inventory model (q, r) with period of grace, quadratic backorder cost and continuous lead time
TRANSCRIPT
American Journal of Engineering Research (AJER) 2015
American Journal of Engineering Research (AJER)
e-ISSN: 2320-0847 p-ISSN : 2320-0936
Volume-4, Issue-5, pp-81-93
www.ajer.org Research Paper Open Access
w w w . a j e r . o r g
Page 81
Inventory Model (Q, R) With Period of Grace, Quadratic
Backorder Cost and Continuous Lead Time
Dr. Martin Osawaru Omorodion Joseph Ayo Babalola University, Ikeji-Arakeji, Nigeria.
ABSTRACT: The paper considers the simple economic order model where the period of grace is operating, the
lead time is continuous and the backorder cost is quadratic. The lead time follows a gamma distribution. The
expected backorder cost per cycle is derived and averaged over all the states of the lead time L. Next we obtain
the expected on hand inventory. The Lead time is taken as a Normal variate. The expected backorder costs are,
derived after which the expected on hand inventory is derived. The cost inventory costs for constant lead times is
then averaged over the states of the lead times which is taken as a normal distribution.
KEY WORDS: Continuous Lead Times, Gamma Distribution, Normal Distribution, Period of Grace, Bessel
Functions, Inventory on hand, Expected Backorder Costs and Chi-Squared Distribution.
Demand
The cost of a backorder )(tC for backorder of length t = )( pzLC =
2
321 )()( pzLbpzLbb where p is the grace period, where the system is out of stock in the
time interval z to z tdz. After the re-order point is reached the R+Y was demanded in the time z and a demand
occurred in time dz. The longer the period of grace the greater the reduction in inventory costs.
SIMPLE INVENTORY MODEL WITH PERIOD OF GRACE, QUADRATIC BACKORDER COST
AND CONTINUOUS LEAD TIME
I. INTRODUCTION
In this inventory model there is a period of grace before backorder costs are incurred. In Hadley (1972),
the general model was stated. This paper derives the backorder costs when the backorder cost is a quadratic cost
depending upon the length of time of backorder after the grace period. The various costs such as expected
backorder costs and expected on hand inventory are derived, to arrive at the total costs of holding inventory.
Demand over the lead time is normally distributed and lead time is a gamma variate.
II. LITERATURE REVIEW Saidey (2001), in his paper Inventory Model with composed shortage considered grace period (perishable delay
in payment) before setting the account with the supplier or producer.
Schemes (2012) also considered credit terms which may include an interest free grace period as much as 30
days in his paper ‘Inventory models with shelf Age an Delay Dependent Inventory costs’.
Hadley and Whitin gave a simple general model for period of grace.
Zipkin (2006), treats both fixed and random lead times and examines both stationary and limiting distributions
under different assumptions.
III. DERIVATION Let the period of grace be p, where the period of grade is the period for which a backorder bears no cost. Let
H(L) be the probability density function of the lead time L.
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)(tC is the cost of a backorder of length t.
H(L) = )(
)(1
k
LespLkk
> 0…………….. (1)
In the analysis k would take on integer values only.
The p.d.f. of demand over the lead time L.
g
L
L
DLxesp
L
DLx
2
2
2
2
2
1
2
x ………………… (2)
Let R + Y, 0 <Y<Q be the inventory position at time O, then if the system is out of stock in the time internal z to
z + dz after the re-order point is reached then R +Y was demanded in the time Z and a demand occurred in time
dz.
Length of time of backorder = L-z length of time of backorder which bears a cost
= L – z – p.
Cost of a backorder = C (L – z – p)
Where the C (t) is the cost of a backorder = b1 + b2 (L – z – p) + b3 (L – z – p)2. (3)
Expected backorder cost per cycle G1 (Q, R).
= D dzdydLL
DzYRgLH
L
pzLCPLQ
).().(.)(
20 200
…………….. (4)
Let v = z + p
G1 (Q,R) = D dvdYdL
L
DvDpYRgLH
L
vLCLQ
)()()(
0 2200
……………...(5)
Making use of equation (2)
G1(Q,R) = D dvdYdLL
DvDpYRespLH
L
vLCLQ
)(2
1)(.
2
)(
20 200
…. (6)
Substituting for C (L-v) from 3
G1(Q,R) = LQ
D000
(b1+b2(L – v) + b3 (L – v)2 + H(L) ………………..(7)
esp -1/2
L
DVDPYR
2
)(
dvdYdL
Noting that
D dvL
DvDpYResp
L
vLbvLbb
s
L2
2
2
321
0)(2/1
)()((
……………. (8)
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=
L
DLDpYRF
L
DLDpYR
D
bL
L
DLDpYR
D
Lb
D
Lbb
22
2
2
2
22
2
3
2
2
31 )((
L
DLDpYRg
L
DLDpYR
D
Lb
D
Lb
222
2
32
2
………………... (9)
Substituting into G1(Q,R) of (7) and changing the range of Y
=
dxdLL
DLxF
L
DLx
D
Lb
L
DLx
D
Lb
D
LbbLH
DpQR
DpR 22
222
22
2
3
2
2
31
0)(
= dxdLL
DLxg
L
DLx
D
LbLbLH
DpQR
DpR
222
2
32
2
0)(
………. (10)
Let G1(Q,R) =
DpQR
DpRdxxG )(2 ……………………………………………… (11)
Hence from equation (10)
G2 (x) = dLL
DLxFDLx
D
b
L
DLx
D
Lb
D
LbbLH
2
2
22
2
32
2
31
0)(
- dLL
DLxFDLx
D
b
2
2
dLL
DLxg
L
DLx
D
Lb
D
LbLH
222
2
3
2/1
0
0)(
………………. (12)
Substituting for H (L) from 1 and simplifying
G2 (x) = 2
12
3
2
2
3
1
01(
)( D
Lxb
D
LbLb
K
kkK
k
)2 2
1
2
2
1
33 k
kkK
LbD
xLb
D
Lb
D
xLb
esp ( L ) F dLL
DLx
2
0
2/1
32
2/1
3
2
2/1
D
Lbx
D
Lb
D
bL
k
kkkk esp ( L)g dL
L
DLx
2……. (13)
Re-arranging terms
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G2 (x) =
D
xb
D
xbbL
k
kk
2
2
2
3
1
1
)(22
1
3
2
3
2
2
3 LespD
Lbb
D
xb
D
bL
kk
2
32
2/1
2 D
xb
D
bL
kdL
L
DLxF
kk
dLL
DLxgLesp
D
Lb k
2
2/1
3 )(
……………………………….……….. (14)
Integrating
dLL
DLxg
kL
LLespdL
L
DLxgLH
L
kk
20 2
1
20 2 .
)()(
1
=
2
22
2
2
20
2/3
2
2
22
DL
L
xesp
DxespL
k
kk
=
2/122
22/1
)2/1(2/1
22
22
22
22
Dx
KD
x
k
Dx
espk
Kk
If k is an integer then
jk
j
k zjkj
jkzkzK
)2(
)!1(!
1)()(
1
0
2/12/1
Where )()(2
)( 2/1
2/1 zespzzK
Hence )()2()!1(!
)1()( 2/1
1
0
2/1 Zespzjkj
jkzK j
k
j
k
And Letting
2 = 2 2
+D2
G2 (x) = esp
D
xb
D
xbb
k
Dx k2
2
2
3
122)(
2
2/12/1
12
2)!(
)1(
x
zkZk
k
z
Kx
Dzk
k
2
2
2/1
2/12
xK
xx
zk
zk
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1
1
2/3
2
3
2
2
3 2)!1(
!2 k
z
zk
z
xD
zk
kb
D
xb
D
b
2
2/1
2/1
2/3
2
xK
xx
xK
Zk
zk
Zk
22/11
2/11
22/21
2/212
12
3 22)!2(
)!1(
xK
xx
xK
xD
zk
k
D
b
zk
zk
zk
zkk
zz
22/11
2/113
22/1
2/1
2
322 2
22)(
xK
x
D
bxK
x
D
xb
D
b
k
Dxesp
k
k
k
k
k
Re-arranging terms
k
z
zk
k
xD
zk
k
k
b
Dxesp
xG1
2/1
2
12
2 2)!(
)!1(
)(2)(
2
2/1
2/1
22/1
2
xK
xx
xK
zk
zk
zk
xK
xD
zk
k
D
b
k
Dxesp
zk
zkk
zz
k
2/1
2/11
1
22
.2)!(
)!1(
)(22
22/1
2/112
xK
x
zk
zk
22/1
2/1121
1
2()!1(
!
xK
xD
zk
k
zk
zk
z
k
z
22/1
2/112
xK
xD
zk
zk
22/11
2/11222
xK
x
k
k
22/1
2/21
222/120
2/21232 22
)!(
)!1(
)(22
xK
x
D
xK
x
Dzk
k
k
bDx
esp
zk
zk
k
k
z
zk
z
k
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22/112
2/11211
1
2
)!1(
!
xK
x
Dzk
kk
zkk
zz
22/1
2/11
2
22
xK
x
D Zk
zk
1
12
2/1
2/21
2
4
)!1(
!k
z zk
zk
z
xK
x
Dzk
k
22/1
2/2124
xK
x
D zk
zk
22/11
2/21
222/21
2/212
1
22
)!2(
)!1(
xK
x
D
xK
x
Dzk
k
zk
zk
zk
zkk
zz
22/11
2/11
222/1
2/11
2
2
22)(
2
xK
x
D
xK
x
Dk
Dxesp
k
k
k
k
k
(16)
From equation 11
DpQR
DpRdxxGRQG )()( 211
Where G1(Q,R) is the expected backorder cost per cycle
Expanding
DpQR
DpQR
DpRdxxGdxxGRQG )()(),( 221 …………………..…….……….(17)
Integrating G2(x) with respect to x from z to and applying
dL
L
DLxgLH
2)(
= dLk
LespL
L
DLxesp
L
kk
)(
)(2/1
2
112
20 2
………………… (18)
dxxGQ
)(2
and letting
2
D
=
zk
j
k
zz
k
jzkj
jzkD
zk
k
k
b
01
1
)!(
)!(2
)!(
)1(
)(4
2
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)!()(
1
)1(2
2.
21
2
2jzk
jzk
Qzk
j
1(2
22
)(
1
)!1(
)!1( 2
21
1
0 jzk
Q
jzkj
jzk j
zk
zk
j
zk
jz
k
z
k
jzkj
jzkD
zk
k
kD
b
01
2
)!(
)!(
)!(
)!1(
)(4
2
)1(2
221 2
22 jzk
Qj
zk
)1(2
22
)(
)!(
)!1(!
)!1( 2
22
1
0 jzk
Qjzk
jzkj
jzk j
zk
zk
j
jzk
jzk
k
zz
jzk
jzkj
jzkD
zk
k2
1
02
21
1
2
)(
)!1(
)!1(!
)!1(
)!1(
!
20
2
)(
)!1(
)!(!
)!(2
2(2
2
zk
zk
j
jzk
jzkj
jzkD
jzk
Q
)2(2
22 2
2 jzk
Qj
)1(2
22
)(
)!(
)!(!
)!(2
2
210
2
jk
Qjk
jkj
jk j
k
k
j
k
z
zk
jz
k
jzkj
jzk
Dzk
k
k
b
0 0
23
)!(!
)!(2
)!(
)!1(
)(2
)3(2
22
)(
)!2( 2
23 jzk
Qjzk j
zk
)3(2
22
)(
)!2(
)!1(!
)1(2 2
23
1
02
3
jzk
Qjzk
jzkj
jzk
D
j
zk
zk
j
1
1
1
02
22
2
21)!1(
)!1(!
)!1(2
)!1(
!k
z
jzk
j
zk
jzkjzkj
jzk
Dzk
k
)2(2
22
jzk
Q
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20
2
2
)(
1)!1(
)!(!
)!(2
zk
zk
j
jzkjzkj
jzk
D
1
1
2
02
2 4
)!1(
!
)2(2
2 k
z
zk
jz Dzk
k
jzk
Q
j
zk
jzk
jzkj
jzk
23
2
)(
1)!2(
)!2(!
)!2(
1
0
22
)!1(!
)!1(4
)3(2
2 zk
j jzkj
jzk
Djzk
Q
)3(2
221)!2(
2
2
3
jzk
Qjzk
jzk
)!2()!2(!
)!2(2
)!2(
)!1( 2
0
2
1
jzkjzkj
jzk
Dzk
k zk
j
k
zz
)3(2
221 2
2
3
jzk
Qj
zk
j
zkzk
j jzkj
jzk
D
2
31
02
21
)!1(!
)!2(2
)3(2
22
jzk
Q
)2(2
22
)(
)!1(
)!1(!
)!1(2 2
22
1
0
2
jk
Qjk
jkj
jk
D
j
k
k
j
)2(2
22
)(
)!1(
)!(!
)!(2 2
220
2
jk
Qjk
jkj
jk
D
j
k
k
j
……………………(19)
Let G3 (z) equal the b1 factor of (19) ………………………………………….(20)
Let G4 (z) equal the b2 factor of (19) …………………………………………..(21)
Let G5 (z) equal the b3 factor of (19) …………………………………………..(22)
Hence from (17)
The expected backorder cost per cycle
= b1 (G3(R + Dp) – G3 (R + Q+ Dp) + b2 (G4 (R + Dp)
- G4 (R +Q +Dp) + b3 (G5 (R + Dp) – G5 (R + Q +Dp)) ………………..…(23)
Hence the expected cost per year
=
)(())(
)(()()(
55
3
4
4
2
33
1
DpQRGDpRGQ
bDpQRG
DpRGQ
bDpQRGDpRG
Q
b
………………… (24)
Next we obtain the expected on hand inventory at any time.
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Let (D, R) be the expected on hand inventory at any time
Let G6 (x) be the probability density function of the quantity on hand x at anytime t. x control be less or above
the re-order level R. 0<x<R
R QR
RdxxxGdxxxGRQD
066 )()(),(
If x lies above R
Where dxL
DLxvesp
QxG
QR
R
2
26 2/1
1)(
If x lies above R
dvt
DLxvesp
QxG
RR
206
2
11)(
Given that the system was in state v and v-x was demanded.
Which gives, expressing in k standard deviations of stock
RxL
xQkF
L
xkF
QxG
0
)((
1)(
221
QRxRL
xQkF
QxG
21
)(1
1)(
Hence D(Q, R)=
R
dxL
DLxQRxF
L
DLxRxF
Q 0 22
1
dxL
DLxQRxFx
Q
QR
R
2
1
Simplifying
dxL
DLxQRxF
Qdx
L
DLxRxF
QRQD
QRR
0 20 2
11),(
QR
Rxdx
Q
1
Let V=
L
DLxR
2
and
L
DLxQR
2
In each integral
Then dVVFDLRVLQ
LRQD L
DL
L
DLRR
)())( 2
2
22
1
L
DL
L
DLQRQ
RQRdVDLQRLV
Q
L2
22
)()(
222
2
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Remembering that
L
DLRk
2
Then expressing R in terms of standard deviations of stock
L
DL
L
QkL
DL
k Q
LdVVFkV
Q
LKQD 2
2
2)(
)()(),(22
LKQ
dVVFL
QkV 2
2 2)()((
Simplifying
L
DL
kL
DL
kdVVkF
Q
LdVVVF
Q
LKQD 22 )(
)()(),(
22
L
DL
L
QkL
DL
L
QkdVVF
L
Qk
Q
LdvVvF
Q
L2
2
2
2
)()()(
)(2
22
LkQ 2
2
Hence D(Q, k), noting that F 12
L
DL
and
k
kgkkFdvvF )()()(
and
kkkgkFkdvvvF )()()1(2/1)( 2
Hence D(Q, k)
)()()1(
2)()(( 2
2
222
2
kkgkFkQ
L
L
DLg
L
DLF
L
DLkgkFk
Q
L
L
QKF
L
Qk
Q
L
L
DLg
L
DLF
L
DL
22
2
22
2
21
LkQ
L
DLg
L
DLF
L
DL
L
Qkg
L
Qk 2
22
2
222 21
2
Which gives
)()()1(22
),( 22
2 kkgkkLQ
LkkQD
L
QkF
L
QkL
22
22 )(1
2
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L
Qkg
L
Qk
22 …………………………………………………….(25)
Remembering that k =
L
DLR
2then
Substituting for k then
2
2
2
122
),,(L
DLRLDLR
QLRQD
2
2
2
2221
2
L
DLQRL
L
DLRg
L
DLR
L
DLRF
L
DLQRg
L
DLQR
L
DLQRF
222
But
L
DLRg
L
DLR
L
DLRF
L
DLRL
222
2
2
2
12
………………..…(26)
dVL
DLVFDLV
L
DLVgL
R
22
2 )(
…………………………….…(27)
Hence
L
DLVgLDLR
QLRQD
QR
R 2
2
2),,(
dVL
DLVFDLV
2)(
Where D(Q,R,L) is the inventory on hand for a given lead time L.
Hence expected on hand inventory
=
0),,()( DLLRQDLH ……………………………………………………..….(28)
0 0
2
0)(
1)()(
2
QR
RLLH
QdLLHLDDLLHR
Q
dVdL
L
DLVFDLV
L
DLVg
22)(
……………………………………(29)
Noting that
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0 22
2 )()( dVdLL
DLVFDLV
L
DLVgLLH
QR
Ris the coefficient of b2 in (iv)
Then integrating (29)
We have
)()(1
2),( 44 QRGRG
Q
DkR
QRQD
Cost of ordering =
Q
DS
Inventory costs for mode (Q1R)
)()((
1)( 44
4
1
1 QRGRGQ
DpRGQ
bRQhcD
Q
DSC
Cost of ordering = Q
DS……………………………………………………………(30)
Inventory costs for mode (Q1R)
DpQRGDpRGQ
bRQhcD
Q
DSC ())( 33
1
1
DpQRGDpRGQ
b()( 44
2 )()((553 DpQRGDpRGb …….(31)
Substituting for D(Q1R) from (30)
DpQRGDpRGbDk
RhcQhc
Q
DSC ())(
2331
))()(()()( 55344
2 DpQRGDpRGbDpQRGDpRGQ
b
)()(( 44 QRGRGQ
hc ……………………………………….…….(34)
The corresponding cost when no period of grace p is operating is
QRGRGbDk
RhcQhc
Q
DSC ())(
2331
)()(()()( 55
3
44
2 QRGRGQ
bQRGRG
Q
bhc
IV. IMPACT OF THE STUDY The study will enable industries or organizations having thousands of items in their warehouses located
in various locations of the world and items supplied by different manufacturers with accurately use realistic lead
times in arriving at their inventory cost.
In many of such cases lead times are not constant.
Expressing lead time as continuous gives a more realistic estimate of inventory costs.
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REFERENCES [1]. Hadley and Whitin, (1972), Analysis of Inventory System, John Wiley and Sms, Inc.
[2]. Saidey Hesam, (2011), “Inventory model with composed shortage and permissible delay in payments linked to order quantity, Journal of Industrial Engineering International, (15), Ed 1 – 7 Fall 2011.
[3]. Schemes. F., (2012), ‘Inventory model with Shelf Age and Delay Dependent Inventory Costs, Academic Commons. Columbia. Edu.
[4]. Zipkin P., (2006), ‘Stochastic lead times in continuous – time inventory models’ Naval Research Logistics, 21 Nov. 2006.