dr.anubha goyal department of mathematics, krishna college ... · dr.anubha goyal department of...

AN INVENTORY MODEL WITH MULTIVARIATE DEMAND WEIBULL DISTRIBUTION

DETERIORATION UNDER THE EFFECT OF INFLATION AND TRADE CREDIT

Dr.Anubha Goyal

Department of Mathematics, Krishna College, Bijnor

[email protected]

Abstract: A deteriorating inventory model with

multivariate demand is developed as in present scenario

it is imperative to take multi-trends while considering

customer’s demand. These trends are stock

dependency, relying on selling price and time demand.

Two or more parameters dependent demand is more

authentic and this model has time, stock and selling

price based demand rate. Weibull rate of deterioration

is considered here while shortages inflationary

environment is included in the model with its

applicability for trade credit to customer.

Keywords: Inventory Model, Multivariate Demand,

Inflation, Trade credit

1. Introduction

The inventory manager has to look out for a new

method, by which he can easily assimilate the changed

inclination of the customers towards the policy

existence. The chain to supply the article is directly

proportional to demand, as demand increases supply

increases and as demand decreases supply of items

decreases. Earlier researchers obtained the solution of

standard inventory models after incorporating the

dependence on variety of demand rates i.e., constant

and linear but there are numerous limitations in all

these demand rates. Stable demand rate occurs very few

in sensible situations. A demand rate of linear time-

dependence deduces a steady change in the demand

rate as per time. This hardly happens in the case of any

commodity in the market.

It was very obvious a fact that given some time, every

item can have a suitable position for itself in the

customer’s mind, thereby raising its demand as time

advances. Later with the advent of outlets, it was

commonly acknowledged that huge displays of stocks

persuade the customer into large purchasing. Also it

was noted that a decline in the level of displayed stock

witnessed a decline in the customer’s demand for that

item. This concept is known as stock dependent

demand. For a long time, stock dependent demands

were explored, researched to a greater extent while new

aspects were being added to the study. The dependence

of the sale of any item on its selling price is not a new

concept, but a common sense conclusion. It is a general

observation that a hike in the selling price of the

commodity discourages its customers from choosing

that item in future. However, a selling price dip, in

whatever form it may come, always notices a sudden

increase in the demand rate, as reduced prices always

encourages the customers to buy more. Therefore, we

have taken the two or more parameter dependent

demand. Such a combination of two or more factors

grants more authenticity to the formulation of the

model and makes it more close to reality.

2. Literature Survey

Levin et al. (1972) observed that “heap of consumer

goods displayed in large outlets lead customers to

more. Haley and Higgins (1973) formulated an

inventory policy with trade credit financing.

Donaldson (1977) developed an optimal algorithm for

solving classical no-shortage inventory model

analytically with linear trend in demand over fixed time

horizon. Silver (1979) developed an approximate

solution procedure for a linearly time-varying demand

by using the Silver–Meal heuristic (1969). The effect

of payment rules on ordering and stock holding in

purchasing was suggested by Kingsman (1983). Silver

and Peterson (1985) developed a model in which sales

at the retail level tend to proportional to inventory

displayed and a large piles of goods displayed in a

supermarket will lead the customer to buy more. Gupta

and Vrat (1986) considered demand rate to be function

of initial stock level. Baker and Urban (1988) were

the first to establish an economic order quantity model

(EOQ) for a power–form inventory-level demand

pattern. Dave, U. (1989) proposed a deterministic lot

size inventory model with shortages and a linear trend

in demand. Mandal and Phaujdar (1989) then

developed a production inventory model for

deteriorating items with uniform rate of production and

linearly stock dependent demand. Goswami and

Chaudhuri (1991) discussed different types of

inventory models with linear trend in demand.

Aggarwal and Jaggi (1995) developed ordering

policies of deteriorating items under permissible delay

in payments. The demand and deterioration were

assumed as constant. Bhunia, A.K. and Maiti, M.

(1998) gave deterministic models of perishable

International Journal of Pure and Applied MathematicsVolume 118 No. 22 2018, 1309-1323ISSN: 1314-3395 (on-line version)url: http://acadpubl.eu/hubSpecial Issue ijpam.eu

1309

inventory with stock-dependent demand rate and

nonlinear holding cost. Liao et al. (2000) developed an

inventory model for stock dependent demand rate when

a delay in payment is permissible. Chang and Dye

(2001) developed an inventory model for deteriorating

items with partial backlogging and permissible delay in

payments. Khanra and Chaudhuri (2003) discussed

an order level decaying inventory model with such time

dependent quadratic demand. Balkhi and Benkherouf

(2004) developed an inventory model for deteriorating

items with stock dependent and time varying demand

rates over a finite planning horizon. Teng et al. (2005)

presented an optimal pricing and ordering policy under

permissible delay in payments. Deterioration rate was

taken as constant and shortages were not allowed in the

study. Wu et al. (2006) proposed an optimal

replenishment policy for non-instantaneous

deteriorating items with stock-dependent demand and

partial backlogging. Panda et al. (2007) considered an

EOQ model with time dependent demand and Weibull

distribution deterioration. Singh, S.R. et al. (2008)

developed an inventory model for deteriorating items

having stock dependent demand. They were allowed

shortages with partial backlogging in their study.

Singh, S.R. et al. (2009) proposed an inventory system

for perishable items with stock dependent demand and

time dependent partial backlogging. In their model,

constant holding cost has been taken. Tripathy, C.K.

and Mishra, U. (2010) developed a deterministic

inventory model for deteriorating items with constant

demand. Shortages were allowed with constant partial

backlogging rate in their study.

Our purpose in this paper is to establish a set of guiding

principles for the effective design and execution of an

inventory system. In this study, we have taken a more

realistic demand rate that depends on more than two

factors; one is the stock level available, second is time

and third is the selling price of the item. Deterioration

rate is taken as three parameter Weibull distribution.

The effect of life time, inflation and permissible delay

in payments are also taken into account. Shortages are

allowed in inventory and partially backlogged.

We believe that our study will provide a solid

foundation for the further study of this sort of important

inventory models with multi-variate demand rate in an

inventory model.

3 .Assumptions and Notations

Assumptions:

The following assumptions have been adopted for the

proposed model to be discussed:

1. Demand rate is dependent on stock selling price and

time.

2. Deterioration rate is a three parameter Weibull

distribution.

3. Inflation rate is also considered.

4. Shortages are allowed. Demand during the stock-out

period is partial backlogged.

5. Permissible delay in payments is taken into account.

Notations:

D(t) = a+ bt+ cI (t) -ds , demand rate in unit per unit

time

θ(t) = αβ (t -λ ) β-1 , deterioration rate in per unit of

time, α << 1

r: Inflation rate

C: Purchase cost per unit time per unit item

So: Selling cost per unit time per unit item

C1: Holding cost per unit time per unit item

C2: Shortage cost per unit time per unit item

C3: Shortage cost per unit time per unit item

A: set up cost

δ: The fraction of the demand during the stock -out

period that will be backordered,0 ≤ δ ≤ 1

H: Planning horizon

T: Replenishment time

λ: Life time

M: Permissible delay

Ie : Interest rate

S: Inventory level

4. Mathematical Formulation and Solution

Let us assume we get an amount S (>0) as initial

inventory. During the period (0, λ) the inventory level

gradually diminishes due to market demand only. After

life time deterioration can take place, therefore during

the period (λ, T1) the inventory level decreases due to

the market demand and deterioration of items and falls

to zero at time t1 .The period (T1 , T) is the period of

shortage which is partially backlogged.

))(()(

dstcIbtadt

tdI−++−= 0 ≤ t ≤ λ (1)

))(()()()( 1

dsscIbtatItdt

tdI−++−=−+ −βλαβ

λ ≤ t ≤ T1 (2)

)()(

dsbtadt

tdI−+−= δ T1 ≤ t ≤ T (3)

With the boundary conditions:

I (0) = S, I (T1) =0, I (T) = 0

Solution of equations (1), (2) and (3) are:

)

21()1(

)()( ctctct

ec

b

c

bte

c

dsaSetI

−−− −+−−−

+=

0 ≤ t ≤ λ (4)

International Journal of Pure and Applied Mathematics Special Issue

1310

))()((1

)()((1

)()(

2))([()( 11

11

11

1

22

11

++++ −−−+

+−−−+

−+−+−−= ββββ λλ

β

αλλ

β

αttTT

btT

dsatT

btTdsatI

))((33

1

22

1

22

1 )](2

)(2

))()(()2)(1(

cttetT

bctT

actT

b +−−++ −+−+−−−++

−βλαββ λλ

ββ

α

λ ≤ t ≤ T1 (5)

and

)](2

))([()( 22

11 tTb

tTdsatI −+−−= δ

T1 ≤ t ≤ T (6)

Equating the equation (4) and (5) at t = λ, one can get

1

11

1

1

22

11 )((1

)((1

)()(

2))([( ++ −

++−

+

−+−+−−= ββ λ

β

αλ

β

αλλ TT

bT

dsaT

bTdsaS

)1()(

)](2

)(2

)()((

)2)(1(

33

1

22

1

2

1

λβ λλλββ

α ce

c

dsaT

bcT

cdsaT

b−

−−−+−

−+−

++− +

)1(

2 2

λλλ c

c

ec

beb−++

−

(7)

4.1 Present Worth Purchase Cost

P.C. = ∫−− +

T

T

rtrM dttDeCTeCS

1

)(0

= 22

10

111 )()([

r

be

r

be

r

bTe

r

ebT

r

edsa

r

edsaCeCS

rTrTrTrTrTrTrM

−−−−−−− +−−+

−−

−+ (8)

4.2 Present Worth Holding Cost

H.C. = ∫ ∫+−− +

λ

λ

λ

0

)(

211

1

])()([

T

trrtdtetIdtetIC

2

)()(

1 )1

)()(

1(

)()([

cr

be

cr

eb

r

e

rrc

e

rcc

a

rc

Se

rc

SC

rrrrcrc λλλλλ λ −−−+−+−

−++−+

−+

++

−+

)22

)(()

22)([(]

)()(

2

1

2

11

)(

222

bcdsaT

TdsaC

rc

b

rc

be

rc

b

rc

be

cr

brcr

+−

++−−++

−+

++−−+−− λ

λλλ

)2(

)()(

)1)(2)(3(

)(2

)1)(2(

)()

33

2(

2

1

3

1

2

11

32

1

3

1

+

−−+

+++

−+

++

−++−

+++

β

λα

βββ

λα

ββ

λαβλλ

βββTdsaTbTTb

TT

)1)(2(

)()(

)1)(2(

)()()

44

3(

3)1(

)( 2

11

2

1

43

1

3

1

3

1

++

−−−

++

−−−+−+

+

−−

+++

ββ

λα

ββ

λαλλ

β

λα βββTcTdsaTdsa

TTbcTb


1311

3

))((

)3)(2(3

)(2

)1)(2(

)(

)1)(2)(3(

)()( 33

11

3

1

2

1

3

1 λ

ββ

λα

ββ

λα

βββ

λα βββrcdsaTcTbTcTbTcdsa +−

−++

−−

++

−−

+++

−−+

+++

4

)(

8

)(

2

))((

6

))((

)4)(2)(3(3

)(2 22

1

4

1

2

1

3

1

4

1 λλ

βββ

λα βTrcbTrcbTrcdsaTrcdsaTcb +

++

−+−

++−

−+++

−+

+

)3)(2)(1(2

))(55)(()(

)1(2

))()(()(

8

)( 3

1

22

11

4

+++

−+++−+

+

−++−−

+−

++

βββ

λββα

β

λλαλ ββTrcdsaTTrcdsarcb

)2)(2)(1(

)()()72(

)2)(1(2

))()((3

)1(2

))(()( 3

11

3

11

2

111

+++

−++−

++

−+++

+

−++−

+++

βββ

λαβ

ββ

λλα

β

λλα βββTTrcbTTrcbTTTrcb

8

)()(

4

)()(

8

)()(

)4)(3)(2)(1(

))(()32( 422

1

4

1

4

1 λλ

ββββ

λαβ βrccdsaTrccdsaTrccdsaTrcb +−

−+−

++−

−++++

−+++

+

15

)(

6

)(

16

)( 523

1

5

1 λλ rcbcTrcbcTrcbc +−

++

+−

(9)

4.3 Present Worth Set Up Cost

SAC = A (10)

4.4 Present Worth Shortage Cost

S.C. = ∫−−

T

T

rt dtetIC

1

)(32

= r

ebT

r

ebT

r

eTdsa

r

edsa

r

edsa

r

edsaC

rTrTrTrTrTrT 11 2

1

2

1

222

)()()()([

−−−−−−

+−−

+−

−−

−−

δ

332

1

2

11

r

be

r

be

r

ebT

r

bTerTrTrTrT −−−−

+−+− (11)

4.5 Present Worth Lost Sale Cost

L.C. = ∫−−+−

T

T

rtdtedsbtaC

1

))(1([3 δ

= )]()()(

)[1(22

13

111

r

e

r

e

r

eT

r

Teb

r

edsa

r

edsaC

rTrTrTrTrTrT −−−−−−

++−−−

−−

− δ (12)

Now regarding the permissible delay period M for settling the accounts, there arise two cases

M ≤ T1 or M > T1


1312

Case I: When M ≤ T1

4.6 Present Worth Interest Earned

I.E. = ])()([1

1

14 ∫−−

T

T

rt

r dtetDtTIC

= )())()(

1)((([{

)(

14c

bdsacS

cr

b

cr

be

rc

e

rcc

bdsacSTIC

rrc

e −−+−+−+

−+

−−+−+− λλ

(r

ebT

r

eTdsa

r

e

r

e

rc

b

rc

e

rc

e

rc

rTrrrrcrc 12

11

22

)(

2

)(

2

)({)}

1()

)()()(

1 −−−−+−+−

−−

+−−−+

−+

−+

λλλλλ λλ

483

)()()(

22

1

43

12

12

1

2

111 λλλ

λλ λλ

bcTbcTrccdsacTdsa

r

ebT

r

ebT

r

eTbrrTr

−−+−

−−−+−+−−−

2

)()()()(22

3

)( 22

1

2

2

22

3 λλλλλ λλλλλTrccdsa

r

edsa

r

edsa

r

eb

r

eb

r

becdsarrrrr +−

+−

−−

−−−−−

−−−−−−

)3)(1(

)(

)1(

)(

)1(2

)()(

6224

5 1

3

1

3

1

1

1

2

1

1

1

3

1

3

1

4

1

++

−+

+

−+

+

−−++−+

+++

ββ

λα

β

λα

β

λαλλ βββTTbcTTbcTTcdsabcTbcTbcT

3

)(22)(

6

)()(

)2)(1(

)()( 3

1

22

1

2

1

2

4

1

2

11111 cTdsa

r

be

r

ebT

r

ebT

r

edsaTrccdsaTcdsarTrTrTrT −

++++−

++−

−++

−−−

−−−−+

ββ

λα β

)3)(2)(1(

)()(2

)1(2

)(

)3)(2(

)()(

)2)(1(

)()( 3

11

1

1

3

1

3

1

2

1

+++

−−++

+

−−

++

−−−

++

−−+

++++

βββ

λα

β

λα

ββ

λα

ββ

λα ββββTTcdsbaTTbcTcdsaTcdsa

}])2)(1(2

)(

)4)(3)(2)(1(

)(3 2

1

2

1

4

1

++

−−

++++

−+

++

ββ

λα

ββββ

λα ββ TTbcTbc (13)

4.7 Present Worth Interest Payable

I.P. = ])([1

4 ∫−

T

M

rt

e dttIeIC

=

)1(

))(()()

2

)()(

33

2)(

22)([(

1

11

32

1

3

1

2

1

2

1

4+

−−−+

+−+−+−−

+

β

λα βTMTdsabcdsaMMT

TMMT

TdsaIC r

)2)(1(

)(3

)1(

)(

)1(

)(

)2(

)()(

)2(

)()( 2

11

1

11

1

1

2

1

22

1

++

−−

+

−−

+

−+

+

−−+

+

−−−

+++++

ββ

λα

β

λα

β

λα

β

λα

β

λα βββββTTbTMTbTTbMdsaTdsa

)2)(1(

)(2

)3)(2)(1(

)(3

)3)(2)(1(

)(3

)2)(1(

)( 2

1

33

1

2

++

−+

+++

−−

+++

−+

++

−+

++++

ββ

λα

βββ

λα

βββ

λα

ββ

λα ββββ TMbMbTbMMb

)2)(1(

)(

)2)(1(

)(

)1(

))(()(()

44

3(

3

22

1

1

11

44

1

4

1

++

−+

++

−−

+

−−−++−+

+++

ββ

λα

ββ

λα

β

λα βββMTTMT

dsaM

MTTbc


1313

)3)(2)(1(

)(

)2)(1(

)(

)2)(1(

)(

)1(2

))(( 322

11

122

1

+++

−−

++

−+

++

−−

+

−−+

++++

βββ

λα

ββ

λα

ββ

λα

β

λα ββββMbMMbTTbMMTb

)2)(1(

)(

)3)(2)(1(

)(

)2)(1(

)(

)1(2

))(()((

2232122

1

++

−+

+++

−−

++

−+

+

−−−+

++++

ββ

λα

βββ

λα

ββ

λα

β

λα ββββMcMbMcMcMMMTc

dsa

2

)()(

)4)(3)(2(3

)(2

)3)(2(3

)(2

)1(3

))(( 2

1

43133

1 MrcTdsaMcMbMcMbMMTcb +−+

+++

−−

++

−+

+

−−+

+++

βββ

λα

ββ

λα

β

λα βββ

)1(2

)()()(

8

)(

4

)(

3

))(( 1

1

2422

1

3

+

−+−+

+−

++

+−−

+

β

λα βTMrcdsaMrcbMTrcbMrcdsa

)1(2

)()(

)3)(2)(1(

))(()(

)1(

)()()(1

1

2

1

32

+

−++

+++

−+−+

+

−+−−

+++

β

λα

βββ

λα

β

λα βββ TMTrcbMrcdsaMMrcdsa

8

)()(

)2)(1(

)()( 422MrccdsaMMrcb +−

−++

−+−

+

ββ

λα β

15

)(

)3)(2)(1(

)()(

)4)(3)(2(

))((2

)2)(1(

)()(2 5343MrcbcMMrcbMrcbMMrcb +

−+++

−++

+++

−+−

++

−++

+++

βββ

λα

βββ

λα

ββ

λα βββ

]6

)(

4

)()(

)4)(3)(2)(1(

))((

)2)(1(2

)()( 23

1

22

1

42

1

2MTrcbcMTrccdsaMrcbTMrcb +

++−

−++++

−++

++

−+−

++

ββββ

λα

ββ

λα ββ

(14)

Hence, the total average cost of the system (TC1) is

given by

TC1= (SAC+H.C. + S.C. + L.C. + P.C. + I.P. - I.E. )/T

(15)

To minimize total average cost per unit time, the

optimal values of t1 and T can be obtained by solving

the following equations simultaneously.

0),(

1

11 =∂

∂

T

TtTC (16)

And

0),( 11 =

∂

∂

T

TtTC (17)

provided, they satisfy the following conditions:

0),(

,0),(

2

11

2

2

1

11

2

>∂

∂>

∂

∂

T

TtTC

T

TtTC

0),(),(),(

1

11

2

2

11

2

2

1

11

2

>

∂∂

∂−

∂

∂

∂

∂

TT

TtTC

T

TtTC

T

TtTC

(18)


1314

Case II: When M > T1

4.8 Present Worth Interest Earned

I.E’= ∫∫−− −+−

11

0

1

0

14 ])()(])()([

T

rt

T

rt

e dtetDTMdtetDtTIC

= )())()(

1)((([{

)(

14c

bdsacS

cr

b

cr

be

rc

e

rcc

bdsacSTIC

rrc

e −−+−+−+

−+

−−+−+− λλ

(r

ebT

r

eTdsa

r

e

r

e

rc

b

rc

e

rc

e

rc

rTrrrrcrc 12

11

22

)(

2

)(

2

)({)}

1()

)()()(

1−−−−+−+−

−−

+−−−+

−+

−+

λλλλλ λλ

483

)()()(

22

1

43

12

12

1

2

111 λλλ

λλ λλ

bcTbcTrccdsacTdsa

r

ebT

r

ebT

r

eTbrrTr

−−+−

−−−+−+−−−

2

)()()()(22

3

)( 22

1

2

2

22

3 λλλλλ λλλλλTrccdsa

r

edsa

r

edsa

r

eb

r

eb

r

becdsarrrrr +−

+−

−−

−−−−−

−−−−−−

)3)(1(

)(

)1(

)(

)1(2

)()(

6224

5 1

3

1

3

1

1

1

2

1

1

1

3

1

3

1

4

1

++

−+

+

−+

+

−−++−+

+++

ββ

λα

β

λα

β

λαλλ βββTTbcTTbcTTcdsabcTbcTbcT

22

1

2

1

2

3

1

4

1

2

11111 22

)36

)(

)2)(1(

)()((

r

be

r

ebT

r

ebT

r

ecTTrccTcdsa

rTrTrTrT −−−−+

++++++

−++

−−−

ββ

λα β

)3)(2)(1(

)()(2

)1(2

)(

)3)(2(

)(

)2)(1(

)()((

3

11

1

1

3

1

3

1

2

11

+++

−++

+

−−

++

−−

++

−−+

++++

βββ

λα

β

λα

ββ

λα

ββ

λα ββββTTcbaTTbcTcTTc

dsa

r

eTMdsaTTbcTbc r )1)()((}

)2)(1(2

)(

)4)(3)(2)(1(

)(3 1

2

1

2

1

4

1 −−−−

++

−−

++++

−+

−++ λββ

ββ

λα

ββββ

λα

)}()()1

(){(22

1

221

111

r

e

r

e

r

e

r

eTb

r

e

r

ea

r

e

r

e

rbTM

rTrrrTrrTrr −−−−−−−−

+−−+−+−−−+λλλλλ λλ

2

)()(

)1

)()(

1(

)(

)()({

cr

be

cr

be

r

e

rrc

e

rcc

dsa

rc

Se

rc

SC

rrrrcrc λλλλλ λ −−−+−+−

−++−+

−+

−+

+−

++

22

)(()

22)([(]

)()(

2

1

2

11

)(

222

bcdsaT

TdsaC

rc

b

rc

be

rc

b

rc

be

cr

brcr

+−

++−−++

−+

++−−+− λ

λλλ

)2(

)()(

)3)(2)(1(

)(2

)2)(1(

)()

33

2(

2

1

3

1

2

11

32

1

3

1

+

−−+

+++

−+

++

−++−

+++

β

λα

βββ

λα

ββ

λαβλλ

βββTdsaTbTTb

TT

)2)(1(

)()(

)2)(1(

)()()

44

3(

3)1(

)( 2

11

2

1

43

1

4

1

3

1

++

−−−

++

−−−+−+

+

−−

+++

ββ

λα

ββ

λαλλ

β

λα βββ TcTdsaTdsaT

TbcTb

3

))((

)3)(2(3

)(2

)2)(1(

)(

)3)(2)(1(

)()( 33

11

3

1

2

1

3

1 λ

ββ

λα

ββ

λα

βββ

λα βββrcdsaTcTbTcTbTcdsa +−

−++

−−

++

−−

+++

−−+

+++

4

)(

8

)(

2

))((

6

))((

)4)(2)(3(3

)(2 22

1

4

1

2

1

3

1

4

1 λλ

βββ

λα β TrcbTrcbTrcdsaTrcdsaTcb ++

+−

+−+

+−−

+++

−+

+


1315

)3)(2)(1(2

))(55)(()(

)1(2

))()(()(

8

)( 3

1

22

11

4

+++

−+++−+

+

−++−−

+−

++

βββ

λββα

β

λλαλ ββTrcdsaTTrcdsarcb

)3)(2)(1(

)()()72(

)2)(1(2

))()((3

)1(2

))(()( 3

11

3

11

2

111

+++

−++−

++

−+++

+

−++−

+++

βββ

λαβ

ββ

λλα

β

λλα βββTTrcbTTrcbTTTrcb

4

)()(

4

)()(

)4)(3)(2)(1(

))(()32( 22

1

4

1

4

1 λ

ββββ

λαβ βTrccdsaTrccdsaTrcb +−

++−

−++++

−+++

+

15

)(

6

)(

16

)(

8

)()( 523

1

5

1

4 λλλ rcbcTrcbcTrcbcrccdsa +−

++

+−

+−− (19)

Hence, the total average cost of the system (TC2) is

given by

TC2= (SAC+H.C.+S.C.+L.C.+P.C.-I.E.’)/T

(20)

To minimize the total average cost per unit time, the

optimal values of t1 and T can be obtained by solving

the following equations simultaneously.

0),(

1

12 =∂

∂

T

TtTC

(21)

And

0),( 12 =

∂

∂

T

TtTC

(22)

provided, they satisfy the following conditions:

0),(

,0),(

2

12

2

2

1

12

2

>∂

∂>

∂

∂

T

TtTC

T

TtTC

0),(),(),(

1

112

2

2

12

2

2

1

12

2

>

∂∂

∂−

∂

∂

∂

∂

TT

TtTC

T

TtTC

T

TtTC

(23)

5. Numerical Illustration

To illustrate the model numerically the following

parameter values are considered.

Demand rate a=25 units, b = 15 units, c = 9 units, d=2

Set up cost A= Rs. 200

Holding cost C1 = Rs. 4 per unit per year

Shortages cost C2= Rs. 9 per unit

Lost sale cost C3= Rs. 3 per unit per year

Backlogging rate δ = 0.5 unit.

Deterioration rate α =5, β = 0.02 units,

Interest earned I.E. = Rs. 0.2 per year,

C4 = Rs. 4.0 per unit, λ = 2 month

Interest payable I.P. = Rs. 0.15 per year,

For case I: permissible delay period is M=1

For case II: permissible delay period is M=10


1316

Case I: When M ≤ T1

Table 1: Variation in Parameters

N T1 T Initial Inv.:

S

TC1

1 2.55467 4.8325 455.4465 24901.6

2 2.33081 4.7923 423.2992 23760.3

3 2.20953 4.4833 321.4844 23673.4

4 2.04875 4.2035 256.1098 22869

5 1.96539 3.9635 210.2748 22796.2

6 1.83514 3.5172 185.3554 21839.9

7 1.75162 3.0934 174.7435 21482.1

8 1.63516 2.8348 153.9355 20287.9

9 1.43804 2.6529 128.7299 19508.1

10 1.32355 2.4216 96.4258 18673.4

Fig 1: Variation in T1 and T w.r.t. N

Fig 2: Variation in S w.r.t. N

Fig 3: Variation in TC1 w.r.t. N


1317

Table 2: Sensitivity Analysis of Optimal Solution

Variation

Parameter

Percentage Variation in Parameters

% -20% -10% 10% 20%

a

T1 0.1179 0.0745 -0.0742 -0.1187

S -19.764 -9.9102 9.9013 19.823

TC1 -19.0123 -9.4892 9.5154 19.0308

b

T1 -0.1329 -0.0587 0.0593 0.1335

S -0.1673 -0.0756 0.0763 0.1677

TC1 -0.55442 0.02776 0.02774 0.55483

Ie

T1 -2.2317 -0.6084 0.6084 1.2318

S -1.2355 -0.6102 0.6104 1.2358

TC1 -0.3511 -0.17564 0.17569 0.3513

r

T1 5.1378 2.5821 -2.5823 -5.1498

S 5.1652 2.588 -2.59 -5.1653

TC1 0.79524 0.38835 -0.38838 -0.79526

Fig 4: Variation in T1 w.r.t. a

Fig 5: Variation in S and TC1 w.r.t. a

Fig 6: Variation in T1 w.r.t. b

Fig 7: Variation in S and TC1 w.r.t. b


1318

Case II: When M > T1

Table 3: Variation in Parameters

N T1 T Initial I. : S TC2

1 5.51618 15.1664 680.3743 15657.2

2 5.82763 14.9812 363.8103 15442.53

3 6.08592 14.2887 255.8913 14520.93

4 6.45351 13.9489 201.4149 14296.37

5 7.24855 13.0631 168.6791 13993.51

6 7.82498 12.8736 145.1577 13528.34

7 8.3447 11.5458 131.6471 13213.68

8 8.84915 10.7967 121.5399 12207.53

9 9.34876 10.0853 104.6293 12042.81

10 9.36723 9.6574 96.925 11793.84

Fig 8: Variation in T1 and T w.r.t. N

Fig 9: Variation in S w.r.t. N

Fig 10: Variation in TC1 w.r.t. N


1319

Table 4: Percentage Variation in Different Parameters

Variation

Parameter

Percentage Variation in Parameters

% -20% -10% 10% 20%

A

T1 0.1175 0.0578 -0.0589 -0.1178

S -19.7894 -9.90233 9.91804 19.8242

TC2 -19.0254 -9.4893 9.5131 19.0263

b

T1 -0.147 -0.0732 0.0736 0.1473

S -0.18173 -0.09082 0.09087 0.18181

TC2 -0.05556 -0.02775 0.02787 0.05574

Ie

T1 -1.3258 -0.6774 0.6777 1.326

S -1.32034 -0.6793 0.6799 1.33042

TC2 -0.27867 -0.1672 0.16722 0.27871

r

T1 5.0387 2.5334 -2.5342 -5.039

S 5.05432 2.54217 -2.54218 -5.05432

TC2 0.85465 0.41802 -0.41806 -0.8547

Fig 11: Variation in T1 w.r.t. Ie

Fig 12: Variation in S and TC2 w.r.t. Ie

Fig 13: Variation in T1 w.r.t. r

Fig 14: Variation in S and TC2 w.r.t. r


1320

5 Conclusion In this paper, we have attempted to develop a

deteriorating inventory model with a very realistic and

practical demand rate. In present scenario, where

market trends change to a large extent, it is crucial that

more than one trend in account is taken while

considering customer’s demand. The proposed model is

very useful in the present market situation as almost

every item can be identified as having a time, stock and

selling price dependent demand rate.

The inventory is allowed to deteriorate during the

period it is stored and during this time period it

undergoes Weibull deterioration rate. This deterioration

rate has the advantage as it is being able to account for

more than one factor affecting deterioration. These

different factors might be humidity, temperature, lack

of proper lighting, etc. Weibull rate can easily account

for these different kinds of factors.

However, in real life, most of goods would have a span

of maintaining quality and the original condition and

during that period, no deterioration occurs. The item is

allowed a definite life time since no article in real life

can be expected to start deteriorating as soon as it is

produced. Deterioration sets in afterwards and it has

been made more realistic and practical by taking two or

three parameter Weibull distribution function.

Shortfalls are allowed and partially backlogged with an

inflationary environment using DCF approach to

impact economic feasibility to the model. Large cycle

length is suggested by the presence of inflation in cost

and its impact on demand. This model is also

applicable when supplier gives the trade credit to the

customer.

Further cases for stochastic demand and in more

realistic conditions can be developed.

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1322

dr.anubha goyal department of mathematics, krishna college ... · dr.anubha goyal department of...

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