A note on profit maximization and monotonicity for inbound call centers

Transcription

1 A note on profit maximization and monotonicity for inbound call center Ger Koole & Aue Pot Department of Mathematic, Vrije Univeriteit Amterdam, The Netherland 23rd December 2005 Abtract We conider an inbound call center with a fixed reward per call and communication and agent cot By controlling the number of line and the number of agent we can maximize the profit Abandonment are included in our performance model Monotonicity reult for the maximization problem are obtained, which lead to an efficient optimization procedure We give a counterexample to the concavity in the number of agent, which i equivalent to aying that the law of diminihing return doe not hold Numerical reult are given 1 Introduction Traditionally call center are een a cot center Thi mean that a certain ervice level ha to be obtained for minimal cot The ervice level i often taen a follow: 80% of the call hould be anwered by a call center agent within 20 econd In thi buine model a call cot money but i neceary to the company Another buine model i the one where the call center i conidered to be a profit center There i a reward aigned to a call, and the objective of the call center i to maximize it profit, defined a reward minu cot The variable cot in a call center mainly conit of alary cot and communication cot, in cae the call center pay for (part of) that Such a buine model can lead to coniderable aving (ee [2] for an example) In thi paper we analyze a model for a call center in which we determine the number of line and agent for which the profit i maximized It i our objective to find the global maximum for thi two-dimenional profit function We derive propertie of the profit function a to avoid having to earch exhautively all poible combination of parameter value Baed on thi we formulate an algorithm that find the global maximum The optimization procedure can be een a a local earch procedure in which the function value i the profit for given parameter value Thee value can be obtained through Marov chain method We give ome numerical reult The intereted reader can alo experiment with a tool that i freely available on the internet In addition to the poitive reult that lead to the optimization procedure we alo offer a counterexample that how that (given the optimal number of line i determined for each poible 1

2 number of agent) the profit function i neither concave nor unimodal in the number of agent Thi how that the law of diminihing return doe not hold for thi model Our model i one of the model tudied in Helber et al [4] (ee alo [3]) In thi paper the author urvey the German call center maret and come up with a number of buine model Next to our ingle period profit maximization model they alo conider a multi-period model with a contraint on the number of available agent hour Our monotonicity reult how how to find the optimal olution in the ingle-period model of [4], and our counterexample how that finding the optimal olution for the multi-period model i, in theory, a non-trivial problem A different call center profit maximization model (with multiple call clae and a hared reource) ha been decribed in Aşin and Harer [1] In the next ection we decribe the model After that we preent the propertie of the profit function and the reulting optimization procedure We alo give numerical reult The ection after that i devoted to the counterexample and it implication In the final ection the propertie of the profit function are derived, mainly uing dynamic programming Thi i done by inductively proving certain propertie, related to concavity, of the dynamic programming value function 2 Model decription and reult The call center model that we conider i commonly called an M/M//n+M ytem That i, it ha Poion arrival and exponential ervice time, with additional feature exponential abandonment and a finite number of line meaning that the total number of call waiting and in ervice i retricted There are no redial of bloced or abandoned call The cutomer arrival rate i λ, and the rate of the ervice time ditribution i µ (In practice uually the expected call duration i taen, which we denote with β = 1/µ) The number of agent i variable, with an upper bound of S, the number of eat in the call center Alo the number of line i limited, to + N if there are agent (We tae + N intead of imply N for reaon that will become clear later) A call that i waiting abandon with rate γ We have communication cot c per call per unit of time, and cot 1 per cheduled agent per unit of time There are expected reward r per handled call (Note that the actual reward per call might vary, but a we are only intereted in expected reward, and a we have no prior information on the reward of a call, we only need r) We define g,n a the average long-run expected profit for agent or erver and n additional waiting line For fixed and n g,n i the tationary reward in a birth-death proce with tate x {0,, + n} (indicating the number of call in the ytem), tranition rate α(, ), and immediate reward δ( ), given by: α(x,x + 1) = λ for 0 x < + n, α(x,x 1) = min{x,}µ+ (x min{x,})γ for 0 < x + n; δ(x) = min{x,}µr xc 2

3 Uing tandard argument for birth-death procee it follow that g,n i given by (tae a = λβ): g,n = x=0 n a x a x(µr c) + x!! x=1 a x x! + a! x=0 λ x xy=1 ((µr c) xc) (µ+ yγ) n x=1 λ x xy=1 (µ+ yγ) (1) Now define g = max 0 n N g,n (with n = argmax 0 n N g,n ) and g = max 0 S g (with = argmax 0 S g ) In Section 4 we how the following reult Theorem 21 g,0 g,n for all 0 S and n n for all 0 < S Theorem 21 tell u the following: for a fixed number of agent the reward i non-decreaing in the number of line up to the optimal number of line, and if the number of agent i increaed than the optimal number of line for that many erver doe not decreae The theorem lead to a imple algorithm for finding and n To avoid trivialitie we chec firt that (1 + c)β < r: if thi i not the cae then the cot for the agent and communication of a call that i directly connected are higher than the profit, and it i better to reject all call and to chedule no agent at all Algorithm for finding (,n ): 0 Tae (,n) = (,n ) = (0,0) 1 If (1 + c)β r then: top 2 For = 1 to S do 3 Compute g,n (uing Equation (1)) 3 If n < N then: Compute g,n+1 4 While g,n < g,min{n+1,n} 5 n n If n < N then: Compute g,n+1 7 If g,n > g,n then: (,n ) (,n) Thu we ee that for each value of we increae n until we have found the optimal value n ; then we increae, and we tart increaing n again, from the value n According to Theorem 21 we certainly encounter the optimal olution, but we can only identify it after having determined n for all value of, becaue g need not be unimodal, a the counterexample in the next ection how In Figure 1 we ee a typical example of how the algorithm travere the (, n)-grid, from the lower left corner to the upper right The correponding parameter are S = 10, N = 30, λ = 5, µ = 1, c = 5, r = 3, γ = 5, and the price of an agent 1 per time unit It tae at maximum S + N tep, while there are SN point in the grid Thi illutrate well the efficiency of the algorithm a compared to enumeration It mae it uitable for routine application to typical call center 3

4 n Figure 1: (,n)-grid for S = 10, N = 30, λ = 5, µ = 1, c = 5, r = 3, and γ = 5 optimization problem with ten of different interval with different parameter per day The optimum i (,n ) = (6,13) The tool by which the numerical reult were obtained can alo be found on our web ite, ee wwwmathvunl/ apot/oftware/erlangprofit/ Note that in the tool the number of line include the number of agent, which i more uual in practice Alo the average ervice and abandonment time have to be entered, not the rate 3 Counterexample and implication In cae g were unimodal then we could top earching a oon a g would decreae after ome We how by a counterexample that thi i not alway the cae Conider the model with the parameter λ = 15, µ = 1, c = 039, r = 152, γ = 1/29 and an agent cot 1 per unit of time, a we defined earlier To analyze the concavity we vary from 0 to 15 In Table 1 the value of g can be found We ee that g increae for up to = 8, then it decreae for = 9, to tae it maximum value at = 10 We conclude that the function g,n i non-unimodal and thu neither convex nor concave in Thi counterexample how the neceity of increaing up to S in the algorithm The intuition behind it i a follow Computation how that n 8 = 1 and n 9 = n 10 = 2 Thu when adding the 9th erver it i optimal to add an additional line for waiting However, for n = 2, it i better to have 10 intead of 9 erver Thu = 9 doe not jutify completely the econd line, but with a ingle line the productivity and thu the reward i too low (g 9,1 = 03824) Thu due 4

5 Table 1: Value of g for variou for λ = 15, µ = 1, c = 039, r = 152, and γ = 1/29 g to the dicrete nature of n we ee a drop in profit at = 9 The counterexample ha further implication Conider we have two (or more) interval and a limited number of agent hour, a in [4] How to allocate the agent hour in an optimal way? A greedy algorithm would find the optimal allocation, auming that the law of diminihing return hold Thi law tate that when the number of erver i increaed the additional return for adding erver decreae Thi i equivalent to concavity Indeed, for a um of concave function the optimal allocation can be found by tarting empty and adding agent hour one by one, each time to the interval with the highet additional return It follow clearly from the counterexample that the concavity doe not hold, and thu the greedy algorithm i not guaranteed to give an optimal olution in the multi-period model of [4] 4 Monotonicity reult In thi ection we prove Theorem 21 Proof of Theorem 21 We tart with proving g,0 g,n for ome with 0 S Aume that n > 0 (otherwie there i nothing to prove) Firt note that g,n+1 = pg,n + (1 p)δ( + n + 1) for ome 0 < p < 1 Thi equation hold for all birth-death procee Now uppoe that g,n > g,n+1 for ome n < n Thi mean that δ( + n + 1) < g,n Note alo that δ(+n ) < < δ(+n+1) Becaue g,n i a convex combination of δ(+n ),,δ(+n+1) and g,n, thi mean that g,n < g,n, which i in contradiction with the optimality of n 5

6 The proof of the econd aertion of Theorem 21 i more involved We ue dynamic programming in it proof We formulate the value function for fixed and admiion control It i well nown that a threhold policy i optimal (Lippman [6]; ee Koole [5] for an overview of thi type of monotonicity reult) We called thi threhold n, meaning that an arrival i rejected if and only if the number of cutomer exceed + n To prove n n for ome 0 < S we need to how that when admiion i optimal in the ytem with erver and a total of x cutomer, then admiion i alo optimal in the ytem with + 1 erver and a total of x + 1 cutomer (giving the ame number of waiting cutomer) Let u now formulate the dynamic programming value function We cale time uch that λ + Sµ+ Nγ = 1 Then the tranition rate can alo be een a tranition probabilitie of the embedded uniformized chain (ee Lippman [6]) The dynamic programming value function V of thi embedded chain, with the epoch, i now given by V+1 (x) = µrmin{,x} cx + λmax{v (x),v (x + 1)}+ [µmin{,x} + γ(x min{,x})]v (x 1) + [µ(s min{,x}) + γ(n x + min{,x})]v (x) if x < + N and > 0, V +1 ( + N) = µr c( + N) + λv ( + N) + [µ+ γn]v ( + N 1)+ µ(s )V ( + N) if > 0, and V0 (x) = 0 for all and x Note that final term in both equation come from the uniformization procedure To prove that admiion i optimal in the ytem with + 1 erver and x + 1 cutomer if it i optimal in the ytem with erver and x cutomer, it uffice to how that V (x + 1) +V (x + 1) V (x) +V (x + 2) (2) for all 0, 0 < S, and 0 x < +N 1 Indeed, if admiion i optimal in x when having erver, and thu V (x) V (x + 1), then according to Equation (2) alo V (x) V (x + 1), and admiion i alo optimal with x call and + 1 erver or agent From Marov deciion theory it follow that the ame hold for the long-run limiting average cae (becaue tate and action pace are finite: ee, eg, Puterman [7]) In the proof of inequality (2) we will ue the well-nown concavity of V in x, ie, V (x) +V (x + 2) V (x + 1) +V (x + 1) (3) Note that inequalitie (2) and (3) ummed give upermodularity (or convexity) V (x + 1) +V (x) V (x) +V (x + 1) (4) We are now ready to prove (2) We do thi by induction to For = 0 the inequality trivially hold Aume that it hold up to ome Now conider the correponding term in V +1 and 6

7 V+1 i one by one (a method formalized in [5]) Conider firt the reward, the inequality to how min{,x + 1} + min{ + 1,x + 1} min{,x} + min{ + 1,x + 2} It i readily hown that thi hold indeed for all value of x and The ame hold for the cot Now conider the term with coefficient λ We have to loo at a number of cae Aume firt that the maximizing action in both V (x + 1) and V (x + 1) i admiion Then max{v V (x + 1),V (x + 2)} + max{v (x + 1),V (x + 2)} = V (x + 2) +V (x + 2) (x + 1) +V (x + 3) max{v (x),v (x + 1)} + max{v (x + 2),V (x + 3)}, the firt inequality i obtained by induction A imilar argument hold in the cae that rejection i optimal in tate x+1 for the ytem with and erver If the optimal action are different, then it mut be that admiion i the optimizing action in V (x + 1), by induction Then max{v (x + 1),V (x + 2)} + max{v (x + 1),V (x + 2)} = V (x + 1) +V (x + 2) max{v (x),v (x + 1)} + max{v (x + 2),V (x + 3)} Conider next the term with coefficient µ, the departure term There coefficient um up to Sµ, a if there are in total S erver For value function V of thee are preent, and in tate x min{,x} of thee are active We number the erver, and aume that in tate x erver 1 upto min{,x} are active We conider the erver one by one Aume firt that x + 1 Then all call in (2) are being erved Of particular interet are erver x + 1 and x + 2, the term related to all other erver hold trivially by induction Server x + 1 lead on the lh to V (x) +V (x), on the rh to V (x)+v (x+1) Server x+2 lead on the lh to V (x+1)+v (x+1), on the rh to V (x)+v (x + 1) Both lh ummed are maller than the rh ummed, becaue of Equation (4), which hold by induction Next aume that x Then all erver are buy, and the + 1th erver give again Equation (4) Finally conider the abandonment Aume that x, otherwie there are no abandonment Taing abandonment of the extra cutomer in queue in V (x + 1) and V (x + 2) into account lead to an equality Reference [1] OZ Aşin and PT Harer Capacity izing in the preence of a common hared reource: Staffing an inbound call center European Journal of Operational Reearch, 147: , 2003 [2] B Andrew and H Paron Etablihing telephone-agent taffing level through economic optimization Interface, 23(2):14 20, 1993 [3] S Helber and R Stolletz Call Center Management in der Praxi Springer,

8 [4] S Helber, R Stolletz, and S Bothe Erfolgzielorientierte Agentenalloation in inbound call Centern Zeitchrift für Betriebwirtchaftliche Forchung, page 3 32, 2005 (February) [5] GM Koole Structural reult for the control of queueing ytem uing event-baed dynamic programming Queueing Sytem, 30: , 1998 [6] SA Lippman Applying a new device in the optimization of exponential queueing ytem Operation Reearch, 23: , 1975 [7] ML Puterman Marov Deciion Procee Wiley,