Statistical Monitoring of Service Levels and ... - Wiley Online Library

Research Article (wileyonlinelibrary.com) DOI: 10.1002/qre.1966

Published online 25 February 2016 in Wiley Online Library

Statistical Monitoring of Service Levels and Staffing Adjustments for Call Centers Junxiang Li,a*† Yafen Liu,b Fugee Tsung,c Jiazhen Huod and Qiang Sud Call centers are an indispensable part of many businesses, and their economic role is important and fast growing. To remain competitive, call centers must provide high-quality services while keeping the number of agents and hence labor costs down. It is thus vital to monitor the service level (SL), which is a common measure of the quality of service, and make suitable staffing adjustments. In this paper, we describe an engineering process control strategy to monitor SL. Staffing is adjusted according to changes in SL. An exponentially weighted moving average for SL based on upper and lower limits is determined from historical data and employed to monitor changes in SL during different time intervals so as to keep the frequency of adjustments to a minimum. Only if SL exceeds its lower or upper limit do we increase or decrease the number of agents for a given interval. Numerical tests show that we can strike a balance between SL and staffing by using our proposed method. Copyright © 2016 John Wiley & Sons, Ltd. Keywords: call center; engineering process control (EPC); service level monitoring; staffing adjustment

1. Introduction call center is a collection of resources (typically agents) for handling customer contacts by telephone. Call centers have become a convenient and widely used channel through which organizations communicate with their customers. The importance of call centers in many industries means that improvements to the management of call center operations can have substantial–financial implications. Service level (SL) is the most common measure of quality of service. As the waiting time distribution is viewed as a very important quantity, the most usual way to define SL is by taking the fraction of calls whose time in queue is shorter than a given threshold, called the acceptable waiting time (AWT).1–4 The ‘industry standard’ is that 80% of all calls should be answered within 20 s, but other numbers are possible. SL can simply be calculated by dividing the number of calls handled before the AWT by the total number of calls. For those call centers providing public services, violations of SL may result in serious penalties imposed by external authorities.5 There are often many measures of SL: for a given time period of the day, for a given call type, for a given combination of call types, and so on. The objective of a call center is usually to deliver a given SL for minimum cost. In general, approximately 80% of a call center’s operating costs pertains to workforce salaries.6–8 Thus, it is important for call centers to schedule a suitable number of agents to meet a certain SL. We incorporate economic considerations into the design of engineering process control (EPC) to calculate the minimum number of agents needed to maintain a given SL. EPC, also known as feedback (closed-loop) control3,9,10 or automatic process control,6,11–13 can be used to minimize the variation to a target through process adjustment. The EPC scheme refers to an algorithm that describes how variables of a process should be adjusted. In this paper, EPC may be incorporated into a model that generates data to adjust the number of agents in a call center by monitoring changes in SL during different time intervals. The process objective is to minimize the deviation of SL from its target level. In using the EPC method, a cycle is defined consisting of the time when the process is in control and the time when the process is an out of control so that the necessary process adjustment can be made. A process control model will be considered in which loss is associated with any deviation from the target. An SL that is too low for too long will seriously affect the degree of satisfaction of callers and may even result in loss of customers or penalties imposed by external authorities. At the same time, an SL that is too high for too

A

a

Business School, University of Shanghai for Science & Technology, Shanghai, China School of Management, Tianjin Polytechnic University, Tianjin, China Department of Industrial Engineering and Logistics Management, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong d School of Economics and Management, Tongji University, Shanghai, China *Correspondence to: Junxiang Li, Business School, University of Shanghai for Science & Technology, Shanghai, China. † E-mail: [email protected] b c

Qual. Reliab. Engng. Int. 2016, 32 2813–2821

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Copyright © 2016 John Wiley & Sons, Ltd.

J. LI ET AL. long suggests overstaffing and unnecessary operating expenses. Because the staffing should not be adjusted too often, in our process adjustment model, we make no staffing adjustment until a control statistic index of SL exceeds given limits. These limits are used for assessing the degree to which SL has deviated from its target and are set according to the best and worst predicted values for SL in a call center. Now, suppose that SL deviates because of unreasonable staffing levels. If the deviation exceeds the limits, staffing is increased or decreased to restore SL to the target level. Assume that an integrated moving average time-series model is generating disturbance in the process model for call centers. We apply to past information an exponentially weighted moving average (EWMA) strategy9,10,14–21 with a discount factor to estimate the next value in the series with minimal mean square error. An adjustment is made only if the magnitude of a predicted value exceeds a fixed limit. Crowder11 and Runger et al.22 adopted optimal univariate and multivariate process adjustments, respectively, to minimize the cost of adjustment. A more detailed analysis of this strategy was presented by Box and Luceno.3 The rest of this paper proceeds as follows. In Section 2, we describe the model and estimation. In Section 3, we give the adjustment policy. Section 4 provides numerical results and makes comparisons. We conclude with several remarks in the last section.

2. Model description and estimation Even after an initial agent schedule is established for a call center, forecasts will continue to be updated, and changes will be made to the schedule until the day for which the schedule is designed has come and gone. When the schedule is executed, a supervisor monitors the SL at all times and adjusts the deployment of agents based on real-time operating conditions. During the day, data are fed back to the workforce management tool, forecasts are updated, and the process repeats itself. In this section, we give and describe a process monitoring and adjustment model for monitoring the SL and adjusting the staffing, where the objective is to keep the SL from straying too far from the standard value. The original unadjusted SL for a call center can be written as follows: y t ¼ Z t þ T;

(2:1)

where yt is the original unadjusted SL at time t, T stands for the standard SL that is an expected value for SL for a given call center, and Zt is a disturbance (noise, error) defined as the deviation of yt from its target value T. If yt deviates far from T, which means an unreasonable staffing level, a staffing adjustment should be made. Yet staffing adjustments, which mainly involve increasing or decreasing the number of agents in a call center, should not be made too often. Hence, an upper and a lower limit should be set for SL. In general, the upper limit is the best expected value of SL that the call center hopes to reach, while the lower limit is the worst expected value of SL that the call center can tolerate. Based on these limits, we can assess whether staffing should be adjusted or not. Because the schedule must be decided in advance, staffing requirements for every time interval would need to be predicted from historical data. The staffing level is adjusted according to changes in SL in the process of schedule execution. We use an EWMA of past values of SL to calculate the number of agents needed. The EWMA estimation process for SL is as follows. Let y ′l be an adjusted value of the initial SL yl, l = t 1, t 2, …. Then the EWMA estimate y t of yt is y t ¼ λ y ′t1 þ θ y ′t2 þ θ2 y ′t3 þ ⋯ ; (2:2) where 0 ≤ θ ≤ 1 and λ = 1 θ. From (2.2), we obtain y t ¼ λy ′t1 þ θy t1 . In (2.2), we need to choose a suitable θ or λ for estimating y t because using a smaller θ (or a larger λ) may result in weights that die out more quickly as the exponents of θ are increased. On the other hand, placing more emphasis on recent observations while choosing a larger θ (or a smaller λ) can produce a stabilizing and averaging effect, but reaction to changes is slower. In general, λ should be set between 0.1 and 0.3.3 After setting the lower and upper limits of the series fy t g, say, L1 and L2, according to the worst expected value of SL that the call center can tolerate and the best expected value of SL that the call center hopes to reach, respectively, monitoring of changes in y t begins. No adjustment is made until some y t exceeds one of the limits. If y t is below the given lower limit, agent numbers are boosted to restore SL to the standard value T. However, if y t is above the given upper limit, agent numbers are reduced to save on workforce costs and make more efficient use of human resources. That is, an adjustment is made only if at some time t, y t ≤ L1 or y t ≥ L2 :

(2:3)

3. Adjustment policy

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When the EWMA of SL exceeds its limits, that is, (2.3) holds, the next step is to determine the level of staffing adjustment needed to restore SL to the standard value T. The Erlang C formula,2 which is at the heart of call center management and plays a crucial role in workforce management, has many important properties that can help us design an optimal call center. We can use it to translate an estimate of the call volume to the number of agents required to meet a given SL. But before the Erlang C formula can be executed, call volume and other parameters must be forecasted first. The M/M/N (Erlang C) queueing model is used to estimate stationary system performance during short—half-hourly or hourly—intervals. The model Copyright © 2016 John Wiley & Sons, Ltd.


J. LI ET AL. assumes constant arrival and service rates, as well as a system that achieves a steady state quickly within each interval. The arrival process is assumed to be Poisson, service times are assumed to be exponentially distributed and independent of each other (as well as everything else in the system), and the service discipline is assumed to be first-come, first-served. Blocking, abandonment, and retrials are ignored. The calculations using the Erlang C formula proceed as follows. Let Nt be the number of the agents on duty at time t, γ be the arrival rate defined by the number of calls arriving on average per time unit (e.g., per minute), and β be the service rate. Put differently, the average service time of calls or average holding time is 1/β. The Erlang C formula is as follows: E t ðN t ; uÞ ¼

u Nt =Nt !

Nt 1 uNt =Nt ! þ ð1 ρÞ ∑ u Nt =k !

:

k¼0

Then SL can be calculated from the following: y t ¼ 1 E t ðNt ; uÞeðNt uÞAWT β ;

(2:4)

where u = γ/β is the offered load and ρ = u/Nt, the traffic intensity, is the associated average system utilization or occupancy. However, if the SL yt is known, Nt cannot be directly obtained from (2.4) because it is difficult to derive the inverse form of the Erlang C formula (2.4). We can, however, obtain the number of agents Nt using the Erlang C formula by trial and error. That is, we start b t such that N b t > γ=β and then increase or decrease the number of agents gradually until we arrive at just the right number with some N that would satisfy the given SL yt. The following is the EPC adjustment strategy to keep the SL within limits in a call center for K time intervals. Algorithm 1: (EPC Adjustment Strategy). Compute the original SL series fy t gKt¼1 according to (2.4) with the current number of agents fNt gKt¼1 , the arrival rate fγt gKt¼1 , the service rate fβt gKt¼1 and the acceptable waiting time, AWT. Given an initial SL after adjustment y ′1 ¼ y 1 and its initial EWMA estimation y 1 ¼ T . Then the initial staffing adjustment effect is g1 = 0, the initial accumulated staffing adjustment effect ν1 = 0 and the initial forecasting error e1 ¼ y ’1 y 1 . For every time t, t = 1, 2, …, K, we do as follows: Compute y tþ1 such that y tþ1 ¼ λy ′t þ θy t , where θ = 1 λ. If (2.3) is not satisfied, set gt + 1 = 0 and νt + 1 = νt + gt + 1, then return step 3. Otherwise, return step 6. Compute y ′tþ1 according to (2.4) with N′tþ1 ¼ Ntþ1 þ νtþ1 , γt + 1, βt + 1 and AWT at time t + 1. Compute forecasting error etþ1 ¼ y ′tþ1 y tþ1 . Replace t by t + 1 and go back to step 1. Compute the number of agents Ntþ1 and Ñt + 1 according to (2.4) with y tþ1 and T, respectively, and the EWMA arrival rate γtþ1 , the EWMA service rate βtþ1 and AWT at time t + 1. e tþ1 Ntþ1 and νt + 1 = νt + gt + 1. Go back to step 3. Step 7 Set y tþ1 ¼ T, gtþ1 ¼ N

Step 1 Step 2 Step 3 Step 4 Step 5 Step 6

4. Numerical results



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We illustrate the EPC strategy with an example, using ‘Anonymous Bank’ call-center data from the documentation by Mandelbaum et al.23,24 We consider two consecutive weekdays (November 1 and 2). In these data, there are many call types such as ‘PS’ (regular activity), ‘PE’ (regular activity in English) and ‘NW’ (potential customers getting information). For convenience, only the ‘PS’ type is considered, which reached the center between 0700 and 2400. Abandonment and retrials are not considered. The data based on half-hourly intervals are shown in Table I. In Table I, γt is the arrival rate, which is the number of calls entering the systems of the call center each minute at time t, 1/βt is the average holding time in seconds at time t, and Nt is the number of agents on duty at time t. The SL monitoring and the corresponding staffing adjustment process using the EPC strategy for the 68 observations in Table I are set out in Table II. In Table II, we set the lower limit L1 = 0.7, the upper limit L2 = 0.95, T = 0.8, and AWT = 20 s, which means that the call center’s objective is to answer 80% of all calls within 20 s. The second column is the unadjusted original SL yts, t = 1, 2, …, 68, which are calculated from (2.4) using data in Table I. The third column gives the SL y ′t after adjustment, which are computed from (2.4) using γt and 1/βt in Table I and the adjusted number of agents, Nt + νt. The EWMA forecast for SL with λ = 0.28 is used to estimate the next SL y tþ1 shown in column 4 of Table II, where the initial value y 1 is taken as the target value T = 0.8 and λ = 0.28 is an empirical value chosen to minimize the sum of squares of forecasting errors.10 The control rule is as follows: An adjustment is made when some of the EWMA forecasts y tþ1 exceed the upper limit L2 or fall below the lower limit L1. The values y tþ1s are calculated according to (2.2) with λ = 0.28 until one of them exceeds one of the limits. When some y tþ1 exceeds a limit, a suitable staffing adjustment value gt is entered into column 5 to restore y tþ1 to the standard SL T according to (2.4) by inputting γt and 1/βt estimated by the EWMA method with λ = 0.28 for the series {γt} and {1/βt}, respectively. As the number of agents is discrete, it is chosen to satisfy approximately the

J. LI ET AL. Table I. The original data of a call center November 1

November 2

Time

No.

γt

1/βt

Nt

No.

γt

1/βt

Nt

7:00–7:30 7:30–8:00 8:00–8:30 8:30–9:00 9:00–9:30 9:30–10:00 10:00–10:30 10:30–11:00 11:00–11:30 11:30–12:00 12:00–12:30 12:30–13:00 13:00–13:30 13:30–14:00 14:00–14:30 14:30–15:00 15:00–15:30 15:30–16:00 16:00–16:30 16:30–17:00 17:00–17:30 17:30–18:00 18:00–18:30 18:30–19:00 19:00–19:30 19:30–20:00 20:00–20:30 20:30–21:00 21:00–21:30 21:30–22:00 22:00–22:30 22:30–23:00 23:00–23:30 23:30–24:00

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

0.011 0.012 0.018 0.015 0.029 0.032 0.033 0.033 0.025 0.018 0.022 0.019 0.022 0.021 0.025 0.027 0.017 0.027 0.023 0.023 0.011 0.016 0.017 0.016 0.021 0.013 0.013 0.017 0.016 0.017 0.015 0.013 0.012 0.009

121.1 175.0 146.4 134.9 169.4 207.0 193.2 165.3 201.9 148.1 189.3 229.2 163.4 167.6 220.2 151.1 198.1 187.6 179.2 160.3 92.8 127.4 154.1 198.9 121.6 149.8 217.0 142.5 138.7 110.8 115.0 165.3 162.6 132.0

3 3 5 6 9 9 8 9 10 9 9 9 9 8 8 7 8 9 8 8 5 5 6 7 6 6 6 6 6 5 4 4 4 4

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68

0.013 0.017 0.024 0.019 0.022 0.022 0.022 0.022 0.035 0.023 0.024 0.020 0.022 0.029 0.023 0.027 0.016 0.020 0.023 0.017 0.014 0.017 0.026 0.018 0.015 0.017 0.015 0.012 0.016 0.014 0.016 0.016 0.017 0.007

132.7 121.2 96.8 115.3 164.5 218.3 176.1 183.1 160.3 277.5 145.0 224.7 132.2 151.8 220.9 200.8 216.1 223.4 214.4 217.5 190.8 324.1 150.5 179.1 149.0 122.5 154.2 187.8 159.3 139.3 143.1 85.4 78.1 143.8

3 3 4 5 6 6 6 7 7 7 5 8 8 8 8 8 6 9 10 8 6 6 6 6 4 4 5 5 5 5 5 3 3 3

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corresponding standard SL T. A value of 0 in column 5 means that no adjustment is made to the number of agents. The accumulated adjustment effects νt of agents are shown in column 6. Column 7 gives the forecasting errors et ¼ y ′t y t . Now, look at the 14th observation. The first adjustment is made, in which the number of agents is reduced by one. So only seven agents will actually be on duty during that interval. Note that the staffing adjustment is still calculated according to the Erlang C model (2.4). In a specific interval, the number of agents after adjustment is equal to the original number Nt plus the corresponding number νt in column 6. Thus, the SL y ′t after adjustment could be calculated from the Erlang C model (2.4) with the adjusted number of agents, Nt + νt. Now, look at the values after the 13th interval in Table II. The EWMA forecast y 14 first exceeds the upper limit L2. By calculation, an adjustment of g14 = 1 is made to restore the EWMA of SL as closely to the target T as possible. In the 14th interval and the following several intervals, we calculated the adjusted SL with νt = 1. The second change in y t is made after the 35th interval, which calls for g36 = 1. Thus, the accumulated adjustment effect ν36 of agents is equal to 0, which means no change is made to the original schedule for the next several intervals. In Table II, four adjustments are made in total. The empirical average adjustment interval (AAI) is equal to (14 + 22 + 9 + 8)/4 = 13.25. As the data are calculated based on a half-hourly interval, changing the number of agents by one unit will lead to changes in manpower costs by half an hour. We calculated the sum of the accumulated adjustments of agents (i.e., 14) in column 6 of Table II to represent the change in overall costs, which means that the overall costs are reduced by 14 half-hours. The whole adjustment process is shown in Figures 1–5. Figure 1 is the unadjusted original SL. Figure 2 is the EWMA of the SL monitoring process in which four critical adjustments are made. Figure 3 is the SL after adjustment. Figure 4 is the staffing adjustment made. Figure 5 is the forecasting error series or residuals series, whose mean is 0.01809 and variance is 0.02958. Copyright © 2016 John Wiley & Sons, Ltd.


J. LI ET AL. Table II. Engineering process control strategy of the call center SL after Exponentially weighted adjustment moving average forecast for Original y ′t SL y t No. SL yt 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

0.877 0.536 0.892 0.989 0.950 0.749 0.630 0.922 0.977 0.999 0.981 0.971 0.994

0.984 0.799 0.906 0.986 0.947 0.958 0.978 0.998 0.959 0.961 0.965 0.970 0.990 0.935 0.977 0.983 0.971 0.924 0.839 0.867 0.978 0.744

0.954 0.600 0.775 0.960 0.878 0.894 0.940 0.988 0.862 0.883 0.904 0.904 0.960 0.827 0.924 0.941 0.896 0.741 0.552 0.613 0.899 0.257

0.614 0.816 0.950 0.844 0.560 0.811 0.906 0.589 0.307

0.614 0.816 0.950 0.844 0.560 0.811 0.906 0.589 0.307

0.704 0.924 0.995 0.937 0.868 0.827 0.880 0.970

0.880 0.969 0.999 0.976 0.942 0.921 0.955 0.989

0.800 0.822 0.742 0.784 0.841 0.872 0.838 0.780 0.819 0.863 0.901 0.924 0.937 [0.953] 0.912 0.924 0.833 0.817 0.857 0.863 0.872 0.891 0.918 0.902 0.897 0.899 0.900 0.917 0.892 0.901 0.912 0.907 0.861 0.775 0.729 0.777 [0.631] 0.925 0.838 0.832 0.865 0.859 0.775 0.785 0.819 0.755 [0.629] 0.871 0.874 0.900 0.928 0.941 0.941 0.936 0.941

Accumulated adjustment of agents νt

Forecasting errors et ¼ y ′t y t

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

0.077 0.285 0.151 0.206 0.109 0.123 0.207 0.142 0.157 0.136 0.080 0.047 0.057

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0.041 0.323 0.058 0.143 0.021 0.031 0.068 0.097 0.056 0.020 0.007 0.006 0.060 0.090 0.032 0.040 0.016 0.166 0.308 0.162 0.169 0.520

1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0.311 0.022 0.118 0.021 0.299 0.036 0.121 0.231 0.448

1 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1

0.008 0.096 0.098 0.048 0.001 0.021 0.019 0.048 (Continues)



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46 47 48 49 50 51 52

0.877 0.536 0.892 0.989 0.950 0.749 0.630 0.922 0.977 0.999 0.981 0.971 0.994

Adjustment of agents gt

J. LI ET AL. Table II. (Continued)

No.

Original SL yt

SL after adjustment y ′t

Exponentially weighted moving average forecast for SL y t

Adjustment of agents gt

0.981 0.971 0.950 0.206 0.814 0.897 0.813 0.852 0.929 0.927 0.895 0.966 0.941 0.879 0.889 0.924

0.981 0.971 0.950 0.206 0.814 0.897 0.813 0.852 0.929 0.927 0.895 0.966 0.941 0.879 0.889 0.924

[0.954] 0.888 0.914 0.930 0.935 0.731 0.754 0.794 0.799 0.814 0.846 0.869 0.876 0.901 0.912 0.903 0.899

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68

Accumulated adjustment of agents νt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Forecasting errors et ¼ y ′t y t 0.093 0.057 0.020 0.729 0.082 0.142 0.019 0.052 0.115 0.081 0.026 0.090 0.039 0.034 0.014 0.025

SL, service level.

Figure 1. Unadjusted original service level yt

Figure 2. Exponentially weighted moving average process monitoring and adjustment of service level y t

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Figure 3. Service level y ′t -adjusted process



J. LI ET AL.

Figure 4. Accumulated staffing adjustments νt

Figure 5. Forecasting error series et



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From Tables I and II and Figures 1–5, it can be seen that the SL after adjustment is more stable than the original SL after compensating for the SL deviations from the desired value by adjusting the staffing in a timely manner. Next, we alter L1, L2, and λ to show the monitoring and adjustment effect of the method. The performance of several adjustment schemes adopting various parameters is shown in Tables III and IV. Table III is a comparison of half-hourly interval process schemes. Table IV is a comparison of hourly interval process schemes. In Tables III and IV, the SL target T = 0.8. The cost is based on an agent’s wage for half an hour. Column 7 shows the average squared deviation from the target when the staffing is unadjusted (ASD). Column 8 shows the average squared deviation from the target when the staffing is adjusted (ASDA). Column 9 gives the sum of squares of errors. Column 10 gives the mean of forecasting errors after adjustment, and Column 11 gives the variance of forecasting errors after adjustment. Schemes 1 and 2 are two repeated adjustment examples. The upper and lower limits are both set at 0.80. In Scheme 1, AAI and sum of squares of errors are 2.5 and 18.6534, respectively. Accordingly, 128 in Scheme 1 means that comparing with the original schedule, after adjustment, there are 128 half-an-hour agents decrease in the continuous 68 half-an-hour schedule, and hence, 128 half-an-hour agent salary costs are reduced. Scheme 3 has tight control limit, resulting in a small AAI. In Schemes 1–3, however, the adjustment is close to continuous. In practice, continuous adjustment is not recommended because the staffing should not be frequently adjusted. Schemes 4–8 test other pairs of limits, which have feasible empirical AAIs and smaller sums of squares of errors than Schemes 1–3. Table IV compares the performance of adjustment schemes for monitoring data collected on an hourly basis. From Tables III and IV, it can also be seen that the magnitude of the tolerable worst value of SL L1, the best expected value of SL L2, and the smooth factor λ that we set will directly affect the frequency of staffing adjustment and the adjustment cost. Hence, these parameters must be chosen carefully. The choice of L1 and L2 depends on how important these phone calls are to the enterprise or organization. For example, an enterprise that sells its products over the telephone should set larger values of L1, T, and L2 because customers could be calling to place an order. However, for non-urgent enquiries, smaller values of L1, T, and L2 may be used because customers can always call at another time. Luceno17 describes how to choose the parameter λ in more detail.

J. LI ET AL. Table III. Comparison of parameter changes in half-hourly interval process schemes Average adjustment interval ASD ASDA Scheme L1 L2 λ Cost 1 2 3 4 5 6 7 8

0.80 0.80 0.75 0.70 0.65 0.80 0.75 0.70

0.80 0.80 0.85 0.90 0.95 0.95 0.95 0.95

0.20 0.98 0.61 0.15 0.28 0.15 0.18 0.28

128 68 70 22 14 12 4 14

2.5 1.5 2.1 18.5 13.3 18.0 13.8 13.3

0.0290 0.0290 0.0290 0.0290 0.0290 0.0290 0.0290 0.0290

0.2162 0.0749 0.0578 0.0404 0.0325 0.0325 0.0332 0.0325

Table IV. Comparison of parameter changes in hourly interval process schemes Average adjustment interval ASD ASDA Scheme L1 L2 λ Cost 1 2 3 4 5 6 7 8

0.80 0.80 0.75 0.70 0.65 0.80 0.75 0.70

0.80 0.80 0.85 0.90 0.95 0.95 0.95 0.95

0.20 0.42 0.33 0.17 0.29 0.38 0.39 0.29

132 84 84 60 16 16 18 16

2.3 1.6 2.4 4.8 9.5 4.2 6.3 9.5

0.0252 0.0252 0.0252 0.0252 0.0252 0.0252 0.0252 0.0252

0.1886 0.0991 0.0938 0.0711 0.0360 0.0356 0.0355 0.0360

Sum of squares of errors

Mean

Variance

18.6534 6.2963 4.7621 2.5833 2.0047 1.9579 1.9078 2.0047

0.4813 0.7262 0.7340 0.8290 0.8453 0.8485 0.8541 0.8453

0.1146 0.0695 0.0534 0.0396 0.0305 0.0302 0.0303 0.0305

Sum of squares of errors

Mean

Variance

7.8805 4.1096 3.4476 2.3178 0.9863 0.8632 0.9051 0.9863

0.5582 0.6992 0.7185 0.7696 0.8779 0.8739 0.8735 0.8779

0.1302 0.0889 0.0871 0.0702 0.0300 0.0302 0.0301 0.0300

5. Conclusions The paper describes an EPC strategy that call centers may adopt to monitor changes in SL and make relevant staffing adjustments. A staffing adjustment is made only if the trend of changes in the whole SL process is determined to be abnormal by EWMA forecasting. That is, the strategy is to do nothing unless the SL deviates so far from the desired level as to exceed the upper and lower limits. The advantage of the method is that frequent adjustments are avoided. Meanwhile, a stable SL can be maintained near the standard level. The method may also be applied to other service systems such as banking and hospital systems to monitor changes in customer or patient numbers and make corresponding staffing adjustments.

Acknowledgements Tsung’s research was supported by RGC General Research Fund (grant nos. 619612 and 619913). Besides, this research was also sponsored by the National Natural Science Foundation of China (grant nos. 71572113, 71301117, 71371140, 71432007, 11171221, 71271138, 71303157, and 71202065), Shanghai Gaofeng Academic Discipline Projects (grant no. XTKX2012), the Innovation Program of Shanghai Municipal Education Commission (grant nos. 14YZ088 and 14YZ089) and the Innovation Fund Project for Undergraduate Students of China and Shanghai (grant nos. 151025213, SH2014054, SH2014062, SH2015056, and XJ2015083).

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Authors' biographies Junxiang Li obtained his M.S. and Ph. D degrees from Dalian University of Technology, Liaoning Province, China in 2005 and 2008, respectively. From 2010 to 2011, he was a Postdoctoral Researcher in the School of Economics and Management, Tongji University, Shanghai, China. In 2011, he was also a visitor Scholar in the Department of Industrial Engineering and Logistics Management, Hong Kong University of Science and Technology, Hong Kong. Now he is an associate professor and supervisor of postgraduate in Business School, University of Shanghai for Science and Technology, Shanghai, China. His major is management science and engineering and his research interests include quality control and systems engineering management and optimization. Yafen Liu is an Associate Professor in the School of Management of Tianjin Polytechnic University, China. She obtained her PhD in 2011 from the Tianjin University, China. Her current research work focuses on statistical process control (SPC). Fugee Tsung is Professor of the Department of Industrial Engineering and Logistics Management (IELM), Director of the Quality Lab, at the Hong Kong University of Science & Technology (HKUST). He is a Fellow of the Institute of Industrial Engineers (IIE), Fellow of the American Society for Quality (ASQ), Academician of the International Academy for Quality (IAQ) and Fellow of the Hong Kong Institution of Engineers (HKIE). He is Editor-in-Chief of Journal of Quality Technology (JQT), Department Editor of the IIE Transactions, and Associate Editor of Technometrics. He has authored over 100 refereed journal publications, and is the winner of the Best Paper Award for the IIE Transactions in 2003 and 2009. He received both his MSc and PhD from the University of Michigan, Ann Arbor and his BSc from National Taiwan University. His research interests include quality engineering and management to manufacturing and service industries, statistical process control and monitoring, industrial statistics and data analytics. Jiazhen Huo received the B.S., M.S., and Ph.D. degrees from Tongji University, Shanghai, China, in 1985, 1987 and 2001, respectively, all in management science. Since 1987, he has worked in School of Economics and Management, Tongji University, Shanghai, China. Since 2009 and 2015, he has been a chair professor of DHL and Bosch company, Germany, respectively. Now he is a professor and director in the school. His research interests include production and service management, logistics and supply chain management, and industrial engineering optimization. Qiang Su received the B.S., M.S., and Ph.D. degrees from Xi’an Jiaotong University, Xi’an, China, in 1991, 1994 and 1997, respectively, all in manufacturing engineering. From 1998 to 2000, he was a Postdoctoral Researcher with the Department of Management Science and Engineering, Tsinghua University, Beijing, China. From 2000 to 2004, he was a Postdoctoral Researcher and then a Senior Research Fellow with the Department of Industrial and Systematic Engineering, Florida International University, Miami. He once worked in the Department of Industrial Engineering and Logistics Management, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, China, in 2004, as an Associate Professor. Since 2010, he joined the Department of Management Science & Technology, School of Economics and Management, Tongji University, Shanghai, as a Full Professor. His research interests include production and service system optimization, quality and cost relationship analysis, and healthcare service improvement. Dr. Su is a senior member of the IEEE, ASQ, CAQ and a member of the POMS.

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