International Journal of Mathematical Archive-6(11), 2015, Available online through ISSN

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1 International Journal of Mathematical Archive-6(), 05, 0-07 Availale online through ISSN EOQ MODEL WIH SHORAGES IN BEGINNING AND PERIODIC DEMAND FOR DEERIORAING IEMS UAM KUMAR KHEDLEKAR* Department of Mathematics an Statistics, Dr. Harisingh Gour Vishwaviyalaya Sagar, Mahya Praesh, Inia, (A Central University) (Receive On: 8-0-5; Revise & Accepte On: 8--5) ABSRAC Many proucts exists in nature which follows repetitions of eman pattern. For example, eman of water cooler is high in summer an low in winter an again increases as temperature increases an hence the eman pattern is perioic. So, we nee to evelop a new inventory moel for such kin of proucts. In this paper an economic orer quantity moel has een presente for eteriorating item with perioic eman. Also one can start usiness with shortage like avance ooking of proucts which coul e fulfille after time uration. We have incorporate the shortage at the eginning of the sale season. A new moel is evelope an optimal shortage uration is otaine along-with some other results. Simulation stuy is performe along with managerial insights. Propose moel is foun useful for proucts having perioic eman pattern an earing shortage at the starting. Keywors: Inventory, Perioic Deman, Deterioration, Shortage, Replenishment time.. INRODUCION In past, Harris (95) an many researchers have een suggeste inventory moels with variety of eman functions. here are many proucts that follow perioic eman pattern an nee to evelop new inventory moel. Also a usiness coul e starte with shortage like avance ooking of LPG gas, electricity supply an pre pulic offer of equity share efore proper functioning of a company. We incorporate two features: one is perioic eman an other is start of usiness with shortage in the propose moel. Few items in the market are of high nee for people like sugar, wheat, oil whose shortage reak the customer s faith an arrival pattern. his motivates retailers to orer for excess units of item for inventory in spite of eing eteriorate. Moreover, eterioration is manageale for many items y virtue of moern avance storage technologies. Inventory moel presents a real life prolem (situation) which helps to run the usiness smoothly. Our aim is to solve the prolem of the usiness which start with shortage an in which the eman of the proucts follow the perioic eman. Burwell et al. (997) solve the prolem arising in usiness y proviing freight iscounts an presente an economic lot size moel with price-epenent eman. Shin (997) etermine an optimal policy for retail price an lot size uner ay-term supplier creit policies ase on constant eman where after maturing the prouct in market, it follows linear eman. Matsuyama (00) presente a general EOQ moel consiering holing costs, unit purchase costs, an setup costs that are time-epenent an continuous general eman functions. he prolem has een solve y ynamic programming so as to fin orering point, orering quantity, an incurre costs. A research overview y Emaghary an Keskinocak (003) is for etermining the ynamic pricing an orer level. eng an Chang (005) presente an economic prouction quantity (EPQ) moel for eteriorating items when the eman rate epens not only on the on-isplay stock, ut also on the selling price per unit consiering market eman. he manipulation in selling price is the est policy for the organization as well as for the customers. Wen an Chen (005) suggeste a ynamic pricing policy for selling a given stock of ientical perishale proucts over a finite time horizon on the internet. he sale ens either when the entire stock is sol out, or when the ealine is over. Here, the ojective of the seller is to fin a ynamic pricing policy that maximizes the total expecte revenues. Shukla an Khelekar (009) introuce the concept of three warehouses in which one oriente an two rente warehouses to store the eteriorating items as per requirements. Corresponing Author: Uttam Kumar Khelekar* International Journal of Mathematical Archive- 6(), Nov. 05 0

2 Uttam Kumar Khelekar* / EOQ Moel with Shortages in Beginning an Perioic Deman for / IJMA- 6(), Nov.-05. he EOQ moel esigne y Hou an Lin (006) reflects how a eman pattern which is price, time, an stock epenent affects the iscount in cash. hey iscusse an EOQ inventory moel which takes into account the inflation an time value of money of the stock-epenent selling price. Existence an uniqueness of the optimal solution has not een shown in this article. Lai et al. (006) algeraically approache the optimal value of cost function rather than the traitional calculus metho an moifie the EPQ moel earlier presente ue to Chang (004) in which he consiere variale lea time with shortage. Chanel an Khelekar (03) also use multiple warehouses on location ase an orere quantity as centralise asis an minimize the incurre cost. Recent contriution in EPQ moels is a source of esteem importance like Biril et al. (007), Hou (007), Khelekar an Shukla (05), Bhaskaran et al. (00), Jogelekar et al. (008), Roy (008), Kumar (0a,, c) an You (005). Motivation is erive ue to Wu (00) an Shukla et al. (00, 04), Khelekar et al. (03) for consiering the shortages in eginning of a usiness an henceforth evelope the propose moel. Section consists of the assumption an notations of the moel while a section 3 eal with the mathematical formulation of moel, section 4 is for numerical example an simulation.. ASSUMPIONS AND NOAIONS Suppose eman of a prouct is a sin(. In starting suppose shortage accumulate till time t so that on han shortage is I ( t ). he orer receives to the company y venor at t an so the shortage ens an inventory reaches up to level I (t ). his inventory level is sufficient to fulfil the eman till time. Our aim is to fin the optimal time t to optimize the total inventory cost. Inventory eplete ue to eman an eterioration as shown in fig. he followings notations are use to evelop the propose moel. D( eman of prouct is D ( = asin( where a an > are positive real values. θ rate of eterioration of prouct, θ <. c holing cost unit per unit time. c shortage cost unit per unit time. c 3 eterioration cost. cycle time. * t optimal time for accumulate shortages. I ( on han shortage of the prouct (I ( > 0). I ( on han inventory of the prouct (I ( > 0). C(t ) optimal inventory cost. D eteriorate units. S shortage units in the system. S C shortage cost. H C holing cost. eterioration cost. D C 3. MAHEMAICAL MODEL Suppose on han shortage is y I( an this accumulates until t. Management has place the orer which fulfille at time t an thus on han inventory is I ( t ). After time t, the inventory eplete ue to eman an eterioration then it reuces to zero at time (see Fig. ). Fig. : Inventory Cycle 05, IJMA. All Rights Reserve 0

3 Uttam Kumar Khelekar* / EOQ Moel with Shortages in Beginning an Perioic Deman for / IJMA- 6(), Nov.-05. t I( = asin(, where 0 t t, I(0) = 0 () I( +θ I( = asin(, where t t () t Bounary conitions for aove two ifferential equations are I (0) = 0, I ( ) = 0, solving equation () we get a I( = ( cos( ) (3) Solving equation () we get ( a θ a I = e θ sin( ) + cos( ) e θ sin( + cos( a + a + Deteriorate units in time ( t, ] is D D = I ( t ) a t ( ) ( ) sin( = ac sin( t t Holing cost HC over time ( t, ] will e θ a θ e e e ( asin( ) + cos( )) a θ + HC = h a ( ) ( sin( ) sin( t ) cos( ) cos( ) t θ + θ a + (4) (5) (6) a ( t ac = ( cos( t ) ) (7) Shortages I t ) = ( cos( )) S C Numer of units incluing shortage in usiness scheule is Q = I t ) + I ( ) ( t (8) otal average inventory cost will e H C + SC + DC C( t ) = (9) θ ac a θ e e = ( cos( ) + h e ( asin( ) + cos( )) a + θ h + ac sin( t a t ( ) ( sin( ) sin( ) cos( ) + cos( ) ) t t θ θ a + t (0) o optimize the total cost function ( C t ) ) first erivative equating to zero t ( θ ahe C = a + ( a + ) θ ac3e ac3 ( asin( ) + cos( )) ( θ ncos( sin( ) a + a + ac ac + ac3sin( + cos( ah ( asin( ) + cos( )) e + ( cos( t ) + θ sin( t )) ( () 05, IJMA. All Rights Reserve 03

4 Uttam Kumar Khelekar* / EOQ Moel with Shortages in Beginning an Perioic Deman for / IJMA- 6(), Nov.-05. On equating C( t ) = 0, we get equation for optimality value of say which is t C( t ) > 0 t * t hus Average total cost is optimum at t =. * t = t Conition for optimality is () Variation in Parameter Ө c c A ale-: Sensitivity of ifferent parameters a c c C 3 ө t C Q() I (t ) Holing Cost Shortage Cost D 4. NUMERICAL EXAMPLE AND SIMULAION Let us assume that moel parameters are a = 0 units, = 0., c = $ per unit per month, c = $ per unit per month, C 3 = $ per unit per month, θ = 0.0, = 4 ays an eman of the prouct is D( = asin(. Uner the given parameter values an y equation (6) to (0) we get output parameters t = 4.3ays, average total inventory cost C = $.43.93, Q = 0 units, average holing cost H C = $ Now, in this section, we stuy how the input parameters change significantly to the output parameters. We change the value of one input parameter, keeping other parameters constant. he output parameter is valuate for ecision making. he ata use for this purpose is in section 4. otal inventory cost increases as the time cycle length increases (see fig ) an same followe y economic orer quantity (tale ). Increments in shortage cost provie an increment in EOQ (see fig 3). he eterioration rate (θ) uplifts the level of eterioration cost an total cost oth. So they are irectly proportional an have a linear tren (see fig 4 an 5). 05, IJMA. All Rights Reserve 04

5 Uttam Kumar Khelekar* / EOQ Moel with Shortages in Beginning an Perioic Deman for / IJMA- 6(), Nov.-05. Management nees to e aware aout the eterioration cost an holing cost oth an tries to keep it as low as possile. High initial eman (parameter a) increases the EOL an EOQ oth (tale ), ut optimal time interval of these two remain unchange. From tale, it is oserve that the optimal time is highly sensitive for eterioration an holing cost C Figure. Effect of time cycle on total cost Q c Figure 3. Effect of shortage cost on total quantity DC C3 Figure 4. Effect of eterioration ratte on eteriorete cost 05, IJMA. All Rights Reserve 05

6 Uttam Kumar Khelekar* / EOQ Moel with Shortages in Beginning an Perioic Deman for / IJMA- 6(), Nov C Figure 5. Effect of eterioration rate on total cost ө 5. CONCLUSIONS A mathematical inventory moel is suggeste in the content for a usiness that starts with shortages an free to place the orer accoring to the eman an customer s response. Inventory managers shoul keep the eterioration rate as low as possile ecause it increases the wastage of quantity as well as the total cost. However one can negotiate the shortage cost to customers, which may keep lower to the incurre cost. Suggeste moel is sensitive for eterioration an holing cost oth as compare to shortage cost for a prouct having perioic eman. his moel can e further extene for variale eterioration, ramp type eman an for finite rate of replenishment. his moel may also e formulate in the fuzzy environment. REFERENCES Bhaskaran S, Ramachanran K an Semple J, A ynamic inventory moel with the right of refusal. Management Science, 56 () (00) Biril SI, Frenk JBG & Bayinir ZP, A eterministic inventory/prouction moel with general inventory cost rate function an piecewise linear concave prouction costs. European Journal of Operational Research, 79() (007) Burwell, Dave DS, Fitzpatrick K E & Roy M R, Economic lot size moel for price epenent eman uner quantity an fright iscount. International Journal of Prouction Economics, 48() (997) Chanel RPS & Khelekar UK, A new inventory moel with multiple warehouses. International Research Journal of Pure Algera Research, 3(5) (03), Chang HC, A note on the EPQ moel with shortages an variale lea time. International Journal of Information Management Sciences, 5 (004) Emaghary & Keskinocak P, Dynamic pricing in the presence of inventory consierations: research overview- Current practice an future irection. Management Science, 49(0) (003) Harris FW, What quantity to make at once the lirary of factory management. Operations an Costa (A. W. Shaw Company, Chicago), 5 (95), Hou KL & Lin LC, An EOQ moel for eteriorating items with price an stock epenent selling price uner inflation an time value of money. International Journal of System Science, 37(5) (006) Hou KL, An EPQ moel with setup cost an process quality as functions of capital expeniture. Applie Mathematical Moeling, 3() (007) Joglekar P, Lee P & Farahani AM, Continuously increasing price in an inventory cycle: An optimal strategy for E-tailors. Journal of Applie Mathematics an Decision Science, 8 (008) -4. Khelekar UK & Shukla D, Dynamic pricing moel with logarithmic eman. Opsearch, 50() (03) -3. Khelekar UK & Shukla D, Simulation of economic prouction quantity moel for eteriorating item, American Journal of Moeling an Optimization, (3) (03), Khelekar UK, Shukla D & Chanel RPS, Computational stuy for isrupte prouction system with time epenent eman. Journal of Scientific an Inustrial Research, 73(5) (04) Khelekar UK, Shukla D & Chanel RPS, logarithmic inventory moel with shortage for eteriorating items. Yugoslav Journal of Operations Research, 3(3) (03) Kumar R & Sharma SK, An M/M//N queuing moel with retention of renege customers an alking. American Journal of Operational Research, () (0a) , IJMA. All Rights Reserve 06

7 Uttam Kumar Khelekar* / EOQ Moel with Shortages in Beginning an Perioic Deman for / IJMA- 6(), Nov Kumar R & Sharma SK, Formulation of prouct replacement policies for perishale inventory systems using queuing theoretic approach. American Journal of operational Research, (4) (0) Kumar, R., an Sharma, S.K., Prouct replacement strategies for perishale inventory system using queuing theory, Journal of Prouction Research an Management, (3) (0c) Lai C S, Huang Y F & Hung H F, he EPQ moel with shortages an variale lea time. Journal of Applie Science, 6(4) (006) Matsuyama K, he general EOQ moel with increasing eman an costs. Journal of Operations Research, 44() (00) Roy A, An inventory moel for eteriorating items with price epenent eman an time-varying holing cost. Avance Moeling an Optimization, 0() (008) Shin S W, Determining optimal retail price an lot size uner ay-term supplier creit. Computers & Inustrial Engineering, 33(3) (997) Shukla D & Khelekar UK, An inventory moel with three warehouses. Inian Journal of Mathematics an Mathematical Sciences, 5() (009), Shukla D & Khelekar UK, Inventory moel for convertile item with eterioration. Communication in Statistics- heory an Methos, Accepte in-press (05) Availale online through, 4 Shukla D, Khelekar UK, Chanel RPS & Bhagwat S, Simulation of inventory policy for prouct with price an time epenent eman for eteriorating item. International Journal of Moeling, Simulation, an Scientific Computing, 3() (00) Shukla D, Khelekar UK, Chanel RPS, ime an price epenent eman with varying holing cost inventory moel for eteriorating items. International Journal of Operations Research an Information Systems, 4(4) (03) eng, J, an Chang C, Economic prouction quantity moel for eteriorating items with price an stock epenent eman. Computer & Operation Research, 3() (005) Wen UP & Chen YH, Dynamic pricing moel on the internet market. International. Journal of Operations Research, () (005) Wu KS, Deterministic inventory moel for items with time varying eman Weiull istriution eterioration an shortages. Yugoslav Journal of Operations Research, () (00) You SP, Inventory policy for prouct with price an time-epenent eman. Journal of the Operational Research, 56(7) (005) Source of support: Nil, Conflict of interest: None Declare [Copy right 05. his is an Open Access article istriute uner the terms of the International Journal of Mathematical Archive (IJMA), which permits unrestricte use, istriution, an reprouction in any meium, provie the original work is properly cite.] 05, IJMA. All Rights Reserve 07

LOGARITHMIC INVENTORY MODEL WITH SHORTAGE FOR DETERIORATING ITEMS

LOGARITHMIC INVENTORY MODEL WITH SHORTAGE FOR DETERIORATING ITEMS Yugoslav Journal of Operations Research 3 (03) Number 3, 43-440 DOI: 0.98/YJOR095005K LOGARIHMIC INVENORY MODEL WIH SHORAGE FOR DEERIORAING IEMS Uttam Kumar KHEDLEKAR and Diwakar SHUKLA Department of Mathematics