Minimize trim waste by determine trim dimensions and reducing the number of exceeding-spec jobs through the application of a new production planning model that considers setup time.. A
INTRODUCTION
Thesis rationale
The corrugated board packaging industry is a sector that involves the production and utilization of corrugated cardboard materials for packaging purposes Corrugated board, often simply called corrugated board, is used for making corrugated boxes, and it is composed of two flat outer sheets (liners or facings) of puncture-resistant paper with a central layer of corrugated paper (fluted paper or “medium”) that gives the packaging resistance to crushing, and protection of the contents of the packaging [1] These parts are bonded by starch adhesives derived from corn, wheat or potatoes [1]
In Vietnam, the corrugated board packaging industry has experienced significant growth in recent years, driven by various factors such as the expansion of manufacturing and export- oriented industries, the rise of e-commerce, and increasing consumer demand for environmentally friendly packaging goods Because of the dynamic market of the packaging industry, companies in the field employ various strategies to enhance competitive advantages, including mergers and acquisitions, cost reduction, partnerships, joint ventures, license agreements, and new product launches One of the most important operational strategies these companies is to minimize material waste Among various types of material waste, trim waste stands out as a priority because of its significant proportion in material waste during the production process
Trim waste during production process, generated due to variations between the width of paper reels and the specified dimensions of cardboard This material waste represents a major cost loss for many companies in the industry Addressing this issue is closely tied to the production plan, as planners must select paper rolls from a variety of rolls with different widths and allocate them to the production sequence of jobs This process closely resembles the well- known one-dimensional cutting stock problem [2] Numerous endeavours have been made to solve the trimming problem, focusing on the utilization of computer models, heuristic approaches [3-7] However, the production planning for corrugated boards is exceptionally intricate for several reasons Firstly, the nature of the products, which are bulky and vulnerable to weather damage, necessitate an efficient production schedule to ensure timely delivery and minimize spoilage [8] Secondly, there exists a critical need to carefully balance setup losses against those incurred due to trim waste This delicate balance poses a significant challenge in the corrugated board manufacturing industry, yet this aspect remains relatively unexplored in
2 current research endeavors Despite its profound importance, there have not been enough studies exploring this important topic, leaving a notable gap in the existing literature
This study is conducted at a well-established corrugated board manufacturing company in Vietnam, where trim waste is a prevalent issue in production process This waste primarily results from production schedules aimed at minimizing setup time Therefore, motivated by the practical needs of the company, the research titled "An Empirical Study on Reducing Trim
Waste at a Corrugated Board Production Company" was conducted to address the company's issues and contribute to the diversity in this field of research.
Thesis objective
Firstly, analyze and evaluate the production process at the company, to determine causes that creates a high amount of trim waste
Secondly, minimize trim waste by determining trim dimensions and reducing the number of exceeding-spec jobs through the implementation of a new production planning model that considers setup time
The evaluation measurement is based on two criteria: %Waste and cost.
Thesis scope and limitations
The study only focuses on the planning function and does not take into account other functions Additionally, certain factors are not considered in this research, such as stock-out of paper reels and variability in paper structure.
Thesis structure
Thesis is constructed in 5 chapters, including:
Chapter 1 Introduction: outline the rationale behind selecting the topic, summarize the research objectives, scope, and limitations, and present an overview of the report's structure.
LITERATURE REVIEW AND METHODOLOGY
Literature review
Genetic algorithms (GA) originated from the principles of natural selection and genetics, aiming to solve optimization problems through evolutionary techniques such as crossover, mutation, selection John Holland is credited as the pioneer of the original Genetic Algorithm, developed in 1970, inspired by Charles Darwin's concept of random search within a defined search space [9]
GA transforms an initial population of individuals into a group of high-quality individuals, where each individual represents a potential solution to the problem at hand The efficacy of each solution is gauged by a fitness function, quantitatively measuring its suitability for a given environment Beginning with a randomly generated initial population, GA iteratively applies three fundamental genetic operators to each individual: selection, crossover, and mutation [9] Within GA, the search space comprises strings, known as chromosomes, each representing a potential solution to the problem The fitness value of each chromosome is determined by its objective function value A population consists of chromosomes alongside their corresponding fitness values, with generations representing populations generated in each iteration of the GA [10]
Pearson correlation is a statistical technique used to measure the degree of relationship between two variables It is the most popular and widely used correlation coefficient Pearson correlation coefficient takes values between -1 and 1, where 1, 0, and -1 indicate a perfect match, no correlation, and perfect negative correlation, respectively It is defined as the covariance of the variables that are to be compared, divided by their standard deviations [11]
The null hypothesis and the alternative hypothesis in Pearson correlation are thus:
Null hypothesis: The correlation coefficient is not significantly different from zero (There is no linear relationship)
Alternative hypothesis: The correlation coefficient deviates significantly from zero (there is a linear correlation)
The P value in Pearson correlation is used to measure the significance of the correlation analysis It is a standard method to determine whether the correlation coefficient is statistically
5 significant or not The P value is typically set at 0.01 or 0.05 A P value less than the cutoff indicates that the correlation coefficient is statistically significant, while a P value greater than the cutoff indicates that the correlation coefficient is not statistically significant [12].
Related studies
The corrugator trim problem is a well-studied problem in the literature, with many papers published on the topic since the early 1980s One of the pivotal contributions in this field is from Haessler and Talbot [13], who introduced a 0-1 model, providing a systematic and foundational approach to addressing the corrugator trim problem Their work not only presented a mathematical framework but also served as the cornerstone for subsequent research endeavors in this domain Accordingly, various methods for this problem has been developed diversely, collectively known as the cutting stock problem Many studies have focused on cutting stock, scheduling, and production planning, with most of them being addressed through heuristic or linear programming approaches For example, Dyckhoff [14] presented the use of linear programming to solve 1-D cutting stock which is NP-complete problem Afshar [5] also used linear programming model to solve 1-D cutting stock with trim loss Rodriguez and Veccietti [15] proposed an integrated approach to solve the cutting stock problem and scheduling in the production of corrugated board boxes Traditionally, these two problems have been addressed independently due to their complexity However, in industries like corrugated board boxes, where scheduling decisions are directly influenced by cutting patterns and setup times, an integrated approach is essential The proposed approach utilizes a disjunctive technique to transform the original nonconvex formulation into a mixed-integer linear programming (MILP) model
Minimizing trim loss is just one of the various significant considerations involved in devising the optimal corrugator schedule Other critical factors include maximizing corrugator width utilization, managing cutting pattern changes (order changes), preventing split orders, and minimizing shutdown costs Due to the inherent complexity of the production process, the corrugator scheduling problem cannot be entirely modeled and resolved using linear or integer programming formulations Chiong and Beng [4] provide an introductory review on how Genetic Algorithm and Evolutionary Algorithm tackle the one-dimensional cutting stock and draw comparison on the effectiveness of these two algorithms in solving cutting stock problem Other approaches to modeling and solving the Cutting Stock Problem have included the swarm particle algorithm approach [3], genetic algorithms [16], and column generation approaches [6]
6 The complexity of the problem has also prompted the development of sequential heuristic procedures, which break down the problem into smaller subproblems that are solved sequentially Several authors have proposed sequential heuristic procedures for addressing the corrugator trim problem For instance, Haessler [17] used a sequential search heuristic with Linear Programming (LP) to generate cutting patterns and minimize waste Subsequently, sequential heuristic procedures were employed in studies by Gradišar et al [18] and Cui et al [19] It is worth noting that in the research conducted by Zouein et al [8], authors put forward a sequential heuristic approach that incorporates a suboptimal three-step procedure This approach is particularly noteworthy as it takes into account the dual objective of minimizing both trim waste cost and setup time cost where setup cost is quantified as the loss of production incurred due to changes in stock roll sizes This innovative approach represents a significant contribution to the field, as it addresses the complex interplay between trim waste and setup time costs in corrugated board manufacturing processes
The field of research dedicated to the corrugator trim problem boasts a rich and well- established history A vast array of effective methodologies exists to tackle this challenge, encompassing both sequential heuristic procedures and more intricate algorithms Selecting the optimal approach hinges on the specific requirements of the corrugator operation, including the dimensions and intricacy of the problems encountered However, a notable shortcoming of the aforementioned research lies in its primary focus on minimizing material waste, neglecting other significant cost factors such as setup expenses This oversight presents a critical consideration in real-world applications, as both material waste and setup times significantly impact the profitability of a corrugator operation
In light of this gap in knowledge, this paper proposes a comprehensive mathematical model that incorporates both trim costs and setup costs within the framework of the corrugator trim problem To effectively address this multifaceted challenge, a genetic algorithm will be introduced as the most suitable method Genetic Algorithms possess significant power and wide-ranging applicability within stochastic search and optimization methods, drawing upon principles derived from evolutionary theory [20] This approach offers efficient and effective solutions due to its ability to handle complex problems with multiple objectives, as demonstrated in the work of Germán et al [16]
By incorporating these additional details, the paper provides a more comprehensive picture of the research landscape, highlights the limitations of existing methods, and strengthens the justification for the proposed approach.
PROBLEM ANALYSIS
Introduction to the company
The study was conducted at one of the largest manufacturers of corrugated boxes in Vietnam The main products include printing corrugated boxes using flexographic technology (Figure 3.1)
Figure 3.1 Example of main products
The company annually produces 40,000 to 45,000 tons of products through continuous production across three shifts The production model is Make-To-Order, targeting the B2B market
There are two main sections involved in producing final products: corrugating and converting (Figure 3.2)
The production process commences with paper reels as the primary input material, which are fed into the corrugator to undergo processing in the corrugating section, resulting in the creation of corrugated boards These boards then progress to the converting section, where they are printed using flexo printing technologies Following this, the boards undergo further processing in automatic or semi-automatic folding, taping, stitching, and gluing machines Ultimately, the finished products are bundled using a tying machine before being delivered to customers
The focus of this study centers on the corrugating section, which inherently faces the predominant issue of waste
3.1.2 Corrugated board and corrugating process
“Corrugated board”, or “corrugated cardboard”, is a type of paperboard that consists of 3/5/7 layers of liners (facings) or flutes (medium), as referred in the Figure 3.3
Produced through a process known as "corrugating process", these cardboard sheets are formed using input materials such as paper reels and adhesive
Figure 3.4 Corrugated board production process
The corrugating section represents a complex process that integrates various sets of machinery to produce the final product – corrugated boards Initially, reels of paper are fed into the corrugators Subsequently, the paper undergoes a hardening process using heat and steam as it passes between corrugating rolls, resulting in the formation of a flute shape (wavy layer) within the single facer, as illustrated in Figure 3.5 The paper is then pulled between a pair of gear-like cylinders, which forms the paper into a series of particular waves Glue is strategically applied to the tips of the flutes at specific locations, after which the flute tips are pressed against a flat liner, culminating in the creation of a continuous sheet of flat paper with fluted paper
10 adhered to it Following this, this continuous sheet undergoes a drying process facilitated by a system of heating plates until it reaches a level of stiffness that renders it unrollable Finally, the dried corrugated board is scored and cut into flat sheets according to the required dimensions at the slitter scorer machine
Figure 3.5 Illustration of corrugator machine (source: [25])
The corrugating process generates a considerable amount of paper waste overall This waste can be categorized into three main types: peeling waste, resulting from dirty or low- quality paper reels (Figure 3.6); defect waste, arising from the production of low-quality products that can not be used anymore (Figure 3.7); and trim waste, refering to the excess material generated due to discrepancies between the width of a paper roll and the required width of the board (Figure 3.8)
Figure 3.6 Example of peeling waste
Figure 3.7 Example of defect waste
Figure 3.8 Example of trim waste
Current analysis
Data on waste components generated during the corrugating process from January to July 2023 was collected, and the chart below illustrates the percentage of waste attributed to these components
Figure 3.9 The proportion of corrugating waste components (%)
12 According to the chart, trim waste constitutes the largest portion of corrugating waste components, ranging from 3.23% to 3.75% Defect waste follows closely behind, accounting for 1.61% to 2.20% of the total waste, while peeling waste is the lowest, ranging from 0.32% to 0.47% On the other hand, based on the factory's waste target, peeling waste and defect waste are within acceptable limits, but trim waste fails to meet the desired target (See Figure 3.10)
Figure 3.10 Target for each waste component
13 Due to trim waste comprising the majority of paper waste components and failing to meet the target, the subsequent section of the thesis will delve into a detailed analysis of the factory's current trim waste status
Trim waste originates from a machine section known as the "slitter scorer," as depicted in Figure 3.11
Figure 3.11 Slitter scorer creates trim waste
There exists a strong correlation between trim waste and the average trim dimension The Figure 3.12 shows the scatter plot and Pearson correlation testing of trim waste and trim dimension The result highlights the relationship between these two variables (p-value is less than the significance level of 0.05, we reject the null hypothesis r=0, indicating a correlation between the two factors), emphasizing the significance of managing trim dimensions effectively to mitigate trim waste
Figure 3.12 Correlation analysis between trim waste and average trim dimension
In the factory, the specification limit of trim dimension ranges from 30 mm to 80 mm Jobs with trim dimensions outside this range require approval from the technical department before proceeding with production In addition, jobs having trim dimension over 80 mm is
14 categorized as an “exceeding-specification job” (ES jobs), indicating instances where the material produced generates trim waste beyond the acceptable limits Identifying and addressing these ES jobs is crucial in optimizing production processes and minimizing unnecessary material waste within carton manufacturing operations
Gather data on trim dimension for 50 days, with 10 random samples per day (refer to Appendix for the dataset) Next, conduct descriptive statistics on the collected data and create a control chart The results of this analysis will be illustrated in Figure 3.13
Figure 3.13 Descriptive statistics, control chart, and process capability for company’s trim dimension
Several data points exceed the Upper Control Limit (UCL), indicating a prevalent occurrence of high trim dimensions The calculated mean trim dimension of 65.2 mm highlights a substantial occurrence of ES jobs within the factory's production processes (total 80 ES jobs/500 jobs) In addition, the resulting Cpk value of 0.16 signifies a notable issue: the current manufacturing procedure generates a considerable level of variation in trim dimensions, falling considerably outside the specification limits
15 Based on the analysis above, it's evident that the factory is facing the problem of high trim waste, exceeding the expected level This issue could lead to significant financial losses for the company, estimated between approximately 850 MVND to 1,200 MVND per month (refer to Appendix for a summary of costs attributed to trim waste provided by the accounting department).
Causes analysis
Using the 5-Whys analysis tool along with consulting expert opinions, the root cause analysis diagram is presented in Figure 3.14 below:
Figure 3.14 5-Whys analysis to find root causes
After the analysis process, three root causes have been identified: planner choose high trim dimension larger than necessary, no suitable supplier, and technical department do not set optimal value of minimum trim dimension However, this thesis only focuses the first cause of job schedulling, the others (regarding technical and purchasing department) will be discussed in future studies.
PRODUCTION SCHEDULING MODEL
Current scheduling procedure
Currently, production scheduling primarily relies on manual methods The current scheduling process has been visually depicted in the flowchart presented in Figure 4.1 below
Figure 4.1 The current scheduling procedure
The existing procedure commences with the compilation of a daily set of job orders essential for timely delivery These orders are then meticulously categorized into discrete job groups, distinguished by their unique characteristics such as flute type and paper composition Within each designated job group, the jobs are organized in a descending order of width Subsequently, a critical step in the procedure entails the selection of a suitable paper reel for each job group This selection process is carefully executed to ensure that the chosen reel possesses a width surpassing that of the board width for every individual job within the respective group
Opting for a single paper reel width across an entire job group holds significant potential for reducing setup time for subsequent jobs within that group This streamlined approach not only results in decreased setup costs but also contributes to minimizing production running time However, as discussed earlier, this method may result in larger trim dimensions for the final jobs within these groups Therefore, while this approach offers clear advantages in terms of setup efficiency, it necessitates careful consideration to strike a balance between these benefits and the potential increase in trim waste
Developing Production Scheduling Model
In this part, a production scheduling model that considers both setup time and trim costs has been developed, aligning with the current operational procedure To ensure accuracy, trim cost and setup expenses were precisely sourced directly from the accounting department Based on current procedure, two interrelated problems must be solved simultaneously Firstly, sequencing jobs on corrugator, and secondly, allocating the most suitable paper reels to each job The goal is to achieve an ideal sequence of jobs paired with the appropriate paper reel allocation This sequence should minimize the overall cost, encompassing both setup cost and trim cost
Model Assumptions Certain assumptions have been carefully crafted to simplify the model without compromising its integrity or deviating from the core aspects of the problem, which are:
Full of type of paper reels and the length of paper reels sufficiently supports job orders
The loss of paper reel leftover and paper reel picking time are not considered
Average running speed of corrugator is 100 m/min
Width of paper reels’ ranges from 1,200 to 2,000 mm
Sets and Parameters Let i be the job index, and j be the index of paper reel width
Each job i ∈ {1,2, … , N} contains the following data: sheet length set Li; sheet width set Wi; number of cuts set Hi; number of sheets per cut set Vi; and job characteristic set Chari The paper reel width set Kj = {2000, 1950, 1900, 1850, 1800, 1750, 1700, 1650, 1600, 1550, 1500,
Xi: binary variable, Xi = 1 if there is setup at job i, and Xi = 0 if otherwise
Yij: binary variable, Yij = 1 if roll width j assigned to job i, and Yij = 0 if otherwise
Mi: binary variable, Mi = 1 if all characteristics of job i and job i-1 is similar (Chari ≡ Chari-1), and Mi = 0 if otherwise
Paper reel allocation model (PRAM)
Equation (4.1) defines the objective function aimed at minimizing the total cost, which encompasses both trim cost and setup cost The selling price of paper is 1,500 VND per square meter, while the setup cost amounts to 32,000 VND per job Additionally, there is a waste of the first 6 meters of paper when operators change new paper reels
Equation (4.2) imposes a constraint on the minimum trim dimension, ensuring that it remains at or above 30 mm Equation (4.3) is dedicated to ensuring that each order is allocated only one paper reel width Equation (4.4) encompasses the constraints related to the operating time within a single working shift Equations (4.5) through (4.7) dictate that only two jobs with identical characteristics can be executed using the same paper reels, which guarantees the conditions of setup Finally, Equations (4.8) through (4.10) outline constraints associated with binary variables
Job Sequencing It's important to note that a single job sequence can have multiple objective values, and the PRAM model assists in identifying the best among these values Additionally, considering multiple sequences, the goal is to identify the sequence that achieves
19 the lowest objective value The job sequencing model aims to determine the most optimal job sequence based on the objective function TC derived from the PRAM model.
Numerical Experiment & Discussion
Chapter 4 Production scheduling model: Create a production planning model based on the existing procedure, explain the algorithm used, and analyze the findings obtained from implementing the new model
Chapter 5 Conclusion and future work: Summarize the key findings and achievements of the study, then providing recommendations for future researches
CHAPTER 2 LITERATURE REVIEW AND METHODOLOGY
Genetic algorithms (GA) originated from the principles of natural selection and genetics, aiming to solve optimization problems through evolutionary techniques such as crossover, mutation, selection John Holland is credited as the pioneer of the original Genetic Algorithm, developed in 1970, inspired by Charles Darwin's concept of random search within a defined search space [9]
GA transforms an initial population of individuals into a group of high-quality individuals, where each individual represents a potential solution to the problem at hand The efficacy of each solution is gauged by a fitness function, quantitatively measuring its suitability for a given environment Beginning with a randomly generated initial population, GA iteratively applies three fundamental genetic operators to each individual: selection, crossover, and mutation [9] Within GA, the search space comprises strings, known as chromosomes, each representing a potential solution to the problem The fitness value of each chromosome is determined by its objective function value A population consists of chromosomes alongside their corresponding fitness values, with generations representing populations generated in each iteration of the GA [10]
Pearson correlation is a statistical technique used to measure the degree of relationship between two variables It is the most popular and widely used correlation coefficient Pearson correlation coefficient takes values between -1 and 1, where 1, 0, and -1 indicate a perfect match, no correlation, and perfect negative correlation, respectively It is defined as the covariance of the variables that are to be compared, divided by their standard deviations [11]
The null hypothesis and the alternative hypothesis in Pearson correlation are thus:
Null hypothesis: The correlation coefficient is not significantly different from zero (There is no linear relationship)
Alternative hypothesis: The correlation coefficient deviates significantly from zero (there is a linear correlation)
The P value in Pearson correlation is used to measure the significance of the correlation analysis It is a standard method to determine whether the correlation coefficient is statistically
5 significant or not The P value is typically set at 0.01 or 0.05 A P value less than the cutoff indicates that the correlation coefficient is statistically significant, while a P value greater than the cutoff indicates that the correlation coefficient is not statistically significant [12]
The corrugator trim problem is a well-studied problem in the literature, with many papers published on the topic since the early 1980s One of the pivotal contributions in this field is from Haessler and Talbot [13], who introduced a 0-1 model, providing a systematic and foundational approach to addressing the corrugator trim problem Their work not only presented a mathematical framework but also served as the cornerstone for subsequent research endeavors in this domain Accordingly, various methods for this problem has been developed diversely, collectively known as the cutting stock problem Many studies have focused on cutting stock, scheduling, and production planning, with most of them being addressed through heuristic or linear programming approaches For example, Dyckhoff [14] presented the use of linear programming to solve 1-D cutting stock which is NP-complete problem Afshar [5] also used linear programming model to solve 1-D cutting stock with trim loss Rodriguez and Veccietti [15] proposed an integrated approach to solve the cutting stock problem and scheduling in the production of corrugated board boxes Traditionally, these two problems have been addressed independently due to their complexity However, in industries like corrugated board boxes, where scheduling decisions are directly influenced by cutting patterns and setup times, an integrated approach is essential The proposed approach utilizes a disjunctive technique to transform the original nonconvex formulation into a mixed-integer linear programming (MILP) model
Minimizing trim loss is just one of the various significant considerations involved in devising the optimal corrugator schedule Other critical factors include maximizing corrugator width utilization, managing cutting pattern changes (order changes), preventing split orders, and minimizing shutdown costs Due to the inherent complexity of the production process, the corrugator scheduling problem cannot be entirely modeled and resolved using linear or integer programming formulations Chiong and Beng [4] provide an introductory review on how Genetic Algorithm and Evolutionary Algorithm tackle the one-dimensional cutting stock and draw comparison on the effectiveness of these two algorithms in solving cutting stock problem Other approaches to modeling and solving the Cutting Stock Problem have included the swarm particle algorithm approach [3], genetic algorithms [16], and column generation approaches [6]
6 The complexity of the problem has also prompted the development of sequential heuristic procedures, which break down the problem into smaller subproblems that are solved sequentially Several authors have proposed sequential heuristic procedures for addressing the corrugator trim problem For instance, Haessler [17] used a sequential search heuristic with Linear Programming (LP) to generate cutting patterns and minimize waste Subsequently, sequential heuristic procedures were employed in studies by Gradišar et al [18] and Cui et al [19] It is worth noting that in the research conducted by Zouein et al [8], authors put forward a sequential heuristic approach that incorporates a suboptimal three-step procedure This approach is particularly noteworthy as it takes into account the dual objective of minimizing both trim waste cost and setup time cost where setup cost is quantified as the loss of production incurred due to changes in stock roll sizes This innovative approach represents a significant contribution to the field, as it addresses the complex interplay between trim waste and setup time costs in corrugated board manufacturing processes
The field of research dedicated to the corrugator trim problem boasts a rich and well- established history A vast array of effective methodologies exists to tackle this challenge, encompassing both sequential heuristic procedures and more intricate algorithms Selecting the optimal approach hinges on the specific requirements of the corrugator operation, including the dimensions and intricacy of the problems encountered However, a notable shortcoming of the aforementioned research lies in its primary focus on minimizing material waste, neglecting other significant cost factors such as setup expenses This oversight presents a critical consideration in real-world applications, as both material waste and setup times significantly impact the profitability of a corrugator operation
In light of this gap in knowledge, this paper proposes a comprehensive mathematical model that incorporates both trim costs and setup costs within the framework of the corrugator trim problem To effectively address this multifaceted challenge, a genetic algorithm will be introduced as the most suitable method Genetic Algorithms possess significant power and wide-ranging applicability within stochastic search and optimization methods, drawing upon principles derived from evolutionary theory [20] This approach offers efficient and effective solutions due to its ability to handle complex problems with multiple objectives, as demonstrated in the work of Germán et al [16]
By incorporating these additional details, the paper provides a more comprehensive picture of the research landscape, highlights the limitations of existing methods, and strengthens the justification for the proposed approach
The research methodology employed in this study adopts a problem-solving approach, structured into 6 distinct steps These steps are visually depicted in Figure 2.1, providing a clear roadmap for navigating and addressing the company's specific challenges comprehensively
To begin with, the methodology starts by defining the problem faced by the company During this step, observations are made and initial data is collected and analyzed to clearly identify the issue
Following the problem definition stage, the next step involves data collection and analysis This phase aims to gather empirical evidence and insights to gain clarity on the problem and its underlying factors
Once the data analysis and problem statement are complete, the focus shifts to identifying potential causes of the issue This involves a thorough exploration of the root causes, utilizing 5 Whys to delve deeper into the underlying factors contributing to the problem
After identifying the root cause, the methodology advances to the solution-building phase In this step, a production scheduling model based on the current procedure is proposed
To solve this model, a genetic algorithm is employed
In next step, proposed solution is then subjected to numerical experiments, allowing for empirical testing and validation of its effectiveness in addressing the problem
Finally, the proposed solution is implemented and planners are trained to use it The actual run results are then monitored and evaluated for effectiveness in comparison to previous methods
The study was conducted at one of the largest manufacturers of corrugated boxes in Vietnam The main products include printing corrugated boxes using flexographic technology (Figure 3.1)
Figure 3.1 Example of main products
The company annually produces 40,000 to 45,000 tons of products through continuous production across three shifts The production model is Make-To-Order, targeting the B2B market
There are two main sections involved in producing final products: corrugating and converting (Figure 3.2)
The production process commences with paper reels as the primary input material, which are fed into the corrugator to undergo processing in the corrugating section, resulting in the creation of corrugated boards These boards then progress to the converting section, where they are printed using flexo printing technologies Following this, the boards undergo further processing in automatic or semi-automatic folding, taping, stitching, and gluing machines Ultimately, the finished products are bundled using a tying machine before being delivered to customers
The focus of this study centers on the corrugating section, which inherently faces the predominant issue of waste
3.1.2 Corrugated board and corrugating process
“Corrugated board”, or “corrugated cardboard”, is a type of paperboard that consists of 3/5/7 layers of liners (facings) or flutes (medium), as referred in the Figure 3.3
Produced through a process known as "corrugating process", these cardboard sheets are formed using input materials such as paper reels and adhesive
Figure 3.4 Corrugated board production process
The corrugating section represents a complex process that integrates various sets of machinery to produce the final product – corrugated boards Initially, reels of paper are fed into the corrugators Subsequently, the paper undergoes a hardening process using heat and steam as it passes between corrugating rolls, resulting in the formation of a flute shape (wavy layer) within the single facer, as illustrated in Figure 3.5 The paper is then pulled between a pair of gear-like cylinders, which forms the paper into a series of particular waves Glue is strategically applied to the tips of the flutes at specific locations, after which the flute tips are pressed against a flat liner, culminating in the creation of a continuous sheet of flat paper with fluted paper
CONCLUSION AND FUTURE WORK
Conclusion
In spite of the rise in setup costs, the total cost sees a notable reduction of 12.16% when scheduling by using Genetic Algorithm This decrease in total costs reflects the comprehensive efficiency achieved by Genetic Algorithm, considering both trim and setup expenses Besides, the Cpk value, a measure of process capability, notably increases from 0.16 in the existing method to 0.44 with Genetic Algorithm implementation This substantial improvement indicates the genetic algorithm's ability to significantly reduce the number of exceeding-spec jobs, emphasising its potential in enhancing production In addition, it's evident that trim cost consistently outweighs setup cost across most scenarios, often correlated with specific job lengths This observation further underscores the significance of effective trim waste management strategies in overall cost reduction efforts
In scenarios characterised by high demand, the current procedure of production scheduling emerges as the favoured option, effectively managing running time to ensure precise operational control On the contrary, the production schedule created by Genetic Algorithm presents a more effective solution in reducing trim waste This advantageous reduction in waste, however, is accompanied by prolonged operational time, impacting overall efficiency Thus, while the Genetic Algorithm excels in minimising trim waste, its trade-off involves an extension in operational duration, necessitating a meticulous assessment of priorities when deciding between the two approaches This decision-making process should weigh the imperative of waste reduction against the importance of operational efficiency in meeting production demands
With only a limited number of studies addressing this specific topic within the field, this article stands out for its contribution of a novel production scheduling method This method takes into account the delicate balance between trim cost and setup cost, addressing a crucial aspect often overlooked in existing literature By incorporating this multifaceted approach, the study provides valuable insights and practical solutions for optimizing production processes in corrugated board manufacturing Furthermore, the findings underscore the importance of considering both material and setup costs in production scheduling, highlighting avenues for future research and industry implementation
Limitation and future work
The scope of this study is limited to the issue of production planning aimed at minimizing trim waste However, it's crucial to acknowledge that the cause of large trim dimension can also stem from machinery and technological factors As such, future research endeavors will aim to address and mitigate trim waste associated with technical challenges In particular, there is a need for research aimed at reducing the minimum trim dimension, which currently stands at 30mm By addressing this technical aspect, the stability of the manufacturing process can be enhanced, leading to improvements in Cpk values and overall operational efficiency
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[10] A Radwan et al., “Using Genetic Algorithm to Improve Information Retrieval
Systems,” World Academy of Science and Engineering Technology, vol 17, iss 2, pp 6-13, 2006
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[12] P Krishna Kumar et al., “State-of-the-art review on automated lumen and adventitial border delineation and its measurements in carotid ultrasound,” Computer Methods and
Programs in Biomedicine, vol 163, pp 155-168, 2018
[13] R.W Haessler and F.B Talbot “A 0-1 Model For Solving The Corrugator Trim Problem,” Working Paper No 205, Division of Research, Graduate School of Business Administration, The University of Michigan 1980
[14] H Dyckhoff, “A new linear programming approach to the cutting stock problem,”
[15] M A Rodriguez and A Vecchietti “Integrated Planning and Scheduling with Due Dates in the Corrugated Board Boxes Industry,” Industrial & Engineering Chemistry Research, vol 52, iss 2, pp 847-860, 2013
33 [16] A Germán et al., “A multi-objective approach based on soft computing techniques for production scheduling in Corrugator manufacturing plants,” Ingeniería y Desarrollo, vol 21, pp 73-92, 2007
[17] R W Haessler, “One-dimensional cutting stock problems and solution procedures,”
Mathematical and Computer Modelling, vol.16, iss 1, pp 1–8, 1992
[18] M Gradisar et al., “A sequential heuristic procedure for one-dimensional cutting,” European Journal of Operational Research, vol 114, no.3 , pp 557–568, 1999
[19] Y Cui et al., “A heuristic for the one-dimensional cutting stock problem with pattern reduction,” in Proceedings of The Institution of Mechanical Engineers Part B-journal of Engineering Manufacture, vol 222, pp 677-685, 2008
[20] M Gen and R.Cheng, Genetic Algorithms and Engineering Design New York: John Wiley&Sons
[21] H.S Ismail and K.K.B Hon, “Nesting of two-dimensional shapes using genetic algorithms,’ in Proceedings of the Institution of Mechanical Engineers, Part-B: Journal of Engineering Manufacture, vol 209, pp 115-124, 1995
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Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Sample 6 Sample 7 Sample 8 Sample 9 Sample 10
Unit Jan-23 Feb-23 Mar-23 Apr-23 May-23 Jun-23 Jul-23
APPENDIX B DATA OF 10 WORKING SHIFTS
Job Flute L H W V Demand Paper Reel Structure
Job 19 AB 1630 1525 692 2 3,050 TKA150 MCA120 MCB180 MCA120 TKA170
Job 25 AB 1486 520 1255 1 520 MCC120 MCC120 MCC120 MCC120 TKA120
Job 26 AB 231 75 1160 1 75 TKA170 MCA120 MCC120 MCC100 TKA170
Job 27 AB 289 2105 1272 1 2,105 TKA170 MCA120 MCC120 MCC100 TKA170
Job 42 AB 315 51 610 3 153 TKA200 MCA180 TKA200 MCA180 TKA200
Job 43 AB 985 184 1752 1 184 TKA200 MCA180 TKA200 MCA180 TKA200
Job 44 AB 1013 56 617 3 168 TKA200 MCA180 TKA200 MCA180 TKA200
Job 45 AB 834 56 617 3 168 TKA200 MCA180 TKA200 MCA180 TKA200
Job 46 AB 1346 180 550 3 540 TKA170 MCA120 TKA150 MCA100 TPH230
Job 47 AB 1414 428 862 2 856 TKA170 MCA120 TKA150 MCA100 TPH230
Job 48 AB 683 256 540 3 768 TLB150 MCA150 MCA100 MCA120 TKA170
Job 49 AB 733 356 552 3 1,068 TLB150 MCA150 MCA100 MCA120 TKA170
Job 55 AB 1501 230 782 2 460 TKA170 MFA120 TKA150 MCA100 TPH230
Job Flute L H W V D AL AM BL BM C
Job 3 AB 868 65 474 4 260 TLB150 MCB150 MCC100 MCC100 TKA170
Job 4 AB 2142 1570 783 2 3,140 TKH175 MCA150 TKA170 MCA120 TKZ175
Job 7 AB 2403 70 841 2 140 TKH175 MCA150 TKH175 MCA120 TKZ175
Job 10 AB 1284 440 758 2 880 TLB170 MCB120 MCB120 MCB120 WCB170
Job 14 AB 722 65 429 4 260 TKA200 MCA150 MCA120 MCA120 TKA170
Job 21 AB 1794 175 708 2 350 TLB170 MCB120 MCB120 MCB120 WCB170
Job 24 AB 1477 220 862 2 440 TKA170 MCA120 TKA150 MCA100 TPH230
Job 25 AB 697 35 460 3 105 TLB120 TLB120 TLB120 TLB120 TLB120
Job Flute L H W V D AL AM BL BM C
Job 22 AB 1572 425 860 2 850 TPH230 MCA120 TKA250 MCA100 TPH230
Job 28 AB 710 215 710 2 430 TLB150 MCA150 MCC100 MCA150 TKA150
Job 29 AB 2080 55 683 2 110 TLB150 MCA150 MCC100 MCA150 TKA150
Job 30 AB 1168 355 503 3 1,065 TLB150 MCA150 MCC100 MCA150 WCB140
Job 33 AB 1364 201 472 4 804 TKA150 MCC100 MCC100 MCC100 TDL150
Job 34 AB 1247 350 635 3 1,050 TKA150 MCC100 MCC100 MCC100 TKA150
Job 35 AB 588 230 922 2 460 MCC100 MCC100 MCC100 MCC100 MCC100
Job 38 AB 1717 70 562 3 210 TKA150 MCA100 MCA100 MCA100 TKA150
Job 39 AB 1977 440 837 2 880 TKA150 MCA100 MCA100 MCA100 TKA150
Job 41 AB 1422 68 509 3 204 TTC150 MCC100 MCC100 MCC100 TDL150
Job 42 AB 1212 91 423 4 364 TTC150 MCC100 MCC100 MCC100 TDL150
Job 43 AB 1422 75 509 3 225 TTC150 MCC100 MCC100 MCC100 TDL150
Job 44 AB 1502 65 507 3 195 TTC150 MCC100 MCC100 MCC100 TDL150
Job 45 AB 1212 375 413 4 1,500 TTC150 MCC100 MCC100 MCC100 TDL150
Job Flute L H W V D AL AM BL BM C
Job 6 AB 1412 285 603 3 855 TKA120 MCB120 MCA100 MCB120 TKA150
Job 9 AB 1107 142 790 2 284 TKA150 MCA120 MCA120 MCA120 TDL150
Job 10 AB 1107 112 790 2 224 TKA150 MCA120 MCA120 MCA120 TDL150
Job 11 AB 1218 185 538 3 555 TKA150 MCA120 MCA120 MCA120 TKA150
Job 14 AB 675 520 1520 1 520 TKA200 MCA180 TKA170 MCA180 TKA250
Job 15 AB 1627 520 743 2 1,040 TKA120 MCB120 MCC100 MCC100 TLB150
Job 16 AB 1127 30 1367 1 30 TLB150 MCB120 MCB120 MCB120 TLB150
Job 17 AB 1457 30 1367 1 30 TLB150 MCB120 MCB120 MCB120 TLB150
Job 21 AB 1408 30 1245 1 30 TKA150 MCA120 MCA100 MCA120 TPH160
Job 27 AB 1544 945 841 2 1,890 TKH175 MCA150 TKA170 MCA120 TKZ175
Job 28 AB 1544 899 841 2 1,798 TKH175 MCA150 TKA170 MCA120 TKZ175
Job 29 AB 779 1275 564 3 3,825 MCB120 MCA120 MCB120 MCA120 MCB120
Job 30 AB 1163 1090 556 3 3,270 TKP175 MCB120 MCB120 MCB120 TKP175
Job 39 AB 595 765 670 2 1,530 MCB120 MCA120 MCB180 MCA120 MCB120
Job 41 AB 1317 55 472 4 220 TKA200 MCA100 TKA150 MCC100 TKA250
Job 42 AB 1567 1250 477 4 5,000 TKA200 MCA100 TKA150 MCC100 TKA250
Job Flute L H W V D AL AM BL BM C
Job 4 AB 1567 1157 477 4 5,000 TKA200 MCA100 TKA150 MCC100 TKA250
Job 6 AB 1357 447 1218 1 447 TLB150 MCC100 MCC100 MCC100 TKA150
Job 8 AB 1502 102 563 3 306 TKA170 MCA100 MCC100 MCA100 TDL150
Job 12 AB 922 71 338 4 284 TDL150 MCA100 MCC100 MCA100 TDL150
Job 15 AB 686 82 925 2 164 MCC100 MCA100 MCC100 MCA100 MCC100
Job 16 AB 686 82 925 2 164 MCC100 MCA100 MCC100 MCA100 MCC100
Job 17 AB 1640 351 900 2 702 MCC100 MCA100 MCC100 MCA100 MCC100
Job 18 AB 595 765 670 2 1,530 MCB120 MCA120 MCB180 MCA120 MCB120
Job 21 AB 686 100 925 2 200 MCC100 MCA100 MCC100 MCA100 MCC100
Job 25 AB 655 70 1750 1 70 TKA170 MCA180 MCC100 MCA100 TKA170
Job 26 AB 1640 737 900 2 1,474 MCC100 MCA100 MCC100 MCA100 MCC100
Job 28 AB 1567 802 478 4 15,432 TKA200 MCA100 TKA150 MCC100 TKA250
Job 29 AB 1666 90 560 3 270 TKA170 MCA120 MCB120 MCA120 TKA170
Job 32 AB 1565 406 1140 1 1,310 TKA170 MCA150 MCA100 MCA100 TKA170
Job 33 AB 1294 593 421 4 2,372 TDL150 MCA100 MCC100 MCA100 TDL150
Job 35 AB 1271 1965 604 3 3,066 TKA170 MCA120 MCB120 MCA120 TKA170
Job 38 AB 1627 90 825 2 180 TKA150 MCA150 MCC100 MCA120 TKA170
Job 43 AB 1707 1887 563 3 7,170 TKA200 MCA100 TKA150 MCC100 TKA250
Job Flute L H W V D AL AM BL BM C
Job 1 AB 1767 1010 568 3 3,030 TKA200 MCA100 TKA150 MCC100 TKA250
Job 2 4AB 1326 340 896 2 680 TLB170 MCA120 MCC100 MCC100 0
Job 15 AB 2077 730 1613 1 730 TKA170 MCA180 MCB150 MCA150 TKZ175
Job 17 4AB 640 760 1400 1 760 TKA170 MCB150 MCB150 MCA150 0
Job 21 AB 1357 120 1143 1 120 TKA170 MCC100 MCC100 MCC100 TKA170
Job 22 4AB 896 620 1326 1 620 MCB120 MCC100 MCC100 MCC100 0
Job 23 AB 950 267 645 2 534 TKA250 MCA180 MCA180 MCA180 TKA250
Job 24 AB 567 515 645 2 1,030 TKA250 MCA180 MCA180 MCA180 TKA250
Job 27 AB 1960 730 1090 1 730 TKS150 MCA150 MCB120 MCA150 TKA150
Job 32 AB 1504 1300 472 4 5,200 TKA200 MCA100 TKA150 MCC100 TKA250
Job 33 AB 1567 1300 478 4 5,200 TKA200 MCA100 TKA150 MCC100 TKA250
Job Flute L H W V D AL AM BL BM C
Job 5 AB 1303 80 596 3 240 TKA200 MCA100 TKA150 MCA100 TKA250
Job 14 AB 1170 235 550 3 705 TKA170 MCA120 MCA100 MCA100 TDL150
Job 15 AB 1109 115 553 3 345 TLB150 MCA120 MCA100 MCA100 TLB170
Job 16 AB 1480 270 803 2 540 TLB150 MCA120 MCA120 MCA120 TKA150
Job 17 AB 697 105 533 3 315 TLB150 MCA120 MCA120 MCA120 WCB140
Job 26 AB 603 115 460 3 345 TKA150 MCA100 MCA100 MCA100 WCB140
Job 29 AB 1906 166 1130 1 166 TKA150 MCA120 MCA100 MCA100 TKA150
Job 30 AB 1805 272 1136 1 272 TKA150 MCA120 MCA100 MCA100 TKA150
Job 42 AB 2145 899 841 2 1,798 TKH175 MCA150 TKA170 MCA120 TKZ175
Job 44 AB 1313 1130 563 3 5,160 TKA200 MCA100 TKA150 MCC100 TKA250
Job 45 AB 1487 1324 567 3 7,155 TKA200 MCA100 TKA150 MCC100 TKA250
Job 54 AB 1806 130 1436 1 260 TKH175 MCA150 TKA170 MCA120 TKZ175
Job 55 AB 2108 1087 783 2 7,420 TKH175 MCA150 TKA170 MCA120 TKZ175
Job 58 AB 2160 175 1512 1 175 TKA250 MCA100 TKA250 MCA100 TKA250
Job Flute L H W V D AL AM BL BM C
Job 4 AB 1438 160 736 2 320 TTC150 MCA120 MCC120 MCB120 TLB150
Job 5 AB 985 260 727 2 520 TTC150 MCA120 MCC120 MCB120 TLB150
Job 6 AB 1614 260 727 2 520 TTC150 MCA120 MCC120 MCB120 TLB150
Job 7 AB 1447 75 633 2 150 TKA170 MCA150 MCA120 MCA120 TKA170
Job 8 AB1K 1751 1692 645 3 5,076 TKA170 MCA150 MCA150 MCA150 TKA170
Job 11 AB 1418 1525 603 3 4,575 TKA120 MCA100 MCA100 MCA100 TLB150
Job 15 AB 2225 2025 841 2 4,050 TKH175 MCA150 TKA170 MCA120 TKZ175
Job 27 AB 1322 315 512 3 945 TKA170 MCA150 MCB150 MCB150 TKA170
Job 31 AB 1010 52 421 4 208 TDL150 MCC100 MCC100 MCC100 TDL150
Job 32 AB 1073 20 1245 1 20 TKA250 TKA250 TKA200 MCA180 TPH230
Job 33 AB 1102 20 1250 1 20 TKA250 TKA250 TKA200 MCA180 TPH230
Job 34 AB 1081 20 1250 1 20 TKA250 TKA250 TKA200 MCA180 TPH230
Job Flute L H W V D AL AM BL BM C
Job 14 AB 1608 240 1204 1 240 TKA120 MCB120 MCC120 MCC120 TLB150
Job 15 AB 1607 644 1202 1 644 TKA120 MCB120 MCC120 MCC120 TLB150
Job 17 AB 1247 208 635 3 624 TKA150 MCC100 MCC100 MCC100 TKA150
Job 18 AB 1342 550 458 4 2,200 TKA200 MCA100 TKA150 MCC100 TKA250
Job 23 AB 1610 1790 519 3 5,370 TTC150 MCB120 MCB120 MCB120 TKA150
Job 25 AB 1917 245 1352 1 245 TKA150 MCA100 MCA100 MCA100 TKA150
Job 26 AB 2447 547 1352 1 547 TKA150 MCA100 MCA100 MCA100 TKA150
Job 27 AB 2977 545 1352 1 545 TKA150 MCA100 MCA100 MCA100 TKA150
Job Flute L H W V D AL AM BL BM C
Job 10 AB 1006 72 1637 1 72 TLB170 MCB150 TLB150 MCB120 TLB200
Job 11 AB 450 70 1637 1 70 TLB170 MCB150 TLB150 MCB120 TLB200
Job 12 AB 1951 70 1637 1 70 TLB170 MCB150 TLB150 MCB120 TLB200
Job 13 AB 1912 70 1637 1 70 TLB170 MCB150 TLB150 MCB120 TLB200
Job 34 AB 300 65 640 2 130 TKA200 MCA180 TKA250 MCA180 TKA250
Job 35 AB 1090 848 478 4 5,256 TKA200 MCA100 TKA150 MCC100 TKA250
Job 45 AB 1032 853 841 2 3,580 TKH175 MCA150 TKA170 MCA120 TKZ175
Job 56 AB 1123 44 1406 1 44 TKA250 MCA150 MCA150 MCA150 TKA250
Job 64 AB 1043 120 678 2 240 TKA170 MCB150 MCA100 MCA100 TKA170
Job Sequence: ['Job 11', 'Job 24', 'Job 4', 'Job 20', 'Job 2', 'Job 27', 'Job 8', 'Job 13', 'Job 17', 'Job 10', 'Job 21', 'Job 12', 'Job 15', 'Job 28', 'Job 26', 'Job 1', 'Job 7', 'Job 19', 'Job 23', 'Job 9', 'Job 16', 'Job 5', 'Job 14', 'Job 22', 'Job 6', 'Job 18', 'Job 3', 'Job 25']
Job Sequence: ['Job 3', 'Job 4', 'Job 2', 'Job 1', 'Job 5', 'Job 17', 'Job 18', 'Job 6', 'Job 7', 'Job 13', 'Job 14', 'Job 8', 'Job 9', 'Job 11', 'Job 10', 'Job 12', 'Job 15', 'Job 16', 'Job 19', 'Job 20', 'Job 21', 'Job 24', 'Job 31', 'Job 22', 'Job 23', 'Job 25', 'Job 26', 'Job 27', 'Job 28', 'Job 29', 'Job 30', 'Job 32', 'Job 33', 'Job 34', 'Job 35', 'Job 37', 'Job 36', 'Job 38', 'Job 39', 'Job 40', 'Job 42', 'Job 45', 'Job 41', 'Job 43', 'Job 44', 'Job 46', 'Job 47']
Job Sequence: ['Job 1', 'Job 2', 'Job 5', 'Job 3', 'Job 4', 'Job 6', 'Job 7', 'Job 8', 'Job 9', 'Job 10', 'Job 11', 'Job 35', 'Job 12', 'Job 13', 'Job 14', 'Job 15', 'Job 16', 'Job 17', 'Job 18', 'Job 19', 'Job 20', 'Job 21', 'Job 22', 'Job 23', 'Job 24', 'Job 25', 'Job 26', 'Job 27', 'Job 28', 'Job 29', 'Job 30', 'Job 31', 'Job 32', 'Job 33', 'Job 34', 'Job 36', 'Job 37', 'Job 38', 'Job 39', 'Job 40', 'Job 42', 'Job 41']
Job Sequence: ['Job 22', 'Job 1', 'Job 36', 'Job 23', 'Job 2', 'Job 3', 'Job 11', 'Job 13', 'Job 46', 'Job 19', 'Job 28', 'Job 4', 'Job 43', 'Job 5', 'Job 27', 'Job 30', 'Job 6', 'Job 7', 'Job 8', 'Job 9', 'Job 10', 'Job 33', 'Job 12', 'Job 15', 'Job 16', 'Job 21', 'Job 17', 'Job 26', 'Job 18', 'Job 20', 'Job 31', 'Job 24', 'Job 25', 'Job 35', 'Job 29', 'Job 32', 'Job 37', 'Job 38', 'Job 39', 'Job 41', 'Job 45', 'Job 42', 'Job 44']
Job Sequence: ['Job 33', 'Job 32', 'Job 1', 'Job 2', 'Job 3', 'Job 5', 'Job 6', 'Job 4', 'Job 7', 'Job 8', 'Job 9', 'Job 10', 'Job 13', 'Job 11', 'Job 12', 'Job 14', 'Job 15', 'Job 16', 'Job 17', 'Job 18', 'Job 19', 'Job 20', 'Job 28', 'Job 21', 'Job 22', 'Job 23', 'Job 24', 'Job 25', 'Job 26', 'Job 27', 'Job 29', 'Job 30', 'Job 31']
Job Sequence: ['Job 2', 'Job 1', 'Job 3', 'Job 4', 'Job 5', 'Job 6', 'Job 11', 'Job 9', 'Job 10', 'Job 13', 'Job 12', 'Job 14', 'Job 15', 'Job 16', 'Job 17', 'Job 18', 'Job 19', 'Job 20', 'Job 22', 'Job 23', 'Job 24', 'Job 25', 'Job 26', 'Job 28', 'Job 30', 'Job 29', 'Job 31', 'Job 36', 'Job 49', 'Job 48', 'Job 50', 'Job 39', 'Job 42', 'Job 55', 'Job 54', 'Job 45', 'Job 44', 'Job 46', 'Job 53', 'Job 56', 'Job 58', 'Job 62', 'Job 63', 'Job 68']
Job Sequence: ['Job 1', 'Job 2', 'Job 3', 'Job 4', 'Job 5', 'Job 6', 'Job 7', 'Job 8', 'Job 9', 'Job 10', 'Job 11', 'Job 12', 'Job 13', 'Job 14', 'Job 15', 'Job 16', 'Job 17', 'Job 19', 'Job 21', 'Job 18', 'Job 20', 'Job 22', 'Job 23', 'Job 24', 'Job 25', 'Job 26', 'Job 27', 'Job 28', 'Job 30', 'Job 29', 'Job 31', 'Job 33', 'Job 34', 'Job 32']
Job Sequence: ['Job 1', 'Job 4', 'Job 2', 'Job 3', 'Job 5', 'Job 6', 'Job 7', 'Job 8', 'Job 9', 'Job 10', 'Job 11', 'Job 12', 'Job 13', 'Job 14', 'Job 15', 'Job 16', 'Job 17', 'Job 18', 'Job 19', 'Job 20', 'Job 21', 'Job 22', 'Job 23', 'Job 24', 'Job 25', 'Job 26', 'Job 27', 'Job 28', 'Job 29']
Job Sequence: ['Job 1', 'Job 2', 'Job 4', 'Job 5', 'Job 6', 'Job 7', 'Job 3', 'Job 8', 'Job 9', 'Job 10', 'Job 11', 'Job 12', 'Job 13', 'Job 15', 'Job 14', 'Job 16', 'Job 17', 'Job 18', 'Job 20', 'Job 19', 'Job 21', 'Job 23', 'Job 24', 'Job 25', 'Job 30', 'Job 29', 'Job 31', 'Job 32', 'Job 33', 'Job 34', 'Job 35', 'Job 37', 'Job 38', 'Job 39', 'Job 40', 'Job 41', 'Job 43', 'Job 44', 'Job 45', 'Job 46', 'Job 48', 'Job 55', 'Job 56', 'Job 63', 'Job 64', 'Job 66', 'Job 65', 'Job 67']
APPENDIX C GENETIC ALGORITHM CODE import copy import numpy as np import random import pandas as pd from pulp import * import time
# Define the objective function (makespan) def objectiveFunction(job_set,schedule,Fi,AMi,BMi,Ci,ALi,BLi,paper_roll_width_set,Li,Hi,Vi,Wi): def sort_by_job_sequence(job, job_sequence, v): v_sequence = [] for job_name in job_sequence: index = job.index(job_name) v_sequence.append(v[index]) return v_sequence
F = sort_by_job_sequence(job_set,schedule,Fi)
AM = sort_by_job_sequence(job_set,schedule,AMi)
BM = sort_by_job_sequence(job_set,schedule,BMi)
AL = sort_by_job_sequence(job_set,schedule,ALi)
C = sort_by_job_sequence(job_set,schedule,Ci)
BL = sort_by_job_sequence(job_set,schedule,BLi) cuts_set = sort_by_job_sequence(job_set,schedule,Hi) sheet_length_set = sort_by_job_sequence(job_set,schedule,Li) sheets_per_cut_set = sort_by_job_sequence(job_set,schedule,Vi) sheet_width_set = sort_by_job_sequence(job_set,schedule,Wi)
# def solve_LP(job_set,sheet_length_set,sheet_width_set,cuts_set,sheets_per_cut_set,paper_roll_width_set,F,AM,AL,BM,BL,C): problem = LpProblem('Paper_roll_allocation', LpMinimize)
Y = LpVariable.dicts('decisionVariable_Yij', ((i, j) for i in range(0,len(job_set)) for j in range(0,len(paper_roll_width_set))), cat='Binary')
X = LpVariable.dicts('decisionVariable_Xi', (i for i in range(0,len(job_set))), cat='Binary')
M.append(0) for i in range (1,len(job_set)): if (F[i] == F[i-1]) and (AM[i] == AM[i-1]) and (AL[i] == AL[i-1]) and (BL[i] == BL[i-1]) and (BM[i] == BM[i-1]) and (C[i]
# Objective Function objective1 = lpSum((lpSum(Y[i,j]*paper_roll_width_set[j] for j in range(0,len(paper_roll_width_set))) - sheet_width_set[i]*sheets_per_cut_set[i])*0.0015*sheet_length_set[i]*cuts_set[i] for i in range(0,len(job_set))) objective2 = lpSum((32000 + 1.5 * 6 * sheet_width_set[i]*sheets_per_cut_set[i]) * X[i] for i in range(0,len(job_set))) problem += objective1+objective2
# Constraint 1 for i in range(0,len(job_set)): problem += (lpSum(paper_roll_width_set[j] * Y[i,j] for j in range(0,len(paper_roll_width_set))) - sheets_per_cut_set[i] * sheet_width_set[i] >= 30)
#Constraint 2 for i in range(0,len(job_set)):
58 problem += (lpSum(Y[i, j] for j in range(0,len(paper_roll_width_set))) == 1)
#Constraint 3: problem += (lpSum((cuts_set[i]*sheet_length_set[i])/(100*1000) + 2*X[i] for i in range(0,len(job_set))) = 1 - M[i] problem += constraint
M = 1000000 for i in range(1,len(job_set)): for j in range(0,len(paper_roll_width_set)): problem += Y[i, j] - Y[i-1,j]