3.4 Step 4: statement and solution of the mathematical model

With the parameters calculated above, we state the mathematical model. The model was then solved using the Extended LINGO© version 8.0 software, getting an optimal global solution in 2 h and 32 min, after 121 iterations, giving as a result 5 manufacturing cells and 9 work stations, 3 of them multicellular, because they process exceptional products. This solution is represented in Figure 7.

## 3.5 Step 5: assigning the product models to the production cells

With the groups of machines obtained thanks to the model's solution, the product models were assigned to each of these production cells. This solution is represented in Table 9.

The proposed cellular manufacturing system that results from the application of the methodology shows the resultant cells and the product families assigned to them, where it is seen that cell 1 processes products of the beam and slotted angle type, cells 2 and 3 process only accessories, cell 4 processes only beams, and cell 5 processes only pillars. In this way, it is easier to improve the obtained balance by sequencing mixed product models, because in a cell there are no different types of products that compete for the same resources (machines), but they rather process the same types of products, grouped in families. Only work stations 1, 2, and 8 turn out being multicellular, because they process exceptional products that undergo intercellular movements, most of which are associated with the operation of the cutter type machines, since they are those that are mostly shared by the different products.

To analyze the results, it is necessary to have performance measures of the solution, but since the present study has taken up a new problem that is part of the manufactured cell formation problem (MCFP) and the general assembly line balancing problem (GALBP), we must use measures of performance commonly employed for both problems separately, such as group capability index (GCI), for

i

i

cii' (MM\$)

> 27

28 29 30 31 Table 5. Intercellular

 transport costs between machines i and i´ (Cii´).

0

 0

 0 0000

 0.043

 0.043

 0

 0.043 0000

0.11

 0

 0

 0

 0

 0

 0

 0

 0

 0.043

 0.043

 0

 0.043

 0

0

 0

 0.353

 0

 0

 0.203

 0

 0.203

 0

 0

 0

 0.353

 0

0

 0

 0

 0.338

 0.253

 0

 0.185

 0

 0

 0.185

 0.338

 0

Mass Production Processes

0

 0

 0.353

 0

 0

 0.203

 0

 0.203

 0

 0

 0

0

56

example, proposed by Seifoddini and Hsu [23], and the grouping efficacy (GE) proposed by Kumar and Chandrasekharan [24], as well as balancing measures like "total line imbalance" proposed by Thomopoulos [12]. Table 10 presents a comparison of results of the case study by the proposed methodology versus the model of Won and Currie [25], considering the latter as an MCFP-type problem. The

model of [25] was chosen because it considers a number of production factors and a complexity level relatively similar to the proposed methodology, and it can be

Won and Currie [23]

(31 67) 152 140 121 114 89.45 85.45 86.45 84.25

Number of iterations GCI (%) GE (%)

Proposed methodology

Won and Currie [23]

Proposed methodology

Won and Currie [23]

In Table 10, it is seen that the proposed methodology takes more time to get the global optimal solution as well as a greater number of iterations compared to the model of [25]. This can be explained because in the proposed model of the methodology the first component of the formulation is of a quadratic type, so the LINGO© software uses an algorithm specialized in these types of problems which makes each iteration take longer time; on the other hand, the model of [25] is of the p-media type, which although linear, in terms of the quality of the solution, it is seen that the proposed methodology gives a group capability index (GCI) greater than the model of [25]. We believe that this is due to the fact that the proposed formulation puts emphasis in minimizing the costs of intercellular product movements, in contrast with the model of Won and Currie [25], which aims to maximize the similarity between the machines that constitute the production cells. On the other hand, with respect to the grouping efficacy (GE), the proposed methodology is better than the model of [25]. Making a deeper analysis of the results and getting more general conclusions is risky because it is necessary to do further research with study cases in which the indicators and models used can be compared. In relation to the comparison of indicators such as the total imbalance of the line proposed by Thomopoulos [12] under the viewpoint of the general assembly line balancing problem (GALBP), it was not done, and it is expected to be dealt with in an

This work has proposed a new way of approaching balancing and cell formation problems, which were previously studied independently. In this way, it is possible to consider aspects that were previously avoided, such as production volumes, processing times, and operation sequences for the MCFP, and the fact that the production cells are not established yet and that they also share information among themselves, for balancing the lines. In the proposed methodology, a model is presented that has advantages at the time of solving it with a commercial software, because it does not need as many variables as other proposals. This approach delivers cells that are more amenable to be used in practice, although it will remain for future research to deliver immediately the layout of the cells as well as to integrate the problem of sequencing mixed models and, in that way, to improve the

In relation to the preliminary results obtained by comparing with another mathematical model from the viewpoint of a type MCFP problem, it must be pointed out

solved in Lingo© in a reasonable computation time.

Won and Currie [23]

Proposed methodology

A Methodology to Design and Balance Multiple Cell Manufacturing Systems

extension of the present work.

4. Conclusions

Problem number (m n)

Table 10.

Computation time

DOI: http://dx.doi.org/10.5772/intechopen.89463

(min)

Proposed methodology

Comparison of the measures for the MCFP.

balance of each station.

59

Figure 7. Representation of the resultant cells and stations.


Table 9. Assignment of products to cells.


#### Table 10.

example, proposed by Seifoddini and Hsu [23], and the grouping efficacy (GE) proposed by Kumar and Chandrasekharan [24], as well as balancing measures like "total line imbalance" proposed by Thomopoulos [12]. Table 10 presents a comparison of results of the case study by the proposed methodology versus the model of Won and Currie [25], considering the latter as an MCFP-type problem. The

Table 9.

58

Figure 7.

Mass Production Processes

Representation of the resultant cells and stations.

Assignment of products to cells.

Comparison of the measures for the MCFP.

model of [25] was chosen because it considers a number of production factors and a complexity level relatively similar to the proposed methodology, and it can be solved in Lingo© in a reasonable computation time.

In Table 10, it is seen that the proposed methodology takes more time to get the global optimal solution as well as a greater number of iterations compared to the model of [25]. This can be explained because in the proposed model of the methodology the first component of the formulation is of a quadratic type, so the LINGO© software uses an algorithm specialized in these types of problems which makes each iteration take longer time; on the other hand, the model of [25] is of the p-media type, which although linear, in terms of the quality of the solution, it is seen that the proposed methodology gives a group capability index (GCI) greater than the model of [25]. We believe that this is due to the fact that the proposed formulation puts emphasis in minimizing the costs of intercellular product movements, in contrast with the model of Won and Currie [25], which aims to maximize the similarity between the machines that constitute the production cells. On the other hand, with respect to the grouping efficacy (GE), the proposed methodology is better than the model of [25]. Making a deeper analysis of the results and getting more general conclusions is risky because it is necessary to do further research with study cases in which the indicators and models used can be compared. In relation to the comparison of indicators such as the total imbalance of the line proposed by Thomopoulos [12] under the viewpoint of the general assembly line balancing problem (GALBP), it was not done, and it is expected to be dealt with in an extension of the present work.

### 4. Conclusions

This work has proposed a new way of approaching balancing and cell formation problems, which were previously studied independently. In this way, it is possible to consider aspects that were previously avoided, such as production volumes, processing times, and operation sequences for the MCFP, and the fact that the production cells are not established yet and that they also share information among themselves, for balancing the lines. In the proposed methodology, a model is presented that has advantages at the time of solving it with a commercial software, because it does not need as many variables as other proposals. This approach delivers cells that are more amenable to be used in practice, although it will remain for future research to deliver immediately the layout of the cells as well as to integrate the problem of sequencing mixed models and, in that way, to improve the balance of each station.

In relation to the preliminary results obtained by comparing with another mathematical model from the viewpoint of a type MCFP problem, it must be pointed out that although the grouping efficacy indicators are better, they are not conclusive if a comparative study with more study problems as well as a sensitivity analysis of the parameters of the proposed model is not made. The same comparative study must be made from the viewpoint of a type GALBP problem in relation to the balancing of the resultant line, in such a way that the indicators can be compared with the proposed methodology with the techniques or methods applied separately for the same sets of study problems.

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Finally, we believe that the proposed methodology responds to a problem of integrating two problems like the MCFP and GALBP under a same approach which is perfectible insofar as the results can be validated in future comparative work, as well as extending the proposed problem, integrating in the design, and balancing of manufacturing cells configured in "U" the design of the family of products that one wishes to manufacture.
