Workforce scheduling: A new model incorporating human factors
Mohammed Othman, Gerard J. Gouw, Nadia Bhuiyan
Concordia University (Canada)
Received: January 2012
Accepted: September 2012
Othman, M., Gouw, G.J., & Bhuiyan, N. (2012). Workforce scheduling: A new model incorporating human factors. Journal of Industrial Engineering and Management, 5(2), 259-284. http://dx.doi.org/10.3926/jiem.451
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Abstract:
Purpose: The majority of a company’s improvement comes when the right workers with the right skills, behaviors and capacities are deployed appropriately throughout a company. This paper considers a workforce scheduling model including human aspects such as skills, training, workers’ personalities, workers’ breaks and workers’ fatigue and recovery levels. This model helps to minimize the hiring, firing, training and overtime costs, minimize the number of fired workers with high performance, minimize the break time and minimize the average worker’s fatigue level.
Design/methodology/approach: To achieve this objective, a multi objective mixed integer programming model is developed to determine the amount of hiring, firing, training and overtime for each worker type.
Findings: The results indicate that the worker differences should be considered in workforce scheduling to generate realistic plans with minimum costs. This paper also investigates the effects of human fatigue and recovery on the performance of the production systems.
Research limitations/implications: In this research, there are some assumptions that might affect the accuracy of the model such as the assumption of certainty of the demand in each period, and the linearity function of Fatigue accumulation and recovery curves. These assumptions can be relaxed in future work.
Originality/value: In this research, a new model for integrating workers’ differences with workforce scheduling is proposed. To the authors' knowledge, it is the first time to study the effects of different important human factors such as human personality, skills and fatigue and recovery in the workforce scheduling process. This research shows that considering both technical and human factors together can reduce the costs in manufacturing systems and ensure the safety of the workers.
Keywords: fatigue; human factors; personality; workforce scheduling
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1. Introduction
Effective
workforce scheduling
is one of the most critical tasks affecting performance of
manufacturing
systems. It is important to assign the right job to the right person at
the
correct time. Also, it is very important to have a close match between
workers’
skills, attitudes and strength and his/her tasks he/she performs (for
simplicity, we will use he/him hereafter). This needs an effective
workforce scheduling
system. This system aims to reduce waste in
employing people, lessen uncertainty about current personnel levels and
future
needs, and avoid worker and skills shortages or surpluses by hiring the
right
workers in appropriate numbers. Traditional workforce scheduling
tools are limited and cumbersome. They are concerned with ‘head count’
rather
than ‘head content’, which prevents the resulting schedule from being
flexible
enough to follow the growing demand of fast changing business dynamics (Birch, O’Brien-Pallas, Alksnis, Tomblin Murphy
&
Thomson, 2003; Castley, 1996; Jensen,
2002). A major problem with existing models is the absence of the most
important human factors inherent in the production system. As one of
the main
elements in a production system, human issues cannot be ignored without
significantly reducing the benefits of the production system.
Considering human
factors in production planning has the potential to improve both injury
risk
and production performance (Neumann & Medbo, 2009; Udo &
Ebiefung,
1999). It is important to integrate human factors early in the
production
planning phase because early changes to the product and work are less
costly
and easier to make than are late changes. Workforce planning is a
systematic
identification and analysis of what a company is going to need in terms
of the
size, type, and quality of workforce to achieve its strategic
objectives. It
determines the right number of the right people in the right place at
the right
time. In this paper, a new model for workforce scheduling to support
production
planning is developed to achieve better production performance while
reducing
risks to operator health. The paper is organized as follows: Section 2
presents
a literature review of human factors and their relation to the planning
process. Section 3 describes the workforce scheduling model formulation
and the
notation used. Next, Section 4 presents the results and insights
generated from
the proposed model. Finally, conclusions and suggestions for future
research
directions are summarized in Section 5.
2. Literature review
Human
Factors (HF), or ergonomics, has been defined as “the theoretical and
fundamental understanding of human behavior and performance in
purposeful
interacting socio-technical systems, and the application of that
understanding
to the design of interactions in the context of real settings” (Wilson,
2000).
During the last decades, ergonomics have been considered minimally in
building
production systems. Most business managers have accepted the idea that
ergonomics are working as protectors of workers, rather than creators
of
systems (Dul & Neumann, 2009; Perrow, 1983).
They generally associate ergonomics with health and safety issues
rather than
with the effectiveness of organizations (Jenkins & Rickards, 2001).
Ergonomics
is considered too late in the production system development process,
making
most managerial decisions hard to change (Helander 1999; Jensen, 2002;
Neumann
& Medbo,
2009). Perrow (1983) mentioned that
the main problem is that human factors specialists have limited
influence and
control within the organizational context. Also, they have no control
of
strategic resources and a weak network in and outside of the
organization.
However, it is shown that ergonomics can contribute to
different company strategies and support the objectives of different
business
functions in the organization (Dul & Neumann,
2009). On the other hand,
many ergonomics models have been developed without a clear
understanding of how
they could be implemented in a specific company (Butler,
2003; Hagg, 2003). Berglund and Karltun (2007) studied the effects of
the
human, technology and organizational aspects on the outcome of the
production
scheduling processes. Based on their study,
schedulers need to consider uncertainty, their experience, problem
solving,
workers’ differences, technical system limitations, the degree of
proximity
between employees and their informal authority. Jensen (2002) presents
approaches and tools developed in Scandinavian countries. He explained
that the
changes in the ergonomics role inside a company require understanding
the
organizational prerequisites. He proposed a political agent in
order to complement the roles of an expert and a facilitator.
He also suggested developing studies on the management of ergonomics and organizational development.
There
are many reasons for not considering human issues early into production
planning. Helander (1999) discussed seven common reasons for not
considering
ergonomics early in the production system development process. Some of
the
common misconceptions regarding ergonomics are that many people think
that it
is for the design of chairs and that it is just common sense; the
research in
ergonomics is too abstract to be useful; people are adaptive, so there
is no
need for ergonomics; and the technical system should be designed first
before
considering ergonomics. Bidanda, Ariyawonggrat, Needy, Norman, and
Tharmmaphornphilas (2005) mentioned that the major reason is that human
issues
are typically difficult to quantify. However, none of these are reasons
to not
consider human factors early in the production process.
In
reality, there is a tremendous variability in individual capabilities.
The
result is that most production system designs ignore the effects of the
human
differences in production system design. Buzacott (2002) indicates that
individual differences can result in substantial loss in throughput. Worker
differences are a fundamental element to consider when assigning
workers to a
workstation on the assembly floor. On the other hand,
Broberg (2007) has pointed out that human factors tools to integrate
ergonomics
into the design process are not known by engineers. Some tools for
handling
human factors in planning are creating digital human models,
integrating
ergonomics into predetermined motion time systems and integrating
ergonomics
into discrete event simulation (DES). However, DES has been considered
to be an
appropriate tool that can incorporate human aspects at the earliest
planning
stage for optimal performance (Neumann & Medbo,
2009). Some ideas on how to integrate human
performance modeling with DES in assembly lines are suggested (Siebers,
2004;
2006). Due to the variation in human performance, there is a need for
non-deterministic models of worker performance. Dul and Neumann
(2009) provided a conceptual framework to help
ergonomists in research, education and practice to understand how to
support
the strategic objectives of a company. This framework helps ergonomics
experts
to focus on ergonomics from the point of view of business performance
rather
than occupational health and safety.
There have been many interesting developments on the
technical
side of planning and scheduling processes. Many researchers considered
a few
human aspects in their quantitative models. Da
Silva, Figueira, Lisboa, and Barman (2006)
developed an aggregate production planning model that includes workers’
training, legal restrictions on workload and workforce size. Jamalnia and Soukhakian (2009) have developed a
fuzzy multi-objective nonlinear programming model for aggregate
production
planning problem in a fuzzy environment. Learning curve effects have
been
considered in formulating the model. Wirojanagud, Gel,
Fowler, and Cardy (2007) used
the general cognitive ability metric to model individual difference in
efficacy
of cross-training and worker productivity. Azizi, Zolfaghari
and
Liang (2010) considered workers
motivation,
learning and forgetting factors and workers' skills to measure
employees’
boredom and skill variations during a production horizon. Corominas,
Olivella and Pastor (2010) have taken into
account learning curves
and workers experience in modeling a scheduling problem. Also, researchers utilized mathematical models,
heuristics and
simulation to study the impact of cross-training on system performance.
Stewart,
Webster, Ahmad and Matson (1994) developed
four
optimization models for different cross-training scenarios to assist
managers
in deciding optimum tactical plans for training and assigning a
workforce
according to the skills required by a forecasted production schedule. Felan
and Fry (2001) investigated the concept
of a multi-level flexibility workforce using simulation. The results
indicate
that it is better to have a combination of workers with high
flexibility and
workers with no flexibility rather than employing all workers with
equal
flexibility. Blumberg and
Pringle (1982) developed a model that can link between worker
motivation and
productive performance. In their paper, they suggested that expected
work
performance of individuals is determined by three factors: Capacity,
Opportunity and Willingness. Jaber and Neumann (2010) developed a mixed-integer
linear programming (MILP) model that describes fatigue and recovery in
dual-resource constrained systems. The results obtained from their
model
suggest that short rest breaks after each task, short cycle times and
faster
recovery rates improve the system’s performance. Fatigue may be defined
as a
physical and mental weariness existing in a person and harmfully
affecting the
ability to perform work. Worker fatigue
can greatly
impact system performance in terms of quality (Eklund, 1997).
It can
significantly affect human productivity (Oxenburgh, Marlow, &
Oxenburgh,
2004). Inordinately long working
hours and
poorly planned shift work can result in employee fatigue.
As
discussed above, the literature review demonstrated that most of the
work on
workforce planning and scheduling assumed that workers are identical. The
problem seems to be systemic and there is an obvious need to integrate
ergonomics processes into the organization early so that underlying
principles
can be incorporated. Our research will contribute to the literature by
extending existing models of service workforce planning and scheduling
beyond
current capabilities. This model will incorporate human issues such as skills,
training, worker personalities, worker recovery and
worker fatigue. Four objective functions are
considered in the proposed model. The first one is cost minimization
and the
second one is top performance workforce firing minimization, the third
one is
idle time minimization and the last one is fatigue rate minimization.
In
summary, ergonomics must be implemented concurrently with production
planning
in order to improve planning process performance. The problem
description,
assumptions and formulation are given in the next
section.
3. Mathematical modelling of the multiple-objective
workforce scheduling problem
In this paper, we analyze the scheduling problem in a job shop environment consisting of different machines types, which are grouped into several machine levels depending on many factors such as the complexity and sophistication of the machine, the quantity of the process plans available and training budget. For example, if we have three machine levels, machine level one is the less complicated one and machine level three is the most complicated level. Worker flexibility can be achieved by using overtime and training. Workers are grouped according to different human skills and personalities and we have made the assumption that the number of worker skill levels is equal to the number of machine levels. Personality can be defined as a dynamic and organized set of characteristics possessed by a person that uniquely influences his or her cognitions, motivations, and behaviors in various situations. We assume that each worker will have at least one personality level that can be assigned to a certain machine level depending on his personal traits such as constructive, creative, dynamic, educated, efficient, etc. They are grouped within the categories of an individual's miscellaneous attributes and skills. We divided the skill levels and the personality levels into three levels: level 1 indicates the lowest level, level 2 indicates the middle level, and level 3 indicates the highest level. In contemporary psychology, the dimensions of personality which are used to describe human personality are openness, conscientiousness, extraversion, agreeableness, and neuroticism. Openness includes characteristics such as curiosity, novelty, imagination, insight and variety. Conscientiousness is a tendency to show self-discipline and being organized, and achievement-oriented. Extraversion includes characteristics such as sociability, excitability, assertiveness, and talkativeness. Agreeableness includes characteristics such as morality, trust, cooperation, kind and sympathy. Finally, Neuroticism is the tendency to experience emotional instability, anger, anxiety, sadness and depression. In this paper, these traits are measured based on percentile scores. Level 1 indicates the range from 0 to 33.3th percentile, level 2 indicates the range from 33.4 to 66.6 percentile, and level 3 indicates the range from 66.7 to 100th percentile. For example, people with high scores on conscientiousness tend to be responsible, organized and mindful of details, whereas people with low scores on openness tend to have less curiosity and more traditional interests. However, people with similar characteristics are grouped into personality levels, which reduce the variability of considering individual personality profiles. Special questionnaires can be developed and validated for use in applied research settings to measure the Big Five domains. If, for example, a worker wants to improve his skills, training can be used. It can also help the person to grow and develop his personality traits. Layoffs or hiring new workers affect the performance of the present workers because they need to be trained to the same level as the previous fired workers. Workers have a certain capacity during work, which is the maximum endurance time, defined as the length of time that workers can continue to work without becoming fatigued. It is assumed that endurance time increases as the personality level is increased. When the productive time increases, the average workload on the worker increases, so that rest breaks have to be given for the physiological recovery of a worker. Relaxation allowance is used to assist recovery from fatigue. It is an addition to the basic time intended to provide the worker with the opportunity to recover from the physiological and psychological effects of carrying out specified work under specified conditions. The amount of allowance will depend on the nature of the job, personality attributes and environment. The proposed mathematical programming model is based on the following assumptions:
· All the objective functions and constraints are linear equations.
· The demand in each period is deterministic over time.
· Fatigue accumulation and recovery curves are linear over time.
· The fraction of maximum load capability is applied continuously by the worker when performing a task for a period equivalent to the task’s duration.
· The length of the break between tasks is not long enough to result in full recovery.
· The top performers have skill and personal levels greater than or equal to 2.
· The length of the shift work of a worker is less than 12 hours including overtime.
The model presented herein is deterministic and in order to satisfy the total demand of each period, we are interested in determining:
· How many workers to assign to each machine level in each period?
· How many workers, with which skill levels, to hire or fire in each period?
· How many workers to train from lower skill level to higher one in each period?
· How many hours a worker with specific skill and personality level can work on overtime basis?
· How long a worker spends on a task in each period?
· How long a break time following any task a worker can take?
Model
characteristics
The
model developed is a multi-objective integer programming model that
allows a
number of different staffing decisions to be made (e.g. hire, train,
fire and
overtime) in order to minimize the sum of hiring, firing, training and
overtime
costs and minimize the top performance workers fired over all periods,
minimize
idle (unproductive) time and minimize the physical load on the
workforce.
Notation
and model variables
In
presenting the model, the following notations are used:
The objective function
aims to minimize: all costs incurred including worker hiring and
firing,
training costs and overtime costs; the top performer layoffs; idle
(unproductive) time; and the weighted average fatigue rate. The purpose
of
optimization is to minimize the deviations from specific goals based on
the
importance of each one. Constraints (1), (2), (3) and (4) represent the
cost
goal, top performance goal, unproductive time goal and fatigue level
goal
constraints, respectively. Constraint (5) shows that the
total regular time a worker spends on a task plus the total overtime
hours are
equal to the number of hours required for each skill in each period.
Constraint (6) shows
that the total regular time a worker spends on a task plus the total
breaks and
interruptions during should not be greater than the available labour
capacity.
Constraint (7) ensures that the fatigue rate at the
end of a period has to be less than the maximum fatigue load a worker
can
accumulate in any task. Constraint
(8) ensures that the workforce in any period should equal the workforce
in the
previous period plus the new hires and is trained to the upper level
minus the
layoffs. Constraint
(9) ensures the overtime workforce available should be less than the
maximum
overtime workforce available in each period.
Constraint (10) ensures that the total number of
workers who are assigned to machine level x in period t-1 and
now
fired or trained for upper skill levels should not be greater than the
number
of workers required in the previous period. Constraint
(11) ensures that workers can be fired if and only if the assignment is
possible. Constraint
(12) denotes that workers can be hired if and only if the assignment is
possible. Constraint
(13) ensures that training for better skills is possible if and only if
the
previous assignment is possible. Constraint
(14) ensures that training for better skills is possible if and only if
the
latter assignment is possible. Constraint
(15) ensures that training for better skills is possible if and only if
training to that skill is possible. Constraints
(16) and (17) guarantee the workers who are trained for skill level j
should not be fired in the same period. Constraints
(18) and (19) ensure that either hiring or firing workers occurs but
not both.
Constraint (20) ensures that the processing time for any task cannot
exceed the
maximum endurance time for any individual performing that task.
Constraint (21)
states that the worker can perform any task if and only
if the worker assignment to that task is possible. Constraint (22)
ensures that
the break time following any task is to be less than or equal to the
recommended recovery duration for that task.
Constraint (23) calculates the value of maximum endurance time based on
the
fraction of the maximum load capability applied when performing certain
task.
Constraint (24) calculates the total limit for maximum fatigue index. Finally,
constraints (25) and (26) are the non-negativity constraints.
Goal
programming can be used to solve the multi-objective functions. It
provides a
way of striving towards conflicting objectives simultaneously. The
basic
approach of goal programming is to establish a specific target for each
of the
objectives, formulate an objective function for each objective, and
then seek a
solution that minimizes the (weighted) sum of deviations of these
objective
functions from their targets. There are two methods for solving goal
programs:
the non-preemptive method (weights method) and the preemptive method.
The
weights methods form a single objective function consisting of the
weighted sum
of the goals, where all goals are roughly comparable of importance. On
the
other hand, the preemptive method organizes the goals one at a time
starting
with the highest priority goal and terminating with the lowest one
without
degrading the quality of a higher-priority goal (Hillier &
Lieberman,
2010). In this paper, the non-preemptive method is used to solve the
problem.
The decision maker must determine penalty weights that reflect his
preferences
regarding the relative importance of each goal. For example, penalty
weights
equal to 1 signify that all goals carry equal weights. The
determination of the
specific values of these weights is subjective. Different methods have
been
developed to estimate the weight values (Tamiz, Jones,
& Romero, 1998; Cohon, 1978). The solution
procedure considers one goal at a time, starting with the costs
minimization
goal, and terminating with the fatigue minimization goal. The process
is
carried out such that the solution obtained from a first goal never
degrades
the other goals solutions. However, weighted goal programming considers
all
goals simultaneously within a composite objective function comprising
the sum
of all deviational variables of the goals from their targets. One of
the
drawbacks of this method is the use of different units of deviational
variables
in an objective function where the sums of unwanted deviational
variables are
minimized. This different measurement unit may damage the relative
importance
of the objective to the decision maker or cause an unintentional bias
towards
the objectives with a larger magnitude (Tamiz et al., 1998). This
problem can
be solved by the use of a normalization procedure or simply using same
unit for
all deviational variables in the objective function. Different
normalization
techniques are suggested (De Kluyver, 1979; Jones, 1995; Masud &
Hwang,
1981; Wildhelm, 1981). In this research, the
following steps are used to handle multi-objective functions:
· Define LP1 as the first Linear programming model with objective function: minimize goalc; LP2 is the second linear programming model with objective function: minimize goalp; LP3 is the third linear programming model with objective function: minimize goalB; LP4 is the fourth linear programming model with objective function: minimize goalF.
· Identify the goal values of each model in step 1, and add these values to the right hand side of each constraint (1), (2), (3) and (4), respectively, to ensure the goals are satisfied.
· Add penalty weights to reflect the decision maker's preferences regarding the relative importance of each goal; for example: in order to minimize total costs (goal C), its penalty weight should be multiplied by the amount over the costs target determined in step 2. Also, in order to minimize total number of top performers fired (goal P), its penalty costs should be multiplied by the amount under the desired number that can be achieved, and so on.
· Solve the combined objective function that minimizes the deviational variables which represents all goals.
A
normalization scheme technique is presented to scale all unwanted
deviations to
a 0-1 range. The value zero represents a deviation of zero and the
value one
represents the worst (highest) possible value of the deviation within
the
feasible set. The one value can be found by a single-objective
maximization or
minimization depending on the objective function. However, it is not
possible
to find this value when the objective function is unbounded. Table 1
illustrates the worst possible values of unwanted deviational variables.
Unwanted Deviation |
Maximum Value |
d+C |
455,995.4 |
d+P |
1,142.3 |
d+B |
4,098.2 |
d+F |
27.1 |
Table 1. the Worst Possible of Deviational Variables
This leads to the following
objective function with the same set of constraints given previously.
The next section presents the
resulting solution for the given problem.
4. Computational results
In
this section, the feasibility of applying the proposed method is
demonstrated
to assess the effect of workers’ differences on the workforce schedule.
Insights
on the effect of various human factors on workforce scheduling
decisions are
presented. The sensitivity of decision parameters to the variations of
relevant
conditions based on the numerical example is tested to show the effects
of
fatigue level and personality levels on workforce decisions and
performance.
Numerical example
Model
validation ensures that the model addresses the right problem, provides
accurate information about the real system being modelled, and makes
the model
actually usable. In this section, a numerical example is given in order
to
demonstrate the application of the model; we assume a company produces
its
products to fulfil known demand along an 8-period planning horizon.
Also, it is assumed that the worker is available for 8
hours a day (160 hours per month) at regular time and for 2 hours a day
(80
hours per month) at overtime. However, it is assumed that a worker is
not
productive during daily breaks and interruptions. Also, the maximum
fatigue
load a worker can accumulate in any task depends on the personality
level. Many
jobs require human effort, and some recovery allowance must be made
from fatigue
for relaxation. We assume that a worker with a high personality level
and in
top physical condition requires a smaller allowance to recover from
fatigue
than a low personality level worker. However, other factors such as the
factors
related to the nature of the work itself and the environment might
affect the
amount of relaxation allowances needed. Moreover,
input data is shown in Tables 2 to 7. The known demand of worker skills
in
worker-hours in each period is summarized in Table 2. Table 3 shows
workers’
availabilities. Table 4 shows the available workforce at period zero.
Next,
Table 5 shows the cost of training from skill level to another skill
level in
each period. Workers daily
salary, hiring costs, lay-off costs, overtime costs and workers’
capacities are
shown in Table 6. Finally, Table 7 shows the values of the maximum
endurance
time, fatigue fractions and the recovery rates for different workers.
These
values are estimated based on the formulas, which are adapted from Jaber
and Neumann (2010). Using the input data presented, the
model consists of 7,364 variables and 12,929 constraints and the
optimal
solution for the problem can be easily obtained using LINGO 13.0
software in
less than a minute of program running.
|
D1a |
D2 |
D3 |
D4 |
D5 |
D6 |
D7 |
D8 |
Worker Skill 1 |
320.0 |
160.0 |
320.0 |
320.0 |
320.0 |
320.0 |
320.0 |
320.0 |
Worker Skill 2 |
400.0 |
320.0 |
320.0 |
320.0 |
400.0 |
160.0 |
320.0 |
480.0 |
Worker Skill 3 |
400.0 |
480.0 |
480.0 |
480.0 |
320.0 |
160.0 |
320.0 |
320.0 |
aD1 represents Day 1
Table 2. Demand of Worker Skills in Each Week (worker-hours)
|
|
D1 |
D2 |
D3 |
D4 |
D5 |
D6 |
D7 |
D8 |
Worker Skill 1 |
Availability (regular time) |
8.0 |
8.0 |
8.0 |
8.0 |
8.0 |
8.0 |
8.0 |
8.0 |
Availability (overtime) |
2.0 |
2.0 |
2.0 |
2.0 |
2.0 |
2.0 |
2.0 |
2.0 |
|
Worker Skill 2 |
Availability (regular time) |
8.0 |
8.0 |
8.0 |
8.0 |
8.0 |
8.0 |
8.0 |
8.0 |
Availability (overtime) |
2.0 |
2.0 |
2.0 |
2.0 |
2.0 |
2.0 |
2.0 |
2.0 |
|
Worker Skill 3 |
Availability (regular time) |
8.0 |
8.0 |
8.0 |
8.0 |
8.0 |
8.0 |
8.0 |
8.0 |
Availability (overtime) |
2.0 |
2.0 |
2.0 |
2.0 |
2.0 |
2.0 |
2.0 |
2.0 |
Table 3. Workers’ Availabilities (worker-hours)
|
|
Machine Level 1 |
Machine Level 2 |
Machine Level 3 |
Worker Skill 1 |
P1b |
20.0 |
0.0 |
0.0 |
P2
|
10.0 |
0.0 |
0.0 |
|
P3
|
5.0 |
0.0 |
0.0 |
|
Worker Skill 2 |
P1 |
5.0 |
10.0 |
0.0 |
P2
|
10.0 |
5.0 |
0.0 |
|
P3
|
5.0 |
10.0 |
0.0 |
|
Worker Skill 3 |
P1 |
5.0 |
0.0 |
10.0 |
P2
|
0.0 |
10.0 |
0.0 |
|
P3
|
10.0 |
5.0 |
10.0 |
bP1 represents Personality level 1
Table 4. Initial Workforce Available in Each Machine Level (workers)
From |
|
To |
D1 |
D2 |
D3 |
D4 |
D5 |
D6 |
D7 |
D8 |
Worker Skill 1 |
P1 |
Skill 2 |
4.0 |
4.0 |
4.0 |
4.0 |
4.0 |
4.0 |
4.0 |
4.0 |
P2 |
Skill 2 |
5.0 |
5.0 |
5.0 |
5.0 |
5.0 |
5.0 |
5.0 |
5.0 |
|
P3 |
Skill 2 |
6.0 |
6.0 |
6.0 |
6.0 |
6.0 |
6.0 |
6.0 |
6.0 |
|
Worker Skill 2 |
P1 |
Skill 3 |
4.0 |
4.0 |
4.0 |
4.0 |
4.0 |
4.0 |
4.0 |
4.0 |
P2 |
Skill 3 |
5.0 |
5.0 |
5.0 |
5.0 |
5.0 |
5.0 |
5.0 |
5.0 |
|
P3 |
Skill 3 |
6.0 |
6.0 |
6.0 |
6.0 |
6.0 |
6.0 |
6.0 |
6.0 |
Table 5. Training Costs in Each Period ($/worker-days)
Results from the model are shown in
Table 8 and 9. In this paper, many human factors such as workers’
training,
skills, overtime, workers’ availabilities, workers’ breaks, workers’
personalities and workers’ fatigue are considered to show their
importance at
the early planning stages. However, the results from the model offer
staffing
decisions on what, how and when to hire, fire and train. Also, the
number of
worker-hours during regular time and overtime and the number of hours
during
breaks workers can take are determined. The
optimal plan is obtained based on the present input data; if
the prioritization of the goals and initial settings is modified, the
results
are likely to be different.
|
|
|
D1 |
D2 |
D3 |
D4 |
D5 |
D6 |
D7 |
D8 |
Worker Skill 1 |
P1 |
Salary |
100.0 |
100.0 |
100.0 |
100.0 |
100.0 |
100.0 |
100.0 |
100.0 |
Hiring Costs |
80.0 |
80.0 |
80.0 |
80.0 |
80.0 |
80.0 |
80.0 |
80.0 |
||
Firing Costs |
95.0 |
95.0 |
95.0 |
95.0 |
95.0 |
95.0 |
95.0 |
95.0 |
||
Overtime |
18.5 |
18.5 |
18.5 |
18.5 |
18.5 |
18.5 |
18.5 |
18.5 |
||
P2
|
Salary |
110.0 |
110.0 |
110.0 |
110.0 |
110.0 |
110.0 |
110.0 |
110.0 |
|
Hiring Costs |
85.0 |
85.0 |
85.0 |
85.0 |
85.0 |
85.0 |
85.0 |
85.0 |
||
Firing Costs |
100.0 |
100.0 |
100.0 |
100.0 |
100.0 |
100.0 |
100.0 |
100.0 |
||
Overtime |
20.5 |
20.5 |
20.5 |
20.5 |
20.5 |
20.5 |
20.5 |
20.5 |
||
P3
|
Salary |
120.0 |
120.0 |
120.0 |
120.0 |
120.0 |
120.0 |
120.0 |
120.0 |
|
Hiring Costs |
90.0 |
90.0 |
90.0 |
90.0 |
90.0 |
90.0 |
90.0 |
90.0 |
||
Firing Costs |
115.0 |
115.0 |
115.0 |
115.0 |
115.0 |
115.0 |
115.0 |
115.0 |
||
Overtime |
22.5 |
22.5 |
22.5 |
22.5 |
22.5 |
22.5 |
22.5 |
22.5 |
||
Worker Skill 2 |
P1 |
Salary |
130.0 |
130.0 |
130.0 |
130.0 |
130.0 |
130.0 |
130.0 |
130.0 |
Hiring Costs |
95.0 |
95.0 |
95.0 |
95.0 |
95.0 |
95.0 |
95.0 |
95.0 |
||
Firing Costs |
120.0 |
120.0 |
120.0 |
120.0 |
120.0 |
120.0 |
120.0 |
120.0 |
||
Overtime |
24.5 |
24.5 |
24.5 |
24.5 |
24.5 |
24.5 |
24.5 |
24.5 |
||
P2
|
Salary |
140.0 |
140.0 |
140.0 |
140.0 |
140.0 |
140.0 |
140.0 |
140.0 |
|
Hiring Costs |
100.0 |
100.0 |
100.0 |
100.0 |
100.0 |
100.0 |
100.0 |
100.0 |
||
Firing Costs |
125.0 |
125.0 |
125.0 |
125.0 |
125.0 |
125.0 |
125.0 |
125.0 |
||
Overtime |
26.5 |
26.5 |
26.5 |
26.5 |
26.5 |
26.5 |
26.5 |
26.5 |
||
P3
|
Salary |
150.0 |
150.0 |
150.0 |
150.0 |
150.0 |
150.0 |
150.0 |
150.0 |
|
Hiring Costs |
115.0 |
115.0 |
115.0 |
115.0 |
115.0 |
115.0 |
115.0 |
115.0 |
||
Firing Costs |
140.0 |
140.0 |
140.0 |
140.0 |
140.0 |
140.0 |
140.0 |
140.0 |
||
Overtime |
28.5 |
28.5 |
28.5 |
28.5 |
28.5 |
28.5 |
28.5 |
28.5 |
||
Worker Skill 3 |
P1 |
Salary |
160.0 |
160.0 |
160.0 |
160.0 |
160.0 |
160.0 |
160.0 |
160.0 |
Hiring Costs |
120.0 |
120.0 |
120.0 |
120.0 |
120.0 |
120.0 |
120.0 |
120.0 |
||
Firing Costs |
145.0 |
145.0 |
145.0 |
145.0 |
145.0 |
145.0 |
145.0 |
145.0 |
||
Overtime |
30.0 |
30.0 |
30.0 |
30.0 |
30.0 |
30.0 |
30.0 |
30.0 |
||
P2
|
Salary |
170.0 |
170.0 |
170.0 |
170.0 |
170.0 |
170.0 |
170.0 |
170.0 |
|
Hiring Costs |
125.0 |
125.0 |
125.0 |
125.0 |
125.0 |
125.0 |
125.0 |
125.0 |
||
Firing Costs |
140.0 |
140.0 |
140.0 |
140.0 |
140.0 |
140.0 |
140.0 |
140.0 |
||
Overtime |
32.5 |
32.5 |
32.5 |
32.5 |
32.5 |
32.5 |
32.5 |
32.5 |
||
P3
|
Salary |
180.0 |
180.0 |
180.0 |
180.0 |
180.0 |
180.0 |
180.0 |
180.0 |
|
Hiring Costs |
140.0 |
140.0 |
140.0 |
140.0 |
140.0 |
140.0 |
140.0 |
140.0 |
||
Firing Costs |
145.0 |
145.0 |
145.0 |
145.0 |
145.0 |
145.0 |
145.0 |
145.0 |
||
Overtime |
33.5 |
33.5 |
33.5 |
33.5 |
33.5 |
33.5 |
33.5 |
33.5 |
Table 6. Salary, Hiring, Firing, and Hourly Overtime Costs ($)
|
|
Fmax |
T1c |
T2 |
T3 |
T4 |
T5 |
T6 |
T7 |
T8 |
T9 |
Fatigue fraction |
P1 |
0.88 |
0.80 |
0.80 |
0.80 |
- |
- |
- |
- |
- |
- |
P2 |
0.60 |
0.50 |
0.50 |
0.50 |
0.50 |
0.50 |
0.50 |
- |
- |
- |
|
P3 |
0.13 |
0.10 |
0.10 |
0.10 |
0.10 |
0.10 |
0.10 |
0.10 |
0.10 |
0.10 |
|
Recovery rate |
P1 |
0.88 |
0.52 |
0.52 |
0.52 |
- |
- |
- |
- |
- |
- |
P2 |
0.60 |
0.51 |
0.51 |
0.51 |
0.51 |
0.51 |
0.51 |
- |
- |
- |
|
P3 |
0.13 |
0.49 |
0.49 |
0.49 |
0.49 |
0.49 |
0.49 |
0.49 |
0.49 |
0.49 |
|
Endurance time |
P1 |
0.88 |
1.10 |
1.10 |
1.10 |
- |
- |
- |
- |
- |
- |
P2 |
0.60 |
1.17 |
1.17 |
1.17 |
1.17 |
1.17 |
1.17 |
- |
- |
- |
|
P3 |
0.13 |
1.27 |
1.27 |
1.27 |
1.27 |
1.27 |
1.27 |
1.27 |
1.27 |
1.27 |
cT1 represents Task 1
Table 7. Fatigue Levels and Recovery Rates
|
D1 |
D2 |
D3 |
D4 |
D5 |
D6 |
D7 |
D8 |
||
|
Demand (workers) |
40.0 |
20.0 |
40.0 |
40.0 |
40.0 |
40.0 |
40.0 |
40.0 |
|
Worker Skill 1 |
P1 |
Workers used on level 1 |
29.1 |
22.1 |
44.2 |
44.2 |
44.2 |
44.2 |
44.2 |
44.2 |
Workers hired on level 1 |
23.9 |
0.0 |
22.1 |
0.0 |
11.1 |
0.0 |
27.7 |
22.1 |
||
Workers fired from level 1 |
0.0 |
7.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
||
Workers trained to Level 2 |
14.8 |
0.0 |
0.0 |
0.0 |
11.1 |
0.0 |
27.7 |
22.1 |
||
P2
|
Workers used on level 1 |
10.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
|
Workers hired on level 1 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
||
Workers fired from level 1 |
0.0 |
10.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
||
Workers trained to Level 2 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
||
P3
|
Workers used on level 1 |
5.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
|
Workers hired on level 1 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
||
Workers fired from level 1 |
0.0 |
5.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
||
Workers trained to Level 2 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
||
|
Demand (workers) |
50.0 |
40.0 |
40.0 |
40.0 |
50.0 |
20.0 |
40.0 |
60.0 |
|
Worker Skill 2 |
P1 |
Workers used on level 1 |
5.0 |
5.0 |
5.0 |
5.0 |
5.0 |
0.0 |
0.0 |
0.0 |
Workers used on level 2 |
19.9 |
8.9 |
8.9 |
8.9 |
19.9 |
0.0 |
27.7 |
49.8 |
||
Workers hired on level 1&2 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
||
Workers fired from level 1&2 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
25.0 |
0.0 |
||
Workers trained to Level 3 |
4.8 |
11.1 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
||
P2
|
Workers used on level 1 |
10.0 |
10.0 |
10.0 |
10.0 |
10.0 |
10.0 |
10.0 |
10.0 |
|
Workers used on level 2 |
5.0 |
5.0 |
5.0 |
5.0 |
5.0 |
5.0 |
1.4 |
1.4 |
||
Workers hired on level 1&2 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
||
Workers fired from level 1&2 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
||
Workers trained to Level 3 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
3.6 |
0.0 |
||
P3
|
Workers used on level 1 |
5.0 |
5.0 |
5.0 |
5.0 |
5.0 |
5.0 |
5.0 |
5.0 |
|
Workers used on level 2 |
10.0 |
10.0 |
10.0 |
10.0 |
10.0 |
10.0 |
0.0 |
0.0 |
||
Workers hired on level 1&2 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
||
Workers fired from level 1&2 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
||
Workers trained to Level 3 |
0.5 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
10.0 |
0.0 |
||
|
Demand (workers) |
50.0 |
60.0 |
60.0 |
60.0 |
40.0 |
20.0 |
40.0 |
40.0 |
|
Worker Skill 3 |
P1 |
Workers used on level 1 |
5.0 |
5.0 |
5.0 |
5.0 |
5.0 |
0.0 |
0.0 |
0.0 |
Workers used on level 2 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
||
Workers used on level 3 |
14.8
|
25.9 |
25.9 |
25.9 |
8.8 |
0.0 |
0.0 |
0.0 |
||
Workers hired on level 1,2&3 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
||
Workers fired from level 1&2&3 |
0.0 |
0.0 |
0.0 |
0.0 |
17.1 |
13.8 |
0.0 |
0.0 |
||
P2
|
Workers used on level 1 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
|
Workers used on level 2 |
10.0 |
10.0 |
10.0 |
10.0 |
5.0 |
5.0 |
5.0 |
5.0 |
||
Workers used on level 3 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0 |
3.6 |
3.6 |
||
Workers hired on level 1,2&3 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
||
Workers fired from level 1&2&3 |
0.0 |
0.0 |
0.0 |
0.0 |
5.0 |
0.0 |
0.0 |
0.0 |
||
P3
|
Workers used on level 1 |
10.0 |
10.0 |
10.0 |
10.0 |
10.0 |
10.0 |
10.0 |
10.0 |
|
Workers used on level 2 |
5.0 |
5.0 |
5.0 |
5.0 |
5.0 |
5.0 |
5.0 |
5.0 |
||
Workers used on level 3 |
10.0 |
10.0 |
10.0 |
10.0 |
10.0 |
10.0 |
20.0 |
20.0 |
||
Workers hired on level 1,2&3 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
||
Workers fired from level 1&2&3 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
Table 8. Resulting Workforce Plan in Number of Workers
|
D1 |
D2 |
D3 |
D4 |
D5 |
D6 |
D7 |
D8 |
||
|
Demand (hours) |
320.0 |
160.0 |
320.0 |
320.0 |
320.0 |
320.0 |
320.0 |
320.0 |
|
Worker Skill 1 |
P1 |
Regular time on level 1 |
152.3 |
115.8 |
231.6 |
231.5 |
231.5 |
231.5 |
231.5 |
231.5 |
Breaks 1 |
80.3 |
61.0 |
122.0 |
122.0 |
122.0 |
122.0 |
122.0 |
120.0 |
||
Overtime Hours |
58.1 |
44.2 |
88.4 |
88.4 |
88.4 |
88.4 |
88.4 |
88.4 |
||
P2
|
Regular time on level 1 |
52.8 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
|
Breaks 1 |
27.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
||
Overtime Hours |
20.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
||
P3
|
Regular time on level 1 |
26.7 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
|
Breaks 1 |
13.3 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
||
Overtime Hours |
10.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
||
|
Demand (hours) |
400.0 |
320.0 |
320.0 |
320.0 |
400.0 |
160.0 |
320.0 |
480.0 |
|
Worker Skill 2 |
P1 |
Regular time on level 1 |
26.2 |
26.2 |
26.2 |
26.2 |
26.2 |
0.0 |
0.0 |
0.0 |
Regular time on level 2 |
104.5 |
46.6 |
46.6 |
46.6 |
104.5 |
0.0 |
145.2 |
261.0 |
||
Breaks 1&2 |
68.9 |
38.4 |
38.4 |
38.4 |
68.9 |
0.0 |
76.5 |
137.5 |
||
Overtime Hours |
49.8 |
27.8 |
27.8 |
27.8 |
49.8 |
0.0 |
55.4 |
99.6 |
||
P2
|
Regular time on level 1 |
52.8 |
52.8 |
52.8 |
52.8 |
52.8 |
52.8 |
52.8 |
52.8 |
|
Regular time on level 2 |
26.4 |
26.4 |
26.4 |
26.4 |
26.4 |
26.4 |
7.1 |
7.1 |
||
Breaks 1&2 |
40.7 |
40.7 |
40.7 |
40.7 |
40.7 |
40.7 |
30.8 |
30.8 |
||
Overtime Hours |
30.0 |
30.0 |
30.0 |
30.0 |
30.0 |
0.6 |
22.7 |
22.7 |
||
P3
|
Regular time on level 1 |
26.7 |
26.7 |
26.7 |
26.7 |
26.7 |
26.7 |
26.7 |
26.7 |
|
Regular time on level 2 |
53.4 |
53.4 |
53.4 |
53.4 |
53.4 |
53.4 |
0.0 |
0.0 |
||
Breaks 1&2 |
39.8 |
39.8 |
39.8 |
39.8 |
39.8 |
39.8 |
13.3 |
13.3 |
||
Overtime Hours |
30.0 |
30.0 |
30.0 |
30.0 |
30.0 |
0.0 |
10.0 |
10.0 |
||
|
Demand (hours) |
400.0 |
480.0 |
480.0 |
480.0 |
320.0 |
160.0 |
320.0 |
320.0 |
|
Worker Skill 3 |
P1 |
Regular time on level 1 |
26.2 |
26.2 |
26.2 |
26.2 |
0.0 |
0.0 |
0.0 |
26.2 |
Regular time on level 2 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
||
Regular time on level 3 |
77.7 |
135.6 |
135.6 |
135.6 |
46.2 |
0.0 |
0.0 |
0.0 |
||
Breaks 1&2&3 |
54.8 |
85.3 |
85.3 |
85.3 |
38.1 |
0.0 |
0.0 |
0.0 |
||
Overtime Hours |
39.6 |
61.7 |
61.7 |
61.7 |
27.6 |
0.0 |
0.0 |
0.0 |
||
P2
|
Regular time on level 1 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
|
Regular time on level 2 |
52.8 |
52.8 |
52.8 |
52.8 |
26.4 |
26.4 |
26.4 |
26.4 |
||
Regular time on level 3 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
0.0 |
19.3 |
19.3 |
||
Breaks 1&2&3 |
27.2 |
27.2 |
27.2 |
27.2 |
13.6 |
13.6 |
23.5 |
23.5 |
||
Overtime Hours |
20.0 |
20.0 |
20.0 |
20.0 |
10.0 |
0.0 |
17.3 |
17.3 |
||
P3
|
Regular time on level 1 |
53.4 |
53.4 |
53.4 |
53.4 |
53.4 |
53.4 |
53.4 |
53.4 |
|
Regular time on level 2 |
26.7 |
26.7 |
26.7 |
26.7 |
26.7 |
26.7 |
26.7 |
26.7 |
||
Regular time on level 3 |
53.4 |
53.4 |
53.4 |
53.4 |
53.4 |
53.4 |
106.9 |
106.9 |
||
Breaks 1&2&3 |
66.4 |
66.4 |
66.4 |
66.4 |
66.4 |
66.4 |
128.9 |
86.9 |
||
Overtime Hours |
50.0 |
50.0 |
50.0 |
50.0 |
50.0 |
0.0 |
70.0 |
70.0 |
Table 9. Resulting Workforce Plan in Worker-hours
This
research shows that workers’ differences can be used to predict hiring,
firing
and training workers and total break time. Table 8 shows the number of
workers
hired, fired and trained in each period for different personality
levels. Also,
Table 9 shows the time workers spend on all the tasks to satisfy the
demand and
the amount of break they take due to the fatigue level for each worker.
From
Tables 8 and 9, it can be seen that the workers who are not working
during
regular time have no breaks. Also, we can notice from running the model
with
different objectives that a worker at higher personality level required
less
amount of break to recover than a worker with low personality level.
Moreover,
the most of the workers hired and trained have low personality level,
which
represents the normal scenario in practice since it is assumed the
lower
personality levels workers costs less than workers with high
personality
levels. However, these results will be different if the input data are
changed
or when the company goals are changed, as shown in Table 10. For
example, if we
considered different productivities for different workers personality
levels,
the model will prefer to hire and use workers with higher personality
levels
because of their high performance.
Model
implementation and results analysis
All workers have the right
to take breaks. The actual amount of break a worker receives is usually
set out
in his contract of employment. Although there are some kinds of jobs
that do
not allow workers to take breaks such as air or sea transport and
working part
time during busy peak periods, not taking a break can result in
overloaded,
stressed, and unproductive workers. Rest breaks are one of break types
that
workers can take under special rules written in the employment contact.
This
model can help to estimate the amount of break a worker can take during
a
working day in order to minimize the risk caused by worker fatigue. In
the previous section, a simple numerical example is given to illustrate
the
performance of the model. In this section, we will study the effects of
fatigue
level and worker differences on workforce decisions. Table 10 shows a
comparison between the two cases with different goals. Also, it shows a
comparison between two cases; the first one represents the case where
fatigue
level is different and the second one represents the case where the
fatigue
level is the same. However, considering human differences that exist
between
workers results in more accurate workforce decisions. In Table 10, it
is
assumed that the fractions of maximum workers’ capability are set to be
the
average values. This fraction can be used to determine the values of
maximum
endurance time, recovery rate and maximum fatigue. Also, it is assumed
that we
decision maker is looking to achieve three goals; costs minimization,
the
number of top performers fired minimization and idle time minimization.
Total |
Goal 1 |
Goal 2 |
Goal 3 |
Goal 4 |
Equal weights (Different fatigue) |
Equal weights (Same fatigue) |
Objective Value |
223,618.2 |
1.9 |
2846.6 |
0.0 |
0.05 |
0.03 |
Demand (Wd.days) |
8,080.0 |
8,080.0 |
8,080.0 |
8,080.0 |
8,080.0 |
8,080.0 |
Regular Time (hrs) |
6,270.8 |
7,883.5 |
5,894.5 |
8,080 |
5,967.5 |
5,962.3 |
Overtime (hrs) |
1,810.223 |
206.0 |
2,185.3 |
0.0 |
2,112.5 |
2,117.6 |
Breaks (W.hrs) |
3,295.5 |
3,957.5 |
2,846.5 |
4,021 |
2,955.2 |
2,984.1 |
Workers (W.days) |
1,195.7 |
1,478.9 |
1092.7 |
1,512.6 |
1,115.3 |
1,118.3 |
Training (W.days) |
77.2 |
21.8 |
82.9 |
54.9 |
101.4 |
115.9 |
Hiring (W.days) |
70.4 |
265.5 |
249.5 |
344.8 |
104.9 |
105.8 |
Firing (W.days) |
46.9 |
186.6 |
227.9 |
264.7 |
82.5 |
82.6 |
Fatigue (%.hr) |
68.4 |
45.3 |
76.3 |
0.0 |
72.9 |
83.9 |
Costs ($) |
223,618.2 |
273,341.2 |
286,859.8 |
304,191.0 |
229,798.0 |
230,056.5 |
d W represents Worker
Table 10. Comparisons Between the Different Goals
This table shows the
comparisons between the two cases regarding the importance of
considering
fatigue level differences between workers to generate a better
solution. The
total cost when fatigue level is same for all workers is $230,056.5,
but when we consider different fatigue level between workers, the total
cost is
$229,798.0. The results show that by
considering worker differences in the model the costs differences are
not
significant. Also, the present study found that fatigue
is not significantly important for scheduling day workers from the
economics
perspective, but it help to determine the amount of the break workers
can take
depending on his personality, and salaries profiles. Further research
should be
done on the effects of the fatigue on worker scheduling with different
shifts.
Moreover, if the initial number of workers
is changed, the number of hired, fired or trained workers is changed
which will
change the total costs. Also, in Table 9, we can notice that the
company can
use this model in planning process by selecting the specific goals
based on its
policy and budget. For example, we can assign a target value for each
goal so
that we can determine the number of workers needed in each period to
satisfy
the demand without exceeding the predefined goals.
Sensitivity
Analysis
Realistic
mixed integer programming models require large amounts of data.
Accurate data
are expensive to collect, so we will generally be forced to use data in
which
we have less than complete confidence. This section discusses the
actual
implementation of the proposed model by manipulating different
alternatives and
analyzing the sensitivity of decision parameters to the variation of
relevant
conditions, based on the preceding numerical example.
Implications
Regarding Different Model Goals
A
user of a model should be concerned with how the recommendations of the
model
are altered by changes in the input data. Table
11 illustrates the comparisons between different scenario problems and
the
effects of changing the weights of the company goals on the total costs
and
utilization of the work. In this Table, we
implement 10 scenarios to compare between the final results in terms of
workers’ utilization, workers’ fatigue, and the total costs. The worker
utilization is calculated by dividing the total productive time for all
the
workers by the total available hours. Worker break percentages
represent the
amount of break workers can take in average during a working day. Also,
workers'
fatigue represents the total physical load on the workforce during a
working
day. We change each scenario by changing the weights of
the unwanted deviational variables in the objective function to show
its
effects on the final objective value. For example, in scenario 1, all
goals
have the same importance in the objective function.
In
the weighted goal programming method, we can use a set of preference
weights
assigned to the penalisation of unwanted deviations to provide
solutions that
are of practical use to the problem owner. In this weight space
analysis, it is
assumed that all weighting vectors have been normalized and hence sum
to one.
Note that in practice the weight of an unwanted deviational variable
has to be
greater than zero to avoid the possibility of generating
Pareto-inefficient
solutions. Tamiz and Jones (1996) defined Pareto inefficiency as an
objective
that can be improved without worsening the value of any other
objective.
Therefore, a small weight (e.g. 0.005) is suggested to replace a zero
weight.
Heuristic method and sensitivity analysis are developed to find the
weight
values in the weighting space (Jones & Tamiz, 2010). By
comparing scenario 1 through 10, we can see that if we add
more weight to the cost goal the total costs are reduced. Also, the
total
fatigue of the workers will be increased if more weight is added to the
breaks
minimization goal. However, increasing the physical
load of the workers may not be desirable due to desired quality levels
or
occupational health and safety issues. Therefore,
the determination of the weight values is a process of interaction with
the
decision maker(s). By doing this sensitivity analysis we can find the
solution
that fit with any company requirements. For example, scenarios 3 and 7
give a
relatively high value of utilization compared to the other scenarios
(e.g.
scenario 2). This means that putting more weight on idle time
minimization or
hiring workers with fast rate recovery can increases workforce
utilization. So
the company can choose which scenario is best based on its policies and
rules.
However, sensitivity analysis can reveal which pieces of information
should be
estimated most carefully.
Goal # |
W1 |
W2 |
W3 |
W4 |
Obj. value |
Utilization |
Fatigue |
Costs ($) |
1 |
0.25 |
0.25 |
0.25 |
0.25 |
0.019 |
66.0% |