Optimal Time and EOQ for Inventory of Deteriorating Items with Variation and Leading Times ABSTRACT The proposed model represents the optimal time

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Optimal Time and EOQ for Inventory of Deteriorating Items with Variation and Leading Times


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The proposed model represents the optimal time, EQO and optimal total cost, for two different times interval as components of first run time under considered constant demand, inventory is non-contact within the first component time run, constant within second, purchasing cost is more than holding, the finite horizon planning, without shortage cost, replenishment required after the second component which is equal the first leading time of first run time. The inventory level is non-zero within length time of horizon. Sensitivity analysis for the proposed model has represented the many values for vary demand, deterioration rate lies in assumed range. The represented figures explained the performance of optimal quantity and optimal total cost within the components of the first runtime (required time), the difference between the optimal total cost and the actual total cost was proposed.

Optimal of First run component time -Deterioration rate –Optimal leading time-optimal total Cost


The inventory modeling, under special assumptions, was represented by many researchers as Abd (1996) proposed optimal pricing and lot sizing under conditions of perish-ability and partial back ordering. Bierman and Thomas (1977) urbanized inventory decisions under inflationary conditions. Chern, Teng, and Chan (1999) considered that compersion among various inventory shortage models for deteriorating items on the foundation of maximizing profit. Dave, Patel (1981) well-thought-out Policy inventory model for deteriorating items with time proportional demand. Friedman (1982) accessible inventory lot size models with general time-dependent and carrying cost function. Haneveld, Teunter (1992) further extended inventory model by developed EOQ within the effects of discounting and demand rate. Misra (1979) offered optimal inventory management under inflation. Padmanabhan, Var (1995) EOQ wrote a model to manage the models for perishable items under stock dependent selling rate. Pal, Bhunia, and Mukherjee (2005) proposed a model for marketing oriented with three component demand rate dependent on the displayed stock level. Rong, Mahapatra, and Maiti (2088), Sachan (1984) residential the model policy for deteriorating items with time proportional demand. This is paper considered the level of inventory to be not zero within variant-time of finite horizon to obtain the optimal components of the first runtime with realistic assumption purchasing cost is greater than holding cost when the inventory level is constant within second components of first run time before the replenishment ,because in real life the level of inventory has to be nonzero with new technical software gave alarm about inventory when total inventory became 10% to order new quantities as replenishment this consideration adopted in this paper, optimal time and optimal quantity for the level inventory should stay within length of horizon with taking the demand rate and required quantity as EOQ, leading time in the contemplation .

2. Material and methods

2.2. Assumptions and notions

2.2.1. Assumptions
In this paper, the mathematical model is developed with the following assumptions
The Planning horizon is finite.
Replenishment rate is infinite.
Single item inventory control.
Demand and deterioration rate are constant.
Deteriorating occurs as soon as the items are received into inventory within 0,t_n.
The Shortage is not allowed.
The leading time is nonzero.
The purchasing cost is more than the holding cost.
The inventory level is constant within the second component of first run time, non zero within the planning horizon.
The total relevant cost consists of fixed ordering, purchasing and holding cost.

2.3. Notation

D= The demand rate quantity in period 0,t_n.
Q_1= The quantity within A, B.
Q_M= The quantity within 0, A
t_1= The first component o f the first runtime.
t_w1= The first leading time.
t_m1= The second component of the first runtime.
Q = The total quantity within 0, B.
TC_A = The total fixed ordering cost during 0, t_n.
I_h=The holding cost.
TC_h = The total holding cost during 0, t_1 .
TC_P = The total purchasing cost during 0,t_1.
TC= The total relevant cost during 0,t_1.

2.4. Parameters

T = The length of the finite planning horizon.
I_1 (t) = The inventory level at time 0,t_m.
I_2 (t) = The inventory level at time t_m ,t_1 .
? t?_1 = The first run time of replenishment.
? = The constant deteriorating rate units/unit time during 0,t_1 .

3. Mathematical model
Let I(t) is the inventory level at any time

(dI(t))/dt+?I(t)=-D_1,0?t?t_(1 ), 0???1 , I_0 (t)=e^(-?t) (1)

Fig.1Graphical demonstration of inventory control diagram

I_1 (t) ?=I?_0 (t)?_t^(t_1)??De^?u-b_1 t_1=? D/? (e^?(t_1-t) -1) ,I_0 (t)=e^(-?t_1 ) (2) \
I_2 (t) ?=I?_01 (t)?_(t_m)^(t_1)??De^?u-Q_m t_m1=? D/? (e^?(t_1-t_m1 ) -1)-?Q_m t_m1 , I_01 (t)=e^(-?t_m1 ) (3)



Based in ?ot_m1 A, ?ot_1 B

Q_1=Q/2,Q_m=Q_1, t_m=t_1/2

I(t)=D/? (e^?(t_1-t) -1)-D/? (e^?(t_1-t_m1 ) -1)-?Q_m t_m1 (4)

I(0)=D/? (e^(?t_1 )-1)-D/? (e^?(t_1-t_m1 ) -1)-?Q_m t_(m1 )

Q_1=D/? (e^(?t_1 )-e^((?t_1)/2) )-(??Q?_1 t_1)/2 , series Talyors for e^(?t_1 )=1+?t_(1 ) ,e^((?t_1)/2)=1+(?t_(1 ))/2 about orginal point.

t_1=(2Q_1)/(D-?Q_1 )

3.1. Fixed ordering cost

The fixed ordering cost in the length of a finite horizon 0, t_1

TC_A=A (5)

3.2. Purchasing cost

According to fig.1 of inventory level the purchasing cost of
?TC?_P=(CD(e^(??t?_1 )-e^((?t_1)/2)))/?-(?Q_1 t_1)/2 (6)

3.3. Holding cost excluding interest cost

We locate the average inventory quantity to obtain holding cost
?TC?_h=I_h ?_0^(t_1)??I(t)dt-I_(hQ_1 t_1 )/2=?_0^(t_1)??(D(e^(??t?_1 )-e^((?t_1)/2)))/?dt-I_(hQ_1 t_1 )/2=(I_h D)/?^2 (e^(?t_1 )-2e^((?t_1)/2)+1)-(???I?_h Q?_1 t_1)/2 ?? (7)

3.4. Optimal inventory level and optimal time

To obtain the EOQ by optimizing the total cost

TC=?TC?_A+?TC?_h+?TC?_P (8)

By subsisting Eq. (5, 6, 7) in Eq. (8)
TC=A +(CD(e^(??t?_1 )-e^((?t_1)/2)))/?-(??Q?_1 t_1)/2+(I_h D)/?^2 (e^(?t_1 )-2e^((?t_1)/2)+1)-(??I_h Q?_1 t_1)/2 (9)

Deputy the t_1=(2Q_1)/(D-?Q_1 ) in Eq. (9)

TC=A +(CDQ_1/(D-??Q?_1 ))-C(?Q_1?^2/(D-??Q?_1 ))-(?I_h ?Q_1?^2)/((D-??Q?_1)) (10)

Deviating Eq. (10) with respect to Q_1

(d?TC?_1)/(dQ_1 )=CD(D/(D-??Q?_1 )^2 )-C( (2DQ_1-???Q?_1?^2)/(D-?Q_1 )^2 )-?I_h ((2DQ_1-???Q?_1?^2)/(D-??Q?_1 )^2 )=0

?Q_1?^*=D/? 1-(1-?C/(C+?I_h ))^(1/2)

?Q_1?^*=D/? 1+(1-?C/(C+?I_h ))^(1/2)

Lemma (1):

a) If 0?k_1?1
i) ??D?Q?_1?^* If ?Q_1?^*=D/? 1-(1-?C/(C+?I_h ))^(1/2)

When k_1?? , 0?k_1?1
ii) ??D>Q?_1?^* If ?Q_1?^*=D/? 1-(1-?C/(C+?I_h ))^(1/2)

When k_1


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