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International Journal of Mechanical Engineering and Technology (IJMET)
Volume 10, Issue 08, August 2019, pp. 232-243, Article ID: IJMET_10_08_020
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=8
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication
MINIMIZATION OF EXPECTED MAKESPAN IN
FLOWSHOP USING CBM
D. Sanjeeva Rao
Part time research scholar, Department of Mechanical Engineering, College of
Engineering(A), Andhra University, Visakhapatnam, India
M. Srinivasa Rao
DGM (Maintenance), Department of MMSM, Visakhapatnam Steel Plant, India
V.V.S. Kesavarao
Professor, Department of Mechanical Engineering, College of Engineering(A),
Andhra University, Visakhapatnam, India
ABSTRACT
Manufacturing units have to meet concurrently several requirements such as:
quick reaction to market demand, high product quality, justifiable manufacturing costs
and well-timed deliveries, etc. Generally, the shop floor level personnel treat
maintenance and processing times in segregation for the purpose of scheduling, joint
consideration of equipment maintenance and scheduling on the performance of the
manufacturing system is taken into consideration in this paper. In practice machines
may be temporarily not available for many reasons, such as unforeseen scheduled
preventive and breakdown maintenance, operator non availability, spare parts
damage, etc. An effort is made to schedule a flowshop problem with time deterioration
under “Condition Based Maintenance” (CBM) constraints to minimize the expected
makespan. The randomness of the problem is tackled by simulation. Genetic and tabu
search algorithms applied to tackle such hard problem. Experimental studies
conducted and the results are promising in nature.
Key words: Scheduling, simulation, condition based maintenance, time deterioration
Cite this Article: D. Sanjeeva Rao, M. Srinivasa Rao and V.V.S. Kesavarao,
Minimization of Expected Makespan in Flowshop Using CBM. International Journal
of Mechanical Engineering and Technology 10(8), 2019, pp. 232-243.
http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&IType=8
1. INTRODUCTION
Due to cut throat competition difficult challenges are being faced by manufacturing
organizations. Automation and modernization of equipment is the path chosen by different
manufacturing companies to increase the product-variety and for better product quality
resulting into raise of the cost of manufacturing processes. To achieve operational excellence
in this new environment, companies need to focus on shop floor efficiency and effectiveness.
Minimization of Expected Makespan in Flowshop Using CBM
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Appropriate maintenance of production systems and machinery, adoption of proper
production scheduling programs, and implementation of quality techniques will yield
effective and efficient shop floor operation. As a practice the scheduler prepares schedule for
set of orders on a number of machines in such a way the timing, order sequence, and machine
assignment of these orders are optimal with respect to one or more objectives. A large number
of performance measures for evaluating scheduling objectives/performance have been in
vogue at present. Such measures are job flow time, machine utilization, makespan, tardiness,
job handling cost, labor utilization, etc.
More overscheduling has become vital in order to meet customer requirements as
promptly as possible while maximizing the profits to stay in the world of global competition.
Scheduling in manufacturing systems is classically associated with scheduling a set of jobs on
a set of machines in order to maximize the profit. Manufacturing system is classified as job
shop, flow shop and open shop. A common job shop problem consists of n jobs {j1, j2,
j3,…,jn} to be processed through m machine {m1, m2, m3, …, mm}. Technological
constraints demand that each job should be processed through the machines in a particular
order and gives a significant special case named as flow shop. For a general job shop
problem, the number of possible sequences are (n!)m, where n is number of jobs and m is the
number of machines. With the technological constraints in case of flow shop, number of
different sequences reduces to (n!). This reduced number is quite large for even temperate size
problems and recognized to be NP hard problems (Pinedo, 2012).
In classical scheduling problems, machines are assumed to be available through the whole
planning horizon, on the contrary in reality machines may not be available during certain
periods of planning horizon, due to breakdown or for attending preventive maintenance jobs.
Preventive maintenance is the main cause of unavailability of machines. Some researches
addressed this problem by machine scheduling with availability constraints where the number
of preventive maintenance periods and their intervals are fixed and known in advance,
without change in the performance measure. In fact, the constraints are formulated in a way to
plan the jobs in the available periods of time. Apart from scheduled preventive maintenance,
there are many uncertainties in process industry such as machine breakdown, operator-stock
condition, changes in availability date and latest completion times; we must consider them to
ensure the production running successfully.
A condition based on preventive and corrective maintenance policy is a technique adopted
in amass production units. The condition of the system/machinery is assumed to deteriorate
with time. The proposed model in this paper incorporates both deterioration as well as random
common cause failures. The deterioration stages are modeled as discrete state processes. The
system is put to random inspection to know the condition. The mean times between
inspections are exponentially distributed. If the observed condition at an inspection exceeds
the threshold value due to deterioration, the system calls immediate attention for maintenance.
Maintenance costs are included operational cost in firms hence are to be minimized without
compromising life of machine. Hence preventive maintenance, inspections, and predictive
maintenance are done properly to reduce maintenance expenses. Collection failure data and
optimizing maintenance schedules will reduce maintenance expenses.
Machine gets stopped because of maintenance. If the job continues processing
immediately where it was left earlier i.e., before stoppage of machine when once the machine
becomes operable after maintenance, the problem was called “resumable”. On the other hand,
the problem was called “non-resumable” if the job has to start afresh from the beginning after
maintenance of machine.
Scheduling of maintenance operations and production sequencing are normally treated
separately. In most of these researches, it is assumed that job processing times are fixed and
D. Sanjeeva Rao, M. Srinivasa Rao and V.V.S. Kesavarao
http://www.iaeme.com/IJMET/index.asp 234 editor@iaeme.com
known in advance while deteriorating jobs exists in practice. Such deterioration appears in,
e.g., scheduling of maintaining job in case of fire fighting, cleaning assignments, issues
involving worker forgetfulness, tool wear, etc. In order to model these practical problems,
processing time of each job is considered as a function of its starting time. In other words, the
jobs processed later consume more time than the jobs processed earlier. The scheduling
problem with deteriorating jobs is mostly found in a metallurgical units.
2. LITERATURE SURVEY
Extensive research has been under progress starting from Johnson to current researchers for
optimization of flow shops scheduling. Their approaches are broadly two types; optimization
algorithms and heuristic algorithms. Constructive techniques are proposed for solving
flowshop problems (Nawaz et al., 1983) or improvement techniques are proposed for solving
flowshop problems (Kirkpatrik, 1983, Taillard, 1990, 1993, Ishibuchi et al. 1995). Cost-
discounted Markov decision process formulated for single machine problem by Glazebrook,
(1984). Birge et al., (1990) considered more general breakdown processes. Allahverdi et al.,
(1994) presented that a problem with parallel machines subject to random breakdowns could
be converted into parallel-machine problem with modified processing time. Constructive
scheduling generates a schedule from beginning. Improvement methods start from a initial
feasible solution improves this solution in the process of reaching global solution.
Improvement methods are generally based on meta-heuristic approaches such as, tabu search,
Genetic Algorithms (GA), Artificial Bee Algorithm (ABA), ant colony algorithm, etc.
The concept of preventive maintenance in scheduling was treated by several researchers:
two heuristic algorithms of time complexity O (n log n) are proposed by Lee, (1994),
considering preventive e maintenance, and provided their error bound analysis. Schmidt,
(2000) described scheduling problems with limited machine availability for one flowshops.
Further analyzed some results from heuristics. The problem with some periods of preventive
maintenance on two machines in resumable case was considered by Blazewicz et al., (2001)
local search based heuristic algorithms are analysed by them. Allaoui et al., (2004) solved
flow shop scheduling problem with maintenance constraints applying simulation and
optimization methods and compared the results with NEH heuristics. Ruiz et al., (2006)
considered two popular preventive maintenance methods in flow shop scheduling problem.
Antonio and Maria,(2008) has described system failures are assumed to occur at the first
instant in which a random constant threshold is reached by (a) the sum of received shocks, (b)
the minimum of shocks, (c) the maximum of shocks. Safariet al., (2009) have considered
condition based maintenance in flow shop scheduling and proposed model by integrating
heuristics and simulation for fixed processing times. Cheng, (2011) illustrated usage of
simulation for finding the Near-Optimal Preventive Maintenance Policies for a Repairable
System. Many of the scheduling problems are NP-hard in nature (Pinedo, 2012).Simulation
study of dispatching rules in stochastic job shop dynamic scheduling was proposed by Edna,
B.S. et al., (2014).
It is clear from the literature review presented in the preceding section that the joint
consideration of scheduling, and maintenance is gaining increasing attention from the
researchers in recent years. While available approaches are mostly limited to simple problems
like single machine, single product, single quality characteristic, etc. Multiple machines and
multiple products, each having multiple quality characteristics are quite often are more
relevant in actual production system. Further, nature and extent of integration required in case
of flow shop need to be optimal. Further, the demand of the jobs/tasks, variety of products,
and availability of resources, production rates, etc. may vary with time. All these factors make
Minimization of Expected Makespan in Flowshop Using CBM
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the actual production systems are dynamic in nature and in turn increase the complexity of
mathematical modeling.
Considering the literature an effort is made to develop a model incorporating the
preventive maintenance and break down maintenance activities in the flow shop scheduling
problem which is solved by integrating simulation as well as heuristics.
The reminder of this paper is organized as follows: A study of maintenance techniques
problem assumptions are described in section-3. Section-4 describes the heuristics that are
applied in this paper. In section-5, simulation algorithm is described. Further model is
presented combining simulation and heuristics in section-6 along with analysis of results and
finally, conclusions are drawn in section-7.
3. STUDY OF MAINTENANCE TECHNIQUES
In a continuous growing global market productivity is playing a key role to stay competitive,
for any manufacturing company. Productivity can be achieved through availability of
machines and availability can be increased through adopting the efficient maintenance
practices, by focusing on different types of maintenance and strategies. In his seminal book
reliability centered maintenance, John Moubry suggested that three distinct generation of
maintenance. In the first generation i.e., upto Second World War the expectation of
maintenance is to fix when machine fails. In second generation maintenance i.e., upto 1970
the expectation of maintenance is high equipment life follow bath tub curve. In third
generation maintenance after 1970 the expectation of maintenance is high and reliability level
is high. Six failure patterns are observed. Condition monitoring is one technique that is
practiced.
Condition Based Maintenance (CBM) or predictive maintenance, uses primarily
nondestructive testing techniques, visual inspection, and performance data to assess
machinery condition. It replaces arbitrarily timed maintenance tasks with appropriate
maintenance task at only when warranted by equipment condition. CBM includes improved
knowledge of failure mechanisms, advancements in failure forecasting techniques,
advancements in monitoring and sensor devices, advancements in diagnostic and prognostic
software, computer networking technologies.
In The time based preventive maintenance, maintenance are performed at fixed periodic
intervals regardless of the dynamic health status of machinery machine breakdown is one of
the disorder commonly found on the production floor. This problem decreases the profit
margin of the due to production loss and maintenance cost. Single component or multiple
failures are the reasons for the machine breakdown. Hence the tendency is to do more
preventive maintenance. This makes the cost of PM keeps on increasing. Where as CBM
recommends maintenance actions based on the information collected through condition
monitoring. So that unnecessary maintenance can be avoided. Depending on the health
condition of machinery, maintenance can be done which ultimately minimizes maintenance
cost.
In condition-based maintenance framework, a deterioration indicator that correctly
describes the dynamic of the failure process is to be defined first. Usually this efficient
metrics can be constructed from collected information on various deterioration-related
monitoring parameters, such as vibration, temperature, lubricating oil contamination, and
noise levels. Deterioration models are suggested by many people considering deterioration
processes in dynamic environments with stochastic approaches. It is assumed that condition
based maintenance in which each machine suffers degradation due to shocks. When total
degradation for such machines reaches to a threshold value maintenance has to be carried out.
Upon the inspection of the system and its condition, inspectors need to decide whether to take
D. Sanjeeva Rao, M. Srinivasa Rao and V.V.S. Kesavarao
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an action such as Nominal Preventive Maintenance (NPM) and Essential Preventive
Maintenance (EPM) or no action is needed. In spite of practicing CBM, some times machines
will cease to perform because of unforeseen breakdowns, which are called random break
down. It is assumed that a machine keeps processing the jobs sequentially until it break down
or it has finished all the jobs, and machine breakdowns may arise at any time in working
periods because of unforeseen situations. Reactive maintenance has to be carried out to bring
back the machine into workable condition. Maintenance policies, which are commonly
considered by simulation are reactive/corrective maintenance and preventive maintenance are
addressed in this paper. Maintenance policies address as the components of the framework for
reactive maintenance follows Weibull chance failures and the shape of function is greater than
1.
4. ASSUMPTIONS
A job consists of several operations, each one has to be performed on a particular machine at
any instant job can be processed on one machine only.
At the start time, all jobs are available for processing. One has to choose the start job among
many. No preference among the jobs. Preemption of jobs is not allowed.
Setup times between operations are included in the processing times.
Two jobs cannot be processed simultaneously on any particular machine.
Jobs are always processed without any defect. Basic job processing times and constant
deterioration rate known in advance.
There is only one type of machine and no restrictions on buffering between machines.
Machines are not available at all times due to NPM and EPM operations.
Machine inspections are planned at periodic times nTn(n = 1, 2, …).
All machines have the same mean values shocks, degradation for each shocks, maintenance
times, recovery value and the maintenance threshold values and inspection will not interrupt
machine production.
Random break downs are considered in this paper using Weibull distribution. This paper
adopts the CBM policy for a cumulative degradation model where a machine suffers
degradation due to shocks, and does EPM when the total amount of additive degradation
exceeds a level called k. The inspections occur in a pre-specified intervals nTI (n = 1, 2, …) to
prevent failures, where TI (> 0) is the inspection interval time. If the total degradation exceeds
a threshold level Z(0 < Z ≤ K) at time nT, the NPM is performed. Otherwise, no PM is
performed.
Shocks are the main reason of degradation for each machine. The number of shocks
occurrence injth
inspection interval Nj has Poisson distribution. In additional, random
variables {Wj} (j = 1, 2, …) denotes the amount of degradation due to jth
shock and is
considered to have an exponential distribution. Degradation of each machine in a period dp is
also considered as:
pN
p jj 1d W
Recovery value (Decreasing the degradation value after PM operations) Δdeg when
operation maintenance is executed, has lognormal distribution. Also degradation values after
NPM and EPM operations d0 are equal to d−Δdeg and 0, respectively. The amount of time
needed for NPM and EPM operations are Tm, Tb respectively, have lognormal distribution,
Minimization of Expected Makespan in Flowshop Using CBM
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where time needed for operations of NPM are more than BPM and we have dt = d0 + dp.
Where d0 is identified at the end of each period for the next period as follows:
deg0 = dt – deg for Z dt < K
deg0 = dt for 0 dt K
deg0 = 0 for dt K
When random shocks occur the degradation process may receive two types of impacts
where the first type is a sudden increment jump and the second one is degradation rate
acceleration. Cha and Finkelstein (2009,18) extended the brown-proschan model by assuming
that random shocks will result in an immediate system failure with a probability of p(t) but
accelerate the system aging process by a certain increment with a probability.
5. SIMULATION ALGORITHM
5.1. Notations
n : number of jobs
d : degradation value when machine is inspected
m : number of machines
Finishi,j : finish time of job j on machine i
Sequence of jobs:
k : kth
job of sequence
W : degradation value of each level
tij : processing time of job j on machine i
bij : basic processing time of j on machine i
aij : linear deterioration coefficient
tij : iij i, ij(a *start ) b
rmni : remaining time to next inspection
N : number of shock in each inspection period
starti,j : start time of job j on machine i
wran = Weibull random number
cwran = Weibull random number value
The permutation flowshop scheduling problem consists of scheduling n independent and
non preemptive jobs gathered in setj = {1,2,3,…,n} and mmachines = {1,2,3,…,m} all jobs
should follow a fixed route of machines to be completed and due to the assumption sequence
of jobs on all machine are identical. We assume processing time tij is of job j on machine i is
given linear function of its starting time t, i.e., tij = (aij * starti,j) + bij where aij 0 denotes the
deterioration rate of job j on machine i and bij 0 is the fixed process time of job j on
machine i.
Degradation value related to each shocks and cumulative value of degradation are calculated
at each inspection interval after determining the number of shocks for each machine.
In each inspection, it is needed to compare degradation value with thresholds.
Based on maintenance strategy, PM or essential repair time are calculated.
After preventive maintenance operation, recovery value is subtracted from degradation value.
When random shocks occur, the degradation process may receive two types of impacts where
the first type is a sudden increment jump and the second one is degradation rate acceleration.
D. Sanjeeva Rao, M. Srinivasa Rao and V.V.S. Kesavarao
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Some times random shocks will result in an immediate system failure with a probability of
p(t) and calls for maintenance immediately and some times accelerate the system aging
process by a certain increment with a probability.
After basic repair, degradation value set to zero.
It is necessary to note where inspection is occurred i.e. in middle or end time of particular
processing job.
Because of the fact that the problem has probabilistic nature, it is necessary to replicate the
computation of simulator several times for each sequence when all the features of the problem
remain constant. After that all replication is done, mean of obtained makespan is considered as
expected makespan.
It is also necessary to note that in some cases in spite of following conditioned base
maintenance inspections, certain surprises happen due to sudden failures and unforeseen
technological disruptions before inspection interval, generally the trend of such failures follow
Weibull distribution where shape of curve is greater than 1. Such conditions are incorporated
in this model.
The simulator algorithms employed in heuristics has the following steps for each
sequence.
Step-0: Initialize i = 1, ii,start = 0, finish0,j = 0, deg0 = 0.
Step-1: If i m {go to step-2}; else {finish the simulator procedure}.
Step-2: Setdeg0 = 0, j = 1; Run degradation algorithm with initial degradation deg0.
Step-3: If j n {go to step-4}; else (go step-14}.
Step-4: Set g = σj, rmni = rmni – tig, starti,g = max(finishi–1g starti,g).
Step-5: If rmni > 0 {go step-6}; else {go step-7}.
Step-6: Setfinishi,g = starti,g + ti,g, j,1i,start = finishi,g and go step-13.
Step-7: If d k {if rmni < 0 {set nmni = rmni + tig; go to step-8}; else {go to step-11}}; else
if d Z {if rmni < 0 {set rmni = rmni + tig; go to step-9}; else {go to step-12}} else {go to
step-10}.
Setrmni = rmni + TI, j 1i,start = finishi,g; Go to step-13.
Step-9: (a) Generate two log normal random numbers as MPM time Tm, recovery value
_deg; Setfinishi,g = starti,g + Tm +tig; deg0 = d – deg;
(b) Run degradation algorithm with initial degradation d0;
Set rmni + TI, j 1i,start = finishi,g; Go to step-13.
5.1.1. Degradation algorithm with initial degradation deg0
The degradation algorithm is used in simulator algorithm which has the following steps:
Step-1: Generate a Poisson random number as degradation level N.
Step-2: d = deg0.
Step-3: for k = 1 to N.
a) Generate an exponential random number as degradation for degradation level k,W.
b) d = d + W
End for (k)
Step-4: Generate Weibull random number wran with shape of curve > 1 and cdf // to address
random shocks because of unforeseen situations.
Minimization of Expected Makespan in Flowshop Using CBM
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If wran cwran
{//sudden failure; d = K + l; goto step-8(a) of simulated algorithm
else goto simulated algorithm}
6. COMBINATION OF SIMULATION AND HEURISTICS
The majority optimization problems are too difficult to be solved by mathematical
programming. Combining optimization and simulation is one kind of approach to solve such
problems. The concept of intelligent agents to simulate the manufacturing process was
adopted by Nadoli et al. (1993). Programming techniques of object-oriented approach,
constraints programming, and simulation techniques were integrated by A.D. Talbi et al.
(1995). Since the problem considered in this paper is strongly NP-hard, Heuristics along with
simulation technique (called simulator) is adopted to obviate the complexity of the problem at
hand. The flow chart Fig.1 is depicted below.
Figure 1 Flow chart
6.1. Analysis of Results
The computational results obtained from our proposed algorithms employed in this paper are
tabulated. The empirical data is collected from a local industry. The CBM parameters are
determined first. In practice the parameter related to stochastic distribution of the number of
shocks in inspection period, EPM and NPM operation times, recovery value and degradation
value for each shock are specified according to the information collected through condition
monitoring devices i.e., shock pulse meters. Also threshold values are determined according
to life distribution of machines based on empirical data.
The number of shocks in each inspection period and degradation value for each shock
have Poisson and exponential distribution with 25 and 0.6 mean values, respectively. Also
recovery value after NPM operations is fixed using lognormal distribution with mean 3 and
variance value of 0.2. This parameter is selected such that the generated random number from
this distribution is less than NPM threshold and more than zero. Shape of Weibull distribution
curve is 2 and scale of curve is taken as 100 based on empirical data. Cumulative distribution
value of survival is taken as 0.02.
Since, the lengths of both periods of inspection and preventive maintenance might
influence the algorithm performance, different levels of these parameters are considered.
Number of jobs n{20, 30, 40} and machines = {20, 30}, therefore there are 6 combinations of
n and m where the processing times bij is distributed as a U[1, 99] and deterioration constant
D. Sanjeeva Rao, M. Srinivasa Rao and V.V.S. Kesavarao
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aij as U[0, 3]. For each combination of jobs and machines, different values of inspection
interval and necessary times of operations of minimal and basic preventive maintenance are
considered. Operation times of minimal and basic maintenance with lognormal distribution
have means 3.0 and 3.65 and variances of 0.1. Inspection period is set {150.250, 350}.
Once the expected makespan of each instance has been obtained for each algorithm, the
best solution obtained for each instance of same size is selected and it is calledbest value of
instance. With this, calculated the relative percentage deviation (RPD) with respect to this
best solution with the following expression:
RPD = (average value of instance – best value of instance)/average value*100 where
average value of instance is average of all expected makespan attained for considered
instances. RPD help us to compare algorithms because the RPD values denote relative
distance of average algorithm solution from best solution obtained for special instance,
clearly, lower values of RPD are preferred.
The worst percentage deviation (WPD) with respect to this best solution with the
following expression is also calculated:
WPD = (maximum value of instance – best value of instance)/
average value*100
where maximum value of instance is maximum (worst)of all expected makespan attained
for considered instances. WPD help us to compare algorithms because the WPD values
denote relative distance of worst algorithm solution from best solution obtained for special
instance, clearly, lower values of WPD are preferred. Statistical analysis performed on the
results for both resumable and non resumable cases.
Multi factor ANOVA performed in which factors such as number of jobs, number of
machines, inspection intervals and response variable of relative percentage deviation are taken
into consideration. Preliminary experiments shown that ANOVA’s hypothesis of
homogeneity of variance not true and hence Box-cox transformation is applied to all RPD.
Mean plot and least significant difference for the algorithms at 95% confidence have shown
us that there is no statistically significant difference.
7. CONCLUSION
The problem is run in C++ and on a PC with 1.5 GHz Intel Core 2 Duo and 2 GB of RAM
memory. In this paper, we had studied the problem of scheduling a flowshop to minimize
expected makespan based on condition based maintenance constraints and sudden unexpected
failure because of breakdown maintenance. In fact many real life industry problems represent
the double complexity (algorithmic and structural-functional). We have illustrated the
approach of combining the simulation and the optimization to deal with this problem. Tabu
and GA and a simulator were used to construct this approach. The obtained results strongly
manifest the superiority of GA over others in both cases resumable and nonresumable. GA
achieved better results with respect to other algorithms in terms of CPU time, RPD and WPD
values. Further to this we investigated the problem of scheduling deteriorating jobs in a flow
shop environment. The job deterioration is a very promising issue and extensions of the same
to other flow-shops, including hybrid and flexible flow-shops, job-shops and open-shops can
be explored as well. Moreover, using more than one objective and developing other meta-
heuristic algorithms, especially population-based meta-heuristics can be regarded as some of
the other future research directions.
Minimization of Expected Makespan in Flowshop Using CBM
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Average RPD values are shown in the table.
Table 1 Average RPD values in non-resumable cases
Instance GA Tabu
20*20 0.62 0.65
20*30 0.6 0.65
20*40 0.48 0.5
Average 0.558 0.576
Table 2 Average RPD values in resumable cases
Instance GA Tabu
20*20 0.72 0.75
20*30 0.7 0.8
20*40 0.64 0.7
30*20 0.8 0.8
30*30 0.62 0.7
30*40 0.65 0.62
Average 0.68 0.72
WPD values are shown in the table.
Table 3 Average WPD values in non-resumable cases
Instance GA Tabu
20*20 0.72 0.75
20*30 0.65 0.76
20*40 0.5 0.65
30*40 0.65 0.67
Table 4 Average WPD values in resumable cases
Instance GA Tabu
20*20 0.76 0.78
20*30 0.72 0.82
20*40 0.66 0.73
30*20 0.81 0.82
Average 0.71 0.75
Table 5 Time in minutes
Instance GA Tabu
20*20 16.5 17.5
20*30 18 18.3
20*40 20.2 21.2
30*20 27.5 28.7
30*30 39.3 39.8
30*40 40 45.1
D. Sanjeeva Rao, M. Srinivasa Rao and V.V.S. Kesavarao
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