An adaptive neuro-fuzzy inference system for makespan estimation of flexible manufacturing system assembly shop: a case study

Abstract

This paper considers the use of combination of neural networks and fuzzy system i.e. adaptive neuro-fuzzy inference system (ANFIS) applied to the n job, m machine real flexible manufacturing system assembly shop problem with the objective of prediction of makespan. Assembly shop makespan is calculated by Nawaz, Enscor, and Ham (NEH) algorithm. On the basis of this algorithm, adaptive neuro-fuzzy inference system model is made to predict the makespan of the jobs. The purpose of this study is to find the makespan estimation in advance if processing time of machines is known. The purpose of this research is to gain the advantage of the capabilities of both Fuzzy systems, which is a rule-based approach and neural network which focus on the network training. This model has been verified by testing and actual data set with the average percentage accuracy achieved is 95.97%. Coefficient of determination and Correlation coefficient is 0.9310 and 0.9649 respectively. The derived values of ANFIS model output are found within the range after being verified practically. Therefore, it can be concluded that makespan calculation of the production system, by the proposed adaptive neuro-fuzzy inference system, can be used as a reliable approach in estimating the makespan of flexible manufacturing system assembly shop.

1 Introduction

Many large industries have tried to introduce flexible manufacturing systems in today’s manufacturing environment as their strategy. It enables them to adapt to the ever-changing competitive market requirements based on quality of machining products, and to reduce the machining costs and to enhance the productivity (Cus and Balic 2003). Flexible manufacturing systems (FMSs) have been developed with the hope that they will be able to tackle new challenges like reduced cost, improved quality, improve delivery speed to satisfy different market segments (Jain and Raj 2016a). A flexible manufacturing system assembly shop schedule is one in which all jobs must visit all machines in the same sequence. Processing of the job should not be started on a succeeding machine before completing processing of a job on a current machine. Although all jobs are available in the beginning but only one job can be performed at any particular time by a machine (Onwubolu 1996). The other machines are left idle queued by other jobs because the first machine has to visit first by each job. Although queuing of jobs is prohibited in just-in-time manufacturing environments, production flow-shop manufacturing continues to find applications in manufacturing (Wittrock 1985), and has attracted much research work (Campbell et al. 1970; Gupta 1972; Nawaz et al. 1983).

An important aspect of scheduling is sequencing. The sequencing is the process in which order jobs visit a machine. Johnson (1954) Johnson’s algorithm is apt for a two-machine problems and can be applied on three-machines. A generalization of Johnson’s algorithm is that proposed by Campbell et al. (1970) for solving general n-jobs m-machine problems in which m − 1 two-machine problems are solved and the sequence having the least makespan is selected. Nawaz et al. (1983) proposed a Nawaz, Enscor, and Ham (NEH) algorithm to construct a jobs sequence in an iterative manner. The production flow shop scheduling of assembly problem is the problem of defining order over a set of jobs as they proceed from one machine (processor) to another in minimum time i.e. makespan of the jobs or assembly.

Scheduling outputs are generally graphically displayed by Gantt charts. Machine processing times for each job is used to draw them. It is also ensured that delay times are taken into consideration.

A minimum makespan, which represents the minimum time required to complete all the jobs, if not found, this process is repeated for different sequences. The obtained sequence is considered to be optimal. The manual method for scheduling is tedious and prone to error. So, soft computing technique is used to find the makespan of the production flow shop. The makespan of the jobs can be calculated by neuro and fuzzy system.

An adaptive neuro-fuzzy inference system (ANFIS) for makespan estimation of flexible manufacturing system assembly shop for five to ten jobs and five machines is presented by this research work. The manufacturing sequences of parts are flexible. Alternative sets of resources may be selected for a manufacturing operation. The characteristics such as resource sharing, concurrency, routing flexibility, mutual exclusion, lot sizes, and synchronization which are difficult to study (Der Jeng et al. 1999).

The main objectives of this research work are as follows:

• To find the makespan of the FMS assembly shop.

• To make a model with the help of neural network and fuzzy rules i.e. ANFIS model.

• To discuss the ANFIS model verification.

In the remainder of this paper, literature review is presented in Sect. 2 for makespan estimation, NEH heuristic, and ANFIS methodology. In Sect. 3, problem description. The Sect. 4 gives the NEH heuristic. Makespan calculation by NEH Algorithm is discussed in Sect. 5. The Sect. 6 gives the ANFIS methodology. Model verification and Conclusion are followed in Sects. 7 and 8 respectively.

2 Literature review

The literature has been reviewed from the perspectives of makespan estimation with neural network and fuzzy rules, NEH heuristic and ANFIS modeling. Cheng and Gupta (1989) used neural networks approaches for estimating the makespan. Yih et al. (1991) proposed a hybrid model to solve a crane scheduling problem. Philipoom et al. (1994) compared a non-linear regression analysis with neural networks of job scheduling problem. Fransoo et al. (1995) compare a makespan estimation based on the analysis of a stochastic queueing network model of the FMS and a makespan minimizing algorithm based on a combinatorial algorithm. Sabuncuoglu and Gurgun (1996) combined neural network and algorithmic approaches to solve the job-shop scheduling problem with minimum makespan. Chen and Muraki (1997) used back-propagation neural network for online rescheduling.Sabuncuoglu (1998) presented a review of the literature and future directions of scheduling approaches using neural network mainly scheduling problems involving artificial neural network (ANN) applications. Ivanescu et al. (2002) used regression analysis to estimate makespan in a batch process shop. Raaymakers et al. (2001) also estimated models based on regression. Raaymakers and Weijters (2003) found that in batch process industries, estimation of makespan is difficult because jobs interact at the shop floor. So, used two different techniques for estimating the makespan of job i.e. regression models and neural networks. Wilson et al. (2004) estimated the minimum makespan for scheduling non similar groups of jobs on a two-stage flow line. Akyol (2004) used ANN models for the prediction of the completion times for each job processed on each machine. Li et al. (2007) proposed a back-propagation network model combined with genetic algorithms for estimation of makespan. Ahmadizar et al. (2010) found a job schedule which minimizes the expected makespan based on ant colony optimization algorithm and a heuristic algorithm. Shokrollahpour et al. (2011) discussed two-stage assembly flowshop scheduling problem with minimisation of weighted sum of makespan and mean completion time by imperialist competitive algorithm. Verma et al. (2012) designed a job schedule that minimizes the makespan. González et al. (2013) tackled the job shop scheduling problem with sequence dependent setup times and maximum lateness minimization by means of a tabu search algorithm. Moradinasab et al. (2013) discussed no-wait two-stage flexible flow shop scheduling problem with setup times aiming to minimize the total completion time by adaptive imperialist competitive algorithm and genetic algorithm. NEH heuristics review as Taillard (1990) Compare the NEH heuristic with taboo search algorithm. Zheng and Wang (2003) used NEH algorithm for flow shop scheduling. Kalczynski and Kamburowski (2007) used NEH algorithm for minimizing the makespan in permutation flow shops. Kalczynski and Kamburowski (2008) used improved NEH algorithm to minimize makespan in permutation flow shops. Dong et al. (2008) also used improved NEH algorithm to minimize makespan in the permutation flow shops. Yagmahan and Yenisey (2008) used NEH algorithm to compare ant colony optimization for multi-objective flow shop scheduling problem. Shafaei et al. (2011) used NEH algorithm with an adaptive neuro fuzzy inference system for estimating the makespan.

In this section, the study of ANFIS work are summarize as the used by other researchers. Some of review as Mar and Lin (2001) defined an ANFIS controller for the car-following collision prevention system. Ho et al. (2009) used an ANFIS to predict the work piece R_a for the end milling process with the hybrid Taguchi-genetic learning algorithm. Samanta (2009) used adaptive neuro-fuzzy inference system for a surface roughness model in end milling with genetic algorithms. Talei et al. (2010) evaluated rainfall by ANFIS in rainfall–runoff modeling. Güneri et al. (2011) used ANFIS model for supplier selection. Mellit and Kalogirou (2011) used ANFIS model for photovoltaic power supply system. Shafaei et al. (2011) used an adaptive neuro fuzzy inference system to solve a no-wait two stage flexible flow shop for minimizing makespan. Heddam et al. (2012) studied an adaptive neuro fuzzy inference system based modelling for coagulant dosage in drinking water treatment plant. Pousinho et al. (2012) proposed an adaptive neuro fuzzy inference system approach for electricity prices forecasting in a competitive market. Chen (2013) developed a hybrid ANFIS model for business failure prediction by utilizing particle swarm optimization and subtractive clustering. Heddam (2014) made a ANFIS model for hourly dissolved oxygen concentration by using two different adaptive neuro-fuzzy inference systems. Chen et al. (2014) proposed ANFIS for an active magnetic bearing system with unbalance mass. Ay and Kisi (2014) used modelling of chemical oxygen demand by using ANNs, ANFIS and k-means clustering techniques. Özkan and İnal (2014) determined that ANFIS algorithm can be used in multi-criteria decision making problems for supplier evaluation and selection with more precise and reliable results. Maher et al. (2014) Investigated the effect of machining parameters on the surface quality of machined brass (60/40) in CNC end milling by ANFIS modeling. Çevik and Çunkaş (2015) presented a short-term load forecasting models, which was developed by using fuzzy logic and ANFIS. Vasileva-Stojanovska et al. (2015) presented a Quality of Experience prediction model in a student-centered blended learning environment, equipped with appropriate technologically enriched classroom.Framinan and Perez-Gonzalez (2015) used heuristic solutions for the stochastic flowshop scheduling problem. Maher et al. (2015) made a ANFIS model based on cutting force for accurate surface roughness prediction in end milling operation for intelligent machining.Azadeh et al. (2015) used a hybrid computer simulation-adaptive neuro-fuzzy inference system algorithm for optimization of dispatching rule selection in job shop scheduling problems under uncertainty. Abdulshahed et al. (2015) applied ANFIS as a prediction models for thermal error compensation on CNC machine tools. Jung and Choi (2015) ANFIS method was used to predict the composite suitability index for the physical habitat simulation of a 2.5 km long reach of the Dal river in Korea. Jain and Raj (2016b) used ANFIS for tool life management for unmanned production system.

From the literature we have found that researchers focused on optimization or minimizing the makespan and have not discussed estimation of makespan which is necessary for good scheduling, product delivery. So, a model for prediction of makespan is developed which is helpful to any manufacturing system to maintain good scheduling system internally to get reliable product delivery.

3 Problem description

The production shop of flexible manufacturing system assembly shop problem formulated as given below. Each of n jobs from the jobs set i = [1,2….,n], for n > 1, has to processed on m machine j = [1,2,….m] in the order given by the indexing of the machine being t_i,j to find the minimum makespan and make a model to predict or estimate the makespan of the assembly jobs.

The following assumptions are considered in this problem:

• 1. All jobs are independent and available at zero time.

• 2. Machines are also available at zero time.

• 3. Processing time of jobs is formerly specified.

• 4. No job has priority over any other job.

• 5. The transportation time between machines and set up time are included in the processing time.

• 6. Assembly of parts is also included in the processing time.

• 7. One job can only be processed on one machine at a time.

• 8. One machine can only process one job at a time.

• 9. No preemption is allowed, i.e. the processing of a job i on a machine j cannot be interrupted.

In this study, the operations set-up times are assumed to be independent of the job sequences, and hence is added to the operation times. The performance of the proposed heuristic algorithm is studied in terms of minimum makespan.

Here, taking a case study of flexible manufacturing system assembly shop. This is the case of a large multi nation organization X engaged in the manufacture of a wide variety of automobile components in India, with an estimated turnover of Rs. 350 crores per year. That is one of the largest automobile component supplier in the country. The product range includes different car manufacturing company like Maruti Suzuki, Hyundia, Honda, Toyota etc. with different models. The organization has to increase the good quality and supply the product with variations of models with minimum time frame.

So, a model is prepared to predict the makespan of the components with different variants (i.e. five to ten jobs) on five machines or workstations including machining and assembly processes. A sample assembly shop line is shown in Fig. 1. The final assembly is completed to pass five machines or workstations including machining and assembly process.

Fig. 1

Five machine FMS assembly shops

In this research work, the framework of the proposed ANFIS-based soft computing intelligent system is described in the ANFIS methodology section for consisting of five machines which are capable of handling a five to ten numbers of jobs.

4 NEH algorithm

An overview of the NEH algorithm can be stated as follows.

Step l:

Calculate total process times for each job i

$$ T_{i} = \sum\limits_{j = 1}^{j = m} {t_{i,j} } $$

(1)

where t_i,j is the process time of job i on machine j.

Step 2:

The jobs are arranged according to descending order of total processing time T_i.

Step 3:

The two jobs are picked from the first and second position of the list of Step 2, and the best sequence is found for these two jobs by calculating makespan for the two possible sequences. The relative positions of these two jobs should remain same with respect to each other in the remaining steps of the algorithm. Set i = 3.

Step 4:

Next the job is picked in the ith position of the list generated in Step 2 and the best sequence is found by placing it at all possible i positions in the partial sequence found in the previous step without changing the relative positions to each other of the already assigned jobs. The number of enumerations at this step equals i.

Step 5:

If n = i, then STOP, otherwise set i = i + 1 and go to Step 4.

5 Makespan calculation by NEH algorithm

Considering 5 machine and 5 jobs for calculation of makespan by NEH algorithm (see Table 1).

Table 1 Processing time of Jobs

Step 1:

Calculate total process times for each job i (see Table 2)

Table 2 Total processing time of individual Jobs

Step 2:

Sort in the decreasing order of processing times (see Table 3)

Table 3 Descending order of Jobs based on total processing time

Step 3:

Take J₄ & J₃

Iteration 1

Possible combinations: J₄–J₃ & J₃–J₄.

For J₄–J₃ (see Table 4):

Table 4 Makespan for partial sequence of 4-3 Jobs

where c is makespan

For J₃–J₄ (see Table 5):

Table 5 Makespan for partial sequence of 3-4 Jobs

C_max for J₄–J₃ < J₃–J_4, therefore we choose J₄–J₃.

Step 4:

Then we take the next job in the sequence i.e., J₂

Now J₂ can be squeezed in three ways i.e., J₂–J₄–J₃, J₄–J₂–J₃, J₄–J₃–J₂

Iteration 2

For J₂–J₄–J₃ (see Table 6):

Table 6 Makespan for partial sequence of 2-4-3 Jobs

For J₄–J₂–J₃ (see Table 7):

Table 7 Makespan for partial sequence of 4-2-3 Jobs

For J₄–J₃–J₂ (see Table 8):

Table 8 Makespan for partial sequence of 4-3-2 Jobs

C_max for J₄–J₃–J₂ < J₄–J₂–J₃, J₂–J₄–J₃ J₃–J_4, therefore we choose J₄–J₃–J₂.

Step 5:

Then we take the next job in the sequence i.e., J₁

Now J1 can be squeezed in 4 ways i.e., J₁–J₄–J₃–J₂, J₄–J₁–J₃–J₂, J₄–J₃–J₁–J₂, J₄–J₃–J₂–J₁.

Iteration 3

For J₁–J₄–J₃–J₂ (see Table 9):

Table 9 Makespan for partial sequence of 1-4-3-2 Jobs

For J₄–J₁–J₃–J₂ (see Table 10):

Table 10 Makespan for partial sequence of 4-1-3-2 Jobs

For J₄–J₃–J₁–J₂ (see Table 11):

Table 11 Makespan for partial sequence of 4-3-1-2 Jobs

For J₄–J₃–J₂–J₁ (see Table 12):

Table 12 Makespan for partial sequence of 4-3-2-1 Jobs

C_max for J₄–J₃–J₂–J₁ < J₁–J₄–J₃–J₂, J₄–J₁–J₃–J₂, J₄–J₃–J₁–J₂, therefore we choose J₄–J₃–J₂–J₁.

Step 6:

Then we take the next job in the sequence i.e., J₅

Now J₅ can be squeezed in 5 ways i.e., J₅–J₄–J₃–J₂–J₁, J₄–J₅–J₃–J₂–J₁, J₄–J₃–J₅–J₂–J₁, J₄–J₃–J₂– J₅–J₁, J₄–J₃–J₂–J₁–J_5.

Iteration 4

For J₅–J₄–J₃–J₂–J₁ (see Table 13):

Table 13 Makespan for partial sequence of 5-4-3-2-1 Jobs

For J₄–J₅–J₃–J₂–J₁ (see Table 14):

Table 14 Makespan for partial sequence of 4-5-3-2-1 Jobs

For J₄–J₃–J₅–J₂–J₁ (see Table 15):

Table 15 Makespan for partial sequence of 4-3-5-2-1 Jobs

For J₄–J₃–J₂–J₅–J₁ (see Table 16):

Table 16 Makespan for partial sequence of 4-3-2-5-1 Jobs

For J₄–J₃–J₂–J₁–J₅ (see Table 17):

Table 17 Makespan for partial sequence of 4-3-2-1-5 Jobs

C_max for J₄–J₃–J₂–J₁–J₅ < J₅–J₄–J₃–J₂–J₁, J₄–J₅–J₃–J₂–J₁, J₄–J₃–J₅–J₂–J₁, J₄–J₃–J₂–J₅–J₁, therefore we choose J₄–J₃–J₂–J₁–J₅ and final makespan is 463.

Hence the makespan can be calculated for 5–10 jobs on 5 machines through NEH algorithm. We have taken five machine and jobs from five to ten according to the requirement of production schedule and makespan is shown in Table 18. In this table N stand for no. of jobs, M₁–M₅ are processing time on each machine.

Table 18 Makespan for five machine and jobs from five to ten

6 ANFIS method

Jang (1993) proposed Adaptive neuro-fuzzy inference system (ANFIS) to construct an input–output mapping based on both i.e. human knowledge (in the form of fuzzy if–then rules) and stipulated input–output data pairs. It is known as an adaptive network, a network of nodes and directional links. This network is connected with a learning rule—for example back propagation or hybrid algorithm. ANFIS can predict data using Sugeno FIS (Fuzzy Inference System) to relate membership and tune it using either back propagation or hybrid method. ANFIS model will simulate the inputs to the outputs correctly. In this research, the various input variables are trained and tested by ANFIS method. They are evaluated on the base of testing performances.

ANFIS schematic diagram is shown in Fig. 2. There are five network layers which are used by ANFIS to perform the following fuzzy inference steps: (a) input fuzzification, (b) fuzzy set database construction, (c) fuzzy rule base construction, (d) decision making, and (e) output defuzzification (Maher et al. 2014).

Fig. 2

Schematic diagram of ANFIS (Shafaei et al. 2011)

To explain this model simply, consists of five layers of adaptive network with two inputs (x and y) with two linguistic values and output f. Basically, inference system is constructed by five layers (Fig. 2) and each ANFIS layer consists of several nodes described by the node function. The present layers’ inputs are derived from the nodes in the previous layers. The rule base of ANFIS contains fuzzy IF–THEN rules of the Sugeno type. For a first-order Sugeno fuzzy inference system, the two rules may be stated as:

$$ \begin{aligned} & {\text{Rule}}\, 1 :\;{\text{IF}}\;{\text{x}}\;{\text{is}}\;{\text{A}}_{ 1} \;{\text{AND}}\;{\text{y}}\;{\text{is}}\;{\text{B}}_{ 1} ,\;{\text{THEN}}\;{\text{f}}\;{\text{is}}\;{\text{f}}_{ 1} \left( {\text{x,y}} \right) \\ & {\text{Rule}}\, 2 :\;{\text{IF}}\;{\text{x}}\;{\text{is}}\;{\text{A}}_{ 2} \;{\text{AND}}\;{\text{y}}\;{\text{is}}\;{\text{B}}_{ 2} ,\;{\text{THEN}}\;{\text{f}}\;{\text{is}}\;{\text{f}}_{ 2} \left( {\text{x,y}} \right) ,\\ \end{aligned} $$

where x and y are the inputs of ANFIS, A_i and B_i are the fuzzy sets, and f_i (x,y) is a first order polynomial and represents the outputs of the first order Sugeno fuzzy inference system. The structure of ANFIS is shown in Fig. 2, and the node function in each layer is described below. Represent the parameter sets that are adjustable in these nodes are presented by adaptive nodes, denoted by squares, whereas fixed nodes, denoted by circles, represent the parameter sets that are fixed in the system (Svalina et al. 2013).

Layer 1

this layer contains adaptive nodes with node functions like i explained as below:

$$ {\text{Q}}_{{ 1 , {\text{ i}}}} = \upmu_{\text{Ai}} \left( x \right)\quad {\text{ for}}\quad {\text{i}} = 1, \, 2 $$

(2)

$$ {\text{Q}}_{{2,{\text{i}}}} = \upmu_{\text{Bi - 2}} \left( {\text{y}} \right)\quad {\text{for}}\quad {\text{i}} = 3,4 $$

(3)

where x and y are the input to node i, A_i and B_i are the linguistic labels such as small or large, µ (x) and µ (y) are the membership functions. Many sorts of the membership functions which are there can be used. However, a Gaussian membership function has been chosen to represent the linguistic terms because the relationship between the processing time and makespan is not linear, so this function assured a smooth transition between 0 and 1. It can be written as follows:

First parameter membership functions

$$ \upmu A_{i} \left( x \right) = \exp \left[ { - 0.5\left( {\frac{{\left( {x - a_{i1} } \right)}}{{b_{i1} }}} \right)^{2} } \right] $$

(4)

Second parameter membership functions

$$ \upmu {\rm B}_{i} \left( y \right) = \exp \left[ { - 0.5\left( {\frac{{\left( {y - a_{i2} } \right)}}{{b_{i2} }}} \right)^{2} } \right] $$

(5)

where a_i,1, a_i,2, b_i,1, and b_i,2 are the parameter set. The bell-shaped functions vary while the values of this parameter are changing.

Layer 2

In this layer every node is a fixed node, which is marked by a circle and the node function has to be multiplied by input signals so that it can serve as output for every node. The nodes of this layer are called rule nodes. Each node computes the firing strength of the associated rule i.e.w₁.

$$ {\text{Q}}_{{ 2 , {\text{i}}}} = {\text{w}}_{1} = \upmu A_{i} \left( x \right) \times \upmu {\rm B}_{i} \left( y \right) $$

(6)

Layer 3

Every node in this layer is also a fixed node, marked by a circle and labeled N to show the normalization of the firing levels.

$$ {\text{Q}}_{{3,{\text{i}}}} = \overline{{w_{i} }} = \frac{{w_{i} }}{{\sum {w_{i} } }}\quad {\text{for}}\quad {\text{i}} = 1, \, 2 $$

(7)

Layer 4

Every node i in this layer is an adaptive node with a node function and marked by a square:

$$ {\text{Q}}_{{4,{\text{i}}}} = \overline{{w_{i} }} \times f_{i} \quad {\text{for}}\quad {\text{i}} = 1, \, 2 $$

(8)

Here f₁ and f₂ are the fuzzy IF–THEN rules as follows:

$$ \begin{aligned} & Rule\, 1 :\;{\text{IF}}\;{\text{x}}\;{\text{is}}\;{\text{A}}_{ 1} \;{\text{AND}}\;{\text{y}}\;{\text{is}}\;{\text{B}}_{ 1} , {\text{THEN}}\;{\text{f}}_{ 1} \;{\text{is}}\; = {\text{p}}_{1} {\text{x}} + {\text{q}}_{1} {\text{y}} + {\text{r}}_{1} \\ & Rule\, 2 :\;{\text{IF}}\;{\text{x}}\;{\text{is}}\;{\text{A}}_{ 2} \;{\text{AND}}\;{\text{y}}\;{\text{is}}\;{\text{B}}_{ 2} , {\text{THEN}}\;{\text{f}}_{ 2} \;{\text{is}}\; = {\text{p}}_{2} {\text{x}} + {\text{q}}_{2} {\text{y}} + {\text{r}}_{2} ,\\ \end{aligned} $$

where $\overline{{w_{i} }}$ is normalised firing strength from layer 3 and [p_i,q_i,r_i] is the parameter set of this node and marked as the consequent parameters.

Layer 5

One fixed node of this layer is marked by a circle. The node has to compute the overall output as the summation of all incoming signals:

$$ {\text{Q}}_{{5,{\text{i}}}} = {\text{f}}_{\text{out}} = \sum {\overline{{w_{i} }} \times f_{i} } = {\text{overall}}\;{\text{output}}. $$

(9)

The first layer and the fourth layer are the two adaptive layers with square nodes in this ANFIS architecture. In the first layer, there are two modifiable parameters known as premise parameters [a_i,b_i] which relates to the input membership functions. In the fourth layer, there are also three modifiable parameters known as consequent parameters [p_i,q_i,r_i] pertaining to the first-order polynomial.

MATLAB is used for ANFIS model development. ANFIS command window is used for training and testing. Gaussian bell membership function was used in input and output. In ANFIS a hybrid learning method is applied for updating the FIS parameters. The training process continues till the desired number of training steps (epochs) or the desired root mean squared error (RMSE) between the desired and the generated output is achieved.

Steps of ANFIS model for makespan estimation of FMSAS are explained as follows:

Step 1:

Normalize the training and test data

Because the range of data is different, so normalized the data as

$$ x_{i}^{'} = \frac{{x_{i} - x_{i,\hbox{min} } }}{{x_{i,\hbox{max} } - x_{i,\hbox{min} } }} $$

(10)

where x_i,min and x_i,max are the minimum and maximum values of ith input data.

Step 2:

Load Input training data and test data into the ANFIS model

Input data are a number of jobs, summation of processing times for one to five machines, whereas the output data is the makespan or the completion time of jobs.

Step 3:

Set the input and output parameters and membership function

The output and input parameters for ANFIS are defined. Membership function i.e. Gaussian bell shape is defined and used evalfis command for this.

Step 4:

Define the optimal parameter values for optimization

The parameters are optimized in which radii parameter is most important.

Step 5:

Define the epochs of the FIS for training

The epochs are set for the training of the model.

Step 6:

Trained the ANFIS model

The training of the model is started.

Step 7:

Testing the ANFIS model

The model is tested after the training.

Step 8:

Find the test output of the ANFIS model

Table 19 shows the parameter values used in testing with the output of the model. Finally, the obtained test output results with ANFIS model are compared with the measured values.

Table 19 Comparison of measured and predicted makespan

Step 9:

Plot correlation coefficient between measured and predicted makespan

Correlation coefficient is a statistical process for estimating the relationships among variables, i.e. prediction of ANFIS model and the measured data used for the testing. Correlation coefficient is widely used for prediction. After obtaining the output of ANFIS model, a plot is drawn between the predicted data of ANFIS model and measured data set. Correlation coefficient of ANFIS model is shown in Fig. 3.

Fig. 3

Correlation Coefficient of ANFIS data

7 Model verification

Twenty-four random readings were used as the testing data set (Table 19). The plot of 24 measured makespan values versus predicted makespan using the ANFIS model is shown in Fig. 4. This figure presents a comparison of the measured makespan and predicted makespan of the testing data set of 24 following training using ANFIS. Appropriate assent is evident between the measured and ANFIS-predicted makespan values. This close assent obviously displays that the ANFIS model can be used to predict the makespan under consideration. Thus, the proposed ANFIS model offers a promising solution to predicting makespan values in the specific range of parameters.

Fig. 4

Measured makespan versus predicted makespan

To assess the ANFIS model, the percentage error E_i and average percentage error E_av defined in Eqs. (11) and (12), respectively, were used.

$$ E_{i} = \frac{{\left| {measured\;makespan } \right. - \left. {predicted\;makespan} \right|}}{measured\;makespan } \times 100 $$

(11)

$$ E_{av} = \frac{1}{m}\sum\limits_{i = 1}^{m} {E_{i} } $$

(12)

where E_i is the percentage error of sample number i; and E_av is the average percentage error of m sample data.

From Table 19 and Fig. 5 show that the average percentage error for predicting makespan is 4.03%. Figure 5 presents the percentage error between the predicted and measured makespan. The highest percentage of error for ANFIS model prediction is 9.7%. The low error level signifies that the makespan results predicted by ANFIS are very close to the actual results. The error and accuracy values mean that the proposed model can predict makespan satisfactorily.

Fig. 5

The error percentage

8 Conclusion

In this study, ANFIS was used to develop an empirical model for predicting the makespan of flexible manufacturing system assembly shop in a manufacturing plant. An ANFIS model was developed based on NEH heuristics for makespan calculation as a scheduling problem. The ANFIS model was developed into two phases, namely training phase and test phase. In the training phase, about 90 values, i.e. 79% of the problems were used and 24 values, i.e. 21% of the problems used for the testing phase. This model was verified by test data, and the 95.97 average percentage of accuracy was achieved. Therefore, it can be concluded that makespan calculation of the production system, by the proposed ANFIS with NEH heuristic rules can be used as a reliable approach in estimating the job completion time of the problem studied. ANFIS shows a good performance with a coefficient of determination is 0.9310 and root-mean-square error (RMSE) of 0.0731. The RMSE is a frequently used measure of the differences between values predicted by a model or an estimator and the values actually observed, and coefficient of determination, describes how much of the variance between the two variables is described by the linear fit. Coefficient of determination of 0.9310 means that 93.10% of the variance is predictable. Correlation coefficient between measured and predicted makespan is also shown in a graphical way (Fig. 3). The value of Correlation coefficient is 0.9649. The results mutually differ less than ± 10%. The correlation coefficient is close to 1 i.e. 0.9649, it would indicate that the variables are positively linearly related and the scatter plot falls almost along a straight line with positive slope. The derived values of ANFIS model output are found within the range after being verified practically. Therefore, it can be concluded that makespan calculation of the production system, by the proposed adaptive neuro-fuzzy inference system, can be used as a reliable approach in estimating the makespan of flexible manufacturing system assembly shop.