CN104850907A - New product pricing method based on backstepping method and dynamic pricing model - Google Patents

Abstract

The invention discloses a new product pricing method of brand manufacturers considering reservation and returning situations. The pricing method is suitable for the brand manufacturers with larger market share to launch a new product under the situation of allowing reservation and returning of consumers. The optimal pricing of the product is calculated via a quantitative method on the basis of historical scales of the manufacturers and the reservation requirements of the consumers through considering reservation canceling rate and returning rate after purchasing of the consumers. The output results are favorable for making the optimal new product pricing decision for the well-know brand manufacturers under the situation of considering the market requirements and the brand market share so that a scientific basis is provided to enterprises for better realizing a new product launching plan and rapid grabbing of the market share.

Description

A kind of new product pricing method based on backstepping method and dynamic pricing models

Technical field

The present invention relates to a kind of based on the new product pricing method under the customer buying behavior of backstepping method and dynamic pricing models, be applicable to affect by customer buying behavior, to the new product optimal pricing behavior that customer buying behavior reacts.

Background technology

The development of market economy exacerbates the competition between enterprise, by product iteration constantly, manufacturing enterprise guarantees that self product keeps competitive power in the market, to meet the individual demand that consumer constantly changes, it is most important that effective new product pricing strategy seizes rapidly market acquisition superiority for enterprise.In addition, enterprise is in order to promote Customer Experience more and more, in the forward direction consumer open reservation function that puts out a new product, provides simultaneously purchase rear returns policy to consumer.But existing new product pricing model does not often consider reservation and the return of goods situation of consumer, the new product pricing strategy of decision maker is difficult to accurately.

Summary of the invention

Goal of the invention: for the deficiency existed in existing new product pricing method, subscribe on the basis of demand data at historical product sales volume and consumer, the invention provides a kind of replacement new product pricing method that a kind of reservation can be cancelled, the normal sale stage can return goods based on backstepping method and dynamic pricing models.

Technical scheme: a kind of new product pricing method based on backstepping method and dynamic pricing models, consider to subscribe and return of goods situation, for the two benches new product Dynamic Pricing problem that consumer can cancel a reservation and return goods, solve the seller's optimal pricing under maximize revenue principle according to backstepping analytical approach.

Specifically comprise the steps:

1 platform database information extraction

Predefined phase consumer cannot observe product and experience and the shortage of new product information causes it uncertain often to the true value of product.In order to impel consumer to select variation, this product takes the sales mode that reservation can be cancelled, the normal sale stage can return goods.According to this marketing model, choose the former generation product of this product or like product as a reference, now choose 1,2 ..., n} (n > 3) secondary historic sales data is as the parameter definition base of this pricing model.

2 sell segmentation according to information determination new product

According to product performance and market study, determine that the sales cycle T of product (comprises presell time [0, T ₁] and normal sale time [T ₁, T]), wherein, [0, T ₁] be subscription phase, subscription price is p ₁, [T ₁, T] and be the normal sale stage, normal sale price is p ₂.Seller takes Dynamic Pricing strategy sell goods, T ₁it is the moment that price changes.

3 determine different segmentation consumer demand

Market exists two class consumers, a class customer buying behavior guards, and never pays close attention to subscription information, and only buy when product normal sale, this kind of consumer is called ordinary consumer.An other class consumer, always expects lower price during shopping, and has buy product enough suitable opportunitys that bears with, and this kind of consumer is called rational consumer.

1. the consumer of subscription phase consists of rational consumer, consumer demand amount d ₁(p ₁)=a ₁-bp ₁, wherein a ₁represent the potential a maximum demand of rational consumer, b represents the sensitivity of consumer to price.A ₁with the computing formula of b be:

a ₁＝max{A ₁,A ₂,...,A _n}

$b=\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\Σ}}}\frac{A_i-d_{1i}}{p_{1i}}$

Wherein, { A ₁, A ₂..., A _nbe a maximum demand of all previous former generation product or like product predefined phase, A _ibe a maximum demand of i-th product predefined phase, d _1ibe the actual demand amount of i-th product predefined phase, p _1iit is the actual set price of i-th product predefined phase.

2. the consumer in normal sale stage consists of ordinary consumer and selects the rational consumer of wait because of discontented subscription price, and only has when normal sale price just can not be bought higher than rational consumer during subscription price, otherwise from Jian market.Therefore, this stage demand function is d ₂(p ₁, p ₂)=a ₂-bp ₂-δ (p ₂-p ₁), wherein a ₂represent the potential a maximum demand of consumer in the normal sale stage, parameter δ reflects that rational consumer is to price deviation sensitivity, be called price deviation sensitivity coefficient, because price is the principal element affecting consumer behaviour, so there is 0 < δ < b.A ₂with the computing formula of δ be:

a ₂＝max{D ₁,D ₂,...,D _n}

$\mathrm{\&delta;}=\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\frac{D_i-d_{2i}-{\mathrm{bp}}_{2i}}{p_{2i}-p_{1i}}$

Wherein, { D ₁, D ₂..., D _nbe corresponding all previous former generation product or a maximum demand in like product normal sale stage, D _ibe a maximum demand in i-th product normal sale stage, d _2ibe the actual demand amount in i-th product normal sale stage, p _2iit is the actual set price in i-th product normal sale stage.

4 determine different segmentation purchase level according to consumer behaviour

Use I ₁t () represents that consumer is at [0, T ₁] purchase level, in this stage, consumer can cancel a reservation at any time, but cancels a reservation and need to pay the Cain r ₁.η/t is [0, T ₁] cancellation rate, consumer's cancellation rate is the subtraction function about the time, η ∈ [0,1).Use I ₂t () represents [T ₁, T] and the consumer in stage buys level, can directly perception and to-be-experienced product normal sale stage consumer, if consumer expects not mate can return goods with actual products, return of goods rate θ, 0 < θ < 1.The normal sale stage returns goods and need to pay the Cain r ₂.The computing formula of η/t and θ is:

$\mathrm{\&eta;}/t=\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i$

$\mathrm{\&theta;}=\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&theta;}}_i$

$r_1=\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{r}_i$

$r_2=\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{R}_i$

Wherein, η _ibe i-th product predefined phase quit the subscription of rate, θ _ibe the return of goods rate in i-th product normal sale stage, r _ibe the cancellation fine of i-th product predefined phase, R _iit is the return of goods fine in i-th product normal sale stage.

5 set up dealer's maximum return model

Because the variable quantity of consumer's purchase level equals consumer demand this moment, so have:

dI ₁(t)/dt＝a ₁-bp ₁-ηt ^-1I ₁(t)，0≤t≤T ₁；

dI ₂(t)/dt＝a ₂-bp ₂-δ(p ₂-p ₁)-θI ₂(t)，T ₁≤t≤T。

Purchase volume due to consumer's beginning is 0, and the purchase volume in the end of term is N, so the boundary condition of consumer's purchase level is I ₁(0)=0, I ₂(T)=N.Can obtain the time dependent purchase level of two benches respectively according to this boundary condition, the purchase level of predefined phase and normal phase is as follows:

$I_1\left(t\right)=\frac{a_{1}t}{\mathrm{\&eta;}+1}-\frac{{\mathrm{bp}}_{1}t}{\mathrm{\&eta;}+1};$

$I_2\left(t\right)=\frac{a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)}{\mathrm{\&theta;}}+e^{\mathrm{\&theta;}(T-t)}[N-\frac{a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)}{\mathrm{\&theta;}}].$

6 calculate total sales revenue according to purchase volume

Use n ^s ₁and n ^s ₂represent that consumer is at [0, T respectively ₁] and [T ₁, T] purchase volume, the purchase volume n of consumer at predefined phase can be obtained according to above required purchase level ^s ₁, total purchase volume is set to N, thus the consumer purchase volume n in normal sale stage ^s ₂=N-n ^s ₁, then purchase volume n can be calculated ^s ₁and n ^s ₂for:

${n^s}_1=\underset{0}{\overset{T_1}{\&Integral;}}{d}_1\left(p_1\right)-{\mathrm{\&eta;t}}^{-1}{I}_1\left(t\right)\mathrm{dt}=-\frac{{\mathrm{bp}}_1{T}_1}{\mathrm{\&eta;}+1}+\frac{a_1{T}_1}{\mathrm{\&eta;}+1};$

${n^s}_2=N-{n^s}_1=N+\frac{{\mathrm{bp}}_1{T}_1}{\mathrm{\&eta;}+1}-\frac{a_1{T}_1}{\mathrm{\&eta;}+1}.$

Thus, can obtain the total sales revenue of dealer is:

$R_s={n^s}_1{p}_1+{n^s}_2{p}_2=(\frac{a_1{T}_1}{\mathrm{\&eta;}+1}-\frac{{\mathrm{bp}}_1{T}_1}{\mathrm{\&eta;}+1})p_1+(N+\frac{{\mathrm{bp}}_1{T}_1}{\mathrm{\&eta;}+1}-\frac{a_1{T}_1}{\mathrm{\&eta;}+1})p_2.$

7 calculate the income obtained of canceling a reservation and return goods

N ^c ₁represent [0, T ₁] stage consumer subscribes cancellation amount, n ^c ₂represent [T ₁, T] return of goods amount of stage consumer, consumer buys level, subscribe cancellation rate and return of goods rate known, can n be obtained as calculated ^c ₁and n ^c ₂for:

${n^c}_1=\underset{0}{\overset{T_1}{\&Integral;}}{\mathrm{\&eta;t}}^{-1}{I}_1\left(t\right)\mathrm{dt}=\frac{{\mathrm{\&eta;a}}_1{T}_1}{\mathrm{\&eta;}+1}-\frac{{\mathrm{\&eta;bp}}_1{T}_1}{\mathrm{\&eta;}+1};$

${n^c}_2=\underset{T_1}{\overset{T}{\&Integral;}}{\mathrm{\&theta;I}}_2\left(t\right)\mathrm{dt}=N(e^{\mathrm{\&theta;}(t-T_1)}-1)+(T-T_1+\frac{1-e^{\mathrm{\&theta;}(T-T_1)}}{\mathrm{\&theta;}})[a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)].$

Thus, the income that dealer brings because of consumer's cancellation and the return of goods is:

$R_c={n^c}_1+{n^c}_2{r}_2=(\frac{\mathrm{\&eta;}{a}_1{T}_1}{\mathrm{\&eta;}+1}-\frac{\mathrm{\&eta;}{\mathrm{bp}}_1{T}_1}{\mathrm{\&eta;}+1})r_1+\{N(e^{\mathrm{\&theta;}(T-T_1)}-1)+(T-T_1+\frac{1-e^{\mathrm{\&theta;}(T-T_1)}}{\mathrm{\&theta;}})[a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)\left]\right\}{r}_2$

8 set up Dynamic Pricing backstepping Optimized model

3. normal sale phase optimal pricing is calculated based on local maximize revenue

The normal sale stage is when starting, subscription price p ₁determine, consumer is also according to p ₁make corresponding decision.Now the criterion of seller formulates rational selling price p ₂make this stage Income Maximum, normal sale income consists of consumer and buys commodity and obtain income two parts composition that income and consumer's return of goods obtain, then normal sale stage total revenue R ₂=n ^s ₂p ₂+ n ^c ₂r ₂, can be obtained by above result:

$R_2=(N+\frac{{{\mathrm{bp}}_{1}T}_1}{\mathrm{\&eta;}+1}-\frac{a_1{T}_1}{\mathrm{\&eta;}+1})p_2+\{N(e^{\mathrm{\&theta;}(T-T_1)}-1)+(T-T_1+\frac{1-e^{\mathrm{\&theta;}(T-T_1)}}{\mathrm{\&theta;}})[a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)\left]\right\}{r}_2$

The optimal conditions in this stage is firm sale price p ₂make this stage income R ₂maximum.

4. maximize based on total revenue and calculate presell phase optimal pricing

Predefined phase wants pricing p ₁and making following total revenue maximum, predefined phase income consists of consumer and subscribes income two parts composition that income acquired by commodity and consumer's cancellation obtain, and its income is

$R_1={n^s}_1{p}_1+{n^c}_1{r}_1=\frac{a_1{T}_1-{\mathrm{bp}}_{1}T}{\mathrm{\&eta;}+1}{p}_1+\frac{\mathrm{\&eta;}{a_{1}T}_1-\mathrm{\&eta;}{\mathrm{bp}}_1{T}_1}{\mathrm{\&eta;}+1}{r}_1$

Total revenue R=R ₁+ R ₂, the optimal conditions in this stage of income determines set price p ₁make total revenue R maximum.

$\begin{array}{c}R=\frac{a_1{T}_1-{\mathrm{bp}}_{1}T}{\mathrm{\&eta;}+1}{p}_1+\frac{{\mathrm{\&eta;a}}_1{T}_1-\mathrm{\&eta;}{\mathrm{bp}}_1{T}_1}{\mathrm{\&eta;}+1}{r}_1+(N+\frac{{\mathrm{bp}}_1{T}_1}{\mathrm{\&eta;}+1}-\frac{a_1{T}_1}{\mathrm{\&eta;}+1})p_2\\ +\{N(e^{\mathrm{\&theta;}(T-T_1)}-1)+(T-T_1+\frac{1-e^{\mathrm{\&theta;}(T-T_1)}}{\mathrm{\&theta;}})[a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)\left]\right\}{r}_2\end{array}$

9 Optimized models solve

1. outright purchase

For convenience of calculating, make T ₁=λ T, 0 < λ < 1, is changed in time from consumer's purchase volume, has I ₁(T ₁)=I ₂(T ₁), because have:

$I_1\left(t\right)=\frac{a_{1}t}{\mathrm{\&eta;}+1}-\frac{{\mathrm{bp}}_{1}t}{\mathrm{\&eta;}+1}$

$I_2\left(t\right)=\frac{a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)}{\mathrm{\&theta;}}+e^{\mathrm{\&theta;}(T-t)}[N-\frac{a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)}{\mathrm{\&theta;}}]$

So have

$\frac{a_1\mathrm{\&lambda;T}}{\mathrm{\&eta;}+1}-\frac{{\mathrm{bp}}_1\mathrm{\&lambda;T}}{\mathrm{\&eta;}+1}=\frac{a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)}{\mathrm{\&theta;}}+e^{\mathrm{\&theta;T}(1-\mathrm{\&lambda;})}[N-\frac{a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)}{\mathrm{\&theta;}}]$

Thus can in the hope of consumer's outright purchase N:

$N=\frac{{\mathrm{\&lambda;Te}}^{\mathrm{\&theta;T}(\mathrm{\&lambda;}-1)}(a_1-{\mathrm{bp}}_1)}{\mathrm{\&eta;}+1}+\frac{[1-e^{\mathrm{\&theta;T}(\mathrm{\&lambda;}-1)}][a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)]}{\mathrm{\&theta;}}$

2. normal sale stage sells price

Calculating for simplifying, making e ^{θ T (λ-1)}-1=φ, because normal sale stage total revenue R ₂for

$R_2=(N+\frac{{{\mathrm{bp}}_{1}T}_1}{\mathrm{\&eta;}+1}-\frac{a_1{T}_1}{\mathrm{\&eta;}+1})p_2+\{N(e^{\mathrm{\&theta;}(T-T_1)}-1)+(T-T_1+\frac{1-e^{\mathrm{\&theta;}(T-T_1)}}{\mathrm{\&theta;}})[a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)\left]\right\}{r}_2$

By T ₁=λ T and N brings above formula into and can obtain:

$\begin{array}{c}{R}_2=(\frac{\mathrm{\&lambda;\&phi;T}(a_1-{\mathrm{bp}}_1)}{\mathrm{\&eta;}+1}-\frac{\mathrm{\&phi;}[a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)]}{\mathrm{\&theta;}})p_2\\ +\{\frac{\mathrm{\&phi;}[a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)]}{\mathrm{\&theta;}}-\frac{\mathrm{\&lambda;\&phi;T}(a_1-{\mathrm{bp}}_1)}{\mathrm{\&eta;}+1}+T[1-\mathrm{\&lambda;}\left]\right[a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)\left]\right\}{r}_2\end{array}$

Can draw as calculated as normal sale stage total revenue R ₂have time maximum:

$p_2=-\frac{\mathrm{\&lambda;T\&phi;\&theta;}(a_1-{\mathrm{bp}}_1)-\mathrm{\&phi;}(\mathrm{\&eta;}+1)(a_2+{\mathrm{\&delta;p}}_1)+\mathrm{T\&theta;}(\mathrm{\&lambda;}-1)(\mathrm{\&eta;}+1)(b+\mathrm{\&delta;})r_2}{\mathrm{\&phi;}(\mathrm{\&eta;}+1)(b+\mathrm{\&delta;})}-r_2$

3. predefined phase set price

The optimal conditions in this stage determines set price p ₁make total revenue R maximum

$\begin{array}{c}R=\frac{a_1{T}_1-{\mathrm{bp}}_{1}T}{\mathrm{\&eta;}+1}{p}_1+\frac{{\mathrm{\&eta;a}}_1{T}_1-\mathrm{\&eta;}{\mathrm{bp}}_1{T}_1}{\mathrm{\&eta;}+1}{r}_1+(N+\frac{{\mathrm{bp}}_1{T}_1}{\mathrm{\&eta;}+1}-\frac{a_1{T}_1}{\mathrm{\&eta;}+1})p_2\\ +\{N(e^{\mathrm{\&theta;}(T-T_1)}-1)+(T-T_1+\frac{1-e^{\mathrm{\&theta;}(T-T_1)}}{\mathrm{\&theta;}})[a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)\left]\right\}{r}_2\end{array}$

By T ₁=λ T, N and e ^{θ T (λ-1)}-1=φ substitutes into above formula abbreviation and can obtain

$\begin{array}{c}R=\frac{\mathrm{\&lambda;T}(a_1-{\mathrm{bp}}_1)}{\mathrm{\&eta;}+1}{p}_1+\frac{\mathrm{\&lambda;\&eta;T}(a_1-{\mathrm{bp}}_1)}{\mathrm{\&eta;}+1}{r}_1+(\frac{\mathrm{\&lambda;\&phi;T}(a_1-{\mathrm{bp}}_1)}{\mathrm{\&eta;}+1}-\frac{\mathrm{\&phi;}[a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)]}{\mathrm{\&theta;}})p_2\\ +\{\frac{\mathrm{\&phi;}[a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)]}{\mathrm{\&theta;}}-\frac{\mathrm{\&lambda;\&phi;T}(a_1-{\mathrm{bp}}_1)}{\mathrm{\&eta;}+1}+T[1-\mathrm{\&lambda;}\left]\right[a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)\left]\right\}{r}_2\end{array}$

The optimal condition p that the normal sale stage is obtained ₂bring total revenue R into, by R (p ₁, p ₂) be converted into R (p ₁) can in the hope of the subscription price p of optimum according to first-order condition ₁.

Beneficial effect: compared with prior art, the present invention is applicable to the larger brand manufacturer of the market share in the pricing method allowing consumer to subscribe and to put out a new product in return of goods situation.Its feature is divided into by the push of new product subscribing and two stages of normal sale, according to the consumer demand in two stages with quit the subscription of, return goods and carry out Dynamic Pricing respectively, be convenient to brand manufacturer and better judge and hold risk.According to dynamic pricing models, more objectively can know consumer experience and the demand of different phase, solve new product demand uncertain problem to a certain extent, make businessman while lifting consumer experience, make and fixing a price more accurately.

Accompanying drawing explanation

Fig. 1 is method flow diagram of the present invention;

Fig. 2 is the new product selling period block plan of the embodiment of the present invention.

Embodiment

Below in conjunction with specific embodiment, illustrate the present invention further, these embodiments should be understood only be not used in for illustration of the present invention and limit the scope of the invention, after having read the present invention, the amendment of those skilled in the art to the various equivalent form of value of the present invention has all fallen within the application's claims limited range.

(1) platform database information extraction

The former generation product of this product or like product is chosen as a reference from database, now choose 1,2 ..., n} (n > 3) secondary historic sales data is as the parameter definition base of this pricing model, and the reference data concrete condition that the present embodiment is chosen is as table 1.

Table 1 former generation or like product historic sales data

(2) segmentation is sold according to information determination new product

See accompanying drawing 2, according to product performance and market study, determine in the present embodiment product sales cycle T=45 (my god).Wherein, subscription phase [1,18] sky, subscription price is p ₁, normal sale stage [18,45] sky, normal sale price is p ₂.Seller takes Dynamic Pricing strategy sell goods, T ₁=18 is moment that price changes, i.e. λ=T ₁/ T=0.4.

(3) different segmentation consumer demand is determined

1. the consumer demand of subscription phase is d ₁(p ₁)=a ₁-bp ₁, wherein a ₁represent the potential a maximum demand of rational consumer, b represents the sensitivity of consumer to price.Have according to formula:

a ₁＝max{A ₁,A ₂,...,A _n}＝A ₂＝1500

$b=\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\frac{A_i-d_{1i}}{p_{1i}}=3$

2. the consumer demand function in normal sale stage is d ₂(p ₁, p ₂)=a ₂-bp ₂-δ (p ₂-p ₁), wherein a ₂represent the potential a maximum demand of consumer in the normal sale stage, parameter δ reflects that rational consumer is to price deviation sensitivity.Have according to formula:

a ₂＝max{D ₁,D ₂,...,D _n}＝D ₄＝3500

$\mathrm{\&delta;}=\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\frac{D_i-d_{2i}-{\mathrm{bp}}_{2i}}{p_{2i}-p_{1i}}=1.5$

(4) different segmentation purchase level is determined according to consumer behaviour

Use I ₁t () represents that consumer is at [0, T ₁] purchase level, in this stage, consumer can cancel a reservation at any time, but cancels a reservation and need to pay the Cain r ₁:

η/t is [0, T ₁] cancellation rate:

Use I ₂t () represents [T ₁, T] and the consumer in stage buys level, can directly perception and to-be-experienced product normal sale stage consumer, if consumer expects not mate can return goods with actual products, return of goods rate θ: $\mathrm{\&theta;}=\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&theta;}}_i=0.1$

The normal sale stage returns goods and need to pay the Cain r ₂:

So, comprehensive above calculate result, the basic parameter situation of the present embodiment product is as shown in table 2:

The basic parameter situation that table 2 the present embodiment product is calculated according to formula

Because the variable quantity of consumer's purchase level equals consumer demand this moment, so have:

dI ₁(t)/dt＝a ₁-bp ₁-ηt ^-1I ₁(t)，0≤t≤T ₁；

dI ₂(t)/dt＝a ₂-bp ₂-δ(p ₂-p ₁)-θI ₂(t)，T ₁≤t≤T。

Purchase volume due to consumer's beginning is 0, and the purchase volume in the end of term is N, so the boundary condition of consumer's purchase level is I ₁(0)=0, I ₂(T)=N.Can obtain the time dependent purchase level of two benches respectively according to this boundary condition, the purchase level of predefined phase and normal phase is as follows:

$I_1\left(t\right)=\frac{a_{1}t}{\mathrm{\&eta;}+1}-\frac{{\mathrm{bp}}_{1}t}{\mathrm{\&eta;}+1};$

$I_2\left(t\right)=\frac{a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)}{\mathrm{\&theta;}}+e^{\mathrm{\&theta;}(T-t)}[N-\frac{a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)}{\mathrm{\&theta;}}].$

(5) total sales revenue is calculated according to purchase volume

Use n ^s ₁and n ^s ₂represent that consumer is at [0, T respectively ₁] and [T ₁, T] purchase volume, the purchase volume n of consumer at predefined phase can be obtained according to above required purchase level ^s ₁, total purchase volume is set to N, thus the consumer purchase volume n in normal sale stage ^s ₂=N-n ^s ₁, then purchase volume n can be calculated ^s ₁and n ^s ₂for:

${n^s}_1=\underset{0}{\overset{T_1}{\&Integral;}}{d}_1\left(p_1\right)-{\mathrm{\&eta;t}}^{-1}{I}_1\left(t\right)\mathrm{dt}=-\frac{{\mathrm{bp}}_1{T}_1}{\mathrm{\&eta;}+1}+\frac{a_1{T}_1}{\mathrm{\&eta;}+1};$

${n^s}_2=N-{n^s}_1=N+\frac{{\mathrm{bp}}_1{T}_1}{\mathrm{\&eta;}+1}-\frac{a_1{T}_1}{\mathrm{\&eta;}+1}.$

Thus, can obtain the total sales revenue of dealer is:

$R_s={n^s}_1{p}_1+{n^s}_2{p}_2=(\frac{a_1{T}_1}{\mathrm{\&eta;}+1}-\frac{{\mathrm{bp}}_1{T}_1}{\mathrm{\&eta;}+1})p_1+(N+\frac{{\mathrm{bp}}_1{T}_1}{\mathrm{\&eta;}+1}-\frac{a_1{T}_1}{\mathrm{\&eta;}+1})p_2.$

(6) income obtained of canceling a reservation and return goods is calculated

N ^c ₁represent [0, T ₁] stage consumer subscribes cancellation amount, n ^c ₂represent [T ₁, T] return of goods amount of stage consumer, consumer buys level, subscribe cancellation rate and return of goods rate known, can n be obtained as calculated ^c ₁and n ^c ₂for:

${n^c}_1=\underset{0}{\overset{T_1}{\&Integral;}}{\mathrm{\&eta;t}}^{-1}{I}_1\left(t\right)\mathrm{dt}=\frac{{\mathrm{\&eta;a}}_1{T}_1}{\mathrm{\&eta;}+1}-\frac{{\mathrm{\&eta;bp}}_1{T}_1}{\mathrm{\&eta;}+1};$

${n^c}_2=\underset{T_1}{\overset{T}{\&Integral;}}{\mathrm{\&theta;I}}_2\left(t\right)\mathrm{dt}=N(e^{\mathrm{\&theta;}(t-T_1)}-1)+(T-T_1+\frac{1-e^{\mathrm{\&theta;}(T-T_1)}}{\mathrm{\&theta;}})[a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)].$

Thus, the income that dealer brings because of consumer's cancellation and the return of goods is:

$R_c={n^c}_1+{n^c}_2{r}_2=(\frac{\mathrm{\&eta;}{a}_1{T}_1}{\mathrm{\&eta;}+1}-\frac{\mathrm{\&eta;}{\mathrm{bp}}_1{T}_1}{\mathrm{\&eta;}+1})r_1+\{N(e^{\mathrm{\&theta;}(T-T_1)}-1)+(T-T_1+\frac{1-e^{\mathrm{\&theta;}(T-T_1)}}{\mathrm{\&theta;}})[a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)\left]\right\}$

(7) Dynamic Pricing backstepping Optimized model is set up

1. normal sale phase optimal pricing is calculated based on local maximize revenue

The normal sale stage is when starting, subscription price p ₁determine, consumer is also according to p ₁make corresponding decision.Now the criterion of seller formulates rational selling price p ₂make this stage Income Maximum, normal sale income consists of consumer and buys commodity and obtain income two parts composition that income and consumer's return of goods obtain, then normal sale stage total revenue R ₂=n ^s ₂p ₂+ n ^c ₂r ₂, can be obtained by above result:

$R_2=(N+\frac{{{\mathrm{bp}}_{1}T}_1}{\mathrm{\&eta;}+1}-\frac{a_1{T}_1}{\mathrm{\&eta;}+1})p_2+\{N(e^{\mathrm{\&theta;}(T-T_1)}-1)+(T-T_1+\frac{1-e^{\mathrm{\&theta;}(T-T_1)}}{\mathrm{\&theta;}})[a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)\left]\right\}{r}_2$

The optimal conditions in this stage is firm sale price p ₂make this stage income R ₂maximum.

2. maximize based on total revenue and calculate presell phase optimal pricing

Predefined phase wants pricing p ₁and making following total revenue maximum, predefined phase income consists of consumer and subscribes income two parts composition that income acquired by commodity and consumer's cancellation obtain, and its income is

$R_1={n^s}_1{p}_1+{n^c}_1{r}_1=\frac{a_1{T}_1-{\mathrm{bp}}_{1}T}{\mathrm{\&eta;}+1}{p}_1+\frac{\mathrm{\&eta;}{a_{1}T}_1-\mathrm{\&eta;}{\mathrm{bp}}_1{T}_1}{\mathrm{\&eta;}+1}{r}_1$

Total revenue R=R ₁+ R ₂, the optimal conditions in this stage of income determines set price p ₁make total revenue R maximum.

$\begin{array}{c}R=\frac{a_1{T}_1-{\mathrm{bp}}_{1}T}{\mathrm{\&eta;}+1}{p}_1+\frac{{\mathrm{\&eta;a}}_1{T}_1-\mathrm{\&eta;}{\mathrm{bp}}_1{T}_1}{\mathrm{\&eta;}+1}{r}_1+(N+\frac{{\mathrm{bp}}_1{T}_1}{\mathrm{\&eta;}+1}-\frac{a_1{T}_1}{\mathrm{\&eta;}+1})p_2\\ +\{N(e^{\mathrm{\&theta;}(T-T_1)}-1)+(T-T_1+\frac{1-e^{\mathrm{\&theta;}(T-T_1)}}{\mathrm{\&theta;}})[a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)\left]\right\}{r}_2\end{array}$

(8) Optimized model solves

1. outright purchase

According to above model, can in the hope of consumer's outright purchase N:

$N=\frac{{\mathrm{\&lambda;Te}}^{\mathrm{\&theta;T}(\mathrm{\&lambda;}-1)}(a_1-{\mathrm{bp}}_1)}{\mathrm{\&eta;}+1}+\frac{[1-e^{\mathrm{\&theta;T}(\mathrm{\&lambda;}-1)}][a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)]}{\mathrm{\&theta;}}$

2. normal sale stage sells price

Calculating for simplifying, making e ^{θ T (λ-1)}-1=φ, can draw as calculated as normal sale stage total revenue R ₂have time maximum:

$p_2=-\frac{\mathrm{\&lambda;T\&phi;\&theta;}(a_1-{\mathrm{bp}}_1)-\mathrm{\&phi;}(\mathrm{\&eta;}+1)(a_2+{\mathrm{\&delta;p}}_1)+\mathrm{T\&theta;}(\mathrm{\&lambda;}-1)(\mathrm{\&eta;}+1)(b+\mathrm{\&delta;})r_2}{\mathrm{\&phi;}(\mathrm{\&eta;}+1)(b+\mathrm{\&delta;})}-r_2$

3. predefined phase set price

The optimal conditions in this stage determines set price p ₁make total revenue R maximum

$\begin{array}{c}R=\frac{\mathrm{\&lambda;T}(a_1-{\mathrm{bp}}_1)}{\mathrm{\&eta;}+1}{p}_1+\frac{\mathrm{\&lambda;\&eta;T}(a_1-{\mathrm{bp}}_1)}{\mathrm{\&eta;}+1}{r}_1+(\frac{\mathrm{\&lambda;\&phi;T}(a_1-{\mathrm{bp}}_1)}{\mathrm{\&eta;}+1}-\frac{\mathrm{\&phi;}[a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)]}{\mathrm{\&theta;}})p_2\\ +\{\frac{\mathrm{\&phi;}[a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)]}{\mathrm{\&theta;}}-\frac{\mathrm{\&lambda;\&phi;T}(a_1-{\mathrm{bp}}_1)}{\mathrm{\&eta;}+1}+T[1-\mathrm{\&lambda;}\left]\right[a_2-{\mathrm{bp}}_2-\mathrm{\&delta;}(p_2-p_1)\left]\right\}{r}_2\end{array}$

The optimal condition p that the normal sale stage is obtained ₂bring total revenue R into, by R (p ₁, p ₂) be converted into R (p ₁) can in the hope of the subscription price p of optimum according to first-order condition ₁.

Bring basic parameter data in table 2 into above Optimization Solution model, can be regarded as to obtain N ≈ 2514, p ₂≈ 33, p ₁≈ 43, namely the total sales volume of the present embodiment new product product epicycle selling period is approximately 2514, and the optimum subscription price of subscription phase is for should be decided to be 43 yuan, and the optimum selling price of formal sales stage should be decided to be 33 yuan, and the total revenue of now seller's acquisition is maximum.

Claims (1)

1., based on a new product pricing method for backstepping method and dynamic pricing models, it is characterized in that, comprise the steps:

1) platform database information extraction

Choose the former generation product of product or like product as a reference, now choose 1,2 ..., n} (n > 3) secondary historic sales data is as the parameter definition base of this pricing model;

2) segmentation is sold according to information determination new product

Determine the sales cycle T of product, wherein, [0, T ₁] be subscription phase, subscription price is p ₁, [T ₁, T] and be the normal sale stage, normal sale price is p ₂;

3) different segmentation consumer demand is determined

1. the consumer of subscription phase consists of rational consumer, consumer demand amount d ₁(p ₁)=a ₁-bp ₁, wherein a ₁represent the potential a maximum demand of rational consumer, b represents the sensitivity of consumer to price;

2. the consumer in normal sale stage consists of ordinary consumer and selects the rational consumer of wait because of discontented subscription price, and only has when normal sale price just can not be bought higher than rational consumer during subscription price, otherwise from Jian market; Therefore, this stage demand function is d ₂(p ₁, p ₂)=a ₂-bp ₂-δ (p ₂-p ₁), wherein a ₂represent the potential a maximum demand of consumer in the normal sale stage, parameter δ reflects that rational consumer is to price deviation sensitivity, be called price deviation sensitivity coefficient, because price is the principal element affecting consumer behaviour, so there is 0 < δ < b;

4) different segmentation purchase level is determined according to consumer behaviour

Use I ₁t () represents that consumer is at [0, T ₁] purchase level, in this stage, consumer can cancel a reservation at any time, but cancels a reservation and need to pay the Cain r ₁; η/t is [0, T ₁] cancellation rate, consumer's cancellation rate is the subtraction function about the time, η ∈ [0,1); Use I ₂t () represents [T ₁, T] and the consumer in stage buys level, can directly perception and to-be-experienced product normal sale stage consumer, if consumer expects not mate can return goods with actual products, return of goods rate θ, 0 < θ < 1; The normal sale stage returns goods and need to pay the Cain r ₂; The computing formula of η/t and θ is:

Wherein, η _ibe i-th product predefined phase quit the subscription of rate, θ _ibe the return of goods rate in i-th product normal sale stage, r _ibe the cancellation fine of i-th product predefined phase, R _iit is the return of goods fine in i-th product normal sale stage;

5) dealer's maximum return model is set up

Because the variable quantity of consumer's purchase level equals consumer demand this moment, so have:

dI ₁(t)/dt＝a ₁-bp ₁-ηt ^-1I ₁(t)，0≤t≤T ₁；

dI ₂(t)/dt＝a ₂-bp ₂-δ(p ₂-p ₁)-θI ₂(t)，T ₁≤t≤T。

The boundary condition of consumer's purchase level is I ₁(0)=0, I ₂(T)=N; Can obtain the time dependent purchase level of two benches respectively according to this boundary condition, the purchase level of predefined phase and normal phase is as follows:

6) total sales revenue is calculated according to purchase volume

Use n ^s ₁and n ^s ₂represent that consumer is at [0, T respectively ₁] and [T ₁, T] purchase volume, the purchase volume n of consumer at predefined phase can be obtained according to above required purchase level ^s ₁, total purchase volume is set to N, thus the consumer purchase volume n in normal sale stage ^s ₂=N-n ^s ₁, then purchase volume n can be calculated ^s ₁and n ^s ₂for:

Thus, can obtain the total sales revenue of dealer is:

7) income obtained of canceling a reservation and return goods is calculated

N ^c ₁represent [0, T ₁] stage consumer subscribes cancellation amount, n ^c ₂represent [T ₁, T] return of goods amount of stage consumer, consumer buys level, subscribe cancellation rate and return of goods rate known, can n be obtained as calculated ^c ₁and n ^c ₂for:

Thus, the income that dealer brings because of consumer's cancellation and the return of goods is:

8) Dynamic Pricing backstepping Optimized model is set up

1. normal sale phase optimal pricing is calculated based on local maximize revenue

Normal sale income consists of consumer and buys commodity and obtain the income two that income and consumer's return of goods obtain

Part composition, then normal sale stage total revenue R ₂=n ^s ₂p ₂+ n ^c ₂r ₂, can be obtained by above result:

the optimal conditions in this stage is firm sale price p ₂make this stage income R ₂maximum;

2. maximize based on total revenue and calculate presell phase optimal pricing

Predefined phase wants pricing p ₁and making following total revenue maximum, predefined phase income consists of consumer and subscribes income two parts composition that income acquired by commodity and consumer's cancellation obtain, and its income is

Total revenue R=R ₁+ R ₂, the optimal conditions in this stage of income determines set price p ₁make total revenue R maximum;

9) Optimized model solves

1. outright purchase

For convenience of calculating, make T ₁=λ T, 0 < λ < 1, is changed in time from consumer's purchase volume, has I ₁(T ₁)=I ₂(T ₁), because have:

So have

Thus can in the hope of consumer's outright purchase N:

2. normal sale stage sells price

Calculating for simplifying, making e ^{θ T (λ-1)}-1=φ, because normal sale stage total revenue R ₂for

By T ₁=λ T and N brings above formula into and can obtain:

Can draw as calculated as normal sale stage total revenue R ₂have time maximum:

3. predefined phase set price

The optimal conditions in this stage determines set price p ₁make total revenue R maximum

By T ₁=λ T, N and e ^{θ T (λ-1)}-1=φ substitutes into above formula abbreviation and can obtain

The optimal condition p that the normal sale stage is obtained ₂bring total revenue R into, by R (p ₁, p ₂) be converted into R (p ₁) can in the hope of the subscription price p of optimum according to first-order condition ₁.