WORKFORCE SIZING AND SCHEDULING FOR A SERVICE CONTRACTOR USING INTEGER PROGRAMMING

Operations Research is perceived to be on the verge of demise as a decision support tool in industry. However, this is not true, since the relevancy and interdisciplinary nature of Operations Research makes it an indispensable part of operations management. What should rather be asked is how Operations Research is introduced to undergraduate industrial engineering students. This paper reports on one of the optimisation initiatives undertaken for a service contractor using integer programming as part of a semester project. Although the model addresses specific business issues related to the contractor, it can easily be generalised for other applications.


INTRODUCTION
Operations Research is taught at numerous South African engineering and management faculties at tertiary level, but acceptance of the optimisation techniques are often questioned at shop-floor level in industry.Leinbach and Stansfield [3] address numerous complaints from the operational level with regard to industrial engineering (IE) professionals.Many industrial engineers have lost touch with the action in companies and spend a lot of time on complicated models where the assumptions are so plentiful that it is impossible to challenge the validity of these models over time.
In discussing the denouement of Operations Research, Ackoff [1] identifies three major effects on the practice of Operations Research as a result of academics' obsession with techniques.Firstly, problematic situations are frequently sourced, selected and distorted so as to favour the application of a specific technique.A second effect is the diluted application of techniques as a result in the techniques being introduced to diverse professionals with little background of the fundamentals of operations research.The third, and probably the most detrimental effect is the classification of operations research as an isolated discipline, as opposed to the original interdisciplinary characteristic of Operations Research.
The thrust towards identifying and solving real-life problems using Operations Research is expressed in Joubert and Steyn [2], as their paper reports on a project undertaken for an undergraduate module in Operations Research at third-year level.The emphasis of their endeavour is to enable students to identify an opportunity for improvement where basic modelling techniques can be employed.The opportunity need not require complex solution methods, as the purpose of the project is to foster students confidence with regard to modelling and optimisation initiatives, and to add value to real-life situations in the process.Thie paper reports on the initiative, and the solution, from one of the projects submitted for the third-year Operations Research module in the Department of Industrial and Systems Engineering at the University of Pretoria.

PROBLEM ENVIRONMENT
The Ground Services function oversees the overall tidiness of the University of Pretoria's campuses, and the responsibility for the main campus grounds is contractually outsourced to a private company.In an attempt to assist the contractor to ensure a favourable outcome when the contract is reviewed at the end of 2004, the authors have embarked on an optimisation exercise.
The outsourced contract requires the contractor to fulfil the complete maintenance function for five distinct areas, namely lawns, flowerbeds, parking areas, pavements, and roads.
The result of the current scenario, however, is inadequate service levels as not all work can be completed on a weekly basis with the current work allocation.Currently 27 workers are employed during the week, and 14 workers on Saturdays, resulting in a weekly wage expenditure of R12,480.Workers are randomly assigned to tasks and areas, and the allocation of tasks follows certain general business rules: • workers should preferably perform at least three tasks per 12-period week, • no worker may do the same task for more than two consecutive time periods, • a constant number of workers are required from Mondays to Fridays: time periods 1 through 10.For example, if twenty workers are assigned for weekdays, then the total number of weekday workers must add up to twenty for every week, although the twenty workers may be assigned to different tasks on different days in different weeks; • a different number of workers may be employed on Saturdays: time periods 11 and 12.
Weekend workers are treated as contract or piece-workers, and it would be possible to hire them on an ad-hoc basis.Weekend workers may, or may not include weekday workers, as they are treated independently.The allocation of duties may vary, but the number of weekend workers must be constant over the various weeks.
• the total number of workers employed should remain constant for the duration of the contract.This holds true for both the total number of weekday workers, as well as the total number of weekend workers.

Further constraints are
• that all work must be completed, and • equipment availability limits the number of workers that can be assigned to specific tasks.
Information regarding the area sizes ( 2 m ), and the 'process times' ( 2 min/ m ) is obtained and used to determine the number of periods that are required for each task.This calculation is done separately for each season.

WORKFORCE SIZING AND SCHEDULING
Schrage [6] states that covering problems tend to rise in service industries, as the crucial feature is the set requirements that must be covered.Various activities are available, each of which helps cover some, but not all, the requirements.In this paper the requirements are the number of periods that are required for each ground task to be completed, while the activities refer to the shift patterns or schedules.Although this type of problem is also referred to as Shift Scheduling Problems [4] or Staffing Problems [6], the authors find the term Workforce Sizing and Scheduling to be the most descriptive [7].The objective is to determine the minimum number of workers required to perform all the necessary work, and to schedule them accordingly.An upper limit of 50 workers is arbitrarily chosen, since the current 41 workers do not yield a feasible solution (all work cannot be completed).The upper limit will only impact on the size of the model (number of variables and constraints).The computational time to find the optimal solution, however, proved insensitive to the upper limit chosen.

Current equipment levels
In modelling the workforce-sizing and scheduling model, the main decision variable, , determines whether, and when a specific worker is assigned to a job.ijkm x http://sajie.journals.ac.za if worker k does job i in period j of season m , where be the number of people (workers) required to perform task i in period j of season m , where subject to The objective function indicated in (1) minimizes the total number of workers employed, where 1 c and 2 c denote the daily wage for a worker on weekdays, and on a Saturday respectively.The balancing factor β attempts to level the workload during weekdays (the first 10 periods of the work-week).Constraints (2) through (5) serve to calculate the total number of workers employed during the week and on Saturdays, based on the model's assignment of workers over the two seasons.The minimum time allocation per task is enforced by (6).Limited tools are addressed in (7) since it influences the number of workers assigned to a task requiring limited resources.The introduction of a balancing factor, β , in (8) and ( 9), attempts to level the workload.The matching of workers and task requirements are addressed in (10).Repetition of tasks is limited by (11), while (12) ensures that each worker is only assigned one task during a given period.
Job rotation is the process of shifting workers from one task to another, in an attempt to alleviate "over-routinisation". Robbins [5] indicates that the strengths of job rotation are that it reduces boredom and increases motivation through diversifying the worker's activities.The model attempts to assign workers to tasks in a multi-skilled fashion in (13) and ( 14).The value M represents a sufficiently large number such that .In the model a value of is used.
The solution of the proposed model indicates a workforce of 28 workers working from periods 1 through 10, and an additional 11 workers working periods 11 and 12.The optimal /sajie.journals.ac.za