Inernaional Journal of Scienific and Research Publicaions, Volume 5, Issue 6, June 05 ISSN 50-5 Invenory Models wih Weibull Deerioraion and ime- Varying Holding Cos Riu Raj *, Naresh Kumar Kaliraman *, Dr Shalini Chandra **, Dr Harish Chaudhary *** * Research Scholar, Cenre for Mahemaical Sciences, Banashali Universiy, P.O. Banashali Vidyapih, Banashali 040, Rajashan, India ** Associae Professor, Cenre for Mahemaical Sciences, Banashali Universiy, P.O. Banashali Vidyapih, Banashali 040, Rajashan,India *** Assisan Professor, Deparmen of Managemen Sudies, Indian Insiue of echnology, Delhi, New Delhi-006, India Absrac- his paper develops invenory models for deerioraing iem; he rae of deerioraion is Weibull disribuion deerioraion wih wo parameers. I considers ramp-ype demand. Shorage is allowed and i is compleely backlogged. In hese models, we consider ime dependen holding cos. hese models are developed under wo differen replenishmen policies; (i) wihou shorage (ii) wih shorage. he aim of hese models is o find he opimal soluion for minimizing he oal invenory coss for boh he above menioned policies. o opimize he models numerical illusraions have been carried ou and sensiiviy analysis occurred o evaluae he resul of parameers on assessmen variables and he enire cos of hese models. Index erms- Holding Cos, Ramp-ype Demand, Shorage, Weibull Deerioraion. I I. INRODUCION nvenory is a fundamenal par of manufacuring, disribuion, reail infrasrucure and demand plays an imporan role in choosing he beneficial invenory sraegy. Researchers were developed he invenory models assuming he demand of he objecs o be consan, sock dependen, linearly increasing, linearly decreasing, exponenially increasing and exponenially decreasing wih ime ec. afer ha, i has been noiced ha he above menioned demand policies do no accuraely describe he demand of assured objecs such as newly launched cosmeics, garmens, fashion iems, elecronics ec., for which he demand increases wih ime as hey are launched ino he marke and afer some ime i becomes sable. Ramp-ype demand paern is inroduced o consider he demand of such ypes of maerials. Mosly he ramp- ype demand rae is increasing for some ime when new variey of consumer goods comes o he markeplace. In case of ramp-ype demand paern, he demand increases linearly a he beginning and when he marke comes ino a seady sae, he demand becomes sable unil he end of he invenory cycle. For example, he demand of fesival greeings likes swees, gods and goddess s picures, phaakhes, painings ec., follows ramp-ype demand rae. A he beginning demands of hese iems increases linearly from saring of Ocober o he end of November monh, afer ha demands of hese iems becomes consan. he invenory of deerioraing iems is a big problem for any organizaion in he supply chain sysem. he finished goods are deerioraing wih ime in invenory. Some iems used in daily life like fruis, vegeables, milk producs, mea ec., are deerioraing a higher rae insead of oher iems. I is necessary ha consume hese iems wihin ime limi o avoid deerioraion. he invenory model wih ramp-ype demand rae was projeced by Mandal and Pal [9]. Wu and Ouyang [] developed a replenishmen invenory model for deerioraing iems wih ramp-ype demand rae. Wu [] developed an economic order quaniy invenory model for deerioraing iems. He considered weibull disribuion deerioraion, parial backlogging and rampype demand. he invenory model wih Weibull deerioraion and relaed issues have been sudied by many researchers. Silver [6] presened a heurisic for deerioraing iems invenory model wih ime dependen linear demand. Goyal and Giri [5] exended review on deerioraion invenory model. Cover and Philip [] presened an invenory model where he ime o deerioraion is described wih wo parameers Weibull disribuion deerioraion. Ghosh and Chaudhuri [7] developed an invenory model for deerioraing iems wih wo parameers Weibull disribuion deerioraion, shorages and quadraic demand rae. Sanni [8] developed an invenory model for hree parameers Weibull deerioraing iems, quadraic demand rae and shorages are allowed. Shorages are permied and cusomers will wai ill he nex order arrives i.e. called a backorder model. All demands are fulfilled insanly. Ghare and Schrader [4] developed an exponenially deerioraing invenory model. In his paper, hey considered sable rae of deerioraion wih no shorage. here afer a lo of research work has been done. Holding cos is considered as known and consan in mos of he models. Now, holding cos may no be considered as consan, i is varying wih regular change in ime value of money and change in price index. here is lo of compeiions in globalizaion, so holding cos may no remain consan over ime. Naddor[0], Van Deer Veen [], Weiss [4] assumed holding cos as a non-linear funcion over ime for which he iems are held in invenory and he amoun of he on-hand invenory. Roy [] proposed an EOQ model for deerioraing iems in which deerioraion rae and holding cos are considered as linearly increasing funcion of ime, selling price is dependen on demand rae and shorage is allowed and compleely backlogged. ripahi [] presened an invenory model for non-deerioraing iems under permissible delay in paymens where holding cos is a funcion of ime. he aim of his paper is developing invenory models where rae of deerioraion is wo parameers weibull disribuion deerioraion. Holding cos as a linearly increasing funcion of ime. hese models have been sudied under wo differen invenory policies such as wihou shorage and wih shorage. he main purpose of his paper is o show ha here exis a unique opimal cycle ime o minimize he oal invenory cos
Inernaional Journal of Scienific and Research Publicaions, Volume 5, Issue 6, June 05 ISSN 50-5 per uni ime. o opimize hese models numerical illusraions have been carried ou and sensiiviy analysis occurred o sudy he resul of parameers on assessmen variables and he enire cos of hese models. II. ASSUMPIONS We consider he following assumpions for developing mahemaical model:. he invenory sysem consider single iem.. is he fixed uni of ime for each ordering cycle.. he replenishmen rae is infinie. F( ) 4. he demand rae F D, <, = D,, is a ramp ype funcion: D [ 0, ] is posiive and coninuous for 5. he lead ime is zero. 6. Shorage is allowed and compleely backlogged. H 7. Holding cos c per uni is ime dependen: Hc ( ) = a + b, where a > 0 and b > 0. 8. he rae of deerioraion is wo parameers weibull deerioraion denoed by θ = α, where α > 0, > 0 and 0. 9. Q is he ordering quaniy.. III. NOAIONS We consider he following noaions for developing mahemaical model: D. c is he deerioraion cos per uni per iem.. is he parameer of he ramp ype demand funcion.. is he ime when invenory level reaches zero. 4. is he opimal poin for he replenishmen policy. S 5. c is he shorage cos per uni per iem. I( ) 6. is he on hand invenory level a ime over he ordering cycle [ ] 0,. 7. is considered annually. IV. MAHEMAICAL MODEL hese models are developed for he following wo differen replenishmen policies: (i) Wihou Shorage. (ii) Wih Shorage. V. MODELS WIHOU SHORAGE he invenory model is saring wih no shorage. A his sage, he invenory level reaches is maximum level and hen producion 0, = and falls zero a.hus, is sopped. he invenory deplees o zero due o demand and deerioraion during he ime inerval [ ]
Inernaional Journal of Scienific and Research Publicaions, Volume 5, Issue 6, June 05 ISSN 50-5 [ ] shorage occurs during, equaions: di a dib d d ( ) ( ) a, which is compleely backlogged. herefore, he invenory is described by he sysem of differenial,0 Ia ( ) = 0... wih boundary condiion = θ I F, = F Ib ( ) = 0... wih boundary condiion We considered wo cases, i < and ii for solving hese differenial equaions: 5.Case (i): < From equaion (), we ge α Ia De + α + + = ( ) + ( )...( ) Equaion (), solve by he following wo ways: dib ( ) = D, d... i dib ( ) = D,, I = I + d... ii We have D Ib... 4 = ( ) D D Ib = D +... 5 and [ 0, ] oal deerioraion amoun during α α D Dc = De d Dd = + 0 0 +... 6 [ 0, ] oal holding cos during is α α + + HC = Hc( ) Ia( ) d = ( a + b) De ( ) + ( ) d + 0 0 4 4 b 4 HC = D P + + Q + + R + + S + + +... 7 8
Inernaional Journal of Scienific and Research Publicaions, Volume 5, Issue 6, June 05 4 ISSN 50-5 See Appendix-A is [ ] Los sale amoun during, D D D Sc = Ib( ) d = Ib( ) d + Ib( ) d = ( ) d + D d D Sc = A + 6 Order quaniy is ( )...( 8) α ( ) OQ = e Dd D d D d = D + α d + D d + D d 0 0 α OQ = D + + A + + See Appendix-A... 9 See Appendix-A [ 0, ] C oal cos during = holding cos + los sale amoun + deerioraion cos ( ) = HC + Sc + Dc + 4 + + 4 + α + b 4 A C ( ) = D P + Q + R + S + + + +...( 0) + 8 6 5.. Soluion: ( + 4) + ( + ) + ( + 4) + ( + ) P Q R S + + + + C = D + b + α + + + + ( + 4)( + ) + ( + )( + ) + ( + 4) P Q R C = D + + b ( + ) + S( + )( + ) + α( + ) + + 4...... Main objecive o minimize he oal relevan cos for he model saring wihou shorage, he necessary condiion o minimize he oal relevan cos is C = 0, We ge + + + + DP ( + 4) + DQ ( + ) + DR ( + 4) + DS ( + ) + Db + Dα + + D D = 0...
Inernaional Journal of Scienific and Research Publicaions, Volume 5, Issue 6, June 05 5 ISSN 50-5 C ( ) by equaion () and he opimal value Using he sofware Mahemaica, we can calculae he opimal value of of he oal relevan cos is deermined by equaion (0). he opimal value of saisfy he sufficien condiion for minimizing oal relevan C C > 0... ( 4) is cos he sufficien condiion is saisfied. 4.0 Invenory Cos 6.9 C 0.7 4.6 8.4. ( 0.46,.8) 6. ime Figure : Graphical represenaion of oal relevan cos wihou shorage. 5.. Numerical Example: Le us consider a = $/ uni, b = $0.5 / uni, Sc = $/ uni / year, dc = $5 / uni, α = 0. 0, =, = year, D = 00, = 0.8 year. P= 0.0000065, Q= 0.0000, R= 0.0004, S = 0.00, A = 0. 99, A = 6 hus, he opimal value of 0.46 < is. he opimal ordering quaniy is OQ = 6.0.he minimum relevan cos is C =.8 5.. Sensiiviy Analysis: o know, how he opimal soluion is affeced by he values of parameers, we derive he sensiiviy analysis for some parameers. he paricular values of some parameers are increased or decreased by + 5%, 5% and + 50%, 50%. Afer ha, we derive he value of C and wih he help of increased or decreased values of Sc and d c. he resul of he minimum relevan cos is exising in he following able. able: Parameers Acual 50% 50% Decreased 5% Increased 5% Values Increased Decreased Sc.5 0.5.5 0.75 dc 5 7.5.5 6.5.75
Inernaional Journal of Scienific and Research Publicaions, Volume 5, Issue 6, June 05 6 ISSN 50-5 0.46 0.55 0. 0.5 0.4 C.8.8 7.4 4.96 0. OQ 6.0 6.045 6.005 6.04 6.0 From he resul of above able, we observe ha oal relevan cos and ordering quaniy is much affeced by deerioraion cos and shorage cos, oher parameers are less sensiive. 5. Case II: Equaion (), solve by he following wo ways: dia ( ) + θ Ia = D, 0, I = I +... i d dia ( ) + θ Ia = D,, I = 0... ii d We have ( ) dia + α Ia ( ) = D, 0,I( ) = I( + ) d α α + + α + + Ia ( ) = De ( ) + ( ) + ( ) + ( )... ( 5) + + and α Ia e D + α + + = ( ) + ( ),...( 6) From equaion (), we ge dib d ( ) [ 0, ] oal deerioraion amoun during = D,, I = 0 b b = ( ) I D c 0 α α α + + D = D e d + D e d D d = D S + [ 0, ] oal holding cos during is HC = H I d = a + b I d + a + b I d a a a 0 0 α + + + + + HC = ad δ+ γ ( ) + ( ) + ε + γ + ψ + η+ κ + α + + + + + bd π + ρ + + ρ+ ξ + τ + ϑ+ Ω + 6... 8 See Appendix-B See Appendix-B
Inernaional Journal of Scienific and Research Publicaions, Volume 5, Issue 6, June 05 7 ISSN 50-5 + α γ + αγ + + HC = ad δ γ + ε + + + η + ψ + κ + + + + + αρ + bd π + ξ + τ + + ϑ + Ω+ + 6 HC ad + + + bd + + κ ψ τ + = + + + Π+R + + Ω+ + ϒ+Ψ...( 9) 6 See Appendix-B is Sc = Ib d = D d = D D... 0 = + [ ] Los sale amoun during, Order quaniy is α OQ = De d + D d + D ( ) d = D + + Z... 0 [ 0, ] oal cos during = holding cos + los sale amoun + deerioraion cos + + + a + κ + ψ + Π+R + + C D + + + α + + b τ S C( ) = HC + Sc + D + Ω+ + ϒ+Ψ + + c 6 + =... 5.. Soluion: + + a + a( + ) Π+ a( + ) ψ + a( + ) κ C + + b = D+ b( + ) τ + b( + ) ϒ+ b( + ) Ω+... + α + ( ) + ( + ) Π+ ( + )( + ) + ( + )( + ) + + ( )( ) τ ( ) ( )( )...( 4) a a a ψ a κ C = D b + + + + b + ϒ+ b + + Ω+ b + + α + Main objecive o minimize he oal relevan cos for he model saring wihou shorage, he necessary condiion o minimize he oal relevan cos is C = 0, we ge + + + bd bdω ( + ) + adκ( + ) + bdτ ( + ) + + + adψ + + D aπ+ bϒ + + α + D a + D = 0... 5
Inernaional Journal of Scienific and Research Publicaions, Volume 5, Issue 6, June 05 8 ISSN 50-5 C ( ) by equaion (5) and he opimal value Using he sofware Mahemaica, we can calculae he opimal value of of he oal relevan cos is deermined by equaion (). he opimal value of saisfy he sufficien condiion for minimizing oal relevan C C > 0... ( 6) is cos he sufficien condiion is saisfied. 5.. Numerical Example: Le us consider a = $/ uni, b = $0.5 / uni, Sc = $/ uni / year, dc = $5 / uni, α = 0.0, =, = year, D = 00, = 0.8 year. δ = 0.7, γ = φ = 0.67, ε = 0.5, ψ = 0.00, η = 0.004, κ = 0.00008, π = 0.5, ξ = 0.6, ρ = 0.5, τ = 0.006, ϑ = 0.004, Ω= 0.0000, R= 0. 09, Π= 0.000047, ϒ= 0.007, Ψ= 0.07, S = 0., Z = 0.08, S = 0. hus, he opimal value of 0.9 > is. he opimal ordering quaniy is OQ = 4.06.he minimum relevan cos is C = 9.9 VI. MODELS WIH SHORAGE [ 0, ] Now, invenory model saring wih shorage during he period and is compleely backlogged. Replenishmen brings he invenory level up o Q afer ime. he invenory level deplees and falls o zero a. wo cases occur < [ ] during, (i) (ii) 6. Case (i) < dq dq = because of demand and deerioraion herefore, he invenory I [ ] is described by he sysem of differenial equaions during ( ) = D( ), 0, q( 0) = 0...( 7) d ( ) + θ q( ) = D, q ( ) = q ( + )...( 8) d ( ) + θ q( ) = D, q = 0...( 9) dq d From equaion (7), we have D q D d = =, 0...( 0) 0 From equaion (8), we have 0, :
Inernaional Journal of Scienific and Research Publicaions, Volume 5, Issue 6, June 05 9 ISSN 50-5 α q e D e d e d e D Z + See Appendix-C From equaion (9), we have α α α α + = + =,...( ) α q e D e d e D Z4,... + α α α + = = [ ] oal deerioraion amoun during, See Appendix-C α ( )...( ) α Dc e α De α d D e + = + d = D α + + C + [ ] oal holding cos during, is HC = a + b q d See Appendix-C a b αb αb N + N az bz + + + 6 8 ( )( 4) ( 4) + + + aα aα + αbz + aαz + = D + + + ( + )( + ) ( + ) ( + ) + bα + 4 aα + ( + )( + 4) ( + )( + ) 4 + 4 See Appendix-C a b 4 + 4 + N+ N az bz + + + N + N4 6 8 HC = D... 4 αbz + aαz + + 4 + + + N 5 N 6 ( + ) + [ 0, ] Los sale amoun during D Sc D d D = = = 6 0 is...( 5) Order quaniy is α OQ D d e = + D ( + α ) d + D ( + α ) d 0 α + + ( ) ( ) + + OQ = D + ( α )... 6 α + + + ( ) + ( ) +
Inernaional Journal of Scienific and Research Publicaions, Volume 5, Issue 6, June 05 0 ISSN 50-5 [ 0, ] oal cos during =holding cos + los sales amoun + deerioraion cos C HC S D ' = + c + c a b 4 + 4 + αbz + N+ N az bz + + + N + N4 + 6 8 ( + ) C ' = D... 7 aα Z 4 + + + α α + + N 5 N 6 + + e + + C + 6 + 6.. Soluion: a b + + az bz + + + N ( + 4) + N4 ( + ) 4 C ' + + + = D+ αbz + aαz N5 ( + 4) N6 ( + )... 8 + α + + + ( α )( + α ) α + + C + b N ( 4)( ) + N 4( ) + + + + + + + 4 + + αbz ( + ) + aαz N5 ( + 4)( + ) C ' + = D N 6( + )( + ) + ( α )( + α( + ) ) + + α ( + α ) α ( + α ) bz + a α + α ( ) + + C + +...( 9) Main objecive o minimize he oal relevan cos for he model saring wih shorage, he necessary condiion o minimize he oal relevan cos is C ' = 0, we ge a b + + az bz + + + N ( + 4) + N4 ( + ) 4 D + + + + αbz + aαz N5 ( + 4) N6 ( + ) = 0...( 40) + α + + + ( α )( + α ) α + + C + C '( ) by equaion (40) and he opimal value Using he sofware Mahemaica, we can calculae he opimal value of of he oal relevan cos is deermined by equaion (7). he opimal value of saisfy he sufficien condiion for minimizing oal relevan C ' > 0... C' ( ) ( 4) cos is he sufficien condiion is saisfied.
Inernaional Journal of Scienific and Research Publicaions, Volume 5, Issue 6, June 05 ISSN 50-5 6.. Numerical Example: Le us consider a = $/ uni, b = $0.5 / uni, Sc = $/ uni, dc = $5 / uni, α = 0.0 =, = year, D = 00, δ =, = 0. 8 year. Z = 0.48, Z4 =.006, C = 0.6, N = 0.5, N = 0.08, N = 0.0004, N4 = 0.00, N = 0.0000065, N = 0. 00004 5 6 hus, he opimal value of ' is 0.099 <. he opimal ordering quaniy is OQ = 48.5.he minimum relevan cos is C ' = 8. 4 6. Case (ii) herefore, he invenory I [ ] is described by he sysem of differenial equaions during dq ( ) = D( ), 0, q( 0) = 0...( 4) d dq ( ) = D( ),, q( ) = q( + )...( 4) d dq ( ) + θ q = D,, q = 0...( 44) d From equaion (4), we have D q D d = =, 0...( 45) From equaion (4), we have 0 q D d D d D = =,...( 46) 0 From equaion (44), we have α q e D e d e D M α α α + = =,...( 47) + [ ] oal deerioraion amoun during, See Appendix-D ( )...( 48) D e α α + = D e d d = D α M c [ ] oal holding cos during, α + is HC = ( a + b) q( ) d = a q( ) d+ b q( ) d αm + + + D+ a M + + D + D ( ) + HC = D M + αm + + b D 4 D + + + + 5 ( + )... 49 0, :
Inernaional Journal of Scienific and Research Publicaions, Volume 5, Issue 6, June 05 ISSN 50-5 [ 0, ] Los sale amoun during 0 is Sc = q d + q d = D +...( 50) 6 Ordered quaniy is α α OQ = D d + D d + e D e d 0 OQ = D + + e + α α + + +...( 5) [ 0, ] oal cos during = holding cos + los sale amoun + deerioraion cos C = HC + S + D ' c c b b D + 5 b D 4 + a D + α + + + + αbm am + + + ad + + α + ( + ) ( + ) + C' ( ) = D... α M + ( + a bm) am+ D M + + + 6 ( 5) 6.. Soluion: + + b b D5( + ) + b D4( + ) a D( + ) + α ( + ) αbm + + + ( + ) ad + + α + ( ) + C ' = D am ( + ) + α M + ( + a bm) ( + ) + am + + + α ( + ) ( )( ) b b D5 + + + b D4 + + a D + + + + + + C ' + = D αbm am ad + + α + + + + + α M + + a bm... 5... 54 Main objecive o minimize he oal relevan cos for he model saring wih shorage, he necessary condiion o minimize he oal relevan cos is
Inernaional Journal of Scienific and Research Publicaions, Volume 5, Issue 6, June 05 ISSN 50-5 C ' = 0, we ge + + + b b D5( + ) + b D4( + ) a D( + ) α ( + ) αbm + + + ( + ) ad + ( ) + + D am 0... 55 = + α ( + ) + α M + ( + a bm) ( + ) + am + Using he sofware Mahemaica, we can calculae he opimal value of of he oal relevan cos is deermined by equaion (5). he opimal value of saisfy he sufficien condiion for minimizing oal relevan C ' > 0...( 56 ) C' is cos he sufficien condiion is saisfied. C '( ) by equaion (55) and he opimal value 4.0 C' 6.9 Invenory Cos 0.7 ( 0.8,8.5) 4.6 8.4. 6. Figure : Graphical represenaion of oal relevan cos wih shorage. 6.. Numerical Example: Le us consider a = $/ uni, b = $0.5 / uni, Sc = $/ uni, dc = $5 / uni, α = 0.0 = = year, D = 00, δ =, = 0.8year. M =.007, D = 0.587, D = 0.00, D = 0.0000, D = 0.006, D = 0. 00009 4 5
Inernaional Journal of Scienific and Research Publicaions, Volume 5, Issue 6, June 05 4 ISSN 50-5, which does no saisfy 0.6 We find ' is 0.8 =. he opimal ordering quaniy is OQ ' = 48. 05, so he maximum poin is max { } C.he minimum relevan cos is ' = 8.5, =. hus, he opimal value of 6.. Sensiiviy Analysis: o know, how he opimal soluion is affeced by he values of parameers, we derive he sensiiviy analysis for some parameers. he paricular values of some parameers are increased or decreased by + 5%, 5% and + 50%, 50%. Afer ha, we derive he value of C and wih he help of increased or decreased values of Sc and d c. he resul of he minimum relevan cos is exising in he following able. able: Parameers Acual +50% -50% +5% -5% Values Increased Decreased Increased Decreased S.5 0.5.5 0.75 c dc 5 7.5.5 6.5.75 0.8 0.8 0.8 0.8 0.8 C 8.5 4.5 5.5 5.0.0 OQ 48.05 48.05 48.05 48.05 48.05 From he resul of above able, we observe ha oal relevan cos and ordering quaniy is much affeced by deerioraion cos and shorage cos, oher parameers are less sensiive. VII. CONCLUSION We developed order level invenory models wih ime varying holding cos for weibull deerioraing iems. We considered rampype demand rae and holding cos is no consan varying over ime. he deerioraing iem is deerioraed wih wo parameers weibull disribuion deerioraion. he models are developed under wo differen policies (i) wihou shorage and (ii) wih shorage, which is compleely backlogged. he oal relevan cos wih consan holding cos is less han he oal relevan cos wih ime varying holding cos. he oal invenory cos wih consan holding cos is more realisic han he oal invenory cos wih ime varying holding cos. Advance research in his way can be carried ou such as sochasic demand and finie replenishmen rae, quaniy discouns and permissible delay in paymens. Appendix-A: α b αb bα αb αb P=, R= + + + + + 4 + + 4 α α α α α α a a a a a a Q=, S = + + + + + + + + + + Where, D A = ( + ) A = + D ( ), Appendix-B: + α + α S = + Where, + +
Inernaional Journal of Scienific and Research Publicaions, Volume 5, Issue 6, June 05 5 ISSN 50-5 + + + α + + α α α δ = +, γ =, ε = + + + + + + + + α α α α α α +, ψ =, η =, + + + + + + + ( + ) 4 + 4 + 4 α α α α + α κ = π = + ( + )( + ) ( + ) 8 ( + )( + 4) ( + ) ( + ) + + + 4 + 4 α α α α α +, ϑ = ρ =, ξ = + + + + + + + + + 4 + 4 α α α α α τ = +, + ( + )( + ) ( + ) + + α α Ω= ( + )( + ) ( + )( + ) + α γ αγ αρ R= δ γ + ε, Π= + η, ϒ= + ϑ, Ψ= π + ξ + + + Z α = + + Appendix-C: Where + α α Z = + + + + + + + + Z α 4 + α = + C = + + +, + + N N + + + + a aα aα Z aα aα az + + 6 ( + )( + ) + ( + ) ( + )( + ) = + + + 8 ( + )( + 4) ( + ) ( + 4) ( + )( + 4) 4 + 4 + + 4 + 4 bz b α b α bz α b bα + + + + + + aα aα Z4 aα az4 ( ) + ( + )( + ) ( + ) ( + ) + + + + aα ( ) bz4 ( ) b( ) bα ( ) = + + ( + )( + ) ( + )( + ) α bz α b bα a + + + + + + 4 + + + + + + N = αb αb, N = aα aα, N = bα, N = aα ( + )( + 4) ( + 4) ( + )( + ) ( + ) ( + )( + 4) ( + )( + ) 4 5 6
Inernaional Journal of Scienific and Research Publicaions, Volume 5, Issue 6, June 05 6 ISSN 50-5 See Appendix-D: α M Where, + = + + α αm α α + + + + D = a M + + + ( + )( + ) ( + ) ( + ) ( + )( + ) + + + + α α α α α = = 4 = 5 M α αm α α b + + ( + )( + ) ( + ) ( + ) ( + )( + ) D, D, D, D = ( + )( + ) ( + ) ( + )( + ) ( + )( + ) ( + ) α ( + )( + ) ACKNOWLEDGMEN he auhors would like o very hankful o he referees for heir valuable guidance. [] R.P. Cover, and G.C. Philip, An EOQ model for iems wih Weibull disribuion deerioraion, AIEE ransacions, 5, 97, -6. [] K.S.Wu, and L.Y. Ouyang, A replenishmen policy for deerioraing iems wih ramp ype demand rae, Proc, Na. Sci Counc.ROC (A), 4, 000, 79-86. [] K.S.Wu, An EOQ invenory model for iems wih Weibull disribuion deerioraion, ramp ype demand rae and parial backlogging, Producion Planning and Conrol,, 00, 787-79. [4] P.M. Ghare, and G.H.Schrader, An invenory model for exponenially deerioraing iems, Journal of indusrial Engg. 4, 96, 8-4. [5] S.K. Goyal, and, B.C. Giri, Recen rends in modeling of deerioraing invenory, European Journal of Operaional Research, 4, 00, -6. [6] E.A. Silver, A simple invenory replenishmen decision rule for a linear rend in demand, J. Oper. Res. Q, 0, 979, 7-75. [7] S.K.Ghosh, and K.S. Chaudhuri, An order-level invenory model for a deerioraing iem wih Weibull disribuion deerioraion, ime quadraic demand and shorages, Adv. Model. Opim. 6(), 979, -5. [8] S.S, Sanni, An economic order quaniy invenory model wih ime dependen weibull deerioraion and rended demand, M.Sc. hesis, Universiy of Nigeria, 0. [9] B.Mandal, and A.K. Pal, Order level invenory sysem wih ramp ype demand rae for deerioraing iems, Journal of Inerdisciplinary Mahemaics,, 998, 49-66. [0] E. Naddor, Invenory Sysem, John Wiley and Sons, New York, 966. [] A. Roy, An invenory model for deerioraing iems wih price dependen demand and ime varying holding cos, Advanced Modeling and Opimizaion, 0, 008, 5-7. [] R. P. ripai, Invenory model wih cash flow oriened and ime-dependen holding cos under permissible delay in paymens, Yugoslav Journal of Operaions Research, (), 0. [] B. Van deer Veen, Inroducion o he heory of Operaional Research, Philip echnical Library, Springer-Verlag, New York, 967. [4] H.J. Weiss, EOQ models wih non-linear holding cos, European Journal of Operaional Research, 9, 98, 56-60. [5] S. Jain, and M. Kumar, An EOQ invenory model for iems wih rampype demand hree parameer Weibull disribuion deerioraion and saring wih shorage, Yugoslav Journal of Operaions Research, 0(), 00, 49-59. [6] A.K. Jalan, B.C. Giri, and K.S.Chadhuri, EOQ model for iems wih weibull disribuion deerioraion, shorages and ramp ype demand, Recen Developmen in Operaions Research, Narosa Publishing House, New Delhi, India, 00, 07-. [7] S. Pal, A. Goswami, and K. S. Chaudhuri, A deerminisic invenory model for deerioraing iems wih sock-dependen demand rae, Inernaional Journal of Producion Economics, (), 99, 9-99. [8] G.P.Samana, and A. Roy, A Producion invenory model wih deerioraing iems and shorages, Yugoslav Journal of Operaion Research, 4(), 004, 9-0. [9] K.Skouri, and I.Konsanaras, Order level invenory models for deerioraing seasonable producs wih ime dependen demand and shorages, Hindawi Publishing Corporaion Mahemaical Problems in Engineering aricle ID 67976, 009. [0] K.Skouri, and I.Konsanaras, S.Papachrisos, and I.Ganas, Invenory models wih ramp ype demand rae, parial backlogging deerioraing and weibull deerioraion rae, European Journal of Operaional Research, 9, 009, 79-9. [] A.K. Jalan, R.R. Giri, and K.S. Chaudhuri, EOQ model for iems wih weibull disribuion deerioraion, shorages and rended demand, Inernaional Journal of Sysem Science, 7, 996, 85-856. [] S.Jain, and M. Kumar, An EOQ invenory model wih ramp-ype demand, Weibull disribuion deerioraion and saring wih shorage, Opsearch 44, 007, 40-50. [] B. Karmakar, and K.D. Choudhury, Invenory models wih ramp-ype demand rae for deerioraing iems wih parial backlogging and imevarying holding cos, Yugoslav Journal of Operaions Research, 4, Number, 04, 49-66. AUHORS Firs Auhor Riu Raj, Research Scholar, Cenre for Mahemaical Sciences, Banashali Universiy, P.O. Banashali Vidyapih, Banashali 040, Rajashan, India, e-mail: rriuii@gmail.com, Mob:09777006. Second Auhor Naresh Kumar Kaliraman, Research Scholar, Cenre for Mahemaical Sciences, Banashali Universiy, P.O. Banashali Vidyapih, Banashali 040, Rajashan, India,email:dr.nareshkumar@gmail.com, Mob:0989008069. hird Auhor Dr Shalini Chandra, Associae Professor, Cenre for Mahemaical Sciences, Banashali Universiy, P.O. Banashali Vidyapih, Banashali 040, Rajashan, India, e-mail: chandrshalini@gmail.com, Mob: 078944849. Fourh Auhor Dr Harish Chaudhary, Assisan Professor, Deparmen of Managemen Sudies, Indian Insiue of echnology, Delhi, New Delhi-006, India, e-mail: hciid@gmail.com, Mob: 099555875.
Inernaional Journal of Scienific and Research Publicaions, Volume 5, Issue 6, June 05 7 ISSN 50-5