INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,  
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)  
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIV, Issue XI, November 2025  
Parallel Machine Scheduling with Constraints in a Labelling  
Industry Case Study  
Tran Ngoc Tu, Ha Thanh Tung, Vo Thuy Thao Vy, Do Ngoc Hien  
Department of Industrial & Systems Engineering, Ho Chi Minh City University of Technology  
(HCMUT), 268 Ly Thuong Kiet Street, Dien Hong Ward; Vietnam National University Ho Chi Minh  
City (VNU-HCM), Linh Xuan Ward, Ho Chi Minh City, Vietnam.  
DOI: https://doi.org/10.51583/IJLTEMAS.2025.1411000024  
Received: 21 November 2025; Accepted: 28 November 2025; Published: 03 December 2025  
ABSTRACT  
This paper would present a case study on solving a parallel machine scheduling problem in the printing  
department of a global leader in labeling solutions. Motivated by low-season conditions with below-capacity  
order volumes, the study prioritizes accelerating job completions to reduce resource idle time in downstream  
processes. It would focus on minimizing the total weighted completion time while addressing real-world  
constraints including machine-job compatibility, shift boundaries, and resource limitations in color matching  
processes. The problem was formulated as a three parallel machines with different speeds model and used  
Weighted Shortest Processing Time dispatching rule to find out solutions. The scheduling objective is to  
minimize total weighted completion time. Our approach progressively incorporates operational constraints and  
achieves around 11.11% improvement in objective value compared to the current manual scheduling method.  
Some discussions on implementation of research results, limitations, and future optimization opportunities and  
real-time production data integration would be mentioned.  
Keywords— Parallel machine scheduling, Production constraints, Total weighted completion time, Dispatching  
rules, Weighted Shortest Processing Time rule.  
INTRODUCTION  
Production scheduling plays a crucial role in manufacturing environments, impacting resource utilization,  
operational costs, and customer satisfaction through delivery reliability. Efficient scheduling is particularly  
important in the printing industry, where machines with different capabilities must process various job types  
while respecting operational constraints.  
Scheduling is defined as the allocation of resources (machines, labor, materials) over time to achieve specific  
production goals [1]. Manufacturing scheduling objectives typically include reducing makespan, optimizing  
inventory, adhering to deadlines, and improving resource utilization. Common challenges include balancing  
multiple conflicting objectives, accommodating complex production constraints, and adapting to dynamic  
changes in production conditions [2, 3].  
This paper examines a real-world scheduling problem at AD label printing department. The current scheduling  
process relies on manual assignment by three separate planners using Excel spreadsheets, taking 30-45 minutes  
daily. This process is inefficient, lacks systematic dispatching rule application, and results in poor resource  
utilization.  
This research would aim to improve this process by developing and implementing a systematic scheduling  
approach that minimizes the total weighted completion time while respecting key operational constraints. Our  
contributions include:  
A practical implementation of the Weighted Shortest Processing Time rule (WSPT rule) in constrained  
parallel machines with different speeds environment  
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Progressive incorporation of operational constraints into the scheduling model  
Quantitative analysis of performance improvements over the manual method  
Identification of further optimization opportunities  
The rest of this paper is organized as follows: Section II provides a literature review of relevant scheduling  
concepts. Section III describes the problem definition and mathematical formulation. Section IV presents the  
methodology and implementation approach. Section V discusses the results and performance comparison.  
Section VI concludes the paper and outlines future research directions.  
LITERATURE REVIEW  
Parallel machine scheduling problems have been extensively studied in operations research literature. The  
problem of minimizing total weighted completion time (∑wᵢCᵢ) on parallel machines is known to be NP-hard  
problems in the strong sense [4].  
For identical parallel machines (P₃||∑wᵢCᵢ), Smith [5] proved that the Weighted Shortest Processing Time  
(WSPT) rule provides an optimal solution. This rule sequences jobs in non-decreasing order of pⱼ/wⱼ, where pⱼ is  
the processing time and wⱼ is the weight of job j.  
For uniform parallel machines (Q₃||∑wᵢCᵢ), where machines have different speeds, the problem becomes more  
complex. Horowitz and Sahni [6] showed that WSPT is not generally optimal but can provide good  
approximations.  
When additional constraints are introduced, such as machine eligibility constraints, release dates, or sequence-  
dependent setup times, heuristic approaches are typically used. Dispatching rules like WSPT are often modified  
to accommodate these constraints [7].  
In practical applications, Kochhar and Morris [8] demonstrated that WSPT-based heuristics can significantly  
improve scheduling performance in manufacturing environments. Similarly, Weng et al. [9] applied weighted  
dispatching rules to semiconductor manufacturing with machine eligibility constraints.  
This study would extend these approaches by applying WSPT in a printing production environment with multiple  
real-world constraints, including shift boundaries and resource limitations.  
PROBLEM DEFINITION  
Company background  
AD is a global leader in labeling solutions and functional materials. The company's customer portfolio consists  
of 81 brands divided into three segments: Market Leading Accounts (MLA, 12 major brands), Key Retailers  
(medium priority), and Others (flexible-schedule). Their production involves seven distinct product families  
following a flow-shop manufacturing process in series as Printing, Inspection, and Packing.  
The production facility houses 30 printing machines distributed across three areas: 14 machines at area 3A, 10  
machines at area 3B, and 6 machines at area 1A. These machines are categorized as either camera-equipped (for  
quality-critical products) or non-camera (for standard products).  
As mentioned above, the scheduling process at AD was inefficiently handled by three separate planners. Job  
assignments follow a priority hierarchy based on customer segmentation, with machine capability constraints  
(camera vs. non-camera) also considered when matching products to appropriate machines.  
This fragmented approach leads to several operational challenges, including highly repetitive manual scheduling  
tasks, limited holistic planning effectiveness, inconsistent application of dispatching rules, and inflexible  
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prioritization logic. These issues result in poor utilization of labor and machinery resources, as the current system  
cannot effectively reflect actual constraints and shifting priorities over time.  
Problem discription  
To evaluate the effectiveness of applying dispatching rules compared to the current manual scheduling method,  
a demonstration using a scaled-down model reflecting the actual proportion of machine types in the HTL  
department was conducted. The demo focused on 3 machines (2 non-camera machines and 1 camera-equipped  
machine), which maintains the same ratio of machine types as the full production environment of 30 machines  
(18 non-camera and 12 camera-equipped).  
In addition to testing the basic dispatching rules, realistic operational constraints that exist in the actual  
production environment were incorporated. These include shift boundary limitations, where jobs must start and  
finish within the same shift (Shift 1: 06:00-14:00; Shift 2: 14:00-22:00), and resource constraints related to color  
matching processes, which limit the number of jobs that can undergo color matching simultaneously due to  
limited colorist availability.  
In the case study, parallel machines could be clustered into relative groups, in which the scheduling problem  
would be solved similarly. Therefore, in this study, the scheduling problem focused on assigning 25 jobs to 3  
parallel machines (1 camera-equipped and 2 non-camera) while minimizing the total weighted completion time.  
Key characteristics of this problem include as followings, where Table 1 shows the machines characteristics,  
and Table 2 shows the priorities and parameters.  
Machine characteristics (Table 1):  
Camera machine (Sakurai 1): Speed of 900 IMP/hr, better scrap rate, can process both YY-01 and HD-  
1096 products  
Non-camera machines (Sakurai 2 and 3): Speed of 1200 IMP/hr, higher scrap rate, can only process HD-  
1096 products  
Table I. Machine Characteristics  
i
Machine  
Sakurai 1  
Sakurai 2  
Sakurai 3  
Group  
Conversed speed  
YY-01 HD-1096  
1
2
3
Camera  
0.75  
1
✅
✅
✅
✅
Non-camera  
Non-camera  
1
Table II. Job List  
No. Job_ID  
IMP  
Product  
HD-1096 Yes  
YY-01 No  
HD-1096 Yes  
HD-1096 Yes  
YY-01 No  
RB  
wj  
camera  
time matching  
1
2
3
4
5
6
7
8
9
J1  
J2  
J3  
J4  
J5  
J6  
J7  
J8  
J9  
450  
Decathlon 5  
0.2  
0.2  
0.2  
TRUE  
TRUE  
FALSE  
TRUE  
TRUE  
FALSE  
TRUE  
FALSE  
TRUE  
FALSE  
900  
Adidas  
Walmart  
Nike  
8
1800  
300  
1
10 0.2  
10 0.2  
600  
Nike  
900  
HD-1096 Yes  
HD-1096 Yes  
HD-1096 Yes  
HD-1096 Yes  
Walmart  
GAP  
1
3
4
0.2  
0.2  
0.2  
0.2  
0.2  
3300  
1200  
600  
Jako  
Decathlon 5  
10 J10  
J11 --> J21  
22 J22  
23 J23  
24 J24  
25 J25  
2400  
YY-01  
No  
Puma  
7
300  
300  
900  
1200  
HD-1096 Yes  
HD-1096 Yes  
HD-1096 Yes  
GAP  
3
0.2  
0.2  
0.2  
0.2  
FALSE  
TRUE  
TRUE  
TRUE  
Decathlon 5  
Adidas  
Puma 7  
8
YY-01  
No  
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Job priorities:  
High priority: Market Leading Accounts (w = 8-10)  
Medium priority: Key Retailers (w = 4-7)  
Low priority: Others (w = 1-3)  
Job parameters:  
Processing time (pⱼ)  
Weight value (wⱼ)  
Setup time (Sᵢ'ᵢ): 0.2 hours for all job transitions  
Color matching requirement: 0.3 additional hours if required  
Operational constraints:  
Machine-job compatibility constraints  
Shift boundary constraints: Jobs must start and finish within single shifts (Shift 1: 06:00–14:00; Shift 2:  
14:00–22:00)  
Color matching resource constraint: Maximum 2 jobs can undergo color matching simultaneously  
Mathematical Fomulation  
The production process at AD follows a specific workflow as illustrated in the flowchart (Fig. 1). After receiving  
the job jacket from the Planning department and checking materials (ink, substrate), operators proceed with  
machine setup. For products requiring color matching, the process includes additional complex steps: the  
machine runs a trial and samples are submitted to the Colorist for checking. If the colors do not meet  
requirements, the ink formula is adjusted, and the trial process is repeated until standards are met. Only when  
colors are approved can mass printing begin. Upon completion, products are transferred to the Inspection station.  
Color matching is a technical process requiring a Colorist's expertise to ensure color accuracy according to  
customer specifications. This process takes an additional 0.3 hours and requires specialized resources - each  
Colorist can only handle a maximum of two color matching jobs simultaneously, creating an important constraint  
in the scheduling process.  
The total time to complete a job (Total Time - TTji) is calculated as the sum of: machine setup time (Setup Time  
- 0.2 hours) + color matching time (if required - 0.3 hours) + processing time (dependent on job volume). This  
calculation formula accurately reflects the actual time needed to complete each job in the production  
environment and serves as the basis for developing an optimized scheduling model.  
A uniform parallel machine with different speeds scheduling problem (Q₃||∑wᵢCᵢ) with additional constraints  
was formulated. The mathematical model objective aims to minimize the total weighted completion time  
(∑wᵢCᵢ), where each job’s weight reflects its relative priority in production. By minimizing ∑wᵢCᵢ, the  
completion of high-priority jobs was accelerated. It could help reduce the overall work-in-process (WIP)  
inventory. This objective directly supports operational goals during the low season by enabling faster job  
transitions to subsequent stages, minimizing resource idle time, and improving overall throughput and labor  
productivity.  
∑
Objective = minimize  
(1)  
=1  
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Where:  
wⱼ = weight (priority) of job j  
Cⱼ = completion time of job j  
Subject to:  
Machine eligibility constraints  
Shift boundary constraints  
Color matching resource constraints  
Non-preemption constraints  
The total time for processing job j on machine i (TTⱼᵢ) is calculated as:  
=
+
ℎ
+
(2)  
Fig. 1. Flow chart  
PROGRAME DEVELOPMENT AND IMPLEMENTATION  
Data collection  
Data was collected through:  
Real scheduling data from company records  
Interviews with planners and operational staff  
The collected data included job specifications (processing times, setup times, priority weights) and machine  
characteristics (speed, capability profiles, shift availability).  
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Implementation Approach  
A Python-based scheduling solution was developed following these steps:  
Simple WSPT Implementation:  
Classify jobs based on product type compatibility with machines  
For each job j and eligible machine i, compute priority index = TTⱼᵢ/wⱼ  
Assign job j to machine i with lowest TTⱼᵢ/wⱼ  
Group jobs by assigned machine  
Sequence jobs on each machine in increasing order of TTⱼ/wⱼ  
Shift Constraint Integration:  
Add shift boundary constraints (06:00-14:00 and 14:00-22:00)  
Check if job fits within current shift; if not, defer to next shift  
Color Matching Constraint Implementation:  
Limit simultaneous color matching to maximum 2 jobs  
If more than 2 color-matching jobs are in queue:  
o
o
Move up a job that does not require color matching  
If no such job is available, delay the third color-matching job  
The scheduling solution generates visualizations (Gantt charts) and numerical results for analysis.  
RESULTS & DISCUSSIONS  
Performance Comparison  
Four scheduling scenarios were considered:  
Manual scheduling: Currently used by planners  
The current manual scheduling process at AD operates across two shifts: Shift 1 from 06:00 to 14:00 and Shift  
2 from 14:00 to 22:00. Planners begin by reviewing jobs in order of priority, with highest weight (wj) jobs  
considered first. For each job, in which they identify all compatible machines based on technical requirements.  
They then select the machine that will become available earliest from this eligible list. Before finalizing the  
assignment, planners verify that the machine's available time plus the job's total processing time does not exceed  
the shift end. If this condition is met, the job is assigned, and the machine's availability is updated accordingly.  
If the job cannot be completed within the remaining shift time, planners attempt to assign it to the next eligible  
machine. Any jobs that cannot be accommodated in Shift 1 are reconsidered for Shift 2 following the same  
prioritization and assignment rules. This manual scheduling method results (Fig. 2) in a total weighted  
completion time (objective function) of C0 = ∑wᵢCᵢ = 720.68.  
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Fig. 2. Gantt chart – manual scheduling  
WSPT rule: Basic implementation with Job and Machine constraints  
In proposed scheduling approach, first jobs are classified according to their product specifications. Products of  
type YY-01 must be processed exclusively on Sakurai 1 because of the only camera-equipped machine capable  
of handling these products. Products of both types YY-01 and HD-1096 can technically be processed on all three  
machines (Sakurai 1, 2, and 3), but HD-1096 products are preferred to assign to non-camera machines (Sakurai  
2 or 3) whenever possible due to their faster processing speed.  
For each job j that can be processed on machine i, the total processing time (TTji) is calculated as the sum of  
setup time, color matching time (if required), and the actual processing time. A priority index for each job-  
machine combination is then computed by dividing the total time by the job's weight (TTji/wj). Jobs are assigned  
to machines that yield the lowest priority index value, effectively implementing the Weighted Shortest  
Processing Time (WSPT) rule. After assignment, jobs on each machine are sequenced in increasing order of  
their priority index as on Fig. 3. This optimized scheduling method achieves a total weighted completion time  
of C1 = ∑wᵢCᵢ = 612.93, demonstrating significant improvement over the manual approach.  
Fig. 3. Gantt chart – WSPT job & machine constraints  
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The Gantt chart displays a WSPT scheduling solution with three Sakurai machines running parallel job  
sequences, clearly showing setup periods (diagonal stripes), processing times (solid colors), and idle periods  
(crosshatched patterns). While the solution successfully demonstrates job scheduling with corresponding setup  
times and color matching requirements while adhering to technological constraints, it fails to account for shift  
end times—a critical practical consideration in manufacturing environments where shifts have defined  
boundaries and operators leave at scheduled times, making this solution theoretically sound but impractical for  
real-world implementation.  
WSPT rule with shift constraints: No job printed between two shifts  
When implementing shift constraints into the scheduling model, two 8-hour shifts per day were enforced, in  
which Shift 1 operates from 06:00 to 14:00 and Shift 2 operates from 14:00 to 22:00. A critical constraint was  
added ensuring that no job could span across shifts—each job must start and finish within a single shift. This  
was achieved by checking if the remaining time in a shift was sufficient to complete a job; specifically, if the  
shift end time minus the latest job completion time was less than the total processing time (TTij) of the new job,  
that job would be deferred to the next available shift. The results (Fig. 4) show that our objective value for this  
shift-constrained schedule is C2 = ∑wᵢCᵢ = 630.70, with a total idle time of 9.48 hours across all machines. In  
Shift 1, a total weighted completion time of 328.75 with 1.73 hours of idle time, distributing jobs optimally  
across the three machines (Sakurai 1: J17-J5-J2-J25-J11; Sakurai 2: J4-J21-J23-J1-J8-J15-J6; Sakurai 3: J20-  
J22-J24-J9-J19-J12) was determined. Shift 2 had a weighted completion time of 301.95 with a higher idle time  
of 7.75 hours, with jobs scheduled as Sakurai 1: J14-J10; Sakurai 2: J13-J16-J3; and Sakurai 3: J18-J7. This  
approach represents a practical improvement over the baseline model while respecting real-world operational  
constraints.  
Fig. 4. Gantt chart – WSPT shift constraints  
The WSPT scheduling approach shown in both Gantt charts achieves an objective value of 630.70 with 9.48  
hours of total idle time, divided across two shifts. The solution successfully implements dispatching rules and  
technical constraints while strictly enforcing the requirement that no job can run across shifts (Shift 1: 06:00-  
14:00 and Shift 2: 14:00-22:00). Notable observations include the presence of unavoidable idle time at the end  
of shifts when remaining time is insufficient to complete the next job, requiring deferral to the next available  
shift. The charts also highlight a resource constraint situation where three machines require color matching  
simultaneously, demonstrating the handling of resource limitations in the ink mixing process. The scheduling  
effectively balances workload with Shift 1 showing 1.73 hours idle time and Shift 2 showing 7.75 hours idle  
time, while maintaining adherence to all operational constraints.  
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WSPT with shift and color matching constraints  
The solution (Fig. 5) shows scheduling data with an objective value of 640.60 and total idle time of 9.48 hours,  
divided between two cases. Case 1 has a lower idle time of 1.73 hours with an objective value of 338.65, featuring  
three Sakurai machine schedules with job sequences (J17-J5-J11-J2-J25), (J4-J21-J23-J1-J8-J15-J6), and (J20-  
J22-J24-J9-J19-J12). Case 2 shows significantly higher idle time at 7.75 hours with an objective value of 301.95,  
containing three different Sakurai machine schedules: (J14-J10), (J13-J16-J3), and (J18-J7). The scheduling  
problem includes important constraints where no jobs can be printed between shifts and a maximum of two jobs  
can undergo color matching simultaneously. When more than two color-matching jobs appear next in the queue,  
either a non-color-matching job is moved up to fill the gap, or if none are available, the third color-matching job  
is delayed until one of the two running color-matching jobs completes. This scheduling approach achieves an  
overall objective value of C3 = ∑wᵢCᵢ = 640.60.  
Fig. 5. Gantt chart – WSPT shift and color-matching constraints  
The WSPT Gantt charts demonstrate a successful job scheduling implementation with an objective value of  
640.60 hours and total idle time of 9.48 hours across two shifts. This solution effectively addresses the color-  
matching constraint, ensuring no more than two jobs undergo color matching simultaneously as required. When  
examining the charts, it's clear that the solution properly handles situations where three potential color-matching  
jobs might have occurred at the same time by either advancing non-color-matching jobs or delaying the third  
color-matching job until resources become available. However, opportunities for further optimization remain  
evident, particularly in the arrangement of jobs across machines. Future improvements could explore applying  
local search techniques (such as Tabu Search or Simulated Annealing algorithms) to better distribute jobs from  
Sakurai 2 and 3 to Sakurai 1, potentially reducing idle time. Additionally, the solution could benefit from  
strategies to insert appropriate jobs into idle periods near shift ends, which would require a larger data pool to  
properly implement. These refinements could further enhance the scheduling efficiency beyond the current  
solution.  
Here is a comprehensive comparison of different scheduling methods under various operational constraints. The  
Table III presents four scheduling scenarios, including the manual method and three WSPT-based approaches  
with progressively increasing constraints. Each scenario is evaluated based on the objective function value  
(ΣwjCj) and practical realism. The baseline WSPT approach (Scenario 1) achieves the lowest objective value of  
612.93 but has low practical realism as it ignores real-world constraints. Scenario 2 incorporates shift constraints,  
resulting in a slightly higher objective value of 630.7 with medium practical realism. The final model (Scenario  
3) implements both shift and color-matching constraints, achieving an objective value of 640.6, which represents  
an 11.11% improvement over the manual method (720.68) while maintaining high practical realism by  
addressing all relevant operational constraints.  
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Table III. Performance Comparison of Scheduling Methods  
Analysis of Constraints Impact  
Shift Constraints Impact:  
The implementation of shift constraints significantly impacted the scheduling solution, increasing the objective  
function from 612.93 to 630.70, representing a 2.9% increase. This constraint created unavoidable idle time  
periods at shift endpoints when insufficient time remained to complete scheduled jobs. Despite this increase in  
the objective function value, the incorporation of shift constraints substantially enhanced the practical  
applicability of the scheduling model by respecting the organizational shift structure that exists in real  
manufacturing environments.  
Color Matching Constraints Impact:  
Additionally, the introduction of color matching constraints further increased the objective function to 640.60,  
an additional 1.6% increase. This constraint necessitated job resequencing to ensure compliance with the  
limitation of maximum two simultaneous color-matching operations at any given time. While this constraint  
further increased the completion time, it successfully eliminated resource constraint violations that were present  
in previous models, resulting in a practically implementable solution that balances efficiency with operational  
feasibility. The final model with both constraints still achieved an 11.11% improvement over the manual  
scheduling method.  
Implementation Considerations  
The proposed scheduling approach offers immediate practical benefits:  
Systematic application of dispatching rules  
Automatic consideration of all operational constraints  
Reduced scheduling time (from 30-45 minutes to seconds)  
Improved objective function (11.11% reduction in total weighted completion time)  
However, several limitations and improvement opportunities are identified:  
Optimization Techniques:  
Apply Local Search algorithms to find better solutions beyond standard dispatching rules  
Implement local search to optimize job sequence on each machine  
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Resource Utilization:  
Fill end-of-shift idle time with short jobs  
Consider job splitting for better utilization (if operationally feasible)  
Organizational Improvements:  
Centralize scheduling data with a single planner having comprehensive access  
Implement real-time integration with production signals for dynamic adjustments  
CONCLUSION AND FUTURE WORK  
This paper presented a case study on improving scheduling efficiency at AD printing department through the  
application of the WSPT dispatching rule with real-world constraints. The proposed approach achieved an  
11.11% improvement in total weighted completion time compared to the current manual method.  
The key contributions include:  
Successful adaptation of the WSPT rule to a constrained printing environment  
Quantification of the impact of different operational constraints on scheduling performance  
Development of a practical scheduling solution that respects all production requirements  
Future research directions should focus on:  
Implementing heuristics algorithms to further optimize job sequences  
Investigating dynamic scheduling approaches that can adapt to production disruptions  
Integrating real-time production data for adaptive scheduling  
Expanding the model to include all 30 machines across the three production areas  
This case study demonstrates that even with the application of relatively simple dispatching rules, significant  
improvements can be achieved when operational constraints are properly incorporated into the scheduling  
process.  
ACKNOWLEDGEMENTS  
We acknowledge Ho Chi Minh City University of Technology (HCMUT), VNU-HCM for supporting this study.  
REFERENCES  
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