Deteriorating Inventory Model When Demand Depends on Advertisement and Stock Display

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1 Inernaional Journal of Operaions Research Inernaional Journal of Operaions Research Vol. 6, No. 2, (29) Deerioraing Invenory Model When Demand Depends on Adverisemen and Sock Display Nia H. Shah, and Poonam Pandey 2 Deparmen of Mahemaics, Gujara Universiy Ahmedabad Gujara, India Corresponding Auhor: Dr. Nia H. Shah 2 Globsyn Business School Campus 97/A, Uvarsad, Off NH-8, Dis:Gandhinagar, Gujara Pin India. Received December 6 h 28; Acceped Ocober 2 h 29 Absrac In his sudy, a mahemaical model is developed o obain opimal ordering policy of ime dependen deerioraing iem when demand rae is dependen on displayed sock level and frequency of adverisemen hrough media. Shorages are no allowed. The objecive is o minimize oal cos. The significan feaures and he resuls are sudied wih he help of a numerical example. The effec of changes in he demand parameer, deerioraion rae ( α -consan deerioraion, - ime dependen deerioraion), rae of frequency of adverisemens, sock dependen parameer and salvage parameer for deerioraed iems on oal cycle ime, oal cos and on procuremen quaniy is sudied numerically. Keyword Lo size, Time dependen deerioraion, Adverisemen frequency, Procuremen quaniy and oal cos.. INTRODUCTION Classical invenory Economic order quaniy (EOQ) model is based on he assumpion ha an iem in sock remains o is % efficiency for infinie ime. Acually iems like volaile liquids, blood, X ray plaes, medicines, elecronic componens, fashion goods, fruis and vegeable ec. looses heir uiliy afer some ime. The mos of he mos of he aricles were based on he fac ha hese deerioraed unis are complee loss o he invenory sysem. See review aricles by Raafa (99), Shah and Shah (2), Goyal and Giri (2). These days adverisemens and display of producs plays a vial role in aracing mass cusomer. The adverisemen hrough elecronic media, news paper, inerne, using innovaive ways of display of produc in he show room is he bes ool for he promoion of a produc. This araced researcher o analyze he invenory problem when demand depends on sock displayed. Refer o Baker and Urban (988), Mandal and Phaujdar (989), Daa and Pal (99), Padmanabhan and Vra (996), Giri e al. (996), Sarkar e al. (997) ec. The effec of adverisemen on he demand of he produc is sudied by Bhunia and Maii (997), Goyal and Gunasekaran (995), Pal e al. (996, 26) developed an invenory model for deerioraing iems by aking ino accoun he impac of markeing sraegies viz pricing, adverisemen and he displayed sock level of he demand rae of he sysem. These aricles opimize oal cos or ne profi per ime uni of an invenory sysem. Misra (979 a) sudied he invenory decisions under inflaionary condiions for EOQ model. Misra (979 b) derived an inflaion model for he EOQ, in which he ime value of money and differen inflaion raes were considered. Gurunani (983) gave he economic analysis of invenory sysems and claimed ha he discouning effecs on EOQ were subsanial. Relaed aricles are by Queyranne (985), Roundy (986), Federgruen e al. (992), Shah e al. (23, 24) In his paper, a mahemaical invenory model for ime dependen deerioraing iems is developed by considering demand o be funcion of adverisemen and he displayed sock level. Shorages are no allowed. The sorage capaciy of he invenory sysem is finie. The objecive is o minimize oal cos. 2. ASSUMPTIONS AND NOTATIONS The Mahemaical model is derived using he following assumpions and noaions 2. Noaions T : Cycle ime (decision variable) Corresponding auhor s niahshah@gmail.com, shahniah@gmail.com 83-73X Copyrigh 29 ORSTW

2 34 Shah and Pandey: Deerioraing Invenory Model When Demand Depends on Adverisemen and Sock Display IJOR Vol. 6, No. 2, (29) : ime poin when he sock level is S : ime poin when he sock level is S( S > S ) C : purchase cos per uni iem h : invenory holding cos per uni per uni ime G : cos of adverisemen O : ordering cos per order : rae of change of frequency of adverisemen γ : salvage parameer for deerioraed unis A : frequency of adverisemen in he cycle a : fixed demand b : rae of change of demand q (): invenory level a any insan of ime during cycle ime θ( ): θ() = α ( Weibull disribuion ) where α >, >. Here > is considered which means deerioraion increases wih ime. S : order quaniy per cycle K :oal cos per ime uni. 2.2 Assumpions. The invenory sysem deals wih a single iem. 2. Replenishmen rae is infinie. 3. Shorages are no allowed and lead ime is zero or negligible. 4. The planning horizon is finie. 5. The deerioraion rae of unis in invenory follows he Weibull disribuion funcion given by θ() = α where α >, >. Here > is considered which means deerioraion increases wih ime. 6. Deerioraed unis can neiher be repaired nor replaced during he cycle ime. 7. The demand rae RAq (, ) is a funcion of he frequency of adverisemen A and he displayed invenory level in he super mall funcional form as RAq (, ) = A( a+ bs ) for q > S = A ( a+ bq( )) for S < q S = A ( a+ bs ) for q > S where ab, >, a > b. 3. MATHEMATICAL MODEL In his model, he cycle sars wih on hand invenory level S a = afer clearing shorages. Then invenory level deplees coninuously and reaches o zero a = T due o demand and consan rae of deerioraion of unis. Meanwhile invenory reaches o S a some ime and i reaches o S a some ime as shown in he Figure.

3 35 Shah and Pandey: Deerioraing Invenory Model When Demand Depends on Adverisemen and Sock Display IJOR Vol. 6, No. 2, (29) S Invenory level S S T Time Figure. Time invenory saus. The following differenial equaions represen he invenory level q () a any insan of ime. dq() d dq2() d ( ) q () A a bs, + α = + ( ) q () A a bq (), + α 2 = + 2 () (2) dq3() d ( ) + α 3() = +, q A a bs T wih boundary condiions q() = S, q2( ) = S, q3( ) = S, q3( T ) = (4) The soluion of differenial Eq () (3) using (4) is + α q() = S( α ) A ( a+ bs) (5) + (3) () ( α ( ) ( )) q = S ba aa α + + ( ) + α ( ) + + ( ) α + ba (6) () ( ( )) ( ) α + + = α + + ( ) α ( ) q3 S A a bs (7) + Using boundary condiion (4), we have + α S = S( + α ) + ( a+ bs) + (8) + Now oal invenory from (, T) is T IT = qd+ qd+ qd (9) 2 3 Number of deerioraed during posiive invenory ime inerval is = = + 2 DU S A a A b( S S ( T )) A b q ( ) d ()

4 Shah and Pandey: Deerioraing Invenory Model When Demand Depends on Adverisemen and Sock Display IJOR Vol. 6, No. 2, (29) Hence, oal cos per ime uni T is KT ( ) = ( Invenory holding cos + Adverisemen cos + ordering cos + cos due o Deerioraion salvage value of he deerioraed unis)/t. = ( h IT + A G + OC + C DU C γ DU ) T The necessary condiion for KT ( ) o be minimum is KT ( ) = and solving i for T by a suiable mahemaical T sofware For obained T, KT ( ) is minimum only if 2 K > for all T >. 2 T 4. COMPUTATIONAL ALGORITHM The following seps are o be compleed for he opimal soluion Sep : sar wih A =. Sep 2: compue,, T, K( T ). Sep 3: Incremen A by. Sep 4: Perform sep 2 unil K( A,,, T) K( A,,, T) K( A+,,, T ). Sep 5: sop. 5. NUMERICAL EXAMPLE Consider an invenory sysem wih following parameers in proper unis : [ OhCG,,,,, ab,, γ, α,, S, S ] = [,,,,.3, 2,.3,.2,.,3.5,4,5]. The opimum value of frequency A is 3, =.898 years, = 2.52 years, T = years, KT ( ) = $ , S = unis (see Figure 2). The sensiiviy analysis is carried ou o sudy he effec of deerioraion of unis; α and, sock dependen parameer; b, rae of change of frequency;, and salvage parameer; γ of deerioraed unis on he objecive funcion are exhibied in following ables: 36 Figure 2. Convexiy of KT ( ).

5 37 Shah and Pandey: Deerioraing Invenory Model When Demand Depends on Adverisemen and Sock Display IJOR Vol. 6, No. 2, (29) Table. Effec of T K S Table 2. Effec of α α T K S Table 3. Effec of T K S Table 4. Effec of γ γ T K S Table 5. Effec of b b..5.2 T K S

6 38 Shah and Pandey: Deerioraing Invenory Model When Demand Depends on Adverisemen and Sock Display IJOR Vol. 6, No. 2, (29) Table 6. Effec of α & γ on decision variables γ α T KT ( ) S T KT ( ) S T KT ( ) S Table 7. Effec of & on decision variables T KT ( ) S T KT ( ) S T KT ( ) S

7 39 Shah and Pandey: Deerioraing Invenory Model When Demand Depends on Adverisemen and Sock Display IJOR Vol. 6, No. 2, (29) Table 8. Effec of γ & γ on decision variables T KT ( ) S T KT ( ) S T KT ( ) S Table 9. Effec of b b & on decision variables T KT ( ) S T KT ( ) S T KT ( ) S

8 4 Shah and Pandey: Deerioraing Invenory Model When Demand Depends on Adverisemen and Sock Display IJOR Vol. 6, No. 2, (29) Table. Effec of b & b on decision variables T KT ( ) S T KT ( ) S T KT ( ) S Table. Effec of α & on decision variables α T KT ( ) S T KT ( ) S T KT ( ) S

9 4 Shah and Pandey: Deerioraing Invenory Model When Demand Depends on Adverisemen and Sock Display IJOR Vol. 6, No. 2, (29) Table 2. Effec of γ α..2.4 α & γ on decision variables T KT ( ) S T KT ( ) S T KT ( ) S Table 3. Effec of b α..2.4 α & b on decision variables T KT ( ) S T KT ( ) S T KT ( ) S

10 42 Shah and Pandey: Deerioraing Invenory Model When Demand Depends on Adverisemen and Sock Display IJOR Vol. 6, No. 2, (29) Table 4. Effec of γ & γ on decision variables T KT ( ) S T KT ( ) S T KT ( ) S Table 5. Effec of γ & b on decision variables b γ T KT ( ) S T KT ( ) S T KT ( ) S

11 43 Shah and Pandey: Deerioraing Invenory Model When Demand Depends on Adverisemen and Sock Display IJOR Vol. 6, No. 2, (29) Observaions As deerioraion parameer α and increases oal cycle ime decreases whereas oal cos increases. Toal cos and cycle ime boh decreases as salvage parameer γ increases. This is because here is some savings by selling deerioraed iems insead of jus hrowing i away. As sock displayed parameer b increases oal cos and cycle ime boh decreases. The reailer will have o pu orders frequenly resuling increase in he oal cos. Cycle ime decreases whereas oal cos increases as rae a which adverisemen is displayed increases. All hese are criical facors in deciding opimal ordering policy o minimize he objecive funcion oal cos. This sensiiviy analysis done here shows ha sock dependen demand increases oal cos decreases significanly. Frequen adverisemen is going o increase in oal cos bu if salvage parameer increases significanly incremen in oal cos can be checked. 6. CONCLUSIONS In marke, he unis deeriorae due o vaporizaion; damages while loading and unloading, perishabliliy and many oher facors. So he deerioraion effec should no be ignored while compuing invenory cos of he reailer. For producs like food grains, blood componens, vegeables and fruis, pharmaceuicals ec. uiliy decreases wih passage of ime and hence reailer needs o find rade off among invenory carrying cos and deerioraion cos. On he oher hand, reailer uses media for he sale of he produc hrough adverisemen and displayed sock o arac he cusomers. This concep encourage auhor o develop propose model. This model can be exended o he incorporae selling price and adverisemen dependen demand. Acknowledgemens The auhors are hankful o anonymous reviewers for consrucive suggesions. REFERENCES. Baker, R. C. and Urban, L. A. (988). Deerminisic invenory sysem wih an invenory level dependen demand rae, Journal of he Operaional Research Sociey, 29: Bhunia, A. K., Maii, M An invenory model for decaying iems wih selling price, frequency of adverisemen and linearly ime dependen demand wih shorages. IAPQR Transacions, 22: Daa, T. K. and Pal, A. K. (99). A noe on an invenory model wih invenory - level - dependen demand rae, Journal of he Operaional Research Sociey, 4: Federgruen, A., Queryranne, M. and Zheng, Y. S. (992). Simple power of wo policies are close o opimal in general class of producion / disribuion neworks wih general join seup coss, Mahemaics of Operaions Research, 7(4): Giri, B. C., Pal, S., Goswami, A., Chaudhuri, K. S An invenory model for deerioraing iems wih sock dependen demand rae. European Journal of Operaions Research, 95: Goyal, S. K., Gunasekaran, A An inegraed producion- invenory-markeing model for deerioraing iems. Compuers and Indusrial Engineering, 28: Goyal, S. K. and Giri, B. C. (2). Recen rends in modeling deerioraed invenory. European Journal of Operaional Research, 34: Gurnani, C. (983). Economic analysis of invenory sysems, Inernaional Journal of Producion Research, 2 (2): Mandal, B. N. and Phaujdar, S. (989). An invenory model of deerioraing iems and sock dependen consumpion rae, Journal of Operaional Research Sociey, 4: Misra, R. B. (979 - a). A sudy of inflaion effecs of invenory sysem, Logisics Specrum, 9: Misra, R. B. (979 - b). A noe on opimal invenory managemen under inflaion, Naval Research Logisics, 26: Padmanabhan, G. and Vra, P. (995). EOQ models for perishable iems under sock dependen selling rae, European Journal of Operaional Research, 86: Pal, S., Goswami, A., Chaudhuri, K. S., Pal, A. K. (996). An invenory model wih wo componen demand rae and shorages. Journal of he Operaional Research Sociey, 47: Pal, A. K., Bhunia, A. K. and Mukharjee, R. N. (26). Opimal lo size model for deerioraing iems wih demand rae dependen on displayed sock level (DSL) and parial backordering. European Journal of Operaional Research, 75: Queryanne, M. (985). A polynomial - ime sub-modular exension o Roundy s 98 % effecive heurisic for producion / invenory sysems, working paper no.36, Faculy of Commerce and Business Adminisraion, Universiy of Briish Columbia, Vancouver, B. C. Canada.

12 Shah and Pandey: Deerioraing Invenory Model When Demand Depends on Adverisemen and Sock Display IJOR Vol. 6, No. 2, (29) 6. Raafa, F. (99). Survey of lieraure on coninuously deerioraing invenory models, Journal of he Operaional Research Sociey, 4: Roundy, R. (986). A 98 % - effecive lo sizing rule a muli produc, muli sage producion invenory sysem, Mahemaics of Operaions Research, : Sarker, B. R., Mukharjee, S. and Balan, C. V. (997). An order level of lo size invenory model wih invenory level dependen demand and deerioraion, Inernaional Journal of Producion Economics, 48: Shah, Nia H., and Shah, Y. K. (2). Lieraure survey on invenory model for deerioraing iems, Economic Annals, XLIV: Shah, Nia H., Shah B. J. and Shah, Y. K. (23). Presen value formulaion of economic lo size model for invenory sysem wih variable deerioraing iems, Modeling Measuremen and Conrol (France), 24(2): Shah, Nia H., Shah B. J. and Shah, Y. K. (24). An order level lo size model wih sock dependen demand for ime dependen rae of deerioraion of iems under permissible delay in paymens, For Eas Journal of Theoreical Saisics, 2(2):