THEORETICAL ARTICLE

Inventory model with learning effect and imprecisemarket demand under screening error

Dharmendra Yadav & S. R. Singh & Rachna Kumari

Accepted: 16 December 2012# Operational Research Society of India 2013

Abstract In this paper, an inventory model is developed to deal the impre-ciseness present in market demand. It is assumed that the received items arenot of perfect quality and after screening; imperfect items are withdrawn frominventory and sold at discounted price. However, in practice, errors occur inscreening test. So, the screening process fails to be perfect. Due to acquain-tance with handling methodology and system, holding cost and ordering costare gradually decreases from one shipment to another. So, learning effect isincorporated on holding cost, ordering cost and number of defective itemspresent in each lot. This type of situation arises in many industries especiallyfor that industry where productivity is influenced by human labour needed forfinal assembly such as: cars, ships, machines, aircrafts, electronics, and stabi-lizer. Due to impreciseness in market demand, profit expression is fuzzy innature. To fuzzify the profit expression, Extension Principle is used and fordefuzzification centroid method is applied. Mathematically, it is shown thatprofit expression is concave in nature. Finally, the feasibility of proposedmodel and the effect of learning on optimal solution are shown through numericalexample.

Keywords Inventory model . Screening error . Extension principle . Learning effects .

Imperfect lot . Centroid method

OPSEARCHDOI 10.1007/s12597-012-0118-x

D. Yadav (*)Department of Mathematics, Keshav Mahavidyalaya, Delhi-34, Indiae-mail: dharmendrayadav3580@gmail.com

S. R. SinghDepartment of Mathematics, D.N. (P.G) College, Meerut 250001, (U.P), Indiae-mail: shivrajpundir@gmail.com

R. KumariDepartment of Mathematics, Meerut College, Meerut 250001, (U.P), Indiae-mail: kumarn_inde@yahoo.com

1 Introduction

Harris [10] first proposed the Economic Order Quantity (EOQ) model and after that itwas modified by several researchers, changing the basic assumptions, with theobjective to make it more realistic and sensible. Several researchers studied inventorymodels with market demand as crisp, stochastic or fuzzy parameter. Salameh andJaber [22], Jaber and Bonney [13], Papachristos and Konstantaras [19], Wee et al.[26], Teng [24] developed the inventory model by taking market demand as crispparameters. It is observed that in practical situations, crisp market demand can bejustified only for the maturity phase of the product. Hadley and Whitin [9] developedthe model by taking stochastic demand. However, stochastic market demand isapplicable only for old product where probability distribution function can be deter-mined by past data. For many products like clothes, fashion accessories, mobilephones etc., the market demand cannot be predicted accurately and the probabilitydistribution function cannot be determined. To develop the inventory models of suchproducts, a major difficulty faced by researchers is to forecast the demand. It is notpossible for researchers to decide the exact market demand in such a complex,changing and uncertain environment. So, it is often found that market demand isimprecise in nature. Very few researchers have worked in this direction such as Chang[2], Das et al. [3], Dutta et al. [6], Dey and Chakraborty [4], Singh et al. [23], Dey andChakraborty [5], Wang et al. [25].

In most of the classical economic order/ production quantity models, the itemsreceived/ produced are implicitly assumed that items are of perfect quality. However,in practice, it is not so. Presence of defective items in lot is due to natural disasters,damage or breakage in transit and many more reasons. Therefore, the lot received/produced may contain some defective items. In order to make some more realisticinventory policies, several researchers considered the above scenarios in formulatingthe inventory/ production models and studied the effect of imperfect quality on lotsizing policy. Rosenblatt and Lee [20] assumed that the defective items could bereworked instantaneously at a cost and the presence of defective products motivatesdecision maker to order smaller lot sizes. Salameh and Jaber [22] assumed that thedefective items could be sold as a single batch at a discounted price prior to receivingthe next shipment. They observed that as the average percentage of imperfectquality items decreases, the economic lot size quantity tends to increase. Goyaland Cádenas-Barrόn [8] reconsidered the work done by Salameh and Jaber [22] andpresented a simple approach for determining the optimal lot size. Chang [2] determinedthe optimal order lot size to maximize the total profit when lot contains imperfect qualityitems. Papachristos and Konstantaras [19] extended the work of Salameh and Jaber’s[22] model focusing on the timing of withdrawing the imperfect quality units fromstock. Wee et al. [26] developed an optimal inventory model for items with imperfectquality and shortages backordering. They assumed that number of defective itemspresent in each lot is random variable. Eroglu and Ozdemir [7] studied the effect ofpercentage of defective items present in each lot on optimal solution. Maddah and Jaber[18] enriched the work of Salameh and Jaber [22] by applying renewal theory to obtainthe expected profit per unit time. Singh et al. [23] developed the model in which theyconsidered that received lot contains defective items. The work of Salameh and Jaber[22] was explored by Hsu and Yu [11] related to quality issues. Ma et al. [17] developed

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the Economic Lot Scheduling Problem to determine an optimal common productioncycle in order to minimize the expected average cost per unit of time. They assumed thatproduction process may shift to an out-of-control state at a random time and conse-quently produces a fixed percentage of defective items. Khan et al. [15] obtained optimalproduction quantity/order quantity by using the similar approach as Salameh and Jaber[22] take care of imperfect processes. An imperfect inspection process is utilized todescribe the defective proportion of the received lot. The defective items, classified bythe inspector and the buyer would be salvaged as a single batch that is sold at a lowerprice. Konstantaras et al. [16] developed as EOQ model with imperfect quality andshortages and assumed that the fraction of imperfect quality in each shipment reducesdue to learning. They observed that managers of company get benefited from thelearning opportunities in their logistics and inventory systems, especially comparingwith different production strategies and making choices regarding the selection ofthe suppliers or the processes. Sarkar [21] developed an economic manufacturingquantity model with price and advertising dependent demand pattern in an imperfectproduction process. Author assumed that if the machine goes through a long run-process it may shift from in-control state to out-of-control state and as a result,system produces imperfect items. In order to reduce the production of the imperfectitems, author assumed that production cost, development cost, material costs aredependent on reliability parameters. Author carried out this study in crisp environ-ment without considering learning phenomenon.

Above survey reveals that many researchers developed the inventory model byconsidering imperfect production system. Note that all of the previous researchesassumed that the screening process is perfect. However, in practice, screening processfails to be perfect, two types of errors (Type 1 and 2) can occur. For the former, gooditems mistakenly taken as defectives and thus result in the necessity of purchasingadditional items with extra cost; the latter arise when defective items mistakenly takenas good items and then incur a penalty cost. It is also observed that most of theresearchers assumed that market demand as crisp parameter. But it is not always thecase because due to high complexities and uncertainty in market, market demandcannot consider as crisp parameter. Most of the researchers assumed that number ofdefective items present in each lot is random variable which follows the knownprobability distribution function while it is not possible to find the probabilitydistribution function for newly launched product due to the non availability ofhistorical data. So, above models are not best suited for newly launched product.Practically, the cost of the system engaged in repetitive operations decrease due to thelearning effect. Therefore, this natural phenomenon cannot be ignored while developingthe inventory model.

In this paper, generalized mathematical model is developed to complement theshortcoming of all the previous models. The primary difference of this paper ascompared to previous studies are

1. Demand rate is imprecise in nature to cope up with the uncertainty in marketdemand.

2. Effects of learning on holding cost and ordering cost are vital and relevant.3. Number of defective items present in each lot follows the learning curve.4. Screening process is not 100 % perfect.

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In addition, it is proved that the optimal order quantity in each lot exists uniquelyand the total profit associated with the inventory system is a concave function of thelot size. Model is illustrated through numerical examples. Finally, summary of thework and suggestions for future research are provided.

2 Preliminaries

To develop the inventory model, it would be useful to mention few mathematicalprerequisite. We state them as follows:

2.1 Learning curve

Geometric progression is the earliest learning curve representation that expresses thedecreasing cost required to accomplish any repetitive operation. That theory states thatas total produced units doubles, the cost per unit declines by some constant percentage(e.g., [12, 28]). The power versus exponential form of the learning curve has beendebated by several authors; refer to Jaber [12] for discussion. There is almost unanimousagreement among academicians and practitioners that the learning curve is best de-scribed by a power as suggested byWright [27]. It is worth noting that the learning curvein practice is an ‘S’-shaped curve (Carlson [1], Jordan [14]), as described in Fig. 1. Thefirst phase (incipient) is the phase during which the worker is getting acquainted with theset-up, the tooling, instruction, blueprints, the workplace arrangement, and the condi-tions of the process. In this phase improvement is slow. The second phase (learning) iswhere most of the improvement, e.g., reduction in errors, changes in the distance movedtakes place. The third and last phase (maturity) represents the learning of the curve.

S-shaped logistic learning curve is of the form

P nð Þ ¼ a

gþ ebnð1Þ

Where a, b, and g>0 are the model parameters, n is the cumulative number ofshipments, and P(n) is the number of defective items present in each shipment.

Hou

rs p

er U

nit

Out

put

Incipient

LearningMaturity

Phase-1 Phase-2 Phase-3

Units

Three phase of learning Curve

Fig. 1 Three phases of learning curve

....

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2.2 Extension principle

One of the most important notions in fuzzy set theory is the Extension Principle. TheExtension Principle provides a general method for combining non-fuzzy and fuzzyconcepts of all kinds, e.g., for combining fuzzy sets and relations, but also for theoperation of a mathematical function on fuzzy sets. Fuzzy sets can also be interpretedas fuzzy numbers. In this case one can use the Extension Principle to add or multiplythese fuzzy numbers. The Extension Principle of Zadeh is a very important tool in thefuzzy set theory which provides a procedure to fuzzify a crisp function.

Let f:X→Y be a crisp function and F(X) (respectively F(Y)) be the set of all fuzzysets (called fuzzy power set) of X (respectively Y). The function f:X→Y induces twofunctions f:F(X)→F(Y) and f:F−1(X)→F−1(Y) and the Extension Principle of Zadehgives formulas to compute the membership function of fuzzy set f(A) in Y(respectively f-1(B) in X) in terms of the membership function of the fuzzy set A inX (respectively B in Y). The Extension Principle of Zadeh states that:

1. μf ðAÞð yÞ ¼ Supx2X ; f ðxÞ¼y

ðμAðxÞÞ; 8A 2 FðX Þ

2. μf �1ðBÞðxÞ ¼ μBð f ðxÞÞ; 8B 2 FðY Þ:

2.3 Centroid method

This method is used to obtain the equivalent crisp expression for fuzzy expression.By doing so, we can analyze the developed model.

Let us consider a fuzzy set eA ¼ a; b; cð Þ , where a<b<c. The membership function

of eA is given by

μeAðxÞ ¼x� að Þ b� að Þ=c� xð Þ c� b=

0

8<:

a � x � b;b � x � c;otherwise:

The centroid of eA can be derived as

C eA� �¼

R1�1

xμeAðxÞdxR1�1

μeAðxÞdx¼ 1

3aþ bþ cð Þ

3 Assumptions and notations

3.1 Assumptions

The following assumptions are made for development of proposed mathematical model:

1) Market demand is imprecise in nature.2) Shortages are not allowed.

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3) The replenishment is instantaneous, i.e., lead time is zero.4) 100 % screening of items is done in each shipment.5) Screening errors may occur. Therefore, a penalty cost is incurred for each

defective unit delivered to the customers.6) Defective items are sold at a discounted price.7) Time horizon is finite.8) Holding cost and ordering cost follows the learning curve.

3.2 Notations

The following notation has been used to develop the model:

eD : Fuzzy demand rate measured in units per unit of timeyn : Order quantity in units for the nth shipment, where n≥1c : Unit purchasing costK(n) : Ordering cost per order is partly constant and partly decreases in each

cycle due to learning effect of employees ¼ k0 þ k1 na1=ð Þ where k0,k1>0 and 0<α1<1.

h(n) : Holding cost per order is partly constant and partly decreases in each cycledue to learning effect of employees ¼ h0 þ h1 na2=ð Þ where h0, h1>0 and0<α2<1.

P(n) : Percentage of defective items in each shipment follows the learning curve¼ a g þ ebn

� ��� �, a,b,g>0

s : Unit selling price per good quality unitv : Unit discounted price per defective unit, v<cTn : Cycle time per shipmentx : Screening rate measured in units per unit of timed : Unit screening costtn : Time to screen yn units, where tn=yn/x<Tn

m : Penalty cost due to error in screening processα : Type 1 errorβ : Type 2 error

4 Formulation of mathematical model

The behavior of the inventory level is shown in Fig. 2. Tn is the cycle time

for the nth shipment, where Tn ¼ yn 1� PðnÞ 1�bð Þþ 1�PðnÞð Það Þð ÞD . Screening of yn

units is performed and imperfect quality items are withdrawn from inventoryand sold as a single batch at a discount price by the end of the screeningperiod tn.

However, according to the assumption, since inspection errors may occurduring the 100 % screening process, the inventory level should be modified asfollows:

N ynð Þ ¼ yn 1� PðnÞ 1� bð Þ � 1� PðnÞð Það Þ

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The total revenue per cycle, R(yn), is the sum of total sales volume of ‘true’ goodquality, imperfect quality items and negative opportunity gains.

TR ynð Þ ¼ yns� 1� bð Þ s� vð ÞynPðnÞ � a s� vð Þyn 1� PðnÞð Þ ð2ÞAnd the total cost per cycle is the sum of ordering cost, purchasing cost, screening

cost, holding cost and penalty cost. The penalty cost is caused by Type 2 error (β).

TC ynð Þ ¼ KðnÞ þ cyn þ dyn þ hðnÞ y2n 1� PðnÞ 1�bð Þþ 1�PðnÞð Það Þð Þ22D þ PðnÞ 1�bð Þþ 1�PðnÞð Það Þy2n

x

� �þ PðnÞynbm

. . . ð3ÞThe net profit per cycle is the total revenue per cycle less the total cost per cycle.

Thus, the average profit per unit time can be written as:

TPU ynð Þ ¼ TR ynð Þ�TC ynð ÞTn

¼ sDþ vD PðnÞ 1�bð Þþ 1�PðnÞð Það Þ1� PðnÞ 1�bð Þþ 1�PðnÞð Það Þ � KðnÞD

1� PðnÞ 1�bð Þþ 1�PðnÞð Það Þð Þyn� cþdð ÞD

1� PðnÞ 1�bð Þþ 1�PðnÞð Það Þð Þ � hðnÞ yn 1� PðnÞ 1�bð Þþ 1�PðnÞð Það Þð Þ2

�þ PðnÞ 1�bð Þþ 1�PðnÞð Það ÞynD

x 1� PðnÞ 1�bð Þþ 1�PðnÞð Það Þð Þ�þ PðnÞbDm

1� PðnÞ 1�bð Þþ 1�PðnÞð Það Þð Þ¼ F ynð ÞD� G ynð Þ

. . . ð4Þwhere

FðynÞ ¼ sþ v PðnÞ 1�bð Þþ 1�PðnÞð Það Þ1� PðnÞ 1�bð Þþ 1�PðnÞð Það Þ � KðnÞ

1� PðnÞ 1�bð Þþ 1�PðnÞð Það Þð Þyn� ðcþdÞ

1� PðnÞ 1�bð Þþ 1�PðnÞð Það Þð Þ � hðnÞ PðnÞ 1�bð Þþ 1�PðnÞð Það Þynx 1� PðnÞ 1�bð Þþ 1�PðnÞð Það Þð Þ

��þ

PðnÞbm1� PðnÞ 1�bð Þþ 1�PðnÞð Það Þð Þ

G ynð Þ ¼ hðnÞ yn 1� PðnÞ 1�bð Þþ 1�PðnÞð Það Þð Þ2

��

Inventory Level

yn

tn

Tn Time

p(n)yn

Fig. 2 Behavior of inventory level over time

....

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4.1 Formulation of fuzzy inventory model

Since the market demand is considered as imprecise in nature so the objectivefunction can be redefined as

Maximize TPU (yn) ¼ FðynÞeD� GðynÞWhere yn>0.wavy bar (~) denotes the fuzzification of the parameters. We express the fuzzy

demand rate eD as the triangular fuzzy number (D0−δ1,D0,D0+δ2). Suppose, the

membership function of the fuzzy demand rate eD is as follows:

μeDðDÞ ¼D� D0�d1ð Þ

d1D0 � d1 � D � D0

D0þd2ð Þ�Dd2

D0 � D � D0 þ d20 elsewhere

8><>:

Here, 0< δ1<D0, 0< δ2and D0 are given fixed numbers. δ1 and δ2 are determined bythe decision maker based on the given uncertainty. From Eq. (4), for each yn, letHynðDÞ ¼ FðynÞD� GðynÞ

and y ¼ HynðDÞthen, we have D ¼ Hyn ðDÞþG ynð Þ

F ynð ÞBy the Extension Principle

μHyn ðeDÞð yÞ ¼ Sup

D2Hyn ð yÞμeDðDÞ ¼ μeD Hyn ðDÞþGð ynÞ

Fð ynÞ� �

¼Hyn ðDÞþGð ynÞ�ðD0�d1ÞFð ynÞ

Fð ynÞd1 ðD0 � d1ÞFð ynÞ � Gð ynÞ � HynðDÞ � D0Fð ynÞ � GðynÞD0þd2ð ÞFð ynÞ�Hyn ðDÞ�Gð ynÞ

Fð ynÞd2 D0Fð ynÞ � Gð ynÞ � HynðDÞ � D0 þ d2ð ÞFð ynÞ � Gð ynÞ0 elsewhere

8>><>>:

Now, we find the centroid of μHyn

eD� �ðyÞ .R1�1 μ

Hyn ðeDÞðyÞdy ¼ RD0FðynÞ�GðynÞðD0�d1ÞFðynÞ�GðynÞ

yþGðynÞ�ðD0�d1ÞFðynÞFðynÞd1 dy

þ R D0þd2ð ÞFðynÞ�GðynÞD0FðynÞ�GðynÞ

D0þd2ð ÞFðynÞ�y�GðynÞFðynÞd2 dy

¼ d1þd22 F ynð Þ ¼ P sayð Þ

and

R1�1 yμ

Hyn ðeDÞðyÞdy ¼ RD0FðynÞ�GðynÞðD0�d1ÞFðynÞ�GðynÞ y

yþGðynÞ�ðD0�d1ÞFðynÞFðynÞd1 dy

þ R D0þd2ð ÞFðynÞ�GðynÞD0FðynÞ�GðynÞ y D0þd2ð ÞFðynÞ�y�GðynÞ

FðynÞd2 dy

¼ D0FðynÞ � GðynÞ � FðynÞðd1�d2Þ3

� �ðd1þd2ÞFðynÞ

2 ¼ RðsayÞ

The centroid of μHyn

eD� �ðyÞ is

M yn; d1; d2ð Þ ¼ R P ¼= D0F ynð Þ � G ynð Þ � F ynð Þ d1 � d2ð Þ3

ð5Þ

where, 0< δ1< D0, 0< δ2.

....

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M(yn,δ1,δ2) is the estimate of total cost in fuzzy sense.If δ1=δ2, then Eq. (5) reduces to Eq. (4).Note that n in Eq. (5) is an input parameter and not a decision variable and P(n) is

constant. Therefore,

d2M

dy2n¼ 2KðnÞ D0 � 1 3= d1 � d2ð Þð Þ

1� PðnÞ 1� bð Þ þ 1� PðnÞð Það Þð Þy3n< 0; 8yn > 0;

This means that Eq. (5) is concave. By setting the first derivative of Eq. (5) equalto zero, i.e., dM

dyn¼ 0 , and solving for yn we get

y*n ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2KðnÞ D0 � 13 d1 � d2ð Þ� �

hðnÞ 1� PðnÞ 1� bð Þ þ 1� PðnÞð Það Þð Þ2 þ 2 PðnÞ 1�bð Þþ 1�PðnÞð Það Þ D0�13 d1�d2ð Þð Þ

x

� vuuut

. . . ð6Þ

where P(n)= agþebn and P(n) is a continuously decreasing function over n since

dP(n)/dn<0, n>0.

5 Numerical example

Following parameters are used for numerical calculation.D0=50,000 units/year, d1 ¼ 15; d2 ¼ 10 , x=175,200 units/year, d=$0.5/unit, c=

$25/unit, s=$50/unit k0=90, k1=10, h0=4, h1=1, α1=0.2, α2=0.2 and v=$20/unit, a=40and g=999, α=0.02, β=0.03, m=$100/unit.

From Table 1, it is observed that as the value of ‘b’ increases the profit of theorganization increases. If the value of ‘b’ increases rapidly then slope of the profitcurve also increases and hence saturation level of profit achieve faster. For b=1, from

Table 1 Effect of learning on profit

Number of Shipment(n) Learning rate

b=1 b=1.2 b=1.4 b=1.6 b=1.8 b=2

1 1257560 1257570 1257570 1257580 1257580 1257590

2 1257730 1257750 1257770 1257810 1257870 1257950

3 1257860 1257940 1258080 1258320 1258680 1259220

4 1258080 1258360 1258890 1259700 1260670 1261520

5 1258510 1259300 1260460 1261560 1262230 1262550

6 1259330 1260730 1261910 1262480 1262680 1262750

7 1260510 1261930 1262550 1262740 1262780 1262800

8 1261630 1262520 1262760 1262810 1262820 1262820

9 1262320 1262740 1262820 1262830 1262840 1262840

10 1262650 1262820 1262850 1262850 1262850 1262850

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Fig. 3, it is observed that saturation level of profit is not achieved till 10th shipmentwhile it can achieved in 10th shipment if b=2. So it clear that if the nature of manageris of high learning from the past company will grow faster. If ‘b’ is considered as aproxy for attitude and managerial capacity of a manager then it observes that a higherlevel of ‘b’ helps in faster growth of the company.

1 2 3 4 5 6 7 8 9 101.257

1.258

1.259

1.26

1.261

1.262

1.263x 10

6

Number of Shipment

Pro

fit

b=1.0

b=1.2

b=1.4b=1.6

b=1.8

b=2.0

Fig. 3 Effect of learning on profit

1 2 3 4 5 6 7 8 9 101.257

1.258

1.259

1.26

1.261

1.262

1.263x 10

6

Number of Shipment

Pro

fit

No Learning Effect

Learning Effect on Holding CostLearning Effect on Orderning Cost

Learning Effect on % of DefectiveItems Present in Each Lot

Learning Effect on all Three

Fig. 4 Profit vs Learning Effect

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Figure 4/ Fig. 5/Table 2 present the effect of learning on holding cost, orderingcost and percentage of defective items present in each lot on profit and on lot sizeindependently and simultaneously. On the analysis following observations aremade:

a. From Fig. 5 it is clear that lot size increases as the holding cost decreases due tolearning effect. Thus in this case it is advisable to hold more inventories to securemore profit (Fig. 4).

b. From Fig. 5 it is clear that lot size decreases as the ordering cost decreases due tolearning effect. So it is advisable for decision-maker to order the lot in smaller insize when ordering cost decreases.

c. From Fig. 5 it is clear that lot size decreases when there is effect of learning onpercentage of defective items present in each lot. In this scenario manager ordersmaller lot so that by learning the number of defective items decreases in each lotand they can earn more profit for organization.

d. From Fig. 4 it is observed that profit is highly affected when we consider theeffect of learning on all three simultaneously.

From above observation we can say that a manager should focus to learn in all thethree directions rather then be a specialization of a particular field.

From Table 3/Fig. 6 it is observed that as the demand changes from +20 % to -20 % the change in lot size in our proposed model is less than that of Papachristos andKonstantaras [19] and Salameh and Jaber [22]. This shows that our proposed model isvery stable with respect to change in demand. This is due to incorporating thevagueness in market demand. So, it is important to consider vague nature of marketdemand in model.

1 2 3 4 5 6 7 8 9 101360

1380

1400

1420

1440

1460

1480

Lot

Siz

e in

Eac

h S

hipm

ent

Number of Shipment

No Learning

Learning Effect on Holding Cost OnlyLearning Effect on Ordering Cost Only

Learning Effect on % of Defective ItemsPresent in Each Lot

Learning Effect in all Three

Fig. 5 Lot Size vs Learning Effect

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From Fig. 7, as workers moves from first phase (incipient) to the secondphase (learning), learning effect due to acquaintance with the system increasesso that the number of defective items present in each shipment graduallydecreases.

Table 2 Effect of learning on lot size and profit in different situation

Number of shipment When there is noLearning Effect

LearningEffect onlyon HoldingCost

LearningEffect onlyon orderingCost

Learning Effectonly on % ofdefective itemin lot

Learning Effecton all threesimultaneously

1 Lot Size 1418.25 1418.25 1418.25 1418.15 1418.15

Profit 1257560 1257560 1257560 1257570 1257570

2 Lot Size 1418.25 1436.97 1409.04 1417.85 1427.24

Profit 1257560 1257650 1257560 1257760 1257750

3 Lot Size 1418.25 1447.08 1404.19 1416.88 1431.35

Profit 1257560 1257700 1257630 1257730 1257940

4 Lot Size 1418.25 1453.89 1400.97 1413.96 1431.84

Profit 1257560 1257730 1257640 1258100 1258360

5 Lot Size 1418.25 1458.97 1398.59 1406.84 1427.18

Profit 1257560 1257760 1257660 1259000 1259300

6 Lot Size 1418.25 1462.99 1396.73 1395.59 1417.78

Profit 1257560 1257780 1257670 1260410 1260730

7 Lot Size 1418.25 1466.31 1395.20 1386.05 1409.73

Profit 1257560 1257790 1257670 1261590 1261930

8 Lot Size 1418.25 1469.11 1393.91 1381.38 1406.37

Profit 1257560 1257810 1257680 1262160 1262520

9 Lot Size 1418.25 1471.54 1392.80 1379.69 1405.85

Profit 1257560 1257820 1257690 1262360 1262740

10 Lot Size 1418.25 1473.67 1391.83 1379.15 1406.36

Profit 1257560 1257830 1257690 1262430 1262820

Table 3 Variation in market demand and its effect on lot size

S.No. % Changein demand

% Change in lot sizein our proposed model

% Change in lot size inPapachristos and Konstantaras [19]

% Change in lot size inSalameh and Jaber [22]

1 -20 -2.1162 -9.9911 -11.5040

2 -15 -1.9829 -7.6017 -8.7633

3 -10 -0.9591 -5.3119 -6.1034

4 -5 -0.0348 -3.1136 -3.5275

5 5 0.5487 3.0352 3.4543

6 10 0.9923 4.9975 5.8497

7 15 1.4254 6.8917 8.1902

8 20 2.7633 8.7224 9.4794

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-20 -15 -10 -5 0 5 10 15 20-15

-10

-5

0

5

10

% Variation in Demand

% C

hange in lot

siz

e

Comprasion between our proposed model with others

Our proposed model

Papachristos and Konstantaras (2006)Salameh and Jaber (2000)

Fig. 6 Comparison between our proposed model with others

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

Number of Shipment

Num

ber

of D

efe

ctiv

e it

em

in p

er

lot

Defective Item vs Learning Effect

b=1b=1.2

b=1.4

b=1.6

b=1.8b=2

Fig. 7 Defective Item vs Learning Effect

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6 Conclusion

In this paper, fuzzy inventory model with imperfect-quality items subjected to thelearning effect on holding cost, ordering cost and number of defective items presentin each received lot is developed. Market demand is imprecise and represented bytriangular fuzzy number. Centroid method is used as the method of defuzzification tofind the estimate of total profit per unit time in the fuzzy sense. Optimal lot size isderived by using the calculus method to maximize the total profit expression.Through numerical example, behavior and utility of developed model is investigated.

We considered the inventory problem in which demand parameter is consider asfuzzy variable to handle the uncertainty of market. This model helps decision-makerto take more stable decision in this uncertain environment. He/she has not to changetheir decision by the vagueness of market demand.

It is also real practice that when the repetitive process goes on then we learn fromthe past and our efficiency to take decision and to calculate the various cost ofinventory gradually increases. Due to this, the number of defective items presentsin each lot, holding cost and ordering cost not remains same in each cycle. So it isrelevant to consider learning effect on these three. This model provides the directionto decision-makers to take account of learning effect while taking decision. By doingso, he/she earns more profit for the organization. Thus we can say that manager kepthis learning process in all direction for the benefit of organization.

A future extension is to discuss model in more realistic situation by considerimpreciseness in different inventory related cost and taking different form of demandpattern.

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