Two-warehouse fuzzy inventory model under the

78 Int. J. Operational Research, Vol. 11, No. 1, 2011

Copyright © 2011 Inderscience Enterprises Ltd.

Two-warehouse fuzzy inventory model under the conditions of permissible delay in payments

S.R. Singh* Department of Mathematics, D.N. College, Meerut (UP), India E-mail: [email protected] *Corresponding author

Neeraj Kumar Department of Applied Science, IIMT Engineering College, Meerut (UP), India E-mail: [email protected]

Rachna Kumari Department of Mathematics, Meerut College, Meerut (UP), India E-mail: [email protected]

Abstract: In this paper, a two-warehouse inventory model under the conditions of permissible delay in payments in fuzzy environment is developed. A rented warehouse is used when the ordering quantity exceeds the limited capacity of the owned warehouse. During the permissible delay period, no interest is charged by the supplier, but beyond this period interest is charged under the terms and conditions agreed upon and, moreover, interest can be earned on the revenue received during the credit period. Here, we have discussed four different cases to represent realistic situations of market: (1) when the inventory system has both the warehouse facilities, (2) when the owned warehouse has large capacity to store the inventory, (3) when simple EOQ model of single storage system and (4) when one does not wish to take RW services and OW has unlimited capacity. Finally, numerical example is given to illustrate feasibility of the model.

Keywords: inventory; two-warehouse system; permissible delay; fuzzy environment.

Reference to this paper should be made as follows: Singh, S.R., Kumar, N. and Kumari, R. (2011) ‘Two-warehouse fuzzy inventory model under the conditions of permissible delay in payments’, Int. J. Operational Research,Vol. 11, No. 1, pp.78–99.

Biographical notes: S.R. Singh is an Associate Professor in the Department of P.G. study and Research in Mathematics in D.N. (P.G.) College, Meerut (UP), India. His PhD is from C.C.S. University, Meerut (UP), India. His research

Two-warehouse fuzzy inventory model 79

interests include inventory control, supply chain management and fuzzy set theory. His research papers are published in Opsearch, Int. J. Operational Research (IJOR), Fuzzy Sets and Systems, International Review of Pure and Applied Mathematics, Int. J. Operations and Quantitative Management, Indian Journal of Mathematics and Mathematical Sciences, Ultra Scientist of Physical Sciences, etc.

Neeraj Kumar is a Lecturer in the Department of Applied Sciences in IIMT Engineering College, Meerut (UP), India. His research interests include inventory control with two storage facilities. His research papers are published in Ultra Scientist of Physical Sciences, International Transactions in Mathematical Sciences and Computer and International Transactions in Applied Sciences, Reflections des ERA Journal of Mathematical Sciences.

Rachna Kumari is a Reader in Mathematics in the Department of Mathematics in Meerut College, Meerut (UP), India. His PhD is from C.C.S. University, Meerut (UP), India. His research interests include inventory control, reliability theory. His research papers are published in, Ultra Scientist of Physical Sciences, International Transactions in Mathematical Sciences and Computerand International Transactions in Applied Sciences.

1 Introduction

In the busy markets such as super market, corporation market, municipality market, etc. the storage area of items is limited. When an attractive price discount for bulk purchase is available or when the item under consideration is a seasonal product such as the output of harvest or the cost of procuring goods is higher than the other inventory related cost or demand of items is very high or there are some problems in frequent procurement, management decide to purchase a large amount of items at a time. These items cannot be accommodated in the existing store-house (viz. the own warehouse (OW)) located at busy market place. In this situation, for storing the excess items, one (sometimes more than one) additional warehouse (viz., rented warehouse (RW)) is hired on rental basis, which may be located little away from it. We assume that the rent (holding cost for the item) of RW is greater than OW and hence the items are stored first in OW and only excess stock is stored in RW, which are emptied first by transporting the stocks from RW to OW in a continuous release pattern for reducing the holding cost. The demand of items is met up at OW only.

In most of the classical EOQ models, it is implicitly assumed that the quantity received per order matches the quantities requisitioned and that the supplier must be paid for the goods as soon as they are received by the system. In practice, however, supplier allows some fixed delay in settling the account and no interest charges are payable on the outstanding amount as long as the account is settled within the specified delay period. The supplier will obviously charge higher interest if the account is not settled by the end of the permissible delay period. Over the years, same research has appeared in the literature that treats inventory problems with varying conditions under permissible delay in payments.

In most of the inventory models that had been proposed in the early literature, the associated costs are represented as real numbers, although the real world inventory costs

80 S.R. Singh, N. Kumar and R. Kumari

usually exist with imprecise components. When uncertainty becomes a matter of debate, conventional approaches to treating uncertainty in inventory control focus on probability theory. In this case, customer demand as one of the key parameters and source of uncertainty have been most often treated by a probability distribution. However, the probability-based approaches may not be sufficient enough to reflect all uncertainties that may arise in a real world inventory system. Modelers may face some difficulties while trying to build a valid model of an inventory system, in which the related costs cannot be determined precisely. For example, costs may be dependent on some foreign monetary unit. In such a case, due to the change in the exchange rates, the costs are often not known precisely. Another source of uncertainty may arise because of the difficulty of determining exact cost components. In some cases trying to determine the precise values of such cost components may be very difficult and costly, if not impossible. For example, inventory-carrying cost is often dependent on some parameters like current interest rate and stock keeping the market price of the unit, which may not be known precisely.

This paper examines the effect of permissible delay in payment for a system with two storage facilities for single item. In this study, we propose to derive a two-warehouse inventory model with permissible delay in fuzzy environment where the system is buying quantity Q = DT in bulk order, which is larger than the capacity W of OW. Consequently, W units are stored at OW and the excess (DT W) units are stored at RW. The capacity of RW is assumed to be sufficiently large. Initially, demands are satisfied from RW in order to bring down the overall holding cost. The holding cost at RW is higher as compared to OW. In two-warehouse inventory control system, we consider demand rate, holding cost, purchasing cost and ordering cost are fuzzy in nature. At the end, four special cases in different scenarios are discussed in length. Finally, numerical example is given to illustrate feasibility of the discussed different cases in model.

2 Literature review

Fuzzy set theory, originally introduced by Zadeh (1965), provides a framework for considering parameters that are vaguely or unclearly defined or whose values are imprecise or determined based on subjective beliefs of individuals. Later on, Kaufmann and Gupta (1985) introduced to fuzzy arithmetic: theory and application. The application of fuzzy theory to inventory problem has been proposed by Kacprzyk and Staniewaski (1982). Chen (1985) gave the operations on fuzzy numbers with function principle. Roy and Maiti (1997) developed a fuzzy EOQ model with demand-dependent unit cost under limited storage capacity. Chen and Hsieh (1999) presented graded mean integration representation of generalised fuzzy numbers. Chen and Hsieh (2000) established a fuzzy economic production model to treat the inventory problem with all the parameters and variables, which are fuzzy numbers. Lee and Yao (1998) and Lin and Yao (2000) also wrote some papers about fuzzy production model. Yao and Chiang (2003) presented an inventory without backorder with fuzzy total cost and fuzzy storing cost defuzzified by centroid and signed distance. Dutta et al. (2005) developed a single-period inventory model with fuzzy random variable demand. In this study, they have applied graded mean integration representation method to find the optimum order quantity. Chen and Ouyang (2006) developed a fuzzy inventory model with permissible delay in payment. Dutta et al. (2007) presented an inventory model for single-period products with reordering opportunities under fuzzy demand. Chen and Chang (2008) presented an optimisation of


fuzzy production inventory model. In this study, they have considered a fuzzy opportunity cost and trapezoidal fuzzy costs under crisp production quantity or fuzzy production quantity in order to extend the traditional production inventory model to the fuzzy environment. They have use function principle as arithmetical operations of fuzzy total production inventory cost and also use the graded mean integration representation method to defuzzify the fuzzy total production and inventory cost.

Goyal (1985) established a single-item inventory model under permissible delay in payments. Mandal and Phaujdar (1989) studied the above mentioned author model by including interest earned from the sales revenue on the stock remaining beyond the settlement period. Aggarwal and Jaggi (1995) extended Goyal’s model to the case of deterioration. Jamal et al. (1997) extended Aggarwal and Jaggi (1995) with allowable shortages. Teng (2002) amended Goyal’s model by considering the difference between unit price and unit cost and established an easy analytical closed-form solution to the problem. De et al. (2003) developed an economic production quantity inventory model involving fuzzy demand rate and fuzzy deterioration rate. Chung et al. (2005) presented optimal inventory policies under permissible delay in payments depending on the ordering quantity. Chen and Ouyang (2006) extended Jamal et al. (1997) by fuzzifying the carrying cost rate, interest paid rate and interest earned rate simultaneously, based on the interval-valued fuzzy numbers and triangular fuzzy numbers to fit the real world. Recently, Chung and Huang (2009) developed a model to determine an optimal ordering policy under conditions of allowable shortages and permissible delay in payment.

Such type of system has been studied by Hartley (1976). Sarma (1983) has derived an inventory model with two storage facilities and an optimum release rule. An extension of Sarma’s (1983) work with a finite replenishment rate has been considered by Murdeshwar and Sathe (1985). They have considered an ordinary release pattern and bulk release pattern from RW to OW. Dave (1988) has modified the models Murdeshwar and Sathe (1985) and Sarma (1983) by taking into consideration the transportation cost, which depends on the quantity to be transported, as well as the distance between two warehouses. Zhou and Yang (2005) presented a two-warehouse inventory model with stock-level-dependent demand rate. Huang (2006) developed an inventory model under two levels of trade credit and limited storage space derived without derivatives. They all have considered deterministic models. Kumari et al. (2008) developed a two-warehouse inventory model for deteriorating items with partial backlogging under the conditions of permissible delay in payments. Chung et al. (2009) discussed a two-warehouse inventory model.

Above literature survey revealed that no researcher developed the model in which they discussed the effect of permissible delay of payment in fuzzy environment with two storage facilities. It is first time that we are considering the effect of permissible delay in payment in fuzzy environment with two storage facilities to make model more realistic.

3 Assumptions and notations

Following assumptions are made to develop the mathematical model.

1 The demand rate is fuzzy variable.

2 Holding cost at RW, holding cost at OW, ordering cost per order, purchase cost is fuzzy variable.


3 Capacity to store number of units in OW is also fuzzy variable.

4 Interest that can be earned on the sales revenue of units sold during the permissible delay period is less than interest charges per rupee per year.

5 Lead-time is zero.

6 During the time period, when the account is not settled, generated sales revenue is deposited in an interest bearing account. At the end of this period, the account is settled and interest charges are payable on the items in stock.

7 The time horizon is infinite.

8 Storage capacity of OW is W and that of RW is infinite. If the order quantity exceeds W, then excess units are kept in RW.

Following notations are used in the construction of mathematical model.

Q = Ordering quantity

D = Fuzzy demand rate per unit time (d1, d2, d3, d4)

F = Fuzzy unit stock holding cost at RW excluding interest charges (f1, f2, f3, f4)

H = Fuzzy unit stock holding cost at OW excluding interest charges (h1, h2, h3, h4)

A = Fuzzy ordering cost per order (a1, a2, a3, a4).

W = Capacity to store number of units in OW fuzzy type (w1, w2, w3, w4)

C = Fuzzy purchase cost per unit (c1, c2, c3, c4)

cI = Interest charges per rupee per year

eI = Interest that can be earned on the sales revenue of units sold during the permissible delay period ( )e cI I

T* = Permissible delay period in settling the accounts

T = Cycle time

4 Fuzzy sets, membership function, defuzzifying approach and arithmetical operations

4.1 Fuzzy sets

A class that admits the possibility of partial membership in it. Let X = { }x denote a space of objects. Then, a fuzzy set A in X is a set of ordered pairs:

, ( ) ,AA x x x X


where ( )A x is termed ‘the grade of the membership of x in A’. For simplicity, ( )A xis a number in the interval [0,1] , with the grades of unity and zero, respectively, full membership and non-membership in the fuzzy set. An object (point) P contained in a set (class) Q is an element of ( )Q P Q .

4.2 Membership function

We first introduce the concept of membership function. There are many different shapes of membership function in the inventory management such as the triangle and trapezoid. The shapes of the triangle membership function and the trapezoid membership function are shown in Figures 1 and 2.

Figure 1 Membership function for triangle number

Figure 2 Membership function for trapezoid number


We assume that Ã is a fuzzy number. If Ã is a triangle number, Ã can be represented as Ã = 1 2 3[ , , ]k k k subject to the constraint 1 2 30 k k k . While Ã is a trapezoid fuzzy number, Ã = 1 2 3 4[ , , , ]k k k k subject to the constraint that 1 2 3 40 k k k k . Membership function of the triangle and trapezoid fuzzy numbers can be defined as follows.

1 3

11 2

2 1

32 3

3 2

0 ,

( ) ( )

( )

A

x k x k

x kx L x k x k

k kk x

R x k x kk k

1 4

11 2

2 1

2 3

43 4

4 3

0 ,

( )( )

1

( )

A

x k x k

x kL x k x k

k kxk x k

k xR x k x k

k k

where ( )A x is a membership function.

4.3 Graded mean integration representation method

In this paper, we denote generalised fuzzy number Ã in Figure 1 as Ã = ( , , , , )LRAc a b d .

When 1A , we simplify the notation as ( , , , )LRA c a b d Chen and Hsieh (1999) introduced the graded mean integration representation method of generalised fuzzy number based on the integral value of graded mean h-level of generalised fuzzy number. Here, we describe its meaning as follows:

Let 1L and 1R are inverse function of L and R, respectively, then the graded mean h-level value of generalised fuzzy number ( , , , , )A LRA c a b d W is h 1 1( ( ) ( ))L h R h /2 as Figure 3.

Figure 3 The graded mean h-level of generalised fuzzy number A = (c, a, b, d, WA)LR


Then, the graded mean integration representation of Ã is

1 1

0

0

( ) ( ) / 2 d( )= , where 0 and 0 1

d

A

A

W

A AW

h L A R h hP Ã h W W

h h

In this paper, we use generalised trapezoidal fuzzy number as the type of all fuzzy parameters in our proposed inventory models. The very popular generalised trapezoidal fuzzy number B is a special case of generalised fuzzy number and be denoted as

( , , , , )BB c a b d W its’ corresponding graded mean integration representation is

0

0

( ( ( ) / ) h) / 2( )

d ( 2 2 ) / 6

B

B

W

B

W

h c d a c d b h W dP B

h h c a b d

where a, b, c, d are any real numbers.

4.4 Properties of second function principle

Chen (1985) proposed second function principal to be as the fuzzy arithmetical operations between generalised trapezoidal fuzzy numbers. Because it does not change the type of membership function of generalised fuzzy number after arithmetical operations. It reduces the trouble and tediousness of operations. Furthermore, Chen already proved the properties of fuzzy arithmetical operations under second function principle. Here, we describe some properties of fuzzy arithmetical operations as follows:

Suppose 1 1 1 1 1( , , , )A c a b d and 2 2 2 2 2( , , , )A c a b d are two generalised trapezoidal fuzzy numbers. Then

1 The addition of Ã1and Ã2 is 1 2 1 2 1 2 1 2 1 2( , , , )Ã Ã c c a a b b d d .

2 The multiplication of Ã1 and Ã2 is 1 2 1 2 1 2 1 2 1 2( , , , )Ã Ã c c a a b b d d .

3 2 2 2 2 2( , , , )A d b a c . Then, the subtraction of Ã1 and Ã2

is 1 2 1 2 1 2 1 2 1 2( , , , )Ã Ã c d a b b a d c .

4 122 2 2 2 21/ = (1/ ,1/ ,1 / ,1 / )Ã A d b a c where 2 2 2, ,c a b and d2 are all positive real

numbers. If 11 1 1 2 2 2, , , , , ,c a b d c a b and d2 are all non-zero positive real numbers, then

the division of 1A and Ã2 is 1 2 1 2 1 2 1 2 1 2( / , / , / , / )ØA A c d a b b a d c .

5 Construction of cost function

Initially, the system is buying quantity Q = DT in bulk order, which is larger than the capacity W of OW. Consequently, W units are stored at OW and the excess (DT W)


units are stored at RW. The cost function is constructed by taking into account fuzzy set up cost, fuzzy holding cost and interest earned and paid. Two cases may arise here, viz.

1 T T

2 T T .

It can easily be seen from Figure 4 that per unit time cost consists of the following fuzzy costs.

1 fuzzy cost of placing an order per unit time is /A T

2 total fuzzy holding cost at RW per unit time is [ ( )( )] 2 ]F D T W D T W D T

3 total fuzzy holding cost at OW per unit time is ~ ~ ~

( )[1 (2 )]~

H W W D T

4 fuzzy interest earned per unit time is *2 *e / 2 , ifTD C I T T T and

* *e ( / 2 ) , ifTD C I T T T

5 fuzzy interest payable per unit time is * *c ( ) / 2 , ifT TD C I T T T and is zero,

if *T >T.

Note that the interest earned should be subtracted from other fuzzy costs in order to get the net total fuzzy costs per unit time.

5.1 Case-I: determination of order quality when T T

Here, we discuss the fuzzy inventory model with fuzzy parameters of annum fuzzy demand 1 2 3 4[ , , , ]D d d d d , fuzzy holding cost 1 2 3 4[ , , , ]F f f f f at RW, fuzzy holding

cost 1 2 3 4[ , , , ]H h h h h at OW, fuzzy ordering cost per order 1 2 3 4[ , , , ]A a a a a , fuzzy

purchase cost 1 2 3 4[ , , , ]C c c c c and fuzzy capacity to store number of units

1 2 3 4[ , , , ]W w w w w in OW. In this case, the total fuzzy cost per unit time is given by

Figure 4 Two-warehouse inventory system


1( ) Fuzzy cost of placing an orderTotal fuzzy holding cost at RW Total fuzzy holding cost at OWFuzzy interest payable Fuzzy interest earned

Z T

1

2*

*2

( ) T 2

T T1 2

2T

2

c

e

AZ T F D W D T W D T

T

IH W W D T D C

I TD C

T

*21

*

( ) 2

[ (2 )]2 2

c e

e

c

Z T A D F H W W D D C T I I

TITD T D F D C

F H W D C I T

(1)

By Second Function Principal, we have

2 2 *21 1 1 1 4 1 1 1

4

*1 11 14 1 4 4 4

2 2 *22 2 2 2 2 2 2

3

*2 22 23 2 3 3 3

2 2 *23 3 3 3 3 3 3

2

3

1( ) 22

2 21 2

2

,2 21 2

2

c e

cc

c e

cc

c e

Z T a d f h w d c T I Id T

d c TId f Tf h w d c I T

a d f h w d c T I Id T



d *3 3 32 3 2 2 2

2 2 *24 4 4 1 4 4 4

1

*4 44 41 4 1 1 1

,2 21 2

2

2 2

cc

c e

cc

f T d c TIf h w d c I T



(2)


Now, we defuzzify the total cost per unit time, using graded mean integration representation method, the result is

2 2 *21 1 1 1 4 1 1 1

4

1 11 14 1 4 4 4

2 2 *22 2 2 3 2 2 2

3*

2 2 2 2 3 2 3 3 3

2 2 *23 3 3 2 3 3 3

2

3

1 1( ) 26 2

*2 2

1 2

2

1 2

c e

cc

c e

c c

c e

P Z T a d f h w d c T I Id T



d f T d c TI f h w d c I T


d

1 1

*3 3 3 2 3 2 2 2

2 2 *24 4 4 4 4 4 4

1

*4 44 44 1 1

2

1 22

2 2

c c

c e

cc

f T d c TI f h w d c I T



(3)

For minimum total cost per unit time, the optimum value of T = T10 will be solution of the

1d (Z (T)) / d 0P T , which gives

2 2 *210 1 1 1 4 1 1 1

4

2 2 *22 2 2 3 2 2 2

3

2 2 *23 3 3 2 3 3 3

2

2 2 *24 4 4 1 4 4 4

1

1 1 1 2 2 2

3 3 3 4 4 4

1 2

2 2

2 2

1 2

2

2

c e

c e

c e

c e

c c

c c

T a d f h w d c T I Id

a d f h w d c T I Id



d f c I d f c I

d f c I d f c I1/ 2

(4)

And minimum total cost per unit time 1 10( ( ))P Z T is

3

2 2 *21 10 1 1 1 4 1 1 1

4

2 2 *22 2 2 3 2 2 2

2 2 *23 3 3 2 3 3 3

2

1 1 26

2 2

2 2

c e

c e

c e

P Z T a d f h w d c T I Id




1

2 2 *24 4 4 1 4 4 4

1/ 2

1 1 1 2 2 2 3 3 3 4 4 4

* *4 1 4 4 4 2 3 2 2 2

*1 4 1 1 1

1 2

2 2

2

c e

c c c c

c c

c


d f c I d f c I d f c I d f c I

f h w d c I T f h w d c I T

f h w d c I T

(5)

Then, the optimum order quantity is

1 10 10 1 10 2 10 3 10 4 101 2 26

Q T D T d T d T d T d T (6)

Special case 1: when costs, demand and capacity to store number of units in OW are real numbers, they are d1 = d2 = d3 = d4 = D, f1 = f2 = f3 = f4 = F, h1 = h2 = h3 = h4 = H,a1 = a2 = a3 = a4 = A, c1 = c2 = c3 = c4 = C, w1 = w2 = w3 = w4 = W. Then, the optimum of T = T10 is

1/22 2 *2c e

10 2c

2AD F H W D CT I IT

D F CI (7)

Then, optimum order quantity is 1/22 2 *2

1 10

2 ( ) c e

c

AD F H W D CT I IQ T

F CI (8)

And minimum cost Z1(T10) per unit is 1/22 2 *2

1 10

*

2 ( )

( )

c e c

c

Z T AD F H W D CT I I F CI

F H W DCI T (9)

These equations are same as those given by Shah and Shah (1992).

Special case 2: when we take holding cost in both warehouses are same (H = F),optimum values of T

1

2 *2 2 *210 1 1 1 1 2 2 2 2

4 3

2 *23 3 3 3

2

2 *24 4 4 4 1 1 1

1 / 2

2 2 2 3 3 3 3 4 4 4

1 22 2

2 2

1 2

2 2

c e c e

c e

c e c

c c

T a d d c T I I a d d c T I Id d

a d d c T I Id

a d d c T I I d h c Id

d h c I d h c I d h c I

(10)


1

2 *21 10 1 1 1 1

4

2 *2 2 *22 2 2 2 3 3 3 3

3 2

2 *24 4 4 4 1 1 1 2 2 2

1/2

3 3 3 4 4 4

* *1 1 2 2

1 1 26

2 22 2

12 2

2

2

c e

c e c e

c e c c

c c

c c

P Z T a d d c T I Id

a d d c T I I a d d c T I Id d

a d d c T I I d h c I d h c Id

d h c I d h c I

d c I T d c I T * *3 3 4 42 c cd c I T d c I T

(11)

11 10 1 10 2 10 3 10 4 106( ) [ 2 2 ]Q T d T d T d T d T , where 10T is given above.

When all costs real numbers, we have 1/22 *2

10 2

2 c e

c

AD D CT I IT

D H CI (12)

1/22 *2

1 10

1/22 *2*

1 10

2and

2

c e

c

c ec

c

AD D CT I IQ T

H CI

AD D CT I IZ T DCI T

H CI

(13)

These equations are same as those given by Goyal (1985).

Special Case 3: when we take holding cost in both warehouses is same, that is, H = F,Ic = Ie = 0, F = H and T* = 0, then optimum values of T is

1/2

1 1 2 2 3 3 4 44 3 2 1

101 1 2 2 3 3 4 4

1 2 2 12 2 2 2

2 2

a d a d a d a dd d d d

Td h d h d h d h

(14)

1 10 1 1 2 2 3 3 4 44 3 2 1

1/21 1 2 2 3 3 4 4

1 1 2 1 12 2 2 26

2 2

P Z T a d a d a d a dd d d d

d h d h d h d h

(15)

and 11 10 1 10 2 10 3 10 4 106 2 2Q T d T d T d T d T , where 10T is given above.


When all costs real numbers, we have 1/2 1/2

1/210 1 10 1 10

2 2, , and [2 ]A ADT Q T Z T ADH

DH H (16)

These results are similar to those of single storage EOQ model, that is, these equations are same as those of classical EOQ of Naddor (1966).

Special case 4: suppose that we do not wish to use RW at all and then we order W units per replenishment, that is, we take 1 WDT , where 1 /T W D is cycle time. In this case, the total cost per unit time due to OW is

*21 1

*

2 [2 ]

1 ( )2

c e

e c

Z T A D D D C T I I W

W H W C I D C I T (17)

By second function principle, we have

*2 11 1 1 1 1 1 1 1

1

* *2 21 1 2 2 2 2 2 2

2

* 2 32 2 c 3 3 3 3 3 3

3

2 43 3 4 4 4 4 4 4 4 4

4

1= 2 + + +2 2

1, 2 + + +2 2

1 22 2

1, 22 2

c e c

c c e c

c e c

c c e c c

wZ T a d d c T I I h c I

w

wd c I T a d d c T I I h c I

ww

d c I T a d d c T I I h c Iw

wd c I T a d d c T I I h c I d c I T

w

(18)

Now, we defuzzify the total cost per unit time, using graded mean integration representation method, the result is

2 11 1 1 1 1 1 1 1 4 4

4

22 2 2 2 2 2 2

3

23 3 3 3 3 3

2

03 3 3 2 2 4 41

44 4 1 1

24 4

12

2 2

12

12 2

12 2

2

2

c e c c

c e c

c c e

c

c cd c T I Ic e

wP Z T a d d c T I I h c I d c I T

w

a d d c T I I W h c Iw

d c I T a d d c T I Iw

W h c I d c I T a dw

wh c I d c I T

(19)

If the cost in (5) is less than (19), it is better to hire RW services.


5.2 Case II: determination of order quality when T* > T

In this case, interest charges are not paid for the items kept in stock. The total fuzzy cost per unit time is given by

2

*

( ) [2 ( ) ] [ (2 )] [ ( )]

( )2

e

e

Z T A D F H W W D T F C I

TD F H W D C I T

(20)

Now, we defuzzify the total cost per unit time using graded mean integration representation method the result is

2 22 1 1 1 4 1 2 2 2 3 2

4 3

2 23 3 3 2 3 4 4 4 1 4

2 1

1 41 1 2 2 2 3 3 3 4 4

* *4 1 4 4 4 3 2 3 3 3

1 1 1 1( ) 2 26 2

1 12 22

2 2

2

e e e e

e e

P Z T a d f h w a d f h wT d d

a d f h w w d f h wd d

d dT f c I d f c I d f c I f c I

f h w d c I T f h w d c I T

* *2 3 2 2 2 1 4 1 1 12 – –e ef h w d c I T f h w d c I T

(21)

For minimum total cost per unit time, the optimum value of T = T20 will be the solution of: 2d ( ( )) / d 0P Z T T , which gives

2 220 1 1 1 4 1 2 2 2 3 2

4 3

2 2 13 3 3 2 3 4 4 4 1 4 1 1

2 11/2

42 2 2 3 3 3 4 4

1 12 221 12 2

2 2

2

e

e e e

T a d f h w a d f h wd d

da d f h w a d f h w f c I

d d

dd f c I d f c I f c I

(22)

The optimum order quantity

~2 2

2 20 1 1 1 4 1 2 2 2 3 24 3

2 23 3 3 2 3 4 4 4 1 4

2 1

1 1 1 e 2 2 2 e 3 3 3 e

4 4 4 e

1 1Q T 2a d f h w 2w d f h w2d d

1 12a d f h w 2a d f h wd 2d

1d f c I 2d f c I 2d f c I

2

d f c I

(23)


And minimum cost 2 20( ( ))P Z T per unit time is

2 22 20 1 1 1 4 1 2 2 2 3 2

4 3

2 23 3 3 2 3 4 4 4 1 4

2 1

1 1 1 2 2 2 3 3 3

1/2*

4 4 4 4 1 4 4 4

3 2 3 3

1 1 22 26

2 12 2 ()

2 2

–

2

e e e

e e

P Z T a d f h w a d f h wd d

a d f h w a d f h wd d

d f c I d f c I d f c I

d f c I f h w d c I T

f h w d c * *3 2 3 2 2 2

*1 4 1 1 1

2e e

e

I T f h w d c I T

f h w d c I T

(24)

Special case 1: when costs, demand and capacity to store number of units in OW are real numbers, they are d1 = d2 = d3 = d4 = D, f1 = f2 = f3 = f4 = F, h1 = h2 = h3 = h4 = H,a1 = a2 = a3 = a4 = A, c1 = c2 = c3 = c4 = C, w1 = w2 = w3 = w4 = W then the optimum of T = T2 is

1/22

20 22 ( )

e

AD F H WT

D F CI (25)

Then, optimum order quantity is 1/22

2 20 202 ( )

e

AD F H WQ T DT

F CI (26)

And minimum cost Z2(T2) per unit is 1/22 *

2 20 2 ( ) ( )e eZ T AD F H W F CI F H W DCI T (27)

These equations are same as those given by Shah and Shah (1992).

Special case 2: when we take holding cost in both warehouses are same (H = F),optimum values of T is

1/ 2

1 1 2 2 3 3 4 44 3 2 1

201 4

1 1 2 2 2 3 3 3 4 4

1 1 1 12 2 2 22 2

2 2e e e e


Td d

h c I d h c I d h c I h c I (28)

1 1 2 2 3 3 4 44 3 2 1

2 20

1 1 1 2 2 2 3 3 3 4 4 4

1 1 1 12 2 2 22 2

1 2 22 e e e e


Q Td h c I d h c I d h c I d h c I

(29)


2 20 1 1 2 2 3 3 4 44 3 2 1

1/2

1 1 1 2 2 2 3 3 3 4 4 4

* * * *1 1 2 2 3 3 4 4

1 1 2 2 12 2 2 26

2 2

2 2

e e e e

e e e e


d h c I d h c I d h c I d h c I

d c I T d c I T d c I T d c I T

(30)

When all costs real numbers, we have 1/2 1/2

20 2 20

1/2 *2 20

2 2

and ( ) [2 ( )] –

e e

e e

A ADT Q T

D H CI H CI

Z T AD H CI DCI T

(31)

These equations are same as these given by Goyal (1985).

Special case 3: when we take holding cost in both warehouses are same, that is, H = F,Ie = 0 and T = 0, optimum values of T is

1/2

1 1 2 2 3 3 4 44 3 2 1

201 1 4 4

2 2 3 3

1 1 1 12 2 2 22 2

2 2


Td h d h

d h d h (32)

1 1 2 2 3 3 4 44 3 2 1

2 20

1 1 2 2 3 3 4 4

1 1 1 12 2 2 22 2

1 2 22


Q Td h d h d h d h

(33)

2 20 1 1 2 2 3 3 4 44 3 3 4

1/2

1 1 2 2 3 3 4

1 1 2 2 12 2 2 26

2 2 4


d h d h d h d h

(34)

when all costs real numbers, we have 1/2 1/2

1/220 2 20 2 20

2 2, Q and Z 2A ADT T T ADH

DH H (35)

These results are similar to those of single storage EOQ model, that is, these equations are same as those of classical EOQ model of Naddor (1966).

Special case 4: if we do not wish to use RW, then taking 2DT W , where 2 /T W D is cycle time. In this case, the total cost per unit time due to OW is

*2 2 2

cc

IZ T A D W H W W C D C I T (36)


By second function principle, we have

1 1 1 2 2 22 2 1 1 4 4 2 2

1 2

3 3 3 4 43 3 3 3 2 2

3 4

44 4 1 1

,2 2

, ,2

2

c c c

c c c

c c

a d w a d wZ T h c I d c I T h c I

w w

a d w a dd c I T h c I d c I T

w w

wh c I d c I T

(37)

Now we defuzzify the total cost per unit time, using graded mean integration representation method, the result is

1 1 1 2 22 2 1 1 4 4 2 2 2

1 2

3 33 3 3 3 3 2 2

3

4 4 44 4 1 1

4

22

2 2 2

2

c c c

c c c

c c

a d w a dP Z T h c I d c I T W h c I

w w

a dd c I T W h c I d c I T

w

a d wh c I d c I T

w

(38)

If the cost in (38) is less than (24), it is better to hire RW services.

6 Numerical example

Let us consider for the inventory system that has: Holding cost in RW = Rs. 2.5, holding cost in OW = Rs. 1.5, purchasing cost =

Rs. 20, Ic = 0.30, Ie = 0.15, capacity to store number of units in OW = 1,000 units, ordering cost = 500, demand rate = 15,000 units per annum. Values of time interval, order quantity and total minimum cost with prescribed permissible delay period and increasing order cost are given by Tables 1 and 2 and Figures 5–8: Table 1 When T* = 0.0634 < T

For two storage facilities When DT10 = W Ratio

A T10 DT10 Z1(T10) T101 2Z T Z1(T10)/ 1 2( )Z T

500 0.0989 1,483.50 5,905.98 0.0667 6,900.60 0.8559 1,000 0.1327 1,991.54 10,222.15 0.0667 14,400.60 0.7098 1,500 0.1595 2,392.50 13,642.52 0.0667 21,900.60 0.6229

Table 2 When T* = 0.2536 > T

For two storage facilities When DT10 = W Ratio

A T10 DT10 Z2(T10) T10 2 2( )Z T Z2(T10)/ 2 2( )Z T

500 0.1454 2,181.00 417.05 0.667 6,748.50 0.0615 1,000 0.1702 2,553.00 3,585.60 0.0667 14,248.50 0.2516 1,500 0.1918 2,877.00 6,347.17 0.0667 21,748.50 0.2918


Figure 5 Two storage facilities

Figure 6 When DT = W

Figure 7 Two storage facilities


Figure 8 When DT = W

7 Observations

Table 1 / fig. 5, 6 show that the values of T10, DT10, Z1 (T10) for two storage facilities and When DT10 = W, with increasing in ordering cost. For a system with two storage facilities, we find the Q10 = DT10 and Z1 (T10) increase as A increase. Moreover, it can be observed that the total optimum cost for the system with two storage facilities is less than that where RW is not used, which suggests that whenever necessary it is economical to hire RW. Similar observations can also be made from table 2 / fig. 7, 8 when T* > T. But comparing the two cases we find that the total minimum cost when T* >T is very low as compared to the case when T* < T.

8 Conclusion

This paper proposed a policy for a two-warehouse inventory system with fuzzy inventory costs under the conditions of permissible delay in payments. This paper develops a method to find the membership function of the total fuzzy inventory cost when the demand quantity, the unit holding cost, ordering cost, purchasing cost and the capacity to store number of items are fuzzy numbers. In this paper, we examine the conditions of permissible delay in payments, and its effect on optimal ordering quantity of two-warehouse facilities models for realistic inventory system. It can observe from the table that an EOQ model with permissible delay period with single storage facility in compare to two-warehouse facilities has a considerable change. We observe that total optimum cost for the system with two storage facilities is less as compared to single storage facilities.

The model is best fit for real life situations in which there is high demand in the beginning of the period and the single storage facilities is insufficient to accommodate the order level and as a result an additional warehouse is used. For future research, it is challenging to consider the factors of shortage, inflation, deterioration and two stage interest payable criterions under fuzzy environment.


Acknowledgements

The authors express their heartfelt gratitude and boundless regards to the anonymous referees for their useful comments and valuable suggestions on the earlier version of this paper. Best efforts have been made by the authors to revise this paper abiding by the comments of the referees.

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