OPERATIONS RESEARCH Vol. 59, No. 5, September October 2011, pp. 1304 1308 in 0030-364X ein 1526-5463 11 5905 1304 http://dx.doi.org/10.1287/opre.1110.0990 2011 INFORMS TECHNICAL NOTE INFORMS hold copyright to thi article and ditributed thi copy a a courtey to the author(). Additional information, including right and permiion policie, i available at http://journal.inform.org/. A Note on Profit Maximization and Monotonicity for Inbound Call Center Ger Koole Department of Mathematic, VU Univerity Amterdam, 1081 HV Amterdam, The Netherland, oole@few.vu.nl Aue Pot CCmath, 1181 BH Amtelveen, The Netherland, pot@ccmath.com 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. Subject claification: call center; monotonicity. Area of review: Manufacturing, Service, and Supply Chain Operation. Hitory: Received July 2008; reviion received January 2009, Augut 2009; accepted October 2009. 1. Introduction Traditionally, call center are een a cot center. Cot center add to the cot but not directly to the profit of the company concerned. Therefore, reducing cot i a prime goal in cot center, given that it product meet the objective of the company. In the etting of call center, thi mean that a certain ervice level ha to be obtained for minimal cot. Thi 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. Lately there ha been ome criticim of conidering call center a cot center. The focu on, for example, the reduction of handling time lead to low-quality contact with (potential) cutomer. To value the cutomer contact, call center hould not be treated a cot center but a profit center. A profit center add directly to the profit of a company. In a call-center etting, thi mean that each contact add value. 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 profit increae. A perfect example where replacing cot minimization by profit maximization lead to coniderable cot aving i decribed in Andrew and Paron (1989), Andrew and Paron (1993). In Andrew and Paron (1989) the author compare for a particular cae the profit of different agent cheduling tool, who themelve ue a ervice-level contraint in the Erlang C model to obtain taffing level. In Andrew and Paron (1993) the agent taffing at interval level i conidered. The optimal taffing level i determined by profit maximization: every handled call give a reward, waiting call lead to (communication) cot, and every cheduled agent cot a fixed amount. Increaing the number of agent increae agent cot, but decreae waiting cot and decreae the abandonment rate, thereby increaing the reward. It i remarable that the Erlang C model i ued, becaue abandonment (which are not modeled by Erlang C) play a crucial role in the paper. Thi i jutified by the low abandonment rate in the cae tudy; it i aumed that in that cae the influence on the ervice level i limited (note that thi ha been contradicted by recent wor on the Erlang A model; ee, for example, Gan et al. 2003, 4.2.2). The abandonment percentage i obtained by linear regreion with the ervice-level a explanatory variable. In thi paper we tudy a ituation where, in addition to the number of agent, the number of line can alo be determined. In practice thi mean that arriving call are bloced and aed to call again at another moment. Thi avoid exceively long waiting time for call for which there i no capacity anyhow. It i our objective to find the global maximum for thi two-dimenional profit function, uing a model that include abandonment. We derive propertie of the profit function o 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 1304
Koole and Pot: Profit Maximization and Monotonicity for Inbound Call Center Operation Reearch 59(5), pp. 1304 1308, 2011 INFORMS 1305 INFORMS hold copyright to thi article and ditributed thi copy a a courtey to the author(). Additional information, including right and permiion policie, i available at http://journal.inform.org/. 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 that the optimal number of line i determined for each poible 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. (2005) (ee alo Helber and Stolletz 2003). In thi paper the author urvey the German call-center maret and come up with a number of buine model. Beide, to our ingleperiod profit-maximization model they alo conider a multiperiod 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 Helber et al. (2005), and our counterexample how that finding the optimal olution for the multiperiod model i, in theory, a nontrivial problem. A different call-center profit-maximization model (with multiple call clae and a hared reource) ha been decribed in Aşin and Harer (2003). There i a ingle condition that i required in part of our proof: the average ervice time hould be horter than the average patience (or willingne to wait ) of a call. The reaon i a technical one related to convexity of dynamic programming value function. Independently, Armony et al. (2009) obtained imilar reult. 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. We ue the idea of event-baed programming in thi part of the proof (ee Koole 2006). In thi ection we alo go into more detail about the retriction on the patience time. 2. Model Decription and Reult The call-center model that we conider i commonly called an M/M//n + M ytem. That i, a in the Erlang C model it ha Poion arrival, exponential ervice time, and a finite number of erver (). However, it ha two additional feature: cutomer who find all erver occupied leave the queue after an exponentially ditributed amount of time (their patience), and there i 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, and the rate of the ervice time ditribution i. (In practice, uually the expected call duration i taen, which we denote with = 1/.) 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. 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 we are only intereted in the expected reward r becaue we have no prior information on the reward of a call.) 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, and immediate reward, given by: x x + 1 = for 0 x < + n x x 1 = min x + x min x for 0 < x + n x = min x r xc Uing tandard argument for birth-death procee, it follow that g n i given by (tae a = ): ( g n a x = x r c x=0 x! ) n + a x! x y=1 + y r c xc ( x=0 Now define g = max 0 n N g n and g = max 0 S g x=1 a x x! + a! n x=1 x x y=1 + y ) 1 (1) with n = arg max g n 0 n N with = arg max 0 S g In 4 we how the following reult. Theorem 2.1. g 0 g n for all 0 S and, if, n n for all 0 < S. Theorem 2.1 tell u the following: for a fixed number of agent the reward i nondecreaing in the number of line up to the optimal number of line, and if the number of agent i increaed, then the optimal number of line for that many erver doe not decreae. The latter tatement hold only in the cae that the ervice rate i not lower than the abandonment rate. The theorem lead to a imple algorithm for finding and n. To avoid trivialitie, we chec firt that 1 + c < 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.
Koole and Pot: Profit Maximization and Monotonicity for Inbound Call Center 1306 Operation Reearch 59(5), pp. 1304 1308, 2011 INFORMS INFORMS hold copyright to thi article and ditributed thi copy a a courtey to the author(). Additional information, including right and permiion policie, i available at http://journal.inform.org/. Algorithm for finding n : 0. Tae n = n = 0 0 1. If 1 + c r, then: top 2. For = 1 to S do 3. Compute g n (uing Equation (1)) n+1 3. If n < N, then: Compute g 4. While g n min n+1 N < g 5. n n + 1 n+1 6. If n < N, then: Compute g 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 2.1 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, = 5, = 1, c = 0 5, r = 3, = 0 5, and the price of an agent 1 per time unit. It tae at maximum S + N tep, wherea 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 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 webite; ee http://obp.math.vu.nl/callcenter/erlangp. 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. Figure 1. n -grid for S = 10, N = 30, = 5, = 1, c = 0 5, r = 3, and = 0 5 n 30 20 10 0 0 2 4 6 8 10 An intereting pecial cae i c = 0. Then there i no reaon to refue any cutomer, and thereby the number of line hould be a big a poible. Thu, all that need to be determined i the optimal number of erver. Now we can apply Theorem 11.7 from Koole (2006), which how that the reward i concave in, again under the aumption that. Thu, it uffice to increae until the reward tart decreaing. When c > 0, the optimal number of line i alway finite: if the queue i long enough, then the holding cot c outweigh the ervice reward r. A final remar concern the ituation where the holding cot are different in ervice than when waiting. Thi, however, can be tranlated to the ituation where holding cot are equal by changing r appropriately. 3. Counterexample and Implication If 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 = 15, = 1, c = 0 39, r = 1 52, = 1/2 9 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 nonunimodal 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 completely jutify the econd line, but with a ingle line the productivity and thu the reward i too low (g 9 1 = 0 3824). Thu, due to the dicrete nature of n we ee a drop in profit at = 9. Table 1. Value of g for variou for = 15, = 1, c = 0 39, r = 1 52, and = 1/2 9. Abandonment Bloced SL g n (in %) (in %) (in %) 0 0 0000 0 0 00 100 00 100 00 1 0 0594 0 0 00 93 75 100 00 2 0 1105 0 0 00 87 55 100 00 3 0 1521 0 0 00 81 40 100 00 4 0 1825 0 0 00 75 32 100 00 5 0 2396 1 4 47 66 05 86 91 6 0 3147 1 3 45 59 99 91 42 7 0 3665 1 2 70 54 05 94 41 8 0 3907 1 2 15 48 28 96 39 9 0 3855 2 4 02 39 79 90 53 10 0 3993 2 3 26 34 36 93 56 11 0 3636 2 2 64 29 22 95 71 12 0 2951 3 3 54 21 94 92 46 13 0 1771 3 2 81 17 62 94 97 14 0 0033 4 3 13 11 86 93 21 15 0 2561 5 3 09 7 29 92 49
Koole and Pot: Profit Maximization and Monotonicity for Inbound Call Center Operation Reearch 59(5), pp. 1304 1308, 2011 INFORMS 1307 INFORMS hold copyright to thi article and ditributed thi copy a a courtey to the author(). Additional information, including right and permiion policie, i available at http://journal.inform.org/. The counterexample ha further implication. Conider we have two (or more) interval and a limited number of agent hour, a in Helber et al. (2005). 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 multiperiod model of Helber et al. (2005). It i intereting to note that thi i not the only ituation where admiion control detroy the concavity. Tae the regular M/M/ queue: it can be hown that the value function i concave in the arrival rate for many common ituation. However, a counterexample to concavity exit a oon a admiion control i added. For more detail, ee Varga (2002) and the lat part of 11.2 of Koole (2006). 4. Monotonicity Reult In thi ection we firt prove Theorem 2.1. Then we dicu ome iue related to convexity. Proof of Theorem 2.1. 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 + n + 1 for ome 0 < p < 1. Thi equation hold for all birth-death procee. Now uppoe that g n n+1 > g for ome n < n. Thi mean that + n + 1 < g n. Note alo that + n < < + n + 1. Becaue g n i a convex combination of + n + n + 1 and g n, thi mean that g n < g n, which i in contradiction with the optimality of n. The proof of the econd aertion of Theorem 2.1 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 1975; ee Koole 2006 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 + S + N = 1. Then the tranition rate can alo be een a tranition probabilitie of the embedded uniformized chain (ee Lippman 1975). Cot and reward are implemented a follow. When a cutomer i allowed to enter, we receive a reward of r. When a cutomer abandon, we incur a cot of r. In tate x thi happen at rate x +. Next to that there are communication cot and erver cot cx +. The dynamic programming value function V of the embedded chain, with the epoch, i now given by V +1 x = r x + cx + max V x r + V x + 1 + min x + x + V x 1 + S min x + N x + V x V +1 + N = rn c + N + V + N if x < + N and > 0 + + N V + N 1 + S V + N if > 0 and V 0 x = 0 for all and x Note that the 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 r +V x +1, then according to Equation (2), alo V x + 1 r + V x + 2, and admiion i alo optimal with x + 1 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, e.g., Puterman 1994). In the proof of inequality (2) we need concavity of V in x, i.e., V x + V x + 2 V x + 1 + V x + 1 (3) Thi reult i valid only if ; ee Koole (2006, Theorem 9.3) and the dicuion in the lat paragraph on p. 50. (Note that in Koole 2006 a cot etting i ued, interchanging convexity and concavity for the value function.) Summing inequalitie (2) and (3) with replaced by +1 lead to upermodularity: 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 V+1 one by one (a method firt formalized in Koole 1998, ee alo Koole 2006). We tart with the firt
Koole and Pot: Profit Maximization and Monotonicity for Inbound Call Center 1308 Operation Reearch 59(5), pp. 1304 1308, 2011 INFORMS INFORMS hold copyright to thi article and ditributed thi copy a a courtey to the author(). Additional information, including right and permiion policie, i available at http://journal.inform.org/. term of the dynamic programming equation, the one related to the reward. The inequality to how i min x + 1 + min + 1 x + 1 min x + min + 1 x + 2 It i readily hown that thi indeed hold for all value of x and. The ame hold for the cot. Now conider the term with coefficient. 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 x+1 V x+2 +max V x+1 V x+2 =V x+2 +V x+2 V 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 + 1 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, the departure term. There coefficient um up to S, 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 up to 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 left-hand ide to V x +V x, on the right-hand ide to V x + V x + 1. Server x + 2 lead on the left-hand ide to V x + 1 + V x + 1, on the right-hand ide to V x + V x + 1. Both lefthand ide ummed are maller than the right-hand ide ummed becaue of Equation (4), which hold by induction. Next aume that x. Then all erver are buy, and the + 1th erver again give Equation (4). Finally, conider the abandonment. Aume that x; otherwie there are no abandonment. Note that V x + 1 and V x + 2 have one more cutomer in queue than V x + 1 and V x. We can combine the abandonment term for the firt x cutomer in queue, leading again to (2), but with one cutomer le. The term concerning the abandonment of the extra cutomer in queue in V x + 1 and V x + 2 lead to an equality. The reult a given here i valid only if. Thi lead to two intereting quetion. The firt i: Are there counterexample to the reult in cae <? If thi would be the cae, then the algorithm could lead to nonoptimal olution. A econd intereting quetion i the following: i the value function till concave if <? The anwer to the econd quetion i no: we will give a counterexample in the next paragraph. Thi how that if the reult i true in general, then another line of proof would be needed. We did extenive numerical experimentation to try to find a counterexample to the firt quetion. On the bai of thi we conjecture that Theorem 2.1 hold for all value of and. The implet counterexample to the concavity of the value function V i a follow. Tae = 0, = 1, = 1, = 2, c = 1, r = 0. Then V1 x = x, and the departure term of V2 i convex. Thi, ummed with a linear cot term, mae V2 convex. It i eay to find numerical example with > 0. A an example, tae = 15, = 1, = 10, = 3, c = 0 2, and r = 0 1. Then the direct reward are trictly concave, but numerical experiment how that the average reward value function i trictly convex. In fact, it can be hown that in the cae of linear (or, in general, convex) reward and a convex total departure rate (in our cae, ervice completion plu abandonment), the value function i convex in the tate (Beer 2008). Thi doe not apply to our ituation: the direct reward are trictly concave due to the reward for ervice completion. 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