Multi-place inventory network optimization method and device, equipment and storage medium

Abstract

The embodiment of the invention provides a multi-place inventory network optimization method and device, equipment and a storage medium, and relates to the technical field of display. The method comprises the following steps: acquiring demand correlation and historical data, constructing a first support set or a second support set based on the correlation, constructing at least one first uncertainty set or second uncertainty set corresponding to the support set according to the historical data, acquiring scheduling parameters of an inventory network, and constructing an inventory optimization model based on the scheduling parameters and the uncertainty sets, and reconstructing the constraint condition set as a linear constraint condition set by using the first uncertainty set, or reconstructing the constraint condition set as a second-order cone programming condition set by using the second uncertainty set, and solving the target function based on the linear constraint condition set or the second-order cone programming condition set to obtain inventory optimization data. For demand uncertainty, different support sets are constructed according to correlation driving, so that an inventory optimization model is not limited to a single constraint of a moment condition any more, the correlation is subjected to hierarchical processing, and the inventory cost is optimized.

Description

Methods, apparatus, equipment and storage media for optimizing multi-location inventory networks Technical Field

[0001] This application relates to the field of inventory management technology, and in particular to methods, apparatus, equipment and storage media for optimizing multi-location inventory networks. Background Technology

[0002] For power plants and generating units distributed across different regions, during power plant maintenance, situations often arise where multiple power plants within the same region share inventory, and spare parts are borrowed from across cities (e.g., between location A and location B). Therefore, it is necessary to address the inventory management problem of spare parts in multiple locations under uncertain demand. To address this issue, it is necessary to optimize inventory strategies to cope with the randomness of power plant maintenance needs, thereby reducing maintenance waiting time, lowering inventory costs, and ensuring the efficiency of power plant operation and maintenance.

[0003] Related technologies construct moment-based partial blob optimization models for optimization and perform inventory scheduling based on the solution results. However, these models only focus on satisfying partial moment conditions and cannot fully utilize the correlation information of demand between multiple locations. Especially when the demand from different locations is correlated, this model is difficult to accurately reflect this relationship, leading to a decline in the decision quality of multi-location inventory networks, easily generating suboptimal inventory strategies, and failing to meet the needs of efficient management of power plant spare parts. Summary of the Invention

[0004] The main objective of this application is to propose a method, apparatus, device, and storage medium for optimizing multi-location inventory networks, thereby improving the accuracy of multi-location inventory network scheduling.

[0005] To achieve the above objectives, a first aspect of this application proposes a multi-location inventory network optimization method, comprising: acquiring the correlation of cross-location item scheduling demand and historical data of the inventory network; constructing a support set based on the correlation, the support set including a first support set or a second support set; if the correlation is small, constructing the first support set based on quantiles of the historical data; if the correlation is large, constructing the second support set based on Mahalanobis distance of the historical data; constructing at least one uncertain set corresponding to the support set based on the historical data, the uncertain set including a first uncertain set or a second uncertain set; acquiring scheduling parameters of the inventory network; constructing an inventory optimization model based on the scheduling parameters and the uncertain set, the inventory optimization model including an objective function and a set of constraints; reconstructing the set of constraints into a linear constraint set using the first uncertain set, or reconstructing the set of constraints into a second-order cone programming condition set using the second uncertain set; solving the objective function based on the linear constraint set or the second-order cone programming condition set to obtain inventory optimization data.

[0006] In some embodiments, the inventory network includes a first number of demand locations, and the construction of the first support set based on quantiles of the historical data includes: obtaining a coverage parameter; calculating a quantile index based on the coverage parameter; performing sampling with replacement on the data corresponding to each demand location in the multiple historical data sets to obtain a sample set for each demand location; calculating a first quantile and a second quantile for each sample set based on the quantile index; repeating the sampling multiple times to obtain a first set corresponding to the first quantile and a second set corresponding to the second quantile; estimating the upper confidence boundary corresponding to the first set to obtain the upper quantile, and estimating the lower confidence boundary corresponding to the second set to obtain the lower quantile; for each demand location, setting the corresponding random demand to be located between the corresponding upper quantile and the lower quantile, and constructing the first support set.

[0007] In some embodiments, the inventory network includes a first number of demand locations. Constructing the second support set based on Mahalanobis distance from the historical data includes: calculating the corresponding data mean and data covariance for each demand location in the historical data, and performing sampling with replacement to obtain a sample set for each demand location; calculating the squared Mahalanobis distance of each data point in the sample set based on the data mean and the data covariance; obtaining a corresponding reference threshold based on the quantiles of the squared Mahalanobis distance; obtaining a threshold set corresponding to the reference threshold; selecting a target threshold from the threshold set; and for each demand location, setting the squared Mahalanobis distance between the corresponding random demand and the data mean to be less than or equal to the target threshold, thereby constructing the second support set.

[0008] In some embodiments, constructing at least one uncertain set corresponding to the support set based on the historical data includes: obtaining an empirical distribution corresponding to the historical data; calculating a metric distance between the probability distribution and the empirical distribution for a probability distribution corresponding to random demand in the support set; obtaining a fuzzy set radius; defining the probability distributions whose metric distance is less than the fuzzy set radius as the fuzzy set; when the support set is the first support set, the fuzzy set is the first fuzzy set, otherwise it is the second fuzzy set; and discretizing the fuzzy set into at least one uncertain set.

[0009] In some embodiments, discretizing the fuzzy set into at least one uncertain set includes: when the fuzzy set is the first fuzzy set, acquiring a first number of historical data points located in the first support set; for each historical data point, setting the distance between the random demand at each demand location and the corresponding center point to be less than or equal to the radius of the fuzzy set, to obtain a first uncertain set; when the fuzzy set is the first fuzzy set, acquiring a first number of historical data points located in the first support set; for each historical data point, acquiring the dimension interval of the random demand corresponding to each demand location, and combining the dimension intervals to obtain a first uncertain set; when the fuzzy set is the second fuzzy set, for the mean vector and mean squared error vector corresponding to all the historical data points, acquiring a second number of historical data points located in the second support set; for each historical data point, calculating the difference vector between the random demand at each demand location and the mean vector; setting the product of the transpose of the difference vector, the inverse of the mean squared error vector, and the difference vector to be less than or equal to the target threshold; and setting the random demand to be located within the interval formed by the value of the historical data and the radius of the fuzzy set, to obtain a second uncertain set.

[0010] In some embodiments, the inventory network includes a first number of demand locations and a second number of distribution centers. The scheduling parameters include at least the unit product procurement cost and the current inventory level of the distribution centers. Constructing a target constraint function based on the scheduling parameters and the uncertainty set includes: calculating the inventory procurement cost according to the unit product procurement cost and the current inventory level of the distribution centers; obtaining auxiliary decision variables for each uncertainty set; calculating the mean of all the auxiliary decision variables; minimizing the sum of the inventory procurement cost and the mean of the variables; and obtaining the target constraint function.

[0011] In some embodiments, the scheduling parameters include at least: unit product inventory holding cost, linear decision variables, unit product stockout loss cost, actual demand quantity at the demand location, combined cost coefficient from the distribution center to the demand location, and upper limit of transportation capacity from the distribution center to the demand location. Constructing a set of constraints based on the scheduling parameters and the uncertainty set includes: obtaining the linear decision variable corresponding to each uncertainty set, the linear decision variable being obtained by summing initial variables and decision variables, the decision variable being obtained from the decision coefficient of each demand location and the corresponding random demand; calculating the inventory holding cost based on the unit product inventory holding cost and the current inventory level of the distribution center; calculating the distribution cost based on the combined cost coefficient and the initial variable; calculating the stockout penalty cost based on the stockout loss cost, the combined cost coefficient, the decision variable, and the random demand; setting the inventory holding cost, distribution cost, and stockout penalty cost to be less than or equal to the auxiliary decision variable to obtain the cost. Upper bound constraints are defined as follows: First, the initial variables are summed to obtain a first reference value. Based on the first reference value, the decision variables, and the random demand, the total delivery volume is obtained. The total delivery volume is set to be less than or equal to the current inventory level of the corresponding distribution center, thus obtaining an inventory balance constraint. Demand response parameters are obtained based on the initial variables, the decision variables, and the random demand. The demand response parameters are set to be less than or equal to zero, thus obtaining a demand response constraint. Actual delivery volume is obtained based on the decision variables and the random demand. The actual delivery volume is set to be less than or equal to the initial variables, thus obtaining a non-negative delivery volume constraint. Transportation capacity parameters are obtained based on the initial variables, the decision variables, and the random demand. The transportation capacity parameters are set to be less than or equal to the corresponding upper limit of transportation capacity, thus obtaining a transportation capacity constraint. Finally, the set of constraints is obtained based on the upper bound cost constraint, the inventory balance constraint, the demand response constraint, the non-negative delivery volume constraint, and the transportation capacity constraint.

[0012] To achieve the above objectives, a second aspect of this application proposes a multi-location inventory network optimization device, comprising: a data acquisition module: configured to acquire the correlation of cross-location item scheduling demand and historical data of the inventory network, construct a support set based on the correlation, the support set including a first support set or a second support set; if the correlation is small, constructing the first support set based on quantiles of the historical data, and if the correlation is large, constructing the second support set based on Mahalanobis distance of the historical data; an uncertainty set construction module: configured to construct at least one uncertainty set corresponding to the support set based on the historical data, the uncertainty set including a first uncertainty set or a second uncertainty set; a model construction module: configured to acquire scheduling parameters of the inventory network, construct an inventory optimization model based on the scheduling parameters and the uncertainty set, the inventory optimization model including an objective function and a set of constraints; and a model solving module: configured to reconstruct the set of constraints into a linear constraint set using the first uncertainty set, or reconstruct the set of constraints into a second-order cone programming condition set using the second uncertainty set, and solve the objective function based on the linear constraint set or the second-order cone programming condition set to obtain inventory optimization data.

[0013] To achieve the above objectives, a third aspect of this application provides an electronic device, which includes a memory and a processor. The memory stores a computer program, and the processor executes the computer program to implement the method described in the first aspect.

[0014] To achieve the above objectives, a fourth aspect of the present application provides a storage medium that stores a computer program, which, when executed by a processor, implements the method described in the first aspect.

[0015] The multi-location inventory network optimization method, apparatus, device, and storage medium proposed in this application acquire the correlation of cross-location item scheduling demand and historical data of the inventory network. Based on the correlation, a support set is constructed, including a first support set or a second support set. If the correlation is low, the first support set is constructed based on quantiles of the historical data; if the correlation is high, the second support set is constructed based on Mahalanobis distance. At least one uncertain set corresponding to the support set is constructed based on the historical data, including a first uncertain set or a second uncertain set. Scheduling parameters of the inventory network are acquired. An inventory optimization model is constructed based on the scheduling parameters and the uncertain set. The inventory optimization model includes an objective function and a set of constraints. The constraint set is reconstructed into a linear constraint set using the first uncertain set, or into a second-order cone programming constraint set using the second uncertain set. The objective function is solved based on the linear constraint set or the second-order cone programming constraint set to obtain the inventory optimization data. This application's embodiments construct different support sets based on correlation: when the correlation is low, the first support set is constructed based on quantiles, focusing on the independent distribution characteristics of single-location demand; when the correlation is high, the second support set is constructed using Mahalanobis distance to explore the correlation between demands from different locations. This approach expands the inventory optimization model beyond simple moment constraints, enabling hierarchical processing of correlations. Furthermore, it generates corresponding uncertainty sets for different support sets to optimize the modeling and solution process: the constraint sets are reconstructed into linear constraints suitable for low-correlation scenarios or second-order cone programming constraints suitable for high-correlation scenarios, ensuring that the objective function fully considers the impact of demand correlations on inventory scheduling during the solution process. Therefore, this embodiment significantly improves the scheduling accuracy of multi-location inventory networks, especially suitable for scenarios like power plant spare parts where high scheduling accuracy is required. It better addresses demand uncertainties arising during power plant maintenance, reduces maintenance waiting time, and optimizes inventory costs. Attached Figure Description

[0016] Figure 1 is a flowchart of the multi-location inventory network optimization method provided in an embodiment of this application.

[0017] Figure 2 is a flowchart of constructing a first support set based on quantiles of historical data according to an embodiment of this application.

[0018] Figure 3 is a schematic diagram of the construction process code of the first support set provided in the embodiment of this application.

[0019] Figure 4 is a flowchart of constructing a second support set based on Mahalanobis distance from historical data, provided in an embodiment of this application.

[0020] Figure 5 is a schematic diagram of the construction process code of the second support set provided in the embodiments of this application.

[0021] Figure 6 is a flowchart of constructing at least one uncertain set corresponding to the support set based on historical data, provided in an embodiment of this application.

[0022] Figure 7 is a flowchart of discretizing a fuzzy set into at least one uncertain set according to an embodiment of this application.

[0023] Figure 8 is a flowchart of constructing a set of constraints based on scheduling parameters and uncertainties according to an embodiment of this application.

[0024] Figure 9 is a structural block diagram of a multi-location inventory network optimization device provided in another embodiment of this application.

[0025] Figure 10 is a schematic diagram of the hardware structure of the electronic device provided in an embodiment of this application. Detailed Implementation

[0026] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0027] It should be noted that although functional modules are divided in the device schematic diagram and the logical order is shown in the flowchart, in some cases, the steps shown or described may be performed in a different order than the module division in the device or the order in the flowchart.

[0028] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs. The terminology used herein is for the purpose of describing embodiments of this application only and is not intended to limit this application.

[0029] For power plants and generating units distributed across different regions, during power plant maintenance, situations often arise where multiple power plants within the same region share inventory, and spare parts are borrowed from across cities (e.g., between location A and location B). Therefore, it is necessary to address the inventory management problem of spare parts in multiple locations under uncertain demand. To address this issue, it is necessary to optimize inventory strategies to cope with the randomness of power plant maintenance needs, thereby reducing maintenance waiting time, lowering inventory costs, and ensuring the efficiency of power plant operation and maintenance.

[0030] These models can be viewed as a newsboy problem involving multiple items and locations. Related technologies construct moment-based partial bloc optimization models for optimization and solve them, then use the results for inventory scheduling. However, while demand may be correlated, these models only focus on satisfying some moment conditions and cannot fully utilize the correlation information between multiple locations. Especially when demand from different locations is correlated, these models struggle to accurately reflect this relationship, leading to a decline in the decision-making quality of multi-location inventory networks. This can easily result in suboptimal inventory strategies and fail to meet the needs of efficient power plant spare parts management.

[0031] This is mainly due to the following key shortcomings in solving moment-based bibliometric optimization models in related technologies. First, there's the reliance on a predefined, conservative set of uncertainty. This requires decision-makers to know the support set of the demand distribution beforehand, but in practical applications, such as power plant scheduling, the support set is often unknown. Therefore, when historical data is insufficient to represent a rapidly changing market, the predefined conservative support set may not accurately reflect the true demand range, leading to an overly conservative or overly optimistic model, affecting decision quality. Second, there's the use of a single linear decision rule to approximate the remedial decision in the second stage. While computationally feasible, this method also leads to suboptimal solutions compared to fully adaptive formulas. Especially in data-driven settings, single linear decision rules often fail to guarantee asymptotic optimality. As the sample size increases, the gap between the approximate solution of the single linear decision rule and the true optimal solution may still be significant, limiting the method's effectiveness in big data environments. Third, the sample averaging approximation method performs poorly with limited data. When the number of available data points is limited, the sample averaging approximation method often produces overly optimistic cost estimates, resulting in unreliable out-of-sample performance. Therefore, when historical data is limited, the sample average approximation method usually exhibits poor out-of-sample performance, leading to suboptimal decisions.

[0032] Based on this, this application provides a method, apparatus, device, and storage medium for optimizing a multi-location inventory network. Different support sets are constructed based on correlation: when the correlation is low, a first support set is constructed based on quantiles, focusing on the independent distribution characteristics of demand at a single location; when the correlation is high, a second support set is constructed using Mahalanobis distance to explore the correlation between demands at different locations. This allows the inventory optimization model to move beyond single moment constraints and perform hierarchical processing of correlation. Furthermore, corresponding uncertainty sets are generated for different support sets to optimize the modeling and solution process: the constraint sets are reconstructed into linear constraints suitable for low-correlation scenarios or second-order cone programming constraints suitable for high-correlation scenarios, ensuring that the objective function fully considers the impact of demand correlation on inventory scheduling during the solution process. Therefore, this embodiment can significantly improve the scheduling accuracy of multi-location inventory networks, especially suitable for scenarios like power plant spare parts where high scheduling accuracy is required, better addressing demand uncertainty during power plant maintenance, reducing maintenance waiting time, and optimizing inventory costs.

[0033] This application provides a method, apparatus, device, and storage medium for optimizing multi-location inventory networks, which are specifically described through the following embodiments. First, the method for optimizing multi-location inventory networks in this application is described.

[0034] The multi-location inventory network optimization method provided in this application relates to the field of inventory management technology. This method can be applied to a terminal, a server, or a computer program running on either the terminal or the server. For example, the computer program can be a native program or software module in an operating system; it can be a native application (APP), i.e., a program that needs to be installed in the operating system to run, such as a client supporting multi-location inventory network optimization, i.e., a program that only needs to be downloaded to a browser environment to run; or it can be a small program that can be embedded in any APP. In short, the above-mentioned computer program can be any form of application, module, or plugin. The terminal communicates with the server via a network. This multi-location inventory network optimization method can be executed by the terminal or the server, or by the terminal and the server working together.

[0035] In some embodiments, the terminal can be a smartphone, tablet, laptop, desktop computer, or smartwatch, etc. The server can be a standalone server, or a cloud server providing basic cloud computing services such as cloud services, cloud databases, cloud computing, cloud functions, cloud storage, network services, cloud communication, middleware services, domain name services, security services, content delivery networks (CDNs), and big data and artificial intelligence platforms; it can also be a service node in a blockchain system, where the service nodes form a peer-to-peer (P2P) network. The P2P protocol is an application layer protocol running on top of the Transmission Control Protocol (TCP). The terminal and server can connect via Bluetooth, Universal Serial Bus (USB), or a network, etc., and this embodiment does not impose any limitations.

[0036] This application can be used in a wide variety of general-purpose or special-purpose computer system environments or configurations. Examples include: personal computers, server computers, handheld or portable devices, tablet devices, multiprocessor systems, microprocessor-based systems, set-top boxes, programmable consumer electronics, network PCs, minicomputers, mainframe computers, and distributed computing environments including any of the above systems or devices. This application can be described in the general context of computer-executable instructions executed by a computer, such as program modules. Generally, program modules include routines, programs, objects, components, data structures, etc., that perform specific tasks or implement specific abstract data types. This application can also be practiced in distributed computing environments where tasks are performed by remote processing devices connected via a communication network. In distributed computing environments, program modules can reside in local and remote computer storage media, including storage devices.

[0037] It should be noted that in all specific embodiments of this application, when processing data related to user identity or characteristics, such as user information, user behavior data, user historical data, and user location information, user permission or consent is obtained first. Furthermore, the collection, use, and processing of this data comply with relevant laws, regulations, and standards. In addition, when embodiments of this application require access to sensitive personal information of users, separate permission or consent from the user is obtained through pop-ups or redirection to confirmation pages. Only after obtaining the user's separate permission or consent is the necessary user-related data required for the proper functioning of these embodiments acquired.

[0038] The following describes a multi-location inventory network optimization method in the embodiments of this application.

[0039] Figure 1 is an optional flowchart of a multi-location inventory network optimization method provided in an embodiment of this application. The method in Figure 1 may include, but is not limited to, steps 110 to 140. It is also understood that this embodiment does not specifically limit the order of steps 110 to 140 in Figure 1, and the order of steps can be adjusted or some steps can be reduced or added according to actual needs.

[0040] Step 110: Obtain the correlation of cross-location item dispatching demand and historical data of the inventory network. Construct a support set based on the correlation. If the correlation is small, construct a first support set based on quantiles of the historical data. If the correlation is large, construct a second support set based on Mahalanobis distance of the historical data.

[0041] In one embodiment, in a power plant maintenance scenario, the power plant can be a thermal, hydro, or photovoltaic power plant, along with its subordinate generating units. Power plants are typically distributed across geographically vast areas with significant environmental differences; for example, location A might be a plains thermal power plant, while location B might be a mountainous hydropower plant. Their operation and maintenance directly impacts the stability of the power supply. In actual maintenance scenarios, there are frequent instances of shared inventory among multiple power plants or cross-city spare parts borrowing within a region. Coupled with the highly random nature of power plant maintenance needs, multi-location spare parts inventory management becomes a crucial link in ensuring operational efficiency. It is essential to avoid prolonged maintenance waiting times due to spare parts shortages while also preventing excessive inventory costs caused by excessive stockpiling of spare parts. Therefore, multi-location inventory network optimization is necessary to address this scenario.

[0042] In one embodiment, the correlation of cross-location dispatching demand is used to measure the degree of correlation between dispatching demands for the same type of spare parts between power plants (or between power plants and the central warehouse) in different locations. Therefore, it can be measured from several dimensions, including power plant type, spare part type, and maintenance scenario. Taking the spare part type "sealing gasket" as an example, for instance, the monthly dispatching demand sequences for "sealing gaskets" from power plants in locations A and B over the past 12 months can be extracted, and the correlation can be calculated using the Pearson correlation coefficient. If the absolute value of the correlation coefficient is less than 0.3, it indicates that the dispatching demand for this spare part in both locations is affected by independent factors, and is judged as "low correlation". If the absolute value of the correlation coefficient is greater than 0.6, it indicates that the demand in both locations is driven by common factors, and is judged as "high correlation". It is understood that the correlation of cross-location dispatching demand can be obtained based on the actual situation.

[0043] In one embodiment, when performing inventory allocation and demand forecasting, it is necessary to construct a corresponding support set. This support set is used to filter samples highly correlated with the current scheduling demand from massive historical data, providing data support for subsequent scheduling decisions. In this embodiment, different construction strategies are used to construct different support sets for different levels of correlation. The support set includes a first support set or a second support set. If the correlation is low, the first support set is constructed based on quantiles of the historical data; if the correlation is high, the second support set is constructed based on Mahalanobis distance of the historical data. Here, "high" or "low" correlation is determined by comparing the correlation with a preset value, which can be 0.5.

[0044] In one embodiment, the inventory network includes a first number of demand locations and a second number of distribution centers. In a power plant maintenance scenario, the demand location is the power plant, and the distribution centers are different spare parts warehouses. Referring to Figure 2, which is a flowchart of constructing a first support set based on quantiles of historical data according to an embodiment of this application, the process specifically includes the following steps: Step 210: Obtain coverage parameters and calculate quantile indexes based on the coverage parameters.

[0045] In one embodiment, a pre-set coverage parameter, confidence level parameter, and bootstrap sample size are first obtained. The coverage parameter... It can be set to =0.05, making 1- =95%, used to indicate that the support set should contain at least 95% probability quality. Confidence level parameter. It can be set to =0.10, making 1- =90%, indicating that the confidence level of the support set estimate is 90%. The number of bootstrap samples B can be 1000, representing the total number of samples that enhance the robustness of the support set estimate through bootstrap.

[0046] In one embodiment, for the k-th demand location, two quantile indices are calculated based on the coverage parameter: an upper quantile index and a lower quantile index. Wherein: the upper quantile index is... The lower quantile index is ,and It is understandable that there are two corresponding indexes for each demand location.

[0047] Step 220: Perform sampling with replacement on the data corresponding to each demand location in multiple historical data sets to obtain a sample set for each demand location. Calculate the first quantile and the second quantile for each sample set based on the quantile index. Repeat the sampling multiple times to obtain the first set corresponding to the first quantile and the second set corresponding to the second quantile. Estimate the upper confidence boundary corresponding to the first set to obtain the upper quantile, and estimate the lower confidence boundary corresponding to the second set to obtain the lower quantile.

[0048] In one embodiment, the number of historical data points is N, where N is an integer greater than 1. Specifically, the historical data can be the actual demand of each demand location at a historical moment, so each historical data point is a K-dimensional vector containing the actual demand of each demand location at the same historical moment. Taking the nth historical data point as an example... For example, it represents the actual demand corresponding to each demand location at the nth historical moment.

[0049] This allows us to obtain the corresponding data for each demand location in each historical dataset, forming a data sequence of length N. Sampling with replacement is then performed from this data sequence; that is, a data point is randomly selected from the dataset, recorded, and then replaced before selecting the next one. This means the same data point may be selected multiple times. Here, from b=1 to b=B, a sample set for each b is formed for that demand location.

[0050] Next, for the b-th sample set of the i-th demand location, the first and second quantiles are calculated based on the quantile index. Assuming N is 1000 and the quantile index is 0.5, the upper quantile index is 0.268 and the lower quantile index is 0.732. Then, the sample set is sorted, and the total number of data points N in the sample set is multiplied by the quantile index, resulting in 268 and 732. Based on the sorting result, the 268th data point can be selected as the second quantile. The 732nd data point was selected as the first quantile. .

[0051] This process is represented as:

[0052]

[0053] in, Let b be the sample set of the i-th demand location. This indicates the calculation of quantiles.

[0054] Then, for each b, the above process of calculating quantiles is repeated multiple times to obtain the first set consisting of B first quantiles. And the second set corresponding to the B second quantiles Then, estimate the upper confidence boundary corresponding to the first set to obtain the upper quantile corresponding to the i-th demand location. And estimate the lower confidence boundary corresponding to the second set to obtain the lower quantile corresponding to the i-th demand location. , is represented as:

[0055]

[0056] in, Used for verification, that is, according to Select the corresponding upper or lower quantile from the first or second set. For example, when When k=0.1 and k=2, the corresponding quantiles are selected using the 2.5% and 97.5% positions.

[0057] Step 230: For each demand location, set the corresponding random demand to be located between the corresponding upper quantile and lower quantile, and construct the first support set.

[0058] In one embodiment, the quantile intervals corresponding to each demand location are obtained according to the above process, denoted as [ For the i-th demand location, assume its random demand is... Set the corresponding random requirements Located between the corresponding upper and lower quantiles, and then with the same settings applied to each required location, the first support set is constructed. , is represented as:

[0059] In one embodiment, referring to Figure 3, which is a schematic code diagram of the construction process of the first support set provided in this application embodiment, it can be seen that the algorithm takes historical data, coverage parameters, confidence level parameters, and the number of bootstrap samples as input, and then outputs the first support set after calculation. .

[0060] Next, the process of constructing the second support set corresponding to the historical data with high correlation is described. Referring to Figure 4, which is a flowchart of constructing the second support set based on Mahalanobis distance for historical data according to an embodiment of this application, the process includes the following steps: Step 410: Calculate the corresponding data mean and data covariance for each demand location in the historical data, and perform sampling with replacement to obtain a sample set for each demand location. Calculate the squared Mahalanobis distance of each data point in the sample set based on the data mean and data covariance. Obtain the corresponding reference threshold based on the quantiles of the squared Mahalanobis distance, and obtain the threshold set corresponding to the reference threshold. Select the target threshold corresponding to each demand location from the threshold set.

[0061] In one embodiment, following the same calculation process as described above, sample sets containing different amounts of data are obtained from b=1 to b=B. The mean of all data in a sample set is calculated, and the covariance is calculated. Therefore, a set of data means can be calculated for each sample set. and data covariance Taking the b-th sample set as an example, it can be represented as:

[0062] Next, for each sample set, calculate the squared Mahalanobis distance for each data point in the sample set based on the data mean and data covariance. , is represented as:

[0063] in, This represents the data in the b-th sample set.

[0064] Then, the corresponding reference threshold is obtained based on the quantiles of the squared Mahalanobis distance. Here As an index of quantiles, it is used to retain the top quantiles in the sample set. The upper bound of the proportional distance, the reference threshold Represented as:

[0065] Then, for b=1 to b=B, we obtain B reference thresholds to form a threshold set. Next, select the target threshold from the threshold set. , is represented as:

[0066] Among them, with As an index for quantiles.

[0067] Step 420: For each demand location, set the squared Mahalanobis distance of the corresponding random demand to be less than or equal to the target threshold, and construct the second support set.

[0068] In one embodiment, the random demand corresponding to each demand location is represented as: Therefore, the squared Mahalanobis distance of the random requirement is set to be less than or equal to the target threshold, and the second support set is constructed. , is represented as:

[0069] in, , This represents the mean vector and covariance vector corresponding to all historical data.

[0070] In one embodiment, referring to Figure 5, which is a schematic code diagram of the construction process of the second support set provided in this application embodiment, it can be seen that the algorithm takes historical data, coverage parameters, confidence level parameters, and the number of bootstrap samples as input, and then outputs the second support set after calculation. .

[0071] Step 120: Construct at least one uncertain set corresponding to the support set based on historical data.

[0072] In one embodiment, a fuzzy set can be constructed before building the uncertain set. The fuzzy set acts as a bridge connecting the support set and the optimization model, defining a set containing all possible demand probability distributions. This set is centered on the empirical distribution of historical data and allows for some deviation. In other words, the actual demand distribution is unknown, but some historical data exists, so it can be assumed that the actual distribution should be "similar" to, but not exactly the same as, the distribution shown by historical data, allowing for some uncertainty. The fuzzy set is this set of "possible" probability distributions. The inventory optimization model then optimizes for the worst-case scenario within this set, resulting in a strategy that is robust to all possible distributions.

[0073] In one embodiment, referring to FIG6, FIG6 is a flowchart of constructing at least one uncertain set corresponding to a support set based on historical data according to an embodiment of the present application, specifically including the following steps: Step 610: Obtain the empirical distribution corresponding to the historical data, and calculate the metric distance between the probability distribution and the empirical distribution for the probability distribution corresponding to the random demand in the support set.

[0074] In one embodiment, the empirical distribution corresponding to the historical data is obtained based on all historical data. , is represented as:

[0075] in, This indicates a focus on historical data. The Dirac measure assumes that future demand will be exactly equal to... This value is rather extreme, so probability assignment is necessary. The empirical distribution is used to assign the same probability 1 / N to each historical data point. Therefore, we can consider historical data as the starting point of knowledge, which is the best guess of the true distribution, and we can construct a corresponding fuzzy set centered on it.

[0076] Then, for the probability distribution P corresponding to the support for centralized random demand, calculate the probability distribution P and the empirical distribution. Measuring distance between , is represented as:

[0077] The distance metric is the ∞-type Wasserstein distance. yes and The joint distribution of the two numbers has marginal distributions P and P, respectively. , The calculations are from P and respectively. Norm distance between two random variables, such as Euclidean distance. Represents the support set for random demand. Indicates in joint distribution The upper bound of the "worst-case scenario" is the largest value after excluding the extreme case where the measure is 0. For example, in a power plant maintenance scenario, the distance metric is used to represent "transforming the true distribution P of random demand into an empirical distribution". The minimum total transportation cost required under the worst-case joint distribution.

[0078] Step 620: Obtain the radius of the fuzzy set, and define the probability distribution of the metric distance being less than the radius of the fuzzy set to form the fuzzy set.

[0079] In one embodiment, the fuzzy set radius This parameter is set according to actual needs and is used to control the conservatism of the subsequent inventory optimization model. The larger the parameter, the more distributions are included in the fuzzy set; the smaller the parameter, the closer the fuzzy set is to the empirical distribution. For example, the fuzzy set radius can be set to... ,in, >0 is a constant. It can be 10.

[0080] Therefore, the fuzzy set is represented as: Since the support set includes either the first support set or the second support set, the fuzzy set is the first fuzzy set when the support set is the first support set, and the fuzzy set is the second fuzzy set otherwise.

[0081] Step 630: Discretize the fuzzy set into at least one uncertain set.

[0082] In one embodiment, after obtaining the fuzzy set, it is also necessary to generate an uncertain set based on the fuzzy set. Correspondingly, the uncertain set includes a first uncertain set or a second uncertain set. Since the fuzzy set is a continuous and abstract set, it is discretized, and a finite set of concrete sets is used to cover the main uncertainty region described by the fuzzy set.

[0083] In one embodiment, referring to FIG7, FIG7 is a flowchart of discretizing a fuzzy set into at least one uncertain set provided by an embodiment of the present application, specifically including the following steps: Step 710: When the fuzzy set is a first fuzzy set, obtain a first number of historical data located in the first support set. For each historical data, obtain the dimension interval of the random demand corresponding to each demand location, and combine the dimension intervals to obtain the first uncertain set.

[0084] In one embodiment, the uncertain set is a subset of the fuzzy set, which defines all possible probability distributions, while the uncertain set defines the specific range of possible probability distributions and is a definite set. In the power plant maintenance scenario, the role of the uncertain set is to quantify the fluctuation range of random demand at demand locations. Since spare parts demand is affected by random factors such as unit aging and maintenance frequency, it cannot be described by fixed values. Therefore, it is necessary to construct an uncertain set containing the "possible demand range" by combining fuzzy sets with historical data features, providing a risk-controllable decision boundary for subsequent inventory strategy formulation.

[0085] Therefore, when the fuzzy set is the first fuzzy set, the historical data is traversed to determine whether it falls within the first support set. After the traversal is complete, the number of first support sets located in the first support set is obtained. For each historical data point, a corresponding first uncertainty set is constructed.

[0086] Using the nth historical data For example, first, obtain the upper and lower bounds of the interval for each demand location in the historical data. The upper bound of the interval... It is obtained by adding the fuzzy set radius to the historical data value of the demand location, while the upper bound of the interval is... It is obtained by subtracting the radius of the fuzzy set from the historical data value of the location of the demand. Then, the center point of the interval is obtained based on the upper and lower bounds of the interval. and interval radius , is represented as:

[0087] Therefore, taking the j-th demand location as an example, its random demand The corresponding dimensional interval is represented as: Combining the dimensional intervals, we obtain the first uncertain set.

[0088] Taking the nth historical data as an example, the first uncertainty set is represented as:

[0089] It's understandable that although fuzzy sets don't appear to directly participate in the calculation of uncertain sets, the transformation from fuzzy sets to uncertain sets represents a shift from probability to reality. If a probability distribution P belongs to a fuzzy set, then the random demand vector drawn from this probability distribution P... It is highly likely that the probability will fall into some uncertain set. Therefore, fuzzy sets essentially limit the range of a large probability distribution.

[0090] In one embodiment, conditional empirical distribution can also be used. Instead of the empirical distribution in the aforementioned fuzzy set, uniform weights are assigned to the historical data within the support set. Ignoring out-of-set data, it is represented as:

[0091] Step 720: When the fuzzy set is the second fuzzy set, calculate the corresponding data mean and data standard deviation for each demand location in all historical data, obtain the second number of historical data in the second support set, and for each historical data, calculate the difference vector between the random demand and the data mean for each demand location, set the transpose of the difference vector, the inverse vector of the data standard deviation, and the product of the difference vector to be less than or equal to the target threshold, and set the random demand to be located within the interval formed by the value of the historical data and the radius of the fuzzy set, to obtain the second uncertain set.

[0092] In one embodiment, when the fuzzy set is the second fuzzy set, historical data is traversed to determine whether it falls within the second support set. After the traversal is completed, the number of second support sets located in the second support set is obtained. For each historical data point, a corresponding second uncertainty set is constructed.

[0093] Next, obtain the mean vector when generating the second support set. and mean squared error vector For historical data located in the second support set, calculate the difference vector between the random demand and the data mean for each demand location. Set the transpose of the difference vector The inverse vector of the mean square error of the data Sum and difference vectors The product of is less than or equal to the target threshold. Meanwhile, the random demand is set to be located within the interval formed by the value in the historical data and the radius of the fuzzy set, thus forming a second uncertain set.

[0094] Taking the nth historical data as an example, the second uncertainty set is represented as:

[0095] Step 130: Obtain the scheduling parameters of the inventory network, and construct an inventory optimization model based on the scheduling parameters and the uncertainty set.

[0096] In one embodiment, the inventory optimization model includes an objective function and a set of constraints. First, the objective constraint function is constructed using scheduling parameters and uncertainties. Specifically, this involves: calculating the inventory procurement cost based on the unit product procurement cost and the current inventory level at the distribution center; obtaining auxiliary decision variables for each uncertainty set; calculating the mean of all auxiliary decision variables; and minimizing the sum of the inventory procurement cost and the variable mean to obtain the objective constraint function.

[0097] The scheduling parameters include at least the unit product procurement cost c, the current inventory level y at the distribution center, the unit product inventory holding cost h, the linear decision variable x, and the unit product stockout loss cost. d) of the actual demand quantity at the demand location, and the combined cost coefficient from the distribution center to the demand location. and the maximum transportation capacity from the distribution center to the demand location. wait.

[0098] Specifically, the collection of distribution centers is represented as: The set of demand locations is represented as: The unit product delivery cost from distribution center i to demand location j is expressed as: The maximum inventory capacity of distribution center i is: The maximum transportation capacity from distribution center i to demand location j: The current inventory of distribution center i is: The actual quantity delivered from distribution center i to demand location j is a linear decision variable, expressed as: The random demand at location j is represented as: The actual implementation requirement at location j is a known actual value, expressed as: Among them, the combined cost coefficient from the distribution center to the demand location is used. Based on unit product delivery cost, unit product inventory holding cost (h), and unit product stockout loss cost. Therefore, the combined cost coefficient from distribution center i to demand location j is calculated as follows: .

[0099] Therefore, the inventory procurement cost calculated based on the unit product procurement cost and the current inventory level at the distribution center is expressed as follows: Obtain the auxiliary decision variables for each uncertain set. Calculate the mean of all auxiliary decision variables. The objective constraint function obtained by minimizing the sum of inventory procurement costs and the mean of variables is expressed as:

[0100] Among them, auxiliary decision variables This is used to represent the upper bound of the worst-case cost that may occur when future stochastic demand falls into the nth uncertain set. Here, the auxiliary decision variable is an abstract concept without specific values, but its specific value will be obtained after solving the model.

[0101] Next, referring to Figure 8, which is a flowchart of constructing a set of constraints based on scheduling parameters and uncertainties according to an embodiment of this application, specifically including the following steps: Step 810: Obtain the linear decision variables corresponding to each uncertainty set.

[0102] In one embodiment, taking the first uncertainty set as an example, the linear decision variables from the i-th distribution center to the j-th demand location are... For example, it can be further expressed as:

[0103] in, This indicates that when the observed random demand is At that time, and the demand belongs to the nth uncertain set. At that time, the actual delivery volume from the i-th distribution center to the j-th demand location is a function that needs to be optimized.

[0104] Since linear decision variables are obtained by summing initial and decision variables, and the decision variables are derived from the decision coefficients and corresponding stochastic demands for each demand location, therefore... This represents the base delivery volume when the random demand at all demand locations is 0, and is used as the initial variable. This represents the actual delivery volume from the i-th distribution center to the j-th demand location when future random demand falls into the n-th uncertain set. Random needs of a customer The response coefficient. The uncertain set here can be viewed as a delivery scenario.

[0105] For example, This represents the additional quantity that should be shipped from distribution center 1 to demand location 1 when the demand falls within the first uncertain set, and the random demand at demand location 1 increases by 1 unit. This indicates the additional quantity that should be delivered from distribution center 2 to demand location 1 when the random demand at demand location 1 increases by 2 units. Meanwhile, This indicates the impact of demand fluctuations at demand location 2 on the volume of goods delivered from distribution center 1 to demand location 1. Constraints: That is, changes in demand at demand location 2 should not affect the quantity of goods delivered to demand location 1.

[0106] Step 820: Calculate the inventory holding cost based on the unit product inventory holding cost and the current inventory level of the distribution center; calculate the distribution cost based on the combination cost coefficient and initial variables; calculate the stockout penalty cost based on the stockout loss cost, combination cost coefficient, decision variables, and stochastic demand; set the inventory holding cost, distribution cost, and stockout penalty cost to be less than or equal to the auxiliary decision variables to obtain the upper bound constraint condition of the cost.

[0107] In one embodiment, the inventory holding cost, calculated based on the unit product inventory holding cost and the current inventory level at the distribution center, is expressed as follows: The delivery cost calculated based on the combination cost coefficient and initial variables is expressed as follows: The stockout penalty cost, calculated based on stockout loss cost, combination cost coefficient, decision variables, and stochastic demand, is expressed as follows: Therefore, by setting inventory holding costs, distribution costs, and stockout penalty costs to be less than or equal to the auxiliary decision variables, the resulting upper bound constraint on costs is expressed as follows:

[0108] in, This indicates that the solution result is valid for each first uncertainty set. All possible random requirements must be met.

[0109] Step 830: Accumulate the initial variables to obtain the first reference value. Based on the first reference value, decision variables, and random demand, obtain the total delivery volume. Set the total delivery volume to be less than or equal to the current inventory of the corresponding distribution center to obtain the inventory balance constraint.

[0110] In one embodiment, the first reference value obtained by summing the initial variables is represented as: The total delivery volume obtained based on the first reference value, decision variables, and stochastic demand is expressed as follows: Therefore, by setting the total delivery volume to be less than or equal to the current inventory level of the corresponding distribution center, the resulting inventory balance constraint is expressed as:

[0111] Step 840: Obtain the demand response parameters based on the initial variables, decision variables, and stochastic demand. Set the demand response parameters to be less than or equal to zero to obtain the demand response constraints.

[0112] In one embodiment, the demand response parameters obtained based on the initial variables, decision variables, and stochastic demand are expressed as follows: By setting the demand response parameter to be less than or equal to zero, the resulting demand response constraints are expressed as follows:

[0113] Step 850: Obtain the actual delivery volume based on the decision variables and random demand, set the actual delivery volume to be less than or equal to the initial variable, and obtain the non-negative constraint condition for the delivery volume.

[0114] In one embodiment, the actual delivery volume obtained based on decision variables and stochastic demand is expressed as: Setting the actual delivery volume to be less than or equal to the initial variable, the resulting non-negative constraint condition for the delivery volume is expressed as follows:

[0115] Step 860: Obtain the transportation capacity parameters based on the initial variables, decision variables, and stochastic demand. Set the transportation capacity parameters to be less than or equal to the corresponding transportation capacity upper limit to obtain the transportation capacity constraints.

[0116] In one embodiment, the transport capacity parameter obtained based on the initial variables, decision variables, and stochastic demand is expressed as follows: By setting the transport capacity parameter to be less than or equal to the corresponding transport capacity upper limit, the resulting transport capacity constraint is expressed as follows:

[0117]

[0118] Step 870: Obtain the set of constraints based on the upper bound constraint of cost, the inventory balance constraint, the demand response constraint, the non-negativity constraint of delivery volume, and the transportation capacity constraint.

[0119] In one embodiment, taking the first uncertainty set as an example, the set of constraints is represented as: st

[0120]

[0121]

[0122]

[0123]

[0124] At this point, the inventory optimization model has been obtained based on the objective constraint function and the set of constraint conditions. The next step is to proceed with the model solution process.

[0125] Step 140: Reconstruct the constraint set into a linear constraint set using the first uncertainty set, or reconstruct the constraint set into a second-order cone programming condition set using the second uncertainty set. Solve the objective function based on the linear constraint set or the second-order cone programming condition set to obtain the inventory optimization data.

[0126] In one embodiment, since the set of constraints must hold for all points in the uncertainty set, this is a semi-infinite constraint, which is difficult to solve. Therefore, for the maximum value problem of each linear function under the box constraint, a corresponding Lagrange dual problem is constructed, and the expression of the corresponding dual problem is used to equivalently replace the corresponding constraint.

[0127] Since there are two types of uncertain sets, the solution is obtained by transforming each type of uncertain set into a dual problem. First, the constraint set is reconstructed into a linear constraint set using the first uncertain set. Then, the five constraints—cost upper bound constraint, inventory balance constraint, demand response constraint, delivery volume non-negativity constraint, and transportation capacity constraint—are transformed using Lagrange duality to obtain the linear constraint set.

[0128] The dual of the cost upper bound constraint in the linear constraint set is transformed into:

[0129] The dual of the inventory balance constraint in the set of linear constraints is transformed into:

[0130] The dual of the demand response constraints in the set of linear constraints is transformed into:

[0131] The dual of the non-negativity constraints on delivery volume in the set of linear constraints is transformed into:

[0132] The dual of the transport capacity constraints in the linear constraint set is transformed into:

[0133] in, , , , , , Variables with subscripts and superscripts are all related dual variables, used to perform dual transformations on the constraint set based on the first support set, and must satisfy the following constraints:

[0134] In one embodiment, the second uncertainty set can also be used to reconstruct the constraint set into a second-order cone programming condition set. Since the second uncertainty set is essentially an ellipsoidal constraint, its dualization will generate a second-order cone constraint, thus yielding a second-order cone programming condition set. For example, in power plant spare parts management, if the demand for turbine spare parts for two power plants is highly correlated, the optimal spare parts allocation can be calculated using second-order cone programming constraints. This avoids the problem of one power plant having a surplus of spare parts while the other suffers a shortage due to ignoring this correlation, thus generating a suboptimal strategy.

[0135] First, for the second uncertainty set, the corresponding set of constraints is represented as: st

[0136]

[0137]

[0138]

[0139]

[0140] Therefore, after dual transformation, the dual of the cost upper bound constraint in the second-order cone programming condition set is transformed into:

[0141]

[0142] According to duality theory, the maximization problem can be transformed into a constrained minimization problem. This process naturally introduces a constraint, namely the dual variable. The absolute value must have an upper bound. The upper bound is given a specific value by the actual dual process. These are auxiliary variables introduced during the duality process.

[0143] The dual of the inventory balance constraint in the second-order cone programming condition set is transformed into:

[0144]

[0145] The dual of the demand response constraints in the second-order cone programming condition set is transformed into:

[0146]

[0147] The dual of the non-negativity constraint on delivery quantity in the second-order cone programming condition set is transformed into:

[0148]

[0149] The dual of the transport capacity constraints in the second-order cone programming condition set is transformed into:

[0150]

[0151] in, , , , , Variables with subscripts and superscripts are all related dual variables, used for dual transformation of the constraint set based on the second support set, and must satisfy the following constraints:

[0152] In one embodiment, the above process yields a set of linear constraints or a set of second-order cone programming conditions. Then, the objective function is solved based on these sets to obtain inventory optimization data. Specifically, the inventory optimization data can be obtained through the solution process. This parameter determines the required inventory level for each demand location. Based on the current inventory level, the replenishment quantity can be determined, and the actual delivery volume from each distribution center to each demand location can be determined, generating inventory and fulfillment strategy recommendations.

[0153] Therefore, this embodiment does not require a predefined conservative support set. Instead, it directly constructs a support set estimate from historical data and dynamically estimates the demand support set using quantile and Mahalanobis distance methods. This approach is suitable for multi-location inventory management scenarios where demand distribution is unknown and demand is assumed to be independent, such as procurement and distribution, and spare parts allocation to independent target areas. This dynamic fuzzy set construction method enables this embodiment to provide reliable decision support even when historical data is limited or demand distribution is unknown. Furthermore, by utilizing Mahalanobis distance support estimation to capture cross-location correlations, it overcomes the limitation of plural bar optimization models in related technologies, which cannot effectively handle correlations. This allows for a more accurate reflection of demand dependencies in multi-location inventory networks, especially when there is a strong correlation between customer locations, significantly improving decision quality.

[0154] Furthermore, employing multilinear decision rules not only maintains computational tractability but also ensures that the solution almost inevitably converges to the optimal solution of the random newsboy network problem as the sample size increases. Compared to single-linear decision rule methods in related technologies, multilinear decision rules provide a more flexible decision space, significantly improving solution quality while maintaining computational efficiency. During the solution process, the complex NP-multiplexed bar optimization problem is reformulated into a computable linear programming or second-order conical programming model using convex analysis techniques. In particular, the model based on Mahalanobis distance support set estimation can be reformulated as a second-order conical programming problem, allowing for efficient solution with acceptable computation time. This reformulation significantly reduces computational complexity, making this embodiment applicable to practically scaled problems while maintaining theoretical optimality. When an appropriate radius parameter is selected… When the sample size N approaches infinity, the data-based solution is close to the optimal solution in a probabilistic sense. When the sample size N approaches infinity, the data-based solution almost inevitably converges to the optimal solution of the random newsboy network problem.

[0155] In one embodiment, under various testing scenarios, especially with limited sample sizes, the multi-location inventory network optimization method of this application reduces the average cost by 15-20% compared to algorithms of related technologies. It maintains good out-of-sample performance even with limited historical data, and its optimization gap and prediction error are significantly lower than those of algorithms of related technologies. Furthermore, it provides different optimization schemes for cost-oriented and service-oriented scenarios to meet diverse business needs, combining limited sample guarantees and asymptotic optimality, while ensuring computational feasibility through tractable restatements. These advantages make the embodiments of this application applicable to multi-location inventory network optimization scenarios, particularly suitable for practical applications with strong correlations and limited historical data.

[0156] In one embodiment, the multi-location inventory network optimization method of this application is described using spare parts inventory scheduling in a power plant maintenance scenario as an example.

[0157] Suppose a power group owns two thermal power plants, Power Plant A and Power Plant B, located in City A (plains area) and City B (mountainous area), respectively. Both power plants are equipped with the same type of steam turbines, generators, and other core equipment. During routine operation and maintenance, it is necessary to dispatch spare parts such as steam turbine gaskets and generator bearings across cities.

[0158] First, obtain scheduling correlations and historical data. For example, collect monthly scheduling demand data for turbine gaskets from power plants A and B over the past two years to calculate the correlation. Assuming that the correlation between the two is 0.25 (indicating low demand correlation) by calculating the Pearson correlation coefficient, and 0.7 (indicating high demand correlation) for generator bearings, the historical data of the inventory network includes spare parts inventory levels, scheduling volumes, and demand fulfillment rates for the two power plants and the group's regional central warehouse.

[0159] Next, due to the low correlation of turbine gaskets, a first support set was constructed based on quantiles of the historical data. Specifically, quantiles were calculated for the historical scheduling data of turbine gaskets for power plants A and B, and the middle 50% (25th to 75th quantiles) of historical data were selected. For power plant A, the 25th quantile was 4 sets, and the 75th quantile was 7 sets; for power plant B, the 25th quantile was 3 sets, and the 75th quantile was 6 sets. Thus, historical data with scheduling volumes of 4-7 sets for power plant A and 3-6 sets for power plant B were selected to form the first support set, containing 30 valid historical records. Simultaneously, considering the high correlation of generator bearings, a second support set was constructed based on Mahalanobis distance for the historical data. Historical data on generator bearings is transformed into a multi-dimensional vector containing indicators such as scheduling volume, remaining inventory, and demand growth rate. The Mahalanobis distance between the current scheduling demand vector (assuming power plant A expects to schedule 4 sets and power plant B expects to schedule 5 sets, combined with inventory and other factors) and each historical sample vector is calculated. The 20 historical data points with the smallest distances are selected to form the second support set. Then, based on the actual data, a first uncertainty set is constructed corresponding to the first support set, and a second uncertainty set is constructed for the second support set.

[0160] Finally, the actual scheduling parameters of the inventory network are obtained, and an inventory optimization model is constructed and solved based on these parameters and the uncertainty set. Specifically, for the turbine gaskets, the model solution utilizes a set of linear constraints; for the generator bearings, it utilizes a set of second-order cone programming constraints. The resulting optimized inventory data is then obtained. For example, power plant A maintains an inventory of 6 sets of turbine gaskets, while power plant B maintains 5 sets; power plant A maintains an inventory of 5 sets of generator bearings, while power plant B maintains 6 sets. This method effectively reduces inventory costs and largely avoids stockouts, ensuring the operational efficiency of the power plants.

[0161] It is understood that the above data is for illustrative purposes only and does not represent actual data.

[0162] The technical solution provided in this application involves acquiring the correlation of cross-location item scheduling demand and historical data of the inventory network. A support set is constructed based on the correlation, including a first support set or a second support set. If the correlation is low, the first support set is constructed based on quantiles of the historical data; if the correlation is high, the second support set is constructed based on Mahalanobis distance. At least one uncertain set corresponding to the support set is constructed based on the historical data, including a first uncertain set or a second uncertain set. Scheduling parameters of the inventory network are obtained. An inventory optimization model is constructed based on the scheduling parameters and the uncertain set. The inventory optimization model includes an objective function and a set of constraints. The constraint set is reconstructed into a linear constraint set using the first uncertain set, or into a second-order cone programming constraint set using the second uncertain set. The objective function is solved based on the linear constraint set or the second-order cone programming constraint set to obtain the inventory optimization data. This application's embodiment constructs different support sets based on correlation: when the correlation is low, the first support set is constructed based on quantiles, focusing on the independent distribution characteristics of single-location demand; when the correlation is high, the second support set is constructed using Mahalanobis distance to explore the correlation between demands from different locations. This approach expands the inventory optimization model beyond simple moment constraints, enabling hierarchical processing of correlations. Furthermore, it generates corresponding uncertainty sets for different support sets to optimize the modeling and solution process: the constraint sets are reconstructed into linear constraints suitable for low-correlation scenarios or second-order cone programming constraints suitable for high-correlation scenarios, ensuring that the objective function fully considers the impact of demand correlations on inventory scheduling during the solution process. Therefore, this embodiment significantly improves the scheduling accuracy of multi-location inventory networks, especially suitable for scenarios like power plant spare parts where high scheduling accuracy is required. It better addresses demand uncertainties arising during power plant maintenance, reduces maintenance waiting time, and optimizes inventory costs.

[0163] This application embodiment also provides a multi-location inventory network optimization device, which can implement the above-mentioned multi-location inventory network optimization method. Referring to FIG9, the device includes: a data acquisition module 910: used to acquire the correlation of cross-location item scheduling demand and historical data of inventory network, and construct a support set based on the correlation. The support set includes a first support set or a second support set. If the correlation is small, the first support set is constructed based on quantiles of historical data. If the correlation is large, the second support set is constructed based on Mahalanobis distance of historical data.

[0164] Uncertain set construction module 920: used to construct at least one uncertain set corresponding to the support set based on historical data, the uncertain set including a first uncertain set or a second uncertain set.

[0165] Model building module 930: Used to obtain the scheduling parameters of the inventory network, and to build an inventory optimization model based on the scheduling parameters and the uncertainty set. The inventory optimization model includes an objective function and a set of constraints.

[0166] Model Solver Module 940: Used to reconstruct the constraint set into a linear constraint set using the first uncertainty set, or to reconstruct the constraint set into a second-order cone programming condition set using the second uncertainty set, and solve the objective function based on the linear constraint set or the second-order cone programming condition set to obtain inventory optimization data.

[0167] The specific implementation of the multi-location inventory network optimization device in this embodiment is basically the same as the specific implementation of the multi-location inventory network optimization method described above, and will not be repeated here.

[0168] This application also provides an electronic device, including: at least one memory; at least one processor; and at least one program; the program is stored in the memory, and the processor executes the at least one program to implement the multi-location inventory network optimization method described above. This electronic device can be any smart terminal, including mobile phones, tablets, personal digital assistants (PDAs), and in-vehicle computers.

[0169] Please refer to Figure 10, which illustrates the hardware structure of an electronic device according to another embodiment. The electronic device includes: a processor 1001, which can be implemented using a general-purpose central processing unit (CPU), microprocessor, application-specific integrated circuit (ASIC), or one or more integrated circuits, for executing related programs to implement the technical solutions provided in the embodiments of this application; and a memory 1002, which can be implemented using a read-only memory (ROM), static storage device, dynamic storage device, or random access memory (RAM), etc. The memory 1002 can store the operating system and other applications. When the technical solutions provided in the embodiments of this specification are implemented through software or firmware, the relevant program code is stored in the memory 1002 and is called and executed by the processor 1001 to implement the multi-location inventory network optimization method of this application embodiment. The input / output interface 1003 is used to realize information input and output. The communication interface 1004 is used to realize communication interaction between this device and other devices. Communication can be realized through wired means (such as USB, network cable, etc.) or through wireless means (such as mobile network, WIFI, Bluetooth, etc.). The bus 1005 transmits information between various components of the device (such as processor 1001, memory 1002, input / output interface 1003 and communication interface 1004). The processor 1001, memory 1002, input / output interface 1003 and communication interface 1004 realize communication connection between each other within the device through the bus 1005.

[0170] This application embodiment also provides a storage medium that stores a computer program, which, when executed by a processor, implements the above-described multi-location inventory network optimization method.

[0171] Memory, as a non-transitory storage medium, can be used to store non-transitory software programs and non-transitory computer-executable programs. Furthermore, memory may include high-speed random access memory, and may also include non-transitory memory, such as at least one disk storage device, flash memory device, or other non-transitory solid-state storage device. In some embodiments, memory may optionally include memory remotely located relative to the processor, and these remote memories can be connected to the processor via a network. Examples of such networks include, but are not limited to, the Internet, intranets, local area networks, mobile communication networks, and combinations thereof.

[0172] The multi-location inventory network optimization method, apparatus, device, and storage medium proposed in this application acquire the correlation of cross-location item scheduling demand and historical data of the inventory network. Based on the correlation, a support set is constructed, including a first support set or a second support set. If the correlation is low, the first support set is constructed based on quantiles of the historical data; if the correlation is high, the second support set is constructed based on Mahalanobis distance. At least one uncertain set corresponding to the support set is constructed based on the historical data, including a first uncertain set or a second uncertain set. Scheduling parameters of the inventory network are acquired. An inventory optimization model is constructed based on the scheduling parameters and the uncertain set. The inventory optimization model includes an objective function and a set of constraints. The constraint set is reconstructed into a linear constraint set using the first uncertain set, or into a second-order cone programming constraint set using the second uncertain set. The objective function is solved based on the linear constraint set or the second-order cone programming constraint set to obtain the inventory optimization data. This application's embodiments construct different support sets based on correlation: when the correlation is low, the first support set is constructed based on quantiles, focusing on the independent distribution characteristics of single-location demand; when the correlation is high, the second support set is constructed using Mahalanobis distance to explore the correlation between demands from different locations. This approach expands the inventory optimization model beyond simple moment constraints, enabling hierarchical processing of correlations. Furthermore, it generates corresponding uncertainty sets for different support sets to optimize the modeling and solution process: the constraint sets are reconstructed into linear constraints suitable for low-correlation scenarios or second-order cone programming constraints suitable for high-correlation scenarios, ensuring that the objective function fully considers the impact of demand correlations on inventory scheduling during the solution process. Therefore, this embodiment significantly improves the scheduling accuracy of multi-location inventory networks, especially suitable for scenarios like power plant spare parts where high scheduling accuracy is required. It better addresses demand uncertainties arising during power plant maintenance, reduces maintenance waiting time, and optimizes inventory costs.

[0173] The embodiments described in this application are for the purpose of more clearly illustrating the technical solutions of the embodiments of this application, and do not constitute a limitation on the technical solutions provided by the embodiments of this application. As those skilled in the art will know, with the evolution of technology and the emergence of new application scenarios, the technical solutions provided by the embodiments of this application are also applicable to similar technical problems.

[0174] Those skilled in the art will understand that the technical solutions shown in the figures do not constitute a limitation on the embodiments of this application, and may include more or fewer steps than shown, or combine certain steps, or different steps.

[0175] The device embodiments described above are merely illustrative. The units described as separate components may or may not be physically separate; that is, they may be located in one place or distributed across multiple network units. Some or all of the modules can be selected to achieve the purpose of this embodiment according to actual needs.

[0176] Those skilled in the art will understand that all or some of the steps in the methods disclosed above, as well as the functional modules / units in the systems and devices, can be implemented as software, firmware, hardware, or suitable combinations thereof.

[0177] The terms “first,” “second,” “third,” “fourth,” etc. (if present) in the specification and accompanying drawings of this application are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments of this application described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms “comprising” and “having,” and any variations thereof, are intended to cover a non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0178] It should be understood that in this application, "at least one (item)" means one or more, and "more than" means two or more. "And / or" is used to describe the relationship between related objects, indicating that three relationships can exist. For example, "A and / or B" can represent three cases: only A exists, only B exists, and both A and B exist simultaneously, where A and B can be singular or plural. The character " / " generally indicates that the preceding and following related objects are in an "or" relationship. "At least one (item) of the following" or similar expressions refer to any combination of these items, including any combination of single or plural items. For example, at least one (item) of a, b, or c can represent: a, b, c, "a and b", "a and c", "b and c", or "a and b and c", where a, b, and c can be single or multiple.

[0179] In the several embodiments provided in this application, it should be understood that the disclosed apparatus and methods can be implemented in other ways. For example, the apparatus embodiments described above are merely illustrative; for instance, the division of the units described above is only a logical functional division, and in actual implementation, there may be other division methods. For example, multiple units or components may be combined or integrated into another system, or some features may be ignored or not executed. Furthermore, the coupling or direct coupling or communication connection shown or discussed may be through some interfaces; the indirect coupling or communication connection between apparatuses or units may be electrical, mechanical, or other forms.

[0180] The units described above as separate components may or may not be physically separate. The components shown as units may or may not be physical units; that is, they may be located in one place or distributed across multiple network units. Some or all of the units can be selected to achieve the purpose of this embodiment according to actual needs.

[0181] Furthermore, the functional units in the various embodiments of this application can be integrated into one processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit. The integrated unit can be implemented in hardware or as a software functional unit.

[0182] If the integrated unit is implemented as a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of this application, in essence, or the part that contributes to the prior art, or all or part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes multiple instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods of the various embodiments of this application. The aforementioned storage medium includes various media capable of storing programs, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0183] The preferred embodiments of the present application have been described above with reference to the accompanying drawings, but this does not limit the scope of the claims of the present application. Any modifications, equivalent substitutions, and improvements made by those skilled in the art without departing from the scope and substance of the embodiments of the present application shall be within the scope of the claims of the present application.

Claims

1. A method for optimizing a multi-location inventory network, characterized in that, include: Obtain the correlation of cross-location item dispatching demand and the historical data of the inventory network; construct a support set based on the correlation, the support set including a first support set or a second support set; if the correlation is small, construct the first support set based on quantiles of the historical data; if the correlation is large, construct the second support set based on Mahalanobis distance of the historical data; construct at least one uncertain set corresponding to the support set based on the historical data, the uncertain set including a first uncertain set or a second uncertain set; Obtain the scheduling parameters of the inventory network, and construct an inventory optimization model based on the scheduling parameters and the uncertainty set. The inventory optimization model includes an objective function and a set of constraints. The constraint set is reconstructed into a linear constraint set using the first uncertainty set, or into a second-order cone programming condition set using the second uncertainty set. The objective function is then solved based on the linear constraint set or the second-order cone programming condition set to obtain inventory optimization data.

2. The multi-location inventory network optimization method according to claim 1, characterized in that, The inventory network includes a first number of demand locations. The step of constructing the first support set based on quantiles from the historical data includes: obtaining a coverage parameter; calculating a quantile index based on the coverage parameter; performing sampling with replacement on the data corresponding to each demand location in the multiple historical data sets to obtain a sample set for each demand location; calculating a first quantile and a second quantile for each sample set based on the quantile index; repeating the sampling multiple times to obtain a first set corresponding to the first quantile and a second set corresponding to the second quantile; estimating the upper confidence boundary corresponding to the first set to obtain the upper quantile, and estimating the lower confidence boundary corresponding to the second set to obtain the lower quantile; for each demand location, setting the corresponding random demand to be located between the corresponding upper quantile and the lower quantile, thus constructing the first support set.

3. The multi-location inventory network optimization method according to claim 1, characterized in that, The inventory network includes a first number of demand locations. The step of constructing the second support set based on Mahalanobis distance from the historical data includes: calculating the corresponding data mean and data covariance for each demand location in the historical data, and performing sampling with replacement to obtain a sample set for each demand location; calculating the squared Mahalanobis distance of each data in the sample set based on the data mean and the data covariance; obtaining a corresponding reference threshold based on the quantiles of the squared Mahalanobis distance; obtaining a threshold set corresponding to the reference threshold; selecting a target threshold from the threshold set; and for each demand location, setting the squared Mahalanobis distance between the corresponding random demand and the data mean to be less than or equal to the target threshold to construct the second support set.

4. The multi-location inventory network optimization method according to claim 1, characterized in that, The step of constructing at least one uncertain set corresponding to the support set based on the historical data includes: obtaining the empirical distribution corresponding to the historical data; calculating the metric distance between the probability distribution and the empirical distribution for the probability distribution corresponding to the random demand in the support set; obtaining the fuzzy set radius; setting the probability distributions whose metric distance is less than the fuzzy set radius to constitute the fuzzy set; when the support set is the first support set, the fuzzy set is the first fuzzy set, otherwise it is the second fuzzy set; and discretizing the fuzzy set into at least one uncertain set.

5. The multi-location inventory network optimization method according to claim 4, characterized in that, Discretizing the fuzzy set into at least one uncertain set includes: when the fuzzy set is the first fuzzy set, acquiring a first number of historical data points located in the first support set; for each historical data point, setting the distance between the random demand at each demand location and the corresponding center point to be less than or equal to the radius of the fuzzy set, to obtain a first uncertain set; when the fuzzy set is the first fuzzy set, acquiring a first number of historical data points located in the first support set; for each historical data point, acquiring the dimension interval of the random demand corresponding to each demand location, and combining the dimension intervals to obtain a first uncertain set; when the fuzzy set is the second fuzzy set, for the mean vector and mean squared error vector corresponding to all the historical data points, acquiring a second number of historical data points located in the second support set; for each historical data point, calculating the difference vector between the random demand at each demand location and the mean vector; setting the product of the transpose of the difference vector, the inverse of the mean squared error vector, and the difference vector to be less than or equal to the target threshold; and setting the random demand to be located within the interval formed by the value of the historical data and the radius of the fuzzy set, to obtain a second uncertain set.

6. The multi-location inventory network optimization method according to claim 1, characterized in that, The inventory network includes a first number of demand locations and a second number of distribution centers. The scheduling parameters include at least the unit product procurement cost and the current inventory level of the distribution centers. Based on the scheduling parameters and the uncertainty set, a target constraint function is constructed, including: calculating the inventory procurement cost according to the unit product procurement cost and the current inventory level of the distribution centers; obtaining auxiliary decision variables for each uncertainty set; calculating the mean of all the auxiliary decision variables; minimizing the sum of the inventory procurement cost and the mean of the variables; and obtaining the target constraint function.

7. The multi-location inventory network optimization method according to claim 6, characterized in that, The scheduling parameters include at least: unit product inventory holding cost, linear decision variables, unit product stockout loss cost, actual demand quantity at the demand location, combined cost coefficient from the distribution center to the demand location, and upper limit of transportation capacity from the distribution center to the demand location. Based on the scheduling parameters and the uncertainty set, a set of constraints is constructed, including: obtaining the linear decision variables corresponding to each uncertainty set, whereby the linear decision variables are obtained by summing initial variables and decision variables, and the decision variables are obtained from the decision coefficient of each demand location and the corresponding random demand; calculating the inventory holding cost based on the unit product inventory holding cost and the current inventory level of the distribution center; calculating the distribution cost based on the combined cost coefficient and the initial variables; calculating the stockout penalty cost based on the stockout loss cost, the combined cost coefficient, the decision variables, and the random demand; and setting the inventory holding cost, distribution cost, and stockout penalty cost to be less than or equal to the auxiliary decision variables to obtain an upper cost bound constraint. The conditions are as follows: First, an initial variable is summed to obtain a first reference value. Based on the first reference value, the decision variable, and the random demand, the total delivery volume is obtained. The total delivery volume is set to be less than or equal to the current inventory level of the corresponding distribution center, thus obtaining an inventory balance constraint. A demand response parameter is obtained based on the initial variable, the decision variable, and the random demand. The demand response parameter is set to be less than or equal to zero, thus obtaining a demand response constraint. The actual delivery volume is obtained based on the decision variable and the random demand. The actual delivery volume is set to be less than or equal to the initial variable, thus obtaining a delivery volume non-negative constraint. A transportation capacity parameter is obtained based on the initial variable, the decision variable, and the random demand. The transportation capacity parameter is set to be less than or equal to the corresponding transportation capacity upper limit, thus obtaining a transportation capacity constraint. Finally, the constraint set is obtained based on the cost upper bound constraint, the inventory balance constraint, the demand response constraint, the delivery volume non-negative constraint, and the transportation capacity constraint.

8. A multi-location inventory network optimization device, characterized in that, include: Data acquisition module: used to acquire the correlation of cross-location item scheduling demand and the historical data of the inventory network, and construct a support set based on the correlation. The support set includes a first support set or a second support set. If the correlation is small, the first support set is constructed based on quantiles of the historical data. If the correlation is large, the second support set is constructed based on Mahalanobis distance of the historical data. Uncertain set construction module: used to construct at least one uncertain set corresponding to the support set based on the historical data. The uncertain set includes a first uncertain set or a second uncertain set. Model building module: used to obtain the scheduling parameters of the inventory network, and build an inventory optimization model based on the scheduling parameters and the uncertainty set. The inventory optimization model includes an objective function and a set of constraints. Model solving module: used to reconstruct the constraint set into a linear constraint set using the first uncertainty set, or to reconstruct the constraint set into a second-order cone programming condition set using the second uncertainty set, and to solve the objective function based on the linear constraint set or the second-order cone programming condition set to obtain inventory optimization data.

9. An electronic device, characterized in that, The electronic device includes a memory and a processor, the memory storing a computer program, and the processor executing the computer program to implement the multi-location inventory network optimization method according to any one of claims 1 to 7.

10. A storage medium storing a computer program, characterized in that, When the computer program is executed by a processor, it implements the multi-location inventory network optimization method according to any one of claims 1 to 7.

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