MANUFACTURING CELL FORMATION CONSIDERING VARIOUS PRODUCTION FACTORS USING MODIFIED ART1 NETWORKS  

MANUFACTURING CELL FORMATION CONSIDERING VARIOUS PRODUCTION FACTORS USING MODIFIED ART1 NETWORKS

5.1 OBJECTIVE

Manufacturing cell formation is one of the initial and essential steps in Group Technology (GT) / Cellular Manufacturing System (CMS). In the third stage of the work, we develop a binary input Adaptive Resonance Theory (ART1) neural network used for cell formation (CF). Numbers of research chapters have been published in cell formation using ART1 neural networks. But those chapters do not deal with production factors. They sell only part-machine incidence matrix (PMIM) optimization. The modified ART1 (MART1) deals with binary data, non-binary data, operation sequences, and operation sequences with production volume. The MART1 is tested with a set of literature problems and compared. The MART1 results are best or equal when compared with all other algorithms. The comparison of results and consistency of the MART1 presented in this chapter.

Significance: Cell formation using ART1 is the best alternative approach when compared to the known algorithms/heuristics. The selection of the vigilance parameter in the ART1 is problematic. That problem solved by using this modified procedure of ART1. This modified ART1 also deals with the production factors of the CF.

5.2 INTRODUCTION

GT is a manufacturing philosophy in which similar parts identified and its associated machines grouped to form a GT cell. The objective of the GT cell is that all the similar parts (called part family) processed in its associated machine group (called machine cells). CMS defined as an application of GT to production. There are significant benefits that can achieved by implementing the CMS. They set up time reduction, work-in-process inventory reduction, material handling cost reduction, and equipment cost reduction, direct/indirect labor cost reduction, improvement of machine utilization, improvement in quality, and improvement in space utilization and improvement in employee morale.

The process of identification of Part Families (PF) and Machine Cells (MC) are called CF. There number of research work done in CF for the last three decades. The researchers developed several algorithms for CF using production flow analysis. In the production flow analysis CF, the PF and MC formed by the PMIM, which obtained from the route card. Some of the familiar production flow analyses that followed below,

1. Array-based methods
2. Clustering methods
3. Mathematical programming-based methods
4. Graph-theoretic approach
5. Search methods
6. Neural network-based methods

The ART network is an unsupervised vector classifier that accepts input vectors that classified according to the stored pattern they most resemble. It also provides for a mechanism-adaptive-expansion of the output layer of neurons until an adequate size is reached based on the number of classes, inherent in the observation. The ART network can adaptively create a new neuron corresponding to an input pattern if it is determined to be “sufficiently” different from existing clusters. This determination, called the vigilance test,  incorporated into the adaptive backward network. Thus, the ART architecture allows the user to control the degree of similarity of patterns placed in the cluster.

A number of research has already done. The performance of ART1 network-based CF has investigated. In the earlier ART1 based CF, the input of ART1 was only the PMIM. But in this MART1, the CF is done by using the binary data (i.e., PMIM), non-binary matrix data, operation sequence data, and operation sequence with production volume data. For each data input, the output of the MART1 is compared with the set of literature problems, and the conclusions are presented in this chapter.

5.2.1 Group Technology

GT is a manufacturing philosophy in which similar parts identified and its associated machines grouped to form a GT cell. The objective of the GT cell is that all the similar parts (called part family) processed in its associated machine group (called machine cells). The application of Group Technology (GT) in today’s automated manufacturing systems plays an important role, especially in batch-type production systems, where identification of part families and machine cells has simplified the layout design and production flow processes. Application of GT in production systems caters to a lot of advantages, e.g., reduction of material handling cost, time, labor requirement, paper works, in-process inventories, process lead time, frequency of set-ups. It also increases the quality of the product, productivity, customers’ satisfaction, and efficient management.

There are some popular measures like grouping efficiency and group efficacy for measuring the goodness of the diagonal block structure of the output matrix in CF problems. However, all these measures treat all the operations equally and are suitable only for the binary matrix. These measures cannot be adopted for generalized cell formation problems where operational sequence and time of the parts are a consideration. Therefore, Group Technology Efficiency (GTE) and modified grouping efficiency can conveniently use to measure the performance considering the sequence of parts and operational times of the parts, respectively. Group technology efficiency is defined as the ratio of the difference between the maximum numbers of inter-cell travels possible and the numbers of inter-cell travel actually required by the system to the maximum numbers of inter-cell travel possible.

The grouping efficiency redefined as grouping efficiency for ratio level data, which is given by the ratio between the total processing time inside the cells to the summation of exceptional elements, the total processing time of the cells, and total value of the voids present in the cells. Voids factor for the cell calculated by multiplying the number of voids in the cell to the average time of machine in a cell. The void factor of the cell multiplied by the total processing time of a cell.

5.2.2 Cellular Manufacturing System (CMS)

CMS defined as an application of GT to production. There are significant benefits that can achieved by implementing the CMS. They are set up time reduction, work-in-process inventory reduction, material handling cost reduction, equipment cost reduction, direct/indirect labor cost reduction, improvement of machine utilization, improvement in quality, improvement in space utilization, and improvement in employee morale. The process of identification of Part Families (PF) and Machine Cells (MC) are called CF. There is a number of research work done in CF. Cellular Manufacturing System (CMS) is an application of GT in manufacturing processes. GT identifies part families and allocate them to machine groups or machine cells for a minimum number of intercellular movements of parts. Part families identified base on their design, manufacturing processes, sequences, parts volume, process routings. For machine cells, different machines in functions grouped into a machine cell. Machine cells should formed so that it can process operations with the minimum intercellular movement of parts or family. However, similar types of machines may be required for different cells to cater to similar processing operations of different part families. This leads to an increase in a number of similar machines, thereby reducing the process flexibility and utilization of machines.

The majority of the cell formations techniques applied for a single process route, equal production volume, and without any sequence of the process. But in CMSs (or in batch-type production systems), apart can be processed in multiple process routings, the unequal production volume of parts and parts should process on a particular sequence, so management can plan for alternative machines to perform operations. Consideration of smallest intercellular movement of routings may reduce capital investment in machines and increase machine utilizations.

5.2.3 Drawbacks of the standard ART1

ART1 does not provide satisfactory results when the stored patterns grow sparser. This effect can minimized by changing the vigilance parameter of the network, but the optimization still becomes difficult as the number of input patterns increases. The classification process is also dependent on the order in which the input vectors applied. This effect can minimized by changing the vigilance parameter of the network, but optimization still becomes difficult as the number of input patterns increases. The classification process is also dependent on the order in which the input vectors applied. In the ARTI network, determining the proper vigilance value can be problematic.

One more major drawback in the standard ART1 is the learning process. In the learning of top-down weights vector (t_ij) is the “logical AND” and is applied between input vector (x_i) and top-down weights. The major disadvantage of the standard ART1 is that for CF, the group degraded during this operation. The stored neuron degraded by the increase of ‘0’ bits in the vector. The degraded group may lead to an improper classification in the future iterations. The is in a stored neuron and Os in an input vector Xi are ‘0’ due to the” logical AND” operation between the stored neuron and input vector. Once the value is set to ‘0’, it can never be set to ‘1’ because the “logical AND” for ‘0’ by any value is always equal to V.

The following input and cost parameters describe the quality of the cell formation manufacturing process. The input and cost parameter factors followed below,

5.2.4 Input Parameters

The input parameter value describes the quality of the cell formation manufacturing process that must supplied in the planning horizon.

1. Product mix: A set of part types to produced in the CMS in each Machines Parts combination. The product mix varies for various Machines-Parts combinations as new parts introduced,, and old parts discontinued. The product mix in each Machines-Parts combination known with certainty.
2. Product demand: The quantity of each part type in the product mix to be produced in each Machines-Parts combination. The product demand of each part type expected to vary across the planning horizon.
3. Operation sequence: An ordered list of operations that the part type must performed. The operation sequence of the parts known as ordinal level data and the operation time of the elements known as ratio level data or workload data, which are obtained through the route sheets to group the elements into part families and machines into machine cells. The proposed algorithm employs the principle of the modified ART1 network. The ART1 network classifies a set of binary vectors into the groups based on their similarity. The ART1 recognizes patterns and clusters the binary vectors with the known model based on the comparison mechanism.
4. Operating time: Time required by a machine to operate on a part type.
5. Machine capability: The ability of a machine to perform operations.
6. Machine capacity: The amount of the time a machine is available for 28 productions in each Machines-Parts combination.
7. Available machines: The available machines are the set of machines that will used to form manufacturing cells. The number of machines available must specified at the beginning of the CMS design.
8. Several cells: The number of groups of machines that need to be pre-specified.
9. Lower/ bounds: The minimum/maximum number of machines in each cell.
10. Batch size: The number of part types to transferred. It is constant for every part type.

5.2.5 Cost Parameters

The following system cost parameters  used to establish the CMS design concerning the objective of minimization of total cost: machine cost, operating cost, inter-cell material handling cost, and machine relocation cost

1. Machine cost: The cost of a total number of machines of each type used in the CMS.
2. Operating cost: The cost of running machines for producing parts. This cost depends on the cost of operating each machine type per hour and the number of hours required for each machine type.
3. Inter-cell material handling cost: The cost of transferring parts between cells when the parts cannot be produced entirely in a single cell. The inter-cell material handling cost incurred when batches of elements have to transferred between cells. This occurs when parts need to be processed in multiple cells, because all the machine types required to handle the parts are either not available in the cell to which the components allocated or because the cell does not have sufficient capacity. Inter-cell moves decrease the efficiency in the cellular manufacturing system by increasing material handling requirements and flow time and complicating production control.
4. Machine relocation cost: The cost of relocating machines from one cell to another in different Machines-Parts combinations. In dynamic and stochastic production environments, the best cellular manufacturing design for one Machines-Parts combination may not be an efficient design for subsequent Machines-Parts combinations. By rearranging the manufacturing cells, the CMS can continue operating efficiently as the product mix and demand change. However, there are some drawbacks to the rearrangement of manufacturing cells. Moving machines from cell to cell requires effort and could lead to the disturbance in normal production.

The inter-cell material handling cost can minimized by duplicating the machines, and the machine investment cost can minimized at the expense of increased inter-cell material handling costs. Similar arguments can be used for the combination of other costs as well. Therefore, the decisions associated with these costs need to be made simultaneously in an integrated manner.

5.2.6 Terminologies used in CMS

Processing time (t_j): It is the time required to process job (j) on any machine. The processing time (t_j) will generally include both actual processing time and set-up time.

Due date (d_j): It is the time at which job (j) is to complete.

Completion Time (C_j): It is the time at which the job (j) completed in a sequence. Performance measures like flow time, lateness, and tardiness for evaluating schedules are usually a function of job completion time.

Flow time (F_j): It is the amount of time job (j) spends in the system. Flow time is a measure of actual time spent by a job in the system. This, in turn, gives some idea about the in-process inventory of the shop floor.

Lateness time (L_j): It is the amount of time by which the completion time of job (j) differs from the due date (L_j = C_j – d_j). Lateness is a measure of non-conformity to the due date. It may either be positive lateness or negative lateness. Positive lateness of the job is said to be lateness or tardiness, and the negative lateness called as earliness. Therefore, it is often desirable to minimize tardiness.

Tardiness (T_j): Tardiness is the lateness of job (j) if it fails to meet its due date and it may either be zero or otherwise T_i = max {C_j – d_j}

• PROBLEM STATEMENT

Cell Formation Manufacturing contains the part families that are grouped and processed by the groups of machines in a cell. The determination of producing part families with groups of machines is known as cell formation problem (CFP). CM has several advantages, including more flexibility in creating of new parts, reducing the work-in-process (WIP), reducing required times for production, reducing material handling costs and setup time, and increasing the production volume, profitability, and quality of cell formation. The challenges may arise when CF cannot complete without considering the number of demands such as coefficient incorporation with process routes, operation sequence, processing time, and production volume. With the finding of all of the above aspects, an extended linear integer programming represented for solving the cell formation problem (CFP)

To overcome these problems, to develop the novelty of Modified Adaptive Resonance Theory (MART1) Neural Network Algorithm to enhance the operation sequences and production volume.

• PROPOSED METHODOLOGY

5.4.1 CF considering various production factors

The objective of this chapter is to formulate a new ART1 paradigm that incorporates process routings, operation sequences, non-binary incidence matrix values (i.e., part demands, machine capacities, processing times), and operation sequence with production volume considered.

5.4.1.1 PMIM CF

The PMIM (a_ij) consists of 0, 1 entry only; a one entry in row i, column j indicates that the machine corresponding processes part corresponding to the ith row to the j^th column. A 0 entry means that the part is not required by the corresponding machine. The rearrangement of the rows and columns of the matrix will provide a block diagonal form. If there is anyone entry is in the off-diagonal block then the corresponding part to processed in more than one cell is called an exceptional element. The CF often recognized as an NP (Non-polynomial) complete problem in the literature.

A very first performance measure in CF problem developed along with some related researches. The performance measure was grouping efficiency. One other performance measurement is the grouping efficacy, which is a new performance measure. The higher grouping efficiency and efficacy will result in a better grouping. The effectiveness of the MART1 CF measured by the total number of an exceptional element (E), the grouping efficiency (η) and grouping efficacy (t).

• Non-binary incidence matrix CF

A PMIM also consists of analog data (called nonbinary incidence matrix NBIM) that will include part demands, machine capacities, processing times, etc. NBIM is used to represent the relationships between machines and parts. Block diagonalization considered the best approach to form machine cells and part families. In the best solution of a CF problem, all the non-zero elements will remain in the diagonal blocks and all other zeros in the off-diagonal blocks.

In the MART1 CF of the NBIM, the primary measure is to maximize the density of cells by rearranging the significant non-zero elements in cells. The exceptional element (E) and the sum of unique element values or objective function value (OFV) considered as the performance measure to measure the MART1 CF.

5.4.1.3 Operation sequences CF

Another critical factor in the design of CMS is the operation sequence of parts. The series of machines visited by a component recorded in the matrix (s_ij), called Operation Sequence Incidence Matrix (OSIM).

k is an integer representing the operation for which part ‘i’ visits machine ‘j’.

(5.1)

is the inter-cell move matrix

The above representation enables us to capture routing sequence information, which is useful in determining material flow between machine pairs. In this case, if the first or last operation is an inter-cell movement, then the particular part has only one inter-cell move. Otherwise, there will be two inter-cell moves. For the operation sequences, CF using MART1, the minimization of inter-cell movement (unusual movement) considered.

5.4.1.4 Operation sequences with production volume

In addition to the PMIM, the data on the operation sequences and production volume of each part given. Since an intermediate operation on a part performed outside its cell involves two inter-cell flows while the first or last operation requires one such flow, the actual flows to or from machine ‘j’ by part ‘i’ accompanied by the production volumes of each part  calculated.

(5.2)

(5.3)

r is the index of operation sequence number, r = 1, . . . . , n_i and

is the set of the operation sequence number along which part ‘_i’ visits machine ‘j’ and

is the production volume of part i

is the inter-cell move matrix

From the equation (5.2), note that part processing, which consists of a single operation in a single machine, is assigned the flows equal to the demand for the specific part even though such processing requires no inter-machine moves by part.

5.4.2 MODIFIED ART1

5.4.2.1 Procedure for PMIM CF

The MART1 consists of two phases. The first phase is almost similar to the standard ART1 except for the cluster learning process. In the second phase, the weights fixed, and there is no change in the masses.

5.4.2.1.1 First phase

The architecture of the first phase of MART1 has two main layers. One is the input layer, also called the comparison layer, and the other one is the output layer, also called the recognition layer. Every input (bottom) neuron connected to every output (top) layer neurons. There are bottom-up weights (b_ij) associated with the arcs from the input neurons to the output neurons and top-down weights (t_ji) associated with the arches from the output neurons to the input neurons. The bottom-up weights used for cluster competition and top-down weights used for cluster verification. The first phase of the MART1 architecture shown in Figure 5.1.

The first phase of the ART1 procedure consists of three processes. The first one is the cluster search process, in which the network computes a matching score to reflect the degree of similarity of the present row-wise input vector (X_i) to the existing stored neurons.

Figure 5.1 First phase of MART1

The initial t_ji and b_ij weights initialized by using the following equations:

(5.4)

(5.5)

The matching score for neuron j, denoted by, is defined by

(5.6)

where is output of the “logical AND” operator applied between X_i and t_ji.

is a constant > 1 and

is the number of input neurons ( i.e total number of parts).

The largest net_j, say net_J, implies that the most like group and the associated group J is the candidate of the group.

The next process in the first phase is the cluster verification process. Even though J is the “most like” group, it does not guarantee that the (X_i) will pass the vigilance test. The vigilance parameter (ρ), 0 ≤ ρ≤ 1, determines the degree of the required similarity between the current input and a neuron already stored. The vigilance test passed means S > ρ, where S is the similarity ratio. The S is the ratio of the number of 1s in the c_i to the number of 1s in the X_i. If the input X_i passes the test, it included as a member of group J. Otherwise, the process returns to the cluster search process and tries the next largest net_j.

The above two methods are similar to the standard ART1 except for the last cluster learning process.  If the similarity between the X_i and the group J is good enough, then the vector X_i is accepted as a member of group J.  The learning process updates b_ij and t_ji.  For the new group, the t_ji is identical to the X_i. But for the already stored neuron, the “logical OR” is applied between X_i and the t_ji. The bottom-up learning weights are

(5.7)

where y_i is the “logical OR” operator applied between X_i and t_ji. The weights will be updated only for the weights associated with group J.

In the standard ART1, the learning of top-down weights vector t_Ji is the “logical AND” and applied between X_i and top-down weights.  The major disadvantage of the standard ART1 is that for CF, the group degraded during this operation.  That eliminated by using the “logical OR” operation.

Due to the “logical OR” operation in the learning process, if the number of 1s increases in t_Ji when compared to the previous t_Ji vector, then the new t_Ji is removed from t_Ji, and y_i is given as input to the network. If any other stored neuron wins, then the entire group is merged with the winner group, and the weights are updated. But if the new neuron wins, then the same stored patterns are maintained. The final output of the first phase of the MART1 is machine groups.

5.4.2.1.2 Second phase

In this phase, there will be three processes called cluster search process, cluster tuning process, and constraint verification process. In the cluster search process, the top-down weights are fixed, called fixed top-down weights (T_ji) based on the previous grouped rows, and fixed bottom-up weights (B_ij) initialized by using equation (5.8).

(5.8)

Where  is the number of input neurons

The column-wise inputs applied to the network, which computes a matching score to reflect the degree of similarity of the present column-wise input vector (X_i) to the existing cluster. The matching score for node j, denoted by fnet_j defined as

(5.9)

The maximum fnet_j, say fnet_J, implies that the most like group and the associated cluster J is the candidate cluster.  This step will ensure the minimization of the exceptional element. For every row or column input, e calculated by using the equation (10)

(5.10)

Where G is the total number of cells ( i.e. total number of neurons in the recognitio layer).

If there is more than one largest, then for the most significant equal values the  is computed

(5.11)

Where l is the total number of similar most significant value vectors.

Is the “logical OR” applied between the and.

The smallest ffnet_j, say ffnet_J, that the most like group and associated cluster J is the candidate cluster. If it is more then one, then the first one is chosen as the related cluster J. This step will ensure the minimization of voids.

After all the column-wise inputs made, the column groups identified. The output of this process is part of families. In the second cluster tuning process, the weights T_ji is fixed based on the previous part groups, and row-wise inputs applied to the second phase of the network. The output grouped rows. If the last row groups are identical to these current row groups, then the tuning process ends. Otherwise, once again, the weights are fixed based on the current row groups and column-wise inputs are applied to the second phase of the network. The current row and column groups are the best groups.

In the last, constraint verification process, the maximum number of cells permitted constraint is verified; if the constraint satisfied, then the MART1 gives the final machine and part groups. If it not filled, later each cell block-diagonal density is confirmed, the lowest cell block-diagonal frequency identified. That cell diagonal machine-part gave as input to the network and the new cells found, which added to the original result. The weights are fixed based on the final grouped rows and column-wise data applied to the same second phase. The second stage continues, and the output is the last grouped columns. Similarly, the weights are fixed based on the grouped columns, and row-wise inputs are applied, and the final row groups identified.

5.4.2.2 Procedure for CF with Production Factors

Few more modifications made in the MART1. The first phase is similar to MART1 CF. The given matrix converted into a machine part incidence matrix (MPIM), and then it given as input to the first phase of the MART1 CF networks. After the completion of the first and second phases of the MART1 CF, there is one more process in the second phase called the objective identification process.

In this process, once again, row-wise inputs are applied to the network. The top-down weight is fixed based on the final part groups, and there is no bottom-up weight for this process. In this process, row-wise non-binary incidence matrix inputs applied to the network and the row, which gives minimum inet_j, say inet_J, and implies that the most like group and the associated cluster J is the candidate cluster. After the all row-wise input, finally, the row groups (i.e., part family) are identified.

(5.12)

Where X_i is the input from NBIM

For Process sequence, the X_i is from the matrix A

For a sequence with production volume, the X_i is from the matrix IM

The total objective function value calculated by using the following equation.

(5.13)

Similarly, the top-down weight only fixed based on the previous output machine cells make the tuning process, and the column-wise inputs applied to the network. The column which gives minimum inet_j, say inet_J, implies that the most like group and the associated cluster J is the candidate cluster. After all row-wise input, finally, the row groups are identified (i.e., machine cells), and the total objective function value also calculated.

Now the row-wise input total OFV and the column-wise input OFV are compared, and the group which gives minimum total OFV is the best part family or machine cell. Based on that group, the top-down weights fixed, and the other part family or machine cell identified. They are the final machine cells and part families.

5.4.3 VIGILANCE PARAMETER SELECTION

In the standard ART1, the selection of the vigilance parameter (ρ) is problematic. For the PMIM problem is too high, a vigilance value will result in more similar groups, at the expense of creating too many groups. Too low a vigilance value will result in everything placed into just a few groups, essentially performing no real classification.

For the selection of the vigilance parameter in the MART1 network, the seven datasets tested with different ρ values. The first four datasets are well-structured, and all the other three data sets are not well-structured datasets. For all the seven datasets are tested with the vigilance parameter value between 0.1 and 1.0. The ZODIAC algorithm for CF with a constraint of the total number of cells (G) of 7. The MART1 tested with the same limitation of G =7. The results obtained from MART1 for various ρ values shown in Table 5.1.

Table 5.1 Differentresults for Chandrasekharan and Rajagopalan (1989) problems

It observed that for the well structured and ill-structured datasets, the ρ =0.1 gives the best the results. The same best result obtained from the different ρ values shown in Table 1 with a ‘√’ mark. In the table, the letter ‘I’ indicates the MART1 solution, which is inferior to the corresponding ρ value. That the solution is inferior means the η and t are low, and E is high when compared to the best result. Table 1 also shows ‘-,’ which indicates that for the particular ρ value the MART1, does not give any output (i.e., all the machines and parts grouped into a single cell).

5.5 RESULTS AND DISCUSSION

5.5.1 For PMIM CF

For CF, using MART1 was tested with different benchmark datasets, which collected from the literature. In all the tested problems, the ρ value is taken as 0.1. The results compared with the different CF approaches with various performance measures.

The datasets already tested for the vigilance parameter selection. The comparative results shown in Table 5.2. For the first two datasets, the MART1 results are equal to the solutions. For the dataset 3 and 4 E are similar, the MART1 grouping efficiency is high when compared to the grouping efficiency. For all other datasets, the E is low, and η is high when compared to the values of E and η.

Table 5.2 Comparison of results with Chandrasekharan and Rajagopalan (1989)

5.5.2 For CF with Production Factors

The first dataset of NBIM adapted from Peker and Kara (2004). They tested their Fuzzy ART network approach by using 7 X 7 NBIM. The same problem considered for the evaluation of the MART1. First of all, the NBIM converted into a PMIM, and it is given as input to the MART1 CF network to get the final output of grouped parts and machines. Then the NBIM CF procedure is followed, and the final output obtained. The OFV (i.e., the sum of exceptional value) is 2.4. The result is identical to the Peker, and Kara (2004) results.

The next NBIM data taken from Vohra et al. (1990). They used machining time NBIM of size 7 X 7. Vohra et al. tested the dataset with a constraint of two and three cells, and their objective function is to minimize the total machining times of the inter-cellular movement. For two cells, the MART1 result shown in Table 5.3. The MART1 considered for the minimization of exceptional elements and minimization of total inter-cellular machining time. In the MART1, the number of a unique component is 1, and the entire inter-cellular movement is 8. But Vohra et al. (1990) used a network approach, and their number of an exceptional element is three, and their total inter-cellular machining time is only 4. Vohra et al. (1990) are not considering the minimization of extraordinary items.

Table 5.3 Result for Vohra et al. (1990) NBIM problem

For three cells, the MART1 number of an exceptional element is two, and the total inter-cellular machining time is 14. But Vohra et al. (1990) give only 9 total inter-cellular machining time, and the number of an exceptional element is 4.

             The next dataset is OSIM. The dataset taken from Suresh et al. (1999). They used 15 X 15 OSIM. The OSIM is converted into a PMIM and it is given as input to the MART1 and the final output is grouped machines and parts. Based on the output, the inter-cell move matrix obtained by using the equation (5.1), and the MART1 with production factors procedures followed. The result is identical to the Suresh et al. (1999) result.

The next two OSIM data adapted from Nair and Narendran (1998). The size of the first dataset is 7 X 7. The result is identical to the Nair, and Narendran (1998) findings. They used the CASE algorithm. The second Nair and Narendran (1998) dataset is 20 X 20, and the number of groups is 3. The MART1 result is identical to Nair and Narendran (1998).

The next dataset is an operational sequence with a production volume dataset, which adapted from Youk Yung Won and Kun Chang Lee (2002). The operation sequences and production volumes for the parts shown in Table 5.4.

Table 5.4 Operation sequences with the production volume problem

The table converted into PMIM, and it given as input to the MART1 network and the final output grouped machines and parts. The inter-cell flows calculated by using equation (5.2). The MART1 with production factor steps followed, and the final output is identical to the Youk Yung Won and Kun Chang Lee (2001) solution.

5.6 SUMMARY

In the third stage of the work, the MART1 neural network has successfully implemented for cell formation problems. The MART1 gives PF and MC and the number of exceptional elements with the constant vigilance parameter. The MART1 algorithm itself straightaway gives the number of unique features. The computational effort is meager in the MART1 when compared to all other algorithms. The MART1 network is suitable for any size of PMIM. This MART1 deals with non-binary data, operation sequences, and operation sequences with production volume. The modified approach is tested with a set of literature problems and compared. The MART1 results are best or equal when compared with all other algorithms. The modified ANN approaches coded with Mat Lab 6.5. The test datasets tested with the Pentium IV 900 MHz, Systems. Several improvements to the MART1 network are also possible for CF. The scope of this MART1 restricted for the CF with an objective of minimization of exceptional elements and maximization of the grouping efficiency. Some of the issues, like more constraints, multi-objectives, etc. can be implemented in these MART networks.