Spare parts shared inventory management method based on horizontal transfer

Abstract

The present invention discloses a method for managing partially shared inventory of spare parts based on horizontal transfer, comprising the following steps: real-time monitoring of the degradation status of equipment, triggering preventive or post-event maintenance according to the equipment degradation status; after maintenance is triggered, if the local warehouse has sufficient spare parts inventory, directly performing the maintenance; if the inventory is insufficient, initiating a horizontal transfer request to other warehouses; using a Markov model to describe a multi-warehouse system; and using a decomposition approximation algorithm to solve the Markov model to derive key performance indicators of the multi-warehouse system. Compared with the prior art, the present invention optimizes critical thresholds based on the derived key indicators to achieve the lowest-cost inventory sharing strategy, thereby providing an efficient and scalable solution for maintenance management of modern industrial equipment.

Description

Technical Field

[0001] The present invention relates to the technical field of industrial equipment maintenance and spare parts inventory management, in particular to a spare parts partially shared inventory management method based on horizontal allocation. Background Art

[0002] In modern industrial systems, efficient equipment operation and maintenance are critical factors in ensuring production continuity and product quality. With the advancement of Industry 4.0, manufacturing productivity has significantly increased, but this has also placed higher demands on equipment health. To ensure stable equipment operation, traditional preventive and post-event maintenance strategies are no longer sufficient in complex and volatile industrial environments. Therefore, condition-based maintenance (CDM) has emerged as a more advanced and efficient maintenance strategy. The core of CDM lies in real-time monitoring of equipment operating status, analyzing equipment health data to predict potential failures and implementing repair measures before they occur. This approach not only effectively reduces equipment failures but also significantly reduces unnecessary maintenance activities. However, the effective implementation of CDM relies on the timely supply of spare parts. Especially when equipment fails or requires preventive maintenance, spare part availability directly determines the efficiency of maintenance activities. In traditional spare parts inventory management, spare parts are typically allocated between vertical levels, while horizontal transfers within the same level are often overlooked. Horizontal transfers allow spare parts to be dispatched from other warehouses on the same level in the event of a failure, thereby reducing equipment downtime caused by stock-outs. Although lateral transshipment has significant advantages in theory, in practical applications, how to effectively integrate condition-based maintenance and spare parts inventory management, especially how to achieve a balance between cost and efficiency in lateral transshipment strategies, remains an urgent problem to be solved. Summary of the Invention

[0003] The technical problem to be solved by the present invention is to propose a spare parts shared inventory management method based on horizontal allocation, which reduces maintenance costs and improves equipment availability and overall system operation efficiency.

[0004] In order to solve the above technical problems, the technical solution adopted by the present invention is:

[0005] A method for managing a partially shared inventory of spare parts based on horizontal transfers comprises the following steps:

[0006] S1 Equipment Status Monitoring and Maintenance Triggering: Real-time monitoring of equipment degradation status, triggering preventive or post-event maintenance based on equipment degradation status. Equipment status is divided into three levels: Status 0, Status 1, and Status 2. Status 0 indicates normal equipment, Status 1 indicates equipment degradation, and Status 2 indicates equipment failure.

[0007] S2 Dynamic Inventory Allocation: When maintenance is triggered, if the local warehouse has sufficient spare parts inventory, the maintenance is carried out directly. If the inventory is insufficient, a horizontal transfer request is initiated to other warehouses. If other warehouses have inventory, the subsequent maintenance spare parts demand is immediately met. When the inventory level of other warehouses exceeds a critical threshold, the spare parts demand generated by preventive maintenance is met.

[0008] S3 multidimensional Markov modeling: state space Describe a multi-warehouse system where , represents the spare parts inventory of warehouse P, where P is the number of warehouses; , represents the number of devices in state 1 in base i, , represents the number of devices in state 2 in base i, where I is the number of bases;

[0009] S4 uses a decomposition approximation algorithm to solve the Markov model: it decomposes the multi-warehouse system into independent subsystems and iteratively updates the overflow spare parts demand and steady-state probability distribution until convergence. Key performance indicators of the multi-warehouse system are derived based on the iteratively updated steady-state probability distribution.

[0010] S5 Strategy Optimization: Optimize critical thresholds based on key performance indicators of multi-warehouse systems to achieve the lowest-cost spare parts inventory sharing.

[0011] In step S1, the equipment status monitoring and maintenance triggering step includes:

[0012] When the equipment degrades from state 0 to state 1, preventive maintenance is triggered; when the equipment suddenly fails in state 0 or degrades from state 1 to state 2, post-event maintenance is triggered; preventive maintenance refers to maintenance before the equipment fails, that is, in the degraded state, aimed at preventing the occurrence of equipment failure; post-event maintenance refers to the corrective operation requirements performed after the equipment breaks down or fails, aimed at restoring the normal operating state of the equipment.

[0013] In step S1, the time of device degradation state transition follows an exponential distribution. The transition rates of the device in base i from state 0 to state 1, from state 0 to state 2, and from state 1 to state 2 are respectively 、 and .

[0014] In step S2, the dynamic allocation of inventory includes:

[0015] In the horizontal transfer request, for spare parts demand for preventive maintenance, the warehouse Only when inventory level Exceeding critical threshold Time Response Warehouse For spare parts needs for subsequent maintenance, the warehouse Only inventory levels are required Time Response Warehouse transfer request.

[0016] In step S4, the method of solving the Markov model using a decomposition approximation algorithm includes:

[0017] Step S41: Decompose the multi-warehouse system into multiple independent subsystems with one warehouse each, and each warehouse operates independently; each base i is modeled as a three-dimensional Markov process ;

[0018] Step S42: Given the overflow preventive maintenance spare parts demand rate and the post-maintenance spare parts demand rate, establish the state transition rate matrix of base i: , and obtain the steady-state probability distribution of base i;

[0019] Step S43: Iteratively update the overflowed preventive maintenance spare parts demand rate and post-maintenance spare parts demand rate according to the steady-state probability distribution of base i;

[0020] Step S44: Alternately update steps S42 and S43 to obtain a final steady-state probability distribution; calculate the key performance indicators of the multi-warehouse system based on the final steady-state probability distribution.

[0021] Step S42: Given the overflow preventive maintenance spare parts demand rate and the post-maintenance spare parts demand rate, establish the state transition rate matrix of base i: , and obtain the steady-state probability distribution of base i; including:

[0022] Step S421: Given the preventive maintenance overflow demand rate and the corrective maintenance overflow demand rate, establish the state transition rate matrix of base i: , state transition rate matrix Each element of is the transfer rate between states of base i;

[0023] Step S422: Construct a linear equation system about the state transition rate matrix: ;

[0024] Where, represents the steady-state probability distribution of base i, and Ω represents the set of all possible system state combinations;

[0025] Step S423: Solve the above-constructed linear equations to obtain the steady-state probability distribution of base i. .

[0026] In step S421, the state transition rate matrix The transfer rate in is determined according to the warehouse inventory level of base i, and the specific method is:

[0027] Base i models the horizontal transfer request from base j through a Poisson process, and They correspond to the preventive maintenance spare parts demand rate and post-maintenance spare parts demand rate overflowing from base j to base i respectively; the initial values ​​of the preventive maintenance spare parts demand rate and post-maintenance spare parts demand rate are set to 0;

[0028] The current inventory status of base i is divided into two categories: inventory and no inventory:

[0029] The warehouse of base i has inventory. There are two situations:

[0030] Case 1: In addition to the preventive and post-maintenance spare parts required by base i, there are also spare parts demands from other warehouses, which will consume the inventory at base i; the status of base i changes from Transfer to , the corresponding transfer rate is: ;

[0031] Case 2: A spare part is delivered, and the inventory of base i warehouse increases by 1; the status of base i changes from Transfer to , the corresponding transfer rate is: ;

[0032] The warehouse of base i has no inventory. There are six situations:

[0033] Case 1: The number of devices in state 2 in base i increases, and the state of base i changes from Transfer to ,in , , the corresponding transfer rate is: ;

[0034] Case 2: The number of devices in state 1 in base i increases, and the state of base i changes from Transfer to ,in , , the corresponding transfer rate is: ;

[0035] Case 3: The number of devices in state 1 in base i decreases, and the number of devices in state 2 increases. The state of base i changes from Transfer to ,in , , the corresponding transfer rate is: ;

[0036] Case 4: When a spare part arrives and no equipment needs repair, the spare part inventory increases by 1 and the state of base i changes from Transfer to ,in , the corresponding transfer rate is: ;

[0037] Case 5: When the spare part arrives and replaces the device in state 2, the number of devices in state 2 decreases by 1, and the state of base i changes from Transfer to , the corresponding transfer rate is: ;

[0038] Case 6: When the spare parts arrive and meet , when the number of equipment waiting for preventive maintenance decreases by 1, the state of base i changes from Transfer to , the corresponding transfer rate is: ;

[0039] In the above formula, 、 and They represent the transition rates of the equipment in base i from state 0 to state 1, state 0 to state 2, and state 1 to state 2, respectively. represents the total number of devices in base i, represents the number of devices in state 1 in base i, represents the number of devices in state 2 in base i, is the inventory level of warehouse p at base i, is the critical threshold of warehouse p, and They represent the probability that base j meets the spare parts demand for preventive maintenance and post-maintenance of base i, is the replenishment rate from the central warehouse to base i, is the baseline inventory level of base i; It is an indicator function, and the value in the brackets is 1 if it holds, and 0 if it does not hold; It is an indicator function, and the value in the brackets is 1 if it holds, and 0 if it does not hold;

[0040] The conversion rate does not belong to the two cases of inventory and the six cases of no inventory mentioned above. is 0.

[0041] Step S43: Iteratively updating the overflowed preventive maintenance spare parts demand rate and post-maintenance spare parts demand rate according to the steady-state probability distribution of base i, including:

[0042] Based on the steady-state probability distribution of base i obtained in step S42, the probability that base i meets the preventive maintenance spare parts demand of base j is determined. , the probability that base i meets the post-maintenance spare parts demand of base j , update the preventive maintenance spare parts demand rate overflowing from base j to base i and the subsequent maintenance spare parts demand rate overflowing from base j to base i :

[0043] ;

[0044] ;

[0045] ;

[0046] ;

[0047] Where, is the critical threshold; represents the total number of devices in base j; represents the number of devices in state 1 in base j, represents the number of devices in state 2 in base j; 、 and are the transition rates of the equipment in base j from state 0 to state 1, from state 0 to state 2, and from state 1 to state 2, respectively;

[0048] Step S44: Alternately update steps S42 and S43 to obtain the final steady-state probability distribution and calculate the key performance indicators of the multi-warehouse system, including:

[0049] Update steps S41 and S42 alternately until the steady-state probability changes by no more than 10 -4 , the calculation stops, and the final steady-state probability distribution of base i is obtained;

[0050] After obtaining the final steady-state probability distribution, determine the long-term average number of transitions from state 0 to state 1, state 0 to state 2, and state 1 to state 2 per time unit for the equipment in base i. 、 and : ;

[0051] ;

[0052] ;

[0053] Where, represents the total number of devices in base i;

[0054] According to the final steady-state probability distribution, the proportion of spare parts requests that base i can directly meet from its own inventory is: ;

[0055] The ratio of base j meeting the preventive maintenance spare parts demand of base i is: ;

[0056] The proportion of base j meeting the post-maintenance spare parts demand of base i is: ;

[0057] Where, is the proportion of spare parts requests that are directly satisfied by base i's own inventory, are the proportions of preventive maintenance spare parts demand and post-maintenance spare parts demand of base i satisfied by base j, respectively;

[0058] They represent the transition rates of the equipment in base i from state 0 to state 1, state 0 to state 2, and state 1 to state 2, respectively. 、 、 They represent the long-term average transition rates from state 0 to state 1, state 0 to state 2, and state 1 to state 2 per time unit for devices in base i, respectively. is the inventory level of warehouse p at base i;

[0059] Finally, calculate the key performance indicators of the multi-warehouse system:

[0060] The long-run average inventory level at base i is: ;

[0061] The average number of times that base j meets the preventive maintenance spare parts demand of base i is: ;

[0062] The average number of times that base j meets the post-maintenance spare parts demand of base i is: ;

[0063] The expected number of devices in faulty state in base i is: ;

[0064] Where, is the long-term average inventory level at base i, and are the average number of times that base j meets the preventive maintenance and post-maintenance spare parts needs of base i, is the expected number of devices in faulty state in base i.

[0065] In step S5, the method for optimizing the critical threshold value according to the key performance indicators of the multi-warehouse system is:

[0066] Replace critical thresholds by enumeration Calculate the key performance indicators of the multi-warehouse system, including the long-term average inventory level, the long-term average number of horizontal transfers, and the number of faulty equipment, and select the critical thresholds for the optimal performance indicators. , for the lowest cost spare parts inventory management.

[0067] The proposed method for managing partially shared spare parts inventories based on horizontal transfers aims to improve spare parts inventory management efficiency, reduce costs, and minimize downtime caused by equipment failures. Compared to conventional spare parts inventory management methods, this proposed method prioritizes maintenance activities based on the actual status of the equipment and, combined with a partial inventory sharing strategy, allows for horizontal transfers when necessary, thereby optimizing spare parts usage and allocation.

[0068] The beneficial effects of the present invention are as follows: by introducing a spare parts inventory management strategy based on real-time equipment status monitoring and on-demand maintenance, the equipment status can be monitored in real time and repairs can be carried out in a timely manner, thus avoiding long-term equipment downtime. Combined with the lateral transfer mechanism, even when the warehouse inventory is insufficient, the timely supply of spare parts can still be ensured, thereby minimizing the losses caused by equipment failure. The invention adopts the lateral transfer of spare parts between warehouses. Even if a warehouse is out of stock, the inventory of other warehouses can be used to meet its spare parts needs, effectively reducing the downtime of equipment waiting for replenishment due to insufficient inventory, and significantly improving the availability of equipment. By setting critical thresholds and lateral transfer strategies, the present invention can avoid the situation of too many or too few spare parts in a single warehouse while ensuring timely maintenance, achieve the optimal inventory configuration, reduce inventory backlogs and expiration losses, and thus save storage costs. The computational model based on approximate methods proposed in the present invention can effectively decompose the complex state space in a multi-warehouse system and reduce the computational burden. By reducing the number of iterations in the calculation process, the computational efficiency in large-scale equipment and multiple warehouse scenarios is significantly improved, solving the problem of high computational complexity of traditional methods.

[0069] The present invention is compared with the invention with publication number CN119477009A, entitled "Spare Parts Inventory Optimization Sharing Module in a New Energy Power Generation Equipment Health Management Platform Based on a Large Model." The comparison is as follows:

[0070] Core Technology: This invention utilizes condition-based maintenance (CBM) and lateral transfer (LT) to dynamically adjust inventory strategies through real-time monitoring of equipment status information. This approach emphasizes the application of equipment status information in inventory management to reduce inventory costs and equipment downtime. The latter approach leverages real-time monitoring and IoT technology to optimize spare parts inventory management based on regional characteristics (such as deserts and coastal areas). This module emphasizes the use of IoT technology to achieve real-time monitoring and rapid response to spare parts inventory.

[0071] Inventory Management Strategy: This invention utilizes a partial sharing strategy, controlling lateral transfers by setting critical thresholds. When warehouse inventory levels exceed the critical threshold, lateral transfers for preventive maintenance are permitted. When inventory levels exceed 0, lateral transfers for post-maintenance repairs are permitted, thus avoiding the increased costs associated with excessive transfers. The latter invention utilizes intelligent search and matching algorithms to rapidly locate spare parts in other areas when shortages occur, and coordinates logistics distribution. This strategy emphasizes real-time response and logistics optimization to ensure the rapid supply of spare parts.

[0072] Computational efficiency: This invention significantly improves computational efficiency through state-space decomposition and approximation algorithms, making it suitable for large-scale systems. By decomposing the multidimensional state space, this method reduces computational complexity and improves system scalability. While the latter invention does not explicitly address computational efficiency, its real-time monitoring and intelligent algorithms may require high computing resources. In particular, the application of IoT technologies and intelligent algorithms may require high computing power to support real-time data processing and analysis.

[0073] Application Scenarios: This invention is applicable to complex industrial systems with multiple warehouses and multiple devices, particularly those requiring horizontal transfers. This method is particularly well-suited for environments requiring collaborative inventory management across warehouses. The latter invention focuses on the health management of new energy power generation equipment, emphasizing the impact of regional characteristics (such as deserts and coastal areas) on inventory management. This module is particularly well-suited for specific scenarios such as distributed photovoltaic power generation sites, emphasizing the impact of regional environments on equipment operation and maintenance.

[0074] This paper proposes a partially shared inventory management method that combines condition-based maintenance (CBM) and lateral transfer (LT) of spare parts. This method, for the first time, integrates equipment status information into inventory management with LT. By dynamically adjusting LT strategies, it optimizes inventory management across multiple warehouses and multiple devices. It also employs multidimensional Markov chain modeling to accurately describe the dynamic changes in spare parts inventory levels and equipment status. State-space decomposition and approximation algorithms significantly improve computational efficiency, addressing the "curse of dimensionality" problem in large-scale systems. Furthermore, this method dynamically adjusts inventory strategies by monitoring equipment status in real time, reducing inventory costs and equipment downtime. A series of performance metrics (such as inventory levels, number of LTs, and number of equipment failures) are proposed for comprehensive evaluation of system operating efficiency and cost. Compared to the latter method, this method offers significant advantages in flexibility, real-time performance, and cost optimization, particularly in the collaborative management of multiple warehouses and complex systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0075] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following briefly introduces the drawings required for use in the description of the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0076] Figure 1 A schematic diagram of the overall flow of a method for managing a shared inventory of spare parts based on horizontal allocation under condition-based maintenance provided by an embodiment of the present invention.

[0077] Figure 2 A Markov chain state transition diagram of a method for managing a shared spare parts inventory based on horizontal allocation under condition-based maintenance provided by an embodiment of the present invention.

[0078] Figure 3 A schematic diagram of a partial process of an approximate calculation method for a method of managing a partially shared spare parts inventory based on horizontal allocation under condition-based maintenance provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0079] To make the above-mentioned objects, features, and advantages of the present invention more clearly understood, the following detailed description of the specific embodiments of the present invention is given in conjunction with the accompanying drawings. It is obvious that the described embodiments are only part of the embodiments of the present invention, not all of them. Based on the embodiments of the present invention, all other embodiments obtained by ordinary persons in this field without creative work should fall within the scope of protection of the present invention.

[0080] Example 1

[0081] Reference Figure 1-Figure 3 This embodiment provides a method for managing a shared inventory of spare parts based on horizontal allocation under condition-based maintenance, including:

[0082] S1 Equipment Status Monitoring and Maintenance Triggering: Real-time monitoring of equipment degradation status and triggering of preventive or post-event maintenance based on the equipment status transition rate;

[0083] S2 Dynamic inventory allocation: After maintenance is triggered, if the local warehouse has sufficient inventory, the maintenance will be carried out directly; if the inventory is insufficient, a horizontal transfer request will be initiated to other warehouses. If other warehouses have inventory, the subsequent maintenance spare parts demand can be immediately met. When the inventory level of other warehouses exceeds the critical threshold, the spare parts demand generated by preventive maintenance can be met.

[0084] S3 multidimensional Markov modeling: state space Describe a multi-warehouse system where represents the spare parts inventory of warehouse P, where P is the number of warehouses; , represents the number of devices in state 1 in base i, , represents the number of devices in state 2 in base i, where I is the number of bases;

[0085] S4 uses a decomposition approximation algorithm to solve the Markov model. To efficiently solve the Markov model, the multi-warehouse system is decomposed into independent subsystems. The overflow spare parts demand and steady-state probability distribution are iteratively updated until convergence. The key performance indicators of the multi-warehouse system are derived based on the steady-state probability distribution.

[0086] S5 Strategy Optimization: Optimize critical thresholds based on key performance indicators of multi-warehouse systems to achieve the lowest-cost spare parts inventory sharing.

[0087] The S1 equipment status monitoring and maintenance triggering step includes classifying the equipment status into three levels: status 0, status 1, and status 2. Status 0 indicates normal equipment, status 1 indicates equipment degradation, and status 2 indicates equipment failure. When the equipment degrades from status 0 to status 1, preventive maintenance is triggered.

[0088] When a device suddenly fails in state 0 or degrades from state 1 to state 2, post-event maintenance is triggered; preventive maintenance refers to maintenance before the device fails, that is, in the degraded state, to prevent the occurrence of equipment failure; post-event maintenance refers to the corrective operation requirements performed after the device fails or fails, to restore the normal operating state of the device.

[0089] The time of equipment degradation state transition follows an exponential distribution. The transition rates of equipment in base i from state 0 to state 1, from state 0 to state 2, and from state 1 to state 2 are respectively 、 and .

[0090] The S2 inventory dynamic allocation step includes: in the horizontal transfer request, for the spare parts demand for preventive maintenance, warehouse P only has the inventory level Exceeding critical threshold Time Response Warehouse For spare parts needs for subsequent maintenance, warehouse P only needs to have an inventory level of Time Response Warehouse transfer request.

[0091] The steps of the S4 decomposition approximation algorithm include:

[0092] Step S41: Decompose the multi-warehouse system into multiple independent subsystems with one warehouse each, and each warehouse operates independently; each base i is modeled as a three-dimensional Markov process ;

[0093] Step S42: Given the overflow preventive maintenance spare parts demand rate and the post-maintenance spare parts demand rate, establish the state transition rate matrix of base i: , and obtain the steady-state probability distribution of base i;

[0094] Step S43: Iteratively update the overflowed preventive maintenance spare parts demand rate and post-maintenance spare parts demand rate according to the steady-state probability distribution of base i;

[0095] Step S44: Alternately update steps S42 and S43 to obtain the final steady-state probability distribution and calculate the key performance indicators of the multi-warehouse system.

[0096] Step S42: Given the overflow preventive maintenance spare parts demand rate and the post-maintenance spare parts demand rate, establish the state transition rate matrix of base i: , and obtain the steady-state probability distribution of base i; including:

[0097] Step S421: Given the overflow preventive maintenance spare parts demand rate and the post-maintenance spare parts demand rate, establish the state transition rate matrix of base i: , state transition rate matrix Each element of is the transfer rate between states of base i;

[0098] Step S422: Construct a linear equation system about the state transition rate matrix: ;

[0099] Where, represents the steady-state probability distribution of base i, and Ω represents the set of all possible system state combinations;

[0100] Step S423: Solve the above-constructed linear equations to obtain the steady-state probability distribution of base i. .

[0101] In step S421, the state transition rate matrix The transfer rate in is determined according to the warehouse inventory level of base i, and the specific method is:

[0102] Base i models the horizontal transfer request from base j through a Poisson process, and They correspond to the preventive maintenance spare parts demand rate and post-maintenance spare parts demand rate overflowing from base j to base i, respectively. The initial values ​​of the preventive maintenance spare parts demand rate and post-maintenance spare parts demand rate are set to 0.

[0103] The current inventory status of base i is divided into two categories: inventory and no inventory:

[0104] The warehouse of base i has inventory. There are two situations:

[0105] Case 1: In addition to the spare parts demand for preventive and post-maintenance maintenance at base i, there are also spare parts demands from other warehouses, which will consume the inventory at base i; the status of base i changes from Transfer to , the corresponding transfer rate is: ;

[0106] Case 2: A spare part is delivered, and the inventory of base i warehouse increases by 1; the status of base i changes from Transfer to , the corresponding transfer rate is: ;

[0107] The warehouse of base i has no inventory. There are six situations:

[0108] Case 1: The number of devices in state 2 in base i increases, and the state of base i changes from Transfer to ,in , , the corresponding transfer rate is: ;

[0109] Case 2: The number of devices in state 1 in base i increases, and the state of base i changes from Transfer to ,in , , the corresponding transfer rate is: ;

[0110] Case 3: The number of devices in state 1 in base i decreases, and the number of devices in state 2 increases. The state of base i changes from Transfer to ,in , , the corresponding transfer rate is: ;

[0111] Case 4: When a spare part arrives and no equipment needs repair, the spare part inventory increases by 1 and the state of base i changes from Transfer to ,in , the corresponding transfer rate is: ;

[0112] Case 5: When the spare part arrives and replaces the device in state 2, the number of devices in state 2 decreases by 1, and the state of base i changes from Transfer to , the corresponding transfer rate is: ;

[0113] Case 6: When the spare parts arrive and meet When the number of equipment waiting for preventive maintenance decreases by 1, the state of base i changes from Transfer to , the corresponding transfer rate is: ;

[0114] In the above formula, They represent the transition rates of the equipment in base i from state 0 to state 1, state 0 to state 2, and state 1 to state 2, respectively. represents the total number of devices in base j, represents the number of devices in state 1 in base j, represents the number of devices in state 2 in base j, is the inventory level of warehouse p at base i, is the critical threshold of warehouse p, and They represent the probability that base j meets the spare parts demand for preventive maintenance and post-maintenance of base i, is the replenishment rate from the central warehouse to base i, is the baseline inventory level of base i; It is an indicator function, and the value in the brackets is 1 if it holds, and 0 if it does not hold; It is an indicator function, and the value in the brackets is 1 if it holds, and 0 if it does not hold;

[0115] If the product does not fall into the two cases of inventory and the six cases of no inventory mentioned above, the conversion rate is 0.

[0116] Step S43: Iteratively updating the overflowed preventive maintenance spare parts demand rate and post-maintenance spare parts demand rate according to the steady-state probability distribution of base i, including:

[0117] Based on the steady-state probability distribution of base i obtained in step S42, the probability that base i meets the preventive maintenance spare parts demand of base j is determined. , the probability that base i meets the post-maintenance spare parts demand of base j , update the preventive maintenance spare parts demand rate overflowing from base j to base i and the subsequent maintenance spare parts demand rate overflowing from base j to base i :

[0118] ;

[0119] ;

[0120] ;

[0121] ;

[0122] Where, is the critical threshold; represents the total number of devices in base j; represents the number of devices in state 1 in base j, represents the number of devices in state 2 in base j; 、 and are the transition rates of the equipment in base j from state 0 to state 1, from state 0 to state 2, and from state 1 to state 2, respectively;

[0123] Step S44: Alternately update steps S42 and S43 to obtain the final steady-state probability distribution and calculate the key performance indicators of the multi-warehouse system, including:

[0124] Update steps S41 and S42 alternately until the steady-state probability changes by no more than 10 -4 , the calculation stops, and the final steady-state probability distribution of base i is obtained;

[0125] After obtaining the final steady-state probability distribution, determine the long-term average number of transitions from state 0 to state 1, state 0 to state 2, and state 1 to state 2 per time unit for the equipment in base i. 、 and : ;

[0126] ;

[0127] ;

[0128] Where, represents the total number of devices in base i;

[0129] According to the final steady-state probability distribution, the proportion of spare parts requests that base i can directly meet from its own inventory is: ;

[0130] The ratio of base j meeting the preventive maintenance spare parts demand of base i is: ;

[0131] The proportion of base j meeting the post-maintenance spare parts demand of base i is: ;

[0132] Where, is the proportion of spare parts requests that are directly satisfied by base i's own inventory, are the proportions of preventive maintenance spare parts demand and post-maintenance spare parts demand of base i satisfied by base j, respectively;

[0133] They represent the transition rates of the equipment in base i from state 0 to state 1, state 0 to state 2, and state 1 to state 2, respectively. 、 、 They represent the long-term average transition rates from state 0 to state 1, state 0 to state 2, and state 1 to state 2 per time unit for devices in base i, respectively. is the inventory level of warehouse p at base i;

[0134] Finally, calculate the key performance indicators of the multi-warehouse system:

[0135] The long-run average inventory level at base i is: ;

[0136] The average number of times that base j meets the preventive maintenance spare parts demand of base i is: ;

[0137] The average number of times that base j meets the post-maintenance spare parts demand of base i is: ;

[0138] The expected number of devices in faulty state in base i is: ;

[0139] Where, is the long-term average inventory level at base i, and are the average number of times that base j meets the preventive maintenance and post-maintenance spare parts needs of base i, is the expected number of devices in faulty state in base i.

[0140] The S5 strategy optimization steps include: replacing critical thresholds by enumeration Calculate the key performance indicators of the multi-warehouse system, including the long-term average inventory level, the long-term average number of horizontal transfers, and the number of faulty equipment, and select the critical thresholds for the optimal performance indicators , for the lowest cost spare parts inventory management.

[0141] like Figure 2 As shown in the figure, a state transition model based on Markov chain is shown when base 1 has one device in state 1, base 2 has two devices in state 2, and both warehouses are out of stock. In the figure, the device enters state 1 from state 0. If the device is not repaired in time, it may further enter state 2. This model provides a basis for equipment maintenance decisions by describing the transfer rate of the device from one state to another. When the status of the equipment changes, the system will trigger the corresponding maintenance operation to prevent the equipment from continuing to degrade. At the same time, the figure also reflects the spare parts transfer mechanism when the warehouse inventory is insufficient. In the case of out-of-stock inventory, the system will allocate spare parts from other warehouses through horizontal allocation to meet the equipment's maintenance spare parts needs. This process controls the flow of spare parts through equipment status and existing inventory levels to ensure the timeliness of equipment maintenance. Two scenarios are selected for detailed description. When the machine of base 1 in state 0 is impacted, the system state (0,0,1,0,0,2) is The rate of transfer to state (0,0,1,0,1,2). When the replenishment of warehouse 1 arrives, the machine waiting for maintenance in state 1 is replaced first. The machine in state 1 returns to state 0 after maintenance. The system state changes from (0,0,1,0,0,0,2) at a rate of Transfer to state (0,0,0,0,0,2), is the replenishment rate from the central warehouse to base i. This model effectively combines equipment status changes with spare parts inventory management, helping to achieve dynamic and flexible inventory management and equipment maintenance management.

[0142] like Figure 3 As shown in Figure 2, the process of the approximate calculation method can be summarized into the following main steps:

[0143] Problem decomposition: Decompose the system's multidimensional state space into multiple independent subsystems (usually each warehouse). This reduces the complex overall problem to multiple simple subproblems that can be solved more efficiently.

[0144] Initialization state: Initialize the state for each subsystem (warehouse). This typically includes setting information such as the base inventory level, inter-state transfer rate, and replenishment rate for each warehouse. Initially, the overflow demand for spare parts requests is set to zero.

[0145] Calculate steady-state probability: For each independent warehouse, use the Markov chain model to calculate the steady-state probability distribution to describe the spare parts inventory changes and equipment status transitions within the warehouse.

[0146] Update overflow demand: Calculate the overflow demand of each warehouse based on the steady-state probability and calculate the probability that other warehouses can meet the overflow demand. These overflow demands are usually assumed to be a Poisson process.

[0147] Iterate until convergence: Update the demand rate of each warehouse under each state based on overflow demand, and then recalculate the steady-state probability distribution of each subsystem. This process is gradually optimized through multiple iterations until the change in steady-state probability is less than a set threshold, at which point the iteration stops.

[0148] Calculate performance indicators: Finally, calculate the system's key performance indicators, such as long-term expected inventory levels, average number of lateral transfers, and average number of failures, to evaluate the overall performance of the system.

[0149] Table 1 shows the comparison of the own spare parts satisfaction rate and the satisfaction rate of preventive and post-maintenance spare parts demand by horizontal allocation in the asymmetric case under the approximate method and the exact method.

[0150] Table 1 Results of asymmetric examples

[0151]

[0152] The results show that the deviation between the approximate method and the exact method is very small. Specifically, the maximum mean absolute error in the self-satisfaction rate is 0.0065, while the mean absolute errors in the proportion of preventive and post-maintenance spare parts demand satisfied by lateral transfers do not exceed 0.0165 and 0.0020, respectively. These findings demonstrate the accuracy of the approximate method.

[0153] Table 2 shows the average computation time of the exact and approximate methods for 5, 6, 7, 8 devices and 13, 14, 15 devices, respectively, under all symmetric instances.

[0154] Table 2 Computation time of all symmetric instances

[0155]

[0156] The results show that the average computation time for the exact and approximate methods is 2.88 seconds and 0.08 seconds, respectively. As can be seen from the table, the average computation time for both methods increases with the number of devices. The advantage of the approximate method becomes increasingly apparent as the number of devices increases.

[0157] Table 3 Exact and approximate methods of cost-related performance measurement

[0158]

[0159] As shown in Table 3, there are slight differences between the exact and approximate methods for cost-related performance indicators and total expected cost. The long-term average inventory level for the expected inventory level is the expected number of devices in a faulty state, and C is the cost. The approximate method shows high agreement with the exact method, with absolute deviations below 0.11 in all cases, indicating a slight but systematic overestimation. For the number of devices in a faulty state, the approximate method achieves high precision, with an absolute error within 0.0061, although exhibiting consistent underestimation. Furthermore, the approximation of the total cost shows excellent accuracy, with an absolute error never exceeding 1.8 in any case.

[0160] Table 4 Cost of partial shared inventory strategy with and without CBM

[0161]

[0162] Table 4 shows the cost of the partially shared inventory strategy with and without CBM under different parameter sets. is the baseline inventory level of base i, is the total number of devices in base i, is the replenishment rate from the central warehouse to base i, is the critical threshold of warehouse p at base i, 、 、 are the transition rates of equipment from state 0 to state 1, state 0 to state 2, and state 1 to state 2 for base i, respectively. The results clearly show that across all parameter sets, implementing CBM consistently results in lower total costs than not using CBM. Specifically, in all cases, the cost of using CBM is at least 15% lower than not using CBM, with the maximum reduction reaching approximately 27%. This highlights the significant cost savings that can be achieved when CBM is applied, especially when the number of equipment increases or the base inventory level decreases, resulting in a lower fill rate. Overall, these results indicate that the partial shared inventory strategy with CBM is more cost-effective than the traditional partial shared inventory strategy without CBM. The consistent cost reduction across all parameter sets emphasizes the effectiveness of CBM in optimizing maintenance and inventory management decisions.

[0163] It should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that the technical solutions of the present invention may be modified or replaced by equivalents without departing from the spirit and scope of the technical solutions of the present invention, which should all be covered by the scope of the claims of the present invention.

Claims

1. A method for managing spare parts shared inventory based on horizontal allocation, characterized in that: The following steps are involved: S1. Equipment status monitoring and maintenance triggering: Real-time monitoring of equipment degradation status, triggering preventive or post-event maintenance based on equipment degradation status. Equipment status is divided into three levels: Status 0, Status 1, and Status 2. Status 0 indicates normal equipment, Status 1 indicates equipment degradation, and Status 2 indicates equipment failure. S2. Dynamic inventory allocation: After a maintenance is triggered, if the local warehouse has sufficient spare parts inventory, the maintenance is carried out directly; If inventory is insufficient, a horizontal transfer request is initiated to other warehouses. If other warehouses have inventory, the demand for spare parts for subsequent repairs is immediately met. When the inventory level of other warehouses exceeds a critical threshold, the demand for spare parts generated by preventive maintenance is met. S3. Multidimensional Markov modeling: state space Describe a multi-warehouse system where , represents the spare parts inventory of warehouse P, where P is the number of warehouses; , represents the number of devices in state 1 in base i, , represents the number of devices in state 2 in base i, where I is the number of bases; S4. Solve the Markov model using a decomposition approximation algorithm: Decompose the multi-warehouse system into independent subsystems, iteratively update the overflow spare parts demand and steady-state probability distribution until convergence; derive the key performance indicators of the multi-warehouse system based on the iteratively updated steady-state probability distribution; S5. Strategy optimization: Based on the key performance indicators of the multi-warehouse system, optimize key thresholds to achieve the lowest cost spare parts inventory sharing; In step S4, the method of solving the Markov model using a decomposition approximation algorithm includes: Step S41: Decompose the multi-warehouse system into multiple independent subsystems with one warehouse each, and each warehouse operates independently; each base i is modeled as a three-dimensional Markov process ; Step S42: Given the overflow preventive maintenance spare parts demand rate and the post-maintenance spare parts demand rate, establish the state transition rate matrix of base i: , and obtain the steady-state probability distribution of base i; Step S43: Iteratively update the overflowed preventive maintenance spare parts demand rate and post-maintenance spare parts demand rate according to the steady-state probability distribution of base i; Step S44: Alternately update steps S42 and S43 to obtain a final steady-state probability distribution; calculate the key performance indicators of the multi-warehouse system based on the final steady-state probability distribution; Step S44: Alternately update steps S42 and S43 to obtain the final steady-state probability distribution and calculate the key performance indicators of the multi-warehouse system, including: Update steps S41 and S42 alternately until the steady-state probability changes by no more than 10 -4 , the calculation stops, and the final steady-state probability distribution of base i is obtained; After obtaining the final steady-state probability distribution, determine the long-term average number of transitions from state 0 to state 1, state 0 to state 2, and state 1 to state 2 per time unit for the equipment in base i. 、 and : ; ； ； Where, represents the total number of devices in base i; According to the final steady-state probability distribution, the proportion of spare parts requests that base i can directly meet from its own inventory is: ； The ratio of base j meeting the preventive maintenance spare parts demand of base i is: The proportion of base j meeting the post-maintenance spare parts demand of base i is: Where, is the proportion of spare parts requests that are directly satisfied by base i's own inventory, are the proportions of preventive maintenance spare parts demand and post-maintenance spare parts demand of base i satisfied by base j, respectively; They represent the transition rates of the equipment in base i from state 0 to state 1, state 0 to state 2, and state 1 to state 2, respectively. 、 、 They represent the long-term average transition rates from state 0 to state 1, state 0 to state 2, and state 1 to state 2 per time unit for devices in base i, respectively. is the inventory level of warehouse p at base i; Finally, calculate the key performance indicators of the multi-warehouse system: The long-run average inventory level at base i is: ; The average number of times that base j meets the preventive maintenance spare parts demand of base i is: ； The average number of times that base j meets the post-maintenance spare parts demand of base i is: ； The expected number of devices in faulty state in base i is: ; Where, is the long-term average inventory level at base i, and are the average number of times that base j meets the preventive maintenance and post-maintenance spare parts needs of base i, is the expected number of devices in faulty state in base i.

2. The method for managing a partially shared spare parts inventory based on horizontal allocation according to claim 1, characterized in that: In step S1, the equipment status monitoring and maintenance triggering step includes: When the equipment degrades from state 0 to state 1, preventive maintenance is triggered; when the equipment suddenly fails in state 0 or degrades from state 1 to state 2, post-event maintenance is triggered; preventive maintenance refers to maintenance before the equipment fails, that is, in the degraded state, aimed at preventing the occurrence of equipment failure; post-event maintenance refers to the corrective operation requirements performed after the equipment breaks down or fails, aimed at restoring the normal operating state of the equipment.

3. The method for managing spare parts shared inventory based on horizontal allocation according to claim 2, characterized in that: In step S1, the time for the device degradation state transition follows an exponential distribution, based on The transfer rates of the device from state 0 to state 1, from state 0 to state 2, and from state 1 to state 2 are 、 and .

4. The method for managing spare parts shared inventory based on horizontal allocation according to claim 1, characterized in that: In step S2, the dynamic allocation of inventory includes: In a horizontal transfer request, for spare parts required for preventive maintenance, warehouse P can only Exceeding critical threshold Time Response Warehouse For spare parts needs for subsequent maintenance, warehouse P only needs to have an inventory level of Time Response Warehouse transfer request.

5. The method for managing spare parts shared inventory based on horizontal allocation according to claim 1, characterized in that: Step S42: Given the overflow preventive maintenance spare parts demand rate and the post-maintenance spare parts demand rate, establish the state transition rate matrix of base i: , and obtain the steady-state probability distribution of base i; include: Step S421: Given the overflow preventive maintenance spare parts demand rate and the post-maintenance spare parts demand rate, establish the state transition rate matrix of base i: , state transition rate matrix Each element of is the transfer rate between states of base i; Step S422: Construct a linear equation system about the state transition rate matrix: ; Where, represents the steady-state probability distribution of base i, and Ω represents the set of all possible system state combinations; Step S423: Solve the above-constructed linear equations to obtain the steady-state probability distribution of base i. .

6. The method for managing spare parts shared inventory based on horizontal allocation according to claim 5, characterized in that: In step S421, the state transition rate matrix The transfer rate in is determined according to the warehouse inventory level of base i, and the specific method is: Base i models the horizontal transfer request from base j through a Poisson process, and They correspond to the preventive maintenance spare parts demand rate and post-maintenance spare parts demand rate overflowing from base j to base i respectively; the initial values ​​of the preventive maintenance spare parts demand rate and post-maintenance spare parts demand rate are set to 0; The current inventory status of base i is divided into two categories: inventory and no inventory: The warehouse of base i has inventory. There are two situations: Case 1: In addition to the spare parts demand for preventive and post-maintenance maintenance required by base i, there are also spare parts demands from other warehouses, which will consume the inventory at base i; the status of base i changes from Transfer to , the corresponding transfer rate is: ; Case 2: A spare part is delivered, and the inventory of base i warehouse increases by 1; the status of base i changes from Transfer to , the corresponding transfer rate is: ; The warehouse of base i has no inventory. There are six situations: Case 1: The number of devices in state 2 in base i increases, and the state of base i changes from Transfer to ,in , , the corresponding transfer rate is: ; Case 2: The number of devices in state 1 in base i increases, and the state of base i changes from Transfer to ,in , , the corresponding transfer rate is: ; Case 3: The number of devices in state 1 in base i decreases, and the number of devices in state 2 increases. The state of base i changes from Transfer to ,in , , the corresponding transfer rate is: ; Case 4: When a spare part arrives and no equipment needs repair, the spare part inventory increases by 1 and the state of base i changes from Transfer to ,in , the corresponding transfer rate is: ; Case 5: When the spare part arrives and replaces the device in state 2, the number of devices in state 2 decreases by 1, and the state of base i changes from Transfer to , the corresponding transfer rate is: ; Case 6: When the spare parts arrive and meet When the number of devices waiting for preventive maintenance decreases by 1, the state of base i changes from Transfer to , the corresponding transfer rate is: ; In the above formula, They represent the transition rates of the equipment in base i from state 0 to state 1, state 0 to state 2, and state 1 to state 2, respectively. represents the total number of devices in base j, represents the number of devices in state 1 in base j, represents the number of devices in state 2 in base j, is the inventory level of warehouse p at base i, is the critical threshold of warehouse p, and are the probabilities of spare parts demand for preventive maintenance and post-maintenance at base j, is the replenishment rate from the central warehouse to base i, is the baseline inventory level of base i; It is an indicator function, and the value in the brackets is 1 if it holds, and 0 if it does not hold; It is an indicator function, and the value in the brackets is 1 if it holds, and 0 if it does not hold; The conversion rate does not belong to the two cases of inventory and the six cases of no inventory mentioned above. is 0.

7. The method for managing spare parts shared inventory based on horizontal allocation according to claim 6, characterized in that: Step S43: Iteratively updating the overflowed preventive maintenance spare parts demand rate and post-maintenance spare parts demand rate according to the steady-state probability distribution of base i, including: Based on the steady-state probability distribution of base i obtained in step S42, the probability that base i meets the preventive maintenance spare parts demand of base j is determined. , the probability that base i meets the post-maintenance spare parts demand of base j , update the preventive maintenance spare parts demand rate overflowing from base j to base i and the subsequent maintenance spare parts demand rate overflowing from base j to base i : ； ； ； ； Where, is the critical threshold; represents the total number of devices in base j; represents the number of devices in state 1 in base j, represents the number of devices in state 2 in base j; 、 and are the transition rates of the devices in base j from state 0 to state 1, from state 0 to state 2, and from state 1 to state 2, respectively.

8. The method for managing spare parts shared inventory based on horizontal allocation according to claim 1, characterized in that: In step S5, the method for optimizing the critical threshold value according to the key performance indicators of the multi-warehouse system is: Replace critical thresholds by enumeration Calculate the key performance indicators of the multi-warehouse system, including the long-term average inventory level, the long-term average number of horizontal transfers, and the number of faulty equipment, and select the critical thresholds for the optimal performance indicators , for the lowest cost spare parts inventory management.

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