Computers & Industrial Engineering 62 (2012) 953–971

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Capability-based virtual cellular manufacturing systems formation in dual-resource constrained settings using Tabu Search Maryam Hamedi a,⇑, G.R. Esmaeilian b, N. Ismail a, M.K.A. Ariffin a a b

Department of Mechanical and Manufacturing Engineering, Faculty of Engineering, University Putra Malaysia, 43400 Serdang, Selangor, Malaysia Department of Industrial Engineering, Payame Noor Universiti, PO BOX 19395-3697, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 20 May 2011 Received in revised form 14 December 2011 Accepted 15 December 2011 Available online 27 December 2011 Keywords: Group technology Capability-based virtual cellular manufacturing systems Dual-resource constrained setting Resource elements Goal programming Tabu Search

a b s t r a c t Formation of Virtual Cellular Manufacturing Systems (VCMSs), as one of the main applications of Group Technology (GT), by presentation of unique and shared capability boundaries of machine tools through defining Resource Elements (REs) creates a good solution for Capability-Based VCMSs (CBVCMSs), which increases opportunities to create systems with more efficient utilizations. Considering workers as the second important resources in Dual-Resource Constraint (DRC) settings makes this problem more serious and critical to research because, in reality, jobs cannot be processed if workers, machines, or both are not available. This paper attempts to form CBVCMSs with DRC settings using a multi-objective mathematical model with a Goal Programming (GP) approach. Using the developed model, parts, machines, and workers are grouped and assigned to the generated virtual cells at the same time. The proposed model is solved through a multi-objective Tabu Search (TS) algorithm to find optimum or near-to-optimum solutions. The validity of the developed model is illustrated by two numerical examples taken from the literature and evaluated through comparing the performance of the CBVCMSs and the original classical CMSs in the System Capacity Utilization (SCU) point of view. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction To accommodate a wide range of production requirements, manufacturing companies are interested in flexible layouts because of maintaining low material handling costs despite fluctuations in the product demand levels and the material handling flows (Benajaafar, 1995; Lahmar et al., 2005). One of the implied assumptions in the modeling and development of Cellular Manufacturing Systems (CMSs) is that the product mix remains stable over the time (Bhandwale et al., 2008; ErhanKesena et al., 2009) and a major deterrent to implement CMSs is changing the layout by entering new demands or variability of them. In the recent decades, it has been tried to develop new layouts with more flexibilities. Many researchers have suggested Virtual Cellular Manufacturing Systems (VCMSs), proposed by National Bureau of Standards (NBS) in the 1980s in USA (McLean et al., 1982). According to McLean et al. (1982), a virtual cell is not identifiable as a fixed physical grouping of workstations, but as data files and processes in a controller. In the both classical and virtual cells, machines are dedicated to a product or a product family, but in ⇑ Corresponding author. Tel.: +60 17339 9418; fax: +60 38656 7122. E-mail addresses: [email protected] (M. Hamedi), g.reza.e@gmail. com (G.R. Esmaeilian), [email protected] (N. Ismail), [email protected]. edu.my (M.K.A. Ariffin). 0360-8352/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2011.12.020

VCMSs the machines are not physically relocated close to each other and there is no need to move the machines to respond to changed demand patterns. Therefore, there is no re-arrangement costs (Balakrishnan et al., 2005). In addition to avoiding the relayout costs, VCMSs are appropriate for small and medium-sized enterprises where physical separation of machines may be constrained by changing practical, technical, and organizational factors (Babu et al., 2000). Cell Formation (CF) is one of the most important stages in the establishment of a CMS (classical, dynamic, and virtual) and includes the formation of manufacturing cells in order to find out which machines dedicated to each cell and part families corresponding to these machines (Moghaddam et al., 2009). Since a virtual cell allows flexible reconfiguration of shop floors in response to changing requirements, a systematic approach is needed for the formation of VCMSs (Fung et al., 2008). Characteristics of manufacturing systems, including design layouts, process plans, and worker skills can be presented in two ways: first, machine-based in such a way that machines are considered as entities; second, capability-based so that machining capabilities include entities and groups of machine tools contained in a manufacturing facility are defined based on the concept of Resource Elements (REs). For the first time, Gindy et al. (1996) illustrated that in a capability-based approach, all exclusive and shared capabilities of machine tools, available in a manufacturing facility, are identified uniquely as the name of Form Generating

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Fig. 1. Representing machines capabilities using REs (Gindy et al., 1996).

Schemas (FGSs). FGSs act as basic capability patterns that can be used for describing the individual/group of machining operation(s). Moreover, Gindy et al. (1996) showed the processing capabilities of a vertical machining center in terms of its FGSs and REs in the form of an example presented in Fig. 1, which demonstrates the relations among FGSs, REs, and machines. The concept of REs helps to representation of unique and shared capability boundaries of machine tools and increases the opportunity to form independent manufacturing cells and efficient utilization of them (Baykasoglu, 2003). In this method, product requirements are generally defined in terms of their processing needs. Once capabilities of available machines are properly defined in terms of REs, parts requiring processing for a RE can equally satisfy it from any machine that has the corresponding RE (Baykasoglu et al., 2001). By using REs in CF applications, it is possible to realize the overlapping capabilities of machines to take advantages of this in configuring cells. To develop VCMSs, if two constraining resources are considered for the outputs, the system will have a Dual-Resource Constrained (DRC) setting in such a way that a part can be processed only if both resources of concern are available. In another word, in a machine-worker DRC system, capacity constraints on output come from both machines and workers (Hottenstein et al., 1998). Since a cell will be constrained to process because of trained worker shortages as well as the lack of available machines, the problem of assigning workers to machines or cells is critical for classical cells in CMSs or virtual cells in VCMSs. In a DRC system, the operators can be reassigned from one machine to another as needed. According to Treleven (1989), two issues can be considered for DRC systems. Design issues, including worker flexibility or crosstraining as the single most important design factor and operational issues involving decisions on dispatching rules as well as due date assignment methods, and decisions on worker allocation rules. Since in DRC systems, the number of machines exceeds the number of workers (Yue et al., 2008), workers need to be multi-skilled to perform more than one function and have to transfer between machines (Bokhorst et al., 2004). Flexible workers can do multifunction or can be cross-trained in different tasks (Cesanı’ et al., 2005). In general, the worker flexibility means the responsiveness of a system to variations in the supply and demand of workers. Yue et al. (2008) argued that the worker flexibility can be viewed from a variety of perspectives, including the level of multi-functionality, the pattern of skill overlaps, and the distribution of skills. Based on Cesanı’ et al. (2005), the worker flexibility contains decisions regarding the number of workers needed, the number of skills for which workers should be cross-trained, and the assignment of workers to machines. This paper focuses on the design issue, and flexible workers are considered. In cellular systems, including VCMSs and classical CMSs, two types of the worker flexibility are identified; inter-cell and intra-cell, whereas in VCMSs workers have more flexibility regarding the inter-cell flexibility (Cesanı’ et al., 2005). From the worker flexibility point of view, workers are different from each other in the case of:

(1) Number of skills – Single-level flexibility, which workers assigned to cells are assumed to have the same degree of cross-training or multi-functionality. In these systems, every worker is trained to operate a machine in a similar number of departments (Felan et al., 2001). – Multi-level flexibility, which each worker can operate a different number of tasks. (2) Task proficiencies – Homogeneous worker flexibility: if it is assumed that workers assigned to a cell or a shop has the same level of proficiency at performing the assigned task (Felan et al., 2001). – Heterogeneous worker flexibility: if it is assumed that workers have a different level of proficiency at performing their assigned tasks (Felan et al., 2001). Considering the above descriptions, the aim of this research is to develop a multi-objective mathematical model to form CBVCMSs focused on the design issue of DRC settings. Workers as the second important resources, which can affect the performance of the proposed system are considered with a multi-level and heterogeneous flexibility. 2. Literature review To cover the previous researches related to CBVCMSs and DRC settings, the literature is reviewed in two main subsections. 2.1. Formation of CBVCMSs Research on VCMSs has attracted considerable attentions in recent years and gained momentum during the last decade. Nomden et al. (2006) surveyed the prior researches in the area of virtual cells and introduced several definitions of VCMSs offered by various researchers. Mostly, researchers agree that VCMSs are generally more efficient than CMSs, especially with respect to the flow performance, production control, quality, average and maximum throughput time, mean and maximum work-in-process, mean and maximum tardiness, and the total marginal cost for a given horizon. In the literature, shorter material traveling distance has been mentioned as the priority of CMSs in comparing with VCMSs for a set of products with deterministic demands, but this problem can be solved using a suitable layout such as distributed layouts as the basic for virtual cells (Hamedi et al., 2011). In addition, according to Baykasoglu (2003), forming a distributed layout prior forming a VCMS helps to improve its performance. Moreover, they presented an example in order to give an idea about the superiority of a capability-based distributed layout over the functional layouts in forming virtual manufacturing cells, but they did not present any method to form a VCMS. The CF is a complex problem, which considers various production factors, such as alternative process routings, operational sequences,

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production volumes, machine capacities, tooling times, and others. This concept has attracted the attention of many researchers in the field of VCMSs design such as Drolet (1989), Babu et al. (2000), Ko and Egbelu (2003), Mak et al. (2005), Slomp et al. (2005), Xambre and Vilarinho (2007) and Fung et al. (2008). (Babu et al., 2000; Drolet, 1989; Fung et al., 2008; Ko and Egbelu, 2003; Mak et al., 2005; Slomp et al., 2005; Xambre and Vilarinho, 2007). In the last decade, the RE based approach has been implied in a few researches to define and solve the problems in manufacturing systems. At first, Gindy et al. (1996) reported the RE approach as a new methodology to describe the capabilities of machine tools and machining facilities. They used REs to capture the processing requirements of components and assessing their similarity. In comparison the RE-based and the conventional machine-based approaches, they showed that using the RE approach leads to component grouping more compact with better matching between processing requirements of components and the capabilities of the machine tools selected for their processing. Baykasoglu et al. (2000) developed a preemptive Goal Programming (GP) model to part-machine cell formation problem using the RE approach and developed a multi-objective Tabu Search (TS) based algorithm for the solution of the model which programmed in Fortran-90. Baykasoglu et al. (2001) used the RE based approach to the formation of classical CMSs not virtual cells in the form of a multi-objective mixed-integer non-linear Problem.. One of the researches, which aimed to present the possibility of improving performance of a CMS by reconfiguring it through virtual cells, belongs to Saad et al. (2002). They defined the reconfiguration as the process of rapid adjustment of the production facility in response to new market conditions. To obtain the performance of the current configuration and possible reconfigurations, they developed a parametric simulation system and introduced a new approach for the reconfiguration of CMSs by virtual cells, which is mainly based on multiple-objective simulation optimization in terms of REs to minimize the interaction between cells. Another attempt in this matters belongs to Fung et al. (2008), who tried to form VCMSs by the RE approach through a multi-stage CF methodology. In their proposed methodology, firstly, appropriate resources for completing production tasks are selected by a Linear Programming (LP) model considering the available capacity and the costs. In order to form virtual cells in a logical mode, the overlapping and interacting functions of machines are analyzed based on the RE approach. 2.2. VCMSs and DRC settings The researches on DRC settings have been conducted and published for over thirty years with the first study in 1967 by Rosser T. Nelson, who considered worker and machine limited production systems (Hottenstein et al., 1998). From that date, a large number of researchers focused on the value of considering workers as the second important constraining resources in DRC systems mostly on job shops and mentioned the impact of worker flexibility, cross-training, degree of centralized control, size of workforce, worker efficiency, and worker assignment rules. Over the past twenty-five years, numerous techniques have been proposed to the cell formation problem. In the most of them, researchers focused just on the single technical objective of developing cells by identifying similar parts and their corresponding machines or considering workers only in terms of their capacities or rate of producing parts, and not in terms of the skills they possess (Normany et al., 2002). In other words, workers have been neglected and manufacturing cells have been considered just in terms of their respective parts and machines, and regards the machines capacities as the factors that limit production. In fact, the research on DRC settings for CMSs has barely begun and a limited number of

955

studies is available in this area (Bokhorst et al., 2004). Bidanda et al. (2005) showed that there was a significant improvement in cells performance if human skills were explicitly considered in the worker training plans and assignment strategies. Supposedly, since VCMSs are newer than CMSs, this shortage in VCMSs is more obvious. Given the fact that workers involve the second major constraining resource and considering the substantial amount of interaction between labor skills and machining technology in a CMS and since many of the advantages associated with cellular manufacturing, classical or virtual, are derived from the worker flexibility, it becomes necessary to extend the researches to DRC systems (Suresh et al., 2005). Bidanda et al. (2005) completed a comprehensive overview on human issues involved in CMSs and determined the importance of eight different human issues in CMSs, including worker assignment strategies, skill identification, training, communication, autonomy, reward/compensation system, teamwork, and conflict management. Suresh et al. (2005) investigated the performance of VCMSs and compared them with functional layouts and classical CMSs in a DRC system context. They considered factors such as ensuring balanced loads for workers, minimization of inter-cell movements of workers, and providing adequate levels of worker flexibility in a pragmatic manner, but in addition to these factors other related factors such as cross-training, worker assignment rules, and worker flexibility should be considered, and their impact needs to be investigated systematically through controlled experiments. Min et al. (1993) and Suresh et al. (2005) proposed cell design procedures in which the complex cell formation problem was solved in two or more phases. The last phase in the both procedures concerned workforce requirements. A basic assumption in the problem formulation of Min et al. (1993) was that workers are linked with the various parts through skill matching factors, which indicated to what extent a worker is able to produce a part. These factors are used for the optimization of the worker assignment problem. Cross training issues were not considered in this work. Suresh et al. (2005), in the last phase of their procedure, addressed various workforce requirements such as the partitioning of functionally specialized worker pools and the required additional training of workers. The need for cross-training was predetermined in their approach by setting minimum and maximum levels for the multi-functionality of workers and the redundancy of machines. They did not determine the need for cross-training analytically. Suer (1996) presented a two-phase hierarchical methodology for the worker assignment and cell loading in worker-intensive manufacturing cells. In that research, the major concern is the determination of the number of workers in each cell and the assignment of workers to specific operations in such a way that workers productivity is maximal. A functional arrangement of tasks was assumed in each cell without considering training and multi functionality problems. Hottenstein et al. (1998) made a survey to determine the effects of cross-trained and flexible workforces on the performance of manufacturing systems during sixteen simulation studies of DRC systems and proposed several offers dealing with the worker flexibility, centralization of control, worker assignment rules, queue discipline, costs of transferring workers, and other selective findings. Askin et al. (2001) focused on the relocation of workers into cells and the training needed for effective CMSs. They proposed a mixed integer GP model for guiding the worker assignment and training process. The model integrated psychological, organizational, and technical factors. They presented greedy heuristics to solve the problem and assumed that the required skills are cell dependent and that workers may need some additional training, again without considering cross-training issues. Felan et al. (2001) focused on the situation where workers receive different levels of training across the various departments as well as possessing different levels of proficiency at each task of shops

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and researched over several major research questions. Norman et al. (2002) presented a mixed integer programming formulation for the assignment of workers to operations in a manufacturing cell. Their formulation permitted the ability to change the skill levels of workers by providing them additional training and taking decisions to balance the productivity and output quality of a cell and the training costs. Slomp et al. (2005) presented a framework for the design of VCMSs, specifically accounting for the limited availability of workers and worker skills. They proposed a GP formulation that first grouped jobs and machines and then assigned workers to the groups to form virtual cells. The objective was to use the capacity as efficiently as possible besides minimizing cells overlapping. In worker grouping, they partitioned the functionally specialized worker pools regrouped into virtual cells and considered factors such as ensuring balanced loads for workers, minimization of inter-cell movements of workers, and adequate levels of worker flexibility in a pragmatic manner. In theory, if all objectives are considered in solution methodologies, problems become more difficult especially when a large number of objectives are considered. In a few cases, all objectives are transformed into a single objective but in the most situations, it is impossible. Therefore, a number of researchers focus only on one objective and do not pay attention to others or relax them, and another group of researchers keeps the nature of problems and applies a multi-objective optimization method based on an approach. In this paper, a GP approach is proposed to form CBVCMSs considering design issues of DRC setting by multi-level and heterogeneous workers flexibility while the most seminal studies have considered machine-based systems involving DRC settings assuming a single level and homogeneous flexibility for workers.

3. The proposed mathematical model A virtual cell formation algorithm in a DRC setting determines the number of virtual cells and the assignment of components, machines, and workers to the cells. Parts are grouped through entering dissimilarity between each two parts to the model. Part dissimilarity level between parts P and P0 considering RE-based processing sequences is determined by using a dynamic programming procedure proposed by Tam (1990) and used by Baykasoglu et al. (2000) to group parts into classical CMSs not VCMSs. Several general characteristics of the proposed model, which are based on the general concept of VCMSs, the RE approach, and the DRC setting are summarized as follows:

 REs are used to define processing capabilities of virtual cells and processing requirements of products.  The similarity among components have been found based on calculating dissimilarity between them and entered to the model as the parameter.  Parts, machines, and workers grouping and virtual cells generation are performed simultaneously.  The minimum and maximum numbers of machines and workers, which can be placed in each cell, are predetermined.  Only multi-skilled workers, who can handle more than one machine, are considered.  Each worker has the different level of job skills (multi-level flexibility).  Workers have a different level of proficiency at performing their assigned tasks (heterogeneous worker flexibility).  The maximum number of virtual cells, which each worker can be assigned, is predefined.  The total capacity of workforces can be changed by hiring and firing of workers.  The cross-training, hiring, and firing costs can be different between workers.  The cross-training cost will be equal to zero if the worker is currently capable of performing the function and a value of one if the worker is incapable of being cross-trained to perform the function.  All data belonging to capacities and times are known and deterministic.  The considered goals are weighted based on decision makers’ opinions and can get different priorities. A goal with priority equal to zero means that goal has been eliminated from the model.  The inputs need to be updated by changing the demands and capacities. 3.1. Model notations 3.1.1. Indices i: Index for resource elements (RE), j: Index for virtual cells (V), p: Index for components (C), m: Index for machines (M), l: Index for workers (W), q: Index for goals (GO),

i 2 I, I = {1, 2, . . . , jIj} j 2 J, J = {1, 2, . . . , jJj} p 2 P, P = {1, 2, . . . , jPj} m 2 M, M = {1, 2, . . . , jMj} l 2 L, L = {1, 2, . . . , jLj} q 2 Q, Q = {1, 2, . . . , jQj}

3.1.2. Parameters  The machine sharing is possible in VCMSs.  There is no limitation regarding the first arrangement of machines and the number and size of families, which are being to process.  Machines are fixed and only their memberships among defined cells are changed.  The number of workers is less than the number of machines (DRC definition).  Each machine can contain one or more capabilities (REs). Before developing a mathematical model for the virtual cell formation problem described in the previous section, the following underlying assumptions are made:  The factory floor is divided into equal grids, and machines are assumed equal space in such a way that each machine just occupies one grid.  Each machine can have multiple copies with different indices in the model.



ami ¼

 bpi ¼

cli ¼



em: el: stpi: tp: tpi:

1; if ith RE is available on machine m 0; otherwise

1; if pth component needs ith RE 0; otherwise

1; if lth worker can work on ith RE 0; otherwise

Capacity of mth machine Capacity of lth worker Setup time needed by the pth part on ith RE Total time available for doing processes of all components Time needed for the pth component on ith RE by machine

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t 0pi : mmin: mmax: wmin: wmax: w0max : maxloadl: crli: hl: fl:

-q : @: p, p0 :

Time needed for the pth component on ith RE by worker Minimum number of machines in a cell Maximum number of machines in a cell Minimum number of workers in a cell Maximum number of workers in a cell Maximum number of cells, which a worker can be assigned Maximum number of machin-REs which lth worker can work in a cell Associated cost of lth worker cross-training to perform ith RE Hiring cost of lth worker Firing cost of lth worker Weight of deviation from qth goal A big number Determined costs by decision makers

3.1.3. Decision variables



X pj ¼

0; otherwise

Y 0mj ¼

Y 00lj ¼



1; if mth machine is assigned to virtual cell j 0; otherwise



1; if lth worker is assigned to virtual cell j 0; otherwise

 Uj ¼  Zl ¼

1; if pth component is assigned to virtual cell j

1; if virtual cell j is generated 0; otherwise

1; if worker l is utilized 0; otherwise

3.1.4. Deviation variables

3.2. Objective functions and constraints 3.2.1. Objective function

minimize

-1 ðDds þ Dþds Þ þ -2

jJj X

Dmj þ -3

j¼1

jJj X jIj X j¼1

!

Dmij

i¼1

jMj jLj X X  þ    þ -4 Dm þ Dm þ -5 Dlom þ Dþlom þ -6 ðDþl þ Dl Þ m¼1

l¼1

jJj  jJj X jIj  jLj   X X X  þ  þ  þ -7 Dl j þ Dl j þ - 8 Dlij þ Dlij þ -9 Dl l j¼1

j¼1

i¼1

 þ  þ -10 Dlol þ Dllol þ -11 ðDþe Þ þ -12 ðDþcr Þ

l¼1

ð1Þ

As mentioned in the Section 2.2, in the developed mathematical model a GP approach is implied to consider objectives and the purpose of optimization is minimizing the deviations from the defined weighted goals, which are prioritized based on decision makers’ ideas through value of weights. Eq. (1) represents the objective function of the proposed model, which includes twelve goals to form CBVCMSs in the DRC setting as minimization of: – – – – – – – – – – – –

the the the the the the the the the the the the

dissimilarity among parts assigned to a virtual cell machine capacity shortage in a virtual cell machine-RE capacity shortage in a cell machine sharing load unbalances at machines in cells worker sharing worker capacity shortage in a cell worker-RE capacity shortage in a cell worker capacity shortage load unbalances at workers in cells workers employing costs worker cross-training costs.

3.2.2. Constraints jJj X jPj X jPj X

dspp0  X pj  X p0 j þ Dds  Dþds ¼ 0

ð2Þ

j¼1 p¼1 p0 ¼1 p0 –p

þ D ds ; Dds þ Dm j ; Dmj þ Dm ij ; Dmij þ D m ; Dm þ D lom ; Dlom þ D l ; Dl þ Dl l ; Dll  Dlj ; Dlþ j þ Dl ij ; Dlij þ D lol ; Dllol þ D e ; De þ D cr ; Dcr

: under and over achivement of dissimilarity goal; : deviations from toal machine capacity in jth cell; : deviation from total machine capacity of ith RE in jth cell; : deviation from number of cells which mth machine is assigned; : under and over achievements of cell load at machines unbalance goal; : deviation from number of cells which lth worker is assigned; : deviation from the capacity of lth worker;

jMj X

Y 0mj  em 

m¼1

X pj  bpi  tpi  dp þ Dmj  Dmþj

p¼1 i¼1

P 0 8j jMj X

ð3Þ

Y 0mj  em  ami 

m¼1

jPj X

X pj  bpi  tpi  dp þ Dmij  Dmþij

p¼1

P 0 8i; j jJj X

ð4Þ

Y 0mj þ Dþm  Dm ¼ 1 8m

ð5Þ

j¼1

: deviation from total worker capacity in jth cell; : deviation from total worker capacity of ith RE in jth cell; : under and over achievements of cell load at workers unbalance goal; : under and over achievements of employee cost goal; : under and over achievements of cross-training cost goal;

jPj X jIj X

PjJj PjPj PjIj

v clj ¼ PjJj

j¼1

j¼1

v clj  PjJj

p¼1

i¼1 X pj  bpi  t pi 0 m¼1 Y mj  em

PjMj

 dp

8j

ð6Þ

!2 PjJj v clj Pj¼1 jJj

j¼1 U j

j¼1

Uj

þ Dlom  Dþlom ¼ 0

ð7Þ

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Fig. 2. The heuristic algorithm to find the initial solution.

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M. Hamedi et al. / Computers & Industrial Engineering 62 (2012) 953–971 Table 1 The data belonging to machines of Example 1 (Baykasoglu et al., 2000). Machine types

1

2

3

4

Copies

1

2a

2b

2c

3a

3b

4a

4b

5a

Capacity

68,000

66,000

66,000

66,000

64,000

64,000

64,000

64,000

64,000

PjJj

j¼1

5

6

7

5b

6

7

64,000

65,000

64,000

!2 PjJj v cllj j¼1 v cllj  PjJj j¼1

PjJj

j¼1 U j

Uj

þ Dlol  Dþlol ¼ 0

jLj X ðHl  Z l Þ þ F l  ð1  Z l Þ þ Dem  Dþem 6

ð13Þ

p

ð14Þ

l¼1 jLj X jIj X jMj X jJj X l¼1

Fig. 3. The functional layout for the Example 1 (Baykasoglu et al., 2000). jJj X

Y 0lj

þ

Dþl



Dl

¼ 1 8l

ð8Þ

j¼1 jLj X jMj X jIj X

0 Y j000 lim  kli  ami  el 

l¼1 m¼1 i¼1 

jPj X jIj X

X pj  bpi  dp  t0pi

p¼1 i¼1 þ

þ Dl j  Dl j ¼ 0 8j jLj X jMj X

ð9Þ

0 Y j000 lim  kli  ami  el 

l¼1 m¼1

jPj X



X pj  bpi  dp  t 0pi þ Dlij

p¼1

þ

 Dlij

¼ 0 8i; j e0l 

ð10Þ

jJj X jMj X jIj X



þ

Y j000 8l lim  t li þ Dll  Dll P 0

ð11Þ

j¼1 m¼1 i¼1

PjJj PjPj PjIj

v cllj ¼

j¼1

p¼1

i¼1 X pj 00 l¼1 Y lj

PjLj

 bpi  t 0pi  dp  e0l

8j

ð12Þ

 þ crli  Y j000 lim þ Dcr  Dcr 6

p0

ð15Þ

i¼1 m¼1 j¼1

Eqs. (2)–(15) demonstrate the goal constraints in such a way that Eq. (2) belongs to the dissimilarity goal and constrains the model to group the most similar components in term of their RE requirements and sequencing, which have been entered to the model. Eqs. (3) and (4) have been taken from Baykasoglu et al. (2000) used for classical CMSs formation. In this model, Eq. (3) presents machine-capacity goal constraint to calculate the deviation from the total available capacity in each virtual cell. In ideal conditions, deviations are equal to zero (i.e. satisfaction of the capacity goal). This means that there is potentially no excess capacity in the system (i.e. total capacity in each cell is equal to total processing time requirement from each cell). Negative deviation means that potentially available capacity in the cell is not sufficient (i.e. total capacity in some cells is less than total processing time requirements). Machine RE-capacity goal constraint is illustrated in Eq. (4) to calculate deviations from total available capacity for individual processing capability units (REs) in each cell. Eq. (5) limits the machine-sharing goal. Based on the VCMS definition, the machine sharing among cells is possible but to avoid increasing set up times, assigning each machine to one cell is preferred. Eq. (6) calculates loads assigned to each virtual cell in respect of the total capacity of that cell. By use of the Eq. (6), Eq. (7) is the constraints belonging to load balancing on cells based on machines goal constraint. To balance the load among formed cells, the load of each cell should be higher than a percent of overall load of all cells.

Table 2 The data belonging to parts for the Example 1 (Baykasoglu et al., 2000). Parts

RE1

RE2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 1

1 1

RE3

RE4

RE6

1

RE8

RE9

RE10

RE11

1 1 1

1

1

1

1 1

1 1 1

1 1

1 1

1

1 1

1 1

1

1 1 1

RE7

1

1

1

RE5

1

1

1 1 1

1 1

1 1

1 1

1 1

1

1

1 1

1

1

1

1 1

1

1

Demand 3000 1000 2500 1520 1480 3500 1000 2000 3000 2000 4500 1000 3000 2500 2500 1900 2400 1200 1300 3000

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M. Hamedi et al. / Computers & Industrial Engineering 62 (2012) 953–971

Table 3 The primary data belonging to workers for the Example 1. Li

RE1

RE2

W1 W2 W3 W4 W5 W6 W7 W8 W9

1 1

1

RE3

RE4

1

1

RE5

RE6

RE7

1

1

1

1

1 1

1 1 1

1 1 1

1 1

1 1

1

1

RE10

RE11

Capacity

1

1

50,000 50,000 50,000 50,000 50,000 50,000 50,000 50,000 50,000

1 1

1 1

1 1

1

RE9

1 1

1 1

1 1

1

1 1

RE8

1 1

1 1 1 1

1

1 1 1 1

1 1

1

Table 4 The costs belonging to workers for the Example 1. CR

RE1

RE2

W1 W2 W3 W4 W5 W6 W7 W8 W9

100 0

400

RE3

RE4

RE5

RE6

RE7

0 300

250

0

0

200

0 200 100

350 200 0

0

200 0

300 150

0

100 0

200 100 150

0 100

0 400

Y 0mj

X pj  bpj 6

jMj X

ð16Þ ð17Þ

Y 0mj

 ami

8p; j; i

ð18Þ

Y 000 limj

6

Y 0mj

 ami

8l; i; m; j

ð19Þ

m¼1 0 Y 000 limj 6 Y mj  ami jLj X jMj X

Y j000 lim 6

l¼1 m¼1 jMj X jIj X

8l; i; m; j

jPj X

ð20Þ

Firing cost 100 100 100 100 100 100 100 100 100

0 150

0 100 350 300

150 0 100 150

0 200

150

Eq. (20) ensures that a worker can be assigned to a machine if that machine has the common RE. Eq. (21) presents that to assign a worker to a RE in a cell, a part that needs that RE necessarily must be existed. Eq. (22) guarantees that if a worker is assigned to a cell, it can be allocated to available REs of that cell. At the end of this part of constraints, Eq. (23) constrains that an assigned worker to a machine is a cell can be allocated to a RE, which has the required skill to do that. Eqs. (24)–(34) constrain the quantity of components of the proposed VCMS. jJj X

X pj ¼ 1 8p

ð24Þ

j¼1 jJj X

Uj 6

jMj mmin

ð25Þ

Uj P

jMj mmax

ð26Þ

Uj 6

jLj wmin

ð27Þ

Uj P

jLj wmax

ð28Þ

j¼1 jJj X j¼1 jJj X j¼1

j¼1 jMj X

Y 0mj  ðU j  mmax Þ 6 0 8j

ð29Þ

m¼1

X pj  bpi  @

8i; j

ð21Þ

jMj X

Y 0mj 6 0 8j

ð30Þ

Y 00lj  ðwmax  U j Þ 6 0 8j

ð31Þ

ðU j  mmin Þ 

m¼1

8l; j

ð22Þ

m¼1 i¼1 00 Y j000 lim 6 Y lj  cli

Hiring cost 1100 1100 1100 1100 1100 1100 1100 1100 1100

0

p¼1 00 Y j000 lim P Y lj

RE11 200

100

jJj X

m¼1 jMj X

RE10 0

100

300 350

However, in the proposed model workers are allowed to share between more than one virtual cell, to avoid increasing the traveled time and delays, assigning each worker to one cell is preferred that this set of constraints have been presented in Eq. (8). Eq. (9) belongs to the virtual cell capacity goal constraint, which presents the capacity of workers should be near the required capacity by parts. After that, cell capacity goal constraints based on RE-labor are illustrated in Eq. (10). Eq. (11) controls the capacity of each worker individually. Eq. (13), such as machines load balancing on virtual cells, by use of the definition of variance in Eq. (12) tries to balance loads on virtual cells based on labors capacities. Worker-employee cost goal constraints based on hiring and firing costs have been demonstrated in Eq. (14). Finally, Eq. (15) belongs to worker-employee cost goal constraints in terms of cross-training costs. Eqs. (16)–(23) represent the dependency constraints of the components in a CBVCMS, which is going to be formed:

8p; j P U j 8m; j

RE9

300 0

200

X pj 6 U j

RE8 0 500

jLj X l¼1

8l; i; m; j

ð23Þ

Eq. (16) guarantees that each part can be assigned to a virtual cell only if that cell is formed. Eq. (17) presents the dependency of assigning of each machine to a cell on the formation of that cell. Eq. (18) ensures that a part is assigned to a virtual cell if the RE corresponding to each required operation of that part is available in the cell. Eq. (19) controls that a worker can be assigned to a RE of a cell if at least one machine in that cell has that RE.

ðwmin  U j Þ 

jLj X

Y 00lj 6 0 8j

ð32Þ

l¼1 jJj X

Y 00lj 6 wmax  Z l

8l

ð33Þ

j¼1 jJj X jIj X j¼1

i¼1

Y j000 lim 6 max loadl

8l; j

ð34Þ

961

3 2 3

4 3 6 5

5

4 7

4 5

7 6

4 8

5 7

3 4

5

4 5

8 5

3 6

6 3 8 5 6 8

5 4

t 0pi

p

X pj

i

2

2 1

1

3 2

2 3 3 1 2 2

3 3 2 2

3 2

ð35Þ

jJj X jMj X jIj X



þ

0 Y j000 8l lim  t li  el þ Dll  Dll 6 0

ð36Þ

5 4

4

3

5

1 3 2 5

5 4 4 3 3 2 1 5

4

3 6 5 3 2 5

5 4 3 3

5 3

3

4

5

5

3

2 f0; 1g

ð40Þ

U j 2 f0; 1g

ð41Þ

Z l 2 f0; 1g Dds ; Dþds ; Dmj ; Dmþj ; Dmij ; Dmþij ; Dm ; Dþm P 0

ð42Þ

 þ  þ  þ Dlo ; Dþlo ; Dl ; Dþl ; Dlj ; Dlj ; Dlij ; Dlij ; Dll ; Dll ; De ; Dþe ; Dcr ; Dþcr ;  þ Dloadl ; Dloadl P 0

ð43Þ ð44Þ

Eqs. (37)–(42) present that the main decision variables can accept only binary values. At the end, Eqs. (43) and (44) present that all deviation values of goals are positive.

5

4 9

4

5

3 3 3

ð39Þ

Y j000 lim

2 2 2

Y 00lj 2 f0; 1g

2 4 6

ð37Þ ð38Þ

stpi

6

5

6

4

5

4

7 5

7 3 5

6 7

5

5 4 4 3 8 5

4 3 8 5 4

t 0pi tpi

Eq. (35) presents that all components must be processed in a time less than or equal to the due date considered for doing processes of all components. Eq. (36) guarantees that the total time, which each worker spends in all REs, must not exceed the capacity of that worker.

Y 0mj 2 f0; 1g

t0pi

4

RE10

tpi stpi t0pi

RE9

tpi stpi t0pi

RE8

tpi stpi t0pi tpi stpi

RE7

j¼1 m¼1

X pj 2 f0; 1g

tpi

RE6

jJj X jMj X jPj X jIj X

j¼1 m¼1 i¼1

t0pi

stpi

RE5

j¼1 p¼1 i¼1

X pj  bpi  ðtpi þ t 0pi Þ  dp þ

 bpi  Y 0mj  a0mi  st pmi 6 tp

tpi

RE4

jJj X jPj X jIj X

2 5

3

6 4 3 2 2 6

3

1 7

2

5

2

3

4

6

4

7

5

5 4 4 2 3 1 1 8 6 2 3 1 2 6 5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

RE3 RE2 RE1

tpi

t 0pi

stpi

tpi

t 0pi

stpi

tpi

t0pi

stpi

4. Solution method

Component

Table 5 Data belonging to times needed by each component to machines (tpi), workers ðt 0pi Þ, and set up (stpi) for the Example 1.

Eq. (24) controls that each part can be assigned to only one virtual cell. Eqs. (25)–(28) present that the number of formed virtual cells must be among maximum and minimum number of virtual cells which have been determined by minimum and maximum number of allowed machines and allowed workers in each cell. Eqs. (29) and (30) limited the number of assigned machines to each cell between two values. Eqs. (31) and (32) present that if a cell is generated, the number of assigned workers to a cell will be constrained between two maximum and minimum numbers of allowed workers to each cell. Eq. (33) present that if a worker is employed, the number of cells, which he/she can be assigned, will be limited. Finally, Eq. (34) limits the number of assigned REs, which each worker in each cell can do.

2

4 9

tpi stpi

RE11

t0pi

5

stpi

M. Hamedi et al. / Computers & Industrial Engineering 62 (2012) 953–971

To solve any type of Non-Linear Programming (NLP) models such as the model developed to form CBVCMS in DRC settings in this research, using exact methods such as Lingo is not helpful, because, even if they do not have any limitation regarding the size of the problem, they will not guarantee finding a global optimization solution. In other words, these methods give a local optimum solution that there is no certainty to be a global optimum or near-toglobal optimum solution also. Since the developed model is MINLP, to find the optimum solution the best method is using meta-heuristic algorithms because they guarantee finding a near-to-global optimum solution through decreasing the problem complexity without any limitations regarding the problem size and can produce strong and efficient algorithms that compute approximate solutions of high quality in realistic computation times (Ibaraki et al., 2005). There are several meta-heuristic methods available in the literature to solve CF problems, including Simulated Annealing (SA), Genetic Algorithm (GA), and TS. SA based algorithms have appeared in the cell formation literature more often than the other two approaches but TS because of its potential abilities to solve multi-objective solutions selected as the most appropriate method

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M. Hamedi et al. / Computers & Industrial Engineering 62 (2012) 953–971

Table 6 Dissimilarity between components for the Example 1. P1

P2

P3

P4

P5

P6

P7

P8

P9

P10

P11

P12

P13

P14

P15

P16

P17

P18

P19

P20

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20

0 0.75 2 2 1.9 2 2 2 1.25 1.375 2 2 2 2 0.667 1.375 2 2.5 1.375 2

0.75 0 2 2 2 2 2 2 1.25 1.875 2 2 2 2 0.667 1.875 2 2.5 1.375 2

2 2 0 1.87 1.75 0.75 2 2 1.4 2 0.75 2 1.4 1.75 2 2 1.25 2.5 1.375 1.9

2 2 1.87 0 1.375 1.375 1.375 2 1.875 1.5 1.875 1.375 1.167 0.667 1.5 1.5 1.875 1.9 1.333 1.375

1.9 2 1.75 1.37 0 1.9 2 2 1.9 1.375 1.9 2 1.9 1.25 2 1.375 1.9 2.5 1.875 1.4

2 2 0.75 1.37 1.9 0 1.4 2 2 2 1.4 1.9 1.9 1.25 2 2 1 1.917 2 1.75

2 2 2 1.37 2 1.4 0 1.25 2 2 1.9 1 1.25 1.4 2 2 1.9 0.625 2 1.25

2 2 2 2 2 2 1.25 0 2 2 1.9 1.75 1.9 2 2 2 2 0.625 2 1.9

1.25 1.25 1.4 1.87 1.9 2 2 2 0 2 1.4 2 1.4 1.9 0.667 2 2 2.5 0.667 2

1.37 1.87 2 1.5 1.375 2 2 2 2 0 2 2 2 2 1.5 0 2 2.5 1.5 2

2 2 0.75 1.87 1.9 1.4 1.9 1.9 1.4 2 0 1.4 1.4 1.9 2 2 1.9 2.417 1.375 1.4

2 2 2 1.37 2 1.9 1 1.75 2 2 1.4 0 1.25 1.9 2 2 1.9 1.625 2 0.75

2 2 1.4 1.16 1.9 1.9 1.25 1.9 1.4 2 1.4 1.25 0 1.75 2 2 1.9 1.8 1.375 1.4

2 2 1.75 0.66 1.25 1.25 1.4 2 1.9 2 1.9 1.9 1.75 0 2 2 1.25 1.917 1.875 1.75

0.66 0.66 2 1.5 2 2 2 2 0.667 1.5 2 2 2 2 0 1.5 2 2.5 0.833 2

1.37 1.87 2 1.5 1.375 2 2 2 2 0 2 2 2 2 1.5 0 2 2.5 1.5 2

2 2 1.25 1.87 1.9 1 1.9 2 2 2 1.9 1.9 1.9 1.25 2 2 0 2.417 2 1.25

2.5 2.5 2.5 1.9 2.5 1.917 0.625 0.625 2.5 2.5 2.417 1.625 1.8 1.917 2.5 2.5 2.417 0 2.5 1.8

1.37 1.37 1.37 1.33 1.875 2 2 2 0.667 1.5 1.375 2 1.375 1.875 0.833 1.5 2 2.5 0 2

2 2 1.9 1.37 1.4 1.75 1.25 1.9 2 2 1.4 0.75 1.4 1.75 2 2 1.25 1.8 2 0

Objective function

DS

360000 350000 340000 330000 320000 310000

Initial objectives TS objectives

300000 290000 9

8

7

6

5

4

3

Maximum number of worker Fig. 4. Analyzing the effect of maximum allowed number of workers on the objective function for the Example 1.

to solve the developed model. Therefore, it is employed to solve the model of this paper over different data sets.

Table 7 The CMS developed by Baykasoglu et al. (2000). Cells

Parts: Xpj

Machines: Y 0mj

Workers: Y 00lj

CPU

Cell 1 Cell 2 Cell 3

4, 7, 8, 12, 13, 18 1, 2, 9, 10, 15, 16, 19 3, 5, 6, 11, 14, 17, 20

1, 2, 12 5, 9, 7, 3 4, 8, 10, 11, 6

– – –

0.74 0.76 0.89

Table 8 The report of the initial heuristic algorithm for the Example 1. Cells

Parts: Xpj

Machines: Y 0mj

Workers: Y 00lj

CPU

SCU

Cell 1 Cell 2

3 4 5 6 7 8 18 1 9 10 12 13 14 15 2 11 16 17 19 20

23469 45689

12345 46789

0.6755 0.7360

0.9012

2 4 8 10 12

2349

0.5355

Cell 3

4.1. Multi-objective TS In general, TS is started with an initial solution and performs a neighborhood search of the current solution. The local neighborhood search technique most commonly used in TS algorithms is the steepest going down pair wise exchange meta-heuristic. Some candidate solutions will be generated from the neighborhood search technique by exchanging pairs of the areas. All possible exchanges are considered and defined as the neighborhood of the current solution. Then each neighbor is evaluated (or moved), and the best acceptable moves are chosen. The best admissible move is defined as tabu (tabu restricted) for the next iterations so that it is recorded in the tabu list, to keep away from cycling back to a local optimum. The admissible move is either a move that is non-tabu or tabu, and has an objective function value better than

Table 9 The report of TS for the Example 1. Cells

Parts: Xpj

Machines: Y 0mj

Workers: Y 00lj

CPU

SCU

Cell 1 Cell 2 Cell 3

2 3 6 7 8 18 1 5 9 10 12 15 4 11 14 16 17 19 20

23469 45689 2 4 8 10 12

12345 46789 459

0.5933 0.7550 0.5993

0.9182

963

M. Hamedi et al. / Computers & Industrial Engineering 62 (2012) 953–971 Table 10 Outputs of the mathematical model for the Example 1. Initial solution Machines

Optimum or near-to-optimum solution by the TS Worker-RE

Machines

Workers

The required extra capacity for machines, workers, and REs M1 0 W1 10,600 RE1 0 M2 0 W2 63,520 RE2 6500 M3 0 W3 93,400 RE3 11,400 M4 81,700 W4 20,200 RE4 4700 M5 0 W5 6000 RE5 12,960 M6 0 W6 8000 RE6 0 M7 0 W7 0 RE7 12,000 M8 23,200 W8 2500 RE8 53,300 M9 12,960 W9 23,000 RE9 17,000 M10 0 – RE10 0 M11 0 – RE11 0 M12 0 – –

0 21,600 0 2320 37,300 65,200 21,600 21,600 41,000 1000 15,600

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12

0 0 0 81,700 0 0 0 23,200 0 0 0 0

W1 W2 W3 W4 W5 W6 W7 W8 W9 – – –

Total

227,220

Total

104,900

Total

Total

Objective function

117,860

Workers

REs

227,220

345000 340000 335000 330000 325000 320000 315000 310000 305000 300000 295000

1

Total

6

11

Machine-RE

117,860

16

21

26

31

36

41

46

51

56

61

66

71

REs

Machine-RE

Worker-RE

0 0 0 109,080 56,700 7840 0 0 32,900

RE1 RE2 RE3 RE4 RE5 RE6 RE7 RE8 RE9 RE10 RE11 –

0 6500 11,400 4700 0 0 12,000 53,300 17,000 0 0

0 14,340 11,300 9500 40,700 47,480 21,600 31,600 30,000 0 0

206,520

Total

104,900

206,520

76

81

86

91

96

Number of iteration Fig. 5. The convergence graph of the TS algorithm for the CBVCMS of Example 1.

the best solution found so far. The solution obtained from the best admissible move becomes the current solution and is the starting solution at the next iteration. This procedure is repeated until a stopping criterion is met. The TS-based solution procedure of any problem, which is going to be solved by TS, including the CBVCMSs typically consists of two phases:  A construction phase to generate an initial solution, which can be a random feasible solution that satisfies hard constraints or a known initial feasible solution.  An improvement phase to improve the initial solution. Heuristic algorithms have become tools for designing good meta-heuristics. To construct the initial solution, a heuristic algorithm is used without any limitation regarding the size of the problem is terms of the number of variables and constraints, which results in an initial feasible solution. The proposed initial heuristic algorithm has been presented in Fig. 2. After finding the initial solution, elements of the proposed TS algorithm for solving the CBVCMSs based on the improvement phase are considered as follows: 4.1.1. Generation of neighborhood solutions or moves Since each solution is generated by a move, generation of neighborhood solutions requires generation of moves. Because the virtual cell formation and assigning the resources are based on parts processing requirements, moves are defined based on changing the assignment of parts from the current virtual cell, which is referred to as the move’s source virtual cell, to another virtual cell, which is referred to as the move’s destination virtual cell The start-

ing cell is a cell involving the highest load based on the defined in the mathematical model. After selecting the first cell, the combination of all parts and possible destination cells involve the neighborhood solutions and will be checked. Moving a part may require changing the membership of several machines to virtual cells or changing the assigning workers based on the process plane of the moved part. Therefore, any possible way to move a part leads to generation of one or more neighborhood solutions. All neighborhood solutions satisfy the mathematical model constraints, including the goal constraints, the dependency constraints, and the variable constraints as presented in the mathematical model Dependent on the constraints and objectives, it may be required just one single-mover or more than one single-move in which in each move all important factors must be considered. For example, the second move can be carried out if the upper bound of the size of the destination cell of the first move is exceeded by a machine just moved in. The destination cell of the second move is the source cell of the first single-move.

4.1.2. Selection of the current best solution vector The objective function defined in the mathematical model is the main criterion to evaluate each configuration. Based on the GP logic, selection of the current best solution vector from neighborhood solutions is performed in the following manner: 4.1.2.1 For each neighborhood solution vector, the goal deviations are computed by using the goal constraint equations. 4.1.2.2 The summation of all weighted goal deviations is calculated for each neighbor’s solution vector, and a solution with the minimum result is chosen. If there is more than one alternative neighbor solution with the same amount, the deviation of a goal with the biggest weight is checked, and so on.

1 1 1 1 1 1 1

1 1

1 1

1 1

1 1 1

1 1 1 1

1 1

1 1 1

1 1

1 1 1

1 1

1 1 1

1 1

1 1 1

1

140,000 140,000 140,000 140,000 140,000 140,000 140,000 140,000 140,000 140,000 140,000 140,000 140,000 140,000 140,000 140,000 140,000 140,000 140,000 140,000 140,000 140,000 1 1

1

1 1 1 1

1 1 1

1 1 1 1

1 1 1

1

1 1 1

1

4.1.4. Tabu length Before forming the tabu list, tabu length must be defined. There is not any general guideline to determine the optimal size of the tabu list. In practice, the search process may be constructed with different tabu list sizes. The output of this research presents that tabu list sizes are proportional to the problem sizes. Accordpffiffiffiffiffi ing to the literature, l ¼ M is a suitable length for tabu lists in cellular systems (Moghaddam et al., 2005). Therefore, l previous current solution vectors are considered as tabu because to allow one of them may trap the algorithm into cycling through recent previous moves. The tabu list is circular, and when it is full, a new part from the selected virtual cell replaces the head of the list.

1 1 1 1 1 1

1

1 1 1 1 1 1

1 1

1 1

1 1

1 1

1 1

1 1 1

1 1 1

1 1

4.1.3. Updating the best-known solution vector The initial feasible solution vector is also recorded as the bestknown initial solution vector. By applying the methodology described above, the best known solution vector is updated in each iteration if improved. In another word, the best move is selected from the set of candidate moves, and a new solution is generated. Then this new allocation is updated and stored as the new best solution. In addition, the selected move is set as tabu. The process of selecting the best move from each candidate arc occurs at the neighborhood level where the improvement level of each move is computed and the best move selected. When a move can lead to a better solution than the best solution obtained so far, the aspiration criterion will be applied, and the move is allowed, even if it is in a tabu status. In the case of this paper methodology, part p in the source virtual cell j is allowed to move to the destination virtual cell j0 notwithstanding is in tabu status, if this move decreases the objective function (of): {(p, j) – tabujof0 > of}.

4.1.5. Tabu list After completing each iteration of the TS algorithm, the best admissible move is defined as tabu (tabu restricted). If the number of tabu moves exceeds from a certain number, which is tabu length, the oldest move should leave the tabu list. The tabu list is a set of two-dimensional arrays denoted as tabu for a part, which is used to keep a record of the status of the tabu restriction. If the current move is exchanging part i from jth virtual cell (the source virtual cell) to j0 th virtual cell (the destination virtual cell), this move will be defined as tabu. In this research approach, single attributes are set tabu when their complements have been part of a selected move. A shift (p, j, j0 ) may be described by the attributes ‘‘part p leaves the virtual cell j00 and enters to the virtual cell j0 . To avoid reversing the move, the attribute (p, j, j0 ) is set to tabu and is written into a cyclic tabu list following the FIFO (first-in first-out) strategy. Moves in the tabu list are forbidden (tabu). Therefore, the algorithm selects an admissible move in each iteration, which leads to the best possible cycle time among all moves that are not tabu.

1

1 1

RE1 RE2 RE3 RE4 RE5 RE6 RE7 RE8 RE9 RE10 RE11 RE12 RE13 RE14 RE15 RE16 RE17 RE18 RE19 RE20 RE21 RE22 RE23 RE24 RE25 RE26 RE27 RE28 RE29 RE30 RE31 RE32 Capacity

M. Hamedi et al. / Computers & Industrial Engineering 62 (2012) 953–971

trM1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 M13 M14 M15 M16 M17 M18 M19 M20 M21 M22

Table 11 Machines capability and capacity for the Example 2 (Baykasoglu, 1999).

964

4.1.5.1. Aspiration criteria. The aspiration criterion is a very important rule in TS in which a move is allowed to get out of tabu status temporarily and makes the quality of the resulting solution less dependent on the tabu size. According to an aspiration criterion, any move that improves the best-known solution is accepted, even if the move is tabu. Aspiration criteria are introduced in TS to determine when tabu activation rules can be overridden, thus removing a tabu arrangement and applicable to a move. 4.1.6. Termination The stopping criteria play a major role in determining the computation time and the quality of solution obtained by TS. On TS algorithms, if a previously determined number of iterations is reached, or if there is no improvement in the best-known solution

M. Hamedi et al. / Computers & Industrial Engineering 62 (2012) 953–971

4 RE(24,25,26,27,28,29,30,31) M5 RE(24,25,26,27,28,29,30,31) M17 RE(25,26,28,29,30,31) M1 RE(3) M6 RE(8,9,11,13) M11 RE(5)

M19 RE(24,25,26,27,28,29,30,31) M18 RE(24,25,26,27,28,29,30,31) M15 RE(18,19,20,22,23) M13 RE(12,16) M7 RE(8,9,11,13) M12 RE(5)

965

M9 M10 RE(1,2,4,6,7) RE(1,2,4,6,7) M20 M21 RE(1,2,4,6,7) RE(1,2,3,4,6,7) M16 M2 RE(18,19,20,22,23) RE(18,19,22,23) M22 M14 RE(21) RE(17) M8 RE(8,9,11,13) M3 RE(9,10,14,15)

Fig. 6. The functional layout for the Example 2.

in the last predetermined number of iterations, the algorithm terminates. In this paper, both of two mentioned criteria are considered to terminate the algorithm and announce the nearto-optimum solution found. Since we do not have any idea beforehand about the number of iterations required to reach a good solution for the problem developed in this paper, the number of iterations is achieved by simulation for each case. The quality of the best solution found by the TS depends on several factors, including the initial solution of the problem and the size of the Tabu list, which both can make large differences in the final solution. 4.2. TS coding The customized TS algorithm developed for the proposed model in this paper is programmed in MatlabÒ software and different test problems with various sizes can be solved for testing the model validity and the efficiency of the program. 5. The numerical examples Developing a CBVCMS in DRC setting is a new problem and needs special data, which cover worker’s data in addition to machines and parts. The most similar examples are belonging to Baykasoglu (1999) and Baykasoglu et al. (2000) who tried to form classical CMSs based on machines capabilities but in SRC settings. 5.1. Example 1 The first example is from Baykasoglu et al. (2000) which presents a shop containing seven types of machines with one or multiple copies, which in total includes twelve machines. Table 1 presents the number of copies, machine-RE relations, and capacity of each machine. Fig. 3 illustrates the first physical arrangement of machines in the form of a functional layout. In this figure, heavy solid lines draw the borders of the same departments and the number of machines presents the machines located in each functional area. In addition to the machine-RE matrix and the first location of machines, the data belonging to the parts requirements based on REs, parts demand, and machines capacity are available in the case taken from Baykasoglu et al. (2000). In the example, there are twenty components proceed by eleven REs. Table 2 demonstrates the data of parts, including the part-RE relation and the demand per each par. Since in this paper the DRC setting is considered, the authors had to add the data belonging to the workers just to present how the proposed methodology works. Table 3 presents the matrix of worker-REs in such a way that values of one (1) illustrates a worker

is currently capable of performing a certain function, or able to being cross-trained to perform the function and the values of zero present that the worker is currently incapable of performing that function and also cannot be cross-trained to perform the function. The associated costs of the cross-training are presented in Table 4. Among values equal to one in Table 3, for functions which each worker is currently capable of performing them, the cross-training costs will be zero and for functions that workers can perform them just after trainings, the cross-training costs are more than zero. Moreover, since the considered workers include both current workers and potential hires, for each worker, the hiring/firing cost (h/f) is provided in Table 4, too. In CBVCMSs, since all requirements and capabilities in terms of components, machines, and workers are presented in the form of REs, for each requirement, there are some options to perform, which are chosen based on the factors of cost and time. Table 5 presents all times, which each part needs to perform each operation on required RE, including times needed to machines, workers, and set up. Parts are grouped considering their similarities (presented in Table 6) and constraints of the model simultaneously with grouping machines and assigning them to cells, which is done through developing the TS. All relations between part-RE, machine-RE, and worker-RE are entered in the form of 0–1 matrixes. First, the initial solution is found based on the initial heuristic algorithm coded in MatlabÒ and second, the initial solution is improved by the TS to find the near-to- global optimum solution. To solve the problem in DRC setting by TS, the characteristics defined by Baykasoglu et al. (2000) are followed, and the developed TS is run with tabu length of 15, maximum and minimum number of parts and machines equal to 9, 4 and 5, 3 respectively. For the TS proposed in this paper, the problem is converged to the optimum or nearto-optimum solution after 100 iterations about one minute. To find the maximum number for workers, which are allowed to assign to a cell, the objective function is analyzed based on different numbers of workers. Since in this case, nine workers have been considered, objective function values are calculated by the initial solution and the TS to find the best number. As it would appear from Fig. 4, the best value for the maximum number in a virtual cell is equal to 5, which resulted in the minimum objective function in the initial and the TS algorithms. Before presenting the output of the initial heuristic algorithms and the TS, to have a comparison between the output of CBVCMS developed in this paper and CMS proposed by Baykasoglu et al. (2000) for this example, Table 7 presents the result of the CMS proposed by them. In the output, they used Cell Capacity Utilization (CCU) as the main criterion to present their model validity. According to Baykasoglu et al. (2000), CCU is calculated by dividing the total load in a cell by its capacity. The numerator in each cell is the summation of all processing time requirements from each

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Table 12 Parts-RE matrix and demand per each part for the Example 2 (Baykasoglu, 1999). RE1 RE2 RE3 RE4 RE5 RE6 RE7 RE8 RE9 RE10 RE11 RE12 RE13 RE14 RE15 RE16 RE17 RE18 RE19 RE20 RE21 RE22 RE23 RE24 RE25 RE26 RE27 RE28 RE29 RE30 RE31 RE32 Demand 1 1 1

1

1 1

1

1

1 1

1 1 1

1

1

1

1

1

1 1

1

1 1

1 1

1 1

1

1

1 1 1

1

1 1 1

1 1 1

1

1

1 1 1

1

1

1

1

1 1

1

1

1

1

1

1 1

1 1

1

550 400 800 800 520 540 510 520 400 500 500 600 780 850 400 502 515 540 400 400

Table 13 The machining and setup times of parts for the Example 2 (Baykasoglu, 1999). tpi + stpi RE1 RE2 RE3 RE4 RE5 RE6 RE7 RE8 RE9 RE10 RE11 RE12 RE13 RE14 RE15 RE16 RE17 RE18 RE19 RE20 RE21 RE22 RE23 RE24 RE25 RE26 RE27 RE28 RE29 RE30 RE31 RE32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

118 86 86 194

97

97 85

85

128

27

56 194

67 21 72

36

99

35

35

60

66 18

14

16 55

96 118

68 12

27

92

110

128 46 19

66

46 198 36

160

42 13 75

174 44 44 24 75

36

33

129

174 85

85

67

53

53

84

13 106

160 198

13

M. Hamedi et al. / Computers & Industrial Engineering 62 (2012) 953–971

P1 P2 P3 1 P4 P5 1 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 1 P20

Table 14 The primary data belonging to workers for the Example 2. Worker RE1 RE2 RE3 RE4 RE5 RE6 RE7 RE8 RE9 RE10 RE11 RE12 RE13 RE14 RE15 RE16 RE17 RE18 RE19 RE20 RE21 RE22 RE23 RE24 RE25 RE26 RE27 RE28 RE29 RE30 RE31 RE32 Capacity 1

1

1 1

1

1

1

1

1 1

1

1

1

1

1 1

1

1

1 1

1

1

1 1

1 1 1

1 1

1 1

1

1

1

1

1

1

1

1

1 1

1

1 1

1

1

1

1 1

1

1

1

1

1 1 1

1

1

1 1

1

1 1

1

1

1

1

1

1 1

1

1

11,000 11,000 11,000 11,000 11,000 11,000 11,000 11,000 11,000 11,000 11,000 11,000 11,000 11,000 11,000

1 1

1

1

1 1 1

1

1

1 1

1

1

1

1

1

1

1

1 1

1

1

1

1

1

1

1

1

1

Table 15 The costs belonging to workers for the Example 2. Worker RE1 RE2 RE3 RE4 RE5 RE6 RE7 RE8 RE9 RE10 RE11 RE12 RE13 RE14 RE15 RE16 RE17 RE18 RE19 RE20 RE21 RE22 RE23 RE24 RE25 RE26 RE27 RE28 RE29 RE30 RE31 RE32 Hiring Firing 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

100

100

200

100

100

100

80

50

80

50 120

200 100

100

100

120 150

60

50

50

200 120

100

300

220 200

120

60 80

100

100

150 100

220

80 110

100

50

120 100

1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100

100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

M. Hamedi et al. / Computers & Industrial Engineering 62 (2012) 953–971

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

967

2 2

3

3

2 2

2

2

3

3

3

2

2 3 2

RE28 RE27

3 3

2

2 2 2

3

4

3

3

2

2

4 3 2 3 3

2 5

2 2 2 2

2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2

2 3

4

2

3

3 2

3

3

2

2

4

2

3

4

3

2 2

4

3

3

3

3

2

4 3 3 2 3

4

4

2

1 2

3

3

2

2

2

3

2

2

3

2

3 3 2 3

3

2 2 2 2

1 3

RE26 RE25 RE24 RE23 RE22 RE21 RE20 RE19 RE18 RE17 RE16 RE15 RE14 RE13 RE12 RE11 RE10 RE9 RE8 RE7 RE6 RE5 RE4 RE3 RE2 RE1 t 0pi

Table 16 The times belonging to workers for the Example 2.

2

3

3 2

3

RE31 RE29

RE30

2

M. Hamedi et al. / Computers & Industrial Engineering 62 (2012) 953–971

RE32

968

machine. The capacity of a cell is computed by summing the capacity of its machines. Table 8 presents the output of the initial algorithm for the CBVCMS formed over the functional layout (see Fig. 3) in DRC setting in the forms of parts, machine, and workers assignment to virtual cells. Since the initial solution is a feasible solution of the developed model, the objective function is measured for each initial solution to present its improvement through applying the TS. This case resulted to an objective equal to 345188. After that, through the developed TS the initial solution improved. Table 9 belongs to the output of the TS, which led to an objective function equal to 311526. Since in the VCMSs and CBVCMSs, a number of machines have been shared among more than one cell, CCU is not a correct criterion to compare the outputs of the TS because the capacity of shared machines is countered more than once. Therefore, in such circumstances, to have an accurate comparison between CMSs and VCMSs, a new criterion needs to be defined to cover overall capacity of employed machines and required machines’ overtime. In this matter, System Capacity Utilization (SCU) is defined as an independent criterion to compare the mentioned systems, and is calculated by dividing the total load in the system by the capacity of utilized machines plus the machines’ overtime. Based on the above definition, the SCU for the initial heuristic algorithm and the TS are sequentially equal to 0.9012 and 0.9182, while for the CMS developed by Baykasoglu et al. (2000), this value is equal to 0.81. The detail of remaining capacity and required overtime for each machine, worker, and RE for the initial and heuristic algorithm are presented in Table 10. The convergence graphic of the proposed TS algorithm to the solution for the developed CBVCMS over the functional layout of Example 1 in DRC setting is plotted in Fig. 5. 5.2. Example 2 This example, taken from Baykasoglu (1999), includes 22 machines, 20 different parts, and is based on 32 various REs. The capability and capacity of machines are presented in Table 11. Fig. 6 presents the functional division of the factory with capabilities of all machine tools in the form of REs. Data belonging to parts properties are presented in Table 12. Table 13 shows the required times by parts to be processed including the machining and set up times. Same with the example 1, since the model is considered in DRC setting, the required data based on the mathematical model input are defined by the author and presented in Tables 14–16. Since the parts are grouped based on their similarity and the model tries do not put parts with high dissimilarity in a cell, Table 17 presents the dissimilarity between parts calculated based on the procedure proposed by Tam (1990). To solve the problem in DRC setting by TS, the characteristics defined by Baykasoglu (1999) are followed and the maximum and minimum number of Machines are considered equal to 10 and 3, respectively. Since they developed a CMS in SRC setting the number of maximum number of workers must be found for this paper. To find the best number, the effect of maximum allowed number of workers on the objective function is analyzed. As presented in Fig. 7, the maximum number of allowed workers in a virtual cell equal to 7 is resulted in the minimum value for the objective function for both of solutions found by the initial heuristic and the TS algorithms. Considering the mentioned parameters, the developed initial heuristic algorithm, as presented in Table 18, resulted in a feasible solution with an objective function equal to 951,621, which the TS improved that to achieve the near-to-optimum solution equal to 427208. Table 19 provides the output of the TS and Table 20 presents the detail of the initial and the TS solution regarding the maintaining capacities, machines, and workers overtimes.

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M. Hamedi et al. / Computers & Industrial Engineering 62 (2012) 953–971 Table 17 Dissimilarity between parts for the Example 2. Parts P1

P2

P3

P4

P5

P6

P7

P8

P9

P10

P11

P12

P13

P14

P15

P16

P17

P18

P19

P20

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20

2 0 1.375 2 1.9 3 2.5 2 2.5 2.5 2.5 2 2 1.1667 2 2 2.5 2.5 1.9 2

2 1.375 0 1.5 0.6667 3 2.5 2 2.5 2.5 2.5 2 2 1.5 1.5 2 2.5 2.5 0.6667 1.5

2 2 1.5 0 2 3 2.5 2 2.5 2.5 1.25 0.6667 1.375 1.5 0.8333 2 2.5 2.5 2 1.5

2 1.9 0.6667 2 0 3 2.5 2 2.5 2.5 2.5 2 2 2 2 2 2.5 2.5 0.75 2

3 3 3 3 3 0 2.8571 3 2.25 1.75 3 3 3 3 3 3 2.25 3 3 3

2.5 2.5 2.5 2.5 2.5 2.8571 0 2.5 1.2 2.4286 2.5 2.5 2.5 2.5 2.5 2.5 1.2 2.5 2.5 2.5

1.9 2 2 2 2 3 2.5 0 2.5 2.5 2.5 2 2 2 2 1.9 2.5 1.8 2 1.1667

2.5 2.5 2.5 2.5 2.5 2.25 1.2 2.5 0 2.3333 2.5 2.5 2.5 2.5 2.5 2.5 0 2.5 2.5 2.5

2.5 2.5 2.5 2.5 2.5 1.75 2.4286 2.5 2.3333 0 2.5 2.5 2.5 2.5 2.5 2.5 2.3333 2.5 2.5 2.5

2.5 2.5 2.5 1.25 2.5 3 2.5 2.5 2.5 2.5 0 0.625 1.8 2.5 1.9 2.5 2.5 2.5 2.5 2.5

2 2 2 0.667 2 3 2.5 2 2.5 2.5 0.625 0 1.25 2 1.375 2 2.5 2.5 2 2

2 2 2 1.375 2 3 2.5 2 2.5 2.5 1.8 1.25 0 2 1.375 2 2.5 2.5 2 2

2 1.1667 1.5 1.5 2 3 2.5 2 2.5 2.5 2.5 2 2 0 1.5 2 2.5 2.5 2 1.5

2 2 1.5 0.8333 2 3 2.5 2 2.5 2.5 1.9 1.375 1.375 1.5 0 2 2.5 2.5 2 1.5

1.4 2 2 2 2 3 2.5 1.9 2.5 2.5 2.5 2 2 2 2 0 2.5 0.625 2 2

2.5 2.5 2.5 2.5 2.5 2.25 1.2 2.5 0 2.3333 2.5 2.5 2.5 2.5 2.5 2.5 0 2.5 2.5 2.5

1.9167 2.5 2.5 2.5 2.5 3 2.5 1.8 2.5 2.5 2.5 2.5 2.5 2.5 2.5 0.625 2.5 0 2.5 1.9

2 1.9 0.6667 2 0.75 3 2.5 2 2.5 2.5 2.5 2 2 2 2 2 2.5 2.5 0 2

1.375 2 1.5 1.5 2 3 2.5 1.1667 2.5 2.5 2.5 2 2 1.5 1.5 2 2.5 1.9 2 0

0 2 2 2 2 3 2.5 1.9 2.5 2.5 2.5 2 2 2 2 1.4 2.5 1.9167 2 1.375

3000000.0 Initial objectives

Objective function

2500000.0

TS objectives

2000000.0 1500000.0 1000000.0 500000.0 0.0 3

4

5

6

7

8

9

10

Maximum number of workers Fig. 7. Analyzing the effect of maximum allowed number of workers on the objective function for the Example 2.

Table 18 The report of the initial heuristic algorithm for the Example 2. Cells

Parts: Xpj

Machines: Y 0mj

Workers: Y 00lj

CPU

SCU

Cell Cell Cell Cell Cell

2 7 9 4 1

3 6 9 10 11 12 13 2 6 13 18 20 22 3 4 6 13 18 14 16 22 1 4 5 10 12 18 19

2 1 6 5 1

0.5754 0.5964 0.7120 1.6242 0.4714

0.8821

1 2 3 4 5

356 8 14 15 10 17 18 11 12 13 16 19 20

3 6 8 9 7

45 7 9 11 13 14 15 10 15 10 12 15

Table 19 The report of TS for the Example 2. Cells

Parts: Xpj

Machines: Y 0mj

Workers: Y 00lj

CPU

SCU

Cell Cell Cell Cell Cell

3 4 9 8 1

1 3 6 9 10 11 12 13 15 22 2 6 13 14 22 3 4 6 13 14 16 18 22 17 18 19 21 4 5 12 19 20

12345 6789 7 9 10 11 13 15 11 12 15 1 2 7 13

0.5660 0.4572 0.5343 0.7202 0.6052

0.84

1 2 3 4 5

5 6 13 19 7 12 15 10 11 17 18 14 20 2 16

The detailed output of the program proposed by Baykasoglu (1999), as presented in Table 21, includes a CMS involving four classical cells. In their solution, they reported three types of REs (RE5, RE10, and RE21) needed overtimes in total equal to 205units. Since in the outputs of the mathematical model in this paper a number of machines have been shared between virtual cells, the

comparison between the CBVCMS and the original classical CMS cannot be based on the CCU. For that reason, in the same manner with the previous example, the definition of SCU is applied. According to the SCU definition and Table 20, SCU for the output of the generated CBVCMS for this example calculated as 0.84. In the original classical system, all machines have been employed to

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M. Hamedi et al. / Computers & Industrial Engineering 62 (2012) 953–971

Table 20 Outputs of the mathematical model to develop CBVCMSs for the Example 2. Initial solution

Optimum or near-to-optimum solution by the TS

Machines

Worker- RE

Machines

The required extra capacity for machines, workers, and REs M1 0 W1 0 RE1 0 M2 0 W2 0 RE2 0 M3 123,990 W3 0 RE3 0 M4 97,900 W4 0 RE4 0 M5 0 W5 0 RE5 49,600 M6 0 W6 0 RE6 7900 M7 0 W7 0 RE7 37,400 M8 0 W8 0 RE8 0 M9 0 W9 0 RE9 16,995 M10 0 W10 0 RE10 93,495 M11 49,600 W11 0 RE11 0 M12 0 W12 0 RE12 6695 M13 8715 W13 0 RE13 0 M14 0 W14 0 RE14 13,500 M15 0 W15 0 RE15 0 M16 187,680 RE16 2020 M17 0 RE17 0 M18 56,080 RE18 0 M19 0 RE19 62,880 M20 45,300 RE20 124,800 M21 0 RE21 120,100 M22 120,100 RE22 0 RE23 0 RE24 23,920 RE25 28,620 RE26 0 RE27 0 RE28 0 RE29 45,360 RE30 0 RE31 0 RE32 56,080

Workers

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 M13 M14 M15 M16 M17 M18 M19 M20 M21 M22

0 48,400 123,990 52,540 0 0 0 0 0 0 15,200 0 8715 0 12,880 0 0 0 0 0 45,300 120,100

W1 W2 W3 W4 W5 W6 W7 W8 W9 W10 W11 W12 W13 W14 W15

Total

0

Total

427,125

Total

689,365

Total

REs

0

Machine-RE

Total

689,365

Workers

REs

Machine-RE

Worker- RE

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

RE1 RE2 RE3 RE4 RE5 RE6 RE7 RE8 RE9 RE10 RE11 RE12 RE13 RE14 RE15 RE16 RE17 RE18 RE19 RE20 RE21 RE22 RE23 RE24 RE25 RE26 RE27 RE28 RE29 RE30 RE31 RE32

0 0 0 0 15,200 7900 37,400 0 16,995 93,495 0 6695 0 13,500 0 2020 0 0 0 61,280 120,100 0 0 23,920 20 0 0 0 0 0 0 28,600

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0

Total

427,125

0

Table 21 Properties of the CMS developed by Baykasoglu (1999). a: Part assignment Cell Cell Cell Cell

1 2 3 4

2 6 4 1

b: Machine assignment 3 7 11 8

5 9 12 16

14 10 13 18

19 17 15 20

8 3 2 1

9 6 5 4

11 10 7 15

12 13 14 18

Total

respond to all demands. Therefore, their suggested system has a SCU equal to0.82. Regardless of investigation and rearrangement costs, which are not seen in the developed methodology, the utilization of the classical CMS is less than both CBVCMSs, initial solution and improved solution by the TS.

6. Conclusion This paper focused on the formation of Capability-Based Virtual Cellular Manufacturing Systems (CBVCMSs) by a multi-objective mathematical model in the form of a Goal-Programming (GP) approach. To consider overlapping capabilities between machines and alternative machines for part processing during the cell formation automatically, a RE-based approach was used to define the shared and unique capabilities of machines. Three main advantages of the capability-based approach in comparison to the machine-based approach include high machine utilization, more flexibility, and less sensitivity to variable demands. Considering the CBVCMS in a Dual-Resource Constrained (DRC) setting is another contribution of this paper, which mostly has been ignored

19 17 16 20

21 22

CCU

Extra capacity (unit)

Required overtime (unit)

%88 %79 %92 %92

99,060 149,915 70,620 52,996

8440 (RE5) 77,295 (RE-I0) 120,100 (RE-21) –

372,591

205,835

in proposed methodologies. Since in DRC settings, number of machines exceeds the number of workers, some of the workers should be multi-skilled. In this research, the design issue of DRC systems was aimed and workers who can do multi-function by cross-training in different tasks, which is referred to worker flexibility, are considered. The worker flexibility involves decisions regarding the number of workers needed, the number of skills that workers should be cross-trained for them, and the assignment of workers to machines. In this research, to have a significant impact on the performances of systems, workers assumed with multi-level and heterogeneous flexibility. In this research, formulating CBVCMSs in the DRC setting made a Mixed Integer Non- Linear Programming (MINLP) model with twelve different goals at the objective function. Because of the non-linear nature of the problem, multi-objective TS proposed so that, at first, an initial solution is found by a heuristic algorithm, and then the near-to-optimum solution is achieved by the developed TS. To present the model validity, the data of two numerical examples, which aimed to form CMSs, taken from the literature and the outputs of the CBVCMSs were compared based on the System Capacity Utilization (SCU). The results presented the priority of the

M. Hamedi et al. / Computers & Industrial Engineering 62 (2012) 953–971

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