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Scheduling preventive maintenance in a parallel machine
environment with workforce dependent processing times
Master’s thesis, MSc Technology & Operations Management
University of Groningen, Faculty of Economics and Business
June 17, 2014
Abstract:
In literature, studies on scheduling techniques with workforce constraints often divide the workforce in classes based on skillsets. Whenever personnel are not within a skillset, they are unable to perform the task. In practice, these constraints do not fully represent reality. For example, less skilled personnel may
be able to do the task, but with a longer duration. These workforce dependent processing times create a critical resource allocation problem. This research aims at exploring this problem, and tries to create an approach generally applicable to scheduling problems with these type of constraints. The technical service
department at Royal Friesland Campina B.V., a cheese factory in the northern part of the Netherlands, has workforce dependent processing times and is therefore used as a worked example. Solving the problem using an extension of the Apparent Tardiness Cost (ATC) rule deemed highly complex, as the problem is NP-hard. For this reason, two heuristics are proposed. One heuristic aims at minimizing the makespan,
while the other aims at doing the high priority tasks first. Both heuristics seem to perform well when compared to the current performance of the schedule, but further research is needed to thoroughly experiment with them. Furthermore, even though the scheduling problem is NP-hard, two integer linear
programs (ILP) are given. One provides a solution with a limited amount of tasks and the objective to minimize the makespan. The optimal solution serves as a comparison for the performance of the first heuristic. The second ILP gives trivial results. The research explores the scheduling problem in a parallel
machine environment with workforce dependent processing times, with priorities, no set-up times, no sequence dependencies, and no due dates.
Tim de Wolf
Student Number: s1796623
Supervisors:
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Contents
Preface ... 3
1. Introduction ... 4
2. Methodology ... 6
3. Preliminary problem exploration ... 7
3.1 Current scheduling technique ... 7
3.2 Potential improvements and objectives ... 7
4. Theoretical framework ...10
4.1 Addressing maintenance and maintenance policies in general ...10
4.2 Gantt Charts ...10
4.3 Maintenance scheduling techniques ...11
4.3.1 Minimizing the makespan ...11
4.3.2 Workforce constraints ...12
4.3.3 Priority rules ...12
4.4 Significance of variance in maintenance scheduling...13
5. Scheduling preventive maintenance at Royal Friesland Campina Bedum B.V. ...15
5.1 Formalizing the workforce ...15
5.2 Proposed solution ...15
5.2.1 Evaluation and validation ...15
5.2.2 User feedback ...16
5.2.3 Concluding remarks ...17
6. Scheduling preventive maintenance tasks with workforce dependent processing times ...18
6.1 Using priority rules to efficiently schedule tasks ...18
6.2 Two scheduling heuristics for tasks with resource dependent processing times ...19
6.2.1 A scheduling heuristic for tasks with resource dependent processing times without
priorities ...19
6.2.2 A scheduling heuristic for tasks with resource dependent processing times with
priorities ...20
6.3 Integer linear program formulations for both heuristics ...20
7. Discussion of the results ...23
8. Conclusions ...24
9. Suggestions for further research ...25
References...26
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Preface
This paper is primarily addressed to members of technical service departments who pursue
improvements in scheduling maintenance. Furthermore, it is addressed to anyone interested in solving the critical resource allocation problem with workforce dependent processing times.
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1. Introduction
In industrial companies, maintenance strategies prove to be crucial when the company tries to ensure maximum availability of their production equipment at the lowest cost (Sameta, Chelbia, and Al Ben Hmidab, 2011). Creating a maintenance strategy, tailored specifically to the needs of the company is difficult, as conflicting goals of the company result in a tradeoff between the costs of maintenance staff and the availability of maintenance staff (Koochaki, Bokhorst, Wortmann, and Klingenberg, 2013). Therefore, to ensure the limited maintenance staff is deployed optimally, a maintenance schedule which allocates all available resources in a time efficient manner is necessary.
Preventive maintenance is often used, as it aims to do maintenance where the task is performed before a system fails (Muchiri, Pintelon, Gelders, and Martin, 2011). Scheduling preventive maintenance effectively is not straightforward however. Variables needed to create an efficient schedule are often difficult to determine. Firstly, estimating the average duration and variance of sets of preventive maintenance tasks is crucial to the schedule. These will determine what paths, within the sets of available preventive
maintenance operations, are critical (e.g. the sequence of operations which add up to the longest overall duration) (Sarin, Nagarajan, and Liao, 2010). Secondly, workforce constraints are often present. The workforce constraints refer to a specific set of available maintenance engineers which limit the number of feasible outcomes of a schedule, since each task requires for its execution a given number of personnel (Koochaki et al., 2013). Thirdly, maintenance personnel usually differ in their respective skillsets, therefore the average duration and variance of maintenance tasks may differ between personnel.
Similarly, the maintenance personnel may sometimes not even be able to execute certain tasks if these are not within their skillset (Gopalakrishnan, and Ahire, 1997). Deciding what preventive maintenance tasks to do, when, and by whom, is a critical resource allocation and scheduling problem.
In literature, numerous linear (integer) programs (i.e. methods for solving linear program problems, or optimization problems, in which the goal function and constraints are linear) have been developed that try to schedule preventive maintenance efficiently (Yin, Wu, Cheng, and Wu, 2014; Wong, Chan, and Chung, 2013; Hsu, Ji, Guo, and Yang, 2013; Cui, and Li, 2004; Gopalakrishnan et al., 1997). Models have been developed that address continuously operating systems, as well as intermittently operating systems with variable processing times and workforce constraints. While these programs achieve their respective goals relatively well, no models have been developed which incorporate workforce dependent processing times. Often, the workforce consists of a variety of features. For example experienced, inexperienced, single-skilled, multi-skilled personnel, young and old. These attributes influence the duration of
preventive maintenance tasks and therefore change the scheduling problem fundamentally. In literature, no models have been developed which incorporate workforce dependent processing times, while the scheduling problem is very relevant for numerous real-life scheduling scenarios.
This is a design science research at the company Royal friesland Campina B.V., a cheese factory located in Bedum, Groningen, the Netherlands. The company has limited maintenance personnel, while the factory runs twenty-four hours a day, seven days a week. Performing maintenance during production is
inconvenient, because whenever maintenance personnel detect that production equipment needs maintenance, stopping production to do the required task is costly, because the ingredients being
processed are perishable. Whenever production stalls for more than 15 minutes, the quality of the cheese produced will deviate significantly from the standards, resulting in costly re-inspections and waste. Thus, whenever a maintenance task is not highly urgent, it is scheduled for a later point in time. To minimize machine downtime and corrective maintenance (i.e. maintenance where the task is performed once a system has failed), each year several weeks are selected for doing preventive maintenance. These revision weeks are specifically meant to do preventive maintenance tasks, for that reason no production is
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project. Task durations, which vary between engineers, are not taken into account. As a result, no insight into the completion times of the revision weeks is available, nor is it possible to determine what tasks caused delays. The research therefore focuses on creating an approach to create a flexible preventive maintenance schedule for these revision weeks. The goal is to explore the resource allocation problem and create an approach that will be generally applicable to workforce dependent processing time scheduling problems in a resource dependent parallel machine environment. The data from Royal Friesland Campina B.V. will be used as a worked example.First, section 2 will present the methodology used throughout the research. Section 3 will explore the problem and formalize it. Based on the preliminary problem exploration, section 4 will address literature relevant for solving the problem. Section 5 will propose a for Royal Friesland Campina B.V. Furthermore, the solution will be validated in this section with user feedback. Section 6 will explore ways to generalize it. Two potential heuristic solutions for the scheduling problem are presented in section 6.2. Section 7 and 8 contain the discussion of the results and the conclusions respectively. Finally, section 9 presents
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2. Methodology
The research aims at developing an approach to create a flexible preventive maintenance schedule for the revision weeks at RFC. A new, more efficient, scheduling technique is needed, and this can be seen as a design science problem, because it requires a practical solution for changing the current situation (Van Strien, 1997). In design research, two types should be distinguished. Type 1 aims to improve a specific situation which leads to new theoretical results. With these theoretical findings one can create theoretical frameworks to generalize the situation. Design research of type 2 first experiments in building a particular artefact or system, which will then lead to validated design principles. Methodological principles of both types will be used in this research. First, a scheduling technique will be created for RFC, in order to improve their current approach. This is design research of type 2, as it tries to find a practical solution, and does not aim to improve, and expand existing literature. Then, the scheduling problem will be
generalized and formulated formally using heuristics and linear programs. This is design science of type 1, as it leads to new theoretical results regarding the scheduling problem.
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3. Preliminary problem exploration
3.1 Current scheduling technique
Royal Friesland Campina Bedum B.V. (RFC) uses a manual scheduling technique to allocate the candidate preventive maintenance tasks for the revision week to the maintenance personnel. The scheduler makes use of a simple heuristic. Tasks which require a specific time slot (when for example, an operator is needed to assist the maintenance engineer or the supplier of the machine needs to be present) are scheduled first. The scheduler then assesses based on his own professional experience, in coordination with another engineer, which tasks are important, and allocates these to engineers he believes are capable of performing the task within that week without major complications. The other tasks are also allocated based on priority, and availability, of the remaining personnel. It is for this reason that the tasks which are not very urgent are postponed to a later point in time or a different revision week. A typical schedule for a revision week is added in Appendix 1, the names of the 12 engineers, the operators, and suppliers are removed for reasons of confidentiality. The same applies for the names of the tasks, which are numbered 1 to 37.
The need for creating a more clear and efficient scheduling technique becomes evident when the
makespan (i.e. the time elapsed when the last task is completed) of the revision week is computed, based on estimations of task durations. The estimations in hours are depicted in Appendix 2 and are provided by experienced members of the technical service department of RFC. They also estimated, on a scale of 1 to 10, how likely the engineer is to have complications or delays while performing the task. If, for illustration purposes, all engineers would perform as a class B engineer (i.e. less experienced engineers), preliminary results show that the makespan of the project is 16 hours. A makespan of 16 hours, given that there are no complications, implies that, of a 40 hour work week, only 40% of the time is used for the tasks. Indeed, some tasks can only be performed later in the week because, for example, the supplier of the machine must be present during the task, but it shows that either production can start earlier, or more preventive maintenance tasks can be performed during the week. The schedule to achieve the 16 hour makespan is shown in Appendix 3 in a Gantt Chart. The different colors of the scheduled tasks each represent a different engineer, furthermore the long stretches of duration for a task which pass through a light grey area from 05:00h to 07:00h depict tasks that have to be continued the following morning. The light gray area represents the non-working hours.
3.2 Potential improvements and objectives
It seems that the current scheduling technique is relatively straightforward and suboptimal. The scheduling is done manually without using a scheduling tool. For this reason, the duration of the
maintenance tasks is not taken into account and there is no insight into the completion time of the project or the occurrence of any delays. Furthermore, the duration of the tasks depend on the assigned engineer. Without taking these workforce dependent processing times into account, it is impossible to create a schedule which allocates the available resources optimally. Using a scheduling tool to allocate the tasks efficiently, while taking the durations of the tasks into account, will create a schedule that provides insight in the project graphically. In addition, it also increases flexibility, because scheduling tools allow for immediate adjustments during the project. Finally, task prioritizing is done based on the professional experience of the scheduler. While these estimates may approach the actual values, formalizing it by means of priority rules will benefit scheduling efficiency, because the prioritizing rules will automatically (via script programming) generate a schedule which approaches optimality.
In order to improve the scheduling technique, two objectives must be distinguished. Mainly, because the durations of the tasks do not only depend on the assigned engineer, but the durations itself vary
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as quickly as possible. However, priorities may influence the allocation of the tasks. Consider, for example, several small duration tasks which have a high priority. If we allocate the more urgent tasks to class A engineers (which seems desirable as it ensures they are completed early on), longer tasks may be allocated to a class B engineer. This will unduly prolong the makespan. One can ask, therefore, if it is necessary to account for priorities, when due dates are not relevant. Indeed, when there is enough time to complete all tasks, and production is scheduled to cease during the project, the makespan of the project should be minimized. Therefore, priorities do not play a role in allocating the tasks for the schedule and will not be incorporated in the objective. On the other hand, when there is enough time available, more tasks can be added to the schedule. In theory, it is even possible to have an infinite number of candidate tasks, and use the full work week to schedule as many tasks as possible. In that case, priorities do matter (note that due dates are still not relevant here, as the end of the week only depicts the end of the project). Then, the objective is to schedule as many tasks as possible, where the more urgent tasks are scheduled early on.In summary, the scheduling problem consists of a parallel machine model with multiple tasks. Here, the machines refer to the engineers. It is assumed that there are no set-up times. For example, if a task is in progress, but will be completed the following morning, the total duration will not be affected. No sequence dependencies exist (i.e. precedence constraints which enforce one task to precede another). Priorities, however, may influence the sequence of the tasks, where the priority indicates the relative importance of a task. Due dates are irrelevant, because the project has a fixed end time and the tasks are scheduled based on their priority rating. In addition the tasks are for preventive maintenance, the objective is either to do as many as possible, or only the available tasks (i.e. the tasks are only preventive, not a necessity). Durations of tasks have a variance which is independent of their duration and priority. The variance does, however, depend on the resource allocated to that task. In this case, a class B engineer has a higher probability of complications than a class A engineer. Finally, it is assumed that all engineers (selected for the project) are available and must be allocated to tasks, so all available resources are used.
Formally, the problem can be described as follows: Sets
• Ԑ denotes the set of employees indexed by e. |Ԑ | = 12.
Кe denotes the employee type of employee e. Кe ∈ {A, B} ∀ e ∈ Ԑ.
• J denotes the set of tasks indexed by j. |J| = 37. Parameters
• is the expected duration in hours of task j when executed by employee e. ∀ j ∈ J , e ∈ Ԑ. Note that is dependent of the employee type Кe.
• is the priority of task j on a scale of 1 to 10. Variables
• is a binary variable indicating whether employee e executes task j. For j ∈ J , ∀ e ∈ Ԑ it is defined as:
{ as s e e ed e ee e se e e
• Weis the total time in hours that employee e s ends n e e ng as s. F ∀ e ∈ Ԑ it is defined
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∑∈
• C a den es e a es an e s ed e nde ns de a n. I s defined as: a
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4. Theoretical framework
4.1 Addressing maintenance and maintenance policies in general
Maintenance is defined as a combination of all technical and associated administrative activities required to keep equipment, installations and other physical assets in the desired operating condition or restore them to this condition (Muchiri, Pintelon, Gelders, and Martin, 2011). In general, maintenance is categorized into two classes, corrective maintenance (CM) and preventive maintenance (PM). CM is maintenance where the task is performed once a system has failed. PM is maintenance where the task is performed before a system fails, with the intention of returning the state of the machine back to its new condition. (Wong, Chan, and Chung, 2013).
In industrial companies assets usually consist of production equipment. Maintenance policies prove to be crucial when the company tries to ensure maximum availability of their production equipment, and thus their entire production facility, at the lowest cost (Sameta, Chelbia, and al Ben Hmidab, 2011). To maximize plant availability, the timely presence of maintenance workers is necessary (Koochaki, Bokhorst, Wortmann, and Klingenberg, 2013). However, maximizing plant availability typically is one of two goals of a maintenance policy. Minimizing costs is the other goal, generally resulting in an unceasing pursue to minimize personnel, including maintenance staff. Tailoring a maintenance policy and strategy specifically to the needs of the company, proves to be a key challenge, since finding an equilibrium between these two conflicting goals is difficult (Koochaki et al., 2013).
Much has been written on maintenance policies, where implementing cost-effective PM strategies proves crucial (Sameta et al., 2012; Harison, and Barkai, 2013; Debasis Das, Goutam Kumar, Dipankar, and Souren, 2013). These strategies aim to achieve the smallest amount of CM (relatively expensive) as possible, by structurally planning PM of the equipment. With limited maintenance personnel, however, structurally planning PM may not be feasible. An opportunistic maintenance approach may then prove useful, where the unplanned maintenance is combined with PM (Samhouri, Al-Ghandoor, Fouad, and Alhaj Ali, 2009). With such an approach, when PM is done, the maintenance engineer finds other components of the machine that require maintenance as well. This gives the maintenance staff an opportunity to replace or repair those items, which are found to be defective or need replacement in the immediate future, during the maintenance of a machine or component. In order to be able to complete these newly found tasks without disrupting the schedule, a flexible PM schedule is needed.
4.2 Gantt Charts
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Fig. 4.1, A typical Gantt Chart representing a project graphically.
A Gantt Chart is a powerful visual reporting tool for project management (Kumar, 2005). Gantt Charts are simple to understand and easy to construct, making it one of the most preferred tools by most of the project managers for all their projects, except the most complex ones. Geraldi et al. (2012), are, however, also skeptical of its usability. The Gantt Chart was developed in the early twentieth century, yet the chart is used for a wide range of projects, while there is little adaption of the tool for the specific circumstances projects sometimes call for.
4.3 Maintenance scheduling techniques
4.3.1 Minimizing the makespan
In the literature on PM, researchers usually consider maintenance as a single task, and schedule it together with the production schedule (Wong et al., 2013). In reality, however, PM usually consists of a wide range of tasks, e. g. lubrication, cleaning, inspection, adjustment, alignment, and/or replacement. Since different kinds of PM tasks have different maintenance intervals and require different durations, scheduling it together with the production schedule as a single task with a predetermined fixed duration, usually results in poor predictions. Wong et al. (2013), recognize this in their paper, and present a scheduling approach to maintenance tasks, where the goal is to minimize the makespan of a set of tasks. The makespan (Cmax) is defined as the time elapsed when the last task is completed, i.e.,
a , . , (1)
where is the completion time of task (Notation taken from Pinedo M. L., 2009: 28). Using the minimization of the makespan as the objective allows the user to have multiple time intervals and durations for different tasks, and find the sequence of tasks which minimizes the total duration of the project.
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4.3.2 Workforce constraints
Minimization of the makespan as an objective allows the user to schedule tasks in the most efficient sequence. However, in many real world scheduling settings there are personnel or workforce constraints. These constraints limit the number of feasible outcomes for a schedule, since each task requires for its execution a given number of personnel, which is often limited. Furthermore, maintenance engineers often differ in their respective skill sets (Gopalakrishnan et al., 1997), some maintenance tasks can therefore only be executed by certain engineers. In literature, the linear programs which try to solve complex scheduling problems with workforce constraints do not incorporate workforce dependent processing times. Often, the workforce consists of experienced, inexperienced, single-skilled and multi-skilled personnel. These attributes influence the duration of maintenance tasks per maintenance engineer and therefore influence the scheduling problem significantly.
Gopalakrishnan et al. (1997), present in their paper an approach to generate an adaptive PM schedule which maximizes the net savings from PM subject to workforce constraints. In a binary integer program, they incorporate both single skilled and multi skilled workforce constraints. The objective is to maximize the total expected PM effectiveness contribution (which depends on task prioritizing) from all the tasks. Their model does not incorporate the duration of maintenance tasks, and hence the objective is not to minimize the makespan. They do, however, pool the workforce based on their respective skill set. This “pooling” of the workforce can be implemented in an integer program, to estimate the effect of workforce constraints on the makespan of a set of maintenance tasks (Pinedo M. L., 2009:72,73). The objective of the program is to minimize the makespan, while ensuring that the precedence constraints are enforced (maintenance tasks usually consist of a set of predetermined operations, which should be performed in a specific sequence) and that the total demand for the workforce pool at a specific time does not surpass the availability of the workforce in that pool. Pinedo (2009) does however note two drawbacks of this linear integer program. Firstly, it assumes that the durations of all tasks are fixed and known, which in reality is not the case. Secondly the program is very hard to solve if the number of tasks is large and the time horizon is long (it is NP-hard). It is typically solved via heuristics when complexity increases.
4.3.3 Priority rules
Consider a single machine environment (i.e. one machine with multiple tasks) witha set of Jtasks, where each task j has a processing time pj, a release date rj and a due date dj. The processing of task j is basically
unconstrained if rjequals 0, and dj is infinite. It is clear that then the makespan Cmax does not depend on
the schedule (i.e. each sequence of tasks generates the same makespan). In many real world scenarios however, the release dates are not always equal to zero, and the due dates differ between tasks and are not infinite. Then, certain priority rules generate optimal schedules. Pinedo (2009) presents several priority rules. If, as is the case for RFC, it is a parallel machine environment (i.e. multiple parallel independent machines), the objective is to minimize the total weighted completion time, i.e., ∑
(where is a given weight for task j and is the completion time of task j), and the processing of the
tasks is unconstrained, then the Weighted Shortest Processing Time first (WSPT) rule, which schedules the tasks in decreasing order of , is optimal. If the objective is the maximum lateness Lmax (i.e. schedule
the tasks in increasing order of the due date), and the tasks are all released at time 0, then the Earliest Due Date first (EDD) rule is optimal. These are examples of static priority rules, which are independent of time. A dynamic priority rule is the Apparent Tardiness Cost first (ATC) rule. The advantage of using a dynamic priority rule is the fact that it is then possible to minimize the makespan, because the tasks can each be scheduled to a specific timeslot (hence, it is then possible to minimize the completion time of the final task). If a task is completed, the ATC rule selects among the remaining tasks the task with the highest value of
e (
( , )13
where K is a scaling parameter and ̅ is the average of the processing times of the tasks that remain to be scheduled. The ATC rule is a weighted mixture of the WSPT and MS priority rules. When the scaling parameter K is adjusted, the rule can be made to operate either more like the WSPT rule or more like the MS rule. For example, if K is increased, the right term in brackets converges to 0 from below. This increases the value of the numerator which is multiplied with. H
ence, more weight will lay on the shortest processing time ( ). Similarly, when K is decreased, the rule will convert more to the EDD rule, e a se e e g a n e d e da e s n e e den na “̅”
decreases),
the goal is then to minimize the maximum lateness.The variable in the ATC rule represents the priority of task j. Deciding what priority to assign to each task is not straightforward. In practice, this is often determined by an expert in the field which will assess, to his best knowledge, the urgency of each task. This may not always be the optimal value. In the paper by Gopalakrishnan et al. (1997), the priority is determined by a key parameter; the task priority Pi. It
represents the relative importance of each PM task and is used to select the PM task which contributes the most total expected PM effectiveness. They modeled the priority of a task to be proportional to its
expected contribution to PM effectiveness (i.e. the expected net savings of scheduling the task at a specific moment in time). The task priority is based on five independent variables which measure operational factors that affect the likelihood of machine failure. Indeed, in industrial companies different production systems use different types of equipment. According to Gopalakrishnan et al. (1997), however, from a PM point of view, the following five factors represent the most generic variables which can affect failure probabilities of machines:
X1 = Cumulative machine utilization (an indicator of the cumulative machine wear)
X2 = Current machine utilization (a ratio which is related to the operational life of the
machine. When the cumulative utilization increases, the cumulative deterioration of the machine increases as well)
X3 = PM delay (the failure probability of the machine as a function of the time elapsed since
the last PM task)
X4 = Comparative machine failure rate with the PM task (the percentile cumulative machine
failure rate associated with a specific PM task, i.e. the total number of failures associated with the task divided by the cumulative machine operation duration)
X5 = Severity of the last repair action (the probability of a machine failure as a function of the
severity of the last repair action taken a e e e a ne’s as ea d n These five independent variables, based on actual data on machine history, provide the machine failure probabilities (qj) for task j. The failure probabilities derived for each PM task are used to develop
priorities for execution, and with a task rescheduling model Gopalakrishnan et al. (1997), are able to allocate, at the beginning of each specified time bucket, candidate PM tasks for execution.
4.4 Significance of variance in maintenance scheduling
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when schedule 2 is selected. Had, in this example, the variance not been included, schedule 2 would never have been identified.Fig. 4.2, Representation of two schedules, consisting of four jobs with different processing times, and variances.
In scheduling literature, the following probability density function is well known for estimating variance, mainly due to its flexibility He e ı´as-Velasco, He e ı´as-Pleguezuelo and van Dorp, 2011):
| , , ,
,(3)
It suffers, however, from some difficulty in specifying the parameters p and q. When data is unavailable to estimate those using statistical tools, one has to rely on the estimation of p and q by experts. Malcolm et al. (1959), recognized this, and created a technique to determine the expectedmakespan of a set of
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5. Scheduling preventive maintenance at Royal Friesland Campina Bedum B.V.
5.1 Formalizing the workforce
Based on interviews with two members of the technical department at RFC., the workforce can be divided into two classes; class A and class B engineers. Class A engineers are highly experienced and are able to do maintenance tasks efficiently, with few complications and delays. Class B engineers are less experienced, and therefore lack efficiency, resulting in increased task durations. Based on these two classes, two highly experienced maintenance engineers estimated the durations for the PM tasks of a revision week in 2013 in hours. They also estimated, on a scale of 1 to 10, how likely the engineer is to have complications or delays while performing the task. A weighted average of the estimations by both engineers is taken. To avoid excessively large tables, for both classes, the estimations are given in Appendix 2.
The difference between the two classes shows immediately. On average, for a given PM task, the duration for a class B engineer is 1.558 times longer than a class A engineer. Furthermore, on average, a class B engineer is 1.657 times more likely to have complications or delays while performing the task. However, the average chance of having complications is quite low for both class A and class B, namely 19,2% and 26,9% respectively. According to the members of the technical department, there are 4 class A engineers, and 8 class B engineers. There are AB engineers as well, which excel in electrical engineering but lack experience in mechanical engineering, but for simplicity reasons we assume the engineers are either of class A or B. This will not affect the scheduling problem significantly, because primarily the tasks must be allocated to an engineer who is able to complete it.
5.2 Proposed solution
For RFC, the Gantt Chart offers an efficient tool for task scheduling in the revision weeks. In order to be able to add workforce dependent processing times, adaptation of the standard Gantt Chart is necessary. The Gantt Chart should preferably be able to recognize which engineers are what class, and be able to automatically compute the duration of the task assigned to a class, based on a predefined default duration. Us ng s a e “d ag and d ” e PM as s from a list of candidate tasks, to allocate the tasks to engineers, will benefit scheduling efficiency. This is not a necessity though, as doing it by hand does not require a lot of work if the number of tasks is not excessively large.
Currently, there is no Gantt Chart software available which allows the user to enter variable processing times of tasks based on the reso e e as s a a ed . H eve , “S a D a ” s a s ness s a e package which allows the user to use multiple templates to create different variations of Gantt Charts. The creators of the software should be contacted to determine whether the proposed adjustments can be implemented in the software. Microsoft Visio, the software used to create the Gantt Chart in Appendix 3, 4, 5, and 6 is very suitable as well, as the Gantt Chart functionality is pre-programmed in the software.
5.2.1 Evaluation and validation
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week, the user can easily assess whether the schedule is still feasible, and if necessary adjust it. If, for example, an engineer has a high amount of complications which cause delays, future tasks which were allocated to that engineer can easily be transferred to engineers who have already completed their tasks, or have available time in a later timeslot during the project. In addition, when, during the execution of a PM task, an engineer finds that other components of the machine are in need of replacement or repairs, these tasks can easily be added to the schedule. The Chart will immediately update the total duration of the project, and if necessary the user can re-allocate tasks to alternative engineers. Fourthly, in practice, the flexibility of the PM schedule is somewhat limited due to the nature of the PM that is required at RFC. For example, machine suppliers have to assist during some maintenance tasks, and should therefore be scheduled weeks before the actual start of the revision week. The Gantt Chart allows the user to fix these tasks in advance, and will create a clear overview of the tasks that depend on the presence of the supplier. Finally, the end of the project will vary depending on the completion times of the tasks. This will allow the user to estimate when the project will be finished, and production can initiate again.Disadvantages of the proposed solution, compared to the current scheduling technique, are not evident. However, for more complex situations, the use of Gantt Charts is limited. In cases, as mentioned in the second objective in the preliminary problem statement, where the objective is to schedule as many tasks as possible, or in cases where there are hundreds of maintenance tasks and engineers, complexity grows. Determining what tasks have priority, ensuring optimal use of both classes of engineers, and scheduling by hand is then not feasible. Scheduling rules or linear programs which generalize the scheduling problem based on input parameters (i.e. priority and task duration), and automatically generate a schedule which approaches optimality, are needed. Section 5 aims at developing such rules to generalize the scheduling problem, and presents two heuristics with corresponding linear programs.
5.2.2 User feedback
As part of the regulative cycle, a member of the technical service department at RFC, responsible for planning revision weeks, was asked to assess whether the advantages of the proposed solution mentioned in section 5.2.1 are relevant to RFC, and if there are disadvantages which were not identified in the research previously. He immediately recognized the main benefit, the potential decrease of the makespan of the revision week, but wants to implement it first, to see if it can be achieved in practice. Table 5.2.1 displays the results of the assessment. Each proposed advantage is assessed based on its relevancy with a ++, +, +/-, -, --, for highly relevant, relevant, indifferent, irrelevant, and highly irrelevant respectively.
Proposed advantage Assessed Relevancy
(1) Creating large project requires few training and little time + (2) Delays can be implemented in the schedule immediately and documented for
future scheduling
-
(3) Increased flexibility ++
4 F “s e a ” as s and d s a ese g a a +/
-(5) The end of the project is displayed clearly ++
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Advantage (1) is deemed relevant, because the creation of a larger project, with this scheduling technique, is now feasible in a time efficient manner. Advantage (2) is deemed irrelevant. Firstly becausecomplications do not arise often, and secondly because tasks usually differ between revision weeks, thereby eliminating the advantage of documenting the delays to benefit future scheduling. Advantage (3) is assessed as highly relevant, as the increased flexibility allows the user to combine PM with corrective maintenance. If, during the execution of a PM task, engineers find that other components of the machine require maintenance, the scheduling technique allows the user to add this task to the schedule efficiently. Advantage (4) is neither relevant nor irrelevant, because currently the scheduler is able to fix these special tasks as well. Finally, advantage (5) is deemed highly relevant. This will allow the scheduler to assess when the project finishes, and is recognized as one of the major advantages of the scheduling approach.
Disadvantages are not yet identified by the member of the technical service department. Not necessarily because they do not exist, but because he feels the approach should be implemented and used first. Then, if there are disadvantages, they will most likely surface.
5.2.3 Concluding remarks
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6. Scheduling preventive maintenance tasks with workforce dependent processing times
6.1 Using priority rules to efficiently schedule tasks
The scheduling tool suggested in section 4.2 greatly benefits scheduling efficiency. However, there is a drawback as well. The tasks still have to be added to the schedule manually, where they are prioritized based on the professional experience of the scheduler. For more important or urgent tasks this may be feasible, because the scheduler can then decide when, and by whom, the tasks are done. For the majority of the tasks however priorities are less clear and differences between the priorities will be smaller. Then, ideally, to create a schedule which approaches optimality, an automated priority rule which schedules the candidate tasks based on certain parameters is needed. The ATC rule presented by Pinedo (2009) selects, among the candidate tasks, the task with the highest value of
e ( ( , )
̅ ) ∀ ∈
As the rule suggests, the value of not only depends on the priority of the task ( ), but also on the processing time , the due date ( , the average duration of the remaining tasks ( ̅), and the scaling parameter K. Basically, the rule creates a schedule based on the value of for task j at time t. See Pinedo (2009) for an extensive explanation of the rule. While the rule is able to schedule the tasks efficiently, it does not account for resource dependent processing times. In order to implement these, the ATC rule should be adjusted and created separately for both classes of engineers.
Class A:
e (
( ̅ , ))
∀ ∈ (4) Class B:e (
( , )̅
)
∀ ∈ (5)and are the durations of task j for class A and B engineers respectively, similarly ̅ ̅ are the average durations of the remaining tasks which have to be scheduled, for class A and class B respectively. Both rules will schedule the tasks efficiently, but having two separate priority rules creates a new problem. Having two priority rules requires some sort of distribution between the two. For example, consider the simple case with two engineers, one of each class, and two tasks. Task 1 has a duration of 3 and 6 hours, and task 2 has a duration of 4 and 8 hours, for class A and B respectively. Task 1 is more urgent than task 2. If we were to schedule these tasks by hand, task 1 would be allocated to the class B engineer and task 2 to the class A engineer. The resulting makespan is 6 hours, and is optimal. If we were to schedule these tasks via the priority rules (4) and (5), both rules would suggest the schedule 1-2. The problem arises in the decision what task to allocate to what engineer. There are 4 possible ways to distribute the two tasks, and it is possible to compute the makespan for each schedule. However, the number of possible schedules grows exponentially with the number of tasks. Pinedo (2009) also recognized this. It is a special case of the project scheduling problem with workforce constraints presented in section 4.3.2. The linear program formulations of the problem are NP-hard, so heuristics must be used to find a solution which approaches optimality.
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schedule. Removing the due dates form the ATC rule results in the weighted processing time, i.e.∀ ∈ (6)
Equation (6) behaves as the Weighted Shortest Expected Processing Time first (WSEPT) rule. Removing the right term of equation (4) also deletes the t parameter. Therefore, scheduling the tasks based on equation (6) would no longer account for completion times, and hence it is impossible to minimize the makespan. Equation (6) will assign the tasks with a low duration but high priority to a class A engineer, and the longer tasks, which have slightly lower priority, to class B (assuming that the more urgent tasks are allocated to class A engineers to ensure their completion as quickly as possible). This is not feasible, because then the difference in duration between the two classes is not used optimally (i.e. allocating the longer tasks to class B engineers may unduly prolong the makespan). However, the fact that due dates are irrelevant is an important finding in developing a heuristic, and will be discussed further in section 6.2.
6.2 Two scheduling heuristics for tasks with resource dependent processing times
Section 6.1 shows that the ATC rule is not suitable for this specific scheduling problem. The resource dependent processing times create a highly complex NP-hard scheduling problem, because a distribution of tasks between class A and B engineers has to be generated. Typically, heuristics are then used to generate possible solutions. Currently, these heuristics are not available in literature. Scheduling methods which incorporate workforce constraints usually simplify the problem. These methods thus do not incorporate differences in processing times based on respective workforce classes. In this section, two heuristics are created based on the two objectives presented in section 3.2. The first heuristic minimizes the makespan without priorities, the second heuristic does incorporate priorities.
6.2.1 A scheduling heuristic for tasks with resource dependent processing times without priorities
When priorities are considered irrelevant, the difference between the duration of a task between a class A engineer and a class B engineer is determinative for the allocation of the task. When the difference is relatively high, a class A engineer should be allocated to that task to ensure that the makespan is not unnecessarily prolonged. The heuristic which tries to minimize the makespan is presented below and the algorithm is summarized as follows.Algorithm:
The differences in task durations between a class A engineer and a class B engineer determine what tasks have more potential benefit if allocated to a class A engineer. To ensure that the two classes are used optimally, the task with the highest difference in duration has the most potential benefit, and should therefore be allocated to a class A engineer. A low difference implies less potential benefit, and should therefore be allocated to a class B engineer. To ensure that the makespan is not unnecessarily prolonged e “d s n” e a se e ATC e , the tasks are allocated to class A engineers until the next potential task added will increase the makespan of the entire project, but will not increase the makespan if it were allocated to a class B engineer. Then, tasks will be allocated to class B engineers, starting with the smallest differences in duration, until the makespan is prolonged if allocated to a class B engineer. This cycle continues until all tasks are scheduled.
Step 1. ~ Compute for all the tasks the difference between the duration for a class A and a class B engineer. Then, sort all tasks on these differences.
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Step 3. ~ Select the candidate task with the lowest difference in duration and the lowest overallduration, allocate this task to the class B engineer which has the earliest completion time for the next task in his schedule (i.e. the smallest value for t). Continue until the completion time of the next candidate task added surpasses the completion time of the task if it was allocated to a class A engineer (with the corresponding duration for a class A engineer).
Step 4. ~ If the week is fully scheduled, or if all tasks are scheduled, STOP. Otherwise go back to step 2.
The schedule resulting from this heuristic is shown in Appendix 4. The makespan of the project is 15 hours.
6.2.2 A scheduling heuristic for tasks with resource dependent processing times with priorities
To ensure that the makespan is not unduly prolonged, longer tasks should be allocated to class Aengineers and tasks with a smaller duration to class B engineers (whenever the completion time dictates so). Furthermore, priorities of the tasks should be implemented in the heuristic. A heuristic which accounts for both constraints is presented below, and the algorithm is summarized as follows. Algorithm:
Since priorities dictate the urgency of a task, the high priority tasks should be completed first. The heuristic will thus start with the high priority tasks (by categorizing the tasks on a scale of 1 to 10), and descend 1 priority point once all tasks within that category are scheduled. Within a priority category, the tasks will be allocated as described in heuristic 6.2.1. This cycle continues until all tasks are scheduled and will ensure that the differences in engineer classes are used optimally, while accounting for the task priorities.
Step 1. ~ Determine, for each task, its priority between 0 and 10. Sort all tasks based on their respective priority, and categorize them in increments of 1 priority point.
Step 2. ~ Select the candidate tasks corresponding to the highest priority available. Sort the tasks on two levels, the duration for class A and the duration for class B respectively.
Step 3. ~ Select the candidate task with the highest class A duration available. Allocate the task to the class A engineer which has the earliest completion time for the next task in his schedule (i.e. the smallest value for t). Continue until the completion time of the next candidate task added is equal to or surpasses the completion time of the task if it was allocated to a class B engineer (with the corresponding duration for a class B engineer). Step 4. ~ From the remaining tasks, select the longest candidate task available. Allocate the
task to the class B engineer which has the earliest completion time for the next task in his schedule (i.e. the smallest value for t). Continue until all tasks within this priority class are scheduled.
Step 5. ~ If the week is fully scheduled, or if all tasks are scheduled, STOP. Otherwise go back to step 2.
The schedule resulting from this heuristic is shown in Appendix 5. The makespan of the project is 16 hours, which is mainly due to the fact that tasks with a higher priority are scheduled earlier in the week. Because of this, class A engineers are more likely to do tasks which have a smaller duration. This increases the makespan, but ensures that tasks with high priority are done first.
6.3 Integer linear program formulations for both heuristics
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given below will not be feasible if more constraints (f.e. sequence dependencies), engineers or tasks are added, but it will allow for immediate performance comparisons with heuristic 6.2.1 presented above. Based on the sets, parameters and variables denoted in the preliminary problem statement, we formulate the ILP for heuristic 6.2.1 as follows:ILP formulation for heuristic 6.2.1 minimize Cmax s.t.
∑
∈ Ԑ∀ ∈ J
∑
∈ J4
∀ ∈ Ԑ 2∀ ∈ Ԑ ∈
{
,}
∀ ∈ J , ∈ Ԑ 4In the above formulation constraints (1) ensure that each task is executed by one employee. Constraints (2) ensure that no employee works more than 40 hours. Constraints (3) are needed to linearize the definition of the makespan. Finally constraints (4) ensure that is a binary variable for all ∈ J , ∈ Ԑ. Solving the ILP results in a makespan of 12. The schedule is shown in figure 6.1. The vertical axis
represent the 12 engineers, where A denotes class A engineers, and B class B engineers. The horizontal a s den es e 7 as s. A as s a ed an eng nee en a “ ” s assigned to it. For comparison purposes, the schedule is added in Appendix 6 in a Gantt Chart.
Fig. 6.1, the resulting schedule from solving the ILP formulation for heuristic 6.2.1 which minimizes the makespan.
Heuristic 6.2.2, which tries to find the optimal schedule with the objective to maximize the weighted makespan (i.e. the makespan multiplied with the task priorities), can be formulated as an ILP formulation as follows:
ILP formulation for heuristic 6.2.2 maximize ∑ ∈
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∑
∈ Ԑ∀ ∈ J ∑ ∈
4
∀ ∈ Ԑ 2 ∈{
,}
∀ ∈ J , ∈ ԐIn the above formulation constraints (1) ensure that each task is executed by one employee, or the task is not executed at all. Constraints (2) ensure that no employee works more than 40 hours. Finally
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7. Discussion of the results
This research presents a first step in solving the resource allocation problem with workforce dependent processing times. The heuristics presented are based on data and insights found during the execution of the research. The heuristics only differ in makespan by one hour. This is mainly due to the small amount of tasks. Comparing both schedules does show that the heuristic 6.2.1. shows a more uniformly distributed workload, where the shorter tasks are done by class B engineers, and the longer tasks by class A
engineers. According to the data provided by the maintenance engineers, class A engineers are less likely to have complications during the tasks. Thus, this may be feasible, if possible delays grow in line with the durations of tasks. If this is not the case, the high amount of short tasks allocated to class B engineers can cause a relatively high amount of delays. Priorities should then be taken into account when allocating these shorter tasks, to ensure that the urgent tasks are completed before any delays can jeopardize the completion of them. Because of the objective to complete the project in minimal time, while allocating all available resources, the heuristics do not take into account the working time left for each engineer on the final day of the project. For example, class A engineer # 1 in Appendix 4 finishes at 10:00 in the morning on day 2. Since employees usually have to work 8 hours a day, this implies that the engineer has spare time when the project finishes. The makespan may thus be only 15 hours, but the employees will be available the full 16 hours, regardless of the makespan. Solving the problem by incorporating this in the heuristics may thus result in cost savings, if, as a result, less engineers are used.
Because, in literature, workforce dependent processing times have not yet been incorporated in
scheduling problems, it is difficult to assess whether the proposed heuristics work in practice with more extensive and complex scheduling projects. We can compare the schedule resulting from heuristic 6.2.1 with the schedule resulting from the ILP formulation which minimizes the makespan. The heuristic clearly is not optimal, as the solution of the ILP formulation has a makespan of 12 hours, which is 3 hours less than the solution of the heuristic. Furthermore, the schedule resulting from the ILP formulation allocates a high amount of small duration tasks to the first two class A engineers. This is noteworthy when comparing it to the principle of heuristic 6.2.1; allocate the longest duration tasks to class A engineers and the
shortest to class B engineers. The ILP is able to weigh the potential benefit of allocating more short tasks to a class A engineer with the potential loss in time of allocating longer tasks to class B engineers. Experimenting with the heuristics to test their practical value would therefore be the next step in future research. Furthermore, the heuristics can be extended with several aspects regarding the scheduling problem by, for example, incorporating the variance in task durations. Suggestions for further research will be addressed more elaborately in section 9.
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8. Conclusions
The goal of this research is to improve the current scheduling technique used for PM at RFC, and to explore the resource allocation problem with workforce dependent processing times. Furthermore, the goal is to generate an approach which is generally applicable to workforce dependent processing time scheduling problems in a resource dependent parallel machine environment. The research shows that the use of Gantt Charts by RFC is feasible and will benefit the PM scheduling. In cases where complexity grows, the use of Gantt Charts is limited, and scheduling rules or programs are needed.
In order to be able to automatically create a schedule, using such rules or programs, based on certain parameters (e.g. number of engineers, task duration per workforce class, and task priority), the ATC prioritizing rule has been explored. Creating a scheduling technique which incorporates both workforce dependent processing times while prioritizing tasks deemed too complex, as the problem is NP-hard. As a result, two heuristics have been presented which attempt to solve the problem based on two objectives. The first objective is to minimize the makespan, disregarding task priorities. The second while taking into account the priorities of each task. Both heuristics seem to perform relatively well compared to the current makespan of the schedule. The revision week in 2013 consist of 37 tasks which were completed in 40 hours. Heuristic 6.2.1, with the objective to minimize the makespan, has a makespan of 15 hours, and shows a more uniformly distributed workload, where the shorter tasks are done by class B engineers, and the longer tasks by class A engineers. The optimal schedule presented by the ILP formulation with the objective to minimize the makespan is 12 hours. Heuristic 6.2.1 thus does not perform optimally, but the difference is not alarming. Experimenting with more tasks with varying parameters should show whether heuristic 6.2.1 will perform well in different scenarios. Heuristic 6.2.2 ensures that the high priority tasks are done first, and has a makespan of 16 hours. This heuristic enables the scheduler to add more
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9. Suggestions for further research
Further research is required to both test and expand the current heuristics. This section suggests several experiments and improvements to the heuristics, which will help to increase the practical use of them. To test whether the heuristics achieve their respective goals, they should be programmed in script language to allow for experiments with multiple tasks. Currently, 37 tasks are scheduled. Increasing the number of tasks to, for example, 50, 100, 200 and 500 tasks which have, for each iteration of the experiment, random parameters (duration and priority), will show whether the heuristics perform well with an increased number of tasks, and where potential problems lay which impede the scheduling problem. Experimenting with the ILP formulations for both heuristics may not be feasible, as the problem is NP-hard.
The heuristics currently do not incorporate variance. This may have a significant impact on the proposed schedules. For heuristic 6.2.2 variance may seem mostly irrelevant, as the priorities of tasks dictate the start time of a task. If these were to delay, they may influence the allocation of the other tasks during the execution of the project, but any delays will not fundamentally change the order of the scheduled tasks. However, if the objective is to maximize the total weighted priorities (i.e. task durations multiplied with their priority), it can influence the schedule. Consider, for example, the situation where a task with both high variance and high priority is not executed to create time for two shorter low variance high priority tasks. For this reason, variance should be incorporated to see what effect it has on the outcome of the schedule. Sarin et al. (2010), describe a method for estimating variance in parallel machine environments. They compute the variance of the makespan by assuming that the makespan is normally distributed with ean µ and va an e σ. D e en s ed es a e en ea ed e es e ve e e ed a es an and variance of the makespan, which allows the user to select the schedule that has the highest probability of being finished before a certain threshold. This does not solve the problem, because ideally the scheduling heuristic immediately accounts for variance and allocates tasks based on this. It is, however, a good starting point for further research.
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Appendices
Appendix 1
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Appendix 2
Estimated durations in hours of the tasks during the revision week, and their corresponding
probability of complications (on a scale of 1 to 10).
Class A Class B
Task # Duration Chance of complications Duration Chance of complications
