When Promotions Meet Operations: Cross-Selling and Its Effect on Call-Center Performance

When Promotions Meet Operations:
Cross-Selling and Its Effect on Call-Center Performance
Mor Armony1
Itay Gurvich2
July 27, 2006
Abstract
We study cross-selling operations in call centers. The following question is addressed: How many
customer service representatives are required (staffing) and when should cross-selling opportunities be
exercised (control) in a way that will maximize the expected profit of the firm while maintaining a
pre-specified service level target. We tackle these questions by characterizing scheduling and staffing
schemes that are asymptotically optimal in the limit, as the system load grows to infinity. Our main
finding is that a threshold priority (TP) control, in which cross-selling is exercised only if the num-
ber of callers in the system is below a certain threshold, is asymptotically optimal in great generality.
The asymptotic optimality of TP reduces the staffing problem to the solution of a simple deterministic
problem, in some cases, and to a simple search procedure in others. Our asymptotic approach estab-
lishes that our staffing and control scheme is near-optimal for large systems. In addition, we numerically
demonstrate that TP performs extremely well even for relatively small systems.
Acknolwedgment
We thank Assaf Zeevi for his helpful advice on the proofs of asymptotic optimality.
1Stern School of Business, New York University, marmony@stern.nyu.edu
2Graduate School of Business, Columbia University, ig2126@columbia.edu

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1 Introduction
Call Centers are in many cases the primary channel of interaction of a firm with its customers. Historically,
call centers were mostly considered a service delivery channel. As an example, in the banking industry
customers would contact the call center to obtain information or perform transactions in their accounts.
Although many issues are still open, the operations of these service driven call centers have gained a lot of
attention in the literature. Typically, service driven call centers plan their operations based on delay related
performance targets. Examples of such performance measures are average speed of answer (ASA), the
fraction of customers whose call is answered by a certain time and the percentage of customer abandonment.
Most companies, however, are not purely service providers. Rather, customer service is a companion
to one or several main products. For example - the core business of computer hardware companies, like
Dell, is to sell computers. They do, however, have a call center whose main purpose is to provide customer
support after the purchase. Most banks nowadays have call centers that give customer support while their
main business is selling financial products. For these companies, the inbound call center can be a natural and
very convenient sales channel. To differ from outbound telemarketing calls, where the timing of the call is
usually arbitrary and not necessarily convenient for the customer, the interaction in the inbound call center
is initiated by the customer. Once the customer calls the center, a sales opportunity is generated and the
agent might choose to exercise this cross-selling opportunity by offering the customer an additional service
or product.
From a marketing point of view, a call center has a potential of becoming an ideal sales environment.
Modern Customer Relationship Management (CRM) systems have dramatically improved the information
available to Customer Service Representatives (CSR’s) about the individual customer in real time. Specif-
ically, in call centers, once the caller has been identified, the CRM system can inform the agent regarding
this customer’s transaction history, her value to the firm and specific cross-selling opportunities. As a result,
cross-sales offerings can be tailored to the particular customer, making modern call centers a perfect channel
for customized sales. Many companies have identified the revenue potential of inbound call centers. Indeed,
as suggested by a recent McKinsey report [14], call centers generate up to 25 percent of total new revenues
for some credit card companies and up to 60 percent for some telecom companies. Moreover, [14] estimates
that cross-selling in a bank’s call center can generate a significant revenue, equivalent to 10% of the revenue
generated through the bank’s entire branch network.
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Although the benefits of running a joint service and sales call center seem clear - there are various
challenges involved in operating such a complex environment. An immediate implication of incorporating
sales is the increase in customer handling times caused by cross-sales offerings. Unless staffing levels are
adjusted, the increased handling times will inevitably lead to service level degradation in terms of waiting
times experienced by the customers. Does this imply that more cross-selling will necessarily lead to dete-
rioration in service levels? What are the appropriate operational tradeoffs that one should examine in the
context of combined service and sales call centers? In purely service driven call centers, the manager typi-
cally attempts to minimize the staffing level while maintaining a pre-determined performance target. Hence,
in this pure service context the operational tradeoff is clear: Staffing Vs. Service Level. When sales and
promotions are introduced, however, one should add a third and important component to this tradeoff. This
component is the potential revenue from promotions. Clearly, if the potential revenue is very high in com-
parison to the staffing cost, it would be in the interest of the company to increase the staffing level and allow
for as much cross-selling as possible. In this case, we show that with appropriate staffing, incorporating
cross-selling actually reduces delays. There are cases, however, where the relation between staffing costs
and potential revenues is more intricate and a careful analysis is required.
Beyond staffing, in a cross-selling environment there is also a component of dynamic control of incom-
ing calls and cross-sales offerings. Specifically, the call center manager needs to determine when should
the agent exercise a cross-selling opportunity. This decision should take into account not only the charac-
teristics of the customer in service but also the effect on the waiting times of other customers. For example,
in order to satisfy a waiting time target, it would be natural to stop all promotion activities in the presence
of heavy congestion. Indeed, a common heuristic, that is used in practice to determine when to exercise a
cross-selling opportunity, is to cross-sell upon service completion only when the number of callers in the
queue is below a certain threshold. Optimal rules, however, are typically hard to find and to implement. The
staffing and control issues are strongly related since even with seemingly adequate staffing levels, the actual
performance might be far from satisfactory when one does not make a careful choice of the dynamic con-
trol. Yet, because of the complexity involved in addressing both issues combined, they have been typically
addressed separately in the literature. To our knowledge, this paper is the first to consider the staffing and
dynamic control in a cross-selling environment jointly, in one single framework.
The purpose of this work is to carefully examine operational tradeoffs that are crucial to the cross-
selling environment. This is done by specifying how to adjust the staffing level and how to choose the
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control in order to balance staffing costs and cross-selling revenue potential while satisfying quality of
service constraints associated with delay performance. Specifically, we provide joint staffing and dynamic
control rules as explicit functions of the quality of service constraints, the potential value of cross-selling
and the staffing costs. The control we propose is a Threshold Priority (TP) rule in which cross-selling is
exercised only when the number of callers in the system is below a certain threshold. In contrast with the
commonly used heuristics we identify cases in which cross-selling should not be exercised even if there are
idle agents in the system, in anticipation for future arrivals.
The rest of the paper is organized as follows: We conclude the introductory part with a literature review.
Section 2 provides the problem formulation. Section 3 outlines the main results of the paper through an
informal description of our proposed solution for the cross-selling problem. We formally introduce the
cross-selling problem in Section 4 where we define the asymptotic optimality framework. The asymptotic
optimality results are stated in Sections 4 and 5. In Section 6 we present some numerical result to support
our proposed solution. The paper is concluded in Section 7 with a discussion of the results and directions
for future research.
For expository purposes, our approach in the presentation of the results is to state them formally and
precisely in the body of the paper, together with some supporting intuition, while leaving some of the formal
proofs to the technical appendix [6]. Accordingly, all proofs that do not appear in the body of the paper are
deferred to the technical appendix.
1.1 Literature Review
A successful and comprehensive treatment of cross-selling implementation in call centers would clearly
require an inter-disciplinary effort combining knowledge from marketing and operations management as
well as human resource management and information technology. An extensive search of the literature
shows, however, that while the marketing literature on this subject is quite rich, very little has been done
from the operations point of view (the reader is referred to Aks¸in and Harker [1] for a survey of some of the
marketing literature). The marketing perspective of cross-selling employment is obviously very significant,
but unless it is backed up by appropriate operational decisions, cross-selling cannot be successful.
Although the operations literature on this subject is still scarce, the topic of cross-selling has received
some attention. In the context of cross-selling in call centers - a significant contribution is due to Aks¸in with
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different co-authors. In Aks¸in and Harker [1] the authors consider qualitatively and empirically the problems
of cross-selling in banking call centers. They also suggest a quantitative framework to evaluate the effects
of cross-selling on service levels, using a processor sharing model, but they do not attempt to find optimal
control or staffing levels. ¨
Ormeci and Aks¸in [25], on the other hand, do pursue the goal of determining
the optimal control, while assuming that the staffing level is given. In their framework, customers’ cross-
selling value follows a certain distribution. The realization of this value can be observed by the call center
before the cross-selling offer is made. Hence, the agent can base the decision on the actual realization of
this value and not only its expected value. However, due to computational complexity, the results in [25] are
limited to multi-server loss systems (customers either hang-up or are blocked if their call cannot be answered
right away) and to structural results that are then used to propose a heuristic for cross-selling. G¨unes and
Aks¸in [17] do not consider the problem of optimal cross-selling policies but rather analyze the problem of
providing incentives to agents in order to obtain certain service levels and value generation goals. This is
indeed a crucial issue in cross-selling environments where the decision whether to cross-sell or not is often
made at the discretion of the individual agents.
Simplicity of the dynamic control is clearly an important factor for a successful implementation of cross-
selling. The simplicity of the control might result, however, in decreasing revenues from cross-selling. For
example, it is intuitive that one can increase revenues by allowing the control to be based on the identity of
the individual customer in addition to the number of customers in the system. Byers and So [13] examine the
value of customer identity information by comparing cross-selling revenues under several control schemes
that differ by the amount of the information they use. Exact analysis is performed for single server systems
and numerical results are given for multi-server systems. The analysis in [13] does not consider an optimal
choice of control and staffing levels but assumes a fixed set of control schemes and a given staffing level.
To position our paper in the context of the literature introduced above, note that our analysis is the first
to consider how to optimally choose both the staffing level and the control scheme in a cross-selling envi-
ronment. If the staffing decision is ignored and the staffing level is assumed to be fixed, the only relevant
tradeoff is between service level (expressed in terms of delay) and the extent to which cross-selling oppor-
tunities are exercised. In this setting then, more cross-selling necessarily causes service level degradation.
Moreover, the existing literature suggests that, when the staffing level is assumed fixed, it is difficult to come
up with simple and practical control schemes for cross-selling. As we show in this paper, however, when
one adds the staffing component, the solution is sometimes simplified tremendously while emphasizing that
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a lot of cross-selling does not necessarily lead to low service levels. To differ from the papers above, we are
able, thanks to our solution approach, to obtain a simple and rather practical control for cross-selling.
Our modelling assumptions of the customers are relatively simple; customers are assumed to belong to
one of two-types. They are either potential cross-selling candidates or not. Also, the potential revenue from
a customer is expressed only through its expected value rather than the whole distribution. This simplicity
allows us to isolate the staffing and cross-selling decisions from other issues such as customer segmentation
and customized cross-sales offerings and pricing. In particular, it allows us to establish asymptotic optimality
in a strong sense of the threshold priority (TP) rule. A follow-up paper [5] uses the results of the current
paper to establish a somewhat weaker notion of asymptotic optimality but for a more elaborate customer
choice model.
Our solution approach follows the many-server asymptotic framework, pioneered by Halfin and Whitt
[18]. In particular, we follow the asymptotic optimality framework approach first used by Borst et al. [12],
and adapted later to more complex settings ([3], [4], [7], [8], [9] and [22]). The asymptotic regime that we
use has been shown to be extremely robust also for relatively small systems (see Borst et al. [12]); Consistent
with this finding we give a strong numerical evidence to support the claim that this robustness is also typical
in our setting.
To conclude this review, we should mention that, while outside the context of call centers, there is a
stream of operations management literature that deals with the implications of cross-selling on the inventory
policy of a firm. Examples are the papers by Aydin and Ziya [10] and Netssine et. al. [24].
2 Problem Formulation
Consider a call center with calls arriving according to a Poisson process with rate λ. An agent-customer
interaction begins with the service phase, whose duration is assumed to be exponentially distributed with rate
µs. Upon service completion, if cross-selling is exercised, this interaction will enter a cross-selling phase,
whose duration is assumed to be exponentially distributed with rate µcs. If cross-selling is not exercised,
either intentionally or due to the customers refusal to listen to a cross-selling offer, the customer leaves the
system. It is assumed that all inter-arrival, service and cross-selling times are independent and that the call
center has an infinite buffer.
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We do not assume that all customers are cross-selling candidates. Indeed, the company might segment
the customer population into two segments such that only a portion ¯
p of the customers are cross-selling
candidates. Moreover, even if an agent decides to cross-sell to a caller, she will not necessarily agree to
listen to the cross-selling offer. We assume that a customer that is presented the option to listen to a cross-
selling offer will agree to do so with probability ¯
q > 0. It is plausible that in practice, the probability that the
caller will agree to listen to a cross-selling offer depends on the customer experience up to that point (such
as his waiting time, service time, service quality, etc.). This dependence introduces analytical complications
because the state space required to describe such a system is very large (in particular, it would need to
include for each customer her current waiting time and service time). Given this complexity we assume in
this paper that the probability of agreeing to listen to a cross-selling offering is independent of the customer
service experience. This assumption is reasonable for systems in which waiting times are not too long and
service quality is uniformly high. The independence assumption is relaxed in Armony et al. [5]). Assuming
that different customers are statistically identical, we have that p = ¯
p¯
q is the probability that a customer is a
cross-selling candidate and agrees to listen to the cross-selling offer. The combined parameter p is sufficient
for our analysis so that we will not make additional references to the parameters ¯
p and ¯
q. We assume that a
cross-selling offer has an expected revenue of r, and revenues from different customers are independent. A
schematic illustration of the system is given in Figure 1, in which N is the number of CSRs.
Cross-
No
N
Sell ?
Yes
Figure 1: A Schematic Description of a Call Center with Cross-Selling
We assume that the staffing cost function, which we denote by C(·), is convex increasing in the staffing
level N . We make stronger assumptions on the cost function in Section 4, where we construct our asymptotic
framework. For simplicity we will say that a customer is in phase 1 of the customer-agent interaction if he
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is in the service phase and in phase 2 if he is in the cross-selling phase. Let π be a control policy which
determines upon a phase 1 completion of a cross-selling candidate whether or not to exercise this cross-
selling opportunity. Accordingly, we let Zπi(t) be the number of servers providing phase i service at time
t, i = 1, 2. Then, Zπ(t) = Zπ1(t) + Zπ2(t) is the total number of busy agents (CSRs) at time t, and
Iπ(t) = N − Zπ(t) is the number of idle agents at time t in a system with N agetns. Also, let Qπ(t)
be the number of customers waiting in queue at time t and Y π(t) be the overall number of customers
in the system at time t, that is Y π(t) = Zπ(t) + Qπ(t). We denote by W π(t) the virtual waiting time
encountered by a customer that arrives to the system at time t, and by P π(cs)(t) the probability that cross-
selling will be exercised for a customer that arrives at time t (that is the probability that the customer
will be asked to listen to a cross-selling offer and he will agree). In all of the above, we omit the time
index t when referring to steady state variables. Also, we omit the superscript π whenever the control
is clear from the context. Note that under any stationary policy, all transition rates in the system can be
determined using the number of agents busy providing either phase of service and the queue length. In
particular, Sπ(t) = {Zπi(t), i = 1, 2; Qπ(t)} is a sufficient state descriptor for a Markovian characterization
of the system under any stationary control. The profit maximization problem formulation we consider is as
follows:
maximize rλP π(cs) − C(N )
subject to E[W π] ≤ ¯
W ,
(1)
N ∈ Z+, π ∈ Π(λ, µs, µcs, N).
Here the average steady-state waiting time is constrained to be less than a pre-determined bound ¯
W . Note
that customers do not abandon, or balk, nor are they being blocked. The control policy π is picked from the
following set of admissible controls Π(λ, µs, µcs, N).
Definition 2.1 Admissible Controls: Given a staffing level N , and parameters λ, µs, µcs, we say that π is
an admissible policy if it is non-preemptive, non-anticipative and
E[Q
lim
π(t)] → 0.
(2)
t→∞
t
Loosely speaking, Π(λ, µs, µcs, N) is the set of stabilizing policies under the given parameters. Definition
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2.1 takes into account the fact that the set of admissible policies depends on the parameters of the model
through the stability conditions of the system. For simplicity of notation, when the parameters λ, µs and
µcs are fixed, we will omit them from the notation and instead use just the notation Π(N). N will also be
omitted whenever the staffing level is clear from the context. One should note that we used the maximization
formulation (1) although the maximum need not exist. We choose the word “maximize” for convenience of
presentation while formally referring to taking the supremum over all staffing levels and admissible policies.
Standard stability considerations imply that R := λ/µs constitutes a lower bound on possible staffing
levels. To allow for a more refined analysis it makes sense to normalize the cost around its lower bound.
Hence, one may re-write (1) as follows:
maximize rλP (cs) − (C(N ) − C(R))
subject to E[W ] ≤ ¯
W
(3)
N ∈ Z+, π ∈ Π(N),
where the only change from (1) is the normalization of the cost around the constant C(R). As an alternative
to the constraint in (3), a common Quality of Service (QoS) constraint used in practice is of the form
P {W > ¯
W } ≤ δ. That is, one requires that a fraction 1 − δ of the customers will be answered within
¯
W units of time. Hence, it is worthwhile mentioning that all the insights of our analysis go through for
constraints of this form under the assumption that customers are served in a First Come First Served (FCFS)
manner. In particular, the structure of the asymptotically optimal staffing and control scheme we propose
remains the same under both types of constraints.
The following is an immediate consequence of Little’s Law and Markov Chain Ergodic theorems.
Lemma 2.1 For any π ∈ Π(N ) that admits a stationary distribution we have that
1. E[Zπ1] = R, and
2. λP π(cs) = µcs · E[Zπ2] = µcs · (E[Zπ] − R) ≤ µcs · (N − R) ∧ λp ,
µcs
where for two real numbers x and y, x ∧ y = min{x, y}.
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Having Lemma 2.1 we can re-write (3) as
maximize rµcs(E[Zπ] − R) − (C(N) − C(R))
subject to E[W ] ≤ ¯
W ,
(4)
N ∈ Z+, π ∈ Π(N).
Within the set of policies, we propose the following control:
Definition 2.2 The Threshold Priority (TP) control is defined as follows:
• An agent that completes a phase 1 service with a customer at a time t will exercise cross-selling if this
customer is a cross-selling candidate and (Y (t) − N ) ≤ K (where K is a pre-determined integer).
• An arriving customer will enter service immediately upon arrival if there are any idle agents.
• Upon a customer departure, the customer at the head of the queue will be admitted to service if the
queue is non-empty.
For brevity, we use the notation T P [K] to denote T P with threshold K (where K may take negative as
well as positive values). One should note the following: Whenever K ≤ 0, T P [K] is a control that uses a
threshold on the number of idle agents. Specifically, upon service completion with a cross-selling candidate,
the agent will exercise cross-selling if there are no customers waiting in queue and the number of idle agents
is at least |K|. Whenever K > 0, T P [K] is a control that uses threshold on the number of customers in
queue. Specifically, upon service completion with a cross-selling candidate, the agent will exercise cross-
selling if the number of customers in queue is at most K.
Note that T P [K] uses only information on the overall number of customers in the system at the time
of service completion. In particular, T P [K] is a stationary policy so that the system can be modelled as a
Markov chain. Specifically, we claim that the state descriptor {Z2(t), Y (t)} Where Z2(t) is the number of
servers working on phase 2 service (cross-selling) and Y (t) is the overall number of customers in system, is
sufficient for a Markovian description of the system under TP. Indeed, since TP disallows a positive queue
when there are idle agents, we have that Q(t) = [Y (t) − N ]+ and Z1(t) + Z2(t) = [Y (t) − N]−. Since our
optimization problem is given in steady state terms it is necessary to have the following simple result for the
Markov chain we have just constructed.
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