Neutrosophic EOQ Model with Price Break

2018·0.54 MB·English

by  M. Mulla, R. Surya;

Neutrosophic Sets and Systems, Vol. 19, 2018 24
University of New Mexico
Neutrosophic EOQ Model with Price Break
M. Mullai1
Assistant Professor of Mathematics, Alagappa University, Karaikudi,
Tamilnadu, India.
E-mail: mullaimalagappauniversity.ac.in
R. Surya2
Department of Mathematics, Alagappa University, Karaikudi,
Tamilnadu, India.
E-mail: [email protected]
Abstract—Inventorycontrolofanidealresourceisthemostimportantonewhichfulfilsvariousactivities(functions)of
anorganisation.Thesuppliergivesthediscountforaniteminthecostofunitsinordertomotivatethebuyers(or)
customerstopurchasethelargequantityofthatitem.Thesediscountstaketheformofpricebreakswherepurchase
costisassumedtobeconstant.InthispaperanEOQmodelwithpricebreakininventorymodelisdevelopedtoobtain
itsoptimumsolutionbyassumingneutrosophicdemandandneutrosophicpurchasingcostastriangularneutrosophic
numbers.Anumericalexampleisprovidedtoillustratetheproposedmodel.
Keywords: Price break, neutrosophic demand, neutrosophic purchase cost, neutrosophic sets, triangular neutrosophic number.
1 INTRODUCTION by M. Mullai and S. Broumi[3].
In this paper, we introduce the neutro-
Bai and Li[1] have discussed triangular and sophic inventory models with neutrosophic
trapezoidal fuzzy numbers in inventory model price break to find the optimal solution of the
for determining the optimal order quantity and model for the optimal order quantity. Also the
the optimal cost. The quantity discount prob- neutrosophic inventory model under neutro-
lem has been analyzed from a buyers perspec- sophic demand and neutrosophic purchasing
tive. Hadley and Whintin[2], Peterson and Sil- cost at which the quantity discount are offered
ver[3], and Starr and Miller[6] considered vari- to be triangular neutrosophic number. Also the
ous discount polices and demand assumptions. optimal order quantity for the neutrosophic to-
Yang and Wee[7] developed an economic tal cost is determined by defining the accuracy
ordering policy in the view of both the sup- function of triangular neutrosophic numbers.
plier and the buyer. Prabjot Kaur and Mahuya
Deb[5]developedanintuitionisticapproachfor
2 NOTATIONS:
price breaks in EOQ from buyer’s perspec-
tive. Smarandache[5] introduced neutrosophic QN = Number of pieces per order
set and neutrosophic logic by considering the
non-standard analysis. Also, neutrosophic in- CN = Neutrosophic Ordering cost for each
0
ventory model without shortages is introduced order
M. Mullai, R. Surya: Neutrosophic EOQ Model with Price Break
Neutrosophic Sets and Systems, Vol. 19, 2018 25
QNP3NIN)(D(cid:48)NP(cid:48)N + D1(cid:48)NC0N + QNP1(cid:48)NIN,DNPN +
2 1 1 QN 2 2 2
CNh = Neutrosophic Holding cost per unit D2NC0N + QNP2NIN,D(cid:48)NP(cid:48)N + D3(cid:48)NC0N +
QN 2 3 3 QN
per year
QNP3(cid:48)NIN)(D(cid:48)(cid:48)NP(cid:48)(cid:48)N+D1(cid:48)(cid:48)NC0N +QNP1(cid:48)(cid:48)NIN,DNPN+
2 1 1 QN 2 2 2
DN = Neutrosophic Annual demand in D2NC0N + QNP2NIN,D(cid:48)(cid:48)NP(cid:48)(cid:48)N + D3(cid:48)(cid:48)NC0N + QNP3(cid:48)(cid:48)NIN)
QN 2 3 3 QN 2
units
The defuzzified total neutrosophic cost
using accuracy function is given by
3 NEUTROSOPHIC EOQ MODEL WITH
PRICE BREAK: D(TC)N = 18[(D1NP1N + D1QNNC0N + QNP21NIN) +
2(DNPN +D2NC0N +QNP2NIN)+(DNPN +D3NC0N +
The Neutrosophic inventory model with 2 2 QN 2 3 3 QN
neutrosophic price break is introduced to QNP3NIN) + (D(cid:48)(cid:48)NP(cid:48)(cid:48)N + D1(cid:48)(cid:48)NC0N + QNP1(cid:48)(cid:48)NIN) +
2 1 1 QN 2
find the optimal solutions for the optimal 2(DNPN + D2NC0N + QNP2NIN) + (D(cid:48)(cid:48)NP(cid:48)(cid:48)N +
2 2 QN 2 3 3
neutrosophic order quantity. Here we
D(cid:48)(cid:48)NCN QNP(cid:48)(cid:48)NIN
3 0 + 3 )]
assume that there is no stock outs, no QN 2
backlogs, replenishment is instantaneous,
To find the minimum of D(TC)N by taking
the neutrosophic ordering cost involved to
the derivative D(TC)N and equating it to zero,
receive an order are known and constant and
purchasing values at which discounts are
(i.e) 1 [(DNCN + 2DNCN + DNCN) +
offered as triangular neutrosophic numbers. 8Q2N 1 0 2 0 3 0
(D(cid:48)(cid:48)NCN + 2DNCN + D(cid:48)(cid:48)NCN)] + 1 [(PNIN +
Consider the following variables: 1 0 2 0 3 0 16 1
DN: Neutrosophic yearly demand, 2P2NIN+P3NIN)+(P1(cid:48)(cid:48)NIN+2P2NIN+P3(cid:48)(cid:48)NIN)] =
0, we get
(cid:114)
PN: Neutrosophic purchasing cost QN = 2[(D1NC0N+2D2NC0N+D3NC0N)+(D1(cid:48)(cid:48)NC0N+2D2NC0N+D3(cid:48)(cid:48)NC0N)]
[(PNIN+2PNIN+PNIN)+(P(cid:48)(cid:48)NIN+2PNIN+P(cid:48)(cid:48)NIN)]
1 2 3 1 2 3
Neutrosophic Price Break:
LetDN =(DN,DN,DN)(D(cid:48)N,DN,D(cid:48)N)(D(cid:48)(cid:48)N,DN,D(cid:48)(cid:48)N)
1 2 3 1 2 3 1 2 3
S.No. Quantity Price Per Unit (Rs)
PN = (P1N,P2N,P3N) (P1(cid:48)N,P2N,P3(cid:48)N) (P1(cid:48)(cid:48)N,P2N,P3(cid:48)(cid:48)N) 1 0≤ QN1 ≤ b P1N
2 b ≤ QN PN(< PN)
2 2 1
PN = (PN,PN,PN) (P(cid:48)N,PN,P(cid:48)N) (P(cid:48)(cid:48)N,PN,P(cid:48)(cid:48)N)
1 11 12 13 11 12 13 11 12 13
4 ALGORITHM FOR FINDING NEUTRO-
PN = (PN,PN,PN) (P(cid:48)N,PN,P(cid:48)N) (P(cid:48)(cid:48)N,PN,P(cid:48)(cid:48)N)
2 21 22 23 21 22 23 21 22 23 SOPHIC OPTIMAL QUANTITY AND NEU-
TROSOPHIC OPTIMAL COST:
are non negative triangular neutrosophic
numbers. Step I:
Consider the lowest price PN and determine
2
Now, we introduce the neutrosophic QN by using the economic order quantity
2
inventory model under neutrosophic demand (EOQ) formula:
(cid:114)
and neutrosophic purchasing cost at which QN = 2[(D1NC0N+2D2NC0N+D3NC0N)+(D1(cid:48)(cid:48)NC0N+2D2NC0N+D3(cid:48)(cid:48)NC0N)]
[(PNIN+2PNIN+PNIN)+(P(cid:48)(cid:48)NIN+2PNIN+P(cid:48)(cid:48)NIN)]
the quantity discounts are offered. Total 1 2 3 1 2 3
neutrosophic inventory cost is given by IfQN liesintherangespecified,b ≥ QN then
2 2
QN is the EOQ .The defuzzified optimal total
(TC)N = DN ⊗PN ⊕ DQNNC0N ⊕ QNPN2⊗IN co2st (TC)N associated with QN is calculated as
follows:
Thenthetotalneutrosophicinventorycostis
(TC)N = DN ∗PN + DNC0N + bP2N∗IN,
(TC)N = (D1NP1N + D1QNNC0N + QNP21NIN,D2NP2N + by using th2e abccuracy 2 function
D2NC0N + QNP2NIN,DNPN + D3NC0N + AN = (a1+2a2+a3)+(a(cid:48)1(cid:48)+2a2+a(cid:48)3(cid:48))
QN 2 3 3 QN 8
M. Mullai, R. Surya: Neutrosophic EOQ Model with Price Break
26 Neutrosophic Sets and Systems, Vol. 19, 2018
Step 2: TC(P1 = 800) = Rs.296039
TC(b=100) = Rs.224250, which is lower
(i) If QN < b, we cannot place an order at
2 than the total cost corresponding to Q
2
the lowest price PN.
2
(ii) We calculate QN with price PN and the (ii) Fuzzy Case:
1 1
corresponding total cost TC at QN.
(iii) If (TC)Nb > (TC)NQN, then EOQ is Given D(cid:101) = (300, 350, 400), P(cid:101)1 = (750, 800,
1 850)
Q∗N = QN, Otherwise Q∗N = b is the required
1
EOQ. P(cid:101) = (550, 600, 650),C = Rs.1500,I = 0.3
2 0
The EOQ in crisp, fuzzy and intuitionistic Q(cid:101)∗2 = 88.192
fuzzy sets are discussed detail in [5]. They are
T(cid:103)C(P = 800) = Rs.297785.95
1
(i) Crisp:
(cid:113)
Q∗ = 2DC0 T(cid:103)C(b=100) = Rs.225500, which is lower
2 P2I than the total cost corresponding to Q .
2
(ii) Fuzzy:
(cid:113) (iii) Intuitionistic Fuzzy Case:
Q(cid:101)∗ = 2(D1C0+2D2C0+D3C0)
2 P1I+2P2I+P3I Given D = (300, 350, 400) (250, 350, 450)
(iii) Intuitionistic fuzzy:
Q∗ = (cid:113)2(D1C0+4D2C0+D3C0+D1(cid:48)C0+D3(cid:48)C0) P1 = (750, 800, 850) (700, 800, 900)
2 P1I+4P2I+P3I+P1(cid:48)I+P3(cid:48)I
P = (550, 600, 650) (500, 600, 700),
2
Using these formula, the numerical example C = Rs.1500, I = 0.3
0
for neutrosophic set is illustrated as follows. ∗
Q = 88.19
2
TC(P = 800) = Rs.299660.85
1
5 NUMERICAL EXAMPLE:
TC(b = 100) = Rs.227375, which is lower
A manufacturing company issues the supply of than the total cost corresponding to Q .
2
a special component which has the following
price schedule: (iv) Neutrosophic Case:
0 to 99 items: Rs.800 per unit Given DN = (300, 350, 400) (250, 350, 450)
(150, 350, 550)
100 items and above: Rs.600 per unit
PN = (750, 800, 850) (700, 800, 900)(600,
1
The inventory holding costs are estimated 800, 1000)
to be Rs.30/- of the value of the inventory. The
procurement ordering costs are estimated to be PN = (550, 600, 650) (500, 600, 700)(400,
2
Rs.1500 per order. If the annual requirement of 600, 800)
the special component is 350, then compute the
economic order quantity for the procurement CN = Rs.1500, IN = 0.3
0
of these items.
We calculate Q∗N corresponding to the lowest
2
Solution: price 600,
(cid:114)
Q∗N = 2[(D1NC0N+2D2NC0N+D3NC0N)+(D1(cid:48)(cid:48)NC0N+2D2NC0N+D3(cid:48)(cid:48)NC0N)]
(i) Crisp Case: 2 [(PNIN+2PNIN+PNIN)+(P(cid:48)(cid:48)NIN+2PNIN+P(cid:48)(cid:48)NIN)]
1 2 3 1 2 3
=76.376,which is less than the price break point.
Given D = 350, P = Rs.800, P = Rs.600, Therefore,wehavetodeterminetheoptimal
1 2
C = Rs.1500, I = 0.3 total cost for the first price and the total cost
0
at the price- break corresponding to the second
Q∗ = 76 price and compare the two.
2
M. Mullai, R. Surya: Neutrosophic EOQ Model with Price Break
Neutrosophic Sets and Systems, Vol. 19, 2018 27
The defuzzified optimal total cost (TC)N asso-
3.5x 105
ciated with PN is calculated as follows:
1 S.No. 1
3 SS..NNoo.. 23
(TC)N(PN =800) = DN ∗PN + DNC0N + QN2 P1N ∗IN 2.5 324354.86 331337.2 SS..NNoo.. 45
1 1 QN 2
2
= Rs.306664.13 2 316127.02 323115.96
1.5 299660.85 306664.13
DNCN bPN ∗IN
(TC)N(b=100) = DN ∗PN + 0 + 2 1 283181.28 290198.96
2 b 2
= Rs.173812.5 0.5 274936.56 281961.24
which is lower than the total cost 0 Intuitionistic (TC)(p1) Neutrosophic (TC)N(p1)
corresponding to QN.
2
Figure 2. Analysis of first price between intuitionistic fuzzy set
andneutrosophicset
6 SENSITIVITY ANALYSIS
2.5x 105
In this section, the analysis between intuitionis-
S.No. 1
S.No. 2
ticset and neutrosophic setistabulated andthe 2 S.No. 3
S.No. 4
S.No. 5
results are compared graphically. 245825 187650
1.5
239675 183037.5
1 227375 173812.5
S.No. IntuitionisticDemand NeutrosophicDemand
1 (270,320,370)(220,320,420) (270,320,370)(220,320,420)(120,320,520) 215075 164587.5
2 (280,330,380)(230,330,430) (280,330,380)(230,330,430)(130,330,530)
0.5
3 (300,350,400)(250,350,450) (300,350,400)(250,350,450)(150,350,550) 208925 159975
4 (320,370,420)(270,370,470) (320,370,420)(270,370,470)(170,370,570)
5 (330,380,430)(280,380,480) (330,380,430)(280,380,480)(180,380,580) 0 Intuitionistic (TC)(b) Neutrosophic(TC)N(b)
Figure3.Analysisofpricebreakcorrespondingtosecondprice
betweenintuitionisticfuzzysetandneutrosophicset
100
90 S.No. 1
S.No. 2
80 SS..NNoo.. 34
S.No. 5
70 Conclusion
91.89 79.58
60 In this paper, EOQ model with price break
50 90.68 78.53
in neutrosophic environment is introduced. An
40
88.19 76.38 inventory model is developed for price breaks
30
20 85.63 74.16 and its optimum solution is obtained by using
10 84.33 73.03 triangular neutrosophic number. An algorithm
0 Intuitionistic (Q) Neutrosophic (Q)N for solving neutrosophic optimal quantity and
neutrosophic optimal cost is also developed.
Figure 1. Analysis of economic order quantity (EOQ) between Thiswillbeanadvantageforthebuyerwhocan
intuitionisticfuzzysetandneutrosophicset
easily decrease the bad cases and increase the
better ones. Hence, the neutrosophic set gives
the better solutions to the real world problems
than fuzzy and intuitionistic fuzzy sets. In fu-
ture, the various neutrosophic inventory mod-
els will be developed with various limitations
such as lead time, backlogging, back order and
deteriorating items, etc.
M. Mullai, R. Surya: Neutrosophic EOQ Model with Price Break
28 Neutrosophic Sets and Systems, Vol. 19, 2018
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M. Mullai, R. Surya: Neutrosophic EOQ Model with Price Break