Three-Level Service Contract between Manufacturer, Agent and Customer (Game Theory Approach)

Transcription

1 Proceedings of the 0 International Conference on Industrial Engineering and Oerations Management Istanbul, Turkey, July 6, 0 Three-evel Service Contract between Manufacturer, Agent and Customer (Game Theory Aroach) Nafiseh Shamsi Gamchi Deartment of Industrial Engineering Alzahra University, Tehran, Iran Maryam Esmaeili Deartment of Industrial Engineering Alzahra University, Tehran, Iran Mohammad-Ali Saniee Monfared Deartment of Industrial Engineering Alzahra University, Tehran, Iran Abstract Warranty as a kind of service contract today lays a role key in business and legal transactions. In this aer, we resent a manufacturer, an agent and a customer's model under different service contracts suggestions. The manufacturer's rofit is maximized by determining sale rice, warranty eriod and warranty rice. In addition we obtain otimal maintenance cost or reair cost for the agent's model to maximize the agent's rofit. Moreover, the customer maximizes his/her rofit by choosing one of the otions that are suggested by the manufacturer and the agent. We also consider the interaction between the manufacturer, the agent and the customer by using game theory aroach. Nash equilibrium is obtained under non-cooerative game. In the non-cooerative game the manufacturer, the agent and the customer choose their best strategy searately and simultaneously. In the semi-cooerative game the manufacturer and the agent make their best strategy together and then they dominate the customer in a conventional way. Keywords Exected rofit, Free relacement, Game theory, Reair, Service contract. Introduction Nowadays, warranty has an imortant effect on urchasing roducts. Each customer refers to buy a roduct which has a reliable warranty during its lifetime which can make the customers sure about their urchase. In other words, the customers are interested in buying the roducts with free or low reair charge whenever their urchased roduct needs reairs. This subject can be formed as service contracts which are attached to the roducts. Therefore, the service-roviders such as agent and manufacturer recently have offered some service contracts to increase their rofit. In addition, the customer is interested in using the suggested service contracts. Different service contracts may be suggested by service roviders. In the literature, there are different tyes of service. We categorize them in four grous as follow: Some researchers study about free relacement, in which the manufacturer relaces the failed item by new one free in charge for the first time and reairs it in low cost during warranty eriod. (Rinaska and Sandoh [4], Zhou, et.al.[] and Wu, et.al.[]). There is a situation in which the manufacturer has to comensate the consumers by refunding the rice aid (money back warranty) in addition to free relacement (Boom []). Moreover, in the second grou outsourced services are considered. Based on this assumtion, when a failure occurs, an external agent rectifies all failure under different otions. (Asgharizadeh and Murthy [], Murthy and 8

2 Asgharizadeh [8], Murthy and Yeung [7], Jackson and Pascual [6]). Game theory aroach is used to determine an otimal structure for the agent s rice. Maintenance otions are considered as a tye of service contract in the third grou. For instance some researchers resent models that the customers can negotiate about the arameters of maintenance such as availability, reliability, time intervals between controls and etc. (Wang [5], Maronick [0]). ifetime warranty olicies and models for redicting failures and estimating costs for lifetime warranty olicies are resented in some researches.(chattoadhyay and Rahman []). In addition to above aers, Hartman and aksana [9] discussed some warranty contracts which include restrictions on reairs and renewals. A significant shortcoming of all these aers is that they model service contracts only with regard to the manufacturer or the agent. In other words, they consider the interaction of the customer with the manufacturer or the agent. But recently the interaction of the customers with manufacturers and agents has been considered simultaneously. Therefore each customer could arbitrarily choose one their suggested otions. For instance, most of high-tech roducts are suorted by the third art (agent) warranty in addition to the manufacturer warranty. ipad, one of the roducts of Ale factory, is a known examle which is suorted by both Alecare (as the manufacturer warranty) and Squaretrade (as the agent warranty). In this aer, we roose the manufacturer, the agent and the customer s models under different warranty otions. The manufacturer suggests two otions to the customer: () free relacement er failure during warranty eriod, () no warranty. The manufacturer rofit is maximized by determining otimal sale rice, warranty rice and warranty eriod under his/her otions. On the other hand, the agent offers three otions: () to reair the failed roducts after warranty exiration for a fixed cost er failure, () to reair all failures occurred during lifetime of roduct for a fixed rice, () to reair the failed roducts for a fixed cost er failure during lifetime. In the agent s model under his/her warranty otions, we determine otimal maintenance cost or reair cost by maximizing the agent s rofit. Moreover, the customer maximizes his/her rofit by choosing one of the otions that are suggested by the manufacturer and the agent. We consider the interaction among the manufacturer, the agent and the customer by using game theory aroach. In addition, the model is considered under the non-cooerative and semi-cooerative games. In the non-cooerative game we consider static and dynamic games. In the static game the manufacturer, the agent and the customer choose their best strategy searately and simultaneously. On the other hand, in the dynamic game we assume the manufacturer has more ower than the agent and the agent has more ower than the customer. Therefore, the agent makes the best strategy by dominating customer in a conventional way. In addition, the manufacturer also makes the best strategy by dominating the agent in a conventional way. In the semi-cooerative game we consider the manufacturer and the agent cooerating together as one service-rovider. In such situation they have more ower than the customer. Therefore we obtain the otimal solution under the dynamic game. For each scenario, numerical examles are resented to illustrate our models. The remainder of this aer is organized as follows. We give the notation and assumtions underlying our model in Section. In Section, we formulate the roblem, including models of the manufacturer, the agent and the customer, and discuss about their otimal solution. Section 4 resents a game theory aroach including noncooerative static and dynamic game. Semi-cooerative game theory aroach is discussed in Section 5. Finally, the aer concludes in Section 6 with some suggestions for further research in this area.. Notation and Assumtion This section introduces the notation and formulation of our model. Here, we state decision variables and inut arameters and assumtions underlying our models.. Decision Variables P i Sale rice of the manufacturer under otion i P a Maintenance cost received by the agent Tw Warranty eriod for the manufacturer C ir Reair cost of the agent er unit under otion i P w Warranty rice of the manufacturer R i Revenue er time unit under otion i (customer). Parameters ifetime of roduct X Working time of roduct after last reair 9

3 C Production cost er unit λ Failure rate P s Salvage value of failed roduct r Aging rate of roduct C' r Reair cost er unit for agent itself Π Mi Profit of manufacturer under otion i N Number of failed items during roduct lifetime Π Ai Profit of agent under otion i N Number of failed items during warranty eriod Π Ci Profit of customer under otion i N Number of failed items after warranty exired Π SP Profit of service rovider X i Oerational time of roduct after ith reair and before (i+)th failure. Assumtion The roosed models in this aer are based on the following assumtions:. There is just one customer, one agent and one manufacturer.. The intensity of failure is increasing function of time based on Jackson and Pascual s model [6]. This kind of aging behavior by a failure hazard, λ(t), given by λ(t)=λ +rt., where λ is the initial failure rate when the 0 0 roduct is new, at t=0, and t is the age of roduct.. The number of roduct failures in the secified time interval follows Poisson distribution and the failure rate is λ(t). 4. Warranty s rice, P w, is a linear function of warranty eriod, T w, P w =βt w, where β>0 deends on the number of failures which is estimated by the manufacturer.. The Proosed Model In this section we roose models of the manufacturer, the agent and the customer. Regarding decision variables of manufacturer and agent, the customer will find the best otion to maximize the rofit.. The manufacturer's model The manufacturer offers the following two otions to the customer: M: The manufacturer has to relace the failed item free in charge during the warranty eriod. M: No warranty... The best strategy of the manufacturer's model under otion M The manufacturer's rofit= Sale rice+ Warranty rice- Production cost- (Production cost- Salvage value)*number of failures during warranty eriod () Π M = P + Pw Cm ( Cm Ps) E( N ) According to () the roblem is to determine sale rice, P, warranty eriod, T w, and warranty rice, P w. Since N is a random variable by exonential distribution (N ~ ex(λ )), we use its exected value. Therefore, N () e λ λ Tw E ( N ) = N = λ = ( λ 0 0t rt ) dt λ0t w rtw N = 0 N! + = + We ut the first derivation of the manufacturer s model equal to zero, to obtain T w. Since the manufacturer s model is concave, we have: () P ( ) * w Tw C Ps λ0 β ( C Ps) λ0 Tw = = rc ( Ps) rc ( Ps) Please note that according to assumtion (4), P w is a linear function of T w, P w =βt w. Solving π M =0 for zero rofit regard to P gives: (4) P = C ( βtw ) + ( C Ps)( λ0t w + rtw ) Since (4) is an increasing linear function of P, the otimal P * occurs at the highest rice that is ossible for the manufacturer to charge the customer. Therefore, (5) * P = k [ C ( βtw ) + ( C Ps)( λ0t w + rtw )] for some k >. 40

4 .. The best strategy of the manufacturer's model under otion M The manufacturer's model under otion M would be: The manufacturer's rofit= Sale rice- Production cost (6) Π M ( P) = P C Solving π M =0, for zero rofit regard to P gives: (7) P Since (7) is an increasing linear function of P, the otimal P * occurs at the highest rice that is ossible for the manufacturer to charge the customer. Therefore, * (8) where k >. P = C = kc. The agent's model The agent has three otions to offer to the customer that they are: A: To reair the roduct when a failure occurs for fixed cost er failure (after exiration of warranty). A: To reair all failures occurred during lifetime of roduct for a fixed rice. A: To reair the roduct when a failure occurs for fixed cost er failure (during lifetime)... The best strategy of the agent's model under otion A The agent's rofit= (Revenue of reairing failed roduct after exiration of warranty- Reair cost) *Number of failed roduct after T w. (9) Π A( C r) = ( C r C r) E( N ) The number of failed roduct after exiration of warranty, N, is a random variable with exonential distribution (N ~ex(λ )). Therefore, the exected value of N is: T (0) w E( N ) = λ = λ( t) dt = λ 0 0( Tw ) + r( Tw ) Solving π A =0 for zero rofit regard to C r gives: () ' Cr = Cr Since (9) is an increasing linear function of C r, the otimal C * r occurs at the highest rice that is ossible for the agent to charge the customer. Therefore, () * ' C r = kc r for some k >... The best strategy of the agent's model under otion A The agent s model under this otion would be: The agent s rofit= Maintenance rice- Reair cost for agent*number of failures during lifetime of roduct- Penalty cost due to delay in reairing queue. N () α Π ( P) = P NC ( max{0, Y τ}) A a a r i i = According to assumtion one, there is not any queue and the enalty cost is ignored. Then () would be: (4) Π A( Pa) = Pa NC r The number of failures in the lifetime of roduct, N, is a random variable with exonential distribution (N ~ex(λ )). We use the exected value of that. (5) E ( N ) = λ = λ() t dt = λ r Solving π A =0 for zero rofit regard to P gives: a (6) Pa = C r( λ0 + r ) Since (6) is an increasing linear function of P a, the otimal P * a occurs at the highest rice that is ossible for the agent to charge the customer. Therefore, 4

5 (7) * for some k 4 >. Pa = k 4C r( λ0 + r ).. The best strategy of the agent's model under otion A The agent s model under this otion is as follow: The agent s rofit= (Revenue of reairing failed roducts- Cost of reair for the agent)*number of failed roducts during lifetime. (8) Π A( Cr) = ( Cr C r) E( N ) Solving π A =0 for zero rofit regard to C r gives: (9) Cr = C r Since (4) is an increasing linear function of P a, the otimal P * a occurs at the highest rice that is ossible for the agent to charge the customer. Therefore * (0) Cr = kc 5 r for some k 5 >.. The Customer s Model The customer would choose one of the agent or manufacturer s otion to maximize his/her rofit. Therefore the customer has three otions: C: (M and A), Paying P w to the manufacturer for free relacement er failure during T w, and after that, aying C r cost er failure to the agent. C: (M and A), Paying P a to the agent, therefore each failure would be reaired free in charge. C: (M and A), Paying C r cost er failure to the agent during lifetime... The best strategy of the customer s model under otion C Under otion C, the customer s model would be as follow: The customer s rofit= Revenue obtained by the roduct- Sale rice- Warranty rice- Reair cost after T w. N () Π ( R ) = R ( T + X + X ) P P C N C w i w r i = We note that the time of new roduct relacement is ignorable. Also the total downtime of roduct for the customer is negligible, because the mean total (waiting + reair) time is very small in relation to mean time to failure []. Therefore: () N R( T + X + X ) R. w i = where N reresents the number of failures. According to () and (), () would be: () Π C( R) = R P Pw C r( λ0( Tw ) + r( Tw ) ) Solving π C =0 for zero rofit regard to R gives: (4) R = ( P + Pw + C r( λ0( Tw ) + r( Tw ) ) Since (4) is an increasing linear function of R, the otimal R * occurs at the highest rice that is ossible for the customer. Therefore, (5) * k 6 R = ( P + Pw + C r( λ0( Tw ) + r( Tw ) ) for some k 6 >... The best strategy of the customer s model under otion C When the customer chooses otion C, his/her rofit would be as follow: The customer s rofit= Revenue obtained by roduct- Sale rice- Cost of agent s service contract. (6) Π C ( R) = R P Pa Solving π C =0 for zero rofit regard to R gives: i 4

6 (7) R = ( P + Pa ) Since (7) is an increasing linear function of R, the otimal R occurs at the highest rice that is ossible for the customer. Therefore, (8) * k 7 R = ( P + Pa) for some k 7 >... The best strategy of the customer s model under otion C The customer s model under otion C is: The customer s rofit= Revenue obtained by roduct- Sale rice- Reair cost during the roduct s lifetime. (9) Π C( R) = R P CrE( N ) Solving π C =0 for zero rofit regard to R gives: (0) R = ( P ( )) + C re N Since (0) is an increasing linear function of R, the otimal R * occurs at the highest rice that is ossible for the customer. Therefore, () * k 8 R = ( P + C r( λ0 + r )) for some k 8 >. 4. The Non-Cooerative Static Game We have a set of layers (N=), and for each layer, we also secify the set of strategies and corresonding rofit available to that layer []. The comonents of the three-erson game are described as follow: N= {M, A, C} where the manufacturer, the agent and the customer are reresented by M, A and C resectively. The set of strategies available to the layer i, S i, i= M, A, C, are S M = {M, M} S A = {A, A, A} S C = {C, C, C} The ayoff of each layer i, u(i), i=m,a,c, is u(m)= {π M (P, P w, T w ), π M (P )} u(a)= { π A (C r ), π A (P a ), π A (C r )} u(c)= { π C (R ), π C(R ), π C (R )} In this section we resent the static and dynamic scenarios as a non-cooerative static game. In the static game the layers obtain their best strategy simultaneously and searately. Therefore, Nash equilibrium is obtained according to Table. Please note that in Table all strategies of layers are resented. The numerical examles are also resented to illustrate our model. 5. The Semi-cooerative Game In this section we assume that the manufacturer and the agent cooerate together and act as a unit, service-rovider. Therefore the customer faces the following otions: (SP) During the warranty eriod, when a failure occurred, the service-rovider tries to reair the fault free of charge, in the secified time (defined in the contract), otherwise the service-rovider relaces it without charge. Please note, after exiration of the warranty, reair er each fault will aear on cost of C r (SP) The customer refers not to buy the service contract. The customer s rofit under otion : () Π C = R P Pw Cr( λ0 ( Tw ) + r( Tw ) ) 4

7 () * k 9 R = ( P + Pw + Cr( λ0 ( Tw ) + r( Tw ) )) Since the manufacturer and the agent cooerate together, the service-rovider has more ower than the customer. We will regard the interaction between the service rovider and the customer as a Stackelberg game. Based on the customer s decision variable, the rofit of the service-rovider is maximized. Therefore the model would be: (4) Max Π = P + P ( C P ) max{0,sgn( Y τ + ε)} N N s w s i i = C C max{0,sgn( τ Y )} + ( C C ) E( N ) st.. r i r r i = * k 9 R = ( P + Pw + Cr( λ0 ( Tw ) + r( Tw ) )) k 9 >. We would illustrate our model by resenting numerical examles. 6. Numerical Examles Examle : We set the arameters of model as below: λ 0 =0.6(eryear), r=0. (er square year), =6 (years), C' r =00(0 )$, C =800(0 )$, β=450(0 )$ and P s =500(0 )$. We also suose, k =., k =., k =., k 4 =., k 5 =., k 6 =., k 7 =. and k 8 =.. We obtain the otimal solution of the manufacturer, the agent and the customer s model under their strategies (Table-). Table : Profits of manufacturer, agent and customer, (Π M,Π A,Π C )(0 )$ to Obtain Nash equilibrium-examle. A A A C M (45,67.5,4.6) (45,80,4.6) (45,90,4.6) M (80,67.5,4.6) (80,80,4.6) (80,90,4.6) C M (45,67.5,.6) (45,80,.6) (45,90,.6) M (80,67.5,.6) (80,80,.6) (80,90,.6) C M (45,67.5,4.) (45,80,4.) (45,90,4.) M (80,67.5,4.) (80,80,4.) (80,90,4.) According to Table, Nash equilibrium would be (M, A, C). It means that the manufacturer refers to suggest his/her first otion. The agent s reference is to offer his/her second otion. And the customer has the most rofit by choosing his/her first otion. The first otion which is selected by the customer is also suggested by the manufacturer. Therefore, the agent wouldn t obtain his/her maximum rofit. Therefore: Π M *=45(0 ) $, T w *= year, P w *=450(0 ) $, P *=870(0 ) $ Π A *=67.5(0 ) $, P a *=080(0 ) $ and Π C *=4.6(0 ) $, R *=46(0 ) $. Now we illustrate the model (4) by resenting the following examle. Examle : Consider the arameters of the examle. We have: N N Max Π = P + P (500) max{0, sgn( Y ε )} max{0, sgn(0.007 Y )} + ( C 00) E ( N ) s w i i r i = i = 44 6 C R P P * r = ( ( )) + w (0.6(6 Tw ) + 0. (6 Tw ).

8 We substituting the Cr in the above function and we reach the following results: T w * =.5 year, P w * =675(0 ), P = 697(0 )$, C r = 05(0 )$, R * = 9 (0 )$ 7. Conclusion In this aer we have resented a three level warranty between the manufacturer, the agent and the customer by using game theory aroach. Under different service contracts suggestions, the manufacturer s rofit is maximized by determining sale rice, warranty eriod and warranty rice. In addition, we obtain otimal maintenance cost or reair cost for the agent s model to maximize the agent s rofit. Moreover, the customer maximizes his/her rofit by choosing one of the otions that are suggested by the manufacturer and the agent. We also consider the interaction among the manufacturer, the agent and the customer under non-cooerative and semi cooerative games. In the noncooerative game, Nash equilibrium is obtained in the static scenario. We also consider the manufacturer and the agent cooerate together and act as one service-rovider which has more ower than the customer under the semi cooerative game. By resenting some numerical examles, we illustrate the obtained results of each scenario. There is much scoe in extending the resent work. For examle, each customer can buy more than one roduct, therefore he/she can select different ortfolio of service contracts for each roduct. In such case, queue may be created, therefore the agent has to ay enalty if the reair of failed roduct is delayed. Also, risk arameter can be considered for the manufacturer, the agent and the customer that have a considerable effect on their decisions. References. Zhou, Zh., i, Y., Tang, K., 009, Dynamic ricing and warranty olicies for roducts with fixed lifetime, Euroean Journal of Oerational Research 96, Wu, Ch.-Ch., in, P.-Ch., Chou, Ch.-Y., 006, Determination of rice and warranty length for a normal lifetime distributed roduct, International Journal of Production economics 0, Asgharizadeh, E., Murthy, D.N.P., 000, Service contracts: A stochastic model, Mathematical and comuter modeling, Rinsaka, K., Sandoh, H., 006, A stochastic model on an additional warranty service contract. Comuters and mathematics with alications 5, Wang, W., 00, A model for maintenance service contract design, negotiation and otimization, Euroean Journal of Oerational Research 0, Jackson, C., Pascual, R., 008, Otimal maintenance service contract negotiation with aging equiment, Euroean Journal of Oerational Research 89, Murthy, D.N.P., Yeung, V., 995, Modeling and analysis of maintenance service contracts, Mathematical and comuter modeling, Murthy, D.N.P., Asgharizadeh. E., 996, A stochastic model for service contracts, International Journal of Reliability, Quality and Safety. 9. Hartman, J.C., aksana, K., 008, Designing and Pricing menues of extended warranty contracts, Wiley InterScience. ( DOI 0.00/nav Maronick, T.J., 008, Consumer ercetions of extended warranties, Journal of Retailing and Consumer Services 4, 4-.. Boom, A., 998, Product risk sharing by warranties in a monooly market with risk-averse consumers, Journal of Economic Behavior and Organization, Chattoadhyay, G., Rahman, A., 008, Develoment of lifetime warranty olicies and models for estimating costs, Reliability Engineering and System Safety 9, Esmaeili, M., Aryanezhad, M.B., Zeehongsekul, P., 009, A game theory aroach in seller-buyer suly chain, Euroean Journal of Oerational Research 95,