AN INVENTORY MODEL WITH MULTIVARIATE DEMAND WEIBULL DISTRIBUTION
DETERIORATION UNDER THE EFFECT OF INFLATION AND TRADE CREDIT
Dr.Anubha Goyal
Department of Mathematics, Krishna College, Bijnor
Abstract: A deteriorating inventory model with
multivariate demand is developed as in present scenario
it is imperative to take multi-trends while considering
customer’s demand. These trends are stock
dependency, relying on selling price and time demand.
Two or more parameters dependent demand is more
authentic and this model has time, stock and selling
price based demand rate. Weibull rate of deterioration
is considered here while shortages inflationary
environment is included in the model with its
applicability for trade credit to customer.
Keywords: Inventory Model, Multivariate Demand,
Inflation, Trade credit
1. Introduction
The inventory manager has to look out for a new
method, by which he can easily assimilate the changed
inclination of the customers towards the policy
existence. The chain to supply the article is directly
proportional to demand, as demand increases supply
increases and as demand decreases supply of items
decreases. Earlier researchers obtained the solution of
standard inventory models after incorporating the
dependence on variety of demand rates i.e., constant
and linear but there are numerous limitations in all
these demand rates. Stable demand rate occurs very few
in sensible situations. A demand rate of linear time-
dependence deduces a steady change in the demand
rate as per time. This hardly happens in the case of any
commodity in the market.
It was very obvious a fact that given some time, every
item can have a suitable position for itself in the
customer’s mind, thereby raising its demand as time
advances. Later with the advent of outlets, it was
commonly acknowledged that huge displays of stocks
persuade the customer into large purchasing. Also it
was noted that a decline in the level of displayed stock
witnessed a decline in the customer’s demand for that
item. This concept is known as stock dependent
demand. For a long time, stock dependent demands
were explored, researched to a greater extent while new
aspects were being added to the study. The dependence
of the sale of any item on its selling price is not a new
concept, but a common sense conclusion. It is a general
observation that a hike in the selling price of the
commodity discourages its customers from choosing
that item in future. However, a selling price dip, in
whatever form it may come, always notices a sudden
increase in the demand rate, as reduced prices always
encourages the customers to buy more. Therefore, we
have taken the two or more parameter dependent
demand. Such a combination of two or more factors
grants more authenticity to the formulation of the
model and makes it more close to reality.
2. Literature Survey
Levin et al. (1972) observed that “heap of consumer
goods displayed in large outlets lead customers to
more. Haley and Higgins (1973) formulated an
inventory policy with trade credit financing.
Donaldson (1977) developed an optimal algorithm for
solving classical no-shortage inventory model
analytically with linear trend in demand over fixed time
horizon. Silver (1979) developed an approximate
solution procedure for a linearly time-varying demand
by using the Silver–Meal heuristic (1969). The effect
of payment rules on ordering and stock holding in
purchasing was suggested by Kingsman (1983). Silver
and Peterson (1985) developed a model in which sales
at the retail level tend to proportional to inventory
displayed and a large piles of goods displayed in a
supermarket will lead the customer to buy more. Gupta
and Vrat (1986) considered demand rate to be function
of initial stock level. Baker and Urban (1988) were
the first to establish an economic order quantity model
(EOQ) for a power–form inventory-level demand
pattern. Dave, U. (1989) proposed a deterministic lot
size inventory model with shortages and a linear trend
in demand. Mandal and Phaujdar (1989) then
developed a production inventory model for
deteriorating items with uniform rate of production and
linearly stock dependent demand. Goswami and
Chaudhuri (1991) discussed different types of
inventory models with linear trend in demand.
Aggarwal and Jaggi (1995) developed ordering
policies of deteriorating items under permissible delay
in payments. The demand and deterioration were
assumed as constant. Bhunia, A.K. and Maiti, M.
(1998) gave deterministic models of perishable
International Journal of Pure and Applied MathematicsVolume 118 No. 22 2018, 1309-1323ISSN: 1314-3395 (on-line version)url: http://acadpubl.eu/hubSpecial Issue ijpam.eu
1309
inventory with stock-dependent demand rate and
nonlinear holding cost. Liao et al. (2000) developed an
inventory model for stock dependent demand rate when
a delay in payment is permissible. Chang and Dye
(2001) developed an inventory model for deteriorating
items with partial backlogging and permissible delay in
payments. Khanra and Chaudhuri (2003) discussed
an order level decaying inventory model with such time
dependent quadratic demand. Balkhi and Benkherouf
(2004) developed an inventory model for deteriorating
items with stock dependent and time varying demand
rates over a finite planning horizon. Teng et al. (2005)
presented an optimal pricing and ordering policy under
permissible delay in payments. Deterioration rate was
taken as constant and shortages were not allowed in the
study. Wu et al. (2006) proposed an optimal
replenishment policy for non-instantaneous
deteriorating items with stock-dependent demand and
partial backlogging. Panda et al. (2007) considered an
EOQ model with time dependent demand and Weibull
distribution deterioration. Singh, S.R. et al. (2008)
developed an inventory model for deteriorating items
having stock dependent demand. They were allowed
shortages with partial backlogging in their study.
Singh, S.R. et al. (2009) proposed an inventory system
for perishable items with stock dependent demand and
time dependent partial backlogging. In their model,
constant holding cost has been taken. Tripathy, C.K.
and Mishra, U. (2010) developed a deterministic
inventory model for deteriorating items with constant
demand. Shortages were allowed with constant partial
backlogging rate in their study.
Our purpose in this paper is to establish a set of guiding
principles for the effective design and execution of an
inventory system. In this study, we have taken a more
realistic demand rate that depends on more than two
factors; one is the stock level available, second is time
and third is the selling price of the item. Deterioration
rate is taken as three parameter Weibull distribution.
The effect of life time, inflation and permissible delay
in payments are also taken into account. Shortages are
allowed in inventory and partially backlogged.
We believe that our study will provide a solid
foundation for the further study of this sort of important
inventory models with multi-variate demand rate in an
inventory model.
3 .Assumptions and Notations
Assumptions:
The following assumptions have been adopted for the
proposed model to be discussed:
1. Demand rate is dependent on stock selling price and
time.
2. Deterioration rate is a three parameter Weibull
distribution.
3. Inflation rate is also considered.
4. Shortages are allowed. Demand during the stock-out
period is partial backlogged.
5. Permissible delay in payments is taken into account.
Notations:
D(t) = a+ bt+ cI (t) -ds , demand rate in unit per unit
time
θ(t) = αβ (t -λ ) β-1 , deterioration rate in per unit of
time, α << 1
r: Inflation rate
C: Purchase cost per unit time per unit item
So: Selling cost per unit time per unit item
C1: Holding cost per unit time per unit item
C2: Shortage cost per unit time per unit item
C3: Shortage cost per unit time per unit item
A: set up cost
δ: The fraction of the demand during the stock -out
period that will be backordered,0 ≤ δ ≤ 1
H: Planning horizon
T: Replenishment time
λ: Life time
M: Permissible delay
Ie : Interest rate
S: Inventory level
4. Mathematical Formulation and Solution
Let us assume we get an amount S (>0) as initial
inventory. During the period (0, λ) the inventory level
gradually diminishes due to market demand only. After
life time deterioration can take place, therefore during
the period (λ, T1) the inventory level decreases due to
the market demand and deterioration of items and falls
to zero at time t1 .The period (T1 , T) is the period of
shortage which is partially backlogged.
))(()(
dstcIbtadt
tdI−++−= 0 ≤ t ≤ λ (1)
))(()()()( 1
dsscIbtatItdt
tdI−++−=−+ −βλαβ
λ ≤ t ≤ T1 (2)
)()(
dsbtadt
tdI−+−= δ T1 ≤ t ≤ T (3)
With the boundary conditions:
I (0) = S, I (T1) =0, I (T) = 0
Solution of equations (1), (2) and (3) are:
)
21()1(
)()( ctctct
ec
b
c
bte
c
dsaSetI
−−− −+−−−
+=
0 ≤ t ≤ λ (4)
International Journal of Pure and Applied Mathematics Special Issue
1310
))()((1
)()((1
)()(
2))([()( 11
11
11
1
22
11
++++ −−−+
+−−−+
−+−+−−= ββββ λλ
β
αλλ
β
αttTT
btT
dsatT
btTdsatI
))((33
1
22
1
22
1 )](2
)(2
))()(()2)(1(
cttetT
bctT
actT
b +−−++ −+−+−−−++
−βλαββ λλ
ββ
α
λ ≤ t ≤ T1 (5)
and
)](2
))([()( 22
11 tTb
tTdsatI −+−−= δ
T1 ≤ t ≤ T (6)
Equating the equation (4) and (5) at t = λ, one can get
1
11
1
1
22
11 )((1
)((1
)()(
2))([( ++ −
++−
+
−+−+−−= ββ λ
β
αλ
β
αλλ TT
bT
dsaT
bTdsaS
)1()(
)](2
)(2
)()((
)2)(1(
33
1
22
1
2
1
λβ λλλββ
α ce
c
dsaT
bcT
cdsaT
b−
−−−+−
−+−
++− +
)1(
2 2
λλλ c
c
ec
beb−++
−
(7)
4.1 Present Worth Purchase Cost
P.C. = ∫−− +
T
T
rtrM dttDeCTeCS
1
)(0
= 22
10
111 )()([
r
be
r
be
r
bTe
r
ebT
r
edsa
r
edsaCeCS
rTrTrTrTrTrTrM
−−−−−−− +−−+
−−
−+ (8)
4.2 Present Worth Holding Cost
H.C. = ∫ ∫+−− +
λ
λ
λ
0
)(
211
1
])()([
T
trrtdtetIdtetIC
2
)()(
1 )1
)()(
1(
)()([
cr
be
cr
eb
r
e
rrc
e
rcc
a
rc
Se
rc
SC
rrrrcrc λλλλλ λ −−−+−+−
−++−+
−+
++
−+
)22
)(()
22)([(]
)()(
2
1
2
11
)(
222
bcdsaT
TdsaC
rc
b
rc
be
rc
b
rc
be
cr
brcr
+−
++−−++
−+
++−−+−− λ
λλλ
)2(
)()(
)1)(2)(3(
)(2
)1)(2(
)()
33
2(
2
1
3
1
2
11
32
1
3
1
+
−−+
+++
−+
++
−++−
+++
β
λα
βββ
λα
ββ
λαβλλ
βββTdsaTbTTb
TT
)1)(2(
)()(
)1)(2(
)()()
44
3(
3)1(
)( 2
11
2
1
43
1
3
1
3
1
++
−−−
++
−−−+−+
+
−−
+++
ββ
λα
ββ
λαλλ
β
λα βββTcTdsaTdsa
TTbcTb
International Journal of Pure and Applied Mathematics Special Issue
1311
3
))((
)3)(2(3
)(2
)1)(2(
)(
)1)(2)(3(
)()( 33
11
3
1
2
1
3
1 λ
ββ
λα
ββ
λα
βββ
λα βββrcdsaTcTbTcTbTcdsa +−
−++
−−
++
−−
+++
−−+
+++
4
)(
8
)(
2
))((
6
))((
)4)(2)(3(3
)(2 22
1
4
1
2
1
3
1
4
1 λλ
βββ
λα βTrcbTrcbTrcdsaTrcdsaTcb +
++
−+−
++−
−+++
−+
+
)3)(2)(1(2
))(55)(()(
)1(2
))()(()(
8
)( 3
1
22
11
4
+++
−+++−+
+
−++−−
+−
++
βββ
λββα
β
λλαλ ββTrcdsaTTrcdsarcb
)2)(2)(1(
)()()72(
)2)(1(2
))()((3
)1(2
))(()( 3
11
3
11
2
111
+++
−++−
++
−+++
+
−++−
+++
βββ
λαβ
ββ
λλα
β
λλα βββTTrcbTTrcbTTTrcb
8
)()(
4
)()(
8
)()(
)4)(3)(2)(1(
))(()32( 422
1
4
1
4
1 λλ
ββββ
λαβ βrccdsaTrccdsaTrccdsaTrcb +−
−+−
++−
−++++
−+++
+
15
)(
6
)(
16
)( 523
1
5
1 λλ rcbcTrcbcTrcbc +−
++
+−
(9)
4.3 Present Worth Set Up Cost
SAC = A (10)
4.4 Present Worth Shortage Cost
S.C. = ∫−−
T
T
rt dtetIC
1
)(32
= r
ebT
r
ebT
r
eTdsa
r
edsa
r
edsa
r
edsaC
rTrTrTrTrTrT 11 2
1
2
1
222
)()()()([
−−−−−−
+−−
+−
−−
−−
δ
332
1
2
11
r
be
r
be
r
ebT
r
bTerTrTrTrT −−−−
+−+− (11)
4.5 Present Worth Lost Sale Cost
L.C. = ∫−−+−
T
T
rtdtedsbtaC
1
))(1([3 δ
= )]()()(
)[1(22
13
111
r
e
r
e
r
eT
r
Teb
r
edsa
r
edsaC
rTrTrTrTrTrT −−−−−−
++−−−
−−
− δ (12)
Now regarding the permissible delay period M for settling the accounts, there arise two cases
M ≤ T1 or M > T1
International Journal of Pure and Applied Mathematics Special Issue
1312
Case I: When M ≤ T1
4.6 Present Worth Interest Earned
I.E. = ])()([1
1
14 ∫−−
T
T
rt
r dtetDtTIC
= )())()(
1)((([{
)(
14c
bdsacS
cr
b
cr
be
rc
e
rcc
bdsacSTIC
rrc
e −−+−+−+
−+
−−+−+− λλ
(r
ebT
r
eTdsa
r
e
r
e
rc
b
rc
e
rc
e
rc
rTrrrrcrc 12
11
22
)(
2
)(
2
)({)}
1()
)()()(
1 −−−−+−+−
−−
+−−−+
−+
−+
λλλλλ λλ
483
)()()(
22
1
43
12
12
1
2
111 λλλ
λλ λλ
bcTbcTrccdsacTdsa
r
ebT
r
ebT
r
eTbrrTr
−−+−
−−−+−+−−−
2
)()()()(22
3
)( 22
1
2
2
22
3 λλλλλ λλλλλTrccdsa
r
edsa
r
edsa
r
eb
r
eb
r
becdsarrrrr +−
+−
−−
−−−−−
−−−−−−
)3)(1(
)(
)1(
)(
)1(2
)()(
6224
5 1
3
1
3
1
1
1
2
1
1
1
3
1
3
1
4
1
++
−+
+
−+
+
−−++−+
+++
ββ
λα
β
λα
β
λαλλ βββTTbcTTbcTTcdsabcTbcTbcT
3
)(22)(
6
)()(
)2)(1(
)()( 3
1
22
1
2
1
2
4
1
2
11111 cTdsa
r
be
r
ebT
r
ebT
r
edsaTrccdsaTcdsarTrTrTrT −
++++−
++−
−++
−−−
−−−−+
ββ
λα β
)3)(2)(1(
)()(2
)1(2
)(
)3)(2(
)()(
)2)(1(
)()( 3
11
1
1
3
1
3
1
2
1
+++
−−++
+
−−
++
−−−
++
−−+
++++
βββ
λα
β
λα
ββ
λα
ββ
λα ββββTTcdsbaTTbcTcdsaTcdsa
}])2)(1(2
)(
)4)(3)(2)(1(
)(3 2
1
2
1
4
1
++
−−
++++
−+
++
ββ
λα
ββββ
λα ββ TTbcTbc (13)
4.7 Present Worth Interest Payable
I.P. = ])([1
4 ∫−
T
M
rt
e dttIeIC
=
)1(
))(()()
2
)()(
33
2)(
22)([(
1
11
32
1
3
1
2
1
2
1
4+
−−−+
+−+−+−−
+
β
λα βTMTdsabcdsaMMT
TMMT
TdsaIC r
)2)(1(
)(3
)1(
)(
)1(
)(
)2(
)()(
)2(
)()( 2
11
1
11
1
1
2
1
22
1
++
−−
+
−−
+
−+
+
−−+
+
−−−
+++++
ββ
λα
β
λα
β
λα
β
λα
β
λα βββββTTbTMTbTTbMdsaTdsa
)2)(1(
)(2
)3)(2)(1(
)(3
)3)(2)(1(
)(3
)2)(1(
)( 2
1
33
1
2
++
−+
+++
−−
+++
−+
++
−+
++++
ββ
λα
βββ
λα
βββ
λα
ββ
λα ββββ TMbMbTbMMb
)2)(1(
)(
)2)(1(
)(
)1(
))(()(()
44
3(
3
22
1
1
11
44
1
4
1
++
−+
++
−−
+
−−−++−+
+++
ββ
λα
ββ
λα
β
λα βββMTTMT
dsaM
MTTbc
International Journal of Pure and Applied Mathematics Special Issue
1313
)3)(2)(1(
)(
)2)(1(
)(
)2)(1(
)(
)1(2
))(( 322
11
122
1
+++
−−
++
−+
++
−−
+
−−+
++++
βββ
λα
ββ
λα
ββ
λα
β
λα ββββMbMMbTTbMMTb
)2)(1(
)(
)3)(2)(1(
)(
)2)(1(
)(
)1(2
))(()((
2232122
1
++
−+
+++
−−
++
−+
+
−−−+
++++
ββ
λα
βββ
λα
ββ
λα
β
λα ββββMcMbMcMcMMMTc
dsa
2
)()(
)4)(3)(2(3
)(2
)3)(2(3
)(2
)1(3
))(( 2
1
43133
1 MrcTdsaMcMbMcMbMMTcb +−+
+++
−−
++
−+
+
−−+
+++
βββ
λα
ββ
λα
β
λα βββ
)1(2
)()()(
8
)(
4
)(
3
))(( 1
1
2422
1
3
+
−+−+
+−
++
+−−
+
β
λα βTMrcdsaMrcbMTrcbMrcdsa
)1(2
)()(
)3)(2)(1(
))(()(
)1(
)()()(1
1
2
1
32
+
−++
+++
−+−+
+
−+−−
+++
β
λα
βββ
λα
β
λα βββ TMTrcbMrcdsaMMrcdsa
8
)()(
)2)(1(
)()( 422MrccdsaMMrcb +−
−++
−+−
+
ββ
λα β
15
)(
)3)(2)(1(
)()(
)4)(3)(2(
))((2
)2)(1(
)()(2 5343MrcbcMMrcbMrcbMMrcb +
−+++
−++
+++
−+−
++
−++
+++
βββ
λα
βββ
λα
ββ
λα βββ
]6
)(
4
)()(
)4)(3)(2)(1(
))((
)2)(1(2
)()( 23
1
22
1
42
1
2MTrcbcMTrccdsaMrcbTMrcb +
++−
−++++
−++
++
−+−
++
ββββ
λα
ββ
λα ββ
(14)
Hence, the total average cost of the system (TC1) is
given by
TC1= (SAC+H.C. + S.C. + L.C. + P.C. + I.P. - I.E. )/T
(15)
To minimize total average cost per unit time, the
optimal values of t1 and T can be obtained by solving
the following equations simultaneously.
0),(
1
11 =∂
∂
T
TtTC (16)
And
0),( 11 =
∂
∂
T
TtTC (17)
provided, they satisfy the following conditions:
0),(
,0),(
2
11
2
2
1
11
2
>∂
∂>
∂
∂
T
TtTC
T
TtTC
0),(),(),(
1
11
2
2
11
2
2
1
11
2
>
∂∂
∂−
∂
∂
∂
∂
TT
TtTC
T
TtTC
T
TtTC
(18)
International Journal of Pure and Applied Mathematics Special Issue
1314
Case II: When M > T1
4.8 Present Worth Interest Earned
I.E’= ∫∫−− −+−
11
0
1
0
14 ])()(])()([
T
rt
T
rt
e dtetDTMdtetDtTIC
= )())()(
1)((([{
)(
14c
bdsacS
cr
b
cr
be
rc
e
rcc
bdsacSTIC
rrc
e −−+−+−+
−+
−−+−+− λλ
(r
ebT
r
eTdsa
r
e
r
e
rc
b
rc
e
rc
e
rc
rTrrrrcrc 12
11
22
)(
2
)(
2
)({)}
1()
)()()(
1−−−−+−+−
−−
+−−−+
−+
−+
λλλλλ λλ
483
)()()(
22
1
43
12
12
1
2
111 λλλ
λλ λλ
bcTbcTrccdsacTdsa
r
ebT
r
ebT
r
eTbrrTr
−−+−
−−−+−+−−−
2
)()()()(22
3
)( 22
1
2
2
22
3 λλλλλ λλλλλTrccdsa
r
edsa
r
edsa
r
eb
r
eb
r
becdsarrrrr +−
+−
−−
−−−−−
−−−−−−
)3)(1(
)(
)1(
)(
)1(2
)()(
6224
5 1
3
1
3
1
1
1
2
1
1
1
3
1
3
1
4
1
++
−+
+
−+
+
−−++−+
+++
ββ
λα
β
λα
β
λαλλ βββTTbcTTbcTTcdsabcTbcTbcT
22
1
2
1
2
3
1
4
1
2
11111 22
)36
)(
)2)(1(
)()((
r
be
r
ebT
r
ebT
r
ecTTrccTcdsa
rTrTrTrT −−−−+
++++++
−++
−−−
ββ
λα β
)3)(2)(1(
)()(2
)1(2
)(
)3)(2(
)(
)2)(1(
)()((
3
11
1
1
3
1
3
1
2
11
+++
−++
+
−−
++
−−
++
−−+
++++
βββ
λα
β
λα
ββ
λα
ββ
λα ββββTTcbaTTbcTcTTc
dsa
r
eTMdsaTTbcTbc r )1)()((}
)2)(1(2
)(
)4)(3)(2)(1(
)(3 1
2
1
2
1
4
1 −−−−
++
−−
++++
−+
−++ λββ
ββ
λα
ββββ
λα
)}()()1
(){(22
1
221
111
r
e
r
e
r
e
r
eTb
r
e
r
ea
r
e
r
e
rbTM
rTrrrTrrTrr −−−−−−−−
+−−+−+−−−+λλλλλ λλ
2
)()(
)1
)()(
1(
)(
)()({
cr
be
cr
be
r
e
rrc
e
rcc
dsa
rc
Se
rc
SC
rrrrcrc λλλλλ λ −−−+−+−
−++−+
−+
−+
+−
++
22
)(()
22)([(]
)()(
2
1
2
11
)(
222
bcdsaT
TdsaC
rc
b
rc
be
rc
b
rc
be
cr
brcr
+−
++−−++
−+
++−−+− λ
λλλ
)2(
)()(
)3)(2)(1(
)(2
)2)(1(
)()
33
2(
2
1
3
1
2
11
32
1
3
1
+
−−+
+++
−+
++
−++−
+++
β
λα
βββ
λα
ββ
λαβλλ
βββTdsaTbTTb
TT
)2)(1(
)()(
)2)(1(
)()()
44
3(
3)1(
)( 2
11
2
1
43
1
4
1
3
1
++
−−−
++
−−−+−+
+
−−
+++
ββ
λα
ββ
λαλλ
β
λα βββ TcTdsaTdsaT
TbcTb
3
))((
)3)(2(3
)(2
)2)(1(
)(
)3)(2)(1(
)()( 33
11
3
1
2
1
3
1 λ
ββ
λα
ββ
λα
βββ
λα βββrcdsaTcTbTcTbTcdsa +−
−++
−−
++
−−
+++
−−+
+++
4
)(
8
)(
2
))((
6
))((
)4)(2)(3(3
)(2 22
1
4
1
2
1
3
1
4
1 λλ
βββ
λα β TrcbTrcbTrcdsaTrcdsaTcb ++
+−
+−+
+−−
+++
−+
+
International Journal of Pure and Applied Mathematics Special Issue
1315
)3)(2)(1(2
))(55)(()(
)1(2
))()(()(
8
)( 3
1
22
11
4
+++
−+++−+
+
−++−−
+−
++
βββ
λββα
β
λλαλ ββTrcdsaTTrcdsarcb
)3)(2)(1(
)()()72(
)2)(1(2
))()((3
)1(2
))(()( 3
11
3
11
2
111
+++
−++−
++
−+++
+
−++−
+++
βββ
λαβ
ββ
λλα
β
λλα βββTTrcbTTrcbTTTrcb
4
)()(
4
)()(
)4)(3)(2)(1(
))(()32( 22
1
4
1
4
1 λ
ββββ
λαβ βTrccdsaTrccdsaTrcb +−
++−
−++++
−+++
+
15
)(
6
)(
16
)(
8
)()( 523
1
5
1
4 λλλ rcbcTrcbcTrcbcrccdsa +−
++
+−
+−− (19)
Hence, the total average cost of the system (TC2) is
given by
TC2= (SAC+H.C.+S.C.+L.C.+P.C.-I.E.’)/T
(20)
To minimize the total average cost per unit time, the
optimal values of t1 and T can be obtained by solving
the following equations simultaneously.
0),(
1
12 =∂
∂
T
TtTC
(21)
And
0),( 12 =
∂
∂
T
TtTC
(22)
provided, they satisfy the following conditions:
0),(
,0),(
2
12
2
2
1
12
2
>∂
∂>
∂
∂
T
TtTC
T
TtTC
0),(),(),(
1
112
2
2
12
2
2
1
12
2
>
∂∂
∂−
∂
∂
∂
∂
TT
TtTC
T
TtTC
T
TtTC
(23)
5. Numerical Illustration
To illustrate the model numerically the following
parameter values are considered.
Demand rate a=25 units, b = 15 units, c = 9 units, d=2
Set up cost A= Rs. 200
Holding cost C1 = Rs. 4 per unit per year
Shortages cost C2= Rs. 9 per unit
Lost sale cost C3= Rs. 3 per unit per year
Backlogging rate δ = 0.5 unit.
Deterioration rate α =5, β = 0.02 units,
Interest earned I.E. = Rs. 0.2 per year,
C4 = Rs. 4.0 per unit, λ = 2 month
Interest payable I.P. = Rs. 0.15 per year,
For case I: permissible delay period is M=1
For case II: permissible delay period is M=10
International Journal of Pure and Applied Mathematics Special Issue
1316
Case I: When M ≤ T1
Table 1: Variation in Parameters
N T1 T Initial Inv.:
S
TC1
1 2.55467 4.8325 455.4465 24901.6
2 2.33081 4.7923 423.2992 23760.3
3 2.20953 4.4833 321.4844 23673.4
4 2.04875 4.2035 256.1098 22869
5 1.96539 3.9635 210.2748 22796.2
6 1.83514 3.5172 185.3554 21839.9
7 1.75162 3.0934 174.7435 21482.1
8 1.63516 2.8348 153.9355 20287.9
9 1.43804 2.6529 128.7299 19508.1
10 1.32355 2.4216 96.4258 18673.4
Fig 1: Variation in T1 and T w.r.t. N
Fig 2: Variation in S w.r.t. N
Fig 3: Variation in TC1 w.r.t. N
International Journal of Pure and Applied Mathematics Special Issue
1317
Table 2: Sensitivity Analysis of Optimal Solution
Variation
Parameter
Percentage Variation in Parameters
% -20% -10% 10% 20%
a
T1 0.1179 0.0745 -0.0742 -0.1187
S -19.764 -9.9102 9.9013 19.823
TC1 -19.0123 -9.4892 9.5154 19.0308
b
T1 -0.1329 -0.0587 0.0593 0.1335
S -0.1673 -0.0756 0.0763 0.1677
TC1 -0.55442 0.02776 0.02774 0.55483
Ie
T1 -2.2317 -0.6084 0.6084 1.2318
S -1.2355 -0.6102 0.6104 1.2358
TC1 -0.3511 -0.17564 0.17569 0.3513
r
T1 5.1378 2.5821 -2.5823 -5.1498
S 5.1652 2.588 -2.59 -5.1653
TC1 0.79524 0.38835 -0.38838 -0.79526
Fig 4: Variation in T1 w.r.t. a
Fig 5: Variation in S and TC1 w.r.t. a
Fig 6: Variation in T1 w.r.t. b
Fig 7: Variation in S and TC1 w.r.t. b
International Journal of Pure and Applied Mathematics Special Issue
1318
Case II: When M > T1
Table 3: Variation in Parameters
N T1 T Initial I. : S TC2
1 5.51618 15.1664 680.3743 15657.2
2 5.82763 14.9812 363.8103 15442.53
3 6.08592 14.2887 255.8913 14520.93
4 6.45351 13.9489 201.4149 14296.37
5 7.24855 13.0631 168.6791 13993.51
6 7.82498 12.8736 145.1577 13528.34
7 8.3447 11.5458 131.6471 13213.68
8 8.84915 10.7967 121.5399 12207.53
9 9.34876 10.0853 104.6293 12042.81
10 9.36723 9.6574 96.925 11793.84
Fig 8: Variation in T1 and T w.r.t. N
Fig 9: Variation in S w.r.t. N
Fig 10: Variation in TC1 w.r.t. N
International Journal of Pure and Applied Mathematics Special Issue
1319
Table 4: Percentage Variation in Different Parameters
Variation
Parameter
Percentage Variation in Parameters
% -20% -10% 10% 20%
A
T1 0.1175 0.0578 -0.0589 -0.1178
S -19.7894 -9.90233 9.91804 19.8242
TC2 -19.0254 -9.4893 9.5131 19.0263
b
T1 -0.147 -0.0732 0.0736 0.1473
S -0.18173 -0.09082 0.09087 0.18181
TC2 -0.05556 -0.02775 0.02787 0.05574
Ie
T1 -1.3258 -0.6774 0.6777 1.326
S -1.32034 -0.6793 0.6799 1.33042
TC2 -0.27867 -0.1672 0.16722 0.27871
r
T1 5.0387 2.5334 -2.5342 -5.039
S 5.05432 2.54217 -2.54218 -5.05432
TC2 0.85465 0.41802 -0.41806 -0.8547
Fig 11: Variation in T1 w.r.t. Ie
Fig 12: Variation in S and TC2 w.r.t. Ie
Fig 13: Variation in T1 w.r.t. r
Fig 14: Variation in S and TC2 w.r.t. r
International Journal of Pure and Applied Mathematics Special Issue
1320
5 Conclusion In this paper, we have attempted to develop a
deteriorating inventory model with a very realistic and
practical demand rate. In present scenario, where
market trends change to a large extent, it is crucial that
more than one trend in account is taken while
considering customer’s demand. The proposed model is
very useful in the present market situation as almost
every item can be identified as having a time, stock and
selling price dependent demand rate.
The inventory is allowed to deteriorate during the
period it is stored and during this time period it
undergoes Weibull deterioration rate. This deterioration
rate has the advantage as it is being able to account for
more than one factor affecting deterioration. These
different factors might be humidity, temperature, lack
of proper lighting, etc. Weibull rate can easily account
for these different kinds of factors.
However, in real life, most of goods would have a span
of maintaining quality and the original condition and
during that period, no deterioration occurs. The item is
allowed a definite life time since no article in real life
can be expected to start deteriorating as soon as it is
produced. Deterioration sets in afterwards and it has
been made more realistic and practical by taking two or
three parameter Weibull distribution function.
Shortfalls are allowed and partially backlogged with an
inflationary environment using DCF approach to
impact economic feasibility to the model. Large cycle
length is suggested by the presence of inflation in cost
and its impact on demand. This model is also
applicable when supplier gives the trade credit to the
customer.
Further cases for stochastic demand and in more
realistic conditions can be developed.
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