Int J Adv Manuf Technol (2009) 41:329–342 DOI 10.1007/s00170-008-1479-4
ORIGINAL ARTICLE
Dynamic cell formation and the worker assignment problem: a new model M. B. Aryanezhad & V. Deljoo & S. M. J. Mirzapour Al-e-hashem
Received: 1 May 2007 / Accepted: 10 March 2008 / Published online: 7 May 2008 # Springer-Verlag London Limited 2008
Abstract In this paper a new model is developed to deal with a simultaneous dynamic cell formation and worker assignment problem (SDCWP). Part routing flexibility and machine flexibility and also promotion of workers from one skill level to another are considered. The proposed model is formulated as a single objective nonlinear integer programming which is converted to a linear one. The objective function consists of two separate components. The first part of the objective function is related to machine-based costs such as production cost, intercell material handling cost, machine costs in the planning horizon. The second part is related to human issues and consists of hiring cost, firing cost, training cost and salary. It is the first time that worker assignment and dynamic cell formation are considered simultaneously. To verify the performance of the proposed model, some numerical examples are presented. Computational and sensitivity analysis results imply the significance of SDCWP. Keywords Dynamic cell formation . Worker assignment . Route and machine flexibility . Training M. B. Aryanezhad : S. M. J. Mirzapour Al-e-hashem Department of Industrial Engineering, Iran University of Science and Technology, 16846113114 Tehran, Iran V. Deljoo Department of Industrial Engineering, Bu Ali Sina University, Hamedan, Iran S. M. J. Mirzapour Al-e-hashem (*) Unit 6, no. 7, Bahar Azadi Alley, Abolfazl St., Marzdaran Blvd., Tehran, Iran e-mail:
[email protected]
1 Introduction Cell formation (CF) is a part of the cellular manufacturing system (CMS) that is actually the implementation of group technology in manufacturing and production systems with the goal of classifying parts in a way that the physical or operational similarities of the parts are used in different aspects of design and production of parts. The advantages derived from cellular manufacturing in comparison with traditional manufacturing systems in terms of system performance have been discussed in (Fry et al. [1], Collet and Spicer [2], Levasseur et al. [3], Singh and Rajamaani [4]). These benefits have been established through simulation studies, analytical studies, surveys, and actual implementations. They can be summarized as follows: 1. Setup time is reduced. 2. Lot sizes are reduced. 3. Work-in-process (WIP) and finished goods inventories are reduced. 4. Throughput times are reduced. 5. Working flexibility is improved. Dynamic production requirements imply multiple periods when designing a CMS. In this case, the entire planning horizon is divided into multiple periods according to the differences in product mix and/or demand in each period. In each period, product mixes and demands can be deterministic or stochastic (Mungwatanna [5]). In the case of deterministic product mixes and demands, they are known in each period. When they are stochastic, the possible product mixes and demands in each period are known with certain probabilities. Literatures in the design of CMSs are
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reviewed according to the above classifications which are as follows: 1. Design of CMSs for dynamic and deterministic production requirements (Seifoddini [6]) 2. Design of CMSs for static and stochastic production requirements. 3. Design of CMSs for dynamic and stochastic production requirements (Harhalakis [7]). In this research dynamic and deterministic demand and production requirement condition are discussed.
2 Dynamic and deterministic production requirements Song and Hitomi [8] developed a methodology to design flexible manufacturing cells. The method integrates production planning and cellular layout via a long-run planning horizon. The integrated planning model is formulated as a mixed-integer problem which contains two types of integer programming problems: determining the production quantity for each product and the timing of adjusting the cellular layout in a finite planning horizon with dynamic demand. Chen [9] developed a mathematical programming model for system reconfiguration in a dynamic cellular manufacturing environment. A mixed integer programming model is developed to minimize intercell material and machine costs as well as reconfiguration cost in a dynamic cellular manufacturing environment with anticipated changes of demand or production process for multiple time periods. Wicks [10] proposed a multi-period formation of the part family and machine cell formation problem. The dynamic nature of production environment is addressed by considering a multi-period forecast of the product mix and resource availability during the formation of part families and machine cells. The design objectives are the minimization of intercell material handling cost, the minimization of investment in additional machines, and the minimization of the cost of system reconfiguration over the planning horizon. Mungwatanna [5] proposed a new model for dynamic cell formation problem considering routing flexibility. The planning horizon divided into some period. Demand and production requirement are given in each period and can be different from one period to another. In addition each operation can be produced in some machine with different process times and production costs. Mungwatanna [5] first presented a nonlinear mixed integer programming then, with some changes, converted it to a linear model and solved with simulated annealing (SA). Tavakoli et al., [11] modified the model that introduced by Mungwatanna
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[5] and presented three Meta heuristic methods consist of genetic algorithm (GA), SA, and Tabu search (TS).Then compared their efficiencies with each other and also with LINGO 6. Safaei et al. [12] developed an extended model for DCF problem, considering the batch inter/intra cell material handling by assuming the sequence of operations. They also presented a hybrid simulated annealing algorithm to solve their model. Safaei et al. [13] developed a fuzzy model for dynamic cell formation problem which is mainly derived from the model presented by Mungwattana [5] and reformulated by a fuzzy approach. In this model the fluctuation in part demands and the availability of manufacturing facilities in each period is considered as fuzzy variables. In this research, models presented by Mungwatanna [5], Tavakoli et al. [11], Safaei et al. [12] and Safaei et al. [13] are reconsidered and with some changes are used to develop a new DCF model.
3 Human related issues in CMS One of the main points in cellular manufacturing is considering human issues since ignoring this factor can considerably reduce benefits of the utility of the cell manufacturing. Bidanda et al. [14] state that it is important for the successful implementation of cellular manufacturing, which one focuses both on technical issues (cell formation and design) and human issues. But unfortunately, human issues are typically not examined as rigorously as often as technical issues: “While cellular manufacturing is a popular research area, there is a singular absence of articles that deal with the human element in cellular manufacturing. There are a variety of reasons for this, including that these issues are typically difficult to quantify. It has been well documented that there is an absence of research in the area of worker placement based on both their technical and human skills.” In another research Balakrishnan and Cheng [15] have reviewed research that has been done to address cell manufacturing under condition of multi period planning horizons, with demand and resource uncertainties. After review of literature they made some directions for future studies. They claimed that in spite of the fact that several research papers have highlighted the importance of interactions between human resources management and operation management and the need to incorporate organizational behavior issues in operations management in recent years, there has not been much research on organizational behavior issues in cellular manufacturing.
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4 Worker assignment in CMS
7 Problem description
In some of the previous research papers this issue is discussed, and according to their assignment strategies they can be divided into two categories:
In this research we consider a cell formation environment consisting of several machine groups called machine levels, each with a number of various machine types. Machines in each machine level need the same worker ability to do jobs containing them. Parts move from one machine type to another according to their flexible routing. Routing flexibility means that each operation of parts can be processed on more than one machine and maybe with different process times. Workers are grouped according to their skills and assigned to skill levels. Workers in each skill level have the same skills to do the jobs related to machine types. Newly hired workers are less efficient than experienced ones and thus their productivity is less than them. Each worker has at least one skill and exactly belongs to one skill level and can be assigned to certain machine levels depending on their skill levels. In each period, workers can be trained to improve their working abilities to operate other machine levels. Being aware of demand estimates for a number of future periods and an initial number of machines from each type located in each cell as well as initial number of workers of each skill level who are assigned to their related machine levels, we would like to determine the optimal strategy (minimum cost) to staff this cell manufacturing in the planning horizon. In particular, in order to satisfy the total demand of each period, we are interested in determining: 1) How many machines and of which types to purchase, relocate, install or remove in each cell and in each period; 2) How many workers, with which skill levels to hire or fire in each period; 3) How many workers to train, promoting from which skill levels to which, in each period. 4) How many workers with which skill levels to assign to each machine level in each period. The machine-related part of proposed model is derived from previous DCF models in the literature, some improvement is applied to this part and human-related issues are added to it. In order to explain with more details, an exemplar condition is considered where there are three machine types as follows:
1) Post-cell formation worker assignment (Suer and Bera [16], Askin and Huang [17], Norman et al. [18]). 2) Simultaneous formation of cells and worker assignment. In this paper, simultaneous formation of cells and worker assignment is discussed.
5 Simultaneous formation of cells and worker assignment Min and Shin [19] stated that cellular manufacturing will not be successful unless, human and machine cells are formed simultaneously. They also state that a new approach is required to handle multiple objectives instead of single objectives, such as maximization of part similarities, maximization of total productivity, minimization of total bottleneck cost, minimization of inter cell material handling costs, etc. In the model developed by Min and Shin [19] assumes that multi-skilled workers with different levels of job skills are available for assignment. Due to the computational difficulty in solving the model, they proposed a sequential heuristic that decomposes the model into two smaller problems: formation of the machine cells and assignment of workers to cells to form the corresponding human cell. The first step of the model is to solve the reduced goal programming formulation for the cell formation. The second step of the model is to assign appropriate part-machine values and solve the reduced formulation for worker assignment. The model developed by them does not address cost of training, worker training and worker skill levels.
6 Training increases the production flexibility Training, the process by which workers become multiskilled, has been recognized as a tool for increasing production flexibility (Park [20]). Balakrishnan and Cheng [15] claimed that regarding to the importance of crosstraining; it would be useful to investigate its effect in cell manufacturing systems. Essentially in CMSs the workers are able to operate several machines. For this reason in this research a new model presented that both cell formation and worker assignment considered simultaneously.
– – –
Machine type1: turning machine Machine type2: milling machine Machine type3: welding machine
And workers in the system have the following skill levels: – – –
Skill level1: Workers can work the turning and milling machine types. Skill level2: Workers can work the welding machine types. Skill level3: Workers can work all of the machine types.
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According to these descriptions, two machine levels are defined as follows:
– –
Machine level1: including turning and welding machine types. Machine level2: including welding machine type.
A worker according to his/her skill level can work some machine levels in the manufacturing system. It means that in this example a worker with skill level1 can only work the machine level1 , and a worker with skill level2 can work the machine level2 , and finally a worker with skill level3 can work the machine level1 and level2. And about promotion, it is obvious that workers with skill level1 and skill level2 can be promoted only to skill level3, and other cases are impossible. A possible cell formation and worker pattern in two consecutive periods can be something as shown in Fig. 1.
8 Assumptions 1. The demand for each part type in each period is known. 2. Parts are moved between cells in batches. The intercell material handling cost per batch between cells is known and constant (regardless of the distance traveled). 3. The machine purchasing cost is known and machines are delivered at the usage place in the manufacturing system (purchasing cost = machine price + freight charge), which means that no removal cost from storage to installation place for newly purchased machines are incurred. 4. The maximum number of cells used should be specified in advance and it remains constant over time. 5. Bounds and quantity of machines in each cell need to be specified in advance and remain constant over time. 6. The machine relocation cost of each machine type is known and it is independent of where the machines
Machine level 1
Machine level 2
Period 1
Period 2
Cell1 Machine type 1
Cell2 Machine type 1
Cell1 Machine type 1
Machine type 2
Machine type 2
Machine type 2
Cell2 Machine type 1
Machine type 2 Relocated
Machine type 3
Machine type 3
Machine type 3
Machine type 4
Machine type 4
Machine type 4
Machine type 3 Relocated Machine type 4 Relocated
Skill level 2
Skill level 2
Can work only on machine level 2
Skill level 1
Skill level 1 Possible Promotion
Can work only on machine level 1
Possible Promotion
Can work on both of machine levels
Promoted
Skill level 3
Skill level 3 Promoted Fired Hired
Fig. 1 Cell formation and worker pattern in two consecutive periods for exemplar condition
Purchased
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Table 1 The skill-machine possibility and the initial number of workers Skill/Machine
m1
m2
m3
m4
sk1 sk2 sk3 sk4 sk5 sk6 sk7 sk8 sk9
(0 0) (0 0) (0 0) (1 0) (1 10) (1 1) (0 0) (0 0) (1 0)
(1 (1 (1 (1 (1 (0 (0 (1 (1
(0 0) (0 0) (1 1) (1 11) (0 0) (1 2) (1 3) (1 0) (1 0)
(0 0) (1 10) (0 0) (0 0) (1 3) (1 2) (1 2) (1 0) (1 0)
10) 20) 10) 1) 1) 0) 0) 2) 0)
For example in (m n)ij, m=1 means that worker with skill level i can work the machine level j and m=0 means that he/she cannot work that. And n is the initial number of workers with skill level i in the beginning of the planning horizon.
7.
8. 9. 10. 11. 12. 13.
14.
are actually being relocated. It is performed between periods and requires zero time. Each machine type can perform one or more operations (machine flexibility). Likewise each operation can be done on one or more machine type with different times (routing flexibility). Workers can only be assigned to machine levels with which they are able to work. Salary is merely dependent on worker’s skill level and not depending on machine levels. All of the machine types which need the same skill levels assumed to be similar in worker assignment. Cost of hiring and firing are given and they merely depend on skill levels. Each machine needs just one worker. To process a specific operation, the related machine and worker must be available concurrently. Training, which is done to promote workers to upper levels, is performed between periods and it takes zero time. The productivity of experienced workers is assumed to be equal to 100%.
15. The productivity of newly trained workers is assumed to be fewer than that of experienced ones, and it depends on the skill level to which they are trained. 16. Productivity of newly hired workers is assumed to be fewer than that of experienced ones and it depends on the skill level for which they are hired. 17. Cost of training from one skill level to another is given and it depends on both skill levels. 18. Learning curve is not regarded explicitly in the model formulation. Indices and their upper bounds C M P H Op ψ MS c m p h j α,β k, k′
Number of cells Number of machine types Number of part types Number of time periods Number of operations required by part p Number of skill levels Number of machine levels Index for manufacturing cells (c=1, …, C) Index for machine types (m=1, …, M) Index for part types (p=1, …, P) Index for time periods (h=1, …, H) Index for operations required by part p (j=1, …, Op) {1,…,ψ} Index for skill levels Index for machine levels{1,…, MS}
Input parameters Tj,p,m Dp,h αm Bm γ INSm UNSm MTm
Time required to perform operation j (j=1,..., op) of part type p (p=1,..., P) on machine type m (m=1,..., M) Demand for product p in period h (h=1, H) Purchase cost of machine type m(plus freight cost) Operating cost per hour of machine type m Inter-cell material handling cost per batch Installing cost of machine type m Removing cost of machine type m Capacity of each machine type m
Table 2 The worker skills promotion possibility and their productivity when promoted from one skill level to another From/to sk
1
sk1 sk2 sk3 sk4 sk5 sk6 sk7 sk8 sk9
(0 (0 (0 (0 (0 (0 (0 (0 (0
2 0) 0) 0) 0) 0) 0) 0) 0) 0)
(1 (0 (0 (0 (0 (0 (0 (0 (0
3 .95) 0) 0) 0) 0) 0) 0) 0) 0)
(1 (0 (0 (0 (0 (0 (0 (0 (0
4 .75) 0) 0) 0) 0) 0) 0) 0) 0)
(1 (0 (1 (0 (0 (0 (0 (0 (0
5 .80) 0) .95) 0) 0) 0) 0) 0) 0)
(0 (0 (0 (0 (0 (0 (0 (0 (0
6 .85) .95) 0) 0) 0) 0) 0) 0) 0)
(0 (0 (0 (0 (0 (0 (1 (0 (0
7 0) 0) 0) 0) 0) 0) .85) 0) 0)
(0 (0 (0 (0 (0 (0 (0 (0 (0
8 0) 0) 0) 0) 0) 0) 0) 0) 0)
(1 (1 (1 (1 (0 (0 (1 (0 (0
9 .85) .75) .80) .70) 0) 0) .75) 0) 0)
(1 (1 (1 (1 (1 (1 (1 (1 (0
.85) .85) .85) .80) .85) .85) .85) .85) 0)
For example in (m n)ij, m=1 means that worker with skill level i can be trained for level j and m=0 means that he/she can not. And n is the productivity of workers after training for skill level j.
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Table 3 The cost of training from skill level i to skill level j in period h From skill Period/to skill 1 2
3
4
5
6
7 8
9
55 65 75 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
60 70 80 0 0 0 50 60 70 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
90 100 110 50 60 70 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 0 0 0 0 0 0 55 60 70 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
120 130 140 100 110 120 100 110 120 100 110 120 80 90 100 80 90 100 50 60 70 50 60 70 0 0 0
Ca;b;h Sa;h
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
sk1
sk2
sk3
sk4
sk5
sk6
sk7
sk8
sk9
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
50 60 70 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
100 110 120 80 90 100 90 100 110 80 90 100 0 0 0 0 0 0 60 65 70 0 0 0 0 0 0
Fa;h la
0 la 1 0 dab 1
d a;b B A R
Decision Variables Nm,
c, h
þ Km;c;h Km;c;h
Number of machines of type m used in cell c (c=1,…, C) during period h Number of machines of type m added in cell c during period h Number of machines of type m removed from cell c during period h ( 1
Xj;p;m;c;h ¼
0 ( Zj;p;c;h ¼
LB UB
1 0
Number of α-level workers who are hired and assigned to machine level k in period h. Number of existing α-level workers who are assigned to machine level k in period h. Number of α–level workers who were assigned to machine level k in period h-1 and they are fired in period h. Number of α–level workers who were assigned to machine level k in period h-1 and now are trained to skill level β and assigned to machine level k′ in period h.
Ua;h;K
if operation j of part p is done on machine type m; otherwise:
Ea;h;K
1 if training from skill level α to skill level β is possible; 0 otherwise; 1 if machine of type m belongs to machine level k: ¼ 0 otherwise
UGν;α ¼
MSLm;k
WPα;k ¼
ha;h
Wa;h;K
UXa;b;h;k;k0
1 if working on machine level k with the skill level is possible 0 otherwise
Cost of hiring of a worker with skill level α in period h.
Table 4 The process times
*0.8 is the processing time of 2nd operation of product 1 using machine type 1 and this operation can’t be processed with other machines.
Product 1
if operation j of part type p is done on machine type min cell c in period h; : otherwise; if operation j of part type p is done in cell c in period h; otherwise;:
1 0
Lower bound of the cell size Upper bound of the cell size
aj;p;m ¼
Training cost of each α-level worker trained for skill level β in period h Salary of each α-level worker in period h Firing cost of each α-level worker fired in period h Productivity of each newly α-level worker hired in period h Training productivity of α-level worker trained for skill level β Batch size for inter-cell material handling Available working time per worker in hours per time period An arbitrary big number
Product 2
Product 3
Operation/machine
1
2
3
4
1
2
3
4
1
2
3
4
1 2 3 4
0.1 0.80* 0.16 0
0 0 0 0
0 0 0 0
0 0 0 0
0.12 0.10 0 0.2
0 0 0.16 0.16
0.10 0 0 0
0 0.14 0.12 0
0 0 0 0
0.16 0.15 0 0
0 0 0 0
0 0 0 0
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Table 5 The human costs and the productivity of newly hired workers Skill level/period
Salary
sk1 sk2 sk3 sk4 sk5 sk6 sk7 sk8 sk9
Hiring 2
3
1
2
3
1
2
3
1500 1600 1600 1600 1700 1700 1800 1800 2500
1600 1700 1700 1700 1800 1800 1900 1900 2600
1700 1800 1800 1800 1900 1900 2000 2000 2700
1500 1600 1600 1600 1700 1700 1800 1800 2500
1600 1700 1700 1700 1800 1800 1900 1900 2600
1700 1800 1800 1800 1900 1900 2000 2000 2700
15 16 16 16 17 17 18 18 19
16 17 17 17 18 18 19 19 20
17 18 18 18 19 19 20 20 21
þ
Multiple costs are considered in the design of objective function in an integrated manner. All costs involved in the design of CF and worker assignment are incorporated; however, it is not possible to consider all costs in the model due to the complexity and computational time required. In this paper, costs are limited to those, which are related to dynamic and deterministic production conditions and the use of part routing and machine flexibility. The objective is to minimize the sum of the following costs:
Min
αm Nm;c;h
H X C X M X P X OP X
þ 12
h¼1
h
Dp;h B
i
ð3 Þ
þ
h¼1 α¼1 k¼1
Dph Tjpm Xj;p;m;c;h Bm þ
H P M P C P h¼1 m¼1 c¼1
ð2Þ
OP1 M P P j¼1 m¼1
ð6Þ
UXα;β;h;k;k 0 þ
ψ P H P MS P h¼1 α¼1 k¼1
ψ PP ψ H P MS P P h¼1 α¼1 β k¼1 k 0 ¼1
Cα;β;h
ð7Þ
Sα;h Eα;h;k þ
ð8Þ
ψ P h P MS P h¼1 α¼1 k¼1
Fα;h Wα;h;k
ð9Þ
Subject to: C X M X
aj;p;m Xj;p;m;c;h ¼ 1
8j; p; h
ð10Þ
c¼1 m¼1
H P M P C P h¼1 m¼1 c¼1
M X
Dp;h Tj;p;m Xj;p;m;c;h MTm Nm;c;h
8m; c; h ð11Þ
Nm;c;h LB
8c; h
ð12Þ
Nm;c;h UB
8c; h
ð13Þ
m¼1 M X
γ Zðjþ1Þ;p;c;h Zj;p;c;h
Kþ m;c;h INSm þ ð4 Þ
hα;h Uα;h;k þ
0.85 0.85 0.85 0.80 0.80 0.85 0.85 0.90 0.95
p¼1 j¼1
h¼1 c¼1 m¼1 p¼1 j¼1 H P
ψ P H P MS P
P X OP X
h¼1 m¼1 c¼1 ð1Þ
þ
Productivity
1
9 Mathematical formulation
H X M X C X
Firing
m¼1
K m;c;h UNSm
Nm;c;h1 þ Kþ m;c;h K m;c;h ¼ Nm;c;h M X
ð5Þ
Xj;p;m;c;h ¼ Zj;p;c;h
8m; c; h
ð14Þ
8j; p; c; h
ð15Þ
m¼1
Table 7 Machine-based costs
Table 6 The demand of products in each period Product/Period
1
2
3
1 2 3
200 50 100
50 50 10
100 200 100
Cost item/machine #
1
2
3
4
Purchasing cost* Machine cost (per hour) Installing cost Removing cost
1500 4 40 40
1200 1 25 25
1500 3 30 30
1000 2 20 20
*It means that purchasing cost for machines 1, 2, 3 and 4 are equal to $1500, $1200, $1500 and $1000, respectively.
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Table 8 Optimum values of human-related variables of SDCWP for numerical example
Table 10 Machine pattern of model1 against that of SDCWP for numerical example
UX (αν h k k′) Value
Model1
UX UX UX UX UX UX UX UX
(2 (2 (3 (3 (5 (6 (7 (7
5 5 4 4 9 9 6 6
1 1 1 1 1 1 1 1
2 4 2 3 4 4 3 4
1) 1) 1) 1) 1) 1) 1) 1)
U (αh k) Value
W (αh k) Value
20.00000 U (4 1 1) 10.0000 W (4 1 3) 10.00000 U (9 1 1) 1.00000 W (6 1 3) 10.00000 W (6 1 4) 1.000000 W (8 2 2) 3.000000 W (9 2 1) 1.000000 3.000000 2.000000
11.00000 2.000000 1.000000 1.000000 5.000000
UX is the number of α-level workers who worked on machine level k in period (h-1) and are trained in period h for skill level ν and assigned to machine level k′ U is the number of α-level workers is hired in period h to work on machine level k W is the number of α-level workers worked on machine level k in previous period and they are fired in current period (h)
Ea;h;k ¼ Ea;h1;k þ Ua;h;k Wa;h;k þ
y X MS X UXu;a;h;k 0 ;k UXa;u;h;k;k 0
8a; h; k
u¼1 k 0 ¼1
ð16Þ 2
3 Eα;h1;k þ 1α Uα;h;k Wα;h;k ψ 6 P MS 7 X 6 ψ P 7 6þ δυ;α UXυ;α;h;k 0 ;k UXα;υ;h;k;k 0 7 A 6 υ¼1 k 0 ¼ 1 7 5 α¼1 4 0 k 6¼ k
C X M X
Nm;c;h MTm MSLm;k
8h; k
N (m c h) N N N N N N N N N
(1 1 1)* (1 1 2) (1 1 3) (1 2 1) (1 2 2) (1 2 3) (2 1 1) (2 1 2) (2 1 3)
Period
1
2
3
(Pperation, product) (1, 1)* (1, 2) (1, 3) (2, 1) (2, 2) (2, 3) (3, 1) (3, 2) (4, 2)
(Machine, cell) (1, 1) (1, 1) (2, 1) (1, 2) (1, 1) (2, 1) (1, 2) (2, 1) (1, 1)
(Machine, cell) (1, 2) (1, 1) (2, 1) (1, 2) (1, 1) (2, 1) (1, 2) (2, 1) (1, 2)
(Machine, cell) (1, 2) (1, 1) (2, 1) (1, 2) (1, 1) (2, 1) (1, 2) (2, 1) (1, 2)
*It means that the 1st operation of the 1st product is processed in machine type 1 and cell 1 during 1st period, in machine type 1 and cell 2 during 2nd period and in machine type 1 and cell 2 during last period, respectively.
N (m c h)3
Value
26.00000 27.00000 27.00000 40.00000 40.00000 40.00000 14.00000 13.00000 13.00000
N N N N N N N N N N N N
33.00000 39.00000 37.00000 39.00000 39.00000 33.00000 7.000000 1.000000 7.000000 3.000000 1.000000 1.000000
(1 1 1) (1 1 2) (1 1 3) (1 2 1) (1 2 2) (1 2 3) (2 1 1) (2 1 2) (2 2 3) (4 1 3) (4 2 1) (4 2 2)
Wa;h;k R WPa;k
8a; h; k
ð18Þ
Ua;h;k R WPa;k
8a; h; k
ð19Þ
UXa;u;h;k;k 0 R WPa;k
8n; a; h; k; k 0
ð20Þ
UXa;u;h;k;k 0 R WPu;k 0
8n; a; h; k; k 0
ð21Þ
UXu;a;h;k;k 0 R UGn;a
8n; a; h; k; k 0
ð22Þ
y X MS X
Table 9 Optimum values of machine-related variables of SDCWP for numerical example
Value
*It means that the number of machine type 1 in 1st cell and in 1th period, are equal to 26 and 33 obtained from solving model1 and SDCWP, respectively
c¼1 m¼1
ð17Þ
SDCWP 1
u
UXa;u;h;k;k 0 þ Wa;h;k Ea;h1;k
8a; h; k
ð23Þ
k0
y X MS X
UXu;a;h;k;k 0 R ya;h;k 0
8a; h; k 0
ð24Þ
u¼1 k¼1
Wα;h;k 0 R 1 yα;h;k 0
8α; h; k 0
þ Nm;c;h ; Km;c;h ; Km;c;h ; Uα;h;k ; UXα;β;h;k;k 0 ; Wα;h;k ; Eα;h;k
Xj;p;m;c;h ; Zj;p;c;h ; yα;h;k ¼ 0 or 1
(25) 0; Integer
ð25Þ
10 Objective functions The objective function consists of several cost items as follows: (1) Machine cost: It is the cost per period to procure machines. This cost is calculated based on the number of machines of each type used in the CF for a specific period.
Cell formation obtained from solving SDCWP for numerical example
Fig. 2 Comparison of cell formations obtained via solving model1 and SDCWP
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Cell formation obtained from solving Model1 for numerical example
Int J Adv Manuf Technol (2009) 41:329–342 M1
M2
Cell 1
Cell 1
Cell 1
M1:26
M1:27
M1:27
M2:14
M2:13
M2:13
Cell 2
Cell 2
Cell 2
M1:40
M1:40
M1:40
M2: 0
M2: 0
M2: 0
Period1
Period2
Period3
1×M2
6×M1
6×M2
1×M2
Fig. 3 Criterion1 for different values of θ in the numerical example
2×M1
3×M4
Cell 1
Cell 1
Cell 1
M1:33
M1:39
M1:37
M2: 7
M2: 1
M2: 0
M4: 0
M4: 0
M4: 3
Cell 2
Cell 2
Cell 2
M1:39
M1:39
M1:33
M2: 1
M2: 0
M2: 7
M4: 0
M4: 1
M4: 0
Period1
Period2
Period3
1◊M 4
1◊M 4
6◊M 1
7◊M 2
Fig. 4 Criterion2 for different values of θ in the numerical example
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(2) Operating cost: The cost of operating machines for producing parts. This cost depends on the cost of operating each machine type per hour and the number of hours required for them. (3) Inter-cell material handling costs: The cost of transferring parts between cells, when parts cannot be produced completely by a machine type or in a single cell. This cost is incurred, when batches of parts have to be transferred between cells. Inter-cell moves decrease the efficiency of CM by complicating production control and increasing material handling requirements and flow time. OP1 H M XX Dph 1X g Zðjþ1Þ;p;c;h Zj;p;c;h 2 h¼1 B j¼1 c¼1 This is a nonlinear integer equation because of the absolute terms. To transform it into a linear mathematical model, non-negative variables (ypjpmch and ymjpmch) are introduced and allowing reformulation into a linear model as follows. In the linear model, the objective function is rewritten as: OP1 H M XX Dp;h 1X g ypj;p;c;h þ ymj;p;c;h 2 h¼1 B j¼1 c¼1 Considering the following set of constraints: Zðjþ1Þ;p;c;h Zj;p;c;h ¼ ypjpch ymj;p;c;h
;
ypj;p;c;h ; ymj;p;c;h 0
(4), (5). Machine relocation cost: The cost of relocating machines from one cell to another between periods. In dynamic and deterministic production conditions, the best CF design for one period may not be an efficient design for subsequent periods. By rearranging the manufacturing cells, the CF can continue operating efficiently as the product mix and demand change. In this paper it is assumed that when a machine is relocated from one cell to another one, it is removed from its current position and is relocated to another place and installed to a new cell or stored somewhere out of manufacturing system. When a machine is removed from one cell, a removal cost (removal plus relocation) is incurred, and if it is installed in another cell an installation cost is incurred too. This cost is calculated in this model by expressions (4), (5). (6). Hiring cost: This cost is incurred, when some workers have to be hired and assigned to specific machine level because available workers are not sufficient.
This cost depends on the cost of hiring per each worker and the number of workers who are hired. (7). Training cost: This cost is incurred when some workers with their skill levels have to be trained to upper ones to improve their abilities to operate other machine levels. This cost is calculated based on the cost of training per each worker and the number of workers who are trained. (8). Salary: The cost of workers payment in the planning horizon. This cost calculated based on the number of workers existing in each period and the cost of payment per each worker. (9). Firing cost: This cost is incurred when some workers have to be fired because they are not required right now and training of them is not advantageous too. This cost depends on the cost of firing per each worker and the number of workers who are fired.
11 Constraints Constraint (10) ensures that each part operation is assigned to one machine and one cell. Constraint (11) ensures that machines capacities are not exceeded and can satisfy the demand. Constraints (12) and (13) specify the lower and upper bounds of cells. Constraint (14) ensures that the number of machines in the current period is equal to the number of machines in the previous period plus the number of machines being moved in and minus the number of machines being moved out. In other words, this ensures the conservation of machines over the horizon. In constraint (15) if at least one of the operations of part p is processed in cell c in period h, then the value of zjpch will be equal to one; otherwise it is equal to zero. constraint (16) is the worker balance equation and represent that the number of α–level workers who are assigned to machine level k in period h is equal to the number of workers with the same characteristics existed in previous period plus the number of α–level workers who are employed or upgraded from lower skill levels and assigned to machine level k, minus the number of workers with the same characteristics who are fired or trained for upper skill levels. Constraint (17) ensures that in order to satisfy the total demands in each period the total available worker hours which are assigned to available machine hours are provided in a way such as hiring, firing or training. Constraints (18) and (19) ensure that, an α–level worker can be fired or hired, if and only if this assignment is possible. Constraints (20), (21) and (22) ensure that, training for skill level ν, is possible, just if the former assignment and the latter are being possible and also training from initial skill level to skill level ν being possible too. Constraint (23) ensures that the total number of α– level workers who are assigned to machine level k in period
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Table 11 Dimension of examples P1–P4 Problem
P1 P2 P3 P4
Number of
Max number of
Skill levels
Machine levels
Cells
Machine types
Periods
Products
Operations
5 6 7 8
4 5 4 5
3 4 5 5
5 6 5 6
3 4 4 3
3 5 4 5
5 5 5 5
h-1 and now, fired or trained for upper skill levels have not to be greater than the number of workers with the same characteristics existed in previous period. Constraints (24) and (25) ensure that the workers who are trained for skill level α and assigned to machine level k in period h have not to be fired in the same period.
12 SDCWP versus hierarchical method As mentioned before, the objective function of the proposed model is sum of two main components. The first part is related to machine-based costs and consists of the
machine cost (Eq. 1), operating cost (Eq. 2), inter-cell material handling costs (Eq. 3) and machine relocation cost (Eqs. 4 and 5) which is named here f1. The second part is related to human issues and consists of hiring cost (Eq. 6), training cost (Eq. 7), cost of salary (Eq. 8) and firing cost (Eq. 9), which is named f2. Suppose that we have three separate models, entitled model1, model2 and SDCWP (proposed model) as follows:
Model1 :
Fig. 5 Criterion1 for different values of θ in four examples with different dimensions
Minimize Z1 ¼ f1 ðf1 ¼ summation of equations ð1Þ; ð2Þ; ð3Þ; ð4Þ and ð5ÞÞ Subject to : Eqs: ð10Þ; ð11Þ; ð12Þ; ð13Þ; ð14Þ and ð15Þ;
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Model2 : Minimize Z2 ¼ f2 ðf2 ¼ summation of equations ð6Þ; ð7Þ; ð8Þ and ð9ÞÞ Subject to : Eqs: ð16Þ ð25Þ;
To measure the deference between cell formations obtained via model1 and SDCWP, two criteria are introduced as follows: f f Criterion1 ¼ 1 13 ; f 13
SDCWP : Minimize Z3 ¼ f3 f3 ¼ f1 þ f2 Subject to : Eqs: ð10Þ ð25Þ:
where model1 tries to form cells regardless of human issues And model2 tries to form workers pattern for the cell design which is obtained from solving model1, while SDCWP tries to form cells considering machine and human cost simultaneously. In SDCWP, formation of cells and human pattern is done concurrently. Another alternative to do this job is using hierarchical method in which firstly cell design is formed via solving model1 and then human pattern is formed for this obtained design, by solving model2. model1 is the same as models introduced in the literature (Mungwatanna [5] and Tavakoli et al., [11], Safaei et al. [12], Safaei et al. [13]). In this section we want to compare performance of the proposed model (SDCWP) and hierarchical method (solving model1 and then model2). To prove the performance of the proposed model, we use some notations as follows: f1 : f2 : f3 : f13 : 1 Nmch : 3 Nmch :
is the optimum value of f1 obtained via solving model1, is the optimum value of f2 obtained via solving model2, is the optimum value of f3 obtained via solving SDCWP, is the optimum value of f1 obtained via solving SDCWP, number of machines of type m used in cell c during period h obtained via solving model1, number of machines of type m used in cell c during period h obtained via solving SDCWP.
It can be shown that f3 f1 þ f2 (see Appendix). It means that, considering machine and human-based costs, simultaneously (solving SDCWP) results in better solution rather than solving model1 and model2 consecutively (hierarchical method) and cell formations obtained via model1 and SDCWP can be different. The rationale behind this idea is that model1 merely considers machine-based costs, but model3 considers human cost and machine-based cost, simultaneously. To show this difference a numerical example is illustrated. The following tables (Tables 1, 2, 3, 4, 5, 6 and 7) show the problem data.
H X C X M X N 1 N 3 Criterion2 ¼ mch mch h¼1 c¼ m¼1
When the values of these criteria are not considerable, it means that f3 ’ f1 þ f2 and it is more reasonable to solve model1 and model2 consecutively (hierarchical method) instead of solving model3 (because of computational difficulties of solving a larger model with more variables and constraints). In other words, proportion of human costs to machine-based cost is not enough considerable to affect f1; thus there is no meaningful difference between the answers obtained from two strategies (hierarchical method and solving SDCWP). Note that computational difficulties of solving SDCWP can persuade managers to prefer a near optimal solution of the hierarchical method to the optimal solution of SDCWP. On the other hand, when the values of these criteria are considerable, it means that f3*¡ f1*+ f2*, and the advantages of using SDCWP can persuade managers to solve it despite the above-mentioned difficulties. The numerical example has been solved by LINGO 8 twice, one time with model1 and another time with SDCWP. The optimal solution for numerical example obtained from solving SDCWP is presented in Tables 8 and 9: The required number of machines resulted from model1 and SDCWP are compared in Table 10 and also a comparison of their related cell formations is depicted in Fig. 2. As shown in obtained results, the location of machines in each cell and in each period which are obtained from model1 and SDCWP are completely different and this means that human cost plays a significant role in cell formation. This fact is demonstrated by calculating aforementioned criteria: f f Criterion1 ¼ 1 13 ¼ 3:5%; f 13
Criterion2 ¼
H X C X M X
N 1 N 3 ¼ 82 mch mch
h¼1 c¼ m¼1
These values represent that if the human issues are taken in to account; total cost could be decreased near to 3.5%
Int J Adv Manuf Technol (2009) 41:329–342
and this reduction is a result of location alteration, which is obtained from solving SDCWP. Moreover a sensitivity analysis is done for the ratio of machine-based cost to human cost as follows: Let us assume: Model1 :
SDCWP :
Minimize Z1 ¼ f1 Subject to : Eqs: ð10Þ; ð11Þ; ð12Þ; ð13Þ; ð14Þ and ð15Þ;
Minimize Z3 ¼ f3 f3 ¼ f1 þ θ f2 Subject to : Eqs: ð10Þ to ð25Þ:
where θ is a parameter which is used for sensitivity analysis. The numerical example is repeated for various quantity of θ then solved by LINGO 8. According to their results, values of those criteria are calculated and depicted in Figs. 3 and 4. As shown in Fig. 3, by increasing the ratio of human cost to machine-based cost, variation of cell formation cost for model1 and SDCWP becomes more considerable. Nearly the same way as explained for Fig. 3, Fig. 4 shows the variation of cell formation design for model1 and SDCWP. As these figures demonstrate, the effect of f2 on f1 is enhanced by increasing the quantity of θ. In other words, whatever the proportion of human cost to the machine-based cost increases in the same rate considering SDCWP seems more reasonable. To verify this result, four different examples with different dimensions are designed which are named P1 to P4 and this sensitivity analysis repeated for them. Dimension of these examples is shown in Table 11. As expected, the same results are obtained which is shown in Fig. 5.
13 Conclusions and future research In this paper a new model is developed to deal with simultaneous dynamic cell formation and worker assignment. The importance of human issues on cell formation is discussed. We demonstrate that by incorporating the worker assignment into the traditional dynamic cell formation, we can not only improve the cell formation quality, but also reduce the total cost of manufacturing system. To show the performance of the proposed model (SDCWP) a numerical example is solved, and also two criteria are introduced. The results obtained from solving numerical example using SDCWP and hierarchical method regarding to these criteria verifies the ability of the proposed model. Finally a sensitivity analysis is done for four different problems which confirm SDCWP in more cases. SDCWP contains many assumptions which restrict its applicability. Removing these assumptions can be a promising area of work in future research. Moreover it seems that proposed model is drastically NP-hard and cannot be solved in
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reasonable time using traditional methods for large scale problems. Thus developing efficient solving methodologies for SDCWP can be a good subject for succeeding research. Doubtlessly, SDCWP has not come to an end and the path is open for researchers to make the model more comprehensive in a way that it considers learning curve and human issues concurrently and more studies are required to further develop this line of work.
Appendix Theorem: Let us assume: P1: Minimize F3 ðX1 ; X2 Þ ¼ F1 ðX1 Þ þ F2 ðX2 Þ St : G1 ðX1 Þ 0 G2 ðX2 Þ 0 HðX2 Þ WðX1 Þ
ð1Þ
We want to solve this problem with a hierarchical method. Firstly a problem P2, formed as follows: P2 : Minimize F1 ðX1 Þ St : G1 ðX1 Þ 0 Then it is solved and being aware of the best solution of P2 (named: X1*) the second problem can be formed as follows: P3 : Minimize F2 ðX2 Þ St : G2 ðX2 Þ 0 HðX2 Þ W X1
ð2Þ
And finally by solving P2 the best solution obtained via hierarchical method for the main problem (X1 , X2 ) is found. Suppose that F1 , F2 and F3 are the best objective function values of P1, P2 and P3, respectively, and X1 , X2 and X3 are their related optimum solutions. It can be shown that F3 F1 þ F2 Proof: Let us assume: R1 : All feasible solutions for P1
R2 : ðX1 ; X2 ÞjX1 ¼ X1 ; ðX1 ; X2 Þ 2 R1 It is clear that R2⊆R1. Let us define S as S=R1-R2, then for each (X1 , X2 )
∈ S,
one of the following states is possible:
1: F3 X1 ; X2 F X1 þ F X2
It means that there is no better solution for P1 in S thus the best solution is which method. is found by hierarchical In other words: X3 ¼ X1 ; X2 ¼ X1 ; X2 2: F3 X1 ; X2 < F X1 þ F X2
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It means that (X1 , X2 ) is a better solution than (X1 , X2 ), obtained via hierarchical method. This implies that a better solution for P1 is found in S. In other words: X3 ¼ X1 ; X2 6¼ X1 ; X2 According to above descriptions It can be concluded that F3 F1 þ F2
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