Development And Analysis Of Some Inventory Control Systems...

5.1 INTRODUCTION

As inventory

various situations, especially inflation. In present times, inflation is a

global phenomenon. Inflation is defined as that state of disequilibrium in

which an expansion of purchasing power tends to cause or is the effect of

an increase in the price level. A period of prolonged, persistent and

continuous inflation results in the economic, political, social and moral

disruption of society. Inflation and time value of money have also

attracted attention of researchers. So, we have considered the concept of

inflation in an inventory system.

In recent years many researchers assuming that the production rate

of a manufacturing system is often assumed to be constant, but in fact

production rate is a variable under managerial control. Production rate

may be influenced due to demand.

An attempt has been made to study a more realistic situation, for

assuming that production rate is decision variable. Volume flexibility of a

manufacturing system is its ability to be operated profitably at different

overall output levels. Volume flexibility permits a manufacturing system

to adjust production upwards or downwards within wide frontier period to

Chapter 5 An EPQ Model for Decaying I

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the start of production of a lot. It helps to reduce the rate of production to

avoid rapid accrual of inventories.

Demand is the most volatile of all the market forces, as it is least

controlled by management personnel. Even a slight change in the demand

pattern for any particular item causes a lot of havoc with the

manufacturing unit concerned. Overall, it means that every time the

demand for any commodity undergoes a noticeable change, the inventory

manager has to reformulate the complete logistics of management for that

item. In such a situation, the inventory manager has to look out for a new

method, by which he can easily assimilate the changed inclination of the

customers into the existing policy. It is perceived that the steady demand

after its increment with the time is never continued indefinitely. The

demand pattern for many products, which initially increases with time for

some period, becomes steady after some period rather than increasing.

Rather it would be followed by exponential decrement with respect to

time after some period of time and becomes asymptotic in nature. Thus,

the demand would be illustrated by three successive time period-

classified time dependent ramp type functions, in which in the first phase

the demand increases with time and after that it becomes steady and

towards the end in the final phase it decreases and becomes asymptotic.


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Most of the researches on real market oriented time dependent

demand is very restrictive. Goyal (1986) and Goswami and Chaudhuri

(1991) discussed different types of inventory models with linear trend in

demand. Hill (1995) considered a product subject to a period of

increasing demand, according to a general power law, followed by a

period of level demand. The characteristic of ramp type demand can be

found in Mandal and Pal (1998) who has taken order level inventory

system with ramp type demand rate for deterioration items. Wu et al.

(1999) developed an EOQ model with ramp type demand rate for items

with deterioration. Wu and Ouyang (2000) developed an order-level

inventory system for deteriorating items with a ramp type demand

function of time. It not only provides the solution procedure for the

problem, but also enumerate two possible shortage models. Wu (2001)

presented an EOQ inventory model which depleted not only by demand

but also by Weibull distribution deterioration, in which the demand rate is

assumed that with a ramp type function of time. In the model, shortages

are allowed partial backlogging and the backlogging rate is variable and

is dependent on waiting time for the next replenishment. Giri et al.

(2003) solved a single-item single-period Economic Order Quantity

model for deteriorating items with a ramp type demand and Weibull

deterioration distribution is considered. The shortages in inventory are


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allowed and backlogged completely. The model is developed over an

infinite planning horizon. Manna and Chaudhuri (2006) also solved an

order level inventory system for deteriorating items and developed with

demand rate as a ramp type function of time. The finite production rate is

proportional to the demand rate and deterioration rate is time

proportional. The unit production cost is inversely proportional to the

demand rate. Deng et al. (2007) considered the inventory models for

deteriorating items with ramp type demand rate. Singh and Singh (2007)

developed an EOQ inventory model with Weibull distribution

deterioration, ramp type demand and partial backlogging. A single item

order level inventory model for a seasonal product is discussed where the

demand rate is represented by a ramp-type time dependent function.

Shortages are not allowed over a finite time horizon by Panda et al.

(2008). Skouri et al. (2009) determined an inventory model with general

ramp type demand rate, time dependent (Weibull) deterioration rate and

partial backlogging of unsatisfied demand is considered.

An inventory model for decaying items with ramp type demand

rate under volume flexibility has been discussed. Inventory deteriorates

over time at a time dependent deterioration rate. Production rate is taken

as demand dependent. Holding cost is taken as variable and it is a linear

increasing function of time. Shortages in inventory are allowed and


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partially backlogged. The backlogging rate is an exponential decreasing

function of time. The effect of inflation has also been considered for

various costs associated with the inventory system. A numerical example

is presented to illustrate the model with cost minimization technique.

5.2 ASSUMPTIONS AND NOTATIONS

In developing the mathematical model of the inventory system the

following assumptions are being made:

1. A single item is considered over a prescribed period T units of

time.

2. Production is considered which depends on the demand,

P(D)=kD(t).

3. The unit production cost is a function of production rate.

4. Production rate is considered to be decision variable.

5. The demand rate D(t) at time t is deterministic and taken as a ramp

type function of time i.e. ,

where H(t- s function defined as :

0

1

, tH(t

, t

6. The replenishment rate is infinite and lead-time is zero.


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7. Holding cost is taken as linear.

8. When the demand for goods is more than the supply. Shortages

will occur. Customers encountering shortages will either wait for

the vendor to reorder (backlogging cost involved) or go to other

vendors (lost sales cost involved). In this model shortages are

allowed and the backlogging rate is exp(- t), when inventory is in

shortage. The backlogging parameter is a positive constant.

9. The variable rate of deterioration is taken as (t) = t, where 0 <

<< 1 and only applied to on hand inventory.

10. No replacement or repair of deteriorated items is made during a

given cycle.

Notations

I(t) : The inventory level at any time t, t 0.

T : Planning horizon.

r : Inflation rate.

(g+ht) : The holding cost per unit per unit time.

C2 : The deterioration cost per unit.

C3 : The shortage cost per unit per unit time.


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C4 : The opportunity cost due to lost sales.

C : The replenishment cost per order.

K : Production rate

The unit production cost is given by:

PG

C N H KK

, where N, G, H are all positive constants. This

cost function is based on the following factors:

The material cost N per unit item is fixed.

As the production rate increases, some costs like labour and energy

costs are equally distributed over a large no. of units. Hence, the unit

production cost decreases, because (G/K) decrease as the production

rate (K) increases.

The third term (HK) associated with tool or die costs is proportional to

the production rate.

5.3 FORMULATION AND SOLUTION OF THE MODEL

Initially, the inventory level is zero. The production starts at time t

= 0, when the inventory level become q deterioration can take place and

after time demand becomes steady, it reaches to maximum inventory

level S at time t1. After this production stops and at time t2, the inventory


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becomes zero. At this time shortage starts developing and at time t3. After

this time fresh production starts to clear the backlog by the time T. Here,

our aim is to find the optimum values of t1, t2, t3, t4 that minimize the total

average cost (K) over the time horizon (0, T). The depletion of inventory

is given in the figure 5.1

Fig. 5.1: Representation of production model.

The inventory system at any time t can be represented by the

following differential equations:

( 1)I (t) 0 t . (5.1)

( 1)I (t) 1 . 5.2)

I (t) 1 2 . 5.3)


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I (t) Ae e , . (5.4)

( 1)I (t) K Ae , 3 . 5.5)

with the boundary conditions,

I(0) =0, I(t1) = S, I(t2) = 0 and I(T) = 0. (5.6)

The solutions of the equations are given by:

2 22 2 2 2 2

3( 1) ( 1) 2 2 22

AI(t) K Se K e ,

22e ( , 0 t . 5.7)

and 2

3 321

1 6

(t t )I(t) Ae (t t) . 5.8)

3 2 31 1 1( ) [ ]

6 2 3rtI t t t t t t t e 1 2 5.9)

2A

I(t) e e e , . 5.10)

( 1)

( )K A

I(t) e e 3 . 5.11)

Due to continuity of I(t) at point t = , it follows from equations (5.7) and

(5.8), one has


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3 3

2 2 211 3

1(1 ) 2 2 2

6 2S A K (t

23

(1 )A

e K ( . 5.12)

The total average cost consists of following elements:

(i) Production cost per cycle (PC)

1

30( )[ ]

t Trt rt rt

t

GN HK Ke dt Ke dt Ke dt

K. 5.13)

(ii) Holding cost per cycle (CH)

1 2

10

[ ( ) ( ) ( ) ( ) ( ) ( ) ]t t

rt rt rt

t

g ht I t e dt g ht I t e dt g ht I t e dt .

5.14)

(iii) Cost of deteriorated units per cycle (CD)

1 2

1

20

[ ( ) ( ) ( ) ]t t

rt rt rt

t

C tI t e dt tI t e dt tI t e dt . 5.15)

(iv) Shortages cost per cycle (CS)

3

2 3

3 [ ( ( )) ( ( )) ]t T

rt rt

t t

C I t e dt I t e dt . 5.16)

(v) Opportunity cost due to lost sales per cycle (CLS)


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3

2

4 [ 1 ]t

t

C ( e ) Ae e dt . 5.17)

Therefore, the total average cost per unit time of our model is

obtained as follows

K(t1, t2, t3)=[Production cost + Holding cost + Deterioration cost +

Shortage cost + Opportunity cost]/T. (5.18)

To minimize the total cost per unit time, the optimal values of t1

and T can be obtained by solving the following equations simultaneously

1

0K

t. (5.19)

2

0K

t. (5.20)

And 3

0K

t. (5.20)

provided, they satisfy the following conditions :

2 2 2

2 2 21 2 3

0 0, 0K K K

,t t t

. 5.21)

Equations (5.19), (5.20) and (5.21) are highly nonlinear equations.

Therefore, numerical solution of these equations is obtained by using the

software.


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5.4 NUMERICAL ILLUSTRATION

To illustrate the model numerically the following parameter values

are considered:

K = 150 units, h= 0.4. r = 0.05 unit,

= 0.2 unit, C2 = Rs. 10.0 per unit, =0.002 unit,

= 0.2 year, C4 = Rs. 4.0 per unit, t2 =0.6,

t3 =0.8, = 0.1 unit, T = 1 year,

G=2000, N=150, H=0.01,

g= Rs. 3.0 per unit per year,

C3 = Rs. 12.0 per unit per year,


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Table 5.1: Effect of various parameters.

% Change t1 S K

-20 0.398067 38.597235 157.696583

-10 0.398901 38.597235 157.882913

0 0.399224 38.597235 158.115354

+10 0.400593 38.597235 158.272559

+20 0.401448 38.597235 158.524118

% Change t1 S K

-20 0.399333 38.600815 158.095851

-10 0.399279 38.599028 158.106061

0 0.399224 38.597235 158.115354

+10 0.399170 38.595447 158.124164

+20 0.399116 38.593661 158.132515

Effects of Ramp Parameter ( )

% Change t1 S K

-20 0.399032 38.826936 158.586542

-10 0.399133 38.710155 158.346963

0 0.399224 38.597235 158.115354

+10 0.399307 38.488159 157.886035

+20 0.399345 38.129344 156.782943


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Fig. 5.2: Variation in t1 with respect to .

Fig. 5.3 with respect to .


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Fig. 5.5: Variation in t1 with respect to .


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Fig. 5.7: Variation in Total with respect to .


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Fig. 5.8: Variation in t1 with respect to µ .

Fig. 5.9: Variation in with respect to µ .


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Fig. 5.10: Variation in with respect to µ .

5.5 OBSERVATIONS

From the numerical illustration of the model, it is observed that the

period in which inventory holds increases with the increment in

backlogging and ramp parameters while inventory period decreases with

the increment in deterioration. Initial inventory level decreases with the

increment in deterioration and ramp parameters while inventory level

increases with the increment in backlogging parameter. The total average

cost of the system goes on increasing with the increment in the

backlogging and deterioration parameters while it decreases with the

increment in ramp parameters.


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5.6 CONCLUSIONS

A production system for deteriorating items is developed with

ramp type demand rate. Demand rate is taken as exponential ramp type

function of time, in which demand decreases exponentially for the some

initial period and becomes steady later on. The effects of inflation are not

considered in some inventory models. However, from a financial point of

view, an inventory represents a capital investment and must complete

capital funds. Thus, it is necessary to

consider the effects of inflation on the inventory system. Therefore, this

concept is also taken in this model. Shortages are allowed in inventory

with the concept of partial backlogging. Backlogging rate is exponential

decreasing function of time. Holding cost is also taken as time dependent.

A numerical assessment of the theoretical model has been done to

illustrate the theory. The solution obtained has been checked for

sensitivity with the result that the model is found to be quite suitable and

stable.

The proposed model can be extended to stochastic demand pattern.

Also, we could extend the model to incorporate some more realistic

features, such as quantity discount or the unit purchase cost, the inventory

holding cost and others are also fluctuating with time.


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REFERENCES

[1] Deng P.S., Lin R.H.J. and Chu P. (2007). A note on the inventory

models for deteriorating items with ramp type demand, E.J.O.R., 178,

112-120.

[2] Giri B.C., Jalan A.K. and Chaudhuri K.S. (2003). Economic order

quantity model with Weibull distribution deterioration, shortage and

ramp-type demand, I.J.S.S., 34(4), 237-243.

[3] Goswami A. and Chaudhuri K.S. (1991). An EOQ model for

deteriorating items with a linear trend in demand, J.O.R.S., 42(12),

1105-1110.

[4] Goyal S.K. (1986). On improving replenishment policies for linear

trend in demand, Engineering costs and production Economics

(E.C.P.E.), 10, 73-76.

[5] Hill R.M. (1995). Inventory model for increasing demand rate followed

by level demand, J. of Operational Research Society, 46, 1250-1259.

[6] Mandal B. and Pal A.K. (1998). Order level inventory system with

ramp type demand rate for deteriorating items, Journal of

interdisciplinary Mathematics, 1, 49-66.

[7] Manna S.K. and Chaudhuri K.S. (2006). An EOQ model with ramp

type demand rate, time dependent deterioration rate, unit production cost

and shortages, E.J.O.R., 171, 557-566.

[8] Panda S., Senapati S. and Basu M. (2008). Optimal replenishment

policy for perishable seasonal products in a season with ramp-type time

dependent demand, Computers and Industrial Engineering, 54, 2, 301-

314.


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[9] Singh S.R. and Singh T.J. (2007). An EOQ inventory model with

Weibull distribution deterioration, ramp type demand and partial

backlogging, Indian Journal of Mathematics and Mathematical Science,

3, 2, 127-137.

[10] Skouri K. et al. (2009). Inventory models with ramp type demand rate,

partial backlogging and Weibull deterioration rate, European Journal of

Operational Research, 192, 1, 79-92.

[11] Wu J.W., Lin C., Tan B. and Lee W.C. (1999). An EOQ inventory

model with ramp-type demand rate for items with Weibull deterioration,

Information and Management Science, 3, 41-51.

[12] Wu K. S. and Ouyang L.Y. (2000). A replenishment policy for

deteriorating items with ramp type demand rate (Short Communication),

Proceedings of National Science Council ROC (A), 24, 279-286.

[13] Wu K.S. (2001). An EOQ inventory model for items with Weibull

distribution deterioration, ramp type demand rate and partial

backlogging, P.P.C., 12(8), 787-793.

Development And Analysis Of Some Inventory Control Systems...

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