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Design of robust cellular manufacturing system for dynamic part population considering multiple processing routes using genetic algorithm
Journal of Manufacturing Systems 35 (2015) 155–163
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Journal of Manufacturing Systems journal homepage: www.elsevier.com/locate/jmansys
Design of robust cellular manufacturing system for dynamic part population considering multiple processing routes using genetic algorithm Kamal Deep a , Pardeep K. Singh b,∗ a b
Department of Mechanical Engineering, Guru Jambheshwar University of Science & Technology, Hisar, HR, India Department of Mechanical Engineering, Sant Longowal Institute of Engineering & Technology, Longowal, PB, India
a r t i c l e
i n f o
Article history: Received 7 October 2012 Received in revised form 15 November 2013 Accepted 22 September 2014 Keywords: Cellular manufacturing Robust design Multiple process-routes Subcontracting part operation Genetic algorithm
a b s t r a c t In this paper, a comprehensive mathematical model is proposed for designing robust machine cells for dynamic part production. The proposed model incorporates machine cell configuration design problem bridged with the machines allocation problem, the dynamic production problem and the part routing problem. Multiple process plans for each part and alternatives process routes for each of those plans are considered. The design of robust cell configurations is based on the selected best part process route from user specified multiple process routes for each part type considering average product demand during the planning horizon. The dynamic part demand can be satisfied from internal production having limited capacity and/or through subcontracting part operation without affecting the machine cell configuration in successive period segments of the planning horizon. A genetic algorithm based heuristic is proposed to solve the model for minimization of the overall cost considering various manufacturing aspects such as production volume, multiple process route, machine capacity, material handling and subcontracting part operation. © 2015 Published by Elsevier Ltd on behalf of The Society of Manufacturing Engineers.
1. Introduction Product diversity, short product life cycle and intense pressure to increase the productivity, etc. lead to global competition among manufactures. A wide product variety as quickly as possible to cater the need of customer with low production cost and maintained quality level signify a capable production system. The traditional manufacturing systems such as job shop and flow lines do not satisfy such requisite. In job shops, products are manufactured in different shops hence jobs spend 95% of the time in non productive activity; much of the time is spent in waiting in queue and the remaining 5% is split between lot setup and processing [1]. In a flow line manufacturing, machines are arranged according to operation sequence of product. It is suitable for high production volume. A major limitation of flow line is the lack of flexibility to produce product for which they are not designed. This is because
∗ Corresponding author. Tel.: +91 94174 62320. E-mail addresses: [email protected] (K. Deep), [email protected] (P.K. Singh).
the system involves specialized machines to perform limited operations with no chance of reconfiguration. The changes in product design and demand fluctuation often require reconfigurable manufacturing systems. An innovative realistic manufacturing methodology is required to suppress the demerits of the traditional manufacturing systems in practice. The Group Technology (GT) is an innovative manufacturing strategy that sorts out similar parts and groups them together into families to take advantage of their similarities in design and manufacturing. The GT build on the concept that single solution can be found to solve a set of problems sharing common principle and tasks, so that time and effort can be saved [2]. The cellular manufacturing (CM) is an application of Group Technology. It takes advantage of the similarity among parts, through standardization and common processing. The CM groups machines into machine cells and parts into part families [3]. The CM suppresses the demerits of job shops and flow lines by increasing the flexibility and variety in production. The major advantage is in terms of material flow which is significantly improved, with reduction in inventory level and distance traveled by the material. This finally results in reduction in cumulative lead time. Various approaches ranging from simple to sophisticated have been suggested for the formation of manufacturing cells and part
http://dx.doi.org/10.1016/j.jmsy.2014.09.008 0278-6125/© 2015 Published by Elsevier Ltd on behalf of The Society of Manufacturing Engineers.
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families. The simple technique usually manipulates part machine matrices. The sophisticated ones can handle many constraints, such as maximum cell size, demand size for different products, number of cells and set up cost [4]. Most of the approaches assume that part demand stays constant over long periods of time [5]. In recent decades, a few studies have been reported considering today’s market based dynamic environment [6–9]. A brief review in this regard has been reported in literature [10,11]. The dynamic conditions prevail in the real world manufacturing environment. Thus the design of CMS, which involves allocation of resources in different manufacturing cells and formation of part families, is difficult. This is due to the fact that the variable product demand size and the machine breakdown affect the performance of cellular manufacturing system design from one period to other. Designers often need to obtain a robust manufacturing system design to handle changes in product demand size, processing times and equipment failure & repairs without any interruption in the production system. This paper suggests a robust model for dynamic production in a cellular manufacturing environment. Multiple process plans for each part and alternatives process routes for each of them are considered. The dynamic part demand can be satisfied from internal production or through subcontracting part operation without affecting the machine cell configuration in successive period segments of the planning horizon. A Genetic algorithm based heuristic is applied to optimize the number of machines in different manufacturing cells in a manufacturing system producing multiple products. The main constraints are the machine-time capacity and the maximum cell size. The computational performance of the proposed approach has been evaluated by testing the algorithm on a few test benchmark problem collected from literature. The robust manufacturing cell configuration so obtained can effectively cope up with the product demand variability, over various periods of planning horizon without production interruption and machine relocation in comparison with adaptive cell configuration of the problems tested.
2. Literature review Different approaches have been proposed to solve PF/MC problems over the decades [2,12–17]. The authors presented numerous formulations and solution strategies to attempt the problem. Wemmerlov and Hyer [18] reviewed over 70 research contributions and classified them based on descriptive and analytical procedures for the part family/machine group identification problem. In another effort Joines et al. [19] reviewed and classified more than 230 methodologies for forming part families and machine cells. Individual techniques are aggregated into methodological groups including array based clustering, hierarchical clustering, non-hierarchical clustering, graph theoretic approach, artificial intelligence, math programming, and other heuristic approaches. The majority of PF/MC formation methods presume a stable part demand and product mix. In reality, the demand for the product varies over its life cycle, new products are introduced, and the production of older products is discontinued. Hence the performance of CMS is adversely influenced by varying value of part demands and product mix. The main input to the PF/MC formation is machine part combinations in cells which represents processing requirements of parts in a given products mix. Hence the product mix variation affects the cell structure in terms of machine part combination and thus performance of the CMS [20]. Rosenblatt and Lee [21] developed a robustness approach to deal with uncertain product demand for single period plant layout. They considered a stochastic environment in which exact values of probability of various scenarios are unknown. The performance of
robust of layout is measured in terms of flexibility to handle various scenarios. Seifoddini [22] presented a probabilistic machine cell formation model to deal with the random nature of product demand for a single period. The machine cell formation is accomplished by employing the similarity coefficient method. The inter-cell material handling cost is used as criteria for selecting the best machine cell configuration. Vakharia and Kaku [23] incorporated long-term demand changes into their 0–1 mathematical programming cell design method by altering parts in part families to regain the benefits of cellular manufacturing system. Similarly, new parts are allocated to existing manufacturing cells. Hence, machine cells composition remains unchanged in their multi-period design. Harhalakis et al. [24] obtained robust cellular configuration using mathematical programming based on the expected values that would be effective over the certain ranges of demand during multiple periods. Once the cells are designed they are expected to remain unchanged during the multi period horizon. Seifoddini and Djassemi [25] carried out sensitivity analysis on the performance of the CMS by considering the product mix variations. The simulation results show that changes in the product mix may lead to the deterioration of the performance of the system. The determination of the sensitivity of a cellular manufacturing system to product mix variation is an important step in the decision making about conversion from job shop to cellular manufacturing. The performance measures used in this analysis are the mean flow time and the work-in-process inventory. Askin et al. [26] proposed algorithm based cell formation method to deal with variation in product mix. The part operation processing has been considered feasible on more than one machine type. Subsequent phases allocate the part operation to specific machines, identify manufacturing cell, and improve the cell design. The machine cells once designed are expected to remain unchanged during the planning horizon. Wicks and Reasor [7] compared the multi period CMS design method with a static CM approach using an illustrative problem. In the static method the cell is designed using the first-period demand only and there are no further rearrangements. When the part mix changes, the new parts are introduced into the existing CMS based on minimizing the inter-cell transfers of parts. Results show that the multi-period approach (adaptive design) involving a planned rearrangement of cells performs better than the static situation. Mungwattana [8] designed the CMS for dynamic and stochastic production requirements, employing routing flexibility. In this work, the differences between robust design and adaptive design strategies of cellular manufacturing systems are described. Arzi et al. [27] proposed a new methodology for the design of CMS working under lumpy demand conditions. Production capacity shortages or machines idleness are observed due to stochastic and unstable external demands. The manufacturing cell reconfiguration is avoided by reducing the variability in the capacity requirements using an appropriate product mix in each period. Kouvelis et al. [28] proposed an algorithm for design of robust layout in cellular manufacturing system. The approach is applicable to single and multi period problems involving variable product mix and demand size. The algorithms, executed in a heuristic fashion, can be effectively used for layout design of large size manufacturing systems. Cao and Chen [29] presented an optimization model integrating cell formation and part allocation with product demand size expressed in number of probabilistic scenario to generate a robust system configuration. Single process plan for each part type is considered. The objective is to minimize the material handling cost and machine cost incurred in the system. A tabu search based heuristic algorithm is proposed to solve the problem.
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Pillai and Subbarao [30] used periodic average part demand to avoid cellular reconfiguration. A mixed integer formulation for robust cell design has been presented to minimize the machine duplication and inter-cell part movements. A genetic algorithm based heuristic is applied to solve the problem. Pillai et al. [31] suggested total penalty cost (TPC) to test the suitability of the layout to be a robust or dynamic layout. TPC values indicate that the suggested layout is suitable as robust layout for the given data sets. The results obtained show that robust plant layout is advantageous in terms of static configuration of facilities in different period segments of planning horizon with no disruptions of the operations. A simulated annealing algorithm is used as a solution procedure for the proposed model. Sharda and Banerjee [32] evaluated candidate machine cell configurations against possible demand variations in future planning periods. The design configuration which results in lowest overall cost at the end of all the planning periods is selected as a robust configuration. The objective function aims at minimizing make span, mean work in progress and number of machines. Uncertainties in processing times, equipment failure & repairs, and product demand are considered. A multi objective GA coupled with Petri net is used to find candidate configurations. Current literature reveals that performance of dynamic cellular manufacturing system usually suffers from variable product demand size, capacity shortage, and sudden machine break down. On the other hand periodic cellular reconfiguration interrupts production, and is associated with machine relocation cost. The discontinued production may cause the loss of profit and customer satisfaction. A robust manufacturing system maintains the performance by effective utilization of production resources, operates consistently, accurately, plans and controls production schedules and delivery deadlines. It minimizes the inventory size and other investments prepared for dealing with uncertainties in dynamic production. In this study an extended problem on robust machine cell formation for dynamic part production capable of producing part families for all the periods has been attempted. Consideration has been made to various real life manufacturing aspects such as multiple part process route, flexible part operation processing, material handling, machine capacity, maximum cell size and subcontracting part operation. Conventional optimization methods for the optimum cell formation problem require substantial amount of time and large memory space. Hence a genetic algorithm based heuristic method has been developed for solving the proposed model. The next section presents the problem formulation and mathematical model of robust manufacturing system. Section 4 describes the genetic algorithm based solution procedure for robust design. An illustrative example with comparison of robust and adaptive CMS design is provided in Section 5. Finally, Section 6 provides the concluding remarks and future research direction.
3. Problem formulation A robust manufacturing system design problem in which different part types undergo multiple manufacturing sequences in dynamic environment is considered. The manufacturing sequences are executed in different production cells. The cells contain identical machines with multiple operational capabilities and limited capacity to process part families. It is assumed that a candidate part operation is processed internally within the machine capacity or through subcontracting part operation to satisfy the dynamic part demand. The proposed approach is able to determine optimal process route instead of user specifying predetermined multiple routes before clustering the machine cell. The machines aggregation in
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the form of cells is based on periodic average part demand of the given planning horizon. The periodic average demand for different segments of planning horizons ensure that each part type is considered for cell design at the beginning [30]. The proposed model is able to form the fixed arrangement of machines in manufacturing cells at the start of planning horizon and determines optimal production plan (i.e. production/subcontracting part operation) for each part type at each period of the planning horizon. The objective of the model is to minimize the overall cost including machine acquisition cost, internal part production cost, subcontracting part operation cost, inter and intra-cellular material handling cost. A mixed-integer mathematical formulation for the robust cell design is presented below. 3.1. Notations
(a) Index sets P {p = 1, 2, 3, . . . P} part types. Op {k = 1, 2, 3, . . ., Op} operation k for part type p. M {m = 1, 2, 3, . . . M} machine types. {c = 1, 2, 3, . . . C} manufacturing cells. C T {t = 1, 2, 3, . . ., T} time periods. (b) Model parameters Am number of machine type m available at the start of planning horizon. number of machine type m assigned to cell c. Amc BU upper cell size limit. lower cell size limit. BL Dp (t) demand for part type p at time period t. DP (1) + DP (2) + . . . + DP (t)/T average demand of part type DPe p. IEp intercellular material handling cost per part type p. IAp intracellular material handling cost per part type p. time required to perform operation k (k = 1, . . ., Op) of part tkpm type p (p = 1, . . ., p) on machine type m(m = 1, . . ., M). ˛m acquisition cost of machine type m. operating cost per hour of machine type m. ˇm kp production cost for operation k of part type p. okp subcontracting cost of operations k of part of type p. Tm capacity of each machine type m in hours. OPekp average demand of part of type p processed for operation k through subcontracting. Nm availability of total number of machine of type m. (c) Decision variables 1, if operation k of parts of type p processed on Xkpmc (t) machine type m in cell c in period t. 0, otherwise. 1, if operation k of parts of type p processed on akpmc machine type m in cell c. 0, otherwise. XPkpmc (t) number of part type p processed by operation k on machine type m in cell c at time t. OPkp (t) number of part type p processed for operation k through subcontracting at time t. 3.2. Mathematical model Minimize
C1 =
Z = C1 + C2 + C3 + C4 + C5 + C6
C M m=1 c=1
(Amc − Am )˛m
(1)
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C2 = T ×
op M C P
C1 : The machine acquisition cost, it is assumed that one unit of each machine type is available at the start of planning horizon C2 : Machine operating cost C3 : Production cost for part operation C4 : Intercellular material handling cost. This cost is sustained whenever the successive operations of the same part type are carried out in different cell. The cost is directly proportional to number of parts moved between two cells. In this model unit intercellular movement is expressed only as a function of part type being handled. C5 : Intracellular material handling cost. The cost upholds the consecutive operations of candidate part processing on different machines in the same cell. It is assumed that unit intracellular cost depends upon the type of part being handled. C6 : Subcontracting cost for part operation. The cost is incurred whenever part operation is subcontracted due to limiting machine capacity or sudden machine breakdown. The model considers unit subcontracting cost for part type being handled.
Dpe. akpmc. tkpm. ˇm
p=1 k=1 m=1 c=1
C3 = T ×
op M C P
Dpe. akpmc. kp
p=1 k=1 m=1 c=1
C4 = T ×
M C P
IEp .Dpe
p=1 m=1 c=1
C5 = T ×
M C P
C6 = T ×
op P
(1 − ak+1,pmc .akpmc )
k=1
op −1
IAp Dpe
p=1 m=1 c=1
op −1
ak+1,pmc .akpmc
k=1
Constraints. The constraints of the problem are shown in equation set (2)–(13) as discussed below.
Opekp okp
p=1 k=1
Subjected to: op P C M T
akpmc · Xkpmc (t) = 1
(2)
XPkpmc (t) + OPkp (t) ≤ Dp (t)
(3)
tkpm. XPkpmc (t) ≤ Tm Amc (t)
(4)
t=1 c=1 k=1 p=1 m=1 op C P M T t=1 c=1 k=1 p=1 m=1 op C P T M t=1 c=1 k=1 p=1 m=1
op P C M T
XPk+1,pmc (t) + opk+1,p (t) = XPkpmc (t) + opkp (t)
t=1 c=1 k=1 p=1 m=1
(5)
C M
Amc ≤ Nm
(6)
Amc ≤ BU
(7)
m=1 c=1 M C m
In addition to these constraints, restrictions represented by Eqs. (9)–(13) denote the logical binary and non negative integer requirement on decision variables.
4. Genetic algorithm based heuristic solution procedure
c=1
M C m
Eq. (2): Each part operation is assigned to one machine, and one cell in period t. Eq. (3): Each part demand can be satisfied in time period t objectively through internal production or subcontracting part operation. More specifically the term ‘XPkpmc ’ represents internal processing of part operations, based a sub-set of operation sequence of part type p are assigned to machines in the cells. Since limiting machine capacity or sudden machine break down results subcontracting of part operation. Eq. (4): Internal part operation processing to be limited to available machine capacity. Eq. (5): The material flow conservation – all the consecutive operations of part type consist of equal production quantities, thus a part operation can be internally processed or subcontracted to satisfy the part demand. Eq. (6): Total number of machine type available in the cells is equal to or less than the total number of machine of same type. Eqs. (7) and (8): The cell size lies within the upper and lower limits.
Amc ≥BL
(8)
c=1
XPkpmc (t)≥0 and integer
(9)
OPkp (t)≥0 and integer
(10)
Amc (t)≥0 and integer
(11)
akpmc (t) ∈ {0, 1}
(12)
Xkpmc (t) ∈ {0, 1}
(13)
Objective function. The model objective function consists of six cost components, explained blow.
The genetic algorithm(s) (GA) are well known search and optimization techniques, having several applications to combinatorial optimization. The basic concept of the GA has its origin in the survival of the fittest principle first laid down by Charles Darwin. The literature on the GA includes quite a large number of papers and a few texts [33–36]. An emblematic GA is based on the controlled growth of population, recombination operators and the schema formation and propagation over generations. It has some unique features such as independence of the gradient information and flexibility to hybridize with domain-dependent heuristics or some other techniques that make it a preferable choice over traditional heuristics [37]. A genetic algorithm based procedure is adopted for solution of the robust design of CMS modeled in previous Section. The GA based solution procedure is presented in following sub-sections.
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cost) value. Thus necessary transformation from objective function to the fitness function is carried out in the following manner [30].
⎧ ⎨ Tmin , i = 1, 2, .....K. T > 0 i Ti Zi = ⎩ 0.1 Ti = 0
Fig. 1. Chromosome macroscopic structure.
Zi : fitness value of string i, Ti : objective function value of the string i. Tmin : the smallest objective function value in the current generation.
4.1. Chromosome representation of solution
4.4. Genetic operators
The chromosomal representation of solution applied to the proposed CMS model comprises of the following genes.
4.4.1. Reproduction or parent selection After evaluating the fitness value of chromosomes in the population, better performing chromosomes (parents) are selected to produce the descendants. Chromosomes with higher fitness value have a higher chance of being selected more often while the poor performing one will be rejected. Different selection schemes have been presented by Goldberg [34]. Binary tournament selection scheme has been adopted for the purpose.
(a) The gene representing the “part operation assigned to machine” is denoted by matrix [PMpk]. The alleles are limited to 1, 2, 3, . . ., M. For instance, the term “PM12 = 4 means ‘operation 2 on ‘part 1 is assigned to machine 4. (b) The gene representing “part operation assigned to cell” is represented by matrix [PCpk]. The alleles are limited to 1, 2, 3, . . ., C. For instance, the term “PC12 = 3 means that ‘operation 2’ of ‘part 1 is assigned to cell 3. The chromosome representation of solution is obtained combining the two matrices described above as shown in Fig. 1. 4.2. Initialization of population The initial population of preferred volume is generated randomly in steps. In first step, the segment [PMpk] of the chromosome is generated randomly considering feasibility of performing part operation on machines. In second step segment [PCpk] of the chromosome is filled randomly. Solution for given problem is represented by the embedded segments (genes) structure known as chromosome. A strategy is applied to minimize the number of inter-cell move of parts. The part operations associated to each part type are assigned to machines existing in the cells. This process is repeated until all the parts are assigned to machines. Given a candidate partmachine assignment solution, the heuristic computes the number of inter-cell part movements that would result minimum number of inter-cell movements (if a part operation is assigned to cell C1, each operation of the part is assigned to cell C1 ). Occasionally cells may not have minimum and maximum number of specified parts that contravene lower or upper limit condition as per Eqs. (7) and (8). For the cell size to remain within the specified lower and upper limits, parts family is to be adjusted by moving parts from the cell having maximum number of parts to the cell having less than minimum number of parts specified.
4.4.2. Crossover or recombination Crossover is performed between two selected parent solutions which create two new child solutions by exchanging segments of the parent solutions, thus child solutions retain partial properties of the parent solutions. Fig. 2 depicts the chromosomes parent 1 and parent 2 selected for crossover. There are two segments in the chromosome, one each for machine and cell. For crossover, the selection of segments can be row-wise or column-wise following the matrix limits and the crossover probabilities. 4.4.3. Mutation Mutation performs a secondary role in functioning of genetic algorithms. Even though crossover operator makes an effective search and recombines chromosome, yet it may cause loss of some useful genetic properties. The mutation operator safeguards against such an irretrievable defeat. The mutation operator performs local search with a low probability. The mutation operator can be implemented by inverting part of a gene in a parent chromosome to obtain child chromosome, as shown in Fig. 3. 4.5. Repair function The crossover and mutation operation may distort chromosome structure so as to yield infeasible solutions i.e. every cell may not have minimum number of machine type. The repair function is used to repair the distorted chromosome such that no machine type is left unassigned, and every cell gets minimum number of machines as per Eqs. (7) and (8). 4.6. Termination of genetic algorithm
4.3. Fitness assessment The fitness value is a decisive factor to measure the quality of a candidate solution or chromosome with reference to the designed objective function (Eq. set (1)) subjected to constraints (Eq. set (2)–(8)) and restrictions (9)–(13). The fitness values are used to select the parent solutions to obtain the next generation of solutions. The descendants or new solutions are selected with higher fitness value obtained by playing binary tournament between parent solutions. The objective function of the CMS design problem is to minimize the total cost. However, genetic algorithm is applicable to the maximization of functions. Hence the maximization of the fitness value corresponds to the minimization of the objective function (total
The genetic algorithm continues to create population of child solutions until a criterion for termination is met. A single criterion or a set of criteria for termination can be adopted. In this case the termination criterion is the maximum number of generation, i.e. the algorithm stops functioning when a specified number of generation is reached. 4.7. Heuristic to control machine duplication The solution obtained in terms of part family and machine cell is based upon the perception that no inter-cell move is allowed. In other words independent cells are created. It leads machine duplication in cells which has to be minimized to drive down
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Fig. 2. Crossover or recombination (a) row wise crossover and (b) column wise crossover.
Fig. 3. Mutation.
the investment. The reduction in machine duplication may lead to increase in inter-cell material handling cost. However in certain circumstances, it is more economical to have inter-cell moves instead of having extra machines [38]. In this segment, tradeoff of having extra machines versus having inter-cell move is considered. If eliminating extra machines in lieu of inter-cell movement results in reduction of total cost, the extra machine will be eliminated. In order to determine the elimination of extra machine from a cell the following algorithm is used. 1. Select a machine type to be considered. Calculate the total number of machines allocated in different cells to meet the production requirements. 2. To eliminate extra machines of the machine type selected, calculate the work load (to be assigned) to the machine type in each cell. The work load is defined as the quantity of the part type to be produced. 3. Compare the ‘saving’ in eliminating a unit of the machine type and the inter-cell material handling cost. If the ‘saving’ in eliminating a unit of the machine type is greater than the inter-cell material handling cost, eliminate the unit of machine in the cell which has the minimum work load. 4. Repeat the steps 1, 2 & 3 for all the machine types. 4.8. Algorithm Step 1. Initialize parameters K (the number of chromosomes in each generation), G (the number of generation), pc (percentage of crossover), and pm (percentage of mutation). Step 2. Generate initial population of machine assignment solug g g g tions, PMi PCi ....PMK PCK and apply the part assignment heuristic to form the part family for each cell explained in Section 4.2. Step 3. Initialize the generation counter g = 1.
Step 4. Generate population fitness as g g g g g g Z1 (PM1 PC1 ), Z2 (PM2 PC2 ), . . ., ZK (PMi PCK ). Step 5. Select individuals from the current population to become parents of the next generation according to their fitness value explained in Section 4.4.1. g g Step 6. Choose mating pool (solutions PMi PCi , in which Zi ≤ 0). Step 7. Generate descendants for new generation: randomly g g g g choose two parents solutions (PMi PCi ) (PMj PCj ) from current mating pool. Step 8. Randomly mate the segments of parent solutions and generate descendant by applying the genetic operator of crossover and mutation explained in Sections 4.4.2 and 4.4.3. Step 9. Use the repair function presented in Section 4.5 to repair the distorted off springs. Step 10. Evaluate the fitness of each descendant solution. Step 11. Increment the generation counter, g = g + 1. Step 12. If g ≤ G, go to step 5, otherwise terminate. Fig. 4. 5. Numerical examples For illustration of the proposed approach for design of the robust CMS, the data sets have been taken from the research reported by Wicks and Reasors [7], Mungwattana [8], Defersha and Chen [9], and Jayakumar and Raju [39] as summarized in Table 1. Since the proposed mathematical model is different from those reported in the research literature, (additional features include internal and subcontracted part operation cost, etc.), the unknown cost parameters such as internal production and subcontracted part operation cost were extracted by cross referencing among the data sets containing them. Therefore all of the data sets used in each numerical example contain values within the same range in term of unit costs. Emphasis is given on number of part types, machine types, number
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Table 1 Summary of computational data sets. Problem no.
Source
Number of part types
Number of M/c types
Number of cells
Number of operation
Number of periods
1 2 3 4 5 6 7
Small-scale Small-scale Small-scale Mungwattana [8] Wicks and Reasor [7] Defersha and Chen [9] Jayakumar and Raju [39]
1 2 3 11 25 25 12
3 3 3 10 11 10 8
2 2 3 3 3 3 3
2 2–3 2–3 2–4 2–3 4–9 3
2 5 5 2 3 2 3
Number of variables 232 1310 2685 42,464 162,015 211,480 560,160
Number of constraints 340 915 2580 33,236 125,460 185,302 435,744
Time elapsed (s) 0.00 0.03 4.25 128.6680 222.2950 409.4677 557.3258
Table 2 The optimum cellular configuration for adaptive design by Jayakumar and Raju [39]. Period
Cell number
Machine type in the cell (number of machines)
Part families
1
1 2 3
M3(2), M6(1), M7(1), M8(1) M1(2), M2(1), M4(1), M7(1) M2(1), M4(2), M5(1), M6(1)
P1, P7, P10 P2, P4, P12 P5, P8, P9
2
1 2 3
M3(1), M7(1), M8(1) M1(2), M4(1) M2(1), M4(1), M5(1), M6(1)
P1, P10 P4, P12 P3, P6, P9
3
1 2 3
M2(1), M3(1), M6(1), M7(1) M4(2), M7(1) M2(1), M4(1), M5(1), M6(1), M7(1)
P5, P10 P4 P2, P6, P7, P8, P11
Fig. 4. Progress of algorithm for search of optimal solution corresponding to the lowest overall cost.
of operations and number of cells. Each one of the numerical example used is solved as an integrated model. It is evident that the required computational time increases with the problem size in terms of variables, constraints and number of period (Table 1). A set of seven problems have been attempted to demonstrate the proposed approach. Problem 7 is considered to be large scale one with 560,160 and 435,744 numbers of variables and constraints respectively. A detailed discussion on solution of the problem 7 has been presented in next section. All the computational experiments were performed on 1.86 GHz Pentium-IV workstation with Windows XP. The mathematical model was solved using MATLAB-2009. 5.1. Result and discussion The largest problem that was attempted for an optimal solution was the one selected from Jayakumar and Raju [39]. This illustrates various features of the solution for the proposed robust CMS model. Jayakumar and Raju [39] used extended lingo version 8.0+ [40] to solve the problem. Using this problem instance and a linearized version of their model, an optimal solution has
been obtained making use of the genetic algorithm. Based on the computational experience the following values are considered for the parameters: probability of crossover (Pc ) = 0.7, probability of mutation (Pm ) = 0.025, population size (K) = 300 and number of generation (G) = 50. The objective function value obtained in the proposed work cannot be compared meaningfully because of inherent difference in the objective cost value as obtained by Jayakumar and Raju [39]. The proposed approach is comparably more general for taking up (i) the internal part operation manufacturing cost considering machine capacity, and (ii) the subcontracting part operation cost. Despite of a few differences in the mathematical model, a meaningful comparison can be made in terms of the decision aspects of the CMS design resulting from two different solutions. The solution obtained with the proposed model on problem 7 (Table 1) is detailed out in the rest of this section and simultaneously compared with solution obtained for adaptive cell design (see Table 2) in Jayakumar and Raju [39]. Table 3 shows the robust manufacturing cell design and allocated part families for three successive period segments in a planning horizon. Table 4 summarizes the optimal process route obtained for each part type by proposed algorithm, applicable to successively formed part family in three periods segment of planning horizon. Tables 5–7 represents the part operation trade-offs in different production mode considering limited resource capacity. Table 8 demonstrates the selected routings for part type P5 (machine and cell) in time period 1. For instance demand for part P5 during period 1 is 550. The part undergoes three different operations to complete the production. The first two operations are performed on machine type m6 and m7 in cell C3 whereas third operation assigned to machine number 6 in cell C3 cannot be performed as machine capacity is over limit. Hence part operation 3 of part type P5 is subcontracted from cell C3 . The solution obtained from Jayakumar and Raju [39] does not involve any flexible internal production for part P5 in period 1. In this paper, the possible alternative process routing for each part type is defined in terms of internal production and subcontracting part operation. Thus, the co-existence of multiple possible source routing (in-house production/subcontracting) for all parts are allowed and it is a tangible advantage during unexpected machine breakdown and production capacity shortage occurring in real world.
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Table 3 The optimum cellular configuration for robust design by proposed algorithm. Period
Cell Number
Machine type in the Cell (Number of Machines)
Part Families
1
1 2 3
M1(1),M4(1),M7(1) M2(1),M4(1) M3(1),M5(1),M6(1),M7(1),M8(1)
P4, P12 P2, P7, P9 P1,P5, P8, P10
2
1 2 3
M1(1),M4(1),M7(1) M2(1),M4(1) M3(1),M5(1),M6(1),M7(1),M8(1)
P4, P12 P3, P9 P1, P6, P10
3
1 2 3
M1(1),M4(1),M7(1) M2(1),M4(1) M3(1),M5(1),M6(1),M7(1),M8(1)
P4 P2,P7 P5, P6, P8, P10, P11
Table 7 Part operation assigned to machine (t = 3).
Table 4 The optimal process route for each part type by proposed algorithm. Part type
Optimal part process route
Part no
P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12
M8–M7–M8 M2–M2–M4 M2–M4–M7 M1–M4–M7 M6–M7–M6 M3–M5–M4 M4–M6–M2 M5–M5–M3 M4–M6–M4 M3–M6–M6 M8–M7–M8 M1–M4–M1
2 4 5 6 7 8 10 11 *
1 2 4 5 7 8 9 10 12 *
Operations 1
2
3
8 2 1 6 4 5 4 3* 1
7 2 4 7 6 5 6* 6 4
8 4* 7 6* 2* 3 4* 6* 1*
Subcontracted part operations.
Table 6 Part operation assigned to machine (t = 2). Part no
1 3 4 6 9 10 12 *
1
2
3
2 1 6 3* 4 5 3 8
2 4 7 5 6 5 6 7
4 7 6* 4 2 3* 6* 8*
Subcontracted part operations.
Table 8 Comparison routing for part type 5 in period 1.
Table 5 Part operation assigned to machine (t = 1). Part no
Operations
Operations 1
2
3
8 2 1 3 4 3* 1
7 4 4 5 6 6 4
8 7 7 4* 4 6 1*
Subcontracted part operations.
Table 9 illustrates the parts demand met for part type P7 and P10 in period 1, part type P4 and P6 in period 2 and part type P10 and P11 in period 3 through internal production considering limited production capacity and subcontracted part operation during three period segments of planning span. The option for subcontracting part operation is not present in the model proposed by Jayakumar and Raju [39]. Hence no comparison can be made in terms of different source routings. The simultaneous consideration of in-house production and subcontracted part operation processing gives a realistic view of manufacturing system. The proposed model
Part route
1
2
3
Proposed model Jayakumar and Raju [39]
550 M6 /C3 550 M6 /C3
550 M7 /C3 550 M7 /C2
550 M6 /C3 * 550 M2 /C3
*
Subcontracted part operation.
Table 9 Comparison of production plan for parts P7 and P10 for (t = 1), P4 andP6 for (t = 2), P10 and P11 for (t = 3). Period 1 Part/operation Internal production Subcontracted Demand
Propose model P7 P10 1,2 2 3 1,3 450 900
Jayakumar and Raju [39] P7 P10 1,2,3 1,2,3 nil nil 450 900
Period 2 Part/operation Internal production Subcontracted Demand
Propose model P4 P6 1,2,3 1,2 Nil 3 500 500
Jayakumar and Raju [39] P4 P6 1,2,3 1,2,3 nil nil 500 500
Period 3 Part/Operation Internal Production Subcontracted Demand
Propose model P10 P11 1,2 1,2 3 3 700 700
Jayakumar and Raju [39] P10 P11 1,2,3 12,3 nil nil 700 700
presents the possible production plan for each part type during each period given in Tables 5–7, showing higher flexibility in meeting the part demand size. In this study the tradeoff is made between intracellular, intercellular part movement and machine duplication by simultaneously minimizing all the three costs within the objective function. This is essential since, high intracellular movement cost for successive part operations implies large cell size, reducing the effectiveness of manufacturing system. On the other side, minimum inter-cell movements lead to increase in number of duplicate machines in the cells and adversely affect the benefits of CMS by inappropriate workload distribution among cells. In this research a few authors (Nsakanda et al. [41]; Ahkioon et al. [42] and Defersha and Chen [9]) addressed the subcontracting/outsourcing only as a
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subset of the part demand size. These proposed approaches make a useful contribution to the existing knowledge. 6. Concluding remarks and future research direction This paper presents a novel integrated mathematical model for design of robust cellular manufacturing system (RCMS) considering dynamic production and multi period production planning. An extended problem is presented based on the basis of RCMS model proposed in literature [30]. The proposed model offers flexibility in production planning (production/subcontracting) that can be achieved by producing product mixes at each period of planning horizon within the limited capacity without affecting the manufacturing cell configuration. This would make the problem more practical and realistic. The proposed approach is demonstrated on a manufacturing system configuration design problem considering different part types undergoing multiple manufacturing sequences in a dynamic environment. The algorithm aggregates resources into different manufacturing cells based on selected optimal process route from user specifying multiple routes. The model is computable with single part routing as well as multiple part routings. The proposed approach can also be readily used where limits are imposed on the cell sizes and/or number of cells. The performance of the model has been verified by different bench mark problems reported in literature. The results obtained show that the co-existence of multiple possible source routings (in-house production/subcontracting) builds up flexibility in production and it is a tangible advantage during unexpected machine breakdown and production capacity shortage occurring in real world. The research reported in this paper is a part of the major research project on robust design of CMS. The authors are working to further improve the mathematical model for design of CMS incorporating more real world aspects of the manufacturing system, such as, lot splitting and machine adjacency requirements to widen its area and make the study more useful. The study on comparison of the benchmark results obtained using other techniques is very important research assignment, and shall be taken up in due course. References [1] Askin R, Standridge C. Modeling and analysis of manufacturing systems. New York: John Wiley & Sons; 1993. [2] Selim H, Askin R, Vakharia A. Cell formation in group technology: review, evaluation and directions for future research. Comput Ind Eng 1998;34(1): 3–20. [3] Hu L, Yasuda K. Minimizing material handing cost in cell formation with alternative processing routes by grouping genetic algorithm. Int J Prod Res 2006;44:2133–67. [4] Venkumar P, Sekar KC. Design of cellular manufacturing system using non-traditional optimization algorithms. In: Modrák V, Pandian RS, editors. Operations management research and cellular manufacturing systems: innovative methods and approaches. Hershey, PA: IGI Global; 2012. p. 99–139 [chapter 6]. [5] Burbidge JL. Production flow analysis. Prod Eng 1963;42:742–52. [6] Rheault M, Drolet J, Abdulnour G. Physically reconfigurable virtual cells: a dynamic model for a highly dynamic environment. Comput Ind Eng 1995;29(1–4):221–5. [7] Wicks EM, Reasor RJ. Designing cellular manufacturing systems with dynamic part populations. IIE Trans 1999;31:11–20. [8] Mungwattana A. Design of cellular manufacturing system for dynamic and uncertain production requirement with presence of routing flexibility [Ph.D. thesis]. Blackburg, VA: Faculty of the Vargina Polytechnic Institute and State University; 2000.
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[9] Defersha F, Chen M. A comprehensive mathematical model for the design of cellular manufacturing systems. Int J Prod Econ 2006;103:767–83. [10] Balakrishnan J, Cheng CH. Dynamic cellular manufacturing under multi-period planning horizons. J Manuf Technol Manag 2005;16(5):516–30. [11] Deep K, Singh PK. Multi period planning in cellular manufacturing: a discussion. Int J Eng Stud 2010;2(3):251–62. [12] Greene T, Sadowski R. A review of cellular manufacturing assumptions, advantages, and design techniques. J Oper Manag 1984;4(2):85–97. [13] Singh N. Design of cellular manufacturing systems: an invited review. Eur J Oper Res 1993;69(3):284–91. [14] Kamrani A, Parsaie H, Chaudhry M. A survey of design methods for manufacturing cells? Comput Ind Eng 1993;25(1–4):487–90. [15] Offodile O, Mehrez A, Grznar J. Cellular manufacturing: a taxonomic review framework? J Manuf Syst 1994;13(3):196–220. [16] Joines J, Culbreth C, King R. A comprehensive review of production oriented cell formation techniques. Int J Fact Autom Inf Manag 1996;3(3–4):225–65. [17] Agarwal A, Sarkis J. A review and analysis of comparative performance studies on functional and cellular manufacturing layouts? Comput Ind Eng 1998;34(1):77–89. [18] Wemmerlov U, Hyer NL. Procedure for the part family/machine group identification problem cellular manufacturing. J Oper Manag 1986;6(2):125–47. [19] Joines JA, King RE, Culbreth CT. A comprehensive review of productionoriented manufacturing cell formation techniques. Int J Flex Autom Intell Manuf 1996;3(3–4):161–7. [20] King J, Nakornchai V. Machine component group formation in group technology: review and extension? Int J Prod Res 1982;20(2):117–33. [21] Rosenblatt M, Lee H. A robustness to facilities design? Int J Prod Res 1987;25(4):479–86. [22] Seifoddini H. A probabilistic model for machine cell formation? J Manuf Syst 1990;9(1):69–75. [23] Vakharia AJ, Kaku BK. Redesigning a cellular manufacturing system to handle long-term demand changes: a methodology and investigation. Decis Sci 1993;24(5):909–30. [24] Harhalakis G, Ioannou I, Minis I, Nagi R. Manufacturing cell formation under random product demand. Int J Prod Res 1994;32(1):47–64. [25] Seifoddini H, Djassemi M. Sensitivity analysis in cellular manufacturing system in the case of product mix variation. Comput Ind Eng 1996;31:163–7. [26] Askin R, Selim H, Vakharia A. A methodology for designing flexible cellular manufacturing systems. IIE Trans 1997;29(7):599–610. [27] Arzi Y, Bukchin J, Masin M. An efficiency frontier approach for the design of cellular manufacturing system in a lumpy demand environment. Eur J Oper Res 2001;134:346–64. [28] Kouvelis P, Kurawarwala AA, Gutiérrez GJ. Algorithms for robust single and multiple period layout planning for manufacturing systems. Eur J Oper Res 2003;63(2):287–303. [29] Cao D, Chen M. A robust cell formation for varying product demands. Int J Prod Res 2005;43(8):1587–605. [30] Pillai. VM, Subbarao K. A robust cellular manufacturing system design for dynamic part population using a genetic algorithm. Int J Prod Res 2007;46(18):5191–210. [31] Pillai VM, Hunagund IB, Krishnan KK. Design of robust layout for dynamic plant layout problems. Comput Ind Eng 2011;61(3):813–23. [32] Sharda B, Banerjee A. Robust manufacturing system design using multi objective genetic algorithms, petri nets Bayesian uncertainty representation. J Manuf Syst 2013;32:315–24. [33] Holland JH. Adaption in natural and artificial systems. Ann Arbor, MI: The University of Michigan Press; 1975. [34] Goldberg DE. Genetic algorithms in search, optimization and machine learning. Reading, MA: Addision-Wesley; 1989. [35] Reeves CR. Genetic algorithm for the operation researcher. INFORMS J Comput 1997;9(3):231–50. [36] Reeves C. Genetic algorithms. In: Glover F, Kochenberger GA, editors. Handbook of meta heuristics. Dordrecht: Kluwer Academic; 2003. p. 55–82. [37] Gen M, Cheng R. Genetic algorithms and engineering design. New York: Wiley Interscience Publications; 1997. [38] Deep K, Singh PK. Machine cell formation: using genetic algorithm based heuristic considering alternative route. Int J Oper Res 2013 [in press]. [39] Jayakumar V, Raju R. An adaptive cellular manufacturing system design with routing flexibility and dynamic system reconfiguration. Eur J Sci Res 2010;47(4):595–611. [40] LINDO Systems Inc. Extended Lingo: Version 8.0+ users guide. Chicago: LINDO Systems Inc.; 2005. [41] Nsakanda AL, Diaby M, Price WL. Hybrid genetic approach for solving large scale capacitated cell formation problems with multiple routings. Eur J Oper Res 2006;171:1051–70. [42] Ahkioon S, Bulgak AA, Bektas T. Integrated cellular manufacturing system design with production planning and dynamic system reconfiguration. Eur J Oper Res 2009;192:414–28.
Contents lists available at ScienceDirect
Journal of Manufacturing Systems journal homepage: www.elsevier.com/locate/jmansys
Design of robust cellular manufacturing system for dynamic part population considering multiple processing routes using genetic algorithm Kamal Deep a , Pardeep K. Singh b,∗ a b
Department of Mechanical Engineering, Guru Jambheshwar University of Science & Technology, Hisar, HR, India Department of Mechanical Engineering, Sant Longowal Institute of Engineering & Technology, Longowal, PB, India
a r t i c l e
i n f o
Article history: Received 7 October 2012 Received in revised form 15 November 2013 Accepted 22 September 2014 Keywords: Cellular manufacturing Robust design Multiple process-routes Subcontracting part operation Genetic algorithm
a b s t r a c t In this paper, a comprehensive mathematical model is proposed for designing robust machine cells for dynamic part production. The proposed model incorporates machine cell configuration design problem bridged with the machines allocation problem, the dynamic production problem and the part routing problem. Multiple process plans for each part and alternatives process routes for each of those plans are considered. The design of robust cell configurations is based on the selected best part process route from user specified multiple process routes for each part type considering average product demand during the planning horizon. The dynamic part demand can be satisfied from internal production having limited capacity and/or through subcontracting part operation without affecting the machine cell configuration in successive period segments of the planning horizon. A genetic algorithm based heuristic is proposed to solve the model for minimization of the overall cost considering various manufacturing aspects such as production volume, multiple process route, machine capacity, material handling and subcontracting part operation. © 2015 Published by Elsevier Ltd on behalf of The Society of Manufacturing Engineers.
1. Introduction Product diversity, short product life cycle and intense pressure to increase the productivity, etc. lead to global competition among manufactures. A wide product variety as quickly as possible to cater the need of customer with low production cost and maintained quality level signify a capable production system. The traditional manufacturing systems such as job shop and flow lines do not satisfy such requisite. In job shops, products are manufactured in different shops hence jobs spend 95% of the time in non productive activity; much of the time is spent in waiting in queue and the remaining 5% is split between lot setup and processing [1]. In a flow line manufacturing, machines are arranged according to operation sequence of product. It is suitable for high production volume. A major limitation of flow line is the lack of flexibility to produce product for which they are not designed. This is because
∗ Corresponding author. Tel.: +91 94174 62320. E-mail addresses: [email protected] (K. Deep), [email protected] (P.K. Singh).
the system involves specialized machines to perform limited operations with no chance of reconfiguration. The changes in product design and demand fluctuation often require reconfigurable manufacturing systems. An innovative realistic manufacturing methodology is required to suppress the demerits of the traditional manufacturing systems in practice. The Group Technology (GT) is an innovative manufacturing strategy that sorts out similar parts and groups them together into families to take advantage of their similarities in design and manufacturing. The GT build on the concept that single solution can be found to solve a set of problems sharing common principle and tasks, so that time and effort can be saved [2]. The cellular manufacturing (CM) is an application of Group Technology. It takes advantage of the similarity among parts, through standardization and common processing. The CM groups machines into machine cells and parts into part families [3]. The CM suppresses the demerits of job shops and flow lines by increasing the flexibility and variety in production. The major advantage is in terms of material flow which is significantly improved, with reduction in inventory level and distance traveled by the material. This finally results in reduction in cumulative lead time. Various approaches ranging from simple to sophisticated have been suggested for the formation of manufacturing cells and part
http://dx.doi.org/10.1016/j.jmsy.2014.09.008 0278-6125/© 2015 Published by Elsevier Ltd on behalf of The Society of Manufacturing Engineers.
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families. The simple technique usually manipulates part machine matrices. The sophisticated ones can handle many constraints, such as maximum cell size, demand size for different products, number of cells and set up cost [4]. Most of the approaches assume that part demand stays constant over long periods of time [5]. In recent decades, a few studies have been reported considering today’s market based dynamic environment [6–9]. A brief review in this regard has been reported in literature [10,11]. The dynamic conditions prevail in the real world manufacturing environment. Thus the design of CMS, which involves allocation of resources in different manufacturing cells and formation of part families, is difficult. This is due to the fact that the variable product demand size and the machine breakdown affect the performance of cellular manufacturing system design from one period to other. Designers often need to obtain a robust manufacturing system design to handle changes in product demand size, processing times and equipment failure & repairs without any interruption in the production system. This paper suggests a robust model for dynamic production in a cellular manufacturing environment. Multiple process plans for each part and alternatives process routes for each of them are considered. The dynamic part demand can be satisfied from internal production or through subcontracting part operation without affecting the machine cell configuration in successive period segments of the planning horizon. A Genetic algorithm based heuristic is applied to optimize the number of machines in different manufacturing cells in a manufacturing system producing multiple products. The main constraints are the machine-time capacity and the maximum cell size. The computational performance of the proposed approach has been evaluated by testing the algorithm on a few test benchmark problem collected from literature. The robust manufacturing cell configuration so obtained can effectively cope up with the product demand variability, over various periods of planning horizon without production interruption and machine relocation in comparison with adaptive cell configuration of the problems tested.
2. Literature review Different approaches have been proposed to solve PF/MC problems over the decades [2,12–17]. The authors presented numerous formulations and solution strategies to attempt the problem. Wemmerlov and Hyer [18] reviewed over 70 research contributions and classified them based on descriptive and analytical procedures for the part family/machine group identification problem. In another effort Joines et al. [19] reviewed and classified more than 230 methodologies for forming part families and machine cells. Individual techniques are aggregated into methodological groups including array based clustering, hierarchical clustering, non-hierarchical clustering, graph theoretic approach, artificial intelligence, math programming, and other heuristic approaches. The majority of PF/MC formation methods presume a stable part demand and product mix. In reality, the demand for the product varies over its life cycle, new products are introduced, and the production of older products is discontinued. Hence the performance of CMS is adversely influenced by varying value of part demands and product mix. The main input to the PF/MC formation is machine part combinations in cells which represents processing requirements of parts in a given products mix. Hence the product mix variation affects the cell structure in terms of machine part combination and thus performance of the CMS [20]. Rosenblatt and Lee [21] developed a robustness approach to deal with uncertain product demand for single period plant layout. They considered a stochastic environment in which exact values of probability of various scenarios are unknown. The performance of
robust of layout is measured in terms of flexibility to handle various scenarios. Seifoddini [22] presented a probabilistic machine cell formation model to deal with the random nature of product demand for a single period. The machine cell formation is accomplished by employing the similarity coefficient method. The inter-cell material handling cost is used as criteria for selecting the best machine cell configuration. Vakharia and Kaku [23] incorporated long-term demand changes into their 0–1 mathematical programming cell design method by altering parts in part families to regain the benefits of cellular manufacturing system. Similarly, new parts are allocated to existing manufacturing cells. Hence, machine cells composition remains unchanged in their multi-period design. Harhalakis et al. [24] obtained robust cellular configuration using mathematical programming based on the expected values that would be effective over the certain ranges of demand during multiple periods. Once the cells are designed they are expected to remain unchanged during the multi period horizon. Seifoddini and Djassemi [25] carried out sensitivity analysis on the performance of the CMS by considering the product mix variations. The simulation results show that changes in the product mix may lead to the deterioration of the performance of the system. The determination of the sensitivity of a cellular manufacturing system to product mix variation is an important step in the decision making about conversion from job shop to cellular manufacturing. The performance measures used in this analysis are the mean flow time and the work-in-process inventory. Askin et al. [26] proposed algorithm based cell formation method to deal with variation in product mix. The part operation processing has been considered feasible on more than one machine type. Subsequent phases allocate the part operation to specific machines, identify manufacturing cell, and improve the cell design. The machine cells once designed are expected to remain unchanged during the planning horizon. Wicks and Reasor [7] compared the multi period CMS design method with a static CM approach using an illustrative problem. In the static method the cell is designed using the first-period demand only and there are no further rearrangements. When the part mix changes, the new parts are introduced into the existing CMS based on minimizing the inter-cell transfers of parts. Results show that the multi-period approach (adaptive design) involving a planned rearrangement of cells performs better than the static situation. Mungwattana [8] designed the CMS for dynamic and stochastic production requirements, employing routing flexibility. In this work, the differences between robust design and adaptive design strategies of cellular manufacturing systems are described. Arzi et al. [27] proposed a new methodology for the design of CMS working under lumpy demand conditions. Production capacity shortages or machines idleness are observed due to stochastic and unstable external demands. The manufacturing cell reconfiguration is avoided by reducing the variability in the capacity requirements using an appropriate product mix in each period. Kouvelis et al. [28] proposed an algorithm for design of robust layout in cellular manufacturing system. The approach is applicable to single and multi period problems involving variable product mix and demand size. The algorithms, executed in a heuristic fashion, can be effectively used for layout design of large size manufacturing systems. Cao and Chen [29] presented an optimization model integrating cell formation and part allocation with product demand size expressed in number of probabilistic scenario to generate a robust system configuration. Single process plan for each part type is considered. The objective is to minimize the material handling cost and machine cost incurred in the system. A tabu search based heuristic algorithm is proposed to solve the problem.
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Pillai and Subbarao [30] used periodic average part demand to avoid cellular reconfiguration. A mixed integer formulation for robust cell design has been presented to minimize the machine duplication and inter-cell part movements. A genetic algorithm based heuristic is applied to solve the problem. Pillai et al. [31] suggested total penalty cost (TPC) to test the suitability of the layout to be a robust or dynamic layout. TPC values indicate that the suggested layout is suitable as robust layout for the given data sets. The results obtained show that robust plant layout is advantageous in terms of static configuration of facilities in different period segments of planning horizon with no disruptions of the operations. A simulated annealing algorithm is used as a solution procedure for the proposed model. Sharda and Banerjee [32] evaluated candidate machine cell configurations against possible demand variations in future planning periods. The design configuration which results in lowest overall cost at the end of all the planning periods is selected as a robust configuration. The objective function aims at minimizing make span, mean work in progress and number of machines. Uncertainties in processing times, equipment failure & repairs, and product demand are considered. A multi objective GA coupled with Petri net is used to find candidate configurations. Current literature reveals that performance of dynamic cellular manufacturing system usually suffers from variable product demand size, capacity shortage, and sudden machine break down. On the other hand periodic cellular reconfiguration interrupts production, and is associated with machine relocation cost. The discontinued production may cause the loss of profit and customer satisfaction. A robust manufacturing system maintains the performance by effective utilization of production resources, operates consistently, accurately, plans and controls production schedules and delivery deadlines. It minimizes the inventory size and other investments prepared for dealing with uncertainties in dynamic production. In this study an extended problem on robust machine cell formation for dynamic part production capable of producing part families for all the periods has been attempted. Consideration has been made to various real life manufacturing aspects such as multiple part process route, flexible part operation processing, material handling, machine capacity, maximum cell size and subcontracting part operation. Conventional optimization methods for the optimum cell formation problem require substantial amount of time and large memory space. Hence a genetic algorithm based heuristic method has been developed for solving the proposed model. The next section presents the problem formulation and mathematical model of robust manufacturing system. Section 4 describes the genetic algorithm based solution procedure for robust design. An illustrative example with comparison of robust and adaptive CMS design is provided in Section 5. Finally, Section 6 provides the concluding remarks and future research direction.
3. Problem formulation A robust manufacturing system design problem in which different part types undergo multiple manufacturing sequences in dynamic environment is considered. The manufacturing sequences are executed in different production cells. The cells contain identical machines with multiple operational capabilities and limited capacity to process part families. It is assumed that a candidate part operation is processed internally within the machine capacity or through subcontracting part operation to satisfy the dynamic part demand. The proposed approach is able to determine optimal process route instead of user specifying predetermined multiple routes before clustering the machine cell. The machines aggregation in
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the form of cells is based on periodic average part demand of the given planning horizon. The periodic average demand for different segments of planning horizons ensure that each part type is considered for cell design at the beginning [30]. The proposed model is able to form the fixed arrangement of machines in manufacturing cells at the start of planning horizon and determines optimal production plan (i.e. production/subcontracting part operation) for each part type at each period of the planning horizon. The objective of the model is to minimize the overall cost including machine acquisition cost, internal part production cost, subcontracting part operation cost, inter and intra-cellular material handling cost. A mixed-integer mathematical formulation for the robust cell design is presented below. 3.1. Notations
(a) Index sets P {p = 1, 2, 3, . . . P} part types. Op {k = 1, 2, 3, . . ., Op} operation k for part type p. M {m = 1, 2, 3, . . . M} machine types. {c = 1, 2, 3, . . . C} manufacturing cells. C T {t = 1, 2, 3, . . ., T} time periods. (b) Model parameters Am number of machine type m available at the start of planning horizon. number of machine type m assigned to cell c. Amc BU upper cell size limit. lower cell size limit. BL Dp (t) demand for part type p at time period t. DP (1) + DP (2) + . . . + DP (t)/T average demand of part type DPe p. IEp intercellular material handling cost per part type p. IAp intracellular material handling cost per part type p. time required to perform operation k (k = 1, . . ., Op) of part tkpm type p (p = 1, . . ., p) on machine type m(m = 1, . . ., M). ˛m acquisition cost of machine type m. operating cost per hour of machine type m. ˇm kp production cost for operation k of part type p. okp subcontracting cost of operations k of part of type p. Tm capacity of each machine type m in hours. OPekp average demand of part of type p processed for operation k through subcontracting. Nm availability of total number of machine of type m. (c) Decision variables 1, if operation k of parts of type p processed on Xkpmc (t) machine type m in cell c in period t. 0, otherwise. 1, if operation k of parts of type p processed on akpmc machine type m in cell c. 0, otherwise. XPkpmc (t) number of part type p processed by operation k on machine type m in cell c at time t. OPkp (t) number of part type p processed for operation k through subcontracting at time t. 3.2. Mathematical model Minimize
C1 =
Z = C1 + C2 + C3 + C4 + C5 + C6
C M m=1 c=1
(Amc − Am )˛m
(1)
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C2 = T ×
op M C P
C1 : The machine acquisition cost, it is assumed that one unit of each machine type is available at the start of planning horizon C2 : Machine operating cost C3 : Production cost for part operation C4 : Intercellular material handling cost. This cost is sustained whenever the successive operations of the same part type are carried out in different cell. The cost is directly proportional to number of parts moved between two cells. In this model unit intercellular movement is expressed only as a function of part type being handled. C5 : Intracellular material handling cost. The cost upholds the consecutive operations of candidate part processing on different machines in the same cell. It is assumed that unit intracellular cost depends upon the type of part being handled. C6 : Subcontracting cost for part operation. The cost is incurred whenever part operation is subcontracted due to limiting machine capacity or sudden machine breakdown. The model considers unit subcontracting cost for part type being handled.
Dpe. akpmc. tkpm. ˇm
p=1 k=1 m=1 c=1
C3 = T ×
op M C P
Dpe. akpmc. kp
p=1 k=1 m=1 c=1
C4 = T ×
M C P
IEp .Dpe
p=1 m=1 c=1
C5 = T ×
M C P
C6 = T ×
op P
(1 − ak+1,pmc .akpmc )
k=1
op −1
IAp Dpe
p=1 m=1 c=1
op −1
ak+1,pmc .akpmc
k=1
Constraints. The constraints of the problem are shown in equation set (2)–(13) as discussed below.
Opekp okp
p=1 k=1
Subjected to: op P C M T
akpmc · Xkpmc (t) = 1
(2)
XPkpmc (t) + OPkp (t) ≤ Dp (t)
(3)
tkpm. XPkpmc (t) ≤ Tm Amc (t)
(4)
t=1 c=1 k=1 p=1 m=1 op C P M T t=1 c=1 k=1 p=1 m=1 op C P T M t=1 c=1 k=1 p=1 m=1
op P C M T
XPk+1,pmc (t) + opk+1,p (t) = XPkpmc (t) + opkp (t)
t=1 c=1 k=1 p=1 m=1
(5)
C M
Amc ≤ Nm
(6)
Amc ≤ BU
(7)
m=1 c=1 M C m
In addition to these constraints, restrictions represented by Eqs. (9)–(13) denote the logical binary and non negative integer requirement on decision variables.
4. Genetic algorithm based heuristic solution procedure
c=1
M C m
Eq. (2): Each part operation is assigned to one machine, and one cell in period t. Eq. (3): Each part demand can be satisfied in time period t objectively through internal production or subcontracting part operation. More specifically the term ‘XPkpmc ’ represents internal processing of part operations, based a sub-set of operation sequence of part type p are assigned to machines in the cells. Since limiting machine capacity or sudden machine break down results subcontracting of part operation. Eq. (4): Internal part operation processing to be limited to available machine capacity. Eq. (5): The material flow conservation – all the consecutive operations of part type consist of equal production quantities, thus a part operation can be internally processed or subcontracted to satisfy the part demand. Eq. (6): Total number of machine type available in the cells is equal to or less than the total number of machine of same type. Eqs. (7) and (8): The cell size lies within the upper and lower limits.
Amc ≥BL
(8)
c=1
XPkpmc (t)≥0 and integer
(9)
OPkp (t)≥0 and integer
(10)
Amc (t)≥0 and integer
(11)
akpmc (t) ∈ {0, 1}
(12)
Xkpmc (t) ∈ {0, 1}
(13)
Objective function. The model objective function consists of six cost components, explained blow.
The genetic algorithm(s) (GA) are well known search and optimization techniques, having several applications to combinatorial optimization. The basic concept of the GA has its origin in the survival of the fittest principle first laid down by Charles Darwin. The literature on the GA includes quite a large number of papers and a few texts [33–36]. An emblematic GA is based on the controlled growth of population, recombination operators and the schema formation and propagation over generations. It has some unique features such as independence of the gradient information and flexibility to hybridize with domain-dependent heuristics or some other techniques that make it a preferable choice over traditional heuristics [37]. A genetic algorithm based procedure is adopted for solution of the robust design of CMS modeled in previous Section. The GA based solution procedure is presented in following sub-sections.
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cost) value. Thus necessary transformation from objective function to the fitness function is carried out in the following manner [30].
⎧ ⎨ Tmin , i = 1, 2, .....K. T > 0 i Ti Zi = ⎩ 0.1 Ti = 0
Fig. 1. Chromosome macroscopic structure.
Zi : fitness value of string i, Ti : objective function value of the string i. Tmin : the smallest objective function value in the current generation.
4.1. Chromosome representation of solution
4.4. Genetic operators
The chromosomal representation of solution applied to the proposed CMS model comprises of the following genes.
4.4.1. Reproduction or parent selection After evaluating the fitness value of chromosomes in the population, better performing chromosomes (parents) are selected to produce the descendants. Chromosomes with higher fitness value have a higher chance of being selected more often while the poor performing one will be rejected. Different selection schemes have been presented by Goldberg [34]. Binary tournament selection scheme has been adopted for the purpose.
(a) The gene representing the “part operation assigned to machine” is denoted by matrix [PMpk]. The alleles are limited to 1, 2, 3, . . ., M. For instance, the term “PM12 = 4 means ‘operation 2 on ‘part 1 is assigned to machine 4. (b) The gene representing “part operation assigned to cell” is represented by matrix [PCpk]. The alleles are limited to 1, 2, 3, . . ., C. For instance, the term “PC12 = 3 means that ‘operation 2’ of ‘part 1 is assigned to cell 3. The chromosome representation of solution is obtained combining the two matrices described above as shown in Fig. 1. 4.2. Initialization of population The initial population of preferred volume is generated randomly in steps. In first step, the segment [PMpk] of the chromosome is generated randomly considering feasibility of performing part operation on machines. In second step segment [PCpk] of the chromosome is filled randomly. Solution for given problem is represented by the embedded segments (genes) structure known as chromosome. A strategy is applied to minimize the number of inter-cell move of parts. The part operations associated to each part type are assigned to machines existing in the cells. This process is repeated until all the parts are assigned to machines. Given a candidate partmachine assignment solution, the heuristic computes the number of inter-cell part movements that would result minimum number of inter-cell movements (if a part operation is assigned to cell C1, each operation of the part is assigned to cell C1 ). Occasionally cells may not have minimum and maximum number of specified parts that contravene lower or upper limit condition as per Eqs. (7) and (8). For the cell size to remain within the specified lower and upper limits, parts family is to be adjusted by moving parts from the cell having maximum number of parts to the cell having less than minimum number of parts specified.
4.4.2. Crossover or recombination Crossover is performed between two selected parent solutions which create two new child solutions by exchanging segments of the parent solutions, thus child solutions retain partial properties of the parent solutions. Fig. 2 depicts the chromosomes parent 1 and parent 2 selected for crossover. There are two segments in the chromosome, one each for machine and cell. For crossover, the selection of segments can be row-wise or column-wise following the matrix limits and the crossover probabilities. 4.4.3. Mutation Mutation performs a secondary role in functioning of genetic algorithms. Even though crossover operator makes an effective search and recombines chromosome, yet it may cause loss of some useful genetic properties. The mutation operator safeguards against such an irretrievable defeat. The mutation operator performs local search with a low probability. The mutation operator can be implemented by inverting part of a gene in a parent chromosome to obtain child chromosome, as shown in Fig. 3. 4.5. Repair function The crossover and mutation operation may distort chromosome structure so as to yield infeasible solutions i.e. every cell may not have minimum number of machine type. The repair function is used to repair the distorted chromosome such that no machine type is left unassigned, and every cell gets minimum number of machines as per Eqs. (7) and (8). 4.6. Termination of genetic algorithm
4.3. Fitness assessment The fitness value is a decisive factor to measure the quality of a candidate solution or chromosome with reference to the designed objective function (Eq. set (1)) subjected to constraints (Eq. set (2)–(8)) and restrictions (9)–(13). The fitness values are used to select the parent solutions to obtain the next generation of solutions. The descendants or new solutions are selected with higher fitness value obtained by playing binary tournament between parent solutions. The objective function of the CMS design problem is to minimize the total cost. However, genetic algorithm is applicable to the maximization of functions. Hence the maximization of the fitness value corresponds to the minimization of the objective function (total
The genetic algorithm continues to create population of child solutions until a criterion for termination is met. A single criterion or a set of criteria for termination can be adopted. In this case the termination criterion is the maximum number of generation, i.e. the algorithm stops functioning when a specified number of generation is reached. 4.7. Heuristic to control machine duplication The solution obtained in terms of part family and machine cell is based upon the perception that no inter-cell move is allowed. In other words independent cells are created. It leads machine duplication in cells which has to be minimized to drive down
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Fig. 2. Crossover or recombination (a) row wise crossover and (b) column wise crossover.
Fig. 3. Mutation.
the investment. The reduction in machine duplication may lead to increase in inter-cell material handling cost. However in certain circumstances, it is more economical to have inter-cell moves instead of having extra machines [38]. In this segment, tradeoff of having extra machines versus having inter-cell move is considered. If eliminating extra machines in lieu of inter-cell movement results in reduction of total cost, the extra machine will be eliminated. In order to determine the elimination of extra machine from a cell the following algorithm is used. 1. Select a machine type to be considered. Calculate the total number of machines allocated in different cells to meet the production requirements. 2. To eliminate extra machines of the machine type selected, calculate the work load (to be assigned) to the machine type in each cell. The work load is defined as the quantity of the part type to be produced. 3. Compare the ‘saving’ in eliminating a unit of the machine type and the inter-cell material handling cost. If the ‘saving’ in eliminating a unit of the machine type is greater than the inter-cell material handling cost, eliminate the unit of machine in the cell which has the minimum work load. 4. Repeat the steps 1, 2 & 3 for all the machine types. 4.8. Algorithm Step 1. Initialize parameters K (the number of chromosomes in each generation), G (the number of generation), pc (percentage of crossover), and pm (percentage of mutation). Step 2. Generate initial population of machine assignment solug g g g tions, PMi PCi ....PMK PCK and apply the part assignment heuristic to form the part family for each cell explained in Section 4.2. Step 3. Initialize the generation counter g = 1.
Step 4. Generate population fitness as g g g g g g Z1 (PM1 PC1 ), Z2 (PM2 PC2 ), . . ., ZK (PMi PCK ). Step 5. Select individuals from the current population to become parents of the next generation according to their fitness value explained in Section 4.4.1. g g Step 6. Choose mating pool (solutions PMi PCi , in which Zi ≤ 0). Step 7. Generate descendants for new generation: randomly g g g g choose two parents solutions (PMi PCi ) (PMj PCj ) from current mating pool. Step 8. Randomly mate the segments of parent solutions and generate descendant by applying the genetic operator of crossover and mutation explained in Sections 4.4.2 and 4.4.3. Step 9. Use the repair function presented in Section 4.5 to repair the distorted off springs. Step 10. Evaluate the fitness of each descendant solution. Step 11. Increment the generation counter, g = g + 1. Step 12. If g ≤ G, go to step 5, otherwise terminate. Fig. 4. 5. Numerical examples For illustration of the proposed approach for design of the robust CMS, the data sets have been taken from the research reported by Wicks and Reasors [7], Mungwattana [8], Defersha and Chen [9], and Jayakumar and Raju [39] as summarized in Table 1. Since the proposed mathematical model is different from those reported in the research literature, (additional features include internal and subcontracted part operation cost, etc.), the unknown cost parameters such as internal production and subcontracted part operation cost were extracted by cross referencing among the data sets containing them. Therefore all of the data sets used in each numerical example contain values within the same range in term of unit costs. Emphasis is given on number of part types, machine types, number
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Table 1 Summary of computational data sets. Problem no.
Source
Number of part types
Number of M/c types
Number of cells
Number of operation
Number of periods
1 2 3 4 5 6 7
Small-scale Small-scale Small-scale Mungwattana [8] Wicks and Reasor [7] Defersha and Chen [9] Jayakumar and Raju [39]
1 2 3 11 25 25 12
3 3 3 10 11 10 8
2 2 3 3 3 3 3
2 2–3 2–3 2–4 2–3 4–9 3
2 5 5 2 3 2 3
Number of variables 232 1310 2685 42,464 162,015 211,480 560,160
Number of constraints 340 915 2580 33,236 125,460 185,302 435,744
Time elapsed (s) 0.00 0.03 4.25 128.6680 222.2950 409.4677 557.3258
Table 2 The optimum cellular configuration for adaptive design by Jayakumar and Raju [39]. Period
Cell number
Machine type in the cell (number of machines)
Part families
1
1 2 3
M3(2), M6(1), M7(1), M8(1) M1(2), M2(1), M4(1), M7(1) M2(1), M4(2), M5(1), M6(1)
P1, P7, P10 P2, P4, P12 P5, P8, P9
2
1 2 3
M3(1), M7(1), M8(1) M1(2), M4(1) M2(1), M4(1), M5(1), M6(1)
P1, P10 P4, P12 P3, P6, P9
3
1 2 3
M2(1), M3(1), M6(1), M7(1) M4(2), M7(1) M2(1), M4(1), M5(1), M6(1), M7(1)
P5, P10 P4 P2, P6, P7, P8, P11
Fig. 4. Progress of algorithm for search of optimal solution corresponding to the lowest overall cost.
of operations and number of cells. Each one of the numerical example used is solved as an integrated model. It is evident that the required computational time increases with the problem size in terms of variables, constraints and number of period (Table 1). A set of seven problems have been attempted to demonstrate the proposed approach. Problem 7 is considered to be large scale one with 560,160 and 435,744 numbers of variables and constraints respectively. A detailed discussion on solution of the problem 7 has been presented in next section. All the computational experiments were performed on 1.86 GHz Pentium-IV workstation with Windows XP. The mathematical model was solved using MATLAB-2009. 5.1. Result and discussion The largest problem that was attempted for an optimal solution was the one selected from Jayakumar and Raju [39]. This illustrates various features of the solution for the proposed robust CMS model. Jayakumar and Raju [39] used extended lingo version 8.0+ [40] to solve the problem. Using this problem instance and a linearized version of their model, an optimal solution has
been obtained making use of the genetic algorithm. Based on the computational experience the following values are considered for the parameters: probability of crossover (Pc ) = 0.7, probability of mutation (Pm ) = 0.025, population size (K) = 300 and number of generation (G) = 50. The objective function value obtained in the proposed work cannot be compared meaningfully because of inherent difference in the objective cost value as obtained by Jayakumar and Raju [39]. The proposed approach is comparably more general for taking up (i) the internal part operation manufacturing cost considering machine capacity, and (ii) the subcontracting part operation cost. Despite of a few differences in the mathematical model, a meaningful comparison can be made in terms of the decision aspects of the CMS design resulting from two different solutions. The solution obtained with the proposed model on problem 7 (Table 1) is detailed out in the rest of this section and simultaneously compared with solution obtained for adaptive cell design (see Table 2) in Jayakumar and Raju [39]. Table 3 shows the robust manufacturing cell design and allocated part families for three successive period segments in a planning horizon. Table 4 summarizes the optimal process route obtained for each part type by proposed algorithm, applicable to successively formed part family in three periods segment of planning horizon. Tables 5–7 represents the part operation trade-offs in different production mode considering limited resource capacity. Table 8 demonstrates the selected routings for part type P5 (machine and cell) in time period 1. For instance demand for part P5 during period 1 is 550. The part undergoes three different operations to complete the production. The first two operations are performed on machine type m6 and m7 in cell C3 whereas third operation assigned to machine number 6 in cell C3 cannot be performed as machine capacity is over limit. Hence part operation 3 of part type P5 is subcontracted from cell C3 . The solution obtained from Jayakumar and Raju [39] does not involve any flexible internal production for part P5 in period 1. In this paper, the possible alternative process routing for each part type is defined in terms of internal production and subcontracting part operation. Thus, the co-existence of multiple possible source routing (in-house production/subcontracting) for all parts are allowed and it is a tangible advantage during unexpected machine breakdown and production capacity shortage occurring in real world.
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Table 3 The optimum cellular configuration for robust design by proposed algorithm. Period
Cell Number
Machine type in the Cell (Number of Machines)
Part Families
1
1 2 3
M1(1),M4(1),M7(1) M2(1),M4(1) M3(1),M5(1),M6(1),M7(1),M8(1)
P4, P12 P2, P7, P9 P1,P5, P8, P10
2
1 2 3
M1(1),M4(1),M7(1) M2(1),M4(1) M3(1),M5(1),M6(1),M7(1),M8(1)
P4, P12 P3, P9 P1, P6, P10
3
1 2 3
M1(1),M4(1),M7(1) M2(1),M4(1) M3(1),M5(1),M6(1),M7(1),M8(1)
P4 P2,P7 P5, P6, P8, P10, P11
Table 7 Part operation assigned to machine (t = 3).
Table 4 The optimal process route for each part type by proposed algorithm. Part type
Optimal part process route
Part no
P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12
M8–M7–M8 M2–M2–M4 M2–M4–M7 M1–M4–M7 M6–M7–M6 M3–M5–M4 M4–M6–M2 M5–M5–M3 M4–M6–M4 M3–M6–M6 M8–M7–M8 M1–M4–M1
2 4 5 6 7 8 10 11 *
1 2 4 5 7 8 9 10 12 *
Operations 1
2
3
8 2 1 6 4 5 4 3* 1
7 2 4 7 6 5 6* 6 4
8 4* 7 6* 2* 3 4* 6* 1*
Subcontracted part operations.
Table 6 Part operation assigned to machine (t = 2). Part no
1 3 4 6 9 10 12 *
1
2
3
2 1 6 3* 4 5 3 8
2 4 7 5 6 5 6 7
4 7 6* 4 2 3* 6* 8*
Subcontracted part operations.
Table 8 Comparison routing for part type 5 in period 1.
Table 5 Part operation assigned to machine (t = 1). Part no
Operations
Operations 1
2
3
8 2 1 3 4 3* 1
7 4 4 5 6 6 4
8 7 7 4* 4 6 1*
Subcontracted part operations.
Table 9 illustrates the parts demand met for part type P7 and P10 in period 1, part type P4 and P6 in period 2 and part type P10 and P11 in period 3 through internal production considering limited production capacity and subcontracted part operation during three period segments of planning span. The option for subcontracting part operation is not present in the model proposed by Jayakumar and Raju [39]. Hence no comparison can be made in terms of different source routings. The simultaneous consideration of in-house production and subcontracted part operation processing gives a realistic view of manufacturing system. The proposed model
Part route
1
2
3
Proposed model Jayakumar and Raju [39]
550 M6 /C3 550 M6 /C3
550 M7 /C3 550 M7 /C2
550 M6 /C3 * 550 M2 /C3
*
Subcontracted part operation.
Table 9 Comparison of production plan for parts P7 and P10 for (t = 1), P4 andP6 for (t = 2), P10 and P11 for (t = 3). Period 1 Part/operation Internal production Subcontracted Demand
Propose model P7 P10 1,2 2 3 1,3 450 900
Jayakumar and Raju [39] P7 P10 1,2,3 1,2,3 nil nil 450 900
Period 2 Part/operation Internal production Subcontracted Demand
Propose model P4 P6 1,2,3 1,2 Nil 3 500 500
Jayakumar and Raju [39] P4 P6 1,2,3 1,2,3 nil nil 500 500
Period 3 Part/Operation Internal Production Subcontracted Demand
Propose model P10 P11 1,2 1,2 3 3 700 700
Jayakumar and Raju [39] P10 P11 1,2,3 12,3 nil nil 700 700
presents the possible production plan for each part type during each period given in Tables 5–7, showing higher flexibility in meeting the part demand size. In this study the tradeoff is made between intracellular, intercellular part movement and machine duplication by simultaneously minimizing all the three costs within the objective function. This is essential since, high intracellular movement cost for successive part operations implies large cell size, reducing the effectiveness of manufacturing system. On the other side, minimum inter-cell movements lead to increase in number of duplicate machines in the cells and adversely affect the benefits of CMS by inappropriate workload distribution among cells. In this research a few authors (Nsakanda et al. [41]; Ahkioon et al. [42] and Defersha and Chen [9]) addressed the subcontracting/outsourcing only as a
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subset of the part demand size. These proposed approaches make a useful contribution to the existing knowledge. 6. Concluding remarks and future research direction This paper presents a novel integrated mathematical model for design of robust cellular manufacturing system (RCMS) considering dynamic production and multi period production planning. An extended problem is presented based on the basis of RCMS model proposed in literature [30]. The proposed model offers flexibility in production planning (production/subcontracting) that can be achieved by producing product mixes at each period of planning horizon within the limited capacity without affecting the manufacturing cell configuration. This would make the problem more practical and realistic. The proposed approach is demonstrated on a manufacturing system configuration design problem considering different part types undergoing multiple manufacturing sequences in a dynamic environment. The algorithm aggregates resources into different manufacturing cells based on selected optimal process route from user specifying multiple routes. The model is computable with single part routing as well as multiple part routings. The proposed approach can also be readily used where limits are imposed on the cell sizes and/or number of cells. The performance of the model has been verified by different bench mark problems reported in literature. The results obtained show that the co-existence of multiple possible source routings (in-house production/subcontracting) builds up flexibility in production and it is a tangible advantage during unexpected machine breakdown and production capacity shortage occurring in real world. The research reported in this paper is a part of the major research project on robust design of CMS. The authors are working to further improve the mathematical model for design of CMS incorporating more real world aspects of the manufacturing system, such as, lot splitting and machine adjacency requirements to widen its area and make the study more useful. The study on comparison of the benchmark results obtained using other techniques is very important research assignment, and shall be taken up in due course. References [1] Askin R, Standridge C. Modeling and analysis of manufacturing systems. New York: John Wiley & Sons; 1993. [2] Selim H, Askin R, Vakharia A. Cell formation in group technology: review, evaluation and directions for future research. Comput Ind Eng 1998;34(1): 3–20. [3] Hu L, Yasuda K. Minimizing material handing cost in cell formation with alternative processing routes by grouping genetic algorithm. Int J Prod Res 2006;44:2133–67. [4] Venkumar P, Sekar KC. Design of cellular manufacturing system using non-traditional optimization algorithms. In: Modrák V, Pandian RS, editors. Operations management research and cellular manufacturing systems: innovative methods and approaches. Hershey, PA: IGI Global; 2012. p. 99–139 [chapter 6]. [5] Burbidge JL. Production flow analysis. Prod Eng 1963;42:742–52. [6] Rheault M, Drolet J, Abdulnour G. Physically reconfigurable virtual cells: a dynamic model for a highly dynamic environment. Comput Ind Eng 1995;29(1–4):221–5. [7] Wicks EM, Reasor RJ. Designing cellular manufacturing systems with dynamic part populations. IIE Trans 1999;31:11–20. [8] Mungwattana A. Design of cellular manufacturing system for dynamic and uncertain production requirement with presence of routing flexibility [Ph.D. thesis]. Blackburg, VA: Faculty of the Vargina Polytechnic Institute and State University; 2000.
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