A Partially Backlogging Inventory Model for Deteriorating Items with Ramp Type Demand Rate

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1 American Journal of Operational Reearch 05, 5(): DOI: 0.593/j.ajor A Partially Backlogging Inventory Model for Deteriorating Item with Ramp ype Demand Rate Suhil Kumar *, U. S. Rajput Department of Mathematic & Atronomy, Univerity of Lucknow, Lucknow, U.P. India Abtract hi paper preent an inventory model for deteriorating item with contant deterioration rate and ramp type demand rate. Shortage are allowed and completely backlogged for the next replenihment cycle. he backlogging rate of unatified demand i aumed a a function of waiting time. he purpoe of our tudy i to minimize the total variable inventory cot becaue if the total cot i minimum then profit will be maximum. A numerical example i given to demontrate the developed model. Senitivity analyi of the change of parameter i alo given. Keyword Deterioration, Partial-backlogging, Shortage and ramp type demand rate. Introduction In mot of the inventory model it ha been oberved that the demand rate i influenced by the amount of on hand inventory tock. Reearcher developed their inventory model by conidering demand rate i either contant or increaing, decreaing exponential function of time, tock dependent and linear function of time. he demand of newly launched product uch a fahionable garment, electronic item and mobile phone etc. increae with time and later it become contant. In uch type of product the concept of ramp type demand i introduced. he ramp type demand i a demand which increae up to a certain time and after that it become table or contant. Deterioration i a major problem in any buine organization becaue the product uch a food grain, vegetable, medicine and alcohol etc. that are tored for future ue alway looe part of their value with paage of time then thi phenomenon i known a deterioration, o deterioration cannot be avoided in any buine organization. Hill [995 firt time developed an inventory model with ramp type demand rate. Mandal and Pal [998 extended the Hill [995 model by allowing hortage. Wu and Ouyang [000 developed an inventory model with ramp type demand rate. Wu [00 devoloped an EOQ model for weibull deteriorating item with ramp type demand function of time by allowing hortage and the backlogging rate i a function of waiting time for next replenihment cycle. Giri et al. [003 developed a ingle item ingle period EOQ model for weibull deteriorating item with ramp type demand rate by allowing hortage which are completely backlogged and * orreponding author: uhilmath4444@gmail.com (Suhil Kumar) Publihed online at opyright 05 Scientific & Acemic Publihing. All Right Reerved an infinite planning horizon. Manna and haudhari [006 developed an order level inventory model with finite production rate and time dependent deterioration rate and the demand rate i a ramp type function of time by allowing hortage which are completely backlogged. Deng et al. [007 propoed an inventory model for deteriorating item with ramp type demand rate. Panda et al. [008 conidered an order level inventory model for eaonal product with ramp type demand rate by allowing hortage. Singh and Singh [008 developed a partially backlogging inventory model for deteriorating item with quratic demand rate. Skouri et al. [009 developed a partially backlogging inventory model for weibull deteriorating item with ramp type demand rate. Singh and Singh [00 developed an inflationary partially backlogging upply chain inventory model for deteriorating item with ramp type demand rate. Singh et al. [00 determine a replenihment policy for non-intantaneou deteriorating item with tock dependent demand and partial backlogging. Jain and Kumar [00 conidered an inventory model for three parameter weibull deteriorating item with ramp type demand rate by allowing hortage. hang [0 developed a partially inventory model for two parameter weibull deteriorating item with ramp type demand rate. Yav et al. [0 determine an optimal replenihment policy for a partially backlogging inventory model with ramp type demand rate. Karmakar and houdhuri [03 developed a partially backlogging inventory model with time varying holding cot and ramp type demand rate. Sakrar and hakrabarti [03 developed an order level inventory model with fuzzy type demand under two level of hortage. Banal and Garg [04 conidered a partially backlogging inventory model for non-intantaneou deteriorating item with ramp type demand rate. he table I how that the variation of total average cot with repect to the change in parameter, table II how that the variation of total average cot with

2 40 Suhil Kumar et al.: A Partially Backlogging Inventory Model for Deteriorating Item with Ramp ype Demand Rate repect to the change in parameter, table III how that the variation of total average cot with repect to the change in parameter b and the table IV how that the variation of total average cot with repect to the change in parameter. Figure II, III and IV give the variation of parameter μ, and with repect to the fixed value of (,, ),(,, ) and (,, ), Figure V and VI give the variation of parameter and with repect to the value of (,, ) and (,, ), Figure VII, VIII and IX give the variation of parameter, and with repect to the fixed value of ( b,, ),( b,, ) and ( b,, ). Figure X and XI give the variation of parameter and with repect to the fixed value of (,, ) and (,, ). In thi paper we developed a partially backlogging inventory model for deteriorating item with contant deterioration rate and ramp type demand rate. Shortage are allowed and completely backlogged for next replenihment cycle, the backlogging rate of unatified demand i aumed a a function of waiting time. he ramp type demand i a demand which increae up to a certain time and after that it become table or contant in the cae of real etate, electronic item, fahionable garment and food grain etc. he purpoe of our tudy i to minimize the total variable inventory cot for maximize the profit.. Aumption and Notation We conider the following aumption and notation correponding to the developed model. he ramp type demand rate i Rt () ae b[ t( t) H ( t), where a i the initial demand rate and b a contant governing exponential demand rate and Ht ( ) i the Heaviide unit tep function, t of time defined by Ht ( ) 0, t. he demand dependent production rate i P( t) kd( t), where k i a contant. 3. i the contant deterioration rate. 4. i the backlogging parameter. 5. h i the holding cot. 6. d i the deterioration cot. 7. i the hortage cot. 8. i the time of maximum inventory level. 9. i the time of zero inventory level. 0. i the length of inventory cycle.. he replenihment rate i infinite.. Shortage are allowed and partially backlogged. 3. he le time i zero. 4. It () i the inventory at any time in [0,. Figure 3. Mathematical Formulation Let u conider an inventory ytem in which the production tart at t=0 and top at t. During the period [0, the inventory level grow due to both production and deterioration and during the period [, the maximum inventory level decreae due to both demand and deterioration and become zero at t. he hortage tart at t and hortage interval i the end of current order cycle. he whole proce i repeated and during the hortage interval [, the unatified demand i backlogged at a rate of ( t), where t i the waiting time and the poitive contant i the backlogging parameter. he intantaneou inventory level at any time t in [0, i governed by the following differential equation di I kp( t) R( t) dt di I a( k ) e bt, 0 dt t () With boundary condition I(0) 0 di b I, t dt ae () With boundary condition I( ) 0 di b a e ( i), t (3) dt With boundary condition ( ) 0 I he olution of equation () i

3 American Journal of Operational Reearch 05, 5(): he olution of equation () i he olution of equation (3) i he maximum inventory level S i obtained by putting t ( b ) t (4) I a( k )[ t, 0 t I a[( b) t a[ ( b ), t t I a( b) [ t, t t (6) in equation (4) ( b ) S a ( k )[ (7) he total number of unit deteriorated during the interval [0, i [ S I( t) dt I( t) dt n d n d 0 ( b ) ( b) a ( k )[ a[ 6 3 b b 3 b 3 3 6, (8) 6 n S he total number of unit hortage during the interval [, i he inventory holding cot per cycle in he deterioration cot per cycle i D he hortage cot per cycle i I() t dt 3 3 a ( b)[, (9) S 3 n [ 0, i H [ I ( t ) dt I ( t ) dt h 0 ( b ) [( k ){ } { 6 3 H b b b } 6, (0) 6 D [ S I ( t ) dt I ( t ) dt d 0 ( b ) ( b ) b [( ){ } { 6 3 b k 3 b 3 3 } 6, () 6 he total variable inventory cot per unit cycle i 3 3 a( b) [ 3, () S (5)

4 4 Suhil Kumar et al.: A Partially Backlogging Inventory Model for Deteriorating Item with Ramp ype Demand Rate (,, ) [ H D S a ( b ) b (,, ) h [( k ){ } { b 6 3 b ( b ) ( b ) 3 } [( k ){ } b 3 b 3 3 { } b 6 6 a( b) 3 3 [ 3, (3) he neceary condition for (,, ) to be minimum i that H 0, 0, H H H H and the ufficient condition i the determinant of the principal minor,,... 3 of Heian matrix (H-matrix), of (,, ) are poitive definite. ( b ) b 3b [( k ){ } { } b ( ) [( ){ ( ) b k b } { b b 3b 3 3 } ab [ 3, (4) k b b k b b [( ){ ( ) } 3 [( ){ ( b) } 3 b, (5) b b b b [ [ 3 a( b) b [ t, (6)

5 American Journal of Operational Reearch 05, 5(): [ ( b ) [ ( b ) 3 [, (7) a ( b ) b 3 b 3 3 b ( b ) b 6 3 [( k ){ } b b ( b ) ( b ) 3 [( ){ } 6 k a ( b) 3 a ( b) 3 3 [ [ 3 ( ) 3 [( ){ b k } 3 b 6 b 3 b 3 3 ( ) ( ) 3 b b [( k ){ } b 3 b 3 3 b 6 a ( b) a ( b) 3 [3 [ a( b) 3 3 [ 3 3 [ b [ b ( ) b ab b, (8), (9) ( b ) b 3b ( b ) [( k ){ ( b ) } b b k b [( ){ } 3 b ab 4 3 ab { } { } () 6 3 [ b [ b [ ( ) b b ab 3 b b b b a [ [ a ( b) ( b) [ [ ( ) (3) (0) ()

6 44 Suhil Kumar et al.: A Partially Backlogging Inventory Model for Deteriorating Item with Ramp ype Demand Rate ( b ) b 3b [( k ){ } { } b ( b ) b b [( k ){ ( b ) } { 3 3 3b ab ab ( ) } [ 3 3 [ (4) 4. Numerical Example [ b b [ b 3 ( b) ( b) b [ [ ( ) (5) a Let u conider an inventory ytem with the given data in appropriate unit a follow: 00, 4, 0.05, 3,, 5, 8, a b k h d S 0, and the total variable inventory cot i = able how the variation of total variable inventory cot with repect to the change in parameter. able A we increae the parameter then the value of the parameter, and decreae and the value of alo decreae Figure. (, and are cont.) Figure 3. (, and are cont.) a Figure 4. (, and are cont.) able how the variation of total variable inventory cot with repect to the change in parameter able Figure 5. (, and are cont.) Figure (, and are cont.)

7 American Journal of Operational Reearch 05, 5(): A we increae the parameter then the value of the parameter and remain contant, the value of the parameter decreae and the value of increae. able 3 how the variation of total variable inventory cot with repect to the change in demand parameter b. b able A we increae the parameter b then the value of the parameter, and decreae and the value of alo decreae Figure 7. (b, and are cont.) Figure 8. (b, and are cont.) Figure 9. (b, and are cont.) able 4 how the variation of total variable inventory cot with repect to the change in hortage cot parameter. A we increae the parameter then the value of the parameter and remain contant, the value of the parameter decreae and the value of increae able Figure 0. (, Figure. (, and and are cont.) are cont.) Senitivity Analyi- From the table I we ee that a we increae the deterioration parameter then the value of the parameter,, and decreae. From the table II we ee that a we increae the backlogging parameter then the value of the parameter and remain contant and decreae and increae. From the table III we ee that a we increae the demand parameter b then the value of the parameter,, and decreae. From the table IV we ee that a we increae the hortage cot then the value of the parameter and remain contant and decreae and increae. hu the parameter and are more enitive than the parameter, b, h and d (becaue for the change of the value h and d the value of, and come out to imaginary). 5. oncluion In thi paper we conider a partially backlogging inventory model for deteriorating item with contant deterioration rate and ramp type demand rate. Shortage are allowed and completely backlogged for next replenihment

8 46 Suhil Kumar et al.: A Partially Backlogging Inventory Model for Deteriorating Item with Ramp ype Demand Rate cycle, the backlogging rate of unatified demand i aumed a a function of waiting time. he change of the value of backlogging parameter and hortage cot give the maximum value of total cot. Further thi model can be generalized by conidering time dependent holding and deterioration cot. AKNOWLEDGEMENS he author would like to thank anonymou referee for their valuable comment and uggetion for the improvement of thi paper. REFERENES [ Hill, R.M. (995), Inventory model for increaing demand followed by level demand, Journal of Mathematical Society, Vol. 46, pp [ Mandal, B., Pal, A.K., (998), Order level inventory ytem with ramp type demand rate for deteriorating item Interdiciplinary Mathematic, Vol. (), pp [3 Wu, K.S., Ouyang, L.Y., (000), A replenihment policy for deteriorating item with ramp type demand rate, Proceeding of National Science ouncil RO (A), Vol. 4, pp [4 Wu, K.S. (00), An EOQ model for Weibull deteriorating item with ramp type demand rate and partial backlogging, Production Planning and ontrol. Vol., pp [5 Giri, B.. Jalan, A.K., haudhari, K.S. (003), Economic order quantity model for Weibull deteriorating item with ramp type demand rate, International Journal of Sytem Science, Vol. 34, pp [6 Manna, S.K., haudhari, K.S. (006), An EOQ model with ramo type demand rate, time dependent production cot and hortage, European Journal of Operation Reearch, Vol. 7(), pp [7 Deng, P.S., Lin, R.H., hu, R.H. (007), A note on the inventory model for deteriorating item with ramp type demand rate, European Journal of Operation Reearch, Vol. 78(), pp. -0. [8 Singh S.R. & Singh,.J. (007), An EOQ model for Weibull deteriorating item with partial backlogging, Indian Journal of Mathematic and Mathematical Science, Vol. 3, no., pp.-. [9 Panda, S., Senapati, S., Bau, M. (008), Optimal replenihment policy for perihable eaonal product with ramp type demand rate, omputer and Indutrial Engineering, Vol. 54(), pp [0 Singh, S.R. and Singh,.J. (008), Perihable inventory model with quratic demand, partial backlogging and delay in payment, International Review of Pure and Applied Mathematic, Vol.. Pp [ Skouri, K., Kontantara, L., Papachrito, S., Gana, J., (009), Inventory model for Weibull deteriorating item with ramp type demand rate and partial backlogging, European Journal of Operation Reearch, Vol. 9, pp [ Singh, S.R., Kumari, R. and Kumar, N. (00), Replenihment policy for non- intantaneou deteriorating item time dependent demand and partial backlogging with two torage facility under inflation, International Reearch and Optimization, Vol. (), pp [3 Jain, S., Kumar, M. (00), An EOQ inventory model for three parameter Weibull deteriorating item with ramp type demand rate by allowing hortage, Yugolav Journal of Operation Reearch, Vol. 0, no., pp [4 Singh, S.R., Singh,., (00), Supply chain inventory model with tochatic le time and ramp type demand rate for expiring item under partial backlogging, International Journal of Operation Reearch, Vol. 8(4), pp [5 eng,.j., han, D., hang,.. (0), On inventory model with ramp type demand rate, partial backlogging and Weibull deterioration rate, amui Oxford Journal of Information and Mathematical Science, Vol. 7(), pp [6 Yav, D., Singh, S.R., Kumari, R. (0), Effect of demand booting policy on optimal inventory model with fuzzy environment under the effect of learning, International Journal of Procurement Management Vol. 5(), pp [7 Singh, S.R., and Sharma, S. (03) A global optimizing policy for two level tre credit financing taking account of preervation technology for decaying item with ramp type demand rate, Advance in Deciion Article Id. 6385, page. [8 Karmakar, B., houdhuri, K.D. (03), Inventory model with ramp type demand for deteriorating item with partial backlogging and time varying holding cot, Yugolav Journal of Operation Reearch, Vol. 3, no., pp. -7. [9 Banal, K.K., Garg, M. (04), An inventory model for non-intantaneou decaying item with ramp type demand and partial backlogging, International Journal of Engineering Reearch and Management echnology, Vol., pp. 5-.