Abstract

Manufacturing cell formation and its balance in just-in-time (JIT) type production environments have usually been studied separately in the literature. This practice is unrealistic since both problems interact and affect each other when the cells are operating. This chapter proposes a methodology to design multiple manufacturing cells and simultaneously balance their workload. The cells considered are U-shaped and process mixed models of product families. A nonlinear integer programming mathematical model is proposed, which integrates cell formation and their balancing, considering various production factors. For illustration, the method is applied to the redesign of a rack manufacturing process.

Keywords: manufacturing cells, assembly line balancing, N U-lines, mathematical programming, mixed model production

## 1. Introduction

Group technology (GT) can be defined as a manufacturing philosophy identifying similar parts and grouping them together to take advantage of their similarities in manufacturing and design [1, 2]. Cellular manufacturing (CM) is an application of GT and has emerged as a promising alternative manufacturing system [3]. When a productive system is changed to make it cellular, it implies solving the manufacturing cell formation problem (MCFP), which means identifying groups of machines and associating them with product families so that the intercellular traffic that the products can have within the productive system is minimized. This problem has been approached historically by analyzing the machine-product incidence matrix (A), where each row represents a machine and each column represents a product, with each element aij equal to one if machine i processes product j, and equal to zero otherwise. When this matrix is partitioned arbitrarily, it is usual to have products that remain outside the diagonal blocks (cells), which are called exceptional elements, since they carry out intercellular movements. Papaioannou and Wilson [3] reviewed the approaches between 1997 and 2008 to solve the above problem, proposing taxonomy based on the solution methodologies. It must be kept in mind that the latter approaches have started taking into account production factors other than the incidence matrix, like processing time, demanded production volumes, and operation sequences. Most recent works [4–7] are oriented mainly to the heuristic and metaheuristic approach to solve the problem, without considering other factors that appear when there are exceptional products that carry out intercellular movements, implying that these different production cells are linked with one another due to the precedence restrictions of these exceptional products, without considering their cost and work under way. This means that in most real cases, it is not possible to analyze the manufacturing cells independently at the time of attempting to balance the load of their work stations, and furthermore, since those cells would be related with families of products, it is therefore possible to introduce the balancing concept of multiple manufacturing cells for mixed models. This happens because cellular manufacturing is commonly used in JIT-type productive systems, in which setup times can be reduced in such a way that each cell works by operating over a family of mixed product models. These cells can also be configured in U shape to use the advantages generated by this configuration. An approach to the above situation is the work of Kumar et al. [8], who propose implementing heuristic cell formation, having the capability to handle production data, operation sequence, production volume, and inter-cell cost simultaneously, taking up some of the previously described elements.

precedence diagram. This is possible, because in this type of mixed model assembly lines, the products that are processed in each cell have only small differences in processing times or in the elimination or addition of activities but always keeping the consistency among the precedence of these activities for the different product models. Therefore, the idea is for each manufacturing cell to process a family of products. In turn, using this combined precedence diagram will allow reducing the

First it is necessary to determine the number of machines, qm, required per type

<sup>j</sup>¼<sup>1</sup>djtmj CAPm

� � � � �

= leveled reference capacity used by qm machines of

(1)

P<sup>n</sup>

where qm, number of machines of type m; j product number; dj production volume demanded of each product j within the planning horizon (in units); tmj unit processing time for product j that is processed in a type m machine (in hours per unit produced); CAPm capacity of each type of machine m within the planning

It should be noted that when a machine of a certain type can work simultaneously on a product together with another machine of the same type and qm > 1, the machines will operate as a single "virtual machine," i.e., that qm machines of type m (denoted by i = m) will work simultaneously in it and that the unit processing time of each of these machines will be tij = tmj/qm, provided that the products processed on that virtual machine are similar to each other. On the other hand, for that type of machine in which qm > 1 and which furthermore can operate simultaneously on a product together with another machine of its same type, it is necessary to assign first which products will be processed on each machine of type

� � � � �

number of variables at the time of tackling the model for the formation of manufacturing cells. The methodology proposed for the N-MiMUCFBP problem

qm ≥

considers five consecutive stages that are presented in Figure 1.

A Methodology to Design and Balance Multiple Cell Manufacturing Systems

2.1 Calculation and assignment of the required machines

of machine m. This number is obtained from (Eq. (1)).

Stages in the methodology proposed for the N-MiMUCFBP problem.

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

m. For that purpose, let us use the following notation:

<sup>i</sup> = capacity used by machine i.

horizon (in available machine hours).

• CAPUseful

Figure 1.

• CAPLevel

type m.

41

<sup>m</sup> ¼

P<sup>n</sup> j¼1 d <sup>j</sup>�tmj qm

On the other hand, the problem of balancing N U-shaped lines has been studied mainly by Sparling [9] and Miltenburg [10], both of whom considered that all the cells operate with a common cycle time (C), but they assume that each cell is independent of the rest and furthermore process a single product. The problem of balancing U-shaped lines for mixed product models (denoted by MiMULBP) was proposed for the first time by Sparling and Miltenburg [11], who used the classical combined precedence graph proposed by Thomopoulos [12] and considered as cycle time (C) the quotient of the time period (T) and the total product demand (D). These authors focused only on the problem of balancing a single cell, making in the appendix the observation that it is possible to consider systems with multiple manufacturing cells, although once again they considered that those cells are independent of one another. More recently, in [13, 14], new heuristics are introduced to solve the problem, and Turkay [15] proposes models of integer linear programming (MILP) considering restrictions that express the precedence of the tasks.

A very recent work [16] proposes a novel configuration of assembly lines, namely parallel adjacent U-shaped assembly lines (PAUL), but none of the revised works integrate the balancing of U-lines with the design of the manufacturing cell. In the present chapter, what is being sought is to integrate these problems, proposing a methodology that delivers cells more applicable to reality, thereby introducing the problem of the formation and balancing of N in U-shaped cells for mixed models (denoting it by N-MiMUCFBP). The rest of this chapter is organized as follows: Section 2 introduces a methodology based on a mathematical model for the N-MiMUCFBP; Section 3 illustrates the proposed methodology using a real case, showing its results; and Section 4 gives the conclusions of the study.

## 2. Methodology

The proposed methodology is based on formulating a new model for the problem of balanced formation of production cells. From the viewpoint of formation of the cells, the model must consider the aspects associated with their design, such as processing and preparation of machines, inefficiencies in the handling of materials or inventories of products being processed, and cell imbalance [17], which includes processing times, sequences, and production volumes, directly related with the mixed model assembly line balancing problem (MiMALBP). Because of this, and in agreement with the heuristic proposed by [12] for the MiMALBP, first it is necessary to group the precedence graphs of each final product in a single combined

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

Figure 1.

other factors that appear when there are exceptional products that carry out intercellular movements, implying that these different production cells are linked with one another due to the precedence restrictions of these exceptional products, without considering their cost and work under way. This means that in most real cases, it is not possible to analyze the manufacturing cells independently at the time of attempting to balance the load of their work stations, and furthermore, since those cells would be related with families of products, it is therefore possible to introduce the balancing concept of multiple manufacturing cells for mixed models. This happens because cellular manufacturing is commonly used in JIT-type productive systems, in which setup times can be reduced in such a way that each cell works by operating over a family of mixed product models. These cells can also be configured in U shape to use the advantages generated by this configuration. An approach to the above situation is the work of Kumar et al. [8], who propose implementing heuristic cell formation, having the capability to handle production data, operation sequence, production volume, and inter-cell cost simultaneously,

On the other hand, the problem of balancing N U-shaped lines has been studied mainly by Sparling [9] and Miltenburg [10], both of whom considered that all the cells operate with a common cycle time (C), but they assume that each cell is independent of the rest and furthermore process a single product. The problem of balancing U-shaped lines for mixed product models (denoted by MiMULBP) was proposed for the first time by Sparling and Miltenburg [11], who used the classical combined precedence graph proposed by Thomopoulos [12] and considered as cycle time (C) the quotient of the time period (T) and the total product demand (D). These authors focused only on the problem of balancing a single cell, making in the appendix the observation that it is possible to consider systems with multiple manufacturing cells, although once again they considered that those cells are independent of one another. More recently, in [13, 14], new heuristics are introduced to solve the problem, and Turkay [15] proposes models of integer linear programming

(MILP) considering restrictions that express the precedence of the tasks.

showing its results; and Section 4 gives the conclusions of the study.

2. Methodology

40

A very recent work [16] proposes a novel configuration of assembly lines, namely parallel adjacent U-shaped assembly lines (PAUL), but none of the revised works integrate the balancing of U-lines with the design of the manufacturing cell. In the present chapter, what is being sought is to integrate these problems, proposing a methodology that delivers cells more applicable to reality, thereby introducing the problem of the formation and balancing of N in U-shaped cells for mixed models (denoting it by N-MiMUCFBP). The rest of this chapter is organized as follows: Section 2 introduces a methodology based on a mathematical model for the N-MiMUCFBP; Section 3 illustrates the proposed methodology using a real case,

The proposed methodology is based on formulating a new model for the problem of balanced formation of production cells. From the viewpoint of formation of the cells, the model must consider the aspects associated with their design, such as processing and preparation of machines, inefficiencies in the handling of materials or inventories of products being processed, and cell imbalance [17], which includes processing times, sequences, and production volumes, directly related with the mixed model assembly line balancing problem (MiMALBP). Because of this, and in agreement with the heuristic proposed by [12] for the MiMALBP, first it is necessary to group the precedence graphs of each final product in a single combined

taking up some of the previously described elements.

Mass Production Processes

Stages in the methodology proposed for the N-MiMUCFBP problem.

precedence diagram. This is possible, because in this type of mixed model assembly lines, the products that are processed in each cell have only small differences in processing times or in the elimination or addition of activities but always keeping the consistency among the precedence of these activities for the different product models. Therefore, the idea is for each manufacturing cell to process a family of products. In turn, using this combined precedence diagram will allow reducing the number of variables at the time of tackling the model for the formation of manufacturing cells. The methodology proposed for the N-MiMUCFBP problem considers five consecutive stages that are presented in Figure 1.

## 2.1 Calculation and assignment of the required machines

First it is necessary to determine the number of machines, qm, required per type of machine m. This number is obtained from (Eq. (1)).

$$q\_m \ge \left| \frac{\sum\_{j=1}^n d\_j t\_{mj}}{\text{CAP}\_m} \right| \tag{1}$$

where qm, number of machines of type m; j product number; dj production volume demanded of each product j within the planning horizon (in units); tmj unit processing time for product j that is processed in a type m machine (in hours per unit produced); CAPm capacity of each type of machine m within the planning horizon (in available machine hours).

It should be noted that when a machine of a certain type can work simultaneously on a product together with another machine of the same type and qm > 1, the machines will operate as a single "virtual machine," i.e., that qm machines of type m (denoted by i = m) will work simultaneously in it and that the unit processing time of each of these machines will be tij = tmj/qm, provided that the products processed on that virtual machine are similar to each other. On the other hand, for that type of machine in which qm > 1 and which furthermore can operate simultaneously on a product together with another machine of its same type, it is necessary to assign first which products will be processed on each machine of type m. For that purpose, let us use the following notation:


• wmj = workload to be assigned of model of product j on the type of machine m.

Then the following procedure, shown in Figure 2, is proposed as a formal

different machines, to minimize probable intercellular motions.

A Methodology to Design and Balance Multiple Cell Manufacturing Systems

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

2.2 Preparation of the extended combined precedence diagram

vector tj contains the processing times tij of task i ∈ Vj.

operate simultaneously on the products.

43

In particular, when qm > 1 and the type m machines cannot work simultaneously, the above procedure aims to assign similar products to each of the qm machines, so that they have a workload as close as possible to the leveled load for that type, and it also attempts to have the demand for each product as little fractionated as possible, so that each product is assigned to a single machine when it has sufficient capacity. This procedure is followed in order to not incorporate directly in the mathematical model the alternative processes and routes, because in this way its complexity and number of variables are reduced. Special care must be taken when assigning similar products (with respect to their precedence relations) to the

With the machines required to satisfy the capacity restrictions, and the assignment of each product to them, an extended combined precedence diagram called GUG' must be created, and the weighted average processing times for each machine must be calculated. The process for preparing this diagram will be described now by combining the precedence diagrams of each model in a single precedence diagram where the nodes represent the operations and the arcs represent the precedence restrictions between the operations. A formal description of the combination of n product models in a combined precedence diagram was made by Macaskill [18]. This procedure, adapted to our problem, is summarized as follows: Represent the precedence diagram of product model j by means of the graph Gj = (Vj, Ej, tj), where the set of nodes Vj represents the set of tasks of product model j, the set of arcs Ej represents the precedence relations (a, b) between tasks a, b∈ Vj, and the weighting

As an example, in Figure 3 the precedence diagrams for six models are represented, remarking the virtual machines in which more than one machine

df <sup>j</sup> <sup>¼</sup> dj

G ¼ ð Þ V, E, t , which is derived from the following definitions (Eqs. (3)–(5)):

ti <sup>¼</sup> <sup>X</sup><sup>n</sup> j¼1

E ¼ ∪ n j¼1 V ¼ ∪ n j¼1

df <sup>j</sup> � <sup>d</sup>ð Þ<sup>i</sup>

As a prerequisite for the generation of the combined set of nodes V in Eq. (3), the tasks that are common to different models, even though they have different processing times, receive a consistent number of nodes for all the models. This

Furthermore, by specifying the demanded volumes of each product model within the planning horizon (dj), it is possible to determine the demand fractions df'<sup>j</sup> of each model j with respect to the total demand D of the product mix, where 0 ≤ df'<sup>j</sup> ≤ 1 is fulfilled, and they are calculated by the following equation (Eq. (2)):

D ¼ P

Therefore, the combined precedence diagram can be represented by the graph

dj n <sup>j</sup>¼<sup>1</sup>dj

V <sup>j</sup> (3)

<sup>j</sup> � tij � �∀i∈<sup>V</sup> (4)

Ej∖f g redundant arcs (5)

(2)

assignment rule.


Figure 2. Procedure for assigning products to machines.

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

Then the following procedure, shown in Figure 2, is proposed as a formal assignment rule.

In particular, when qm > 1 and the type m machines cannot work simultaneously, the above procedure aims to assign similar products to each of the qm machines, so that they have a workload as close as possible to the leveled load for that type, and it also attempts to have the demand for each product as little fractionated as possible, so that each product is assigned to a single machine when it has sufficient capacity.

This procedure is followed in order to not incorporate directly in the mathematical model the alternative processes and routes, because in this way its complexity and number of variables are reduced. Special care must be taken when assigning similar products (with respect to their precedence relations) to the different machines, to minimize probable intercellular motions.

#### 2.2 Preparation of the extended combined precedence diagram

With the machines required to satisfy the capacity restrictions, and the assignment of each product to them, an extended combined precedence diagram called GUG' must be created, and the weighted average processing times for each machine must be calculated. The process for preparing this diagram will be described now by combining the precedence diagrams of each model in a single precedence diagram where the nodes represent the operations and the arcs represent the precedence restrictions between the operations. A formal description of the combination of n product models in a combined precedence diagram was made by Macaskill [18]. This procedure, adapted to our problem, is summarized as follows: Represent the precedence diagram of product model j by means of the graph Gj = (Vj, Ej, tj), where the set of nodes Vj represents the set of tasks of product model j, the set of arcs Ej represents the precedence relations (a, b) between tasks a, b∈ Vj, and the weighting vector tj contains the processing times tij of task i ∈ Vj.

As an example, in Figure 3 the precedence diagrams for six models are represented, remarking the virtual machines in which more than one machine operate simultaneously on the products.

Furthermore, by specifying the demanded volumes of each product model within the planning horizon (dj), it is possible to determine the demand fractions df'<sup>j</sup> of each model j with respect to the total demand D of the product mix, where 0 ≤ df'<sup>j</sup> ≤ 1 is fulfilled, and they are calculated by the following equation (Eq. (2)):

$$df\_j = \frac{d\_j}{D} = \frac{d\_j}{\sum\_{j=1}^n d\_j} \tag{2}$$

Therefore, the combined precedence diagram can be represented by the graph G ¼ ð Þ V, E, t , which is derived from the following definitions (Eqs. (3)–(5)):

$$\mathbf{V} = \bigcup\_{j=1}^{n} \mathbf{V}\_{j} \tag{3}$$

$$\overline{t}\_{i} = \sum\_{j=1}^{n} d\mathbf{f}\_{j} \cdot \left( d\_{j}^{(i)} \cdot t\_{i\bar{\jmath}} \right) \forall i \in \mathcal{V} \tag{4}$$

$$E = \bigcup\_{j=1}^{n} E\_j \backslash \{ \text{redundant arcs} \} \tag{5}$$

As a prerequisite for the generation of the combined set of nodes V in Eq. (3), the tasks that are common to different models, even though they have different processing times, receive a consistent number of nodes for all the models. This

• wmj = workload to be assigned of model of product j on the type of machine m.

• Bm = arrangement that contains the models of products similar to each other processed on type m machine, arranged in decreasing order according to wmj.

<sup>j</sup> = fraction assigned to machine i of the demand for product j.

• tij = Unit processing time on machine i due to product j.

• dð Þ<sup>i</sup>

Mass Production Processes

Figure 2.

42

Procedure for assigning products to machines.

prevents assigning these tasks to different stations, which otherwise would need multiple investments in the resources required at each station in which a duplicate task has been assigned. Tasks that are not required by a product model receive a processing time (weight of the node) equal to zero, so the average processing times ti can be calculated simply by weighting every specific task fractionated time according to model dð Þ<sup>i</sup> <sup>j</sup> � tij with its corresponding demand portion df'<sup>j</sup> of the model in Eqs. (4) and (5), which determines the combined precedence restrictions by joining the arc sets of each model. This can lead to redundant arcs (a, b), which represent the transitive precedence relations. An arc is redundant and can therefore be deleted without loss of information, if there is another way from node a to node b by means of more than one arc.

The combined precedence diagram for the example is shown in Figure 4. Note that the redundant arcs are denoted by dotted lines.

A particular action should be considered if there is no consistency among the precedence of the activities, i.e., if there are conflictive precedence relations between different models that lead to a cyclic (that repeats itself over and over) combined precedence graph. To allow a single sequence of task operations, those

loops must be deleted by means of one of the following actions proposed by Ahmadi

• The models must be separated into subsets so that two or more acyclic precedence graphs can be formed. In practice, this leads to machine preparation operations that must be performed every time the production

• With the purpose of assigning the tasks to a single station, the loops in the precedence graphs can be deleted duplicating these nodes. To minimize the number of duplicate nodes and, in this way, reduce the danger of assigning equal tasks to different stations, an optimization problem must be solved [19].

To model what is related to the problem of balancing the N U-shaped lines, we

will consider the concepts developed by Urban [20] to formulate the problem mathematically. For this we set up an auxiliary graph, connecting it with the original combined precedence graph. This is illustrated in Figure 5, denoting with dotted lines the auxiliary combined precedence graphs. If we start in the middle of this extended graph, it is possible to perform assignments to stations forward through the original graph, backward through the auxiliary graph, or simultaneously in both directions, and in this way, it is possible to create stations that have machines at the beginning and at the end of the "U" line. Special care must be taken when joining the auxiliary precedence diagram with the original. For example, final task 15 is joined only with initial task 5, because task 15 is finished only from task 5

2.3 Calculation of the parameters needed for the mathematical model

co <sup>j</sup> � eii<sup>0</sup> <sup>j</sup> � dj � <sup>d</sup>ð Þ<sup>i</sup>

From the input information and the extended combined precedence diagram produced in the previous point, we must calculate the total costs for intercellular transport between machines i and i' (cii'), which are determined by means of

> <sup>j</sup> <sup>þ</sup> <sup>d</sup>ð Þi<sup>0</sup> j 2 !

� aij � ai

<sup>0</sup> <sup>j</sup> (6)

changes from one subset of models to another.

A Methodology to Design and Balance Multiple Cell Manufacturing Systems

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

and Wurgaft [19]:

Diagram of extended combined precedence.

Figure 5.

for model 6 (see Figure 3).

cii<sup>0</sup> <sup>¼</sup> <sup>X</sup><sup>n</sup>

j¼1

Eq. (6):

45

Figure 3. Precedence diagrams for six product models.

Figure 4. Combined precedence diagram for the example.

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

Figure 5. Diagram of extended combined precedence.

prevents assigning these tasks to different stations, which otherwise would need multiple investments in the resources required at each station in which a duplicate task has been assigned. Tasks that are not required by a product model receive a processing time (weight of the node) equal to zero, so the average processing times ti can be calculated simply by weighting every specific task fractionated time

in Eqs. (4) and (5), which determines the combined precedence restrictions by joining the arc sets of each model. This can lead to redundant arcs (a, b), which represent the transitive precedence relations. An arc is redundant and can therefore be deleted without loss of information, if there is another way from node a to node

The combined precedence diagram for the example is shown in Figure 4. Note

A particular action should be considered if there is no consistency among the

precedence of the activities, i.e., if there are conflictive precedence relations between different models that lead to a cyclic (that repeats itself over and over) combined precedence graph. To allow a single sequence of task operations, those

<sup>j</sup> � tij with its corresponding demand portion df'<sup>j</sup> of the model

according to model dð Þ<sup>i</sup>

Mass Production Processes

Figure 3.

Figure 4.

44

b by means of more than one arc.

Precedence diagrams for six product models.

Combined precedence diagram for the example.

that the redundant arcs are denoted by dotted lines.

loops must be deleted by means of one of the following actions proposed by Ahmadi and Wurgaft [19]:


To model what is related to the problem of balancing the N U-shaped lines, we will consider the concepts developed by Urban [20] to formulate the problem mathematically. For this we set up an auxiliary graph, connecting it with the original combined precedence graph. This is illustrated in Figure 5, denoting with dotted lines the auxiliary combined precedence graphs. If we start in the middle of this extended graph, it is possible to perform assignments to stations forward through the original graph, backward through the auxiliary graph, or simultaneously in both directions, and in this way, it is possible to create stations that have machines at the beginning and at the end of the "U" line. Special care must be taken when joining the auxiliary precedence diagram with the original. For example, final task 15 is joined only with initial task 5, because task 15 is finished only from task 5 for model 6 (see Figure 3).

#### 2.3 Calculation of the parameters needed for the mathematical model

From the input information and the extended combined precedence diagram produced in the previous point, we must calculate the total costs for intercellular transport between machines i and i' (cii'), which are determined by means of Eq. (6):

$$\mathbf{c}\_{\text{ii'}} = \sum\_{j=1}^{n} \mathbf{c}o\_{j} \cdot \mathbf{e}\_{\text{ii'}j} \cdot d\_{j} \cdot \left(\frac{d\_{j}^{(i)} + d\_{j}^{(i)}}{2}\right) \cdot a\_{i\mathbf{j}} \cdot a\_{i'j} \tag{6}$$

where

cii' = cost of intercellular transport between machines i and i' (within the planning horizon).

coj = intercellular transport cost of product j.

eii<sup>0</sup> <sup>¼</sup> 1 if machines i and i<sup>0</sup> are directly or indirectly related to the GOUGA graph 0 otherwise �

ti = weighted average processing time of machine i,

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

A Methodology to Design and Balance Multiple Cell Manufacturing Systems

Mmin ¼ ⌊m=Kmax⌋.

Mmax ¼ ⌈m=Kmin⌉.

2.4 Mathematical model

manufacturing cell.

product model.

47

2.4.1 Model assumptions

ti <sup>¼</sup> <sup>X</sup><sup>n</sup> j¼1

GO = original combined precedence diagram. G = (V, E, ti). GA' = auxiliary combined precedence diagram. G´ = (V, E<sup>0</sup>

Mmin = minimum number of machines that a cell must contain,

Mmax = maximum number of machines that a cell must contain,

Smin = minimum number of work stations required, <sup>S</sup>min <sup>¼</sup> <sup>P</sup><sup>m</sup>

Smax = maximum number of work stations required (Smax ≤ m).

The assumptions of the proposed model are presented below:

• The processing times of the tasks are known and constant.

• The operators can work in or out of the "U" cell.

is known with certainty (static problem).

they are capable of operating any machine.

• The setup times of the machines are not significant.

work stations to operate at different production rates.

• Multiple similar models of a product are produced if possible in a single

• Each task of the combined precedence diagram is performed for at least one

• The average time of each task is not greater than the average cycle time C.

• In a manufacturing cell, the precedence restrictions are consistent among the different product models produced in it, i.e., if task a precedes task b in some model, then there is no other model in the cell in which task b precedes task a.

• The precedence graphs of the product models are not fractionated, i.e., all the tasks to produce a product model are joined together directly or indirectly.

• The mix of models, i.e., the demands for models within the planning horizon,

• There are no buffers between the work stations, so it is not possible for the

• The workers are capable of performing any task in the manufacturing cell, i.e.,

• Each machine is assigned to a single work station for each product.

df <sup>j</sup> � tij:

, ti).

i¼1 ti=C.

aij <sup>¼</sup> 1 if machine <sup>i</sup> processes product <sup>j</sup>

0 otherwise �

ai´ j <sup>¼</sup> 1 if machine i´processes product j

0 otherwise �

dj = production volume demanded by product model j.

dð Þi <sup>j</sup> = fraction assigned to machine i of the demand for product j.

dð Þ <sup>i</sup>´ <sup>j</sup> = fraction assigned to machine i´ of the demand for product j.

Other parameters to be used in the model, some of which must be calculated, are the following:

cs = unit cost per work station (within the planning horizon).

V = set of machines of the combined precedence diagram, V ¼ ∪ n V <sup>j</sup>.

j¼1 E = set of precedence relations between the machines that belong to V, E ¼ f g 1, …,e, …, j j E . For example, e = (a, b) is the ordered pair that indicates that machine a precedes machine b immediately,

$$E = \bigcup\_{j=1}^{n} E\_j \backslash \{ \text{redundant } \text{arcs} \}.$$

T = period of time available for planning.

tij = processing time on machine i of product model j.

dj = production volume demanded by product model j.

D = total demand for the product models, <sup>D</sup> <sup>¼</sup> <sup>P</sup><sup>n</sup> j¼1 dj.

C = common average cycle time, C ¼ T=D.

dfj = fraction of total demand for product model j, df <sup>j</sup> ¼ dj=D.

θ = maximum number of production cells to which a multicellular station can belong 2 ≤θ ≤Kmax.

Kmax = maximum number of production cells (Kmax ≤ m). It can be specified based on diagram G, or the maximum bound (K´) proposed by Al Kattan [21] can be used as reference:

$$K' = \frac{W}{\max\_{i=1,\dots,m} \{w\_i\}} \Rightarrow K\_{\max} = \lfloor K' \rfloor$$

where

wi = marginal workload of machine <sup>i</sup>, wi <sup>¼</sup> <sup>P</sup><sup>n</sup> j¼1 df <sup>j</sup> � <sup>d</sup>ð Þ<sup>i</sup> <sup>j</sup> � tij � �

<sup>W</sup> = total workload, <sup>W</sup> <sup>¼</sup> <sup>P</sup><sup>M</sup> <sup>i</sup>¼<sup>1</sup>wi.

Kmin = minimum number of production cells. It can be specified based on diagram G, or a modification of the bound proposed by Al Kattan [21] can be considered:

$$K\_{\min} = \lceil \frac{\max\_{i=1,\dots,m} \{w\_i\}}{(W/m)} \rceil \,.$$

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

ti = weighted average processing time of machine i,

$$
\overline{t}\_i = \sum\_{j=1}^n df\_j \cdot t\_{\vec{\eta}}.
$$

GO = original combined precedence diagram. G = (V, E, ti). GA' = auxiliary combined precedence diagram. G´ = (V, E<sup>0</sup> , ti). Mmin = minimum number of machines that a cell must contain, Mmin ¼ ⌊m=Kmax⌋. Mmax = maximum number of machines that a cell must contain,

$$\mathcal{M}\_{\text{max}} = \lceil m/K\_{\text{min}} \rceil.$$

where

�

Mass Production Processes

�

�

dð Þi

dð Þ <sup>i</sup>´

the following:

planning horizon).

eii<sup>0</sup> <sup>¼</sup> 1 if machines i and i<sup>0</sup>

0 otherwise

0 otherwise

0 otherwise

coj = intercellular transport cost of product j.

aij <sup>¼</sup> 1 if machine <sup>i</sup> processes product <sup>j</sup>

ai´ j <sup>¼</sup> 1 if machine i´processes product j

dj = production volume demanded by product model j.

<sup>j</sup> = fraction assigned to machine i of the demand for product j.

cs = unit cost per work station (within the planning horizon). V = set of machines of the combined precedence diagram, V ¼ ∪

that machine a precedes machine b immediately,

E ¼ ∪ n j¼1

tij = processing time on machine i of product model j. dj = production volume demanded by product model j. D = total demand for the product models, <sup>D</sup> <sup>¼</sup> <sup>P</sup><sup>n</sup>

dfj = fraction of total demand for product model j, df <sup>j</sup> ¼ dj=D.

<sup>K</sup><sup>0</sup> <sup>¼</sup> <sup>W</sup> max i¼1, …, m

wi = marginal workload of machine <sup>i</sup>, wi <sup>¼</sup> <sup>P</sup><sup>n</sup>

T = period of time available for planning.

C = common average cycle time, C ¼ T=D.

belong 2 ≤θ ≤Kmax.

where

46

can be used as reference:

<sup>W</sup> = total workload, <sup>W</sup> <sup>¼</sup> <sup>P</sup><sup>M</sup>

<sup>j</sup> = fraction assigned to machine i´ of the demand for product j.

E = set of precedence relations between the machines that belong to V,

Other parameters to be used in the model, some of which must be calculated, are

E ¼ f g 1, …,e, …, j j E . For example, e = (a, b) is the ordered pair that indicates

θ = maximum number of production cells to which a multicellular station can

Kmax = maximum number of production cells (Kmax ≤ m). It can be specified based on diagram G, or the maximum bound (K´) proposed by Al Kattan [21]

f g wi

<sup>i</sup>¼<sup>1</sup>wi.

Kmin ¼ ⌈

Ej∖f g redundant arcs :

j¼1 dj.

) Kmax ¼ ⌊K<sup>0</sup>

j¼1

f g wi

⌉:

Kmin = minimum number of production cells. It can be specified based on diagram G, or a modification of the bound proposed by Al Kattan [21] can be considered:

> max i¼1, …, m

> > ð Þ W=m

⌋

<sup>j</sup> � tij � �

df <sup>j</sup> � <sup>d</sup>ð Þ<sup>i</sup>

cii' = cost of intercellular transport between machines i and i' (within the

are directly or indirectly related to the GOUGA graph

n j¼1 V <sup>j</sup>. Smin = minimum number of work stations required, <sup>S</sup>min <sup>¼</sup> <sup>P</sup><sup>m</sup> i¼1 ti=C. Smax = maximum number of work stations required (Smax ≤ m).
