CN106960368A - Pricing method of the symmetrical manufacturer with combined recovery business under monopolization take-back model - Google Patents

Abstract

The invention discloses a kind of pricing method of symmetrical manufacturer and combined recovery business under monopolization take-back model, combined recovery business monopolization take-back model pricing strategy betting model is set up；Then the Nash equilibrium solutions of the betting model are solved using reverse induction, exist and under the unique and constraints of market demand function non-negative meeting Nash equilibrium solutions, manufacturer is drawn and has reclaimed the condition and scheme of business's stakeholder's win-win, be that manufacturer and the setting of recovery business's pricing strategy and government set up and provide theoretical foundation on the laws and regulations for promoting China's electronic product to reclaim.The concept of symmetrical manufacturer and combined recovery business are introduced in the research of Waste ammunition reverse supply chain pricing strategy, the pricing strategy of reverse supply chain is have studied from different angles, the development of current reverse supply chain system is more conformed to.

Description

Pricing method of the symmetrical manufacturer with combined recovery business under monopolization take-back model

Technical field

The present invention relates to electronic product recycling field, more specifically, more particularly to a kind of symmetrical manufacturer is with combining back Receive pricing method of the business under monopolization take-back model.

Background technology

With the development and the raising of the people's living standard level of consumption of science and technology and modern industry, electric appliance and electronic production Product largely change the life style of the mankind.All kinds of electric appliance and electronic products such as household electrical appliance, mobile phone, computer not only structure Outward appearance is constantly weeded out the old and bring forth the new, intelligent level also more and more higher, causes the life cycle of present electronic product to greatly shorten, largely Waste electrical equipment electronic product produce.But these waste electrical equipment electronic products also have certain salvage value, if by this Directly abandon a bit, not only result in the huge waste of resource, but also living environment can be polluted.

On the one hand, substantial amounts of waste electrical equipment electronic product brings serious environmental problem.These waste products not only contain Have substantial amounts of reusable parts, and also have to environment very big harm, containing the heavy metals such as substantial amounts of mercury, lead with And other noxious materials, if dealt with improperly, ecological environment and human health can be constituted a serious threat.On the other hand, it is modern Consumption of the industry to raw material and the energy make it that natural resources is increasingly in short supply.The existing amount of storage of mineral resources in the whole world is non- It is often limited, and the heavy metal in waste electrical equipment electronic product is not only the regenerated resources of preciousness, also Waste ammunition Reclaim the important component of interests.If these valuable resources can be used, it can not only alleviate environmental pressure, moreover it is possible to Enough turn waste into wealth, alleviate industrial development demand and the contradiction of resource scarcity.

In face of severe shortage of resources and problem of environmental pollution, many relevant laws and regulations have been put into effect in countries in the world in succession, So that manufacturer must be born by more social responsibilities, the available money in waste electrical equipment electronic product is developed as much as possible Source, makes its immense value contained be recycled recycling.The requirement that junk recycling is not only policy is carried out, is also The trend of sustainable development, while cost can also be saved for manufacturing enterprise.With directly fabricating into new product from raw material Compare, the resources costs for carrying out recycling to waste product are lower, and energy resource consumption is smaller, according to statistics, recycling The general manufacturing process than new product of process saves 40%-65% cost.However, because the recovery of waste and old product is related in many ways The game of dom, is a quite cumbersome and uncertain process, if rational way of recycling can not be selected, enterprise very may be used It can be tied down by the recovery of waste product.Therefore, enterprise needs related decision guidance, and the reverse supply chain based on game theory is ground Study carefully can be engaged in reclaim remanufacture running enterprise or government's regulatory authorities the guiding opinion of some is provided.

Research to reverse supply chain can not only accelerate the process that China builds a resource-conserving and environment-friendly society, And recycling economy this Major Strategic can also be complied with, the integration ability of social resources is improved, promotes the hair of Chinese national economy Exhibition can rapidly step into good and fast development track comprehensively.

Highlighting due to environment and resource Wen Questions in recent years, research of the domestic and foreign scholars to reverse supply chain is more and more. The concern and research on reverse supply chain problem are mostly concentrated mainly on and set up single manufacturer, retailer and consumption at present On the basis of person, to fixed a price between maker and seller or contract formulate in terms of study, and on symmetrically make The research made between business and combined recovery business is less.Symmetrical manufacturer refers to the scale identical manufacture of manufacturing enterprise Business, combined recovery business refers to carry out each other to reclaim business in combination；Symmetrical manufacturer and combined recovery business in reverse supply chain It is two important nodes, each participant in symmetrical manufacturer and combined recovery business, analysis reverse supply chain is considered based on game theory Profit change, this is the game being related between a variety of analysis of strategies and each stakeholder, is a considerably complicated neck Domain, with abundant theoretical research value.

In China, waste electrical equipment number of electronic products is sharply increased but its recovery is extremely lack of standardization, and the enterprise for participating in reclaiming gets over Come more, and reclaim that the overall profit of industry is relatively low, how to improve the profit for reclaiming and participating in enterprise, strengthen the recovery of returned enterprise again It is a urgent problem to manufacture enthusiasm.Generally, reverse supply chain system it is main by consumer, reclaim business, These three nodes of manufacturer are constituted, wherein, first node consumer is the buyer or direct user of electric appliance and electronic product, It is the direct producer of waste and old product, while being also the beginning of reverse supply chain；It is manufacturer and consumption that second node, which reclaims business, Connection hinge between person, is the key node for promoting electronic product to reclaim；3rd node is manufacturer or importer, system The supplier that business is new product is made, the specific of process is still remanufactured and tramples passerby, manufacturer remanufactures efficiency with remanufacturing Smoothly implementation it is closely bound up.

The scope of the enterprise of the node of the two reverse supply chain systems of manufacturer with recovery business, market capacity, efficiency of operation Deng respective cost is not only influenceed, vital effect also is played to the whole model selection for reclaiming industry.Manufacturer with Reclaiming the pricing strategy of business need to be adjusted according to current conditions, and different pricing strategies are selected at different conditions so that reverse to supply Profit maximization of the manufacturer of chain with reclaiming the two nodes of business is answered, so just can ensure that what waste electrical equipment electronic product was reclaimed Reverse supply chain system can health and constantly operate, and accelerate waste electrical equipment electronic product recovery market specifications enter Journey.Therefore, under conditions of symmetrical manufacturer, the pricing method of manufacturer and recovery business under monopolization take-back model are studied, it is right Improve manufacturer has very important meaning with the profit of business, enhancing manufacturer is reclaimed with reclaiming the enthusiasm that business's recovery is remanufactured Justice.

The content of the invention

Weak it is an object of the invention to solve existing domestic consumer's environmental consciousness, reverse supply chain system is not good for also Entirely, manufacturer is limited with reclaiming the profit that business obtains during remanufacturing, and recovery of the manufacturer with reclaiming business is remanufactured actively Property is not high, so that the problem of causing reverse supply chain system slower development, there is provided a kind of symmetrical manufacturer and combined recovery business Pricing method under monopolization take-back model, builds Liang Ge manufacturers under this condition and the best profit letter of two recovery business Number betting model, equilibrium solution and optimal solution are sought using reverse induction, finally draw manufacturer with reclaiming the pricing strategy of business With optimal profit.

The present invention is achieved through the following technical solutions above-mentioned purpose：A kind of symmetrical manufacturer is with combined recovery business in monopolization Pricing method under take-back model, the symmetrical manufacturer refers to the manufacturer of scale identical first and the second manufacture Business, combined recovery business refers to that carry out combined recovery each other first reclaims business and the second recovery business, the pricing method Comprise the following steps：

1) manufacturer j market demand function is set up：Assuming that market is linear function to the demand of product, then manufacturer j Market demand function be：

In formula, j represents manufacturer, and j=1 is to represent that manufacturer j is the first manufacturer, and j=2 is to represent that manufacturer j is the Two manufactures, k represents to represent that take-back model is that the first recovery business and the second recovery business mutually combine during take-back model, wherein k=cs Monopolization take-back model, α_jRepresent manufacturer j market capacity size, α_j>0；β represents the coefficient of elasticity of the market demand, i.e., corresponding The alternative degree of manufacturer's product, 0≤β<1；

Then under monopolization take-back model, the market demand function of the first manufacturer is：

In formula, α₁The market capacity size of the first manufacturer is represented,Represent the first manufacturer under monopolization take-back model Selling price,Represent selling price of second manufacturer under monopolization take-back model；

The market demand function of second manufacturer is：

In formula, α₂Represent the market capacity size of the second manufacturer；

2) set up and reclaim business and second recovery business's cost recovery consumption and the situation of large-scale production factor all same first The best profit function of lower first manufacturer and the second manufacturer：

Under monopolization take-back model, the first recovery business and the second recovery business are t to manufacturer's fee charged^cs, due to First manufacturer and the second manufacturer are symmetrical manufacturer, then α₁=α₂=α, due to reclaiming business for combined recovery business, therefore first Reclaim the cost recovery η of business_AWith the second cost recovery η for reclaiming business_BConsume identical, i.e. η_A=η_B=η, it is assumed that first reclaims business Large-scale production factor θ_AWith the second large-scale production factor θ for reclaiming business_BIt is identical, i.e. θ_A=θ_B=θ, then can draw manufacturer j The best profit function be：

In formula, c_mRepresent manufacturer's manufacture new product cost, c_rRepresent manufacturer's manufacture reproduced goods cost, wherein c_r<c_m, σ Represent that waste and old product remanufacture rate, τ is the rate of recovery of waste and old product, and 0<τ≤1；

3) the recovery flow function for reclaiming business is set up：

Monopolization take-back model under, due to combined recovery business reclaim all manufacturers product, therefore, first reclaim business and The yield of second recovery business is represented by：

4) according to the recovery flow function for reclaiming business, the best profit function for reclaiming business is set up：

Because the number of recovery cost is relevant with product market scale economy, the rate of recovery, therefore, according to cost recovery formulaThe best profit function that combined recovery business can be drawn is：

In formula, η is represented under the influence of not by large-scale production, and first, which reclaims business and second, reclaims the waste and old product institute of business's recovery unit The expense that need to be consumed；θ represents large-scale production factor；

5) manufacturer j is solved respectively with reclaiming business i optimal profit function using reverse induction：

In the game in stage two, manufacturer formulates selling price pricing strategy according to the competition between manufacturer.First The recovery cost t that manufacturer can collect according to a given combined recovery business to manufacturer^cs, solve its profit of sening as an envoy to and reach maximum Selling priceTo the profit function of the first manufacturerOn decision variableFirst derivative is sought, and makes its single order lead Number is 0, i.e.,

To the formula solve can obtain first manufacturer's profit it is optimal when selling priceArbitrarily sold for the second manufacturer PriceThe selling price of first manufacturerThe optimal reaction functions of decision-making is：

Similarly, the recovery cost t that the second manufacturer collects according to given one group of recovery business to manufacturer^csSolution is sent as an envoy to, and it is sharp Profit reaches the selling price of maximumTo the profit function of the second manufacturerOn decision variableFirst derivative is sought, and It is 0 to make its first derivative, i.e.,

Can try to achieve second manufacturer's profit it is optimal when selling priceFor any sale price of the first manufacturer LatticeThe selling price of second manufacturerThe optimal reaction functions of decision-making is

By the first manufacturer and the reaction functions simultaneous of second manufacturer's decision optimization, i.e. simultaneous formula (8) and formula (10) the Nash equilibrium solutions for, trying to achieve the first manufacturer and second manufacturer's selling price are：

In the game in stage one, combined recovery business sets recovery cost according to the waste and old product of recovery, combined recovery business's Yield is：

Formula (11) is substituted into formula (12), the yield for obtaining combined recovery business is：

To formula (13) on t^csDerivation, can be obtained：

To the profit function Π of combined recovery business^csOn decision variable t^csFirst derivative is sought, and it is 0 to make its first derivative, Solve can obtain combined recovery business's profit it is optimal when unit take-back cost t^cs:

Formula (13) and formula (14), which are substituted into formula (15), can obtain the Nash equilibrium solutions of combined recovery business's recovery cost For：

The Nash equilibrium solutions t of obtained combined recovery business's recovery cost will be solved^*csBring into formula (11), can be in the hope of The balanced selling price of Liang Ge manufacturersWithFor：

Formula (16) and formula (17) are substituted into formula (4), arrange abbreviation can in the hope of Liang Ge manufacturers optimal profit ProfitWithRespectively：

Formula (12), formula (16) and formula (17) are brought into formula (6), obtain combined recovery business's after arranging abbreviation Optimal profit Π^*csFor：

Further, existed and unique condition according to Nash equilibrium solutions, object function must be the plan on independent variable Convex function, it is therefore desirable to which the profit function for ensureing manufacturer is the quasiconvex function on selling price, while reclaiming the profit of business Function is the quasiconvex function on recovery cost；From formula (4) as can be seen that the best profit function of manufacturerAll the time it is On decision variableConvex function, therefore, the equilibrium solution of the selling price of manufacturer is uniquely present；Can from formula (6) To find out, reclaim business profit function not be always quasiconvex function, it is necessary to constraints come ensure reclaim business profit function Quasiconvexity；

The assumed condition in business and second recovery business's the best profit function model is reclaimed according to first, it is ensured that reclaim business's It is convex function that the quasiconvexity of profit function, which is equal to the profit function for ensureing to reclaim business, that is, reclaims the profit function of business on reclaiming The second dervative of expense is less than or equal to zero, then the profit function of combined recovery business should be less than on the second dervative of recovery cost Equal to zero,

Understood according to formula (1) and formula (5)：

To ω^cs(t^cs) on t_csDerivation, can be obtained：

Formula (22) is substituted into formula (20), can be obtained after arranging abbreviation：

Wherein, Δ^cs=2- β+2 (β -1) θ τ²And (2- β)²>0、τ²>0、β-1<0, therefore：

Δ^cs=2- β+2 (β -1) θ τ²≥0 (24)

θ span can be solved from formula (24), i.e.,：

I.e. for the quasiconvexity of the profit function that ensures to reclaim business, large-scale production factor θ and market demand coefficient of elasticity β with And waste and old product rate of recovery τ must is fulfilled for the relation of formula (25).

Further, the value of the demand function of product must be non-negative, therefore, under monopolization take-back model, manufacture Business j product consumptionIt must be non-negative；

Because under monopolization take-back model, the market demand function of manufacturer is：

Formula (4) is solvedAnd substitute into formula (26), it can be obtained after arranging abbreviation：

Wherein, Λ=α+(β -1) (c_m-σ(c_m-c_r)-(r-η)τ)；

In above-mentioned inequality (26), according to 0≤β<1, the denominator of inequality meets following relation：

Δ^cs=2- β+2 (β -1) θ τ²≥0 (28)

Therefore want so that inequality (26) is set up, it is necessary to meet following condition：

Λ=α+(β -1) (c_m-σ(c_m-c_r)-(r-η)τ)>0 (29)

Therefore for ensure manufacturer j product consumptionIt must be non-negative, the equilibrium solution of discussion is saved in satisfaction In the presence of under conditions of, parameter alpha, β, c_m、c_r, σ, r, η, τ must also meet the relation of inequality (29).

The beneficial effects of the present invention are：The present invention draws in the research of Waste ammunition reverse supply chain pricing strategy The concept of symmetrical manufacturer and joint manufacturer is entered, the pricing strategy of reverse supply chain has been have studied from different angles, more The development for current reverse supply chain system of fitting；By building the monopolization take-back model of symmetrical manufacturer and combined recovery business, structure Building between Liang Ge manufacturers under this condition and two recovery business most has profit function betting model, using reverse induction Seek equilibrium solution and optimal solution, draw pricing strategy and optimal profit of the manufacturer with reclaiming business under the conditions of this；The present invention is to make The optimal profit of business and recovery business are made as starting point, the result drawn more conforms to manufacturer and recovery business and is personally beneficial to need Ask, be that manufacturer and the price for reclaiming business provide reference, therefore unprofitable mechanism or the reply of relevant Decision department are entirely returned Receipts system plays a part of supervision, according to the big small-scale difference of manufacturer, designs the recovery cost pricing strategy of differentiation Structure so that the overall interests of all stakeholder are all protected, while being set up for government on promoting waste gas electrical equipment electricity The relevant laws and regulations of sub- Product recycling provide foundation and the suggestion of theory.

Brief description of the drawings

Fig. 1 is that the flow of pricing method of the symmetrical manufacturer of the invention with combined recovery business under monopolization take-back model is illustrated Figure.

Fig. 2 is competition take-back model schematic diagram.

Embodiment

The invention will be further described below in conjunction with the accompanying drawings：

As depicted in figs. 1 and 2, the present invention combines the methods such as reality, qualitative analysis, comparative analysis to reverse confession with theory Answer chain pricing strategy model to be studied, with certain theory and practice meaning, theoretical direction base is provided for corresponding industry Plinth, its technology path are as shown in figure 1, first by reading the situation of the current research situation of document summary and practical application, so Afterwards on the basis of the monopolization take-back model for remanufacturing reverse supply chain recovery models based on stackelberg leader-followers games, knot The present situation of domestic and international reverse supply chain is closed, monopolization take-back model pricing strategy betting model is set up；Then reverse induction is utilized The Nash equilibrium solutions of the betting model are solved, the presence of Nash equilibrium solutions and the constraint of unique and market demand function non-negative is being met Under the conditions of, draw under the conditions of different take-back models, the condition and scheme of stakeholder's win-win are that manufacturer decides through consultation valency with recovery The setting of strategy and government set up provides theoretical foundation on the relevant laws and regulations for promoting China's electronic product to reclaim.

A kind of pricing method of the symmetrical manufacturer of the present invention with combined recovery business under monopolization take-back model, the symmetrical system Make business and refer to the manufacturer of scale identical first and the second manufacturer, combined recovery business refers to being joined each other Close the first of recovery and reclaim business and the second recovery business, initially set up the pricing strategy of manufacturer and recovery business under monopolization take-back model Betting model, reclaims pricing strategy as research object with two reverse supply chains for reclaiming business's composition using Liang Ge manufacturers, sets up Two benches principal and subordinate's dynamic game.Wherein to reclaim leader of the business as the leader-followers games, reclaim business first formulate in the first stage to The unit take-back cost that manufacturer collects, and manufacturer then makes the price plan of corresponding selling price according to the pricing strategy for reclaiming business Slightly, it is described in detail below：

In the stage one, the leader of leader-followers games reclaims business's setting and reclaims the recovery that the waste and old product of unit are collected to manufacturer Expense pricing strategy, in the stage two, manufacturer's setting selling price pricing strategy, wherein manufacturer are formulating selling price Consideration is needed to reclaim the influence for the recovery cost that business collects to it during pricing strategy, and exist between manufacturer and manufacturer Competition will also influence its pricing strategy.Using reverse induction to combined recovery business monopolize take-back model in manufacturer j with The optimal profit function for reclaiming business i is solved respectively.In Two-stage Game, first when the first manufacturer sets sale price Lattice, the second manufacturer makes a policy according to the selling price of the first manufacturer, its profit is reached maximum, similarly, when the second system Make business and set selling price, the first manufacturer makes a policy according to the selling price of the second manufacturer, its profit is reached most Greatly, you can draw the pricing strategy of the Nash equilibrium solutions, i.e. manufacturer's selling price of manufacturer's selling price；Then it is same, the One recovery business can draw the Nash equilibrium solutions for the recovery cost collected to manufacturer with the second recovery business, that is, reclaim business and reclaim and take Pricing strategy；The equilibrium solution of equilibrium solution and recovery business's recovery cost finally according to manufacturer's selling price obtains system respectively Make business's profit and reclaim the Nash equilibrium solutions of business's profit, i.e. manufacturer with reclaiming the optimal profit solution of business.

Specific pricing method comprises the following steps：

1) manufacturer j market demand function is set up：Assuming that market is linear function to the demand of product, then manufacturer j Market demand function be：

In formula, j represents manufacturer, and j=1 is to represent that manufacturer j is the first manufacturer, and j=2 is to represent that manufacturer j is the Two manufactures, k represents to represent that take-back model is that the first recovery business and the second recovery business mutually combine during take-back model, wherein k=cs Monopolization take-back model, α_jRepresent manufacturer j market capacity size, α_j>0；β represents the coefficient of elasticity of the market demand, i.e., corresponding The alternative degree of manufacturer's product, 0≤β<1；

Then under monopolization take-back model, the market demand function of the first manufacturer is：

In formula, α₁The market capacity size of the first manufacturer is represented,Represent the first manufacturer under monopolization take-back model Selling price,Represent selling price of second manufacturer under monopolization take-back model；

The market demand function of second manufacturer is：

In formula, α₂Represent the market capacity size of the second manufacturer；

2) set up and reclaim business and second recovery business's cost recovery consumption and the situation of large-scale production factor all same first The best profit function of lower first manufacturer and the second manufacturer：

Under monopolization take-back model, the first recovery business and the second recovery business are t to manufacturer's fee charged^cs, due to First manufacturer and the second manufacturer are symmetrical manufacturer, then α₁=α₂=α, due to reclaiming business for combined recovery business, therefore first Reclaim the cost recovery η of business_AWith the second cost recovery η for reclaiming business_BConsume identical, i.e. η_A=η_B=η, it is assumed that first reclaims business Large-scale production factor θ_AWith the second large-scale production factor θ for reclaiming business_BIt is identical, i.e. θ_A=θ_B=θ, then can draw manufacturer j The best profit function be：

In formula, c_mRepresent manufacturer's manufacture new product cost, c_rRepresent manufacturer's manufacture reproduced goods cost, wherein c_r<c_m, σ Represent that waste and old product remanufacture rate, τ is the rate of recovery of waste and old product, and 0<τ≤1；

3) the recovery flow function for reclaiming business is set up：

Monopolization take-back model under, due to combined recovery business reclaim all manufacturers product, therefore, first reclaim business and The yield of second recovery business is represented by：

4) according to the recovery flow function for reclaiming business, the best profit function for reclaiming business is set up：

Because the number of recovery cost is relevant with product market scale economy, the rate of recovery, therefore, according to cost recovery formulaIt can show that the best profit function in combined recovery business is：

In formula, η is represented under the influence of not by large-scale production, and first, which reclaims business and second, reclaims the waste and old product institute of business's recovery unit The expense that need to be consumed；θ represents large-scale production factor；

5) manufacturer j is solved respectively with reclaiming business i optimal profit function using reverse induction：

In the game in stage two, manufacturer formulates selling price pricing strategy according to the competition between manufacturer.First The recovery cost t that manufacturer can collect according to a given combined recovery business to manufacturer^cs, solve its profit of sening as an envoy to and reach maximum Selling priceTo the profit function of the first manufacturerOn decision variableFirst derivative is sought, and makes its single order lead Number is 0, i.e.,

To the formula solve can obtain first manufacturer's profit it is optimal when selling priceArbitrarily sold for the second manufacturer Price latticeThe selling price of first manufacturerThe optimal reaction functions of decision-making is：

Similarly, the recovery cost t that the second manufacturer collects according to given one group of recovery business to manufacturer^csSolution is sent as an envoy to, and it is sharp Profit reaches the selling price of maximumTo the profit function of the second manufacturerOn decision variableFirst derivative is sought, and It is 0 to make its first derivative, i.e.,

Selling price when the i.e. available second manufacturer's profit of solution is optimalFor any pin of the first manufacturer Price latticeThe selling price of second manufacturerThe optimal reaction functions of decision-making is

By the first manufacturer and the reaction functions simultaneous of second manufacturer's decision optimization, i.e. simultaneous formula (8) and formula (10) the Nash equilibrium solutions for, trying to achieve the first manufacturer and second manufacturer's selling price are：

In the game in stage one, combined recovery business sets recovery cost according to the waste and old product of recovery, combined recovery business's Yield is：

Formula (11) is substituted into formula (12), the yield for obtaining combined recovery business is：

To formula (13) on t^csDerivation, can be obtained：

To the profit function Π of combined recovery business^csOn decision variable t^csFirst derivative is sought, and it is 0 to make its first derivative, Solve can obtain combined recovery business's profit it is optimal when unit take-back cost t^cs:

Formula (13) and formula (14), which are substituted into formula (15), can obtain the Nash equilibrium solutions of combined recovery business's recovery cost For：

The Nash equilibrium solutions t of obtained combined recovery business's recovery cost will be solved^*csBring into formula (11), can be in the hope of The balanced selling price of Liang Ge manufacturersWithFor：

Formula (16) and formula (17) are substituted into formula (4), arrange abbreviation can in the hope of Liang Ge manufacturers optimal profit ProfitWithRespectively：

Formula (12), formula (16) and formula (17) are brought into formula (6), obtain combined recovery business's after arranging abbreviation Optimal profit Π^*csFor：

Further, existed and unique condition according to Nash equilibrium solutions, object function must be the plan on independent variable Convex function, it is therefore desirable to which the profit function for ensureing manufacturer is the quasiconvex function on selling price, while reclaiming the profit of business Function is the quasiconvex function on recovery cost；From formula (4) as can be seen that the best profit function of manufacturerAll the time it is On decision variableConvex function, therefore, the equilibrium solution of the selling price of manufacturer is uniquely present；Can from formula (6) To find out, reclaim business profit function not be always quasiconvex function, it is necessary to constraints come ensure reclaim business profit function Quasiconvexity；

The assumed condition in business and second recovery business's the best profit function model is reclaimed according to first, it is ensured that reclaim business's It is convex function that the quasiconvexity of profit function, which is equal to the profit function for ensureing to reclaim business, that is, reclaims the profit function of business on reclaiming The second dervative of expense is less than or equal to zero, then the profit function of combined recovery business should be less than on the second dervative of recovery cost Equal to zero,

Understood according to formula (1) and formula (5)：

To ω^cs(t^cs) on t_csDerivation, can be obtained：

Formula (22) is substituted into formula (20), can be obtained after arranging abbreviation：

Wherein, Δ^cs=2- β+2 (β -1) θ τ²And (2- β)²>0、τ²>0、β-1<0, therefore：

Δ^cs=2- β+2 (β -1) θ τ²≥0 (24)

θ span can be solved from formula (24), i.e.,：

I.e. for the quasiconvexity of the profit function that ensures to reclaim business, large-scale production factor θ and market demand coefficient of elasticity β with And waste and old product rate of recovery τ must is fulfilled for the relation of formula (25).

Further, the value of the demand function of product must be non-negative, therefore, under monopolization take-back model, manufacture Business j product consumptionIt must be non-negative；

Because under monopolization take-back model, the market demand function of manufacturer is：

Formula (4) is solvedAnd substitute into formula (26), it can be obtained after arranging abbreviation：

Wherein, Λ=α+(β -1) (c_m-σ(c_m-c_r)-(r-η)τ)；

In above-mentioned inequality (26), according to 0≤β<1, the denominator of inequality meets following relation：

Δ^cs=2- β+2 (β -1) θ τ²≥0 (27)

Therefore want so that inequality (26) is set up, it is necessary to meet following condition：

Λ=α+(β -1) (c_m-σ(c_m-c_r)-(r-η)τ)>0 (28)

Therefore for ensure manufacturer j product consumptionIt must be non-negative, the equilibrium solution of discussion is saved in satisfaction In the presence of under conditions of, parameter alpha, β, c_m、c_r, σ, r, η, τ must also meet the relation of inequality (29).

Above-described embodiment is presently preferred embodiments of the present invention, is not the limitation to technical solution of the present invention, as long as The technical scheme that can be realized without creative work on the basis of above-described embodiment, is regarded as falling into patent of the present invention Rights protection scope in.

Claims (3)

1. a kind of pricing method of symmetrical manufacturer with combined recovery business under monopolization take-back model, it is characterised in that：It is described right Manufacturer is claimed to refer to the manufacturer of scale identical first and the second manufacturer, combined recovery business refers to entering each other The first of row combined recovery reclaims business and second and reclaims business, and the pricing method comprises the following steps：

1) manufacturer j market demand function is set up：Assuming that market is to city that the demand of product is linear function, then manufacturer j Demand function is：

$\begin{array}{cc}{d}_j^k={\mathrm{\&alpha;}}_j-p_j^k+{\mathrm{\&beta;p}}_{3-j}^k& j=1,2\end{array}---\left(1\right)$

In formula, j represents manufacturer, and j=1 is to represent that manufacturer j is the first manufacturer, and j=2 is to represent that manufacturer j is the second system Business is made, k represents to represent that take-back model reclaims business for first and the second recovery business is mutually united during take-back model, wherein k=cs Monopolize take-back model, α_jRepresent manufacturer j market capacity size, α_j>0；β represents the coefficient of elasticity of the market demand, i.e., corresponding system Make the alternative degree of business's product, 0≤β<1；

Then under monopolization take-back model, the market demand function of the first manufacturer is：

$d_1^{cs}={\mathrm{\&alpha;}}_1-p_1^{cs}+{\mathrm{\&beta;p}}_2^{cs}---\left(2\right)$

In formula, α₁The market capacity size of the first manufacturer is represented,Represent pin of first manufacturer under monopolization take-back model Price lattice,Represent selling price of second manufacturer under monopolization take-back model；

The market demand function of second manufacturer is：

$d_2^{cs}={\mathrm{\&alpha;}}_2-p_2^{cs}+{\mathrm{\&beta;p}}_1^{cs}---\left(3\right)$

In formula, α₂Represent the market capacity size of the second manufacturer；

2) the is set up in the case where first reclaims business and second reclaims business's cost recovery consumption and large-scale production factor all same One manufacturer and the best profit function of the second manufacturer：

Under monopolization take-back model, the first recovery business and the second recovery business are t to manufacturer's fee charged^cs, due to first Manufacturer and the second manufacturer are symmetrical manufacturer, then α₁=α₂=α, due to reclaiming business for combined recovery business, therefore first reclaims The cost recovery η of business_AWith the second cost recovery η for reclaiming business_BConsume identical, i.e. η_A=η_B=η, it is assumed that first reclaims the rule of business Mould economic factor θ_AWith the second large-scale production factor θ for reclaiming business_BIt is identical, i.e. θ_A=θ_B=θ, then can draw manufacturer j most Good profit function is：

$\begin{array}{c}\underset{p_j^{cs}}{\max }{\mathrm{\&Pi;}}_j^{cs}=\left(p_j^{cs}-c_m\right)\left(1-\mathrm{\&sigma;}\right)\left(\mathrm{\&alpha;}-p_j^{cs}+{\mathrm{\&beta;p}}_{3-j}^{cs}\right)+\left(p_j^{cs}-c_r\right)\mathrm{\&sigma;}\left(\mathrm{\&alpha;}-p_j^{cs}+{\mathrm{\&beta;p}}_{3-j}^{cs}\right)\\ -t^{cs}\mathrm{\&tau;}\left(\mathrm{\&alpha;}-p_j^{cs}+{\mathrm{\&beta;p}}_{3-j}^{cs}\right)\\ =\left(p_j^{cs}-c_m+\mathrm{\&sigma;}\left(c_m-c_r\right)-t^{cs}\mathrm{\&tau;}\right)\left(\mathrm{\&alpha;}-p_j^{cs}+{\mathrm{\&beta;p}}_{3-j}^{cs}\right)\end{array}---\left(4\right)$

In formula, c_mRepresent manufacturer's manufacture new product cost, c_rRepresent manufacturer's manufacture reproduced goods cost, wherein c_r<c_m, σ represents Waste and old product remanufacture rate, and τ is the rate of recovery of waste and old product, and 0<τ≤1；

3) the recovery flow function for reclaiming business is set up：

Under monopolization take-back model, because combined recovery business reclaims the product of all manufacturers, therefore, first reclaims business and second The yield for reclaiming business is represented by：

${\mathrm{\&omega;}}^{cs}\left(t^{cs}\right)=\mathrm{\&tau;}\&CenterDot;(p_1^{cs}+p_2^{cs})---\left(5\right)$

4) according to the recovery flow function for reclaiming business, the best profit function for reclaiming business is set up：

Because the number of recovery cost is relevant with product market scale economy, the rate of recovery, therefore, according to cost recovery formulaThe best profit function that combined recovery business can be drawn is：

$\underset{t^{cs}}{\max }{\mathrm{\&Pi;}}^{cs}=(t^{cs}+r)\&CenterDot;{\mathrm{\&omega;}}^{cs}\left(t^{cs}\right)-\&lsqb;\mathrm{\&eta;}\&CenterDot;{\mathrm{\&omega;}}^{cs}\left(t^{cs}\right)-\mathrm{\&theta;}\&CenterDot;{\left({\mathrm{\&omega;}}^{cs}\right(t^{cs}\left)\right)}^2\&rsqb;---\left(6\right)$

In formula, η is represented under the influence of not by large-scale production, is disappeared needed for the first recovery business and second recovery business's recovery waste and old product of unit The expense of consumption；θ represents large-scale production factor；

5) manufacturer j is solved respectively with reclaiming business i optimal profit function using reverse induction：

In the game in stage two, manufacturer formulates selling price pricing strategy according to the competition between manufacturer.First manufacture The recovery cost t that business can collect according to a given combined recovery business to manufacturer^cs, solve the pin that its profit of sening as an envoy to reaches maximum Price latticeTo the profit function of the first manufacturerOn decision variableSeek first derivative, and make its first derivative be 0, i.e.,

$\frac{\&part;{\mathrm{\&Pi;}}_1^{cs}}{\&part;p_1^{cs}}=\mathrm{\&alpha;}-p_1^{cs}+{\mathrm{\&beta;p}}_2^{cs}-(p_1^{cs}-c_m+\mathrm{\&sigma;}(c_m-c_r)-t^{cs}\mathrm{\&tau;})=0---\left(7\right)$

To the formula solve can obtain first manufacturer's profit it is optimal when selling priceFor any selling price of the second manufacturerThe selling price of first manufacturerThe optimal reaction functions of decision-making is：

$p_1^{cs}\left(p_2^{cs}\right)=\frac{\mathrm{\&alpha;}+c_m-\mathrm{\&sigma;}(c_m-c_r)+{\mathrm{\&beta;p}}_2^{cs}+t^{cs}\mathrm{\&tau;}}{2}---\left(8\right)$

Similarly, the recovery cost t that the second manufacturer collects according to given one group of recovery business to manufacturer^csSolution its profit of sening as an envoy to reaches To maximum selling priceTo the profit function of the second manufacturerOn decision variableFirst derivative is sought, and makes it First derivative is 0, i.e.,

$\frac{\&part;{\mathrm{\&Pi;}}_2^{cs}}{\&part;p_2^{cs}}=\mathrm{\&alpha;}-p_2^{cs}+{\mathrm{\&beta;p}}_1^{cs}-(p_2^{cs}-c_m+\mathrm{\&sigma;}(c_m-c_r)-t^{cs}\mathrm{\&tau;})=0---\left(9\right)$

Can try to achieve second manufacturer's profit it is optimal when selling priceFor any selling price of the first manufacturer The selling price of second manufacturerThe optimal reaction functions of decision-making is

$p_2^{cs}\left(p_1^{cs}\right)=\frac{\mathrm{\&alpha;}+c_m-\mathrm{\&sigma;}(c_m-c_r)+{\mathrm{\&beta;p}}_1^{cs}+t^{cs}\mathrm{\&tau;}}{2}---\left(10\right)$

By the first manufacturer and the reaction functions simultaneous of second manufacturer's decision optimization, i.e. simultaneous formula (8) and formula (10), ask The Nash equilibrium solutions of the first manufacturer and second manufacturer's selling price are：

$p_1^{cs}=p_2^{cs}=\frac{\mathrm{\&alpha;}+c_m-\mathrm{\&sigma;}(c_m-c_r)+t^{cs}\mathrm{\&tau;}}{2}---\left(11\right)$

In the game in stage one, combined recovery business sets recovery cost, the recovery of combined recovery business according to the waste and old product of recovery Measure and be：

$\begin{array}{c}{\mathrm{\&omega;}}^{cs}\left(t^{cs}\right)=\mathrm{\&tau;}\&CenterDot;D\left(p_1^{cs},p_2^{cs}\right)=\mathrm{\&tau;}\&CenterDot;\left(p_1^{cs}+p_2^{cs}\right)\\ =\mathrm{\&tau;}\&CenterDot;\left(\mathrm{\&alpha;}-p_1^{cs}+{\mathrm{\&beta;p}}_2^{cs}+\mathrm{\&alpha;}-p_2^{cs}+{\mathrm{\&beta;p}}_1^{cs}\right)\\ =\mathrm{\&tau;}\&CenterDot;\left(2\mathrm{\&alpha;}-\left(\mathrm{\&beta;}-1\right)p_1^{cs}+\left(\mathrm{\&beta;}-1\right)p_2^{cs}\right)\end{array}---\left(12\right)$

Formula (11) is substituted into formula (12), the yield for obtaining combined recovery business is：

${\mathrm{\&omega;}}^{cs}\left(t^{cs}\right)=2\mathrm{\&tau;}\&CenterDot;(\mathrm{\&alpha;}-(\mathrm{\&beta;}-1\left)\frac{\mathrm{\&alpha;}+c_m-\mathrm{\&sigma;}(c_m-c_r)+t^{cs}\mathrm{\&tau;}}{2-\mathrm{\&beta;}}\right)---\left(13\right)$

To formula (13) on t^csDerivation, can be obtained：

$\frac{\&part;{\mathrm{\&omega;}}^{cs}}{\&part;t^{cs}}=\frac{2(\mathrm{\&beta;}-1){\mathrm{\&tau;}}^2}{2-\mathrm{\&beta;}}---\left(14\right)$

To the profit function Π of combined recovery business^csOn decision variable t^csFirst derivative is sought, and it is 0 to make its first derivative, is solved Can obtain combined recovery business's profit it is optimal when unit take-back cost t^cs：

$\frac{\&part;{\mathrm{\&Pi;}}^{cs}}{\&part;t^{cs}}={\mathrm{\&omega;}}^{cs}+(t^{cs}+r)\frac{\&part;{\mathrm{\&omega;}}^{cs}}{\&part;t^{cs}}-\mathrm{\&eta;}\frac{\&part;{\mathrm{\&omega;}}^{cs}}{\&part;t^{cs}}+2{\mathrm{\&theta;\&omega;}}^{cs}\frac{\&part;{\mathrm{\&omega;}}^{cs}}{\&part;t^{cs}}=0---\left(15\right)$

Formula (13) and formula (14), which are substituted into formula (15), can try to achieve the Nash equilibrium solutions of combined recovery business's recovery cost For：

$\begin{array}{c}{t}^{*cs}=\frac{1}{2\mathrm{\&tau;}(\mathrm{\&beta;}-1)(2-\mathrm{\&beta;}+2(\mathrm{\&beta;}-1\left){\mathrm{\&theta;\&tau;}}^2\right)}\&CenterDot;\left(\left(\mathrm{\&beta;}-2\right)\right(\mathrm{\&alpha;}+\left(\mathrm{\&beta;}-1\right)(c_m-\mathrm{\&sigma;}\left(c_m-c_r\right)\\ +\left(r-\mathrm{\&eta;}\right)\left(\mathrm{\&beta;}-1\right)\mathrm{\&tau;}\left)\right)-4{\mathrm{\&theta;\&tau;}}^2\left(\mathrm{\&beta;}-1\right)\left(\mathrm{\&alpha;}+\left(\mathrm{\&beta;}-1\right)\left(c_m-\mathrm{\&sigma;}\left(c_m-c_r\right)\right)\right)\end{array}---\left(16\right)$

The Nash equilibrium solutions t of obtained combined recovery business's recovery cost will be solved^*csBring into formula (11), can be in the hope of two The balanced selling price of manufacturerWithFor：

$\begin{array}{c}{p}_1^{*cs}=p_2^{*cs}=\frac{1}{-8+6\mathrm{\&beta;}+3{\mathrm{\&beta;}}^2-2{\mathrm{\&beta;}}^3+2{\mathrm{\&theta;\&tau;}}^2(\mathrm{\&beta;}-1)({\mathrm{\&beta;}}^2-2)}\&CenterDot;\left(\left({\mathrm{\&beta;}}^2-2\right)\right(c_m-\\ \mathrm{\&sigma;}\left(c_m-c_r\right)-\left(r-\mathrm{\&eta;}\right)\mathrm{\&tau;})+2\mathrm{\&alpha;}({\mathrm{\&beta;}}^2-3-{\mathrm{\&theta;\&tau;}}^2\left({\mathrm{\&beta;}}^2-2\right)\left)\right)\end{array}---\left(17\right)$

Formula (16) and formula (17) are substituted into formula (4), arrange abbreviation can in the hope of Liang Ge manufacturers optimal profit WithRespectively：

${\mathrm{\&Pi;}}_1^{*cs}={\mathrm{\&Pi;}}_2^{*cs}={\left(\frac{\mathrm{\&alpha;}+(\mathrm{\&beta;}-1)(c_m-\mathrm{\&sigma;}(c_m-c_r)-(r-\mathrm{\&eta;}\left)\mathrm{\&tau;}\right)}{2(2-\mathrm{\&beta;}+2(\mathrm{\&beta;}-1\left){\mathrm{\&theta;\&tau;}}^2\right)}\right)}^2---\left(18\right)$

Formula (12), formula (16) and formula (17) are brought into formula (6), the optimal of combined recovery business is obtained after arranging abbreviation Profit Π^*csFor：

${\mathrm{\&Pi;}}^{*cs}=\frac{{(\mathrm{\&alpha;}+(\mathrm{\&beta;}-1\left)\right(c_m-\mathrm{\&sigma;}\left(c_m-c_r\right)-\left(r-\mathrm{\&eta;}\right)\mathrm{\&tau;}\left)\right)}^2}{2(1-\mathrm{\&beta;})(2-\mathrm{\&beta;}+2(\mathrm{\&beta;}-1\left){\mathrm{\&theta;\&tau;}}^2\right)}---\left(19\right)$

2. a kind of price side of the symmetrical manufacturer according to claim 1 with combined recovery business under monopolization take-back model Method, it is characterised in that：Existed according to Nash equilibrium solutions and unique condition, object function must be the convex letter of plan on independent variable Number, it is therefore desirable to which the profit function for ensureing manufacturer is the quasiconvex function on selling price, while reclaiming the profit function of business It is the quasiconvex function on recovery cost；From formula (4) as can be seen that the best profit function of manufacturerAll the time be on Decision variableConvex function, therefore, the equilibrium solution of the selling price of manufacturer is uniquely present；Can from formula (6) Go out, reclaim business profit function not be always quasiconvex function, it is necessary to constraints come ensure reclaim business profit function plan Convexity；

The assumed condition in business and second recovery business's the best profit function model is reclaimed according to first, it is ensured that reclaim the profit of business It is convex function that the quasiconvexity of function, which is equal to the profit function for ensureing to reclaim business, that is, reclaims the profit function of business on recovery cost Second dervative be less than or equal to zero, then the profit function of combined recovery business should be less than being equal on the second dervative of recovery cost Zero,

$\frac{{\&part;}^2{\mathrm{\&Pi;}}^{cs}}{\&part;{\left(t^{cs}\right)}^2}=2\frac{\&part;{\mathrm{\&omega;}}^{cs}}{\&part;t^{cs}}+2\mathrm{\&theta;}{\left(\frac{\&part;{\mathrm{\&omega;}}^{cs}}{\&part;t^{cs}}\right)}^2\&le;0---\left(20\right)$

Understood according to formula (1) and formula (5)：

${\mathrm{\&omega;}}^{cs}\left(t^{cs}\right)=2\mathrm{\&tau;}\&CenterDot;(\mathrm{\&alpha;}-(\mathrm{\&beta;}-1\left)\frac{\mathrm{\&alpha;}+c_m-\mathrm{\&sigma;}\left(c_m-c_r\right)+t^{cs}\mathrm{\&tau;}}{2-\mathrm{\&beta;}}\right)---\left(21\right)$

To ω^cs(t^cs) on t^csDerivation, can be obtained：

$\frac{\&part;{\mathrm{\&omega;}}^{cs}}{\&part;t^{cs}}=\frac{2(\mathrm{\&beta;}-1){\mathrm{\&tau;}}^2}{2-\mathrm{\&beta;}}---\left(22\right)$

Formula (22) is substituted into formula (20), can be obtained after arranging abbreviation：

$\frac{4(\mathrm{\&beta;}-1){\mathrm{\&tau;}}^2{\mathrm{\&Delta;}}^{cs}}{{(2-\mathrm{\&beta;})}^2}\&le;0;---\left(23\right)$

Wherein, Δ^cs=2- β+2 (β -1) θ τ²And (2- β)²>0、τ²>0、β-1<0, therefore：

Δ^cs=2- β+2 (β -1) θ τ²≥0 (24)

θ span can be solved from formula (24), i.e.,：

$\mathrm{\&theta;}\&le;\frac{2-\mathrm{\&beta;}}{2(1-\mathrm{\&beta;}){\mathrm{\&tau;}}^2}---\left(25\right)$

I.e. for the quasiconvexity of the profit function that ensures to reclaim business, large-scale production factor θ and market demand coefficient of elasticity β and useless Old product rate of recovery τ must is fulfilled for the relation of formula (25).

3. a kind of price side of the symmetrical manufacturer according to claim 1 with combined recovery business under monopolization take-back model Method, it is characterised in that：The value of the demand function of product must be non-negative, therefore, under monopolization take-back model, manufacturer j Product consumptionIt must be non-negative；

Because under monopolization take-back model, the market demand function of symmetrical manufacturer is：

$\begin{array}{cc}{d}_j^{cs}=\mathrm{\&alpha;}+(\mathrm{\&beta;}-1)p_j^{cs}& j=1,2\end{array}---\left(26\right)$

Formula (4) is solvedAnd substitute into formula (26), it can be obtained after arranging abbreviation：

$\begin{array}{cc}{d}_j^{cs}=\frac{\mathrm{\&Lambda;}}{2{\mathrm{\&Delta;}}^{cs}}& j=1,2\end{array}---\left(27\right)$

Wherein, Λ=α+(β -1) (c_m-σ(c_m-c_r)-(r-η)τ)；

In above-mentioned inequality (26), according to 0≤β<1, the denominator of inequality meets following relation：

Δ^cs=2- β+2 (β -1) θ τ²≥0 (28)

Therefore want so that inequality (26) is set up, it is necessary to meet following condition：

Λ=α+(β -1) (c_m-σ(c_m-c_r)-(r-η)τ)>0 (29)

Therefore for ensure manufacturer j product consumptionMust be non-negative, the equilibrium solution that discussion is saved in satisfaction is present Under conditions of, parameter alpha, β, c_m、c_r, σ, r, η, τ must also meet the relation of inequality (29).