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Scheduling part-time personnel with availability restrictions and preferences to maximize employee satisfaction
Mathematical and Computer Modelling 48 (2008) 1806–1813 www.elsevier.com/locate/mcm
Scheduling part-time personnel with availability restrictions and preferences to maximize employee satisfaction Srimathy Mohan School of Global Management and Leadership, Arizona State University, P.O. Box 37100, Phoenix, AZ 85069-7100, United States Received 9 October 2006; received in revised form 5 November 2007; accepted 26 December 2007
Abstract With the economy continually moving towards retail and services, the size of the part-time workforce is increasing constantly. In this paper, we consider the problem of scheduling a workforce consisting entirely of part-time workers. The part-time workers have availabilities, preferences for the shifts, and also a seniority level. We propose an integer programming model to maximize employee satisfaction and to meet the demand requirements for each shift. The model is tested on randomly generated instances. The results demonstrate the effectiveness of branch-and-cut methods in reducing computation time. c 2008 Elsevier Ltd. All rights reserved.
Keywords: Part-time worker scheduling; Employee satisfaction; Integer programming
1. Introduction The current trend of varying economic cyclic patterns and increasing competitiveness in all industries demands versatility in adjusting employee requirements to avoid over- or under-staffing. Also, the new competitive environment dictates customer service beyond traditional weekday hours. The result of these business realities is a heavy reliance on part-time work. Nearly one in five workers in the United States currently works part-time [1] and the number of part-time employees was more than 27 million in August 2006 [2]. Furthermore, part-time employees constitute a major proportion of entire industries such as service and retail [3]. As a result, intelligent scheduling of part-time employees has become essential for reducing the cost of operations and maintaining employee morale. The North American economy has grown in the last fifty years due to advances in technology, but just as important, due to improvements in the systems and techniques used by managers to coordinate these resources. Classic scientific management techniques such as scheduling, statistical quality control, and project management have all contributed to these advances. In addition, competitive pressures to reduce cost while increasing versatility and flexibility are continuously requiring improvements of another management technique: employee scheduling. The area of employee scheduling has been extensively studied in the literature and many different applications in a wide number of fields have been reported. Blochliger [4] offers a tutorial of staff scheduling problems discussing data requirements, cost modeling, hard (inflexible) and soft (flexible) constraints and objectives. Ernst et al. [5]
E-mail address: [email protected]. c 2008 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2007.12.027
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provide a comprehensive review; they also offer a problem classification, discuss the different application areas (transportation systems, call centers, health care systems and others) and solution methods (artificial intelligence, constraint programming, mathematical programming etc.). Employee scheduling problems typically fall into three classes: days off scheduling, shift scheduling, and tour scheduling. Most employee scheduling models typically deal with full-time personnel and develop minimum cost schedules while satisfying constraints. Part-time employees are typically considered as supplemental workforce and most models attempt to minimize the total operational costs by hiring the minimum number of low cost part-time employees. Also, the models only indicate the number of part-time employees required. The problem of actually selecting the part-time personnel based on their availability still remains. Also, the satisfaction level of part-time employees is rarely considered in scheduling models. With the growth of the service economy, there has been an increase in the need for accommodating two specific needs in employee scheduling — (i) determining specific assignments for the supplemental part-time workers, considering their preferences, and (ii) determining employee schedules in applications where the entire workforce consists of part-time workers. Under such circumstances, the problem no longer deals with determining cyclic schedules that can be repeated or determining days off for the employees. The part-time employees’ availability may vary widely from one time period to another and hence, has to be approached differently from the traditional staff scheduling problems. The existence of real world part-time employee scheduling problems, such as the newspaper industry problem explored by Gopalakrishnan et al. [6], serves as our basis for the creation of a more general model to accommodate more industries. Gopalakrishnan et al. [6] implemented a decision support system that uses a scheduling heuristic for a 16% reduction in the number of part-time employees hired, which translates into a $50,000 annual savings for the Birmingham newspaper company. The research by Gopalakrishnan et al. [6] includes general part-time employee scheduling constraints, considerations for part-time employee seniority, availability and preferences, and meeting shift requirements. They develop a constructive, single pass, rule-based heuristic to schedule part-time employees over a given planning horizon. The heuristic does not incorporate any local improvement procedures and there are no penalties associated with hiring a part-timer for a shift that he/she does not prefer. Another application that includes only part-time employees who are hired on a strictly as-needed basis is the scheduling of substitute teachers in many school districts. The school district maintains a database of available substitute teachers. When there are planned absences, the school district determines the need for the entire week and contacts people on the database (according to seniority and availability) to staff the various classrooms. When there are unplanned absences, the scheduling problem is solved just for that day rather than for the entire week. With more and more people opting to work part-time due to the flexibility it offers, consideration of the part-time employees’ preferences is also becoming important. In this research, we consider an employee pool that consists of only part-time workers. Our primary objective is to develop a work schedule that maximizes the employee satisfaction level. Given employee availability, seniority and preferences, maximum and minimum allowable working hours in a day and a week, and employee requirements for each shift, our model seeks to maximize employee satisfaction while meeting the demand requirements for each shift and satisfying other required constraints. We formulate the problem as an integer program. We define employee satisfaction as a function of seniority, preferences, and availability. This model can be used as an aid during second level planning. During the first level, the part-time worker requirements are determined to minimize overall system costs for the organization. Given this requirement for the various shifts, our model seeks to develop a part-timer work schedule that takes employee preferences, seniority and availability into consideration. The rest of the paper is organized as follows. We present some relevant literature in the next section. Section 3 presents the generic integer programming model and Section 4 presents the results of our experimentation. We present the concluding remarks in Section 5. 2. Literature review Most of the existing literature on workforce scheduling deal with full-time workers. Research falls into three types of problems namely, days off scheduling, shift scheduling, and tour scheduling. Days off scheduling deals with assigning the workers work and non-work days over the planning horizon. Shift scheduling problems deal with determining the actual shifts (start and end times, breaks, etc) during the planning horizon. Tour scheduling problems combine features of days off and shift scheduling. Here, typically, work shifts are to be assigned to cover the daily
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demand for each shift, and each worker’s days off schedule has to be determined. As mentioned earlier, a few studies that do deal with part-time workforce do not provide a work schedule for the part-time employees. Part-time workers are extensively used in all types of industries, and provide great flexibility to an organization. Early research in the area is focused on reducing the number of part-time personnel in days-off scheduling problems. Byrne and Potts [7] consider the scheduling of toll-booth operators to meet the forecasted demand. They provide an integer program to determine the levels of both full-time and part-time work. Baker [8] considers the daysoff scheduling problem with both full-time and part-time workers. The goal is to reduce the number of part-time employees. Some researchers have considered the problem of scheduling part-time workers in financial services such as banks, to supplement full-time workers, especially during peak-hours [9,10]. Krajewski et al. [11] discuss part-time employee use to produce more flexible schedules for the processing of checks in large banks. Their shift-scheduling model assigns a mix of full-time and part-time employees to a number of pre-selected shifts. Glover and McMillan [12] present a heuristic based on Tabu search for producing employee schedules for a full-time work force supplemented by part-timers. Their general employee scheduling problem attempts to produce schedules that in aggregate provide adequate service while respecting existing employee scheduling preferences. Their model is similar to the model we examine in our research in that employees may specify the maximum hours they can work in a week and their availability. The model considers seniority. However, the main goal of that work is to minimize the number of part-time workers needed to satisfy all work requirements. Love and Hoey [13] implement a part-time employee tour scheduling model for a fast-food operation based on network methodology. Their model allows employees to specify their availability and preferences and it considers employee skill ratings in separate work areas. Employees are limited to the number of days they may work in a week and to one shift of 3–8 h a day. Also, their model permits over-staffed schedules. Lauer et al. [14] present an interactive decision support system for a tour scheduling problem arising from a computer lab. The system can be highly responsive to various management objectives and allows managerial interactions by overriding existing schedules. Willis and Huxford [15] consider a staff scheduling problem in a call center involving both part-time and full-time staff. They use an integer programming approach for the problem. Schindler and Semmel [16] consider a workforce problem at Pan American World Airways. They add additional constraints to standard set-covering model formulation to ensure certain constraints for part-time staff. Dowsland [17] considers nurse scheduling with an application at a large general hospital. A Tabu search technique is used. The model incorporates ranks, skills, preferences and equity distribution of rosters; it also considers the use of part-time staff to cover demand. Easton and Rossin [18] examine the issue of staffing at a minimum cost while providing acceptable levels of service and employee satisfaction. The scheduling of a mixed work force with an even distribution of over-staffing yielded consistent customer and employee satisfaction levels. Over-staffing provides higher customer satisfaction with better and more consistent service and higher employee satisfaction with the availability of more flexible work hours. They propose that in the service industry staffing at a minimum cost is not as important a factor as customer and employee satisfaction. The maintenance of high satisfaction levels retains customers and minimizes employee turnover. A few papers have recently considered part-time employees in their research. Bard et al. [19] consider a staff scheduling problem at the United States Postal Service (USPS) that involves both full-time and part-time workers and also the principal constraints of the union contract. Their goal is to examine several scenarios on reducing the size of the workforce. They model their problem as an integer programming formulation and solve it using CPLEX. Isken [20] considers applications of the well-known tour scheduling problem for both full-time and part-time employees in healthcare. Another stream of research on scheduling nurses and airline crew deal with developing work schedules that take into account employee requests for certain shifts and days off [21–24]. The algorithms typically work to minimize the overall labor costs (full-time and supplemental part-time workforce) and try to accommodate as many of the “soft” requirements regarding employee preferences as possible. Most of the concepts and algorithms developed for a full-time workforce are not applicable for a part-time workforce since we are not guaranteed continued availability of the part-time workers. The contractual rules are not as strict either. However, it is still to the advantage of the firm to accommodate the preferences of part-time workers. The workers will be more inclined to work for a firm that provides work schedules that are tailored to their individual needs and preferences. This second level manpower planning problem has not been addressed in the literature and the next section proposes an integer programming model for the same.
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3. Problem definition and formulation The mathematical model discussed in this section is concerned with finding the part-time employee schedule that maximizes employee satisfaction while fulfilling all constraints. Demand in the form of employee requirements for each shift is assumed to be given. Shift lengths vary during a day and across the week. Also, certain restrictions are imposed on the number of hours a part-timer can work during a day and in a given week. Employee assignments maximize employee preferences, while considering their seniority and availability. We will first introduce the assumptions and notation that we use throughout the paper. 3.1. Assumptions and notation 1. Schedules are developed using information provided by the part-time employees on their availability and shift preference, and based on the company’s shift requirements. 2. Shift lengths vary during a day and across a week. All shift requirements have to be exactly met. 3. Unavailable employees can be hired for work if necessary, with a penalty to the objective function. 4. If an employee is scheduled for at least one shift in a day, the number of hours assigned to the employee must be within a pre-specified range. 5. Employees can work more than one shift a day and they need not be consecutive. 6. Daily minimum and maximum hours and weekly minimum hours are fixed for a particular schedule. 7. Every employee’s weekly hours, worked, must be within a pre-specified range. 8. Weekly maximum hours for an employee is either the maximum hours allowed in a week or the maximum hours an employee wants to work in a week. Indices i = part-time employee j = shift of the day k = day of the week Parameters wi = seniority of part-time employee i b jk = number of part-time employees needed for shift j on day k t jk = length of shift j on day k ai jk = current availability of part-time employee i for shift j on day k pi jk = preference of part-time employee i for shift j on day k Dmax = maximum allowable working hours for a part-time employee during a day Dmin = minimum required working hours for a part-time employee during a day Wmax =maximum allowable working hours for a part-time employee during a week Wmin = minimum required working hours for a part-time employee during a week Decision variables 1 if part-time employee i is assigned to shift j on day k Yi jk = 0 otherwise 1 if part-time employee i is hired on day k X ik = 0 otherwise.
3.2. Model formulation Maximize
XXX i
subject to
j
k
Yi jk
wi pi jk ai jk + ai jk − 1 wi
(1)
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X
Yi jk = b jk
(2)
∀ j, k
i
Dmin X ik ≤
X
Yi jk ti jk ≤ Dmax
∀i, k
(3)
∀i
(4)
j
Wmin ≤
XX j
Yi jk ti jk ≤ Wmax
k
Yi jk ≤ X ik
(5)
Yi jk , X ik
(6)
∀i, j, k ∈ {0, 1} ∀i, j, k.
The objective function (1) maximizes employee preferences while considering their seniority and availability. The seniority of a part-time employee,wi , is smaller for a less senior employee when compared to the seniority of a more senior employee. The seniority of an employee can either be directly determined by the duration of service or could be a combination of other factors, including prior experience in similar jobs and formal training in a particular field. For example, in many school districts substitute teachers are part-time employees who are hired from a pool of available people. In this case, the applicant’s prior experience in handling a particular subject and grade level along with training is a better indicator of seniority than just the duration of service. The parameter wi used in the model is generic in that it can be used to represent the duration of service directly or if qualitative factors are involved, the following scheme can be used to represent seniority. The wi for the least senior employee could be assigned a value of 1 and the value of wi for other employees could be increased as the seniority level increases. The increase could be linear or some function of the seniority level. The parameter pi jk is numbers that indicate the preference of the various shifts for each part-time employee. Every part-time employee assigns a preference value for every shift during the planning horizon, irrespective of whether he/she is available during that shift. The parameter pi jk can be a value in the range [0, 10]. The higher the preference value, the more preferred the shift. Availability is a binary integer value where zero indicates unavailable and one indicates availability. Thus, the first term in the objective function captures the seniority of a part-time employee assigned to a shift and the employee’s preference for that shift. The term ai jk − 1 wi serves as a penalty if an unavailable employee is hired to work. This term penalizes the objective function more if a more senior employee is assigned to a shift for which he or she is not available. To summarize, the objective function maximizes the overall satisfaction of the part-time employees by assigning as many senior employees to their preferred shifts as possible. Constraints (1), the shift requirement constraints, ensure that every shift is assigned the required number of employees. Constraints (3) force the maximum and minimum number of hours’ requirement for every part-time employee assigned on any given day. Constraints (4) ensure that an employee is assigned an absolute minimum every week and is not forced to work more than the maximum number of hours during a week. Constraints (5) ensure that a part-time employee is assigned to a shift in a day only if he or she is hired on that day. Finally constraints (6) force X ik and Yi jk to be binary integer variables. When a given instance has a part-time employees, b shifts, and c days, the scheduling model contains (abc + ac) binary integer variables and a (bc + 2)+c (2a + b) constraints. Hence, even for a small sized application with 50 parttime employees who have to be scheduled for 5 days, each with 2 shifts, the model has 750 binary integer variables and 1110 constraints. But the constraint matrix is a very sparse matrix and more than half of the non-zero entries have a value of 1. Hence, we conjecture that a generic branch-and-bound algorithm should be able to solve the problem within a reasonable amount of time. We tried to solve reasonably sized problems and present the details and results in the next section. 4. Experimentation The part-time employee scheduling model presented in the previous section can be solved using standard enumeration schemes available in commercial solvers like CPLEX. However, the size of the problem may sometimes make some instances computationally intractable. Hence, we decided to test the effectiveness of solving these models using a standard branch-and-bound algorithm on a set of randomly generated problems. We generated two classes of problems — one with 50 employees and another with 100 employees. For each class, we generated problems with three different overall employee availability levels — 25%, 50% and 75%. An availability
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S. Mohan / Mathematical and Computer Modelling 48 (2008) 1806–1813 Table 1 Effectiveness of adding cuts to the branch-and-bound enumeration procedure Instance
100 employees, 75% availability Cut % LP gap
100 employees, 50% availability Cut % LP gap
100 employees, 25% availability Cuts % LP gap
1 2 3 4 5 6 7 8 9 10
3.02 1.04 2.88 1.11 6.24 1.45 1.12 0.93 0.54 0.67
7.32 4.90 14.99 5.43 15.86 11.19 5.43 4.83 4.79 5.00
0.76 0.17 0.41 0.25 0.12 0.37 0.67 0.90 0.17 0.68
9.70 5.61 13.34 6.75 14.25 10.43 6.52 5.00 5.26 6.33
0.00 0.02 0.20 0.00 0.28 0.14 0.00 0.00 0.00 0.00
7.02 5.90 11.89 7.75 11.57 8.99 5.89 3.97 5.22 6.53
Average
1.90
7.97
0.45
8.32
0.06
7.47
Instance
50 employees, 75% availability Cut % LP gap
50 employees, 50% availability Cut % LP gap
50 employees, 25% availability Cuts % LP gap
1 2 3 4 5 6 7 8 9 10
4.08 0.53 2.28 2.35 1.55 0.56 2.78 0.52 0.99 0.59
10.41 5.80 15.51 10.64 5.50 5.57 9.64 5.24 5.00 5.49
0.39 0.00 0.48 0.63 0.53 0.00 0.50 0.03 0.11 0.00
10.73 7.28 12.93 12.62 7.01 5.77 9.14 5.95 4.91 5.20
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
8.86 6.77 13.96 8.43 6.36 6.11 7.74 6.89 6.58 4.39
Average
1.62
7.88
0.27
8.15
0.00
7.61
Numbers in bold indicate an optimal solution.
level of 25% implies that the ratio of number of shifts for which part-time employees are available to the total number of shifts to be scheduled is 25%. For each combination of number of employees and availability level, we generated 10 feasible instances to test the scheduling model. The other parameters were generated as follows. The time horizon considered for generating a schedule is 1 week (5 days) with 3 shifts per day. Employees are randomly assigned a seniority ranging from 1 to 10, where 10 indicates the most senior position and 1 the least senior. Each employee has to work a minimum of 12 and a maximum of 40 hours per week. If a part-time employee is hired on a given day, he/she has to work for at least 2 shifts that day. The number of people required per shift is generated from a discrete uniform distribution between 30% and 50% of MAX, where MAX = number of employees * availability level. The duration of each shift (in hours) is generated from a uniform distribution between 4 and 6. For each part-time employee, the preference for each shift is a number randomly chosen between [0, 10]. The integer programming model is solved using the basic branch-and-bound enumeration algorithm in CPLEX solver. We allow a maximum limit of 30 min for CPLEX to determine the optimal solution. If an optimal solution is not found during that time, CPLEX reports the best feasible solution. Our preliminary experimentation indicated that even for the 50 employee problems, when availability increased from 25% to 50%, CPLEX failed to produce optimal solutions in 80% of the problems. As availability increased, the solution space also increases and the branchand-bound algorithm has a much bigger tree to search through. Hence, the basic branch-and-bound algorithm failed to produce an optimal solution for several problems within the 30 min time limit. Integer programming solve times can often be improved by generating new constraints (or cuts) based on polyhedral considerations. These additional constraints tighten the feasible region, reducing the number of fractional variables to choose from when CPLEX needs to select a branching variable. The CPLEX program generates several kinds of cuts, some of which are generalized upper bound cuts, clique cuts, cover cuts. The cut generation takes some processing time, but could potentially reduce the overall time required to find an optimal solution. In order to investigate the usefulness of the CPLEX cuts for our problem, we solved the same problems with the additional cuts
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Table 2 Comparison of solution times (in seconds) Instance
100 employees, 75% availability Cuts No cuts
100 employees, 50% availability Cuts No cuts
100 employees, 25% availability Cuts No cuts
1 2 3 4 5 6 7 8 9 10
1800 1.04 1800 40.45 1800 1800 113.13 0.86 0.7 0.7
1800 1800 1800 1800 1800 1800 1800 1800 1800 1800
13.17 1.21 1800 2.15 1800 2.15 1800 26.5 15.4 0.67
1800 1800 1800 1800 1800 1800 1800 1800 1800 1800
0.41 0.12 1800 0.35 1800 0.87 0.2 0.12 0.1 0.6
174.32 1800 1800 64.24 1800 618 1800 23 133.29 100
Average
735.69
1800.00
546.13
1800.00
360.28
831.29
Instance
50 employees, 75% availability Cuts No cuts
50 employees, 50% availability Cuts No cuts
50 employees, 25% availability Cuts No cuts
1 2 3 4 5 6 7 8 9 10
752 1.93 1800 1800 4.24 0.56 1800 0.39 0.58 0.93
1800 1800 1800 1800 1800 1800 1800 1800 1800 1800
16.43 0.23 37.26 2.85 0.3 0.11 1.18 0.96 0.33 0.24
1800 958 1800 1800 1800 4.96 1800 1800 1800 1800
0.18 0.19 0.08 0.12 0.14 0.23 0.33 1.1 0.12 0.06
2 0.56 7.53 38.67 0.44 1.06 11.32 502.03 13.78 0.31
Average
616.06
1800.00
5.99
1536.30
0.26
57.77
during the branch-and-bound enumeration. Our objective here is to illustrate the effectiveness of cuts for this problem and the savings in computation time. We also determined the optimal solution for the linear programming relaxation of each instance, which provides an upper bound (LP upper bound) on the objective function value of the corresponding integer programming model. We tracked the performance of the solution procedures using two measures. (Solution value with cuts − solution value without cuts) ∗ 100% Solution value with cuts (LP upper bound − solution value with cuts) LP Gap = ∗ 100%. LP upper bound Cut % =
The cut % measures the improvement in solution quality by using the branch and bound with cuts and the LP gap provides an overall idea about how effective the solution procedure is. The experimental results are presented in Tables 1 and 2. Table 1 provides the solution vales for all 60 instances using the branch-and-bound procedure, both with and without cuts. The table also provides the LP upper bound for each instance. The results show that adding cuts definitely improves the chances of finding an optimal solution as the problem size and availability level increase. For the 100 employee problem, the branch-and-bound procedure without cuts found the optimal solution for 5 instances when availability was 25%. When the availability increased to 50% and 75%, it did not find any optimal solutions. On the other hand, the branch-and-bound procedure with cuts, found optimal solutions for 7 instances when availability was 50% and for 6 instances when availability was 75%. As availability increases, the solution space also increases and it would naturally take longer to determine the optimal solution. However, when we add the cuts, it effectively eliminates regions of the solution space and the probability of finding the optimal solution within the specified time limit increases. On an average, the solution with cuts is 2% better than the solution obtained without cuts for the 100 employee problems at 75% availability level. The comparative values for the 50% and 25% availability levels are 0.5% and 0.06%. CPLEX produces the improved
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solutions with the cuts in a much shorter time. Table 2 presents the results on the time taken by the two procedures to solve the 60 instances. The branch and bound with cuts takes 735 s on average to solve the 100 employee problems with 75% availability. The regular branch and bound took 1800 s since an optimal solution was not found for any of the instances. Even in the case where both procedures obtained optimal solutions for all 10 instances (50 employees at 25% availability), the branch and bound with cuts took 0.26 s and regular branch and bound took 57 s, on average. An analysis of the gap between the LP upper bound and the solution produced by the branch and bound with cuts indicates that on an average the gap is around 8%. The gap seems to be mainly due to the weak upper bound since the solution procedure obtained optimal solutions for many of the instances. 5. Concluding remarks We have presented a generic model for scheduling part-time workers while considering shift length and other related constraints. The unique aspect of our model is that it aims to maximize the overall satisfaction of a parttime workforce. Typically, part-time workers have been used as a supplemental workforce and the objective has been to minimize overall cost of the additional resources hired. In our research, we present a second level planning model to determine a part-time worker schedule to maximize employee satisfaction. Satisfaction is measured using a combination of employee seniority and preferences. Our experimentation shows that reasonable size instances of the integer programming model can be solved relatively quickly using a branch-and-bound enumeration procedure with additional cuts. Directions for future research include developing specialized solution procedures for the part-timer scheduling model and testing the proposed methodology in real world scenarios. References [1] A. Kalleberg, Nonstandard employment relations: Part-time, temporary and contract work, Annual Review of Sociology 26 (2000) 341–365. [2] Bureau of Labor Statistics, The employment situation news release: August 2006. http://stats.bls.gov/news.release/pdf/empsit.pdf. [3] N. Conway, R.B. Briner, Full-time versus part-time employees: Understanding the links between work-status, the psychological contract, and attitudes, Journal of Vocational Behaviour 61 (2002) 279–301. [4] I. Blochliger, Modeling staff scheduling problems — a tutorial, European Journal of Operational Research 158 (2004) 533–542. [5] A.T. Ernst, H. Jiang, M. Krishnamoorthy, D. Sier, Staff scheduling and rostering: A review of applications, methods and models, European Journal of Operational Research 153 (2004) 3–27. [6] M. Gopalakrishnan, S. Gopalakrishnan, D.M. Miller, A decision support system for scheduling personnel in a newspaper publishing environment, Interfaces 23 (1993) 104–115. [7] J. Byrne, R. Potts, Scheduling of toll collectors, Transportation Science 7 (1973) 224–245. [8] K. Baker, Scheduling full-time and part-time staff to meet cyclic staffing requirements, Operational Research Quarterly 25 (1974) 65–76. [9] V. Mabert, A. Raedels, The detail scheduling of a part-time work force: A case study of teller staffing, Decision Sciences 8 (1977) 109–120. [10] S. Moondra, A linear programming model for work force scheduling for banks, Journal of Bank Research 6 (1976) 299–301. [11] L.J. Krajewski, L.P. Ritzman, P. McKenzie, Shift scheduling in banking operations: A case application, Interfaces 10 (1980) 1–7. [12] F. Glover, C. McMillan, The general employee scheduling problem: An integration of management science and artificial intelligence, Computers and Operations Research 13 (1986) 563–593. [13] R.J. Love, J.R. Hoey, Management science improves fast-food operations, Interfaces 20 (1990) 21–29. [14] J. Lauer, L. Jacobs, M. Brusco, S. Bechtold, An interactive, optimization-based decision support system for scheduling part-time computer lab attendants, Omega 22 (1994) 613–626. [15] R. Willis, S. Huxford, Staffing rosters with breaks: A case study, Journal of the Operational Research Society 42 (1991) 727–731. [16] S. Schindler, T. Semmel, Station staffing at Pan American World Airways, Interfaces 23 (1993) 91–98. [17] K. Dowsland, Nurse scheduling with tabu search and strategic oscillation, European Journal of Operational Research 106 (1998) 393–407. [18] F. Easton, D. Rossin, Overtime schedules for full-time service workers, Omega 25 (1997) 285–299. [19] J.F. Bard, C. Binici, A.H. deSilva, Staff scheduling at the United States postal service, Computers and Operations Research 30 (2003) 745–771. [20] M. Isken, An implicit tour scheduling model with applications in healthcare, Annals of Operations Research 128 (2004) 91–109. [21] E.R. Butchers, P.R. Day, A.P. Goldie, S. Miller, J.A. Meyer, D.M. Ryan, A.C. Scott, C.A. Wallace, Optimized crew scheduling at Air New Zealand, Interfaces 31 (2001) 30–56. [22] G. Felici, C. Gentile, A polyhedral approach for the staff rostering problem, Management Science 50 (2004) 381–393. [23] J.F. Bard, H.W. Purnomo, Preference scheduling for nurses using column generation, European Journal of Operational Research 164 (2005) 510–534. [24] H.W. Purnomo, J.F. Bard, Cyclic preference scheduling for nurses using branch and price, Naval Research Logistics 54 (2007) 200–220.
Scheduling part-time personnel with availability restrictions and preferences to maximize employee satisfaction Srimathy Mohan School of Global Management and Leadership, Arizona State University, P.O. Box 37100, Phoenix, AZ 85069-7100, United States Received 9 October 2006; received in revised form 5 November 2007; accepted 26 December 2007
Abstract With the economy continually moving towards retail and services, the size of the part-time workforce is increasing constantly. In this paper, we consider the problem of scheduling a workforce consisting entirely of part-time workers. The part-time workers have availabilities, preferences for the shifts, and also a seniority level. We propose an integer programming model to maximize employee satisfaction and to meet the demand requirements for each shift. The model is tested on randomly generated instances. The results demonstrate the effectiveness of branch-and-cut methods in reducing computation time. c 2008 Elsevier Ltd. All rights reserved.
Keywords: Part-time worker scheduling; Employee satisfaction; Integer programming
1. Introduction The current trend of varying economic cyclic patterns and increasing competitiveness in all industries demands versatility in adjusting employee requirements to avoid over- or under-staffing. Also, the new competitive environment dictates customer service beyond traditional weekday hours. The result of these business realities is a heavy reliance on part-time work. Nearly one in five workers in the United States currently works part-time [1] and the number of part-time employees was more than 27 million in August 2006 [2]. Furthermore, part-time employees constitute a major proportion of entire industries such as service and retail [3]. As a result, intelligent scheduling of part-time employees has become essential for reducing the cost of operations and maintaining employee morale. The North American economy has grown in the last fifty years due to advances in technology, but just as important, due to improvements in the systems and techniques used by managers to coordinate these resources. Classic scientific management techniques such as scheduling, statistical quality control, and project management have all contributed to these advances. In addition, competitive pressures to reduce cost while increasing versatility and flexibility are continuously requiring improvements of another management technique: employee scheduling. The area of employee scheduling has been extensively studied in the literature and many different applications in a wide number of fields have been reported. Blochliger [4] offers a tutorial of staff scheduling problems discussing data requirements, cost modeling, hard (inflexible) and soft (flexible) constraints and objectives. Ernst et al. [5]
E-mail address: [email protected]. c 2008 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2007.12.027
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provide a comprehensive review; they also offer a problem classification, discuss the different application areas (transportation systems, call centers, health care systems and others) and solution methods (artificial intelligence, constraint programming, mathematical programming etc.). Employee scheduling problems typically fall into three classes: days off scheduling, shift scheduling, and tour scheduling. Most employee scheduling models typically deal with full-time personnel and develop minimum cost schedules while satisfying constraints. Part-time employees are typically considered as supplemental workforce and most models attempt to minimize the total operational costs by hiring the minimum number of low cost part-time employees. Also, the models only indicate the number of part-time employees required. The problem of actually selecting the part-time personnel based on their availability still remains. Also, the satisfaction level of part-time employees is rarely considered in scheduling models. With the growth of the service economy, there has been an increase in the need for accommodating two specific needs in employee scheduling — (i) determining specific assignments for the supplemental part-time workers, considering their preferences, and (ii) determining employee schedules in applications where the entire workforce consists of part-time workers. Under such circumstances, the problem no longer deals with determining cyclic schedules that can be repeated or determining days off for the employees. The part-time employees’ availability may vary widely from one time period to another and hence, has to be approached differently from the traditional staff scheduling problems. The existence of real world part-time employee scheduling problems, such as the newspaper industry problem explored by Gopalakrishnan et al. [6], serves as our basis for the creation of a more general model to accommodate more industries. Gopalakrishnan et al. [6] implemented a decision support system that uses a scheduling heuristic for a 16% reduction in the number of part-time employees hired, which translates into a $50,000 annual savings for the Birmingham newspaper company. The research by Gopalakrishnan et al. [6] includes general part-time employee scheduling constraints, considerations for part-time employee seniority, availability and preferences, and meeting shift requirements. They develop a constructive, single pass, rule-based heuristic to schedule part-time employees over a given planning horizon. The heuristic does not incorporate any local improvement procedures and there are no penalties associated with hiring a part-timer for a shift that he/she does not prefer. Another application that includes only part-time employees who are hired on a strictly as-needed basis is the scheduling of substitute teachers in many school districts. The school district maintains a database of available substitute teachers. When there are planned absences, the school district determines the need for the entire week and contacts people on the database (according to seniority and availability) to staff the various classrooms. When there are unplanned absences, the scheduling problem is solved just for that day rather than for the entire week. With more and more people opting to work part-time due to the flexibility it offers, consideration of the part-time employees’ preferences is also becoming important. In this research, we consider an employee pool that consists of only part-time workers. Our primary objective is to develop a work schedule that maximizes the employee satisfaction level. Given employee availability, seniority and preferences, maximum and minimum allowable working hours in a day and a week, and employee requirements for each shift, our model seeks to maximize employee satisfaction while meeting the demand requirements for each shift and satisfying other required constraints. We formulate the problem as an integer program. We define employee satisfaction as a function of seniority, preferences, and availability. This model can be used as an aid during second level planning. During the first level, the part-time worker requirements are determined to minimize overall system costs for the organization. Given this requirement for the various shifts, our model seeks to develop a part-timer work schedule that takes employee preferences, seniority and availability into consideration. The rest of the paper is organized as follows. We present some relevant literature in the next section. Section 3 presents the generic integer programming model and Section 4 presents the results of our experimentation. We present the concluding remarks in Section 5. 2. Literature review Most of the existing literature on workforce scheduling deal with full-time workers. Research falls into three types of problems namely, days off scheduling, shift scheduling, and tour scheduling. Days off scheduling deals with assigning the workers work and non-work days over the planning horizon. Shift scheduling problems deal with determining the actual shifts (start and end times, breaks, etc) during the planning horizon. Tour scheduling problems combine features of days off and shift scheduling. Here, typically, work shifts are to be assigned to cover the daily
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demand for each shift, and each worker’s days off schedule has to be determined. As mentioned earlier, a few studies that do deal with part-time workforce do not provide a work schedule for the part-time employees. Part-time workers are extensively used in all types of industries, and provide great flexibility to an organization. Early research in the area is focused on reducing the number of part-time personnel in days-off scheduling problems. Byrne and Potts [7] consider the scheduling of toll-booth operators to meet the forecasted demand. They provide an integer program to determine the levels of both full-time and part-time work. Baker [8] considers the daysoff scheduling problem with both full-time and part-time workers. The goal is to reduce the number of part-time employees. Some researchers have considered the problem of scheduling part-time workers in financial services such as banks, to supplement full-time workers, especially during peak-hours [9,10]. Krajewski et al. [11] discuss part-time employee use to produce more flexible schedules for the processing of checks in large banks. Their shift-scheduling model assigns a mix of full-time and part-time employees to a number of pre-selected shifts. Glover and McMillan [12] present a heuristic based on Tabu search for producing employee schedules for a full-time work force supplemented by part-timers. Their general employee scheduling problem attempts to produce schedules that in aggregate provide adequate service while respecting existing employee scheduling preferences. Their model is similar to the model we examine in our research in that employees may specify the maximum hours they can work in a week and their availability. The model considers seniority. However, the main goal of that work is to minimize the number of part-time workers needed to satisfy all work requirements. Love and Hoey [13] implement a part-time employee tour scheduling model for a fast-food operation based on network methodology. Their model allows employees to specify their availability and preferences and it considers employee skill ratings in separate work areas. Employees are limited to the number of days they may work in a week and to one shift of 3–8 h a day. Also, their model permits over-staffed schedules. Lauer et al. [14] present an interactive decision support system for a tour scheduling problem arising from a computer lab. The system can be highly responsive to various management objectives and allows managerial interactions by overriding existing schedules. Willis and Huxford [15] consider a staff scheduling problem in a call center involving both part-time and full-time staff. They use an integer programming approach for the problem. Schindler and Semmel [16] consider a workforce problem at Pan American World Airways. They add additional constraints to standard set-covering model formulation to ensure certain constraints for part-time staff. Dowsland [17] considers nurse scheduling with an application at a large general hospital. A Tabu search technique is used. The model incorporates ranks, skills, preferences and equity distribution of rosters; it also considers the use of part-time staff to cover demand. Easton and Rossin [18] examine the issue of staffing at a minimum cost while providing acceptable levels of service and employee satisfaction. The scheduling of a mixed work force with an even distribution of over-staffing yielded consistent customer and employee satisfaction levels. Over-staffing provides higher customer satisfaction with better and more consistent service and higher employee satisfaction with the availability of more flexible work hours. They propose that in the service industry staffing at a minimum cost is not as important a factor as customer and employee satisfaction. The maintenance of high satisfaction levels retains customers and minimizes employee turnover. A few papers have recently considered part-time employees in their research. Bard et al. [19] consider a staff scheduling problem at the United States Postal Service (USPS) that involves both full-time and part-time workers and also the principal constraints of the union contract. Their goal is to examine several scenarios on reducing the size of the workforce. They model their problem as an integer programming formulation and solve it using CPLEX. Isken [20] considers applications of the well-known tour scheduling problem for both full-time and part-time employees in healthcare. Another stream of research on scheduling nurses and airline crew deal with developing work schedules that take into account employee requests for certain shifts and days off [21–24]. The algorithms typically work to minimize the overall labor costs (full-time and supplemental part-time workforce) and try to accommodate as many of the “soft” requirements regarding employee preferences as possible. Most of the concepts and algorithms developed for a full-time workforce are not applicable for a part-time workforce since we are not guaranteed continued availability of the part-time workers. The contractual rules are not as strict either. However, it is still to the advantage of the firm to accommodate the preferences of part-time workers. The workers will be more inclined to work for a firm that provides work schedules that are tailored to their individual needs and preferences. This second level manpower planning problem has not been addressed in the literature and the next section proposes an integer programming model for the same.
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3. Problem definition and formulation The mathematical model discussed in this section is concerned with finding the part-time employee schedule that maximizes employee satisfaction while fulfilling all constraints. Demand in the form of employee requirements for each shift is assumed to be given. Shift lengths vary during a day and across the week. Also, certain restrictions are imposed on the number of hours a part-timer can work during a day and in a given week. Employee assignments maximize employee preferences, while considering their seniority and availability. We will first introduce the assumptions and notation that we use throughout the paper. 3.1. Assumptions and notation 1. Schedules are developed using information provided by the part-time employees on their availability and shift preference, and based on the company’s shift requirements. 2. Shift lengths vary during a day and across a week. All shift requirements have to be exactly met. 3. Unavailable employees can be hired for work if necessary, with a penalty to the objective function. 4. If an employee is scheduled for at least one shift in a day, the number of hours assigned to the employee must be within a pre-specified range. 5. Employees can work more than one shift a day and they need not be consecutive. 6. Daily minimum and maximum hours and weekly minimum hours are fixed for a particular schedule. 7. Every employee’s weekly hours, worked, must be within a pre-specified range. 8. Weekly maximum hours for an employee is either the maximum hours allowed in a week or the maximum hours an employee wants to work in a week. Indices i = part-time employee j = shift of the day k = day of the week Parameters wi = seniority of part-time employee i b jk = number of part-time employees needed for shift j on day k t jk = length of shift j on day k ai jk = current availability of part-time employee i for shift j on day k pi jk = preference of part-time employee i for shift j on day k Dmax = maximum allowable working hours for a part-time employee during a day Dmin = minimum required working hours for a part-time employee during a day Wmax =maximum allowable working hours for a part-time employee during a week Wmin = minimum required working hours for a part-time employee during a week Decision variables 1 if part-time employee i is assigned to shift j on day k Yi jk = 0 otherwise 1 if part-time employee i is hired on day k X ik = 0 otherwise.
3.2. Model formulation Maximize
XXX i
subject to
j
k
Yi jk
wi pi jk ai jk + ai jk − 1 wi
(1)
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X
Yi jk = b jk
(2)
∀ j, k
i
Dmin X ik ≤
X
Yi jk ti jk ≤ Dmax
∀i, k
(3)
∀i
(4)
j
Wmin ≤
XX j
Yi jk ti jk ≤ Wmax
k
Yi jk ≤ X ik
(5)
Yi jk , X ik
(6)
∀i, j, k ∈ {0, 1} ∀i, j, k.
The objective function (1) maximizes employee preferences while considering their seniority and availability. The seniority of a part-time employee,wi , is smaller for a less senior employee when compared to the seniority of a more senior employee. The seniority of an employee can either be directly determined by the duration of service or could be a combination of other factors, including prior experience in similar jobs and formal training in a particular field. For example, in many school districts substitute teachers are part-time employees who are hired from a pool of available people. In this case, the applicant’s prior experience in handling a particular subject and grade level along with training is a better indicator of seniority than just the duration of service. The parameter wi used in the model is generic in that it can be used to represent the duration of service directly or if qualitative factors are involved, the following scheme can be used to represent seniority. The wi for the least senior employee could be assigned a value of 1 and the value of wi for other employees could be increased as the seniority level increases. The increase could be linear or some function of the seniority level. The parameter pi jk is numbers that indicate the preference of the various shifts for each part-time employee. Every part-time employee assigns a preference value for every shift during the planning horizon, irrespective of whether he/she is available during that shift. The parameter pi jk can be a value in the range [0, 10]. The higher the preference value, the more preferred the shift. Availability is a binary integer value where zero indicates unavailable and one indicates availability. Thus, the first term in the objective function captures the seniority of a part-time employee assigned to a shift and the employee’s preference for that shift. The term ai jk − 1 wi serves as a penalty if an unavailable employee is hired to work. This term penalizes the objective function more if a more senior employee is assigned to a shift for which he or she is not available. To summarize, the objective function maximizes the overall satisfaction of the part-time employees by assigning as many senior employees to their preferred shifts as possible. Constraints (1), the shift requirement constraints, ensure that every shift is assigned the required number of employees. Constraints (3) force the maximum and minimum number of hours’ requirement for every part-time employee assigned on any given day. Constraints (4) ensure that an employee is assigned an absolute minimum every week and is not forced to work more than the maximum number of hours during a week. Constraints (5) ensure that a part-time employee is assigned to a shift in a day only if he or she is hired on that day. Finally constraints (6) force X ik and Yi jk to be binary integer variables. When a given instance has a part-time employees, b shifts, and c days, the scheduling model contains (abc + ac) binary integer variables and a (bc + 2)+c (2a + b) constraints. Hence, even for a small sized application with 50 parttime employees who have to be scheduled for 5 days, each with 2 shifts, the model has 750 binary integer variables and 1110 constraints. But the constraint matrix is a very sparse matrix and more than half of the non-zero entries have a value of 1. Hence, we conjecture that a generic branch-and-bound algorithm should be able to solve the problem within a reasonable amount of time. We tried to solve reasonably sized problems and present the details and results in the next section. 4. Experimentation The part-time employee scheduling model presented in the previous section can be solved using standard enumeration schemes available in commercial solvers like CPLEX. However, the size of the problem may sometimes make some instances computationally intractable. Hence, we decided to test the effectiveness of solving these models using a standard branch-and-bound algorithm on a set of randomly generated problems. We generated two classes of problems — one with 50 employees and another with 100 employees. For each class, we generated problems with three different overall employee availability levels — 25%, 50% and 75%. An availability
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S. Mohan / Mathematical and Computer Modelling 48 (2008) 1806–1813 Table 1 Effectiveness of adding cuts to the branch-and-bound enumeration procedure Instance
100 employees, 75% availability Cut % LP gap
100 employees, 50% availability Cut % LP gap
100 employees, 25% availability Cuts % LP gap
1 2 3 4 5 6 7 8 9 10
3.02 1.04 2.88 1.11 6.24 1.45 1.12 0.93 0.54 0.67
7.32 4.90 14.99 5.43 15.86 11.19 5.43 4.83 4.79 5.00
0.76 0.17 0.41 0.25 0.12 0.37 0.67 0.90 0.17 0.68
9.70 5.61 13.34 6.75 14.25 10.43 6.52 5.00 5.26 6.33
0.00 0.02 0.20 0.00 0.28 0.14 0.00 0.00 0.00 0.00
7.02 5.90 11.89 7.75 11.57 8.99 5.89 3.97 5.22 6.53
Average
1.90
7.97
0.45
8.32
0.06
7.47
Instance
50 employees, 75% availability Cut % LP gap
50 employees, 50% availability Cut % LP gap
50 employees, 25% availability Cuts % LP gap
1 2 3 4 5 6 7 8 9 10
4.08 0.53 2.28 2.35 1.55 0.56 2.78 0.52 0.99 0.59
10.41 5.80 15.51 10.64 5.50 5.57 9.64 5.24 5.00 5.49
0.39 0.00 0.48 0.63 0.53 0.00 0.50 0.03 0.11 0.00
10.73 7.28 12.93 12.62 7.01 5.77 9.14 5.95 4.91 5.20
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
8.86 6.77 13.96 8.43 6.36 6.11 7.74 6.89 6.58 4.39
Average
1.62
7.88
0.27
8.15
0.00
7.61
Numbers in bold indicate an optimal solution.
level of 25% implies that the ratio of number of shifts for which part-time employees are available to the total number of shifts to be scheduled is 25%. For each combination of number of employees and availability level, we generated 10 feasible instances to test the scheduling model. The other parameters were generated as follows. The time horizon considered for generating a schedule is 1 week (5 days) with 3 shifts per day. Employees are randomly assigned a seniority ranging from 1 to 10, where 10 indicates the most senior position and 1 the least senior. Each employee has to work a minimum of 12 and a maximum of 40 hours per week. If a part-time employee is hired on a given day, he/she has to work for at least 2 shifts that day. The number of people required per shift is generated from a discrete uniform distribution between 30% and 50% of MAX, where MAX = number of employees * availability level. The duration of each shift (in hours) is generated from a uniform distribution between 4 and 6. For each part-time employee, the preference for each shift is a number randomly chosen between [0, 10]. The integer programming model is solved using the basic branch-and-bound enumeration algorithm in CPLEX solver. We allow a maximum limit of 30 min for CPLEX to determine the optimal solution. If an optimal solution is not found during that time, CPLEX reports the best feasible solution. Our preliminary experimentation indicated that even for the 50 employee problems, when availability increased from 25% to 50%, CPLEX failed to produce optimal solutions in 80% of the problems. As availability increased, the solution space also increases and the branchand-bound algorithm has a much bigger tree to search through. Hence, the basic branch-and-bound algorithm failed to produce an optimal solution for several problems within the 30 min time limit. Integer programming solve times can often be improved by generating new constraints (or cuts) based on polyhedral considerations. These additional constraints tighten the feasible region, reducing the number of fractional variables to choose from when CPLEX needs to select a branching variable. The CPLEX program generates several kinds of cuts, some of which are generalized upper bound cuts, clique cuts, cover cuts. The cut generation takes some processing time, but could potentially reduce the overall time required to find an optimal solution. In order to investigate the usefulness of the CPLEX cuts for our problem, we solved the same problems with the additional cuts
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Table 2 Comparison of solution times (in seconds) Instance
100 employees, 75% availability Cuts No cuts
100 employees, 50% availability Cuts No cuts
100 employees, 25% availability Cuts No cuts
1 2 3 4 5 6 7 8 9 10
1800 1.04 1800 40.45 1800 1800 113.13 0.86 0.7 0.7
1800 1800 1800 1800 1800 1800 1800 1800 1800 1800
13.17 1.21 1800 2.15 1800 2.15 1800 26.5 15.4 0.67
1800 1800 1800 1800 1800 1800 1800 1800 1800 1800
0.41 0.12 1800 0.35 1800 0.87 0.2 0.12 0.1 0.6
174.32 1800 1800 64.24 1800 618 1800 23 133.29 100
Average
735.69
1800.00
546.13
1800.00
360.28
831.29
Instance
50 employees, 75% availability Cuts No cuts
50 employees, 50% availability Cuts No cuts
50 employees, 25% availability Cuts No cuts
1 2 3 4 5 6 7 8 9 10
752 1.93 1800 1800 4.24 0.56 1800 0.39 0.58 0.93
1800 1800 1800 1800 1800 1800 1800 1800 1800 1800
16.43 0.23 37.26 2.85 0.3 0.11 1.18 0.96 0.33 0.24
1800 958 1800 1800 1800 4.96 1800 1800 1800 1800
0.18 0.19 0.08 0.12 0.14 0.23 0.33 1.1 0.12 0.06
2 0.56 7.53 38.67 0.44 1.06 11.32 502.03 13.78 0.31
Average
616.06
1800.00
5.99
1536.30
0.26
57.77
during the branch-and-bound enumeration. Our objective here is to illustrate the effectiveness of cuts for this problem and the savings in computation time. We also determined the optimal solution for the linear programming relaxation of each instance, which provides an upper bound (LP upper bound) on the objective function value of the corresponding integer programming model. We tracked the performance of the solution procedures using two measures. (Solution value with cuts − solution value without cuts) ∗ 100% Solution value with cuts (LP upper bound − solution value with cuts) LP Gap = ∗ 100%. LP upper bound Cut % =
The cut % measures the improvement in solution quality by using the branch and bound with cuts and the LP gap provides an overall idea about how effective the solution procedure is. The experimental results are presented in Tables 1 and 2. Table 1 provides the solution vales for all 60 instances using the branch-and-bound procedure, both with and without cuts. The table also provides the LP upper bound for each instance. The results show that adding cuts definitely improves the chances of finding an optimal solution as the problem size and availability level increase. For the 100 employee problem, the branch-and-bound procedure without cuts found the optimal solution for 5 instances when availability was 25%. When the availability increased to 50% and 75%, it did not find any optimal solutions. On the other hand, the branch-and-bound procedure with cuts, found optimal solutions for 7 instances when availability was 50% and for 6 instances when availability was 75%. As availability increases, the solution space also increases and it would naturally take longer to determine the optimal solution. However, when we add the cuts, it effectively eliminates regions of the solution space and the probability of finding the optimal solution within the specified time limit increases. On an average, the solution with cuts is 2% better than the solution obtained without cuts for the 100 employee problems at 75% availability level. The comparative values for the 50% and 25% availability levels are 0.5% and 0.06%. CPLEX produces the improved
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solutions with the cuts in a much shorter time. Table 2 presents the results on the time taken by the two procedures to solve the 60 instances. The branch and bound with cuts takes 735 s on average to solve the 100 employee problems with 75% availability. The regular branch and bound took 1800 s since an optimal solution was not found for any of the instances. Even in the case where both procedures obtained optimal solutions for all 10 instances (50 employees at 25% availability), the branch and bound with cuts took 0.26 s and regular branch and bound took 57 s, on average. An analysis of the gap between the LP upper bound and the solution produced by the branch and bound with cuts indicates that on an average the gap is around 8%. The gap seems to be mainly due to the weak upper bound since the solution procedure obtained optimal solutions for many of the instances. 5. Concluding remarks We have presented a generic model for scheduling part-time workers while considering shift length and other related constraints. The unique aspect of our model is that it aims to maximize the overall satisfaction of a parttime workforce. Typically, part-time workers have been used as a supplemental workforce and the objective has been to minimize overall cost of the additional resources hired. In our research, we present a second level planning model to determine a part-time worker schedule to maximize employee satisfaction. Satisfaction is measured using a combination of employee seniority and preferences. Our experimentation shows that reasonable size instances of the integer programming model can be solved relatively quickly using a branch-and-bound enumeration procedure with additional cuts. Directions for future research include developing specialized solution procedures for the part-timer scheduling model and testing the proposed methodology in real world scenarios. References [1] A. Kalleberg, Nonstandard employment relations: Part-time, temporary and contract work, Annual Review of Sociology 26 (2000) 341–365. [2] Bureau of Labor Statistics, The employment situation news release: August 2006. http://stats.bls.gov/news.release/pdf/empsit.pdf. [3] N. Conway, R.B. Briner, Full-time versus part-time employees: Understanding the links between work-status, the psychological contract, and attitudes, Journal of Vocational Behaviour 61 (2002) 279–301. [4] I. Blochliger, Modeling staff scheduling problems — a tutorial, European Journal of Operational Research 158 (2004) 533–542. [5] A.T. Ernst, H. Jiang, M. Krishnamoorthy, D. Sier, Staff scheduling and rostering: A review of applications, methods and models, European Journal of Operational Research 153 (2004) 3–27. [6] M. Gopalakrishnan, S. Gopalakrishnan, D.M. Miller, A decision support system for scheduling personnel in a newspaper publishing environment, Interfaces 23 (1993) 104–115. [7] J. Byrne, R. Potts, Scheduling of toll collectors, Transportation Science 7 (1973) 224–245. [8] K. Baker, Scheduling full-time and part-time staff to meet cyclic staffing requirements, Operational Research Quarterly 25 (1974) 65–76. [9] V. Mabert, A. Raedels, The detail scheduling of a part-time work force: A case study of teller staffing, Decision Sciences 8 (1977) 109–120. [10] S. Moondra, A linear programming model for work force scheduling for banks, Journal of Bank Research 6 (1976) 299–301. [11] L.J. Krajewski, L.P. Ritzman, P. McKenzie, Shift scheduling in banking operations: A case application, Interfaces 10 (1980) 1–7. [12] F. Glover, C. McMillan, The general employee scheduling problem: An integration of management science and artificial intelligence, Computers and Operations Research 13 (1986) 563–593. [13] R.J. Love, J.R. Hoey, Management science improves fast-food operations, Interfaces 20 (1990) 21–29. [14] J. Lauer, L. Jacobs, M. Brusco, S. Bechtold, An interactive, optimization-based decision support system for scheduling part-time computer lab attendants, Omega 22 (1994) 613–626. [15] R. Willis, S. Huxford, Staffing rosters with breaks: A case study, Journal of the Operational Research Society 42 (1991) 727–731. [16] S. Schindler, T. Semmel, Station staffing at Pan American World Airways, Interfaces 23 (1993) 91–98. [17] K. Dowsland, Nurse scheduling with tabu search and strategic oscillation, European Journal of Operational Research 106 (1998) 393–407. [18] F. Easton, D. Rossin, Overtime schedules for full-time service workers, Omega 25 (1997) 285–299. [19] J.F. Bard, C. Binici, A.H. deSilva, Staff scheduling at the United States postal service, Computers and Operations Research 30 (2003) 745–771. [20] M. Isken, An implicit tour scheduling model with applications in healthcare, Annals of Operations Research 128 (2004) 91–109. [21] E.R. Butchers, P.R. Day, A.P. Goldie, S. Miller, J.A. Meyer, D.M. Ryan, A.C. Scott, C.A. Wallace, Optimized crew scheduling at Air New Zealand, Interfaces 31 (2001) 30–56. [22] G. Felici, C. Gentile, A polyhedral approach for the staff rostering problem, Management Science 50 (2004) 381–393. [23] J.F. Bard, H.W. Purnomo, Preference scheduling for nurses using column generation, European Journal of Operational Research 164 (2005) 510–534. [24] H.W. Purnomo, J.F. Bard, Cyclic preference scheduling for nurses using branch and price, Naval Research Logistics 54 (2007) 200–220.