Sampling Bounds

In Section three, I showed that the average solution to the stochastic program provides a point estimate on the lower bound on the true optimal solution, while the average expected outcome of the candidate solution forms a point estimate of the upper bound of the true optimal. In Figure 4- 12 I plot the point estimate of the upper and lower solution bounds for Project J at multiple scenario levels, estimated using five batches at each scenario level.

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Sampling Bounds Point Estimate - Project J

10,000 10,500 11,000 11,500 12,000 12,500 13,000 0 25 50 75 100 Number of Scenarios O p ti m al E xp ecta ti o n

Figure 4-14 Point Estimate of Bounds

Equation (4.23) provides a mechanism to calculate a confidence interval on the optimality gap. In Figure 4-13 I plot the 90% confidence interval on the magnitude of the optimality gap

Sampling Error Optimality Gap - Project J

0 50 100 150 200 250 0 10 20 30 40 50 60 70 80 90 100 Number of Scenarios U p p er B oun d on Ga p

Figure 4-15 Optimality Gap

These graphs show that the mean value problem exhibits significant bias, but that even with a moderate number of scenarios, and a few batches, we are able to generate fairly tight bounds on the true optimal value. The data suggests that solving the problem with as few as 25 scenarios provides reasonably good results, while a 50 or 100 scenario model gives us a tighter bound that may be useful when trying to make detailed comparisons between alternatives.

124 For each project listed in table 6-1 the stochastic program lowers overall expected cost by increasing direct labor. It is somewhat paradoxical that stochastic programs provide better results by calculating worse objective functions. The intuition is however straightforward; deterministic

optimization programs assume away uncertainty and therefore do not adequately hedge for variability.

Figure 4-16 compares the schedules generated from a mean value program and a stochastic program. The stochastic program adds incremental staffing at various points throughout the day. Figure 4-17 shows a 90% confidence interval for the calls received by period. Comparing that graph to Figure 4-16 we see that incremental staffing is added in periods with relatively high volumes and high variability.

Optimal Monday Schedule

0 5 10 15 20 25 Midnight 4:00 AM 8:00 AM 12:00 PM 4:00 PM 8:00 PM Time of Day Ag en ts S ch ed u le d Mean Value Stochastic

125

Call Arrivals - 90% Confidence Level

0 10 20 30 40 50 60 70 Midnight 2:00 AM 4:00 AM 6:00 AM 8:00 AM 10:00 AM 12:00 PM 2:00 PM 4:00 PM 6:00 PM 8:00 PM 10:00 PM Time of Day C all s per H alf H our

Figure 4-17 Confidence Interval for per period calls

Figure 4-16 shows the mean value and stochastic solution for Monday, the busiest day of the week. In Figure 4-18 I plot the incremental staffing generated by the stochastic solution over the course of the week. We see that the stochastic model adds incremental capacity during the busy periods of most days, but reduces staffing in some low volume periods.

Incremental Staffing from Considering Variability

-3 -2 -1 0 1 2 3 4 5

Mon Tue Wed Thu Fri Sat Sun

A d d itio n al A g en ts S ta ffe d

126

4.5.3 Impact of Variability

In the prior analysis I calculated schedules for models of several real world projects and examined the convergence properties of the solution. I examined the differences between the mean value solution and the stochastic solution and showed that the stochastic schedule adds extra capacity to buffer against uncertainty. The analysis showed that VSS varies from project to project and the data suggests that for projects with higher variability the stochastic solution diverges from the Mean Value Solution more significantly.

In this section I conduct a controlled experiment to assess the impact of variability more directly. While I still base the analysis on a specific project, I manipulate key parameters to determine the impact of variability on the resulting schedule. Specifically, I analyze a series of alternative project configurations for which the expected number of calls, and the average seasonality pattern are based on project J, but I manipulate key environment and policy variables.

4.5.3.1 Experimental Design

To assess the impact of variability I will conduct a controlled experiment that adjust factors related to variability as well as the required service level quality, Specifically I will conduct an experiment using the following factors

• Daily CV Scale: the variability of daily arrivals is adjusted by scaling the coefficient of variation for day of week effects; the mean is held constant and standard deviation is adjusted to achieve the scaled CV.

• Time Period CV Scale: the same variability scaling is performed on the Time of Day effect.

• Service Level Requirement: a loose SLA (70/120) and a tight SLA (90/30).

• Service Level Penalty: different penalty costs for failing to achieve the specified service level target.

• Shock Probability: arrivals with and without shocks. In the case of shocks I scale down the non-shock volume so that the expected call volume is constant across all design points.

127 I created an experiment with 16 design points as defined below

A B C D E Factor Definitions - +

1 - - - - + A Daily CV Scale 0.75 1.25

2 + - - - - B Time Period CV Scale 0.75 1.25

3 - + - - - C Service Level Requirement 70/120 90/30

4 + + - - + D Shock Probability 0% 5%

5 - - + - - E Service Level Penalty 50,000 150,000

6 + - + - + 7 - + + - + 8 + + + - - 9 - - - + - 10 + - - + + 11 - + - + + 12 + + - + - 13 - - + + + 14 + - + + - 15 - + + + - 16 + + + + +

Table 4-7 Impact of Variability Experimental Design

This is a 5 1 V

2

resolution of V, which allows us to estimate all the main effects and all the two way interaction effects. Higher level interactions are confounded and can not be estimated independently.

4.5.3.2 Experimental Results

To conduct this experiment I generate 5 batches of 50 scenarios and a single evaluation batch of 500 scenarios at each of the 16 design points. I solve the optimization problems for each batch, calculating a candidate solution which is evaluated against the 500 scenario evaluation batch to calculate expected outcomes. Based on these solutions I calculate the following response variables:

• Labor cost: the cost of direct labor in the candidate solution.

• Expected Outcome: the labor and penalty cost found when evaluating the candidate solution.

• TSF Cushion: the difference between the expected TSF found when evaluating the candidate solution, and the SLA performance goal.

• Confidence: the proportion of evaluation scenarios for which the service level target is achieved.

128 This approach generates 5 samples for each response. The results of this analysis are presented in the following table. Recall that all design points in this experiment have the same expected call volume. A B C D E Labor Cost Expected Outcome TSF Cushion Confidence Labor Cost Expected Outcome 1 - - - - + 8,992 9,097 3.9% 93.3% 75.6 40.9 2 + - - - - 9,048 9,210 3.9% 85.0% 85.6 20.3 3 - + - - - 9,056 9,170 2.1% 83.2% 51.8 16.1 4 + + - - + 9,524 9,616 5.6% 95.9% 91.0 51.3 5 - - + - - 12,404 12,856 -0.4% 41.7% 138.1 53.7 6 + - + - + 13,200 13,520 1.9% 84.3% 154.9 45.7 7 - + + - + 13,440 13,969 0.7% 71.2% 105.8 62.5 8 + + + - - 13,408 14,002 -0.4% 47.5% 100.6 31.7 9 - - - + - 8,836 8,942 2.5% 83.8% 43.4 15.9 10 + - - + + 9,216 9,349 5.6% 94.1% 69.9 7.0 11 - + - + + 9,128 9,347 2.9% 88.7% 22.8 20.0 12 + + - + - 9,248 9,465 3.3% 80.8% 68.7 12.0 13 - - + + + 12,748 13,027 1.2% 80.2% 136.1 56.0 14 + - + + - 12,692 13,117 0.2% 58.9% 156.6 13.3 15 - + + + - 13,168 13,481 0.0% 54.7% 128.5 47.7 16 + + + + + 13,332 13,855 -0.1% 51.8% 136.8 11.8

Average Standard Deviation

Table 4-8 Impact of Variability Experimental Results

4.5.3.3 Analysis of Results

The resolution V experimental design allow us to calculate the main effects; the impact of moving each factor from it’s low to high value, as well as first level interaction terms; the interaction of each unique pair of factors. Given the orthogonal nature of the experimental design all factors are perfectly uncorrelated and we have no issue of multicolinearity in our analysis.

The following table summarizes the estimated main and first level interaction effects for each of the response variables. The main effects represent the average change in the response when the factor is changed from its low value to its high value. The interaction effects estimate the impact of factors that have a coupled influence upon the response beyond their main effects, they are calculated as one half of the average difference in response when both factors change together (Box, Hunter et al. 2005)52_{. Only those effects that are statistically significant at the 0.01 level}

are displayed.

52 _{If both factors are at the same level, the interaction term is added to the estimated outcome. If they are at} opposite levels the interaction term is subtracted from the estimated outcome. So for example, the BC

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Factor Effects Labor

Expected Outcome SL Cushion Confidence Intercept 11,091 11,378 2.0% 74.4% A - Daily CV 259 282 1.0% B - Time Period CV 395 474 -0.6% -6.0%

C - Service Level Req 3,919 4,207 -3.3% -27.4%

D - Shock Prob -87 -104 E - SL Penalty 211 189 1.3% 15.7% A*B -122 -41 -10.9% A*C -0.8% B*C 179 224 -8.3% A*D -126 -64 -8.3% B*D -45 -8.8% C*D -106 A*E -55 B*E -48 C*E 34 11.8% D*E -99 -52 -0.9% -12.3%

Table 4-9 Impact of Variability on Expected Outcome - Main Effects

This table presents a considerable amount of information. Some key observations include: • The average cost of operating this call center is $11,378 per week, but the realized cost varies

considerably.

• The most influential cost driver is the service level requirement, increasing the service level requirement adds about $4,000, or 50% to the cost of operations.

• Variability has a substantial impact on the cost of delivery, but the impact is influenced by the SLA regime.

- In a loose SLA environment (C - , E - ) increased daily variability increases costs by about 6%. Increased time variability increases cost by 2.3%, together they increase cost by 6.6%

- In a tight SLA environment (C - , E - ) increased daily variability increases costs by about 2.5%. Increased time variability increases cost by 8.3%, together they increase cost by 9.4%

• On average the optimal staffing decision staffs the project so that the expected service level is 2% above the requirement, which results in a 74% confidence level. However in the tight SLA regime, the cost meeting a high service level cause the cushion and confidence level to drop significantly.

interaction term increases the labor estimate by $179 if both factors have the same setting. If one is high and the other low the estimate is reduced by $179.

130

4.6 Staffing Flexibility

4.6.1 Overview

One of the operational challenges associated with the type of call center analyzed here is that demand is often more variable then capacity. The arrival pattern shown in Figure 2-24, for example, has a large spike in demand between 8 and 11 AM. In order to efficiently match supply and demand we would like to create a corresponding spike in capacity at the same time. Accomplishing this with full time staffing can be difficult. However, in practice some call centers are often staffed exclusively with full time resources; that is that resources scheduled to 40 hour per week schedules.

Managers have multiple reasons for hiring only full time agents. Full timers are believed to be less expensive to train because their hiring and training cost is amortized more quickly. Many managers also believe that full time agents will learn faster and thus be more productive than one working part time by being exposed to more calls. Some managers also believe that part time agents are more difficult to recruit and retain53_{. The potential savings from using part time}

resources is of interest, from a both a practical and research perspective. I examine that issue in this section.

In Section 4.6.2 I develop alternative scheduling patterns and develop a range of flexibility options. In section 4.6.3 I develop a conceptual framework for evaluating the cost of staffing flexibility. In section 4.6.4 I perform a numerical experiment to calculate the cost of staffing under each scheduling pattern for 3 model projects. In section 4.6.5 I perform a related experiment that look at the implications of limiting the availability of part timers.