algorithm for robust cellular manufacturing

1. DESIGN OF ROBUST CELLULAR MANUFACTURING SYSTEM
FOR DYNAMIC PART POPULATION CONSIDERING MULTIPLE
PROCESSING ROUTES USING GENETIC ALGORITHM
NAME: DWITI KRISHNA PANDA GUIDED BY:
ROLL NO: 34328 DR. DHIREN KU. BEHERA
REGD NO: 1501105293 (ASST. PROFESSOR, MECH ENGG. DEPT.

2. CONTENTS
1. INTRODUCTION
2. PROBLEM FORMULATION
3. GENETIC ALGORITHM BASED HEURISTICS
4. NUMERICAL EXAMPLES
5. CONCLUSION
6. REFERENCE

3. INTRODUCTION
 The cellular manufacturing (CM) is an application of Group Technology.
 The CM groups machines into machine cells and parts into part families.
 The major advantage is in terms of material flow which is significantly improved,
with reduction in inventory level and distance travelled by the material.
 Genetic algorithm is used as it is simpler than conventional heuristic approaches.

4. NOTATIONS

5. MATHEMATICAL MODEL
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.

6. CONSTRAINTS
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.
In addition to these constraints, restrictions represented by Eqs.(9)–(13) denote the logical binary
and non-negative integer requirement on decision variables.

7. CHROMOSOME REPRESENTATION OF
SOLUTION
 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
 The gene representing “part operation assigned to cell” is rep-
resented 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.

8. INITIALISATION 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.
 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
 For instance, if a part operation is assigned to cell C1, each operation of the part is assigned to cell C1
 For the cell size to remain within the specified lower and upper limits, parts family is to be adjusted by movingparts from the
cell having maximum number of parts to the cell having less than minimum number of parts specified

9. 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
 The descendants or new solutions are selected with higher fitness value obtained by playing binary
tournament between parent solutions
 Genetic algorithm is used for maximization but we need to minimize the objective function
 So, the maximization of the fitness value corresponds to the minimization of the objective function (total cost)
value
 Thus necessary transformation from objective function to the fitness function is carried out in the following
manner
 Where, Zi: fitness value of string i, Ti: objective function value of the string i. Tmin: the smallest objective
function value in the current generation

10. Genetic operators
Reproduction or
parent selection
• 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
• Binary tournament selection
scheme has been adopted for the
purpose
Crossover or
recombination
• Crossover is performed between two
selected parent solutions which
create two new child solutions by
exchanging segments of the parent
solutions
• 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
Mutation
• Crossover may cause loss of some
useful genetic properties
• 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

11. Repair function and termination
Repair function
 The crossover and mutation operation
may distort chromosome structure so as
to yield infeasible solutions
 So, 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
termination
 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

12. Heuristic to control machine duplication
 Select a machine type to be considered. Calculate the total number of machines
allocated in different cells to meet the production requirements.
 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.
 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.
 Repeat the steps 1, 2 & 3 for all the machine types

13. 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 solutions,
PMg
iPCg
i....PMg
KPCg
K and apply the part assignment heuristic to form
the part family for each cell explained previously
 Step 3. Initialize the generation counter g = 1.

14. Algorithm (contd.)
 Step 4. Generate population fitness as Z1(PMg
1PCg
1), Z2(PMg
2PCg
2), . . ., ZK(PMg
iPCg
K).
 Step 5. Select individuals from the current population to become parents of the next
generation according to their fitness value.
 Step 6. Choose mating pool (solutions PMg
iPCg
i, in which Zi≤ 0).
 Step 7. Generate descendants
 Step 8. Randomly mate the segments of parent solutions and generate descendant
by applying the genetic operator of crossover and mutation
 Step 9. Use the repair function 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.

15. Numerical example
 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, Mungwattana, Defersha and Chen, and Jayakumar and Raju as summarized in Table 1
 The unknown cost parameters such as internal production and subcontracted part operation cost were extracted by cross
referencing among the data sets containing them
 Emphasis is given on number of part types, machine types, number of operations and number of cells
 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
 All the computational experiments were performed on 1.86 GHz Pentium-IV workstation with Windows XP. The mathematical
model was solved using MATLAB-2009

16. Result
 Jayakumar and Raju used extended lingo version 8.0+ 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 proposed approach is comparably more general for taking up
(i) the internal part operation manufacturing cost considering machine
capacity
(ii) the subcontracting part operation cost

17. Result (contd.)
 Following table shows the robust manufacturing cell design and allocated
part families for three successive period segments in a planning horizon
 Following table 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

18. Result (contd.)
 Following tables represents the part operation trade-offs in
different production mode considering limited resource
capacity
 Following tables 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 trade-off is made between intracellular,
intercellular part movement and machine duplication by
simultaneously minimizing all the three costs within the
objective function

19. Conclusion
 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.
 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 benchmark 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.

20. References
 Deep K., Singh P.K. Design of robust cellular manufacturing system for dynamic part population
considering multiple processing routes using genetic algorithm. Journal of Manufacturing Systems
35 (2015) p.155-163
 Jayakumar V, Raju R. An adaptive cellular manufacturing system design with routing flexibility and
dynamic system reconfiguration. Eur J Sci Res2010;47(4):595–611.
 Wicks EM, Reasor RJ. Designing cellular manufacturing systems with dynamicpart populations. IIE
Trans 1999;31:11–20.
 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.
 Defersha F, Chen M. A comprehensive mathematical model for the design ofcellular manufacturing
systems. Int J Prod Econ 2006;103:767–83.
 LINDO Systems Inc. Extended Lingo: Version 8.0+ users guide. Chicago: LINDO Systems Inc.; 2005.

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