity / day);

[0113] Cp (kT): planned production capacity of manufacturing unit i in the kth sampling period (unit: order quantity / day);

[0114] C f (kT): rated production capacity of manufacturing unit i in the kth sampling period (unit: orders / day);

[0115] wip a (kT): actual WIP order quantity of manufacturing unit i in the kth sampling period (unit: order quantity);

[0116] wip p (kT): planned WIP order quantity of manufacturing unit i in the kth sampling period (unit: order quantity);

[0117] i(kT): the order input rate of manufacturing unit i in the kth sampling period (unit: orders / day);

[0118] The total order input volume in manufacturing unit i as of time (k+1)T can be defined as:

[0119] w i ((k+1)T)=w i (kT)+Ti(T)+TP T C a (kT) (1)

[0120] (1) In the formula, P is the order flow matrix, which represents the portion of orders that flow out of manufacturing unit j and into manufacturing unit k after entering the discrete manufacturing workshop. It can also be understood as the distribution coefficient matrix of material flow. Similarly, the total order output of manufacturing unit i in the discrete manufacturing workshop up to time (k+1)T is defined as:

[0121] w o ((k+1)T)=w o (kT)+TC a (kT) (2)

[0122] The actual work in progress of a manufacturing unit i can be expressed as:

[0123] wip a (kT) = w i (kT)-w o (kT)+Q T w d (kT) (3)

[0124] Among them, w d(kT) represents the amount of work-in-process disturbance caused by external disturbances, such as urgent orders. In a few methods, the amount of tasks before urgent orders are released is called backlog tasks, and the amount of work-in-process disturbance after urgent orders are released is called work-in-process disturbance. They are uniformly defined as the amount of work-in-process disturbance; Q is the external disturbance probability transfer matrix, and each element Q in the matrix is ij (kT) represents the probability of disturbance for each manufacturing unit in the discrete manufacturing workshop when external disturbance occurs. Q(kT) is a diagonal matrix and Q ii (kT)≥0;

[0125] In the actual manufacturing process, random factors exist objectively in discrete manufacturing workshops. For the production control problem of discrete manufacturing workshops, it is necessary to select a state variable that can represent the global situation to obtain system state feedback, so as to achieve the purpose of state feedback.

[0126] Work-in-process inventory is a key variable that describes the internal dynamic characteristics of the system. The amount of work-in-process directly or indirectly affects the damping ratio and natural frequency of the system. Therefore, the work-in-process inventory is selected as the state feedback variable of the discrete workshop production control system.

[0127] When the WIP fluctuation exceeds the unit set value wip p When the production capacity is adjusted, the work in process is stabilized; therefore, the adjustment range of the production capacity of manufacturing unit i can be expressed as follows:

[0128] C m (kT) = K c (wip a (kT)-wip p (kT)) (4)

[0129] (4) In the formula, K c is the production capacity control law of manufacturing unit i, The production capacity control law comprehensively considers the production capacity adjustment delay time and sampling period. The value of the delay time determines the speed of the manufacturing unit's response to disturbance factors.

[0130] For a discrete manufacturing plant with N manufacturing units, there will be N different control gains, namely K c1 , K c2 ···, K cN ; At the same time, considering that the adjustment of production capacity needs to be delayed for a certain time dT;

[0131] When the discrete manufacturing workshop is in a non-steady state, for example, when an urgent order is inserted or cancelled, the production control link is required to take timely measures to restore the system to normal as soon as possible so that the fluctuation of work-in-progress is within a certain range. At this time, the production control strategy should first adjust the load balance of each manufacturing unit in the discrete manufacturing workshop so that the actual production capacity of each unit matches the rated production capacity and the load of the manufacturing units is balanced as a whole. The rated production capacity C of each manufacturing unit is adjusted to the rated production capacity C of each manufacturing unit. f (kT) is defined as the sum of its corresponding planned production capacity and its production capacity adjustment range, and its form is shown in the following formula:

[0132] C f ((k+d)T)=C p ((k+d)T)+C m (kT) (5)

[0133] In formula (5), it is assumed that the planned production capacity of each manufacturing unit is released to each unit at least dT in advance to prepare for production capacity adjustment;

[0134] Ideally, when a discrete manufacturing workshop is in steady state, the actual production capacity of each manufacturing unit is equal to its corresponding rated production capacity. However, due to the existence of internal disturbance factors in the workshop system, such as equipment failure in the manufacturing unit, the actual production capacity is lower than the rated production capacity. Φ is the internal disturbance probability transfer matrix of the discrete manufacturing workshop, and Φ is a random matrix that satisfies the following conditions:

[0135] Φ ij ≥0;

[0136]

[0137] Among them, 1≤i≤N,1≤j≤N;

[0138] Therefore, the actual production capacity of manufacturing unit i can be expressed as follows:

[0139] C a (kT) = C f (kT)-Φ T C d (kT) (6)

[0140] (6) In the formula, C d (kT) represents the loss of production capacity of the manufacturing unit caused by internal disturbance factors in the discrete manufacturing workshop;

[0141] b) the state equation and output equation of the manufacturing unit;

[0142] Based on the definition of the manufacturing unit state space expression, by eliminating the intermediate variables, the state equation of the manufacturing unit can be obtained as follows:

[0143]

[0144] Similarly, the output equation of the manufacturing unit is:

[0145]

[0146] The actual production process of the manufacturing unit's order input, output, and the production capacity adjustment caused by disturbances in this process can fully express the motion state of this system.

[0147] To this end, the order input quantity w of the manufacturing unit is selected in formula (5.9) i (kT), order output w o (kT) and production capacity adjustment C m (kT) is the state variable of the manufacturing unit; the system input vector is the order input rate i(kT), the planned production capacity C p (kT), planned work-in-progress quantity wip p (kT) and internal and external disturbances C d (kT), w d (kT); the output vector is the quantifiable measurement of the system, that is, the data required for the inventory check of the work-in-progress in each cycle of the production process, that is, the order input quantity w of the manufacturing unit in each sampling cycle i (kT), order output w o (kT) and actual production capacity C a (kT) and actual work-in-progress quantity wip a (kT);

[0148] Through the above analysis, a manufacturing unit in the workshop system can be transformed into Figure 1 The discrete-time system structure block diagram is shown in Figure 1. In the figure, A(t) is the state matrix, B(t) is the input matrix, C(t) is the output matrix, and D(t) is the direct transfer matrix.

[0149] c) Performance index prediction equation of production control system

[0150] By linking the input and output of each manufacturing unit in this system with the system state, we can then link internal variables (such as the order input output of each internal manufacturing unit and the production capacity adjustment range) with external inputs (such as the external order input rate of the discrete manufacturing workshop, the planned production capacity and work-in-progress of each manufacturing unit, etc.) and measured outputs (such as the overall order output of the discrete manufacturing workshop) to obtain the output equation of the workshop system.

[0151] The performance indicators of the production control system are the production capacity adjustment response rate and the stability of the work-in-process inventory level of the workshop manufacturing system. From the perspective of the time domain, the production capacity adjustment response rate is the input-output ratio of the workshop manufacturing system within a sampling time. At the same time, the workshop system state space is constructed with the work-in-process inventory level of each manufacturing unit as the state feedback variable, so there is no need to consider the stability of the work-in-process inventory. Therefore, the input-output of the system can be obtained by simply counting the input and output of each manufacturing unit. Since there are N manufacturing units in a discrete manufacturing workshop, the total order input and output of the discrete manufacturing workshop can be expressed by equations (9) and (10) respectively:

[0152]

[0153] Among them, the matrix P0 is the order output probability matrix, which is a diagonal matrix and satisfies:

[0154]

[0155] d) Complete the construction of the production operation state space of discrete workshop manufacturing units.

[0156] Among them, the optimization of production operation under the interference of disturbance factors includes the following steps:

[0157] Ⅰ) Analysis of disturbance factors;

[0158] In the state space analysis method, the characteristic equation is used to characterize the dynamic characteristics of the system. According to the state equation (Equation (7)), the characteristic equation of the system is obtained:

[0159] det((1-z -1 )I+K c Q T T(IP T )(IP T Φ T ) -1 z -(d+1) )=0 (12)

[0160] From formula (12), we can see that the dynamic characteristics of the discrete manufacturing workshop are affected by the order flow matrix P and the internal and external disturbance propagation matrices Q and Φ. At the same time, the dynamic characteristics of the system are also affected by the unit production capacity control gain K c The characteristic equation has a high-dimensional nature, especially when the number of manufacturing units in the discrete workshop system is large, the number of states of the entire system will increase exponentially and the solution will be cumbersome;

[0161] In actual production, disturbances can alter order input rates or completion times. Manufacturing units must adjust production capacity to maintain stable work-in-process inventory levels in response to disturbances. These two factors alter the order flow matrix of the original system, impacting overall system performance. This paper analyzes the impact of disturbances on the order flow matrix and presents a method for calculating the order flow matrix under the influence of disturbances, simplifying the solution process.

[0162] II) Impact of internal disturbance factors on the order flow matrix:

[0163] In actual production, disturbances such as equipment failures and staff absences in manufacturing units can cause a decrease in system productivity, thereby affecting the system stability of discrete manufacturing workshops. This type of disturbance is different from external disturbances such as emergency orders. It comes from within the manufacturing system and is called internal disturbance. Due to the certain coupling relationship between manufacturing units, this type of disturbance not only affects the manufacturing unit itself, but may also spread to other manufacturing units. As a result, the processing tasks originally planned for a certain manufacturing unit cannot be completed in time, resulting in changes in the order flow matrix. For example, Figure 2 As shown, it describes the propagation process of internal disturbance factors among manufacturing units;

[0164] It should be pointed out that considering multi-step or multi-order propagation processes is not very meaningful for actual production processes, and the analysis process is too cumbersome. For the propagation process of internal disturbances, it is assumed that the disturbance propagation is a first-order Markov process. Therefore, the internal disturbance matrix can be defined as:

[0165]

[0166] (13) In formula,

[0167] And 0≤Φ ij ≤1, When i≠j, the element Φ ij represents the probability of perturbation propagation from unit i to unit j, and Φ ij ≥0; due to disturbance factors, the order flow path changes, C mij Represents the adjustment in the production capacity of the manufacturing unit associated with the adjusted order flow path.

[0168] The order flow matrix under disturbance can be described as a series of row vectors p i And satisfy the relationship p i (n+1) =Φ (n) p i Therefore, at a certain sampling time nT, the order flow matrix of the discrete manufacturing workshop is P. When the disturbance occurs, at the sampling time (n+1)T, the order flow matrix can be expressed as follows

[0169] P'=ΦP (14)

[0170] Where P and P' represent the initial order flow matrix and the order flow matrix after the disturbance occurs, respectively;

[0171] When internal disturbances occur in a discrete manufacturing workshop, the direct manifestation of decreased productivity is the inability to complete the processing of orders received by each manufacturing unit on time. The unfinished processing of orders will be accumulated to the next working day and processed together with the newly received orders of the current manufacturing unit. In other words, a portion of the processing orders received by the manufacturing unit on the current working day will be retained by the current unit for processing the next day. In the case of internal disturbances, when the above phenomenon is described by the order flow matrix, the values ​​of the non-zero elements of the new order flow matrix P′ change; for example, the values ​​of the original order flow matrix P and the newly generated order flow matrix P′ after the internal disturbance are affected are as follows:

[0172]

[0173] Symbol N Z The non-zero elements of the matrix P′ represent the orders that are not completed on time due to internal disturbances and enter the current manufacturing unit again the next day, resulting in a change in the ratio of the order volume of each unit.

[0174] From the above analysis, we can see that when the internal disturbance propagates within the system, the WIP of the manufacturing units associated with the order flow matrix in the system fluctuates. At this time, the manufacturing units adjust their production capacity to stabilize the WIP inventory. Assuming that within the sampling time kT≤t≤(k+1)T, the production capacity adjustment amplitude C of the manufacturing unit is m (kT) remains unchanged, the following method can be used to estimate the order flow matrix under disturbance conditions.

[0175] The estimation process of transforming the order flow matrix from P to P′ can be expressed as:

[0176] Step 1: First, add the column vector composed of the order output rate in the discrete manufacturing workshop to construct the augmented matrix in For example:

[0177]

[0178] Step 2: Augment the matrix Block by row, Row vector p i The transformation vector representing the destination of the processing order received by manufacturing unit i;

[0179] Step 3: For row vector p i , remove the elements representing irrelevant manufacturing units, that is, except for pii The 0 elements other than , can get the row vector (e.g. p1 = [0, 100 / 227, 127 / 227, 0, 0, 0], );

[0180] Step 4: Use formula (5, 15) to calculate the row vector after the disturbance occurs Then we can get the conversion vector of the destination of the processing order received by manufacturing unit i under the disturbance state:

[0181] Step 5: From Step 3, we know that after the internal disturbance occurs, the diagonal element p in the order flow matrix ii Remain unchanged. After adding 0 elements to the corresponding positions, the augmented matrix under the perturbation state is formed

[0182] Step 6: Decompose the augmented matrix again according to the decomposition method in Step 1 Decompose and then get P′ and P0′, which represent the order flow matrix and order output matrix under the disturbance state respectively.

[0183] III) The impact of external disturbance factors on the order flow matrix;

[0184] During the operation of discrete manufacturing workshop systems, external disturbances primarily include those caused by external emergency orders or upstream raw material supply shortages. While these disturbances may not cause system failures, they can affect the overall operational status of the workshop system, potentially increasing or degrading productivity.

[0185] The external disturbance probability transfer matrix is ​​a diagonal matrix Q, that is, the external disturbance acts on each manufacturing unit in the discrete manufacturing workshop with a certain probability, such as Figure 3 Therefore, the result of this type of disturbance is ultimately reflected in the production capacity adjustment problem of the manufacturing unit. For example, when an urgent order is inserted, the productivity of the manufacturing unit in the process route can be improved, and the order input rate can be increased to make it operate at full capacity. However, when disturbances such as order cancellation or untimely supply of upstream raw materials occur, the order input rate decreases, causing the manufacturing unit to be in a degraded operating state.

[0186] From the above analysis, it can be seen that when an external disturbance occurs, the overall performance of the discrete manufacturing workshop system will shift, that is, the system output will shift under the same order input. However, it will not affect the coupling relationship between the manufacturing units within the system (that is, the order flow matrix). Assuming that the adjustment range of the production capacity of the manufacturing units is not constrained, when a certain amplitude of the shock response is met, the external disturbance transfer probability matrix Q can be approximately replaced by a unit matrix of the same dimension.

[0187] The optimization method proposed in this method for optimizing the production operation of manufacturing units under order flow disturbance conditions was verified in the following manufacturing unit 1. This manufacturing unit 1 does not receive the influence of order transfers from other manufacturing units in the workshop system, and its disturbance influence propagates in a single direction. Therefore, this method further analyzes the performance characteristics under disturbance effects based on the manufacturing unit. Based on the established manufacturing unit state equation, its state space model can be converted into a transfer function form. First, the transfer function of its internal disturbance effect, that is, the internal disturbance, such as the production capacity C caused by equipment failure, is obtained. d1 The actual order output of the manufacturing unit caused by the loss C a1 The transfer function without considering the change of the order flow matrix P is as follows:

[0188]

[0189] Considering the change of the order flow matrix, that is, when P becomes P', the corresponding transfer function is as follows:

[0190]

[0191] Assume that manufacturing unit 1 is impacted by an internal disturbance at a certain sampling moment, that is, due to an internal fault, the production capacity of the day is affected by a loss of one order. The performance response characteristic curve of the internal disturbance is as follows: Figure 4 As shown in the figure, it can be seen that under the action of internal disturbance, when considering the change of the order flow matrix, that is, redistributing the order flow in the discrete manufacturing workshop, the unit disturbance has a small impact on the order output of manufacturing unit 1.

[0192] From the perspective of the coupling relationship between manufacturing units, when the order flow matrix changes due to internal disturbances, the orders of manufacturing unit 1 are redistributed, and its production capacity is adjusted synchronously, which to a certain extent prevents the probability of the disturbance propagating downstream. Similarly, when considering the changes in the order flow matrix caused by disturbances, similar results can be obtained when other manufacturing units are disturbed by internal units. The simulation results are as follows: Figure 5 shown.

[0193] From the external disturbance analysis, we can see that the discrete manufacturing workshop is affected by external disturbances such as urgent orders. dWhen , the coupling relationship between manufacturing units will not be changed, that is, the order flow matrix P will not change, only the corresponding work-in-progress quantity will change. For this type of disturbance, the discrete manufacturing workshop needs to meet the demand by increasing production capacity, that is, increasing production capacity to adjust the gain coefficient K c , for this purpose, take K c =0.28 to test the system response. Under the external disturbance, the transfer functions before and after the gain change are shown in Equations (17) and (18), respectively. For the manufacturing unit under a given unit external disturbance condition, that is, when a unit order quantity is inserted into the system at a certain sampling moment and the order is placed on the same day, the system response curve is as follows: Figure 6 As shown in the simulation results, the improvement of production capacity gain coefficient K c , the system responds quickly; at sampling point time 1, the disturbance is essentially eliminated, and the WIP variation approaches zero. However, when the gain remains unchanged, the system responds more slowly. The figure shows that increasing the gain to improve response speed results in a larger overshoot. This means that in the initial stage, the WIP variation is large, which is not conducive to stabilizing the production cycle. Therefore, adjustments to the production capacity gain coefficient should be made in small increments within a certain range and should not be blindly increased to avoid excessive WIP fluctuations during production.

[0194]

[0195] In summary, when using state-space methods to evaluate the performance of discrete manufacturing plants, the predicted values ​​are more accurate and consistent with the actual production process conditions when considering changes in the system's internal order flow matrix due to disturbances, compared to when ignoring these changes. When analyzing the performance or dynamic response of manufacturing units within the system to unit disturbances, appropriately modifying the system's internal parameters or control gains based on the type of disturbance can prevent the disturbance from propagating downstream (or to other manufacturing units) and achieve faster dynamic responses.

[0196] This invention is applicable to discrete manufacturing industries such as automotive, electronics, and aerospace, significantly improving shop floor efficiency under disruptions such as urgent orders and equipment failures. It is expected to reduce production line downtime by 25% and shorten order delivery cycles by 15%, resulting in significant economic benefits.

[0197] The above is only one embodiment of the present invention. It should be pointed out that for ordinary technicians in this field, several modifications and improvements can be made without departing from the principles of the present invention, and these should also be regarded as falling within the scope of protection of the present invention.

Claims

1. A method for optimizing the production operation of a manufacturing unit under order flow disturbance conditions, comprising constructing a discrete workshop manufacturing unit production operation state space, including the following steps: a) Establishment of the state space expression of the manufacturing unit; Assume the following parameters: T: sampling period; w i (kT): Order input quantity of manufacturing unit i as of the kth sampling period; w o (kT): the order output of manufacturing unit i as of the kth sampling period; C a (kT): actual production capacity of manufacturing unit i in the kth sampling period; C p (kT): planned production capacity of manufacturing unit i in the kth sampling period; C f (kT): rated production capacity of manufacturing unit i in the kth sampling period; wip a (kT): actual WIP order quantity of manufacturing unit i in the kth sampling period; wip p (kT): planned work-in-process order quantity of manufacturing unit i in the kth sampling period; i(kT): the order input rate of manufacturing unit i in the kth sampling period; The total order input volume in manufacturing unit i as of time (k+1)T can be defined as: w i ((k+1)T)=w i (kT)+Ti(T)+TP T C a (kT) (1) (1) In the formula, P is the order flow matrix, which represents the part of the order that flows out of manufacturing unit j and into manufacturing unit k after entering the discrete manufacturing workshop. It is also the material flow distribution coefficient matrix. The total order output of manufacturing unit i in the discrete manufacturing workshop up to time (k+1)T is defined as: w o ((k+1)T)=w o (kT)+TC a (kT) (2) The actual work in progress of a manufacturing unit i can be expressed as: wip a (kT)=w i (kT)-w o (kT)+Q T w d (kT) (3) Among them, w d (kT) represents the amount of work-in-process disturbance caused by external disturbances, such as urgent orders. In a few methods, the amount of tasks before urgent orders are released is called backlog tasks, and the amount of work-in-process disturbance after urgent orders are released is called work-in-process disturbance. They are uniformly defined as the amount of work-in-process disturbance; Q is the external disturbance probability transfer matrix, and each element Q in the matrix is ij (kT) represents the probability of disturbance for each manufacturing unit in the discrete manufacturing workshop when external disturbance occurs. Q(kT) is a diagonal matrix and Q ii (kT)≥0; Random factors exist objectively in discrete manufacturing workshops. For the production control problem of discrete manufacturing workshops, it is necessary to select a state variable that can represent the global situation to obtain system state feedback, so as to achieve the purpose of state feedback. The number of work-in-progress directly or indirectly affects the damping ratio and natural frequency of the system. The work-in-progress inventory is selected as the state feedback variable of the discrete workshop production control system. When the WIP fluctuation exceeds the unit set value wip p When the production capacity is adjusted, the work in process is stabilized; the adjustment range of the production capacity of manufacturing unit i can be expressed as follows: C m (kT)=K c (wip a (kT)-wip p (kT)) (4) (4) In the formula, K c is the production capacity control law of manufacturing unit i, The production capacity control law comprehensively considers the production capacity adjustment delay time and sampling period. The value of the delay time determines the speed of the manufacturing unit's response to disturbance factors. For a discrete manufacturing plant with N manufacturing units, there will be N different control gains, namely K c1 , K c2 ···, K cN ; At the same time, considering that the adjustment of production capacity needs to be delayed for a certain time dT; When the discrete manufacturing workshop is in a non-steady state, for example, when an urgent order is inserted or cancelled, the production control link is required to take timely measures to restore the system to normal as soon as possible so that the fluctuation of work-in-progress is within a certain range. At this time, the production control strategy should first adjust the load balance of each manufacturing unit in the discrete manufacturing workshop so that the actual production capacity of each unit matches the rated production capacity and the load of the manufacturing units is balanced as a whole. The rated production capacity C of each manufacturing unit is adjusted to the rated production capacity C of each manufacturing unit. f (kT) is defined as the sum of its corresponding planned production capacity and its production capacity adjustment range, and its form is shown in the following formula: C f ((k+d)T)=C p ((k+d)T)+C m (kT) (5) In formula (5), it is assumed that the planned production capacity of each manufacturing unit is released to each unit at least dT in advance to prepare for production capacity adjustment; When a discrete manufacturing workshop is in steady state, the actual production capacity of each manufacturing unit is equal to its corresponding rated production capacity. However, due to the existence of internal disturbance factors in the workshop system, such as equipment failure in the manufacturing unit, the actual production capacity is lower than the rated production capacity. Φ is the internal disturbance probability transfer matrix of the discrete manufacturing workshop, and Φ is a random matrix that satisfies the following conditions: F ij ≥0; Among them, 1≤i≤N,1≤j≤N; Therefore, the actual production capacity of manufacturing unit i can be expressed as follows: C a (kT)=C f (kT)-Φ T C d (kT) (6) (6) In the formula, C d (kT) represents the loss of production capacity of the manufacturing unit caused by internal disturbance factors in the discrete manufacturing workshop; b) State equations and output equations of the manufacturing unit; Based on the definition of the manufacturing unit state space expression, the state equation of the manufacturing unit is obtained as follows: Similarly, the output equation of the manufacturing unit is: The actual production process of the manufacturing unit's order input, output, and the production capacity adjustment caused by disturbances in this process can fully express the motion state of this system. Select the order input quantity w of the manufacturing unit i (kT), order output w o (kT) and production capacity adjustment C m (kT) is the state variable of the manufacturing unit; the system input vector is the order input rate i(kT), the planned production capacity C p (kT), planned work-in-progress quantity wip p (kT) and internal and external disturbances C d (kT), w d (kT); the output vector is the quantifiable measurement of the system, that is, the data required for the inventory check of the work-in-progress in each cycle of the production process, that is, the order input quantity w of the manufacturing unit in each sampling cycle i (kT), order output w o (kT) and actual production capacity C a (kT) and actual work-in-progress quantity wip a (kT); c) Performance index prediction equation of production control system The input and output of each manufacturing unit are linked to the system status, and then the internal variables are linked to the external input and measurement output to obtain the output equation of the workshop system; The performance indicators of the production control system are the production capacity adjustment response rate and the stability of the work-in-process inventory level of the workshop manufacturing system. From the perspective of the time domain, the production capacity adjustment response rate is the input-output ratio of the workshop manufacturing system within a sampling time. At the same time, the workshop system state space is constructed with the work-in-process inventory level of each manufacturing unit as the state feedback variable, so there is no need to consider the stability of the work-in-process inventory. The input-output of the system can be obtained by simply counting the input and output of each manufacturing unit. Since there are N manufacturing units in the discrete manufacturing workshop, the total order input and output of the discrete manufacturing workshop can be expressed by equations (9) and (10) respectively: Among them, the matrix P0 is the order output probability matrix, which is a diagonal matrix and satisfies: d) Complete the construction of the production operation state space of discrete workshop manufacturing units.

2. The method for optimizing manufacturing unit production operations under order flow disturbance conditions according to claim 1, wherein: It also includes the optimization of production operation under disturbance factors, including the following steps: Ⅰ) Analysis of disturbance factors; In the state space analysis method, the characteristic equation is used to characterize the dynamic characteristics of the system. According to the state equation (Equation (7)), the characteristic equation of the system is obtained: det((1-z -1 )I+K c Q T T(IP T )(IP T Φ T ) -1 With -(d+1) )=0 (12) From formula (12), we can see that the dynamic characteristics of the discrete manufacturing workshop are affected by the order flow matrix P and the internal and external disturbance propagation matrices Q and Φ. At the same time, the dynamic characteristics of the system are also affected by the unit production capacity control gain K c and the change of the production capacity adjustment delay time d; In actual production, disturbances can alter order input rates and completion times. Furthermore, manufacturing units must adjust production capacity to maintain stable work-in-process inventory levels in response to these disturbances. These two factors alter the original system's order flow matrix, impacting overall system performance. II) Impact of internal disturbance factors on the order flow matrix: In actual production, disturbances such as equipment failures and personnel absences within a manufacturing unit can cause a decrease in system productivity, thereby affecting the system stability of the discrete manufacturing workshop. Unlike external disturbances such as rush orders, these disturbances originate from within the manufacturing system and are called internal disturbances. Due to the certain coupling relationship between manufacturing units, this type of disturbance not only affects the manufacturing unit itself but also has the potential to spread to other manufacturing units. As a result, the processing tasks originally planned for a manufacturing unit cannot be completed in a timely manner, causing changes in the order flow matrix. Considering multi-step or multi-order propagation processes is of little significance for the actual production process, and the analysis process is too cumbersome. For the propagation process of internal disturbances, assuming that the disturbance propagation is a first-order Markov process, the internal disturbance matrix can be defined as: (13) In formula, And 0≤Φ ij ≤1, When i≠j, the element Φ ij represents the probability of perturbation propagation from unit i to unit j, and Φ ij ≥0; due to disturbance factors, the order flow path changes, C mij It represents the adjustment of the production capacity of the manufacturing unit related to the adjusted order flow path; The order flow matrix under disturbance can be described as a series of row vectors p i And satisfy the relationship p i (n+1) =Φ (n) p i Therefore, at a certain sampling time nT, the order flow matrix of the discrete manufacturing workshop is P. When the disturbance occurs, at the sampling time (n+1)T, the order flow matrix can be expressed as follows: P ' =ΦP (14) Where P and P' represent the initial order flow matrix and the order flow matrix after the disturbance occurs, respectively; When internal disturbances occur in a discrete manufacturing workshop, the direct manifestation of decreased productivity is that the processing volume of orders received by each manufacturing unit cannot be completed on time. The processing volume of unfinished orders will be accumulated to the next working day and processed together with the new orders received by the current manufacturing unit. In other words, a portion of the processing orders received by the manufacturing unit on the current working day will be retained by the current unit for processing the next day. In the case of internal disturbances, when the above phenomenon is described by the order flow matrix, the values ​​of the non-zero elements of the new order flow matrix P′ change. The values ​​of the original order flow matrix P and the newly generated order flow matrix P′ after the internal disturbance are as follows: Symbol N Z The non-zero elements of the matrix P′ represent the orders that were not completed on time due to internal disturbances and enter the current manufacturing unit again the next day, resulting in a change in the ratio of the orders occupied by each unit; When the internal disturbance propagates within the system, the WIP of the manufacturing units associated with the order flow matrix in the system fluctuates. At this time, the manufacturing units adjust their production capacity to stabilize the WIP inventory. Assuming that within the sampling time kT≤t≤(k+1)T, the production capacity adjustment amplitude C of the manufacturing unit is m (kT) remains unchanged, the following method can be used to estimate the order flow matrix under disturbance conditions; The order flow matrix is ​​transformed from P to P′ and can be expressed as: First, add the column vector composed of the order output rate in the discrete manufacturing workshop to construct the augmented matrix in The augmented matrix Block by row, Row vector p i The transformation vector representing the destination of the processing order received by manufacturing unit i; For the row vector p i , remove the elements representing irrelevant manufacturing units, that is, except for p ii The 0 elements other than , can get the row vector (e.g. p1 = [0, 100 / 227, 127 / 227, 0, 0, 0], ); Use the above formula to calculate the row vector after the disturbance occurs Then we can get the conversion vector of the destination of the processing order received by manufacturing unit i under the disturbance state: After the internal disturbance occurs, the diagonal element p in the order flow matrix ii Remain unchanged; vector After adding 0 elements to the corresponding positions, the augmented matrix under the perturbation state is formed According to the decomposition method, the augmented matrix Decompose and then get P′ and P0′, which represent the order flow matrix and order output matrix under disturbance state respectively; III) The impact of external disturbance factors on the order flow matrix; During the operation of discrete manufacturing workshop systems, external disturbances mainly refer to disturbances caused by external emergency orders or upstream raw material supply shortages. When such disturbances occur, although the system does not fail, they will affect the overall operation of the workshop system, causing the system to either increase productivity or degrade operation. The external disturbance probability transfer matrix is ​​a diagonal matrix Q, that is, the external disturbance acts on each manufacturing unit in the discrete manufacturing workshop with a certain probability; When an external disturbance occurs, the overall performance of the discrete manufacturing workshop system shifts, that is, the system output shifts under the same order input, but it will not affect the coupling relationship between the manufacturing units within the system. It is assumed that the adjustment range of the production capacity of the manufacturing units is not constrained. Therefore, when a certain amplitude of the shock response is met, the external disturbance transfer probability matrix Q can be approximately replaced by a unit matrix of the same dimension.

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