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Starting-time decisions in labor tour scheduling: An experimental analysis and case study
European Journal of Operational Research 131 (2001) 459±475
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Case Study
Starting-time decisions in labor tour scheduling: An experimental analysis and case study Michael J. Brusco a
a,*
, Larry W. Jacobs
b,1
Information and Management Sciences Department, College of Business, The Florida State University, Tallahassee, FL 32306-1110, USA b Operations Management and Information Systems Department, College of Business, Northern Illinois University, DeKalb, IL 60115, USA Received 17 February 1999; accepted 18 April 2000
Abstract Many service organizations limit the number of daily planning periods in which employees may begin their shifts to a ®xed number, S. Even for relatively small values of S, which are quite common in practice, there may be hundreds, thousands or millions of possible subsets of starting times. This paper presents the results of a large experimental study that revealed that, in many instances, only a very small portion of starting-time subsets was capable of providing the minimum workforce size. The importance of eective starting-time selection is further supported by a case study that describes a spreadsheet-based program designed for scheduling customer service representatives in the System Support Center, United States and Canada Group, Radio Network Solutions Group, Land Mobile Products Sector, Motorola. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Scheduling; Integer programming; Heuristics; Call centers
1. Introduction Two critical challenges facing service operations managers are (1) ®nding ways to improve service productivity [4], and (2) facilitating the balance of work and family responsibilities for * Corresponding author. Tel.: +1-850-644-6512; fax: +1-850644-8225. E-mail addresses: [email protected] (M.J. Brusco), [email protected] (L.W. Jacobs). 1 Tel.: +815-753-6165; fax: +815-753-3300.
their employees [11,19]. One of the most eective tools available to the operations manager for meeting these challenges is labor scheduling. Since labor schedules determine the number of employees present during dierent hours of the day and dierent days of the week, they have important implications for operating cost and customer service. If the number of employees scheduled is not sucient to satisfy customer demand, then the operating system may choose to let customer service decline (e.g., an increase in customer waiting times), or it may increase labor supply via some
0377-2217/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 0 ) 0 0 1 3 5 - 1
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other means (e.g., the use of overtime). In either case, costs of a labor shortage are encountered. Similarly, if the number of employees scheduled creates a supply of labor that exceeds customer demand, then costs of idle labor are incurred. In the light of these issues, it is not surprising that there has been a substantial research eort devoted to the development of optimal and heuristic procedures for generating ecient labor schedules [2,3,5,6,8,21,22,26±29]. In addition to methodological development, labor-scheduling research has also focused on the evaluation of stang and scheduling policies. For example, Easton et al. [13] evaluated a variety of popular nurse-scheduling policies in terms of their relative scheduling costs. Studies have also been conducted to measure the eects of part-time employees [18], overtime policies [12], the number of workperiods in a daily workshift and the number of workdays in a workweek [15], the overlapping of shifts from one workday to the next [7], overlapping bands on daily starting times for employees [16], cross-utilization of employees [10], and limited employee availability [26,27]. The selection of daily starting times for employee workshifts is another type of scheduling policy and is the focus of this paper. The selection of daily starting times may be important even when shifts do not overlap. For example, a hospital nursing unit might experience higher productivity when using (9 am to 5 pm, 5 pm to 1 am, and 1 am to 9 am) as their three non-overlapping shifts, as opposed to (7 am to 3 pm, 3 pm to 11 pm, and 11 pm to 7 am). Starting-time decisions are even more important for the large number of service organizations that are now deploying multiple overlapping shifts. This is due to the fact that an increase in the number of allowable starting times generally increases the potential number of starting-time subsets. For a 24-hour day with hourly planning intervals, 8-hour shifts, and ®ve shift starting times, there are 245 42,504 dierent possible subsets of shift starting times. Although some of these subsets are infeasible because they fail to cover all hours of the planning horizon, there remain 25,344 feasible subsets. In this context, the selection of daily shift starting times consists of two interrelated deci-
sions: (a) the number, S, of shift starting times permitted during an operating day, and (b) the determination of precisely which of the S planning periods of the day will be allocated as starting times. There are obvious trade-os associated with the ®rst of these decisions. By increasing the value of S, the service manager might be able to cover all labor requirements with fewer employees. More starting times provide greater ¯exibility for covering labor requirements, which may enable less overstang and, therefore, greater labor productivity. Improved labor productivity associated with more starting times might also provide bene®ts from a human-resources standpoint. A better match between demand and labor capacity should promote fewer shortages of employees and a more equitable distribution of workload, which may in turn result in less absenteeism and turnover [13]. There are also disadvantages associated with increasing the number of starting times. Most of these disadvantages stem from an administrative burden associated with a larger number of starting times. Increasing the number of shift starting times results in more time periods during the day at which employees will be beginning shifts, beginning breaks, ending breaks, and ending shifts. With employees coming and going at many different times during the day, it may be dicult to monitor employee punctuality and there can be a serious loss of labor productivity. In addition, many service organizations may use the beginning (or end) of employee shifts for brie®ng (de-briefing) sessions [8,9]. The scheduling of these sessions becomes increasingly more dicult as the number of starting times increases, thus limiting their effectiveness. Finally, a large number of starting times can result in a small number of assignments (perhaps only one or two employees) to particular starting times and this may not be conducive to a team work environment. In short, the service operations manager must be able to evaluate the relative trade-os associated with increasing the number of starting times. There are no de®nitive rules assessing such tradeos and the appropriate number of starting times may vary from one service operating system to the next. However, there does appear to be a
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trend in many service industries to move away from the traditional three starting times. For example, a survey conducted by Siferd and Benton [24] revealed that nearly 30% of hospital nursing units utilized four or ®ve shift starting times per day. Similarly, Taylor and Huxley [25] described an application for the San Francisco Police Department in which increasing the number of starting times from three to four or ®ve was shown to enhance labor productivity. Applications in the airline and other service industries have used ®ve or more shift starting times [2,9,23]. Despite the prevalence of multiple overlapping shifts, little is known about the eect of startingtime decisions for practical problems. An interesting research question is, ``which of the startingtime subsets, and how many of them, will provide the minimum workforce stang cost?'' Related questions might focus on the results associated with mediocre or poor choices of starting-time subsets. To date, there has been no comprehensive evaluation of the impact of starting-time selections on workforce stang costs. This paper is the ®rst to provide such an evaluation via a large computational study using sets of labor requirements from three dierent service operating environments. The study was conducted within the context of a continuous tour-scheduling problem environment [6,20,21]. This problem, which is concerned with the assignment of employees to both daily workshifts and weekly days-o patterns, has formed the basis for practical scheduling applications in hospital nursing units [22,29], telecommunication centers [1,20,28], airport stations [9,23], and police patrols [17,25]. The importance of starting-time decisions in labor tour scheduling is further supported by a case study focusing on the development of a spreadsheet program designed for scheduling customer service representatives at Motorola call centers. This system is currently being used at Motorola to investigate labor stang and scheduling policies. This practical scheduling problem is far more complex than the one considered in the computational study, yet we observed results that are surprisingly consistent with those obtained in the computational study.
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Section 2 presents the integer linear programming (ILP) formulation for the continuous tourscheduling problem with restrictions on the number of shift starting times and also describes previous solution approaches to such problems. Section 3 provides a description of the experimental study used to assess the eects of startingtime decisions. Section 4 describes how the results of the experimental study were applied to the development of a labor scheduling system for the Motorola call centers. A brief summary is presented in Section 5. 2. An ILP formulation for the limited starting-time tour-scheduling problem The labor tour-scheduling formulation presented below is based on a planning horizon of one week and is comparable with formulations used in several previous studies [3,6,8,12,13,18,21]. For the sake of parsimony and consistency of presentation, the scheduling environment is assumed to consist of full-time employees and the single objective is to minimize the total workforce size (this assumption holds for both the experimental and case studies). A speci®c tour is uniquely de®ned by a daily shift starting time (which is consistent for each workday of the tour) and the speci®c days worked during the week. The ILP formulation is as follows: Minimize:
Z
I X J X
Xij
1
i1 j1
subject to I X J X
aijkl Xij P rkl
8k 1; . . . ; I; 8l 1; . . . ; L;
i1 j1
2 J X Xij
mYi 6 0
8i 1; . . . ; I;
3
j1 I X Yi 6 S; i1
4
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Xij P 0 and integer
8i 1; . . . ; I 8j 1; . . . J ; 5
Yi 2 f0; 1g
8i 1; . . . I;
6
where I is the number of planning periods in the operating day, i 1; . . . ; I; J the number of allowable days-o patterns, j 1; . . . J ; L the number of operating days in the weekly planning horizon, l 1; . . . ; L; Xij the number of employees assigned to a tour on days-o pattern j that begins in planning period i; aijkl 1, if planning period k of day l is a workperiod in a tour that begins in planning period i and is associated with days-o pattern j and 0 otherwise; rkl the number of employees required in planning period k of day l; Yi 1, if any employee begins a tour in planning period i and 0 otherwise; m a large positive integer; and S is the maximum allowable number of starting times (as de®ned previously). The objective function (1) of the ILP model, which is consistent with the work of Morris and Showalter [21], Bechtold and Brusco [3], and Brusco and Jacobs [6], is to minimize total workforce size. Constraints (2) ensure that a sucient number of employees are available to satisfy labor requirements in each planning period of the day and on each day of the week. Constraint (3) guarantees that a starting time is recognized as ``used'' if any employee is assigned to a tour associated with that starting time. If the integer programming model is solved directly, then it should be observed that is possible to have one Pit J or more Yi 1 even if j1 Xij 0. However, this does not present a problem from a pragmatic standpoint. Constraint (4) limits the total number of starting times used to be no more than S. Constraints (5) and (6) place the appropriate nonnegativity and integer restrictions on the Xij and Yi variables, respectively. 3. An experimental study 3.1. Problem environment Given a ®xed number of candidate starting times (I) and a ®xed number of starting times to
select from these candidates (S), there are often many subsets of size S to choose from. The purpose of this study was to evaluate, for ``realworld'' sets of labor requirements, the eect of starting-time selection. Speci®cally, research questions addressed in this study include: (1) can a small value of S provide optimal workforce sizes comparable to those obtained when all I starting times are considered? (2) if ZS denotes the optimal workforce size given S starting times, then what proportion of feasible starting-time subsets (of size S) would provide an objective function value equal to ZS ? and (3) just how detrimental is a relatively poor selection of starting times? The successful implementation of a study designed to answer these questions required the solution of a substantial number of ILP formulations. Therefore, it was necessary to select a labor tour-scheduling environment for which integer-optimal solutions could be eciently obtained. The continuously operating scheduling environment used to conduct this study consisted of a one-week planning horizon divided into hourly planning intervals. Full-time employees working ®xed shifts on consecutive days were used to sta the operation and meal/rest breaks were not explicitly considered. Six unique operating policies were investigated by considering all combinations of two workweek alternatives and three startingtime limitations. The workweek alternatives were a ®ve-day, 8 hours per day (5/40) workweek, and a four-day, 10 hours per day (4/40) workweek. The starting-time limitations were associated with 3, 4, or 5 allowable starting times. Hereafter, 5/40-S and 4/40-S will be used to identify operating policies characterized by the respective workweek alternative with S starting times. Each operating policy was tested across three sets of labor requirement patterns. The ®rst set of labor requirement patterns, consisting of six problems from the telephone industry (TEL_1± TEL_6), was originally reported by [20]. These patterns consisted of relatively large labor requirements for individual planning intervals. The magnitude and pattern of the labor requirements diered across certain days of the week. The second set of 12 labor requirement patterns associated with tollway operations (TOLL_1±TOLL_12)
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was taken from Jacobs and BruscoÕs [16] study. Like the McGinnis et al. [20] patterns, the tollway labor requirement patterns diered across days of the week, however, the magnitude of demand in individual planning intervals was much smaller. The third set of labor requirement patterns, consisting of 27 sets of airport station requirements (AIR_1±AIR_27), was based on those used by Brusco and Jacobs [8]. The magnitude of demand for these patterns varied widely across airport stations, but generally contained substantial variability from one planning interval to the next. Unlike the telephone and tollway requirement patterns, each airport station pattern generally exhibited the same shape and magnitude across dierent days of the week. 3.2. Solution method In their application for the San Francisco Police Department, Taylor and Huxley [25] compared the results associated with three, four, and ®ve starting times, but did not describe how the starting times were selected. Schindler and Semmel [23] and Brusco et al. [9] described mathematical programming-based methods for selecting starting times within the context of airport station stang and scheduling applications, and Brusco and Jacobs [8] recently provided an in-depth analysis of column generation-based heuristics for startingtime selection. Unfortunately, none of the methods associated with the aforementioned studies provided guaranteed optimal solutions to the ILP formulation of the limited starting-time tourscheduling problem, nor was there any indication of what proportion of all possible starting-time subsets would lead to optimal solutions. To complete this current study, it was deemed necessary to solve an ILP formulation for all feasible subsets of starting times, for all 45 testproblems. Although there were 24 possible subsets of S starting times, many of these were infeasible because they were not capable of satisfying requirements in all periods of the planning horizon. Nevertheless, the total number of feasible subsets could be quite large. Speci®cally, there were 8, 224, 966, 3798, 11424, and 25344 feasible starting-time
463
subsets for the 5/40-3, 4/40-3, 5/40-4, 4/40-4, 5/405, and 4/40-5 operating policies, respectively. Thus, this experimental study required the solution of 8 224 966 3798 11424 25344 45 1,879,380 integer programs. Fortunately, the solution of these problems was facilitated by the deployment of the dual, all-integer cutting plane [14], which has proven very eective for such problems [3,5]. A computer program was written in Fortran PowerStation (Ver. 1.0) to generate the ILP formulation for all feasible subsets of starting times and subsequently solve them using the cutting plane. Fig. 1 displays the logic of this program. Data were collected regarding the minimum workforce size, the number of subsets of starting times that yielded the minimum workforce size, and the total CPU time to solve all problems for the workforce policy. For benchmarking purposes, the cutting plane was also used to solve, for each labor requirement pattern, an integer linear program associated with all 24 starting times. 3.3. Experimental results The results of the experimental study are depicted in Tables 1±3. Table 1 presents the optimal workforce size under each workforce policy for each labor requirement pattern. Table 1 clearly shows that increasing the number of starting times from three to four, and from four to ®ve, often enables a signi®cant reduction in workforce size. However, the increase from three to four starting times provided greater bene®t than the increase from four to ®ve. Speci®cally, the 5/40-3 (4/40-3) policy was able to match the optimal workforce size provided by the 5/40-4 (4/40-4) policy for only 3 (8) of the 45 test problems, whereas 5/40-4 (4/404) matched 5/40-5 (4/40-5) for 15 (35) test problems. It is also interesting to note that, for the 5/40 workweek, ®ve (four) starting times yielded the same optimal workforce size as 24 starting times for 20 (13) of the 45 test problems. The results were even more impressive for the 4/40 workweek for which ®ve (four) starting times yielded the same optimal workforce size as 24 starting times for 42 (34) of the 45 test problems. These ®ndings suggest that the answer to the ®rst question posed
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Fig. 1. Flowchart for starting-time evaluation program.
in Section 3.1 is ``yes'', very ecient scheduling solutions may often be achieved with a small number of starting times. Table 2 reports, for each demand pattern and each workforce policy, the total CPU time required to solve integer programs associated with all feasible subsets of starting times. These results indicate that the dual all-integer cutting plane provided very ecient solutions for the integer linear programs. For the 5/40-5 policy, the generation and solution of the integer programs associated with the 25,344 feasible starting-time subsets generally required less than 500 CPU seconds (an average of approximately 0.02 CPU seconds per problem). Table 3 presents, for each operating policy, the percentage of starting-time subsets that were capable of providing the minimum workforce size yielded across all subsets. For the 5/40-3 policy, there are only eight feasible subsets of starting times. If one of these starting-time subsets yielded
a workforce size that was less than the workforce sizes associated with the other seven subsets, then the reported percentage was 12.50. If two subsets provided the smallest workforce size, then the reported percentage was 25.0, and so on. For other operating policies, there are many more feasible subsets and thus there is clearly the potential for a much smaller percentage of subsets to provide the minimum workforce size. For example, consider the 5/40-5 policy for the TEL_4 pattern. In this instance, only one of the 11,424 subsets of starting times provided the minimum workforce size of 168, resulting in a reported percentage of 0.009 100 1=11424. Similar ®ndings were observed for many of the test problems. In fact, when four or ®ve starting times were used, no more than 5% of the starting-time subsets were ever capable of providing the minimum workforce size for the telephone or tollway test problems. These results suggest that the answer to the second question posed in Section 3.1 is
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465
Table 1 Optimal workforce sizes Demand pattern TEL_1 TEL_2 TEL_3 TEL_4 TEL_5 TEL_6 TOLL_1 TOLL_2 TOLL_3 TOLL_4 TOLL_5 TOLL_6 TOLL_7 TOLL_8 TOLL_9 TOLL_10 TOLL_11 TOLL_12 AIR_1 AIR_2 AIR_3 AIR_4 AIR_5 AIR_6 AIR_7 AIR_8 AIR_9 AIR_10 AIR_11 AIR_12 AIR_13 AIR_14 AIR_15 AIR_16 AIR_17 AIR_18 AIR_19 AIR_20 AIR_21 AIR_22 AIR_23 AIR_24 AIR_25 AIR_26 AIR_27 a
3 Starting times
4 Starting times
5 Starting times
24 Starting times
5/40
4/40
5/40
4/40
5/40
4/40
5/40
4/40
221 221 187 173 246 198
240 241 236 208 235 221
208 208 185 173 231 197
235 236 214 202 218 206
198 198 185 168 213 197
235 236 214 176 218 195
187 187 185 165 212 194
235 236 214 169a 218 177
35 33 35 34 21 21 21 21 17 17 17 17
38 36 36 36 22 22 22 22 17 17 18 17
34 31 32 32 20 20 20 20 16 16 16 16
35 33 34 34 21 21 20 20 17 16 16 16
32 30 31 31 19 19 19 19 16 15 15 15
34 33 34 34 20 20 20 20 16 16 16 16
29 29 30 29 19 18 18 18 15 14 15 15
34 33 34 34 20 20 20 20 16 16 16 16
24 44 19 29 28 23 315 26 9 68 153 94 218 35 32 14 32 30 452 57 34 14 57 111 304 29 38
23 44 17 24 25 25 374 26 11 85 168 99 232 36 34 13 32 31 479 56 31 14 59 105 325 31 40
22 42 16 25 20 20 301 22 9 68 135 79 211 32 29 11 28 29 435 54 30 11 51 107 300 25 35
21 43 15 22 23 23 335 24 11 85 168 99 231 32 30 13 28 31 453 51 31 13 55 97 305 31 36
20 40 16 25 20 20 298 21 9 68 135 79 199 29 28 11 26 27 383 49 30 11 48 105 275 25 32
20 43 15 22 23 23 332 24 11 85 168 99 231 32 30 13 26 31 452 51 31 13 55 94 305 31 36
19 40 16 25 20 20 285 21 9 68 135 79 186 28 28 11 24 26 365 46 30 11 47 105 259 25 30
20 43 15 22 23 23 332 24 11 85 168 99 231 32 30 13 25 31 452 51 31 13 55 94 305 31 36
Indicates a lower bound on the workforce size, the integer optimal solution was not obtained.
that the proportion of starting-time subsets capable of providing the optimal workforce size is often very small.
Solution quality generally deteriorated very rapidly as less eective starting-time subsets were used. To illustrate this ®nding, consider the tele-
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M.J. Brusco, L.W. Jacobs / European Journal of Operational Research 131 (2001) 459±475
Table 2 Total CPU seconds to solve integer programs for all subsets of starting times Demand pattern
a
3 Starting times
4 Starting times
5 Starting times
24 Starting times
5/40
5/40
4/40
5/40
4/40
5/40
4/40
4/40
TEL_1 TEL_2 TEL_3 TEL_4 TEL_5 TEL_6
0.17 0.17 0.11 0.16 0.11 0.11
3.29 2.85 2.69 2.70 2.69 2.74
13.73 13.24 12.52 12.69 12.46 12.91
56.57 56.84 52.18 56.36 52.73 57.73
190.43 190.82 177.63 191.97 185.48 202.95
468.57 469.83 432.21 540.19 457.04 539.15
2.04 3.52 0.99 0.87 2.59 7.30
0.33 0.38 0.49 N/Sa 0.33 5.16
TOLL_1 TOLL_2 TOLL_3 TOLL_4 TOLL_5 TOLL_6 TOLL_7 TOLL_8 TOLL_9 TOLL_10 TOLL_11 TOLL_12
0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11
2.80 2.80 2.81 2.80 2.69 2.69 2.75 2.75 2.69 2.74 2.69 2.69
12.91 12.80 12.96 12.80 12.63 12.58 12.35 12.42 12.31 12.30 12.25 12.19
55.42 55.53 55.20 54.82 54.05 53.66 52.75 53.11 51.19 52.62 52.12 51.41
179.88 181.03 181.37 177.51 176.97 174.67 168.62 169.66 165.21 170.05 166.92 166.37
473.73 491.86 492.46 469.23 473.19 464.17 447.03 445.28 411.56 445.72 436.71 400.51
3.30 1.92 10.88 2.91 4.51 2.75 2.08 0.82 4.23 3.51 3.46 6.48
0.55 0.61 0.88 0.44 1.26 0.77 0.83 0.77 1.37 0.94 0.77 0.49
AIR_1 AIR_2 AIR_3 AIR_4 AIR_5 AIR_6 AIR_7 AIR_8 AIR_9 AIR_10 AIR_11 AIR_12 AIR_13 AIR_14 AIR_15 AIR_16 AIR_17 AIR_18 AIR_19 AIR_20 AIR_21 AIR_22 AIR_23 AIR_24 AIR_25 AIR_26 AIR_27
0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11
2.69 2.80 2.69 2.75 2.69 2.69 2.96 2.69 2.53 2.80 2.86 2.86 2.91 2.80 2.69 2.69 2.74 2.75 2.91 2.74 2.80 2.63 2.86 2.81 2.91 2.80 2.74
12.69 13.07 12.64 12.63 12.74 12.47 13.84 12.41 11.09 12.80 13.62 13.46 13.74 12.85 12.63 12.30 12.74 12.52 13.95 12.91 12.68 12.09 13.07 13.24 14.12 12.53 12.58
52.83 56.08 52.46 53.39 54.05 52.13 62.24 52.56 44.10 52.51 57.07 56.63 59.87 53.49 52.23 50.75 54.00 55.15 60.70 55.48 54.49 50.80 55.42 58.66 61.46 52.24 54.32
178.12 188.73 178.51 174.39 184.55 173.24 211.41 169.60 136.77 174.28 195.32 191.86 198.55 179.82 175.10 164.83 181.20 175.37 217.72 183.06 175.60 162.19 182.51 198.17 212.24 166.97 171.65
424.36 515.92 404.86 422.54 437.48 425.35 583.03 449.40 308.85 408.70 468.73 470.77 518.60 450.17 410.95 411.60 474.12 480.04 563.97 497.02 474.01 407.10 464.78 534.87 547.28 385.41 445.50
0.22 1.04 0.16 0.27 0.28 0.33 1.10 0.27 0.01 0.22 0.16 0.27 0.55 0.38 0.22 0.11 1.21 0.17 0.94 0.49 0.66 0.16 0.71 0.39 0.83 0.05 0.32
0.28 0.93 0.22 0.27 0.49 0.60 1.70 0.94 0.01 0.16 0.66 0.39 0.60 0.60 0.11 1.48 5.11 0.88 0.88 0.44 0.33 0.72 0.17 0.28 1.54 0.11 1.21
N/S no solution.
phone labor requirements and the 5/40-4 policy. For each of the labor requirement patterns, the workforce sizes for each of the 966 subsets of
starting times were ranked in ascending order and graphically depicted in Fig. 2. This ®gure shows that moving beyond the top 5±10% of starting-
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467
Table 3 Percentage of starting-time subsets that provide the optimal workforce size Demand pattern
3 Starting times
4 Starting times
5 Starting times
24 Starting times
5/40
5/40
5/40
5/40
4/40
4/40
4/40
4/40
TEL_1 TEL_2 TEL_3 TEL_4 TEL_5 TEL_6
12.500 12.500 12.500 12.500 37.500 37.500
0.446 1.339 8.482 8.482 0.446 0.446
0.207 0.207 0.311 2.174 0.207 0.932
0.105 0.421 0.500 0.026 0.026 0.026
0.026 0.018 0.525 0.009 0.035 1.970
0.750 1.602 1.349 0.012 0.110 0.004
N/A N/A N/A N/A N/A N/A
N/A N/A N/A N/A N/A N/A
TOLL_1 TOLL_2 TOLL_3 TOLL_4 TOLL_5 TOLL_6 TOLL_7 TOLL_8 TOLL_9 TOLL_10 TOLL_11 TOLL_12
12.500 25.000 37.500 12.500 12.500 25.000 25.000 25.000 12.500 37.500 37.500 37.500
5.357 4.464 3.125 3.125 3.125 4.464 1.786 4.464 1.339 6.250 8.482 6.250
1.760 0.311 0.207 0.311 0.414 1.346 1.242 1.449 0.207 1.035 1.449 0.414
0.421 0.527 0.316 0.685 0.843 1.553 0.105 0.369 3.607 0.869 0.026 0.948
0.158 0.018 0.018 0.193 0.018 0.053 0.035 0.053 0.639 0.044 0.114 0.026
0.004 1.491 1.018 2.178 0.016 0.189 0.454 1.267 0.004 2.438 0.225 2.525
N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
AIR_1 AIR_2 AIR_3 AIR_4 AIR_5 AIR_6 AIR_7 AIR_8 AIR_9 AIR_10 AIR_11 AIR_12 AIR_13 AIR_14 AIR_15 AIR_16 AIR_17 AIR_18 AIR_19 AIR_20 AIR_21 AIR_22 AIR_23 AIR_24 AIR_25 AIR_26 AIR_27
50.000 12.500 50.000 12.500 37.500 12.500 12.500 12.500 50.000 12.500 12.500 12.500 12.500 12.500 12.500 25.000 37.500 12.500 12.500 12.500 75.000 25.000 12.500 12.500 12.500 25.000 12.500
8.929 3.125 1.339 2.679 0.446 22.321 3.125 16.964 73.214 44.643 13.393 24.107 3.125 10.714 1.339 1.786 3.125 3.125 0.893 8.929 8.929 26.786 6.250 2.679 0.446 20.536 8.929
0.207 0.311 0.414 1.760 0.621 1.863 0.104 0.621 63.458 11.698 0.621 0.104 0.311 3.313 1.035 0.207 0.414 0.518 0.104 0.414 3.416 1.035 0.311 0.414 0.932 4.037 0.518
0.527 0.184 1.264 0.158 1.738 1.422 0.211 2.001 81.043 50.105 24.697 27.278 0.211 1.211 1.869 8.926 0.211 6.793 0.500 0.527 12.507 8.004 0.369 0.263 0.053 38.231 1.632
0.613 0.009 3.746 6.434 1.234 2.871 0.184 0.236 71.183 14.163 1.322 0.184 0.009 0.114 1.120 0.814 0.053 0.044 0.009 0.053 7.931 2.215 0.341 0.683 0.018 7.668 0.079
0.355 0.552 3.196 0.537 5.114 4.321 0.110 5.027 86.261 59.028 36.521 34.916 1.125 4.013 4.968 17.436 0.710 12.567 0.075 1.480 18.316 17.586 2.143 0.103 0.911 52.438 5.761
N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
time subsets results in a substantial deterioration of solution quality. These ®ndings suggest that the answer to the third question in Section 3.1 is that a
poor (or even mediocre) selection of starting times can lead to a substantial penalty in terms of workforce size.
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4. A case study ± scheduling customer service representatives in call-center operations in the System Support Center, US and Canada group, Radio Network Solutions Group, Land Mobile Products Sector (LMPS), Motorola 4.1. An overview of the call-center scheduling environment
Fig. 2. This ®gure shows the workforce size response surface subsets of starting times for the 5/40-4 policy for the TEL_1± TEL_6 labor requirements.
3.4. Summary of major ®ndings of the experimental study One important ®nding of this study was that, for many of the test problems, restricting the number of starting times to four or ®ve did not result in a substantial increase in workforce size. This ®nding is good news for service operations managers because it suggests that minimum cost labor schedules may be achieved with relatively few shift starting times. As observed in previous studies [2,8,9,23,25], a limited number of starting times is desirable because it makes schedules easier to administer. Although service managers may take solace in the ®nding that a small number of starting times is often sucient to yield low-cost solutions, they must take caution regarding a second ®nding of this study. Speci®cally, for the real-world labor requirements patterns considered in this study, only a very small percentage of the starting-time subsets associated with S 4 or S 5 were typically capable of providing the minimum workforce size. This implies that service managers must pay close attention to ensure that an eective subset of starting times is selected. This attention is especially important in light of the third major ®nding of the study that workforce sizes were quite excessive for some subsets. For any given number of starting times, S, the dierence between a good and poor subset of starting times was often considerable. In fact, many good subsets of size S 1 ; S 1 < S 2 , provided smaller workforce sizes than poor subsets of size S2 .
MotorolaÕs LMPS Radio Network Solutions Group Call Center receives and processes inbound calls for customer support. This call center operates continuously and receives several varieties of incoming calls, many of which generate requirements for outbound calls. The call center operates with 30-minute planning intervals and is staed almost exclusively by full-time employees working ®ve consecutive days per week. Shifts are 8.5 hours in length with a 1-hour meal break. Meal-break ¯exibility is provided by allowing breaks to occur either 4, 4.5, or 5 hours into the shift. In addition, the scheduling problem allows for the speci®cation of ``project time'', in which customer service representatives perform other duties and are not available to satisfy service requirements. This project time, which can range from 0 to 60 minutes in length, may occur at the beginning or end of workshift, or it can be attached to the meal break. For example, consider an 8.5-hour shift from 7 am to 3:30 pm that contains a meal break from 11:30 am to 12:30 pm. The project time for this shift may occur from 7:00±8:00 am, 10:30±11:30 am, 12:30±1:30 pm, or 2:30±3:30 pm. Motorola LMPS management has moved towards consolidating additional operations (including those in South America) into the call center. Moreover, as the installed base of Radio Network systems has increased, the call center has experienced signi®cant increases in inbound (and outbound) call volume. Motorola LMPS wanted a system that could be used to assist in sta planning for the present call-center expansion, as well as to provide for future expansions in this and other Motorola call centers. The scheduling system we developed for the Radio Network Solutions Group call center was designed to provide decision support for this problem.
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Speci®cally, we developed a spreadsheet-based decision support system to aid Motorola management in making call-center stang and scheduling decisions. The primary requirements for building this system were that it be developed using Microsoft Excel and VBA macros, and that it provide schedules in a rapid and ecient manner on a microcomputer. The system currently consists of three modules provided in the form of VBA macros: (1) a labor-requirement-generation module; (2) a sta-planning module; and (3) a stascheduling module. The labor-requirement-generation module allows the user to provide information regarding arrival rates of incoming calls, service rates, and desired service levels. The VBA macro uses this information in conjunction with an M/M/s queuing model to generate labor requirements for each 30-minute planning period of the week (this approach is consistent with those previously reported in the literature [1,9,17]). These labor requirements subsequently serve as inputs to the sta-planning and sta-scheduling modules, which are of particular relevance to this paper. The sta-planning and sta-scheduling modules are described in Sections 4.3 and 4.4, respectively. Prior to their description in Section 4.2, we provide a linkage between the experimental analysis in Section 3 and the problem faced by the Motorola call centers.
469
(24 starting times 7 days-o patterns) when all 24 starting times were considered as candidates. For the LMPS call-center problem, the number of Xij variables is 4032 (48 starting times p 4 possible project-time placements b 3 possible meal-break placements 7 days-o patterns). The number of Yi variables in the call-center problem is 48, compared to 24 in the experimental analysis problem. The total number of constraints (2)±(4) in the call-center problem is 385 336 48 1, compared to 193 168 24 1 in the experimental analysis problem. The size and complexity of the LMPS callcenter scheduling problem, in conjunction with managementÕs desire to have a spreadsheet-based system that runs rapidly on a microcomputer, absolutely precluded the use of optimal solution methods (such as the cutting plane) in both the sta-planning and sta-scheduling modules. For this reason, it was necessary to resort to heuristic methods for both of these modules. In order to give LMPS management the opportunity to answer the types of questions we considered in the experimental analysis in Section 3, we built the spreadsheet modules with considerable ¯exibility. That is, even though optimal solutions could not be guaranteed, LMPS management could use the model to estimate the eects of starting-time (and meal-break and project-time) decisions on workforce size.
4.2. Linkage of experimental analysis to the design of the scheduling system
4.3. The sta-planning module
In at least two respects, the problem faced by MotorolaÕs LMPS call centers is more complex than the scheduling environment considered for the experimental analysis in Section 3. First, in addition to starting-time policies, LMPS management wanted to be able to simultaneously examine meal-break and project-time policy decisions. Second, the call centers use 30-minute planning intervals (I 48) instead of hourly intervals (I 24). Together, these two conditions result in an ILP formulation, (1)±(6), that is substantially larger than the one considered in the experimental analysis. For example, the number of Xij variables in the experimental analysis in Section 3 was 168
The sta-planning model was designed to minimize the number of workers necessary to satisfy the labor requirement in each 30-minute planning interval. The inputs for this heuristic included the labor-requirement distribution (developed by the labor-requirement generator described in the previous subsection), the project time, the number of replications of the heuristic to be performed, and the set of allowable starting times. A screenshot of the EXCEL spreadsheet associated with these inputs is shown in Fig. 3. Management desired a system that could rapidly evaluate a variety of starting-time subsets, yet contain sucient ¯exibility to prohibit the use of
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Fig. 3. Screenshot of the sta-planning module input screen.
certain starting times. Further, to increase the ¯exibility of the system, we wanted to provide management with the opportunity to select the precise location of each feasible starting time. Therefore, the starting-time input was designed so that the user could turn starting times on and o. A value of Ô1Õ in Fig. 3 indicates that the starting time may be used, whereas a value of Ô0Õ precludes the use of the starting time. The VBA macro uses this input in conjunction with a greedy construction heuristic to build a schedule. This heuristic begins with Xij 0, for all i and j and adds employees, one at a time, until constraint (2) is satis®ed. At each iteration, an employee is assigned to the tour associated with shift starting time e and days-o pattern f, such that def maxi;j dij , where
dij
I X L X k1 l1
8i and j:
0 aijkl @ max rkl
I X J X
!2 1 aijkl Xij ; 0A
i1 j1
7
In other words, at each iteration, an employee is added to the tour that hits the largest sum-ofsquared remaining labor requirements (ties are broken arbitrarily). Several studies have suggested that construction heuristics of this type are often eective at providing near-optimal workforce sizes [6,8,20,21]. The user can specify multiple replications of the heuristic in order to guard against the potential for an unusually poor individual replicate. Further, because the solutions are constructed in a matter of seconds, call-center management can use the sta-planning module to
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test a wide variety of starting-time con®gurations in a short period of time. As was the case in the experimental study, the parsimonious use of starting times is desirable. If LMPS management is interested in the ®rst research question considered in the experimental analysis (i.e., what are the workforce size implications of using S < I starting times relative to I starting times?), then we suggested they begin by running the sta-planning module with all I 48 (or nearly all) starting times to serve as a benchmark. Next, dierent con®gurations of S starting times can be tested to see how they compare to the benchmark. During this process, answers to the second and third research questions in the experimental analysis are being developed (albeit in a heuristic manner). Although LMPS management certainly is not guaranteed to ®nd the best subset of S starting times, they can at the very least guard
471
against the selection of a very poor subset of starting times. The output of the sta-planning model includes the workforce size, as well as an indication that the ®nal solution is feasible. Speci®cally, Fig. 3 displays a workforce size of 37 and the ``ERROR?'' box displays the word ``NO'', indicating a feasible solution. A schedule summary is also generated as depicted in Fig. 4. This summary displays detailed information regarding the properties of schedules to which employees were assigned. For example, row 14 of the spreadsheet screenshot in Fig. 4 reveals that one employee was assigned to a schedule associated with shifts between 12:30 and 9:00 am from Tuesday±Saturday (i.e., Sunday and Monday o) and a meal break from 4:00 to 5:00 am. Since the project time was zero for this execution of the module, all of the ``Actual Start'' and ``Actual Finish'' times are the same as the ``Starting'' and
Fig. 4. Screenshot of the schedule summary report generated by the sta-planning module.
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``Finishing'' times, respectively. However, if the project time (p) is greater than zero and is assigned to the beginning (end) of the shift, then the Actual Start (Actual Finish) time will be p periods later (earlier) then the Starting (Finishing) time. Thus, the Actual Start and Actual Finish times re¯ect the actual availability of the employees to meet callcenter requirements. When the project time is allocated to a break, the Actual Start and Actual Finish times are the same as Starting and Finishing, respectively, and the dierence between ``Break Start'' and ``Break Finish'' expands by p periods. 4.4. The sta-scheduling module Although the sta-planning module is helpful for determining the eect of labor requirement, project time, and starting-time changes on
workforce size, it is not directly bene®cial for actually constructing employee schedules because the current workforce size may be dierent than the solution provided the model. Therefore, the sta-scheduling module was developed to construct a schedule for a prescribed number of employees (i.e., the current workforce size). This module uses a greedy heuristic comparable to that of the sta-planning module. However, due to the fact that the workforce size is ®xed, it may not be possible to construct a schedule that will satisfy the labor requirement for each period. Therefore, the objective criterion associated with the sta-scheduling module is to minimize total sum-of-squared understang. Fig. 5 depicts a screenshot of the input screen associated with the sta-scheduling module. The module requires input regarding the labor requirements, project time, allowable starting times, and the current workforce size.
Fig. 5. Screenshot of the sta-scheduling module input screen.
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Fig. 6. Screenshot of the labor coverage report generated by the sta-scheduling module.
The sta-scheduling module generates a schedule summary comparable to the one produced by the sta-planning module (see Fig. 4). In addition, sta-scheduling model generates output corresponding to actual labor coverage in each planning interval of the week. Speci®cally, Fig. 6 reveals that information is provided with respect to the number of customer service agents working in each 30-minute interval of each day, as well as the shortage or surplus of agents in each interval. The shortages, which are associated with negative numbers, are accentuated in ``red'', whereas the intervals with exact or surplus coverage are in ``blue''. This enables call-center management to rapidly visualize the coverage associated with the current set of project time and starting-time policies. Like the sta-planning module, the stascheduling module is extremely fast and a wide variety of policies can be tested in a very short period of time.
5. Summary The large computational study presented herein revealed that a small number of starting times was often sucient to provide workforce sizes that were comparable to those obtained when all possible starting times were considered. Although it is acknowledged that one could easily ``contrive'' labor-requirement patterns that would necessitate exactly 3, 4, 5, 6, 7, etc. starting times, it should be observed that this study used three sets of actual (real-world) labor-requirement patterns that were not so contrived. The study also revealed that only a few of the many possible subsets of starting times were capable of providing the minimum workforce size and that many subsets resulted in rather poor scheduling solutions. The case study associated with the scheduling of customer service representatives at Motorola service call centers demonstrated that starting-time
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decisions often must be examined in conjunction with other scheduling policies (e.g., break policies, days-o policies, project-time policies, etc.). Under such conditions, it was not practical to utilize a model that evaluated all possible subsets of starting times. Instead, we developed a ¯exible decision support system that enables call-center management to rapidly evaluate starting time and other policy decisions. This system contains modules that enable it to be used for either planning or operational purposes.
Acknowledgements We would like to extend our gratitude to Mark Hurlbert, Director of System Support Center; John Williams, Call Center Operations Manager; and Keith Toborg, all of the United States and Canada Group, Radio Network Solutions Group, Land Mobile Products Sector, Motorola for their invaluable assistance in completing this project.
References [1] B.H. Andrews, B.L. Parsons, L.L. Bean chooses a telephone agent scheduling system, Interfaces 19 (1989) 1±9. [2] N. Beaumont, Scheduling sta using mixed integer programming, European Journal of Operational Research 98 (1997) 473±484. [3] S.E. Bechtold, M.J. Brusco, Working set generation methods for labor tour scheduling, European Journal of Operational Research 74 (1994) 540±551. [4] M. van Biema, B. Greenwald, Managing our way to higher service-sector productivity, Harvard Business Review 75 (1997) 87±95. [5] M.J. Brusco, Solving personnel tour scheduling problems using the dual all-integer cutting plane, IIE Transactions 30 (1998) 835±844. [6] M.J. Brusco, L.W. Jacobs, A simulated annealing approach to the cyclic sta-scheduling problem, Naval Research Logistics 40 (1993) 69±84. [7] M.J. Brusco, L.W. Jacobs, Cost analysis of alternative formulations for personnel scheduling in continuously operating organizations, European Journal of Operational Research 86 (1995) 249±261. [8] M.J. Brusco, L.W. Jacobs, Personnel tour scheduling when starting time restrictions are present, Management Science 44 (1998) 534±547.
[9] M.J. Brusco, L.W. Jacobs, R.J. Bongiorno, D.V. Lyons, B. Tang, Improving personnel scheduling at airline stations, Operations Research 43 (1995) 741±751. [10] M.J. Brusco, T.R. Johns, Stang a multi-skilled workforce with varying levels of productivity: An analysis of crosstraining policies, Decision Sciences 29 (1998) 499±515. [11] M. Conroy, S. Caldwell, R. Buehrer, W. Wolfe, Flextime revisited: The need for a resurgence of ¯extime, Journal of Compensation and Bene®ts 13 (1997) 36±39. [12] F.F. Easton, D.F. Rossin, Overtime schedules for full-time service workers, Omega 25 (1997) 285±299. [13] F.F. Easton, D.F. Rossin, W.S. Borders, Analysis of alternative scheduling policies for hospital nurses, Production and Operations Management 1 (1992) 159±174. [14] R.E. Gomory, An algorithm for integer solutions to linear programs, in: R.L. Graves, P. Wolfe (Eds.), Recent Advances in Mathematical Programming, McGraw-Hill, New York, 1963, pp. 269±302. [15] L.W. Jacobs, S.E. Bechtold, Labor utilization eects of labor scheduling ¯exibility alternatives in a tour scheduling environment, Decision Sciences 24 (1993) 148±166. [16] L.W. Jacobs, M.J. Brusco, Overlapping start-time bands in implicit tour scheduling, Management Science 42 (1996) 1247±1259. [17] P.J. Kolesar, K.L. Rider, T.B. Crabill, W.E. Walker, A queuing-linear programming approach to scheduling police patrol cars, Operations Research 23 (1975) 1045± 1062. [18] V.A. Mabert, M.J. Showalter, Measuring the impact of part-time workers in service organizations, Journal of Operations Management 9 (1990) 209±229. [19] A.S. McCampbell, Bene®ts achieved through alternative work schedules, Human Resource Planning 19 (1996) 30± 37. [20] L.F. McGinnis, W.D. Culver, R.H. Deane, One- and twophase heuristics for workforce scheduling, Computers and Industrial Engineering 2 (1978) 7±15. [21] J.G. Morris, M.J. Showalter, Simple approaches to shift, days-o, and tour scheduling, Management Science 29 (1983) 942±950. [22] E.S. Rosenbloom, N.F. Goertzen, Cyclic nurse scheduling, European Journal of Operational Research 31 (1987) 19± 23. [23] S. Schindler, T. Semmel, Station stang at Pan American World Airways, Interfaces 23 (1993) 91±106. [24] S.P. Siferd, W.C. Benton, Workforce stang and scheduling: Hospital nursing speci®c models, European Journal of Operational Research 60 (1992) 233±246. [25] P.E. Taylor, S.J. Huxley, A break from tradition for the San Francisco Police: Patrol ocer scheduling using an optimization-based decision support system, Interfaces 19 (1989) 4±24. [26] G.M. Thompson, Shift scheduling in services when employees have limited availability: an LP approach, Journal of Operations Management 9 (1990) 352±370. [27] G.M. Thompson, Optimal scheduling of shifts and breaks using employees having limited time availability, Interna-
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[29] D.M. Warner, Scheduling nursing personnel according to nursing preference: A mathematical programming approach, Operations Research 24 (1976) 842±856.
www.elsevier.com/locate/dsw
Case Study
Starting-time decisions in labor tour scheduling: An experimental analysis and case study Michael J. Brusco a
a,*
, Larry W. Jacobs
b,1
Information and Management Sciences Department, College of Business, The Florida State University, Tallahassee, FL 32306-1110, USA b Operations Management and Information Systems Department, College of Business, Northern Illinois University, DeKalb, IL 60115, USA Received 17 February 1999; accepted 18 April 2000
Abstract Many service organizations limit the number of daily planning periods in which employees may begin their shifts to a ®xed number, S. Even for relatively small values of S, which are quite common in practice, there may be hundreds, thousands or millions of possible subsets of starting times. This paper presents the results of a large experimental study that revealed that, in many instances, only a very small portion of starting-time subsets was capable of providing the minimum workforce size. The importance of eective starting-time selection is further supported by a case study that describes a spreadsheet-based program designed for scheduling customer service representatives in the System Support Center, United States and Canada Group, Radio Network Solutions Group, Land Mobile Products Sector, Motorola. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Scheduling; Integer programming; Heuristics; Call centers
1. Introduction Two critical challenges facing service operations managers are (1) ®nding ways to improve service productivity [4], and (2) facilitating the balance of work and family responsibilities for * Corresponding author. Tel.: +1-850-644-6512; fax: +1-850644-8225. E-mail addresses: [email protected] (M.J. Brusco), [email protected] (L.W. Jacobs). 1 Tel.: +815-753-6165; fax: +815-753-3300.
their employees [11,19]. One of the most eective tools available to the operations manager for meeting these challenges is labor scheduling. Since labor schedules determine the number of employees present during dierent hours of the day and dierent days of the week, they have important implications for operating cost and customer service. If the number of employees scheduled is not sucient to satisfy customer demand, then the operating system may choose to let customer service decline (e.g., an increase in customer waiting times), or it may increase labor supply via some
0377-2217/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 0 ) 0 0 1 3 5 - 1
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other means (e.g., the use of overtime). In either case, costs of a labor shortage are encountered. Similarly, if the number of employees scheduled creates a supply of labor that exceeds customer demand, then costs of idle labor are incurred. In the light of these issues, it is not surprising that there has been a substantial research eort devoted to the development of optimal and heuristic procedures for generating ecient labor schedules [2,3,5,6,8,21,22,26±29]. In addition to methodological development, labor-scheduling research has also focused on the evaluation of stang and scheduling policies. For example, Easton et al. [13] evaluated a variety of popular nurse-scheduling policies in terms of their relative scheduling costs. Studies have also been conducted to measure the eects of part-time employees [18], overtime policies [12], the number of workperiods in a daily workshift and the number of workdays in a workweek [15], the overlapping of shifts from one workday to the next [7], overlapping bands on daily starting times for employees [16], cross-utilization of employees [10], and limited employee availability [26,27]. The selection of daily starting times for employee workshifts is another type of scheduling policy and is the focus of this paper. The selection of daily starting times may be important even when shifts do not overlap. For example, a hospital nursing unit might experience higher productivity when using (9 am to 5 pm, 5 pm to 1 am, and 1 am to 9 am) as their three non-overlapping shifts, as opposed to (7 am to 3 pm, 3 pm to 11 pm, and 11 pm to 7 am). Starting-time decisions are even more important for the large number of service organizations that are now deploying multiple overlapping shifts. This is due to the fact that an increase in the number of allowable starting times generally increases the potential number of starting-time subsets. For a 24-hour day with hourly planning intervals, 8-hour shifts, and ®ve shift starting times, there are 245 42,504 dierent possible subsets of shift starting times. Although some of these subsets are infeasible because they fail to cover all hours of the planning horizon, there remain 25,344 feasible subsets. In this context, the selection of daily shift starting times consists of two interrelated deci-
sions: (a) the number, S, of shift starting times permitted during an operating day, and (b) the determination of precisely which of the S planning periods of the day will be allocated as starting times. There are obvious trade-os associated with the ®rst of these decisions. By increasing the value of S, the service manager might be able to cover all labor requirements with fewer employees. More starting times provide greater ¯exibility for covering labor requirements, which may enable less overstang and, therefore, greater labor productivity. Improved labor productivity associated with more starting times might also provide bene®ts from a human-resources standpoint. A better match between demand and labor capacity should promote fewer shortages of employees and a more equitable distribution of workload, which may in turn result in less absenteeism and turnover [13]. There are also disadvantages associated with increasing the number of starting times. Most of these disadvantages stem from an administrative burden associated with a larger number of starting times. Increasing the number of shift starting times results in more time periods during the day at which employees will be beginning shifts, beginning breaks, ending breaks, and ending shifts. With employees coming and going at many different times during the day, it may be dicult to monitor employee punctuality and there can be a serious loss of labor productivity. In addition, many service organizations may use the beginning (or end) of employee shifts for brie®ng (de-briefing) sessions [8,9]. The scheduling of these sessions becomes increasingly more dicult as the number of starting times increases, thus limiting their effectiveness. Finally, a large number of starting times can result in a small number of assignments (perhaps only one or two employees) to particular starting times and this may not be conducive to a team work environment. In short, the service operations manager must be able to evaluate the relative trade-os associated with increasing the number of starting times. There are no de®nitive rules assessing such tradeos and the appropriate number of starting times may vary from one service operating system to the next. However, there does appear to be a
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trend in many service industries to move away from the traditional three starting times. For example, a survey conducted by Siferd and Benton [24] revealed that nearly 30% of hospital nursing units utilized four or ®ve shift starting times per day. Similarly, Taylor and Huxley [25] described an application for the San Francisco Police Department in which increasing the number of starting times from three to four or ®ve was shown to enhance labor productivity. Applications in the airline and other service industries have used ®ve or more shift starting times [2,9,23]. Despite the prevalence of multiple overlapping shifts, little is known about the eect of startingtime decisions for practical problems. An interesting research question is, ``which of the startingtime subsets, and how many of them, will provide the minimum workforce stang cost?'' Related questions might focus on the results associated with mediocre or poor choices of starting-time subsets. To date, there has been no comprehensive evaluation of the impact of starting-time selections on workforce stang costs. This paper is the ®rst to provide such an evaluation via a large computational study using sets of labor requirements from three dierent service operating environments. The study was conducted within the context of a continuous tour-scheduling problem environment [6,20,21]. This problem, which is concerned with the assignment of employees to both daily workshifts and weekly days-o patterns, has formed the basis for practical scheduling applications in hospital nursing units [22,29], telecommunication centers [1,20,28], airport stations [9,23], and police patrols [17,25]. The importance of starting-time decisions in labor tour scheduling is further supported by a case study focusing on the development of a spreadsheet program designed for scheduling customer service representatives at Motorola call centers. This system is currently being used at Motorola to investigate labor stang and scheduling policies. This practical scheduling problem is far more complex than the one considered in the computational study, yet we observed results that are surprisingly consistent with those obtained in the computational study.
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Section 2 presents the integer linear programming (ILP) formulation for the continuous tourscheduling problem with restrictions on the number of shift starting times and also describes previous solution approaches to such problems. Section 3 provides a description of the experimental study used to assess the eects of startingtime decisions. Section 4 describes how the results of the experimental study were applied to the development of a labor scheduling system for the Motorola call centers. A brief summary is presented in Section 5. 2. An ILP formulation for the limited starting-time tour-scheduling problem The labor tour-scheduling formulation presented below is based on a planning horizon of one week and is comparable with formulations used in several previous studies [3,6,8,12,13,18,21]. For the sake of parsimony and consistency of presentation, the scheduling environment is assumed to consist of full-time employees and the single objective is to minimize the total workforce size (this assumption holds for both the experimental and case studies). A speci®c tour is uniquely de®ned by a daily shift starting time (which is consistent for each workday of the tour) and the speci®c days worked during the week. The ILP formulation is as follows: Minimize:
Z
I X J X
Xij
1
i1 j1
subject to I X J X
aijkl Xij P rkl
8k 1; . . . ; I; 8l 1; . . . ; L;
i1 j1
2 J X Xij
mYi 6 0
8i 1; . . . ; I;
3
j1 I X Yi 6 S; i1
4
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Xij P 0 and integer
8i 1; . . . ; I 8j 1; . . . J ; 5
Yi 2 f0; 1g
8i 1; . . . I;
6
where I is the number of planning periods in the operating day, i 1; . . . ; I; J the number of allowable days-o patterns, j 1; . . . J ; L the number of operating days in the weekly planning horizon, l 1; . . . ; L; Xij the number of employees assigned to a tour on days-o pattern j that begins in planning period i; aijkl 1, if planning period k of day l is a workperiod in a tour that begins in planning period i and is associated with days-o pattern j and 0 otherwise; rkl the number of employees required in planning period k of day l; Yi 1, if any employee begins a tour in planning period i and 0 otherwise; m a large positive integer; and S is the maximum allowable number of starting times (as de®ned previously). The objective function (1) of the ILP model, which is consistent with the work of Morris and Showalter [21], Bechtold and Brusco [3], and Brusco and Jacobs [6], is to minimize total workforce size. Constraints (2) ensure that a sucient number of employees are available to satisfy labor requirements in each planning period of the day and on each day of the week. Constraint (3) guarantees that a starting time is recognized as ``used'' if any employee is assigned to a tour associated with that starting time. If the integer programming model is solved directly, then it should be observed that is possible to have one Pit J or more Yi 1 even if j1 Xij 0. However, this does not present a problem from a pragmatic standpoint. Constraint (4) limits the total number of starting times used to be no more than S. Constraints (5) and (6) place the appropriate nonnegativity and integer restrictions on the Xij and Yi variables, respectively. 3. An experimental study 3.1. Problem environment Given a ®xed number of candidate starting times (I) and a ®xed number of starting times to
select from these candidates (S), there are often many subsets of size S to choose from. The purpose of this study was to evaluate, for ``realworld'' sets of labor requirements, the eect of starting-time selection. Speci®cally, research questions addressed in this study include: (1) can a small value of S provide optimal workforce sizes comparable to those obtained when all I starting times are considered? (2) if ZS denotes the optimal workforce size given S starting times, then what proportion of feasible starting-time subsets (of size S) would provide an objective function value equal to ZS ? and (3) just how detrimental is a relatively poor selection of starting times? The successful implementation of a study designed to answer these questions required the solution of a substantial number of ILP formulations. Therefore, it was necessary to select a labor tour-scheduling environment for which integer-optimal solutions could be eciently obtained. The continuously operating scheduling environment used to conduct this study consisted of a one-week planning horizon divided into hourly planning intervals. Full-time employees working ®xed shifts on consecutive days were used to sta the operation and meal/rest breaks were not explicitly considered. Six unique operating policies were investigated by considering all combinations of two workweek alternatives and three startingtime limitations. The workweek alternatives were a ®ve-day, 8 hours per day (5/40) workweek, and a four-day, 10 hours per day (4/40) workweek. The starting-time limitations were associated with 3, 4, or 5 allowable starting times. Hereafter, 5/40-S and 4/40-S will be used to identify operating policies characterized by the respective workweek alternative with S starting times. Each operating policy was tested across three sets of labor requirement patterns. The ®rst set of labor requirement patterns, consisting of six problems from the telephone industry (TEL_1± TEL_6), was originally reported by [20]. These patterns consisted of relatively large labor requirements for individual planning intervals. The magnitude and pattern of the labor requirements diered across certain days of the week. The second set of 12 labor requirement patterns associated with tollway operations (TOLL_1±TOLL_12)
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was taken from Jacobs and BruscoÕs [16] study. Like the McGinnis et al. [20] patterns, the tollway labor requirement patterns diered across days of the week, however, the magnitude of demand in individual planning intervals was much smaller. The third set of labor requirement patterns, consisting of 27 sets of airport station requirements (AIR_1±AIR_27), was based on those used by Brusco and Jacobs [8]. The magnitude of demand for these patterns varied widely across airport stations, but generally contained substantial variability from one planning interval to the next. Unlike the telephone and tollway requirement patterns, each airport station pattern generally exhibited the same shape and magnitude across dierent days of the week. 3.2. Solution method In their application for the San Francisco Police Department, Taylor and Huxley [25] compared the results associated with three, four, and ®ve starting times, but did not describe how the starting times were selected. Schindler and Semmel [23] and Brusco et al. [9] described mathematical programming-based methods for selecting starting times within the context of airport station stang and scheduling applications, and Brusco and Jacobs [8] recently provided an in-depth analysis of column generation-based heuristics for startingtime selection. Unfortunately, none of the methods associated with the aforementioned studies provided guaranteed optimal solutions to the ILP formulation of the limited starting-time tourscheduling problem, nor was there any indication of what proportion of all possible starting-time subsets would lead to optimal solutions. To complete this current study, it was deemed necessary to solve an ILP formulation for all feasible subsets of starting times, for all 45 testproblems. Although there were 24 possible subsets of S starting times, many of these were infeasible because they were not capable of satisfying requirements in all periods of the planning horizon. Nevertheless, the total number of feasible subsets could be quite large. Speci®cally, there were 8, 224, 966, 3798, 11424, and 25344 feasible starting-time
463
subsets for the 5/40-3, 4/40-3, 5/40-4, 4/40-4, 5/405, and 4/40-5 operating policies, respectively. Thus, this experimental study required the solution of 8 224 966 3798 11424 25344 45 1,879,380 integer programs. Fortunately, the solution of these problems was facilitated by the deployment of the dual, all-integer cutting plane [14], which has proven very eective for such problems [3,5]. A computer program was written in Fortran PowerStation (Ver. 1.0) to generate the ILP formulation for all feasible subsets of starting times and subsequently solve them using the cutting plane. Fig. 1 displays the logic of this program. Data were collected regarding the minimum workforce size, the number of subsets of starting times that yielded the minimum workforce size, and the total CPU time to solve all problems for the workforce policy. For benchmarking purposes, the cutting plane was also used to solve, for each labor requirement pattern, an integer linear program associated with all 24 starting times. 3.3. Experimental results The results of the experimental study are depicted in Tables 1±3. Table 1 presents the optimal workforce size under each workforce policy for each labor requirement pattern. Table 1 clearly shows that increasing the number of starting times from three to four, and from four to ®ve, often enables a signi®cant reduction in workforce size. However, the increase from three to four starting times provided greater bene®t than the increase from four to ®ve. Speci®cally, the 5/40-3 (4/40-3) policy was able to match the optimal workforce size provided by the 5/40-4 (4/40-4) policy for only 3 (8) of the 45 test problems, whereas 5/40-4 (4/404) matched 5/40-5 (4/40-5) for 15 (35) test problems. It is also interesting to note that, for the 5/40 workweek, ®ve (four) starting times yielded the same optimal workforce size as 24 starting times for 20 (13) of the 45 test problems. The results were even more impressive for the 4/40 workweek for which ®ve (four) starting times yielded the same optimal workforce size as 24 starting times for 42 (34) of the 45 test problems. These ®ndings suggest that the answer to the ®rst question posed
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Fig. 1. Flowchart for starting-time evaluation program.
in Section 3.1 is ``yes'', very ecient scheduling solutions may often be achieved with a small number of starting times. Table 2 reports, for each demand pattern and each workforce policy, the total CPU time required to solve integer programs associated with all feasible subsets of starting times. These results indicate that the dual all-integer cutting plane provided very ecient solutions for the integer linear programs. For the 5/40-5 policy, the generation and solution of the integer programs associated with the 25,344 feasible starting-time subsets generally required less than 500 CPU seconds (an average of approximately 0.02 CPU seconds per problem). Table 3 presents, for each operating policy, the percentage of starting-time subsets that were capable of providing the minimum workforce size yielded across all subsets. For the 5/40-3 policy, there are only eight feasible subsets of starting times. If one of these starting-time subsets yielded
a workforce size that was less than the workforce sizes associated with the other seven subsets, then the reported percentage was 12.50. If two subsets provided the smallest workforce size, then the reported percentage was 25.0, and so on. For other operating policies, there are many more feasible subsets and thus there is clearly the potential for a much smaller percentage of subsets to provide the minimum workforce size. For example, consider the 5/40-5 policy for the TEL_4 pattern. In this instance, only one of the 11,424 subsets of starting times provided the minimum workforce size of 168, resulting in a reported percentage of 0.009 100 1=11424. Similar ®ndings were observed for many of the test problems. In fact, when four or ®ve starting times were used, no more than 5% of the starting-time subsets were ever capable of providing the minimum workforce size for the telephone or tollway test problems. These results suggest that the answer to the second question posed in Section 3.1 is
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465
Table 1 Optimal workforce sizes Demand pattern TEL_1 TEL_2 TEL_3 TEL_4 TEL_5 TEL_6 TOLL_1 TOLL_2 TOLL_3 TOLL_4 TOLL_5 TOLL_6 TOLL_7 TOLL_8 TOLL_9 TOLL_10 TOLL_11 TOLL_12 AIR_1 AIR_2 AIR_3 AIR_4 AIR_5 AIR_6 AIR_7 AIR_8 AIR_9 AIR_10 AIR_11 AIR_12 AIR_13 AIR_14 AIR_15 AIR_16 AIR_17 AIR_18 AIR_19 AIR_20 AIR_21 AIR_22 AIR_23 AIR_24 AIR_25 AIR_26 AIR_27 a
3 Starting times
4 Starting times
5 Starting times
24 Starting times
5/40
4/40
5/40
4/40
5/40
4/40
5/40
4/40
221 221 187 173 246 198
240 241 236 208 235 221
208 208 185 173 231 197
235 236 214 202 218 206
198 198 185 168 213 197
235 236 214 176 218 195
187 187 185 165 212 194
235 236 214 169a 218 177
35 33 35 34 21 21 21 21 17 17 17 17
38 36 36 36 22 22 22 22 17 17 18 17
34 31 32 32 20 20 20 20 16 16 16 16
35 33 34 34 21 21 20 20 17 16 16 16
32 30 31 31 19 19 19 19 16 15 15 15
34 33 34 34 20 20 20 20 16 16 16 16
29 29 30 29 19 18 18 18 15 14 15 15
34 33 34 34 20 20 20 20 16 16 16 16
24 44 19 29 28 23 315 26 9 68 153 94 218 35 32 14 32 30 452 57 34 14 57 111 304 29 38
23 44 17 24 25 25 374 26 11 85 168 99 232 36 34 13 32 31 479 56 31 14 59 105 325 31 40
22 42 16 25 20 20 301 22 9 68 135 79 211 32 29 11 28 29 435 54 30 11 51 107 300 25 35
21 43 15 22 23 23 335 24 11 85 168 99 231 32 30 13 28 31 453 51 31 13 55 97 305 31 36
20 40 16 25 20 20 298 21 9 68 135 79 199 29 28 11 26 27 383 49 30 11 48 105 275 25 32
20 43 15 22 23 23 332 24 11 85 168 99 231 32 30 13 26 31 452 51 31 13 55 94 305 31 36
19 40 16 25 20 20 285 21 9 68 135 79 186 28 28 11 24 26 365 46 30 11 47 105 259 25 30
20 43 15 22 23 23 332 24 11 85 168 99 231 32 30 13 25 31 452 51 31 13 55 94 305 31 36
Indicates a lower bound on the workforce size, the integer optimal solution was not obtained.
that the proportion of starting-time subsets capable of providing the optimal workforce size is often very small.
Solution quality generally deteriorated very rapidly as less eective starting-time subsets were used. To illustrate this ®nding, consider the tele-
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Table 2 Total CPU seconds to solve integer programs for all subsets of starting times Demand pattern
a
3 Starting times
4 Starting times
5 Starting times
24 Starting times
5/40
5/40
4/40
5/40
4/40
5/40
4/40
4/40
TEL_1 TEL_2 TEL_3 TEL_4 TEL_5 TEL_6
0.17 0.17 0.11 0.16 0.11 0.11
3.29 2.85 2.69 2.70 2.69 2.74
13.73 13.24 12.52 12.69 12.46 12.91
56.57 56.84 52.18 56.36 52.73 57.73
190.43 190.82 177.63 191.97 185.48 202.95
468.57 469.83 432.21 540.19 457.04 539.15
2.04 3.52 0.99 0.87 2.59 7.30
0.33 0.38 0.49 N/Sa 0.33 5.16
TOLL_1 TOLL_2 TOLL_3 TOLL_4 TOLL_5 TOLL_6 TOLL_7 TOLL_8 TOLL_9 TOLL_10 TOLL_11 TOLL_12
0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11
2.80 2.80 2.81 2.80 2.69 2.69 2.75 2.75 2.69 2.74 2.69 2.69
12.91 12.80 12.96 12.80 12.63 12.58 12.35 12.42 12.31 12.30 12.25 12.19
55.42 55.53 55.20 54.82 54.05 53.66 52.75 53.11 51.19 52.62 52.12 51.41
179.88 181.03 181.37 177.51 176.97 174.67 168.62 169.66 165.21 170.05 166.92 166.37
473.73 491.86 492.46 469.23 473.19 464.17 447.03 445.28 411.56 445.72 436.71 400.51
3.30 1.92 10.88 2.91 4.51 2.75 2.08 0.82 4.23 3.51 3.46 6.48
0.55 0.61 0.88 0.44 1.26 0.77 0.83 0.77 1.37 0.94 0.77 0.49
AIR_1 AIR_2 AIR_3 AIR_4 AIR_5 AIR_6 AIR_7 AIR_8 AIR_9 AIR_10 AIR_11 AIR_12 AIR_13 AIR_14 AIR_15 AIR_16 AIR_17 AIR_18 AIR_19 AIR_20 AIR_21 AIR_22 AIR_23 AIR_24 AIR_25 AIR_26 AIR_27
0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11
2.69 2.80 2.69 2.75 2.69 2.69 2.96 2.69 2.53 2.80 2.86 2.86 2.91 2.80 2.69 2.69 2.74 2.75 2.91 2.74 2.80 2.63 2.86 2.81 2.91 2.80 2.74
12.69 13.07 12.64 12.63 12.74 12.47 13.84 12.41 11.09 12.80 13.62 13.46 13.74 12.85 12.63 12.30 12.74 12.52 13.95 12.91 12.68 12.09 13.07 13.24 14.12 12.53 12.58
52.83 56.08 52.46 53.39 54.05 52.13 62.24 52.56 44.10 52.51 57.07 56.63 59.87 53.49 52.23 50.75 54.00 55.15 60.70 55.48 54.49 50.80 55.42 58.66 61.46 52.24 54.32
178.12 188.73 178.51 174.39 184.55 173.24 211.41 169.60 136.77 174.28 195.32 191.86 198.55 179.82 175.10 164.83 181.20 175.37 217.72 183.06 175.60 162.19 182.51 198.17 212.24 166.97 171.65
424.36 515.92 404.86 422.54 437.48 425.35 583.03 449.40 308.85 408.70 468.73 470.77 518.60 450.17 410.95 411.60 474.12 480.04 563.97 497.02 474.01 407.10 464.78 534.87 547.28 385.41 445.50
0.22 1.04 0.16 0.27 0.28 0.33 1.10 0.27 0.01 0.22 0.16 0.27 0.55 0.38 0.22 0.11 1.21 0.17 0.94 0.49 0.66 0.16 0.71 0.39 0.83 0.05 0.32
0.28 0.93 0.22 0.27 0.49 0.60 1.70 0.94 0.01 0.16 0.66 0.39 0.60 0.60 0.11 1.48 5.11 0.88 0.88 0.44 0.33 0.72 0.17 0.28 1.54 0.11 1.21
N/S no solution.
phone labor requirements and the 5/40-4 policy. For each of the labor requirement patterns, the workforce sizes for each of the 966 subsets of
starting times were ranked in ascending order and graphically depicted in Fig. 2. This ®gure shows that moving beyond the top 5±10% of starting-
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467
Table 3 Percentage of starting-time subsets that provide the optimal workforce size Demand pattern
3 Starting times
4 Starting times
5 Starting times
24 Starting times
5/40
5/40
5/40
5/40
4/40
4/40
4/40
4/40
TEL_1 TEL_2 TEL_3 TEL_4 TEL_5 TEL_6
12.500 12.500 12.500 12.500 37.500 37.500
0.446 1.339 8.482 8.482 0.446 0.446
0.207 0.207 0.311 2.174 0.207 0.932
0.105 0.421 0.500 0.026 0.026 0.026
0.026 0.018 0.525 0.009 0.035 1.970
0.750 1.602 1.349 0.012 0.110 0.004
N/A N/A N/A N/A N/A N/A
N/A N/A N/A N/A N/A N/A
TOLL_1 TOLL_2 TOLL_3 TOLL_4 TOLL_5 TOLL_6 TOLL_7 TOLL_8 TOLL_9 TOLL_10 TOLL_11 TOLL_12
12.500 25.000 37.500 12.500 12.500 25.000 25.000 25.000 12.500 37.500 37.500 37.500
5.357 4.464 3.125 3.125 3.125 4.464 1.786 4.464 1.339 6.250 8.482 6.250
1.760 0.311 0.207 0.311 0.414 1.346 1.242 1.449 0.207 1.035 1.449 0.414
0.421 0.527 0.316 0.685 0.843 1.553 0.105 0.369 3.607 0.869 0.026 0.948
0.158 0.018 0.018 0.193 0.018 0.053 0.035 0.053 0.639 0.044 0.114 0.026
0.004 1.491 1.018 2.178 0.016 0.189 0.454 1.267 0.004 2.438 0.225 2.525
N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
AIR_1 AIR_2 AIR_3 AIR_4 AIR_5 AIR_6 AIR_7 AIR_8 AIR_9 AIR_10 AIR_11 AIR_12 AIR_13 AIR_14 AIR_15 AIR_16 AIR_17 AIR_18 AIR_19 AIR_20 AIR_21 AIR_22 AIR_23 AIR_24 AIR_25 AIR_26 AIR_27
50.000 12.500 50.000 12.500 37.500 12.500 12.500 12.500 50.000 12.500 12.500 12.500 12.500 12.500 12.500 25.000 37.500 12.500 12.500 12.500 75.000 25.000 12.500 12.500 12.500 25.000 12.500
8.929 3.125 1.339 2.679 0.446 22.321 3.125 16.964 73.214 44.643 13.393 24.107 3.125 10.714 1.339 1.786 3.125 3.125 0.893 8.929 8.929 26.786 6.250 2.679 0.446 20.536 8.929
0.207 0.311 0.414 1.760 0.621 1.863 0.104 0.621 63.458 11.698 0.621 0.104 0.311 3.313 1.035 0.207 0.414 0.518 0.104 0.414 3.416 1.035 0.311 0.414 0.932 4.037 0.518
0.527 0.184 1.264 0.158 1.738 1.422 0.211 2.001 81.043 50.105 24.697 27.278 0.211 1.211 1.869 8.926 0.211 6.793 0.500 0.527 12.507 8.004 0.369 0.263 0.053 38.231 1.632
0.613 0.009 3.746 6.434 1.234 2.871 0.184 0.236 71.183 14.163 1.322 0.184 0.009 0.114 1.120 0.814 0.053 0.044 0.009 0.053 7.931 2.215 0.341 0.683 0.018 7.668 0.079
0.355 0.552 3.196 0.537 5.114 4.321 0.110 5.027 86.261 59.028 36.521 34.916 1.125 4.013 4.968 17.436 0.710 12.567 0.075 1.480 18.316 17.586 2.143 0.103 0.911 52.438 5.761
N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A
time subsets results in a substantial deterioration of solution quality. These ®ndings suggest that the answer to the third question in Section 3.1 is that a
poor (or even mediocre) selection of starting times can lead to a substantial penalty in terms of workforce size.
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4. A case study ± scheduling customer service representatives in call-center operations in the System Support Center, US and Canada group, Radio Network Solutions Group, Land Mobile Products Sector (LMPS), Motorola 4.1. An overview of the call-center scheduling environment
Fig. 2. This ®gure shows the workforce size response surface subsets of starting times for the 5/40-4 policy for the TEL_1± TEL_6 labor requirements.
3.4. Summary of major ®ndings of the experimental study One important ®nding of this study was that, for many of the test problems, restricting the number of starting times to four or ®ve did not result in a substantial increase in workforce size. This ®nding is good news for service operations managers because it suggests that minimum cost labor schedules may be achieved with relatively few shift starting times. As observed in previous studies [2,8,9,23,25], a limited number of starting times is desirable because it makes schedules easier to administer. Although service managers may take solace in the ®nding that a small number of starting times is often sucient to yield low-cost solutions, they must take caution regarding a second ®nding of this study. Speci®cally, for the real-world labor requirements patterns considered in this study, only a very small percentage of the starting-time subsets associated with S 4 or S 5 were typically capable of providing the minimum workforce size. This implies that service managers must pay close attention to ensure that an eective subset of starting times is selected. This attention is especially important in light of the third major ®nding of the study that workforce sizes were quite excessive for some subsets. For any given number of starting times, S, the dierence between a good and poor subset of starting times was often considerable. In fact, many good subsets of size S 1 ; S 1 < S 2 , provided smaller workforce sizes than poor subsets of size S2 .
MotorolaÕs LMPS Radio Network Solutions Group Call Center receives and processes inbound calls for customer support. This call center operates continuously and receives several varieties of incoming calls, many of which generate requirements for outbound calls. The call center operates with 30-minute planning intervals and is staed almost exclusively by full-time employees working ®ve consecutive days per week. Shifts are 8.5 hours in length with a 1-hour meal break. Meal-break ¯exibility is provided by allowing breaks to occur either 4, 4.5, or 5 hours into the shift. In addition, the scheduling problem allows for the speci®cation of ``project time'', in which customer service representatives perform other duties and are not available to satisfy service requirements. This project time, which can range from 0 to 60 minutes in length, may occur at the beginning or end of workshift, or it can be attached to the meal break. For example, consider an 8.5-hour shift from 7 am to 3:30 pm that contains a meal break from 11:30 am to 12:30 pm. The project time for this shift may occur from 7:00±8:00 am, 10:30±11:30 am, 12:30±1:30 pm, or 2:30±3:30 pm. Motorola LMPS management has moved towards consolidating additional operations (including those in South America) into the call center. Moreover, as the installed base of Radio Network systems has increased, the call center has experienced signi®cant increases in inbound (and outbound) call volume. Motorola LMPS wanted a system that could be used to assist in sta planning for the present call-center expansion, as well as to provide for future expansions in this and other Motorola call centers. The scheduling system we developed for the Radio Network Solutions Group call center was designed to provide decision support for this problem.
M.J. Brusco, L.W. Jacobs / European Journal of Operational Research 131 (2001) 459±475
Speci®cally, we developed a spreadsheet-based decision support system to aid Motorola management in making call-center stang and scheduling decisions. The primary requirements for building this system were that it be developed using Microsoft Excel and VBA macros, and that it provide schedules in a rapid and ecient manner on a microcomputer. The system currently consists of three modules provided in the form of VBA macros: (1) a labor-requirement-generation module; (2) a sta-planning module; and (3) a stascheduling module. The labor-requirement-generation module allows the user to provide information regarding arrival rates of incoming calls, service rates, and desired service levels. The VBA macro uses this information in conjunction with an M/M/s queuing model to generate labor requirements for each 30-minute planning period of the week (this approach is consistent with those previously reported in the literature [1,9,17]). These labor requirements subsequently serve as inputs to the sta-planning and sta-scheduling modules, which are of particular relevance to this paper. The sta-planning and sta-scheduling modules are described in Sections 4.3 and 4.4, respectively. Prior to their description in Section 4.2, we provide a linkage between the experimental analysis in Section 3 and the problem faced by the Motorola call centers.
469
(24 starting times 7 days-o patterns) when all 24 starting times were considered as candidates. For the LMPS call-center problem, the number of Xij variables is 4032 (48 starting times p 4 possible project-time placements b 3 possible meal-break placements 7 days-o patterns). The number of Yi variables in the call-center problem is 48, compared to 24 in the experimental analysis problem. The total number of constraints (2)±(4) in the call-center problem is 385 336 48 1, compared to 193 168 24 1 in the experimental analysis problem. The size and complexity of the LMPS callcenter scheduling problem, in conjunction with managementÕs desire to have a spreadsheet-based system that runs rapidly on a microcomputer, absolutely precluded the use of optimal solution methods (such as the cutting plane) in both the sta-planning and sta-scheduling modules. For this reason, it was necessary to resort to heuristic methods for both of these modules. In order to give LMPS management the opportunity to answer the types of questions we considered in the experimental analysis in Section 3, we built the spreadsheet modules with considerable ¯exibility. That is, even though optimal solutions could not be guaranteed, LMPS management could use the model to estimate the eects of starting-time (and meal-break and project-time) decisions on workforce size.
4.2. Linkage of experimental analysis to the design of the scheduling system
4.3. The sta-planning module
In at least two respects, the problem faced by MotorolaÕs LMPS call centers is more complex than the scheduling environment considered for the experimental analysis in Section 3. First, in addition to starting-time policies, LMPS management wanted to be able to simultaneously examine meal-break and project-time policy decisions. Second, the call centers use 30-minute planning intervals (I 48) instead of hourly intervals (I 24). Together, these two conditions result in an ILP formulation, (1)±(6), that is substantially larger than the one considered in the experimental analysis. For example, the number of Xij variables in the experimental analysis in Section 3 was 168
The sta-planning model was designed to minimize the number of workers necessary to satisfy the labor requirement in each 30-minute planning interval. The inputs for this heuristic included the labor-requirement distribution (developed by the labor-requirement generator described in the previous subsection), the project time, the number of replications of the heuristic to be performed, and the set of allowable starting times. A screenshot of the EXCEL spreadsheet associated with these inputs is shown in Fig. 3. Management desired a system that could rapidly evaluate a variety of starting-time subsets, yet contain sucient ¯exibility to prohibit the use of
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Fig. 3. Screenshot of the sta-planning module input screen.
certain starting times. Further, to increase the ¯exibility of the system, we wanted to provide management with the opportunity to select the precise location of each feasible starting time. Therefore, the starting-time input was designed so that the user could turn starting times on and o. A value of Ô1Õ in Fig. 3 indicates that the starting time may be used, whereas a value of Ô0Õ precludes the use of the starting time. The VBA macro uses this input in conjunction with a greedy construction heuristic to build a schedule. This heuristic begins with Xij 0, for all i and j and adds employees, one at a time, until constraint (2) is satis®ed. At each iteration, an employee is assigned to the tour associated with shift starting time e and days-o pattern f, such that def maxi;j dij , where
dij
I X L X k1 l1
8i and j:
0 aijkl @ max rkl
I X J X
!2 1 aijkl Xij ; 0A
i1 j1
7
In other words, at each iteration, an employee is added to the tour that hits the largest sum-ofsquared remaining labor requirements (ties are broken arbitrarily). Several studies have suggested that construction heuristics of this type are often eective at providing near-optimal workforce sizes [6,8,20,21]. The user can specify multiple replications of the heuristic in order to guard against the potential for an unusually poor individual replicate. Further, because the solutions are constructed in a matter of seconds, call-center management can use the sta-planning module to
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test a wide variety of starting-time con®gurations in a short period of time. As was the case in the experimental study, the parsimonious use of starting times is desirable. If LMPS management is interested in the ®rst research question considered in the experimental analysis (i.e., what are the workforce size implications of using S < I starting times relative to I starting times?), then we suggested they begin by running the sta-planning module with all I 48 (or nearly all) starting times to serve as a benchmark. Next, dierent con®gurations of S starting times can be tested to see how they compare to the benchmark. During this process, answers to the second and third research questions in the experimental analysis are being developed (albeit in a heuristic manner). Although LMPS management certainly is not guaranteed to ®nd the best subset of S starting times, they can at the very least guard
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against the selection of a very poor subset of starting times. The output of the sta-planning model includes the workforce size, as well as an indication that the ®nal solution is feasible. Speci®cally, Fig. 3 displays a workforce size of 37 and the ``ERROR?'' box displays the word ``NO'', indicating a feasible solution. A schedule summary is also generated as depicted in Fig. 4. This summary displays detailed information regarding the properties of schedules to which employees were assigned. For example, row 14 of the spreadsheet screenshot in Fig. 4 reveals that one employee was assigned to a schedule associated with shifts between 12:30 and 9:00 am from Tuesday±Saturday (i.e., Sunday and Monday o) and a meal break from 4:00 to 5:00 am. Since the project time was zero for this execution of the module, all of the ``Actual Start'' and ``Actual Finish'' times are the same as the ``Starting'' and
Fig. 4. Screenshot of the schedule summary report generated by the sta-planning module.
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``Finishing'' times, respectively. However, if the project time (p) is greater than zero and is assigned to the beginning (end) of the shift, then the Actual Start (Actual Finish) time will be p periods later (earlier) then the Starting (Finishing) time. Thus, the Actual Start and Actual Finish times re¯ect the actual availability of the employees to meet callcenter requirements. When the project time is allocated to a break, the Actual Start and Actual Finish times are the same as Starting and Finishing, respectively, and the dierence between ``Break Start'' and ``Break Finish'' expands by p periods. 4.4. The sta-scheduling module Although the sta-planning module is helpful for determining the eect of labor requirement, project time, and starting-time changes on
workforce size, it is not directly bene®cial for actually constructing employee schedules because the current workforce size may be dierent than the solution provided the model. Therefore, the sta-scheduling module was developed to construct a schedule for a prescribed number of employees (i.e., the current workforce size). This module uses a greedy heuristic comparable to that of the sta-planning module. However, due to the fact that the workforce size is ®xed, it may not be possible to construct a schedule that will satisfy the labor requirement for each period. Therefore, the objective criterion associated with the sta-scheduling module is to minimize total sum-of-squared understang. Fig. 5 depicts a screenshot of the input screen associated with the sta-scheduling module. The module requires input regarding the labor requirements, project time, allowable starting times, and the current workforce size.
Fig. 5. Screenshot of the sta-scheduling module input screen.
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Fig. 6. Screenshot of the labor coverage report generated by the sta-scheduling module.
The sta-scheduling module generates a schedule summary comparable to the one produced by the sta-planning module (see Fig. 4). In addition, sta-scheduling model generates output corresponding to actual labor coverage in each planning interval of the week. Speci®cally, Fig. 6 reveals that information is provided with respect to the number of customer service agents working in each 30-minute interval of each day, as well as the shortage or surplus of agents in each interval. The shortages, which are associated with negative numbers, are accentuated in ``red'', whereas the intervals with exact or surplus coverage are in ``blue''. This enables call-center management to rapidly visualize the coverage associated with the current set of project time and starting-time policies. Like the sta-planning module, the stascheduling module is extremely fast and a wide variety of policies can be tested in a very short period of time.
5. Summary The large computational study presented herein revealed that a small number of starting times was often sucient to provide workforce sizes that were comparable to those obtained when all possible starting times were considered. Although it is acknowledged that one could easily ``contrive'' labor-requirement patterns that would necessitate exactly 3, 4, 5, 6, 7, etc. starting times, it should be observed that this study used three sets of actual (real-world) labor-requirement patterns that were not so contrived. The study also revealed that only a few of the many possible subsets of starting times were capable of providing the minimum workforce size and that many subsets resulted in rather poor scheduling solutions. The case study associated with the scheduling of customer service representatives at Motorola service call centers demonstrated that starting-time
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decisions often must be examined in conjunction with other scheduling policies (e.g., break policies, days-o policies, project-time policies, etc.). Under such conditions, it was not practical to utilize a model that evaluated all possible subsets of starting times. Instead, we developed a ¯exible decision support system that enables call-center management to rapidly evaluate starting time and other policy decisions. This system contains modules that enable it to be used for either planning or operational purposes.
Acknowledgements We would like to extend our gratitude to Mark Hurlbert, Director of System Support Center; John Williams, Call Center Operations Manager; and Keith Toborg, all of the United States and Canada Group, Radio Network Solutions Group, Land Mobile Products Sector, Motorola for their invaluable assistance in completing this project.
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