A computing method of computing power coupon subsidies based on supply chain network equilibrium

Abstract

This invention presents a computational power voucher subsidy calculation method based on supply chain network equilibrium, relating to the fields of digital economy and resource allocation optimization. Addressing the problem of governments formulating computational power voucher subsidy strategies under fiscal budget constraints, a two-layer optimization model is constructed, encompassing upper-level policy decisions and lower-level market responses. The upper layer aims to maximize social welfare, determining the subsidy intensity under budget and product quality constraints. The lower layer describes the computational power resource allocation behavior of enterprises under the influence of subsidies, obtaining equilibrium results by establishing the correlation between computational power input, information generation, product quality, and market demand. Furthermore, by constructing a coupling relationship between policy variables and market behavior, a linked analysis of subsidy strategies and market responses is achieved, and an alternating solution method is used to obtain the optimal solution. This invention can characterize government decision-making and market feedback mechanisms within a unified framework, improving the scientific nature of subsidy strategies and the efficiency of resource allocation, and is applicable to computational power resource allocation and digital economy policy decision-making scenarios.

Description

Technical fields:

[0001] This invention proposes a computing power voucher subsidy calculation method based on supply chain network equilibrium, which relates to the fields of digital economy and resource allocation optimization. Background technology:

[0002] With the rapid development of the digital economy, computing power resources have gradually evolved from traditional information technology support conditions into important basic production factors, playing a crucial role in data processing, artificial intelligence model training, and industrial digital transformation. The acquisition and use of computing power resources are also shifting from traditional local deployment to networked and service-oriented approaches. Enterprises can obtain computing power services on demand through cloud computing centers or edge computing nodes, thereby participating in cross-regional and cross-entity computing power resource allocation systems. Against this backdrop, to reduce the cost for enterprises to acquire computing power resources and promote their effective utilization, some regions are gradually exploring the issuance of computing power vouchers to subsidize enterprises' purchase of computing power services. As a policy tool, computing power vouchers partially compensate enterprises for their computing power usage costs through fiscal funds, thereby guiding enterprises to increase their investment in computing power and promoting the development of artificial intelligence and related industries. However, existing computing power voucher subsidy policies, in practice, often rely on experience-based judgments or simple proportional allocation methods, lacking systematic quantitative analysis methods, making it difficult to achieve optimal allocation of subsidy effects under fiscal budget constraints. On the one hand, excessively low subsidy levels may fail to effectively incentivize enterprises to increase computing power investment, affecting the policy's implementation effectiveness. On the other hand, excessively high subsidy levels may lead to a waste of fiscal resources and even a decline in resource allocation efficiency. Furthermore, enterprises exhibit distinct responsiveness to subsidy policies; computing power investment not only affects enterprise production efficiency but also influences market demand and prices through improved product quality, thus forming a complex market feedback mechanism. Existing technologies rarely offer methods that simultaneously consider the interaction between government subsidy decisions and enterprise behavioral responses, making it difficult to comprehensively assess the combined impact of subsidy policies on market operations and social welfare. Therefore, it is necessary to provide a computing power voucher subsidy optimization method that comprehensively considers the relationship between policy decisions and market responses under fiscal budget constraints, thereby improving the scientific rigor of subsidy policy formulation and the efficiency of resource allocation. Summary of the Invention: Technical issues:

[0003] Existing technologies for computing power voucher subsidies primarily rely on empirical judgment or simple proportional allocation methods, lacking systematic optimization under fiscal budget constraints. This makes it difficult to strike a balance between incentivizing corporate computing power investment and improving resource allocation efficiency. Furthermore, existing methods fail to fully consider the impact of subsidy policies on corporate behavior and market operations, lacking unified modeling and analysis tools for the interaction between policy decisions and market responses, thus hindering overall optimization of subsidy strategies. Therefore, there is an urgent need for a method to optimize computing power voucher subsidies under fiscal budget constraints, comprehensively considering the linkage between government subsidy decisions and corporate behavioral responses, to improve the scientific rigor, rationality, and resource allocation efficiency of subsidy policy formulation. Technical solution:

[0004] To address the aforementioned technical problems, this invention provides a method for optimizing computing power voucher subsidy policies based on bi-level programming. First, policy parameters and market parameters are obtained. The policy parameters include fiscal budget constraints, the range of subsidy intensity values, and constraints. The market parameters include computing power cost parameters, demand function parameters, and quality-related parameters. Second, an upper-level policy decision-making model is constructed, with the government as the decision-maker and social welfare maximization as the goal. Under the conditions of meeting fiscal budget constraints and preset constraints, the intensity of computing power voucher subsidies is optimized. Then, a lower-level market response model is constructed to describe the computing power resource allocation behavior of enterprises under given subsidy policy conditions. By establishing the correlation between computing power input and information generation, product quality, and market demand, the market equilibrium state is determined. Further, a coupling relationship is established between the upper-level policy decision-making model and the lower-level market response model, so that changes in subsidy intensity can affect enterprise computing power input behavior and market equilibrium results. Based on this, a numerical optimization method is used to solve the bi-level programming model. By iteratively searching the subsidy intensity and updating it based on the equilibrium results of the lower-level model, the optimal subsidy strategy that meets the constraints is obtained. Finally, the optimal subsidy intensity and the corresponding market equilibrium results are output, including changes in enterprise computing power input levels, product quality levels, and market demand. 1. Model Method

[0005] Computing power vouchers, a new policy tool that has been gradually promoted in recent years, essentially reduce the marginal cost for enterprises to use computing power resources through subsidies, thereby incentivizing enterprises to increase their investment in computing power. Against this backdrop, the government is introduced as the upper-level decision-making body, and a two-level programming model is constructed. The upper-level problem describes how the government, under given fiscal budget constraints and minimum product quality constraints, determines the optimal computing power voucher subsidy ratio with the goal of maximizing social welfare. The lower-level problem describes the equilibrium of the computing power-enabled supply chain network formed by raw material suppliers, computing power service providers, manufacturers, and the demand market under perfect competition, given the given subsidy policy. This model can systematically characterize the interaction mechanism between government subsidy policies, enterprise computing power usage behavior, product quality changes, market equilibrium, and social welfare, thus providing a unified theoretical framework for evaluating the economic effects of computing power voucher policies. The relevant parameters and their meanings used in the model are shown in Table 1, the relevant variables and their meanings in Table 2, and the relevant functions and their meanings in Table 3. Table 1. Relevant parameters and their meanings Table 2. Relevant Variables and Their Meanings Table 3. Related Functions and Their Meanings

[0006] In this two-tiered planning framework, the upper-level problem characterizes the government's policy-making behavior. The government is viewed as a system-level planner and policymaker, not directly involved in specific market activities such as raw material procurement, computing power trading, or product sales. Instead, it indirectly influences manufacturers' computing power usage decisions in the lower-level network equilibrium by formulating computing power voucher subsidy policies. Under this framework, the government's decision-making objective is not to maximize the profit of a single entity, but rather, from a societal perspective, to maximize social welfare by comprehensively considering both consumer and producer surplus. Specifically, the government needs to address the given fiscal budget constraint D and minimum product quality constraint. θ Under these conditions, the optimal computing power voucher subsidy intensity τ∈[0,1] is selected so that the overall system welfare corresponding to the lower-level market equilibrium result induced by it reaches the optimal level.

[0007] The underlying problem describes the market equilibrium state formed by raw material suppliers, manufacturers, computing power service providers, and the demand market, based on their respective objective functions, under given government subsidy policy parameters. Given technological conditions, cost structures, and demand environments, each entity makes optimal choices regarding decision variables such as raw material flow, product flow, computing power usage, and market prices. Manufacturers gain information gain by purchasing computing power services. This information gain is transformed into product quality through a quality function and influences the inverse demand function on the demand side. Furthermore, it affects manufacturing costs by improving production efficiency, thus forming an endogenous linkage mechanism between computing power usage, quality evolution, and market demand.

[0008] The supply chain network structure empowered by lower-level computing power remains consistent with Chapter 4, still consisting of upstream raw material suppliers and computing power service providers, intermediate manufacturers, and the downstream demand market. The difference lies in that, building upon the network equilibrium model in Chapter 4, this chapter further introduces government computing power voucher subsidies, bringing the manufacturers' computing power usage costs under policy regulation. This allows government subsidies to influence enterprises' computing power investment behavior, further impacting product quality, market equilibrium, and system welfare. The specific network structure is as follows: Figure 1 As shown.

[0009] Therefore, in this two-tiered planning problem, the upper and lower tiers are linked through policy decisions, firm responses, and market equilibrium, and through a welfare feedback mechanism: the upper-level government determines the intensity of subsidies based on fiscal budgets and quality objectives; the lower-level market entities form a network equilibrium through the interaction of prices and flows under the given subsidy policy; and the equilibrium results of the lower level are further fed back into the objective function and constraints of the upper level. Specific interactions are as follows: Figure 2 As shown. 2.1 Upper-level model

[0010] In this two-tiered planning problem, the upper-level government subsidizes manufacturers' actual computing power usage costs by issuing computing power vouchers, thereby reducing their effective marginal cost and guiding computing power adoption. Manufacturers incur computing fees when transacting with computing power service providers. At this point, the government deducts a percentage τ from the computing costs, and the manufacturer's actual payment percentage is (1-τ). The government's decision variable is the computing power voucher subsidy intensity parameter τ, which affects the cost of computing power usage for manufacturers, indirectly changing the scale of computing power usage, system quality level, and market price structure in the lower equilibrium.

[0011] The government does not directly intervene in the production and transaction behavior of market entities; the lower-level equilibrium is still formed endogenously by market mechanisms. The government's optimization goal is to maximize social welfare. Within the framework of welfare economics, social welfare typically consists of two parts: consumer surplus and producer surplus, used to measure the overall economic welfare created by the market under a given resource allocation. Consumer surplus characterizes the net utility obtained by the demand side from market transactions, defined as the difference between a consumer's maximum willingness to pay for a good or service and the price they actually pay. Under continuous demand conditions, consumer surplus can be expressed as the area below the inverse demand function and above the market transaction price, that is, the utility gain obtained by consumers when they obtain a certain quantity of products or services at a uniform market price. Consumer surplus can be expressed as a function:

[0012] Producer surplus characterizes the net profit gained by the supply side in market transactions, defined as the actual income producers receive from the market minus the real production costs incurred in providing the corresponding products or services. Producer surplus reflects the excess profit obtained by production entities under given technological and cost conditions through participation in market transactions. It should be noted that in multi-agent networks or supply chain systems, the social welfare accounting of producer surplus should be based on the income generated by the final demand market, minus the various real resource costs consumed during system operation. Producer surplus can be expressed as a function: Therefore, the total social surplus can be expressed as:

[0013] Regarding constraints, government policies must simultaneously meet both fiscal budget constraints and minimum product quality constraints. The fiscal budget constraint requires that the total subsidy expenditure for issuing computing power vouchers not exceed the predetermined fiscal budget ceiling D. The total subsidy expenditure for computing power vouchers can be expressed as: Right now Spend(τ)≤D.

[0014] The expenditure is jointly determined by the intensity of the unit computing power subsidy and the total amount of computing power used in the lower-level equilibrium; the minimum quality constraint requires that the system quality level induced by the lower-level equilibrium result is not lower than the preset threshold, so as to ensure that the computing power subsidy policy meets the basic requirements of the digital economy for product quality level while improving efficiency. Therefore, the upper-level model can be expressed mathematically as follows: 2.2 Lower-level model

[0015] This section will conduct behavioral analysis and characterize the optimal conditions for the raw material supplier layer, computing power service provider layer, manufacturer layer, and demand market layer under the condition of perfect competition, and establish a model of the equilibrium of the entire supply chain network. (1) Behavior and optimality conditions of raw material suppliers

[0016] In a computing-enabled supply chain network, raw material suppliers produce raw materials and sell them to manufacturers at a certain price. The following analyzes the costs and revenues of supplier i. For the raw material supplier, the main costs are as follows: firstly, the production cost f of the raw materials. i In a perfectly competitive environment, the output depends not only on supplier i's own output, but also on the output of all raw material suppliers in the market. f i =f i (Q 1 ),

[0017] Second, the raw material transaction costs between raw material suppliers and manufacturers. In addition to the purchase cost of the materials themselves, there are also comprehensive transaction costs throughout the procurement, collaboration, contract, and logistics execution processes. Here, we assume that the transaction costs are related to the raw material transaction volume q. ij Positively correlated, that is c ij =c ij (q ij ), The larger the volume of raw material transactions, the higher the transaction costs.

[0018] Raw material supplier revenue The main source of revenue is the sales of raw materials, of which ρ 1ij Let be the wholesale price of raw materials between raw material supplier i and manufacturer j, which is an endogenous variable. Therefore, the profit of raw material supplier i can be expressed as: Theorem 1: In a supply chain network, raw material suppliers at the same level engage in non-cooperative competition, and their cost functions are convex. The equilibrium condition for these raw material suppliers can be expressed by the following variational inequality, with the optimal solution... satisfy Equation (2) has a good explanation in economic terms: if the price that the manufacturer is willing to pay for raw materials is exactly equal to the marginal production of the raw material supplier and the transaction costs associated with the manufacturer, the raw material supplier will ship a positive quantity of raw materials to the manufacturer, and the flow on the corresponding route will be positive; if the marginal production and transaction costs of the raw material supplier exceed the price that the manufacturer is willing to pay for the product, then the flow on that route will be zero. (2) Behavior and optimality conditions of computing power service providers

[0019] In supply chain networks, computing power service providers receive data from manufacturers, process it into effective information that can improve production efficiency and product quality, and act as data element processors in the supply chain network, providing computing power services. They are key intermediaries in the modern supply chain digitalization process. The following is a detailed analysis of the cost and revenue composition of computing power service providers.

[0020] For computing power service provider S, the main cost is the cost of data computation. Assume that the computation cost can be expressed as a function of the sum of all data processed by the computing power provider: Q 3 Let q represent the data transaction volume between the manufacturer and the computing power service provider, assuming that the computing cost is convex and continuously differentiable, and has bounded second-order partial derivatives. Manufacturer data q js After the calculation and processing are completed, it is converted into a certain amount of information (Info). js The relationship between data and information can be represented by a function as follows: Info js =a s q js , Where a s This refers to parameters related to the algorithms and hardware used by the computing power service provider.

[0021] The revenue of computing power service providers mainly comes from the income from providing data processing services. Therefore, in this network, the profit of computing power service provider s can be expressed as: Assuming that computing power service providers compete in a non-cooperative manner, and that the computational cost function is continuous and convex, then based on the first-order optimality condition of convex optimization, the following theorem can be obtained: Theorem 2: The optimality condition for computing power service providers can be expressed as the following variational inequality, with the optimal solution... satisfy Equation (1) gives the equilibrium optimality condition for the computing power service provider. Its economic meaning can be interpreted as the marginal computing cost when the computing power service provider provides services to a certain manufacturer. The unit computing power price ρ payable to the manufacturer 3js Consistent; if the unit transaction price is insufficient to cover the marginal calculation cost, no transaction will occur. (3) Manufacturer behavior and optimality conditions

[0022] Manufacturers, as key players in the network, need to purchase raw materials from raw material manufacturers to produce products and sell them to the demand market. They also need to transact with computing power service providers, transmitting data to them to obtain big data information. The following section analyzes in detail the composition of manufacturers' costs and revenues.

[0023] For manufacturer J, the main costs are as follows: firstly, the cost of purchasing raw materials, using... The second is the product's manufacturing cost f. j This assumes that in a perfectly competitive environment, the manufacturing cost of a product depends not only on the raw materials the manufacturer obtains from its suppliers, but also on the data the manufacturer transmits to the computing service provider for calculation. f j =f j (Q 1 Q 3 ), Among these factors, production costs increase with output and decrease with the amount of information; thirdly, there are costs associated with data transactions with computing power service providers, primarily including the cost of purchasing computing power services. and the cost of calculating response time Assuming that the response time of data computation mainly consists of the time for data transmission to the computing power service provider, the waiting time for data computation, and the time for the completed computation information to be transmitted back to the manufacturer, the data transmission time cost is... For the amount of data to be transmitted q js With transmission rate The ratio, that is in It is the bandwidth between js files B. js A function of signal-to-noise ratio M: Calculated waiting time cost Simultaneously affected by the capacity C of computing power service providers s The amount of data q sent by the manufacturer for calculation js The effect can be represented by a function as: Here, ω is the computational latency cost coefficient, used to reflect the intensity of the impact of computing resource congestion on task waiting time costs. When multiple manufacturers submit computing tasks to the computing service provider simultaneously, the increase in the total number of tasks will intensify competition for computing resources, thus causing a non-linear increase in waiting time costs. The transmission cost of sending the processed information back to the manufacturer is the amount of information transmitted, Info. js With transmission rate The ratio, the conversion relationship between data and information, can be expressed as a function: Info js =a s q js , Right now in It is the bandwidth between sj. sj A function of signal-to-noise ratio M: Therefore, the calculated response time cost It can be represented as:

[0024] Manufacturer's revenue The revenue mainly comes from product sales, of which ρ 2jk Let $\frac{j}{k}$ be the transaction price of the product between manufacturer $j$ and the demand market $k$, which is an endogenous variable. Therefore, manufacturer $j$'s profit can be expressed as: Theorem 3 assumes that manufacturers compete in a non-cooperative manner, and that the product production cost function and the response time cost function are continuous and convex. Define the feasible set of manufacturer j. According to the first-order optimality condition of convex optimization, the manufacturer's optimality condition can be expressed as the following variational inequality, for any (q) ij q jk q js )∈K j ,have As can be seen from the optimality condition given in equation (2), manufacturers cannot simultaneously adjust the scale of raw material procurement, product deployment structure, and the transaction volume of computing power data to obtain greater profits under given resource constraints and non-negativity constraints, thus reaching a stable optimal state. From the perspective of raw material and product allocation, the optimality condition reflects the inherent consistency between production and sales. The quantity of raw material procurement and the quantity of product sales are interconnected through resource balance constraints. The implicit opportunity cost corresponding to this constraint intrinsically affects the manufacturer's allocation decisions among different suppliers and markets in the equilibrium state. Manufacturers allocate resources at the marginal level to paths with higher net returns until all feasible paths achieve a balance between marginal returns and marginal costs. In terms of computing power data transactions, the optimality condition reflects the manufacturer's trade-off mechanism between computing power service costs and production efficiency improvement. In the optimal state, the marginal decrease in manufacturing costs brought about by information is exactly equal to the marginal cost constituted by the computing power service price and response time cost, so that manufacturers have no incentive to further adjust computing power investment. (4) Demand market behavior and optimality conditions

[0025] As the downstream players in the supply chain network, consumers in the demand market not only react to product prices but also to their quality levels. j Respond accordingly. High-quality products typically offer superior functionality, higher reliability, and a better consumer experience, directly increasing consumers' willingness to pay. Assume the product's demand-price function is p. k Regarding θ j Increasing and continuously differentiable, regarding the product demand d k Decreasing. Therefore, the demand price p for product j in the demand market. k This typically depends on the overall consumption pattern and the quality level of the product, where quality θ j It is endogenous, a direct result of manufacturers submitting data to computing power service providers and obtaining processed information. The marginal improvement of quality by information is diminishing, which can be expressed as a function: Where η is the information quality conversion coefficient, used to characterize the efficiency with which information obtained by the manufacturer through the computing power service provider is transformed into improved product quality during the product manufacturing process, and e is the natural logarithm. When the amount of information is 0, the initial quality of product j is η. The inverse demand function can be expressed as: ρ 4k =p k (d k q js ). In addition, consumers also need to bear the transaction costs with manufacturers. jkHere we assume that the transaction costs between manufacturers and the demand market are equal to the transaction volume q between them. jk Positively correlated, that is c jk =c jk (Q 2 ), The larger the transaction volume of a product, the higher the transaction costs. Therefore, the equilibrium condition for consumers in the demand market can be expressed as: Equation (3) shows that if The market price that consumers are willing to pay It must equal the wholesale price of the product plus the transaction cost of the product; if the wholesale price of the product plus the transaction cost of the product is higher than the market price, then the transaction volume of the product in this channel is 0; Equation (4) shows that if the market transaction volume Then the market price must equal the inverse demand price; if the market price is higher than the price consumers are willing to pay... The trading volume is then squeezed to 0.

[0026] Next, we define the feasible set of the demand market k: The variational inequality of the equilibrium state can then be expressed by the following theorem: Theorem 4: In demand market k, let the market decision vector be... Equilibrium state of demand market k The optimality conditions (3) and (4) are satisfied if and only if they satisfy the following variational inequality: for any (q) 1k ,...,q nk , ρ 4k d k )∈K k ,have

[0027] In equilibrium, any possible deviation in the demand market cannot simultaneously improve the consumer's payment structure and the market clearing situation, thus ensuring the stability of market operation. The first term of equation (10) reflects the principle of payment consistency when consumers choose between products produced by different manufacturers: when a manufacturer has a positive transaction volume in demand market k, its generalized payment price must be consistent with the market price at the marginal level; if the generalized payment price of a manufacturer is higher than the market price, its product will not be chosen by consumers in equilibrium, and the transaction volume will be zero. The second term describes the clearing conditions of the demand market and its price adjustment direction: in equilibrium, the total supply must be consistent with the demand scale at the current price and quality level; when supply exceeds demand, there is downward pressure on market prices to suppress demand and restore clearing; conversely, when supply is insufficient to meet demand, there is upward momentum in prices, thereby driving the market to adjust to a new equilibrium state. Product quality enters the demand function endogenously through information input, enabling manufacturers' decisions on computing power services to indirectly affect the equilibrium result of the demand market. Higher levels of information input expand demand by improving product quality, thereby increasing the market's tradable volume at a given price and altering the price-quantity equilibrium. This mechanism demonstrates that the demand market is closely coupled with the decisions of manufacturers and computing power providers through quality channels.

[0028] In the supply chain network, manufacturers acquire big data information resources by transmitting data to computing power service providers and purchasing computing power services. In the absence of policy intervention, the cost of computing power services that manufacturers pay for data transactions is determined by the pricing of the computing power service providers, and the purchase cost of computing power services is... After the government implemented the computing power voucher subsidy policy, due to the existence of a τ-proportion subsidy, the corresponding cost became... Based on this, the profit function of manufacturer j can be expressed as: The manufacturer's optimality condition can be expressed as the following variational inequality:

[0029] Next, we present the equilibrium and variational inequalities for the entire supply chain network. Under the assumptions that all cost and demand functions are continuously differentiable and convex, the system equilibrium solution is... If and only if for any X = (Q) 1 Q 2 Q 3 For any ρ(4, λ) ∈ K, satisfy the following system variational inequality: The variational inequality (5) is rewritten in standard form as follows, for a given X *∈K, satisfying: Where <·, ·> represent the inner product in the mn+no+nr+o+n-dimensional Euclidean space. For the equilibrium solution in the supply chain network, where the system operator The function expressions for each component are as follows: The system variational inequality characterizes the equilibrium operation mechanism in the computing-enabled supply chain network from a unified optimality perspective. Its essence reflects the marginal consistency and constraint coordination results under the conditions of multiple subjects and multiple road segments. 2.3 Bilevel Programming Model

[0030] By combining the aforementioned social welfare function, fiscal budget constraints, and supply chain network equilibrium conditions, a two-level programming model can be constructed between government policymakers and market entities: The bi-level programming model described by equation (6) has an upper-level programming problem that is an investment optimization decision problem and a lower-level programming problem that is a variational inequality problem derived from network equilibrium. 2.4 Solution Algorithm

[0031] To solve the proposed bi-level programming model, this chapter designs a nested iterative algorithm that combines outer-level policy optimal solution search with inner-level network equilibrium solution. The upper-level government uses the computing power voucher subsidy intensity τ as a policy decision variable, selecting the optimal subsidy level with the goal of maximizing social welfare under constraints of fiscal budget and minimum quality. The lower-level government, given τ, forms a supply chain network equilibrium endogenously through market mechanisms involving suppliers, manufacturers, computing power service providers, and the demand market; this equilibrium state is characterized by a variational inequality model. Specifically, the outer layer uses the golden section method for one-dimensional search within τ∈[0,1]; the inner layer, for each candidate τ, uses an improved projection algorithm to solve the corresponding variational inequality problem to obtain the equilibrium solution, and calculates social welfare, fiscal expenditure, and quality level accordingly. If a candidate τ causes the lower-level problem to fail to converge, or if its induced equilibrium result violates budget or quality constraints, then τ is deemed infeasible and excluded during the outer-level search. Through this method, the algorithm can obtain the optimal subsidy intensity in the sense of social welfare under strict policy constraints. The specific steps are as follows.

[0032] Step 1: Parameter Setting and Initialization. Set the maximum number of iterations k for the lower-level projection algorithm. max A convergence threshold ε is used, and a fixed step size a = 1 / L is selected based on the estimated Lipschitz constant L of the operator F(·τ). Given the upper limit of the government budget D and the minimum quality requirements... θAnd set the termination precision ε of the upper-level one-dimensional search. τ and the maximum number of iterations T max Initialize the lower-level decision variable vector. And through the nonnegative projection operator ∏ K (·) Project it onto the feasible region K = {X | X ≥ 0} to obtain the initial feasible point for the lower-level equilibrium solution. At the same time, initialize the upper-level policy search interval as [a, b] = [0, 1], and set the golden ratio constant.

[0033] Step Two: Solving the lower-level network equilibrium given the subsidy intensity τ. Given the policy parameter τ, the lower-level problem is formulated as the following variational inequality problem: Find X. * (τ)∈K, such that Among them, the subsidy intensity τ enters the Q of the operator F(·τ) by affecting the computing power cost and response transmission cost perceived by the manufacturer. 3 The components, specifically, are weighted by a coefficient (1-τ) on relevant marginal cost terms, thereby altering the manufacturer's optimal decision regarding computing power utilization. To solve the above variational inequality, the algorithm employs a fixed-step projection algorithm for iterative updates: X (t+1) =∏ K (X (t) -αF(X (t) ;τ)), With ||X (t+1) -X (t) || 2 <ε is used as the convergence criterion. If the maximum number of iterations k is reached... max If the above conditions are met, the lower-level equilibrium solution under the given τ is considered successful; otherwise, the lower-level problem under τ is determined to have failed to converge, and the solution process continues.

[0034] Step 3: Social Welfare Calculation and Feasibility Verification. This involves calculating the social welfare level and obtaining the equilibrium solution X under a given τ. * After (τ), the corresponding social welfare level is first calculated based on the equilibrium result: SW(τ)=CS(Q 2* (τ), Q 3* )-C(Q 1* Q 2* (τ), Q 3* ), Further calculations of government fiscal expenditure: And calculate the system's minimum mass level θ minIf Spend(τ)≤D, then θ min ≥ θ If the subsidy intensity τ is found to be a feasible policy, the corresponding social welfare SW(τ) is recorded; otherwise, the τ is considered not to meet the policy constraints and is excluded during the upper-level search process.

[0035] Step Four: One-Dimensional Search for Outer Subsidy Intensity. Under the constraint of the feasibility verification mechanism, this chapter uses the golden section method to perform a one-dimensional search for the subsidy intensity τ within the interval [a, b]. Specifically, two internal test points are constructed. c = bg(ba), d = a + g(ba), Steps two and three are then performed on τ=c and τ=d respectively to obtain their corresponding feasibility assessment results and social welfare levels. If the welfare evaluation of point c is not lower than that of point d, the search interval is updated to [a, d]; otherwise, it is updated to [c, b]. The above process is repeated until the interval length satisfies (ba) < ε. r Or reach the maximum number of iterations T max The optimal subsidy intensity τ is finally obtained. * and the lower-level equilibrium solution X induced by it. * (τ * ).

[0036] Step 5: Output Results. Output the optimal policy parameter τ. * , corresponding equilibrium decision variables Social welfare SW(τ) * ), fiscal expenditure Spend(τ) * and minimum quality level

[0037] The above algorithm can be represented in pseudocode as follows: Step 1: Initialization Let the step size α = 1 / L, and define the projection operator ∏ K (X) = max(X, 0). Initialize the golden ratio constant And set the outer interval [a, b] = [0, 1]. A minimum penalty constant, -E = -108, is set for determining whether hard constraints are infeasible. Let the initial value of hot start be X base ←x (0) . Step 2: Calculation and feasibility assessment of the upper-level objective function given the subsidy intensity τ: Eval(τ,X) lnit ) For any candidate τ∈[0,1], perform the following procedure: Lower-level solution (projection method): From X init Start iterating X (t+1) =∏ K (X (t) -aF(X (t) ;τ)), Until ||X (t+1) -X (t) || 2 <ε or t=k max Obtain X * (τ * ), and record whether convergence has occurred. Benefits and Constraints Calculation: Calculate welfare SW(τ)=CS(Q) 2* (τ), Q 3* )-PS(Q 1* Q 2* (τ), Q 3* ); Calculate budget expenditures Calculate the minimum mass θ min . Feasibility determination of constraints: If the lower level converges and Spend(τ)≤D, θ min ≥ θ Then let Φ(τ)=SW(τ); otherwise, set Φ(τ)=-E. Return (τ,X) * (τ), Φ(τ), W(τ), Spend(τ), θ min ). Step 3: Outer Golden Ratio Maximization Search Calculate c = bg(ba) and d = a + g(ba), and call Eval(c, X) respectively. init Eval(d, X) init ), Let the current optimal solution (τ) * X * These are candidate points that allow Φ to take a larger value. For t = 1, 2, ..., T max repeat: If Φ(c)≥Φ(d), let b←d, update d←c, and then calculate the new c=bg(ba); Simultaneously set the hot start point X base ←X * Call Eval(c, X) base ); Otherwise, let a←c, update c←d, and then calculate the new d=a+g(ba); Simultaneously set warm start X base ←X * Call Eval(d, X) base ). If the Φ of the new candidate point is better than the current best, then update (τ). * X * ). If the interval length satisfies (ba) < ε r If the outer iteration is terminated, then the outer iteration will be terminated. Step 4: Output Output τ * And give X * (τ * ), welfare SW (τ) * ) and budget expenditure Spend(τ * ) and minimum quality

[0038] The algorithm is represented by a flowchart as follows: Figure 3 As shown. Beneficial effects:

[0039] Compared with existing technologies, this invention constructs a two-layer optimization model that combines upper-level policy decision-making with lower-level market response. Under fiscal budget constraints, it systematically optimizes the intensity of computing power voucher subsidies, avoiding the subjectivity and uncertainty inherent in traditional methods that rely on experience-based judgment or simple proportional allocation. This makes subsidy strategy formulation more scientific and rational. Simultaneously, by establishing a linkage between subsidy policies and enterprise computing power investment behavior, this invention allows subsidy intensity to directly influence enterprise resource allocation decisions and further impact product quality and market demand. This achieves synergistic optimization between policy decisions and market operation, effectively overcoming the problem of policy formulation and market response being disconnected in existing technologies. Furthermore, by introducing fiscal budget constraints and quality constraints into the model, this invention can improve the efficiency of subsidy fund utilization under limited resource conditions, enhance the incentive effect of policies on enterprise computing power investment, and conduct multi-scenario analysis under different parameter conditions to assess the impact of subsidy strategy changes on social welfare and market operation. This provides quantitative decision-making basis for policy formulation and has good scalability and applicability, making it applicable to fields such as computing power resource allocation optimization and digital economy-related policy design. Attached image description: Figure 1 A diagram of the supply chain network structure; Figure 2 This is a diagram illustrating the two-tiered linkage mechanism under the computing power voucher subsidy scheme. Figure 3 A flowchart for the solution algorithm; Figure 4 The diagram shows a computing network structure as an example. Detailed implementation method:

[0040] This section considers a bilevel programming problem consisting of a top-level government and a bottom-level supply chain network comprising two raw material suppliers, two computing power service providers, two manufacturers, and two demand markets to construct a numerical example. In this bilevel programming problem, the top-level government encourages manufacturers to engage in data transactions with computing power service providers by issuing computing power vouchers that subsidize manufacturers. The subsidy ratio is τ, where τ∈[0,1]. Furthermore, the government's subsidy funds are not unlimited; here, it is assumed that the government's investment budget is constrained to 1 million yuan, and the minimum quality level of products entering the market is... θ =1.5. The upper-level decision problem is how to determine the optimal subsidy ratio to maximize social welfare. In the lower-level problem, each manufacturer can purchase raw materials from any raw material supplier and can choose to submit data to any computing power service provider for calculation; the finished products of the manufacturers can be launched into multiple demand markets. The specific network structure diagram is as follows. Figure 4 As shown.

[0041] The relevant cost settings in the lower-level supply chain network are as follows: First, the raw material suppliers. The raw material production costs of the two raw material suppliers are expressed as follows: The transaction costs for the two raw material suppliers and the two manufacturers are expressed as follows: For computing power service providers, the computational costs of data are expressed as follows: The relationship between data and information can be represented as follows: For manufacturers, the manufacturing cost of raw materials can be expressed as: The response time cost of computation consists of the time it takes for data to be transmitted to the computing power service provider, the waiting time for data computation, and the time it takes for the computation completion information to be transmitted back to the manufacturer, and can be expressed as: In a demand market, the inverse demand function can be expressed as: The transaction costs between manufacturers and the demand market can be expressed as: Based on the above parameter settings, this chapter uses the proposed algorithm to numerically solve the bilevel programming model. First, a one-dimensional search in the outer layer determines the feasible subsidy range that satisfies the fiscal budget and minimum quality constraints, and then further searches for the optimal subsidy intensity in a social welfare sense within this range. For each candidate subsidy level, the inner layer solves the variational inequality equilibrium of the supply chain network using the projection method. This method allows for the simultaneous acquisition of the optimal policy parameters at the upper level and the equilibrium flow, price, and quality levels at the lower level, as shown in Table 1 below. Table 1. Network Equilibrium Results of Computing Power Voucher Subsidies As shown in Table 1, under the given network structure and parameter settings, the optimal subsidy intensity of the upper-level government is τ. * =0.23, corresponding to a social welfare level of 91738.12. The initial social welfare of the system before subsidies was 91572.57. The subsidy policy increased social welfare by approximately 1655, indicating that computing power voucher subsidies can, to some extent, incentivize manufacturers to increase computing power investment by reducing the cost of computing power usage for enterprises, thereby improving the overall system efficiency and welfare level. The total subsidy expenditure in equilibrium is 1 million yuan, and the government budget is fully utilized, demonstrating that under the current parameter environment, computing power voucher subsidies can significantly improve the overall system efficiency, making the investment of fiscal resources have a clear effect on improving social welfare. In equilibrium, the manufacturer's raw material flow from suppliers is 46.40, and the product flow to various demand markets is also 46.40. This result indicates that the manufacturer's production scale is jointly determined by market demand and production costs, forming a stable equilibrium within the network. Simultaneously, when allocating products between markets, the manufacturer uses market price signals for equilibrium distribution, thus maintaining a stable supply and demand relationship between different markets. Regarding computing power resource usage, the manufacturer submits 7.38 units of data computation to the computing power service provider. This computing power investment enters the production stage through the data-information transformation process, further improving the manufacturer's product quality. Ultimately, both manufacturers achieve a product quality level of 12.07, demonstrating that manufacturers utilize computing power resources for data analysis and information processing, thereby improving the efficiency of production decisions and the manufacturing process, leading to improved final product quality. This improved product quality further influences market prices through the demand function, resulting in an equilibrium price of 818.1 for both demand markets. Since the market demand function includes product quality factors, when manufacturers improve product quality through computing power investment, consumers' willingness to pay for the product increases, thus driving up the market equilibrium price. This result reflects the transmission mechanism described by the model, from the input of computing power to the improvement of product quality, which in turn affects demand and leads to price increases. In other words, computing power, as a digital production factor, can indirectly improve manufacturers' revenue and market performance by improving product quality. In summary, the results of this example validate the core economic mechanism depicted by the model: the government incentivizes manufacturers to increase computing power usage through computing power voucher subsidies; once these resources enter the production system, they improve product quality; and this quality improvement, in turn, affects market prices and manufacturer profits through demand-side responses, ultimately leading to increased social welfare. Meanwhile, a comparison with the scenario without subsidies reveals that while subsidies do improve system configuration efficiency under the current parameters, the improvement is relatively limited, indicating that the effectiveness of subsidies is constrained to some extent by budget limits.

Claims

1. A method for calculating computing power voucher subsidies based on supply chain network equilibrium, the method comprising the following technical features: ( 1) Construct a lower-level computing power supply chain network that includes raw material supply nodes, manufacturing nodes, computing power service nodes, and demand market nodes, and define the product flow, data flow, and computing power flow relationships between nodes; (2) Obtain policy parameters and market parameters, wherein the policy parameters include fiscal budget constraints and the range of subsidy intensity values, and the market parameters include computing power cost, demand function parameters and quality parameters; (3) Construct a two-layer optimization model, in which the upper-layer model describes the government subsidy decision-making process. With the goal of maximizing social welfare, the intensity of computing power voucher subsidies is determined under the conditions of fiscal budget constraints and preset constraints. The upper-layer model can be expressed as follows: i j ≥ θ 0≤T≤1, Where SW represents the overall social welfare level of the system; τ represents the subsidy intensity parameter of the computing power voucher, and q js This represents the data transaction volume between JavaScript files; all transaction volumes form an nr-dimensional column vector. Q 1 Q represents the total volume of all raw material transactions between ij; 2 All product transaction volume between JK; p 4k (t,q js f is the product price function in the demand market k; i (Q 1 f represents the production cost function of raw materials; j (Q 1 Q 3 ) is the production cost function of the product; It refers to the cost of calculating response time between JavaScript code; c jk (Q 2 ) is the product transaction cost function between manufacturers and the demand market; c ij (Q 1 This represents the raw material transaction cost function between raw material suppliers and manufacturers. It is the data computation cost function of computing power service provider s; ρ 3js This represents the price of data transactions between enterprises and computing power service providers; D represents the upper limit of the government's fiscal budget. θ j This indicates the product quality level after the computing power voucher subsidy; θ Indicates the minimum quality level of the product; (4) Construct a lower-level market response model to describe the computing power resource allocation behavior of enterprises under a given subsidy policy. By establishing the correlation between computing power input and product quality and market demand, the market equilibrium result is obtained. The lower-level model can be expressed by variational inequalities as follows: Where q ij q represents the volume of raw material transactions between ij; jk Indicates the product transaction volume between j and k; q js This represents the amount of data exchanged between JavaScript files. This represents the raw material transaction volume between ij under equilibrium conditions, and all transaction volumes form an mn-dimensional column vector. This represents the product transaction volume between j and k under equilibrium conditions, with all transaction volumes forming a no-dimensional column vector. This represents the data transaction volume between JavaScript servers under equilibrium conditions. All transaction volumes form an nr-dimensional column vector. f i (Q 1* f represents the production cost function of raw materials; j (Q 1* Q 3* ) is the production cost function of the product; It's a function for calculating response time costs between JavaScript instances; c jk (Q 2* () is a product transaction cost function between manufacturers and the demand market. λ represents the raw material transaction costs between raw material suppliers and manufacturers. j They are Lagrange multipliers; These are the values ​​of the Lagrange multipliers under equilibrium conditions; It is the data computation cost function of computing power service provider s; ρ 4k It is the product price in the demand market, k. It represents the product price in the demand market (k) under equilibrium conditions. It is the market demand for the product under equilibrium conditions; (5) Based on the relationship between the upper-level model and the lower-level model, establish the coupling relationship between policy decision-making and market response; (6) The two-layer optimization model is solved by using a numerical optimization method that alternates between upper and lower layers to obtain the optimal computing power voucher subsidy strategy that satisfies the constraints. (7) Output the optimal subsidy strategy and the corresponding market equilibrium result.

2. The method for calculating computing power voucher subsidies based on supply chain network equilibrium according to claim 1, characterized in that, The numerical optimization method described in step (6) that involves alternating solutions at different levels has the following specific steps: (6.1) Parameter setting and initialization: Set the maximum number of iterations k for the lower-level projection algorithm. max Given a convergence threshold ε and a fixed step size a = 1 / L based on the estimated Lipschitz constant L of the operator F(·τ), and a given upper limit of the government budget D and minimum quality requirements. θ And set the termination precision ε of the upper-level one-dimensional search. τ and the maximum number of iterations T max Initialize the lower-level decision variable vector And through the nonnegative projection operator ∏ K (·) Project it onto the feasible region K = {X | X ≥ 0} to obtain the initial feasible point for the lower-level equilibrium solution. At the same time, initialize the upper-level policy search interval as [a, b] = [0, 1] and set the golden ratio constant. (6.2) Solving the lower-level network equilibrium given the subsidy intensity τ: Given the policy parameter τ, the lower-level problem is formulated as the following variational inequality problem: Find X * (τ)∈K, such that Among them, the subsidy intensity τ enters the Q of the operator F(·r) by affecting the computing power cost and response transmission cost perceived by the manufacturer. 3 The components, specifically, are related marginal cost terms weighted by a coefficient (1-τ), thereby altering the manufacturer's optimal decision regarding computing power utilization. To solve the above variational inequality, the algorithm employs a fixed-step projection algorithm for iterative updates: X (t+1) =∏ K (X (t) -αF(X (t) ;t)), With ||X (t+1) -X (t) || 2 <ε is used as a convergence criterion; if the maximum number of iterations k is reached... max If the above conditions are met, the lower-level equilibrium solution under the given τ is considered successful; otherwise, the lower-level problem under τ is determined to have failed to converge, and the solution process continues. (6.3) Social welfare calculation and feasibility test: When the lower layer converges under a given τ and obtains the equilibrium solution X * After (τ), the corresponding social welfare level is first calculated based on the equilibrium result: SW(τ)6CS(Q 2* (τ),Q 3* )-C(Q 1* ,Q 2* (τ),Q 3* ), Further calculations of government fiscal expenditure: And calculate the system's minimum mass level θ min If Spend(T)≤D, then θ min If the value of the subsidy is greater than or equal to θ, then the subsidy intensity τ is determined to be a feasible policy, and the corresponding social welfare SW(τ) is recorded; otherwise, the τ is considered not to meet the policy constraints and is excluded in the upper-level search process. (6.4) One-dimensional search for the outer subsidy intensity: Under the constraint of the feasibility verification mechanism, this chapter adopts the golden section method to perform a one-dimensional search for the subsidy intensity τ in the interval [a, b]. Specifically, two internal test points are constructed. c = bg(ba), d = a + g(ba), Steps two and three are then performed on τ=c and τ=d respectively to obtain their corresponding feasibility assessment results and social welfare levels. If the welfare evaluation of point c is not lower than that of point d, the search interval is updated to [a, d]; otherwise, it is updated to [c, b]. The above process is repeated until the interval length satisfies (ba) < ε. r Or reach the maximum number of iterations T max The optimal subsidy intensity τ is finally obtained. * and the lower-level equilibrium solution X induced by it. * (τ * ); (6.5) Output Results: Output the optimal policy parameter τ * , corresponding equilibrium decision variables Social welfare SW(τ) * ), fiscal expenditure Spend(τ) * and minimum quality level

Images