2.4.2 Notation

The following points present the notation and the proposed model:

a. Indexes

h = work station (h = 1,…, Smax). i, i' = machine (i = 1,…, m). j = product model (j = 1,…, n). k = production cells (k = 1,…, Kmax).

b. Decision variables

xik <sup>=</sup> 1 if machine i is assigned to cell k 0 otherwise ( . yk <sup>=</sup> 1 if production cell k is used, i:e:, if it is assigned machines 0 otherwise ( . uihG <sup>=</sup> 1 if machine i of graph G is assigned to work station h 0 otherwise � . rh <sup>=</sup> 1 if work station h is used, i:e:, it is assigned machines 0 otherwise � . fkh <sup>=</sup> 1 if cell k is used by work station h 0 otherwise � . gh <sup>=</sup> 1 if work station h is multicellular 0 otherwise � .

#### 2.4.3 Model

#### 2.4.3.1 Objective function

The objective of this formulation is to minimize the total cost of intercellular transport between machines, which will appear every time there are finished products between machines i and i' and they belong to different production cells. Decision variables xik define the set of groups of machines, while the product families will be defined after the solution of this model, supported by the information of the extended combined precedence graph (GOUGA):

$$\min Z = \sum\_{i=1}^{m-1} \sum\_{i'=i+1}^{m} \sum\_{k=1}^{K\_{\text{max}}} c\_{ii'} (1 - \boldsymbol{\varkappa}\_{ik} \cdot \boldsymbol{\varkappa}\_{i'k}) + c\boldsymbol{\varsigma} \cdot \sum\_{h=\lceil \boldsymbol{\Sigma}\_{\text{min}} \rceil + 1}^{\mathcal{S}\_{\text{max}}} r\_h \tag{7}$$

A Methodology to Design and Balance Multiple Cell Manufacturing Systems DOI: http://dx.doi.org/10.5772/intechopen.89463

Restrictions for the formation of manufacturing cells:

• The displacement times of the workers in a manufacturing cell are not

The following points present the notation and the proposed model:

• The machines related by some intercellular product movement must belong, if

.

.

.

The objective of this formulation is to minimize the total cost of intercellular transport between machines, which will appear every time there are finished products between machines i and i' and they belong to different production cells. Decision variables xik define the set of groups of machines, while the product families will be defined after the solution of this model, supported by the informa-

cii<sup>0</sup> 1 � xik � xi

0 <sup>k</sup> ð Þþ cs � <sup>X</sup>

Smax

rh (7)

h¼⌈Smin⌉þ1

.

.

.

yk <sup>=</sup> 1 if production cell k is used, i:e:, if it is assigned machines

uihG <sup>=</sup> 1 if machine i of graph G is assigned to work station h

rh <sup>=</sup> 1 if work station h is used, i:e:, it is assigned machines

significant, but not so out of them.

possible, to the same work station.

h = work station (h = 1,…, Smax).

k = production cells (k = 1,…, Kmax).

xik <sup>=</sup> 1 if machine i is assigned to cell k

fkh <sup>=</sup> 1 if cell k is used by work station h

gh <sup>=</sup> 1 if work station h is multicellular

tion of the extended combined precedence graph (GOUGA):

Xm

K Xmax k¼1

i, i' = machine (i = 1,…, m). j = product model (j = 1,…, n).

0 otherwise

0 otherwise

0 otherwise

0 otherwise

0 otherwise

min <sup>Z</sup> <sup>¼</sup> <sup>X</sup><sup>m</sup>�<sup>1</sup>

i¼1

i 0 ¼iþ1

0 otherwise

b. Decision variables

(

(

�

�

�

2.4.3.1 Objective function

2.4.3 Model

48

�

2.4.2 Notation

Mass Production Processes

a. Indexes

$$\sum\_{k=1}^{K\_{\text{max}}} x\_{ik} = 1; \qquad \forall i = 1, \ldots, m \tag{8}$$

$$\sum\_{i=1}^{m} \mathbf{x}\_{ik} \le \mathbf{M}\_{\text{max}} \cdot \boldsymbol{\upnu}\_{k}; \qquad \forall k = \mathbf{1}, \ldots, K\_{\text{max}} \tag{9}$$

$$\sum\_{i=1}^{m} \mathbf{x}\_{ik} \ge \mathbf{M}\_{\text{min}} \cdot \boldsymbol{\upnu}\_{k}; \qquad \forall k = \mathbf{1}, \ldots, K\_{\text{max}} \tag{10}$$

Restrictions for the balance of U-shaped cells:

$$\sum\_{h=1}^{S\_{\text{max}}} (\mu\_{ihGO} + \mu\_{ihGA}) = \mathbf{1} \qquad \forall i = \mathbf{1},...,m\tag{11}$$

$$\sum\_{i=1}^{m} \overline{t}\_i (u\_{ihGO} + u\_{ihGA}) \le \overline{C} \cdot \eta\_i \qquad \forall h = 1, \dots, S\_{\max} \tag{12}$$

$$\sum\_{h=1}^{S\_{\text{max}}} (\mathbb{S}\_{\text{max}} - h + 1)(u\_{ahGO} - u\_{bhGO}) \ge 0 \qquad \forall (a, b) \in GO \tag{13}$$

$$\sum\_{h=1}^{\mathcal{S}\_{\text{max}}} (\mathcal{S}\_{\text{max}} - h + 1)(u\_{bh\text{GA}} - u\_{ah\text{GA}}) \ge 0 \qquad \forall (a, b) \in \text{GO} \tag{14}$$

Linking restrictions between the formation and the balance of the cells:

$$\sum\_{i=1}^{m} (\mu\_{ihGO} + \mu\_{ihGA}) \cdot \ge\_{ik} \le m \cdot f\_{kh} \quad \forall k = 1, \dots, K\_{\max}; \ h = 1, \dots, \text{S}\_{\max} \tag{15}$$

$$\sum\_{h=1}^{S\_{\text{max}}} \mathbf{g}\_h \cdot f\_{kh} \le 2 \quad \forall k = 1, \dots, K\_{\text{max}} \tag{16}$$

$$\sum\_{k=1}^{K\_{\text{max}}} f\_{kh} \le (\theta - 1) \cdot \mathbf{g}\_h + 1 \quad \forall h = 1, \dots, \mathbf{S}\_{\text{max}} \tag{17}$$

Restrictions for defining binary variables:

$$\{\mathcal{X}\_{ik}, \mathcal{Y}\_{\mathbb{k}}, u\_{ihG}, r\_h, f\_{ih}, \mathcal{g}\_h \in \{0, 1\} \qquad \forall \ h, i, j, k\tag{18}$$

The set of restrictions (8) restricts each machine to a single cell. The set of restrictions (9) restricts each created cell to a maximum of Mmax machines, while the set of restrictions (10) restricts them to a minimum of Mmin machines; the value of yk is equal to one for the first Kmin restrictions, since it is known that these cells are required. As to the balance of the lines, the objective is to minimize the cost per required work station in addition to the theoretical minimum, avoiding the need to have rh variables for stations 1 through ⌈Smin⌉. The set of restrictions (11) ensures that each machine is assigned to only one station, either in the original precedence graph or in the auxiliary one [18]. The set of restrictions (12) ensures that for every station, the sum of the weighted average processing times of their assigned machines does not exceed the average cycle time; the values of rh are equal to one

for the first ⌈Smin⌉ restrictions. The set of restrictions (13) and (14) force the precedence restrictions between the machines; these relations are reversed for the auxiliary graph.

final products are assembled at the customer's facilities, so the work orders are divided considering the final product's components, which are generally pillars, beams, struts, slotted angles (ANRA), and accessories. Each of them can have various modifications in size, processing times, and complexity in its operations (precedence restrictions), so it is possible to identify them previously as families of products, therefore complying with the observations proposed by Burbidge [22], who says that a system can be naturally susceptible to be transformed into one of

A Methodology to Design and Balance Multiple Cell Manufacturing Systems

The company has a factory in the commune of Quilicura, in the Metropolitan Region of Chile, and has 31 machines that can be classified into 17 types, processing 67 different product models. The place where the methodology will be applied has a

This study will consider a time planning horizon of 3 months, in which the plant operates 16 h/d from Monday to Saturday. According to the company's policies, 1 day per month is devoted to preventive maintenance operations of each machine,

m (type of machine) Identification characteristic Present number Simultaneous machine

J (products) Identification characteristic

60–67 Slotted angle (ANRA)

1–4 Strut 5–29 Beam 30–51 Accessory 52–59 Pillar

so it will be considered that each machine has a capacity of 1248 h, within a

 Sheet metal cutter 3 No Strippit 1 No Punching machine 1 No 25 tons press 2 No 45 tons press 2 No 55 tons press 2 No 90 tons press 1 No Pneumatic press 160 tons 2 No Sheet metal bender 2000 m 2 No Sheet metal bender 3000 m 2 No Sheet metal bender 4000 m 2 No Sheet metal bender 160 ton 2 Yes Welder accessories 2 No Beam welder 2 No Connector welder 2 No Pillar welder 2 No Forming machine ANRA 1 No

the cellular manufacturing type.

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

job shop-type configuration.

Table 1.

Table 2.

51

Information on the types of machines.

Information on the types of products.

The set of restrictions (15) makes each variable fkh be equal to one when cell k is used by station h. The set of restrictions (16) allows a maximum of two multicellular work stations for each manufacturing cell, so that there are not many interferences between stations [9]. The set of restrictions (17) limits to θ the number of cells to which a multicellular work station can belong, and it also makes every variable gh equal to one when h is a multicellular station. The set of restrictions (18) defines the decision variables xik, yjk, uih, rh, fkh, and gh as binary.

#### 2.5 Assigning the product models to the obtained production cells

With the groups of machines obtained, we must now assign the product models to each resultant cell. For this, let:

Pkj = Number of machines of cell k that process product j,

$$P\_{k\vec{\eta}} = \sum\_{i=1}^{m} a\_{i\vec{\eta}} \cdot \mathbf{x}\_{ik}$$

Ω<sup>j</sup> = Set of cells that have the maximum number of machines that process j,

$$\Omega\_{\boldsymbol{j}} = \left\{ \boldsymbol{k} | P\_{\boldsymbol{k}\boldsymbol{j}} = \max\_{k=1,\ldots,K} \{ P\_{\boldsymbol{k}\boldsymbol{j}} \} \right\}.$$

Wð Þ<sup>k</sup> <sup>j</sup> = Total workload (in hours) of cell k due to product j,

$$\boldsymbol{W}\_{j}^{(k)} = \sum\_{i=1}^{m} d\_{j} \cdot t\_{\vec{y}} \cdot \boldsymbol{\varkappa}\_{ik}$$

Ψ<sup>j</sup> = Set of cells having maximum total workload due to product model j and belonging to Ωj,

$$\Psi\_j = \left\{ k | \mathcal{W}\_j^{(k)} = \max\_{k \in \Omega\_j} \left\{ \mathcal{W}\_j^{(k)} \right\} \right\}.$$

Then, to assign to which production cell each product j belongs, the following formal procedure is defined, where three cases can occur:

Case 1: If Ω<sup>j</sup> � � � � ¼ 1 ) j∈ Jk⇔Pkj ¼ max k¼1, …,K Pkj � �; assign each product to the cell where it will be processed by more machines.

Case 2: If Ω<sup>j</sup> � � � �>1∧ Ψ<sup>j</sup> � � � � <sup>¼</sup> <sup>1</sup> ) <sup>j</sup><sup>∈</sup> Jk⇔Wð Þ<sup>k</sup> <sup>j</sup> ¼ max k¼1, …, K Wð Þ<sup>k</sup> j n o; if there is a tie it must be assigned to the cell in which the product spends most processing time. Case 3: If Ω<sup>j</sup> � � � �>1 ∧ Ψ<sup>j</sup> � � � �>1 ), assign j arbitrarily to a cell that belongs to Ψj; if a tie occurs, assign the product to the cell with a smaller number of machines or randomly.

#### 3. Application of the methodology: illustrative case

The company in which the proposed methodology will be applied is of the metalworking type, making storage products (racks) for the retail industry. The

### A Methodology to Design and Balance Multiple Cell Manufacturing Systems DOI: http://dx.doi.org/10.5772/intechopen.89463

final products are assembled at the customer's facilities, so the work orders are divided considering the final product's components, which are generally pillars, beams, struts, slotted angles (ANRA), and accessories. Each of them can have various modifications in size, processing times, and complexity in its operations (precedence restrictions), so it is possible to identify them previously as families of products, therefore complying with the observations proposed by Burbidge [22], who says that a system can be naturally susceptible to be transformed into one of the cellular manufacturing type.

The company has a factory in the commune of Quilicura, in the Metropolitan Region of Chile, and has 31 machines that can be classified into 17 types, processing 67 different product models. The place where the methodology will be applied has a job shop-type configuration.

This study will consider a time planning horizon of 3 months, in which the plant operates 16 h/d from Monday to Saturday. According to the company's policies, 1 day per month is devoted to preventive maintenance operations of each machine, so it will be considered that each machine has a capacity of 1248 h, within a


#### Table 1.

for the first ⌈Smin⌉ restrictions. The set of restrictions (13) and (14) force the precedence restrictions between the machines; these relations are reversed for the

decision variables xik, yjk, uih, rh, fkh, and gh as binary.

to each resultant cell. For this, let:

2.5 Assigning the product models to the obtained production cells

Pkj <sup>¼</sup> <sup>X</sup><sup>m</sup>

Ω<sup>j</sup> ¼ kjPkj ¼ max

<sup>j</sup> <sup>¼</sup> <sup>X</sup><sup>m</sup> i¼1

Ψ<sup>j</sup> = Set of cells having maximum total workload due to product model j and

<sup>j</sup> ¼ max k∈ Ω <sup>j</sup>

Then, to assign to which production cell each product j belongs, the following

k¼1, …,K

must be assigned to the cell in which the product spends most processing time.

tie occurs, assign the product to the cell with a smaller number of machines or

The company in which the proposed methodology will be applied is of the metalworking type, making storage products (racks) for the retail industry. The

Pkj

<sup>j</sup> ¼ max k¼1, …, K

� � n o

<sup>j</sup> = Total workload (in hours) of cell k due to product j,

Wð Þ<sup>k</sup>

<sup>Ψ</sup><sup>j</sup> <sup>¼</sup> <sup>k</sup>jWð Þ<sup>k</sup>

formal procedure is defined, where three cases can occur:

where it will be processed by more machines.

�>1∧ Ψ<sup>j</sup> � � �

�>1 ∧ Ψ<sup>j</sup> � � �

� ¼ 1 ) j∈ Jk⇔Pkj ¼ max

3. Application of the methodology: illustrative case

� <sup>¼</sup> <sup>1</sup> ) <sup>j</sup><sup>∈</sup> Jk⇔Wð Þ<sup>k</sup>

i¼1

Ω<sup>j</sup> = Set of cells that have the maximum number of machines that process j,

Pkj = Number of machines of cell k that process product j,

The set of restrictions (15) makes each variable fkh be equal to one when cell k is used by station h. The set of restrictions (16) allows a maximum of two multicellular work stations for each manufacturing cell, so that there are not many interferences between stations [9]. The set of restrictions (17) limits to θ the number of cells to which a multicellular work station can belong, and it also makes every variable gh equal to one when h is a multicellular station. The set of restrictions (18) defines the

With the groups of machines obtained, we must now assign the product models

aij � xik

k¼1, …,K

dj � tij � xik

� � � �

Pkj

Wð Þ<sup>k</sup> j

� �; assign each product to the cell

; if there is a tie it

Wð Þ<sup>k</sup> j n o

�>1 ), assign j arbitrarily to a cell that belongs to Ψj; if a

auxiliary graph.

Mass Production Processes

Wð Þ<sup>k</sup>

belonging to Ωj,

Case 1: If Ω<sup>j</sup> � � �

Case 2: If Ω<sup>j</sup> � � �

Case 3: If Ω<sup>j</sup> � � �

randomly.

50

Information on the types of machines.


#### Table 2. Information on the types of products.

planning horizon of T = 1296 h. Tables 1 and 2 present general information with respect to the types of machines and the different product models, respectively.
