Iteratioal Joural of Mathematical Scieces Vol., No. 3-4, July-December 2, pp. 435-446 Serials Publicatios A FINIE HORIZON INVENORY MODEL WIH LIFE IME, POWER DEMAND PAERN AND LOS SALES Vipi Kumar & S. R. Sigh Abstract: I this paper, the authors attempted to develop a determiistic ivetory model for deterioratig item uder a very realistic ad practical demad rate which depeds upo power of time (Power demad patter). Also i this study we cosider that the item deteriorate after a fixed time period called life time ad takig icremetal holdig cost. Effect of partial backloggig also take i accout. Sesitivity aalyses of some parameter also have bee discussed.. INRODUCION May ivetory models i past were developed uder the assumptios that the holdig cost is costat for the etire ivetory cycle. But this is particularly false i the storage of deterioratig ad perishable items such as food products. he loger these food products are kept i storage, the more sophisticated the storage facilities ad services eeded ad therefore, the higher the holdig cost. he holdig cost is assumed to be varyig over time i oly few ivetory models. he effect of time, storage coditios, weather coditios etc. are those factors which affect the quatity ad quality of stock stored i the warehouse. Various kids of materials are stored i the warehouse. Each material has its ow characteristics. Some of the materials are affected by evirometal coditios, the method of storage or the time of storage. Eve later o, as researchers started takig other factors resposible for the decay of ivetory ito accout, they stuck to comparatively primitive situatios. Mostly, a liear ad time depedet rate of decay was cosidered, which assumed that with time, the rate of decay of ay commodity will go o icreasig. his is still a comparatively better assumptio, sice very ofte it is observed, that oce aythig starts to spoil, its rate of spoilage goes o icreasig by the day. Although it is simple, but still, time depedet liear rate of deterioratio is a bit closer to reality tha a costat rate. Sice, it is ujustified to assume that ay item starts deterioratig as soo as it is produced, hece, the item has bee allowed a defiite lifetime durig which there is o deterioratio ad the item is able to sustai its qualities. After that, deterioratio sets i, ad it has bee made more realistic ad practical by takig Weibull distributio fuctio. he competitive ature of the market has bee accouted for by takig partial, time depedet, backloggig ito cosideratio. he model is developed for a fiite plaig horizo, keepig i mid the short lifespa of the customer s prefereces.
436 Vipi Kumar & S. R. Sigh Cotrollig ivetories of perishable items poses a sigificat challege due to limited useful life of items. hese items if ot used before the expiry date would outdate ad there would be a additioal cost of outdatig of perished items. o maximize total reward over a fiite horizo, price promotios ca be used to clear off the sale of items havig less remaiig useful life. I a price sesitive market price promotio ca be a reasoable optio to stimulate the demad. Problem cosidered here is to determie a optimal time to aouce price promotio ad optimal orderig quatity i each period for a perishable item over a fiite horizo. It is assumed that after the fixed horizo the product has to be withdraw from the market. his is a quite realistic assumptio as due techological advacemet or to be competitive i market, maagemet may withdraw the old products ad itroduce ew oes. Perishability refers to decrease i value or usability of product over time due to the iheret characteristics of product; whereas obsolescece refers to loss i value of product due exteral factors such as, techological iovatios, ew product itroductio by competitor, etc. he literature o perishable ivetory to determie optimal orderig policies, cosidered differet scearios related to demad patters, issuig policies, review periods of ivetory, etc. Gupta ad Vrat (986) were amogst the first few researchers to deliberate the effects of stock depedet cosumptio rate o a EOQ model. I this study they established EOQ for two cases, oe for a istataeous repleishmet ad aother for a fiite rate of repleishmet. Baker ad Urba (988) aalyzed a cotiuous determiistic case of a ivetory system i which the demad rate is a polyomial fuctio of the ivetory level. he algorithm usig separable programmig was employed to fid the optimal solutio. Madal ad Phaujdar (989) wrote a ote a ivetory model with istataeous stock repleishmet ad stock depedet cosumptio rate. Datta ad Pal (99) developed a ivetory model with stock depedet demad util the stock level reached a particular poit, after which the demad became costat. Giri et al., (996) exteded Datta ad Pal by relaxig their restrictio of zero ivetories at the ed of order cycle ad icludig deterioratio effects. Balkhi ad Bekherouf (24) aalyzed a deterioratig stock depedet model for a fiite horizo. Mahapatra ad Maiti (25) set forth a study o multi objective ivetory models with stock ad quality depedet demad. Roy ad Chaudhuri (26) studied a model with stock depedet demad uder iflatio ad costat deterioratio. Chug ad Li (2)preseted a optimal ivetory repleishmet models for deterioratig item takig accout of time discoutig. Papachristos ad Skouri (23) exteded the Wee (999) model with the demad rate is a covex decreasig fuctio of the sellig price ad the backloggig rate is a time-depedet. Yag (25) developed a compariso amog various partial backloggig ivetory lot size models for deterioratig items o the basis of maximum profit. Roy A., (28) itroduced a determiistic ivetory model for deterioratig items with price depedet demad ad time varyig holdig cost.
A Fiite Horizo Ivetory Model with Life ime, Power Demad Patter ad Lost Sales 437 He also cosidered, deterioratio rate is time proportioal ad demad rate is a fuctio of sellig price. Giri et al., (996) developed a EOQ model for deterioratig items with shortages, i which both the demad rate ad the holdig cost are cotiuous fuctios of time. Shao et al., (2) determied the optimum quality target for a maufacturig process where several grades of customer specificatios may be sold. Sice rejected goods could be stored ad sold later to aother customer, variable holdig cost is cosidered i the model. I this model two type time depedet holdig cost step fuctios are cosidered, retroactive holdig cost icrease ad icremetal holdig cost icrease. Most models that cosider the deterioratig items are deteriorate from zero time. But this is ot true i realistic, sice each item deteriorate after a fixed time period called life time. here are few models are developed i which life time has bee take a importat factor. Hwag ad Hah (2) developed a optimal procuremet policy for items with a ivetory level depedet demad rate ad fixed life time. Sigh et al., (25) preseted a ivetory model with life time, radom rate of deterioratio ad partial backloggig. Balkhi ad Bekherouf (24) aalyzed a deterioratig stock depedet model for a fiite horizo. Mahapatra ad Maiti (25) set forth a study o multi objective ivetory models with stock ad quality depedet demad. Roy ad Chaudhuri (26) studied a model with stock depedet demad ad costat deterioratio. Alfares (27) developed a ivetory model wish stock level depedet demad rate ad variable holdig cost. I this paper, a ivetory model is developed for deterioratig item cosiderig a power law form of the time depedece of demad. Also i this study we cosider that the item deteriorate after a fixed time period called life time ad takig icremetal holdig cost. Effect of partial backloggig ad storage are also take i accout. A comprehesive sesitivity aalysis has also bee doe for some parameters. Cost miimizatio techique is used to get the approximate expressios for total cost ad other parameters. he model i developed ad aalyzed with the use of followig assumptio ad otatio. 2. ASSUMPION AND NOAION he followig assumptio ad otatio are made i developig the preset mathematical models of the ivetory system: I (t) is the ivetory level at ay time t, t I (t) is the first derivative with respect to time θ (t) = θ.t is variable rate of deterioratio t is the time at which shortage starts ad is the legth of repleishmet cycle t
438 Vipi Kumar & S. R. Sigh S is the iitial ivetory after fulfillig the backorders. C deote the set up cost for each repleishmet. c s ad c deote the shortage cost for backlogged item ad the uit cost of lost sales respectively ad c d be the deterioratio cost per uit item per uit time µ is the life time of items ad deterioratio of the items is cosidered oly after the life time of the items D (t) = the demad rate at ay time ad D() t = dt Shortages are allowed ad backloggig rate is D () t = dt () + α t Whe ivetory is i shortage. he backloggig parameter α is positive costat ad α <<. 3. FORMULAION AND SOLUION OF HE MODEL: Let us assume after fulfillig backorders we get a amout S (S ) as iitial ivetory. Durig the period [, µ ], the ivetory level decrease due to the market demad oly. After life time, i.e. durig the period [µ, t ] the ivetory level decreases due to market demad ad deterioratig of items ad fall to zero at time t. he period [t, ] is the period of the shortage which is partially backlogged. Also we assume that the holdig cost icrease time to time. S ime(t) µ t Lost Sale
N o w d u r i g t h e p e r i o d [, ] A Fiite Horizo Ivetory Model with Life ime, Power Demad Patter ad Lost Sales 439 he differetial equatio goverig the ivetory level I (t) at ay time t durig the cycle [, ] are di () t dt di () t dt = D() t + θ ()()() t I t = D t t µ () µ t t (2) di ()() t D t = dt () + α t t t (3) he boudary coditio are I (t) = S at t = ad I (t) = at t =. Solutio of equatio (), (2) ad (3) with the help of suitable boudary coditio are give by dt I () t = S θ 2+ 2+ ( ) ( ) 2 d θt I () t = t t + t t 2 2(2 ) d α + + Ad I () t = ( α ) ( t t ) + ( t t ) ( + ) By applyig the boudary coditio, S ca be obtaied as t µ (4) t µ (5) t t (6) 2 θ 2 + 2 + ( ) ( ) ( ) dµ d θµ S = + t + t t 2(2 + ) 2 total holdig cost C h ca be obtaied as (7) C H = µ µ µ 2 µ m = t c I()()() t dt + c...() I t dt + c I t dt + c I t dt 2 m µ µ µ m µ m µ = t r = c I()() t dt + cr I t dt r = µ r
44 Vipi Kumar & S. R. Sigh = c d + Sµ ( + ) + + ( ) + + ( t r ) θ t t t t t 6 m d θ + cr + r = 2(3 + ) + 2(3 + ) 3 + 3 3 ()() r ) r r ( r ) θ 2+ 3+ 3+ t () t r t = d θ 2+ θ 3+ θ + 3 c µ t + µ t + µ t 2(2+ ) 2+ 2 + + + θ 3+ t t µ r + µ r + t + + ( + ). (8) m 3(3 ) + cr 2 r = θ 3+ θ θ 2+ 3 + µ r r t r t + + + (2 )(3 ) 6 2(2 ) otal deteriorated cost C D durig the period [, ] is give by C D = cd t µ θ()() t I t dt t dθ + = ( ) d c t t dt µ dθ 2+ 2 + 2 + 2 = cd ( t t ) ( t ) 2 2 + (Neglectig the power of θ higher the oe) otal shortage deteriorated cost C S durig the period [, ] is give by (9) C S = c s t I () t dt = c s + 2α + + t ( α ) t d + +. () 2 2α 2+ α 2+ + t + + + ( )(2 ) 2
A Fiite Horizo Ivetory Model with Life ime, Power Demad Patter ad Lost Sales 44 otal lost sales cost C L durig the period [, ] is give by C L = c l t () () + α t D t dt = dα + + cl + t t + +. () Now the average cost of the system per uit time is give by K = [ C + C + + + ] H CD CS DH (2) he ecessary coditio for the total cost per uit time to be miimize are K = t (3) Ad K = (4) provided the followig coditio 2 2 K K >, >, ad t he equatio (3) ad (4) are gives 2 2 2 K K K > t t (5) m c µ 2 2 3 3 t {2()} + θ t 6 {()} + 2 cr t r 3 + θ t + θµ r r t 2 6 r = θcd 2 2 + ()()[ t {()} + t ] CS. + α t + cα = 2
442 Vipi Kumar & S. R. Sigh Ad θ 2+ θ 3= θ + 3 C µ t + µ t + µ + µ t 2(2+ ) 2+ 2 + + + + θ 3 + C + d t µ r t + µ r + t + + + m 3(3 ) + cr 2 θ 3+ θ θ 2+ r= 3 r + µ r t r t (2+ )(3 + ) 6 2(2 + ) + 2+ 2 + 2 + 2 + dθcs ( t µ t ) ( t µ + ) 2 + + α + 2 2 2α 2+ α 2+ + dcs + t + ( α ) t + t + + ( )(2 ) 2+ + + + + + + dα c + t t + + 2αc + + s + d ()( cs +α {2c }) + cs α αc t. ( t + ) = + 4. NUMERICAL ILLUSRAION o illustrate the model umerically the followig parameter values are cosidered d = 5 = 2 α =. uits θ =.2 uit C = Rs. 2 per order c = Rs. 3 per uit per order c r = Rs. (c + r) per uit per year c d = Rs. per year µ =.4 year c s = Rs. 2 per uit per order = year c = Rs. 4 per uit akig m =, the for the miimizatio of total average cost, optimal solutio is t =.8592 S = 44.8925 K = 245.2664
v e t o r y p e r i o d t A Fiite Horizo Ivetory Model with Life ime, Power Demad Patter ad Lost Sales 443 able O p t i m a l S o l u t i o f o r V a r i o u s V a l u e s o f D e t e r i o r a t i o R a t e ( θ) Parameter value % Chage t * S * K *.2 4.8697 44.9322 245.44.6 2.8625 44.973 245.5496.2.859 44.8925 245.298.24 + 2.846 44.8695 245.2522.28 + 4.8288 44.83879 245.29693 he followig poit are oted from above table (i) he total average cost K * icreases as the deterioratio rate (θ) icreases. (ii) Ivetory period t * decreases as deterioratio rate icreases. (iii) he iitial ivetory also decreases as deterioratio rate icreases. able 2 Optimal Solutio for Various Values of Life ime (θ) Parameter value % Chage t * S * K *.24 4.784 44.2388 247.7575.32 2.79263 44.54837 246.373.4.859 44.8925 245.298.48 + 2.888 44.24396 244.35958.56 + 4.8363 44.678 243.7784 he followig poits are oted from above table. (i) he total average cost K * decrease as life time of the items icrease. (ii) I * icrease as life time of the items icrease. (iii) he iitial ivetory S * icrease as life time of the items icrease. able 3 Optimal Solutio for Various Values of Backloggig Parameter (α) Parameter value % Chage t * S * K *.6 4.8422 44.8659 245.47.8 2.8466 44.8787 245.684..859 44.8925 245.298.2 + 2.8553 44.9245 245.23284.4 + 4.8596 44.946 245.26364
444 Vipi Kumar & S. R. Sigh he followig poits are oted from above table. (i) he total average cost K * decrease as backloggig parameter icrease. (ii) Ivetory period t * icrease as backloggig parameter icrease. (iii)he iitial ivetory S * icrease as backloggig parameter icrease. 5. CONCLUSIONS I this paper, we have attempted to develop a deterioratio ivetory model with a very realistic ad practical demad rate which depeds upo power fuctio of time. he rate of deterioratio is variable. Also i this paper we cosider that the item deteriorate after a fixed time period called life time ad takig icremetal holdig cost. Allowable shortage is cosidered. Effect of partial backloggig also take i accout. Numerical examples ad sesitivity aalysis has also bee doe for some parameter which show that the optimal productio strategy may vary with chages i system parameters. hus the authors hope that the preset paper has a wide scope i realistic situatios ad ca be applied i may real busiess situatio. REFERENCES [] Giri B. C., Goswami A., ad Chaudhuri K. S., (996), A EOQ Model for Deterioratig Items with ime Varyig Demad ad Costs, Joural of the Operatioal Research Society, 47(), 398-45. [2] Hwag H., ad Hah K. H., (2), A Optimal Procuremet Policy for Items with a Ivetory Level Depedet Demad Rate ad Fixed Life ime, Europea Joural of Operatio Research, 27(3), 537-545. [3] Shao Y. E., Fowler J. W., ad Ruger G. C., (2), Determiig the Optimal arget for a Process with Multiple Markets ad Variable Holdig Costs, Iteratioal Joural of Productio Ecoomics, 65(3), 229-242. [4] Sigh S. R., et al., (25), A Ivetory Model with Life ime, Radom Rate of Deterioratio ad Partial Backloggig, Published i the Proceedig of Fifth Iteratioal Coferece o Operatioal Research for Developmet (ICORD-V), 34-346. [5] Alfares H. K., (27), Ivetory Model with Stock Level Depedet Demad Rate ad Variable Holdig Cost, Iteratioal Joural of Productio Ecoomics, (I Press). [6] Baker R. C., ad Urba. L., (988), A Determiistic Ivetory System with a Ivetory Level Depedet Demad Rate, Joural of the Operatioal Research Society, 39(9), 823-83. [7] Balkhi Z.., ad Bekherouf L., (24), O a Ivetory Model for Deterioratig Items with Stock Depedet ad ime Varyig Demad Rates, Computers ad Operatios Research, 3, 223-24.
A Fiite Horizo Ivetory Model with Life ime, Power Demad Patter ad Lost Sales 445 [8] Che J. M., ad Che L.., (25), Pricig ad Productio Lot Size/Schedulig with Fiite Capacity for a Deterioratig Item Over a Fiite Horizo, Computers ad Operatios Research, 32, 28-289. [9] Datta. K., ad Pal A. K., (99), Determiistic Ivetory Systems for Deterioratig Items with Ivetory Level Depedet Demad Rate ad Shortages, Joural of the Operatioal Research Society, 27, 23-224. [] Dye C. Y., (27), Joit Pricig ad Orderig Policy for a Deterioratig Ivetory with Partial Backloggig, Omega, 35(2), 84-89. [] Giri B. C., Pal S., Goswami A., ad Chaudhuri K. S., (996), A Ivetory Model for Deterioratig Items with Stock Depedet Demad Rate, Europea Joural of Operatioal Research, 95, 64-6. [2] Gor R., ad Shah N., (26), A EOQ Model for Deterioratig Items with Price Depedet Demad ad Permissible Delay i Paymets Uder Iflatio, Opsearch, 43(4), 376-388. [3] Gupta R., ad Vrat P., (986), Ivetory Model for Stock Depedet Cosumptio Rate, Opsearch, 23(), 9-24. [4] Mahapatra N. K., ad Maiti M., (25), Multi Objective Ivetory Models of Multi Items with Quality ad Stock Depedet Demad ad Stochastic Deterioratio, Advaced Modelig ad Optimizatio, 7(), 69-84. [5] Madal B. N., ad Phaujdar S., (989), A Note o a Ivetory Model with Stock Depedet Cosumptio Rate, Opsearch, 26(), 43-46. [6] Roy., ad Chaudhuri K. S., (26), Determiistic Ivetory Model for Deterioratig Items with Stock Level-Depedet Demad, Shortage, Iflatio ad ime Discoutig, Noliear Pheomea i Complex Systems, 9(), 43-52. [7] Subbaih K. V., Rao K. S., ad Satyaaraya B., (24), Ivetory Models for Perishable Items Havig Demad Rate Depedet o Stock Levels, Opsearch, 4(4), 223-236. [8] Wee H. M., ad Law S.., (999), Ecoomic Productio Lot Size for Deterioratig Items akig Accout of the ime Value, Computers ad Operatios Research, 26, 545-558. [9] Wee H. M., (997), A Repleishmet Policy for Items with Price Depedet Demad ad a Varyig Rate of Deterioratio, Productio Plaig ad Cotrol, 8(5), 494-499. [2] Wee H. M., ad Law S.., (2), Repleishmet ad Pricig Policy for Deterioratig Items akig Ito Accout the ime Value of Moey, Iteratioal Joural of Productio Ecoomics, 7, 23-22. [2] Papachristos, ad Skouri, (28), A Ivetory Model with Deterioratig Item Quality Discout Pricig ad ime Depedet Partial Backloggig, Operatioal Research Letters, 27, 75-84. [22] Yag H. L., (25), A Compariso Amog Various Partial Backloggig Ivetory Lot Size Models for Deterioratig Items o the Basis of the Maximum Profit, It. J. of Productio Ecoomics, 96,9-28.
446 Vipi Kumar & S. R. Sigh [23] Chug K. J., ad Li C. N., (2), Optimal Ivetory Repleishmet Models for Deterioratig Items akig Accout of ime Discoutig, Computers ad Operators Research, 28, 678-83. [24] Roy A., (28), A Ivetory Model for Deterioratig Items with Price Depedet Demad ad ime Varyig Holdig Cost, Advaced Modelig ad Optimizatio,, 26-37. Vipi Kumar Dept. of Mathematics, BKBIE, Pilai (Raj.), Idia E-mail: vipibkbiet@yahoo.co.i S. R. Sigh Dept. of Mathematics, D.N.College Meerut, Idia E-mail: shivrajpudir@yahoo.com