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• Open Access   Article

  On (gg)*- Closed Sets in Topological Spaces

  I. Christal Bai, T.Shyla Isac Mary

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.4 , pp.395-403, Aug-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i4.395403

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  Abstract
  In this paper, we introduce a new class of closed sets called generalization of generalized star closed sets (briefly (gg)* - closed) in topological Spaces. A subset A of a topological space (X , τ) is called (gg)* - closed if U contains regular closure of A whenever U contains A and U is gg-open in (X , τ). We studied the relation of this set with some of the other closed and generalized closed sets and some of the characteristics, of (gg)*- closed sets have been investigated and studied.

  Key-Words / Index Term
  (gg)* - closed set, gg - open, regular closure

  References
  [1] N .Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo, 2 , 89- 96, 1970.
  [2] D. E. Cameron, Properties of s-closed spaces proc, Amer. Math. Soc., 72, 581-586 ,1978.
  [3] M. Stone, Application of the theory of Boolean rings to general topology, Trans. Amer. MathSoci, 412, 374-481, 1937.
  [4] Basavaraj M Ittanagi and Govardhana Reddy H G, On gg-Closed Sets in Topological Spaces, International Journal of Mathematical Archive-8(8), 126-133, 2017.
  [5] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70, 36-41, 1963.
  [6] A.S. Mashhour, M.E.Abd El-Monsef and S.N.El-Deeb, On pre-continious and weak pre- continious mappings, Proc. Math and Phys. Soc. Egypt ,53, 47-53,1982.
  [7] D. Andrijevic, Semi-preopen sets, Mat. Vesnik 38 ,24-32, 1986.
  [8] O. Njastad, On some classes of nearly open sets, Pacific J. Math., 15, 961-970, 1965.
  [9] T.Indira and K. Rekha, Applications of *b-open sets and **b-open sets in Topological Spaces, Annals of pure and Applied Mathematics, 1, 44-56, 2012 .
  [10] J. Dontchev, On generalizing semi-pre-open sets, Mem. Fac. Sci. Kochi Univ. ser. A. Math.,
  16, 35-48 ,1995.
  [11] H. Maki, J. Umehara and T. Noiri, Generalized preclosed sets, Mem. Fac. Sci. Kochi Univ.
  Ser. A. Math, 17, 33-42, 1996.
  [12] S.P. Arya and T.M.Nour, Characterizations of s-normal spaces, Indian J. Pure. Appl. Math.,21 (8), 717-719 ,1990.
  [13] H. Maki, R. Devi and K. Balachandran, Associated topologies of generalized α -closed sets and α -generalized closed sets, Mem. Fac. sci. Kochi. Univ. Ser. A. Math.,15, 51-63, 1994.
  [14 ] N. Palaniappan and K.C. Rao, Regular generalized closed sets, Kyungpook Math.J.,33, 211-219, 1993.
  [15] Y. Gnanambal, On generalized pre regular closed sets in topological spaces, Indian J. Pure. Appl. Math., 28 (3),351-360,1997.
  [16] G.B Navalagi and Chandrashekarappa, On gspr closed sets in topological spaces, Vol.2,Nos 1-2, pp.35-44, 2010.
  [17] M.K.R.S.Veerakumar, g*-preclosed sets, Acta Ciencia India, Vol.XXVIII M, 1, 51- 60, 2002.
  [18] A. Pushpalatha, Studies on gereralizations of mappings in topological space, ph.D. Thesis, Bharathiar University Coimbatore, 2000.
  [19] Ahmed A.El.Mabhouh and Hanady A.Al-Faiuomy, tgr-Closed Sets and t*gr-Closed Sets. International Mathematical Forum, Vol. 10, No. 5, 211 – 220, 2015.
  [20] S.S. Benchalli and R.S.Walli, On RW-closed sets in topological spaces, Bull. Malaysian.Math. Sci.Soc., (2)30(2), 99-110, 2007.
  [21] A.Vadivel and K.Vairamanickam, rgα-closed sets and rgα -open sets in topological Spaces, Int. Journal of Math. Analysis, 3 (37), 1803-1819, 2009
  [22] H. Maki, R. Devi and K. Balachandran, generalized α -closed sets in topology, Bull. Fukuoka Univ. Ed. Part (2), 42, 13-21, 1993.
  [23] P. Bhattacharya and B.K. Lahiri , Semi-generalized closed sets in topological spaces,
  Indian J. Math, 29, 376-382, 1987.
  [24] C. Janaki and Renu Thomas, On R* closed sets in topological spaces, International
  Journal of Mathematical Archive, 3(8), 3067-3074 , 2012.
  [25] Basavaraj M Ittanagi and Raghavendra. K, On R#-Closed Sets in Topological Spaces, International Journal of Mathematical Archive-8(8), 134-141, 2017.
  [26] C. Dhanapakyam, k. Indirani, On βg* closed sets in topological Spaces, International journal of Applied Research ,2(4),388-391, 2016.
  [27] D. Savithiri and C Janaki , On Regular^Generalized closed sets in topological spaces International Journal of Mathematical Archive, 4(4), 162-169, 2013.
  [28] M. Pauline Mary Helen, Veronica Vijayan, Ponnuthai Selvarani, g** closed sets in Topological Spaces, IJMA, 3(5), 1-15, 2012.
  [29] P. Sundaram and M. Sheik John, On w-closed sets in topology, Acta, Ciencia Indica, 4, 389-392, 2000.
  [30] S. Bhattacharya, On Generalized Regular Closed Sets, Int. J.Contemp. Math. Sciences, 6 , 145-152, 2011.
  [31] K. Indirani, P.Sathishmohan and V. Rajendran, On gr*-Closed sets in a Topological Spaces, International journal of Mathematics trends and Technology, Vol, feb 2014.
  

  Citation
  I. Christal Bai, T.Shyla Isac Mary, "On (gg)*- Closed Sets in Topological Spaces," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.4, pp.395-403, 2018

• Open Access   Article

  On Intuitionistic sgp-closed sets in Intuitionistic Topological Space

  P. Agrawal and J.K Maitra

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.4 , pp.404-408, Aug-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i4.404408

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  Abstract
  In 2007, Navalagi and Bhat introduced and study a new class of sets, namely, sgp-closed sets in Topological Spaces. He observed that this class is properly placed between the class of closed sets and generalized preclosed sets. In this paper we have introduced intuitionistic sgp-closed sets and intuitionistic sgp-open sets in intuitionistic toplogical spaces and obtained its significant properties. We have constructed some examples which are quite useful in theory of intuitionistic sgp-closed and intuitionistic sgp-open sets.

  Key-Words / Index Term
  Intuitionistic sgp closed set, Intuitionistic sgp-open set

  References
  [1] A. Singaravelan, “On intuitionistic β-open sets”,
  Mathematical science I. J., Vol. 5, 2016.
  [2] A. Singaravelan and G. Ilango, “ Some more
  properties Of intuitionistic β-open sets”, Int. Jour.
  Of pure and Applied Mathematics, vol. 106, no 8,
  13-20, 2016.
  [3] D. Coker, “An introduction to intuitionistic
  topological Spaces”, Preliminary report, Akdeniz
  university, Math. Dept. Turkey, 51-56, 1995.
  [4] D. Coker, “A note on intuitionistic sets and
  intuitionistic Points”,Turkish J. Math., 20, No. 3,
  343-351, 1996.
  [5] D. Coker, An Introduction to intuitionistic
  topological spaces, BUSEFAL, 81, 51-56, 2000.
  [6] G. Ilango and S. Selvanyaki, “Generalized pre
  regular closed sets in intuitionistic topological
  spaces”, IJMA, 5(4), 30-36, 2014.
  [7] G. Navalagi and M. Bhat, “On sgp-closed sets in
  Topological spaces”, Jour. Of Applied
  Mathematical Analysis and Applications, 3(1), 45-
  58, 2007.
  [8] G. Sasikala and M.N. Krishnan, “Study on
  Intuitionistic Semiopen Sets”, IOSR journal of
  Mathematics, vol. 12, Isssue 6 Ver. III, 2016, 79-
  84.
  [9] G. Sasikala and M.N. Krishnan, “On intuitionistic
  preiopen Sets”, Inter. journal of Pure and Applied
  Mathematics, vol. 116, no 24, 281-292, 2017.
  [10] G. Sasikala and M.N. Krishnan, “Study on
  intuitionistic α-open sets and α-closed sets”, Int.
  Jour. of Mathematics Archive, 8(1), 26-30, 2017.
  [11] J.P. Bajpai and S.S. Thakur, “Intuitionistic fuzzy
  sgp-closed sets”, Int. Jour. Of Latest Trends in
  Engineering and Technology”, Vol 8, Issue 1,
  636-642, 2017.
  [12] J.T., Yaseen S. and Abdualbaqi L., “Some
  generalized sets and generalized mapping in
  intuitionistic topological spaces”, J. of al-
  Qadisiyah for computer science and mathematics,
  7, 2, 2015.
  [13] Y.J. Yaseen and A.G. Raouf, “On generalized
  closed sets and generalized continuity on
  intuitionistic topological spaces”, J. of al-anbar
  uni. for pure science, 3,1, 2009.
  

  Citation
  P. Agrawal and J.K Maitra , "On Intuitionistic sgp-closed sets in Intuitionistic Topological Space," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.4, pp.404-408, 2018

• Open Access   Article

  New Class of Closed Sets in Bitopological Spaces

  S.V. Vani, G. Priscilla Pacifica

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.4 , pp.409-413, Aug-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i4.409413

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  Abstract
  The concept of bitopological space was first introduced by J.C.Kelly in 1963 (i.e) a non empty set X equipped with two arbitrary topologies τ_1 and τ_2.The concept of generalized closed sets plays a significant role in general topology and these are the research topics of many Topologists worldwide.In 1970 Norman Levine introduced the concept of generalization of closed sets in topological spaces and he defined the semi-open sets and semi-continuity in bitopological spaces. In this paper we introduce a new class of generalized closed sets namely (i,j) - g^(##) -closed sets in bitopological spaces (i.e)a subset A of a bitopological space (X,τ_1,τ_2 ) is called (i,j)-g^(##)-closed if τ_j-cl(A)⊆U,whenever A⊆U,U is τ_i-g^#-open in (X,τ_1 〖,τ〗_2 ) and some of the properties were discussed. The class of (i,j) - g^(##)-closed sets settled in between the class of (i,j) -g^*- closed sets and the class of (i,j)- gs- closed sets.Some of the basic properties of (i,j)-g^(##)- closed sets are investigated.

  Key-Words / Index Term
  Topological spaces , Bitopological spaces, (i,j)- g^(##)-closed sets.

  References
  [1] Arockiarani,studies on generalizations of generalized closed sets and maps in topological spaces,Ph.d.,Thesis,Bharathiar Univ,Coimbatore,1997.
  [2] Bhattacharyia.P and Lahiri.B.K, semi generalized closed sets in Topology, Rend.circ, Math.Palermo, 19(2) (1970), 89-96.
  [3] R.Devi, K.Balachandran and H.Maki, semi-generalized closed maps and generalized closed maps, Mem.Eac.Sci.Kochi Unvi.Ser.A.Math.,14(1993), 41-54.
  [4] L.Elvina Mary,E.L.Mary Tency,A study on (i,j)-(Gs)^*-closed sets in bitopological spaces,IJOSER,vol 3 issue 1 january-2015.
  [5] J.C.Kelley,Proc.London Math.Sci.,13(1963),71-89.
  [6] Levine.N.Generalised closed sets in topology, Rend, circ, Math.palermo, (19)(2)(1970), 86-96.
  [7] Mashhour.A.S.,AbdEl-deeb.S.N.,α-open mappings, Acta.Math.Hungar, 41 (1983),213-218
  [8] S.Pauline Mary Helen.M, SindhuSurya.R, (i,j) strongly a^(**)-closed sets in Bitopological spaces,IJMTT(2014).
  [9] M.K.R.S.Veerakumar, between closed sets and g-closed sets, Mem.Fac.Kochi Unvi.ser.A, Math., 17(1996), 33-42.
  

  Citation
  S.V. Vani, G. Priscilla Pacifica, "New Class of Closed Sets in Bitopological Spaces," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.4, pp.409-413, 2018

• Open Access   Article

  Common Fixed Point Theorems in Multiplicative metric spaces using concept of CRC-self mappings

  Sushma Devi

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.4 , pp.414-418, Aug-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i4.414418

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  Abstract
  In this paper, we prove common fixed point theorems using the idea of compatibility together with conditionally reciprocal continuity (CRC) in Multiplicative metric spaces.

  Key-Words / Index Term
  Common fixed point, compatible, multiplicative metric space, self maps

  References
  [1] A.E. Bashirov, E.M. Kurplnara, A. Ozyapici, Multiplicative calculus and its applications, J. Math. Anal. Appl., 337 (2008), 36-48. doi:0.1016/j.jmaa.2007.03.081
  [2] Abbas M, Ali B, Suleiman YI. Common fixed points of locally contractive mappings in multiplicative metric spaces with application. Int. J. Math. Math. Sci. 2015;7. Article ID 18683.Available:http://dx.doi.org/10.1155/2015/ 218683
  [3] Abdou AAN. Fixed point theorems for generalized contraction mappings in multiplicative metricspaces. J. Nonlinear Sci. Appl. 2016;9:2347-2363.
  [4] Abodayeh K, Pitea A, Shatanawi W, Abdeljawad T. Remarks on multiplicative metric spaces andrelated fixed points. Arxiv: 1512.03771v1 [math.GN]; 2015.Available:http:// dx.doi.org/10.1186/1687-1812-2014-48
  [5] Bashirov AE, Kurplnara EM, Ozyapici A. Multiplicative calculus and its applications, J. Math. Anal. Appl. 2008;337:36-48.
  [6] Bashirov AE, Misirli E, Tandogdu Y, Ozyapici A. On modelling with multiplicative differential equations. Applied Mathematics-A Journal of Chinese Universities. 2011;26:425-438.
  [7] F. Gu, L.M. Cui, Y.H. Wu, Some fixed point theorems for new contractive type mappings, J. Qiqihar Univ., 19 (2013), 85-89.
  [8] He X, Song M, Chen D. Common fixed points for weak commutative mappings on a multiplicative metric space, Fixed Point Theory Appl. 2014;48:9.
  [9] J.S.Kushwah, Weak compatible and reciprocally continuous maps in Non-Archimedean Menger PM-Space, IJSRMSS, Vol.3 (2),2016 p 9-12.
  [10] Kang SM, Kumar P, Kumar S, Nagpal P, Garg SK. Common fixed points for compatible mappings and its variants in multiplicative metric spaces. Int. J. Pure Appl. Math. 2015;102:383-406. DOI: 10.12732/ijpam.v102i2.14
  [11] Kang SM, Nagpal P, Garg SK, Kumar S. Fixed points for multiplicative expansive mappings in multiplicative metric spaces. International Journal of Mathematical Analysis. 2015;9(39):1939-1946.
  [12] M. ¨Ozavsar, A.C. Cevikel, Fixed point of multiplicative contraction mappings on multiplicative metric space, arXiv:1205.5131v1 [matn.GM], 2012.
  [13] M. Sarwar, R. Badshah-e, Some unique fixed point theorems in multiplicative metric space, arXiv:1410.3384v2 [math.GM], 2014 Available:http://dx.doi.org/10.12988/ijma. 2015.54130.
  [14] Muhammad Sarwar. Badshah-e-Rome: Some unique fixed point theorems in multiplicative metric spaces, arXiv: 1410.3384v2 [math.GM];2014.
  [15] Nagpal P, Kang SM, Garg SK, Kumar S. Several fixed point theorems for expansive mappings in multiplicative metric spaces. Int. J. Pure Appl. Math. 2016;107(2):357-369.Available: http://www.ijpam.euDOI: 10.12732/ Ijpam.V10 7i2.6
  [16] Ozavsar M, Cervikel AC. Fixed points of multiplicative contraction mappings on multiplicative metric spaces. [math.GM]; 2012. Available: http://arxiv.org/abs/ 1205.5131v1
  [17] Ravi P. Agarwal, Erdal Karapınar, Bessem Samet. An essential remark on fixed point results on multiplicative metric spaces. Fixed Point Theory and Applications. 2016;1(2016):1-3.
  [18] S.M.King, P.Kumar, S.Kumar, P.Nagpal and S.Garg, Common fixed points for compatible mappings and its variants in multiplicative metric spaces, I. J of Pure and Applied mathematics, 102(2) 2015, 383-406.
  [19] Sessa S. On a weak commutativity condition of mappings in fixed point considerations. InstitutMath´ematique. 1982;32:149–153.
  [20] Sukh raj S., Manoj U., R.D.Daheriya, Rashmi J.and Suraj S., Coincidence point & common fixed points for multiplicative expansive type mappings, British journal of mathematics & computer science 19(3), 1-14, 2016.
  [21] X. He, M. Song, D. Chen, Common fixed points for weak commutative mappings on a multiplicative metric space, Fixed Point Theory Appl., 48(2014), 9 pages. doi: 10.1186/1687-1812-2014-48.
  

  Citation
  Sushma Devi, "Common Fixed Point Theorems in Multiplicative metric spaces using concept of CRC-self mappings," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.4, pp.414-418, 2018

• Open Access   Article

  Some Common Fixed Point Theorems for Single-Valued and Multi-Valued Mappings in Metric Spaces

  Krishnadhan Sarkar, Kalishankar Tiwary

  Review Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.4 , pp.419-424, Aug-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i4.419424

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  Abstract
  In this paper we prove some common fixed point theorems for single-valued and multi-valued mappings, with the help of CLR property and OWC property in metric spaces. In this paper we prove our main theorems in two parts; in part (A) we use OWC property and in part (B) we use CLR property. We also give some examples, corollaries and remarks. These theorems generalize, extend and improve many existing results available in this literature.

  Key-Words / Index Term
  CLR property, OWC property, Common Fixed Point, Weakly Compatible Mapping

  References
  [1]. A. A. N. Abdou, Common fixed point results for multi-valued mappings with some examples, J. Nonlinear Sci. appl. 9(2016), 787-798.
  [2]. M. A. Ahmed, Common fixed point theorems for weakly compatible mappings, Rocky Mountain J. Math., 33(4) (2003), 1189-1203.
  [3]. N.A Assad and S.Sessa, Common fixed points for nonself-maps on compacta, SEA Bull. Math. 16(1992), 1-5.
  [4]. H. Bouhadjera and A. Djoudi, Common fixed point theorems for single and set-valued maps without continuity, An. St. Univ. Ovidius Constanta, 16(1) (2008), 49-58.
  [5]. N.Chandraet. Al, Coincidences and fixed points of nonexpansive type multi-valued and single-valued maps, Indian J. Pure Appl. Math. 26(1995), 393-401.
  [6]. Tong-Huei Chang, Fixed point theorems of contractive type set-valued mappings, Math. Japon. 38(4)(1993), 675-690.
  [7]. Y.J.Choet.al , A common fixed point theorem of Meir andKeeler type, Internat. J. Math.Math.Sci. 16(1993), 669-674.
  [8]. R. O. Davies and S Sessa, A common fixed point theorem of Gregus type for compatible mappings, Facta Univ. (Nis) Ser. Math. Inform.7(1992), 51-58.
  [9]. A. Djoudi and R. Khemis, Fixed points for set and single valued maps without continuity, DemonstratioMathematica, 38(3) (2005), 739-751.
  [10]. B. Fisher , Common fixed points of mappings and set-valued mappings, Rostock. Math. Kolloq. 18(1981), 69-77.
  [11]. B. Fisher, Common fixed points of mappings and set-valued mappings on a metric spaces, Kyungpook Math. J. 25(1985), 35-42.
  [12]. B. Fisher and S Sessa, Two common fixed point theorems for weakly commuting mappings, Period. Math. Hunger. 20(1989), 207-218.
  [13]. G. Jungck and B. E. Rhodes, Fixed points for set valued functions without continuity, Indian J. Pure Appl. Math., 29(3)(1998), 227-238.
  [14] Liu, Li-Shan, Common fixed points of a pair of single-valued mappings and a pair of set-valued mappings , QufuShifanDaxueXuebaoziranKexue Ban, 18(1992), no. 1, 6-10.
  [15]. S. Sessa and Y. J. Cho, Compatible mappings and a common fixed point theoremof change type, Publ. Math. Debrecen 43(3-4) (1993), 289-296.
  [16]. S. Sessa, M. S. Khan and M. Imdad, Common fixed point theorem with a weak commutativity condition, Glas. Mat. 21(41) (1986), 225-235.
  [17]. S. Sessa, B. E. Rhoades and M. S. Khan, On common fixed points of compatible mappings, Internat. J. Math. Math. Sci. 11(1988), 375-392.
  [18]. W.Sintunavarat, P. Kumam, Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric space, J. Appl. Math. 2011(2011), 14Pages.
  [19]. W.Sintunavarat, P. Kumam, Gregus-type common fixed point theorems for tangential multi-valued mappings of integral type in metric spaces, Internat. J. Math. Math. Sci., 2011(2011), 12 pages.
  [20] W.Sintunavarat, D. M. Lee, Y. J. Cho, Mizoguchi-Takahashi’s type common fixed point theorems without T-weakly commuting condition and invariant approximations, Fixed Point Theory appl., 2014(2014), 10 pages.
  [21] K. Tas, M. Telci and B. Fisher, Common fixed point theorems for compatible mappings, Internat. J. Math. Math. Sci. 19(1996), 451-456.
  [22]. N. Wairojjana, W. Sintunavarat, P. Kumam, Common tripled fixed point theorems for W-compatible mappings along with the CLRg- property in abstract metric spaces, J. Inequal. Appl., 2014(2014), 17 pages.

  Citation
  Krishnadhan Sarkar, Kalishankar Tiwary, "Some Common Fixed Point Theorems for Single-Valued and Multi-Valued Mappings in Metric Spaces," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.4, pp.419-424, 2018

• Open Access   Article

  Inventory Optimization with Stock-Level Dependent Demand, Shortages and Variable Holding Cost

  Sandeep Kumar

  Review Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.4 , pp.425-429, Aug-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i4.425429

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  Abstract
  In the present study the optimal order quantity and the time of optimal cycle of an inventory model are determined .Here the rate of demand depends on the stock. The holding cost per unit of the item per unit time is taken to be linear function of the quantity in storage. Here, it also considered that the shortages in stock are allowed. In traditional inventory models, the rate of demand and holding cost are considered to be constant. There are so many inventory models, where the holding cost was taken to be constant for the whole inventory cycle without shortages. But here the holding cost is considered as a variable which is an increasing step function of the time due to storage. This study provides the basis for further extensions like the incorporation of deteriorating items, shortage cost and stochastic lead time in order to develop more complex and realistic inventory models.

  Key-Words / Index Term
  Inventory, Demand, Stock, Costs, Storage.

  References
  1. Wilson RH (1934) a scientific routine for stock control. Harv Bus Rev 13:116-128
  2. Whitin TM (1957) The theory of inventory management. Princeton: Princeton University Press.
  3. Ghare PM, Schrader GF (1963) A model for an exponentially decaying inventory. Journal of Industrial and Engineering Chemistry 14:238-243.
  4. G. Zoutendijk, Nonlinear Programming: A Numerical Survey, SIAM J. Control. 4 (1966) 194-210.
  5. Baker, R.C., Urban, T.L., 1988a. A deterministic inventory system with an inventory level dependent rate. Journal of the Operation Research Society 39 (9), 823-831.
  6. Baker, R.C., Urban, T.L., 1988b. Single-period inventory dependent demand models. Omega-The International Journal of Management Science 16 (6), 605-607.
  7. B. N. Mondal and S. Phaujder, An inventory model for deteriorating items and stock-dependent demand rate, Opl. Res. Soc. 40 (1989) 483-488.209
  8. Datta, T.K and Pal A.K. (1990). “ A note on an inventory model with inventory-level dependent demand rate”, Journal of the Operational Research Society, 41 (10), 971-975.
  9. T. K. Datta and A. K. Pal, A note on an inventory model with inventory level-dependent demand rate, Opl.Res. Soc. 41 (1990) 971-975.
  10.T. L. Urban, An inventory model with an inventory-level-dependent demand rate and relaxed terminal condition, Opl. Res. Soc. 43 (1992) 721-724
  11.M. Goh, EOQ models with general demand and holding cost functions,Euro. J. Oper. Res.73 (1994) 50-54.
  12. B. C. Giri, S. Pal, A. Goswami and K. S. Chaudhuri, An inventory model for deteriorating items with stock-dependent demand rate, Euro.J.Oper. Res.95 (1995) 604-610.
  13. Wee HM (1995) a deterministic lot-size inventory model for deteriorating items with shortages and a declining market. Computers &Operations Research - Journal 22:345-356 Publisher Full Text
  14. Urban, T.L., 1995. Inventory models with the demand rate dependent on stock and shortage levels. International Journal of Production Economics 40 (1), 2-28.
  15. Ray, J., Chaudhuri, K.S., 1997. An EOQ model with stock dependent demand, shortage, inflation and time discounting. International Journal of Production Economics 53 (2), 171-180
  16. J. Roy and K. S. Chaudhuri, An EOQ model with stock-dependent demand, shortages, inflation and time discounting, Int. J. Prod. Eco. 53 (1997) 171-180.
  17. J. Roy, A. Goswami and K. S. Chaudhuri, on an inventory model with two levels of storage and stock-dependent demand rate, Int. J.Systems. Sci.29 (1998) 249-254.
  18. Chang HJ, Dye CY (1999) An EOQ model for deteriorating items with time varying demand and partial backlogging Journal of the Operational Research Society 50:1176-1182
  19. Hwang, H., Hahn, K.H., 2000. An optimal procurement policy for items with an inventory level dependent demand rate and fixed life time. European Journal of Operation
  20. Goyal SK, Giri BC (2001) recent trends in modelling of deteriorating inventory. European Journal of Operational Research 134:1-16Publisher Full Text International Journal of Production Economics 113:852-861 Publisher Full Text
  21. Ouyang W, Cheng X (2005) an inventory model for deteriorating items with exponential declining demand and partial backlogging.Yugoslav journal of operations research 15(2):277-288Publisher Full Text
  22. Alamri AA, Balkhi ZT (2007).Tithe effects of learning and forgetting on the optimal production lot size for deteriorating items with time varying demand and deterioration rates. International Journal of Production Economics 107:125-138 Publisher Full Text
  23. Dye CY, Ouyang LY, and Hsieh TP (2007) Deterministic inventory model for deteriorating items with capacity constraint and time proportional Backlogging rate. . European Journal of Operational Research178 (3):789-807 Publisher Full Text
  24. Alfares, H. K., 2007. Inventory model with stock-level dependent demand rate and variable holding cost.Int. J. Production Economics 108 (2007) 259-265
  25. Liao JJ (2008) An EOQ model with non instantaneous receipt and exponential deteriorating item under two-level trade credit.and time-varying demand, Quality and Quantity, 38, 457-473.
  26. Mishra VK, Singh LS (2011) Inventory model for ramp type demand, time dependent deteriorating items with salvage value and shortages. International Journal of Applied Mathematics and Statistics 23(D11):84-91
  27. Mishra et al. (2013) An inventory model for deteriorating items with time-dependent demand and time-varying holding cost under partial backlogging. Journal of Industrial Engineering International 2013, 9:4
  28. Inventory Model for Deteriorating Items with Fixed Life under Quadratic Demand and Nonlinear Holding Cost. International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 12, June 2014

  Citation
  Sandeep Kumar, "Inventory Optimization with Stock-Level Dependent Demand, Shortages and Variable Holding Cost," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.4, pp.425-429, 2018