An Intuitionistic Approach for Price Breaks in … intuitionistic approach for price breaks in EOQ...

Applied Mathematical Sciences, Vol. 9, 2015, no. 71, 3511 - 3523

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2015.52105

An Intuitionistic Approach for Price Breaks in

EOQ from Buyer’s Perspective

Prabjot Kaur 1 and Mahuya Deb2

1Dept. of Applied Mathematics

Birla Institute of Technology, Mesra

Jharkhand, India

2Usha Martin Academy, Ranchi

Jharkhand, India

Copyright © 2015 Prabjot Kaur and Mahuya Deb. This article is distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in

any medium, provided the original work is properly cited.

Abstract

Inventory though an ideal resource is considered to be one of the most important

asset which fulfils various functions of an organisation. The classical model

attributed to Ford Whitman Harris ( Harris 1913 ), for inventory, however, does

not take into account , amongst other things , the quantity discounts given by the

supplier, on the cost of the units themselves in order to motivate buyers to

purchase in larger quantities . The occurrence of such problems takes place when

discounts are offered for purchase of large quantities. These discounts take the

form of price breaks (Gupta and Hira, 2011). Unlike the majority of the

inventory literature, where purchase price is assumed to be constant and hence

excluded from the model, suppliers in many real life applications occasionally

offer price discounts.The current paper is an attempt to define a class of inventory

problems for buyers where the demand and purchasing cost is a triangular

intuitionistic fuzzy number. The ultimate objective is to determine the optimal

order quantity and the minimum cost under the proposed model.

Keywords: Price breaks, Inventory, Triangular Fuzzy Numbers, Triangular

Intuitionistic Fuzzy Numbers

3512 Prabjot Kaur and Mahuya Deb

1. Introduction

Classical inventory problems are generally formulated under the assumption

that the cost of the item was not affected by the size of the order. It considers a

unit price which is constant and independent in nature. The supplier’s challenge

lies as to how he can adjust his pricing schedule so as to motivate his customer to

buy in large quantities. To encourage buyers to purchase more units of an item

this adjustment in the price schedule calls for offering quantity discounts .Such a

situation calls for evaluating the trade off between the saving in purchase cost ,

ordering cost and the increased cost of holding the inventory. This inclusion of

purchase price makes the inventory problems different from the classical ones. In

the classical inventory model [9], the optimal order quantity Q* is the point that

minimizes the total annual inventory cost, where total annual inventory cost is a

function of order quantity, and is the sum of the annual carrying cost and the

annual ordering cost. In determining the annual carrying cost, it is convenient to

use the average on-hand inventory level that can be simply calculated as one-half

of the order quantity. Here, we describe the function of total annual inventory cost

of inventory model as following [19]

TC=Q

DC0 +2

hQC

Where

Q = Number of pieces per order,

C0 = Ordering cost for each order,

Ch = Holding or carrying cost per unit per year,

D = Annual demand in units.

Therefore when discount is available, the total cost per unit of inventory system

and items would be,

TC = 𝐷𝐾1 +𝐷∗𝐶0

𝑄+

1

2∗ 𝑄 ∗ 𝐾1 ∗ 𝐼,

Where K1 is the price per unit for Q units, I is the holding cost rate

By using property of minimum, the optimum order quantity is given by

Q*= √2∗𝐶0∗𝐷

𝐾1∗𝐼

An intuitionistic approach for price breaks in EOQ 3513

Suppose the following price discount schedule is quoted by a supplier in which a

price break (quantity discount) occurs at quantity b. This means

Quantity Price per unit (Rs)

0 ≤ Q1< 𝑏 P1

b ≤ Q2 P 2 (<P1)

The optimal purchase quantity can be determined by the following procedure:

The Algorithm:

Step 1: Consider the lowest price (P2) and determine Q2* by using the basic EOQ

formula:

Q2* =

IP

DC

*

2

2

0

If Q2* lies in the range specified,

*

2Qb then Q2* is the EOQ i,e Q*=Q2*. The

optimal cost TC* associated with Q* is calculated as follows:

TC*=.2

*202

IbP

b

DCDP

Step 2: If Q2*< b, we cannot place an order at the reduced price P2. We calculate

Q1* with price P1 and the corresponding total cost TC at Q1

*.If TC(b)>TC(Q1*) ,

then EOQ is Q*=Q1*.

Otherwise Q*=b is the required EOQ.

An extensive study of the literature discusses various inventory models related to

discounts to get deeper insights into relationship between price discounts and

order policy. Traditionally, the quantity discount problem has been analyzed from

a buyer's perspective. Hadley and Whitin [3], Peterson and Silver [16] and

Starr &Miller [8] considered various discount policies and demand assumptions.

On the other hand Monahan [6] has approached the problem from a supplier's

perspective. Lee et al [12 ] in his paper has discussed about the joint problem of

ordering and offering price discount by a supplier to his sole/major buyer.Yang

[14] developed an optimal pricing and ordering policy for a deteriorating item

with price sensitive demand. Inventory management with uncertain prices has

received intense attention in the operations research and management science

community for years. A major shortcoming of the current literature on inventory

models with discount prices is that the value of the discount price is assumed to

be deterministic. Precise information about the inventory parameters are difficult

to tap in the real world. This type of imprecise data is not always well represented


by random variables selected from probability distribution. So, decision making

methods under uncertainty are needed. To deal with this uncertainty and imprecise

data, the concept of fuzziness can be applied. Fuzzy set theory was introduced by

Zadeh [11] and the application of fuzzy set theory to inventory problems has been

proposed by authors like Kacpryzk and Staniewski [5] and Park [9]. Benton

[21] presented an efficient heuristic algorithm for evaluating alternative discount

schedules under multi-item and multi-supplier conditions, considering the

resource limitations. Benton and Park [22] classified the literature on lot sizing

determination under several types of discount schemes and discussed some of the

significant literature in this area. Das et al [7] discussed a multi objective

economic lot size model for both the buyer and the seller for a deteriorating item

with discount in crisp and fuzzy environment. Syed and Aziz [20] developed a

Fuzzy inventory model with price breaks where the model is solved for optimum

results by reformulating it as Fuzzy linear programming problem. One thing the

fuzzy sets lack is non-membership function. The information expressed by fuzzy

sets is not complete in context of decision making because alternatives satisfy the

attributes but no arrangements for alternatives dissatisfying the attributes.

Atanassov [10] characterized the IFS by expressing it in terms of membership

function and non-membership function, such that the sum of both values is less

than one. Thus intuitionistic fuzzy set theory seems to be very useful for

modelling situations with missing information or hesitance. Banerjee and Roy

[17] generalized the application of the intuitionistic fuzzy optimization in the

constrained multi objective stochastic inventory model. Recently Chakraborty et

al [21] gave the solution for the basic EOQ model using intuitionistic fuzzy

optimization technique. Pal and Chandra [13] in their paper studied a periodic

review inventory model with stock dependent demand and When stock on hand

was zero, the inventory manager offers a price discount to customers who are

willing to backorder their demand. Lee and Yang [1] proposed an efficient

genetic algorithm to solve the lot sizing problem with multi supplier and quantity

discount.

This paper is organized as follows: Section 2 describes the basics of Intuitionistic

Fuzzy Set . Section 3 deals with the development of Inventory Model with price

breaks using the demand and price as triangular fuzzy and triangular intuitionistic

fuzzy numbers. In Section 4 a numerical example illustrates our proposed

approach. Finally we concluded in Section 5 that the optimal order quantity for

crisp, fuzzy and intuitionistic fuzzy sets is the same while there exist differences

in the total cost. In addition to the above sensitivity analysis is also carried out.

2. Fuzzy EOQ Model with fuzzy Price break

In the following, we introduce the fuzzy inventory models with fuzzy price break

and find the optimal solutions of the model for the optimal crisp order quantity. In

this paper, we assume that there is no stock outs , no backlogs , replenishment is

instantaneous and the ordering cost involved to receive an order are known and


constant and that purchasing values at which discounts are offered are triangular

fuzzy numbers . In addition, in order to simplify the treatment of this fuzzy

inventory models, we use the following variables:

D~

: fuzzy yearly demand,

P~

: fuzzy purchasing cost

And suppose ),,(~

321 DDDD , ),,(~

321 PPPP , ),,(~

1212111 PPPP and

),,(~

2322212 PPPP are non negative triangular fuzzy numbers.

Now, we introduce the fuzzy inventory model under fuzzy demand and fuzzy

purchasing price at which the quantity discounts are offered. The fuzzy total cost

function is given by –

2

*~~

~*

~~ 0 IPQ

Q

CDPDCT (1)

Then we solve the optimal order quantity of formula (1)

Using the arithmetic operations of triangular fuzzy numbers we calculate the

fuzzy total inventory cost as follows

)2

**,

2

**,

2

**(

~ 30333

20222

10111

IPQ

Q

CDPD

IPQ

Q

CDPD

IPQ

Q

CDPDCT The

total cost after defuzzification as in formula (*) is given by

]2

**)

2

**(2

2

**[

4

1)

~( 303

33202

22101

11

IPQ

Q

CDPD

IPQ

Q

CDPD

IPQ

Q

CDPDCTD

We find the minimization of )~

( CTD by taking the derivative of )~

( CTD and

equating it to zero which after simplification gives

),,(

)2(2~

321

030201

IPIPIP

CDCDCDQ

(2)

Fuzzy price break

Quantity Price per unit (Rs)

0 ≤ Q1< 𝑏 1

~P

b ≤ Q2 12 )~

(~

PP

The Algorithm

Step I: Consider the lowest price ( 2

~P ) and determine 2

~Q by using the basic EOQ

formula:


)(

)2(2~

321

030201

2IPIPIP

CDCDCDQ

If 2

~Q lies in the range specified, 2

~Qb then 2

~Q is the EOQ .The optimal cost

CT~

associated with Q~

is calculated as follows:

CT~

=.2

*~~

~~ 20

2

IPb

b

CDPD

Step 2: If 2

~Q < b, we cannot place an order at the reduced price 2

~P . We calculate

1

~Q with price 1

~P and the corresponding total cost TC at 1

~Q . If )

~(

~)(

~1QCTbCT ,

then EOQ is Q*=. 1

~Q Otherwise Q*=b is the required EOQ.

3. Intuitionistic Fuzzy EOQ Model

We assume in the crisp model the demand and the purchasing price at which

discounts are offered to be triangular Intuitionistic fuzzy numbers[10] and

represented as follows:-

�̿�= ),,)(,,( 321321

DDDDDD

�̿�),,)(,,( 321321

PPPPPP

),,)(,,( 1312111312111

PPPPPPP

),,)(,,(P 2322212322212

PPPPPP

2

*0 IPQ

Q

CDPDTC

Accuracy function for defuzzification in TIFN �̿� / /

1, 2 3 1 2 3( , );( , , )a a a a a a

is defined as


�̿�′ =(𝑎1+2𝑎2+𝑎3)+(𝑎1

′+2𝑎2′+𝑎3

′)

8 (**)

The basic EOQ after defuzzification using equation (**) is given by

)4(

)4(2

31321

0301030201

IPIPIPIPIP

CDCDCDCDCDQ

. (3)

The Algorithm

Step 1: Consider the lowest price ( 2P ) and determine 2Q by using the basic EOQ

formula (3)

If 2Q lies in the range specified, 2Qb then 2Q is the EOQ .The optimal cost

TC associated with 2Q is calculated as follows:

2

*0

IPb

b

CDPDTC

Step 2: If 2Q < b, we cannot place an order at the reduced price 2P . We calculate

1Q with price 1P and the corresponding total cost TC at 1Q . If )()( 1QTCbTC ,

then EOQ is Q*= 1Q Otherwise Q*=b is the required EOQ.

4. Numerical Example

A manufacturing company issues the supply of a special component which has the

following price schedule:

0 to 99 items: Rs 1000 per unit

100 items and above: Rs 950 per unit

The inventory holding costs are estimated to be 25% of the value of the inventory.

The procurement ordering costs are estimated to be Rs 2,000 per order. If the

annual requirement of the special component is 300, compute the economic order

quantity for the procurement of these items.


Solution:

Given D=300, C0=Rs 2000 I =0.25 P1=Rs 1000 P2= Rs 1000

We calculate Q2 corresponding to the lowest price 950.

Q2 = 950*25.0

2000*300*2

= 71 which is less than the price break point.

Therefore, we have to determine the optimal total cost for the first price and the

total cost at the price- break corresponding to the second price and compare the

two

TC (P1=1000) =2

25.0*1000*71

71

2000*3001000*300

= Rs 317325.704

And

TC (b=100) 2

25.0*950*100

100

2000*300950*300

=Rs 302875 which is lower than the total cost corresponding to Q2.

Therefore, the economic quantity for a procurement lot is 100 units (price break

point)

Fuzzy Case:

Let

25.0

2000

)1050,950,850(~

)1100,1000,900(~

)400,300,200(~

0

2

1

I

RsC

P

P

D

According to the algorithm

We calculate 2

~Q corresponding to the lowest price 2

~P


)(

)2(2~

321

030201

2IPIPIP

CDCDCDQ

= 82.078 which is less than the price break point



two.

The optimal cost CT~

associated with 1

~P is calculated as follows:

CT~

=.2

*~~

~

~~~ 12

2

0

1

IPQ

Q

CDPD

=319821(after defuzzification )

Also

CT~

(b=100)= 2

*~~

~~ 20

2

IPb

b

CDPD

= 307875 which is lower than the total cost corresponding to Q2.


point)

Intuitionistic Fuzzy Case:

Let

25.0

2000

)1150,950,750)(1050,950,850(~

)1200,1000,800)(1100,1000,900(~

)500,300,100)(400,300,200(~

0

2

1

I

RsC

P

P

D

According to the algorithm

We calculate 2Q corresponding to the lowest price 2P

)4(

)4(2

31321

0301030201

2

IPIPIPIPIP

CDCDCDCDCDQ

=50.26 which is less than the price break point




two.

The optimal cost associated with 1

~P is calculated as follows

2

** 12

2

0

1

IPQ

Q

CDPDTC

= 330720.3363 (after defuzzification )

Also

2

** 20

2

IPb

b

CDPDTC

=240453.125 (after defuzzification) which is lower than the total cost

corresponding to Q2.


point) which is described in Table 1.

Table 1

Crisp Fuzzy Intuitionistic

Optimal Quantity 100 100 100

Optimal Cost 302875 307875 240452.125

5. Sensitivity Analysis

Now the effect of changes in the system parameters on the optimal values of Q i,e

the economic order quantity when only one parameter changes and others remain

unchanged the computational results are described in Table 2. The EOQ is less

sensitive to the changes in demand.

Table 2 Demand EOQ

Crisp

270 67.43

280 68.67

300 71.08

320 73.40

340 75.67


Table 2 (Continued)

Fuzzy

(250,270,300) 78.23

(260,280,310) 79.65

(200,300,400) 82.08

(300,320,340) 84.77

(320,340,360) 87.38

Intuitionistic

Fuzzy

(250,270,300)(240,270,310) 67.45

(260,280,310)(250,280,320) 68.97

(200,300,400)(100,300,500) 71.08

(300,320,340)(280,320,350) 73.25

(320,340,360)(300,340,380) 75.67

6. Conclusion and further research

This paper contributes to the application of the EOQ model with price break in

crisp, fuzzy and intuitionistic environment. The uncertainty inherent in the price

and demand due to the dynamic business environment is reflected by considering

these parameters as fuzzy and intuitionistic fuzzy numbers. The optimal values for

the problem are obtained and the solution compared with the fuzzy and crisp case.

A numerical example coherent with these settings reflects the same. Although the

addition of a one more degree of freedom in the uncertainty measure has not much

impact on the optimal quantity being ordered but there is substantial savings in

terms of optimum cost. The advantage of the proposed intuitionistic approach is

that it is a robust model which deals with the varying parameters in a general

business inventory consistent with human behaviour by reflecting and modelling

the hesitancy present in real life situations. The sensitivity analysis indicates the

consistency of the crisp case from the fuzzy and intuitionistic fuzzy sense. Even

though the results are same in terms of the optimal quantity the buyers are in a

position to make better decisions even by paying for some uncertainty costs. It

gives an opportunity for decision making with better results with the use of

multiple choice and options. In fact discount level and price consciousness have

an effect on purchasing decisions of buyers. Prior knowledge affects price

acceptability. However in situations where the decision maker does not have

exact knowledge about the values of the coefficients that take part in the problem

and moreover the vagueness is not of probabilistic kind, the inexact values which

are determined by models using IF numbers can be substantially useful. This can

be an advantage for the buyer who can easily minimize the worse cases and

maximize the better ones. Hence this model is executable and useful in the real

world. Further research includes the task to cover more membership functions

than the triangular ones.

Also the difference between different defuzzification methods should be

investigated within these settings.


References

[1] A. Lee and H. Kang, Genetic Algorithm to Solve the Lot Sizing Problem

with Multi Supplier and Quantity Discount, IEEE 2011.

http://dx.doi.org/10.1109/pacrim.2011.6032877

[2] A. Nagoorgani and K. Ponnalagu, A New Approach on Solving Intuitioni-

stic Fuzzy Linear Programming Problem”, Applied Mathematical Sciences, Vol.

6, no. 70(2012), 3467 – 3474.

[3] G. Hadley and T. M. Whitin, Analysis of Inventory Systems, Prentice

Hall, Inc, Englewood Cliffs, N.J 1963.

[4] G. S. Mahapatra and T. K. Roy, Reliability Evaluation Using Triangular

Intuitionistic Fuzzy Numbers Arithmetic Operations. World Academy of Science,

Engineering and Technology, 50(2009), 574-581.

[5] J. Kacpryzk and P. Staniewski, Long-term inventory policy-making

through fuzzy decision-making models. Fuzzy Sets and Systems, 8(1982), 117-

132. http://dx.doi.org/10.1016/0165-0114(82)90002-1

[6] J. P. Monahan, A Quantity Discount Pricing Model to Increase Vendor

Profits, Management Science, Volume 30, 1984.

http://dx.doi.org/10.1287/mnsc.30.6.720

[7] K. Das, T. K. Roy and M. Maiti, Buyer –Seller fuzzy inventory model for

a deteriorating item with discount. International Journal of Systems Science, 35(8)

(2004), 457-466. http://dx.doi.org/10.1080/00207720410001734200

[8] K. M. Starr and D. W. Miller, Inventory Control: Theory and Practice,

Prentice Hall, Inc, Englewood Cliffs, N.J 1962.

[9] K. S. Park, Fuzzy set theoretic interpretation of economic order quantity,

IEEE Transactions on Systems, Man and Cybernetics, (1987) 1082-1084.

http://dx.doi.org/10.1109/tsmc.1987.6499320

[10] K. T. Atanassov, Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20,

(1986), 87-96. http://dx.doi.org/10.1016/s0165-0114(86)80034-3

[11] L. A. Zadeh, Fuzzy Sets. Information and Control, 8(1965), 338-353.

http://dx.doi.org/10.1016/s0019-9958(65)90241-x

[12] L. E. Lee and M. J. Rosenblatt, A Generalized Quantity Discount Pricing

Model to Increase Supplier's Profits, Management Science, Volume 32 Issue 9

(1986), 1177-1185. http://dx.doi.org/10.1287/mnsc.32.9.1177

http://dx.doi.org/10.1109/pacrim.2011.6032877

http://dx.doi.org/10.1016/0165-0114%2882%2990002-1


http://dx.doi.org/10.1080/00207720410001734200

http://dx.doi.org/10.1109/tsmc.1987.6499320

http://dx.doi.org/10.1016/s0165-0114%2886%2980034-3

http://dx.doi.org/10.1016/s0019-9958%2865%2990241-x



[13] M. Pal and S. Chandra. A Periodic Review Inventory Model with Stock

Dependent Demand, Permissible Delay in Payment and Price Discount on

Backorders, Yogoslav Journal Of Operations Research , (2013) ISSN: 0354-0243

EISSN: 2334-6043 24.1 (2014). http://dx.doi.org/10.2298/yjor120512017p

[14] P. C. Yang, Pricing Strategy for deteriorating items using quantity

discount when demand is price sensitive, European Journal of Operational

Research 157(2) (2004), 389-397.

http://dx.doi.org/10.1016/s0377-2217(03)00241-8

[15] P. K. Gupta and S. H. Hira, Operations Research, S. Chand and Company

Ltd, New Delhi, revised and enlarged edition 2012.

[16] R. Peterson and E. A. Silver, Decisions Systems for Inventory

Management and Production Planning, John Wiley &Sons, New York 1979.

[17] S. Banerjee and T. K. Roy, Solution of single and multiobjective

stochastic inventory models with fuzzy cost components by intuitionistic fuzzy

optimization techniques, Advances in Operations Research (2010).

http://dx.doi.org/10.1155/2010/765278

[18] S. Chakraborty, M. Pal, and P. K. Nayak, Intuitionistic Fuzzy

Optimization technique for the solution of an EOQ model, Fifteenth International

Conference on IFS, Burgas, NIFS 17, 2 (2011) 52-64.

[19] S. Kalavathy, Operations Research with C programs, Vikas Publishing

house Pvt. Third edition 2009.

[20] Syed &Aziz, An inventory model with price breaks: Fuzzy Approach,

African Journal of Mathematics and Computer Science Research, vol (10) (2010),

232-236.

[21] W. Benton, Quantity discounts under conditions of multiple items,

multiple suppliers and resource limitations, Int. J. Prod. Res., 29(1991), 1953–

1961. http://dx.doi.org/10.1080/00207549108948060

[22] W. C. Benton and S. Park. "A classification of literature on determining

the lot size under quantity discounts." European Journal of Operational Research

92.2 (1996): 219-238. http://dx.doi.org/10.1016/0377-2217(95)00315-0

Received: February 17, 2015; Published: April 23, 2015

http://dx.doi.org/10.2298/yjor120512017p

http://dx.doi.org/10.1016/s0377-2217%2803%2900241-8

http://dx.doi.org/10.1155/2010/765278

http://dx.doi.org/10.1080/00207549108948060

http://dx.doi.org/10.1016/0377-2217%2895%2900315-0

An Intuitionistic Approach for Price Breaks in … intuitionistic approach for price breaks in EOQ...

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