Inventory Models Involving Lead Time Crashing Cost as an Exponential Function

International Journal of Managing Value and Supply Chains (IJMVSC) Vol. 7, No. 2, June 2016
DOI: 10.5121/ijmvsc.2016.7204 29
INVENTORY MODELS INVOLVING LEAD TIME
CRASHING COST AS AN EXPONENTIAL FUNCTION
M. Vijayashree1*
, R. Uthayakumar2
1*
Full Time Research Scholar, Department of Mathematics, The Gandhigram Rural
Institute – Deemed University, Gandhigram- 624 302, Tamilnadu, India.
2
Professor & Head, Department of Mathematics, The Gandhigram Rural Institute –
Deemed University, Gandhigram- 624302, Tamilnadu, India.
ABSTRACT
Inventory management and control is concerned with the acquisition and storage of materials required for
supporting various business operations. Lead time reduction is another important production activity in an
integrated inventory control. Lead time plays a vital role and has been a topic of interest for many authors
in inventory organization.
Lead time is a necessary aspect in any supply chain management and inventory management system. The
time gap between placing of an order and its actual arrival in the inventory is known as lead time. In most
of the literature dealing with inventory problems, either in deterministic or probabilistic model, lead time is
viewed as a prescribed constant or a stochastic variable, which therefore, is not subject to control. But, in
numerous sensible circumstances, lead time can he reduced by an additional crashing cost; in other words,
it is controllable, so in this article, we have considered the lead time crashing cost is an exponential
function of lead time. A solution procedure is developed to find the optimal solution.
In this article, both lead time and the order quantity are considered as the decision variables. The
behaviour of the model is presented graphically. The result is illustrated with the help of a numerical
example. Finally, Graphical representation is presented to illustrate the model. The solution procedure
with the help of the software Matlab 2008 is furnished to determine the optimal solution.
Keywords: Integrated inventory model, Supply chain management, lead time crashing cost
Subject Classification Code: 90B05, 90C25, 90C30.
*Corresponding author
Tel.: +91-451-2452371, Fax: 91-451-2453071
1. INTRODUCTION
Inventory control is significant in supply chain management. In current years, the majority
inventory problems have focused on the integration between the supplier and the retailer. The
integrated inventory model has become more and more important, because the supplier and the
retailer wish to increase their mutual benefit; therefore, they endeavour to form a strategic
alliance in order to minimize their own cost or maximize their own profit, and then the trading
parties can collaborate and share information to achieve improved benefits. This implies that the
optimal contract quantity and number of deliveries must be determined at the outset of the
contract based on their integrated total cost function.

30
Gholami-Qadikolaei et al. [10] have summarised the number of advantages and benefits has been
associated in the efforts of control of the lead time (which is a goal of JIT inventory management
philosophies that emphasizes high quality and keeps low inventory level and lead time to a
practical minimum). The benefits of JIT purchasing include small lot sizes, frequent deliveries,
consistent high quality, reduction in lead times, decrease in inventory levels, lower setup cost and
ordering cost, and close supplier ties. In recent years, companies have found that there are
substantial benefits from establishing a long-term sole-supplier relationship with supplier
Martinich [24]. In the JIT environment, a close cooperation exists between supplier and purchaser
to solve problems together, and thus maintains stable, long-term relationships.
Among the modern production management, the Japanese successful experiences of using Just-
In-Time (JIT) production show that the advantages and benefits associated with the efforts to
control the lead-time can be clearly perceived. The goal of JIT inventory management
philosophies is the focus that emphasizes high quality, keeps low inventory level and lead-time to
a practical lowest amount. Shortening the lead time is recognized as the feasible and effective
way to achieve the goal of JIT. The objective of JIT inventory management philosophies is the
focus which emphasizes high quality, keeps low inventory level and lead time to a practical
lowest amount.
Ouyang and Wu [33] discuss lead time plays a very important role in inventory organization. In
most of the literature dealing with inventory problems, either in deterministic or probabilistic
model, lead time is viewed as a prescribed constant or a stochastic variable, which therefore, is
not subject to control (Naddor[26], Silver and Peterson [41]). In fact, lead time usually consist of
the following components (Tersine [43]); order training, order shipment, dealer lead time release
time, and setup time. In many practical situations, lead time, we can lower the safety stock,
reduce the loss caused by stockout, improve the service level to the customer, and increase the
spirited ability in business. Recently, some models considering lead-time as a decision variable
have been developed. Liao and Shyu [22] have initiated a study on lead-time reduction by
presenting an inventory model in which lead- time is a decision variable and the order quantity is
predetermined. Ben-Daya and Raouf [3] developed a model that considered both lead-time and
order quantity as decision variables. Later, Ouyang et al. [32] and Ouyang and Wu ([28], [30] and
[29]) have generalized the Ben-Daya and Raouf [3] model by considering shortages in which the
lead-time demand is measured a normal distribution or distribution free. Later, Moon and Choi
[25], Hariga and Ben-Daya [15] have extended the Ouyang et al. [32] model to relax the
assumption of a given service level and treat the reorder point as a decision variable. Other papers
related to this area are Lan et al. [20], Ouyang and Chuang [29], Ouyang and Chang [27], Pan and
Hsiao [34] and others.
Later many researchers (see [4, 6, 7, 8, and 9]) developed lead time reduction inventory models
under various crashing cost function and practical situations. The underlying assumption in the
above studies was that lead time could be decomposed into n mutually independent components,
each with a different but fixed crash cost independent of the lead time. Recently, Hsu and Lee
[19] investigated the decisions of replenishment and lead-time reduction for a single manufacturer
multiple-retailer integrated inventory system with the assumption of lead time crashing cost is a
non-increasing stairstep function of lead time. Later, Yang [47] developed supply chain integrated
inventory model with present value and the crashing cost is determined by the length of lead time,
which is polynomial form. In contrast to existing inventory models, this paper considers that the
crashing cost is an exponential function of lead time. Further, numerical example is provided to
illustrate the benefits of integration.

31
2. LITERATURE REVIEW
Shukla et al. [40] have talk about the Supply Chain Management (SCM) is a network of facilities
that produce raw materials, transform them into intermediate goods and then final products, and
deliver the products to customers through a distribution system. The management of the supply
chain and the roles of various factors involved differ from industry to industry and company to
company.
Supply Chain Management is a network of facilities that produce raw materials, transform them
into intermediate goods and then final products, and deliver the products to customers through a
distribution system. It spans procurement, manufacturing and distribution (Lee & Billington [21])
the basic objective of supply chain management is to “optimize performance of the chain to add
as much value as possible for the least cost possible”. In other words, it aims to link all the supply
chain agents to jointly cooperate within the firm as a way to maximize productivity in the supply
chain and deliver the most benefits to all related parties (Finch [8] ). Adoption of Supply chain
management practices in industries has steadily increased since the 1980s. A number of
definitions are proposed and the concept is discussed from many perspectives. However Cousins
et al. [7]; Sachan and Datta [38]; Storey et al. [42] provided excellent review on supply chain
management literature. These papers define the concept, principals, nature, and development of
SCM and indicate that there is an intense research being conducted around the world in this field
they critically assessed developments in the theory and practice of supply management.
Sara et al. [39] discuss on the importance of supply chain management has created remarkable
interest to make a kind of partnership among supply chain's entities. Traditionally, the buyer and
vendor used a policy according to economic order quantity (EOQ) or economic production
quantity (EPQ) classical methods. To refine these classical methods, in order to satisfy today's
market requirements, the concept of joint economic lot size (JELS) is introduced. In fact, the
concept of JELS is a kind of integration between two different business entities to gain
competitive advantages and lesser total cost.
Yang and Pan [46] explain the supply chain management is a well-liked practice in
manufacturing systems, and just-in-time (JIT) production plays a crucial role in supply chain
environments. Companies are using JIT production to gain and maintain a competitive advantage.
The characteristics of JIT systems are consistent high quality, small lot sizes, frequent delivery,
short lead time, and close supplier ties. Recently, the issue of just-in-time (JIT) manufacturing has
received considerable attention, and one of the most interesting topics on this issue is the
integration of vendor and buyer in the supply chain system (Chang et al. [5]). Many researchers
have shown that improved benefits can be achieved through the coordination of vendor and
buyer. Combination between two different business entities is an important way to gain
competitive advantages as it lowers the joint total cost of the system. Therefore, in modern
enterprises, the integration of vendor–buyer inventory system is an important issue. The
coordination between vendor and buyer for improving the performance of inventory control has
received a great deal of attention and the integrated approach has been studied for years.
Typically, integrated approach focuses on the production-inventory decisions of supply chain
partners while minimizing the total relevant cost of the system.
The combination between vendor and buyer for improving the performance of inventory system
control has been discussed for years. Goyal [12] was an initiate in studying an integrated
inventory model consisting of a vendor and a buyer, who showed a coordinated replenishment
policy is more desirable than each party operating its individual optimal policy. The framework
proposed by Goyal [12] encouraged many researchers to explore various types of integrated
inventory model. Banerjee [2] modified Goyal’s [12] model considering finite production rate of

32
the vendor, and the vendor produces for the buyer on a lot-for-lot basis. Goyal [12] undisturbed
the lot-for-lot policy and suggested that the vendor’s economic production quantity should be an
integer multiple of the buyer’s order quantity. Lu [23] relaxed the assumption of Goyal [13] about
completing a production batch before starting shipments and proposed a model which allows
shipments to take place during production period also. Subsequently, several other shipment
policies (Goyal [14], Hill [17], Hill [18], Goyal and Nebebe [11]) have been proposed to study
the single-vendor single- buyer supply chain.
The Japanese experience of using Just-In-Time (JIT) production shows that there are advantages
and benefits associated with their efforts to control lead time. Lead time plays an important role
and has been a topic of interest for many authors in continuous review inventory system. The
replenishment lead time is usually prescribed whether deterministic or probabilistic. However, in
many practical situations lead time can be controlled and reduced at an added cost (called
crashing cost). The crashing time idea originated in project management where the duration of
some activities can be reduced by allocating more resources at an extra direct costs for these
activities.
Lead time decrease is an additional important production action in an integrated inventory
control. Lead time consists of order preparation, order transmittal, order processing and assembly,
additional stock acquisition time and delivery time Ballou [1]. Recently, Ouyang et al. [25]
extended the (Q, r) model by Ben-Daya and Raouf [3] to think the lead time effect and add in the
partial backordering into the inventory model. Hariga [16] studied the relationship between lot
size and lead time in the process time aspect. Pan and Yang [27] presented an integrated supplier-
purchaser model with controllable lead time. The model has a substantial cost saving when lead
time is controllable. Chen et al. [6] presented a continuous review inventory model when ordering
cost is dependent on lead time. Pan and Yang [35] developed a study of an integrated inventory
model with controllable lead time. Ben-Daya and Hariga [4] developed a continuous review
inventory model where lead time is considered as a controllable variable. Lead time is
decomposed into all its components: set-up time, processing time and non-productive time. Later,
(Ouyang et al. [31] extended Pan and Yang’s [34] model by allowing shortages. Liao and Shyu
[15] state that there is a lack of an appropriate inventory model that treats lead time as a decision
variable. These authors have presented a model which can be used to determine the length of lead
time that minimizes the expected total relevant cost. Lead time is decomposed into n components
each having a different crashing cost for reduced lead time. Consequently, the lead time crashing
cost function is described using a piecewise linear function.
Exponential functions demonstrate exponential development (or decompose). This means that the
functions moreover rise or go down much more radically than polynomials. In a real
surroundings, to reduce lead time, managers may ask workers to work overtime, employ part-time
workers, use individual delivery, and so on. Impulsively, a high additional cost should be paid for
these services. Therefore, crashing costs were observed to produce with lead time by a quantity
which can be approximated by an exponential function of lead time. The best thing about
exponential functions is that they are so useful in real world situations. Exponential functions are
used to model populations, carbon date artifacts, help coroners determine time of death, compute
investments, as well as many other applications.
Ghare and Schrader [9] were the first to develop an EOQ model for an item with exponential
decay and constant demand. The certain commodities were observed to shrink with time by a
proportion which can be approximated by a negative exponential function of time. In the recent
years, Priyan and Uthayakumar [37] have developed Optimal inventory management strategies
for pharmaceutical company and hospital supply chain in a fuzzy–stochastic environment. Here
the author has been used exponential function of lead time crashing cost. Priyan and
Uthayakumar [36] have developed Continuous review inventory model with controllable lead

33
time, lost sales rate and order processing cost when the received quantity is uncertain.
M.Vijayashree and R Uthayakumar [44] have developed integrated inventory model with
controllable lead time involving investment for quality improvement in supply chain system. M.
Vijayashree and R Uthayakumar [45] have developed two-echelon supply chain inventory model
with controllable lead time, here we have used exponential function of lead time crashing cost.
To the best of our knowledge, the author has developed inventory models involving lead time
crashing cost as an exponential function. In this model we have considered the lead time crashing
cost as an exponential function.
The remainder of this chapter includes the following aspects: Literature review is given in section
2. The primary assumption and notations are provided in section 3. Section 4 describes the
mathematical model is developed. The solution procedure is given in the section 5. A numerical
example is provided in section 6, to illustrate the results. Finally, conclusion is given in section 7.
3. NOTATIONS AND ASSUMPTIONS
To establish the mathematical model, the following notations and assumptions are used as follows
3.1. Notations
To develop the proposed model, we adopt the following variables and parameters
D Demand
Q Order quantity
A Ordering cost
R reorder point
h Holding cost
L Lead time in weeks
TC Total cost
3.2. Assumptions
To develop the proposed model, we adopt the following assumptions
1. The inventory is continuously reviewed. The buyers place an order when the on hand
inventory reaches the reorder point R .
2. The buyer places an order when the inventory position reaches the reorder point R . The
reorder point R = expected demand during lead time + safety stock, that is,
LkDLR σ+= where k is a safety factor and σ is the standard deviation.
3. The demand X during lead time L follows a normal distribution with mean Lµ and
standard deviation Lσ .
4. The extra cost incurred by the vendor will be transferred to the buyer if shortened lead time
is requested.
5. If the buyer is not eager to add an extra cost to control the lead time, he can obtain his
items exactly on the existing lead time ( eLL = ) and the crashing cost is zero. Otherwise,
the lead time L of the buyer should be within the interval [ ]se LL , , that is es LLL <≤ .

34
6. The crashing costs were experimental to grow with lead time by a proportion which can be
approximated by an exponential function. Therefore, the lead time crashing cost per order
),(LR is assumed to be an exponential function of L and is defined as



<≤
=
=
es
LC
e
LLLie
Li
LR
f,
Lf0
)( /
where C is a positive constant. The application of the
exponential function lead time crashing cost has been proposed by many authors, for
example Vijayashree and Uthayakumar [45] and Priyan and Uthayakumar[36]
4. MATHEMATICAL MODEL
Based on the above notations and assumptions, the total cost is given by ( )LQTC , =Ordering
cost+ holding cost+ lead time crashing cost.
Since A is the ordering cost per order, the expected ordering cost per year is given by
Q
AD
.
From assumption (2), the reorder point LkDLR σ+= , where k is known as the safety factor.
If we assume a linear decrease over the cycle, then
( ) Lk
Q
DLR
Q
rQI σ+=−+≅=
22
,
Lead time crashing cost )(LR is given by ( )LR
Q
D
( ) ( )LR
Q
D
Lk
Q
h
Q
AD
LQTC +





++= σ
2
,
L
C
e
Q
D
Lk
Q
h
Q
AD
+





++= σ
2
(1)
for es LLL <≤
5. SOLUTION PROCEDURE
Taking the partial derivatives of ( )LQTC , with respect to Q and L in each time interval
[ ]se LLL ,∈ and equating to zero, we obtain
( ) 0
2
,
22
=−+−=
∂
∂
Q
Deh
Q
AD
Q
LQTC L
C
(2)
( ) 0
2
1,
2
2
1
=−=
∂
∂
−
QL
DCe
Lhk
L
LQTC L
C
σ (3)
Notice that for fixed ( )LQTCL ,, , is convex in Q . (See the appendix) However, for fixed
( )LQTCQ ,, is concave in [ ]se LLL ,∈ . Therefore, for fixed Q the minimum occurs at the end
points of the interval. From (2), we have

35










+
=
h
DeDA
Q
L
C
22
(4)
The above solution procedure can be used to find the optimal Q and [ ]se LLL ,∈ .
For each break point [ ]se LLL ,∈ compute Q using (4).
Compute the corresponding integrated total cost from equation (1)
The optimal Q and [ ]se LLL ,∈ will be the values for which the total cost ( )LQTC , is
minimum.
6. NUMERICAL EXAMPLE
In this section, a numerical example is given to illustrate the above solution procedure. Let us
consider the inventory system with the following data ,units/year600=D
order,per200$=A 20,h = ,units/week6=σ 3.2k = and lead time crashing cost
( )





<≤
=
=
61if
6Lif0
LL
C
e
LR where 5=C .
Applying the above solution procedure the computational results are presented in table (1). The
optimal solutions from table (1) can be read off as lead time weeks,2*
=L order quantity
units113*
=Q and the corresponding integrated total cost 2647*
=TC . A graphical
representations is presented to show the convexity of TC in figure (1) and Graphical
representation of the optimal solution in figure (2).
Table (1) optimal values for different values of lead time
L=1 L=2 L=3 L=4 L=5 L=10
( ) 51,5, <≤== LCeLR L
C
( ) 0=LR
Q TC Q TC Q TC Q TC Q TC Q TC
146 3168 113 2647 111 2698 110 2762 110 2823 110 2867

36
Figure (1) Graph representing the convexity of TC
Figure (2) Graphical representation of the optimal solution in TC
7. CONCLUSION
Currently lead time is a fashionable tradition in mechanized systems in the inventory
control and supply chain management. Lead time is a noteworthy factor in any inventory
management scheme. A lead time is the latency between the start and finishing of a
process. Traditional inventory models assumed that lead time is a constant or random
variable which is not a controllable factor. However, in practice, lead time could be
shortened by paying an additional crashing cost; in other words, it is controllable.
In a lot of realistic condition, lead time can be reduced by an extra crashing cost. That is,
lead time is controllable. So in this article presents inventory models that consider order
quantity and lead time as decision variables. Here we have used the lead-time crashing
cost per order ( )LR is assumed to be an exponential function of L . A numerical example
demonstrates the effectiveness of the developed model. The contribution of this article is
developing a mathematical model and an effective solution procedure to find the optimal

37
solution. Moreover, a numerical example is presented to illustrate the important issues
related to the proposed models.
APPENDIX
( ) 0
22,
332
>+=
∂
∂
Q
De
Q
AD
Q
LQTC L
C
ACKNOWLEDGEMENT
The authors wish to express their thankfulness to the editors and reviewers for their
helpful suggestions and corrections to improve the clarity of the present paper. The first
author research work is supported by DST INSPIRE Fellowship, Ministry of Science and
Technology, Government of India under the grant no. DST/INSPIRE
Fellowship/2011/413A dated 22.12.2015 and UGC–SAP, Department of Mathematics,
The Gandhigram Rural Institute – Deemed University, Gandhigram – 624302,
Tamilnadu, India.
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Biographical Notes
M. Vijayashree is a full time research scholar in the Department of Mathematics at
The Gandhigram Rural Institute – Deemed University, Gandhigram, India. She
received her B.Sc in Mathematics from M.V.M Govt Women’s Arts College,
Dindigul, Tamilnadu, India in 2007 and M.Sc. Tech (IMCA) in Industrial Mathematics
with Computer Applications from The Gandhigram Rural Institute-Deemed
University, Gandhigrm, Dindigul, Tamilnadu, India in 2010. Currently, she is a
Senior Research Fellow (SRF) under DST INSPIRE, New Delhi, in the Department
of Mathematics, The Gandhigram Rural Institute – Deemed University, Gandhigram-624302, Dindigul,
Tamilnadu, India. She has published about 7 papers in international and national journals. Her research
interests include the following fields: Operations Research, Inventory Management and Control,
Supply Chain Management.
R. Uthayakumar was born in Dindigul, Tamilnadu, India, in 1967. He received his
B.Sc. degree in Mathematics from G.T.N. Arts College, Dindigul in 1987, M.Sc.
degree in Mathematics from American College, Madurai in 1989, M.Phil., degree in
Mathematics from Madurai Kamaraj University, Madurai in 1991, B.Ed. degree in
Mathematics from Madurai Kamaraj University, Madurai in 1992 and Ph.D. degree in
Mathematics from The Gandhigram Rural Institute – Deemed University,
Gandhigram, India in 2000. Currently, he is professor of the Department of Mathematics at The
Gandhigram Rural Institute – Deemed University, Gandhigram, India. He has published about 133 papers
in International and National journals. His research interests include the following fields: Mathematical
Modeling, Fractal Analysis, Operations Research, Inventory Management & Control and Supply
Chain Management.

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