an optimal replenishment policy for deteriorating items...

International Journal of Statistics and Systems

ISSN 0973-2675 Volume 12, Number 3 (2017), pp. 457-474

© Research India Publications

http://www.ripublication.com

An Optimal Replenishment Policy for Deteriorating

Items with Power Pattern under Permissible delay in

payments

Nalini Prava Behera

P.G. Department of Statistics, Utkal University, India.

Prof. Pradip Kumar Tripathy

P.G. Department of Statistics, Utkal University, India.

Abstract

In this paper, Economic order quantity (EOQ) model based for time dependent

deteriorating items with power demand pattern is presented. Demand is related

to shortage under permissible delay in payments. The optimal cycle time is

determined to minimize the total inventory cost. Furthermore, sensitivity

analysis of the optimal solution is studied with respect to changes in different

parameter values and to draw managerial insights of proposed model.

Keywords: Power Demand Pattern, Salvage Cost, Permissible Delay in

Payment, Partial backlogging.

AMS Mathematics Subject Classification (2010):90B05

INTRODUCTION:

The Economic Order Quantity (EOQ) model proposed by Harris [10] has been widely

used by enterprises in order to reduce the cost of stock. Due to the variability in

458 Nalini Prava Behera and Prof. Pradip Kumar Tripathy

economic circumstances, many scholars constantly modify the basic assumptions of

the EOQ model and consider more realistic factors in order to make the model

correspond with reality. One such modification is the inclusion of the deterioration of

items. The effect of deterioration is very important in many inventory systems.

Deterioration is defined as decay or damage such that the item cannot be used for its

original purpose. Food items, drugs, pharmaceuticals and radioactive substances are

example of items in which sufficient deterioration can take place during the normal

storage period of the units and consequently this loss must be taken into account while

analyzing the system.

During the past few years, many research papers dealing with deteriorating inventory

problems have appeared in various research journals. In this paper the deterioration

rate is time dependent. Inventory of deteriorating items first studied by Whitin [20],

he considered the deterioration of fashion goods at the end of prescribed storage

period. Ghare and Schrader [7] extended the classical EOQ formula to include

exponential decay, wherein a constant fraction of on hand inventory is assumed to be

lost due to deterioration. Covert and Philip [4] then Ghare and Schrader’s model for

variable rate of deterioration by assuming two parameter weibull distribution

function.

In traditional EOQ model, it is assumed that buyer must pay for the items purchased

as soon as the items are received. But supplier permits the buyer a period of time (say

credit period), to settle the total amount owed to him. Usually, interest is not charged

for the outstanding amount if it is paid within the permissible delay period. However,

if the payment is not paid within the permissible delay period, then interest is charged

on the outstanding amount under the previously agreed terms and conditions. The

extensive use of permissible delay has been addressed by Goyal [9] who developed an

EOQ model under the condition of permissible delay in payments. Chand and Ward

[2] studied Goyal’s model under assumptions of the classical EOQ model. Goyal’s

model was extended by Aggarwal and Jaggi [1] for deteriorating items. Chung [3]

presented discounted cash flow (DCF) approach for the analysis of the optimal

inventory policy in the presence of trade credit.

An Optimal Replenishment Policy for Deteriorating Items with Power Pattern… 459

Backlogging occurs due to shortages. Sometimes, researchers assumed partial

backlogging while others considered full backlogging. In reality, if all customers are

prepared to wait until the arrival of the next order, then it is called completely

backlogged else, all the customers leave the system. However, in certain situations,

some customers will be able to wait for the next order in order to satisfy their

demands during the stock out period, while others do not wish to or cannot wait,

hence they meet their demands from other sources (the partial backlogging case).

Dye [6] have developed a optimal selling price and lot size with a varying rate of

deterioration and exponential partial backlogging. Skouri and Papachristors [11] have

developed an inventory model with deteriorating items, time-varying demand, linear

replenishment cost and partially time varying backlogging. Sushil and Rajput [17]

have introduced a Partially Backlogging Inventory Model for Deteriorating Items with

Ramp Type Demand Rate. Dave and Patel [5] proposed an EOQ model for time

proportional demand with constant deterioration. Geol and Aggarwal [19] formulated

an order level inventory system with power demand pattern for deteriorating items.

Datta and Pal [18] proposed an order level inventory system with power demand

pattern for items with variable rate of deterioration.

Giri et al. [8] developed an inventory models with time dependent deterioration.

Mishra and Shah [12] have developed an inventory management of time dependent

deteriorating items with salvage value. Rajeswari and Vanjikkodi [13] proposed an

inventory model for items with two parameter weibull distribution deterioration and

backlogging. Mohan and Venkateswarlu [15] have discussed an inventory

management model with quadratic demand, variable holding cost with salvage value.

With this motivation, this paper attempts to investigate an EOQ model assuming the

existence of a suitable power demand pattern, time dependent deterioration and time

dependent holding cost. The salvage value is associated to the deterioration units.Only

deterministic case of demand is considered. Shortages are allowed. Permissible delay

in payments is also considered for the optimal total cost. Suitable numerical example

and sensitivity analysis is also done.


Table 1. Major characteristic on inventory models on selected areas:

Authors

&Publication

year

Deterioration Varying

Demand

Backlogged

Allowed

Permissible

Delay in

Payments is

allowed

Dave & Patel

(1981)

Constant Time

Proportional

No No

Geol & Aggarwal

(1981)

Constant Power

Demand

No No

T.K. Datta et.al.

(1988)

Variable rate of

Deterioration

Power

Demand

Yes No

Giri et.al. (1996) Linear Time

Varying

No No

Skouri &

Papachristos

(2002)

Constant Time varying Partial No

Dye C. (2007) Variable rate of

Deterioration

Exponential Partial No

Poonam Mishra

et.al (2008 )

Weibull Constant No No

N. Rajeswari et.al.

(2012)

Weibull Power

Demand

partial No

Roy & Chaudhuri

(2012)

Constant Stock

Dependent

No No

Venkateswaralu R.

et. al (2013)

Weibull Quadratic No No

Sushil & Rajput

(2015)

Constant Ramp type partial No

Present Paper

(2017)

Time Dependent Power

Demand

Partial Yes


ASSUMPTIONS AND NOTATIONS:

To develop the mathematical model the following assumptions and notations are

being made:

Assumptions:

(i)The inventory consists of only one type of item.

(ii) The demand up to time t is assumed to be n

Td

1

1

, where d is the demand size

during the fixed cycle time T and n (0 < n < ) is the pattern index. n

nn

nT

dttD1

1

is

the demand rate at time t. Such pattern in the demand rate is called power demand

pattern.

(iii) A variable fraction 𝜃(t) of the on hand inventory deteriorates per unit time. In the

present model, the function 𝜃(t) is assumed in the form

tt 0 ; 10 0 , t > 0

(iv) The demand rate is deterministic and constant.

(v) The lead time is zero.

(vi) The planning horizon is infinite.

(vii) The holding cost is time dependent i.e tth , where 0 .

(viii) Shortages are allowed and demand is backlogged at the rate of

.1

1

tT The

backlogging parameter is a positive constant and (T-t) is the waiting time

.1 Ttt

(ix) During the trade credit period, M, the account is not settled; generated sale

revenue is deposited in an interest bearing account. At the end of the period, the

retailer’s pays off all units bought, and start to pay the capital opportunity cost for the

items in stock.


Notations:

t : Inventory level at time t.

tQ : Order quantity at time t=0.

A: Ordering cost per order.

P: The purchasing cost per unit.

S: The selling price per unit, with S > P.

T: The length of order cycle.

2C : The deterioration cost per unit per year.

3C : The Shortage cost for backlogged per unit per year.

4C : The unit cost of lost sales per unit.

10 : The salvage value associated with deteriorated units during a cycle

time.

eI : The interest earned per dollar per year, where ce II .

cI : The interest charged in stock by the supplier.

M: Trade credit period.

1t : Length of time in which the inventory has no shortage.

TC: The total cost of the system.

Formulation and Solution of the Model:

The inventory system is developed as follows: Q units of items arrive at the inventory

system at the beginning of each cycle. The inventory level is dropping to zero owing

to demand and deterioration during the time interval 1,0 t . Finally, a shortage occurs

due to demand and partial backlogging during the time interval Tt ,1 .

Based on the above description, during the time interval 1,0 t , the differential

equation representing the inventory status is given by


tDtt

dttdI

, 10 tt (1)

With the boundary condition 01 t and QI 0 the solution of (1) is

nn

nnnn

n

ttn

tttT

dt1212

1

011

1

2

01 12221

(2)

and the order quantity is

n

nn

n

tn

tT

dQ12

1

01

11 122

(3)

During the second interval Tt ,1 , shortage occurred and the demand is partially

backlogged.

That is, the inventory level at time t is governed by the following differential

equation:

,1 tT

tDdt

tdI

Ttt 1 (4)

With the boundary condition 01 t , the solution of (4) is

n

nnn

nn

n

ttn

ttTT

dt1

1

11

1

1

1 11

(5)

Ordering cost per cycle time is given by

OC = A (6)


The deteriorating cost DC during the period [ 0,T] is given by

DC= 1

0

t

dttDQ

DC=

21

11

0

2

122

n

n

tnT

dC

(7)

The holding cost HC during the period [0,T] is given by

HC = 1

0

t

dttt

=

41

1

041

10

21

11 122142

1

1428

1

122

1nnn

n

tnn

ntnnt

nn

T

d

(8)

The salvage cost SV during the period [0,T] is given by

SV =

1

0

t

dttDQ =

21

11

0

2

122

n

n

tnT

dC

(9)

The total amount of shortage cost during the period Tt ,1 is given by

SC = T

t

dttC1

3

=

21

21

1

21

1

11

1

1

1

11

2

1

1

1

13

121

111

nn

nnnnn

n Ttnn

n

tTtn

tTn

nn

TntTTtTt

T

dC

(10)


The amount of lost cost during the period (0, T) is given by

LC =

n

nn

T

t nT

dttT

C1

1

4

11

11

=

11

1

11

1

1

14 ntTt

nnT

T

dCnn

nnn

n

(11)

The total average cost of the system per unit time is given by

12

11

,

0,

tMTCtMTC

TC (12)

Where 1TC and 2TC are discussed as follows.

Case 1: 10 tM

In this case the length of delay in payment (M) is less than equal to the period with

positive inventory 1t . The retailers can sale units during (0, M) at a sale price (S) per

unit which he can bring an interest rate eI per unit per annum in an interest bearing

account. So the total interest earned during (0, M) is

Mn

n

e

n

nn

e MnT

dPItdt

nT

dtSIIE0

11

11

1

1

1

(13)


During 1,tM , the supplier will charge the interest to the retailer on the remaining

stock at the rate cI per unit per annum. Hence total interest charges payable by the

retailer during

1,tM is

1

11

2

2

2111

21

21

1

22

1

1

1

11

1

11

1

1

11

1

nMt

n

MttMtn

nn

tMtt

T

dPIdttIPIIC

nn

nnnn

n

ct

Mc

(14)

So, the total variable cost per unit time is

SVIEICLCSCDCHCOCT

TC 111

1 (15)

21

11

0

2

11

1

21

21

1

22

1

1

1

11

11

11

1

1

1

11

11

1

11

14

21

21

1

21

1

11

1

1

1

11

2

1

1

1

13

21

11

0

2

41

1

041

10

21

11

11

1221

1

11

2

2

2

111

11

1211

11

122

122142

1

1428

1

122

1

1,

n

n

n

n

e

nn

n

nnn

n

c

nn

n

n

nnnn

nnn

n

n

n

nnn

n

tnT

dCM

nT

dPI

nMt

nMt

t

Mtn

nn

tMtt

T

dPInt

Ttn

nT

T

dC

Ttnn

ntTtn

tTn

nn

TntTTtTt

T

dCtnT

dC

tnn

ntnnt

nn

T

dA

TTtTC


The solutions for the optimal values of t1 and T can be found by solving the

following equations simultaneously.

0

,

1

11 dt

TtdTC and

0

,11 dT

TtdTC. (16)

Provided

0,

2

1

11

2

dt

TtTCd and

0

,2

11

2

dT

TtTCd (17)

Case 2: 1tM

In this case, the period of delay in payment (M) is more than period with positive

inventory 1t . The retailer earns interest on the sales revenue up to the permissible

delay period and no interest is payable during the period for the item kept in stock.

Interest earned for the time period (0, T) is

1

1

1

1

11

0

11121

1

nn

ne

t

e tn

nMtnT

dSItDttMdttDSIIE (18)

Here, the interest charges is zero i.e. 02 IC (19)

So, the total variable cost per unit time is

SVIEICLCSCDCHCOCT

TC 222

1 (20)


21

11

0

2

11

1

1

11

11

11

1

11

14

21

21

1

21

1

11

1

11

1

11

2

1

1

1

13

21

11

02

41

104

1

10

21

11

12

122111

1211

11

122

122142

1

1428

1

122

1

1,

n

n

nn

ne

nn

n

n

nnnn

nnn

n

n

n

nnn

n

tnT

dCt

nnMt

nT

dPIntTt

nnT

T

dC

Ttnn

ntTtn

tTn

nn

TntTTtTt

T

dCtnT

dC

tnn

ntnnt

nn

T

dA

TTtTC

The solutions for the optimal values of t1 and T can be found by solving the

following equations simultaneously.

0

,

1

12 dt

TtdTC and

0

,12 dT

TtdTC. (21)

Provided

0,

2

1

12

2

dt

TtTCd and

0

,2

12

2

dT

TtTCd

(22)

Numerical Example:

Case 1: 10 tM

Consider an inventory system with the following data: In appropriate units

25.0,12.0,13.0

,08.0,15.0,30,8.0,1,30,01.0,15,10,35,30,100 432

e

c

IMIPndCCCPA

Then we obtained the optimal values as 9041.0*

1 t , 2628.1* T and

848.165,*

11 TtTC


Case 2: 1tM

Consider an inventory system with the following data: In appropriate units

25.0,12.0,13.0

,22.0,15.0,30,17.0,1,30,01.0,15,10,35,30,100 432

e

c

IMIPndCCCPA

Then we obtained the optimal values as 7894.0*

1 t , 2212.2* T and

293.225,*

12 TtTC

Sensitivity Analysis:

On the basis of the data given in example above we have studied the sensitivity

analysis by changing the following parameters one at a time and keeping the rest

fixed.

Table 2: Case 1: (0 ≤ M ≤ 𝒕𝟏)

Parameter % change 1t T TtTC ,11

+50 1.2664 0.4967 637.433

+25 1.5651 2.2388 333.770

0 0.9041 1.2628 168.848

-25 0.9084 1.2723 164.538

-50 0.9130 1.2825 163.155

+50 0.8884 1.2316 170.724

+25 0.8960 1.2466 168.342

0 0.9041 1.2628 168.848

-25 0.9124 1.2787 163.347

-50 0.9214 1.2959 160.683

M +50 0.9313 1.3310 157.909

+25 0.9171 1.2962 161.935


0 0.9041 1.2628 168.848

-25 0.8917 1.2291 169.853

-50 0.8802 1.1965 173.870

+50 1.6164 1.7381 226.117

+25 1.4626 1.0802 211.377

0 0.9041 1.2628 168.848

-25 1.4775 0.9299 162.147

-50 2.1210 0.9585 156.434

+50 1.2648 0.9050 165.572

+25 1.2638 0.9043 165.781

0 0.9041 1.2628 168.848

-25 1.2618 0.9037 165.972

-50 1.2608 0.9032 166.125

Table 3: Case 2: M > 𝒕𝟏

Parameter % change 1t T TtTC ,11

+50 0.4055 2.1366 343.098

+25 0.5259 2.4027 336.453

0 0.7894 2.2212 225.293

-25 0.5626 2.3963 323.851

-50 0.6013 2.4159 313.749

+50 0.5488 2.4247 337.714

+25 0.5531 2.4117 331.804

0 0.7894 2.2212 225.293

-25 0.5614 2.3815 328.819

-50 0.5654 2.3687 322.406

M +50 0.7871 2.4287 245.827

+25 0.6706 2.4139 290.804


0 0.7894 2.2212 225.293

-25 0.4511 2.3793 357.478

-50 0.3594 2.3616 384.331

+50 0.2789 2.7351 452.149

+25 0.3956 2.1025 342.264

0 0.7894 2.2212 225.293

-25 0.1508 2.7883 396.211

-50 0.1348 2.8561 308.465

+50 0.5589 2.3970 325.133

+25 0.5581 2.3971 325.403

0 0.7894 2.2212 225.293

-25 0.5565 2.3974 325.957

-50 0.5557 2.3975 326.227

DISCUSSION:

From the Table 2 we observed that

Decrease in results in decreases in inventory periods, decrease in total cost

per unit time.

Decrease in results in increases in inventory periods, decrease in total cost

per unit time.

Decrease in results in decreases in inventory periods, increase in total cost

per unit time.

Decrease in results in decreases in inventory periods, decrease in total cost

per unit time.

Decrease in results in decreases in inventory periods, marginal increases in

total cost per unit time.


From the Table 3 we observed that


per unit time.


per unit time.

Decrease in results in decreases in inventory periods, increase in total cost

per unit time.


per unit time.

Decrease in results in increases in inventory periods, marginal increases in

total cost per unit time.

CONCLUSION

In this paper, we have developed a deterministic inventory model for time

proportional deteriorating items with associated salvage value. Shortages are allowed.

It is assumed that the retailer generates revenue on unit selling price which is

necessarily higher than the unit purchase cost when a power demand pattern has been

assumed with demand rate. The effect of delay period offered by the supplier to

retailer is analysed. It has been observed from the sensitivity analysis that decreases in

delay period M results in increase in total inventory cost. Decrease in salvage value

result in marginal increase in total inventory cost.

ACKNOWLEDGEMENT

The research work is supported by DST INSPIRE Fellowship, Ministry of Science

and Technology, Government of India, and P.G. Department of Statistics, Utkal

University, India.

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Journal of Operational Research, 5(2), PP. 39-46 DOI:

10.5923/j.ajor.20150502.03.

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