CN106991580A - Pricing method of the asymmetric manufacturer with combined recovery business under monopolization take-back model - Google Patents

Abstract

The invention discloses a kind of pricing method of asymmetric 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 relevant laws and regulations for promoting China's electronic product to reclaim；The concept of asymmetric manufacturer and combined recovery business are introduced in the research of waste electric products 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 asymmetric manufacturer with combined recovery business under monopolization take-back model

Technical field

The present invention relates to electronic product recycling field, in particular, more particularly to a kind of asymmetric manufacturer is with combining Reclaim 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 and electronic equipment produce.But these waste electrical and electronic equipments 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 and electronic equipment brings serious environmental problem.These reject products not only contain Have substantial amounts of reusable parts, and also have very big harm to environment, 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 causes natural resources 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 gas electric appliance and 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 and electronic equipment is developed as much as possible Source, makes its immense value contained be recycled recycling.The requirement that scrap concrete recycles the policy that is not only 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 the mistake of recycling The general manufacturing process than new product of journey saves 40%-65% cost.However, because the recovery of waste and old product is related to multi-party profit The game of beneficial body, is a quite cumbersome and uncertain process, if rational way of recycling can not be selected, enterprise is likely to Tired out by the recovery of waste product.Therefore, enterprise needs related decision guidance, and the reverse supply chain research based on game theory can So that to being engaged in, recovery remanufactures the enterprise of running or government's regulatory authorities provide certain guiding opinion.

Therefore, the research to reverse supply chain is worth with abundant research value and business economic.It can not only promote Enter the process that China builds a resource-conserving and environment-friendly society, moreover it is possible to comply with recycling economy this Major Strategic, improve whole The ability of social resources is closed, recycling economy development track can rapidly be stepped into comprehensively by promoting the development of Chinese national economy.

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 asymmetric Research between manufacturer and combined recovery business is less.Asymmetric manufacturer refers to the different system of the scale of manufacturing enterprise Business is made, combined recovery business refers to reclaim business in combination each other；Asymmetric manufacturer and combined recovery business in reverse supply chain It is two important nodes, considers respectively to participate in asymmetric manufacturer and combined recovery business, analysis reverse supply chain based on game theory The profit change of person, this is the game being related between a variety of analysis of strategies and each stakeholder, is one considerably complicated Field, with abundant theoretical research value.

In China, waste electrical and electronic equipment quantity 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 products, 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.

For manufacturer and the node for reclaiming the two reverse supply chain systems of business, the scope of the enterprise of the two nodes, city Field capacity, efficiency of operation etc. not only influence respective cost, even more the whole model selection for reclaiming industry are played most important Effect.Manufacturer need to grow with each passing hour with reclaiming the pricing strategy of business according to current conditions, select different at different conditions Pricing strategy so that the manufacturer of reverse supply chain and the profit maximization for reclaiming the two nodes of business, so just can ensure that useless Abandon electric appliance and electronic Product recycling supply chain system can health and constantly operate, and accelerate waste electrical and electronic equipment market Standardization process.Therefore, under conditions of asymmetric manufacturer, under research combined recovery business monopolization take-back model manufacturer and The pricing method of business is reclaimed, with the profit for improving manufacturer and reclaiming business, enhancing manufacturer reclaims the product remanufactured with reclaiming business Polarity has very important significance.

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 and reclaim that the profit that is obtained during remanufacturing of business is limited, manufacturer with reclaim the recovery enthusiasm of business by Influence, so that the problem of causing reverse supply chain system slower development, there is provided a kind of asymmetric 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, draw price of the manufacturer with reclaiming business under the conditions of this Tactful and optimal profit.

The present invention is achieved through the following technical solutions above-mentioned purpose：A kind of asymmetric manufacturer is with combined recovery business on ridge Pricing method under disconnected take-back model, the asymmetric manufacturer refers to the first manufacturer and second that scale is differed Manufacturer, wherein the market capacity α of the first manufacturer₁More than the market capacity α of the second manufacturer₂, combined recovery business refers to phase The first of combined recovery is carried out between mutually and reclaims business and the second recovery business, the 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 manufacturers, k represents to represent that take-back model reclaims business for first and the second recovery business is mutual during take-back model, wherein k=acsm United 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., The alternative degree of corresponding manufacturer 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 in monopolization take-back model Under 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 the monopolization take-back model of combined recovery business, first, which reclaims business and second, reclaims business to manufacturer's fee charged It is t^acsm, because the first manufacturer and the second manufacturer are asymmetric manufacturer, the market capacity α of the first manufacturer₁With second The market capacity α of manufacturer₂Differ, due to reclaiming business for combined recovery business, therefore the cost recovery η of the first recovery business_AWith Two reclaim the cost recovery η of business_BConsume identical, i.e. η_A=η_B=η, it is assumed that first reclaims the large-scale production factor θ of business_AWith second Reclaim the large-scale production factor θ of business_BIt is identical, i.e. θ_A=θ_B=θ, then the best profit function that can draw manufacturer j is：

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, t^acsmThe waste and old product of per unit are reclaimed for combined recovery business to receive to the first manufacturer and the second manufacturer The recovery cost taken, 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, in combined recovery business First recovery business and second recovery business yield be represented by：

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

Because the size 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, reclaims the expense that business reclaims consumption needed for the waste and old product of unit；θ tables Show 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 direct competitive between manufacturer； The recovery cost t that first manufacturer can collect according to a given combined recovery business to manufacturer^acsm, solve its profit of sening as an envoy to and reach To maximum selling priceTo the profit function of the first manufacturerOn decision variable p₁ ^acsmFirst derivative is sought, and It is 0 to make its first derivative, 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, combines back

Receive business yield be：

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 makes its first derivative be 0, Solve can obtain manufacturer's profit it is optimal when selling price 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 t^*acsmFor：

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

Formula (16) and formula (17) are substituted into formula (4), arrange abbreviation can in the hope of manufacturer optimal profitFor：

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

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)：

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 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 combined recovery pattern, manufacture Business j product consumptionIt must be non-negative；

Due under combined recovery pattern, 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 formula (27) and 0≤β<1, the denominator of inequality meets following relation：

Δ^acsm=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 have to meet the relations of inequality (29).

The beneficial effects of the present invention are：The present invention draws in the research of waste electric products reverse supply chain pricing strategy The concept of asymmetric manufacturer and joint manufacturer is entered, the pricing strategy of reverse supply chain has been have studied from different angles, more Plus meet the development of current reverse supply chain system；Mould is reclaimed by building asymmetric manufacturer and the monopolization of combined recovery business Formula, builds Liang Ge manufacturers under this condition and two the best profit function betting models reclaimed between business, using reverse Induction seeks equilibrium solution and optimal solution, draws pricing strategy and optimal profit of the manufacturer with reclaiming business under the conditions of this；This hair The bright optimal profit using manufacturer and recovery business is as starting point, and the result drawn more conforms to manufacturer and recovery business is personally sharp Beneficial demand, is that manufacturer and the price for reclaiming business provide reference, 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 asymmetric manufacturer of the invention with combined recovery business under monopolization take-back model is shown It is intended to.

Fig. 2 is monopolization 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 side such as reality, qualitative analysis, comparative analysis, quantitative analysis with theoretical Method is studied reverse supply chain pricing strategy model, and is tested using China's waste electrical and electronic equipment as data instance Card, with certain theory and practice meaning, provides theoretical direction basic for corresponding industry, its technology path as shown in figure 1, First by reading the situation of the current research situation of document summary and practical application, it is then based on stackelberg principal and subordinate and wins On the basis of the combined recovery pattern for remanufacturing reverse supply chain recovery models played chess, with reference to showing for domestic and international reverse supply chain Shape, sets up combined recovery pattern pricing strategy betting model；Then the Nash for solving the betting model using reverse induction is equal Weighing apparatus solution, exists and under the unique and constraints of market demand function non-negative meeting Nash equilibrium solutions, above-mentioned model is carried out Data verification is analyzed, and is drawn under the conditions of different take-back models, the condition and scheme of stakeholder's win-win are manufacturer and recovery The setting of business's pricing 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 asymmetric manufacturer of the present invention with combined recovery business under monopolization take-back model, it is described non- Symmetrical manufacturer refers to the first manufacturer and the second manufacturer that scale is differed, wherein the market of the first manufacturer holds Measure α₁More than the market capacity α of the second manufacturer₂, what combined recovery business referred to carrying out combined recovery each other first reclaims Business and second reclaims business, initially sets up the pricing strategy model of manufacturer and recovery business under combined recovery business monopolization take-back model, Pricing strategy is reclaimed as research object with two reverse supply chains for reclaiming business or combined recovery business composition using Liang Ge manufacturers, built Vertical two benches principal and subordinate's dynamic game.Wherein to reclaim leader of the business as the leader-followers games, reclaim business and first work out in the first stage The unit take-back cost collected to manufacturer, 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 directly affect it and formulate pricing strategy；Using reverse induction to the system in combined recovery business's combined recovery pattern Make business j and solved respectively with reclaiming business i optimal profit function.In Two-stage Game, first when the first manufacturer sets Selling price, the second manufacturer makes a policy according to the selling price of the first manufacturer, its profit is reached maximum, similarly, When the second manufacturer sets selling price, the first manufacturer makes a policy according to the selling price of the second manufacturer, makes its profit Profit reaches maximum, simultaneous association response function, you can draw the Nash equilibrium solutions of manufacturer's selling price, i.e. manufacturer's sale price The pricing strategy of lattice；Then, same first recovery business can draw the recovery cost collected to manufacturer with the second recovery business Nash equilibrium solutions, that is, reclaim the pricing strategy of business's recovery cost；Equilibrium solution and recovery business finally according to manufacturer's selling price The equilibrium solution of recovery cost obtains manufacturer's profit and reclaims the Nash equilibrium solutions of business's profit, i.e. manufacturer with reclaiming business's respectively Optimal profit solution.The 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 manufacturers, k represents to represent that take-back model reclaims business for first and the second recovery business is mutual during take-back model, wherein k=acsm United 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., The alternative degree of corresponding manufacturer 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 in monopolization take-back model Under 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 the monopolization take-back model of combined recovery business, first, which reclaims business and second, reclaims business to manufacturer's fee charged It is t^acsm, because the first manufacturer and the second manufacturer are asymmetric manufacturer, the market capacity α of the first manufacturer₁With second The market capacity α of manufacturer₂Differ, due to reclaiming business for combined recovery business, therefore the cost recovery η of the first recovery business_AWith Two reclaim the cost recovery η of business_BConsume identical, i.e. η_A=η_B=η, it is assumed that first reclaims the large-scale production factor θ of business_AWith second Reclaim the large-scale production factor θ of business_BIt is identical, i.e. θ_A=θ_B=θ, then the best profit function that can draw manufacturer j is：

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, t^acsmThe waste and old product of per unit are reclaimed for combined recovery business to receive to the first manufacturer and the second manufacturer The recovery cost taken, 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, in combined recovery business First recovery business and second recovery business yield be represented by：

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

Because the size 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, reclaims the expense that business reclaims consumption needed for the waste and old product of unit；θ tables Show 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 direct competitive between manufacturer. The recovery cost t that first manufacturer can collect according to a given combined recovery business to manufacturer^acsm, solve its profit of sening as an envoy to and reach To maximum selling priceTo the profit function of the first manufacturerOn decision variableFirst derivative is sought, and It is 0 to make its first derivative, 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^acsmSolution sends as an envoy to it Profit reaches the selling price of maximumTo the profit function of the second manufacturerOn decision variableSeek single order Derivative, and make its first derivative be 0, 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 makes its first derivative be 0 When, solve can obtain manufacturer's profit it is optimal when selling price 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 t^*acsmFor：

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

Formula (16) and formula (17) are substituted into formula (4), arrange abbreviation can in the hope of manufacturer optimal profitFor：

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

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)：

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 and Waste and old product rate of recovery τ must assure that the relation of formula (25).

Further, the value of the demand function of product must be non-negative, therefore, under combined recovery pattern, manufacture Business j product consumptionIt must be non-negative；

Due under combined recovery pattern, 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 formula (27) and 0≤β<1, the denominator of inequality meets following relation：

Δ^acsm=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 (28).

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 asymmetric manufacturer with combined recovery business under monopolization take-back model, the asymmetric manufacturer The first manufacturer and the second manufacturer that scale is differed are referred to, wherein the market capacity α of the first manufacturer₁More than The market capacity α of two manufacturers₂, what combined recovery business referred to carrying out combined recovery each other first reclaims business and second time Receive business, it is characterised in that 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：

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

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 is that the first recovery business and the second recovery business mutually combine during take-back model, wherein k=acsm 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：

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

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：

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

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 the monopolization take-back model of combined recovery business, the first recovery business and the second recovery business are to manufacturer's fee charged t^acsm, because the first manufacturer and the second manufacturer are asymmetric manufacturer, the market capacity α of the first manufacturer₁With the second manufacture The market capacity α of business₂Differ, due to reclaiming business for combined recovery business, therefore the cost recovery η of the first recovery business_AWith second time Receive the cost recovery η of business_BConsume identical, i.e. η_A=η_B=η, it is assumed that first reclaims the large-scale production factor θ of business_AReclaimed with second The large-scale production factor θ of business_BIt is identical, i.e. θ_A=θ_B=θ, then the best profit function that can draw manufacturer j is：

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

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, t^acsmReclaim what the waste and old product of per unit were collected to the first manufacturer and the second manufacturer for combined recovery business Recovery cost, 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, the in combined recovery business The yield of one recovery business and the second recovery business are represented by：

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

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

Because the size 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：

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

In formula, η is represented under the influence of not by large-scale production, reclaims the expense that business reclaims consumption needed for the waste and old product of unit；θ represents rule Mould economic 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^acsm, solve its profit of sening as an envoy to and reach maximum Selling priceTo the profit function of the first manufacturerOn decision variable p₁ ^acsmFirst derivative is sought, and makes its single order Derivative is 0, i.e.,

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

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

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

Similarly, the recovery cost t that the second manufacturer collects according to given one group of recovery business to manufacturer^acsmSolution is sent as an envoy to its profit Reach the selling price of maximumTo the profit function of the second manufacturerOn decision variableSeek first derivative, And make its first derivative be 0, i.e.,

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

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

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

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：

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

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：

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

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

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

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

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

To the profit function Π of combined recovery business^csOn decision variable t^csFirst derivative is sought, and makes its first derivative be 0, is solved Can obtain manufacturer's profit it is optimal when selling price t^cs：

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

Formula (13) and formula (14), which are substituted into formula (15), can obtain the Nash equilibrium solutions t of combined recovery business's recovery cost^*acsm For：

$\begin{array}{c}{t}^{*acsm}=\frac{1}{4(\mathrm{\&beta;}-1)\mathrm{\&tau;}(2-\mathrm{\&beta;}+2(\mathrm{\&beta;}-1\left){\mathrm{\&theta;\&tau;}}^2\right)}\&CenterDot;\left(2\right(\mathrm{\&beta;}-1)\left(\right(\mathrm{\&beta;}-2)(r-\mathrm{\&eta;})\mathrm{\&tau;}+(c_m-\\ \mathrm{\&sigma;}(c_m-c_r)\left)(\mathrm{\&beta;}-2-4(\mathrm{\&beta;}-1\left){\mathrm{\&theta;\&tau;}}^2\right)\right)+(\mathrm{\&beta;}-2-4{\mathrm{\&theta;\&tau;}}^2(\mathrm{\&beta;}-1\left)\right)({\mathrm{\&alpha;}}_1+{\mathrm{\&alpha;}}_2))\end{array}---\left(16\right)$

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

$\begin{array}{c}{p}_j^{*acsm}=\frac{1}{4({\mathrm{\&beta;}}^2+\mathrm{\&beta;}-2)(2-\mathrm{\&beta;}+2(\mathrm{\&beta;}-1\left){\mathrm{\&theta;\&tau;}}^2\right)}\&CenterDot;\left(2\right({\mathrm{\&beta;}}^2+\mathrm{\&beta;}-2)(c_m-\\ \mathrm{\&sigma;}(c_m-c_r)-(r-\mathrm{\&eta;})\mathrm{\&tau;})+(7\mathrm{\&beta;}-10-12(\mathrm{\&beta;}-1\left){\mathrm{\&theta;\&tau;}}^2\right){\mathrm{\&alpha;}}_j+\\ (4{\mathrm{\&beta;}}^2-5\mathrm{\&beta;}-2+4(\mathrm{\&beta;}-1\left)\right(1-2\mathrm{\&beta;}\left){\mathrm{\&theta;\&tau;}}^2\right){\mathrm{\&alpha;}}_{3-j})\end{array}---\left(17\right)$

Formula (16) and formula (17) are substituted into formula (4), arrange abbreviation can in the hope of manufacturer optimal profit For：

$\begin{array}{c}{\&Pi;}_j^{*acsm}=\frac{1}{16{({\mathrm{\&beta;}}^2-4-2({\mathrm{\&beta;}}^2+\mathrm{\&beta;}-2\left){\mathrm{\&theta;\&tau;}}^2\right)}^2}\&CenterDot;\left(2\right({\mathrm{\&beta;}}^2+\mathrm{\&beta;}-2)(c_m-\mathrm{\&sigma;}(c_m-c_r)-\\ (r-\mathrm{\&eta;})\mathrm{\&tau;})+(3\mathrm{\&beta;}-2-4\left(\mathrm{\&beta;}-1\right){\mathrm{\&theta;\&tau;}}^2){\mathrm{\&alpha;}}_j+(6-\mathrm{\&beta;}+4\left(\mathrm{\&beta;}-1\right){\mathrm{\&theta;\&tau;}}^2){\mathrm{\&alpha;}}_{3-j}{)}^2\end{array}---\left(18\right)$

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^*acsmFor：

${\&Pi;}^{*acsm}=\frac{{({\mathrm{\&alpha;}}_1+{\mathrm{\&alpha;}}_2+2(\mathrm{\&beta;}-1\left)\right(c_m-\mathrm{\&sigma;}\left(c_m-c_r\right)-\left(r-\mathrm{\&eta;}\right)\mathrm{\&tau;}\left)\right)}^2}{8(1-\mathrm{\&beta;})(2-\mathrm{\&beta;}+2(\mathrm{\&beta;}-1\left){\mathrm{\&theta;\&tau;}}^2\right)}---\left(19\right)$

2. a kind of price side of the asymmetric 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,

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

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

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

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

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 waste and old Product rate of recovery τ must assure that the relation of formula (25).

3. a kind of price side of the asymmetric 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 combined recovery pattern, manufacturer j Product consumptionIt must be non-negative；

Due under combined recovery pattern, the market demand function of manufacturer is：

$d_j^{acsm}=\mathrm{\&alpha;}+(\mathrm{\&beta;}-1)p_j^{acsm},j=1,2---\left(26\right)$

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

$d_j^{acsm}=\frac{\mathrm{\&Lambda;}}{2{\mathrm{\&Delta;}}^{acsm}},j=1,2---\left(27\right)$

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

In above-mentioned inequality (26), according to formula (27) and 0≤β<1, the denominator of inequality meets following relation：

Δ^acsm=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).