Equipment extension warranty service pricing method for different purchase opportunities

Abstract

The invention discloses an equipment extended warranty service pricing method aiming at different purchase opportunities, and the method provides two strategies according to different time points, and the two strategies comprise an SEW (Security Enforcement Warning) extended warranty strategy and a DEW (Data Enforcement Warning) extended warranty strategy. The extended warranty service pricing method comprises the following specific steps: S1, constructing respective consumer utility functions under the condition that a manufacturer and a retailer provide extended warranty services at the same time under an SEW condition and a DEW condition; s2, further constructing extended insurance demand functions for purchasing extended insurance under the SEW condition and the DEW condition; s3, profit functions are respectively determined according to the demand functions; s4, respectively substituting the functions into a profit function in an SEW condition and a DEW condition, and solving optimal solutions of the functions about decision variables; and S5, comparing the optimal extended insurance prices and profits of the manufacturers and the retailers under the SEW condition and the DEW condition. According to the invention, a traditional sales mode of providing the extended warranty service at a single time point is broken through, and differentiated extended warranty services are provided for different types of consumption subjects in a customized manner.

Description

Technical Field

[0001] The present invention relates to the technical field of equipment extended warranty pricing methods, and in particular to a pricing method for equipment extended warranty services at different purchase times. Background Art

[0002] Among various product services, extended warranty services have quickly gained widespread attention from consumers once they were launched. Extended warranty is a typical additional service plan in which suppliers promise to repair or replace products for consumers. Unlike the basic warranty provided by manufacturers with their products, extended warranty provides consumers with additional insurance and is usually sold separately. As a typical after-sales service, extended warranty has greatly improved the profitability of manufacturers.

[0003] However, in previous studies, most scholars considered the situation where the extended warranty is sold together with the product, and few studies considered the extended warranty pricing decisions at different purchase times. In addition, some studies considered the situation where extended warranty services are provided at the end of the basic warranty period of the product, but only considered the situation where the manufacturer provides the extended warranty alone, and did not study the situation where both the manufacturer and the retailer provide extended warranty services to consumers at the same time. Therefore, considering the situation where both the manufacturer and the retailer provide extended warranty services at the same time, studying different extended warranty purchase time points for the optimal pricing strategy in the supply chain is still an urgent problem to be solved.

[0004] The defects of existing service pricing methods are:

[0005] 1. The pricing method for equipment extended warranty service in the prior art only considers the pricing decision under a single extended warranty sales model, and does not consider the extended warranty pricing decision at multiple purchase time points;

[0006] 2. Application document CN116976929A proposes a service pricing method and device, electronic device, and storage medium. However, the pricing method of the patent document does not take into account the situation where manufacturers and retailers provide extended warranty services to consumers.

[0007] 3. Application document CN114243687A discloses a pricing method for wind power spinning reserve services based on cost and benefit. However, the service pricing method of the patent document does not consider combining the failure rate with the purchasing choices of different types of consumers to make pricing choices. Summary of the invention

[0008] The purpose of the present invention is to provide a pricing method for equipment extended warranty service at different purchase times to solve the problems raised in the above background technology.

[0009] To achieve the above object, the present invention provides the following technical solutions: a pricing method for equipment extended warranty service at different purchase times, wherein the extended warranty service sales supply chain of the extended warranty service pricing method is composed of dual channels of manufacturers and retailers, and the extended warranty service pricing method provides two strategies according to different time points, including SEW extended warranty strategy and DEW extended warranty strategy, wherein the SEW extended warranty strategy is sold together with the product when the product is sold, and the DEW extended warranty strategy is sold just after the basic warranty period of the product ends:

[0010] The specific steps of this extended warranty service pricing method are as follows:

[0011] S1. In the two cases of SEW extended warranty and DEW extended warranty, construct the consumer utility functions when the manufacturer and the retailer provide extended warranty services at the same time;

[0012] S2. Based on the consumer utility function and the product demand function D, further construct the extended warranty demand function for purchasing extended warranty from manufacturers and retailers in the SEW and DEW scenarios respectively;

[0013] S3. Based on the above demand function, determine the profit functions of the manufacturer and the retailer in the SEW case and the DEW case respectively;

[0014] S4. In the SEW and DEW cases, substitute the above demand functions into the profit functions of the manufacturer and the retailer respectively, and find the optimal solution with respect to the decision variables;

[0015] S5. Numerical calculation and numerical simulation experiments are used to compare the optimal extended warranty pricing of manufacturers and retailers under the SEW and DEW scenarios, as well as their respective profits.

[0016] Preferably, if the manufacturer holds a dominant position in the supply chain, the manufacturer first determines its own extended warranty price P when the product starts to be sold. m , the price w of wholesale products to retailers, the basic warranty period L and the total repair period T, TL represents the extended warranty coverage purchased by consumers, and then the retailer determines its extended warranty price P r and the product's sales price P, c represents the cost coefficient of warranty. The higher c means the lower the cost efficiency and the higher the cost. The warranty cost is assumed to increase quadratically during the warranty period. Therefore, the cost of extended warranty is c(T 2 -L 2 ), the consumer’s sensitivity to product prices is λ, the difference coefficient of consumer utility between manufacturers and retailers is μ, the initial product reliability is x, and the consumer’s utility coefficient per unit of extended warranty coverage is θ iAccording to the principle that product demand D decreases linearly with product price P and increases linearly with basic warranty period L, we construct the product demand function: D=1+L-λP.

[0017] Preferably, in S1, the manufacturer's consumer utility in the SEW case is:

[0018]

[0019] The retailer’s consumer utility under the SEW scenario is:

[0020]

[0021] Among them, the division of market demand is mainly determined by consumers' risk preference;

[0022] The manufacturer's consumer utility in the DEW case is:

[0023]

[0024] The retailer’s consumer utility under the DEW scenario is:

[0025]

[0026] In the DEW case, since different extended warranty purchase plans are provided, f(x) is used to represent the expected product failure rate at the end of the basic warranty period. By comparing the actual number of product failures n during the basic warranty period with the expected number of product failures xL, the probability density function F(x) that obeys the Weibull distribution is referenced to construct the probability density function f(x) = x + x(1-F(n));

[0027] Some consumers in the DEW case will buy at the first time point, which is consistent with the SEW case;

[0028] In the DEW case, some consumers will purchase extended warranty services at a second point in time, that is, at the end of the basic warranty period.

[0029] Preferably, in S2, under the SEW scenario, the demand function for purchasing extended warranty from the manufacturer is The demand function for extended warranties purchased from retailers is

[0030] In the DEW case, the extended warranty demand function for purchasing extended warranty from the manufacturer at the second time point is: The demand function for extended warranties purchased from retailers is

[0031] Preferably, in S3, the profit functions of the manufacturer and the retailer in the SEW case and the DEW case are specifically:

[0032] Manufacturer profit function:

[0033]

[0034] Retailer profit function:

[0035]

[0036] Preferably, in S4, in the SEW case, the specific steps for solving the optimal solution of the decision variables are as follows:

[0037] Step 1: Solve the retailer’s profit function with respect to the decision variables product price P and retailer’s extended warranty price P r The first and second derivatives of and the Hessian matrix H1 are obtained;

[0038]

[0039] Step 2: Find the odd-order and even-order determinants of the Hessian matrix H1. If the odd-order determinant is less than zero and the even-order determinant is greater than zero, it is determined that the retailer's profit function is jointly concave with respect to the decision variables and there is an optimal solution. Otherwise, there is no optimal solution.

[0040] Step 3: If there is an optimal solution, set the first-order derivative equal to zero to obtain the functional expression of the retailer's decision variables;

[0041] Step 4: Use the reverse induction method to substitute the equilibrium decision expression into the manufacturer's profit function, and follow steps 1-3 again to determine whether the manufacturer's profit function has an optimal solution for the decision variables and solve the manufacturer's optimal solution.

[0042] Step 5: Substitute into step 3 to get the function expression for the retailer, and finally get the optimal solution for the retailer

[0043] Preferably, in S4, in the DEW case, the specific steps for solving the optimal solution of the decision variables are as follows:

[0044] Step 1: Solve the retailer’s profit function with respect to the decision variables p, p r The first and second derivatives of , we get the Hessian matrix H2;

[0045]

[0046] Step 2: Find the odd-order and even-order determinants of the Hessian matrix H2. If the odd-order determinant is less than 0 and the even-order determinant is greater than 0, it is determined that the retailer's profit function is jointly concave with respect to the decision variables and there is an optimal solution. Otherwise, there is no optimal solution.

[0047] Step 3: If there is an optimal solution, set the first-order derivative equal to 0 to obtain the function expression of the retailer's decision variables;

[0048] Step 4: Use the reverse induction method to bring the equilibrium decision expression into the manufacturer's profit function, and then follow steps 1-3 to determine whether the manufacturer's profit function has an optimal solution for the decision variables and solve the manufacturer's optimal solution.

[0049] Step 5: Substitute into step 3 to get the retailer's function expression, and finally get the retailer's optimal solution

[0050] Preferably, in S5, the manufacturer's extended warranty pricing under the SEW situation is studied by numerical calculation. Extended Warranty Pricing at Retailers and product pricing P S The relationship between the product failure rate x and the optimal extended warranty pricing and optimal profit of manufacturers and retailers in the SEW situation leads to the conclusion that:

[0051] ① and It shows that under the SEW strategy, as the failure rate of the product increases, the retail price of the product will decrease, while the extended warranty price offered by the manufacturer and the retailer will increase;

[0052] ② and It shows that in this case, manufacturers tend to set higher extended warranty prices than retailers, and the optimal profit of the former will also be higher than that of the latter.

[0053] Preferably, in S5, the manufacturer's extended warranty pricing under the SEW situation is studied by numerical calculation. Extended Warranty Pricing at Retailers and product pricing P D The relationship between the expected failure rate of the product f(x) and the profit comparison between the manufacturer and the retailer in the SEW and DEW scenarios lead to the conclusion that:

[0054] ① and It shows that in the case of DEW, product pricing will decrease as the expected failure rate of the product increases, while extended warranty pricing will increase instead;

[0055] ② and This shows that whether it is a manufacturer or a retailer, choosing the DEW strategy to provide extended warranty services will result in higher profits than choosing the SEW strategy.

[0056] Preferably, in S5, the numerical simulation experiment method is to implement the numerical simulation experiment through MATLAB software, and compare the optimal profits under the SEW and DEW strategies based on the MATLAB simulation experiment results.

[0057] Compared with the prior art, the present invention has the following beneficial effects:

[0058] 1. Aiming at the defect of single selection of extended warranty service for products in the current market, the present invention constructs a service model with multiple extended warranty purchase time options to design customized extended warranty sales strategies, provides retailers and manufacturers with new ideas for extended warranty sales, and at the same time meets the needs of consumers and markets with different types of risk preferences, breaks the traditional sales model of providing extended warranty services at a single time point, and realizes customized provision of differentiated extended warranty services for different types of consumers; in the high-end equipment sales service market, it greatly improves consumers' willingness to purchase extended after-sales warranty services, reduces the inventory costs of decision-makers in the supply chain such as manufacturers and retailers, meets the purchasing needs of specific groups, expands the sales volume of extended warranty services, and solves the defects of the single extended warranty sales model existing in actual enterprises, and is widely used.

[0059] 2. The present invention constructs a strategy in which manufacturers and retailers provide extended warranty at the same time and allow the service to be provided at multiple time points (DEW), and innovatively introduces the difference coefficient μ of consumer utility between manufacturers and retailers to take into account the differences in consumers' choice of extended warranty services between the two. μ (0<μ<1) can reflect the difference in extended warranty prices set by manufacturers and retailers. The present invention provides extended warranty sales services based on multiple extended warranty purchase time points, and constructs a corresponding pricing model considering the situation where manufacturers and retailers provide extended warranty at the same time, aiming to solve practical problems and provide solutions and plans for them, which is more realistic and more suitable for actual scenarios.

[0060] 3. The present invention studies the pricing decisions (DEW) at multiple extended warranty purchase time points, and when considering purchasing an extended warranty at the second time point, consumers often make the next choice based on the failure rate of the product within the basic warranty period. Therefore, the present invention specifically proposes an expected failure rate f(x) of the product that is different from the product failure rate x at a single time point. By comparing the actual number of product failures n within the basic warranty period with the expected number of product failures xL, and citing the probability density function F(x) that obeys the Weibull distribution, the consumer's expected probability of product failure f(x) = x+x(1-F(n)) is constructed, which provides a more accurate and differentiated theoretical basis for the study of extended warranty purchase choices of different types of consumers, thereby better helping manufacturers and retailers and even the entire supply chain to make better pricing decisions. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] Figure 1 is a flow chart of the method of the present invention;

[0062] Figure 2 is a supply chain model diagram of the present invention;

[0063] Figure 3 The consumer purchase tendency diagram of the SEW situation of the present invention;

[0064] Figure 4 The DEW scenario consumer purchasing tendency diagram of the present invention;

[0065] Figure 5 This is a diagram showing the effect of SEW wholesale price on pricing in the present invention;

[0066] Figure 6 This is a diagram showing the effect of SEW wholesale price on profit of the present invention;

[0067] Figure 7 It is the influence diagram of the wholesale price of DEW on pricing of the present invention;

[0068] Figure 8 This is a diagram showing the impact of the DEW wholesale price on profits according to the present invention. DETAILED DESCRIPTION

[0069] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0070] The following combination Figure 1-Figure 4 , the technical solution of the present invention is further explained:

[0071] The embodiment provided by the present invention is a pricing method for equipment extended warranty service at different purchase times. The extended warranty service sales supply chain of the extended warranty service pricing method is composed of dual channels of manufacturers and retailers. The extended warranty service pricing method provides two strategies according to different time points, including SEW extended warranty strategy and DEW extended warranty strategy. The SEW extended warranty strategy is sold together with the product when the product is sold, and the DEW extended warranty strategy is sold just after the basic warranty period of the product ends:

[0072] The specific steps of this extended warranty service pricing method are as follows:

[0073] S1. In the two cases of SEW extended warranty and DEW extended warranty, construct the consumer utility functions when the manufacturer and the retailer provide extended warranty services at the same time;

[0074] S2. Based on the consumer utility function and the product demand function D, further construct the extended warranty demand function for purchasing extended warranty from manufacturers and retailers in the SEW and DEW scenarios respectively;

[0075] S3. Based on the above demand function, determine the profit functions of the manufacturer and the retailer in the SEW case and the DEW case respectively;

[0076] S4. In the SEW and DEW cases, substitute the above demand functions into the profit functions of the manufacturer and the retailer respectively, and find the optimal solution with respect to the decision variables;

[0077] S5. Numerical calculation and numerical simulation experiments are used to compare the optimal extended warranty pricing of manufacturers and retailers under the SEW and DEW scenarios, as well as their respective profits.

[0078] Furthermore, in a dual-channel extended warranty service sales supply chain consisting of a manufacturer and a retailer, there are two different extended warranty provision strategies at different time points: one is the SEW extended warranty strategy, and the other is the DEW extended warranty strategy. The SEW extended warranty strategy refers to selling the extended warranty service together with the product when the product is sold. This is the case of providing extended warranty services at a single time point. The DEW extended warranty strategy is to sell the extended warranty service just after the basic warranty period of the product ends. This is the case of providing extended warranty services at multiple time points. Assuming that the manufacturer occupies a dominant position in the supply chain, when the product starts to be sold, the manufacturer first determines the data including the extended warranty price P m , the wholesale price w of the product to the retailer, the basic warranty period L and the total repair period T, where T represents the total coverage of the basic warranty and extended warranty, and TL represents the extended warranty coverage purchased by the consumer; the retailer then determines its extended warranty price P r and the product's sales price P; the cost coefficient of warranty is represented by c. The higher the value of c, the lower the cost efficiency and the greater the cost. Assuming that the warranty cost increases quadratically during the warranty period, the cost of extended warranty is c(T 2 -L 2 ), the consumer's sensitivity to product prices is λ, the difference coefficient of consumer utility between manufacturers and retailers is μ, the initial product reliability is x, and the consumer's utility coefficient per unit of extended warranty coverage is θ i , because product demand D decreases linearly with product price P, and increases linearly with basic warranty period L, that is, D = 1 + L-λP.

[0079] The specific steps of this method are as follows:

[0080] S1. In the two cases of providing extended warranty at a single time point (SEW) and providing extended warranty at multiple time points (DEW), the consumer utility functions of manufacturers and retailers are constructed respectively when they provide extended warranty services at the same time:

[0081] The manufacturer's consumer utility in the SEW case is:

[0082]

[0083] The retailer’s consumer utility under the SEW scenario is:

[0084]

[0085] The division of market demand is mainly determined by the degree of consumer risk preference: in the SEW situation, risk-averse consumers will choose to purchase extended warranty services when they initially purchase the product. Specifically, consumers will purchase extended warranty from retailers or manufacturers; otherwise, they will not purchase extended warranty.

[0086] The manufacturer's consumer utility in the DEW case is:

[0087]

[0088] The retailer’s consumer utility under the DEW scenario is:

[0089]

[0090] In the DEW case, since different extended warranty purchase plans are provided at this time, consumers will consider the number of product failures xL during the basic warranty period when making a choice. Therefore, f(x) is used to represent the expected product failure rate at the end of the basic warranty period. By comparing the actual number of product failures n during the basic warranty period with the expected number of product failures xL, the probability density function F(x) that obeys the Weibull distribution is referenced to construct the probability density function f(x)=x+x(1-F(n)).

[0091] Consumers in the DEW case will buy at the first time point, which is consistent with the SEW case;

[0092] Some consumers will purchase extended warranty services at a second point in time, which is the end of the basic warranty period. Specifically, consumers will purchase extended warranty services from retailers at the second point in time, or they may purchase extended warranty services from manufacturers at the second point in time. In this case, consumers did not choose to purchase extended warranty services at the first point in time, which is when they purchased the product.

[0093] S2, based on the consumer utility function and product demand function D, we can then model

[0094] SEW case: the need to purchase extended warranty from the manufacturer

[0095] The need to purchase extended warranty from retailers

[0096] In the case of DEW:

[0097] Extended warranty needs purchased from the manufacturer at a second point in time

[0098] The need to purchase extended warranty from retailers

[0099] S3. Based on the above demand function, determine the profit functions of manufacturers and retailers in the SEW and DEW scenarios respectively.

[0100] Manufacturer profit function:

[0101]

[0102] Retailer profit function:

[0103]

[0104] S4. In the SEW and DEW cases, the above demand functions are substituted into the profit functions of the manufacturer and the retailer respectively, and the optimal solution for the decision variables is obtained. The specific solution steps are as follows:

[0105] ①SEW situation:

[0106] Step 1: Solve the retailer’s profit function with respect to the decision variables product price P and retailer’s extended warranty price P r The first and second derivatives of and the Hessian matrix H1 are obtained;

[0107]

[0108] Step 2: Find the odd-order and even-order determinants of the Hessian matrix H1. If the odd-order determinant is less than zero and the even-order determinant is greater than zero, it is determined that the retailer's profit function is jointly concave with respect to the decision variables and there is an optimal solution. Otherwise, there is no optimal solution.

[0109] Step 3: If there is an optimal solution, set the first-order derivative equal to zero to obtain the functional expression of the retailer's decision variables;

[0110] Step 4: Use the reverse induction method to substitute the equilibrium decision expression into the manufacturer's profit function, and follow steps 1-3 again to determine whether the manufacturer's profit function has an optimal solution for the decision variables and solve the manufacturer's optimal solution.

[0111] Step 5: Substitute into step 3 to get the function expression for the retailer, and finally get the optimal solution for the retailer

[0112] ②DEW situation:

[0113] Step 1: Solve the retailer’s profit function with respect to the decision variables p, p r The first and second derivatives of , we get the Hessian matrix H2;

[0114]

[0115] Step 2: Find the odd-order and even-order determinants of the Hessian matrix H2. If the odd-order determinant is less than 0 and the even-order determinant is greater than 0, it is determined that the retailer's profit function is jointly concave with respect to the decision variables and there is an optimal solution. Otherwise, there is no optimal solution.

[0116] Step 3: If there is an optimal solution, set the first-order derivative equal to 0 to obtain the function expression of the retailer's decision variables;

[0117] Step 4: Use the reverse induction method to bring the equilibrium decision expression into the manufacturer's profit function, and then follow steps 1-3 to determine whether the manufacturer's profit function has an optimal solution for the decision variables and solve the manufacturer's optimal solution.

[0118] Step 5: Substitute into step 3 to get the retailer's function expression, and finally get the retailer's optimal solution

[0119] S5. Compare the optimal extended warranty pricing of manufacturers and retailers under two different situations, as well as their respective profits. The comparison method uses numerical calculation and numerical simulation experiments.

[0120] (1) Using numerical calculations to study the manufacturer's extended warranty pricing in the SEW case Extended Warranty Pricing at Retailers and product pricing P S The relationship between the product failure rate x and the optimal extended warranty pricing and optimal profit of manufacturers and retailers in the SEW situation leads to the conclusion that:

[0121] ① and It shows that under the SEW strategy, as the failure rate of the product increases, the retail price of the product will decrease, while the extended warranty price offered by the manufacturer and the retailer will increase;

[0122] ② and It shows that in this case, manufacturers tend to set higher extended warranty prices than retailers, and the optimal profit of the former will also be higher than that of the latter.

[0123] (2) Also study the manufacturer’s extended warranty pricing in the SEW case Extended Warranty Pricing at Retailers and product pricing P D The relationship between the expected failure rate of the product f(x) and the profit comparison between the manufacturer and the retailer in the SEW and DEW scenarios:

[0124] ① and It shows that in the case of DEW, product pricing will decrease as the expected failure rate of the product increases, while extended warranty pricing will increase instead;

[0125] ② and This shows that whether it is a manufacturer or a retailer, choosing the DEW strategy to provide extended warranty services will result in higher profits than choosing the SEW strategy.

[0126] (3) MATLAB software is used to implement numerical simulation experiments to compare the optimal profits under SEW and DEW strategies. Combining the actual situation and related assumptions, T = 3, L = 1, λ = 0.5, x = 0.3, c = 0.06, k = 1.5, β = 2.2 are set, and the parameters are entered into MATLAB software to study the models under different strategies. The following results are obtained. Figure 5-8 Through numerical experiments, it can be obtained that in both cases, the extended warranty pricing of manufacturers and retailers will first increase and then decrease with the increase of wholesale price w; and when the wholesale price of the product is too low, the manufacturer is generally in a non-profitable state. However, as w gradually increases, the manufacturer's profit steadily increases, while the retailer's profit gradually decreases or even presents a loss state.

[0127] Through numerical calculation and simulation experiments, the present invention has found that for all parties in the supply chain, setting multiple extended warranty purchase time options is more beneficial. At the same time, the high profit margin brought by the extended warranty not only stimulates manufacturers to continue to provide services other than basic warranty, but also attracts retailers to provide their own extended warranty services. Therefore, the model that considers manufacturers and retailers providing extended warranty services at the same time is absolutely practical and has good practical applicability.

[0128] In studying the service strategy of manufacturers and retailers offering extended warranties at multiple time points at the same time, the present invention focuses on the differentiated choices of consumers in the extended warranty services provided by the two. By introducing the relevant variable, namely the difference coefficient of consumer utility between manufacturers and retailers, and expressing it with the difference in extended warranty prices provided by the two, the types of consumers in the market are further subdivided: choosing to buy from manufacturers or retailers at the first or second time point, or choosing not to buy extended warranty services; the present invention considers the types of market consumers more comprehensively, and the pricing decision model for supply chain entities established based on this can better help manufacturers and retailers in the actual market to price and make decisions on extended warranty services, provide them with new extended warranty service options, and better improve their competitiveness in the market;

[0129] The present invention considers the pricing decision of extended warranty services under the condition that manufacturers and retailers provide extended warranty services at the same time, taking into account multiple extended warranty purchase options and market segmentation, and provides theoretical guidance for the design of the optimal pricing strategy for equipment extended warranty services.

[0130] It will be apparent to those skilled in the art that the invention is not limited to the details of the exemplary embodiments described above and that the invention can be implemented in other specific forms without departing from the spirit or essential features of the invention. Therefore, the embodiments should be considered exemplary and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description, and it is intended that all variations falling within the meaning and scope of the equivalent elements of the claims be included in the invention. Any reference numeral in a claim should not be considered as limiting the claim to which it relates.

Claims

1. A pricing method for equipment extended warranty service at different purchase times, characterized in that: The extended warranty service sales supply chain of the extended warranty service pricing method is composed of dual channels of manufacturers and retailers. The extended warranty service pricing method provides two strategies according to different time points, including SEW extended warranty strategy and DEW extended warranty strategy. The SEW extended warranty strategy is sold together with the product when the product is sold, and the DEW extended warranty strategy is sold just after the basic warranty period of the product ends: The specific steps of this extended warranty service pricing method are as follows: S1. In the two cases of SEW extended warranty and DEW extended warranty, construct the consumer utility functions when the manufacturer and the retailer provide extended warranty services at the same time; S2. Based on the consumer utility function and the product demand function D, further construct the extended warranty demand function for purchasing extended warranty from manufacturers and retailers in the SEW and DEW scenarios respectively; S3. Based on the above demand function, determine the profit functions of the manufacturer and the retailer in the SEW case and the DEW case respectively; S4. In the SEW and DEW cases, substitute the above demand functions into the profit functions of the manufacturer and the retailer respectively, and find the optimal solution with respect to the decision variables; S5. Numerical calculation and numerical simulation experiments are used to compare the optimal extended warranty pricing of manufacturers and retailers under the SEW and DEW scenarios, as well as their respective profits.

2. The pricing method for equipment extended warranty service at different purchase times according to claim 1, characterized in that: If the manufacturer holds a dominant position in the supply chain, when the product starts to be sold, the manufacturer first determines its own extended warranty price P m , the price w of wholesale products to retailers, the basic warranty period L and the total repair period T, TL represents the extended warranty coverage purchased by consumers, and then the retailer determines its extended warranty price P r and the product's sales price P, c represents the cost coefficient of warranty. The higher c means the lower the cost efficiency and the higher the cost. The warranty cost is assumed to increase quadratically during the warranty period. Therefore, the cost of extended warranty is c(T 2 -L 2 ), the consumer’s sensitivity to product prices is λ, the difference coefficient of consumer utility between manufacturers and retailers is μ, the initial product reliability is x, and the consumer’s utility coefficient per unit of extended warranty coverage is θ i According to the principle that product demand D decreases linearly with product price P and increases linearly with basic warranty period L, we construct the product demand function: D=1+L-λP.

3. The pricing method for equipment extended warranty service at different purchase times according to claim 1, characterized in that: In S1, the manufacturer's consumer utility under SEW is: The retailer’s consumer utility under the SEW scenario is: Among them, the division of market demand is mainly determined by consumers' risk preference; The manufacturer's consumer utility in the DEW case is: The retailer’s consumer utility under the DEW scenario is: In the DEW case, since different extended warranty purchase plans are provided, f(x) is used to represent the expected product failure rate at the end of the basic warranty period. By comparing the actual number of product failures n during the basic warranty period with the expected number of product failures xL, the probability density function F(x) that obeys the Weibull distribution is referenced to construct the probability density function f(x) = x + x(1-F(n)); Some consumers in the DEW case will buy at the first time point, which is consistent with the SEW case; In the DEW case, some consumers will purchase extended warranty services at a second point in time, that is, at the end of the basic warranty period.

4. The pricing method for equipment extended warranty service for different purchase occasions according to claim 1, characterized in that: In S2, under the SEW scenario, the demand function for purchasing extended warranty from the manufacturer is The demand function for extended warranties purchased from retailers is In the DEW case, the extended warranty demand function for purchasing extended warranty from the manufacturer at the second time point is: The demand function for extended warranties purchased from retailers is 5. The pricing method for equipment extended warranty service at different purchase times according to claim 1, characterized in that: In S3, the profit functions of manufacturers and retailers in the SEW and DEW scenarios are as follows: Manufacturer profit function: Retailer profit function:

6. The pricing method for equipment extended warranty service at different purchase times according to claim 1, characterized in that: In S4, under the SEW situation, the specific steps for solving the optimal solution of decision variables are as follows: Step 1: Solve the retailer’s profit function with respect to the decision variables product price P and retailer’s extended warranty price P r The first and second derivatives of and the Hessian matrix H1 are obtained; Step 2: Find the odd-order and even-order determinants of the Hessian matrix H1. If the odd-order determinant is less than zero and the even-order determinant is greater than zero, it is determined that the retailer's profit function is jointly concave with respect to the decision variables and there is an optimal solution. Otherwise, there is no optimal solution. Step 3: If there is an optimal solution, set the first-order derivative equal to zero to obtain the functional expression of the retailer's decision variables; Step 4: Use the reverse induction method to substitute the equilibrium decision expression into the manufacturer's profit function, and follow steps 1-3 again to determine whether the manufacturer's profit function has an optimal solution for the decision variables and solve the manufacturer's optimal solution. Step 5: Substitute into step 3 to get the function expression for the retailer, and finally get the optimal solution for the retailer 7. The pricing method for equipment extended warranty service at different purchase times according to claim 1, characterized in that: In S4, under the DEW situation, the specific steps for solving the optimal solution of decision variables are as follows: Step 1: Solve the retailer’s profit function with respect to the decision variables p, p r The first and second derivatives of , we get the Hessian matrix H2; Step 2: Find the odd-order and even-order determinants of the Hessian matrix H2. If the odd-order determinant is less than 0 and the even-order determinant is greater than 0, it is determined that the retailer's profit function is jointly concave with respect to the decision variables and there is an optimal solution. Otherwise, there is no optimal solution. Step 3: If there is an optimal solution, set the first-order derivative equal to 0 to obtain the function expression of the retailer's decision variables; Step 4: Use the reverse induction method to bring the equilibrium decision expression into the manufacturer's profit function, and then follow steps 1-3 to determine whether the manufacturer's profit function has an optimal solution for the decision variables and solve the manufacturer's optimal solution. Step 5: Substitute into step 3 to get the retailer's function expression, and finally get the retailer's optimal solution 8. The pricing method for equipment extended warranty service at different purchase times according to claim 1, characterized in that: In S5, the extended warranty pricing of manufacturers in the SEW case is studied by numerical calculation. Extended Warranty Pricing at Retailers and product pricing P S The relationship between the product failure rate x and the optimal extended warranty pricing and optimal profit of manufacturers and retailers in the SEW situation leads to the conclusion that: ① It shows that under the SEW strategy, as the failure rate of the product increases, the retail price of the product will decrease, while the extended warranty price offered by the manufacturer and the retailer will increase; ② It shows that in this case, manufacturers tend to set higher extended warranty prices than retailers, and the optimal profit of the former will also be higher than that of the latter.

9. The pricing method for equipment extended warranty service at different purchase times according to claim 1, characterized in that: In S5, the extended warranty pricing of manufacturers in the SEW case is studied by numerical calculation. Retailer's extended warranty pricing r D and product pricing P D The relationship between the expected failure rate of the product f(x) and the profit comparison between the manufacturer and the retailer in the SEW and DEW scenarios lead to the conclusion that: ① It shows that in the case of DEW, product pricing will decrease as the expected failure rate of the product increases, while extended warranty pricing will increase instead; ② This shows that whether it is a manufacturer or a retailer, choosing the DEW strategy to provide extended warranty services will result in higher profits than choosing the SEW strategy.

10. The pricing method for equipment extended warranty service at different purchase times according to claim 1, characterized in that: In S5, the numerical simulation experiment method is to realize the numerical simulation experiment through MATLAB software, and compare the optimal profits under the SEW and DEW strategies based on the MATLAB simulation experiment results.

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