Economic lot sizing for unreliable production system with shortages Economic lot sizing for unreliable production system with shortages

Article (PDF Available)inInternational Journal of Mathematics in Operational Research 7(4):464-483 · January 2015with 159 Reads
DOI: 10.1504/IJMOR.2015.070198
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Abstract
The purpose of the present study is to analyse the optimal lot size in an unreliable single-machine production system with shortages. The production system is subject to failure due to machine breakdown. Breakdown times are considered to be according to Weibull distribution. It is assumed that the shortages are allowed and backlogged. During each production, the set-up preventive (regular) maintenance is performed. The corrective (i.e., emergency) maintenance is carried out immediately after breakdown. For the illustration purpose, numerical results are provided for the special cases. To obtain the optimal cost per unit time, we also employ the artificial neuro-fuzzy inference system (ANFIS) approach which has the learning capability of neural network as well as advantages of rule-base fuzzy system. It is noted that the results obtained by neuro-fuzzy technique are at par with the results computed by the analytical techniques.
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464 Int. J. Mathematics in Operational Research, Vol. 7, No. 4, 2015
Copyright © 2015 Inderscience Enterprises Ltd.
Economic lot sizing for unreliable production system
with shortages
Neetu Singh*
Department of Applied Sciences,
World College of Technology and Management,
Gurgaon, India
Email: neetusingh1608@gmail.com
*Corresponding author
Madhu Jain
Department of Mathematics,
IIT Roorkee, Roorkee-247 667, India
Email: drmadhujain.iitr@gmail.com
Nisha Arora
School of Engineering,
GD Goenka University,
Gurgaon, India
Email: nishaarora4@gmail.com
Abstract: The purpose of the present study is to analyse the optimal lot size in
an unreliable single-machine production system with shortages. The production
system is subject to failure due to machine breakdown. Breakdown times are
considered to be according to Weibull distribution. It is assumed that the
shortages are allowed and backlogged. During each production, the set-up
preventive (regular) maintenance is performed. The corrective (i.e., emergency)
maintenance is carried out immediately after breakdown. For the illustration
purpose, numerical results are provided for the special cases. To obtain the
optimal cost per unit time, we also employ the artificial neuro-fuzzy inference
system (ANFIS) approach which has the learning capability of neural network
as well as advantages of rule-base fuzzy system. It is noted that the results
obtained by neuro-fuzzy technique are at par with the results computed by the
analytical techniques.
Keywords: inventory; lot-sizing; unreliable machine; preventive maintenance;
shortage; adaptive network-based fuzzy inference system; ANFIS.
Reference to this paper should be made as follows: Singh, N., Jain, M. and
Arora, N. (2015) ‘Economic lot sizing for unreliable production system with
shortages’, Int. J. Mathematics in Operational Research, Vol. 7, No. 4,
pp.464–483.
Economic lot sizing for unreliable production system with shortages 465
Biographical notes: Neetu Singh is an Associate Professor in the Department
of Applied Sciences at the World College of Technology and Management,
Gurgaon. She received her MSc (Mathematics) and PhD (Mathematics) from
Dr. Bhim Rao Ambedkar University, Agra. Her research specialisation is
operations research. Her research interests include queueing theory, inventory
theory and stochastic modelling. She has published several research papers in
national and international journals and has presented/published papers in
several national and international conferences and has successfully organised
one national conference.
Madhu Jain is an Associate Professor of Mathematics at the IIT Roorkee. She
received her MSc, MPhil, PhD and DSc degrees in Mathematics from the
University of Agra. There are more than 360 research publications in refereed
international/national journals and more than 20 books to her credit. She is the
recipient of the Young Scientist Award and SERC Visiting Fellow of DST,
India, and Career Award of UGC. Her current research interest includes
performance modelling, stochastic models, soft computing, bio-informatics,
reliability engineering and queueing theory. Thirty-five candidates have
received their PhD under her supervision. She has visited more than 30 reputed
universities/institutes in the USA, Canada, Australia, UK, Germany, France,
Holland, Taiwan and Belgium.
Nisha Arora is an Assistant Professor at the School of Engineering, GD Goenka
University, Haryana. She holds an MPhil and PhD in Mathematics. Her
research interests include production, inventory control and supply chain
management. She has published several research papers in refereed
national/international journals. She has participated in various training
programs in the area of mathematics and operations research. Also, she has
presented papers in several national/international conferences.
1 Introduction
In modern highly automated manufacturing systems production unreliability of the
production unit is a frequent problem. The poor electrical connections, over-running
machine capacity, tightened misalignment, improper weather-related use, ignoring of
warning signals, untrained operating personnel, etc., are the main cause of machine
breakdowns in production environment. A majority of the studies in the area of
production and inventory models focus their attention to system consisting of
non-repairable components. But, in the present day practice most of the elements or
components constituting the system are repairable. In the new economy, business
requires their manufacturing systems to maintain low cost levels. Production systems are,
therefore, being studied to address this issue with incorporation of inventory, shortage
and backlogging. Chiu (2010) analysed optimisation for production system subject to
random machine breakdown and failure in rework. This paper includes mathematical
modelling and derivation of the production-inventory cost functions for both systems
with/without breakdowns. Hadidi et al. (2011) studied an integrated cost model for
production scheduling and perfect maintenance.
466 N. Singh et al.
Soft computing methods have shown advantages in predicting the performance of
many systems working in machining environment over usual mathematical techniques.
Diverse applications of these techniques can be found in the areas of power systems,
manufacturing, computer communication networks, medicine, signal processing,
robotics, etc. The concepts of fuzzy logic and neural networks are combined in adaptive
network-based fuzzy inference systems (ANFIS) which forms a hybrid intelligent system
that enhances the ability to automatic learning and adaptability. Gill and Singh (2010)
presented an adaptive neuro-fuzzy inference system modelling for material removal rate.
This paper uses ANFIS technique model which combines modelling function of fuzzy
inference with the learning ability of artificial neural network. Zhou and Dexter (2013)
studied offline identification of nonlinear systems using a neuro-fuzzy modelling
technique presenting a method to generate data for identifying a type of neuro-fuzzy
model.
The neuro-adaptive learning technique provides a method for the fuzzy modelling
procedure to learn information about a data set to automatically compute the membership
function parameters which best allow the associated fuzzy interference system (FIS) to
track the given input/output data. The combination of least square method and back
propagation gradient descent method is used for training of the membership function
parameters. These membership function parameters will be changed through the learning
process and their adjustment is done by a gradient vector. It gives the measure of FIS
modelling of the input/output data for the given set of parameters.
In this paper the main objective is to determine optimum lot-size for the unreliable
production system with the provision of preventive and corrective maintenance. Here, we
have developed a single machine production system by considering that the shortages
arise and backlogged. The rest of the paper is organised in the following manner. In
Section 3, we describe the model by stating the requisite assumptions and notations used.
The expected cost rate and optimum lot size is derived in Section 4. In Section 5, some
special cases are deduced. To obtain optimal cost per unit time, the neuro fuzzy approach
is discussed in Section 6. We employ the neuro-fuzzy approach for evaluating the
expected cost for the model under consideration. Finally in Section 7, we summarise our
findings by highlighting the noble features of the investigation done.
2 Literature review
The considerable amount of research is available on the problem of lot sizing for the
inventory systems and it has been successfully used in various problems such as
production/manufacturing systems. Most of the earlier works in this direction has been
done by assuming that the machining system which produces items never fails however
in real life situations, the machining systems are not completely reliable; the machines
may fail and do not resume production before repair. Almost all manufacturing
environments operate under conditions of uncertainty about jobs processing times and
machine reliability. Many complex production/ inventory systems are characterised by
the uncertain capabilities due to unexpected breakdowns, unplanned repair, unscheduled
maintenance, etc.
The optimal lot sizing for an imperfect production/inventory system has been taken
into consideration by several researchers in different frameworks. Groenvelt et al.
Economic lot sizing for unreliable production system with shortages 467
(1992a) considered the two production control policies, say, no-resumption policy and
abort/resume policy assuming negligible repair time under the effects of corrective
maintenance. Groenevelt et al. (1992b) again explored the lot sizing and safety stock
policies under the assumption that the repair times having a general distribution and
exponential failure time. Dagpunar (1997) studied lot-sizing model in a production
facility with machine breakdowns considering failures to be dealt with minimal repairs,
until the production of required lot. At this point a new cycle is set-up and the machine
restored to ‘as-new’ condition. Ben-Daya (2002) analysed the joint control of EPQ and
preventive maintenance level for deteriorating system. Economic manufacturing quantity
(EMQ) models in different frameworks for imperfect production systems were developed
by several other researchers including Chung and Hou (2003), Wang (2004), Darvish and
Ben-Daya (2007) and Chakraborty et al. (2009). Sarkar et al. (2010) studied optimal
production lot size and safety stock considering variable reliability parameter. Rafiee
et al. (2011) analysed inventory lot sizing in an unreliable cellular manufacturing system.
Chiu et al. (2010) studied optimisation of a production model scrap, rework and random
machine breakdown. Chakraborty et al. (2013) analyzed the problem of the joint
determination of the optimal lot sizing and optimal production control policy for an
unreliable and imperfect manufacturing system. Hajeeh (2012) presented availability of a
system with different repair options like imperfect repair, minimal repair and perfect
repair.
The effect of shortages on inventory cost for deteriorating model linear
demand is investigated by Dye and Chang (2003). Giri and Yun (2005) studied the lot
sizing model with unreliable production system considering at most two failures in a
production cycle, allowing shortages with partial backlogging by resuming the
production run. Roy et al. (2010) analysed an inventory model of a volume flexible
manufacturing system for a deteriorating item with randomly distributed shelf life,
continuous time-varying demand, and shortages over a finite time horizon. Gupta and
Arora (2010) discussed a production inventory model with backlogging of demand
considering alternating replenishment rates. Ghosh et al. (2011) proposed an EOQ
model for deteriorating item with time varying demand. Gupta and Arora (2011)
analyzed EOQ model with power pattern demand, Weibull deterioration and
partial backlogging. Sarkar et al. (2012) proposed an optimal inventory replenishment
policy for a deteriorating item with time-quadratic demand and time-dependent
partial backlogging with shortages in all cycles. Mishra et al. (2013) studied an
inventory model allowing shortages with time proportional deterioration and partial
backlogging. Sharma and Chaudhary (2013) discussed a shortage inventory model
with Weibull distribution deterioration. The ANFIS which was developed by Jang
(1993) is a nonlinear soft computing approach which combines fuzzy logic and system
identification in a hybrid manner that has potential for modelling complex systems.
The ANFIS can rapidly identify important characteristics of the data, which is an
important and useful feature of inventory models. Jang and Sun (1995) introduced the
design methods for ANFIS in both modelling and control applications. The neuro
fuzzy approach has been applied in different frameworks by many researchers;
some worth mentioning works in this direction are due to Nelson and Tham
(2000), Ciaramella et al. (2006), Tozan and Vayvay (2009) and Bahramifar et al.
(2013). Park et al. (2012) studied the adaptive ANFIS model with different membership
functions.
468 N. Singh et al.
3 Model description
In the present study the main objective is to determine optimum lot-size for the unreliable
production system with provision of preventive and corrective maintenance. Here we
consider a manufacturing system consisting of one machine producing one part type. It
starts producing with a constant production rate to satisfy customer’s demand, which is
also assumed to be constant. A regular (preventive) maintenance is used to retain the
healthy condition of the machine. The machine is prone to random breakdowns and
repairs. Failures are instantly repaired (corrective maintenance). Shortages are also
allowed and backlogged.
The basic assumptions used for developing the model under consideration are as
follows:
there is a single-machine production facility which may breakdown at any random
point of time during a production set-up
preventive (regular) maintenance is performed in each production set-up and the cost
of a production set-up includes cost of preventive maintenance
the machine is subject to failure between consecutive production set-ups and is
immediately repaired by providing the corrective (emergency) maintenance
the life time of machine follows Weibull distribution
after every maintenance operation, the system is assumed to be as good as new
the failure and repair times of the machine are independent of each other
the production process is stopped when the requisite lot size is manufactured.
Now, we introduce some notations to formulate the mathematical model as follows:
D demand rate of the consumer
P production rate
t1 period of production and consumption
t2 period of consumption only
t3 period of shortage
t4 period of production and satisfying backorders
Csu expected cost of production set-up
ts known duration of production set-up
Cr expected cost of each corrective maintenance
tr expected duration of corrective maintenance (each repair)
Ch cost of carrying one unit of inventory per unit of time
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