PRODUCTION INVENTORY MODEL WITH DISRUPTION CONSIDERING SHORTAGE AND TIME PROPORTIONAL DEMAND

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1 Yugoslv Journl of Opertions Reserch 28 (2018), Number 1, DOI: PRODUCTION INVENTORY MODEL WITH DISRUPTION CONSIDERING SHORTAGE AND TIME PROPORTIONAL DEMAND U. K. KHEDLEKAR Dr.Hrisingh Gour Vishwvidyly, Deprtment of Mthemtics nd Sttistics, A Centrl University Sgr M.P. Indi , uvkkcm@yhoo.co.in A. NAMDEO Dr.Hrisingh Gour Vishwvidyly, Deprtment of Mthemtics nd Sttistics, A Centrl University Sgr M.P. Indi A. NIGWAL Deprtment of Mthemtics, Government Engineering College, Ujjin MP-Indi Received: November 2016 / Accepted: My 2017 Abstrct: The disruption in production system occurs due to lbor problem, mchines brekdown, strikes, politicl issue, nd wether disturbnce, etc. This leds to dely in the supply of the products, resulting customer to pproch other delers for the products. This pper is n ttempt to develop n economic production quntity model using optimiztion method for deteriorting items with production disruption. We obtined optiml production time before nd fter the system gets disrupted. We hve lso devised the model for optimizing the shortge of the products. This reserch is useful to determine the time for strt nd stop of the production when system gets disrupted. The optiml production nd inventory pln re provided, so tht the mnufcturer cn reduce the loss occurred due to disruption. Finlly grph bsed simultion study hs been given to illustrte the proposed model. Keywords: Inventory, Disruption, Deteriortion, Preservtion Cost, Shortge. MSC: 90B05, 90B30, 90B50.

2 124 U.K.Khedlekr, et l. / Production Inventory Model 1. INTRODUCTION The production system cn lwys be ffected by lbor problem, politicl crises, mchine brekdown, strike, politicl issue, nd undetermined wether. If the production disruption ppers, this leds us to big loss becuse we re unble to fulfill the demnd nd new orders re still being received from the costumers. Other loss is loss of credibility of the firm tht ffects the goodwill, nd the costumer my turn to nother supplier/seller or product. So, the nlyticl study is necessry to mnge the production system. In rel life, the effect of deteriortion is very importnt in every inventory systems. Generlly, deteriortion is defined s decy, dmge, spoilge, evportion, obsolescence, loss of utility, or loss of mrginl vlue of commodity tht results in decresing usefulness. A continuous production control inventory model for deteriorting items with shortge is developed by Smnt nd Roy (2004) nd the optiml verge system costs, stock level, bcklog level nd production cycle time re formulted when the deteriortion rte is very smll. Roy nd Chudhuri (2011) introduced n economic production lot size model, where production rte depends on stock nd selling price per unit. In this model deteriortion is ssumed s constnt frction nd shortges re not llowed. Rosenbltt nd Lee (1986) studied the effect of n imperfect production process on the optiml production cycle time by ssuming tht system gets deteriorte during the production process nd produces some defective items. Chndel nd Khedlekr (2013) presented n integrted inventory model to optimize the totl expenditure of wrehouse set-up. Moon et l. (2005) developed n inventory model by considering both meliortion nd deteriortion over finite plnning horizon with time vrying demnd. Benhdid et l. (2008) developed production inventory model for deteriorting item nd dynmic costs. Shukl et l. (2012) presented n inventory model for deteriorting items by ssuming tht there exists n optiml number of price setting for obtining mximum profit. Khedlekr nd Shukl (2012) pplied the concept for logrithmic demnd nd simulted the results for vrious businesses. The outcomes of the study is tht β is the most significnt prmeter tht ffected optiml profit nd respective number of price setting. Widydn nd Wee (2010) designed n EPQ models for deteriorting items by considering stochstic mchine unvilbility time nd price dependent demnd. In this model lost sles will occur when mchine unvilbility time is longer thn the non production time. They used Genetic Algorithm to solve the model. The price rte is the more sensitive prmeter thn the mchine unvilbility time nd the lost sles cost. Blkhi nd Bkry (2009) considered dynmic inventory model for deteriorting items in which ech of the production, demnd, nd the deteriortion rte, s well s costs prmeters re ssumed to be generl function of time. Both infltion nd time vlue of money re tken into ccount. Wee (1993) devised n economic production quntity model for deteriorting items with prtil bck-ordering. There re numerous studies on inventory models for deteriorting items under different conditions, such s Chung nd Hung (2007), Ouyng et l. (2005), Khedlekr nd

3 U.K.Khedlekr, et l. / Production Inventory Model 125 Nmdeo (2015), Shukl nd Khedlekr (2015), Choudhuri nd Mukherjee (2011), Giri et l. (2003), Kumr nd Shrm (2012, b, c) etc. Priority of ny mnufcturing firm, retiler, nd storekeeper should be preventing the commodity from deteriortion. For this purpose, we my pply the preservtion technology to reduce deteriortion rte. The investment in preservtion technology includes n dditionl cost tht we hve to ber. You nd Hung (2013) developed model for deteriorting sesonl product whose deteriortion rte could be controlled by investing in preservtion efforts. Zhng et l. (2014) developed n inventory model in which demnd is dependent on both selling price nd time; lso, deteriortion could be controlled by preservtion technology. Khedlekr et l. (2016) devised deteriorting inventory model for liner declining demnd where preservtion technology is pplied to preserve the commodity, nd they shown tht the profit is concve function of optiml selling price, replenishment time, nd preservtion cost prmeter. Mishr (2013) devised model for Weibull distribution deteriorting sesonl product by considering constnt demnd rte, shortge nd slvge vlue; lso, the deteriortion rte is reduced by pplying the preservtion technology. Khedlekr et l. (2016) extended his model [Khedlekr nd Shukl (2013)] by incorporting exponentil declining demnd in which prt of inventory ws prevented from deteriortion by preservtion technology. At the beginning of ech cycle, the mnufcturer should decide the optiml production time so tht the production quntity meet both demnd nd deteriortion, nd ll quntity should be sold out in ech cycle, tht is, t the end of ech cycle, the inventory level should rech to zero. However, fter the pln is implemented, the production run is often disrupted by some emergent events, such s supply disruptions, mchine brekdowns, finncil crisis, politicl event, nd policy chnge. Production disruption will led us to hrd decision in production nd inventory pln. In this pper we incorported shortge t the end of time horizon becuse fter the plnning horizon, there is possibility of some disruption before strting the next production run. But s new orders re still receiving, we hve to cop-up this shortge before strting the next plnning horizon. Recently, there is growing literture on production disruptions. He nd He (2010) proposed production-inventory model for deteriorting item with production disruption. In this study, n extension is mde to consider the fct tht some products my deteriorte during their storge. Chen nd Zhng (2010) considered model of three-echelon supply chin system which consists of suppliers, mnufcturer, nd customers under demnd disruptions. Furthermore, n improved Anlyticl Hierrchy Process (AHP) is studied to select the best supplier bsed on quntittive fctors such s the optiml long-term totl cost obtined through the simulted nneling method under demnd disruptions. The objective is to minimize the totl cost under different demnd disruption scenrios. Khedlekr et l. (2014) formulted production inventory model for deteriorting item with production disruption nd nlyzed the system under different situtions. Srkr nd Moon (2011) considered clssicl EPQ model with stochstic demnd under the effect of infltion. The model is described by considering generl distribu-

4 126 U.K.Khedlekr, et l. / Production Inventory Model tion function. Benjfr nd ElHfsi (2006) considered the optiml production nd inventory control of n ssemble-to-order system with m components, one end product, nd n costumer clsses. Therefore, in this model, we devised production inventory model for deteriorting items with production disruption, shortge occurs once t the end of the time horizon, nd preservtion technology is pplied for reducing deteriortion. Once the production rte is disrupted, our object is to find the nswer to following questions: Whether to replenish from spot mrket or not? How to djust the production pln if the production system cn still stisfy the demnd? When to replenish from spot mrket if the new production system no longer stisfy the demnd? How long nd how much quntity we hve to replenish if the shortge occurs t the end of the cycle? 2. ASSUMPTION AND NOTATION In this model we consider time proportionl demnd rte, which is deterministic but not constnt. The norml production rte is lwys greter thn the demnd rte, therefore p t > 0. Suppose tht constnt deteriortion exists in the system. Shortge is llowed t the end of the finite cycle. To reduce deteriortion, we incorporte preservtion technology. The reltion between deteriortion rte nd preservtion technology investment prmeter stisfies α < 0, nd 2 α 2 > 0. Hence, in this pper we ssumed tht = λ 0 e αδ. Here is the deteriortion rte fter investing preservtion technology, λ 0 is the deteriortion rte without preservtion technology investment, nd δ is the sensitive prmeter of investment to the deteriortion rte. In this model the bsic prmeters re s follow: p: norml production rte, t: demnd rte, such tht p > t, > 0, : deteriortion rte, α: cost of preservtion technology investment per unit time, H: norml time horizon, T o : time horizon including shortge, T p : production time without disruption, T d : production disruption time,

5 U.K.Khedlekr, et l. / Production Inventory Model 127 Figure 1: Production system without disruption T d p : production period with disruption, T r : replenishment time, Q r : replenishment quntity t time T r, T s : shortge time, Q s : shortge quntity. 3. MODEL WITHOUT DISRUPTION Suppose mnufcturer produces kind of product nd sells it in mrket. Since the production rte is p > 0, nd demnd rte is D(t) = t (p > t > 0), thus inventory is ccumulted t rte (p t). Inventory mngement need to stop production t time T p nd there fter, inventory is depicted due to demnd rte (t) nd deteriortion (See fig. 1). Now, it is ssumed tht the inventory is sufficient to fulfill the demnd till time H. The inventory level I(t) t ny time t [0, H] is obtined by the following differentil equtions (3.1) nd (3.2). I 1 (t) t I 2 (t) t + I 1 (t) = p t, 0 t T p (3.1) + I 2 (t) = t, T p t H (3.2) using the boundry condition I 1 (0) = 0, nd I 2 (H) = 0, the solution of these differentil equtions re ( p I 1 (t) = + ) ( 2 1 e t) t (3.3) I 2 (t) = ( ) He H t t ( 2 1 e H t) (3.4)

6 128 U.K.Khedlekr, et l. / Production Inventory Model The condition I 1 (T p ) = I 2 (T p ) yields, ( p + ) ( ) 2 1 e Tp T p = ) (He H Tp T p ( ) 2 1 e H Tp (3.5) If << 1, then expnding the exponentil function nd neglecting the second nd higher power of, T p = T p = 2H + H2 + p + 2 (3.6) 2 (H(H 1) ph) 2 + p + 2 > 0, (3.7) Now, we cn get the following corollry. Corollry 3.1. If <<1, then T p is incresing in. Tht implies tht the mnufcturer hs to produce longer in ccording with the deteriortion rte increse. Hence, decresing deteriortion rte is most importnt to reduce the production cost. 4. THE PRODUCTION INVENTORY MODEL WITH PRODUCTION DISRUPTION If Proposition 4.1 p p(1 e H + e Td H ) (1 e H + H) ( ) 1 e T d H then the mnufcturer cn still stisfy the demnd fter system get disrupted. Otherwise, p p < p(1 e H + e Td H ) (1 e H + H) ( ) 1 e T d H then there will be shortge due to the production disruption, therefore the production rte decreses deeply. Proof: Let the new disrupted production rte is p + p, where p < 0, if production rte decreses nd p > 0 if production rte increses. Suppose tht the production disruption time is T d. Then, the differentil equtions in this sitution re: ( p I 1 (t) = + ) ( 2 1 e t) t, 0 t T d. (4.1)

7 U.K.Khedlekr, et l. / Production Inventory Model 129 The boundry condition I 1 (T d ) = I 2 (T d ), yields I 2 (t) = p (1 ) e t + e T d t Hence, + p ( ) 1 e T d t + ( 2 1 e t) t I 2 (H) = p (1 ) e H + e T d H + p ( ) 1 e T d H + ( 2 1 e H) H (4.2) (4.3) If I 2 (H) 0, then p p(1 e H + e Td H ) (1 e H + H) ( ) 1 e T d H This mens tht the mnufcturer cn still stisfy the demnd fter system get disrupted. If I 2 (H) < 0, then p p < p(1 e H + e Td H ) (1 e H + H) ( ) 1 e T d H Therefore, the mnufcturer will fce the shortge since the production rte decrese deeply. If Proposition 4.2 p p(1 e H + e Td H ) (1 e H + H) ( ) 1 e T d H then, the mnufcturer s production time with production disruption is Tp d = 1 ln p (p + (H + 1 p)etd p + p Proof: In this sitution the differentil equtions re I 2 (t) t I 3 (t) t + I 2 (t) = p + p t, T d t T d p + I 3 (t) = t, T d p t H )eh +

8 130 U.K.Khedlekr, et l. / Production Inventory Model using the conditioni 1 (T d ) = I 2 (T d ) nd I 3 (H) = 0, we hve I 2 (t) = p (1 ) e t + e T d t + p ( ) 1 e T d t + ( 2 1 e t) t (4.4) nd I 3 (t) = t + ( H 2 + ) 2 e H t (4.5) since, I 2 (T d p ) = I 3 (T d p ), the production time fter disruption is Tp d = 1 ln p (p p)etd + (H + 1 p + p )eh + (4.6) Thts the proof of the proposition. Now, if << 1, then nd I 2 (t) = p (1 + T d) + p(t T d ) I 3 (t) = H 2 Ht So, I 2 (T d p ) = I 3 (T d p ), revels tht T d p = H2 p (1 + T d) + pt d p + H Now, differentite this with respect to T d, we cn get Tp d p + p = T d p + H Now, we cn get the following corollry. Corollry If p < 0, then Tp d is decresing in T d, nd if p > p > 0, then Tp d is incresing in T d. Similrly, we hve Tp d = p 2 ( p + H) > 0, Corollry For << 1, Tp d is incresing in.

9 U.K.Khedlekr, et l. / Production Inventory Model 131 Figure 2: Production system fter disruption Proposition 4.3 If I 2 (H) < 0, then the production system does not fulfill the time proportionl demnd. The replenishment time T r nd the order quntity Q r re e Tr ((p p)e T d nd Q r = (p + p) ( p + (1 e H Tr ) )) T r + ( (p + p) + (T r He H Tr ) + 2 ( 1 e H Tr ) ) = 0 (4.7) Proof: Suppose T r nd Q r re time of plcing n order nd order quntity respectively (See fig. 2), then I 2 (T r ) = 0. eqution (4.3) leds to e Tr ((p p)e T d ( p + )) T r + p + p + The formultion of differentil eqution in this sitution is I 3 (t) t + I 3 (t) = p + p t, T r t H Boundry condition I 3 (H) = 0, yields (p + p) I 3 (t) = (1 e H t) ( t He H t) + 2 ( 1 e H t) = 0(4.8)

10 132 U.K.Khedlekr, et l. / Production Inventory Model Hence, the order quntity Q r = I 3 (T r ) will be (p + p) ) Q r = (1 e H Tr (T r He H Tr ) + 2 ( 1 e H Tr ) (4.9) The proposition is proved. If I 2 (H) < 0, nd << 1, then nd T r T d = 1 T r T d < 0, Q r T d = (p + p H) T r T d > 0, Now, we cn get the following corollry. Corollry If I 2 (H) < 0, nd << 1, then T r is decresing in T d while Q r is incresing in T d. Proposition 4.4 Suppose there is n dditionl disruption occurs between two successive production cycle nd order re still receiving, so the shortge could exist t time H till time T s, nd then the production strts to cop-up this shortge nd the production grows t the beginning of next cycle time T o. Then shortge time T s nd shortge quntity Q s re nd e To Ts (T o p) + T 2 s 2 Q s = p ) (1 e To Ts T s + p + H2 2 (T s T o e To Ts ) + 2 ( 1 e To Ts ). Proof: If shortge occurs t time H, nd continues till time T s therefter production will strt t time T s, (See fig. 3) nd continues till time T o. If shortge quntity Q s is t time T s then, the differentil eqution will be nd I 4 (t) t I 5 (t) t = 0 = t, H t T s, (4.10) + I 5 (t) = p t (4.11)

11 U.K.Khedlekr, et l. / Production Inventory Model 133 Figure 3: Production system fter disruption with shortge where T s t T o nd I 4 (T s ) = I 5 (T s ), nd I 5 (T o ) = 0. Eqution (4.10) yields I 4 (t) = ( t 2 H 2) 2 Boundry condition I 5 (T o ) = 0 yields I 5 (t) = p (1 e To t) ( t T o e To t) + The condition I 4 (T s ) = I 5 (T s ), revels tht e (To Ts) (T o p) + T 2 s 2 the ordering quntity Q s = I 5 (T s ) will be Q s = p ) (1 e To Ts The proposition is proved. If << 1, then T s T o = ( 2 1 e To t). (4.12) T s + p + H2 2 (T s T o e To Ts ) + 2 ( 1 e To Ts ) T o p (T o T s ) + p > 0, = 0 (4.13) (4.14)

12 134 U.K.Khedlekr, et l. / Production Inventory Model Figure 4: I 2(H) with respect to H nd Q s T o for 0 Ts T o 1. = (T o p)(1 T s T o ) + (T o T s ) > 0, Now, we cn get the following corollry. Corollry For << 1, T s is incresing in T o while Q s is incresing in T o for 0 Ts T o APPLICATION AND SENSITIVE ANALYSIS In this section we ssume tht production rte p = 200 unit/dy, = 10 disruption occur in production system t rte p = 100, rte of deteriortion = 0.22, time horizon H = 20 dys, production system disrupted fter T d = 2 dys, if I 2 (H) = > 0, then production time fter disruption is longer thn the production time before disruption, so the mngement need to mintin more stock to consume for next 7.5 hours. Cse I If I 2 (H) > 0, or p p ( 1 e H + e ) T d H (1 e ( ) 1 e (T d H) H + H

13 U.K.Khedlekr, et l. / Production Inventory Model 135 Figure 5: T d p with respect to T d Figure 6: T r with respect to

14 136 U.K.Khedlekr, et l. / Production Inventory Model If I 2 (H) = 0, nd the orders re still being received, then there could exist the shortge. Hence, the shortge time is T s = 3.15 dys, nd the shortge quntity Q s = 435 unit. Next, we observe how I 2 (H), Tp d, T d, T r would chnge s H, T d, T r, nd, respectively. Figure 4 shows tht I 2 (H) is decresing in H, therefore the on hnd inventory I 2 (H) will decrese if time horizon H is lrge. So the dvice to mnufcturer/retiler is to keep the time horizon s smll s possible. Cse II If I 2 (H) 0, tht is ( ) p(1 e H + e Td H (1 e H + H) p p < ( ) 1 e T d H From figure 5, we cn find tht Tp d is decresing in T d when 2 T d 5. For 0 T d 2, the mnufcturer will hve to replenish inventory from spot mrket. From figure 6, we cn see tht T r is decresing in when > CONCLUSION In this pper, inventory production model hs been developed considering time proportionl demnd for deteriorting items. The shortge hs been incorported t the end of the time cycle. We hve clculted nd grphiclly simulted the time of production disruption nd the quntity of production fter disruption. This pper suggest the production mngement to keep short time spn to produce the product in smll lot nd to keep minimum deteriortion when the replenishment time occurs shorten. For this, mngement cn use preservtion technology to reduce deteriortion rte. In cse of erly disruption, it is difficult for mngement to mnge with time proportionl demnd. We hve considered constnt deteriortion rte in this study, but future reserch in this field my consider vrible deteriortion rte, one cn consider vrible deteriortion with stochstic demnd. Also, one cn formulte the model in fuzzy environments. REFERENCES [1] Blkhi, Z.T., nd Bkry, A.S., A generl nd dynmic production lot size inventory model, Interntionl Journl of Mthemticl Models nd Methods in Applied Science, 3 (3) (2009) [2] Benhdid, Y., Tdj, L., nd Bounkhel, M., Optiml control of production inventory system with deteriorting Items nd dynmic costs, Applied Mthemtics E-Notes, 8 (2008) [3] Benjfr, S. nd ElHfsi, M., Production nd inventory control of single product ssemble-to-order system with multiple customer clsse, Mngement Science, 52 (12) (2006) [4] Chndel, R.P.S. nd Khedlekr U.K., A new inventory model with multiple wrehouses, Interntionl Reserch Journl of Pure Algebr, 3 (5) (2013)

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