Price Game Analysis of Leader-Follower Service Providers with Service Delivery. Time Guarantees. ZHANG Yu-lin. ZHANG Jian-wei.

Transcription

1 Price Game Analyi of eader-follower Service Provider with Service Delivery Time Guarantee ZANG Yu-lin School of Economic and Management, Southeat Univerity(SEU), Nanging, China, ZANG Jian-wei School of Economic and Management, Southeat Univerity, Nanging, China, Chen Ying-Ju IEOR Department,Univerity of California at Berkeley, Berkeley,CA, POMS nd Annual Conference Reno,Nevada,U.S.A April 9 to May,0 Supported By National Nature Science Foundation Proect of China (707706). Correponding Author: Yulin Zhang, Profeor of SEU,Viiting Scholor of UC Berkeley, zhangyl@eu.edu.cn

2 Abtract: Under the ituation that cutomer have the ame time enitivity and obey the uniform ditrbution on a linear city with length one, we tudy the price game of leader-follower ervice provider denoted a p and p) with two type of ervice delivery time guarantee. The tudy how that if the number of potential cutomer i fixed, when the cutomer reervation payment (CRP) i maller, the two ervice provider are both local monopoly. With the increae of the CRP, if p chooe the longer ervice delivery time guarantee, the two ervice provider are till local monopoly and if p chooe the horter ervice delivery time guarantee, they would compete for ome cutomer. When the CRP i larger, no matter which type of ervice delivery time guarantee they chooe, they will compete for ome cutomer. When the two ervice provider compete with each other, the follower ha the econd-move advantage. If the per unit ervice cot and the ervice capacity cot are the ame, the per unit time profit of p i higher than the per unit time profit of p. Keyword: ervice delivery time guarantee; ervice price; ervice capacity; linear city Introduction In traditional indutrie, the price and yield are often the focu of competition between enterprie. Reearcher alway aume that each production of the product i identical. owever, beide price and yield, there are ome other difference. In 99, otelling etablihed a price competition model of the two companie with patial different product []. In the model, he aume that the two companie have different location in the linear city and in order to get the product, conumer hould not only pay the price but alo pend ome tranport expenditure. In the modern ervice indutry, conumer alway pay their attention to price and the waiting time pending on the ervice. In fact, geographical difference of ervice provider alo

3 have ome influence on the cutomer choice. In thi paper, baed on the otelling model, we tudy the price game of leader-follower ervice provider with two type of ervice delivery time guarantee. Our main aim i to anwer the quetion: how do the two ervice provider chooe their ervice time guarantee and price the ervice in order to achieve profit maximization? The Exiting reearch reult related to thi paper mainly have two apect: one i the price, ervice capacity and ervice delivery time guarantee competition of ervice provider; the other i to tudy the ervice price and capacity deciion with conideration the location of ervice provider. On the firt apect, Chen and Wan (003) tudied the price game of two make-to-order firm with different ervice capacity, which indicate that the firm with higher ervice capacity or lower per unit operation cot could take a relatively larger market hare with premium price []. Baed on thi, Chen and Wan (005) tudied the ervice price and ervice capacity competition of two make-to-order firm with fixed market capacity. The tudy proved the exitence of Nah equilibrium and how that in the equilibrium, the number of firm operating in a market depend on the capacity of the market and the duopoly Nah equilibrium i ocially optimal if and only if there i one firm operating in the equilibrium [3].So.KC (000) tudied the ervice price and ervice delivery time guarantee competition of firm with demand enitive to both price and delivery time guarantee. The reearch indicated that firm will exploit their ditinctive firm characteritic to differentiate their ervice. Auming all other factor being equal, the high capacity firm provide better time guarantee, while firm with lower 3

4 operating cot offer lower price, and the differentiation become more acute a demand become more time-enitive [4]. Zhang and Tan (009) tudied the ervice price competition between two web ervice provider offering functionally the ame web ervice with ervice level guarantee. The tudy how that in the long term, the two provider intend to chooe different ervice level and ervice price. owever, in the horter term, they may incline to the imilar ervice level and ervice price [5]. Fan and Kumar (009) tudied the ervice price and oftware quality hort-term and long-term competition between two oftware ervice provider (SaS: oftware a a ervice, SWS: hrink-wrap oftware) with the ervice time guarantee [6]. Pekgun and Griffin 008 analyzed the ervice price and lead-time competition of two firm under centralized deciion making and decentralized deciion making [7]. On the econd apect, Dobon and Stavrulaki (006) developed a model of imultaneou price, location and capacity deciion of a ervice provider with time-enitive cutomer uniformly ditributing on a line city. They worked out the optimal olution with the conideration of the hipping delay [8]. On thi bai, Kwanica and Stavrulaki (008) tudied the ervice capacity and location two-tage competition between the two ervice provider who can chooe their location on a linear city. The tudy indicate that unle the cot of ervice capacity obtained i particularly low, two ervice provider will chooe a limited ervice capacity and focu on the location competition. When the market capacity i large, the two ervice provider will not compete and ervice the cutomer cloer to their own facilitie while the market demand i mall, two ervice provider will compete for ome 4

5 cutomer [9]. Under the aumption that when cutomer face with a congeted facility, they may be balking, reneging and veering, Mohammad and Ebrahim (00) tudied the influence of the cutomer behavior on the market hare of multiple ervice provider. The tudy how that the provider with higher ervice capacity can occupy a higher market hare. If the balking ratio i high, the market hare of all the provider will be reduced. Price volatility will not have great impact on the market hare of each ervice provider [0]. In our reearch, conidering the ituation that two ervice provider (p, p) are located at both end of a traight line with length and the cutomer are time-enitive and uniformly ditribute on the linear city, we tudy the price game of leader-follower ervice provider with two type of ervice delivery time guarantee. The ditinguihing feature of our work i to conider the patial difference of the two ervice provider and two type of ervice delivery time guarantee contraint. Thi ha ignificant difference with the reearch of Zhang (009). Their reearch i baed on fixed ervice capacity and doe not conidering the patial difference of ervice provider. Notation and Aumption Notation: λ i cutomer arrive rate of ervice provideri i =, µ i ervice rate of ervice provideri i =, p i ervice price of ervice provideri i =, γ i the unit operation cot of ervice provideri i =, 5

6 i ervice delivery time guarantee ( i =, and > ) gx the ( ) function of traveling cot, x i the ditance c the unit ervice capacity cot π i the per unit time profit of ervice provideri i =, v cutomer reervation payment c 0 per unit time waiting cot of cutomer Aumption: ()For each ervice provider, the ervice ytem i M/M/ type. Each ervice provider aim to maximize profit and each cutomer i target to maximize utility. ()The two ervice provider are located at the two end of a traight line. Cutomer uniformly ditribute on the linear city with a denity l. Cutomer reervation payment and the per unit time waiting cot of cutomer are the ame for each cutomer. (3)The two type of ervice delivery time guarantee are, ( >, or, they do not include the traveling delay). The two ervice provider mut enure that the probability of meeting the time guarantee for each ervice provider mut be at leat α (α can be 0.95 or o.98). Each ervice provider can not occupy the entire market. 3 Model Formulation In thi paper, we write the cutomer utility function a: U = v c0 g( x) pi ( =,, i=,). Only when it atifie U 0, the cutomer could chooe the ervice. Otherwie, they will not chooe it. We denote the function of tranportation cot a: gx ( ) = ax. x i the ditance that a cutomer hould travel in order to get the ervice and a i the per unit 6

7 travelling cot of each cutomer. A for a M/M/ queuing ytem, the requirement that the probability of meeting the time guarantee for each ervice provider mut be at leatα can be written a follow( So 000): ( µ λ) e α =, () Sp i the leader and p i the follower. Baed on optimal price and ervice time guarantee of p, p draw it own optimal olution. If each ervice provider can not occupy the entire market, then in equilibrium there mut exit a point (denote a A, the ditance from A to p i denoted a x and then the ditance from A to p i x ). Then we can draw an equation a follow: v c ax p = v c a( x) p =, () 0 0 Sp a the leader, he can chooe or, and formulate the optimal ervice price. Sp a the follower, hi choice have a ignificant impact on the equilibrium olution and the profit of p. So p need to analyze the repone of p when he chooe the each type of ervice delivery time guarantee and compare the per unit profit. Therefore,we dicu in two cae. 3. p chooe Baed on the analyi above-mentioned, the optimization problem of p can be written a: max π = ( p γ ) λ ( p, ) cµ (3) p, µ, ( ) e µ λ α (4) t. v c ax p = v c a( x) p (5) 0 0 7

8 v c a( x) p 0 (6) 0 Simplifying (4), we obtain: k µ + λ k = ln( α) =, (7) We do not conider inequality contraint (6) and when π ( p, µ, ) i at optimality, contraint (7) mut be binding. So we have: k µ = + λ k = ln( α) =, (8) Proof: If there i an optimal olution( p, µ, ) which make contraint (7) to be trict inequality, then reducing µ can increae the π. So ( p, µ, ) i not the optimal olution. So when π ( p, µ ) i at optimality, contraint (7) mut be binding. A i either or, we can regard a a contant. When we get the olution, we ubtitute and into π ( ) and have a comparion. So we can get the optimal ervice delivery time guarantee. Subtituting (5) and (8) into (3), we obtain: p p+ c0( ) k π( p) = ( p γ c)( ) l c (9) a It i not difficult to get the olution of π ( p) : p a+ p + γ + c c ( ) 0 = k µ = + λ ( p ) =, p= p A for p, it optimization problem can be written a: max π = ( p γ ) λ ( p ) cµ (0) p, µ ( ) e µ λ α () 8

9 t. v c ax p = v c a( x) p () 0 0 v c ax p 0 0 (3) For the ame method above-mentioned, we can obtain: p p+ c0( ) + a k π( p) = ( p γ c)( ) l c (4) a Subtituting p into (4), we obtain p 3a+ γ + γ + c+ c ( ) 0 =, = (5) x γ γ + 3 a+ c ( ) 0 =, 8a k µ = + λ ( p ) (6) p= p Subtituting (5) into p, we obtain p 5a+ γ + 3γ + 4 c c ( ) 0 = =, 4 Subtituting (5) and (6) into (3) which i equal to (6), we can obtain: v c ax p v c 5γ + 3γ + 5a+ 8c+ 5 c ( ) 0 0 = 0 (7) 5γ + 3γ + 5a+ 8c 5γ + 3γ+ 5a+ 8c+ 5 c0( ) Denoting v = + c0, v = + c0, 8 8 then we have: () If v< v the utility of cutomer on the boundary point i negative which mean (7) i negative. At thi time, the two ervice provider are local monopoly They have no need to compete with each other. The optimization problem of p can be written a: 8 max π = ( p γ ) λ ( p ) cµ p, µ 9

10 ( ) t. e µ λ α v c ax p = 0 0 It i not difficult to get the optimal olution: v c + γ + c 0 pˆ = k ˆ µ = + λ( p) p= pˆ In the ame method, we can get ˆp ˆµ So we can obtain the optimal per unit time profit of p and p: ( v c0 γ c) π = 4a ( v c0 γ c) π = max(, =, ) 4a Becaue of the cutomer reervation payment i mall, the market capacity i large enough for them to realize local monopoly. So the two ervice provider can chooe their optimal olution without conidering the choice of the other. The optimal ˆp ˆp i increaing in v. The profit of two provider are decreaing in the unit ervice capacity cot and the unit operation cot. () If v> v p, p are the optimal ervice price of p and p The per unit time profit are : (3 a+ γ γ+ c0( )) π =, 6a (5 a+ γ γ c0( )) π = max( =, ) 3a In thi cae the two ervice provider compete with each other for ome cutomer and each cutomer can get poitive utility. From the expreion of π, we can ee that if γ i, ac, 0 are unchanged the ervice delivery time guarantee of p depend on ervice capacity cot. If the ervice capacity cot i relatively large p will chooe. Otherwie, he will chooe.from the expreion of π we can 0

11 oberve that when p chooe the per unit time profit of i relatively larger. Thi mean p chooe relatively higher ervice delivery time guarantee i conducive to p. A for p, hi choice of ervice delivery time guarantee depend on the pecific parameter value. From the expreion of p, p we can ee that p i increaing in and p i decreaing in In order to compare the profit of the two ervice provider, we aume that p alo chooe andγ γ =. We can obtain: p = 3/ a p = 5/4 a x= 3/8 π π= 7 al /3 Obviouly, p occupie a relatively large market hare with a lower ervice price and get higher profit. From thi, we can ee that when the condition are the ame, the follower p can obtain a econd-move advantage and it per unit time profit i on le than the profit when it chooe the ame ervice delivery time guarantee with p. (3) If v v v then we have: When p chooe, then the optimal olution are ( ˆp, ˆµ ) and ( ˆp, ˆµ ) and their repective per unit time profit are a follow π ( v c0 γ c) = π 4a ( v c0 γ c) = 4a When p chooe, then the optimal olution are ( their repective per unit time profit are a follow p, µ ) and ( p, µ )and π (3 a+ γ γ) = π 6a (5 a+ γ γ) = 3a In thi cae, if p chooe then they will be in competition and if p chooe, then they will be in local monopoly. Sp would compare the profit of the two condition and make hi deciion. So the per unit time profit of p can be denoted

12 a π = max( π, π ) Comparingπ with π we can obtain : v (5 a γ γ ) / 8 4 cka( ) / l c γ c 0 = ) If v ( v, v ), then we have If v v v < p will chooe If v < v v p will chooe. If v v, p will chooe. 3. p chooe The proce i imilar with the ituation when p chooe. Denoting v 5γ + 3γ + 5a+ 8c+ 5 c ( ) (5γ + 3γ + 5a+ 8 c) = + c, v4 = + c0, then we have () If v< v3 the two ervice provider are local monopoly. The repective per unit time profit of the two ervice provider are a follow: ( v c0 γ c) π = 4a ( v c0 γ c) π = max(, =, ) 4a () If v> v4 the two ervice provider are in competition. The repective per unit time profit of the two ervice provider are a follow: (3 a+ γ γ+ c0( )) (5 a+ γ γ c0( )) π = π = max( =, ) 6a 3a (3) If v 3 v v 4 : When p chooe, the two ervice provider are local monopoly and their repective per unit time profit are a follow π ( v c0 γ c) = π 4a ( v c0 γ c) = 4a When p chooe the two ervice provider are in competition and their

13 repective per unit time profit are a follow (3 a+ γ γ+ c0( )) 6a (5 a+ γ γ + c0( )) 3a π = π = The optimal per unit time profit of p i π = max( π, π) 4 A Numerical Example Given the parameter γ = γ = a = c = c 0 = 0.75 = 4 = 3.6 l = 4.5, we will compare the optimal ervice price and the profit of the two ervice provider. ()When the two ervice provider in local monopoly,their optimal ervice price i the ame due to the ame the unit operation cot. When they are in competition, the optimal ervice price and the equilibrium point (denote a x ) are provided in Table. From Table, we can ee that p a the follower can occupy a relatively large market hare with a lower ervice price. p, p ( p, p, x) (p) (p) (p) (4.5, 4.5, ) (4.65, 4.75,0.43) (p) (4.35,4.35,0.338) (4.5, 4.5, ) () The profit comparion Provided p chooe the repective profit of p and p when p chooe,. the per unit time profit of p v v S S the per unit time profit of p v v v S S

14 ,, From Figure, we can ee that if v> v,then p chooing i beneficial to p.from Figure, we can oberve that if v v or v v,p will chooe in order to attain profit maximization and if v < v< v, there i a point v :if v v v < <, then p will chooe ; if v < v< v, then p will chooe. the per unit time profit of p When p chooe v 3,the repective profit of p and p when p chooe,. v 4 S S the per unit time profit of p cutomer reervation payment v cutomer rervation payment v,, From Figure 3, we can ee that if v> v3 p chooing i beneficial to p From Figure 4, we can oberve that if v v3 or v v4,p will chooe in order v v3 v 4 S S to attain profit maximization and if v3 < v< v4, there i a point v :if v v v 3 < <, then p will chooe ; if v < v< v 4, then p will chooe. We know that p can draw it optimal ervice price, ervice capacity and ervice delivery time guarantee after he know the deciion of p. owever, a for p he hould tudy the cutomer reervation payment, market capacity and other parameter and the optimal repone of p when he chooe different ervice delivery time guarantee. Only in thi way, he can draw hi optimal olution. From the four figure, we can ee that if v v4, the optimal ervice delivery time guarantee of p i. 4

15 Becaue if v v4, form Figure, p mut chooe ; if p chooe, form Figure4, p mut chooe.then we compare the π and p chooe )with π (the profit of p when p ( the profit of p when p chooe and p chooe ). From Figure and Figure three, we can ee π > π.so the optimal ervice delivery time guarantee of p i when v v4. With the ame method, we can find the optimal ervice delivery time guarantee of p given the pecific value of v. If we know the ervice delivery time guarantee, we can draw the optimal ervice price and ervice capacity. 5 Concluion In thi paper, we tudy the price game of leader-follower ervice provider with two type of ervice delivery time guarantee contraint and changeable ervice capacity. We mainly reolve the problem that how the leader-follower ervice provider chooe the ervice delivery time guarantee and pricing the ervice in order to attain maximum profit. Our reearch how that if the number of potential cutomer i fixed, when the cutomer reervation payment i maller, the two ervice provider are both local monopoly. With the increae of the cutomer reervation payment, if p chooe the longer ervice delivery time guarantee, the two ervice provider are till local monopoly and if p chooe the horter ervice delivery time guarantee, they would compete for ome cutomer. When the cutomer reervation payment i larger, no matter which type of ervice delivery time guarantee they chooe, they will compete for ome cutomer. When the two ervice provider compete with each other, the follower ha the econd-move advantage. If the per unit ervice cot and the ervice 5

16 capacity cot are the ame, the per unit time profit of p i higher than the per unit time profit of p. Sp can draw it optimal olution after he know the deciion of p. owever, a for p he hould tudy the cutomer reervation payment, market capacity and other parameter and the optimal repone of p when he chooe different ervice delivery time guarantee. Then he can make hi bet deciion. Reference: [] arold otelling. Stability in competition [J].The Economic Journal, 99,39 (53): [] Chen ong and Wan Yat-wah. Price competition of make-to-order firm [J].IIE Tranaction, 003,35: [3] Chen ong and Wan Yat-wah. Capacity competition of make-to-order firm [J]. Operation Reearch etter,005,33: [4] Kut C.So. Price and time competition for ervice delivery [J]. Manufacture &Service Operation Management, 000, (4): [5] Zhang Zhongu, Tan Yong and Dey Debabrata. Price competition with ervice level guarantee in web ervice [J].Deciion Support Sytem, 009(47): [6] Fan Ming, Kumar Subodha and Whinton Andrew. Short-term and long-term competition between provider of hrink-wrap oftware and oftware a a ervice[j].european Journal of Operation Reearch,009(96): [7] Pekgun Pelin, Griffin Paul M. and Kekinocak,P. Centralized v Decentralized Competition for Price and ead-time Senitive Demand. Working paper,.milton School of Sytem and indutrial Engineering, Georgia Intitute of Technology, Atlanta, GA, USA,006. [8] Gregory Dobon and Euthemia Stavrulaki.Simultaneou Price, location and Capacity Deciion on a line of time-enitive cutomer [J]. Naval Reearch ogitic, 006,54():-0. [9] Anthony M. Kwanica and Euthemia Stavrulaki.Competitive ocation and Capacity Deciion for Firm Serving Time-enitive Cutomer [J]. Naval Reearch ogitic, 008,55(7): [0] Mohammad Saidi-Mehradad, Ebrahim Teimory and Ali Pahlavani.Modeling cutomer reaction to congetion in competitive ervice facilitie[j].journal of Service Science and Management,00,3():