Modeling Of an Inventory System for Non-Instantaneous decaying Items with Partial Backlogging and Time Dependent Demand Rate under Permissible Delay
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The document presents a mathematical model for managing inventory systems of non-instantaneously decaying items with time-dependent demand and partial backlogging. It explores the implications of trade credit and varying costs, and formulates a system of differential equations to describe the inventory dynamics. The findings highlight the model's utility for retailers in optimizing order quantities to minimize inventory costs.
![Modeling of an Inventory System for Non-Instantaneous Decaying items with partial…
DOI: 10.9790/5728-11143841 www.iosrjournals.org 39 |Page
Also the solution of equation (3) by using (4) is given by
q t = −θa(t − t1)t1 ≤ t ≤ T…(9)
The holding cost during the period (0,𝐭 𝟐) is given by
HC =h[ q(t)
μ
0
e−rt
dt + q(t)e−rt
dt]
t1
μ
= h [
mμ3
3
−
cμ4
4
−
2cαμβ+4
β+2 β+4
−
bμ3
6
+
bαβ μβ+3
2 β+2 β+3
+
rcμ5
12
+
3cαrμβ+5
2 β+2 β+5
+
brμ4
8
+
cμ5
15
+
b−m αrμβ+4
β+2 β+4
−
αmμβ+3
2 β+3
+
β+2 αcμβ+4
2 β+1 β+4
+
at1
2
2
+
mt1
3
3
−
mμt1
22mαβt1
β+3
2 β+1 β+3
+
aαβ t1
β+2
β+1 β+2
−
mαβμβ+3
2(β+2)(β+3)
−
art1
3
6
−
mr t1
4
2
−
arαt1
β+3
2 β+3
−
β+6 mαrt1
β+4
2 β+2 β+4
−
mr μ4
8
+
mr t1
2μβ+1
β+1
]…(10)
The deterioration cost during the period (0,𝐭 𝟏) is given by
DC= d[ αβtβ−1
e−rt
q t dt + αβtβ−1
e−rtt1
μ
μ
0
q(t)dt]
=dαβ[
rcμβ+4
6 β+1
−
bμβ+2
2 β+2
−
cμβ+3 β+1
2β β+3
−
rm t1
2μβ+1
2 β+1
+
brμβ+3
2 β+3
+
cμβ+4
3 β+4
+
at1
β+1
β β+1
+
mt1
β+2
β β+2
+
mμβ+2
2(β+2)
−
art1
β+2
(β+2)(β+1)
−
mr t1
β+2
2(β+1)
+
mr t1μβ+1
2(β+1)
+
mr t1
β+3
2(β+3)
−
mr μβ+3
2(β+3)
]…(11)
The shortage cost during the period (𝐭 𝟏, 𝐓) is given by
SC= -s[ q(t)e−rt
dt
T
t1
]
=
sθa
r2 e−rt1 − e−rT
[rT 1 −
t1
T
+ 1] …(12)
The opportunity cost due to lost sales during the period (𝐭 𝟏, 𝐓) is given by
LSC= l a(1 − θ)
T
t1
e−rt
dt
=
la(1−θ)
r
[e−rt1 − e−rT
]…(13)
The ordering cost is given by
OC=A …(14)
The total cost of the system is given by
The total average cost of the inventory system per unit time is given by
CA t1,T =
OC + HC + DC + SC + LSC
T
=
A
T
+
h
T
[
mμ3
3
−
cμ4
4
−
2cαμβ+4
β+2 β+4
−
bμ3
6
+
bαβ μβ+3
2 β+2 β+3
+
rcμ5
12
+
3cαrμβ+5
2 β+2 β+5
+
brμ4
8
+
cμ5
15
+
b−m αrμβ+4
β+2 β+4
−
αmμβ+3
2 β+3
+
β+2 αcμβ+4
2 β+1 β+4
+
at1
2
2
+
mt1
3
3
−
mμt1
22mαβt1
β+3
2 β+1 β+3
+
aαβ t1
β+2
β+1 β+2
−
mαβμβ+3
2(β + 2)(β + 3)
−
art1
3
6
−
mrt1
4
2
−
arαt1
β+3
2 β + 3
−
β + 6 mαrt1
β+4
2 β + 2 β + 4
−
mrμ4
8
+
mrt1
2
μβ+1
β + 1
]
+
dαβ
T
[
rcμβ+4
6 β+1
−
bμβ+2
2 β+2
−
cμβ+3 β+1
2β β+3
−
rm t1
2μβ+1
2 β+1
+
brμβ+3
2 β+3
+
cμβ+4
3 β+4
+
at1
β+1
β β+1
+
mt1
β+2
β β+2
+
mμβ+2
2 β+2
−
art1
β+2
β+2 β+1
−
mr t1
β+2
2 β+1
+
mr t1μβ+1
2 β+1
+
mr t1
β+3
2 β+3
−
mr μβ+3
2 β+3
]
+
sθa
r2T
e−rt1 − e−rT
[rT 1 −
t1
T
+ 1]
+
la(1−θ)
rT
[e−rt1 − e−rT
] ….(15)
III. Approximation Solution Procedure
The total average cost has the two variables t1 and T. To minimize the total average cost , the optimal values of
t1 and T can be obtained by solving the following equations simultaneously
∂CA(t1,T)
∂t1
= 0 … … . (16)](https://image.slidesharecdn.com/f011143841-151126035906-lva1-app6892/85/Modeling-Of-an-Inventory-System-for-Non-Instantaneous-decaying-Items-with-Partial-Backlogging-and-Time-Dependent-Demand-Rate-under-Permissible-Delay-2-320.jpg)
![Modeling of an Inventory System for Non-Instantaneous Decaying items with partial…
DOI: 10.9790/5728-11143841 www.iosrjournals.org 40 |Page
∂CA(t1, T)
∂T
= 0 … . . (17)
Provided, they satisfy the following conditions
∂2CA (t1,T)
∂t1
2 > 0,
∂2CA (t1,T)
∂T2 > 0
(
∂2CA (t1,T)
∂t1
2 )(
∂2CA (t1,T)
∂T2 )−
∂2CA t1,T
∂t1 ∂T
> 0 ……(18)
The numerical solution of the equations (16), (17) and (18) can be obtained by using suitable computational
numerical method.
Numerical Example
a = 50, s =25, b = 20, h = 1, c = 10, d = 20, C = 200, l =10, μ = 1, α = .10, β = 3, r = .2,θ = .56, H = 10 in
appropriate units.
IV. Results
In this study one of the main objectives a volume flexible production inventory model is developed for
perishable items having linear demand. Then with the help of this model demand rate is taken as quadratic
function of time and production rate is decision variable and total inventory cost is obtained. The total cost
obtained than can be used to obtain a total inventory average cost, using by calculus techniques.
V. Discussion
The inventory model is developed for deteriorating items with time dependent demand Rate. The unit
inventory cost is depending upon material cost, labor cost and tool or die Cost. Shortages with partially
backlogged are allowed a very natural phenomenon in Inventory model. The suggested model and study will
help retailers in deciding their Optimal ordering quantity to have minimum inventory cost.
VI. Conclusion
This model is very most useful in their retail business. The result is our model, the fresh product time
increases the order quantity and total cost are decreases. It can we develop an inventory model with time
dependent deterioration, the effect of inflation and time value of money in formulating the inventory
replenishment policy.We have taken shortages in inventory are allowed and fully backlogged. It can be used for
electronic components, fashion apparel etc. Furtherwe have considered demand rate is an exponential increasing
function of time, the small change in time, demand is increasing a lot.
References
[1]. Chang H.J, Hung C.H., Dye C.Y. (2001) ‘An Inventory Model For Deterioration Items With Linear Trend Demand Under The
Condition Of Permissible Delay In Payment’, Production Planning & Control,12-3,274-282.
[2]. Goyal S.K. And Giri B.C. (2003) ‘The Production-Inventory Problem Of A Product With Time Varying Demand, Production And
Deterioration Rates’, European Journal Of Operational Research 147, 549-557.
[3]. Hou K.L. (2006) ‘An Inventory Model For Deteriorating Items With Stock-Dependent Consumption Rate And Shortages Under
Inflation And Time Discounting’, European Journal Of Operational Research 168, 463-474.
[4]. Mandal M. And Maiti M. (1999) ‘Inventory Of Damageable Items With Variable Replenishment Rate, Stock-Dependent Demand
And Some Units In Hand’, Applied Mathematical Modelling 23,799-809.
[5]. Mandal B.N. And Phaujdar S. (1989) ‘An Inventory Model For Deteriorating With Stock- Dependent Consumption Rate’,
Opsearch26, 43-46.
[6]. Mishra R.B. (1979) ‘A Note Of Optimal Inventory Management Under Inflation’, Navel Research Logistics Quarterly 26, 161-165.
[7]. Padmanabhan G. And Vrat P. (1995) ‘EOQ Models For Perishable Items Under Stock Dependent Selling Rate’, European Journal
Of Operational Research 86, 281-292.
[8]. Rau H.L., Wu M.Y. And Wee H.M. (2004) ‘Deteriorating Item Inventory Model With Shortage Due To Supplier In An Integrated
Supply Chain’, International Journal Of System Science 35(5), 293-303.
[9]. Sachan R.S. (1984) ‘On (T,Si) Policy Inventory Model For Deteriorating Items With Time Proportional Demand’, Journal Of The
Operational Research Society 35, 1013-1019.
[10]. Sana S. And Choudhari K.S. (2003) ‘An EOQ Model With Time-Dependent Demand Inflation And Money Value For Ware-House
Enterprises’, Advanced Modelling And Optimization 5, 135-146.
[11]. Sarkar B.R., Mukherjee S. And Balan C.V. (1997) ‘An Order-Level Lot Size Inventory Model Withinventory-Level Dependent
Demand And Deterioration’, International Journal Of Production Economics 48, 227-236.
[12]. Shah Y.K. (1997) ‘An Order-Level Lot Size Inventory Model For Deteriorating Items’, AIIE Transactions 9, 108-112.
[13]. Skouri K. And Papachristos S. (2003) ‘Optimal Stopping And Restarting Production Times For An EOQ Model With Deteriorating
Items And Time-Dependent Partial Backlogging’, International Journal Of Production Economics 81-82, 525-531.
[14]. Teng J.T., Chang H.J., Dye C.Y., And Hung C.H. (2002) ‘ An Optimal Replenishment Policy For Deteriorating Items With Time-
Varying Demand And Partial Backlogging’, Operations Research Letters, Vol. 30, No.6, Pp. 387-393.
[15]. Teng J.T. And Yang H.L. (2004), ‘ Deterministic Economic Order Quantity Models With Partial Backlogging When Demand And
Cost Are Fluctuating With Time’, Journal Of The Operational Research Society, 55(5), 495-503.](https://image.slidesharecdn.com/f011143841-151126035906-lva1-app6892/85/Modeling-Of-an-Inventory-System-for-Non-Instantaneous-decaying-Items-with-Partial-Backlogging-and-Time-Dependent-Demand-Rate-under-Permissible-Delay-3-320.jpg)
![Modeling of an Inventory System for Non-Instantaneous Decaying items with partial…
DOI: 10.9790/5728-11143841 www.iosrjournals.org 41 |Page
[16]. Uthayakumari R. And Geetha K.V. (2009) ‘A Replenishment Policy For Non-Instantaneous Deteriorating Inventory System With
Partial Backlogging’, Tamsui Oxford Journal Of Mathematical Sciences 25(3), 313-332.
[17]. Vasistha P.K. (2011) ‘An Inventory Model With Weibull Distribution Deterioration And Time Dependent Demand’, International
Journal Of Advances In Engineering Research 2.
[18]. Wee H.M. (1997) ‘A Replenishment Policy For Items With A Price-Dependent Demand And A Varying Rate Of Deterioration’,
Production Planning And Control 18(5), 494-499.](https://image.slidesharecdn.com/f011143841-151126035906-lva1-app6892/85/Modeling-Of-an-Inventory-System-for-Non-Instantaneous-decaying-Items-with-Partial-Backlogging-and-Time-Dependent-Demand-Rate-under-Permissible-Delay-4-320.jpg)
Modeling Of an Inventory System for Non-Instantaneous decaying Items with Partial Backlogging and Time Dependent Demand Rate under Permissible Delay
- 1. IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. IV (Jan - Feb. 2015), PP 38-41 www.iosrjournals.org DOI: 10.9790/5728-11143841 www.iosrjournals.org 38 |Page Modeling Of an Inventory System for Non-Instantaneous decaying Items with Partial Backlogging and Time Dependent Demand Rate under Permissible Delay Ruchika Tyagi1 and Prashant Chouhan2 IFTM University, Moradabad, Uttar Pradesh, India Abstract: We will discuss an inventory model is investigates with variable demand rate and time dependent deteriorating items.In this study, we have taken shortages in inventory are allowed and fully backlogged. This model is studied under the condition for decaying items of permissible delay in payments which is most important and an outcome of interaction between product and financial markets which arises. This model based on time-dependent, holding cost, shortages cost and the combination of model is unique and practical. I. Introduction Trade credit would play an important role in the conduct of business for many reasons. For a supplier who offers trade credit, it is also an efficient method to stimulate the demand of the product. For a retailer, it is an efficient method of bonding a supplier when the retailer is at the risk of receiving inferior quality goods or service and is also an effective means of reducing the cost of holding stocks. Teng and Yang (2004) developed a deterministic economic order quantity models with partial backlogging when demand and cost are fluctuating with time. Chang, et al. (2001) developed an inventory model for deteriorating items with linear trend demand under the condition of permissible delay in payments. Teng et al. (2002) has discussed inventory model for deteriorating items with time varying demand and partially backlogging. II. Formulation And Solution Of The Model To discussan inventory model with the same assumptions as adopted by Vashistha (17), except the time dependent demand, the inflation and time-discounting. The inventory system is governed by the following differential equations in the interval (0, T) are q′ t + αβtβ−1 q t = − a + bt + ct2 , 0≤ t ≤ μ…(1) q′ t + αβtβ−1 q t = − a + b + cμ t , μ ≤ t ≤ t1…(2) And q′ t = −aθ, t1 ≤ t ≤ T…(3) With the condition q (0) =S and q (t1) =0 …(4) Solution of equations (1) and (2) by using (4) are giving by q t = at + b t2 2 + c t3 3 + aαtβ+1β+1+bαtβ+2β+2+cαtβ+3β+3e−αtβ+Se−αtβ0≤t≤μ …(5) q t = at1 + m t1 2 2 + aα t1 β+1 (β + 1) + mα t1 β+2 (β + 2) − at − m t2 2 − aα tβ+1 β + 1 − mα tβ+2 β + 2 )e−αtβ μ ≤ t ≤ t1 … (6) From (5) and (6), we get S= at1 + m t1 2 2 + aα t1 β+1 (β+1) + mα t1 β+2 (β+2) − c μ3 6 − cα μβ+3 β+2 β+3 )…(7) Using (7) and (5) becomes q t = at1 + m t1 2 2 + aα t1 β+1 (β+1) + mα t1 β+2 (β+2) − c μ3 6 − cα μβ+3 β+2 β+3 − at − b t2 − c t3 3 − aα tβ+1 β+1 − bα tβ+2 β+2 − cα tβ+3 β+3 )e−αtβ 0 ≤ t ≤ μ…(8)
- 2. Modeling of an Inventory System for Non-Instantaneous Decaying items with partial… DOI: 10.9790/5728-11143841 www.iosrjournals.org 39 |Page Also the solution of equation (3) by using (4) is given by q t = −θa(t − t1)t1 ≤ t ≤ T…(9) The holding cost during the period (0,𝐭 𝟐) is given by HC =h[ q(t) μ 0 e−rt dt + q(t)e−rt dt] t1 μ = h [ mμ3 3 − cμ4 4 − 2cαμβ+4 β+2 β+4 − bμ3 6 + bαβ μβ+3 2 β+2 β+3 + rcμ5 12 + 3cαrμβ+5 2 β+2 β+5 + brμ4 8 + cμ5 15 + b−m αrμβ+4 β+2 β+4 − αmμβ+3 2 β+3 + β+2 αcμβ+4 2 β+1 β+4 + at1 2 2 + mt1 3 3 − mμt1 22mαβt1 β+3 2 β+1 β+3 + aαβ t1 β+2 β+1 β+2 − mαβμβ+3 2(β+2)(β+3) − art1 3 6 − mr t1 4 2 − arαt1 β+3 2 β+3 − β+6 mαrt1 β+4 2 β+2 β+4 − mr μ4 8 + mr t1 2μβ+1 β+1 ]…(10) The deterioration cost during the period (0,𝐭 𝟏) is given by DC= d[ αβtβ−1 e−rt q t dt + αβtβ−1 e−rtt1 μ μ 0 q(t)dt] =dαβ[ rcμβ+4 6 β+1 − bμβ+2 2 β+2 − cμβ+3 β+1 2β β+3 − rm t1 2μβ+1 2 β+1 + brμβ+3 2 β+3 + cμβ+4 3 β+4 + at1 β+1 β β+1 + mt1 β+2 β β+2 + mμβ+2 2(β+2) − art1 β+2 (β+2)(β+1) − mr t1 β+2 2(β+1) + mr t1μβ+1 2(β+1) + mr t1 β+3 2(β+3) − mr μβ+3 2(β+3) ]…(11) The shortage cost during the period (𝐭 𝟏, 𝐓) is given by SC= -s[ q(t)e−rt dt T t1 ] = sθa r2 e−rt1 − e−rT [rT 1 − t1 T + 1] …(12) The opportunity cost due to lost sales during the period (𝐭 𝟏, 𝐓) is given by LSC= l a(1 − θ) T t1 e−rt dt = la(1−θ) r [e−rt1 − e−rT ]…(13) The ordering cost is given by OC=A …(14) The total cost of the system is given by The total average cost of the inventory system per unit time is given by CA t1,T = OC + HC + DC + SC + LSC T = A T + h T [ mμ3 3 − cμ4 4 − 2cαμβ+4 β+2 β+4 − bμ3 6 + bαβ μβ+3 2 β+2 β+3 + rcμ5 12 + 3cαrμβ+5 2 β+2 β+5 + brμ4 8 + cμ5 15 + b−m αrμβ+4 β+2 β+4 − αmμβ+3 2 β+3 + β+2 αcμβ+4 2 β+1 β+4 + at1 2 2 + mt1 3 3 − mμt1 22mαβt1 β+3 2 β+1 β+3 + aαβ t1 β+2 β+1 β+2 − mαβμβ+3 2(β + 2)(β + 3) − art1 3 6 − mrt1 4 2 − arαt1 β+3 2 β + 3 − β + 6 mαrt1 β+4 2 β + 2 β + 4 − mrμ4 8 + mrt1 2 μβ+1 β + 1 ] + dαβ T [ rcμβ+4 6 β+1 − bμβ+2 2 β+2 − cμβ+3 β+1 2β β+3 − rm t1 2μβ+1 2 β+1 + brμβ+3 2 β+3 + cμβ+4 3 β+4 + at1 β+1 β β+1 + mt1 β+2 β β+2 + mμβ+2 2 β+2 − art1 β+2 β+2 β+1 − mr t1 β+2 2 β+1 + mr t1μβ+1 2 β+1 + mr t1 β+3 2 β+3 − mr μβ+3 2 β+3 ] + sθa r2T e−rt1 − e−rT [rT 1 − t1 T + 1] + la(1−θ) rT [e−rt1 − e−rT ] ….(15) III. Approximation Solution Procedure The total average cost has the two variables t1 and T. To minimize the total average cost , the optimal values of t1 and T can be obtained by solving the following equations simultaneously ∂CA(t1,T) ∂t1 = 0 … … . (16)
- 3. Modeling of an Inventory System for Non-Instantaneous Decaying items with partial… DOI: 10.9790/5728-11143841 www.iosrjournals.org 40 |Page ∂CA(t1, T) ∂T = 0 … . . (17) Provided, they satisfy the following conditions ∂2CA (t1,T) ∂t1 2 > 0, ∂2CA (t1,T) ∂T2 > 0 ( ∂2CA (t1,T) ∂t1 2 )( ∂2CA (t1,T) ∂T2 )− ∂2CA t1,T ∂t1 ∂T > 0 ……(18) The numerical solution of the equations (16), (17) and (18) can be obtained by using suitable computational numerical method. Numerical Example a = 50, s =25, b = 20, h = 1, c = 10, d = 20, C = 200, l =10, μ = 1, α = .10, β = 3, r = .2,θ = .56, H = 10 in appropriate units. IV. Results In this study one of the main objectives a volume flexible production inventory model is developed for perishable items having linear demand. Then with the help of this model demand rate is taken as quadratic function of time and production rate is decision variable and total inventory cost is obtained. The total cost obtained than can be used to obtain a total inventory average cost, using by calculus techniques. V. Discussion The inventory model is developed for deteriorating items with time dependent demand Rate. The unit inventory cost is depending upon material cost, labor cost and tool or die Cost. Shortages with partially backlogged are allowed a very natural phenomenon in Inventory model. The suggested model and study will help retailers in deciding their Optimal ordering quantity to have minimum inventory cost. VI. Conclusion This model is very most useful in their retail business. The result is our model, the fresh product time increases the order quantity and total cost are decreases. It can we develop an inventory model with time dependent deterioration, the effect of inflation and time value of money in formulating the inventory replenishment policy.We have taken shortages in inventory are allowed and fully backlogged. It can be used for electronic components, fashion apparel etc. Furtherwe have considered demand rate is an exponential increasing function of time, the small change in time, demand is increasing a lot. References [1]. Chang H.J, Hung C.H., Dye C.Y. (2001) ‘An Inventory Model For Deterioration Items With Linear Trend Demand Under The Condition Of Permissible Delay In Payment’, Production Planning & Control,12-3,274-282. [2]. Goyal S.K. And Giri B.C. (2003) ‘The Production-Inventory Problem Of A Product With Time Varying Demand, Production And Deterioration Rates’, European Journal Of Operational Research 147, 549-557. [3]. Hou K.L. (2006) ‘An Inventory Model For Deteriorating Items With Stock-Dependent Consumption Rate And Shortages Under Inflation And Time Discounting’, European Journal Of Operational Research 168, 463-474. [4]. Mandal M. And Maiti M. (1999) ‘Inventory Of Damageable Items With Variable Replenishment Rate, Stock-Dependent Demand And Some Units In Hand’, Applied Mathematical Modelling 23,799-809. [5]. Mandal B.N. And Phaujdar S. (1989) ‘An Inventory Model For Deteriorating With Stock- Dependent Consumption Rate’, Opsearch26, 43-46. [6]. Mishra R.B. (1979) ‘A Note Of Optimal Inventory Management Under Inflation’, Navel Research Logistics Quarterly 26, 161-165. [7]. Padmanabhan G. And Vrat P. (1995) ‘EOQ Models For Perishable Items Under Stock Dependent Selling Rate’, European Journal Of Operational Research 86, 281-292. [8]. Rau H.L., Wu M.Y. And Wee H.M. (2004) ‘Deteriorating Item Inventory Model With Shortage Due To Supplier In An Integrated Supply Chain’, International Journal Of System Science 35(5), 293-303. [9]. Sachan R.S. (1984) ‘On (T,Si) Policy Inventory Model For Deteriorating Items With Time Proportional Demand’, Journal Of The Operational Research Society 35, 1013-1019. [10]. Sana S. And Choudhari K.S. (2003) ‘An EOQ Model With Time-Dependent Demand Inflation And Money Value For Ware-House Enterprises’, Advanced Modelling And Optimization 5, 135-146. [11]. Sarkar B.R., Mukherjee S. And Balan C.V. (1997) ‘An Order-Level Lot Size Inventory Model Withinventory-Level Dependent Demand And Deterioration’, International Journal Of Production Economics 48, 227-236. [12]. Shah Y.K. (1997) ‘An Order-Level Lot Size Inventory Model For Deteriorating Items’, AIIE Transactions 9, 108-112. [13]. Skouri K. And Papachristos S. (2003) ‘Optimal Stopping And Restarting Production Times For An EOQ Model With Deteriorating Items And Time-Dependent Partial Backlogging’, International Journal Of Production Economics 81-82, 525-531. [14]. Teng J.T., Chang H.J., Dye C.Y., And Hung C.H. (2002) ‘ An Optimal Replenishment Policy For Deteriorating Items With Time- Varying Demand And Partial Backlogging’, Operations Research Letters, Vol. 30, No.6, Pp. 387-393. [15]. Teng J.T. And Yang H.L. (2004), ‘ Deterministic Economic Order Quantity Models With Partial Backlogging When Demand And Cost Are Fluctuating With Time’, Journal Of The Operational Research Society, 55(5), 495-503.
- 4. Modeling of an Inventory System for Non-Instantaneous Decaying items with partial… DOI: 10.9790/5728-11143841 www.iosrjournals.org 41 |Page [16]. Uthayakumari R. And Geetha K.V. (2009) ‘A Replenishment Policy For Non-Instantaneous Deteriorating Inventory System With Partial Backlogging’, Tamsui Oxford Journal Of Mathematical Sciences 25(3), 313-332. [17]. Vasistha P.K. (2011) ‘An Inventory Model With Weibull Distribution Deterioration And Time Dependent Demand’, International Journal Of Advances In Engineering Research 2. [18]. Wee H.M. (1997) ‘A Replenishment Policy For Items With A Price-Dependent Demand And A Varying Rate Of Deterioration’, Production Planning And Control 18(5), 494-499.