AProductionInventoryModelforDeterioratingProductswithMultipleMarketandSellingPricedependentdemand

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1 Global Joural o Sciece Frotier Research: F Matheatics ad Decisio Scieces Volue Issue Versio. Year Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic. (USA Olie ISSN: 49-4 & Prit ISSN: A Productio Ivetory Model or Deterioratig Products with Multiple Maret ad Sellig Price depedet dead By Piy Saxea, S. R Sigh & Isha Sagal Krisha Istitute o Egieerig ad Techology Abstract- This paper presets a ivetory odel or deterioratig ites that have a sigle auacturer but ultiple aret deads or a iite plaig horizo. For the aret, dieret sellig seasos are cosidered. It is a productio ivetory odel, which has a dead rate depedet o the sellig price. Here, we have preseted a solutio-search procedure to id the optial repleishet policy or raw aterial ad optial productio tie. The odel is illustrated usig a uerical exaple. Further; sesitivity aalysis is also perored to chec the stability o the odel. Keywords: productio, deterioratio, ultiple aret, price depedet dead. GJSFR-F Classiicatio: MSC : 9B AProductioIvetoryModelorDeterioratigProductswithMultipleMaretadSelligPricedepedetdead Strictly as per the copliace ad regulatios o :. Piy Saxea, S. R Sigh & Isha Sagal. This is a research/review paper, distributed uder the ters o the Creative Coos Attributio-Nocoercial. Uported Licese perittig all o coercial use, distributio, ad reproductio i ay ediu, provided the origial wor is properly cited.

2 A Productio Ivetory Model or Deterioratig Products with Multiple Maret ad Sellig Price depedet dead Piy Saxea α, S. R Sigh σ & Isha Sagal ρ Abstract- This paper presets a ivetory odel or deterioratig ites that have a sigle auacturer but ultiple aret deads or a iite plaig horizo. For the aret, dieret sellig seasos are cosidered. It is a productio ivetory odel, which has a dead rate depedet o the sellig price. Here, we have preseted a solutio-search procedure to id the optial repleishet policy or raw aterial ad optial productio tie. The odel is illustrated usig a uerical exaple. Further; sesitivity aalysis is also perored to chec the stability o the odel. Keywords: productio, deterioratio, ultiple aret, price depedet dead. I. INTRODUCTION To cosider the eect o deterioratio i the developet o ivetory odels is a essetial eed o ivetory odelig. Igorig the eect o deterioratio isleads the results. Ghare ad Schrader (9 were the irst to itroduce the deterioratio i ivetory odellig. They developed this odel with costat rate o dead ad deterioratio. Mishra (975 preseted productio lot-size odel or a syste havig deterioratig ivetory. Kag ad Ki (98 cae orward with a study o the price ad productio level o the deterioratig ivetory syste. Wee (99 deied the deterioratio o products as decay, daage, spoilage, evaporatio, obsolescece, pilerage, loss o utility or loss o argial value o a coodity that results i decreasig useuless. The deterioratio occurs i ost o the physical products such as edicies, ood products, dairy products ad volatile products, etc. Goyal ad Giri ( proposed a ivetory odel or recet treds i deterioratig ivetory odelig. I today s global aret, there are so ay opportuities or a vedor or auacturer. It is ot ecessary that it will deal with a sigle aret. It ca deal i ay arets or a sigle productio ru. Khoua ( preseted the eect o large order quatities o expected proit i the sigle-period odel. This odel is developed with ultiple aret dead. I this odel, he gave the exaple o a garet idustry which sells its product i dieret arets. It is proitable or the auacturer but very coplicated also. This is because i dieret arets the dead rate will also be dieret. A Ecooic Order Quatity (EOQ odel was Global Joural o Sciece Frotier Research ( F Volue XVI Issue VI V ersio I Year 59 Author α: Applied Scieces Departet, Krisha Istitute o Egieerig ad Techology, Ghaziabad, Idia. e-ail: piysaxea@gail.co Author σ: Matheatics Departet, C.C.S Uiversity, Meerut, Idia. e-ail: shivrapudir@gail.co Author ρ: Matheatics Departet, Baasthali Uiversity, Raastha, Idia. e-ail: isha.sagal@gail.co Global Jourals Ic. (US

3 A Productio Ivetory Model or Deterioratig Products with Multiple Maret ad Sellig Price depedet dead Global Joural o Sciece Frotier Research ( F Volue XVI Issue VI V ersio I Year preseted by Covert ad Philip (97 or ites that have tie-depedet dead ad deterioratio as per Wei-bull distributio. Giri et al. (99 put orward a ivetory odel or deterioratig ites that have a dead rate depedet o the stoc. Yag ad Wee ( preseted a productio ivetory odel with ultiple-buyer or a deterioratig ite. Wu et al. ( proposed, or o-istataeous deterioratig ites that have stoc-depedet dead ad partial bacloggig, a optial-repleishet policy. I this paper, we propose ad develop a productio ivetory odel or deterioratig ites that have ultiple arets ad a dead rate depedet o the sellig price. This odel is illustrated usig a uerical exaple ad sesitivity aalysis is also perored to show the odel s stability. II. ASSUMPTIONS AND NOTATIONS a Assuptios. A sigle product is ro a sigle auacturer.. The odel is or ultiple aret dead.. Dead rate is a uctio o the sellig price. 4. Productio rate is costat ad is greater tha the dead o all the arets. 5. The products beig cosidered are deterioratig i ature.. The rate o deterioratio is assued to be a liear uctio o tie. 7. No, shortages are allowed. 8. Plaig horizo is cosidered iite. 9. No replaceet or repair o deteriorated uits is doe durig a give cycle.. It is assued that the auacturer arrages the ix quatity o raw aterial at a ix iterval o tie. The quatity o iished products produced by the auacturer is eough to eet the occurrig dead ad deterioratio i all the arets. b Notatios A : Productio rate. θ : Deterioratio paraeter. α : Dead paraeter i th tie iterval. : Dead paraeter. p : sellig price per uit tie. T : productio period. K : setup cost h : Holdig cost per uit or iished products. I i (t: Ivetory level at ay tie t i i th iterval where (i=,,. d : Deterioratio cost per uit or iished products. d r : Deterioratio cost per uit or raw aterial ivetory. h r : Holdig cost per uit or raw aterial. µ : Orderig cost per order. q r : Delivery lot size o raw aterial. r : Nuber o deliveries o raw aterial i tie T. III. MODEL DESCRIPTION AND ANALYSIS The odel is developed with a piecewise costat uctio i dead. It is assued that the optial productio ru tie T lies i the iterval [T -, T ]. Let us Global Jourals Ic. (US

4 A Productio Ivetory Model or Deterioratig Products with Multiple Maret ad Sellig Price depedet dead divide the iterval [T -, T ] ito two parts. Here I - (t represets the ivetory level durig t [T -, T] ad I (t represets ivetory durig t [T, T ]. Dead d - d d d d - d d d - The ivetory tie behavior o the iished products at ay tie t is give by the ollowig dieretial equatios: With boudary coditios: Mauacture' s tie v/s dead Fig. : di( t α I ( t = A ; T t T dt ( p di ( t α I ( t = A ; T dt p t T ( di ( t α I ( t = ; T t T dt p ( di ( t α I ( t = ; T dt p t T (4 I (=, I (T - =I - (T -,I - (T - =I - (T -, I (T =, I - (T - =I (T (5 The solutio o these equatios is give by (see appedix A: I t A α t t e ( α α T T e θ i i θ ( = ( ( ( i i p i= p T t T or =,, (- ( Tie Global Joural o Sciece Frotier Research ( F Volue XVI Issue VI V ersio I Year I t A α t t e ( α α T T e θ i i θ ( = ( ( ( i i p i= p T t T (7 α α ( α α I t t t e T T e T T e θ θ i i θ ( = ( ( ( i i p p i= p Global Jourals Ic. (US

5 A Productio Ivetory Model or Deterioratig Products with Multiple Maret ad Sellig Price depedet dead T t T (8 α α ( α α I t t t e T T e T T e θ θ i i θ ( = ( ( ( i i p p i= p T t T or =(, (.. (9 Global Joural o Sciece Frotier Research ( F Volue XVI Issue VI V ersio I Year Now the total ivetory durig [T -, T ] will be calculated as ollows: T I = I( t dt T The total ivetory durig [T -, T] will be: α T T θ 4 4 I = ( A ( ( T T p ( αi αi θ ( θ Ti Ti (( T T ( T T i= p T I = I ( t dt T The total ivetory durig [T, T ] will be: α T T θ 4 4 I = ( A ( ( T T p ( αi αi θ ( θ Ti Ti (( T T ( T T i= p T I = I ( t dt T α T T θ 4 4 I = [ ( ( T T p α θ θ p ( T T (( T T ( T T ( α α θ θ ( (( ( ] ( ( i i Ti Ti T T T T i p ( = Siilarly, the total ivetory durig [T -, T ] or all =( : α T T θ 4 4 I = [ ( ( T T p α θ θ ( T T (( T T ( T T p Global Jourals Ic. (US

6 A Productio Ivetory Model or Deterioratig Products with Multiple Maret ad Sellig Price depedet dead ( α α θ θ ( (( ( ] i i Ti Ti T T T T i p ( = The holdig cost o the syste or iished products is calculated as ollows: = = = (4 HC. h ( I I I I The deterioratio cost or the syste o iished products is give by: α D. C d ( AT ( T T (5 = p = Set up cost or the iished product is coputed as ollows: SC. = K ( Now, the total cost or the iished product ca be calculated as ollows: TC. = HC. DC. SC. (7 a Mauacturer s raw aterial ivetory odel I this odel, the ivetory is repleished at t=. Durig [, T/ r ], the ivetory depletes as a result o the cobied eect o dead ad deterioratio. At t=t/ r, the ivetory level becoes zero ad is agai repleished. The dieretial equatio that govers the trasitio o the syste is represeted as ollows: With boudary coditio I r (T/ r =. The solutio o equatio (8 will be: We ow that I r ( = q dir ( t θrtir( t = r ; t T / r dt t T θ r T θ r r( r{( ( ( } ; / r r r (8 I t = t t e t T (9 T θ T q= ( r r{( ( } r r Now, the holdig cost o the syste or carryig the raw aterial will be: T/ r H. C = h I ( t dt r r r r θ θ H. C = h [ { ( ( }] ( T r T 4 r T 4 r r r r r 8 r 4 r Global Joural o Sciece Frotier Research ( F Volue XVI Issue VI V ersio I Year The cost o deteriorated uits o raw aterial: DC. r = dr( q r rt ( Now, the total cost o raw aterial ivetory ca be calculated as ollows: TC. = HC. DC. OC. r r r r ( Global Jourals Ic. (US

7 A Productio Ivetory Model or Deterioratig Products with Multiple Maret ad Sellig Price depedet dead Ivetory I r (t T/ r T/ r Tie Global Joural o Sciece Frotier Research ( F Volue XVI Issue VI V ersio I Year 4 Fig. : Raw aterial ivetory syste The, the itegrated total cost or whole the syste will be as ollows: TC.. = TC. TC. (4 Ater puttig the values i equatio (4, we observe that T.C. or whole the syste is a uctio o r ad T, where r is a discrete variable ad T is a cotiuous variable. dead Ist aret 5 5 Tie i wees Fig. : Tie v/s dead i each the aret IV. SOLUTION PROCEDURE Step : Here r is a discrete variable. TC Step : Solve the equatio = or all possible values o r startig ro r =. T Step : Put all these values o T ad r i equatio (4 ad id the value o T.C. Step 4: Repeat the step - util the optial value o T.C. is oud. V. NUMERICAL EXAMPLE Here we have discussed oly the two aret situatio. The dead rate is tae as a uctio o sellig price. The value o dead paraeter α or aret oe ad aret two is 5 uits ad uits respectively. The sellig rate is Rs 8 per uit. The value or the deterioratio paraeter or iished products ad raw aterial are. ad.5 respectively. The seaso or sellig ites i aret oe is cosidered ro -5 wees ad or secod aret it is -5 wees. The setup cost ad orderig cost or productio ad orderig the raw aterial is give by Rs 75 ad Rs 5 respectively. The productio cost ad purchasig cost o raw aterial is Rs 5 ad Rs per uit respectively. The holdig cost h ad h r are.5 ad. per uit per tie. The productio rate is 5 uits/ wee. Here =, T =5, T =7, T =5 By applyig the above etioed solutio procedure, we id the ollowig values: T*= wees, r=5.8, T.C. = Rs 85. IId aret r Global Jourals Ic. (US

8 A Productio Ivetory Model or Deterioratig Products with Multiple Maret ad Sellig Price depedet dead Mauacturer s dead i wees d d d Tie i wees Fig. 4: Tie v/s dead i each the aret VI. SENSITIVITY ANALYSIS For the give uerical illustratio, ivestigatios have bee perored to study the ipact o chages o dieret paraeters o the uber o deliveries o raw aterial, productio period alog with the total cost o the syste. Table : Sesitivity aalysis or productio rate A % variatio i A A r T T.C. -% % % % % % % % % Fig. 5: Variatio o total cost T.C. with respect to productio rate A Table : Sesitivity aalysis or deterioratio paraeter θ T.C. % variatio i θ θ r T T.C. -% % % % % A Global Joural o Sciece Frotier Research ( F Volue XVI Issue VI V ersio I Year 5 Global Jourals Ic. (US

9 A Productio Ivetory Model or Deterioratig Products with Multiple Maret ad Sellig Price depedet dead 5% % % % Table : Sesitivity aalysis or holdig cost o iished products h Global Joural o Sciece Frotier Research ( F Volue XVI Issue VI V ersio I Year % variatio i h h r T T.C. -% % % % % % % % % Table 4: Sesitivity aalysis or holdig cost o raw aterial h r % variatio i h r h r r T T.C. -% % % % % % % % % T.C. θ Fig. : Variatio i total cost T.C. w.r.t deterioratio paraeter θ Global Jourals Ic. (US

10 A Productio Ivetory Model or Deterioratig Products with Multiple Maret ad Sellig Price depedet dead Fig. 7: Variatio i total cost. T.C with respect to holdig cost h T.C. h Fig. 8: Variatio i total cost T.C. with respect to holdig cost hr VII. Cocludig Rears A sesitivity aalysis is perored with respect to dieret syste paraeters to chec the stability o the odel.. Fro table, it is observed that as the value o the productio paraeter A icreases, it also results i icrease i the total cost o the syste.. Table shows the variatio i deterioratio paraeter θ. It is show i this table that a icreet i deterioratio paraeter θ results also i a icreet i T.C. o the syste.. I table ad table 4, the variatio i holdig cost h ad h r are show. It is observed ro these tables that with the icreet i holdig paraeters, the T.C. o the syste icreases i both the cases. VIII. Coclusio T.A.C. I this chapter we have preseted a ivetory odel or deterioratig products. I this odel, the auacturer produces the ites at a sigle locatio ad sells it i dieret arets havig dieret deads ad sellig seasos. With the help o a atheatical odel, we calculate the dieret associated costs ad provide a ethod to reduce the total cost o the syste. This is doe by derivig the optial productio tie ad repleishes tie or the raw aterial. Further; perorig a sesitivity aalysis helps to chec the stability o the odel. With the sesitivity aalysis, the odel is oud to be quite stable. For urther research, tie value o oey ad dieret perissible coditios ca be applied or the extesio o this odel. The solutio o equatio ( is give by: Appedix A: α θ I( t ( A ( t t e ce p i =, ro I (=,We get c = = or =,..(- hr Global Joural o Sciece Frotier Research ( F Volue XVI Issue VI V ersio I Year 7 Global Jourals Ic. (US

11 A Productio Ivetory Model or Deterioratig Products with Multiple Maret ad Sellig Price depedet dead Now ro the boudary coditio, we get,i (T - = I - (T - Fro this equatio we get: θ ( c c = ( T T ( α α p θ c c = ( α α ( T T p Global Joural o Sciece Frotier Research ( F Volue XVI Issue VI V ersio I Year 8 Suig up all these equatios gives us: Sice c is zero so: The the solutio o equatio ( will be: θ c c = ( α α ( T T p θ c c = ( α α ( T T 4 4 p θ c c = ( α α ( T T p θ c c ( ( T T = αi αi i i i p = θ c ( ( T T = αi αi i i i p = α θ θ ( = ( ( ( i i ( i i α α p i= p I t A t t e T T e T t T or =,, (- Let =, the solutio o equatio ( will be: α θ θ ( = ( ( ( αi αi ( i i p i= p I t A t t e T T e T t T α I t t t e T T e p p t θ α θ θ ( = ( ( i= θ i αi Ti Ti e ( α ( p or =, T t T Global Jourals Ic. (US

12 A Productio Ivetory Model or Deterioratig Products with Multiple Maret ad Sellig Price depedet dead t α θ α θ θ ( = ( ( I t t t e T T e p p i= θ i αi Ti Ti e ( α ( p T t T Reereces Rééreces Reerecias. Goyal, S.K., Giri, B.C., Recet treds i odelig o deterioratig ivetory, Europea Joural o Operatioal Research,4,,.. Ghare, P.M., Schrader, S.F., A odel or expoetially decayig ivetory. Joural o Idustrial Egieerig, 4, 8 4, 9.. Wee, H.M., Ecooic productio lot size odel or deterioratig ites with partial bac-orderig. Coputers ad Idustrial Egieerig, 4, , Mishra, R.B., Optiu productio lot-size odel or a syste with deterioratig ivetory. Iteratioal Joural o Productio Research,, , Kag, S., Ki, I., A study o the price ad productio level o the deterioratig ivetory syste. Iteratioal Joural o Productio Research,, , 98.. Yag, P.C., Wee, H.M., A sigle-vedor ad ultiple-buyers productio ivetory policy or a deterioratig ite. Europea Joural o Operatioal Research, 4, 57 58,. 7. Khoua, M., The eect o large order quatities o expected proit i the sigleperiod odel. Iteratioal Joural o Productio Ecooics, 7, 7 5,. 8. Covert, R.P., Philip, G.C., A EOQ odel or ites with weibull distributio deterioratio. AIIE Trasactios, 5,, Giri, B.C., Pal, S., Goswai, A., Chaudhuri, K.S., A ivetory odel or deterioratig ites with stoc-depedet dead rate. Europea Joural o Operatioal Research, 95, 4, 99.. Wu, K.S., Ouyag, L.Y., Yag, C.T., A optial repleishet policy or o istataeous deterioratig ites with stoc-depedet dead ad partial bacloggig. Iteratioal Joural o Productio Ecooics,, 9 84,. Global Joural o Sciece Frotier Research ( F Volue XVI Issue VI V ersio I Year 9 Global Jourals Ic. (US

13 A Productio Ivetory Model or Deterioratig Products with Multiple Maret ad Sellig Price depedet dead Global Joural o Sciece Frotier Research ( F Volue XVI Issue VI V ersio I Year 7 This page is itetioally let bla Global Jourals Ic. (US