An Inventory Model for Deteriorating Items with QuadraticDemand and Partial Backlogging

ISSN 2319-8133 (Online)

An Inventory Model for Deteriorating Items with Quadratic Demand and Partial Backlogging

Bidyadhara Bishi^*1and Sudhir Kumar Sahu²

1

Research Scholar P.G. Department of Statistics, Sambalpur University, Orissa-768019, INDIA.

2

P.G. Department of Statistics,

Sambalpur University, Orissa-768019, INDIA.

email: [email protected] & [email protected]

(Received on: November 16, 2018)

ABSTRACT

In this paper, a deterministic inventory model is investigated for deteriorating items in which the demand rate of time quadratic and shortages are allowed and partially backlogged. The backlogging rate is a variable and dependent on the waiting time for the next replenishment. The backlogging rate is assumed to be dependent on the length of the waiting time for the next replenishment. The longer the waiting time is, the smaller the backlogging rate would be. The deterioration rate is assumed to be constant. A numerical example is taken to illustrate the model and the sensitivity analysis is also studied.

Keywords: Inventory; quadratic demand; deteriorating items; partial backlogging.

1. INTRODUCTION

The effect of deterioration of physical goods cannot be overlooked in many inventory

systems. In daily life, the deterioration of goods is a common incident. Deterioration is defined

as decay, spoilage, damage, evaporation, obsolescence, pilferage, loss of utility or marginal

value of a commodity that results in decreasing usefulness from the original one. For example

pharmaceuticals, chemicals, foodstuff etc. deteriorate significantly. The deterioration rate of

inventory in stock during the storage period constitutes an important factor, which has attracted

the researchers. Whitin (1957) is the first author who studied an inventory model for fashion

goods deteriorating at the end of a prescribed storage period. An exponentially decaying

inventory was developed by Ghare and Schrader (1963). Emmons (1968) established a

replenishment model for radioactive nuclide generators. Shah and Jaiswal (1977) established an order-level-inventory model for unpreserved items with a constant rate of deterioration.

However, in some inventory system, for many stocks such as designer produce, the length of the waiting time for the next replenishment becomes main factor for determining whether the backlogging will be accepted or not. The longer the waiting time, the smaller would be the backlogging rate. Therefore, the backlogging rate is variable and is dependent on the waiting time for the after that the replenishment. The partial backlogging rate was assumed to be a fixed fraction of demand rate during the shortage period. In this model, they established the Heuristic solution procedure. Many research articles by Ritchie (1980, 1984, 1985), Deb and Chaudhuri (1986), Hariga (1993), Kim (1995), Jalan and Chaudhuri (1999b), Lin et al. (2000) etc., analyzed linear time-varying demand. Aggarwal and Bahari-Kashani (1991); Hollier and Mak (1983), Hariga and Benkherouf (1994), Wee (1995), Jalan and Chaudhuri (1999a) etc, developed inventory models in which exponential time-varying demand has been taken. With the progress of time, researchers developed inventory models with time-dependent demand rates.

Goyal and Giri (2001) provided an exhaustive review of deteriorating inventory literatures. They indicated: the assumption of constant demand rate is not always applicable to many inventory items like fashionable clothes, electronic goods etc. As they experience fluctuations in the demand rate. Many products experience a period of rising demand during the growth phase of their product life cycle. Time-varying demands were first considered by Silver and Meal (1969). The deterioration occurs as soon as the retailer receives the commodities that have assumed in all inventory models for deteriorating items. But in real life situation, most of goods would have a span of maintaining quality or the original condition.

During that period, there was no occurrence of deterioration. These items are fish, fruit, meat, vegetables, alcohols, blood and gasoline. Consequently, the opportunity cost due to lost sales should be considered in the modeling. Chang and Dye (1999) argued that the backlogging rate should be dependent on the length of the waiting time. They were the first to give a definition for the time-dependent partial backlogging rate. Wee (1995) developed an article in the field of deteriorating items with shortages has revealed the economic order quantity with a known market demand rate.

Later, Ghosh and Chaudhuri (2004) Khanra and Chaudhuri (2003) etc., established

their models in which quadratic time-varying demand was considered. Sana (2010) considered

the case of ameliorating items and deteriorating items in a multi-item EOQ model where the

demand is being influenced by enterprises. In the opinion of many authors, an alternative or

perhaps more realistic approach is to consider quadratic time-dependence of demand. This

demand may represent all types of time-dependence depending on the signs of the parameters

of the time-quadratic demand function. Khanra et al. (2010) considered an EOQ model with

stock and price dependent demand rate. They developed production policy for a deteriorating

item with time-dependent demand and shortages. The present work attempts to model the

situation where the demand rate is a continuous function of time and items deteriorate at a

constant rate. Here shortages are allowed and the backlogging rate is variable and dependent

on the waiting time for the next replenishment. In the present paper, we have extended the work of Sahu et al. (2007) by taking the demand rate to be quadratic function of time. The purpose of the present paper is to give a new approach to the inventory literature or time- dependent demand patterns. Quadratic function of time is the best representation of accelerated growth (or decline) in the demand.

Let’s focus on the inventory models for deteriorating items with stock-dependent

demand rate and nonlinear holding cost. The first assumption considers that the higher the inventory level the greater the demand rate, and it has been used in the literature of inventory models because sometimes an increase in exposed merchandise may bring about increased sales of some items. The second one is useful when the holding cost per item is not proportional to the time held in stock, or the holding cost per unit of time is not proportional to the amount held in inventory. In this context, Giri and Chaudhuri (1998) consider two systems: Model A with time-dependent holding cost and Model B with stock-dependent holding cost. They assume that the rate of deterioration at any instant is a constant small fraction of the on-hand inventory and the demand rate is a concave potential function of the inventory level. Later on, Mahata and Goswami (2009) extend these two models to the case of fuzzy deterioration rate. In these models, the objective is to minimize the total inventory cost per unit time, defined as the sum of ordering, holding and deterioration costs. Sahu and Bishi (2017) extended the deteriorating Items under permissible delay in payments. There are many situations where certain percentage of the buyers is willing to wait for service, depending on the comparative price of the product. The retailer should develop a replenishing and pricing policy which will ensure the largest net profit. Yet, Balakrishnan, Pangburn, and Stavrulaki (2004) show that, for EOQ models with products having demand rates that increase with

inventory levels, “the optimal policy yields significantly higher profits than cost-based

inventory policies, underscoring the importance of profit-driven inventory management”. It therefore seems appropriate to reformulate the EOQ models with deteriorating items and stock-dependent demand rate from the perspective of maximizing profits per unit time, because the optimal policy might be different from the other one with inventory costs minimization.

Recently Begum et al. (2010b) has developed an EOQ model with quadratic demand with Weibull distribution deterioration. During the first three decades, many marketing researchers observed that in some retailer systems like supermarket, the demand of goods might be influenced by the on-hand inventory. For example, Levin et al. (1972) pointed out that at times, the presence of inventory has a motivational effect on the people around it. It is a common belief that a large pile of goods displayed in a supermarket will lead the customer to buy more. Dave and Patel (1981) developed an inventory model for deteriorating items with time-proportional demand.

The rest of the paper is organized as follows. Section 2, introduces the assumptions and notation used throughout the paper. Section 3, presents the analysis of the inventory system and proposes a procedure to calculate the optimal solution for the inventory problem.

In Section 4, numerical example is provided. Section 5, to illustrate the numerical analysis.

Finally, in section 6 we present conclusions.

2. ASSUMPTION AND NOTATION

The mathematical model is developed on the basis of the following assumptions:

2.1 Assumptions

(i) The lead time is zero.

(ii) A single item is considered with a constant rate of deterioration.

(iii) The replenishment occurs instantaneously at an infinite rate.

(iv) A deteriorated unit is neither repaired nor replaced during a given cycle (v) The deterministic demand rate D ( t ) varies quadratic with time, i.e.,

)

2

( t a bt ct

D    , where a , b and c are constants such that a , b , c  0 . (vi) Shortages are allowed and incompletely backlogged.

(vii)Throughout stock out period, the backlogging rate is variable and is dependent on the length of the waiting time for the next replenishment. The backlogging rate is assumed to be

) (

1 1

t T 

  where the backlogging parameter  is a positive constant and ( T  t ) is waiting time t

₁

 t  T .

2.2 Notations

A Ordering cost per order

h Inventory carrying cost per unit per unit time s Shortage cost per unit per unit time

l Opportunity cost due to lost sales per unit r Purchase cost per unit

t

1

the time at which the inventory level reaches zero, t

₁

 0 t

2

the length of period during which shortages are allowed, t

₂

 0

2

1

t

t

T   the length of cycle time )

1

( t

I the level of positive inventory at time t , 0  t  t

₁

)

2

( t

I the level of negative inventory at time t , t

₁

 t  t

₁

 t

₂

 Deterioration rate, a fraction of the on-hand inventory IM Maximum inventory level during [ 0 , T ]

IB Maximum inventory level during shortage period )

( t

D Demand rate at any time t  0

3. MATHEMATICAL MODELS

The stocks on hand are depleted due to demand and deterioration. The instantaneous

states of inventory for both the retailer and the customer at any instant of time ' t ' can be

represented by the inventory level decreases owing to quadratic demand rate during the time interval [ 0 , t

₁

] and [ t

₁

, T ] .

Fig-1: Graphical Presentation of Inventory System

The inventory level I

₁

( t ) and I

₂

( t ) s governed by the following differential equation

1 2

1

1

( ) ( ) ( ), 0

t t ct

bt a t

dt I t

dI         ^(3.1)

and

T t t t

T ct bt a dt

t

dI  









 

² ₁

2

,

) (

1

) (

) (

 ^(3.2)

With the boundary conditions

0 )

( 0

) ( )

(

_{2 1 1}

1

t  I t  at t  t and I t  IM at t 

I

The solution of equation (3.1) I

₁

( t ) ,which yields

2 1 2 1

2 1 1

0 1 , 1

1 1

1 ) 1

(

^{(1 )}

t t c t

b t e a

c t b t

t a

I

^{t t}



 





 

 

 

  



 

 

  

 

 

 

 

 

 



 

  

 



 

  







^



























(3.3)

The solution of equation (3.2) I

₂

( t ) ,which yields

 

 











 



 



      



2 2 1 3

2 2 1 2

2 1

2

1

) (

log 1 )) (

1 ( )) (

1 ) (

( t

t t t t

t c

t t b

t a

I 













2 1 1

2 1 2

1

1

,

2 ) ( 1 3 ) ( )

( t t c t t t t t t t t

b    



 



 

 

 

 



 ^(3.4)

The total cost per replenishment cycle consist of the following components

1. Ordering Cost (OC)=A (3.5) 2. Inventory Holding Cost (HC) per cycle ^ 

¹

0 1

( )

t

dt t I h

 ^{( 2 ( 1 ) 1 )}

) 2 1

(

_{1 1} ² ₁^{2 2}

2 4

1

1

     

     



^

b e

^

t t

t e

h a

t t



) 5 )

2 2 (

3 3 (

3 3 1 1

2 2 1

1

   

 c e

_t^

t  t  t 

(3.6) 3.

Cost of Deterioration(CD)

 ^{( 2 ( 1 ) 1 )}

) 2 1

(

_{1 1} ² ₁^{2 2}

2 3

1

1

     

     







b e t t

t e

rh a

t t



) 5 )

2 2 (

3 3 (

3 3 1 1

2 2 1

1

   

 c e

_t^

t  t  t 

(3.7)

4. Backordered Cost (BC) per cycle ^

¹



^2

^

1

)

2

(

t t

t

dt t I s

 

 





 

 



  

 



  

 

 



2 3 2

2 2 1 2

2 1

1 log 1 1 ))

( 1 ( )) (

1 (

t t t

t c

t t b

s a















 ^{b c t t c} 

t   

 3 2 ( 3 2 )

6

^{2 1 2}

2

2

 

 ^(3.8)

5. Lost Cost (LC) per cycle  ^{a bt ct}  ^dt

t t l t

t t

t







 



 









 



^{1 2}

1

2 2

1

)

( 1 1 1



 _



 



     

 



 



  





_{2 1 2} ^{22 2} ₁² _{2 1 2}^{2 23 1 2 22 2}₂

2 3 2

3

2    

t t t t t t

t t t t c

t t t b at l

 



 



 



      



2 2

2 2 1 2

1

1 log 1 )) (

1 ( )) (

1 ( 1

t t

t c t t a b











 ^(3.9)

When t  0 the level of inventory is maximum and it is donated by IM. Then from equation (3.3), we get

 

 



 



  



 

  

 



 

  

 



 









₂

2

_{3 1 1}

1

₁

1 1

₂



















^

c t

b t e a

c b

IM a

^t

(3.10)

The maximum backordered inventory is obtained at t  t

₁

 t

₂

and it is denoted by IB. Thus

from equation (3.4), we get

) (

_{1 2}

2

t t

I

IB   

2 3

2 2 1 2

2 1

1 log 1 )) (

1 ( )) (

1 (

t t

t c

t t a b











 

 



 



 



      





 

 







^{2 22}

2

^{1 2} ₂²

2

3









ct t ct ct

bt (3.11)

Thus the order size during total time interval ( T 0 , ) is given by IB

IM

Q   (3.12)

6. Purchase Cost (PC) per cycle  rQ

 

 



 



  



 

  

 



 

  

 



 

 



  



₂

2

_{3 1 1}

1

₁

1 1

₂



















^

c t

b t e a

c b

r a

^t

2 3

2 2 1 2

2 1

1 log 1 )) (

1 ( )) (

1 (

t t

t c t t b a











 

 

  

 



  

 

 



 

 



 







^{2 22}

2

¹² ₂²

2

3









ct t ct ct

bt (3.13)

Therefore the Total Variable Cost (TVC) per time unit is given by

TVC=  OC HC CD BC LC PC 

t

t     



₂

1

1

   ^{2 ( 1 ) 1} 

1 2

1

2 2

1 2

1 1

2 4 2

1

1

1

  

    

  

      



^

b e

^

t t

t e h a

t A t

t t

 ^{3 ( 2 2 ) 5}    ¹ 

3

¹

2 3 3

3 1 1

2 2 1

1

1

        

  

 





^



rh a e t

t t

t

c e

t t

   ^{3 ( 2 2 ) 5} 

1 3 1

( 2 2

3 3 1 1

2 2 1 2

2 1 2 1

1

1

       

 



 



  

t t

t c e

t t

b e

_{t t}

 

 





 

 

 

 



 



  



 

2 3 2

2 2 1 2

2 1

1 log 1 1 ))

( 1 ( )) (

1 (

t t t

t c

t t b

s a



 









 

  

 



 



 



  



 



 





 ( 3 2 ( 3 2 ) ) 2 3

6

3 2 2 2 1 2 1 2

2 2 2 1 2 2

2 1 2

2

t

t t t t t c

t t t b at l c t t c t b

 

 

 



 



 

 

 

  



 



 





2 2

2 2 1 2

1 2

2 2 2 2 1

1 log 1 )) (

1 ( )) (

1 ( 1

2 3 2

t t

t c

t t a b

t t t t



















 

  



 



 

 

 

  

 



 

  

 



 



 

 

₂

2

_{3 1 1}

1

₁

1

²

1

₂

( 1 (

₂^{1 2}

))





 









 

 

^

t t b

t a t c

b e a

c b

r a

^t

 

 

 







 



 

 

¹₃ ^{2 2} ₂ ^{2 22}

2

^{1 2} ₂²

2 ) 3

1 )) log(

( 1 (







 



 bt ct ct t ct

t t t

c (3.14)

The necessary condition for the total cost per time unit to be a minimum is 0

1

 

 t

TC and

t

2

TC





and 0

2 1 2

2 2 2

2 1

2

 



 









 

 

 







 



 









t t

TC t

TC t

TC

4. NUMERICAL EXAMPLE

Let A  250 , h  0 . 05 , s  1 . 2 , l  1 . 5 ,   8 ^,   0 . 005 ^, a  25 , b  20 ,

 15

c . The computed the optimal value of TVC is 92 . 53 .

5. SENSITIVITY ANALYSIS

Consider an inventory system with the following parameter in proper units:

Table-5.1

 t

₁

t

₂ ^TC

7.5 5.3 0.05 92.32 8 5.3 0.05 92.53 8.5 5.3 0.04 92.59 9 5.31 0.04 92.66

Table-5.2

 t

₁

t

₂

^TC 0.004 5.32 0.05 91.12 0.0045 5.31 0.05 91.72 0.005 5.3 0.05 92.53 0.0055 5.3 0.05 92.65

Table-5.2

c t

₁

t

₂

^TC

13 5.88 0.05 82.63

14 5.61 0.05 88.21

15 5.3 0.05 92.53

16 5.28 0.05 94.67

Table-5.3

b t

₁

t

₂

^TC

16 5.77 0.05 85.72 18 5.52 0.05 89.16 20 5.3 0.05 92.53 21 5.2 0.05 93.12

Table-5.4

a t

₁

t

₂

^TC

20 5.32 0.05 92.42 23 5.31 0.05 92.48 25 5.3 0.05 92.53 26 5.3 0.05 92.75

From the above tables 5.1, 5.2, 5.3, 5.4 and 5.5 .it is observed that the total variable cost increases if the parameters  ,  , a , b and c are increased. Also the parameters a ,

b and c are more sensitive than  ^and  ^.

6. CONCLUSION

In this paper, we have developed an order quantity model for deteriorating items with time – quadratic demand and partial backlogging. In particular, the backlogging rate is considered to be a decreasing function of the waiting time for the next replenishment an optimal replenishment policy has been considered for non-instantaneous deteriorating items with quadratic demand. Here shortages are allowed and the backlogging rate is variable. The backlogging rate is dependent on the waiting time for the next replenishment. A complete rate or a constant partial rate was used in many studies to describe the backlogging rate. But it is more realistic to assume the backlogging rate to be time proportional with waiting time of backlogging. The proposed model can be extended in several ways. It used in inventory control of certain non-instantaneous deteriorating items such as food items, electronic components, fashionable commodities and others. For instance, we may extend the demand function to stochastic fluctuating demand patterns or stock-dependent demand rate. Finally, we could extend the model to incorporate some more realistic features such as quantity discounts, permissible delay in payments, time value of money, a finite rate of replenishment, inflation, and probabilistic demand etc.

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