Operational Research in Engineering Sciences: Theory and Applications 
 Vol. 5, Issue 3, 2022, pp. 17-39 
ISSN: 2620-1607 
eISSN: 2620-1747 

 DOI: https://doi.org/10.31181/oresta060722075b 

* Corresponding author. 
prasad.bari@fcrit.ac.in (P. Bari), pmkarande@me.vjti.ac.in (P. Karande), 
jaystonmenezes@gmail.com (J. Menezes) 

SIMULATION OF JOB SEQUENCING FOR STOCHASTIC 
SCHEDULING WITH A GENETIC ALGORITHM 

Prasad Bari 1,2*, Prasad Karande 1, Jayston Menezes 2 

1 Department of Mechanical Engineering, Veermata Jijabai Technological Institute, 
Mumbai, India 

2 Department of Mechanical Engineering, Fr. C. Rodrigues Institute of Technology, 
Vashi, Navi-Mumbai, India 

Received: 15 April 2022  
Accepted: 18 June2022  
First online: 06 July 2022 

Research paper 

Abstract: Sequencing is done to determine the order in which the jobs are to be 
processed. Extensive research has been carried out with an aim to tackle real-world 
scheduling problems. In industries, experimentation is performed before an ultimate 
choice is made to know the optimal priority sequencing rule. Therefore, an extensive 
approach to selecting the correct choice is necessary for the management decision-
making perspective. In this research, the genetic algorithm (GA) and working of a 
simulation environment are explained, in which a scheduling operator, under any given 
circumstances, can obtain the appropriate sequence for job scheduling in a shop. The 
paper also explains the stochastic based linguistic, scenarios and probabilistic 
approaches to solve sequencing problem. The simulation environment allows the 
operator to select the tardiness and non-tardiness related performance measures. The 
simulator takes input values such as number of jobs, processing time and due date and 
discovers a near-optimal sequence for scheduling of jobs that minimizes the 
performance measures selected by the operator as per requirement. The case study 
considered is solved using scenarios based stochastic scheduling approach and results 
are shown. The results are compared with the classical method used in the company and 
observed that the proposed approach gives a better result. 

Key words: Stochastic scheduling, Genetic algorithm, Sequencing, Tardiness 

1. Introduction 

Job scheduling defines the order in which jobs are to be completed at one or more 
workplaces in a workshop. Job scheduling is critical to avoid long lines or production 
delays, which can result in financial losses or penalties for the firm. Scheduling of the 

mailto:pmkarande@me.vjti.ac.in
mailto:jaystonmenezes@gmail.com


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18 

 

jobs can be done by keeping the main focus on one of the factors like priority 
sequencing rules (Sharma & Jain, 2015), performance measures (Oyetunji, 2009), cost 
minimization (Bari & Karande, 2020)(Bari & Karande, 2022), preference to weighted 
jobs, etc. Oyetunji (2009) formulated expressions of performance measures for 
computing their values. In this paper, the primary focus is on the minimization of the 
performance measures for scheduling. Performance measures can be minimized by 
arranging jobs in a particular sequence. In order to obtain the optimum sequence of 
the jobs, it is essential to know the number of jobs, their corresponding processing 
time (PT), and their due date (DD).  

A job sequence is created by applying sequencing rules such as shortest processing 
time (SPT), in which the job that requires the least amount of machine processing time 
is scheduled first. It is applicable in the need of reduction of average completion time. 
Under the earliest due date (EDD) rule, jobs are completed in the order of their earlier 
due date. It can be used to cut down on tardiness. Jobs are processed in the order they 
arrive on the machine under the first-come, first-served (FCFS) rule. It is effective at 
lowering the variance in completion time for any dataset. The operator follows one of 
the sequencing rules as per their industry requirement. The operator's primary focus 
is primarily on the performance measure values of the sequence so that he can process 
the operation in the lowest possible time. In the scheduling operations, Kumar et al. 
(2017) discussed about 13 significant performance measures: completion time, flow 
time, total flow time, average job completion time, average number of jobs in the 
system, percentage utilization, lateness, average lateness, tardiness, total tardiness, 
maximum tardiness, average tardiness, and number of tardy jobs. Section 2 explains 
these performance measures in detail. To maximize production efficiency, all 
performance measurements except the percentage utilization from the considered 
performance measures should be minimized. Maximum percentage utilization 
indicates that the processing of the job is being done effectively on the machines in the 
job shop. Applying a PROMETHEE-GAIA technique can optimize these performance 
measures (Bari & Karande, 2021). 

Considering the deterministic scheduling model, the optimum sequence with 
reference to these performance measures can be attained. However, in real-world 
scheduling problems, various factors affect the PT of the jobs that cannot be neglected. 
Therefore, such problems can be solved using the stochastic model. The stochastic 
approaches are more significant because of their capability to handle non-linear and 
multi-objective formulations effortlessly. The stochastic model can be solved using 
either the exact method or heuristic approach. However, the computational time 
required to solve deterministic and stochastic problems using the exact method is 
more than the heuristic approach. One of the popular heuristic methods is the genetic 
algorithm (GA). The trend for using stochastic techniques with GA is increasing for 
solving industrialized problems because GA finds a better objective function value for 
each iteration in less computational time. (Deb et al., 2002), (Sarkar & Modak, 2005), 
(Koratiya et al., 2010), (Ramteke & Srinivasan, 2012). Shrouf et al. (2014) proposed 
the application of GA to solve the single machine scheduling in the time off use tariff 
with respect to the machine's status, which consists of processing, idle, turning on and 
turning off. Roy et al. (2019) introduced novel GA concerning selection and crossover 
operation and found it effective to solve the travelling salesman problem. They have 
also mentioned that the algorithm proved effective for solving other problems like 
network optimization. Kurniawan et al. (2020) implemented GA to determine the job 



Simulation of Job Sequencing for Stochastic Scheduling with a Genetic Algorithm 

19 
 

sequencing, the job assignment, and the starting time of the job. Bayu et al. (2020) 
applied GA by considering stochastic conditions for sequencing the operations of 
gasoline blending and distribution plants to give gasoline a high commercial latent 
without negotiating quality and the clients' demand. Stanković et al. (2020) presented 
a model for solving flexible job shop scheduling problem (FJSP) built on meta-heuristic 
algorithms, tabu search, GA, and ant colony optimization. They have demonstrated the 
effectiveness of the GA method in resolving the FJSP problem, which gives 
commendatory results after testing. Garg (2016, 2019) solved the constraint 
optimization problem by feeding genetic operators of GA with particle swarm 
optimization and a gravitational search algorithm and discovered that these combined 
algorithms were efficient in finding solution to engineering design problems. Souza et 
al. (2022) proposed GA with simulated annealing approach to generate schedules in 
deterministic and stochastic machine unavailability restrictions. As a result, the 
authors demonstrated the significance of GA in their research. Researchers are also 
researching other combinatorial optimization strategies to solve and improve 
solutions to optimization problems. Kundu T., & Garg H.  (2021) combined Harris 
hawks optimization and teaching–learning-based optimization algorithm and found 
to be better to find solution to numerical optimization problems. 

The paper is further divided into the sections listed below. Section 2 discusses 
methodology, including performance measures, deterministic and stochastic 
scheduling, and their approaches. Section 3 describes the optimization of the non-
tardiness and tardiness-related measures, as well as the GA algorithm. Section 4 
includes a case study of a manufacturing company as well as the model developed for 
it. Section 5 discusses the case study results. Finally, section 6 shows the conclusions 
of the article. 

2. Methodology 

The main objective of this article is to develop a model to find the optimal solution 
to scheduling problems using GA. The model was created using the Python 
programming language on a computer with 8 GB of RAM, a 500 GB hard drive, and the 
Windows 10 operating system. This model aids in determining the best sequence for 
minimizing performance measures in accordance with industry standards. For 
deterministic, stochastic linguistic, stochastic probabilistic, and stochastic scenarios-
based scheduling problems, the model finds the best solution. The following are the 
scheduling performance measures: 

1. Completion time (Cj): It is the period necessary to finish a single job j. Consider 
five jobs, j1, j2, j3, j4 and j5, with their respective PT and DD as p1, p2, p3, p4, p5 and d1, d2, 
d3, d4, d5. Additionally, job's In-Time and Out-Time, that is, the time at which the 
processing operation of a specific job, from a list of jobs, begins and ends, respectively, 
is also required. Table 1 shows the completion time of each job under the Out-Time 
column. Çetinkaya & Duman, (2021) developed an approach to minimize completion 
time of the sublots and job lots with a single job and multiple jobs.  

 

 

 



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Table 1. Data of 5 jobs 
Job Number (j) PT (pj) In-Time Out-Time (Cj) DD (dj) 

j1 p1 0 p1 d1 
j2 p2 p1 p1 + p2 d2 
j3 p3 p1 + p2 p1 + p2 + p3 d3 
j4 p4 p1 + p2 + p3 p1 + p2 + p3 + p4 d4 
j5 p5 p1 + p2 + p3 + p4 p1 + p2 + p3 + p4 + p5 d5 

2. Flow time (Fj): It is the period between the finishing time of a job and the starting 
time, that is, the difference between the Out-Time and release time of the job. For the 
deterministic model, rj = 0, where rj is the job's release time or starting time. The 
relationship is defined by equation 1. 

Fj = Cj - rj (1) 

3. Total flow time (TFT): It is the cumulative flow time for all the jobs and 
represented by equation 2. From Table 1, TFT is the summation of the values in the 
Out-Time column. 

𝑇𝐹𝑇 = ∑ 𝐹𝑗

𝑛

𝑗=1

 (2) 

4. Average job completion time (Tavg): It is the ratio of TFT to the number of jobs 
(n) in a given set and shown in equation 3.     

𝑇𝑎𝑣𝑔 =
𝑇𝐹𝑇

𝑛
 (3) 

5. Average number of jobs in the system (Navg): It is the ratio of TFT and the job 
with the maximum flow time, i.e. 𝑚𝑎𝑥(𝐹𝑗 ) formulated as equation 4. 

𝑁𝑎𝑣𝑔 =
𝑇𝐹𝑇

𝑚𝑎𝑥(𝐹𝑗 )
 (4) 

6. Percentage utilization (% Utilization): It is the reciprocal of the average number 
of jobs in the system given in equation 5. It is defined as the number of machines 
available in a job shop used to process a job. 

% Utilization =
𝑚𝑎𝑥(𝐹𝑗 )

𝑇𝐹𝑇
 (5) 

7. Lateness (Lj): It is the difference between the completion time and the DD of a 
job expressed in equation 6. From Table 1, the lateness of a job is the difference 
between the Out-Time and corresponding DD. It can have either a positive, negative 
or zero value. If lateness is positive, the job will be delayed, whereas, if it is negative, 
the job will be completed before the DD. If lateness is zero, the job will be completed 
on time. 



Simulation of Job Sequencing for Stochastic Scheduling with a Genetic Algorithm 

21 
 

𝐿𝑗  =  𝐶𝑗 − 𝑑𝑗    (6) 

8. Average lateness (Lavg): It is the ratio of the lateness of all the jobs in the system 
to the number of jobs shown in equation 7. 

𝐿𝑎𝑣𝑔 =
1

𝑛
∑ 𝐿𝑗

𝑛

𝑗=1

 (7) 

9. Tardiness (Tj): It is the measure of the delay in the completion of a job beyond 
the DD formulated as equation 8. Tardiness can have either a positive or zero value. If 
the difference between completion time and DD is negative, the job is early and not 
tardy; hence, the tardiness in such a case will be 0. 

𝑇𝑗 = 𝑚𝑎𝑥(0,  𝐶𝑗 −  𝑑𝑗 ) (8) 

10. Total tardiness (T): It is the cumulative delay of all the jobs in the set 
represented in equation 9. It is the summation of the tardiness of all jobs. 

𝑇 = ∑ 𝑇𝑗

𝑛

𝑗=1

 (9) 

11. Maximum tardiness(Tmax):  It is the measure of the maximum delay of a job 
beyond the DD.  

12. Average tardiness (Tavg): It is the ratio of total tardiness and the number of jobs 
in the system shown in equation 10. 

𝑇𝑎𝑣𝑔 =   
𝑇

𝑛
 (10) 

13. Number of tardy jobs (Ntj): It is a measure of the number of delayed jobs in the 
system and is expressed in equation 11. 

𝑁𝑡𝑗 = ∑ 𝛿(𝑇𝑗 )            {
𝛿(𝑥) = 1 𝑖𝑓 𝑥 > 0

   𝛿(𝑥) = 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

𝑛

𝑗=1

 (11) 

2.1. Deterministic Scheduling 

The deterministic scheduling operation is done considering only the present 
scenario at hand. This type of scheduling requires only the jobs' PT and DD. The DD 
given by the client remains fixed. The PT changes depending on the nature of the 
factors affecting the jobs. Deterministic scheduling does not take into account these 
factors. Hence, deterministic scheduling can be referred to as an idealistic operation. 
The deterministic model considers assumptions like jobs are available simultaneously 
for processing, a machine can process only one job at a time, set up times are included 
in the PT, input data is deterministic and known in advance, machines are available 



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continuously, and never kept idle, once an operation begins on the machine it proceeds 
without interruption (French, 1982). An illustration of the data required for 
deterministic scheduling is given in Table 2. Consider five jobs, j1, j2, j3, j4 and j5, with 
their respective PT and DD as p1, p2, p3, p4, p5 and d1, d2, d3, d4, d5.  

Table 2. Data format for deterministic scheduling 
Job Number (j) PT (pj) DD (dj) 

j1 p1 d1 
j2 p2 d2 
j3 p3 d3 
j4 p4 d4 
j5 p5 d5 

2.2. Stochastic Scheduling 

In real-world scheduling problems, various factors affect the PT of jobs and cannot 
be neglected. The deviation that occurred in the accuracy of the PT values can hamper 
the efficacy of the job shop. This further can cause a delay in circumstances where 
these factors are in the worst case. To assist the industry in scheduling of jobs with 
real-time data, stochastic scheduling is preferred, which provides more accurate 
results. This also gives an idea to the operators about when and how to sequence the 
jobs and provides the client with a view to set the DD of procurement. The 
assumptions like input data is deterministic and known in advance, and machines are 
available continuously and never kept idle made in deterministic scheduling are 
relaxed in stochastic scheduling. Through the literature review, three stochastic 
scheduling models, as mentioned below (Baker & Trietsch, 2009), have been identified 
that can provide the required results. 

2.2.1. Scenarios Method 

In this method, scheduling is done considering more than one scenario of the jobs 
as mentioned in Table 3. These scenarios are for the same jobs; that is, the end product 
is the same. However, the path to preparing the end product in different scenarios may 
be different. For example, in some scenarios, jobs can be made of different materials 
or the size of the raw material can be different, or the machine used to process can be 
different. These various factors affecting the jobs can increase or decrease the PT of 
the jobs. Thus, different scenarios are created to assist in stochastic scheduling with 
real-time data. 

Consider five jobs, j1, j2, j3, j4, and j5 with the respective PT of p11, p12, p13, p14, p15, (for 
scenario 1) p21, p22, p23, p24, p25, (for scenario 2) and p31, p32, p33, p34, p35. (for scenario 3). 
The DD for the five jobs are d1, d2, d3, d4, and d5. This data is shown in Table 3.  

Table 3. Data format for scenarios scheduling 
Job Number Scenario 1 PT Scenario 2 PT Scenario 3 PT DD 

j1 p11 p21 p31 d1 
j2 p12 p22 p32 d2 
j3 p13 p23 p33 d3 
j4 p14 p24 p34 d4 
j5 p15 p25 p35 d5 



Simulation of Job Sequencing for Stochastic Scheduling with a Genetic Algorithm 

23 
 

In deterministic scheduling, only the PT of one scenario is considered at a given 
time. The scenario can be 1, 2 or 3 from Table 3. However, in the stochastic model, the 
PT of every job is considered by taking the average of the PT from all the possible 
scenarios that can occur and shown in equation 12. Thus, the PT for any job concerning 
the above table can be given as: 

𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑃𝑇 𝑜𝑓  𝑗𝑡ℎ  𝑗𝑜𝑏 = 𝑝𝑗 =
𝑝𝑗

1 + 𝑝𝑗
2 + 𝑝𝑗

3

3
=

1

3
∑ 𝑝𝑗

𝑖

3

𝑖=1

 

Hence, the formula can be given as: 

𝑝𝑗 =
1

𝑛𝑠
∑ 𝑝𝑗

𝑛𝑠

𝑗=1

 (12) 

where, ns – Number of scenarios and j – Scenario number  

2.2.2. Linguistic method 

The linguistic method stands out when selecting real-time data of jobs. In this 
method, the PT of jobs is estimated by considering various factors that affect their 
operation. For instance, to perform a turning operation on a job, some of the factors 
affecting are the condition of the tool, the machine, the material of the job or coolant 
type. These and many other factors can either increase or decrease the PT. If a 
particular factor increases the PT of a job, it may not be in its best form. For example, 
an increase in PT concerning the above factors can be because the tool is blunt, the 
machine is not operating as it should, the material of the job is rough, or the coolant is 
not effective enough. 

Conversely, if the same factors are in their best form, the PT of the job will decrease. 
In the linguistic method, either one or all the factors can be in their best or worst shape 
under different circumstances. Thus, if there are 'nf' factors affecting the PT of a job 
and each factor can have either Good (G) or Bad (B) form, the number of conditions is 
given by 2𝑛𝑓 . In a more general aspect, the number of conditions can be given by  𝑁𝑐 =
𝑥 𝑛𝑓  where, 'nf' is the number of factors and x is the number of forms each factor can 
have. 

Consider there are five jobs, and three factors are taken into account that can affect 
the processing of each job. Each factor can have two forms that is G or B. Hence, there 
will be eight (23) conditions for each job like BBB, which means all factors for jobs are 
in bad conditions, BBG means the first two factors are in bad conditions whereas the 
third factor is in good condition and so on. Table 4 shows the data format of the 
linguistic method with eight conditions for each job. 

Table 4. Data format for linguistic scheduling 
Job Number BBB BBG BGB BGG GBB GBG GGB GGG DD 

1 32 28 27 25 22 20 16 15 23 
2 30 26 23 22 19 17 14 13 21 
3 33 29 25 24 20 19 16 11 17 
4 28 27 26 21 17 16 13 9 13 
5 35 28 25 20 16 15 13 10 15 



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From the table, the real-time data of the jobs can be selected by the form of each 
factor. For example, if the condition chosen for Job Number 1, 2, 3, 4 and 5 is BGB, GBB, 
BBB, GBG and GGG, respectively, the new data for the scheduling operation will be 
shown in Table 5. 

Table 5. Stochastic Scenarios Scheduling data after operator selection 
Job Number (j) PT (pj) DD (dj) 

1 27 23 
2 19 21 
3 33 17 
4 16 13 
5 10 15 

Thus, the PT of the jobs is now based on the form of the individual factors in the 
present scenario.  

2.2.3. Probabilistic Method 

The probabilistic scheduling method is similar to the scenarios method, wherein 
each job has a different PT in each scenario. However, this method does not take an 
average of the PT; instead, a probability of occurring is associated with each scenario. 
Therefore, the sum of the probabilities of each scenario happening is equal to 1.  The 
probabilistic approach is used for repetitive jobs carried out in industry. Ideally these 
jobs should have the same PT, but this PT can be different for different scenarios and 
thus the probability of occurring in the scenarios may vary due to different elements. 
The probability of scenarios is determined by elements such as machine breakdown, 
power failure, worker absenteeism, technology failure, material shortages, 
unavoidable delays, and so on. The decision-maker computes these probabilities by 
analyzing historical data. While calculating the performance measure, each scenario 
has to be arranged in the sequence, and the performance value is then calculated for 
the individual scenarios. Next, the expected value of the performance measure is 
calculated by taking the summation of the product of individual scenarios' 
performance measure and the probability of the scenario. Consider the data of 5 jobs, 
j1, j2, j3, j4, and j5, as shown in Table 6, with the respective PT for Scenario 1 are p11, p12, 
p13, p14, p15 for Scenario 2 are p21, p22, p23, p24, p25 and for Scenario 3 are p31, p32, p33, p34, 
p35. The DD for the five jobs are d1, d2, d3, d4, and d5.  The probability of Scenario 1, 2 
and 3 occurring is P1, P2 and P3, respectively. 

Table 6. Data format for Stochastic Probabilistic Scheduling 
Job Number Scenario 1 PT Scenario 2 PT Scenario 3 PT DD 

j1 p11 p21 p31 d1 
j2 p12 p22 p32 d2 
j3 p13 p23 p33 d3 
j4 p14 p24 p34 d4 
j5 p15 p25 p35 d5 

If the performance measure, TFT calculated for the above 3 Scenarios is TFT1, TFT2 
and TFT3, then the Expected TFT is given in equation 13, 



Simulation of Job Sequencing for Stochastic Scheduling with a Genetic Algorithm 

25 
 

𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑇𝐹𝑇 = 𝑇𝐹𝑇1 ∗ 𝑃1 +  𝑇𝐹𝑇2 ∗ 𝑃2 + 𝑇𝐹𝑇3 ∗ 𝑃3 (13) 

The same approach can be implemented for finding the expected values of all the 
performance measures. 

3. Optimizing Performance Measures 

The performance measures are divided into two categories, non-tardiness 
performance measures and tardiness performance measure. The non-tardiness 
performance measures are optimized with the SPT rule, while for tardiness related 
measures, GA is applied to generate an optimal sequence. 

3.1. Non-Tardiness Performance Measures 

Non-tardiness performance measures are total flow time, average flow time, total 
PT, percentage utilization, the average number of jobs in the system, total lateness and 
average lateness. All these measures, except percentage utilization, can be reduced by 
arranging the jobs in the SPT sequence. If the jobs are arranged in an SPT sequence, 
percentage utilization will have a maximum value. The explanation for this approach 
has been given using the following example. 

Consider five jobs j1, j2, j3, j4 and j5 with PT p1, p2, p3, p4 and p5. The DD of the jobs are 
d1, d2, d3, d4 and d5. This data is summarized in Table 2. To calculate the non-tardiness 
related performance measures, two additional columns must be added, In-time and 
Out-time of each job. In-time indicates when a job arrives on a machine, more 
specifically, after completing the preceding job, if any. Out-time indicates the time at 
which a job leaves the machine that is after completion of its PT.  

1. Total flow time 

It is given by the sum of the values in the Out-Time column. Therefore, in this case,  

𝑇𝐹𝑇 =  𝑝1 + 𝑝1 + 𝑝2 + 𝑝1 + 𝑝2 + 𝑝3 + 𝑝1 + 𝑝2 + 𝑝3 + 𝑝4  +  𝑝1 + 𝑝2 + 𝑝3 + 𝑝4 + 𝑝5  

𝑇𝐹𝑇 = 5𝑝1 + 4𝑝2 + 3𝑝3 + 2𝑝4 + 𝑝5 (14) 

To minimize the value of TFT shown in equation 14, the job with the least PT has 
to be multiplied with the largest coefficient and so on from left to right in increasing 
order of PT. 

2. Average job completion time 

𝑇𝑎𝑣𝑔 =  
5𝑝1 + 4𝑝2 + 3𝑝3 + 2𝑝4 + 𝑝5

𝑛
 (15) 

To reduce Tavg, shown in equation 15, the number of jobs is constant, and hence the 
sequence which will reduce the TFT will also reduce Tavg. 

3. Percentage utilization 



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26 

 

% Utilization  =
max (𝐹𝑙𝑜𝑤 𝑇𝑖𝑚𝑒)

𝑇𝐹𝑇
∗  100 

The performance measure shown in equation 16 requires to be maximum to make 
the utmost utilization of the machine for the given set of jobs. For example, from Table 
1, job number 5 will have the maximum flow time, and the TFT must be minimized to 
maximize the performance measure.  

% Utilization  =
𝑝1 + 𝑝2 + 𝑝3 + 𝑝4 + 𝑝5

5𝑝1 + 4𝑝2 + 3𝑝3 + 2𝑝4 + 𝑝5
∗ 100 (16) 

4. Average number of jobs in the system 

𝑁𝑎𝑣𝑔 =
1

% Utilization
 (17) 

To minimize this performance measure shown in equation 17, the percentage 
utilization performance measure must be maximum. Hence, this value can be 
minimized using the same approach from the previous performance measure. 

5. Total lateness 

Total lateness is the sum of the difference between the Out-time of a job and its DD. 
From the Table 1, the equation obtained is, 

Total lateness =    (𝑝1 − 𝑑1)  + (𝑝1 + 𝑝2 − 𝑑2) + (𝑝1 + 𝑝2 + 𝑝3 − 𝑑3) + (𝑝1 + 𝑝2 + 𝑝3
+ 𝑝4 − 𝑑4) + (𝑝1 + 𝑝2 + 𝑝3 + 𝑝4 + 𝑝5 − 𝑑5) 

Total lateness = (5𝑝1 + 4𝑝2 + 3𝑝3 + 2𝑝4 + 𝑝5) − ( 𝑑1 + 𝑑2 + 𝑑3 + 𝑑4 + 𝑑5) 

Total Lateness = 𝑇𝐹𝑇 − ( 𝑑1 + 𝑑2 + 𝑑3 + 𝑑4 + 𝑑5) 
(18) 

The summation of the DD is the same irrespective of the job sequence. Hence, the 
TFT must be minimum to reduce the lateness shown in equation 18. Therefore, a 
minimum value can be obtained using the same sequence of jobs from the previous 
measures. 

6. Average lateness 

The average lateness is shown in equation 19. The number of jobs is constant. So, 
to minimize the numerator, the same approach from the previous performance 
measure can be used. 

𝐿𝑎𝑣𝑔 =
Total Lateness

𝑛
 (19) 

7. Total PT 



Simulation of Job Sequencing for Stochastic Scheduling with a Genetic Algorithm 

27 
 

 It is the summation of the PT of all jobs in the scheduling problem. The total PT is 
a constant value for any given sequence of jobs and cannot be minimized. 

3.2. Tardiness Performance Measures 

For the tardiness related parameters, a few of the approaches tested to minimize 
the performance measures were the Branch and Bound algorithm (Tyagi et al., 2016), 
Excel Workbook tool, sequentially searching through each of the 'n!' sequences 
possible. However, these approaches were limited by various factors, including 
computing time and data size variation. Hence, a randomized search approach was 
implemented like GA (Bancila & Buzatu, 2008).  

The GA initiates by using random sequences of the jobs as starting population. 
Then, it proceeds with each sequence of the population over a fitness function. Later, 
it chooses the fittest sequence of the population to reproduce using the reproduction 
function of GA and repeats the assessment and reproduction until a selected number 
of iterations. In the end, the algorithm presents the optimal sequence of the population 
according to the fitness function. Following are the steps in the GA used in the 
simulation model:  

Step 1: Select the initial population randomly 

The initial population is a set of randomly generated chromosomes (sequence of 
jobs) as input to the GA. GA chooses a set of samples randomly from n! sequences as 
the initial population. 

For example, the 30 sequences are selected randomly for five jobs 0,1,2,3,4 from 5! 
that is 120 sequences as an initial population shown below. 

[2, 0, 1, 4, 3] [1, 4, 2, 0, 3] [4, 1, 2, 3, 0] [3, 2, 0, 4, 1] [1, 2, 4, 0, 3] 

[3, 4, 1, 0, 2] [0, 4, 2, 1, 3] [2, 4, 1, 3, 0] [4, 1, 0, 3, 2] [3, 0, 2, 4, 1] 

[0, 1, 2, 3, 4] [1, 3, 4, 2, 0] [4, 2, 1, 0, 3] [4, 2, 0, 3, 1] [1, 3, 4, 2, 0] 

[4, 3, 2, 1, 0] [4, 0, 2, 3, 1] [4, 3, 0, 2, 1] [0, 3, 2, 1, 4] [4, 1, 3, 0, 2] 

[2, 4, 1, 3, 0] [1, 3, 2, 4, 0] [4, 1, 0, 3, 2] [2, 1, 4, 0, 3] [2, 1, 3, 4, 0] 

[0, 1, 4, 2, 3] [4, 0, 3, 1, 2] [2, 4, 1, 3, 0] [3, 0, 4, 1, 2] [4, 3, 2, 1, 0] 

Step 2: Application of GA operators 

Crossover operator: Crossover is a genetic operator used to modify a chromosome 
or chromosomes by combining chromosomes from one generation to the next. 
Randomly select two parents, just for manageable iteration, number all the sequences 
and shuffle it for reproduction operation on the population to get offspring.  

0 6 12 18 24 

1 7 13 19 25 

2 8 14 20 26 

3 9 15 21 27 

4 10 16 22 28 

5 11 17 23 29 

[26, 7, 9, 29, 3, 10, 2, 28, 8, 27, 21, 16, 25, 12, 20, 15, 4, 24, 6, 19, 5, 18, 14, 0, 13, 23, 
11, 22, 17, 1] 



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Let's select the first two sequences as parents 26 and 7. 

P1 [1, 3, 4, 2, 0]  

P2 [0, 4, 2, 1, 3] 

Randomly select half of the number of jobs and create the sequences just by 
swapping the jobs at that positions to get children. Now remaining places are filled 
with jobs that are not in children.   

Let’s select [0,4]  

Child_1 = [0, 'na', 'na', 'na', 3] 

Child_2 = [1, 'na', 'na', 'na', 0] 

‘na’ represents the blank place  

Thus in Child_1, jobs [1, 4, 2] are not present. Similarly, in Child_2, jobs [4, 2, 3] are 
not present. Now just put these jobs to get Child_1 and Child_2 as follows 

Child_1 = [0,1,4,2,3] 

Child_2 = [1,4,2,3,0] 

This process needs to repeat for the remaining sequences, producing the different 
sequences as offspring list. 

Mutation: Mutation operators are generally viewed as a random disturbance term 
of the individual chromosome. The children list of sequences produced by the 
crossover function of GA is used as input to this mutation function. First, randomly 
select the sequence and swap the position of jobs that gives the new list of offspring 
sequences. 

Let’s take Child_2 = [1,4,2,3,0] 

Select any two positions and swap the jobs. For example, let's select 2nd and 3rd 

positions and change the jobs to produce a new sequence [1,2,4,3,0]. This process 
repeats for all sequences present in the children list, input for mutation function to get 
new offspring. 

Step3: Evaluation of offspring 

Evaluation of offspring is finding the sequence with the lowest tardiness value. 
Now parent list and offspring list are merged to get the whole list of sequences to find 
optimal sequences. The tardiness performance measures are needed to minimize. The 
total tardiness of all the sequences is calculated, and the sequence with minimum 
tardiness is selected as the optimal sequence. The exact sequence is used to calculate 
other tardiness based measures. 

Step 4: Termination Condition 

The termination condition is the stopping criterion for the algorithm. In this paper, 
the user gives the number of iterations as a termination condition for finding the best 
sequence. After a specified iteration, the algorithm stops and produces the optimal 
sequence for minimizing total tardiness. 



Simulation of Job Sequencing for Stochastic Scheduling with a Genetic Algorithm 

29 
 

Based on the GA implemented, Figure 1 shows a graph that indicates the time 
required to compute total tardiness, average tardiness, number of tardy jobs and 
maximum tardiness in one iteration against the number of jobs in the dataset. 

 

Figure 1. Number of jobs and time required for 1 iteration 

From Figure 1, it is observed that the time required to compute one iteration 
roughly increases as the number of jobs increases. This gives the operator an idea of 
how much time will be required for computing the required number of iterations. 
Another benefit of the implemented algorithm is that if the values reach a minimum 
value, the algorithm will check the value for a few more iterations and exit the loop 
after a few iterations to save the operator's time from computing the remaining 
number of iterations. Hence implementing the SPT approach and a GA, an optimizing 
stochastic model can be developed for any scheduling application.  

4. Case Study 

A case study was performed in association with a manufacturing company 
specializing in pipe fittings and flanges to verify the methodology and compare results 
with the real-world scenario. Figure 2 shows sample data from a datasheet of the 
company. The model developed for the case study uses the scenarios method of 
scheduling explained earlier in section 2.2.1. The model is limited to n jobs one 
machine scenario that is scheduling will be done for any number of jobs as long as they 
are processed on one machine. With respect to Figure 2, the simulation model assigns 
job numbers based on the sizes of the individual components; that is, job numbers are 
assigned based on the 'Size' column. The model also calculates the PT of the respective 
job, in minutes, by taking the product of the values from the column 'Qty.' and 
'Processing time'. The PT is then converted to hours because the DD of the jobs is also 
accepted in terms of hours. Thus, the data required for performing scheduling 
operations is prepared. The model is created using Python programming language on 
a computer system with 8 GB RAM and 500 GB hard disk and Windows 10 operating 
system. This model can also run on other versions of the operating system like 
Windows 7, Windows 8 and other operating systems. Other than this, no other special 
hardware and software requirements are needed.  

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 10 20 30 40 50 60 70 80 90 100 110 120 130

T
im

e
 (

in
 s

e
c
o

n
d

s)

Number of Jobs



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30 

 

 

Figure 2. Sample data of case study 

A similar approach is followed if there are more than one scenario. If there are 
more than one scenario, the simulation model requires the same number and similar 
jobs in all the datasheets. Figure 3 shows the Main Window of the job shop scheduling 
simulator. All four scheduling methods explained earlier are included in this simulator. 
A new window will open for the respective scheduling model on clicking any button. 
The 'About' button provides information about the simulator. For example, the case 
study model developed can be located by clicking the 'Stochastic Scheduling 
(Scenarios) Company Model' button. On clicking the company model button, the main 
window of the model will open up, as shown in Figure 4. The model has two ways of 
data input, manual input or importing an excel file. The 'Enter Number of Iterations' 
box is required only for the tardiness related performance measures, without which 
error will be displayed. If there are more than one scenario, the operator can select the 
individual scenarios by entering in the 'Enter Scenarios of Jobs' box. The 'Instructions' 
give information about the file format and the format of input wherever required. 

 

Figure 3. Main Window of Simulator  



Simulation of Job Sequencing for Stochastic Scheduling with a Genetic Algorithm 

31 
 

 

Figure 4. Main Window of Company Model 

If the operator chooses to import the data, a 'Browse' window will appear to select 
one or more files from the device storage. An error will be displayed if the datasheet 
is not in the proper format. After selecting the file(s), the software will read through 
the 'Process used' column of the datasheet(s) to identify different machines. Now the 
operator has to enter for which machine scheduling has to be performed, as the 
simulation will be done for n jobs one machine. This input will be taken in the 'Select 
Process' window, as shown in Figure 5. The operator can also enter the DD of the jobs, 
in hours, in the same window. On clicking the 'Submit' button, the data will be ready 
for scheduling operation. 

 

Figure 5. Process used and due date 

The operator can also enter the data of the jobs manually, that is, the DD and PT of 
the jobs, by clicking the 'Enter Data' button in the main window. On clicking, a window, 
as shown in Figure 6, will appear. The simulator will automatically assign the job 
number in the order the input is given. The PT of more than one scenario can also be 
added there. The units, however, of both the inputs have to be the same as no units are 
assumed or assigned here. 



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32 

 

   

Figure 6. Window to manually enter data 

The operator must select the performance measures for which the minimizing 
sequence has to be found and the corresponding minimum value. The individual 
performance measures can be chosen, or the 'All Performance Measures' can be 
selected as a whole. If the operator selects 'All Performance Measures' or any of the 
tardiness related performance measures, input will be required from the operator for 
the number of iterations. The computing time will increase if the number of iterations 
are more or if the number of jobs are more. The operator can give the appropriate 
value based on the time at hand. The operator is required to enter the scenarios of the 
jobs. This can be done in the following ways. 

1. For selecting single scenarios, the scenario number can be given, that is, 1 or 
2 

2. For more than one scenario, each scenario should be separated with a comma, 
3, 4, 5, 6, 7. 

3. The hyphen symbol can be used for selecting a range of scenarios; that is, 4-
10 will select all the scenarios within this range, including 4 and 10. 

4. For a combination of individuals and a range of scenarios, the following 
format can be followed that is, 4-10, 12, 13, 16. This will select the scenarios 
from the range of 4 and 10, including the individual scenarios of 12, 13, and 
16. 

The operator can also view the data table, that is, the job number, PT and DD of 
every job and scenario, by clicking on the 'Preview Table' button. On clicking, a 
window, as shown in Figure 7, will appear. The data, here, cannot be manipulated. 
After all the required input is given for the scheduling operation, as shown in Figure 8, 
the scheduling operation is ready to be performed.  



Simulation of Job Sequencing for Stochastic Scheduling with a Genetic Algorithm 

33 
 

 

  Figure 7. Sample Preview Table Window 

  

Figure 8. Performance measure selection 

A new window will appear after clicking the 'Submit’ button, as shown in Figure 9. 
This window will consist of all the names of the performance values selected by the 
operator and the corresponding minimum value in the top table. Next, in the bottom 
table, the job details will be provided. Finally, in the right-hand side section of the 
window, the names of the performance measures will be given and the sequence that 
will minimize performance measures. Thus, the result window consists of all the data 
required to perform the scheduling operation.  



Bari et al. / Oper. Res. Eng. Sci. Theor. Appl. 5(3)2022 17-39 

34 

 

 

Figure 9. Results for due date 700 

The DD given by the client was four months, that is, 700 hours (assuming 25 
working days in a month and seven working hours per day). After running the model 
with DD of 700 hours, the tardiness comes to be 0, as shown in Figure 9. This indicates 
that all the jobs are completed early. The just-in-time approach in manufacturing 
industries stimulates the notion of tardiness and earliness also. In industry, in view of 
due-dates, the primary intent is to complete all the jobs on time. The intent may be 
achieved by permitting the loose DD as 700 hours as DD in the case study. In this case 
of unrestricted DD situation, all jobs can be completed before DD by any sequence. 
Completing the jobs before time affects inventory carrying costs like storage and 
protection costs. However, due dates should be chosen carefully, as DD that is tight or 
restricted invite more customers. The above discussion indicates that the DD should 
be tighter. If the DD is negotiable with the customers and can be made still tighter, it 
will attract more customers, and there will be no need to keep a loose DD. After several 
runs with a different DD, it is found that the DD of 548 hours, approximately 3 months 
and 3 days is optimum for the case study. The DD produces the same result for 
tardiness measures, as shown in Figure 10. It is noted that the sequence of jobs for 
non-tardiness performance measures is the same as it is based on the SPT priority 
sequencing rule; however, the sequence for tardiness performance measure is 
different for both cases shown in Figure 9 and Figure 10. The average lateness is 
reduced from 627.457 to 475.457, which indicates that the inventory level of the 
completed job can be minimized. 



Simulation of Job Sequencing for Stochastic Scheduling with a Genetic Algorithm 

35 
 

 

Figure 10. Results for due date 548 

5. Results and Discussion of Case Study 

The case study is conducted using the DD given by the client with a stochastic 
scenario model. The company sequenced the jobs in the FCFS rule. There were 90 jobs 
for the company to schedule on the lathe machine for sequencing. The performance 
measures were calculated for the dataset, with the DD as four months for the dataset. 
The company followed a deterministic approach to job scheduling. Hence, only one 
datasheet was considered, whereas the model developed uses the stochastic approach 
and therefore uses two datasheets for two scenarios. Table 7 shows the calculated 
values for the dataset and the values calculated by the model. 

Table 7. Deterministic and Stochastic Results for Data Set  

Sr. 
No. 

Performance Measures 
Deterministic 

Approach 
Stochastic 
Approach 

1. Total Flow Time 32967.75 hours 6528.85 hours 
2. Average Job Completion Time 366.30 hours 72.543 hours 
3. Total Processing Time 547.6 hours 547.6 hours 
4. Percentage Utilization 1.442 % 8.387 % 
5. Average Number of Jobs in the 

System 
69.338 11.923 

6. Total Lateness -30032.2 hours -56471.15 
hours 7. Average Lateness -333.69 hours -627.457 hours 

8. Total Tardiness 0 0 
9. Average Tardiness 0 0 

10. Maximum Tardiness 0 0 
11. Number of Tardy Jobs 0 0 

Both the approaches assign job numbers based on the sizes and consider the 
quantity of those sizes as one job. There were 90 jobs in the dataset, and to calculate 
the tardiness performance measures, ten iterations as termination condition were 
given for GA. From Table 7, it is observed that the developed model, which uses SPT 



Bari et al. / Oper. Res. Eng. Sci. Theor. Appl. 5(3)2022 17-39 

36 

 

and GA methods for calculating non-tardiness and tardiness performance measures, 
respectively, provides a better value. Due to the proposed approach, the company will 
have more time after completing a set, which can be used for various tasks like 
maintaining the machines or beginning the operations on the next set of jobs. It is 
further suggested that if the DD is negotiable, that means a tighter DD as compared to 
the given DD by the client; the company can clear out the inventory of finished 
products without affecting performance measures. Finishing the jobs well beforehand 
also reduces inventory costs in terms of storage or protection and improves the 
company's value. By lowering the tardiness performance measures company can 
lessen the delaying cost. The total tardiness for each iteration with different DD is 
shown in Table 8. Figure 11 shows the tardiness with respect to iteration for DD 400, 
450, 500 and greater than equal to 548 hours. Table 8 and Figure 11 suggest that the 
DD can be tighter as 548 hours without affecting tardiness measures. It is observed 
that DD of less than 548 hours affects the total tardiness. Further decrease in DD 
increases the total tardiness. This total tardiness can be improved as the number of 
iterations in GA are increased. After specific iterations, it gives a constant value, but an 
increase in the number of iterations will increase computational time.  

Table 8. Total tardiness for each iteration with different due date 

Number 
of 

Iterations 

DD (In hours) 

700 650 600 550 548 500 450 400 

Tardiness (In hours) 

1 0 0 0 0 0 93 144.9
5 

203.8 
2 0 0 0 0 0 47.6 144.9

4 
203.8 

3 0 0 0 0 0 47.6 144.9
4 

175.2
7 4 0 0 0 0 0 47.6 138.0

4 
140.4

4 5 0 0 0 0 0 47.6 137.5
2 

140.4
4 6 0 0 0 0 0 47.6 137.7

3 
129.9

4 7 0 0 0 0 0 47.6 137.1
7 

129.9
4 8 0 0 0 0 0 47.6 137.1

7 
129.9

4 9 0 0 0 0 0 47.6 137.0
7 

129.9
4 10 0 0 0 0 0 47.6 137.0

7 
129.9

4 

 

Figure 11. Total Tardiness for different due date 

-10
10
30
50
70
90

110
130
150
170
190
210
230

0 1 2 3 4 5 6 7 8 9 10 11

T
a

rd
in

e
ss

Number of Iterations

Due Date>=548

Due Date=500

Due Date=450

Due Date=400



Simulation of Job Sequencing for Stochastic Scheduling with a Genetic Algorithm 

37 
 

In comparison to traditional methods such as an excel workbook or manual 
calculation, the proposed work advocates using GA to reduce the computing time 
required to calculate performance measures and find the best sequence of jobs in a 
scheduling problem. In addition, the proposed work uses stochastic scenarios, 
linguistics, and probabilistic approaches to find job sequences that minimize 
performance measures. When the proposed model is applied to real data from a 
company, it is discovered that the sequence obtained by the model produces the 
lowest performance measure value when compared to the company's method. The 
proposed method is a unique application that allows for faster computation and better 
results. 

6. Conclusions 

The developed simulation model can handle the stochastic scheduling problem 
with linguistic, scenarios and probabilistic data to discover an optimal sequence of 
jobs for scheduling on a single machine. It also helps to solve the problem in a 
deterministic way. If the number of jobs and iterations increases, the computational 
time required to discover the optimal sequence also increases. When the problem was 
solved using the excel solver, it took more time in discovering the near-optimal 
sequence. The developed simulation model not only produces results with lesser time, 
but also improves solution. Stochastic models allow the operator to select real-time 
data, whereas deterministic scheduling does scheduling based on the data at hand. The 
SPT rule minimizes TFT and minimizes all the non-tardiness related performance 
measures. The developed GA model also minimizes the tardiness related performance 
measures. As an outcome, the proposed stochastic technique assisted the company in 
reducing the average completion time of job from 366.30 hours to 72.543 hours. 
Additionally, increased the percentage utilization from 1.442 percent to 8.387 percent. 
While doing the analysis of the company dataset using the developed model, it is 
revealed that a tighter DD is beneficial for reducing inventory costs.  

The work is restricted to a single machine with an unlimited number of jobs. GA 
generates optimal sequences for scheduling problems, however, for different runs, it 
generates different sequences with the minimized performance values. To tackle this 
issue, a combinatorial approach of GA with other optimization techniques can be used.   

Future work on this project may include developing simulation environments for 
n jobs m machines which will provide a broader range of options as per the industry’s 
needs. Modifying the simulation environment from software to web application will 
not require instalment on the device, and the operator can use it remotely on any 
device. This paper focuses on stochastic scheduling and the reduction in time for 
completion of job. However, another factor that majorly affects scheduling is cost, 
which may be further analyzed. Delay of jobs can incur costs to the company of a by 
considerable amount. Hence, developing a simulation environment to minimize both 
time and cost will assist the manufacturing industries with better insight into the 
scheduling and sequencing of jobs. 



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38 

 

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© 2022 by the authors. Submitted for possible open access publication under the 
terms and conditions of the Creative Commons Attribution (CC BY) 
license (http://creativecommons.org/licenses/by/4.0/). 


	SIMULATION OF JOB SEQUENCING FOR STOCHASTIC SCHEDULING WITH A GENETIC ALGORITHM
	Prasad Bari 1,2*, Prasad Karande 1, Jayston Menezes 2
	1. Introduction
	2. Methodology
	2.1. Deterministic Scheduling
	2.2. Stochastic Scheduling
	2.2.1. Scenarios Method
	2.2.2. Linguistic method
	2.2.3. Probabilistic Method


	3. Optimizing Performance Measures
	3.1. Non-Tardiness Performance Measures
	3.2. Tardiness Performance Measures

	4. Case Study
	5. Results and Discussion of Case Study
	6. Conclusions
	References