SOP TRANSACTIONS ON APPLIED MATHEMATICSISSN(Print): 2373-8472 ISSN(Online): 2373-8480

Volume 1, Number 2, July 2014

SOP TRANSACTIONS ON APPLIED MATHEMATICS

An Inventory Model with FuzzyDeterioration and Fully BackloggedShortage under InflationSonia Shabani, Abolfazl Mirzazadeh* , Ehsan Sharifi

Department of Industrial Engineering, College of Engineering, University of Kharazmi, Tehran, Iran.*Corresponding author: a.mirzazadeh@aut.ac.ir

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 deteriora-tion is defined as the damage, spoilage, dryness, vaporization, etc. that results in decrease of usefulness ofthe original one. For example, the commonly used goods like fruits, vegetables, meat, foodstuffs, per-fumes, alcohol, gasoline, radioactive substance, photographic films, electronic components, etc. Nahmias[1] reviewed perishable inventory theory. Raffat [2] Surveyed literature of continuously deterioratinginventory 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 investigatingthese reviews and some other papers [5–20] we found out that deterioration rate is considered constant inmost of previous researches. But, in the real world, deterioration rate is not actually constant and slightlydisturbed from its original crisp value. Deng [21] Improved inventory models with ramp type demand andweibull deterioration. Chen et al. [22] developed an EOQ model with ramp type demand rate and timedependent deterioration rate. Banerjee and Agrawal [23] developed a two-warehouse inventory model foritems with three-parameter weibull distribution deterioration, shortages and linear trend in demand.Dye etal. [24] determined optimal selling price and lot size with a varying rate of deterioration and exponentialpartial backlogging. Saker and Sarker, [25] improved an inventory model with partial backlogging, timevarying 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 fuzzydemand rate and fuzzy deterioration rate. Roya et al. [27] developed two storage inventory model withfuzzy objective function deterioration over a random planning horizon, Mishra and Mishra [15] developeda (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 byBellman and Zadeh [28] in the field of decision-making problems in management sciences as well as ORsciences. 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-linearprogramming method where different types of membership functions for inventory parameters werespecified. They examined the fuzzy EOQ problem with a demand-dependent unit priceand an imprecisestorage area using both fuzzy geometric and non-linear programming methods.

In this paper, we have developed an inflationary inventory model by considering deterioration as aleft-shaped fuzzy number. The main idea is to develop a more practical inventory model by consideringdeterioration 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 thepreservation of goods so that deterioration rate does not exceed its initial approximation q0. To find theextreme order quantity, which minimizes the total fuzzy cost function. Fuzzy cost function is derivedwith 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, mathematicalmodel is presented. Fuzzy model is proposed in section 4. In section 5, an example is provided to provethe validity of the methodology in which the fuzzy model is formulated. A sensitivity analysis is alsocarried out to get the sensitiveness of the tolerance of different input parameters in section 6. Finally abrief 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 eq is given by

µ (q) =

8><

>:

0 f or q � q0

1� q0�qp1

f or q0 � p1 q q0

1 f or q q0 � p1

The initial approximation of the deterioration rate is q0 and p1 its tolerance. This type of fuzzy (left-shaped or L-fuzzy) number is introduced, because, in a fuzzy environment, one can always expect thedecision-maker to be more conscious about the preservation of goods so that deterioration rate does notexceed its initial approximation q0.

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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

eq : The fuzzy deterioration rate 0 eq 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 horizonis 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 fromthe existing stock. Thus, during the time interval [( j� 1)T, jT ], the inventory level leads to zero andshortages occur at time ( j+ k� 1)T . During [( j+ k� 1)T, jT ], we do not have any deterioration andtherefore, shortages level linearly increases by the demand rate.

Shortages are accumulated until jT before they are backordered. The optimal inventory policy yieldsthe 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 demandsandpartly due to deterioration. Hence, the variation of inventory with respect to time can be described bythe following differential equation:

dI1 (t1)dt1

+q I1 (t1) =�D 0 t1 kT (1)

The shortages occur at time kT and accumulated until T before they are backordered. The shortages levelto be represented by

dI2 (t2)dt2

=�D 0 t2 (1� k)T (2)

The solution of the above differential equations after apply the boundary conditions I1(kT)=0163

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Figure 1. Graphical representation of the inventory system

and I2(0)=0, are

I1 (t1) =�Dq

e�q t1

e�qkT �1�

(3)

I2 (t2) =�Dt2 (4)

The total cost in this inventory system includes the replenishment cost, purchase cost, holding cost andshortages costs. The problem has been formulated with the total annual cost. The detailed analysis ofeach 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

0DeqT dt +

Z T

KTDdt = D(eqkT �1)/q +DT (1� k)

The exponential function is expanded in the following way:

Dq

⇣eqkT �1

⌘=

Dq(1+ qkT +

q 2kT 2

2+

q 3kT 3

6+ · · ·�1) = D

✓kT +

qkT2

+q 2kT 3

6+ . . .

◆

⇡ D(

✓kT +

qkT 2

2

◆+q 2

✓kT 3

6+ . . .

◆)⇡ D(kT +

qkT 2

2)

Since q1, so q 2 and higher powers are neglected.

Therefore, the annual cost of purchasing, similarly replenishment cost, is164

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

PC = c

D(kT +qkT 2

2)+DT (1� k)

�1+

i(1�T )2

�(6)

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

I [0,kT ] =Z kT

0

I(t)kT

dt =DT (eqkT �qkT �1)

q 2k

Inventory level is zero during the time interval [k,T ]. Hence, the average inventory level over each cycle,[0,T ], is given by

I [0,T ] = I [0,kT ]kTT

=DT (eqkT �qkT �1)

q 2

Also the exponential function above is expanded

DT�eqkT �qkT �1

�

2=

DTq 2 [

✓1+qkT +

q 2kT 2

2+

q 3kT 3

6+ . . .

◆�1�qkT ]⇡ DT [kT 2 +

qkT 3

6]

Since q1, so q 2 and higher powers are neglected.

Therefore, the annual holding cost is

HC = c1

DT [kT 2 +

qkT 3

6]

�1+

i(1�T )2

�(7)

The shortages linearly increase during the time interval [kT,T ]. Therefore, the average shortages levelequals 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)

T= DT (1� 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 ) = (AT+ cDk+(1� k)+

c1DT2

(kT )2 +c2DT (1� k)2

2)

✓1+

i(1�T )2

◆

f2 (k,T ) =Dk2T

2(c+

c1T 3

3)

✓1+

i(1�T )2

◆

4. FUZZY MODEL

Throughout the development of EOQ models, previous authors have assumed that the deterioration rateis 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 eq . Since the objective function is afunction of fuzzy number eq , so the objective function itself is transformed into a fuzzy number. Usingthe general notations to represent the fuzzy number, the cost functions (10) may be rewritten in a fuzzysense as

gMin Z⇣

k,T, eq⌘= f1 (k,T )+ eq f2(k,T ) (11)

A fuzzy non-linear programming problem (FNLPP) may be defined as

gMing0 (x, y)

x � 0

)(12)

where y=eq T is the fuzzy co-efficient vector ofg0.

From fuzzy set theory the fuzzy objective and co-efficient are defined bytheir membership functionswhich may be linear and/or non-linear. Here we have assumed µ0, µy as the non-increasing and non-decreasing continuous linear membership function, for objective and negative fuzzy co-efficient vectors yof the objective function g0, respectively, and these are

µ0(g0 (x)) =

8><

>:

0 f or g0 (x)< Z0

1� g0(x)�Z0p0

f or Z0 g0 (x) Z0 + p0

1 f or g0 (x)> Z0 + p0

µy(u) =

8><

>:

0 f or u > q0

1� q0�uP1

f or q0 �P1 u q0

1 f or u < q0 �P1

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

Max a subject to

g0(x,µ�1y (a)) µ�1

0 (a), x � 0, a 2 [0, 1] (13)166

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

Table 1. Optimal solutions

Decision variables crisp fuzzyk⇤ 0.603 0.668T⇤ 0.780 0.799Z⇤ 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 :

µ0(Z (k, T )) =

8><

>:

0 f or Z (k, T )< Z0

1� Z(k, T )�Z0p0

f or Z0 Z (k, T ) Z0 + p0

1 f or Z (k, T )> Z0 + p0

µq (u) =

8><

>:

0 f or u > q0

1� q0�uP1

f or q0 �P1 u q0

1 f or u < q0 �P1

where k and T are positive variables,q0 and Z0are the initial assumptions of deterioration rate and objectivegoals, respectively, p1 and P0 are their respective tolerances. for a proper choice of p1 we can always havethe a-level set of eq as q0 � (1�a) p1 � 0. Thus, the fuzzy model given by equation (11) reduce to thefollowing 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 andits 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 correspondingfuzzy model q0 = 0.25, Z0 = 1184, p0 = 100 and p1 = 0.1. Using these parameter values, results areillustrated 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 setof the parameters q0 , A, c1, c2, i, p1and p0 (shown in Table 2) and the parameters D, c and Z0 (shownin Table 3). Table 2 shows the relative changes of k, T and total inventory cost, Z, when each of theparameters is being changed from -50 percent to +50 percent. Table 3 shows the same changes when theparameters are being changed from -10 percent to +10 percent. Table 2 shows that k and T are moderatelysensitive and the optimum system cost Z is less sensitive for changing in the parameters q0 , A, c1, c2 and

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

Parameter % change k⇤ T ⇤ Z⇤

q0

A

c1

c2

i

p1

p0

+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

0.5850.5510.7760.8910.6070.6030.7820.8550.6020.5920.7290.8120.6620.7180.7140.7490.6030.5910.7080.7450.6680.6680.6680.6680.6680.6680.6680.668

0.7790.7650.8540.9220.8390.9050.8520.8930.7620.7310.8530.9240.7750.7240.8280.8480.8250.8880.8060.8120.7990.7990.7990.7990.7990.7790.7790.779

1189.0491196.6001183.9821183.9771195.6031213.4271183.9541183.7491185.8751189.9981183.9031183.8941188.0471193.4851183.9401184.9641185.0731187.6911183.9791183.9191183.9631183.9631183.9631183.9631183.9631183.9631183.9631183.963

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

Parameter % change k⇤ T ⇤ Z⇤

D

c

Z0

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

0.6760.6420.9120.9900.6520.6290.9040.9920.9950.9060.6420.675

0.7870.7870.9420.9990.8210.8040.9350.9990.9960.9380.8000.811

1276.0251230.1001138.6951094.5791271.7631227.7921154.4561123.3261289.7131239.6301174.3991165.507

i. but all the variables, T, k and Z are very insensitive with respect to the changes of the parameters p1 andp0. Table 3 shows that the replenishment time, T, k and the optimum system cost, Z, are very sensitive tothe change of parameters D, c and Z0. Couse the parameters D, c andZ0are too sensitive their parametricrange 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. Thisfactor motivates us to develop an inventory model with a fuzzy deterioration rate and fully backloggedshortage 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 procedureshave been explained.A sensitivity analysis is also carried out to get the sensitiveness of the tolerance ofdifferent input parameters.

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