A study on call/contact centers’ inbound and outbound management process in Mexico

One challenge related to contact center management involves determining which process best serves customers, inbound or outbound. Such decisions impact the number of service agents available for operations, affecting costs. The size of the call centers market worldwide is estimated to reach $337 billion dollars by 2018. This industry employs 670,000 people in Mexico. A series of equations for calculating the difference in the number of service agents required by the two processes is determined using the direct demo method. Developed theorems and corollary may help simplify decision-making processes. The findings demonstrate that the number of agents required for both processes depends on the percentage of customers served at each location and on service agent occupation rates. The study recommends some best practices to the Mexican call center industry in order to improve its profits and quality within the inbound outbound services.

must wait on hold on average until an agent is available to attend to a call (Mehrotra, Grossman, & Samuelson, 2011;Nah & Kim, 2013;Rod & Ashill, 2013;Stolletz, 2012). Vernez, Robert, Massey, and Driscoll (2007) identified that contact centers manage a diversity of operations due to complex processes, thus having difficulties to reach the highest levels of performance at all of them. Most inbound traffic companies create some outbound traffic, and vice versa. Depending on the process established, varying numbers of service agents are required to facilitate operations; due to varying payroll costs for operations, contact center management teams face the challenge of selecting operational processes needed to satisfy customer demand, either inbound or outbound. Tan and Netessine (2014), Zhao, Jin, and Yue (2015) and Afeche (2013) studied the basic operations of contact centers and recommend three ratios that enable both agents and customers to follow homogeneous practices. These ratios are used for both inbound and outbound processes.

Literature Review
a) The average workload R for a time interval is defined based on the ratio between the average customer entry  and exit µ derived from the process employed by all available agents, as expressed in equation 1.
The individual average traffic intensity ρ is defined based on the average workload with active agents N being available to serve customers, as shown in equation 2.
The individual average traffic intensity ρ also represents the percentage of agents occupied at their workstations.

c)
The average length of service time that an agent needs to assist a customer E[S] is commonly known as the average handle time (AHT) for contact center operations and is estimated based on the average number of customers that each agent can serve within a time interval µ, as shown in equation 3.

E[S] = 60 / µ
This quantity in the numerator (60) is fixed to facilitate the development of equations in this article; this implies that transactions are completed over a standard time interval of one hour (60 minutes). When it is necessary to use the equations determined for other time intervals, it is necessary to obtain the corresponding ratio (e.g., to use 3,600 when measuring time in seconds).
From the average workload Rin, it uses the M/M/Nin model of queue theory for inbound contact centers to determine the likelihood that a contact center will be busy according to the Erlang C(Nin, Rin) formula (Baccelli & Brémaud, 2013), which is given by The model assumes the following: a) entrance and service rates are constant over a time interval, b) the entrance process is a Poisson process, c) the service times obey an exponential and independent distribution that d) assumes an order in which the first to call is the first to be served and that e) ignores blockades and assumes that customers do not hang up. Under these assumptions, the Erlang-C formula enables direct calculation of the probability distribution of the time that a customer must wait to access a contact center. In cases for which ρin ≤ 1, Nin must be greater than Rin. Janssen, van Leeuwaarden and Zwart (2008) formally analyzed the "quality and efficiency driven" value of a queuing system M/M/Nin for which the service rate µ is fixed, whereas the probability that a contact center is busy C(Nin, Rin) lies within the range [0, 1].
Under such conditions, some customers must wait to receive services. If in tends to infinity and Nin tends to infinity, then ρin = in / (Nin . Μ) tends to 1. There is a service grade β, where Gans, Koole and Mandelbaum (2003) recommend using a safety margin of additional agents ("Square-Root Safety Staffing Rules") for contact centers. These rules suggest that the number of agents required for an inbound process is equal to the workload plus a "safety staffing" follow-up margin of additional agents, Nin = Rin + (β . (Rin) 0.5 ). The safety staffing margin is Δ = β (Rin) 0.5 , where 0 < Δ < ∞. Thus, the number of agents can be expressed as equation 5.

Nin = Rin + Δ (5)
Additionally, Gans, Koole, and Mandelbaum (2003) noted that in contact center operations, the average time that a customer expects to wait in a queue to receive a service from an agent can be expressed as E [WAIT], which is known as the "average speed answer" (ASA) (alternatively Wq "waiting in the queue"), as formulated in equation 6.  Gans, Koole & Mandelbaum, 2003)). Gans, Koole and Mandelbaum(2003) recommend simplifying the calculations made in contact centers based on an approximation of the "Square-Root Safety Staffing Rules." This approximation is valid for large contact centers and is given by E[WAIT] ≈ 1 / Δ. Borst, Mandelbaum and Reiman (2004) conducted a comprehensive study that revealed an optimal β for when waiting costs and human resource expenses are linear functions of time.
A large body of literature has addressed agent calculations for inbound contact centers. Fewer studies have focused on calculating the number of outbound contact center agents (Gans, Koole & Mandelbaum, 2003;Aksin, Armony, & Mehrotra, 2007). Colledani and Gershwin (2013) noted that outbound processes are similar to industrial processes of continuous production and that "traffic intensity" levels are related to work volumes based on the displacement capacities for such jobs. When ongoing production processes are well configured, all entries for an interval of time out should be equal to all exits (Nout µ) without forming wait queues. Hence, for outbound processes, traffic intensity levels should always take a value of ρ = 1 such that for equation 2, the number of agents Nout is determined from equation 7.
If the productivity equation 3 is substituted into equation 7, the equation for determining the number of service agents needed in an outbound process is obtained from the time of service that each agent must devote to each client, E[s], and from the volume of customers estimated to be processed per hour at a contact center, out, as determined using It is important to note that the average service time, E[S], includes the time required for prior preparation and post-service administrative work.

Research Methods
To determine the difference between the number of available agents required to manage contact centers employing inbound and outbound processes, a mathematical development method of direct proof is used under three different service scenarios that are each described in the following two theorems and a corollary.
Theorem 1: If the volume of services to be processed within an inbound system is the same as the volume of services to be processed in an outbound system, i.e., in = out = , both processes have an identical service time, E[S], and for inbound customers, there is a certain queuing time, E[WAIT], which is defined as the "Average Speed Answer Call Factor" (ASACF) or as the ratio between the waiting time and service time, as shown in equation 9.

ASACF = E[WAIT] / E[S]
(9) Then, the difference between the numbers of service agents required for the two systems is given by equation 10.
The assumption of equal volumes to be processed in both systems holds in only exceptional cases, and Theorem 2 is designed to violate this equality because outbound process require more time than inbound processes owing to the difficulty of reaching customers in the former.

E[s] / 60 ), from equation 8, is substituted into equation 12, when out = in and based on the E[WAIT] / E[S] value for equation 9, it obtained equation 10.
Theorem 2: If the volume of services to be processed by agents within an inbound system, in, is less than the volume of services to be processed by agents in an outbound system, out, (because in the latter process, the target contact is not always reached) and E(L) is the average rate of target contact location for outbound processes according to equation 13, then: For both processes, agents produce a same time of service value, E[S], regardless of whether the target contact was reached, and there is a time that inbound customers must wait in the queue, E [WAIT]. The difference between the numbers of service agents required for both systems is given by equation 14.

Nin = E(L) Nout + C(Nin, Rin) / ASACF
Where C(Nin, Rin) is defined by the Erlang-C formula, equation 4, and where ASACF is the "Average Speed Answer Call Factor" defined by equation 9.
Proof: For the proof, it is necessary to find an equation that determines the difference between (Nin -Nout) given different customer volumes, which are related by in = out . E(L) according to equation 13. If the µ value for equation 3 for an inbound process is input into equation 2, equation 15 is obtained.

Rin = E(L) Nout
When the ρin value for equation 15 is used in equation 6, equation 18 is obtained.   Table 2 illustrates Theorem 2 and Corollary 1. Assume the same contact center data used to exemplify Theorem 1 in addition to a localization rate for outbound processes of 0.50. Equation 10 from Theorem 1 is a particular case of equation 14 of Theorem 2 when the probability of finding effective contacts is E(L) = 1. Equation 20 of Corollary 1 determines the difference between the numbers of service agents required for inbound and outbound processes, and this difference depends on the location rate, E(L), and the individual average occupancy rate, ρin. In particular, if E(L) > ρin, for a fixed volume of effective customer contacts for the same average service time, fewer agents are needed in an outbound process than in an inbound process, and vice versa. Regarding previous research, the equations of Theorem 1 confirm the existence of the "Safety Staffing" Δ margin proposed by Gans et al. (2003) for calculating the number of agents employed in a contact center. If the Nin -Rin value for equation 5 is used in equation 19, the value is determined by the margin Δ according to equation 22.

Δ = C(Nin, Rin) / ASACF
In summary, the use of inbound or outbound contact center processes is dependent on two variables: (a) the probability of effective contact locations, E(L), in outbound processes and (b) the projected workload for inbound processes, ρin.
The results of the present study suggest two lines of future research: a) developing a formula for determining safety margins when E(L) obeys a probability distribution for which the average and standard deviation are known and b) modifying the assumption that the service time, E[S], is the same for inbound and outbound processes.
In spite of the strength that low human resources cost brings to the Mexican call center industry, the use of quantitative methods, as the ones developed in this article, will allow call center managers to optimize the balance of inbound and outbound calls cost composition.