ISSN (e): 2250 – 3005 || Volume, 06 || Issue, 01 ||January – 2016 || International Journal of Computational Engineering Research (IJCER)
EOQ Inventory Models for Deteriorating Item with Weibull Deterioration and Time-Varying Quadratic Holding Cost 1
Naresh Kumar Kaliraman, 2Ritu Raj, 3DrShalini Chandra, 4 Dr Harish Chaudhary
1
Research Scholar, Department of Mathematics & Statistics, Banasthali University, P.O. Banasthali Vidyapith, Banasthali – 304022, Rajasthan, India, 2 Research Scholar, Department of Mathematics & Statistics, Banasthali University, P.O. Banasthali Vidyapith, Banasthali – 304022, Rajasthan, India, 3 Associate Professor, Department of Mathematics & Statistics, Banasthali University, P.O. Banasthali Vidyapith,Banasthali – 304022, Rajasthan, India 4 Assistant Professor, Department of Management Studies, Indian Institute of Technology, Delhi, New Delhi-110016, India,
Abstract This paper develops an EOQ inventory model for deterioratingitemswith two parameters Weibull deterioration. Shortages are permissible andpartially backlogged. In this model we consider time varying quadratic holding cost and ramp-type demand. The model is developed under two different replenishment policies: (i) Starting with no shortages (ii) Starting with shortages.The aim of this study is to find the optimal solutiontominimizing the total inventory costs for both the above mentioned strategies. To elevate the model a numerical example has been carried out and a sensitivity analysis occurred to study the result of parameters on essential variables and the entire cost of this model.
Keywords: Inventory, Partial Backlogging, Quadratic Holding Cost, Ramp-Type Demand, Single Item, Shortages, Weibull Deterioration
I.
Introduction
An economic order quantity model determines the quantity a company or a retailer must order to minimize the total inventory cost by balancing the inventory holding cost and fixed ordering cost. In the usual inventory system, it was considered that the buyer pays to vendor as soon as he receives the goods. Goods deteriorate and their valuesdecreases over time. Electronic equipment may become outdated as technology changes; fashion trends depreciate the value of clothes over time; batteries die out as they are old. The outcome of time is even mostimportantfor consumable goods such as foodstuff and drugs.An inventory model for fashion goods deteriorating at the end of the storage period was developed by Whitin [29]. Ghare and Schrader [7] developed an exponentially deteriorating inventory model. A replenishment inventory model for radioactive nuclide generators presented by Emmons [6].An order- level inventory model for decaying items with a deterministic rate of deterioration was developed by Shah and Jaiswal [24]. The retailer receives the delivery of goods and deterioration starts at that moment in all inventory models for decaying items. In real life situation, some customers would like to wait for backlogging during the shortage phase but the others would not. Therefore, the opportunity cost due to lost sales should be considered in the inventory models. Murdeswar[18]; Goyal et al. [11]; Hariga [15];Chakrabarti and Chaudhuri [2]; Hariga and Alyan [16] developed many inventory models assuming that shortages are completely backlogged. Chang and Dye [3] considered that backlogging shortages depends on lead time. An article in the field of deteriorating items with shortages has provided the economic order quantity with a known market demand rate developed by Wee [28]. Sana [22] and Roy et al. [19, 20, 21] developed many inventory models considering partial backlogging rates. The length of the lead time for the next replenishment becomes main factor for determining whether the backlogging will be accepted or not. A detailed review of deteriorating inventory literatures was provided by Goyal and Giri [12]. They considered variable demand rate for many inventory models. Silver and Meal [25] considered time-varying demand rate in their inventory models.
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EOQ Inventory Models for Deteriorating Item with… In recent research, Covert and Philip [4] presented an inventory model where the time to deterioration is considered with two parameter Weibull distribution deterioration. Ghosh and Chaudhuri [9] developed an inventory model for two parameters Weibull deteriorating items, with quadratic demand rate and shortages were permissible. Haley and Higgins [13] extended the inventory policy for two part trade credit, where the vendors consider cash discount for paying within a specified period and due in a large credit period. Goyal [10] developed the EOQ model under the conditions of permissible delay in payments. Aggarwal and Jaggi [1] considered deteriorating items to develop ordering policy under the conditions of permissible delay in payments.Sarkar et al. [23] developed an EOQ model for deteriorating items with shortages and time value of money.Teng et al. [27] proposed an EOQ model with deteriorating objects, price perceptive demand, and different selling and purchasing prices under trade credit policy and discussed the result obtained by Goyal [10]. Donaldson [5] provided ananalytical solution procedure of the basic inventory policy for a case of positive linear trend in demand.Giri et al. [8] developed an economic order quantity model for single deteriorating item with ramp type demand and Weibull distribution deterioration. Shortage is allowed and backlogged completely over an infinite planning horizon. Skouri et al. [26] provided an inventory model with time dependent Weibull deterioration rate, partial backlogging of unsatisfied demand and general ramp type demand rate. The model is developed under two different replenishment policies: (a) starting with no shortages and (b) starting with shortages. The backlogging rate is non-increasing function of the waiting time up to the next replenishment. Wu [30] presented an economic order quantity model with ramp type demand and Weibull distribution deterioration. Shortages are allowed partial backlogging and the rate of backlogging is dependent on waiting time for the next replenishment. The main purpose of this paper is to show that there exist a unique optimal cycle time to minimize the total inventory cost.A numerical illustrationand sensitivity analysisis presented to show the result of the proposed model.
II.
Assumptions
The following assumptions are mandatory todevelop mathematical model: 1. The Inventory system consider single item. 2. The inventory level defined by I t at time t. i.e. I a t 0 , 0 t t 1 I t , where I t is retailer’s stock level. I b t 0 , t 1 t T
3. The demand rate D t is a ramp type function of time and is defined as: D t, D t D,
t t
, D t is positive and continuous for time t 0 , T
4. The lead time is zero. 5. Shortagesare allowed and partially backlogged. 6. Cycle horizon is considered as T units of time and replenishment rate is infinite. 7. The rate of deterioration is time dependent, which is two parameters Weibulldistribution deterioration, 1
denotedby t ,where 0 1 , 1 and t 0 .A value of t 1 defines that the failure rate decreases with time. This happens if imperfect items are deterioratingfirst and the failure rate decreases with time. A value of t 1 defines that the failure rate is deterministic over time. A value of t 1 definesthat the failure rate increases with time. 8. The ordering quantity level is considered as Q . 9. Holding cost H C t is a time dependent quadratic function and is defined as H C t a bt ct
2
,
where a 0 b 0 and c 0 , .
III.
Notations
The following notations are mandatory to develop mathematical model: H C : The stock holding cost per year. S c : The shortage cost per year.
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EOQ Inventory Models for Deteriorating Item with… T : Length of time is considered as annually. t 1 : Time at which inventory becomes zero.
T C : Total inventory cost. L S : Lost sale cost.
I t :Inventory level at time t .
: Ramp-type demand function of time. D c :Deterioration cost.
IV.
Mathematical Modelwithout Shortages:
The inventory model is starting without shortages. At the start of the cycle, the production starts at t 0 and continuesup to t t 1 . At this time theinventory level reaches its maximum level and then production
is stopped. The inventory lessens to zero due to demand and deteriorationduring 0 , t1 and falls to zero at t t 1 . Thus, shortages occur during t1 , T which is partially backlogged.Therefore, the inventoryis described by the followingdifferential equations: dIa
t
dt
t
I a
t ,0
D
t t1
... 1
t T
... 2
with boundary condition I a t1 0 dIb
t
D
dt
t T
t
, t1
with boundary condition I b t1 0 Where t
D t, D t D,
t
and t
,
1
,
There are two cases arise: i t and ii t
4.1. Case (i) t From Eq. (1), we have Ia t De
t
1 2 2 2 2 t1 t t1 t 2 2
... 3
Eq. (2), solved by the following two ways: dIb
t
D t
dt dIb
t
T
t
T
D
dt
,
t
i
t1 t
,
t T , I
I
ii
We have, Ib t D
T 2
t
2
t1
2
1
t 3
3
t1
3
... 4
and
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EOQ Inventory Models for Deteriorating Item with… 1 1 3 1 2 2 B T t1 t1 t Tt 1 2 3 2
Ib t D
... 5
Where 1
B1
T
2
2
1
T 3
6
Total amount of deterioration during 0 , t1 t1
D te
Dc
t
t1
dt
0
D td t
D
2
0
t1
... 6
2
Total cost of holding during 0 , t1 is t1
H t I t d t
HC
1
a
D
0
a 3 b 4 3 2 3 4 2 4 t t1 B t1 C t1 E t1 F t1 1 3 8
... 7
Where B
a
3
a
a
2
1 3
,C
b
2
1 2 3
,E
2 4
b
2 4
,F
b
2
22 4
The shortage cost during t1 , T is
T
Sc
Ib t dt
t1
T
Ib t dt
I t dt b
D
t1
1 2 B t1 6 T 2 12
2
8 T t1 3 t1
2
... 8
Where B2
T
1
3
6
12
B1 T 4
1 6
T
3
3
T
T
2
2
2
Lost sales cost during t1 , T is
LS
T
1 T t D td t D
1 T
t d t
t1
1
D B 3 2 t1 3 T 1 t1 ... 9 3
6
2
Where B 3 6 T 3 T 3 T 2
2
3
2
3
Ordering quantity during 0 , T is t1
OQ e
t
D td t D
0
T
t T t dt D
T
t1
t dt
3 1 T 2 1 2 O Q D t1 t1 T t1 2 2 2 2 3
2
2
2
T
1 6
3
... 1 0
Total cost during 0 , T is the sum of deterioration cost, holding cost, shortage cost and lost sales cost is given by T C 1 t1 H C S c L S D c
T C 1 t1
b 4 1 2 3 3 2 3 t1 C t1 t1 a 2 T t1 B t1 2 4 3 8 D B 1 4 2 4 2 2 F t1 T T 1 t1 3 B 2 E t1 2 6
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EOQ Inventory Models for Deteriorating Item with… 4.1.1. Solution: b 1 t1 2 D t 2 2 E 1
T C1 t1
T C1 2
t1
2
3 2 2 C 2 3 t1 a 2 T t1 B 3 t1 3 2 3 2 4 t1 F 2 4 t1 T T 1 t1
... 1 2
b 2 1 t1 3 t1 2 a 2 T t1 B 2 2 1 2 1 2 D 3 t1 C 2 2 2 3 t1 E 3 4 t1 2 2 2 T T 1 F 2 3 2 4 t1
... 1 3
Main objective is to minimize the total relevant cost for the inventory model starting without shortages. The essential condition to minimize the total relevant cost is t1 t1
b 3 t1 a 2 T 2
1
2 2
E 4 t1
3
F
2
t1
4 t1
2
T C1 t1
B 3 t1
2 3
2
T T
2
0 , we have
C 2 3
... 1 4
1 t1 0
Using the software Mathematica, we can calculate the optimal value of t 1 by Eq. (14) and the optimal value T C 1 t1 of the total relevant cost is determined by Eq. (11). The optimal value of t 1 satisfy the sufficient condition for minimizing total relevant cost T C 1 t1 is T C1 2
t1
2
... 1 5
0
The sufficient condition is satisfied. TC1
t
Fig.1. Graphical representation of T C 1 and t 1 . 4.1.2. Numerical example: Let us consider a $ 1 / u n it , b $ 0 .5 / u n it , S c $ 0 .5 / u n it , D c $ 5 / u n it , 0 .1, 2 , T 1 y e a r , D 1 0 0 ,
1, 0 .5 y e a r B 1 0 .3 5 , B 2 0 .1 4 , B 3 1 .4 , B 0 .0 2 , C 0 .0 0 0 5 , E 0 .0 0 2 , F 0 .0 0 0 1 5
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EOQ Inventory Models for Deteriorating Item with…
Thus, the optimal value of t 1 is t 1 0 .4 4 . The optimal ordering quantity is O Q 1 7 .5 . The minimum
relevant cost is T C 1 2 8 .6 . 4.1.3. SensitivityAnalysis: To know, how the optimal solution is affected by the values of constraints, we derive the sensitivity analysis for some constraints. The specific values of some constraints are increased or decreased
by 2 5 % , 2 5 % and 5 0 % , 5 0 % .Thus,we computethe values of t 1 and T C 1 with the help of increased or decreased values of S c and D c .The result of the minimum relevant cost is existing in the following table 1. Table: 1 Parameters
Actual Values
5 0 % Increased
5 0 % Decreased
Sc
0.5
0.75
Dc
5
t1
T C1
2 5 % Decreased
0.25
25% Increased 0.625
7.5
2.5
6.25
3.75
0 .4 4
0 .5
0 .2 3
0 .5
0 .3 4
28.6
4 0 .7 5
1 7 .5
18.9
OQ
0.375
2 1 .9 8 1 8 .9
1 5 .9 3
From the result of above table, we observe that total relevant cost and ordering quantity is much affected by deterioration cost and shortage cost, and other parameters are less sensitive. 4.1.4. When Holding Cost is a Quadratic Function: We have t1
HC
t1
H 1 t I a t d t
0
HC D
a b t c t D e 2
t
0
1 2 2 2 2 t1 t t1 t dt 2 2
a 3 c 5 b 4 3 2 3 4 2 4 5 2 5 t1 t1 t1 B t1 C t1 E t1 F t1 F1 t 1 F 2 t1 ... 1 6 8 3 15
Where 1 3 a
B
F
b
1
1
,C
2
22 4
, F1
a
1
b
2
2
3
,E
1 c 1 5 3 3
4
, F2
1 2
1
c
2
,
2
32 5
Total cost is the sum of quadratic holding cost, shortage cost, deterioration cost and lost sales cost during
0,T T C 1 t1 H C S c L S D c T C1
t1
c 4 1 b 2 5 3 3 2 3 t1 t1 C t1 a 2 T t1 B t1 t1 2 15 4 3 8 D 1 1 4 2 4 5 2 5 2 2 F t1 F1 t 1 F 2 t1 T 1T t1 B 2 B3 E t1 2 6
... 1 7
4.1.5. Solution:
T C1 t1
c 4 b 3 1 2 2 2 2 t1 t1 a 2 T t1 B 3 t1 C 2 3 t1 t1 3 2 D E 4 t 3 F 2 4 t 2 3 F 5 t 4 F 2 5 t 2 4 T 1 T 1 1 1 1 2 1
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2
... 1 8 t1
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EOQ Inventory Models for Deteriorating Item with…
T C1 2
t1
2
4c 3 b 2 1 t1 3 t1 2 a 2 T t1 B 2 3 t1 1 t1 3 2 2 1 2 2 2 ... 1 9 D C 2 2 2 3 t1 E 3 4 t1 F 2 3 2 4 t1 3 2 3 2 F 2 2 4 2 5 t1 T 1T F1 4 5 t 1
Main objective is to minimize the total relevant cost of the inventory model starting without shortages. The essential condition to minimize the total relevant cost is t1 t1
1
3
F
b 3 4 t1 t1 a 2 T 3 2 c
2
4 t1
2 3
F1 5 t 1
4
t1
2
T C1
0 , we have
t1
B 3 t1
F 2 2 5 t1
2
2 4
C
2
3 t1
T 1T
2
2 2
t
1
E 4
0
... 2 0
Using the software Mathematica, we can calculate the optimal value of t 1 by Eq. (20) and the optimal value T C 1 t1 of the total relevant cost is determined by Eq. (17). The optimal value of t 1 satisfy the sufficient condition for minimizing total relevant cost T C 1 t1 is T C1 2
t1
2
... 2 1
0
The sufficient condition is satisfied.
TC
t
Fig.2. Graphical representation of T C 1 and t 1 . 4.1.6. Numerical Example: Let us consider a $ 1 / u n it , b $ 0 .5 / u n it , c 1, S c $ 0 .5 / u n it , D c $ 5 / u n it , 0 .1, 2 , T 1 y e a r , D 1 0 0 ,
1, 0 .5 y e a r B 1 0 .3 5 , B 2 0 .1 4 , B 3 1 .4 , B 0 .0 1, C 0 .0 0 0 5 , E 0 .0 0 2 , F 0 .0 0 0 1 5 , F1 0 .0 0 2 , F 2 0 .0 0 0 2
Thus, the optimal value of t 1 is t1 0 .4 2 5 . The optimum ordering quantity is O Q 1 7 .5 . The
minimum relevant cost is T C 1 2 8 .6 9 . 4.1.7. SensitivityAnalysis: To know, how the optimal solution is affected by the values of constraints, we derive the sensitivity analysis for some constraints. The specific values of some constraints are increased or decreased
by 2 5 % , 2 5 % and 5 0 % , 5 0 % .Thus,we computethe values of t 1 and T C 1 with the help of increased or decreased values of S c and D c .The result of minimum relevant cost is existing in the following table 2.
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EOQ Inventory Models for Deteriorating Item with… Table: 2 Parameters
Actual Values
Sc Dc
t1
T C1
50% Decreased 0.25
25% Increased 0.625
2 5 % Decreased
0.5
50% Increased 0.75
5
7.5
2.5
6.25
3.75
0 .4 2 5
0 .5 8
0 .2 3
0 .5 1
0 .3 3
28.7
4 0 .6 1
3 4 .9
2 2 .0 1
1 7 .5
18.9
1 8 .9
1 5 .9 3
OQ
0.375
From the result of above table, we observe that total relevant cost and ordering quantity is much affected by deterioration cost and shortage cost, and other parameters are less sensitive.
4.2. Case (ii): t The differential Eq.(1) becomes
t
dIa
dt
t
dIa
dt
Ia
t
D t,
0 t , I
Ia
t
D,
t t1 ,
I t1
I
0
We have Ia t De
t
1 2 2 1 t t1 t1 C1 t , 2 2 1
0 t
... 2 2
Where 1 1 C1 1 2
and I a t D e
t
2
1
2
2
1 1 t1 t t t1 , 1 1
t t1
... 2 3
The differential Eq. (2) becomes dIb t dt
D T t ,
t1 t T , I b t1 0
t
We have I b t D
T
t dt D
t1
1 2 2 T t t1 t t1 , t1 t T 2
... 2 4
Total amount of deterioration during 0 , t1
Dc
D te
t
t1
dt D
e
t
t1
dt
0
D dt
1 D S t1 1
... 2 5
Where, S
1 1 2 1 2 2
2
Total holding cost during 0 , t1 is
t1
HC
H t I t d t 1
a
0
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t1
a b t I t d t a b t I t d t a
0
a
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EOQ Inventory Models for Deteriorating Item with… HC D
a 2 b 3 1 2 3 2 2 2 3 t1 t1 C 3 C 5 t1 C 4 C 8 t1 C 7 t1 C 6 t1 C 9 t1 C 1 0 t1 C 2 ... 2 6 2 6
Where, 1 C2 2 2 2 1 2 4
C3 a
C5
a
1
a
1
C7 1
2
1
b
3
a
b
2
2
2
2
1
1
2
a
1
b
1
b
2 1
2
2 3
b C 1
2
1
2
1
2
1
1 3
2
b
1
2
1
5
3
bC1
1
a
3
2
a C1,
3
1 2
b
4
3
2 2
b
24
b
3
2 1
a C 1
1 a 2 3
b
2
b
2 1
2
a
2
b
,C6
2 a
1 1 2
,C4
2
b
2
4
3
3
b
2
b
1
1 3
a 1 ,C9 2 2 1 2
2
1 1 2 22 3 2 1
1
a
C8
1
1
2 4
2 4
2
2 b
1
1 2 3
3
,
a 2 , 1 1 1
2
2 1
, C 10
1 2 3
b 2 2 1 1
The shortage cost during t1 , T is T
Sc
T
I t dt b
D
t1
t1
T
t t1
1 2
t
2
t1
2
T t 2 2 dt D T t1 T t1 1 3 3 3
3
... 2 7
Lost sales cost during t1 , T is T
LS D
1 T
t d t
t1
1
D T t1 2 T t1
2
... 2 8
Ordering quantity is during 0 , T is
OQ
D te
t
t1
dt D
T
T
dt D
0
2 D t1 1 T 2
t dt
t1
t1
2
2
2
2
T 2
2
... 2 9
Total cost is the sum of deterioration cost, holding cost, shortage cost and lost sales cost during 0 , T
T C 2 t1 H C S c L S D c
T C 2 t1
2 3 2 2 3 2 C 9 t1 C 6 t1 C 7 t1 C4 C8 C 1 0 t1 1 2 3 a b 2 C3 C5 T 1 D t1 T t1 t 1 ... 3 0 t1 1 T 3 2 6 2 3 T 2 T T C2 S 3 2
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EOQ Inventory Models for Deteriorating Item with… 4.2.1.Solution:
TC 2 t1
C 2 3 t 2 2 C 2 2 t 2 1 C 3 t 2 1 1 1 10 9 6 b 1 2 D C 7 2 t1 C4 C8 1 t1 3 t1 1 3 6 a 2 2 T t1 C 3 C 5 T 1 T 2 2 C 1 0 2 2 2 3 t1 C 9 2 1 2 2 t1 C 6 1 D 2 3 t1 C 7 1 2 t1 C 4 C 8 1 1 t 1 b 2 t a 2 T 1 1 2 1
TC2 2
t1
2
2
... 3 1
... 3 2
Main objective is to minimize the total relevant cost of the inventory model starting without shortages. The essential condition to minimize the total relevant cost is C 1 0 2 3 t1
2 2
C 9 2 2 t1
2 1
TC 2 t1
0 , we have
C 6 3 t1
2
C 7 2 t1
1
b 2 a C4 C8 T t1 C 3 1 t1 3 t1 2 1 3 2 6 2 C5 T
2
1 T
... 3 3
0
Using the software Mathematica, we can calculate the optimal value of t 1 by Eq. (33) and the optimal value T C 2 t1 of the total relevant cost is determined by Eq. (30). The optimal value of t 1 satisfy the sufficient condition for minimizing total relevant cost T C 2 t1 is TC2 2
t1
2
... 3 4
0
The sufficient condition is satisfied.
TC
t
Fig.3. Graphical representation of T C 2 and t 1 .
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EOQ Inventory Models for Deteriorating Item with… 4.2.2. Numerical example: Let us consider a $ 1 / u n it , b $ 0 .5 / u n it , S c $ 0 .5 / u n it , D c $ 5 / u n it , 0 .1, 2 , T 1 y e a r , D 1 0 0 , 1 , 0 .5 y e a r C 1 0 .1 2 , S 0 .1 2 , C 2 0 .0 2 , C 3 0 .2 8 , C 4 0 .0 0 2 , C 5 0 .5 6 , C 6 0 .0 0 2 5 , C 7 0 .0 1 7 , C 8 0 .0 1 8 , C 9 0 .0 0 0 5 , C 1 0 0 .0 0 0 2
Thus, the optimal value of t 1 is t1 0 .5 4 . The optimum ordering quantity is O Q 1 9 .9 5 . The
minimum relevant cost is T C 2 7 7 .0 3 . 4.2.3. When Holding Cost is a Quadratic Function: Total Quadratic holding cost during 0 , t1 is
t1
H 1 t I a t d t
HC
0
a bt ct
2
t1
Ia t dt
a b t c t I t d t 2
a
0
c 4 C 3 1 b 3 a 2 2 t1 t1 t1 C 3 C 4 t1 C 7 t1 C 6 t1 6 2 1 H C D 12 C t 3 C t 4 C t 2 2 C t 2 3 C t 2 4 C 5 1 10 1 8 1 9 1 11 1 2
... 3 5
Where, C 2 a C1
b
4
2 3
2 4
2
2 5
a
a
b
2
a
a
1
3
1 2
b
2
2
1 C 11 2 4
2 3
5
10
2
5
22 3
3
2 3
C1
b
4
3
2 1
b 2
b 2 , C 10 2 1
b
2
1 4
b
b
4
8
b
2
4
2 4
3
c
3
3
4
2 4
1 2 3
c
3
,
3
,C 6 1 3 b
1 3
b 2
3
c
4
2
2
c
b
b
2
2
c
1
2
C1
4
1 3
1 3
2 1
2
2 5
b
b
b
1 2 4
a ,C3 a 1
2
c 2
3
b
,C5
2 3
1
2 5
a
1 2 2
3
2
2
c
a
c
3
3
5
3
2
c
c
cC1
a
a
3
6 a 6 a 3b 2 c 6 1
1 C9 2 3
C1
3
3
2 5
3
1
2 4
c
2
22 4
3
2
1
1
3
b
5
1
2
4
c 1 4 1
C4 a
C7
b
22 5 5
a
3
6
C1 c
c
2
2
a
,C8
c
3
a , 1 2 1
a
2
2 1 c
3 1
2
,
,
c 2 3 1 1
Total cost is the sum of quadratic holding cost, shortage cost, deterioration cost and lost sales cost during
0,T T C 2 t1 H C S c L S D c c 4 1 C 3 a 2 b 3 2 t1 C 7 t1 t1 T t1 C 3 C 4 T T 1 t1 12 6 3 2 2 1 1 D 3 T 1 2 3 4 2 2 2 3 2 4 2 C 5 t1 C 1 0 t1 C 8 t1 C 9 t1 C 1 1 t1 T C 2 T S C 6 t1 3 2
T C 2 t1
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... 3 6
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EOQ Inventory Models for Deteriorating Item with… 4.2.4. Solution: c 3 b 2 2 t 1 t 1 2 T a t 1 C 3 C 4 T T 1 C 3 C 7 1 t 1 3 2 1 2 3 2 1 2 2 D C 6 2 t1 C 5 3 t1 C 1 0 4 t1 C 8 2 2 t1 C 9 2 3 t1 2 3 C 1 1 2 4 t1
TC 2 t1
c t1 2 b 2 t1 2 T a C 3 C 7 1 t1 1 1 2 D C 6 1 2 t1 C 5 2 3 t1 C 1 0 3 4 t1 2 2 1 2 2 C 1 1 2 3 2 4 t1 C 8 2 1 2 2 t1 C 9 2 2 2 3 t1
TC2 2
t1
2
.. . 3 7
... 3 8
Main objective is to minimize the total relevant cost for the inventory model starting without shortages. The essential condition to minimize the total relevant cost is c
TC 2 t1
b 2 3 t1 t1 2 T a t1 C 3 C 4 T 3 2
C 6 2 t1
1
C 1 1 2 4 t1
C 5 3 t1
2 3
2
C 1 0 4 t1
3
2
0 , we have
T 1 C 3 C 7 1 t 1
C 8 2 2 t1
2 1
C 9 2 3 t1
2 2
... 3 9
0
Using the software Mathematica, we can calculate the optimal value of t 1 by Eq. (39) and the optimal value T C 2 t1 of the total relevant cost is determined by Eq. (36). The optimal value of t 1 satisfy the sufficient condition for minimizing total relevant cost T C 2 t1 is TC2 2
t1
2
... 4 0
0
The sufficient condition is satisfied. 4.2.5. Numerical example: Let us consider a $ 1 / u n it , b $ 0 .5 / u n it , c 1, S c $ 0 .5 / u n it , D c $ 5 / u n it , 0 .1, 2 , T 1 y e a r , D 1 0 0 , 1,
0 .5 y e a r C 1 0 .1 2 , C 2 0 .0 2 , C 3 0 .6 , C 4 0 .3 9 , C 5 0 .0 0 2 5 , C 6 0 .0 1 7 , C 7 0 .0 1 3, C 8 0 . 0 0 0 5 , C 9 0 .0 0 0 2 , C 1 0 0 .0 5 3, C 1 1 0 .0 0 0 2 5
Thus, the optimal value of t 1 is t 1 0 .5 . The optimum ordering quantity is O Q 1 9 .9 5 . The minimum
relevant cost is T C 2 8 2 .8 .
V. Mathematical Model with Shortages: The cycle starts with shortages that occur during the period 0 , t1 and shortages are partially backlogged. Replenishment grasps the inventory level up to Q after time t 1 . The inventory level depletes and falls to zero at t T because of demand and deterioration during t1 , T
. Two cases occur
(i) t1 (ii) t1
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EOQ Inventory Models for Deteriorating Item with… 5.1. Case (i) t1 Therefore, during 0 , T d q1
inventory I t is
the
t
t t1
D
t t
t
,
0 t t1 ,
q
t
D t t1 t ,
q
t
D t T ,
dt dq3
by
the
system
of
differential
equations
dt dq2
described
dt
q
0
q
q T
... 4 1
0
q
... 4 2
... 4 3
0
From Eq. (41), we have 1 t 2 D t1 t 3 2 3
q1
t
... 4 4
From Eq. (42), we have q2 t e
t
1 2 2 D D1 t t 2 2
... 4 5
Where, D1
2
2
T
2 From Eq. (43), we have 2
q3 t e
t
1
T
1
1
1 D D2 t t 1
... 4 6
Where,
D2 T
1
1
T
Total amount of deterioration during t1 , T Dc e
t1
t D te d t D t1
D td t T
T
e
t
dt
t1
T
D 1d t
2 2 t1 D c D 1 t1 t1 D3 D4 2
Where, D 3 T 2
T
1
1
1
... 4 7
2
2
, D 4 T
2
T
Total holding cost during t1 , T is H C
a b t q t d t t1
HC D
b 4 a 3 1 2 4 3 2 1 2 3 2 4 D D 6 t1 t1 b D 1 t1 a D 1 t1 D 7 t1 D 8 t1 D 9 t1 D 1 0 t1 D 1 1t1 D 1 2 t1 ... 4 8 5 8 6 2
Where,
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EOQ Inventory Models for Deteriorating Item with… D 5 a D 2 T a 2
T
T
1 2 2
2
2
2
2 2
a D 2
1
bD2
T
a T
2 2
1
2
T
3
3
3
1
,D
6
a D1
1 2
b 2
1 1 T 3 1
b
b
2
1 2 3
3
b D1 2
a
3
3
6
b 8
T
2 3
2 3
a bD2 a 2 1
b
4
b
2
22 4
2 4
4
2 1 4 b D1 a D 1 1 a a 1 1 3 2 3 2 1 , 2 2 2 2 3 3 2 2 2 1
1 1 , D8 4 2 2
b
D7
a
D11
b D1 a D1 1 1 , D9 , D10 , 3 2 2 2 1 a
b
2
22 3
, D12
2
22 4
The shortage cost during 0 , t1 is t1
Sc
t1
t
1
t D t t1 t d t D
0
t t
t dt 2
1
D
t1
12
0
... 4 9
4
The lost sales cost during 0 , t1 is t1
LS
1 t
1
0
1 2 1 3 t D td t D t 1 t 1 6 2
... 5 0
Ordering quantity during 0 , T is t1
O Q1 '
D t
t1
t dt e
t1
t dt D D te t1
0
1 1 3 O Q 1 ' D t1 1 t1 2 6
t1
2
2
2
T
e
t
2
t1
dt
2
T
1
T
1
1
... 5 1
Total cost during 0 , T is the sum of deterioration cost, holding cost, shortage cost and lost sales cost is given by T C 1 t1 H C S c L S D c '
b 4 a t1 8 1 2 6 ' T C1 D 4 3 D 8 t1 D 7 t1 www.ijceronline.com
t1 3
3 b D1 2
t1 a D 1 t1 D 1 2 t1 2
2 4
D 1 1 t1
2 1 D 1 0 t 1 D 3 t1 D9 t1 2
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2 ... 5 2 D3 D4 D5 D6
2 3
2
2 2
t1
Page 41
EOQ Inventory Models for Deteriorating Item with… 5.1.1. Solution:
T C1
b 3 a 2 2 3 2 2 t1 3 b D 1 t1 D 1 2 2 4 t1 D 1 1 2 3 t1 t1 3 2 2 2 2 2 2 1 3 2 1 D t1 D 7 4 t1 D 8 3 t1 D9 2 t .. . 5 3 1 2 2 1 D 1 0 1 t1 D 3 t1 a D1
'
t1
T C1 2
t1
b 2 2 2 2 1 D 1 1 2 2 2 3 t1 t1 a t1 D 1 2 2 3 2 4 t1 3 2 3 2 2 1 2 2 2 2 1 D t1 D 7 3 4 t1 D 8 2 3 t1 3 ... 5 4 2 1 2 D 1 2 t D 1 t D 1 t b D 1 10 1 3 1 1 9 2
'
2
Main objective is to minimize the total relevant cost for the inventory model starting without shortages. The essential condition to minimize the total relevant cost is
T C1 t1
'
0 , we have
3 a 2 b 2 3 2 2 t1 3 b D 1 t1 D 1 2 2 4 t1 D 1 1 2 3 t1 t1 3 2 2 D 7 4 t1 D 3 t1
1
3
D 8 3 t1
2
2
2
2
2
2 1
t1
1 D9 D 1 0 1 t1 2 t1 2 .. . 5 5
a D1 0
Using the software Mathematica, we can calculate the optimal value of t 1 by Eq. (55) and the optimal value T C 1 t1 of the total relevant cost is determined by Eq. (52). The optimal value of t 1 satisfy the sufficient '
condition for minimizing total relevant cost T C 1 t1 is '
T C1 2
t1
2
'
... 5 6
0
The sufficient condition is satisfied. 5.1.2. Numerical Example: Let us consider a $ 1 / u n it , b $ 0 .5 / u n it , S c $ 0 .5 / u n it , D c $ 5 / u n it , 0 .1, 2 , T 1 y e a r , D 1 0 0 ,
1, 0 .5 y e a r D 1 0 .3 6 , D 2 1 .0 3 , D 3 0 .1 4 , D 4 0 .2 5 , D 5 0 .0 9 6 , D 6 0 .2 7 , D 7 0 .0 0 6 2 5 , D 8 0 .0 1 5 , D 9 0 .0 0 4 5 , D 1 0 0 .0 0 6 , D 1 1 0 .0 0 0 3 5 , D 1 2 0 .0 0 0 1 5
Thus, the optimal value of t 1 is t1 0 .0 3 5 . The optimum ordering quantity is O Q
'
3 9 .0 5 . The
'
minimum relevant cost is T C 1 3 3 .9 .
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EOQ Inventory Models for Deteriorating Item with… 5.1.3.When Holding Cost is a Quadratic Function: We have T
HC
a b t c t q t d t 2
t1
1 c 5 b 4 3 2 3 1 t1 t1 D 7 t1 b D 1 t1 a D 1 t1 D 5 D 6 D 8 t 1 D 9 t1 10 8 2 HC D b D1 2 3 4 2 2 4 5 2 5 D 1 0 t1 D 1 1 t1 t1 D 1 2 t1 D 1 3 t1 D 1 4 t1 2
... 5 7
Where,
D5
2 bD2 a 2 cD b c c aD 2 T T 2 2 T 3 3 T 4 4 T 2 4 2 4 2 3 4 1 2 4 2 2 b a c 1 2 3 2 3 4 4 T 1 T T 2 2 2 2 1 2 3 1 2 2 4 1 a D 2 b a 3 3 2 2 b cD 2 T a bD2 T T 1 3 1 2 1 1
D7
a
c D1
6
3
1 4 2
b
D11
D 6 a D1
, D8
a
1 2
2
, D12 2
2
2
2 3
c D1
3
c
3
2
b
4
8
1
b D1
1
, D13
c
5
1
a D1
1
5
, D10
a
1 5 2 2
1
1
2
1 b 4 2
1
,
, D14
2 5
22 5
2
22 3
c
c
10
2
2
a D1
2
22 4
5 2
c D1 , D 9 b
a c D1 3 6
2
a
a
1
b D1
22 3
a 2
a 3 2
, 1
b
c
22 5
2
22 4
2
2
4
2 4
3
,
Total cost is the sum of quadratic holding cost, shortage cost, deterioration cost and lost sales cost during
0,T T C 1 t1 H C S c L S D c '
c 5 b 4 3 1 2 2 5 2 4 2 3 2 2 t1 D 1 2 t1 D 1 0 t1 t1 3 b D 1 t1 a D 1 t 1 D 1 4 t 1 t1 D 7 t1 10 8 1 2 6 2 2 D 5 4 3 2 1 D 1 1 t1 D 8 t1 D 9 t1 D 3 t1 D 3 D 4 D 5 D 6 b D 1 1 t1 D 1 3 t1 2 2
T C1
'
5.1.4. Solution:
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... 5 8
EOQ Inventory Models for Deteriorating Item with…
T C1
c 4 3 2 b 2 4 2 3 D 1 2 2 4 t1 t1 3 D 7 t1 3 b D 1 t1 a D 1 D 1 4 2 5 t1 t1 2 2 3 2 2 2 2 2 1 4 3 2 D D 1 0 2 3 t1 D 1 3 5 t1 D 1 1 4 t1 D 8 3 t1 2 2 t1 2 b D 1 t 1 D 1 t D t 1 1 1 9 1 3 1
'
t1
T C1 2
t1
'
2
... 5 9
2 b 3 2 3 2 2 D 1 2 2 3 2 4 t1 t1 2 3 D 7 t1 3 b D 1 D 1 4 2 4 2 5 t1 2 c t1 3 2 3 2 2 2 2 1 2 3 2 D D 1 0 2 2 2 3 t1 D 1 1 3 4 t1 1 2 1 t1 D 1 3 4 5 t1 2 D 8
2 3 t1
1
1 b D 1 1 t1
D9
1 t1
1
D 3
1 t1
2
... 6 0
Main objective is to minimize the total relevant cost for the inventory model starting without shortages. The T C1
essential condition to minimize the total relevant cost is
3 2 b 4 t1 t1 3 D 7 t1 2 3 2 2 c
t1
2 3
D10
3 t1
2
2
3 t1
b D1
2 2
0 , we have
b D 1 t1 a D 1 D 1 4
2
5 t1
2 4
D12
2
4
2
2
1 t1
3
t1
'
1
2
2 t1
D9
2 1
1 t1
D13
5 t1
D 3 t1
1
4
D11
4 t1
3
D8
... 6 1
0
Using the software Mathematica, we can calculate the optimal value of t 1 by Eq. (61) and the optimal value T C 1 t1 of the total relevant cost is determined by Eq. (58). The optimal value of t 1 satisfy the sufficient '
condition for minimizing total relevant cost T C 1 t1 is '
T C1 2
t1
2
'
... 6 2
0
The sufficient condition is satisfied. 5.1.5. Numerical Example: Let us consider a $ 1 / u n it , b $ 0 .5 / u n it , c $ 1 / u n it , S c $ 0 .5 / u n it , D c $ 5 / u n it , 0 .1, 2 , T 1 y e a r , D 1 0 0 ,
1, 0 .5 y e a r D 1 0 .3 6 , D 2 1 .0 3 , D 3 0 .1 4 , D 4 0 .2 5 , D 5 0 .1 1 2 , D 6 0 .3 , D 7 0 .0 4 6 , D 8 0 .0 2 2 , D 9 0 .0 1 2 , D 1 0 0 .0 0 0 3 , D 1 1 0 .0 0 6 2 5 , D 1 2 0 .0 0 0 1 5 , D 1 3 0 .0 1, D 1 4 0 .0 0 0 3
Thus, the optimal value of t 1 is t1 0 .0 3 5 . The optimum ordering quantity is O Q
'
3 9 .0 5 . The
'
minimum relevant cost is T C 1 3 3 .9 .
5.2. Case (ii) t1 Therefore, during 0 , T
the
inventory I t is
described
by
the
system
of
differential
equations
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EOQ Inventory Models for Deteriorating Item with…
t
d q1
t1
D t
dt dq2 t
t1 t ,
D
dt
dq3 t
,
t
0 t ,
t t1 , q
q t D , t1 t T , q T
dt
0
q
q
... 6 3
0
... 6 4
... 6 5
0
From Eq.(63), we have 1 t 2 D t1 t , 3 2 3
q1
t
... 6 6
0 t
From Eq.(64), we have 1 1 2 q 2 t D t1 t1 t t , t t1 6 2 2 2
... 6 7
FromEq.(65), we have q3
t
e
t
1 D D2 t t , 1
... 6 8
t1 t T
Where,
D2 T
1
1
T
Total amount of deterioration during t1 , T T
Dc e
t1
D
D 1 t1
T
e
t
dt D
t1
dt t1
D
2
t1
1
t1
T t 1 ... 6 9
1
Total holding cost during t1 , T is T
HC
a b t q t d t 3
t1
b 3 2 3 2 1 2 2 2 3 F a D 2 t1 F 2 t1 t1 F 3 t1 F 4 t1 F 5 t1 F 6 t1 F 7 t1 1 3
D
... 7 0
Where, F1 1
b
1
1
1
2
1 2 3
1
T
2
1 , F4
2 3
T
a D 2
1
a
2
2 T
1
a
b 3
1
1
1 T
3
2
bD2
T
2 2
2
1 2
bD2
1 , F5
a T
a D 2
1
2
b 1 3
a D 2T , F 2
, F6
1 2
bD2
2
T
1
3
1
a
bD2
, F3
a
2
1
2
a
2 b
, F7
3 b
2
1 2 3
The shortage cost during 0 , t1 is
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EOQ Inventory Models for Deteriorating Item with…
t1
q1 t d t
Sc
0
3 2 q 2 t d t D t1 t1 t1 2 3 12 3 2
3
4
... 7 1
The lost sales cost during 0 , t1 is
D t 1 t
LS
1
t dt D
t1
1 t
1
1
t dt
6
0
D F 9 F1 0 t1 3 t1
2
... 7 2
Where, F9 3 , F1 0 3 2
Ordering quantity during 0 , T is
OQ
D t 1
t1 t d t D
t1
T
t1 t d t e
t1
D
0
e
t
dt
t1
1 5 3 2 1 1 2 OQ D t1 t1 D 2 t1 t1 1 t1 1 1 2 2 6 2
2
1 2
D2
... 7 3
Total cost is the sum of deterioration cost, holding cost, shortage cost and lost sales cost during 0 , T
T C 1 t1 H C S c L S D c '
1 1 1 3 2 2 F1 0 2 6 a D 2 t 1 D 2 t 1 F 5 b t1 2 F 2 t1 3 2 6 1 ... 7 4 D 2 3 1 1 2 3 2 1 2 2 2 3 F 4 t1 F 3 t1 t1 F 6 t1 F 7 t1 F 9 D 2 F1 T t1 1 12 6
TC2
'
5.2.1. Solution: 1 2 2 1 b t1 2 F 2 t1 F1 0 2 6 a D 2 D 2 t 1 6 2 1 2 2 D F5 1 t F 2 t F 3 t 2 1 t1 1 4 1 3 1 1 1 F 2 2 t 2 1 F 2 3 t 2 2 7 1 6 1
TC 2
'
t1
TC2 2
t1
2
'
... 7 5
2 b t1 2 F 2 D 2 1 t1 2 F 5 1 t1 1 2 2 2 1 2 1 1 D F 4 1 2 t1 F 3 2 3 t1 t1 ... 7 6 1 F 2 1 2 2 t 2 2 F 1 2 3 t 2 1 1 7 1 6
Main objective is to minimize the total relevant cost for the inventory model starting without shortages. The essential condition to minimize the total relevant cost is
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TC 2 t1
'
0 , we have
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EOQ Inventory Models for Deteriorating Item with… b
F4
t1
2
t1
2
2 F2 1
F3
t1
3 t1
1
F
6 2
10
2
6aD2
2
D 2 t1
F5 1
1
1 t1
2
1
2
1 t1
2
F
2
2
6
t1
2 1
F7
2
3 t1
2 2
0
... 7 7
Using the software Mathematica, we can calculate the optimal value of t 1 by Eq. (77) and the optimal value T C 1 t1 of the total relevant cost is determined by Eq. (74). The optimal value of t 1 satisfy the sufficient '
condition for minimizing total relevant cost T C 1 t1 is '
TC2 2
t1
2
'
... 7 8
0
The sufficient condition is satisfied.
TC
0.76,13.92
t
' Fig.4. Graphical representation of T C 2 and t 1 .
5.2.2. Numerical Example: Let us consider a $ 1 / u n i t , b $ 0 .5 / u n i t , S c $ 0 .5 / u n i t , D c $ 5 / u n i t , 0 .1, 2 , T 1 y e a r , D 1 0 0 , 1,
0 .5 y e a r D 2 1 .0 3 , F1 0 .5 9 7 , F 2 0 .2 4 , F 3 0 .0 0 6 7 , F 4 0 .0 1 7 , F 5 0 .0 3 4 , F 6 0 . 0 1 2 , F 7 0 .0 0 0 2 3 , F 9 1 .7 5 , F1 0 7 .5
Thus, the optimal value of t 1 is t1 0 .7 6 . The optimum ordering quantity is O Q
'
2 0 .9 . The
'
minimum relevant cost is T C 2 1 3 .9 2 . 5.2.3. SensitivityAnalysis: To know, how the optimal solution is affected by the values ofconstraints, we derive the sensitivity analysis for some constraints. The specific values of some constraintsare increased or decreased
by 2 5 % , 2 5 % and 5 0 % , 5 0 % .Thus,we computethe values of t 1 and T C 1 with the help of increased or decreased values of
Sc
and D c .The result of minimum relevant cost is existing in the following table 3. Table: 3
Parameters
Sc Dc t1
T C1
OQ
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Actual Values 0.5
+50% Increased 0.75
-50% Decreased 0.25
+25% Increased 0.625
-25% Decreased 0.375
5
7.5
2.5
6.25
3.75
0 .7 6
0 .6 9
0 .8 4
0 .7 2
0 .7 9
1 3 .9 2
1 9 .7 1
7 .5
1 6 .8 8
1 0 .8 2
2 0 .9
2 4 .6
1 6 .9 7
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2 2 .9 8
1 9 .3 7
Page 47
EOQ Inventory Models for Deteriorating Item with… From the result of above table, we observe that total relevant cost and ordering quantity is much affected by deterioration cost and shortage cost, and other parameters are less sensitive. 5.2.4. When Holding Cost is a Quadratic Function: We have T
HC
a b t c t q t d t 2
3
t1
a D 2 1 c 4 2 3 4 3 2 F a D 2 t1 F 2 t1 F 3 t1 t1 F 4 t1 F 5 t1 F 6 t1 t1 1 4 1 D F t 2 2 F t 2 4 F t 2 3 7 1 8 1 9 1
... 7 9
Where, c
F1
1 2 4
1 D2 3 b
F2
F6
1 2
a
b
2
T
2 4
T 1 1
b D 2 , F3
1 3
3
2
1 2 3
T
1 2 a
b c D 2 , F4
2 3
bD2 a 2 2 1 2
T 1 1
2
a D 2
1
T
1
1 1 , F5 4 1
bD 1 a 2 1 , F7 2 2 12 2 1
, F8
c
2 2
T
4
4
c
a
T
1 3
1 1 T 4 1 c
cD 2
b T
cD 2 b 3
c
3
2
1 2 4
bD2
a D 2T
a T
2
, 1 b
b
2
, F9
1
4
2
1 2 3
Total cost is the sum of quadratic holding cost, shortage cost, deterioration cost and lost sales cost during
0,T T C 1 t1 H C S c L S D c '
c 4 3 1 1 2 2 1 F1 0 2 6 a D 2 t 1 D 2 t 1 a D 2 t1 t1 F 3 t1 2 F 2 t1 4 3 2 6 1 ... 8 0 TC2 ' D 2 3 F9 2 3 4 2 1 2 2 2 3 2 4 F 5 t1 F 4 t1 t1 F 7 t1 F 9 t1 F 8 t1 D 2 F1 T F 6 t1 1 12 6
5.2.5. Solution:
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EOQ Inventory Models for Deteriorating Item with… 3 2 c t1 3 F 3 t1 2 F 2 t1 TC 2 ' 1 D a D 2 t1 F 6 2 t1 F5 t1 2 2 1 2 2 1 t1 F 7 2 2 t1 1
6 2 3 3 t1 F 4 4 t1 2 2 2 3 F 9 2 3 t1 F 8 2 4 t1 1
F
2
10
2
6 a D 2 D 2 t1
1
... 8 1
3 c t1 2 3 F 3 t1 2 F 2 D 2 1 t1 a D 2 t1 2 2 2 2 1 TC2 ' 1 2 D F 6 1 2 t1 F 5 2 3 t1 F 4 3 4 t1 2 ... 8 2 t1 1 t 2 1 F 2 1 2 2 t 2 F 2 2 2 3 t 2 1 F 2 3 2 4 t 2 2 7 1 9 1 8 1 1 2
2
1
Main objective is to minimize the total relevant cost for the inventory model starting without shortages. The essential condition to minimize the total relevant cost is
c t1 3 F 3 3
F 6 2 t1
t1
1
F 9 2 3 t1
2
2 F 2 t1
F 5 3 t1
2 2
1 6
F
10
F 8 2 4 t1
2 3
'
t1
2
F 4 4 t1
2
TC 2
3
2
0 , we have
6 a D 2 D 2 t1
2
2
1
1
t1
2
1
a D 2 t1
F 7 2 2 t1
2 1
... 8 3
0
Using the software Mathematica, we can calculate the optimal value of t 1 by Eq. (83) and the optimal value T C 1 t1 of the total relevant cost is determined by Eq. (80). The optimal value of t 1 satisfy the sufficient '
condition for minimizing total relevant cost T C 1 t1 is '
TC2 2
t1
2
'
... 8 4
0
The sufficient condition is satisfied.
TC
t
' Fig.5. Graphical representation of T C 2 and t 1 .
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EOQ Inventory Models for Deteriorating Item with… 5.2.6. Numerical Example: Let us consider a $ 1 / u n it , b $ 0 .5 / u n it , S c $ 0 .5 / u n it , D c $ 5 / u n it , 0 .1, 2 , T 1 y e a r , D 1 0 0 ,
1, 0 .5 y e a r D 2 1 .0 3 , F1 0 .6 8 2 , F 2 0 .2 4 , F 3 0 .1 7 6 , F 4 0 .0 1 1, F 5 0 .0 1 4 , F 6 0 .0 1 6 , F 7 0 .0 1 2 , F 8 0 .0 0 0 4 , F 9 0 .0 0 0 2 4 , F1 0 7 .5
Thus, the optimal value of t 1 is t 1 0 .8 . The optimum ordering quantity is O Q
'
2 0 .9 . The minimum
'
relevant cost is T C 2 2 2 .1 5 . 5.2.7. SensitivityAnalysis: To know, how the optimal solution is affected by the values of constraints, we derive the sensitivity analysis for some constraints. The specific values of some constraints areincreased or decreased
by 2 5 % , 2 5 % and 5 0 % , 5 0 % .Thus,we compute the values of t 1 and T C 1 with the help of increased or decreased values of S c and D c . The result of the minimum relevant cost is existing in the following table 4. Table: 4 Parameters
Sc Dc t1
T C1
OQ
Actual Values 0.5
+50% Increased 0.75
-50% Decreased 0.25
+25% Increased 0.625
-25% Decreased 0.375
5
7.5
2.5
6.25
3.75
0 .8
0 .7 4
0 .8 8
0 .7 7
0 .8 4
2 2 .1 5
3 1 .9 9
1 2 .1 3
2 7 .4 8
1 7 .0 0 4
2 0 .9
2 4 .6
1 6 .9 7
2 2 .9 8
1 9 .3 7
From the result of above table, we observe that total relevant cost and ordering quantity is much affected by deterioration cost and shortage cost, and other parameters are less sensitive.
VI. Conclusion We presented an order level inventory model for deteriorating items with two parameters Weibull deterioration. The model developed under two replenishment policies: (i) With no shortages and (ii) With shortages which are partially backlogged. We considered ramp-type demand rate and time varying linear and quadratic holding costs, and found that total relevant cost with linear holding cost is less than total relevant cost with quadratic holding cost. Therefore, the linear time-dependent holding cost is more realistic than quadratic time-dependent holding cost. The proposed model can be extended in numerous ways like permissible delay in payments, time value of money,quantity discounts etc. Acknowledgement:We are very thankful to the referees for their convenient suggestions.
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