Inernaional Refereed Journal of Engineering and Science (IRJES) ISSN (Online) 39-83X, (Prin) 39-8 Volume 5, Issue 3 (March 6), PP.-7 Producion Invenory Model wih Differen Deerioraion Raes Under Shorages and Linear Demand S.R. Sheikh, Raman Pael Deparmen of Saisics, V.B. Shah Insiue of Managemen, Amroli, Sura, INDIA Deparmen of Saisics, Veer Narmad Souh Gujara Universiy, Sura, INDIA Absrac:- A producion invenory model wih linear rend in demand is developed. Differen deerioraion raes are considered in a cycle. Holding cos is considered as funcion of ime. Shorages are allowed. o illusrae he model numerical example is provided and sensiiviy analysis is also carried ou for parameers. Keywords:- Producion, Invenory model, Varying Deerioraion, Linear demand, ime varying holding cos, Shorages I. INRODUCION Deerioraion of an iem is defined as damage, spoilage, dryness, vaporizaion, ec. ha decreases he uiliy of he iem. Ghare and Schrader [963] firs developed an EOQ model wih consan rae of deerioraion. he model was exended by Cover and Philip [973] by considering variable rae of deerioraion. he model was furher exended by Shah and Jaiswal [977] by considering shorages. Gupa and Vra [986] developed an invenory model wih sock dependen consumpion rae. Giri e al. [996] developed invenory model wih sock dependen demand. he relaed works are found in (Raafa [99], Goyal and Giri [], Ruxian e al. []). Mandal and Phaujdar [989] developed producion invenory model for deerioraing iems wih linear sock dependen demand. Roy and Chaudhary [9] sudied a producion invenory model for deerioraing iems when demand rae is invenory level dependen and producion rae depends boh on sock level and demand. ripahy and Mishra [] developed an order level invenory model wih ime dependen Weibull deerioraion and ramp ype demand rae where producion and demand were ime dependen. Pael and Pael [3] developed a deerioraing iems producion invenory model wih demand dependen producion rae. Sharma and Chaudhary [3] deal wih invenory model for deerioraing iems wih ime dependen demand rae. Shorages were allowed and compleely backlogged. Pael and Sheikh [5] developed an invenory model wih differen deerioraion raes under linear demand and ime varying holding cos. In mos of he producs, iniially here is no deerioraion. Afer cerain ime deerioraion sars and again afer cerain ime he rae of deerioraion increases wih ime. Here we have used such a concep and developed he deerioraing iems invenory models. A producion invenory model wih differen deerioraion raes for he cycle ime under ime varying holding cos is developed in his paper. Shorages are allowed and compleely backlogged. Numerical example is provided o illusrae he model and sensiiviy analysis of he opimal soluions for major parameers is also carried ou. II. ASSUMPIONS AND NOAIONS NOAIONS: he following noaions are used for he developmen of he model: P() : Producion rae is a funcion of demand ( P() = ηd(), η>) D() : Demand rae is a linear funcion of ime (a+b, a>, <b<) A : Replenishmen cos per order c : Purchasing cos per uni p : Selling price per uni : Lengh of invenory cycle I() : Invenory level a any insan of ime, Q : Order quaniy iniially Q : Quaniy of shorages Q : Order quaniy c : Shorage cos per uni θ : Deerioraion rae during μ, < θ< Page
Producion Invenory Model Wih Differen Deerioraion Raes Under Shorages ANS Linear Demand θ : Deerioraion rae during,, < θ< π : oal relevan profi per uni ime. ASSUMPIONS: he following assumpions are considered for he developmen of wo warehouse model. he demand of he produc is declining as a linear funcion of ime. Rae of producion is a funcion of demand Replenishmen rae is infinie and insananeous. Lead ime is zero. Shorages are allowed and compleely backlogged. Deerioraed unis neiher be repaired nor replaced during he cycle ime. III. HE MAHEMAICAL MODEL AND ANALYSIS Le I() be he invenory a ime ( ) as shown in figure. Figure he differenial equaions which describes he insananeous saes of I() over he period (, ) is given by di() = (η - )(a + b), d di() + θi() = (η - )(a + b), d di() + θi() = - (a + b), d di() = - (a + b), d μ () μ () (3) (4) di() = (η - )(a + b), d wih iniial condiions I() =, I(μ ) = S, I( ) = Q, I( ) =, I( ) = -Q, and I() =. Soluions of hese equaions are given by I() = (η - )(a + b ). I() = S + θ μ - (η-) a μ + b μ + aθ μ + bθ μ - aθ μ - bθ μ 3 3 3 4 4 I() = a + b + aθ + bθ - aθ - bθ 6 8 4 3 3 (5) (6) (7) (8) Page
Producion Invenory Model Wih Differen Deerioraion Raes Under Shorages ANS Linear Demand I() = a + b (9) I() = η- a - + b - () (by neglecing higher powers of θ) Puing = µ in equaion (6), we ge S = (η - )(aμ + bμ ) () Puing = in equaions (7) and (8), we have I( ) = S + θμ 3 3 () - (η-) a μ + bμ + aθ μ + bθμ -aθ μ - bθ μ 3 3 3 4 4 I( ) = a + b + aθ + bθ - aθ - bθ. 6 8 4 (3) So, from equaions (8) and (9), we have S ( + θμ ) - (η-)aμ - a = (4) S θ - ηa From equaion (4), we see ha is a funcion of μ, and S, so is no a decision variable. Similarly puing = in equaions (9) and (), we have I( ) = a + b (5) I( ) = η- a - + b - (6) So, from equaions (5) and (6), we have + (η-) = (7) η From equaion (7), we see ha is a funcion of and, so is no a decision variable. Puing = in equaion (7), we ge Q = (η - )(aμ + bμ ) + θμ (8) 3 3 - (η-) a μ + bμ + aθ μ + bθ μ -aθ μ - bθ μ 3 Similarly, puing = in equaion (9), we ge Q = a + b (9) Puing value of S from equaion () in equaion (7), we have I() = (η - )(aμ + bμ ) + θμ () 3 3 - (η-) a μ + bμ + aθ μ + bθ μ -aθ μ - bθ μ 3 Based on he assumpions and descripions of he model, he oal annual relevan profi (π), include he following elemens: (i) Ordering cos (OC) = A () (ii) HC = (x+y)i()d μ = (x+y)i()d + (x+y)i()d + (x+y)i()d μ 3 Page
(iii) DC = c Producion Invenory Model Wih Differen Deerioraion Raes Under Shorages ANS Linear Demand = -x a + b + aθ + bθ + y ηb- b μ + xηa-a μ - x - b- aθ - bθ -ya 6 8 4 3 4 3 4 4 3 3 3 3 + x - b- aθ- bθ -ya -x ηaθμ + ηbθμ - aθμ - bθμ μ + xbθ+ yaθ 3 4 6 6 5 8 3 + xaθ+y- b- aθ - bθ + -xa+y a + b aθ + bθ - xbθ+ yaθ 4 3 4 6 8 5 8 3 4 3 4 4 5 4 + x - ηbθ+ bθ +y- b+ ηb+ aθ- ηaθ + x 4 6 6 + x ηb- b +yηa-a μ 3 3 ηa-a +y ηaθμ + ηbθμ - aθμ - bθμ 6 6 5 3 3 - ybθ - x - ηbθ+ bθ +y - b+ ηb+ aθ- ηaθ μ - -xa+y - b- aθ - bθ 48 4 6 6 4 6 4 4 3 3 3 - xηa-a +y ηaθμ + ηbθμ - aθμ - bθμ μ +x ηaθμ + ηbθμ- aθμ - bθμ 6 6 6 6 - x - b+ ηb+ aθ- ηaθ +yηa-a μ + y- ηbθ+ bθ - y- ηbθ+ bθ μ 5 6 6 5 6 6 3 5 5 3 3 4 6 + x - b+ ηb+ aθ- ηaθ +yηa-a +x a + b aθ + bθ + ybθ 3 6 8 48 (by neglecing higher powers of θ) θi()d + θi()d μ 3 ηaμ + ηbμ -aμ- bμ +ηaθμ μ + ηbθμ μ -aθ μ 3 3 4 -aθμ μ - bθμ μ -ηa μ - ηb μ - bθ μ 3 = cθ 4 3 3 4 3 4 - ηaθ μ - ηbθ μ +ηaθ μ + ηbθ μ 3 3 4 3 4 3 3 3 4 +a μ + bμ + aθμ + bθ μ 3 3 3 4 3 3 3 4 3 4 -cθ ηaμ + ηbμ - aμ - bμ + ηaθμ + ηbθμ - aθμ - bθμ 6 6 3 8 3 8 6 5 4 3 3 4 +cθ bθ + aθ + - b - aθ - bθ - a a + b + aθ + bθ (3) 48 5 4 4 3 6 8 - cθ bθ + aθ 48 5 (iv) Shorage cos is given by SC = - c 3 + - b - aθ - bθ - a a + b + aθ + bθ 4 4 3 6 8 6 5 4 4 3 4 I()d = - c I()d + I()d 3 +η- +η- +η- +η- 3 = - c - b - 3 - a - +a - + b - 6 η η η η 3 3 +η- +η- +η- ηb- b - + ηa-a - -ηa - η η - c +η- +η- +η- - ηb - +a - + b - η η η 3 3 η () (4) 4 Page
Producion Invenory Model Wih Differen Deerioraion Raes Under Shorages ANS Linear Demand (v) SR = p (a+b)d = pa + b (5) he oal profi during a cycle, π(, ) consised of he following: π(, ) = SR - OC - HC - DC - SC (6) Subsiuing values from equaions () o (5) in equaion (6), we ge oal profi per uni. he opimal value of = * and = * (say), which maximizes profi π(,) can be obained by differeniaing i wih respec o and and equae i o zero π(, ) π(, ) i.e. =, = (7) provided i saisfies he condiion π(, ) π(, ) >. π(, ) d π(, ) IV. NUMERICAL EXAMPLE Considering A= Rs., a = 5, b=.5, η=, c=rs. 5, p= Rs. 4, θ=.5, x = Rs. 5, y=.5, c = Rs. 8, μ =.3, in appropriae unis. he opimal value of * =.897, * =.49 and Profi*= Rs. 9596.39. he second order condiions given in equaion (8) are also saisfied. he graphical represenaion of he concaviy of he profi funcion is also given. (8) and Profi and Profi Figure Figure V. SENSIIVIY ANALYSIS On he basis of he daa given in example above we have sudied he sensiiviy analysis by changing he following parameers one a a ime and keeping he res fixed. 5 Page
Producion Invenory Model Wih Differen Deerioraion Raes Under Shorages ANS Linear Demand able Sensiiviy Analysis Parameer % Profi +%.65.4496 3557.874 a +%.765.4694 576.68 -%.349.583 766.87 -%.39.5494 5638.6883 +%.855.4889 9594.68 θ +%.875.495 9595.463 -%.98.4936 9597.3473 -%.94.495 9598.47 +%.583.473 9576.836 x +%.73.485 9586.74 -%.386.563 967.449 -%.333.59 969.855 +%.365.5383 9557.49 A +%.334.557 9576.3934 -%.75.467 967.9 -%.598.447 9639. From he able we observe ha as parameer a increases/ decreases average oal profi and opimum order quaniy also increases/ decreases. Also, we observe ha wih increase and decrease in he value of θ and x, here is corresponding decrease/ increase in oal profi and opimum order quaniy. From he able we observe ha as parameer A increases/ decreases average oal profi decreases/ increases and opimum order quaniy increases/ decreases. VI. CONCLUSION In his paper, we have developed a producion invenory model for deerioraing iems wih linear demand wih differen deerioraion raes and shorages. Sensiiviy wih respec o parameers have been carried ou. he resuls show ha wih he increase/ decrease in he parameer values here is corresponding increase/ decrease in he value of profi. REFERENCES []. Cover, R.P. and Philip, G.C. (973): An EOQ model for iems wih Weibull disribuion deerioraion; American Insiue of Indusrial Engineering ransacions, Vol. 5, pp. 33-38. []. Ghare, P.N. and Schrader, G.F. (963): A model for exponenially decaying invenories; J. Indus. Engg., Vol. 5, pp. 38-43. [3]. Giri, B.C., Pal, S. Goswami, A. and Chaudhary, K.S. (996): An invenory model for deerioraing iems wih sock dependen demand rae; Euro. J. Ope. Res., Vol. 95, pp. 64-6. [4]. Goyal, S.K. and Giri, B. (): Recen rends in modeling of deerioraing invenory; Euro. J. Oper. Res., Vol. 34, pp. -6. [5]. Gupa, R. and Vra, P. (986): Invenory model for sock dependen consumpion rae, Opsearch, pp. 9-4. [6]. Mandal, B.N. and Phujdar, S. (989): A noe on invenory model wih sock dependen consumpion rae; Opsearch, Vol. 6, pp. 43-46. [7]. Pael, R. and Pael, S. (3): Deerioraing iems producion invenory model wih demand dependen producion rae and varying holding cos; Indian J. Mahemaics Research, Vol., No., pp. 6-7. [8]. Pael, R. and and Sheikh, S.R. (5): Invenory Model wih Differen Deerioraion Raes under Linear Demand and ime Varying Holding Cos; Inernaional Journal of Mahemaics and Saisics Invenion, Vol. 3, No. 6, pp. 5-3. [9]. Raafa, F. (99): Survey of lieraure on coninuous deerioraing invenory model, J. of O.R. Soc., Vol. 4, pp. 7-37. []. Roy,. and Chaudhary, K.S. (9): A producion invenory model under sock dependen demand, Weibull disribuion deerioraion and shorages; Inl. rans. In Oper. Res., Vol. 6, pp. 35-346. []. Ruxian, L., Hongjie, L. and Mawhinney, J.R. (): A review on deerioraing invenory sudy; J. Service Sci. and Managemen; Vol. 3, pp. 7-9. 6 Page
Producion Invenory Model Wih Differen Deerioraion Raes Under Shorages ANS Linear Demand []. Shah, Y.K. and Jaiswal, M.C. (977): An order level invenory model for a sysem wih consan rae of deerioraion; Opsearch; Vol. 4, pp. 74-84. [3]. Sharma, V. and Chaudhary, R.R. (3): An invenory model for deerioraing iems wih Weibull deerioraion wih ime dependen demand and shorages; Research Journal of Managemen Sciences, Vol., pp. 8-3. [4]. ripahy, C.K. and Mishra, U. (): An EOQ model wih ime dependen Weibull deerioraion and ramp ype demand; Inernaional J. of Indusrial Engineering and Compuaions, Vol., pp. 37-38. 7 Page