Neutrosophic EOQ Model with Price Break

M.Mullai1

AssistantProfessorofMathematics,AlagappaUniversity,Karaikudi, Tamilnadu,India.

E-mail:mullaimalagappauniversity.ac.in R.Surya2

DepartmentofMathematics,AlagappaUniversity,Karaikudi, Tamilnadu,India.

E-mail:suryarrrm@gmail.com

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.

1INTRODUCTION

BaiandLi[1]havediscussedtriangularand trapezoidalfuzzynumbersininventorymodel fordeterminingtheoptimalorderquantityand theoptimalcost.Thequantitydiscountproblemhasbeenanalyzedfromabuyersperspective.HadleyandWhintin[2],PetersonandSilver[3],andStarrandMiller[6]consideredvariousdiscountpolicesanddemandassumptions.

YangandWee[7]developedaneconomic orderingpolicyintheviewofboththesupplierandthebuyer.PrabjotKaurandMahuya Deb[5]developedanintuitionisticapproachfor pricebreaksinEOQfrombuyer’sperspective.Smarandache[5]introducedneutrosophic setandneutrosophiclogicbyconsideringthe non-standardanalysis.Also,neutrosophicinventorymodelwithoutshortagesisintroduced

byM.MullaiandS.Broumi[3].

Inthispaper,weintroducetheneutrosophicinventorymodelswithneutrosophic pricebreaktoﬁndtheoptimalsolutionofthe modelfortheoptimalorderquantity.Alsothe neutrosophicinventorymodelunderneutrosophicdemandandneutrosophicpurchasing costatwhichthequantitydiscountareoffered tobetriangularneutrosophicnumber.Alsothe optimalorderquantityfortheneutrosophictotalcostisdeterminedbydeﬁningtheaccuracy functionoftriangularneutrosophicnumbers.

M. Mullai, R. Surya: Neutrosophic EOQ Model with Price

2N

QN =Numberofpiecesperorder CN 0 =NeutrosophicOrderingcostforeach

Neutrosophic Sets

, 2018 24

OTATIONS:

order

and Systems, Vol. 19

University of New Mexico

Break

N h =NeutrosophicHoldingcostperunit peryear

DN =NeutrosophicAnnualdemandin units

3NEUTROSOPHIC EOQMODEL WITH PRICE BREAK:

TheNeutrosophicinventorymodelwith neutrosophicpricebreakisintroducedto ﬁndtheoptimalsolutionsfortheoptimal neutrosophicorderquantity.Herewe assumethatthereisnostockouts,no backlogs,replenishmentisinstantaneous, theneutrosophicorderingcostinvolvedto receiveanorderareknownandconstantand purchasingvaluesatwhichdiscountsare offeredastriangularneutrosophicnumbers. Considerthefollowingvariables: DN :Neutrosophicyearlydemand, PN :Neutrosophicpurchasingcost Let DN =(DN 1 ,DN 2 ,DN 3 )(D N 1 ,DN 2 ,D N 3 )(D N 1 ,DN 2 ,D N 3 ) PN =(P N 1 ,P N 2 ,P N 3 )(P N 1 ,P N 2 ,P N 3 )(P N 1 ,P N 2 ,P N 3 ) PN 1 =(P N 11 ,P N 12 ,P N 13 )(P N 11 ,P N 12 ,P N 13 )(P N 11 ,P N 12 ,P N 13 ) PN 2 =(P N 21 ,P N 22 ,P N 23 )(P N 21 ,P N 22 ,P N 23 )(P N 21 ,P N 22 ,P N 23 )

arenonnegativetriangularneutrosophic numbers.

Now,weintroducetheneutrosophic inventorymodelunderneutrosophicdemand andneutrosophicpurchasingcostatwhich thequantitydiscountsareoffered.Total neutrosophicinventorycostisgivenby (TC)N = DN ⊗ P N ⊕ DN CN 0 QN ⊕ QN P N ⊗I N 2

QN P N 3 I N 2 )(D N 1 P N 1 + D N 1 CN 0 QN + QN

P N 3 I N 2 )

Thedefuzziﬁedtotalneutrosophiccost usingaccuracyfunctionisgivenby D(TC)N = 1 8 [(DN 1 P N 1 + DN 1 CN 0 QN + QN P N 1 I N 2 )+ 2(DN 2 P N 2 + DN 2 CN 0 QN + QN P N 2 I N 2 )+(DN 3 P N 3 + DN 3 CN 0 QN + QN P N 3 I N 2 )+(D N 1 P N 1 + D N 1 CN 0 QN + QN P N 1 I N 2 )+ 2(DN 2 P N 2 + DN 2 CN 0 QN + QN P N 2 I N 2 )+(D N 3 P N 3 + D N 3 CN 0 QN + QN P N 3 I N 2 )]

0 + DN 3 C N 0 )+ (D N 1 C N 0 +2DN 2 C N 0 + D N 3 C N 0 )]+ 1 16 [(P N 1 I N + 2P N 2 I N + P N 3 I N )+(P N 1 I N +2P N 2 I N + P N 3 I N )]= 0,weget QN = 2[(DN 1 CN 0 +2DN 2 CN 0 +DN 3 CN 0 )+(D N 1 CN 0 +2DN 2 CN 0 +D N 3 CN 0 )] [(P N 1 I N +2P N 2 I N +P N 3 I N )+(P N 1 I N +2P N 2 I N +P N 3 I N )] NeutrosophicPriceBreak: S.No. Quantity PricePerUnit(Rs) 1 0≤ QN 1 ≤ b P N 1 2 b ≤ QN 2 P N 2 (<P N 1 ) 4ALGORITHM FOR FINDING NEUTROSOPHIC OPTIMAL QUANTITYAND NEUTROSOPHIC OPTIMAL COST: StepI: Considerthelowestprice P N 2 anddetermine QN 2 byusingtheeconomicorderquantity (EOQ)formula: QN = 2[(DN 1 CN 0 +2DN 2 CN 0 +DN 3 CN 0 )+(D N 1 CN 0 +2DN 2 CN 0 +D N 3 CN 0 )] [(P N 1 I N +2P N 2 I N +P N 3 I N )+(P N 1 I N +2P N 2 I N +P N 3 I N )] If QN 2 liesintherangespeciﬁed, b ≥ QN 2 then QN 2 istheEOQ.Thedefuzziﬁedoptimaltotal cost (TC)N associatedwithQN iscalculatedas follows: (TC)N = DN ∗ P N 2 + DN CN 0 b + bP N 2 ∗I N 2 , byusingtheaccuracyfunction AN = (a1 +2a2 +a3 )+(a1 +2a2 +a3 ) 8 Neutrosophic Sets and Systems, Vol. 19, 2018 25

M. Mullai, R. Surya: Neutrosophic EOQ Model with Price Break

C

Thenthetotalneutrosophicinventorycostis (TC)N =(DN 1 P N 1 + DN 1 CN 0 QN + QN P N 1 I N 2 ,DN 2 P N 2 + DN 2 CN 0 QN + QN P N 2 I N 2 ,DN 3 P N 3 + DN 3 CN 0 QN + P N 1 I N 2 ,DN 2 P N 2 + DN 2 CN 0 QN + QN P N 2 I N 2 ,D N 3 P N 3 + D N 3 CN 0 QN + QN P N 3 I N 2 )(D N 1 P N 1 + D N 1 CN 0 QN + QN P N 1 I N 2 ,DN 2 P N 2 + DN 2 CN 0 QN + QN P N 2 I N 2 ,D N 3 P N 3 + D N 3 CN 0 QN + QN

ToﬁndtheminimumofD(TC)N bytaking thederivativeD(TC)N andequatingittozero, (i.e) 1 8Q2N [(DN 1 C N 0 +2DN 2 C N

Step2:

(i)If QN 2 <b,wecannotplaceanorderat thelowestprice P N 2 (ii)Wecalculate QN 1 withprice P N 1 andthe correspondingtotalcostTCatQN . (iii)If (TC)N b> (TC)N QN 1 ,thenEOQis Q∗N = QN 1 ,OtherwiseQ∗N = b istherequired EOQ.

TheEOQincrisp,fuzzyandintuitionistic fuzzysetsarediscusseddetailin[5].Theyare

TC(P1 =800) =Rs.296039

TC(b=100)=Rs.224250,whichislower thanthetotalcostcorrespondingto Q2 (ii) FuzzyCase:

Given D =(300,350,400), P1 =(750,800, 850)

P2 =(550,600,650),C0 =Rs.1500,I=0.3

Q∗ 2 =88.192

TC(P1 =800) =Rs.297785.95

TC(b=100)=Rs.225500,whichislower thanthetotalcostcorrespondingto Q2. (iii) IntuitionisticFuzzyCase:

Given D =(300,350,400)(250,350,450)

P 1 =(750,800,850)(700,800,900)

Usingtheseformula,thenumericalexample forneutrosophicsetisillustratedasfollows.

5NUMERICAL EXAMPLE:

Amanufacturingcompanyissuesthesupplyof aspecialcomponentwhichhasthefollowing priceschedule:

0to99items:Rs.800perunit 100itemsandabove:Rs.600perunit

Theinventoryholdingcostsareestimated tobeRs.30/-ofthevalueoftheinventory.The procurementorderingcostsareestimatedtobe Rs.1500perorder.Iftheannualrequirementof thespecialcomponentis350,thencomputethe economicorderquantityfortheprocurement oftheseitems.

Solution:

(i) CrispCase:

GivenD=350,P1 =Rs.800,P2 =Rs.600, C0 =Rs.1500,I=0.3

Q∗ 2 =76

P 2 =(550,600,650)(500,600,700), C0 =Rs.1500,I=0.3

Q ∗ 2 =88.19

TC(P1 =800) =Rs.299660.85

TC(b =100) =Rs.227375,whichislower thanthetotalcostcorrespondingto Q2. (iv) NeutrosophicCase:

Given DN =(300,350,400)(250,350,450) (150,350,550)

P N 1 =(750,800,850)(700,800,900)(600, 800,1000)

P N 2 =(550,600,650)(500,600,700)(400, 600,800) CN 0 =Rs.1500,IN =0.3

price600,

76.376,whichis lessthanthepricebreakpoint. Therefore,wehavetodeterminetheoptimal totalcostfortheﬁrstpriceandthetotalcost attheprice-breakcorrespondingtothesecond priceandcomparethetwo.

M. Mullai, R. Surya: Neutrosophic EOQ Model with Price Break

0

P2

3

2(D1 C0 +4D2 C0 +D3 C0 +D1 C0 +D3 C0 ) P1 I+4P2 I+P3 I+P1 I+P3 I

(i)Crisp: Q∗ 2 = 2DC0 P2 I (ii)Fuzzy: Q∗ 2 = 2(D1 C0 +2D2 C

+D3 C0 ) P1 I+2

I+P

I (iii)Intuitionisticfuzzy: Q ∗ 2 =

26 Neutrosophic Sets and Systems, Vol. 19, 2018

Wecalculate Q∗N 2 correspondingtothelowest

Q∗ N 2 = 2[(DN 1 CN 0 +2DN 2 CN 0 +DN 3 CN 0 )+(D N 1 CN 0 +2DN 2 CN 0 +D N 3 CN 0 )] [(P N 1 I N +2P N 2 I N +P N 3 I N )+(P N 1 I N +2P N 2 I N +P N 3 I N )] =

Thedefuzziﬁedoptimaltotalcost (TC)N associatedwith P N 1 iscalculatedasfollows:

(TC)N (P N 1 =800)= DN ∗ P N 1 + DN C N 0 QN 2 + QN 2 P N 1 ∗ I N 2 = Rs.306664 13

(TC)N (b =100)= DN ∗ P N 2 + DN C N 0 b + bP N 2 ∗ I N 2 = Rs.173812 5

whichislowerthanthetotalcost correspondingto QN 2 .

6SENSITIVITY ANALYSIS

Inthissection,theanalysisbetweenintuitionisticsetandneutrosophicsetistabulatedandthe resultsarecomparedgraphically.

S.No. Intuitionistic Demand NeutrosophicDemand

1 (270,320,370) (220,320,420) (270,320,370) (220,320,420)(120,320,520)

2 (280,330,380) (230,330,430) (280,330,380) (230,330,430)(130,330,530)

3 (300,350,400) (250,350,450) (300,350,400) (250,350,450)(150,350,550)

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)

Figure3.Analysisofpricebreakcorrespondingtosecondprice betweenintuitionisticfuzzysetandneutrosophicset

Conclusion

Inthispaper,EOQmodelwithpricebreak inneutrosophicenvironmentisintroduced.An inventorymodelisdevelopedforpricebreaks anditsoptimumsolutionisobtainedbyusing triangularneutrosophicnumber.Analgorithm forsolvingneutrosophicoptimalquantityand neutrosophicoptimalcostisalsodeveloped. Thiswillbeanadvantageforthebuyerwhocan easilydecreasethebadcasesandincreasethe betterones.Hence,theneutrosophicsetgives thebettersolutionstotherealworldproblems thanfuzzyandintuitionisticfuzzysets.Infuture,thevariousneutrosophicinventorymodelswillbedevelopedwithvariouslimitations suchasleadtime,backlogging,backorderand deterioratingitems,etc.

Intuitionistic(Q) Neutrosophic 0 10 20 30 40 50 60 70 80 90 100 S.No.1 S.No.2 S.No.3 S.No.4 S.No.5 84.33 88.19 85.63 90.68 91.89 (Q)N 73.03 74.16 76.38 78.53 79.58 Figure1.Analysisofeconomicorderquantity(EOQ)between intuitionisticfuzzysetandneutrosophicset Intuitionistic Neutrosophic 0 0.5 1 1.5 2 2.5 3 3.5 x10 S.No.1 S.No.2 S.No.3 S.No.4 S.No.5 (TC)(p1) (TC)N(p1) 274936.56 283181.28 299660.85 316127.02 324354.86 281961.24 290198.96 306664.13 323115.96 331337.2

Intuitionistic(TC)(b) Neutrosophic 0 0.5 1 1.5 2 2.5 x105 S.No.1 S.No.2 S.No.3 S.No.4 S.No.5 (TC)N(b) 208925 215075 227375 239675 245825 159975 164587.5 173812.5 187650 183037.5

Figure2.Analysisofﬁrstpricebetweenintuitionisticfuzzyset andneutrosophicset

Neutrosophic Sets and Systems, Vol. 19, 2018 27

M. Mullai, R. Surya: Neutrosophic EOQ Model with Price Break

REFERENCES

[1]S.BAIAND Y.LI,Studyofinventorymanagementbasedontri-

angularfuzzynumberstheory, IEEE,978-1-4244-2013-1/08/ 2008.

[2]G.HADLEYAND T.M.WHITIN,AnalysisofInventorySystems,PrenticeHall,Inc,EnglewoodCliffs,N.J1963.

[3]M.MULLAIAND S.BROUMI, NeutrosophicInventoryModel withoutShortages, AsianJournalofMathematicsandComputerResearch,23(4):214-219,2018..

[4]R.PETERSONAND E.A.SILVER,DecisionsSystemsforInventoryManagementandProductionPlanning, JohnWileyand Sons,NewYork1979.

[5]PRABJOT KAURAND MAHUYA DEB,AnIntuitionisticApproachforPriceBreaksinEOQfromBuyer’sPerspective, AppliedMathematicalSciences,Vol.9,2015,no.71,3511-3523

[6]F.SMARANDACHE, Neutrosophicset-ageneralizationofthe intuitionisticfuzzyset,GranularComputing, 2006IEEEInternationalConference,2006,pp.38–42.

[7]K.M.STARRAND D.W.MILLER,InventoryControl:Theory andPractice,PrenticeHall,Inc,EnglewoodCliffs, N.J1962.

[8]YANG,P., AND WEE,H.,Economicorderingpolicyofdeteriorateditemforvendorandbuyer:anintegratedapproach.ProductionPlanningandControl, 11,2000,474–480

Neutrosophic Sets and Systems, Vol.

,

[9] Abdel-Basset, M., Mohamed, M., Smarandache, F., &

Chang, V. (2018). Neutrosophic Association Rule Mining Algorithm for Big Data Analysis. Symmetry, 10(4),106.

[10] Abdel-Basset, M., & Mohamed, M. (2018). The Role of Single Valued Neutrosophic Sets and Rough Sets in Smart City: Imperfect and Incomplete Information Systems. Measurement. Volume 124, August 2018, Pages47-55

[11] Abdel-Basset, M., Gunasekaran, M., Mohamed, M., &

Smarandache, F. A novel method for solving the fully neutrosophic linear programming problems. Neural Computing and Applications,1-11.

[12] Abdel-Basset, M., Manogaran, G., Gamal, A., & Smarandache, F. (2018). A hybrid approach of neutrosophic sets and DEMATEL method for developing supplier selection criteria. Design Automation for Embedded Systems,1-22.

[13] Abdel-Basset, M., Mohamed, M., & Chang, V. (2018). NMCDA:Aframeworkforevaluatingcloudcomputing services. Future Generation Computer Systems, 86,12-29.

Received : January 17, 2018. Accepted : March 5, 2018

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