An Inventory Model with Fuzzy Deterioration and

SOP TRANSACTIONS ON APPLIED MATHEMATICS ISSN(Print): 2373-8472 ISSN(Online): 2373-8480 Volume 1, Number 2, July 2014

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An Inventory Model with Fuzzy Deterioration and Fully Backlogged Shortage under Inflation Sonia Shabani, Abolfazl Mirzazadeh* , Ehsan Sharifi Department of Industrial Engineering, College of Engineering, University of Kharazmi, Tehran, Iran. *Corresponding author: [email protected]

Abstract: Looking through the inventory models with deteriorating items shows that the deterioration rate is considered constant in most of the previous researches. But,in the real world, deterioration rate is not actually constant and slightly disturbed from its original crisp value. In this paper a more realistic inventory model with fuzzy deterioration and fully backlogged shortage under inflation is considered. The mathematical model is formulated to obtain the optimal value of the fuzzy total cost. The numerical example is used to illustrate the computation procedure. A sensitivity analysis is also carried out to get the sensitiveness of the tolerance of different input parameters. Keywords: Inventory; Inflation; Deterioration; Membership Function; Fuzzy Total Cost

1. INTRODUCTION Deterioration is considered in many inventory researches in the last decades. In the literature deterioration is defined as the damage, spoilage, dryness, vaporization, etc. that results in decrease of usefulness of the original one. For example, the commonly used goods like fruits, vegetables, meat, foodstuffs, perfumes, alcohol, gasoline, radioactive substance, photographic films, electronic components, etc. Nahmias [1] reviewed perishable inventory theory. Raffat [2] Surveyed literature of continuously deteriorating inventory models. Goyal and Giri [3] investigated recent trends in modeling of deteriorating inventory. Bakker et al. [4] reviewed the inventory systems with deterioration since 2001. Through investigating these reviews and some other papers [5–20] we found out that deterioration rate is considered constant in most of previous researches. But, in the real world, deterioration rate is not actually constant and slightly disturbed from its original crisp value. Deng [21] Improved inventory models with ramp type demand and weibull deterioration. Chen et al. [22] developed an EOQ model with ramp type demand rate and time dependent deterioration rate. Banerjee and Agrawal [23] developed a two-warehouse inventory model for items with three-parameter weibull distribution deterioration, shortages and linear trend in demand.Dye et al. [24] determined optimal selling price and lot size with a varying rate of deterioration and exponential partial backlogging. Saker and Sarker, [25] improved an inventory model with partial backlogging, time varying deterioration and stock-dependent demand. Also a few papers considered deterioration as a fuzzy

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number. De et al. [26] developed an economic production quantity inventory model involving fuzzy demand rate and fuzzy deterioration rate. Roya et al. [27] developed two storage inventory model with fuzzy objective function deterioration over a random planning horizon, Mishra and Mishra [15] developed a (Q, R) model with fuzzified deterioration under cobweb phenomenon and permissible delay in payment. The fuzzy set theory was developed in the mid-1960s. Later an extension principle was developed by Bellman and Zadeh [28] in the field of decision-making problems in management sciences as well as OR sciences. Dubois and Prade [29] explained various operations on fuzzy numbers. Roy and Maiti [30, 31] solved the classical EOQ problem with a fuzzy goal and fuzzy inventory costs using a fuzzy non-linear programming method where different types of membership functions for inventory parameters were specified. They examined the fuzzy EOQ problem with a demand-dependent unit priceand an imprecise storage area using both fuzzy geometric and non-linear programming methods. In this paper, we have developed an inflationary inventory model by considering deterioration as a left-shaped fuzzy number. The main idea is to develop a more practical inventory model by considering deterioration as a fuzzy number. This type of fuzzy (left-shaped or L-fuzzy) number is introduced because, in a fuzzy environment, one can always expect the decision-maker to be more conscious about the preservation of goods so that deterioration rate does not exceed its initial approximation q0 . To find the extreme order quantity, which minimizes the total fuzzy cost function. Fuzzy cost function is derived with the help of Zimmerman[32] and its solution is obtained with the help of Kaufmann and Gupta [33]. In section 2, assumptions and notations of the proposed model are listed. In section 3, mathematical model is presented. Fuzzy model is proposed in section 4. In section 5, an example is provided to prove the validity of the methodology in which the fuzzy model is formulated. A sensitivity analysis is also carried out to get the sensitiveness of the tolerance of different input parameters in section 6. Finally a brief conclusion is drawn in section 7.

2. ASSUMPTIONS AND NOTATIONS 2.1 Assumptions 1. The demand rate is known and constant. 2. Shortages are allowed and fully backlogged. 3. The replenishment is instantaneous. 4. The initial inventory level is zero. 5. The linear membership function of the deteriorationrate qe is given by 8 > : 1

162

q0 q p1

f or q q0 f or q0 p1  q  q0 f or q  q0 p1

The initial approximation of the deterioration rate is q0 and p1 its tolerance. This type of fuzzy (leftshaped or L-fuzzy) number is introduced, because, in a fuzzy environment, one can always expect the decision-maker to be more conscious about the preservation of goods so that deterioration rate does not exceed its initial approximation q0 .

An Inventory Model with Fuzzy Deterioration and Fully Backlogged Shortage under Inflation

2.2 Notations The following notations are used: i: The inflation rate per unit time D:The demand rate per unit time q : The constant deterioration rate, where 0 q  1 qe: The fuzzy deterioration rate 0  qe  1

c1 : The inventory holding cost per unit at time zero c: The per unit purchase cost of the item at time zero c2 : The shortage cost of the item. A:The ordering cost per order at time zero T:The replenishment time interval k:The proportion of time in any given inventory cycle which orders can be filled from the existing stock Z(k, T ) : The total cost Z*: The minimum total cost

3. Mathematical model The graphical representation of the inventory system is shown in Figure 1. The infinite time horizon is divided into equal parts each of length T. Each inventory cycle, T, can be divided into two parts. k (0 < k  1) is the proportion of time in any given inventory cycle which orders can be filled from the existing stock. Thus, during the time interval [( j 1)T, jT ], the inventory level leads to zero and shortages occur at time ( j + k 1)T . During [( j + k 1)T, jT ], we do not have any deterioration and therefore, shortages level linearly increases by the demand rate. Shortages are accumulated until jT before they are backordered. The optimal inventory policy yields the ordering and shortage points, which minimize the expected inventory cost. During the time interval [0, kT ], the level of inventory I1 (t1 ) gradually decreases mainly to meet demands andpartly due to deterioration. Hence, the variation of inventory with respect to time can be described by the following differential equation: dI1 (t1 ) + q I1 (t1 ) = D dt1

0  t1  kT

(1)

The shortages occur at time kT and accumulated until T before they are backordered. The shortages level to be represented by dI2 (t2 ) = D dt2

0  t2  (1

k) T

(2)

The solution of the above differential equations after apply the boundary conditions I1 (kT)=0

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Figure 1. Graphical representation of the inventory system

and I 2 (0)=0, are I1 (t1 ) =

 D e q e

qt1 q kT

1

(3)

I2 (t2 ) = Dt2

(4)

The total cost in this inventory system includes the replenishment cost, purchase cost, holding cost and shortages costs. The problem has been formulated with the total annual cost. The detailed analysis of each cost function is given below. The present value of the inventory replenishment cost is RC = [1 + i(1

T )/2]A/T

(5)

The total purchase quantity for each cycle equals to Z kT 0

Deq T dt +

Z T

KT

Ddt = D(eq kT

1)/q + DT (1

k)

The exponential function is expanded in the following way: D ⇣ q kT e q

⌘ D q 2 kT 2 q 3 kT 3 1 = (1 + q kT + + +··· q 2 6

✓ ◆ q kT q 2 kT 3 1) = D kT + + +... 2 6

✓ ◆ ✓ 3 ◆ q kT 2 q kT 2 2 kT ⇡ D( kT + +q + . . . ) ⇡ D(kT + ) 2 6 2 Since q 1, so q 2 and higher powers are neglected. 164

Therefore, the annual cost of purchasing, similarly replenishment cost, is

An Inventory Model with Fuzzy Deterioration and Fully Backlogged Shortage under Inflation

 q kT 2 PC = c D(kT + ) + DT (1 2

k)



1+

i (1

T) 2

(6)

The average inventory level during the time interval [0, kT ] is I [0, kT ] =

Z kT I(t) 0

kT

dt =

DT (eq kT q kT q 2k

1)

Inventory level is zero during the time interval [k, T ]. Hence, the average inventory level over each cycle, [0, T ], is given by kT DT (eq kT q kT 1) I [0, T ] = I [0, kT ] = T q2 Also the exponential function above is expanded DT eq kT

q kT 2

1

✓ ◆ DT q 2 kT 2 q 3 kT 3 = 2 [ 1 + q kT + + +... q 2 6

1

q kT ] ⇡ DT [kT 2 +

q kT 3 ] 6

Since q 1, so q 2 and higher powers are neglected. Therefore, the annual holding cost is

HC = c1



q kT 3 DT [kT + ] 6 2



1+

i (1

T) 2

(7)

The shortages linearly increase during the time interval [kT, T ]. Therefore, the average shortages level equals to B[kT, T ] = Bmax /2 = DT (1

k))/2

The average shortages level over each cycle, [0, T ], is B[0, T ] = B[kT, T ]

T (1 k) = DT (1 T

k)2 /2

Hence, the annual shortages cost is

SC = c2

"

DT (1 k)2 2

#

1+

i (1

T) 2

(8)

So, the total annual cost is given by Z(k, T ) = RC + PC + HC + SC

(9)

which reduces, after some elementary manipulation, to the following form: Min Z (k, T, q ) = f1 (k, T ) + q f2 (k, T )

(10) 165

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where

f1 (k, T ) = (

✓ ◆ c1 DT c2 DT (1 k)2 i (1 T ) (kT )2 + ) 1+ 2 2 2 ✓ ◆ Dk2 T c1 T 3 i (1 T ) f2 (k, T ) = (c + ) 1+ 2 3 2

A + cDk + (1 T

k) +

4. FUZZY MODEL Throughout the development of EOQ models, previous authors have assumed that the deterioration rate is constant. But in the present world situation, the deterioration rate is some what more uncertain in nature. That is why we consider the deterioration rate q as the fuzzy number qe. Since the objective function is a function of fuzzy number eq , so the objective function itself is transformed into a fuzzy number. Using the general notations to represent the fuzzy number, the cost functions (10) may be rewritten in a fuzzy sense as ⇣ ⌘ g Z k, T, qe = f1 (k, T ) + qe f2 (k, T ) Min

(11)

A fuzzy non-linear programming problem (FNLPP) may be defined as g 0 (x, y) Ming ˜ x 0

)

(12)

where y= ˜ qeT is the fuzzy co-efficient vector ofg0 .

From fuzzy set theory the fuzzy objective and co-efficient are defined bytheir membership functions which may be linear and/or non-linear. Here we have assumed µ0 , µy as the non-increasing and nondecreasing continuous linear membership function, for objective and negative fuzzy co-efficient vectors y˜ of the objective function g0 , respectively, and these are 8 > : 1 8 > < 0 1 µy (u) = > : 1

g0 (x) Z0 p0

f or g0 (x) < Z0 f or Z0  g0 (x)  Z0 + p0 f or g0 (x) > Z0 + p0

q0 u P1

f or u > q0 f or q0 P1  u  q0 f or u < q0 P1

Now exploiting max-min operator, which was first developed by Bellman and Zadeh [28] long back and subsequently used by Zimmermann [32], etc. the solution of the FNLPP (12) can be obtained from Max a subject to

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g0 (x, µy 1 (a))  µ0 1 (a), x

0, a 2 [0, 1]

(13)

An Inventory Model with Fuzzy Deterioration and Fully Backlogged Shortage under Inflation

Table 1. Optimal solutions Decision variables

crisp

fuzzy

k⇤

0.603

0.668

0.780

0.799

T⇤ Z⇤

1184.06 1185.126

For the proposed fuzzy model given by (11), we define the membership function of fuzzy inventory minimum cost and fuzzy deteriorationrate as follows : 8 > : 1 8 > < 0 µq (u) = 1 > : 1

Z(k, T ) Z0 p0

q0 u P1

f or Z (k, T ) < Z0 f or Z0  Z (k, T )  Z0 + p0 f or Z (k, T ) > Z0 + p0 f or u > q0 f or q0 P1  u  q0 f or u < q0 P1

where k and T are positive variables,q0 and Z0 are the initial assumptions of deterioration rate and objective goals, respectively, p1 and P0 are their respective tolerances. for a proper choice of p1 we can always have the a-level set of qe as q0 (1 a) p1 0. Thus, the fuzzy model given by equation (11) reduce to the following form max a sub ject to :

Z (k, T, a)  Z0 + (1

a) p0 ; Z (k, T, a) = f1 (k, T ) + (q0

(1

a) p1 ) f2 T, k

0ae [0, 1]

(14)

5. NUMERICAL EXAMPLE We have considered here the following numerical example table to illustrate crisp model and its corresponding fuzzy model. Let D=1000units/year, A=$60/order, c=$1/unit, c1=$0.2/unit/year, c2=$0.6/unit/year, q =0.25 and Theinflation rate isi=$0.12/$/year for crisp model and for its corresponding fuzzy model q0 = 0.25, Z0 = 1184, p0 = 100 and p1 = 0.1. Using these parameter values, results are illustrated in the Table 1.

6. SENSITIVITY ANALYSIS In this section, we have examined the sensitiveness of the decision variables k, T and Z in each set of the parameters q0 , A, c1 , c2 , i, p1 and p0 (shown in Table 2) and the parameters D, c and Z0 (shown in Table 3). Table 2 shows the relative changes of k, T and total inventory cost, Z, when each of the parameters is being changed from -50 percent to +50 percent. Table 3 shows the same changes when the parameters are being changed from -10 percent to +10 percent. Table 2 shows that k and T are moderately sensitive and the optimum system cost Z is less sensitive for changing in the parameters q0 , A, c1 , c2 and 167

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Table 2. Sensitivity analysis on q0 , A, c1 , c2 , i, p1 , p0 . Parameter

% change

k⇤

T⇤

Z⇤

q0

+20 +50 -20 -50 +20 +50 -20 -50 +20 +50 -20 -50 +20 +50 -20 -50 +20 +50 -20 -50 +20 +50 -20 -50 +20 +50 -20 -50

0.585 0.551 0.776 0.891 0.607 0.603 0.782 0.855 0.602 0.592 0.729 0.812 0.662 0.718 0.714 0.749 0.603 0.591 0.708 0.745 0.668 0.668 0.668 0.668 0.668 0.668 0.668 0.668

0.779 0.765 0.854 0.922 0.839 0.905 0.852 0.893 0.762 0.731 0.853 0.924 0.775 0.724 0.828 0.848 0.825 0.888 0.806 0.812 0.799 0.799 0.799 0.799 0.799 0.779 0.779 0.779

1189.049 1196.600 1183.982 1183.977 1195.603 1213.427 1183.954 1183.749 1185.875 1189.998 1183.903 1183.894 1188.047 1193.485 1183.940 1184.964 1185.073 1187.691 1183.979 1183.919 1183.963 1183.963 1183.963 1183.963 1183.963 1183.963 1183.963 1183.963

A

c1

c2

i

p1

p0

Table 3. sensitivity analysis on the parameters D, c and Z0 . Parameter

% change

k⇤

T⇤

Z⇤

D

+10 +5 -5 -10 +10 +5 -5 -10 +10 +5 -5 -10

0.676 0.642 0.912 0.990 0.652 0.629 0.904 0.992 0.995 0.906 0.642 0.675

0.787 0.787 0.942 0.999 0.821 0.804 0.935 0.999 0.996 0.938 0.800 0.811

1276.025 1230.100 1138.695 1094.579 1271.763 1227.792 1154.456 1123.326 1289.713 1239.630 1174.399 1165.507

c

Z0

i. but all the variables, T, k and Z are very insensitive with respect to the changes of the parameters p1 and p0 . Table 3 shows that the replenishment time, T, k and the optimum system cost, Z, are very sensitive to the change of parameters D, c and Z0 . Couse the parameters D, c andZ0 are too sensitive their parametric range is -10 percent to +10 percent instead of -50 percent to +50 percent.

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An Inventory Model with Fuzzy Deterioration and Fully Backlogged Shortage under Inflation

7. CONCLUSION Most researchers have assumed the deterioration rate as a constant quantity, but in the real world, deterioration rate may be uncertain in nature, so rate of decay of an item may notalways be constant. This factor motivates us to develop an inventory model with a fuzzy deterioration rate and fully backlogged shortage under inflation. Using fuzzy mathematics, we have derived the fuzzy cost function of the model, and the solution procedure has been discussed. With the help of numerical examples solution procedures have been explained.A sensitivity analysis is also carried out to get the sensitiveness of the tolerance of different input parameters.

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