Economic order quantity model under fuzz

1 Introduction

Usually, the models developed in conventional inventory systems are commonly associated with a lot of vague information or uncertainties which are generally linked with demand, various relevant costs, lead time, etc. These uncertainties are treated as randomness and are commonly handled by probability theory. But, in real life situations, these uncertain parameters may have greater chance to depart from the exact value and there arises a situation where these uncertain parameters do not follow any probability distribution. These features make the models unrealistic which eventually affect in decision-making to an inventory manager of any organisation. Therefore, to make the models more realistic it has become an important task to incorporate the concept of fuzzy set theory because fuzzy set theory proves to be a practical approach and its arithmetic is

a successful tool to solve the problems with uncertain parameters which are advantageous for many complicated problems involving parameters of uncertain nature. The fuzzy set theory was first introduced by Zadeh (1965). Sommer (1981) used fuzzy dynamic programming to solve an inventory and production scheduling problem in which there is

a contract to be fulfilled for providing a product and then withdraw from the market. The concept of fuzzy set theory discussed above clearly shows that this theory is more suitable in a case where the uncertainty in parameters exist and therefore, this theory is highly applicable and advantageous for the inventory models involving marketing parameters especially dealing with new product diffusion concept. The diffusion of new products in the market is mainly based on the theory of innovation-diffusion. Diffusion is defined as the process by which an innovation is communicated through certain channels over time among members of a social system (Rogers, 1995). Rogers (1962) had observed that when a new innovation is introduced in the market consumers show variable buying behaviour and categorise them according to their time to purchase. Diffusion models constitute a wide range of useful tools in assessing the nature of acceptability of the products in all fields of management whether it is of business,

Economic order quantity model under fuzzy sense 363 marketing or inventory management. Diffusion pattern also helps in knowing the specific

characteristics of an innovative behaviour which are useful to the managers while taking important decisions in inventory management field. Therefore, innovation-diffusion theory can be useful process to assist inventory manager for planning and making strategy to determine and control the stock levels within the physical distribution function so as to balance the need for product availability and at the same time minimising the stock holding and handling costs in a situation where models developed in inventory field containing the specific characteristics of innovative behaviour of products. Because of integrated effects of globalisation, scientific advancements and other related factors, the new products are constantly entering into the market and due to its unpredictable nature the life cycles of such products are persistently reducing. Also, the potential market size is not fixed in general and it changes according to the nature of the products introduced in the market. Therefore, developing inventory models based on innovation-diffusion criterion having dynamic potential market size is the need of the hours and it also becomes essential to tackle the problems of uncertain nature of its parameters associated with it to make the inventory models realistic. In this article, an attempt has been made to develop an inventory model with demand follows innovation diffusion criterion having dynamic potential market size. The concept of fuzzy set theory has been incorporated in the model by taking some of its parameters as fuzzy numbers to make the model realistic and to improve the accuracy in decision-making. The structure of this paper has been organised as follows. Section 2 describes the literature survey to provide a theoretical background for research of the model. In Section 3, model assumptions and notations are stated which have been used later to develop the mathematical equations of this model. Section 4 describes the mathematical framework which has been developed to show the model mathematically. In Section 5, a simple solution procedure in the form of algorithm is presented to perform the numerical exercise systematically. Section 6 presents the numerical examples which have been proposed on the basis of mathematical framework developed in Section 4. To well describe the nature, behaviour and applicability of the model the separate subsections such as 6.1, 6.2 and 6.3 containing observations, discussions and managerial implications sections respectively have been well presented. Finally, the model is concluded in Section 7.

2 Literature review

The role of fuzzy set theory in developing inventory models is not only advantageous from the managerial point of view but also its contribution in further research and development cannot be ignored. It is one of the suitable tool which is useful in most of the areas of research and development such as estimation, forecasting, optimisation, etc. The introduction of fuzzy set theory was firstly initiated by Zadeh (1965). Thereafter, various researchers made their contribution in developing inventory models using fuzzy concept which are as follows. Kacprzyk and Staniewski (1982) tackle the problem of controlling inventory over an infinite planning horizon. Park (1987) considered the economic order quantity (EOQ) formula by associating the fuzziness with the cost data in the fuzzy set environment. Petrovic et al. (1996) developed fuzzy models for the newsboy problem. Chen et al. (1996) developed backorder fuzzy inventory model under function principle. Roy and Maiti (1997) considered a fuzzy EOQ model with demand-dependent

364 K.K. Aggarwal et al. unit cost under limited storage capacity. Lee and Yao (1999) discussed EOQ model in

fuzzy sense for inventory without backorder. Yao and Lee (1999) developed a fuzzy inventory model with or without backorder for fuzzy order quantity with trapezoidal fuzzy number. Li et al. (2002) developed fuzzy models for single-period inventory problem. Hojati (2004) explained the probabilistic-parameter EOQ model and the fuzzy parameter EOQ model. Mandal et al. (2005) developed a multi-objective fuzzy inventory model with three constraints. Liu (2007) developed a method to find the membership function of the fuzzy profit when the demand quantity and unit cost are represented as fuzzy numbers. Benyoucef and Canbolat (2007) considered a model of fuzzy AHP-based supplier selection in e-procurement. Chen (2010) developed a model using a membership function approach to lot size re-order point inventory problems in fuzzy environments. Mahata and Goswami (2009) discussed an EOQ model with fuzzy lead time, fuzzy demand and fuzzy cost coefficients. Umap (2010) developed a fuzzy EOQ model for deteriorating items with two warehouses in which he has considered that the deteriorating rate is age specific failure rate. Tripathy et al. (2011) discussed a fuzzy EOQ model with reliability and demand-dependent unit cost. Sandeep et al. (2011) explained the application of fuzzy SMART approach for supplier selection. Chellappan and Natarajan (2011) developed a fuzzy rule-based model for the perishable collection production inventory system.

Also, the application of fuzzy set theory is highly desired when the models are developed using new product diffusion concept because new product diffusion models are based on the innovation diffusion theory and the parameters involved in it have high degree of uncertainty. The various diffusion processes have been studied in the marketing literature, which are as follows. The pure innovative model has been considered by Fourt and Woodlock (1960) whereas Fisher and Pry (1971) explain the pure imitative model. The Bass (1969) model admits both the innovative and imitative aspects of product adoption. The Bass (1969) model, introduced many years ago has been widely used in marketing by Mahajan and Muller (1979), Parker (1994), Rogers (1983, 2003, 1995) and Mahajan et al. (1990, 2000). Kalish (1985) has discussed an adoption model of a new product with price advertising and uncertainty. Mahajan and Robert (1978) have studied innovation diffusion in a dynamic potential adopter population. Sharif and Ramanathan (1982) have considered the application of dynamic potential adopter diffusion model through their study of diffusion of oral contraceptives in Thailand. The Fourt and Woodlock (1960) model explains the diffusion process in terms of number of customers who have bought the product by time ‘t’ by a modified exponential curve (Figure 1). This model captures the innovative characteristic with its coefficient ‘p’ as the coefficient of innovation. Zailani et al. (2007) considered a new product development benchmarking to enhance operation competitiveness. Rajagopal (2010) explained the case of bridging sales and service quality functions in retailing high-technology consumer products.

There are very few inventory models which have been developed using new product diffusion concept. Chern et al. (2001) have formulated the EOQ model in which the demand follows innovation diffusion criterion as considered by Bass (1969) model. Chanda and Kumar (2011) have explained the EOQ model with demand influenced by dynamic innovation effect. Aggarwal et al. (2011) have considered the EOQ model with innovation diffusion criterion having dynamic potential market size. Also, Chanda and Kumar (2012) explain the EOQ model having demand influenced by innovation diffusion criterion under inflationary condition. The periodic review inventory model with demand

Economic order quantity model under fuzzy sense

influenced by promotion decisions has been considered by Cheng and Sethi (1999). Sarkar et al. (2011) have developed an economic production quantity model with stochastic demand in an imperfect production system.

Figure 1 Fourt and Woodlock curve

Cumulative Cumulative Percentage of Percentage of Buyers Buyers

Time Time

Source: Lilien et al. (1999) To develop inventory models using innovation diffusion concept under fuzzy

environment will be advantageous for a number of reasons. Therefore, in this paper, an inventory model is developed where demand of the product is assumed to follow an innovation diffusion process as considered by Fourt and Woodlock (1960) model under a strict assumption that the potential market size is dynamic. To make the model more realistic, an attempt has been made to solve the model in the light of fuzzy set theory under the trapezoidal membership function. A numerical example and a comprehensive sensitivity analysis have been provided to illustrate the results of the proposed model. The results obtained using two methods of defuzzification such as median rule of defuzzification and graded mean integration representation method of defuzzification have been compared to know the practical utility of the model. The calculated results provide significant impetus while taking inventory decisions for new products.

2.1 Purely innovative diffusion process Pure innovative models assume that the diffusion takes place due to innovation or

external influence. Fourt and Woodlock (1960) model is the earliest pure innovative model. The demand model considered in this paper is purely innovative in nature. In this section, we very briefly discuss Fourt and Woodlock model, which will be used later on to conceptualise the demand model to find the optimal EOQ policy. The model is mostly used in the form

Q rQ (1 r ) t − 1 t = − (1)

366 K.K. Aggarwal et al. where

Q t sales at time ‘t’ which is a fraction of the potential sales Q total potential sales as fraction of all buyers

r rate of penetration of untapped potential t time period. The model assumes that the level of adoption capability Q t can be expressed as a function

of time t and generate an exponential declining curve of new-adopter sales over time. The values for the parameter Q and r, can be estimated on the basis of available historical

data points in such a way that the resulting curve fits the data well. On the basis of this curve, future adoption can be forecasted.

3 Assumptions and notations

The model emphasises a case which is concerned with the diffusion models of new product acceptance having objective to represent the level of spread of an innovation among a given set of prospective buyers. Here, the mathematical model developed is based on the framework of innovation diffusion criterion which is influenced by the Fourt and Woodlock (1960) model. The demand model considered here is time dependent and having the functional form as nt () = λ () t = pNt [() − Nt ( )] which is based on Fourt and

Woodlock (1960). The constant parameter ‘p’ mentioned in the demand function represented as a coefficient of innovation and interpreted as reflecting the extent of a consumer’s intrinsic propensity to purchase the product. The coefficient of innovation may also be stated as the likelihood that somebody who is not yet using the product will start using it because of mass media coverage or other external factors. For t ≥ 0, N(t) is the cumulative number of adopters of a product by time ‘t’ in a large target population. The potential market size of total number of adopters is not constant but dynamic in nature which is time dependent and has been denoted here as N t Assume that the time ( ).

interval ‘T’ is given and we are to plan an inventory policy for a certain commodity during time interval ‘T’ by using the demand rate λ(t) as explained above with its certain assumptions which are as follows. The replenishment rate is infinite implies that the replenishments are instantaneous. Lead time is zero and shortages are not allowed. There is only one product bought per new adopter. The innovation’s sales are confined to a single geographical area and it is assumed that there is no seasonality in sales of the new product. It is considered that the impacts of marketing strategies by the innovator are adequately captured by the model’s parameters.

In addition, the following notations will be used in developing the proposed model:

A ordering cost per order

C unit cost p(t) innovation effect at time t

I inventory carrying charge

Economic order quantity model under fuzzy sense

IC inventory carrying cost T

length of the replenishment cycle Q

number of items received at the beginning of the period p coefficient of innovation K(T) the total cost of the system per unit time I(t) on hand inventory at any time t n(t) = λ(t) = S(t) the number of adoptions at time t, i.e., demand at time t

N 0 initial potential market size.

4 Mathematical model

The basic demand model used in this paper is based on the following assumptions:

1 adoptions take place due to innovation-diffusion process and it is influenced by the innovation-effect (mass-media) only

2 the innovation effect is constant throughout the cycle time

3 the potential market size is time dependent (dynamic). The demand model used in this paper is based on the hypothesis that there exists potential

market whose size is time dependent (dynamic). With the passage of time prospective buyers adopt the product. Using (1) and the demand assumptions, the mathematical form of rate adoption at any time ‘t’ can be given as:

ft ϕ () () t = = pt () = p (2) 1 − Ft ()

where ϕ(t) is the hazard rate that gives the conditional probability of a purchase in a small time

interval (t, t + Δt), if the purchase has not occurred till time ‘t’ f(t) is the likelihood of purchase at time ‘t’

Ft () = () f t dt is the cumulative likelihood of purchasing the product at time ‘t’. ∫

Equation (2) can be rewritten as

ft () = p [ 1 − Ft ( ) ; where ] Ft () =

The number of adoptions (S(t)) at time ‘t’ can be derived by multiplying f(t) with the market size Nt ( ). Thus,

368 K.K. Aggarwal et al.

⎦ (4) ⎤ Here, the term ‘ [ ( ) pNt − Nt ( )]’ in equation (4) represents ‘innovators’ or people who

accept a new product independently of the decisions of others in a dynamic potential market size which varies with time.

It is true with the fact that the potential market size in case of new product diffusion does not remain constant with time. It varies with time. It may decrease or increase as time passes. The potential market size of a product may decrease for that product itself if its substitute product enters into the market and it may increase linearly or exponentially or follows any other distribution for a number of reasons such as bringing different kind of promotional efforts like advertising, discounting, etc. and by imparting external influences among people of the society. Here, we have considered a case when potential market size increases exponentially.

Now, when potential market size increases exponentially

Taking, Nt () = Ne gt 0 g > 0

gt

Therefore, λ () t = nt () = pNe ⎣ ⎡ 0 − Nt (), ⎤ ⎦ g > 0

Let p and N 0 be represented in fuzzy sense as p and 0 that is p and 0 are supposed

to be as fuzzy numbers. Therefore, ( ) t nt () pNe gt λ = = ⎣ ⎡ 0 − Nt (), ⎤ ⎦ g > 0 (6) The demand usage λ(t) which is a function of time plays pivotal role to shrink the

inventory size over a period of time. If in the time interval (t, t + dt) the inventory size is dipping at the rate λ(t)dt, then the total reduction in the inventory size during the time interval dt can be given by –dI(t) = λ(t)dt. Thus, differential equation describing the instantaneous state of the inventory level I(t) in the interval (0, T) is given by:

dI t () =− λ ( ), t 0 ≤ ≤ (7) t T dt

Using the equations (6) and (7) we get dI t ()

0 − Nt () ⎤ (8) dt

=− gt pNe ⎡

The solution of the differential equation (8) is:

e gt

p 2 ⎛ gt e e − pt ⎞

g ( g p ⎜ ) ⎜ +⎝ + g p ⎟ ⎟ ⎠

It () = I (0) − Np 0 + N 0 (9)

where t =⇒ 0 It () = I (0) and Nt () = (10) 0 Again, by model assumption, replenishment is instantaneous and shortages are not

allowed. Thus, the inventory level at the initial point of the planning horizon can be assumed to be the cumulative adoption of the product during the cycle time T. Hence,

Economic order quantity model under fuzzy sense

I (0) == Q λ () t dt (11) ∫

⇒ I (0) == Q Np ( ) − N

e gT − 1 2 ⎡ e p gT − 1 e − pT 1 ⎤

0 ⎥ g (12)

Using the equations (9) and (12) we get

= Np 0 (

) − N 0 ⎢ ( − )( − + ) ⎥

() I t dt = N pT 0 ( ) − N 0 ⎢ ( )( +

pT

The total cost of inventory system per unit time consists of the following elements:

Ordering cost per unit time (OC) = (15)

T QC

Material cost per unit time (MC) = (16)

⇒ MC = N Cp 0 ( ) − N 0 ⎢ ( )( +

e gT 1 2 ⎡ e gT

− 1 e pT − 1 ⎤

pC

( g + pT ) ⎢ g p ⎥

gT

IC T

Cost of carrying inventory per unit time (IHC) =

() I t dt (18)

⇒ IHC = ICN p 0 ( ) − ICN 0 ⎢ ( )( +

( g pT

Tg

370 K.K. Aggarwal et al. Let the inventory carrying charge ‘I’ in fuzzy sense be represented by ‘’

supposed to be as fuzzy number. Therefore, equation (19) can be written as follows

IHC = ICN p 0 ( ) − ICN 0 ⎢ (

e gT 1 2 ⎡ e gT 1 e − − pT p 1 ⎤

− ICN p 0 ( )

For the sake of convenience, let the total cost per unit time K(T) in fuzzy sense be represented as KT ( ). Therefore, using (15), (17) and (20) the total cost per unit time,

KT () is given by:

− ( pT ) (

e A gT − 1 p 2 ⎡ e gT

KT () =+ ICN p 0 − ICN 0 ⎢

ICN p ( )

+ N pC 0 ( )

− pT 1 ⎡

− N 0 ⎢ ( )(

e gT 2 gT

pC

( g + pT ) ⎢ g p ⎥

gT

Here, three parameters ,

fuzzy numbers have been used by making use of trapezoidal membership function.

Now the trapezoidal membership function μ A () x for the fuzzy numbers is described

as follows: ⎧ ( xc ) w − c

μ A =⎨ (22) ( x d ⎪ )

bd − ≤≤ ⎪⎩

0 otherwise

where 0 w 1, A A = [ , , , ], , , , cabd abcd are real

numbers and c ≤ a ≤ b ≤ d. The graphical representation of the trapezoidal fuzzy number has been described as follows.

Economic order quantity model under fuzzy sense

Figure 2 Trapezoidal fuzzy number

μ () x

Let ,

defined in equation (22) such as

p = ( p 1 , p 2 , p 3 , p 4 ) (23) N 0 = ( N 01 , N 02 , N 03 , N 04 ) (24)

I = ( II 1 , 2 , I 3 , I 4 ) (25)

Now, in order to make simpler the calculation of trapezoidal fuzzy number, here, we use the Chen’s (1985) function principle to calculate the fuzzy total cost per unit time of our proposed model under the following fuzzy arithmetical operations. We apply this principle for the operation of addition, subtraction, multiplication and division of fuzzy numbers.

Suppose A = (, , , ) a 1 a 2 a 3 a 4 and B = (, , , ) b 1 b 2 b 3 b 4 are two trapezoidal fuzzy

numbers. Then

a The addition of

A ⊕= B ( a 1 + ba 1 , 2 + ba 2 , 3 + ba 3 , 4 + b 4 )

where a 1 ,a 2 ,a 3 ,a 4 ,b 1 ,b 2 ,b 3 ,b 4 are any real numbers.

b The multiplication of

A ⊗= B (, , , ) c 1 c 2 c 3 c 4 where

λ = (a 1 b 1 ,a 1 b 4 ,a 4 b 1 ,a 4 b 4 ), λ 1 = (a 2 b 2 ,a 2 b 3 ,a 3 b 2 ,a 3 b 3 )

c 1 = min λ, c 2 = min λ 1 ,c 3 = max λ 1 ,c 4 = max λ. If a 1 ,a 2 ,a 3 ,a 4 ,b 1 ,b 2 ,b 3 ,b 4 are all non-zero positive real numbers, then

A ⊗= B ( abababab 11 , 22 , 33 , 44 )

c −=− B ( b 4 , − b 3 , − b 2 , − b 1 ),

372 K.K. Aggarwal et al.

A Θ= B ( a 1 − ba 4 , 2 − ba 3 , 3 − ba 2 , 4 − b 1 )

where a 1 ,a 2 ,a 3 ,a 4 ,b 1 ,b 2 ,b 3 ,b 4 are any real numbers.

=⎜ b 4 b 3 b 2 b 1 ⎟ where b 1 ,b 2 ,b 3 ,b 4 are all positive real numbers.

⎠ If a 1 ,a 2 ,a 3 ,a 4 ,b 1 ,b 2 ,b 3 ,b 4 are all non-zero positive real numbers then the division

AB ⎛ 1 , , , 2 3 4 ∅=⎜ ⎞

e Let α ∈ R, then

1 for α ≥ α 0, ( ⊗= A α a 1 , α a 2 , α a 3 , α a 4 )

2 for α < α ⊗= 0, ( A α a 4 , α a 3 , α a 2 , α a 1 )

Therefore, using the function principle method the membership function of KT () can be defined as

KT () = ( kT 1 ( ), kT 2 ( ), kT 3 ( ), kT 4 () ) (26)

where

I CN p 0 i i ( ) − i I CN 0 i

I CN p ( ) N i i

( g pT

( g + pT i ) ⎢ g p i ⎥

gT

The total cost per unit time containing fuzzy numbers is calculated using defuzzification method. Here, we have used two kinds of defuzzification method, one is median rule and the other is graded mean integration representation method to calculate total cost per unit time of the system.

Case 1 Defuzzification of total cost per unit time using median rule To find the minimisation of KT ( ), here, we apply the median rule as shown below in

Figure 3. The median k m of KT () can be derived from the following equation:

⎡ ( k m − k 1 )( + k m − k 2 ) ⎤⎡ ( k 4 k

⎣ k ⎦⎣ − m )( + 3 − k m ) ⎤

Economic order quantity model under fuzzy sense

( k 1 +++ k 2 k 3 ⇒ k k 4

Figure 3 Median rule

μ () x

Now, using equations (27) and (29) we have

1 () T = 1 I CN p 01 1 ( ) − 4 I CN

I CN p (

I CN

1 01 4 ( g + p

4 ( g 2 + 02 p 2 ) ⎢ g ⎥ 4 ( g + p 3 ) ⎢ p 3 ⎥

) − I CN

e gT

gT

I CN p

1 pT − e − 3 ⎤

gT

+ 3 I CN 03 ⎢

4 4 I CN g p p 04 4

gT

− pT 4

I CN

1 01 4 g p

I CN

g 4 04

374 K.K. Aggarwal et al.

− 4 I CN 04 p 4 ( )

e gT − 1 2 ⎡ gT

1 + I Cp N 1

4 Tg 2 4 ( g + p 4 ) T ⎢ g 2 ⎥

) ⎥ − 3 I CN p 03 3 ( )

I Cp 2 ⎡ 1 2 pT 2 ( ) 2 2 ( − e

4 ( g + pT

3 ⎢ g 4 ) 2 ⎥ ( g + pT 3 ⎢ p ⎥

2 02 2 + N 03 ⎢ ( ) ⎥

4 Tg 2 4 ( g + p 2 ) T ⎢ g 2 ⎥

1 I CN p ( 01 1 +

2 ) ⎢ p 2 ⎥ 4 Tg ⎣ 2 ⎦

4 − + e 04 ⎢

I Cp 2 ⎡ e gT

) ⎥ + N 04 ⎢ (

− pT

4 I Cp 2

4 ( g + pT

4 ( g + pT ⎢ p 2 1

+ N Cp ( )

04 4 ( g + pT

4 gT

Cp 2 ⎡ 1 − e − pT

) N Cp ( )

e gT − 1

4 ⎥+ 02 ( 2 g + p 4 ) T ⎢ p 4 ⎥ 4 gT

2 ⎡ e Cp gT − 1 ⎤

2 ⎡ 1 e − pT 2 ⎤

Cp

g ⎢ 02 ⎥ 4 ( g + pT 3 ) ⎢ p 3 ⎥

03 3 ( ) − N 02 ⎢ ( ) 4 ⎥ gT 4

( g + pT 3 ) ⎢ g ⎣ ⎥ ⎦

Cp 2 ⎡ 1 − e − pT

⎥ + N Cp ( )

e gT

4 04 ( 4 g + p 2 ) T ⎢ p 2 ⎥ 4 gT

4 ( 04 g + p 4 ) T ⎢ g ⎥ 4 ( g + pT 1 ⎢ p 1 ⎥

Economic order quantity model under fuzzy sense

I CN p ( ) I CN

I 4 − i CN 0(4 − i ) p (4 − i ) ( )

3 2 ⎡ 1 e − pT i + 1 3 gT

i = 0 4 ( g + p 4 − i ) ⎢ p 4 − i ⎥ ∑ i 0 4 Tg

I i CN

3 ⎡ e I gT Cp 2 − 1 ⎤ 3 I Cp 2 ⎡ 1 e − pT i + 1 ⎤

i = 0 4 ( g 2 0( 1) + + p 4 − i ) T ⎢ g ⎥ ∑ i = 0 4 ( g + p 4 − i ) T ⎢ p

∑ i = 1 4 gT T

Now, again by applying median rule of defuzzification as discussed in equations (28) and (29) and also explained through Figure 3, the lot size Q as obtained in equation (12) represented in fuzzy sense after defuzzification as Q m1 for Case 1 has been described as follows:

gT

1 2 e gT − p ⎡ e − 1 ⎤

Q m Np ( ) N 4 ⎢ ( ) ⎥ N 1 1 ⎢ (

p 2 ⎡ 1 e − pT

4 g − 04

4 ( g + 01 p + 1 ) ⎢ g ⎥ 4 ( g + p 4 ) ⎢ p 4 ⎥

2 ⎡ 1 e − − pT ( 2 ) p −

e gT 1 2 ⎡ e gT 1 ⎤

Np

4 g 4 + ( 02 g + p 2 ) ⎢ g ⎥ 4 ( g + p 3 ) ⎢ p 3 ⎥

4 g − 02 4 ( g + p 3 ) ⎢ g + ⎥ 03 4 ( g + p 2 ) ⎢ p 2 ⎥

+ Np 04 4 ( )

01 ⎢ ( ) ⎥ N

2 ⎡ 1 e − pT − 4 p − ⎤

e gT − 1 p 2 ⎡ e gT 1 ⎤

For optimum total cost, the necessary criterion is dk m 1 () T = (33)

dT dk m () T

Therefore,

1 = 0 dT

376 K.K. Aggarwal et al.

e ∑ − 0(4 )

I 4 − i CN 0(4 ) i p (4 ) i ( )

= 0 4 i − ( g + p 4 − i ) ⎢ p ⎣ 4 − i ⎥ ⎦ ∑ i = 0 4 Tg

⎡ e I gT 2

+ 1 Cp i + 1 ( −

4 Tg ∑ +

N Cp 0 i i ( ) 2 + N Cp 0 i i = 0

Now, for k m1 (T) to be convex dk 2 m 1 () T

> (35) 0 dT 2

dk 2 Therefore,

+ gT 2 N i + 1 ⎢ ( ) − i

i + 1 Cp 2 ⎡ 1 − e ⎤

3 Cp 2

0(4 − i )

Cp i + 1 −

3 Cp 2 3 2 1 pT e − +

gT

2 N 0( 1) i

Economic order quantity model under fuzzy sense 377

4 e gT 1 3

+ gT

∑ i = 1 4 gT ∑ − − i = 0 4 ( g + p i + 1 )

i = 0 ( + 4 − i ) ⎢ ⎣ 4 − i ⎥ ⎦ ∑ i = 0 4 Tg

+ 2 N Cp 0 i ∑ i

4 e gT

The solution of the equation m 1 () T = gives the optimum value of T provided it 0

dk

dT

satisfies the condition m 1 () T > Since the above cost equation (34) is highly 0.

dk 2

dT 2

non-linear, the problem has been solved numerically for given parameter values. The solution gives the optimum value T * of the replenishment cycle time T. Once T * is known the value of optimum order quantity Q m1 and the optimum cost k m1 (T * ) can easily be obtained from the equations (32) and (30), respectively. The numerical solution for the given base value has been obtained by using software packages LINGO and Excel-Solver.

Case 2 Defuzzification of total cost per unit time using graded mean integration representation method

The graded mean integration representation method based on the integral value of graded mean h-level of fuzzy number was proposed by Chen and Hsieh (1999) has been described as follows.

On the basis of trapezoidal membership function as defined and explained in equation (22) and Figure 2, suppose the trapezoidal fuzzy number be A = [ , , , ], cabd

a, b, c, d are real numbers and c ≤ a ≤ b ≤ d. Let

( xc ) L A () x

, c ≤≤ x a

( ac − )

378 K.K. Aggarwal et al. and

( x d ) R A () x

, b ≤≤ x d

( bd − ) where ( ) L A x and R A () x are the functions L and R of the trapezoidal fuzzy number ‘’

respectively. Suppose L − 1 and

A () h R − A () h are the inverse functions of the functions

L A () x and R A () x at h-level respectively as defined below.

L − 1 A () h =+− c ( ach ),0 h 1 and R − ≤≤ 1 A () h =+− d ( bdh ),0 ≤≤ h 1 hL [ − 1 () h − A 1 + R A ( )] / 2 h as

denoted as () is 1 hL ⎡ − 1

A () h + R − 1 h dh 1 1 A 1 () ⎤ hc +− ( ach ) ++− d ( b d h dh )

KA () = ⎣

] / hdh

/ hdh =

( c +++ 2 a 2 b d )

Figure 4 The graded mean h-level of fuzzy number ‘’

L A ~ () x R ~ A () x

L − 1 A ~ 0 c () h a h 1 ~

R [ ~ L − A () h − + 1 A h / 2 b R − ] 1 A () ~ () h d x

Hence, to defuzzify the total cost variable using graded mean integration representation method under trapezoidal membership function the k m has been defined as follows:

( k 1 + 2 k + k 2 2 k 3 + k m 4 = ) (37)

6 where

( kk 1 , 2 , k 3 , k 4 ) ≡ (,,,) cabd

Now, using equations (27) and (37) we have

Economic order quantity model under fuzzy sense 379

− pT () 1

− pT () 2

⎢ ( ) ⎥ + I CN

) 2 g 02

+ 3 I CN p 03 3 ( )

e gT − 1 p 2 ⎡ e gT 1 ⎤

2 ⎡ 1 e − pT 3 ⎤

g ⎥+ 3 3 03 ) ⎢ ⎥ 3 ( g + p 2 ) ⎢ p 2 ⎥

I CN g 2 02

I CN

− 4 I CN p 04 4 ()

6 ( g + pT

4 ) ⎢ g 2 ⎥ 01 6 ( g + pT 4 ) ⎢ p 2 4 ⎥

− I 3 CN p 03 3 ( ) + N

⎢ ( 2 ) 02 ⎥ + N

3 ( g + pT 3 ) ⎢ p 3 ⎥

I CN p () N

− 1 3 2 3 I Cp ⎡ e − 1 ⎤

⎢ () ⎥ N 3 3

3 Tg 2 3 ( g + pT

3 g + pT

⎡ − gT 1 2 e − pT 1 ⎤ 2 ⎡ 1 − e − () 4 4 4 I Cp I Cp

04 ⎢ () ⎥ + N 04 ⎢ (

6 ⎦ 1 ( g + pT

6 Tg

+ N Cp 01 1 (

− N 04 ⎢ ( ) ⎥ + N 01 ⎢ (

6 ( g + pT 1 ) ⎢ g ⎥

6 ( g + pT 4 ) ⎢ p 4 ⎥

+ N Cp 02 2 ( )

e gT − 1 Cp 2 3 ⎡ e gT 1 ⎤

3 ( g + pT 2 ) ⎢ g ⎥

3 ( g + pT 3 ) ⎢ p 3 ⎥

+ N 03 Cp 3 ( ) − N

02 ⎢ () ⎥ N

3 ( g + pT

3 ( g + pT 2 ) ⎢ p 2 ⎥

1 − 1 Cp 2 4 1 − e ⎤ + A N Cp

− pT ( 4 − )

e gT 1 gT

Cp 2 ⎡ e ⎤

04 4 − N 01 ⎢ () ⎥ + N 04 ⎢ (

6 ( g + pT 4 ) ⎢ g ⎥

6 ( g + pT 1 ) ⎢ p 1 ⎥ T

6 gT

Now, again by applying graded mean integration representation method of defuzzification as discussed in equation (37) and also explained through Figure 4, the lot size Q as obtained in equation (12) represented in fuzzy sense after defuzzification as Q m2 for Case 2 has been described as follows:

380 K.K. Aggarwal et al.

m 2 = Np ( ) 01 1 − N

⎢ ( 04 ) ⎥ + N

e gT − 1 p 2 ⎡ e gT − 1 ⎤

2 ⎡ 1 − − pT e 2 ⎤

+ Np

+ Np ( )

3 ( g + p 3 ) ⎢ g ⎥ 03 3 ( g + p 2 ⎢

+ Np 04 4 (

e gT 1 2 ⎡ e gT 1 ⎤

2 ⎡ 1 e − pT 4 ⎤

− N 01 ⎢

For optimum total cost, the necessary criterion is dk m 2 () T =

0 (40) dT

dk m () Therefore,

2 T = 0 dT

⇒ ( 1 I CN p 01 1 + 4 I CN p

) ⎡ I CN

e gT

1 ⎤ e gT

6 − ⎢ 4 ⎢ 04 6 ( g + p + 1 01 1 ⎥ ) 6

e + gT ( 2 02 2 + 3 03 3 ) −

3 ⎢ 3 03 3 + 2 ⎢ 02 ( g + p 2 ) 3 ( g p ⎥

6 ( g + pT 1 )

p 3 ⎡ e − 4 pT 4 2 () 2 2 ⎤ 4 4 I Cp ⎡ e gT

04 2 + 2 I CN 02 N

3 ( ⎢ ( g + p 3 ) ⎢ p ⎥ + 04 6 ( g ⎢ ⎥

6 g + pT 1 ) ⎢ g ⎥

+ pT 1 ) g

Cp 2 ⎡ e gT − 1 ⎤

I Cp 3 − pT 4 Cp 2 ⎡ e ⎡ gT e ⎤ − 1 ⎤

+ 04 2 + 04 ⎢ 2 ⎥ + N

6 ( g + pT 1 ) ⎢ g ⎥

6 ( g pT 1 )

3 ( g + pT 2 ) ⎢ ⎦ g ⎥

2 ⎡ 1 e − pT Cp 2 N

2 2 ⎡ 1 − e pT 1 ⎤

) N Cp ⎥ 02 2 ⎢ ⎤ ( − )

( + 4 ) ⎢ p 4 ⎥ 3 g + pT 3 ⎢ p 3 ⎣ ⎥ ⎦ ( ) ⎣ ⎦

6 g pT 2 2 2

2 2 ⎡ 1 e − pT Cp 4 ⎤ N

2 ⎡ e gT 1 ⎤

4 I Cp 4 ⎢ ( − ) ⎥

Cp 2 ⎥ gT

3 ( g + pT ⎢ g ⎥

6 g pT 2 ⎢ p 3 2 ) ( + 3 ) + 1

3 g pT

Economic order quantity model under fuzzy sense

p 3 ⎡ e − pT 3 2 1 − pT 3 2 ⎤ 2 2 I Cp ⎡ ( − e ⎤

3 g p ⎢ p ⎥ − 02 2 2 + N 01 ⎢ 2 ⎥ ( (41) + 2 )

3 ( g + pT 3 ) ⎢ p 3 ⎥

6 ( g + pT 4 ) ⎢ p 4 ⎥

( g + pT 3 ) ⎢ p 3 ⎥

+ 1 ) p ⎣ ⎢ 1 ⎦ ⎥ 3 ( g + pT 2 ) ⎢ p 2 ⎥

− N 01 ⎢ ()

+ 01 − N 02 ⎢ ()

6 ( g ⎢⎥

6 g pT 2

2 + 2 pT

4 ) g 3 g + pT 3 ⎢ g ⎥

( + pT 3 ) g ⎣⎦

3 ( g + pT 2 ) ⎢ g ⎥

3 ( g + pT 2 ) g

3 ( g + pT ⎢

I Cp

3 ( g + pT 2 ) ⎢ p 2 ⎥

3 ( g + pT 2 ) ⎢ p

( g + pT 4 ) ⎢ p 4 ⎥

6 ( g + pT 4 ) ⎢ p

3 g ⎣ pT ⎦ ⎣ 4 ⎦ ⎥ ( + 4 )

+ pT 4 ) ⎢ g ⎥

3 ( g + pT 2 ) ⎢⎣ p 2 ⎥ 3 ( g + pT

2 ⎡ 1 e − pT N Cp 4

04 4 ( ⎤ ⎢ − ) A

( g + pT 1 ) ⎢ p 1 ⎥ ⎣ T ⎦

Now, for k m2 (T) to be convex dk 2 m 2 () T

dT 2

m 2 () T = gives the optimum value of T provided it 0 dT dk 2 m 2 () T

dk

The solution of the equation

satisfies the condition

2 > Since the above cost equation (41) is highly dT 0.

non-linear, the problem has been solved numerically for given parameter values. The solution gives the optimum value T * of the replenishment cycle time T. Once T * is known the value of optimum order quantity Q and the optimum cost k

m2

m2 (T ) can easily be

obtained from the equations (39) and (38), respectively. The numerical solution for the given base value has been obtained by using software packages LINGO and Excel-Solver.

382 K.K. Aggarwal et al.

5 Solution procedure

Here, the aim of our model is to determine the optimum value of T which minimises the total cost per unit time such as k m1 (T) and k m2 (T) for median rule of defuzzification and graded mean integration representation method of defuzzification respectively for fixed values of other parameters associated with it. This optimum value of T will also help in getting the optimum lot size which eventually assists in mentioning the reorder point. The detailed step-wise solution procedure to calculate the optimum cycle length, optimum total cost and the optimum lot size has been summarised in the following algorithm.

Step 1 Input all parameter values such as different cost parameters, coefficient of innovation, potential market size, etc. with three parameters on which sensitivity analysis is to be performed separately such as inventory carrying charge, coefficient of innovation and potential market size are to be taken in the form of fuzzy numbers for both the cases median rule as well as graded mean integration representation method.

Step 2 Compute all possible values of ‘T’ using equation (34) and equation (41) for Case 1 and Case 2, respectively.

Step 3 Select the appropriate value of ‘T’ separately using equation (35) and equation (42) for Case 1 and Case 2, respectively. Here, the value of T is to be considered as optimal cycle length denoted as T * which will minimise k m1 (T) and k m2 (T) and

dk 2 m 1 () T

0 m 2 () > and T > for median rule and 0

dk 2

will satisfy the conditions

dT 2 dT 2

graded mean integration representation method respectively. Step 4 Compute k (T *

) and k m2 (T * ) for the optimum value T obtained in Step 3 using

m1

equation (30) and equation (38) for Case 1 and Case 2, respectively and hence calculate the corresponding value of Q (T * ) and Q (T * m1 m2 ) using equation (32) and equation (39) for median rule and graded mean integration representation method respectively.

The above steps are used for all replenishment schedules using appropriate parameter values. In order to obtain the values of ‘T’, we need to solve the equation (30) for Case 1 and equation (38) for Case 2 using LINGO and EXCEL-Solver software packages.

6 Numerical examples

The behaviour and application of the proposed model have been shown here by the following numerical analysis. Consider a hypothetical example in an inventory system with the following parameters in an appropriate units as follows:

A = $1,100/order, C = $300/unit, g = 0.20, N 0 = (10, 000, 15, 000, 20, 000, 25, 000)

I = (0.10, 0.15, 0.20, 0.25) and p = (0.005, 0.006, 0.007, 0.008) The results obtained for the above set of parameters have been solved for the values of T

by following the above solution procedure and using the software packages Excel-Solver and LINGO, which are as follows:

Economic order quantity model under fuzzy sense 383

m () =

T * 0.37, k T * = 1 Q

43, 631, m 1 () T = 46 (using median rule)

T * = 0.38, k

m 2 () T =

44, 027, Q m 2 () T * = 46.97 (using graded mean integration

representation method) The effect of changes in the parameters p, N 0 and I on the optimal cycle length, optimum

cost and the optimum order quantity of the inventory model has been shown numerically in the following numerical tables using the above stated solution procedure. Also, to prove the validity of the model numerically and to get the appropriate parameter values, the references have been considered as Chanda and Kumar (2011, 2012), Chandrasekaran and Tellis (2007), Sultan et al. (1990), Talukdar et al. (2002), Van den Bulte and Stremersch (2004), and Aggarwal et al. (2011).

Case 1 Numerical tables based on median rule Table 1 Sensitivity analysis on coefficient of innovation ‘’ p

Notes: For N 0 = (10,000, 15,000, 20,000, 25,000) I = (0.10, 0.15, 0.20, 0.25)

Table 2 Sensitivity analysis on inventory carrying charge ‘ ’

Notes: For p = (0.001, 0.002, 0.003, 0.004) N 0 = (10,000, 15,000, 20,000, 25,000)

384 K.K. Aggarwal et al.

Table 3 Sensitivity analysis on initial potential market size ‘ 0 ’ 0 T * k m 1 (T * )

Notes: For p = (0.001, 0.002, 0.003, 0.004) I = (0.10, 0.15, 0.20, 0.25)

Case 2 Numerical tables based on graded mean integration representation method

Table 4 Sensitivity analysis on coefficient of innovation ‘ ’ p p

Notes: For N 0 = (10,000, 15,000, 20,000, 25,000) I = (0.10, 0.15, 0.20, 0.25)

Economic order quantity model under fuzzy sense 385 Table 5 Sensitivity analysis on inventory carrying charge ‘ ’

Notes: For p = (0.001, 0.002, 0.003, 0.004) N 0 = (10,000, 15,000, 20,000, 25,000) Table 6 Sensitivity analysis on initial potential market size ‘ 0 ’ T 0 * k m 2 (T * )

Notes: For p = (0.001, 0.002, 0.003, 0.004) I = (0.10, 0.15, 0.20, 0.25)

6.1 Observations The results obtained from different numerical tables in the numerical example section

explain the effect of changes in the system parameters on the optimal values of total cost per unit time k

mi (T ), the optimal cycle length T * and the optimal order quantity Q

mi (T ).We have observed the following relationship during the numerical exercise.

a In both the cases median rule as well as graded mean integration representation method, as the average value of coefficient of innovation increases keeping other parameters constant then the optimal cycle length T * decreases while both the optimal order quantity Q mi (T * ) and the optimal total cost k mi (T * ) increases as depicted

386 K.K. Aggarwal et al. in Table 1 and Table 4. This is consistent with the reality as more investment on

promotion will increase the diffusion of a product in the market resulting in shrinkage of the optimal reorder cycle time as a result optimal cost is increased.

b As depicted in Table 2 and Table 5, it has been observed in both the cases that as the average value of inventory carrying charge increases keeping other parameters constant then both the optimal cycle length T * and the optimal order quantity Q mi (T * ) decreases while the optimal total cost k * mi (T ) increases. This is true with the fact that as inventory carrying charge increases, it compels the inventory manager to keep the inventory for reduced time period which leads to shrinkage of ordered quantity and in that process optimum total cost is increased due to increment in the ordering cost because of addition in the number of orders.

c On the basis of results obtained in Table 3 and Table 6, it has been observed in both the cases that as the average value of potential market size increases keeping other parameters constant then the optimal cycle length T * decreases while both the optimal order quantity Q mi (T * ) and the optimal total cost k mi (T * ) increases. This is explained as the growth of potential market size may increase the number of adopters which will force the inventory manager to keep more inventories for short time period to keep itself from fading out from the market as a result optimal cost is increased.

d If we compare the results obtained in the above six tables for both the cases separately, it can be clearly shown that the median rule of defuzzification is more suitable for this model than the graded mean integration representation method of defuzzification. Here, the suitability of defuzzification method has been judged on the basis of comparative results obtained for optimum cycle time, optimum total cost and the optimal order quantity for both the cases separately.

Figure 5 Cost-time graphs showing convexity of the cost functions for (a) Case 1 and (b) Case 2 (see online version for colours)

Cost-Time Graph

Cost-Time Graph

s t 806190

C 769984

C 806170 806150

Time

Time

(a)

(b)

Economic order quantity model under fuzzy sense 387

6.2 Discussion The parameters involved in the inventory models are generally not fixed and cannot be

predicted with certainty. These parameters fluctuate in their domain from the actual values. Hence, assuming constant parameters which are vague in nature and performing sensitivity analysis on a fixed parameter will make the inventory models unrealistic and will not give the desired results. This model is based on the new product diffusion concept and the parameters such as coefficient of innovation, potential market size, etc. associated with it are of uncertain nature which makes the model unrealistic. Therefore, to make the model realistic and to improve the accuracy in the desired results, here, three parameters namely coefficient of innovation, inventory carrying charge and the potential market size have been taken as fuzzy numbers and the sensitivity analysis of the model has been performed on these three parameters. The behaviour and sensitive nature of these parameters have been well explained in the observation section. On the basis of results obtained in the above six numerical tables, the comparative study of the two methods of defuzzification used in this model clearly shows that the median rule of defuzzification is more suitable for this model than the graded mean integration representation method of defuzzification. Therefore, this model will help the decision-makers while taking optimum decisions if the nature of the problems are matched with such model having specific characteristics as discussed above and moreover this model will facilitate in reducing the level of uncertainty in case of those parameters which are of uncertain nature and are calculated by converting them in fuzzy numbers.

6.3 Managerial implications The first and foremost challenge before any organisation is to sustain its growth

by maximising the profit and minimising the cost incurred. This becomes more crucial when the new products are introduced in the market because managing and scheduling of new products is a formidable challenge before the organisation. Here, inventory models are proved to be a great success to meet with such kind of challenges because inventory models give an idea to formulate the optimal policies. These models also suggest several policy implications for inventory forecasting and demand management, lead time analysis and reduction and optimal inventory level strategies. In this paper, an EOQ model for new products diffusion with the application of fuzzy set theory has been developed. Here, the role of fuzzy set theory is to make the model realistic which will be of a great help for the inventory manager while formulating the optimal policies of inventories based on this model. The uniqueness of this model is that how the inventory manager should keep the optimal cycle time of any fixed lot size entering into the inventory system when there is new product diffusion in a dynamic potential market size and the parameters associated with it are of uncertain nature. Also, when realistic models are developed by taking help of fuzzy set theory, it tries to minimise the risk factor associated with it while taking optimal inventory decisions by the inventory manager.

388 K.K. Aggarwal et al.

7 Conclusions

The results obtained through the models developed in the inventory management section are far away from the reality and certainty because of unavailability of adequate data and hence difficulties arise to get the desired results. The kind of uncertainties exist in almost all of the parameters of the inventory models which makes the models unrealistic and eventually these unrealistic inventory models affect the growth of any organisation. Therefore, it becomes an important task for us to make the models realistic and increase the degree of certainty as far as possible. Fuzzy set theory is a tool which addresses the problems arising out of such uncertainties. In this paper, an inventory model has been developed in which demand of the product is assumed to follow an innovation diffusion process as proposed by Fourt and Woodlock (1960) and the application of fuzzy set theory has been incorporated to make the model more realistic. The parameters

associated with it such as coefficient of innovation (p), potential market size (N 0 ), and the inventory carrying charge (I) have been considered in the form of fuzzy numbers. Here, calculations are made under the trapezoidal membership function with the help of function principle and the total cost per unit time has been calculated by using two kinds of defuzzification methods, one is median rule of defuzzification and other is the graded mean integration representation method of defuzzification. The comparative results of the above two methods of defuzzificztion have been well explained and discussed in the observations section. The proposed model acknowledged relationship between the coefficient of innovation, potential market size, inventory carrying charge and the optimal inventory policies. The results obtained are very encouraging, consistent with the reality and showing the relationship of sensitive nature of parameters with the optimal inventory policies and are essential to identify a trend. A simple solution procedure in the form of algorithm is presented to determine the optimal cycle time and optimal order quantity of the cost functions which helps in getting optimum cost of the model. The proposed model can be extended for backorders, quantity discount, partial lost sales, technological substitution, multiple generation, etc. and the future research can also be made by introducing different approaches of getting solutions such as geometric programming approach and genetic algorithm approach. The main limitation of this model is that as the existence and the uniqueness of the cost function in getting optimal solution has been shown numerically because of its highly non-linear nature. Thus, research on some alternative approach to get the optimal analytical solution of the problem is important. Also, this model uses trapezoidal membership function, the research can be extended for other different non-linear membership functions. The unique nature of this model can be explained as follows. As the models developed so far in the inventory field are unfortunately ignorant of the ideas of innovation diffusion along with fuzzy concept. There are few inventory models which have incorporated the ideas of innovation diffusion such as Chanda and Kumar (2011, 2012), and Aggarwal et al. (2011), but these models are not developed under fuzzy environment. The contribution and uniqueness of this paper is to develop an inventory policy with demand follows innovation diffusion criterion having dynamic potential market size under Fuzzy environment. A numerical example with comprehensive sensitivity analysis on various parameters of the model has been performed to show the effectiveness of the proposed inventory model and to demonstrate its practical usage.

Economic order quantity model under fuzzy sense 389

Acknowledgements

The authors are thankful to the anonymous reviewers and the editor for their constructive comments and suggestions. This paper has been revised in light of their suggestions and comments.

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