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...that face this problem. In each case, management's goal is to find the best mix of hourly employees so that demand is satisfied at minimum cost. The problem is complicated by labor laws, union contracts and aperiodic, random fluctuations in demand.
In the most general sense, workforce scheduling can be viewed temporally and decomposed along the time axis. This suggests a hierarchical analytic approach. In the long run, the goal is to find weekly schedules or tours for each employee in a given skill category (Emmons and Burns, 1991; Billionnet, 1999). This means specifying the work days, their length, the daily start time, and the lunch break. The case addressed in this paper is tour scheduling at United State Postal Service (USPS) mail Processing and Distribution Centers (P & DCs) where these specifications constitute a "bid job" (Showalter et al., 1977; Jarrah et al., 1994). The results are applicable, however, to virtually all sectors of the service industry.
At the second level in the hierarchy, adjustments must be made to the regular workforce on a weekly basis to account for planned leave and expected departures from average demand. To accomplish this, critical resources must be tracked and evaluated. The goal is to provide weekly schedules that balance overtime and the use of part-time labor so that all requirements are met at minimum cost and at minimum deviation from the bid jobs. This can be thought of as a replanning problem.
Finally, at the day-to-day level, supervisors must deal with unplanned absenteeism, machine breakdowns, and unexpected spikes in demand; i.e., uncertainty. This is a real-time scheduling problem that falls under the heading of control. In the airline industry where bad weather, for example, can occasion flight delays and cancellations (Bard et al., 2001; Clausen et al., 2001), the goal is to get back on track as soon as possible at minimum cost and with minimum deviation from the published schedule.
In this paper, we focus on long-term planning issues and attempt to quantify the benefits of permitting workers with higher level skills to be assigned to jobs (shifts) associated with lower level skills. The same analysis, however, can be performed for weekly and daily scheduling. The procedure in which higher skilled workers are substituted for lower skilled ones is sometimes called downgrading (Dawid et al., 2001). It is especially relevant at the 275 P & DCs around the US which employ several hundred thousand mail processors.
Emmons and Burns (1991) were among the first to address downgrading and to develop a solution methodology for a simplified version of the problem. They considered a facility that had to be staffed 7 days a week with requirements for m different types of workers. These requirements were constant from day to day. Several bounds on the workforce size were derived, and then shown to be exact. In view of these results, they proposed a series of algorithms to find the optimal size and mix, such that each worker was given 2 days off a week, including A out of B weekends off, and only work-stretches between 2 and 5 days were permitted. In addition, they considered the case in which a minimum number of workers in each skill category is required even when extra higher qualified personnel are on duty.
Malhotra et al. (1992) and Malhotra and Ritzman (1994) were among the first to consider the welter of issues faced by the postal service in scheduling a flexible workforce. Billionnet (1999) formulated the downgrading problem as an Integer Program (IP) and was able to find solutions to randomly generated instances in less than a minute in most cases using the modeling system MPL. He considered values of m between two and five, along with the requirement that each worker be given from 2 to 4 days off. The IP returned the minimum number of workers needed per day, by type. A second IP was used as a post-processor to find weekly schedules that maximized the number of consecutive days off.
The most recent work on downgrading concerns the rostering of airline crews (Dawid et al., 2001). This problem is typically solved once a month and involves the construction of individualized schedules that take into account various pre-assignments, such as training and observer flights, as well as crew requests, such as days off and preferred flights. Rostering fits squarely in the second tier of the personnel scheduling hierarchy. The authors present an efficient branch-and-bound scheme that is able to solve real-world applications, most within 10 seconds. To demonstrate the benefits of downgrading, they investigated a single instance associated with a medium-sized European carrier with two types of crew categories (25 pilots and 30 copilots). Their analysis showed that without the ability to substitute pilots for copilots, it was not possible to obtain a feasible solution to a problem with 280 work stretches.
In the next section, we define the problem to be investigated and state the underlying assumptions. This is followed by a presentation of the integer programming model in Section 3 along with the proposed solution methodology in Section 4. Results for the Dallas P & DC are highlighted in Section 5. We close with a discussion of implementation issues and suggestions for improving the USPS application.
2. General background and problem components
One of the earliest studies on tour scheduling at USPS facilities was undertaken by Ritzman et al. (1976). They used data for a typical week broken down into 84 2-hour periods, and proposed several rules for choosing tours (e.g., add the tour that results in the largest reduction of unprocessed mail). In their algorithm, tours are generated, assigned a fixed weight reflecting the importance of the accompanying rule, and put on a candidate list. A biased random sampling scheme is then used to select the next tour from the list.
In a related study, Showalter et al. (1977) were concerned with specifying weekly tour start times, work center assignments, and mail class responsibility for each employee. Using a construction heuristic with the simultaneous objectives of maximizing the mail throughput rate, minimizing the number of Sunday and night man-hours worked, and minimizing the number of daily tour start times, they solved several hypothetical problems involving the assignment of 800 employees to 20 different work centers handling four mail classes with two to five processing stages.
To put the staff scheduling problem in context, one can think of P & DCs as high volume factories, the same as those commonly found in the processing industry. The primary difference is that the raw material (mail) enters the center in aggregated form and is disassembled as it is worked. The opposite is true in traditional manufacturing. Figure 1 depicts the major mail flows in a P & DC. Arriving mail enters at the left of the diagram, and depending on its attributes and previous level of processing, is channeled through the various workstations. In the process, it is barcoded with an 11-digit number representing the final address, and dispatched to a local Delivery Unit (DU) for hand delivery by a carrier. In the diagram, outgoing and turnaround mail correspond to locally collected mail that is either dispatched outside the area or delivered within the area, respectively. Incoming mail is the mail that arrives from outside the area to be delivered locally.
Like most government organizations, the USPS is highly unionized and consequently bound by a rigid set of rules that limits its options in developing weekly schedules and adjusting daily assignments. The bulk of the workforce at P & DCs are equipment operators known as clerks, and material handlers who are responsible for mail transport within the facility. Our analysis is restricted to the clerk craft which comprises about 80% of that workforce. As currently defined by the national labor agreement, there are nine types of clerks.
Table 1 lists the full set of labor categories as well as abbreviations used to reference the equipment in Fig. 1. Those at the top left of the table are the most highly skilled in terms of the functions they perform and the equipment they operate, as well as the most highly paid. Union rules require that clerks be given assignments that match their skills, but may be downgraded as need be, as long as they are paid their normal wage. Table 2 lists the permissible substitutions for clerks and mail handlers.
A P & DC has three types of career employees: Full-Time Regulars (FTRs), Part-Time Regulars (PTRs), and Part-Time Flexibles (PTFs). A regular employee is given a 5-day a week schedule (a tour), and may...
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