Employment of Set Operations to Improve LMP Assessments Design and Implementation

Received 2018-11-30 Accepted 2018-12-27 Published 2019-01-01 Abstract Objective This paper aims to provide a new mathematical model for the construction of assessment tools for Lean Manufacturing Practices (LMP) through the integration process with the set operations. This study also strives to develop key elements to enhance the effectiveness of LMP assessments and their impact through mathematical interrelationship. Problem Previous studies have shown a lack of clear mathematical methodology to carry out a rigorous assessment of the LMP effects on the firm’s operational and financial performance. Therefore, this paper tries to address this aspect by developing new equations. Design The methodology of this study is based on the conversion of the linguistic description (companies, processes, waste, practices) into numerical measurement model by integrating with set theory. Finding The results show a set of relationships and equations that can be applied to be a fundamental basis for ensuring the effectiveness of the assessment. This paper may contribute to the improvement of LMP’s design and development.


Introduction
According to many studies, the practice of lean manufacturing has been used to effectively improve the operational and financial performance of the organization (Taj and Morosan, 2011;Dora et al., 2013;Fullerton et al., 2014;Godinho Filh et al., 2016;Negrão et al., 2017). The lean can be defined according to a common formula for what has been described in previous studies Tagge et al., 2017;Laureani and Antony, 2017;Randhawa and Ahuja, 2017) as a work without waste that includes a set of practices, thereby contributing to improved performance in the organizations. On the other hand, waste can be defined as anything that has no value within the institution, such as anything other than the minimum amount of equipment, materials, parts, and working time that are essential to production (Pavnaskar et al, 2003;Taj, 2008;Taj, 2011;chen et al., 2013). Common waste in the enterprises includes seven types, namely Transportation, Inventory, Motion, Waiting, Overproduction, Over processing and Defects (Arunagiri Gnanavelbabu, 2014;Qureshi et al, 2015). The elimination of previous waste through lean practices helps to reduce the enterprises excessive spending and over-effort (Siasos et al., 2017). In order to measure these positive effects of the lean practices on performance, there are many studies in the field of evaluation, and most of these studies have followed the descriptive logic of the relationship between performance and lean applications. Most of the results show the depth of the relationship between performance and lean practices (Negrão et al., 2017). However, despite these references, especially in quantitative and quantitative studies there is a lack of actual value of its effect. The previous studies are based on providing descriptive result of the relationship only. Therefore, this issue is considered as the focus of the present study for mathematical models and the development of mathematical models to achieve a methodology that provides a numerical description of the relation between lean and operational and financial performance to make the evaluation process more effective. Over the past decade, 130 studies were conducted at different institutions in the world in the field of LMP Assessment. By analyzing the methodology used in these studies, it was found that they adopted quantitative and qualitative methodologies without focusing on the mathematical logic in explaining the results of measurement. Figures 1-2 shows the distribution of the assessment methodology between (quantity, type, integration of quantity and type, other methods, Mathematical Model), and these differences show no general logic of measurement. Some studies have made references to the development of a digital concepts that indicate the effects of practices in the performance through mathematical models that provides different weights but are not comprehensive (Cil et al., 2013). Some other studies have also attempted a digital measurement of the performance through the use of Analytic Hierarchy Thanki et al., 2016;. Therefore, this study will extend the digital description in evaluating the impact of lean on the performance, thus updating what has been achieved in this field (Albzeirat et al., 2018). The structure of this study will include the Introduction, which provides a preliminary conception of the objectives of the study and the elements of the study in general, followed by the second section, which includes the methodology of the study. The fourth section includes the integration of the basic elements of the study (companies, operations, waste. Finally, the conclusions are discussed in the fifth section.

Method
This study aims to develop the mathematical equations that can give a new perception of the descriptive state of the relationship between LMP, companies, processes, waste and performance. It is known that the engineering application transcends administrative and descriptive science by giving a digital representation of data and relationships. In this context, the description of the relationship between LMP and performance will be exceeded by mathematical equations that reflect a more accurate perception of this relationship. To achieve this goal, this study will include Figure 3.
The methodology will be achieved through the following steps: First: Defining LMP in general and establishing the mathematical representation of the general set of lean practices based on set operations.

Second:
Identifying the general groups of the classification of institutions according to the type of companies and the status of the mathematical representation of these groups based on set operations.
Third: Determining the total waste that can be formed in these companies based on set operations.
Fourth: Identifying the group of practices that represent a solution for the removal of waste in each group of types of waste type according to the type of company based on set operations.

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Fifth: Conversion of representation in previous collections to mathematical equations based on set operations.

General Lean Manufacturing Set
This part of the study defines lean manufacturing practices in general and establishes the mathematical representation of the general set of lean practices. Therefore, the mathematical representation of the meaning of the set is called and applied to obtain the mathematical form of the general set of lean manufacturing practices (GSLMP) by using a mathematical application that includes a set logic known as a group theory. Based on review of the previous studies, it is clear that the LMP represents a set of practices that aim at making a positive change in the company or plant performance. As an illustration, GSLMP can be mathematically defined as follows:   These sets can be expressed mathematically through a group theory as follows: Also SW can be represented as follows: , whereby m is a number of a Company. Also C can be represented as: , whereby z is a number of processes. Also SP can be represented by: Through the descriptive and practical relationships between LMP and Definitions 3-.5, the following descriptive theories can be inferred: Theory 1. From the logic of a difference in the types of waste, there is a difference in the type of appropriate LMP for each type of waste. It is therefore a mathematical logic to have a subset of LMP suitable for each type of waste.

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The relationship between Equations 1 and 2 in Theory 1 can be represented as follows: Theory 2. From the logic of a difference in the types of companies, there is a difference in the type of appropriate LMP for each type of waste in companies. It is therefore a mathematical logic to have a subset of LMP suitable for each type of company. The relationship between Equations 1 and 3 in Theory .2 can be represented as follows: can be applied in this type of companies.
Theory 3. From the logic of a difference in the types of processes, there is a difference in the type of appropriate LMP for each type of processes. It is therefore a mathematical logic to have a subset of LMP suitable for each type of processes.
The relationship between Equations 1 and 4 in theory 3 can be represented as follows: can be applied in this type of companies.
The previous definitions include the following aspects: 1. Classification of practices depending on the type of companies.
2. Classification of practices by the type of operation.
3. Classification of practices according to the type of waste.

Combining Waste with Companies
Different types of companies, industrial, service, logistics, educational, health… etc. This difference in the classification of companies gives a difference in the type of waste, depending on the classification of the company, and this logic can be represented by: CSS: a group of companies that includes a joint description, such as energy companies, mining, education, health ...etc. Each of these Subsets may include a variety of waste types. The SSC can be represented through the Set Theory as follows: Through the Set Theory characteristics, subsets can be formed within specific classifications of a set, which can be formed by a set containing at least one element of the elements of SC.
This requires a description of the classification; namely a system to classify companies and enterprises according to the activities in which they are engaged.
Through the characteristics of groups, the following relationships can be inferred: To identify a set of waste in any group of companies, the following steps are taken: 1. Identifying processes.

Monitoring processes through the observation and digital values.
3. Designing an appropriate assessment to determine the presence of waste.

Combining Processes with Company
Each process has its own conditions that may include the human factor, the technical and the machinery. These diversity factors are shown in the diversity of wastes and the differences according to the tasks in the process. Consequently, each process involves a group of waste, whereby this group can be expressed through Group Theory as follows:

Set of Processes for Set of Companies (SPSC) as:
        In order to achieve the accuracy of the assessment tool, the classification of waste is determined according to the processes within the company. This classification can be obtained through the matrix of the integration of processes and waste, and can be represented as follows: Set Waste of Processes for company (SWPC) by relations 2 and 14:

Combining Waste with Processes and Lean Manufacturing Practices
Different processes and waste variation indicate the difference in the lean practices that can be used to remove wastes in each process. Based on this logic, practices can be classified according to the type of processes. To determine the relationship between practices and waste, the mathematical model must be developed to represent this relationship through the Group Theory as follows: Through the previous matrix " ", the following sets can be found: Depending on the characteristics of the intersection, union and the characteristics of operations, the sets of practices include the following possibilities: 1. Sets can intersect with each other.
2. The output of the intersection may be an empty set  .
3. Each set of practices in any process is a subset of the total set of practices.
4. Each practice includes a certain impact rate on waste.

Mathematical Equations and Financial Values
To demonstrate the effectiveness of the previous logic, the former mathematical logic in sections 3.1-3.4 must be converted to mathematical values, which can be reviewed as follows:

Capacity of Production for Companies
Each company has a capacity Production "Quantity" where the percentage of the amount of production can be calculated by means of this quantity. These values vary relatively from one company to another, and therefore, these values can take multiple ratios for the total amount of actual production. Accordingly, it is possible to derive the following relationships:

 
The

Amount of Production of Processes
Each process has a Production Capacity of Operations in the company. These values vary relatively from one process to another, and therefore, these values can take multiple ratios for the total actual value of production in the company. Accordingly, it is possible to derive the following relationships: Every process in the company i has a productivity value

Actual Performance of Companies
It is known that each company has a real performance value, and this value shows the actual value of production compared to the available human or technical capabilities. Through the indicators of production, performance of companies can determine the actual values that can be achieved by the improvement, and through the actual performance, values can be formulated by the following equations: The true value of the optimum performance of all companies ( SC OP ) representing a percentage of 100%, which is evident through the following equation:

Waste and Performance
It is logical to have a defect in the performance due to the presence of waste in the process. Through this relationship, the financial value of waste (FVW) is given through the reflection in the financial value of the defect in the processes. Consequently, the defect related to any process is a measure of the existence of waste and can be represented in the following relationship:

Actual Values of Impact LMP
LMP is used to remove waste, and due to the different processes and wastes, there is a difference in the impact of the use of LMP on the performance. The relationship between practices and performance varies according to the set of practices that can be applied to waste elimination. Based on this logic, practices can be arranged within the reflections of financial values that ensure prioritization in the application of LMPs to ensure real returns to their application. These values can contribute to the development of effective programs in determining practices and the mechanism of their application. Accordingly, actual values of the impact LMP (AVILMP) applications for performance improvement should be close to the value of loss in performance. It can be mathematically expressed by: .
i C AVILMPs FVD  The value of use and application LMP should not exceed the value of loss in performance.

Make Decision to Apply LMP
The decision to apply LMP must be based on an actual value that ensures the achievement of positive results, and through the values of the previous equations, a set of steps can be taken to make the appropriate decision as follows: