DETERIORATING INVENTORY MODEL WITH LINEAR DEMAND AND VARIABLE DETERIORATION TAKING INTO ACCOUNT THE TIME-VALUE OF MONEY

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1 International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN Vol., Issue Mar -5 JPRC Pvt. Ltd., DEERIORAING INVENORY MODEL WIH LINEAR DEMAND AND VARIABLE DEERIORAION AKING INO ACCOUN HE IME-VALUE OF MONEY NEEU (Asst. Professor) PG Deptt. of Mathematics, APJ College of Fine Arts, Jalandhar. mannat_7ind@yahoo.co.in ABSRAC ARUN KUMAR OMER (Asst. Professor) PG Deptt. of Mathematics, SMDRSD College, Pathankot. tomer4@rediffmail.com Most works on inventory models do not consider deterioration, time-value of money and price-dependent demand factor. A deteriorating inventory model taking into account the time-value of money is developed for linear demand. Shortages are completely backordered and variable rate of deterioration is taken. he objective of this study is to enable the retailer to develop a policy which will ensure the largest net profit. Numerical examples are also given to support the theoretical results.. INRODUCION Ghare and Schrader (96) were the first authors to analyse deteriorating inventory models. hey developed an EOQ model by assuming a constant rate of deterioration, Covert and Philip (97) have developed an EOQ model for items whose deterioration patterns follow the Weibull distribution. Misra (979) simultaneously considered both the inflation and time-value of money for internal as well as external inflation rate and analyzed the influence of interest rate and inflation rate on replenishment strategy. Chandra and Bahner (985) extended the result in Misra to allow for shortages. Abad (988) considered

2 Neetu & Arun Kumar omer pricing and lot-size decisions with incremental quantity discounts for a nondeteriorating product. Sarker and Pan (994) assumed a finite replenishment model and studied the effects of inflation and time-value of money on order quantity when shortages are allowed. Hariga (995) extended the study to analyze the effects of inflation and time-value of money on an inventory model with time-dependent demand rate and shortages. Wee (997) developed a replenishment policy for items with a price-dependent demand and a varying rate of deterioration. eng et al. (997) investigated all possible policies with linearly increasing demand and mathematically identified the least expensive policy among them. Padmanbhan and Vrat (995) developed an inventory model in which the backlogging rate depends upon the total number of customers in the waiting line. Ouyang et al. () extended their model by incorporating the inflation and time-value of money for a finite planning horizon. Zhou () used the same backlogging rate to develop a deterministic replenishment model with multiple warehouses. Deterioration starts as soon as the items are received into inventory. In this paper, we are going to develop an deteriorating inventory model with linear demand and variable deterioration taking into account the time value of money.. ASSUMPIONS () he rate of deterioration is variable i.e. θ ( t) θ t, < θ << () Deterioration occurs as soon as the items are received into inventory. () Shortages are completely back-ordered. (4) he system operates for a prescribed period of planning horizon. (5) Demand rate is a decreasing linear function of the selling price. (6) Replenishment rate is finite.

3 Deteriorating Inventory Model With Linear Demand and Variable Deterioration aking into Account the ime-value of Money (7) here is no replacement or repair of deteriorating items under consideration. (8) he replenishment rate is instantaneous, the order quantity and replenishment cycle is same for each period.. NOAIONS () is replenishment cycle. () H is planning horizon. () N is number of replenishments during the planning horizon, NH/. (4) s is per unit selling price of the item. (5) d(s) is demand rate, d( s) a bs, where (6) (7) positive demand. I t is inventory level at any time t, t I t is inventory level at any time t, t (8) C is per unit cost of the item. (9) r is interest rate. () q is nd, rd.nth replenishment lot size (units). () I is maximum inventory level. s < b and a bs > for a () C is cost per replenishment when t () C is per unit holding cost per unit time. (4) C is per unit shortage cost per unit time. (5) is time of positive inventory.

4 Neetu & Arun Kumar omer 4 4. MODEL DEVELOPMEN At t, initial replenishment I is made. During the period, the inventory level decreases due to demand and deterioration till it becomes zero at t. During the time interval t shortages occur and are accumulated until at t before they are backordered. he inventory system at any time t can be represented by the following differential equations: di t dt di t dt + θ t I t d s ; t d s ; t..()..() and the boundary conditions are I I, I Solution of equation () can be given as t θ t t θ t I( t ) e d( s) e dt, t θt t θt I t e I d s e dt t θ I d s e dt θt I t, t e.. ()

5 5 Deteriorating Inventory Model With Linear Demand and Variable Deterioration aking into Account the ime-value of Money θt Where n n n n I d s e dt d s θ t dt n θ t d( s) n.. (4) n n ( n + ) n ( n + ) d s n n+ n n + θ Since n n < θ <<, and solution of equation () is given by θ I d s +.(5) 6 I t d s t, t..(6) he total cost in this model includes the replenishment cost, material cost, holding cost and shortage cost. he time-value of money with compounding interest rate is taken. he objective is to maximize the total profit. 4. Present-Value Sales Revenue rt r R s d s e dt + se d s dt rt sd s e + e r r r e sd s + e r r r + r sd s r r + ( )( )

6 Neetu & Arun Kumar omer 6 r R sd( s) r + r.. (7) 4. Present-Value Ordering cost Since replenishment in each cycle is done at the start of each cycle; the present value replenishment cost is C C...(8) 4. Present-Value Inventory Cost Inventory occurs during period. Present-Value Inventory cost during the period is rt H C C I t e dt θt t θt e dt e dt rt θt C d s e dt e θt n n+ n+ n ( t ) ( rt) θ C d( s ) dt n n n ( n ) n n + n n θ θt + ( ) 6 C d s t t rt dt {( + θ θ θ ) + + } ( ) 6 C d s t t r t rt t t dt n

7 7 Deteriorating Inventory Model With Linear Demand and Variable Deterioration aking into Account the ime-value of Money 4 r θ Cd( s) +..(9) Present-Value Shortage Cost All shortage during ( ) will be completely backordered at, the present value shortage cost for the period is r( + t ) s ( ) C C I t e dt r rt C d s t e e dt r( t ) + C d s t e dt r( + t t ) r( t ) e + C d( s) e r r C d s r r e + e ( r + r ) r C d s r r r r r r + + ( r + r ) r 6 6 C d s 6 r + r + r 6..()

8 Neetu & Arun Kumar omer Present-Value Item Cost Replenishment is done at t and t the replenished items are consumed by demand as well as by deterioration during.he present-value item cost, C p includes item cost and deterioration cost i.e. r p + C CI Ce d s dt θ Cd s + + C r d s 6 [By equation (4)] Cd( s) r + r + θ 6..() he first cycle present-value net profit is P R C CH Cs CP 4 r r θ P sd( s) r + r C C d( s) + 6 Cd s θ 6 r r r Cd( s) r r () here are N cycles during the planning horizon. Since inventory starts and ends at zero; an extra replenishment at th is required to satisfy backorders of last cycle in the planning horizon. Hence, total number of replenishment N + times the first replenishment lot size I and the nd, rd and N th replenishment lost size

9 9 Deteriorating Inventory Model With Linear Demand and Variable Deterioration aking into Account the ime-value of Money q I + d s dt..() and (N+) th replenishment lot size d s dt..(4) he time-value of money affects all the replenishment periods and hence must be considered separately. he total net present-value profit for the planning N r horizon is ( ) i.e. r r rh P s,, N P e e... e C e N n r n Pe C e rh rn P e C e r e rh P e C e r e rh rh, where H N rh 4 e r r θ P s,, N sd( s) r r C Cd( s) r + + e 6 C d s θ rh 6 r r r Cd( s) r r Ce rh P e r θ sd s r r + r Cd s + e Now (5) Cd s 6 r 6 r Cd( s) θ r (6)

10 Neetu & Arun Kumar omer 4 P rh e r r θ ( sd '( s) + d( s) ) r r d '( s) C s r + + e 6 C θ + ( 6 r + r + r ) C r + r For extreme conditions By equation (6) P P, s..(7) r θ C θ s( r + r) C + ( 6 + r + 6 r ) C r + 6..(8) P rh e r e Cd s sd( s)( r) Cd ( s)( r + θ ) 6 6r Cd s θ 6 ( ) ( ) { } rh ( d( s) ) rs + C ( r + θ ) + C ( r ) + Cθ r { rs C ( r ) C ( r ) C } Hd s + + θ + + θ < [ r < ] Q.(9)

11 Deteriorating Inventory Model With Linear Demand and Variable Deterioration aking into Account the ime-value of Money P H r d ' s r r + s Q d '' s d s a bs, b > d ' s b + < bh r r r [ r < ] Q () P ( ) H r θ sd ' s d s r + + r C d ' s + s θ Cd ' s 6 r 6 r Cd ' s r H r θ d' s s r r C C 6 r 6 r ( ) ( ) θ C r + + d( s)( r + r ) { } H d ' s. + d s r [By equation (8)] H d ( s ) r ( )..() P P P > s s..()

12 Neetu & Arun Kumar omer profit. Hence values of and s given by equations (6) & (7) will maximize the 5. NUMERICAL EXAMPLE Let the values of parameters of inventory model are: C 8/order θ.5 C.6/unit/year r.8 C.4/unit/year C 5/unit/year d(s) 4s/unit/year H years. From table,we conclude that with increase in number of replenishments, the total net present value profit decreases.. If r, i.e. when the time- value of money is not considered then keeping the other parameters same,we observe from table, that the net presentvalue profit is higher than the case when the time-value of money is considered. able No. of Cycles (N) Sales Price (s) ime Interval (year) () ime Interval (year) (-) ime Interval (year) () Present Value Profit

13 Deteriorating Inventory Model With Linear Demand and Variable Deterioration aking into Account the ime-value of Money able No. of Cycles (N) Sales Price (s) ime Interval (year) () ime Interval (year) (-) ime Interval (year) () Present Value Profit CONCLUSIONS In this paper, an Inventory model with variable deterioration rate, taking into account the time-value of money is considered. his type of model is useful for all items which deteriorates with time but not at a constant rate. Emphasis is on profit-maximization. A numerical example is also given to illustrate the theory. We observe that total net present value profit is higher when time-value of money is not considered. REFERENCES. Misra, R.B. (979) : A note on optimal inventory management under inflation. Naval Logistics Quarterly, 6, Chandra, M.J., Bahner, M.L. (985) : he effects of inflation and time value of money on some inventory systems. International Journal of Production Research, (4),7-7.. Sarker, B.R., Pan, H. (994) : Effects of inflation and time value of money on order quantity and allowable shortage. International Journal of Production Economics, 4, 65-7.

14 Neetu & Arun Kumar omer 4 4. Wee, H.M. (995) : A deterministic lot size inventory model for deteriorating items with shortages and a declining market. Computer and Operation Research, Abad, P.L. (996) : Optimal pricing and lot-sizing under conditions of perishability and partial backlogging. Management Science 4(8), Hui-Ming Wee (999) : Deteriorating inventory model with quantity discount, pricing and partial backordering. Int. J. Production Economics 59, Hariga, M.A. (995) : Effects of inflation and time value of money on an inventory model with time-dependant demand rate and shortages. European Journal of Operational Research, 8, Wee, H.M. (997) : A replenishment policy for items with a price dependant demand and a varying rate of deterioration. Production Planning and Control.8(5), Ghare P.M., Schrader, G.F. (96) : A model for exponential decaying inventory. Journal of Industrial Engineering, 4,8-4.. Hui-Ming Wee, Sh-yan Law () : Replenishment and pricing policy for deteriorating items taking into account the time-value of money. Int. J. Production Economics 7, -.. Wang, S. P. () : An inventory replenishment policy for deteriorating items with shortages and partial backlogging. Computers and Operational Research, 9, 4-5.

15 5 Deteriorating Inventory Model With Linear Demand and Variable Deterioration aking into Account the ime-value of Money. eng, J.., Yang, H. L. and Quyang, L.Y. () : On an EOQ model for deteriorating items with time varying demand and partial backlogging. Journal of Operational Research Society, 54, Skouri, K. and Papachristos, S. () : Optimal stopping and restarting production items for an EOQ model with deteriorating items and time dependent partial backlogging. International Journal of Production Economics, 8-8, eng, J.. and Yang, H.L. (4) : Deterministic economic order quantity models with partial backlogging when demand and cost are fluctuating with time. Journal of the Operational Research Society, 55(5), eng, J.., Chern, M.S,Yang, H.L.(997) : An optimal recursive method for various inventory replenishment models with increasing demand and shortages. Naval Research Logistics, 44, Padmanbhan, G., Vrat, P. (995) : EOQ models for perishable items under stock dependent selling rate. European Journal Of Operation Research, 86, Ouyang, L.Y., Heish,.P., Dye, C.Y., Chang, H.C. () : An Inventory model for deteriorating items with stock-dependent demand under the conditions of inflation and time-value of money. he Engineering Economist, 48, Zhou, Y.W. () : A multi-warehouse inventory model for items with time-varying demand and shortages. Computers and Operation Research., 5-4.