Pricing in Service Systems with Strategic Customers

Transcription

1 Pricing in Service Systems with Strategic Customers Refik Güllü Boğaziçi University Industrial Engineering Department Istanbul, Turkey Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 1 / 86

2 Before the talk My professor s advice on queueing theory versus game theory mathematical difficulty versus conceptual maturity How to learn new stuff? teaching, writing a code, thesis supervision Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 2 / 86

3 Before the talk A review from a personal perspective A great place to start reading: Rafael Hassin and Moshe Haviv, To queue or not the queue: equilibrium behavior in queueing systems, Springer, hassin/book.html A follow up survey book, Rational Queueing to appear soon Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 3 / 86

4 Outline A simple example of an unobservable queue Parameter uncertainty Effect of delay information Observable queues: residual service time Multiple customer types: identical price Multiple customer types: differentiation An inventory model Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 4 / 86

5 A Framework Example A student cafeteria in a university University administration regulates the price possibly in the form of a subsidy Students arrive according to a Poisson process Λ is the rate of potential students A single server with exponential rate µ > Λ. Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 5 / 86

6 A Framework Example Cafeteria is sort of far away from the main building Students decide eating there or not before observing the congestion There are other dining facilities on campus Once a decision is made, it can not be changed Students are identical with respect to their valuation of the service, value of time, and their behaviour towards risk all are rational decision makers Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 6 / 86

7 A Framework Example R : the value of service (as judged by students) c : the unit cost for waiting p : fee for dining at the cafeteria The expected utility of a student from the service R p ce[sojourn time] the system is at steady state Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 7 / 86

8 A Framework Example: equilibrium behaviour As students are identical, their equilibrium behaviour is expected to be the same Each student choose to enter the cafeteria with probability q Let U(q tagged, q others ) be the utility of a tagged student when all the others behave with q others Best response against q others : U(q, q others ) U(q, q others ) for all q. Symmetric Nash equilibrium: best response against itself U(q e, q e ) U(q, q e ) for every q Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 8 / 86

9 A Framework Example: equilibrium behaviour 0 q e (p) 1 is the equilibrium probability of joining the cafeteria when the fee is p λ e (p) = Λq e (p) < µ For effective arrival rate λ < µ 3 cases need to be examined w(λ) = 1/(µ λ) Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 9 / 86

10 Case 1 Nobody else is joining and q e (p) = 0 λ e (p) = 0 w(λ e (p)) = 1/µ p + cw(0) > R = R < p + c 1 µ µ < c R p Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 10 / 86

11 Case 2 If everybody else is joining and q e (p) = 1 λ e (p) = Λ w(λ e (p)) = 1/(µ Λ) 1 p + cw(λ) R = R p + c µ Λ µ Λ + c R p Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 11 / 86

12 Case 3 p + cw(0) R < p + cw(λ) R = p + cw(λ e (p)) q e (p) = λ e (p)/λ w(λ e (p)) = 1/(µ λ e (p)) λ e (p) = µ c R p Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 12 / 86

13 Administration s Problem: social optimization The administration cares about the overall performance Solves the following problem max {λ(r c 1 0 λ Λ µ λ )} strictly concave maximum occurs at λ 0 (by R c/µ) λ = µ cµ R Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 13 / 86

14 Administration s Problem: social optimization By considering the constraint λ Λ λ = min{λ, µ cµ R } if Λ µ cµ R optimal objective function value ( Rµ c) 2 w(λ ) = R cµ Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 14 / 86

15 Administration s Problem: social optimization if Λ µ cµ R optimal objective function value w(λ ) = 1 µ Λ Λ(R c µ Λ ) Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 15 / 86

16 By assuming R c/µ λ e (0) = µ c R µ cµ R = λ Individual optimization leads to longer queues than imposed by social optimization Admission fee can regulate this Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 16 / 86

17 Revenue maximization Let p m be the admission fee charged for dining p m = R cw(λ) max p mλ 0 λ Λ Same as the social optimization objective The socially optimal arrival rate can be induced by the fee p m = p = R cw(λ cr ) = R µ λ e (p ) = λ, profit = ( Rµ c) 2 Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 17 / 86

18 Rafael Hassin and Moshe Haviv, To queue or not the queue: equilibrium behavior in queueing systems, Springer, (Chapter 3) Bell, C. E., Stidham, Jr., 1983, Individual versus Social Optimization in the Allocation of Customers to Alternate Servers, Management Science, 29, Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 18 / 86

19 Parameter Uncertainty The preceding model considered an unobservable system The queue length or the waiting times upon arrival are unobservable Need to be careful Many things are known and/or intelligently computable: service rate, expected waiting time, service value, etc. These parameters are known with certainty Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 19 / 86

20 Uncertainty in the Service Rate Suppose µ, the service rate, can take two values { µ1 with probability α µ = µ 2 with probability 1 α µ 1 > µ 2 Do students know the realised service rate? No, they are uninformed Yes, they are informed, and the server charges either a different fee or the same fee Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 20 / 86

21 Service Rate Uncertainty For the uninformed case v = (R p)/c v = α µ 1 λ + 1 α µ 2 λ in equilibrium v is a solution of a nonlinear equation Π un = λ(r cv) Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 21 / 86

22 Service Rate Uncertainty informed case with two prices λ i = µ i Π in 2 = c R p i, p i = R cr µ i α( Rµ 1 c) 2 + (1 α)( Rµ 2 c) 2 if R c µ 2 α( Rµ 1 c) 2 if c µ 1 < R < c µ 2 0 if R c µ 1 Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 22 / 86

23 Service Rate Uncertainty informed case with a single price λ i = µ i c R p (rate of arrivals given the service rate) If p is small enough to attract customers for both values of µ average arrival rate for the single price p λ = α(µ 1 c R p ) + (1 α)(µ 2 = µ c R p c R p ) Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 23 / 86

24 Service Rate Uncertainty Maximizing p λ = p µ (R p) 2 = Rc/ µ Resulting profit pc R p = p = R ( R µ c) 2 Rc µ Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 24 / 86

25 Service Rate Uncertainty But a higher price may be chosen: so that customers opt out when µ = µ 2. p = R Rc µ 1 Resulting profit: α( Rµ 1 c) 2 Two profit terms are equal for ( ) 2 R 1 α c = η = µ µ1 α Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 25 / 86

26 Service Rate Uncertainty Π in 1 = ( R µ c) 2 if R c η α( Rµ 1 c) 2 1 if µ 1 R c η 0 if R c 1 µ 1 Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 26 / 86

27 Parameters Uncertainty The service provider benefits from revealing the service rate, and from pricing accordingly Π in 2 Π in 1 Π un As the variability in service rate increases, Π in 2 increases The server provider may lose (1 α) fraction of the customers but extracts higher revenue from the remaining Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 27 / 86

28 Parameters Uncertainty Hassin, R., 2007, Information and Uncertainty in a Queuing System, Probability in the Engineering and Informational Sciences, 21, waiting cost uncertainty service valuation uncertainty Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 28 / 86

29 Delay Information So far: unobservable systems What if the customers are revealed information on the possible delay before they decide to join or not observable systems Three levels of information 1. No information (same as before) 2. Partial information: how many customers are in front of me? 3. Full information: what is my exact waiting time? Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 29 / 86

30 Delay Information M/M/1 type service system W : waiting time (in the queue) θ: customer type parameter, a random variable, θ [0, 1] with cdf H, pdf h c(t): cost of waiting t time units Previously: θ 1, c(t) = ct U = R θe[c(w )] Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 30 / 86

31 Delay Information Scale R and c(t) so that R = 1 Assume that c(0) > 1 Customers with θ > 1/c(0) balk Scale Λ (ignore them) to λ, and assume (new) c(0) = 1 Customers with θ 1 are also attracted to join when W = 0. U(no waiting) = 1 θ Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 31 / 86

32 Delay Information: who stays in the system? Information I, a random variable Want: U I = 1 θe W [c(w ) I] 0 Given information I = i, an arriving customer stays if Pr{stays I = i} = H(θ i ) θ θ i = 1 E W [c(w ) I = i] Fraction of customers who stay: E I [H(θ I )] Throughput λe I [H(θ I )] Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 32 / 86

33 Average utility Define Average utility J(θ) = 1 θ θ 0 H(x)dx u = E[U + ] = E θ,i [(1 θe W [c(w ) I]) + ] [ θi ] = E I (1 xe W [c(w ) I])h(x)dx 0 = E I [H(θ I ) (1/θ I ) = E I [J(θ I )] θi 0 ] xh(x)dx Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 33 / 86

34 Case 1: No information The equilibrium arrival rate: ( ) λ NI 1 = λh E[c(W NI )] ρ NI = λ NI /µ c(s) = 0 e st c(t)dt LST of c(t). Pr{W NI > t} = ρ NI e µ(1 ρni )t E[c(W NI )] = (1 ρ NI ) + ρ NI µ(1 ρ NI ) c(µ(1 ρ NI )) Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 34 / 86

35 Case 1: No information An example: Uniform customers with Linear Cost c(s) = 1 s + 1 s 2 λ NI = π NI n c(t) = 1 + t λ 1 + ρ NI /(µ(1 ρ NI )) = (1 µ)(ρ NI ) 2 + (µ + λ)ρ NI λ = 0 = ρ NI = (µ + λ) + (λ + µ) 2 + 4λ(1 µ) 2(1 µ) = (1 ρ NI )(ρ NI ) n Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 35 / 86

36 Case 2: Partial information The service provider tells the arriving customer N(t) The customer computes c n = E[c(W ) N(t) = n] Stays if θ θ n = 1/c n Birth-death process with state dependent arrival rate λ n = λh(θ n ) Steady-state probabilities Θ n = n 1 m=0 H(θ m ), Θ = π0 P I = 1/(1 + Θ) πn P I = Θ n (λ/µ) n π0 P I Θ n (λ/µ) n n=1 Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 36 / 86

37 Case 2: Partial information An example: Uniform customers with Linear Cost θ m = 1 1+ m, Θ n = n 1 m=0 µ Θ n (λ/µ) n = Γ(µ)Γ(µ+n) λ n π P I 0 = Γ(x) = 1 1+ m µ c(t) = 1 + t γ(µ, λ)λ 1 µ e λ, πp n I 0 = π P I 0 t µ 1 e t dt and γ(µ, λ) = Γ(µ) Γ(µ + n) λn λ 0 t µ 1 e t dt Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 37 / 86

38 Case 3: Full information The service provider tells the arriving customer workload V (t) = v Critical point: θ v = 1/c(v) Effective arrival rate λ(v) = λh(θ v ) f(v): the pdf of the stationary workload V π F I 0 = under linear cost and uniform customers λe µ µ (λ+1) γ(λ + 1, µ) f F I (v) = λπ F I 0 (1 + v) λ e µv, v > 0 Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 38 / 86

39 Benefit of Information The service provider would like to have large throughput λe I [H(θ I )] Customers would like to have a large average utility E I [J(θ I )] E I [ 1 θ I θi 0 ] H(x)dx What is the impact of information on these measures? Clearly the answer depends on H(x). Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 39 / 86

40 Benefit of Information There are cases where the service provider and the customers are aligned H(x) = x α, α > 0 J(θ) = 1 θ θ 0 xα dx = 1 α+1 θα = 1 α+1 H(θ) Average utility Throughput More information is better for all parties Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 40 / 86

41 Benefit of Information Consider constant θ No information: throughput is λ for sufficiently small λ Partial information: there is a threshold n beyond which customers do not join throughput < λ The service provider may hide information Essentially whether information beneficial to one party or the other depends on the shape of H(θ) Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 41 / 86

42 Benefit of Information If H(1/x) is convex in x 1, then more information benefits the service provider by increasing throughput If J(H 1 (y)) is convex on [0, 1], then more information benefits the customers by increasing the average utility. H(x) = γe γx x [0, 1] 1 e γ = γ 2 (h(x) does not decrease too rapidly) Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 42 / 86

43 Benefit of Information If H(1/x) is convex, then π0 P I π0 NI π0 NI = 1 λ ( ) µ H 1 E[c N NI ] π0 P I = 1 λ ( )] 1 [H µ E c N P I π0 P I > π0 NI = N P I st N NI = E[c N P I ] E[c N NI ] a contradiction ( = H 1 E[c N NI ] ) E [ H ( 1 c N P I )] by convexity of H(1/x) and Jensen s inequality Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 43 / 86

44 Guo, P., Zipkin, P., 2007, Analysis and Comparison of Queues with Different Levels of Delay Information, Management Science, 53, Guo, P., Zipkin, P., 2009, The Effects of the Availability of Waiting-time Information on a Balking Queue, EJOR, 198, Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 44 / 86

45 Observable Queues: Residual Service Time Consider an observable M/G/1 queue The arriving customer observes the queue length before joining If the service time is exponential Customer joins if the number in the system is less than R ce[service Time] With non-exponential service times Residual service time matters Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 45 / 86

46 Observable Queues: Residual Service Time A customer who observes n customers upon arrival joins with probability q n (q 1, q 2,...) The behaviour of others has an effect on the assessment of residual service time E[RST n ] = E[residual ST when the arriving customer finds n in the system] E[RST n ] = f n (q 1, q 2,..., q n ) (a recursive expression) E[RST 1 ] = E[ST ] 1 G(Λq 1 ) 1 Λq 1 Suppose deterministic service time and q 1 is high information about the current service state Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 46 / 86

47 Observable Queues: Residual Service Time Once q 1, q 2,..., q n 1 are known if ne[st ] + f n (q 1,..., q n 1, 1) R/c = q n = 1 if ne[st ] + f n (q 1,..., q n 1, 0) R/c = q n = 0 otherwise ne[st ] + f n (q 1,..., q n 1, q) = R/c = q n = q Intuition: q 1 q 2 q 3 Which turns out to be wrong! Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 47 / 86

48 Observable Queues: Residual Service Time ST = 1 with probability ɛ (small), ST = 0 with probability 1 ɛ 1 E[RST 1 ] = 1 e 1 Λq 1 Λq 1 Solve ɛ + E[RST 1 ] = R/c to find Λq 1. Λ Λ 1 = q 1 = q 2 = 1 Λ 1 < Λ Λ 2 = 0 < q 1 < 1, q 2 = 1 Λ > Λ 2 = 0 < q 1 < q 2 < 1 The fact that there are two customers (one in service) means that the we are probably nearing the end of the current service time, and the service time of the next customer is very likely to be zero anyway Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 48 / 86

49 Observable Queues: Residual Service Time If the service time distribution is of type decreasing mean residual life (DMRL) that is, E[ST t ST > t] is monotone decreasing in t There is n e as the smallest integer satisfying q e [0, 1) satisfying ne[s] + f n (1, 1, 1,..., 1) R/c n e E[S] + f n (1, 1, 1,..., q e ) = R/c 1 n < n e q n = q e n = n e 0 n > n e Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 49 / 86

50 Observable Queues: Residual Service Time Intuition: waiting time difference between states n and n + 1 (n + 1)E[ST ] + E[RST n+1 ] ne[st ] E[RST n ] > E[ST ] E[RST n ] 0 Waiting times are increasing in n There exist at most one n with mixed strategy [RST n ] is increasing in q for (q 1,..., q n 1, q) Hence q e is unique. Avoiding the crowd versus following the crowd Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 50 / 86

51 Naor, P., 1969, The Regulation of Queue Size by Levying Tolls, Econometrica, 37, Kerner, Y., 2011, Equilibrium Joining Probabilities for an M/G/1 queue, Games and Economic Behavior, 71, Haviv, M., Kerner, Y., 2007, On Balking from an Empty Queue, Queueing Systems, 55, Kerner, Y., 2008, The Conditional Distribution of the Residual Service Time in the Mn/G/1 Queue, Stochastic Models, 24, Manou, A., Economou, A., Karaesmen, F., 2014, Strategic Customers in a Transportation Station: When Is It Optimal to Wait?, Operations Research, 62, Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 51 / 86

52 Multiple Customer Classes-identical price Things get complicated with multiple customer classes Consider two classes of customers with M/G/1 type service facility R 1, R 2, c 1, c 2, Λ 1, Λ 2 Suppose that the customers are either indistinguishable to the service provider or price discrimination is not possible Service times are identically distributed Customers are treated in a FCFS manner They can not observe the queue length Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 52 / 86

53 Multiple Customer Classes-identical price λ = λ 1 + λ 2, the equilibrium arrival rate z i = R i (c i /µ), i = 1, 2 u i (λ, p) = R i p c i (w Q (λ) + (1/µ)) = z i p c i w Q (λ) z 1 z 2, and p z i (otherwise does not join) u i (λ, p) is strictly decreasing and concave in λ since w Q (λ) is convex increasing Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 53 / 86

54 Multiple Customer Classes-identical price For any price p and 0 λ < µ, c 1 c 2 = u 1 (λ, p) u 2 (λ, p) If c 1 > c 2, there is a critical value λ so that λ > λ = u 1 (λ, p) < u 2 (λ, p) As system gets more congested, the one with higher sensitivity to delay hurts more Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 54 / 86

55 Multiple Customer Classes-identical price By solving u i (λ, p) = 0 for λ and p λ i (p) = 2µ 2 (z i p) 2µ(z i p) + c i (1 + cv 2 ) p i (λ) = z i λc i(1 + cv 2 ) 2µ(µ λ) maximum arrival rate for i for a given price p maximum price i is willing to pay for total arrival rate λ Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 55 / 86

56 Multiple Customer Classes-identical price How do customers behave for a given price p? Depends on c 1 and c 2 If c 1 c 2, then Λ 1 and λ 2 (p) are compared λ 1 (p) λ 2 (p) Λ 1 λ 2 (p) = (q 1 e, q 2 e) = (min{1, λ 1 (p)/λ 1 }, 0) Λ 1 < λ 2 (p) = (q 1 e, q 2 e) = (1, min{1, (λ 2 (p) Λ 1 )/Λ 2 }) Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 56 / 86

57 Multiple Customer Classes-identical price Suppose Λ 1 λ 1 (p) ( = λ 2 (p)) First: Class-1 enters with rate λ 1 (p) u 2 (λ 1 (p), p) u 1 (λ 1 (p), p) = 0 = qe 2 = 0 Next: Class-2 does not enter the system u 1 (λ, p) > 0 λ < λ 1 (p) u 1 (λ, p) is decreasing in λ = qe 1 = λ 1 (p)/λ 1 Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 57 / 86

58 Multiple Customer Classes-identical price How the server provider sets the price? max p λ(p)p If c 1 c 2, then depends on the market size of Type-1 Λ 1 large = p e = max{p 1, p 1 (Λ 1 )} and (qe, 1 qe) 2 = (min{1, λ 1 (p 1 )/Λ 1}, 0) p 1 = arg max pλ 1 (p) p c1 (1 + cv 1 = z 1 2 )(c 1 (1 + cv 2 ) + 2µz 1 ) c 1 (1 + cv 2 ) 2µ Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 58 / 86

59 Zhou, W., Chao, X., Gong, X., 2014, Optimal Uniform Pricing Strategy of a Service Firm when Facing Two Classes of Customers, POM, 23, Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 59 / 86

60 Multiple Customer Types-differentiation M customer types R i, c i, Λ i Some questions: what will be the control policy what will be the price/delay menu? what is the information structure: who knows what? Incentive compatibility p i + c i w i p j + c i w j j i individual rationality R i p i + c i w i Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 60 / 86

61 Multiple Customer Types-Revenue Maximization Model max p,w,u M i=1 p iλ i λ i = Λ i Pr{R i p i + c i w i } i p i + c i w i p j + c i w j j i M λ i < µ i=1 Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 61 / 86

62 Multiple Customer Types-General Results If the service provider observes the types of the customers Set priorities with a work conserving discipline (cµ rule) If the service provider does not observe the types Set priorities with strategic delays Work conserving discipline may not be optimal Delay cost minimization is not the dominant criterion Strategic delay (for low priority items) deters high priority customers purchasing a low priority menu This accomplishes incentive compatibility Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 62 / 86

63 Multiple Customer Types-An example (Allon, 2010) M/M/1 queue with µ = 1 λ 1 = 0.2, R 1 = 100, c 1 = 20 λ 2 = 0.3, R 2 = 30, c 2 = 4 Under cµ rule, w 1 = 1/(µ λ 1 ) = 1.25, w 2 = 1/(µ(1 ρ 1 )(1 ρ 1 ρ 2 )) = 2.5 p 1 = R 1 c 1 w 1 = (1.25) = 75 p 2 = R 2 c 2 w 2 = 30 4(2.5) = 20 Revenue: 0.2(75) + 0.3(20) = 21 Overall delay cost=2.5(4) (20) = 35 Not IC: (1.25) > (2.5) = 70 Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 63 / 86

64 Multiple Customer Types-An example Highest IC price under cµ: p 1 = p 2 + c 1 w 2 c 1 w 1 = (2.5) 20(1.25) = 45 with revenue=45(0.2) + 20(0.3) = 15 If one can have w = 3.5 Price for Type-2 becomes: p 2 = 30 (3.5)4 = 16 IC Type 1 price p 1 = (3.5) 20(1.25) = 61 Revenue=61(0.2) + 16(0.3) = 17. Overall delay cost=3.5(4) (20) = 39 Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 64 / 86

65 A manufacturing and service system A common part is used for service activity of heterogenous customers AudiA4 and VW Passat use the same engine, transmission and some other features Design for after-sales-service Customers have different sensitivity for waiting and service valuation Service provider keeps a common spare parts inventory Operates with a base stock policy (base stock level y) Parts are replenished through a finite capacity system M/M/1 but can be generalized Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 65 / 86

66 A manufacturing and service system Whenever there is on-hand stock, customer demand is satisfied irrespective of the type If on-hand stock is zero, customers have to wait Non-preemptive priorities A customer is tagged with an incoming part (irrespective of the type) y = net inventory + outstanding parts (N) Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 66 / 86

67 Manufacturer An Illustration N=0 Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 67 / 86

68 Manufacturer An Illustration N=1 W=0 Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 68 / 86

69 Manufacturer An Illustration N=1 Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 69 / 86

70 Manufacturer An Illustration N=2 W=0 Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 70 / 86

71 Manufacturer An Illustration N=2 Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 71 / 86

72 Manufacturer An Illustration N=3 W=0 Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 72 / 86

73 Manufacturer An Illustration N=4 C1 Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 73 / 86

74 Manufacturer An Illustration N=5 C2 C1 Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 74 / 86

75 Manufacturer An Illustration N=5 C2 C1 Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 75 / 86

76 Manufacturer An Illustration N=6 C3 C2 C1 Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 76 / 86

77 Manufacturer An Illustration N=5 C2 C1 W>0 Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 77 / 86

78 The service provider determines base stock level y price menu p = (p 1,..., p M ) the priority scheme Customers react by arriving with λ = (λ 1,..., λ M ) R i = p i + c i E[waiting time] i = 1, 2,..., M Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 78 / 86

79 The distribution of the outstanding parts M/M/1: Pr{N = k} = ρ k (1 ρ) ρ = M i=1 λ i/µ By PASTA property E[waiting with y 0] = Pr{N y}e[waiting in a standard queue] = ρ y E[waiting in a standard queue] Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 79 / 86

80 The service provider s problem { M λ i R i ρ y max y,λ,u i=1 M i=1 u = (u(1), u(2),..., u(m)) the priority order c i λ i w i he[(y N) + ] } Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 80 / 86

81 Main Results Restricted to work-conserving disciplines: cµ rule is optimal: c 1 c 2 c M Given λ, optimal base stock level: y (λ) = min { y 0 : ρ y+1 v = (1 ρ)/ρ, H(λ) = } h h + vh(λ) M c i λ i E[waiting(λ)] Prices given by the first order conditions are incentive compatible i=1 Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 81 / 86

82 A reduction: If R i R j and c i c j, then Type-i dominates Type-j λ j = 0 (p j = R j) c i = c, R k = max{r i } or R i = R, c k = min{c i } { max Rµρ cρ y ρ y 0,ρ [0,1) 1 ρ h(y ρ } 1 ρ (1 ρy )) y(ρ) = min{y 0 : ρ y+1 h/(c + h)} Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 82 / 86

83 A continuous approximation: Pr{N y} e vy Variable Value / / h / c λ µ µk R y E[W ] p K ( ) / RµK K h, / R (µk) / R c R (µk) h log(1 + c/h) Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 83 / 86

84 K = h log(1 + c/h) c as h Compare p above with p = R Optimal profit: ( Rµ K) 2 cr µ = R c R µc K < c = (profit with y > 0) (profit with y = 0) Attracts more demand (with smaller price) and achieves a higher total profit. Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 84 / 86

85 Mendelson, H., Wang, S., 1990, Optimal Incentive-Compatible Priority Pricing for the M/M/1 Queue, Operations Research, 38, Afèche, P., 2013, Incentive-Compatible Revenue Management in Queueing Systems: Optimal Strategic Delay, MSOM, 15, Allon, G., 2010, Pricing and Scheduling Decisions, Wiley Encyclopedia of Operations Research and Management Science edited by James J. Cochran, Wiley. Maglaras, C., Yao, J., Zeevi, A., 2013, Optimal Price and Delay Differentiation in Queueing Systems, Working Paper. Guler, G., Bilgic, T., Gullu, R., 2014, Joint Inventory and Pricing Decisions when Customers are Delay Sensitive, to appear in IJPE. Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 85 / 86

86 Some less explored topics Alternative cost and reward structures willingness to wait : E[(W W tow ) + ] Correlation of R and c Competition and cooperation among service providers Distribution free bounds Estimation errors in parameters max min λe[(r cw(λ)) + ] λ f R Bounded rationality (Huang, Allon and Bassamboo, 2014) Refik Güllü (Boğaziçi University) YEQT VIII Eindhoven 86 / 86