Microsoft Word - 10-ZHANG Yong.doc Advances in Systems Science and Applications (2010), Vol.10, No.1 61-66 ISSN 1078-6236 International Institute for General Systems Studies, Inc. Research on Dynamic Pricing Model Based on e-Supply Chain Management* Yong Zhang1,2, Yingjin Lu1 and Xianglan Jiang1 1school of Management, University of Electronic Science & Technology of China, Chengdu 610054, China 2 College of Information Science and Technology, Chendu University, Chengdu 610054, China Email: luyingjin@uestc.edu.cn, yingjin.lu@changhong.com Abstract With the influence of the uncertain factor in business trade, the research on dynamic pricing considering the random fluctuation of price has become an important project in managerial economics. In this paper, we introduce the uncertain factor, which produced by the random errors, into the pricing model of the dominant manufacturers. By introducing the expectation of retail price, variance and transfer price, we change the game of incomplete information into the game of complete information. Then we resolve the problem of optimal pricing about transfer price with extremum of function according to the optimal production and storage model. Keywords Random fluctuation Dynamic pricing Incomplete information game Extremum of function 1. Introduction Based on the network and electronic commerce (EC), the internet can arrive any place of the world. It results in the substantial increase of demand or supply under non-prediction. At the same time, lower menu cost supports traders to change the price more frequently according to the condition of market. So it is less and less efficient to confirm the optimal price according to establish the static demand price model under the random fluctuation of demand and price. Many scholars researched the dynamic pricing model. The article[4] discussed the dynamic pricing model of the dominant manufacturer after the time factor was added to the demand function. The article[5] suggested introducing uncertain factor into economic model and establishing the price model of rational expectation. The article[6] displayed the optimization of dynamic game and nonlinear pricing and the dominant firm price leadership model completely. Based on the above articles, this paper discuss the pricing model of the dominant manufacturer further with uncertain factor, which produced by the random errors in EC. By introducing uncertain expectation and variance, it changes the game of incomplete information into the game of complete information. Then it solves the problem of optimal pricing about transfer price with the extremum function method according to the optimal production and storage model. 2. Analyzing and Hypothesizing of the Model Assumed that a new kind of production is in monopoly position, and there is only one manufacturer and one seller. Because of the new production, the manufacturer is in dominance and can predict the demand of market. The manufacturer confirms the transfer prices of its production according to the maximum profit. Then, the seller confirms the order quantity according to the maximum profit. This process can be considered as the incomplete information game with the decentralized decision. This paper considers uncertain factor as an endogenous variable and introduce it into the model. Then it assumes the demand in consumption market is * This work was supported by Fund Item for the Doctoral Program of High Education (No. 20030614011), National Scientific Fund Item for Excellent Youth (No. 79725002), Chinese Science Fund Item for Post Doctor (No. 79725002). Zhang: Research on Dynamic Pricing Model Based on e-Supply Chain Management 62 related not only to the price, but also to the price fluctuation which is the uncertain factor of price, so it expresses price fluctuation with price variance. 3. Supply Chain Model 3.1 Optimal Order Strategy of Sellers After manufacturer price the production 1p with maximum profit according to some information, sellers choose the optimal ordering quantity. Supposed the lead time T K− is fixed, and the period of order T is constant. When sellers estimate the ordering quantity according to the profit maximization, they order T K− days in advance. Then, the products will arrive after K days. After sellers introduce uncertain factor into economic model as inner variable, they can price according to the demand of the market in internet. The demand function of consumers in this period is: 1 1( ) ( ( ), , ) ( )u u s s s s sq t D p t p u a bp t p uλ= = − + + , (1) where ( )sp t is the sell price at point t, and ( )sq t is the demand amount at point t. Both of them are random PROCESS. u sp is the variance of price (fixed), which reflects the fluctuation of price. If λ is negative, consumers hate the price undulation. If λ is positive, consumers like the price undulation. 1u is a random disturbance. And the parameters , 0a b > . (And the parameters a and b are greater than zero.) Because of the predomination of manufacturer, transferring price 1p is assured. Seller assures sell price ( )sp t , and market gives the amount of demand ( )sq t . Assumed all the demands are satisfied under ( )sp t , the order quantity in this period is the accumulation of instantaneous demand ( )sq t , that is, 0 ( ) T sQ q t dt= ∫ ， (2) The storage at point x is 0 0 0 ( ) ( ) ( ) ( ) x T x T S s s s sx Q Q q t dt q t dt q t dt q t dt= − = − =∫ ∫ ∫ ∫ ， (3) If the storage cost of unit production is 2c at unit time, the storage cost is 2 2 20 0 ( ) ( ( ) ) ( ) T T T S S s sx D C c Q x dx c q t dt dx c q t dtdx= ⋅ = =∫ ∫ ∫ ∫∫ ， 2 20 0 0 ( ( ) ) ( ) T t T s sc q t dx dt c t q t dt= = ⋅∫ ∫ ∫ , (4) where D is a triangle, surrounded by 0, , ,x x T t x t T= = = = . The purchase cost is 1 1 1 10 ( ) T b sC p Q C p q t dt C= + = +∫ , (5) where 1C is fixed purchase cost, so total cost is s b STC C C= + , (6) The revenue of the seller is sI 1 0 0 1( ) ( ) [ ( ) ] ( ) T T u s s s s s uap t q t dt q t p q t dt b b b b λ = = − + +∫ ∫ Advances in Systems Science and Applications (2010), Vol.10, No.1 63 2 1 0 1[ ( ) ( ) ( ) ( )] T u s s s s s ua q t q t p q t q t dt b b b b λ = − + +∫ , (7) And the profit of the seller in order period is s s s b sR I TC I C C= − = − − 2 1 1 2 10 0 0 1[ ( ) ( ) ( ) ( )] ( ) ( ) T T Tu s s s s s s s ua q t q t p q t q t dt p q t dt c t q t dt C b b b b λ = − + + − − × −∫ ∫ ∫ 2 1 1 2 10 1[ ( ) ( ) ( ) ( ) ( ) ( )] T u s s s s s s s ua q t q t p q t q t p q t c t q t dt C b b b b λ = − + + − − × −∫ . (8) Assumed supply quantity equal demand quantity in every period, the seller predicts and chooses supply quantity ( )sq t and its sell price ( )sp t to get maximize the profit sR . If 2 1 1 2 1( ) ( ) ( ) ( ) ( ) ( )u s s s s s s s uaF q t q t p q t q t p q t c t q t b b b b λ = − + + − − × . We will get maximum profit sR with Euler Equation. Then ( ) 0 [ ] '[ ]s s F d F q t dt q t ∂ ∂ − = ∂ ∂ (9) Solving this differential equation, we can get the sell function at point t when the profit is maximum: 1 2 1 1( ) ( ) 2 u s sq t a bp p bc t uλ= − + − + (10) *Q is the optimal order quantity for sellers after the wholesale price is determined, that is, 2 2 1 2 1 1 10 2 1 1 0.51( ) ( ) 4* 0.5 0 . u T u s s s u s a p bc T u q t dt bc T a bp p u T p bQ a p bc T u p b λ λ λ ⎧ + − + = − + − + + ≤⎪⎪= ⎨ + − +⎪ >⎪⎩ ∫ (11) 3.2 Rational Expectations of the Best Product Plan and Transferring Price That Manufacturer Determined Assumed the demand function of the market and the order interval of the seller are open. Then, the order quantity of seller can be predicted, and the transferring price 1p can be determined under the maximum profit. The manufacturer signs a contract for goods with the seller. According the contract, the seller purchase *Q productions with 1p and the manufacturer deliver at point K . For the manufacturers, they must consider sales revenue, production cost and storage cost. Sales revenue is multiplication of price and order quantity. Production cost depends on production rate, which is the quantity in unit time. And the production rate is higher, the production cost is more. Storage cost is determined by the finished productions and expiration time. ( )x t is the cumulative production of product plan until point t without consideration of assets’ time value. Because the product rate at point t (marginal product) is '( )x t , marginal product cost is ( '( ))f x t , marginal storage cost is ( ( ))g x t and marginal revenue is ( '( ))x tϕ . Then, the total cost ( ( ))C x t is Zhang: Research on Dynamic Pricing Model Based on e-Supply Chain Management 64 0 ( ( )) [ ( '( )) ( ( ))] K C x t f x t g x t dt= +∫ . (12) In order to confirm the actual expression of this function, we assume the following items: (1) The cost of increasing one production in uniform time is in proportion to the product rate, where the coefficient is 12k . So, we get 1 ( '( )) 2 '( ) '( ) df x t k x t dx t = , or, 2 1( '( )) [ '( )]f x t k x t= . (13) (2) The storage cost is in proportion to storage amount in unit time, where the coefficient is 2k , that is, 2( ( )) ( )g x t k x t= (14) The total revenue ( ( ))I x t from point 0 to point K is: 10 0 ( ( )) ( '( )) '( ) K K I x t x t dt p x t dtϕ= =∫ ∫ (15) The profit R from point 0 to point K is: 0 0 ( ( )) ( ( )) ( ( )) ( ( )) [ ( '( )) ( ( ))] K K R R x t I x t C x t x t dt f x t g x t dtϕ= = − = − +∫ ∫ 2 1 1 20 [ '( ) ' ( ) ( )] K p x t k x t k x t dt= − −∫ (16) (0) 0, ( ) *x x K Q= = (17) The problem may be attributed to find the maximum of functional ( ( ))R x t , which can be solved by variation method. If 2 1 1 2( , , ') ' 'MR t x x p x k x k x= − − , we will get the maximum profit R according to Euler Equation, we have to let: ( ) 0 ' MR d MR x dt x ∂ ∂ − = ∂ ∂ Then we can get the differential equation of second order: 2 12 ''( ) 0k k x t− + = Under the constraint condition (17), its solution is: * 2 22 1 2 1 1 4 ( ) 4 4 k k Q k K x t t t k k K − = + . (18) It’s the product plan of maximum profit. Obviously, with ( ) 0,0x t t K≥ ≤ ≤ , we can get 2 * 2 14 k K Q k ≥ . (19) That is, when the order quantity of the sellers satisfies condition (19), the product plan confirmed by condition (18) is optimal. If 2 * 2 14 k K Q k < , we can delay the start time which make the time difference K satisfying equation 2 * 2 14 k K Q k = from t1 to K, so the product plan is better. After we substitute ( )x t in the equation (16), the maximum profit is solved according to 1p and ( )x t : Advances in Systems Science and Applications (2010), Vol.10, No.1 65 3 2 * * 2 *2 2 1 1 1 ( ) 48 2m K k Kk Q k Q R p Q k K = − + − (20) Take non-zero of *Q in equation (11) into equation (20), we get 3 2 22 2 1 2 1 1 1 1 ( ) 48 2 4 u m s K k Kk R p bc T a bp p u T k λ⎛ ⎞ ⎡ ⎤= + − − + − + +⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠ 2 21 2 1 1 1 ( ) 4 u s k bc T a bp p u T K λ⎡ ⎤− − + − + +⎢ ⎥⎣ ⎦ (21) Obviously, R is a quadratic and convex function of price 1p . If we get maximum R with appropriate 1p , that is 1 0dR dp = , then 2 2 2 2 1 1 1 2 2 1 1 1 1 2 4 ( 2 ) 4 ( ) 2 8 ( ) 8 ( ) u u s sb c k T a K bk T K p u bK k bc KT bk T p u p b K bk T λ λ− + + + + + − + + = + (22) 1 ep is the expectation of the best price 1p , 1 1( )ep E p= 2 2 2 1 2 1 2 2 1 1 1 4 ( 2 ) 2 2 2 ( ) 8 ( ) 2 ( ) u s a K bk T b c k T bK k bc KT K bk T E p b K bk T b K bk T λ + − + − + = + + + . Assumed variance u sp of transferring price sp is a constant. Then 2 2 2 1 2 1 2 2 1 1 1 1 4 ( 2 ) 2 2 2 8 ( ) 2 ( ) e u s a K bk T b c k T bK k bc KT K bk T p p b K bk T b K bk T λ + − + − + = + + + (23) So, 1 ep is made up of two terms. One is determined by the structure of model, the other involves retail price undulation. When 0λ > , 1 ep is in proportion to u sp . u sp is higher, 1 ep is higher. It is coincident with the theory, which is the expectation revenue of asset increased with the risk. In a word, under the incomplete information, the expectation of economical people is scaled to uncertainty rather than related to it only. 4. Calculation Examples 1( ) 970 3 ( )s sq t p t u= − + Assumed 1000, 3, 2a b λ= = = − and 15u sp = in equation (1), the demand function of the consumer in order cycle time is: 1( ) 970 3 ( )s sq t p t u= − + If 7T = , we can confirm the lead time is two days according to some conditions, such as transportation, so 2 5K T= − = . The coefficient between rate of change and production rate 1 2 k is 1 when the product rate increase unit product cost, and the coefficient between storage cost and storage amount 2k is 0.05. And the purchase cost 1C is 100 Yuan, holding cost per unit production in unit time 2c is 1 Yuan. Above data can be adjusted and modified according Zhang: Research on Dynamic Pricing Model Based on e-Supply Chain Management 66 to the actual statistic and forecasting. Because zero is the average of 1u , we substitute mathematical expectation ( )sq t for ( )sq t when we calculated. According to equation (23), we can get 1 314.78 0.32 305.31e u sp pλ≈ + = . And with 1 321.58ep ≤ from equation (11), the optimal sale *Q is 360 when the manufacturer determine price. According to equation (21) and (8), the maximum profit of the manufacturers Rm is 58028.6, and the maximum profit of the wholesalers sR is 1295.17. From the data simulation, we can know that the manufacturers have the active power and control the profit mostly because of the open and dominant price under the monopoly and competition. 5. Conclusions Comparing to definite situation，the hypothesis containing fluctuate price is more close to the fact. With the development of the electronic commerce, the forecast and decision should be more reliable and it is more important to study the random situations. 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