An inventory model for Gompertz deteriorating items with time-varying holding cost and price dependent demand

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1 Inernaional Journal of Mahemaics rends and echnology (IJM) Volume 49 Number 3 Sepember 7 An invenory model for Gomperz deerioraing iems wih ime-varying holding cos and price dependen demand Absrac Nurul Azeez Khan, V.S.Verma, Vijay Kumar Deparmen of Mahemaics and Saisics, DDU Gorakhpur Universiy, Gorakhpur-739 he presen paper deals wih a deerminisic invenory model which follows he Gomperz disribuion deerioraion rae of iems. Commodiies such as fruis, vegeables and foodsuffs sufer from depleion by direc spoilage while kep in sore. Holding cos is ime dependen and demand rae is assumed as price dependen in linear form. Shorages allowed and compleely backlogged. Replenishmen is insananeous and lead ime is zero. he model is solved analyically by maximizing he oal profi. he resuls illusraed wih he numerical example and also shown by graphically. he sensiiviy of he soluion wih he changes of he values of he parameers associaed wih he model is discussed. Keywords:- Deerioraing iems, Price dependen demand, shorage, ime varying holding cos..inroducion I is usually seen ha he price dependen demand of he iems affec he delivery of goods. Mos of he cusomers moivaed by he aracive price of he iems o buy more goods and ha siuaion creaes he greaer demand of he goods. Due o his condiion, reailers wan o increase heir order quaniy and he reailers earn he more profi o increase heir revenue bu he condiions become more complex when iems deeriorae. herefore deerioraion of iems is one of he mos imporan facors in any invenory and producion sysem. A large number of work has been repored for invenory wih deerioraing iems in recen years because mos of he physical goods undergo decay or deeriorae over ime.nahimias[] developed a perishable invenory heory by considering deerioraion of iems. An order level invenory sysem for deerioraing iems was developed by Aggarwal and Goel[]. Raffa[3] discussed an invenory model for coninously deerioraing iems. Goyal and Giri[4] developed an invenory model for deerioraing invenory. Rao e.al[5] developed a producion invenory model for deerioraing iems. Invenory models creae lo of ineres due o heir ready applicabiliy a various places like marke yards, w houses, producion processes, ransporaion sysems, ec. Several invenory models have been developed and analysed o sudy various invenory sysems. he mos influencing facors of he invenory sysems holding, demand and replenishmen. In radiional invenory sysems he holding cos is considered as consan bu holding cos varies wih ime. Naddor[6]developed an invenory sysem by aking ime varying holding cos. Muhlemann and Valis Spanopoulus[7] produced an invenory model by aking variable holding cos. Goh[8] and Ajana Roy[9] developed invenory models by considering ime varying holding cos. he mos imporan influencing facor of invenory sysem is demand. In classical invenory model he dmand rae is usually assumed o be consan bu in realiy demand rae for physical goods vary wih ime. Selling price plays an imporan role in invenory sysem. A discoun price aracs more cusomers o by he produc in a super marke. Burwell e.al[] developed economic lo size model for price-dependen demand under quaniy and freigh discouns. An invenory sysem of amelioraing iems for pricedependen demand rae was considered by Mondal e.al[]. You[]developed an invenory model wih price and ime dependen demand. Rao e.al[3] developed an invenory model wih hypo exponenial lifeime having demand is funcion of selling price and ime. Sridevi e.al[4]deermined invenory model for deerioraing iems wih Weibull rae of replenishmen and selling price dependen demand. Chaudhry and Sharma[5] developed and invenory model for deerioraing iems wih ime dependen demand and shorages. An invenory model for deerioraing iems wih shorages and ime varying demand were developed by Sicilia e.al[6]. In his presen paper, we have developed an invenory model by aking a new ype Gomperz ISSN: hp:// Page 83

2 Inernaional Journal of Mahemaics rends and echnology (IJM) Volume 49 Number 3 Sepember 7 disribuion deerioraion rae of iems and demand rae is a funcion of selling price. Holding cos is ime varying. Shorages allowed here and compleely backlogged. We solve he model o opimize he oal profi. Model is illusraed wih numerical examples and verified graphically. Also he sensiiviy analysis is carried ou wih he base of numerical example.. Assumpions and Noaions he fundamenal assumions and noaions of his model as follows: (i) (ii) (iii) (iv) (v) (vi) (vii) he deerioraion of iems follows he Gomperz disribuion ( ) e, and Demand rae is funcion of selling price Shorages allowed and compleely backlogged Holding cos h() per iem per uni ime is ime dependen and is assumed o be h h where h, Selling price s follows an increasing rend where demand rae is f ( s) ( a s) is he complee lengh of cycle Replenishmen is insananeous and lead ime is zero Q is he order quaniy in one cycle A is he cos of placing an order s selling price per uni iem (viii) (ix) (x) (xi) C is he uni cos of an iem (xii) C is he shorage cos per uni per uni ime 3. Mahemaical Formulaions and Soluions During ime,invenory is depleed due o deerioraion and demand of iem. A ime he invenory becomes zero and shorages sar occuring. Le I() be he invenory level a ime. he differenial equaions o describe insananeous sae over (,) given by di() di() e I( ) ( a s), ( a s), Wih I a Neglecing he higher powers of, he soluions of and given as I( ) ( a s) e ( ) e e, 3 I( ) ( a s)( ), 4 Now, oal number of deerioraed iems is given by D e I() D a s e ( ) ( ) Ordering quaniy is given by Q D ( a s) Q ( a s) e ( ) Holding cos is given by H ( ) I( ), using 3 5 H ( a s) ( e ) ( e ) ( a s) e 3 e (3 ) 4 ( e ) 5 3 oal shorage cos is given by S C I() S C a s 6 ( )( ) 8 ISSN: hp:// Page 84

3 Inernaional Journal of Mahemaics rends and echnology (IJM) Volume 49 Number 3 Sepember 7 Now, oal profi per uni ime is given by P(, ) s( a s) ( A CQ H S) A C( a s) e h( e e ) P(, ) s( a s) ( a s) ( e 3 6 e 3 3 e e ) 5 5 C( a s)( ) (9) In order o maximize he oal profi funcion P(, ) he necessary condiions P(, ) P(, ) and P(, ) which gives A C( a s) e h( e e ) ( a s) ( e e e 4 e ) 5 5 C( a s)( ) C ( a s) C ( a s)( ) and and C ( a s) e h( e e ) 3 3 ( e e P(, ) ( a s) e e e e 4 4 ) C( a s)( ) he soluions of and he values of and will give and., so obained, he opimal value P (, ) of he average ne profi is deermined by 9 provided hey saisfy he sufficien condiions for maximizing P(, ) P(, ) P(, ), P(, ) P(, ) P(, ) a and ( ) 3 4. Numerical Example Le us consider he values of parameers in appropriae unis as A=, a=98, C =5, C =5, s=6, h=, =., =, = Based on hese inpu daa, he compuer oupus as follows: Profi P (, ) 8.348,.88,.88 ISSN: hp:// Page 85

4 Inernaional Journal of Mahemaics rends and echnology (IJM) Volume 49 Number 3 Sepember 7 5. Sensiiviy Analysis o sudy he effecs of changes of he parameers on he opimal profi P (, ),, derived by he proposed model, a sensiiviy analysis is performed in view of he numerical example given above. Sensiiviy analysis is execued by changing (decreasing or increasing) he parameers by %, % and 3% and aking one parameer a a ime, keeping he remaining parameers a heir original values. he corresponding changes in shown in below(able). Parameers % Change C A C a s h * P (, ), and * * P*(, ) able A cful sudy of above( able) reveals he following: (i) he values of * P (, ) increases when he values of A decreases while he values of and decreases wih decrease he value of A and increases wih he increase of he value of A. (ii) he values of P (, ) decreases when he values of C increases while he values of and increases wih decrease he value of C. he values of P (, ), and slighly sensiive o change in he values of parameer C. (iii) P (, ), slighly sensiive o change in he values of parameer C while * is moderaely sensiive o increase or decrease in he values of parameer C. (iv) P (, ) is highly sensiive o change in he values of parameer a while and moderaely sensiive o change in he values of parameer a. (v) P (, ), slighly sensiive o change in he values of parameer s while * is moderaely sensiive o decrease and decrease in he values of parameer s. (vi) P (, ), and slighly sensiive o change in he values of parameer h. (vii) he values of P (, ), and increases when he values of parameer decreases and heir values slighly sensiive. (viii) P (, ), and moderaely sensiive o increase and decrease in values of parameer. (ix) P (, ), and slighly sensiive o increase and decrease in values of parameer. ISSN: hp:// Page 86

5 Inernaional Journal of Mahemaics rends and echnology (IJM) Volume 49 Number 3 Sepember 7 6. Conclusion In his paper, we have developed an invenory model for deerioraing iems which follows he Gomperz disribuion deerioraion rae. he demand rae is assumed o be a funcion of selling price. Manager of he indusry always ake c of selling price parameers which affec he profi quickly. Shorages allowed and compleely backlogged in he presen model. he radiional parameers of holding cos is assumed here o be ime varying. As he changes in he ime value of money and in he price, holding cos can no remain consan over ime. Here we assumed ha he holding cos is increasing funcion of ime. Numerical example is given o illusrae he model and also verified graphically(figure). Comprehensive sensiive analysis has been carried ou for showing he effec of variaion in he parameers. he model is solved analyically by maximizing he oal profi. In he numerical example, we found he opimum value of profi P, and Figure-. he presen model is also exened wih shorages by aking parial backlogging rae. References []. Nahimias, S. (98): Perishable invenory heory: a review. Operaions Research Vol.3, No.4,pp []. Aggarwal, S.P. and Goel, V.P. (984): Order level invenory sysem wih power demand paern for deerioraing iems. Operaions Research in Managemen, Vol.4,pp [3]. Raafa,F.(99): Survey of lieraure on coninously deerioraing invenory models. Journal of he Operaional Research sociey, Vol.4, No.,pp [4]. Goyal, S.K. and Giri,B.C.(): Recen rends in modeling of deerioraing invenory. European journal of Operaional Research, Vol.34, No., pp.-6. [5]. Rao, S.V.U.M., Subbaiah, K.V. and Rao, K.S.(): Producion invenory models for deerioraing iems wih sock dependen demand and Weibull decay. IS ransacion of Mechanical Sysems-heory and Applicaions, Vol., No.(), pp.3-3. [6]. Naddor, E.(996): Invenory Sysems. Wiley, New York. [7]. Muhlemann, A.P. and Valis-Spanopoulos, N.P.(98): A variableholding cos rae EOQ model. European Jpurnal of Operaional Research, Vol.4, pp [8]. Goh, M.(994): EOQ models wih general demand and holding cos funcions. European Journal of Operaional Research, Vol.73, pp.5-54 [9]. Rao, Ajana.(8): An invenory model for deerioraing iems wih price dependen demand and ime-varying holding cos. Advanced modeling and opimizaion, Vol., No.. []. Burwell,.H., Dave, D.S. and Fizparick, K.E., Roy, M.R.(997): Economic lo size model for price-dependen demand under quaniy and freigh discouns. Inernaional Journal of Producion Economics, Vol.48, No., pp []. Mondal,B., Bhunia, A.K. and Maii, M.(3): An invenory sysem of amelioraing iems for price dependen demand rae. Compuers and Indusrial Engineering, Vol.45, No.3, pp []. You, S.P.(5): Invenory policy for producs wih price and ime dependen demands. Journal of Operaional Research Sociey, Vol.56, pp [3]. Rao, K.S., Reddy, P.V.G.D.P. and Gopinah, Y.(6): Invenory model wih hypo exponenial lifeime having demand is funcion of selling price and ime. Journal of Ulra Science for physical sciences, Vol.8, No., pp.3-3. [4]. Sridevi, G., Devi, K.N. and Rao, K.S.(): Invenory model for deerioraing iems wih Weibull rae of replenishmen and selling price dependen demand. Inernaional Journal of Operaional Research, Vol.9, No.3, pp [5]. Chaudhry, R.R. and Sharma, V.(3): An invenory model for deerioraing iems wih Weibull deerioraion wih ime dependen demand and shorages. Research journal of managemen sciences. Vol., No.3, pp.-4. [6]. Sicilia, J., Acosa, J.F. and Pablo, D.A.(4): An invenory model for deerioraing iems wih shorages and ime varying demand. Inernaional Journal of Producion Economics. ISSN: hp:// Page 87