Sea lion with enhanced exploration phase for optimization of polynomial fitness with SEM in lean technology

Abstract

With the establishment of lean manufacturing, myriad industries implemented the lean manufacturing principles and guidelines. To train the professionals based on certain policies to achieve continuous improvements in terms of productivity and minimizing wastages. However, the complexities such as variability, sustainability, multi-dimensional views, factory size, to name just a few negatively influences the performance of deploying lean manufacturing in industries. It is, therefore, very important for companies to recognize and understand the critical success factors for successfully implementing lean manufacturing. Hence, this paper plans to develop a model on concerning the analysis of lean manufacturing to find the most important factor regarding the technology among the industries. With this intention, this paper is analyzed through three phases. In the first phase, the prepared questionnaire is distributed to the professionals in various companies. In the questionnaire, all the mandatory questions are included. Then, the professional are recommended to fill the precise information as far as possible. In the second phase, the responses from the concerned practitioners associated with the industries are considered for analysis. Herein, the analysis is carried out based on structural equation modeling approaches with the contribution of higher order statistical analysis, which is performed using the input factors such as the lean awareness, lean technology, organizational support, organizational performance, employee involvement and management commitment among the industries via attaining better maximum likelihood values of the questionnaires. In the third phase, the Prediction of polynomial fitting of the response (i.e. the objective function) is achieved with the aid a novel optimization algorithm SLnO-EE model (Sea Lion with Enhanced Exploration phase), which is the extended version of SLnO (Sea Lion Optimization). Finally, the proposed SLnO-EE model is evaluated over traditional SLnO model in terms of certain performance evaluations to exhibit the improvement in prediction accuracy.

1 Introductions

Generally, lean manufacturing is a technique, which concentrates on reducing waste in the manufacturing organization, along with dealing with increasing the productivity [9]. Lean manufacturing alludes to an approach intended to bring down the costs associated with production with the goal of limiting waste [27]. Research has shown that companies that implement and practice lean manufacturing or lean production see a significant improvement in operational performance [7]. This process is also referred as lean production or LT and the combined socio-technical technique depends on the industries manufacturing system which is continuously utilized by the industry and many others [11, 15]. Moreover, lean manufacturing depends on numerous particular polices like Kaizen (continuous improvement using small changes). Initially, it was established in the Western countries in 1990 through the article “The Machine That Changed the World” [10, 12]. Still now, the lean polices extreme impacts the production ideas all over the world and the companies [28] beyond production such as service organizations, IT (software), health care, etc. Besides, the applications of lean involve minimized production time, minimized operating expense, enhanced product quality, and so on [3, 13, 14].

In order to enhance the benefits of lean manufacturing, a lot of policies and methods were introduced. TOC is an efficient approach to enhance the production system containing policies and training strategies which can be exploited by the professionals [16, 17, 19, 21]. Then, supply chain approach was implemented by the management of multiple fields’ viz., production, place, and transportation for the professionals, stockroom, to name a few for attaining the finest integration of response ability and proficiency in the marketplace it deals [29]. For mass manufacturing scenario, the conditions are simple to determine, whereas for lean manufacturing, these conditions are not available and forced the researchers to devise a new methodology. Moreover, the six sigma method was proposed, which is a statistical approach that helps to minimize variability and eliminate the defects [18, 20, 31].

Massive techniques were presented to achieve successful lean manufacturing principle. However, the complexities due to time consuming and expensiveness still exists [22]. Furthermore, many tools and approaches for analyzing the lean manufacturing by means of decision making with AHP, CFA and SEM were implemented using various approaches [1, 24, 25, 30]. Although, these methods brought out some achievements, the limitations namely, factory size (small, medium, and large), sustainability, determining degree of leanness implemented, variability, etc., influences the results and degrades the productivity [6, 23]. Thus, there exist several challenges to be accomplished, which seek the attention of researcher and scholars to innovate significant and intellectual methodologies to improve the productivity through successful lean manufacturing principles.

The main contribution of this paper is as follows.

• 1. This paper introduces a lean manufacturing model through the efficiency of SEM analysis with ML technique.

• 2. The proposed lean manufacturing model is accomplished through three phases. In the first phase, a questionnaire is prepared and distributed to the professionals in various companies. Then, the professional are recommended to fill the precise information as far as possible.

• 3. In the second phase, the responses from the concerned practitioners associated with the industries are considered. The analysis is based on SEM under six input factors to attain better ML values of the questionnaires.

• 4. In the third phase, the objective function (lessening of prediction errors) in the polynomial expression is achieved by constructing a novel optimization algorithm called SLnO-EE model, which is an improved model of SLnO. The resultant acquired by resolving the objective function in SLnO-EE model evidently exhibits the improvement in prediction accuracy over extant SLnO.

The organization of this paper is in this order: Section II presents the literature regarding lean manufacturing models. Proposed lean manufacturing process via structural equation modelling is illustrated in Section III. SEM investigation with CFA, EFA (Exploratory Factor Analysis) and regression analysis is demonstrated in Section IV. Section V gives the attained results, and Section VI concludes the paper.

2 Literature review

2.1 Related works

In 2019, Kock [1] had presented a SEM approach with the efficiency of PLS named PLSF method to analysis the structural model parameters regarding marketing expenditures. Moreover, the author performed a comparative study to analyse whether factor-based model were better than composite-based models or not. From, the experimental study, it was clear that the factor-based PLSF model attained a minimum type I and II error and achieved better results than composite-based models.

In 2019, Mantilla et al. [2] have investigated a SEM model using a questionnaire for analysing the influences of ISO 9001 standards for schools. In addition to this, they have evaluated the technical behaviors of the apparatus provided and evaluated the reliability, satisfaction, as well as building strength via SEM deployed in software AMOS 24. Through the investigative analysis, the SEM accomplished reliable questionnaire with better Cronbach’s α of 0.985 as well as the values greater than 0.93 for all 6 dimensions.

In 2017, Sims and Wan [3] have presented a case study of a TOC model for lean manufacturing using 3 approaches such as flow constraint, effective utilization and quick effective utilization analysis. The first method used the view as well as analysed user satisfaction, second one to obtain used for identifying the position of system condition for the particular procedure, and the third one was exploited while there was slight/no historical line performance data exists. Finally, a case study was performed via a real production environment.

In 2014, García et al. [4] have examined the efficiency of a supply system using lean manufacturing for the industry via Promodel software to represent the simulated framework of CVJ manufacturing units. Here, 2 supply approaches were utilized like random and clustering supply methods. Moreover, the industry’s objective is to maintain 1% loss. From the experimentation, the random supply system achieved better outcomes and proved its efficiency over the clustering supply system.

In 2012, Nasab et al. [5] have proposed a lean manufacturing system using ANN with AHP called A2 and A3 to measure the success of lean manufacturing, deployment in an automobile industry via analyzing the degree of leanness. Furthermore, they have attained simpler estimations, fast execution and exact decision making. Using the experimental work, the authors proved the efficiency of A2 in terms of time as well as cost and better results over A3 (adaptive AHP) model.

In 2018, Nevil et al. [6] have developed a combined lean manufacturing along with green manufacturing by considering the progress of production in SMEs with the deployment of drivers using TOPSIS and SAW and with fuzzy approach and borda approach. In addition, the authors have implemented five various drivers to influence the industry in terms of decision making and motivated the SMEs to achieve improvements using limited resources.

In 2016, Alhuraish et al. [7] have introduced an AHP approach for lean manufacturing along with six sigma techniques on considering 3 constraints like financial, operational, and innovation performance. Moreover, the authors deployed all these three constraints in the industries and evaluated the results based on the industries implementing lean manufacturing alone and lean six sigma models with limited tools. From the investigative analysis, the novel AHP model achieved better improvements under all the 3 constraints.

In 2015, Susilawati et al. [8] have presented a lean manufacturing model to handle the multidimensional factors, unavailability standards as well as uncertainty on concerning the degree of leanness using fuzzy logic. Besides, they have built the principles through the primary as well as secondary information with a complete literary and verified via interviews. The subjective human judgments for the degree of leanness training was handled via fuzzy number with deployment of lean training size and the exercise of multi-analysers. From this study, the authors demonstrated the applicability and benefits of lean manufacturing.

2.2 Review

Table 1 summarizes the features and challenges of conventional models for lean manufacturing model by various methods which has the potential and efficiency to be deployed in the real-world enterprises. The PLSF [1] model attained minimized type I and II errors and accomplished reliable results. However, complex to perform CFA for some parameters and suffered from overestimation problem. The SEM [2] model ensured the reliability of the tool and attained better validity, but for large sample set, it was time consuming and it required more additional information. Even though, the TOC [3] model needed minimum calculations and data for quick effective utilization approach and revealed the efficiency of TOC, it was time consuming and for effective utilization approach, it needed complex calculations and data. The supply System [4] model attained minimum losses and provided improved performance, yet it was complex to implement a random supply system and required high computational time. The A2 and A3 [5] model acquired low cost model and provided an accurate degree of leanness still, the Crisp analysis limited the performance and Pair wise Weighted Matrix (PWM) was chosen depends on primitive evaluation. Although, the Fuzzy TOPSIS, fuzzy SAW and borda [6] model considered the welfare of the environment and ensured the improvement in productivity, it required expertise skills to use the system and was limited to financial resources. The AHP [7] model attained improved performance and revealed the efficiency of AHP, but it required complex calculations and provided poor improvements for healthcare and services sectors. The Fuzzy Logic [8] model achieved productivity maximization and accomplished minimum wastage. However, the operational complexity increased for multi-criteria decision making process and it was applicable only for a limited number of practices. Thus, the aforementioned limitations and complexities bother the efficiency of lean manufacturing practices in the industries and set the motivation to innovate and devise new effectual methodologies for productivity improvisation.

Table 1 Features and challenges of conventional models for lean manufacturing model by various methods

3 Proposed lean manufacturing process via structural equation modeling

3.1 Proposed architecture

In lean manufacturing, continuous improvement via productivity maximization and decision making plays a significant role. Thereby, this paper plans to devise a model for considering the analysis of lean manufacturing among the industries. With this intention, this paper is analyzed through three phases. In the first phase, the prepared questionnaire is distributed to the professionals in various companies. In the questionnaire, all the mandatory questions are included. Then, the professional are recommended to fill the precise information as far as possible. In the second phase, the responses from the concerned practitioners associated with the industries are considered for analysis. Herein, the analysis is carried out based on SEM approaches with the contribution of higher order statistical analysis, which is performed using the input factors such as the LT, LT, OS, OP, EI and MC among the industries via attaining better ML values of the questionnaires. Total of 42 questions are prepared under this 6 categories, and response for the same is obtained from 600 individual who are in various positions of different organizations. In fact, all these statistical computations can help to find the most important factor regarding the technology among the industries. In the third phase the reliability of prediction under each category is analyzed by diminishing the error in the polynomial expression, which is the objective of the current research work. This objective is achieved here by constructing a novel optimization algorithm referred as SLnO-EE model, which aids in enhancing the prediction accuracy. The detail description of LT, LT, OS, OP, EI and MC are as follows.

3.1.1 Lean awareness

Lean is a key Business Improvement technique to diminish waste and increase productivity. It has transformative effects on organizations around the world and across industries. As Lean keeps on developing in popularity, companies are increasingly searching for Lean awareness from workers.

3.1.2 Lean technology

Lean manufacturing or lean production is a systematic method originated in the manufacturing industry for reducing the waste within a manufacturing system without the loss in productivity, which can cause issues.

3.1.3 Organizational support

Perceived Organizational Assistance (POS) is how representatives recognize that their organization respects their contributions and cares about their prosperity and meets socio-emotional needs.

3.1.4 Organizational performance

Organizational performance includes the actual output or results of an organization as measured against its planned outputs.

3.1.5 Employee involvement

Employee involvement is a philosophy practiced by companies that allow the employees to systematically give their input into decisions that directly affect their own work.

3.1.6 Management commitment

Management commitment is direct participation by the most elevated level officials in a particular and critically significant aspect or program of an organization (Fig. 1).

Fig. 1

Geometrical representation of implemented lean manufacturing model

3.2 Structural equation modeling

In this section, the mathematical model of SEM approach can be given by Eq. (1), in which $x$ specifies the vector of observable indicators of $\mu$, $\Lambda^{x}$ represents factor loading matrix of $x$, $\mu$ points to the vector of latent variables, and $\varepsilon$ indicates error.

$$x = \Lambda^{x} \mu + \varepsilon$$

(1)

Here, the $\mu$ value is calculated by Eq. (2), in which $\mu$ signifies latent variables with latent endogenous constructs, $v$ points to latent exogenous constructs, $G$ represents coefficient matrix demonstrates the effects of endogenous constructs with one another, $\Gamma$ portrays coefficient matrix expressing the effects of exogenous and endogenous constructs, and $\xi$ indicating errors in equations.

$$\mu = G\mu + \Gamma v + \xi$$

(2)

Now, the $v$ value is derived using Eq. (3), in which $\Lambda^{v}$ indicates factor loading of $v$, and $S$ denotes error in measurements.

$$v = \Lambda^{v} v + S$$

(3)

Moreover, the fit indices employed for SEM analysis are path estimates, Chi-square, DF, Probability (P), chi-square statistics (CMIN/DF), GFI, SRMR, NFI, RFI, IFI, TLI, CFI, RMSEA, and RMR. Chi-square $\gamma^{2}$ is a basic metric for fit estimation among several fit metrics. Perceptually, it can be defined as a function of samples length as well as variance among observed covariance matrix with the model covariance matrix. DF is utilized to verify the model as portrayed in Eq. (4), in which $m$ indicates number of samples and $n$ denotes number of different parameters.

$$DF = D = m - n$$

(4)

CMIN shows the minimum value of discrepancy $\gamma^{2}$ and CMIN/DF is $\frac{{\gamma^{2} }}{DF}$. For best results, the ratio must be near to 1 and the values from 2 to 5 are reasonable. In RMSEA, the best fit is indicated by obtaining 0 values and the value below 0.05 are satisfactory and the values 0.08 or less is reasonable. Moreover, the values like 0.1 or above denote poor fit. NFI evaluates the discrepancy among the value of chi-squared model and chi-squared null model. The values below 0.09 signify the best fit as stated in Eq. (5), in which the normed fit index or $\Delta_{1}$ in the notation of Bollen, $\overline{D} = h\overline{E}$ refers to the minimum discrepancy of the model to be analysed and $\overline{D}_{g} = h\overline{E}_{g}$ indicates minimum discrepancy of the baseline model.

$$nfi = \Delta_{1} = 1 - \frac{{\overline{D}}}{{\overline{D}_{g} }} = 1 - \frac{{\overline{E}}}{{\overline{E}_{g} }}$$

(5)

P portrays the probability of obtaining greater discrepancy achieved among the current samples and represented as P value that denotes exact model in the population. TLI is also referred as NNFI and ranges between 0 and 1. Besides, the value near to one specifies best fit. The value of CFI is calculated based on Eq. (6), in which $\overline{D}$ indicates discrepancy, $k$ denotes degree of freedom and $ncp$ signifies non centrality parameter measure of the model to be analysed. Alternatively, $\overline{D}_{g}$ indicates discrepancy, $k_{g}$ represents degree of freedom, and $ncp_{g}$ points to non centrality parameter measure of baseline model. Usually, the values of CFA near to 1 denote best fit.

$$cfi = 1 - \frac{{\max \left( {\overline{D} - k,0} \right)}}{{\max \left( {\overline{D}_{g} - k_{g} ,0} \right)}} = 1 - \frac{ncp}{{ncp_{g} }}$$

(6)

GFI is exploited for ML and ULS evaluation. The values below or equal to one represent good fit. Particularly, the value 1 reveals best fit. SRMR express an absolute fit index and the best fit can be below 0.08 or 0.1. RFI compares chi-square value of hypothesized model and null model, which should be below 0.9. IFI is similar to RFI. RMR states the square root of discrepancy among a sample and model covariance matrix. Furthermore, the various fitness indices utilized to evaluate the lean manufacturing process are discussed. Most particularly, the analysis accomplished a covariance based SEM model. To test the proposed hypotheses, the study applied SEM technique ML estimation. Specifically, the study conducted a covariance based SEM technique. This covariance based SEM is preferable when the researcher tries to test and confirm the proposed model under consideration. The SEM technique has applied to test the study hypotheses because of several reasons. First, the constructs proposed in this study are measured using multiple questions, and therefore the constructs are latent in nature. In this case, SEM technique is more preferable over other techniques, if the objective is to capture the interrelationship between the proposed constructs under consideration. Second, in the hypotheses formulation section, the researcher proposed several relationships, which involved the interrelationship between several variables, in a simultaneous fashion. Finally, it also recommended that SEM is more useful in case of latent variables with multiple items, where the researcher like to capture item-wise error rate. The SEM modeling involves different phases. In the first phase, the researcher made a diagrammatic representation of the relationship between the study variables. In this, it is conceptualized that LA is the exogenous variable, leading to the other endogenous variables, MC and employee involvement. This endogenous variable has connected to lean technology. Further, LT has connected to organizational support. Finally, OS is linked to organizational performance.

In the second phase, we estimated the goodness of fit coefficients of the conceptual model and examined the goodness of fit of the model with the observed data. In this goodness of fit indices, the researcher examined various fit indices. In case of fit indices, there exists a confusion among researchers as to which fit indexes to report. Jaccard and Wan (1996) recommend the use of at least three fit tests, one from each of the first three categories like absolute fit, relative fit and parsimony measures. From the examination of the fit measures, the study found that the model proposed fit well with the collected data.

4 Hybrid optimization insisted error minimization in polynomial fitting

4.1 Objective model

In this paper, the proposed lean manufacturing is implemented using SEM. Here, the objective is to attain maximum prediction reliability by means of diminishing the fitness (i.e., error), so as to enhance the performance and revenue. The objective function is expressed mathematically in Eq. (7), in which $f$ is the sum-squared error function. The mathematical formula for fitness $f$ is expressed in Eq. (8), where $C_{1}$ is the predicted value acquired from the Polynomial expression given in Eq. (9). Here, $a_{0,1,2,...n.}$ are the unknown coefficients of the polynomial expression that needs to be optimized between 0 and 1.

$$Obj = Min\left( f \right)$$

(7)

$$f = \frac{{sum\left( {abs\left( {C_{1} - t\arg et} \right)} \right)^{2} }}{2}$$

(8)

$$C_{1} = a_{0} + a_{1} x + a_{2} x^{2} + ... + a_{n} x^{n}$$

(9)

4.2 Solution encoding

The unknown coefficients $a_{0,1,2,...n.}$ of the polynomial expression need to be optimized between 0 and 1 for an ideal lean manufacturing system. Hence, they ($a_{0,1,2,...n.}$) are fed as input to the proposed optimization algorithm. Figure 2 shows the solution encoding of the unknown coefficients given as input to the proposed model. These unknown coefficients vary on the basis of the count of categories of questions.

Fig. 2

Schematic representation of proposed SEM implemetation in AMOS

For illustration: for question category 1 there is 5 questions and hence the count of unknown coefficient is 5 (i.e., from $a_{0,1,2,...5.}$) (Fig. 3).

Fig. 3

Solution encoding showing hidden neurons optimization

4.3 Traditional SLnO

The traditional SLnO algorithm was developed with the raw motivation acquired from the hunting behavior of Sea lions [26]. The sea lion are gifted with certain fasctianting features like speedy movement, improved hunting property and lucid vision. Further, Whiskers is the super sensitive feature of sea lion that helps in tracking as well as searching the prey. Moerover, these whiskers also aids in defining the position, shape, and size of prey. The major phases of the hunting behavior of sea lions are depcited below:

• Prey tracking as well as chasing with the aid of their whiskers.

• Following the prey and encircling it by means of calling other sea lions that belong to its subgroup.

• Prey attacking

Mathematical model: there are four major phases in the SLnO algorithm, they are tracking, social hierarchy, attacking and encircling prey.

Phase 1—Prey detection and tracking: the whiskers of sea lion help them to sense the existing prey and determine their position, while the direction of whiskers is opposite to the direction of the water waves. They have the ability of identifying the location of the prey and then call the members in its sub-group to join along with it to chase and hunt the prey. For this mechanism of hunting this sea lion which calls others are said to be the leader and the others update their position with correspondence to the target prey. Here, the target prey is assumed to be the current best solution or close to optimal solution. This behaviour is mathematically explained as per Eq. (10). Here, $\overrightarrow {Di}$ indicates the distance between the target prey and the sea lion. In addition, the notation $\overrightarrow {M} (t)$ and $\overrightarrow {L} (t)$ indicates the vector position of sea lion and target prey, respectively. The current iteration and the random vector in the interval [0, 1] is denoted using the symbol $t$ and $\vec{J}$, respectively. Next to the current iteration, the sea lion moves to a position over the target prey to get much closer to the prey. The mathematical expression for this behavior of sea lion in the subsequent iteration is expressed in Eq. (11). The next iteration is symbolized as $(t + 1)$ and over the course of iterations, the value of $\vec{H}$ gradually lessens from 2 to 0.

$$\overrightarrow {Di} = \left| {2\overrightarrow {J} .\overrightarrow {M} (t) - \overrightarrow {L} (t)} \right|$$

(10)

$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {S} (t + 1) = \overrightarrow {M} (t) - \overrightarrow {Di} .\vec{H}$$

(11)

Phase 2—Vocalization: the sea lions have the ability of surviving both in land and in water (amphibians). In water, the sound of the sea lion is four times quicker, while compared to their sound in air. Further, numerous vocalization is being utilized by sea lions to communicate with others during chasing or hunting process. They also use their sound to call other members to stay on the shore. The sea lions have smaller ears to detect sounds both under and above water. Thus, when a sea lion tends to identify a prey, it calls the other members to join it for encircling and attacking the prey. The mathematical expression for this behaviour of sea lion is expressed in Eqs. (12), (13) and (14), respectively. The speed of sea lion is symbolized as $\overrightarrow {{LP_{leader} }}$ and the speed of their sound in water and air is indicated as $\overrightarrow {{R_{1} }}$ and $\overrightarrow {{R_{2} }}$.

$$\overrightarrow {{LP_{leader} }} = \left| {\left( {\overrightarrow {{R_{1} }} (1 + \overrightarrow {{R_{2} }} )} \right)/\overrightarrow {{R_{2} }} } \right|$$

(12)

$$\overrightarrow {{R_{1} }} = \sin \theta$$

(13)

$$\overrightarrow {{R_{2} }} = \sin \varphi$$

(14)

Phase 3—Attacking phase: Here, in the exploration phase, the sea lions get the ability of recognizing the location of the target prey as well as encircling them. The best search agent said to be the leader guide the hunting mechanism. The current candidate best solution is nothing but the target prey. In this phase, two major sub- phases like Dwindling encircling technique and Circle updating position are present.

Dwindling encircling approach: In Eq. (12), this encircling mechanism takes place in correspondence to $\vec{H}$ value. Further, over the course of iterations, the value of $\vec{H}$ _{is lessened} from 2 to 0. This lessening of $\vec{H}$ value aids the sea lion leader to move towards the prey and encircle them. Thus, the position of the incoming sea lion can be localized at any position in the search space in between the search agents premier location and present best agent.

Circle updating position: bait ball of fishes are searched by the sea lion and then start hunting from the edges. On the basis of this searching and hunting mechanism, Eq. (15) is defined. The distance in between the best optimal solution (target prey) and the search agent (sea lion) is symbolized as $\left| {\overrightarrow {M} (t) - \overrightarrow {L} (t)} \right|$. In addition, the absolute value and the random number in [-1, 1] is represented as $| \, |$ and $l$, respectively.

$$\overrightarrow {L} (t + 1) = \left| {\overrightarrow {M} (t) - \overrightarrow {L} (t).\cos (2\pi l)} \right| + \overrightarrow {M} (t)$$

(15)

Phase 4—Prey searching: with the help of the whiskers, the sea lions randomly search and swimming zigzagging to find prey. On the basis of the best search agent, the sea lions tend to update their position in the exploitation phase. On the other hand, with correspondence to the randomly selected sea lion, the position of the search agent is updated in the exploration phase. The global search in SLnO algorithm accomplished when the value of $\vec{H}$ is larger than 1. This process is expressed mathematically in Eqs. (16) and (17), respectively.

$$\overrightarrow {Di} = \left| {2\overrightarrow {J} .\overrightarrow {{L_{rnd} }} (t) - \overrightarrow {L} (t)} \right|$$

(16)

$$\overrightarrow {L} (t + 1) = \overrightarrow {{L_{rnd} }} (t) - \overrightarrow {Di} .\vec{H}$$

(17)

4.4 Proposed algorithm

The traditional SLnO algorithm still requires improvement in position update at the exploration phase and so a novel algorithm called Sea Lion with Enhanced Exploration phase (SLnO-EE) is developed here. The steps involved in the proposed SLnO-EE model are described below:

• Initially, two random numbers $r_{1}$ and $r_{2}$ are initialized.

• Generally, in the exploration phase (i.e., attacking phase), the traditional SLnO updates the position of the current search agent using the mathematical expression depicted in Eq. (17). As a controversy to this, the proposed SLnO-EE algorithm updates the position of the current search agent using Eq. (18).

  $$\begin{gathered} \overrightarrow {S} (t_{1} :) = x(t_{1} :) + r_{1} * (best - abs\left( {x(t_{1} :)} \right) - \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - r_{2} * (worst - abs\left( {x(t_{1} :)} \right) \hfill \\ \end{gathered}$$

  (18)

Where, $r_{1}$ and $r_{2}$ are the two random numbers. The best and the worst solution are indicated as $best$ and $worst$, respectively. The pseudo-code of the proposed SLnO-EE model is shown in Algorithm 1. The flow chart of the proposed SLnO-EE is exhibited in Fig. 4.

Fig. 4

Flow chart of proposed SLnO-EE

5 SEM investigation with CFA, EFA and regression analysis

5.1 CFA analysis

Usually, CFA is a unique way of the factor analysis model, employed for social research. It is mainly utilized to verify whether the metrics of the construct are reliable to the scholars understanding regarding the properties of the construct. Here, the CFA analysis explains the measurement model testing results performed to confirm the validity and reliability of the scale measure. This measurement model is necessary before the test of hypotheses. In this paper, this measurement model testing was conducted through CFA. Validity explains the ability of an instrument (here the scales) to measure what the researcher intended to measure. Similarly, the reliability of an instrument examines the extent of random error in the scale measures. In this stage, study decided to perform two types of validity. First type validity called as convergent validity. This type of validity explains the extent of convergence among the items covering the respective dimension. Similarly, the discriminant validity explains the extent of divergence between the items covering the different dimensions. At this stage, the study analyzed the reliability of the study measurements using a coefficient, called as Composite Reliability (CR).

5.2 EFA analysis

Generally, EFA is utilized to reveal the primary structure of the quite huge set of variables. It is an approach in factor analysis with the aim of discovering the relationships among measured variables. Moreover, it is usually employed by the scholars and research communities while proposing a scale (group of questions exploited to estimate a specific research topic) and helps to discover a group of latent constructs. It is employed while the researchers have no priori constructs regarding the factors of measured variables. Typically, huge set of measured variables is considered for the smaller set of unobserved factors. For this reason, the researchers have to assume the measured variables set to consider in the study. Moreover, fitness procedures can be exploited to calculate the factor loadings as well as specific variances. Factor loadings can be defined as the regression coefficients among items and factors as well as estimate the impact of normal factor in a measured variable. Among the numerous factor analysis approaches, ML is most significant and is implemented in this model. Specifically, ML provides better outcomes, based on the data, whether it is normally distributed or the normality is violated. Moreover, ML permits the researchers to estimate a huge set of indexes for the goodness of fit as well as verify the statistical importance of factor loading, and estimate correlations between factors.

5.3 Regression analysis

It can be defined as a group of statistical methods to calculate the relationships between variables. It comprised of several methods to model and evaluate numerous variables, while concentrating the relationship between a dependent variable with one or many independent variables. Most particularly, it assists in the interpretation of the working process of the variation in the dependent variable values with respect to changes in any 1 of the independent variable, among all independent variables. The mathematical model of regression analysis is given in Eq. (19), where $X$ is dependent variable, $Y$ is independent variable and $\chi$ are unknown parameters.

$$X \approx f\left( {Y,\chi } \right)$$

(19)

Herein, the estimation is commonly given as $E\left( {X|Y} \right) = f\left( {Y,\chi } \right)$. In order to perform the regression, $f$ should be indicated.

5.4 Cronbach’s alpha analysis

In this section, the results of Cronbach’s Alpha in terms of reliability statistics are discussed. Table 2 shows the reliability statistics under EI. Here, Cronbach’s alpha attained 0.883 and Cronbach's Alpha Based on Standardized Items gets 0.884 for 7 items. For MC, Cronbach’s alpha attained 0.953 and Cronbach's Alpha Based on Standardized Items gets 0.954 for 8 items.

Table 2 SEM results for fit indices

6 Results and discussions

6.1 Experimental setup

The proposed lean model was analyzed in AMOS, after providing the attained raw data from the professionals based on the questionnaires. The analytical results are focussed on the drivers like LA, LT, OS, OP, EI and MC and attain the results based on path estimates, Chi-square, DF, P, CMIN/DF, GFI, SRMR, NFI, RFI, IFI, TLI, CFI, RMSEA, and RMR and attained the observed results. In addition, the proposed lean model with SLnO-EE was implemented in MATLAB and the resultants acquired were recorded. Here, the achievement of the objective with SLnO-EE is analysed over traditional SLnO in terms of three types of performance measures, viz. positive, negative and others. The optimal polynomial expression acquired from optimized unknown coefficients for existing SLnO and SLnO-EE are depicted in Table 3. Here, from the table it is clear that most optimized unknown coefficients are achieved with the proposed SLnO-EE model, while compared with existing SLnO. Question set 2, 4 and 5, encloses 8 questions and hence 8 unknown coefficients (from $a_{0,1,2,...8.}$). Then, question set v3 and 6 have 9 questions and hence 9 unknown coefficients (from $a_{0,1,2,...9.}$).

Table 3 POlynomial Expression with optimized unknown Coefficints

6.2 SEM analysis

The proposed lean model attained better results and the betterment is discussed in this section. Table 2 shows the actual and recommended values for the fit indices attained through SEM analysis. For Chi-square and DF, the values attained are 1086.02 and 582 respectively. For P, it gets 0.0 which is better than the recommended value 0.05. For CMIN/DF, it gets 7% better than the recommended value. For, RMR, the actual values are 20% superior to recommended value. In case of GFI, it attained 3.88% superior to recommended values. For SRMR, it gets 9% better than recommended value. From all the fit indices, it obtained superior result and thus it proved its efficiency.

Here, Table 4 tabulates the path estimates of SEM model. Further, the study examined the path coefficients to test the proposed set of study hypotheses. While checking the path coefficients, it was found that that all the estimated path coefficients followed the researchers’ expectation. As reported in Table 5, it provides the estimated path coefficients derived from the model. In this table, the first and second columns show the relationship between the exogenous and endogenous constructs. The third column reported the unstandardized path coefficients. Next, to standardized path estimates are provided. In the final column, the table reported the p values associated with the test values. From the results, the study found support for all the path estimates.

Table 4 Path estimates

Table 5 Results of regression analysis

6.3 Regression analysis

In this subsection, the results of regression analysis are discussed. Table 6 shows the results of regression analysis and the ML between two factors such as LA $\to$ $\to$ EI, LA $\to$ MC, EI $\to$ LTECH, MC $\to$$\to$ LTECH, and OS $\to$$\to$ OP. Here, five models are represented with its corresponding results. Values inside the cell shows unstandardized regression coefficients of stepwise regression. *_P < . 007 items. Similarly, for all analytical parameters, the proposed model accomplished better results and thus proved its significance.

Table 6 Reliability statistics under the input factors

6.4 EFA factor loading analysis

In EFA factor loading analysis, a total of 42 questions are analysed under the factors LA, LT, EI, MC, OS and OP. Now, LA is having 5 questions, LT, EI and OP having 7 questions, MC and OS is having 8 questions are considered for analysis. Table 7 summarizes the EFA factor loading and its corresponding results.

Table 7 EFA factor loading analysis

6.5 CFA analysis

In the section, the study analysed the reliability of the study measurements using CR and the study provided the measurement model testing table derived from CFA analysis and its interpretations. Table 8 tabulates the unstandardized CFA factor loadings. In this study, the researcher tested the convergent validity of the scale measures using two different measures. The first measure used is CFA factor loadings. It was recommended that if the CFA factors loading are above 0.50 and loaded high, then it indicates the evidence of convergent validity. In addition, it is also suggested that if the CR values are above 0.50 it also informs the evidence of convergent validity. Moreover, the notation (***) shows significant at 0.01 level. In the current study, the researcher used these two measures for the purpose of checking the same. Figure 4 shows the observed SEM results under the evaluation parameters with the attained values (Fig. 5).

Fig. 5

Attained SEM results

Table 8 Unstandardized CFA factor loadings

6.6 Performance evaluation

Positive measures: The prediction performance of the proposed SLnO-EE is analysed over the traditional SLnO for each of the 6 categories of questions and the resultant acquired (in %) are shown graphically in Fig. 6. Generally, the advanced lean manufacturing model with SEM using the SLnO-EE model is said to be more reliable in prediction, when it is free from errors or has the lowest errors. The positive performance evaluation is accomplished in term of positive measures like accuracy, specificity, sensitivity and precision. These positive measures need to be higher for as typical lean manufacturing model. Initially, from Fig. 6a, relating to 4th set of question, the proposed SLnO-EE model is 5.26% better than the traditional SLnO in terms of accuracy. In case of sensitivity from Fig. 6c, the proposed SLnO-EE model is 5.2% better than tradition SLnO model at 4 th set of question. Similar to this, the other positive measures (Precision and Specificity) for SLnO-EE are also ahead of traditional SLnO used in lean manufacturing model with SEM. Thus, from the resultant it is obvious that the proposed SLnO-EE model is better than the traditional SLnO model in case of positive performance.

Fig. 6

Positive performance evlaution of proposed SLnO-EE over SLnO showing a Accurcay, b Precision, c Sensitivity and d specificity

Table 9 exhibits the overall performance of the proposed SLnO-EE over SLnO by varying the learning Percentage (LP) from 30, 40, 50, 60 to 80 for each questions. From the Table, it is vivid that the reliability of SLnO-EE increases with higher learning percentage (80). For 1^st set of question, the accuracy of the SLnO-EE at LP = 80 is 0.95833, which is 7.3%, 6.8%, 5.3% and 3.6% better than LP = 30, LP = 40, LP = 50, LP = 60 and LP = 80, respectively. Then, the sensitivity of SLnO-EE is maximum for 1st set of question at LP = 80, which is 1.4%, 0.5%, 0.6% and 0.7% superior to LP = 30, LP = 40, LP = 50, LP = 60 and LP = 80, respectively. Similar to this, in other positive measures concerning different questions, the proposed SLnO-EE is higher at higher learning percentage.

Table 9 Overall performance evaluation of proposed SLnO-EE over SLnO by varying learning percentage: positive measures

Negative measures: Figure 7 exhibits the performance evaluation (in %) of the proposed SLnO-EE model over the existing SLnO model in terms of negative measures like FDR, FNR and FPR. The advanced lean manufacturing model with SEM using SLnO-EE model is said to have achieved the objective function, when it has the lowest negative measures. In Fig. 7a, FDR of the proposed SLnO-EE model for 5th set of question is 14.28% better than the traditional SLnO model. Then, for 2nd set of question in Fig. 7b, the proposed SLnO-EE model is 83.3% better than existing SLnO model in terms of FNR. There is an improvement of 20% in the proposed SLnO-EE model over the existing SLnO model in terms of FPR at 3rd set of question in Fig. 7c. Therefore, the resultant of the evaluation clearly exhibited the enhancement of the proposed SLnO-EE model over the extant SLnO model in terms of negative measures.

Fig. 7

Negative performance evlaution of proposed SLnO-EE over SLnO showing a FDR b FNR and c FPR

The overall performance of SLnO in terms of negative performance measures for varying LP is tabulated in Table 10. In case of 2nd set of question, FPR of SLnO-EE at LP = 80 is 0.33333, which is 30.5%, 31%, 28.5% and 15.15% better than LP = 30, LP = 40, LP = 50, LP = 60 and LP = 80, respectively. Similarly, for FDR in 2nd set of question, the lowest value (0.067568) is achieved at LP = 80 and it is 59.13%, 57.7%, 50.1% and 37.9% better than LP = 30, LP = 40, LP = 50, LP = 60 and LP = 80, respectively. Here, the FPR, FDR and FNR achieve the lowest values at higher learning percentage and here said to be more accurate.

Table 10 Overall performance evaluation of proposed SLnO-EE over SLnO by varying learning percentage: negative measures

Other measures: the other measures include MCC and F1-score, which need to be higher to enhance the reliability of the prediction. Figure 8 exhibits the graphical representation (in %) of the SLnO-EE over SLnO for other measures (MCC and F1-Score). In case of F1-score from Fig. 8a, the proposed SLnO-EE model is 2.1% better than the state-of-art models at 4 th set of question. The overall performance of SLnO-EE for varying count of LP for MCC and F1-Score is tabulated in Table 11. Here, MCC achieves the highest value 0.98378 in 3rd set of question at LP = 80 and the resultant acquired is 6.5%, 6.8%, 6.3% and 5.4% better than LP = 30, LP = 40, LP = 50, LP = 60 and LP = 80, respectively. Then, for the same set of question (i.e. 3rd set of), the proposed SLnO, the highest MCC (0.91707) is achieved at LP = 80, which is 12.95, 12.03%, 9.9% and 7.2% superior to LP = 30, LP = 40, LP = 50, LP = 60 and LP = 80, respectively. Thus, the resultants of the evaluation exhibit the enhancement in the performance of SLnO-EE for distinctive question at varying LP’s.

Fig. 8

Other performance evlaution of proposed SLnO-EE over SLnO showing (s) F1-Score and b MCC

Table 11 Overall performance evaluation of proposed SLnO-EE over SLnO by varying learning percentage: others

7 Conclusion

In this paper, an advanced lean manufacturing model with SEM using SLnO-EE model is introduced. Moreover, the proposed model was analysed through two phases. Firstly, a questionnaire was prepared and distributed to the professionals in various companies. All the significant questions were mentioned in the questionnaire. After that, the professional were asked to fill the exact data as far as possible. Secondly, the responses from the concerned practitioners associated with the industries were considered for analysis. Now, the analysis was carried out based on SEM technique with the contribution of higher order statistical analysis, which was performed using the Regression analysis, EFA, regression and Cronbach’s Alpha. In fact, all these statistical computations can help to find the most important analytical parameters regarding the lean manufacturing. For Chi-square and DF, the values attained are 1086.02 and 582 respectively. For P, it gets 0.0 which is better than the recommended value 0.05. For CMIN/DF, it gets 7% better than the recommended value. Moreover, Cronbach’s alpha attained 0.883 and Cronbach's Alpha Based on Standardized Items gets 0.884 for 7 items. For MC, Cronbach’s alpha attained 0.953 and Cronbach's Alpha Based on Standardized Items gets 0.954 for 8 items. Then, in case of positive performance evaluation (accuracy, sensitivity, specificity, precision, F1-Score and MCC) with SLnO-EE model, the proposed SLnO-EE model had overridden the existing SLnO model. The proposed SLnO-EE model is 5.26% better than the traditional SLnO in terms of accuracy for the 4 th set of question. Simultaneously, the SLnO-EE model is ahead of traditional SLnO models in terms of negative measures (FPR, FNR and FDR). In case of 2nd set of question, FPR of SLnO-EE at LP = 80 is 0.33333, which is 30.5%, 31%, 28.5% and 15.15% better than LP = 30, LP = 40, LP = 50, LP = 60 and LP = 80, respectively. Hence, the proposed lean model acquired better results and confirmed its applicability through higher prediction accuracy and lower errors.