Keywords

1 Introduction

A call center has been playing a vital role in hospital as it is an initial contact point between customers and a hospital. The call center provides convenient services to customers, such as delivering hospital’s service information and appointment rearrangement. While the demand of incoming calls has rapidly increased, many hospitals face a problem to determine the suitable number of operators to handle those calls. Several hospitals have no systematic approach to verify whether or not the current operators are capable of responding the calls (Nah and Kim 2013).

Additionally, the hospital call center also suffered from the staff-arrangement issues due to performing multiple tasks, e.g., returning calls, and administrative work (Moore et al. 2001). The operators’ competencies rely on experiences and intuition to prioritize when and which task has to be performed. The lack of these competencies results in customer’s long waiting time and increased abandonment rate.

Much research focuses on considering only one type of inbound calls. In fact, there are several types of customers’ needs leading to a variety of unpredictable types of inbound calls. For instance, some customers may want to get information and make an appointment, while other customers want to report and complain regarding services of hospitals. Other types of task include dialing outbound calls and other tedious jobs, such as collecting the complaints and comments from customers or other administrative jobs. These extra tasks make the operators not able to response customer calls effectively (Aksin et al. 2007; Atlason et al. 2008; Cezik and L’Ecuyer 2008; Ernst et al. 2004; Gans et al. 2003; Mehrotra 1997). To address the staff planning and allocation problem, several researchers focus on call center problem.

Specially, Nah and Kim (2013) have proposed a mixed-integer nonlinear model to determine the number of staffs and time slot allocation. Their objective function is to minimize the labor cost and penalty cost (both waiting time and abandon rate). However, in their study, a constant time is considered for each outbound calls, which could not represent a real situation where each outbound call has different characteristics, e.g, reminding customers of appointments regular of promotion announcement, keep close relation with special customers (for example, greeting on customer birthday), and respond to special requests or queries (Saltzman and Mehrotra 2007). Therefore, in our study, we will consider the additional constraints of different outbound characteristics as well as the administrative jobs.

The objective of this chapter is to develop a mathematical model to assist call center staff planner to determine a set of number of staffs and how to assign task to the staffs in each shift while observing labor cost, waiting time, and penalty.

The organization of this paper is as follows: We first present call center work process, then in Sect. 3, our mathematical model formation is explained. A numerical example is demonstrated in Sects. 4, and 5 provide conclusion.

2 Hospital Call Center Tasks

Figure 1 shows the hospital call center work process. When customers call, the inbound type is classified into three types, namely information inquiry, rearrangement requests, and complaints.

Fig. 1
figure 1

Hospital call center work process

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In the case of information inquiry, if the operator is capable, he/she will respond to the call immediately; otherwise, they will ask customer to wait in line until the customer receives information asked. However, if the customers wait for a long period of time, they may abandon calls and the operator will return call once he/she contact to other department for more information.

The second type of inbound call is rearranging an appointment. The operator does an online check in hospital database to make appointment changes for the customer. However, in some cases, the operator may require more time to contact other department before making changes. In this case, the operator will call back the customer once the issues have been resolved.

The last type of task is complaint/suggestion handling. If the operator is able to handle these tasks, he/she will collect all commentaries; otherwise, he/she will transfer the call to a responsible person.

3 Mathematical Programming Model Formulation

In this section, we present the mathematical model with all notations: indices, parameters, decision variables, objective functions, and constraints as follows:

3.1 Indices, Parameter, Decision Variable Description

See Table 1.

Table 1 Notation for the model
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Let T be the set of 1 h time period where the call center department is open, and D denotes the set of working day. Then, t and d are the indices of the set T and D. Let m stand for the total number of operating days per week. We use k as a symbol for a shift in set K. Note that, there are several types of inbound call. Let I denote the set of call category; then, i is the indices of the set I.

We assume that there are three kinds of calls, thus i is rearranging, information gathering, and complaints handling. Probability theory used in this research for classifying the sort of inbound calls. Let P i denote the probability of call type i. The working time for administration is represented by F i . Let E i denote an operated time of outbound call in each call type i. The volume of inbound calls is denoted by B td . We used w td and α td to denote observed mean waiting time per caller and observed abandonment rate, respectively. The w td is used for determining the predicted waiting time (W td ) by formulating the relation between observed waiting time and inbound load. Similarly, predicted abandonment rate is denoted as A td for the relation between observed abandonment rate and inbound load (Nah and Kim 2013). Let f 1(·) and f 2(·) stand for the mentioned relations.

In general, there are numerous expenses in the entire operations. Let α k denote the mean daily wage per operator working on shift k. Nah and Kim (2013) present the method to evaluate the penalty cost due to waiting time and abandonment rate, and β and μ denote the penalty cost due to waiting cost per caller per time unit and the mean cost per lost call, respectively. The value of β can be estimated by the value of time based on the living expenses and the average wage. The amount of μ can be evaluated by comprehensively considering the effects of the lost calls, such as the current and potential mean profits generated by a patient being served, the percentage of inbound calls leading to appointments, and the percentage of lost calls resulting in lost customers.

Let N k denote the number of operators in set k (k = 1, 2… K). The total man-hour of each tasks including, inbound call, outbound call, and administrations is indicated by x tdk , y tdk, and v tdk , respectively. The non-value-added time is z tdk .

3.2 Mathematical Model

Here, we present a mixed-integer nonlinear mathematical model of the workforce planning and allocation for a hospital call center. Note that, the successful operations can be defined as a low total costs but maintain high quality of services. Therefore, the objective of the mathematical model can be stated as

Minimize

$$ \varOmega = \alpha m\mathop \sum \limits_{k = 1}^{K} N_{i } + \beta \mathop \sum \limits_{d} \mathop \sum \limits_{t} B_{td} W_{td} + \mu \mathop \sum \limits_{d} \mathop \sum \limits_{t} B_{td} A_{td} + \gamma \mathop \sum \limits_{d} \mathop \sum \limits_{t} B_{td} C_{td}$$

Subject to

$$ l_{td} = \frac{{B_{td} }}{{\sum\nolimits_{k = 1}^{K} {X_{tdk} } }}\quad \forall t,d $$
(1)
$$ W_{td} = f_{1} (l_{td} )\quad \forall t,d $$
(2)
$$ A_{td} = f_{2} (l_{td} )\quad \forall t,d $$
(3)
$$ A_{td} \le u_{td} \quad \forall t,d $$
(4)
$$ \frac{{\sum\nolimits_{t} {B_{td} A_{td} } }}{{\sum\nolimits_{t} {B_{td} } }} \le u_{td} \quad \forall d $$
(5)
$$ \frac{{\sum\nolimits_{d} {\sum\nolimits_{t} {B_{td} A_{td} } } }}{{\sum\nolimits_{d} {\sum\nolimits_{t} {B_{td} } } }} \le u $$
(6)
$$ g_{d} + P_{i} E_{i} \sum \limits_{t} B_{td} \le \sum \limits_{t} \sum \limits_{k} y_{tdk} \quad \forall d $$
(7)
$$ P_{i} F_{i} \sum \limits_{t} B_{td} \le \sum \limits_{k} \sum \limits_{t} v_{tdk} \quad \forall d $$
(8)
$$ N_{k} = x_{tdk} + y_{tdk} + z_{tdk} + v_{tdk} \quad \forall t,d $$
(9)
$$ N_{k} \ge 0 \, {\text{and integer}} $$
(10)
$$ x_{tdk} \ge 0, y_{tdk} \ge 0, v_{tdk} \ge 0 $$
(11)

Our objective function is to minimize total cost which is composed of labor cost, waiting cost, abandonment cost, and loss of opportunity cost. The problem is subject to the following constraints.

The first constraint is an equation to determine workload which will be used to predict abandon rate in Eq. 2 and waiting time in Eq. 3. The expected abandonment rates should satisfy the hourly, daily, and weekly standard of the maximum allowable abandonment rate in Eqs. 4–6. Equation 7 indicates each type of demand of outbound calls. Equation 8 illustrates the calculation of man-hours for administration tasks.

The main purpose of this research is to find the optimal number of operators. We assume that the entire staffs can work at the same constant man-hour level which is 1 h per person per period. The number of operators (N k ) can be interpreted as total man-hours available during each one-hour period. Subsequently, as shown in Eq. 9, the number of operator in each shift can be calculated by the summation of total man-hour for inbound call, outbound call, administration tasks, and non-value added activities.

In Eq. 10, N k is a nonnegativity integer. Also, we require positive man-hours for inbound call, outbound calls, and administrative hours as shown in Eq. 11.

4 A Numerical Example

To illustrate the mathematical model and provide number of results for the problem, we use a simple numerical example mainly based on the data represented in Nar and Kim (2013) as well as some data by consultation with our case study hospital call center.

The observed call center is operated from 8:00 to 18:00, and each interval represents a single hour. We consider only Monday through Friday. There are two shifts of operator which operate from 8:00 to 17:00 and 9:00 to 18:00, respectively. We also use the demand of inbound calls in each period as shown in Fig. 2. Tables 2 and 3 display model input parameters and maximum allowable abandonment rate in each day, respectively.

Fig. 2
figure 2

Observing call demands each day

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Table 2 Summary of input data
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Table 3 Maximum allowable abandonment rate
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From the above information, the mathematical model is shown below; however, the loss of opportunity is not considered

Minimize

$$\varOmega = 80 \times 5 \sum \limits_{1}^{2} N_{i } + 7.2 \times \sum \limits_{t = 1}^{10} \sum \limits_{d = 1}^{5} B_{td} W_{td} + 8.7 \times \sum \limits_{t = 1}^{10} \sum \limits_{d = 1}^{5} B_{td} A_{td}$$

Subject to

$$\begin{aligned}& l_{td} = \frac{{B_{td} }}{{\sum\nolimits_{k = 1}^{2} {X_{tdk} } }}\quad \forall t,d\\ & W_{td} = 0.00249\left( {l_{td} } \right) \\ &A_{td} = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {{\text{if}}\,l^{td} < 20} \hfill \\ {0.011457l^{td} - 0.02261,} \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right.\\ & A_{td} \le u_{td}\\ &\frac{{\sum \nolimits_{t = 1}^{10} B_{td} A_{td} }}{{\sum \nolimits_{t = 1}^{10} B_{td} }} \le u_{d}\\ & \frac{{\sum \nolimits_{d = 1}^{5} \sum \nolimits_{t = 1}^{10} B_{td} A_{td} }}{{\sum \nolimits_{d = 1}^{5} \sum \nolimits_{t = 1}^{10} B_{td} }} \le u \\ & 10 + 0.4 \times 0.05 \times \sum\limits_{t = 1}^{10} {B_{td} } + 0.5 \times 0.05 \times \sum\limits_{t = 1}^{10} {B_{td} } + 0.1 \times 0.05 \times \sum\limits_{t = 1}^{10} {B_{td} } \le \sum\limits_{k = 1}^{2} {\sum\limits_{t = 1}^{10} {y_{tdk} } }\\ & 0.4 \times 0.05 \times \sum \limits_{t = 1}^{10} B_{td} + 0.5 \times 0.05 \times \sum \limits_{t = 1}^{10} B_{td} + 0.1 \times 0.05 \times \sum \limits_{t = 1}^{10} B_{td} \le \sum \limits_{k = 1}^{2} \sum \limits_{t = 1}^{10} y_{tdk}\\ & N_{k} = x_{tdk} + y_{tdk} + z_{tdk} + v_{tdk}\\ & N_{k} \ge 0 \, {\text{and integer}}\\ & x_{tdk} \ge 0, y_{tdk} \ge 0, v_{tdk} \ge 0 \end{aligned}$$

The model is then solved using http://www.neos-server.org/. The results are shown in Tables 4, 5 and 6. It indicates that the optimal number of operators for shift 1 and shift 2 is 22 and 24, respectively.

Table 4 Allocation of inbound tasks over weak
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Table 5 Allocation of outbound task over weak
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Table 6 Allocation of administration task over week
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Tables 4, 5 and 6 display the tasks that should be deployed in each time period. To illustrate the result, for example, during 10:00–11:00 on Wednesday of shift 1, 22 operators are required. According to Table 4, the man-hours for inbound call is 19.8 indicating that 19 operators should be assigned for handling inbound calls over this period and another one operator should be assigned for additional of 48 min. However, to assign for the whole shift during this period 2, operator will be idle as no more jobs waiting for handling.

The advantage of this guideline obtained from the results is that the hospital has systematic approach to determine the number of operators. In addition, those operators have an idea how to decide when to perform their tasks in each period as well they know which day they should take a leave for vocation or personal reason.

In contrast, this guideline may not be useful in some situations. For example, if there is high fluctuation on the nature of each calls resulting in the operators may not finish all tasks in time, e.g., on Thursday during 8:00–9:00, the result shows that there are 6 min to make outbound calls. In fact, the length of operating time for outbound call relies on the unpredicted behaviors of each customer rather than on the operators’ skill. Thus, we may not guarantee that 6 min is sufficient to respond those unmanageable calls.

5 Conclusion and Future Work

The workforce planning and allocation is important in the hospital call center. In this paper, we present a mathematical model by using mixed-integer nonlinear programming to incorporate with some real situations. Experimental results are provided. Not only the number of operators required but also some insights for planner as a guideline for further planning are provided. Since it is time-consuming to solve the model, heuristic methods, genetic algorithm, particle swarm optimization, or differential evolution, will be used to help planning in practical way.

Furthermore, our model is not able to incorporate with the situation where the behavior of each customer is uncertain. The stochastic programming with the consideration of the planning results should be able to withstand the changes or differences of customers’ behavior.