Optimal replenishment policy with variable deterioration for xed-lifetime products

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1 Scientia Iranica E (06) 3(5), 38{39 Sharif University of echnology Scientia Iranica ransactions E Industrial Engineering Otimal relenishment olicy with variable deterioration for xed-lifetime roducts B. Kumar Sett a, S. Sarkar b, B. Sarkar b; and W. Young Yun c a. Deartment of Mathematics, Hooghly Mohsin College, Hooghly, India. b. Deartment of Industrial & Management Engineering, Hanyang University, Ansan Gyeonggi-do, 46 79, South Korea. c. Deartment of Industrial Engineering, Pusan National University, Busan , Korea. Received October 04; received in revised form 9 August 05; acceted 8 December 05 KEYWORDS Ram-tye demand; Fixed lifetime; ime-varying deterioration; ime-deendent holding cost; Shortages. Abstract. Although numerous researchers have develoed dierent inventory models for deteriorating items, very few of them have taken the maximum lifetime of a deteriorating item into consideration. his aer illustrates a mathematical model to obtain an otimal relenishment olicy for deteriorating items with maximum lifetime, ram-tye demand, and shortages. Both holding cost and deterioration function are linear functions of time, which are treated as constants in most of the deteriorating inventory models. A simle solution rocedure is rovided to obtain the otimal solutions. Numerical examles along with grahical reresentations are rovided to illustrate the model. Sensitivity analysis of the otimal solution with resect to key arameters of the model has been carried out and the imlications are discussed. 06 Sharif University of echnology. All rights reserved.. Introduction It is common nature of human behavior that the customer likes to buy a single item, even though the item is chosen from a large amount of stocks, i.e. the demand rate may go u or down if the onhand inventory level increases or decreases. For this reason, the retailers have to store a large amount of items in their store laces. However, certain tyes of commodities either deteriorate or become obsolete in the course of time. herefore, they must have an exiry date, i.e. the roduct will have a maximum lifetime which is time bound; for examle, goods like fruits, vegetables, meat, foodstus, erfumes, alcohol, gasoline, radioactive substances, hotograhic lms, electronic comonents, etc., of which deterioration is usually observed. he deteriorated items cannot be used for their original urose. he loss of inventory *. Corresonding author. el ; Fax address bsbiswajitsarkar@gmail.com (B. Sarkar) due to deterioration cannot be ignored. herefore, it is very essential to know the maximum lifetime of roducts such that deterioration of roducts can be controlled. Very few of the existing inventory models in the literature assume the xed lifetime of roducts. Wee [] considered roduct's outdate or deterioration as decay, damage, soilage, evaoration, obsolescence, ilferage, loss of utility, or loss of marginal value of a commodity that results in decreasing usefulness. In this direction, Sana [] develoed otimal sellingrice and lot size with time-varying deterioration and artial backlogging. In the aer, he introduced a new time-deendent deterioration function that followed the maximum lifetime of roducts. By using the same time-varying deterioration function, Sarkar [3] develoed a deteriorating inventory model with tradecredit olicy. Sett et al. [4] discussed a two-warehouse inventory model with increasing demand and timevarying deterioration. Sarkar et al. [5] formulated a trade-credit olicy with variable deterioration for xedlifetime roducts. Wu et al. [6] discussed an otimal credit-eriod and lot sizing roblem for deteriorating

2 B. Kumar Sett et al./scientia Iranica, ransactions E Industrial Engineering 3 (06) 38{39 39 items with exiration dates under two-level trade credit nancing. Sarkar et al. [7] derived retailer's otimal strategy for xed-lifetime roducts. Sarkar [8] develoed a suly chain coordination with variable backorder, insections, and discount olicy for xedlifetime roducts. Ghare and Schrader [9] were the rst authors to consider deterioration in inventory model. heir model is a basic inventory model with constant rate of deterioration. Several authors worked on determining inventory model. Covert and Phili [0] relaxed the assumtion of constant deterioration rate by considering a twoarameter Weibull distribution. Further, Phili [] extended the model by assuming variable deterioration rate. Dave and Patel [] invented the deteriorating inventory model with increasing linear demand without shortages. Wee [3] formulated an otimal relenishment olicy for deteriorating items with a linear ricefunction of demand. Goyal and Giri [4] resented a detailed review of the deteriorating inventory literature. Yang and Wee [5] considered an inventory model with nite relenishment rate and rice-sensitive demand for short life cycle and erishable electronic roduct. Law and Wee [6] develoed an integrated roductioninventory model for ameliorating and deteriorating items. hey used the Discounted Cash Flow (DCF) aroach and otimization technique to determine the otimal roduction and relenishment olicy. Chung and Wee [7] discussed a deteriorating inventory model for ricing olicy with imerfect roduction, insection lanning, warranty eriod, and stock-level-deendent demand. Hsu et al. [8] addressed a deteriorating inventory olicy when the retailer invested on the reservation technology to reduce the rate of roduct deterioration. Sarkar and Sarkar [9] and Sarkar et al. [0] extended dierent tyes of inventory model with deterioration. Sarkar [] considered a roductioninventory model for three dierent tyes of continuously distributed deterioration functions. Sarkar and Sarkar [] discussed a control inventory roblem with robabilistic deterioration. hey solved the model with the hel of Euler-Lagrange method. Shah and Cardenas-Barroan [3] analyzed the retailer's decision for ordering and credit olicies when a sulier oered its retailer either a cash discount or a xed credit eriod. On the issue of shortage case, Deb and Chaudhuri [4] were the rst authors to incororate shortage into inventory model. Abad [5] discussed a ricing and lot-sizing roblem for deteriorating roduct by assuming shortages and artial backlogging. Wee et al. [6] discussed an inventory model for ameliorating and deteriorating items with artial backordering under ination. Without using the calculations for otimality from dierential calculus, Cardenas-Barroan [7] solved an economic roduction quantity model with basic algebraic rocedure. aleizadeh et al. [8] considered an inventory model to determine the otimal order and shortage quantities for a erishable item when the sulier oered a secial sale. Sometimes, managers refer to use lanned backorders to reduce the total system cost. In this direction, Cardenas-Barroan [9] resented an inventory model with lanned backorders to determine the economical roduction quantity for a single roduct. Sarkar and Sarkar [30] resented an imroved inventory model with artial backlogging, time-varying deterioration, and stock-deendent demand. Most recently, Sakar et al. [3] extended the inventory model with random defective rate, rework rocess, and variable backorders. Vishkaei et al. [3] used 00% screening rocess in an Economic Order Quantity (EOQ) model under shortages and delay-inayments. Sarkar et al. [33] develoed a continuousreview inventory model with backorder rice-discount under controllable lead time. Recently, Sarkar et al. [34] discussed a deteriorating inventory model for high-tech roducts with artial backlogging. Classical inventory model considers constant demand rate. However, it is observed that the demand rate for electronic goods (e.g., hard disk, RAM, rocesser, mobile, etc.), new brands of consumer goods coming to the market, and seasonal roducts (fruits, e.g. mango, orange, etc.) increases linearly at the beginning u to a certain moment as time increases and then stabilizes to a constant rate until the end of the inventory cycle. o reresent such tye of demand attern, the term \ram-tye" is used. Mandal and Pal [35] were the rst authors to introduce ramtye demand in inventory model. Deterministic and robabilistic demand situations were discussed in that model. Giri et al. [36] develoed a single-item singleeriod EOQ model for deteriorating items with a ramtye demand and Weibull distributed deterioration. Manna and Chaudhure [37] develoed an EOQ model with ram-tye demand rate, time-deendent deterioration rate, unit roduction cost, and shortages. hey assumed that the time oint at which the demand was stabilized occurred before the roduction stoing time. Skouri et al. [38] discussed an inventory model with general ram-tye demand rate, time-deendent deterioration rate, and artial backlogging. he model was studied under two dierent relenishment olicies (a) starting with no shortages, and (b) starting with shortages. Skouri et al. [39] extended Manna and Chaudhuri's [39] model by considering that (a) for the model with no shortages, the demand rate was stabilized after the roduction stoing time, and (b) for the model with shortages, the demand rate was stabilized after the roduction stoing time or after the time when the inventory level reached zero or after the roduction restarting time. Sarkar et al. [40] develoed an inventory model for imerfect roduction

3 30 B. Kumar Sett et al./scientia Iranica, ransactions E Industrial Engineering 3 (06) 38{39 with rice- and time-deendent demand. Pal et al. [4] develoed an inventory model for deteriorating items with ram-tye demand rate. hat model was solved under cris and fuzzy environment to evaluate the otimum solution in dierent cases. We summarize our contribution comared with other models in able. his model considers an inventory system for deteriorating items with ram-tye demand. Every deteriorating item has its exiration date (maximum lifetime), i.e. roducts may deteriorate with increasing value of time. An interesting relation between the variable increasing time and the xed lifetime of roducts is considered in this model. Shortages are allowed, which are fully backlogged. he main urose of this aer is to develo an otimal relenishment olicy which minimizes the total cost er unit time. he necessary and sucient conditions of the existence and uniqueness of the otimal solutions are rovided. Finally, several numerical examles are given to illustrate the theoretical results of this model. Sensitivity analysis of the otimal solution with resect to key arameters and their discursion is given. he rest of the aer is designed as follows in Section, notation and assumtions are given. In Section 3, mathematical model and solution rocedure are derived. Numerical exeriments and sensitivity analysis are resented to illustrate the model in Section 4. Finally, conclusions are made in Section 5.. Notations and assumtions o derive the model, following notations and assumtions are used... Notations Q Order quantity er cycle (units); (t) A C h C C d C s ime-deendent deterioration rate; Ordering cost er order ($/order); Unit inventory holding cost er week ($/unit/week); Purchasing cost er unit urchase ($/unit); Deterioration cost ($/unit deteriorated); Shortage cost ($/unit); Ram-tye demand breaking oint arameter (week); I(t) On-hand inventory level at time t; t Length of time in which the inventory level falls to zero (week); Fixed length of each ordering cycle (week); C (t ) C (t ) otal cost for Model I ($/week); otal cost for Model ($/week)... Assumtions. he model is considered for a single tye of item;. he deterioration rate is assumed as (t) = + t, where > t and is the maximum lifetime of roducts at which the total on-hand inventory deteriorates. When t increases, (t) increases and Limt t! (t)! ; 3. he demand rate R(t) is assumed to be a ram-tye function of time, i.e. R(t) = R 0 [t (t )H(t )] ; R 0 > 0; where H(t H(t ) = ) is the Heaviside's function as follows 8 < if t 0 if t < 4. Holding cost is linear function of time, i.e. C h (t) = h 0 + h t, where h 0, h > 0; 5. Shortages are allowed and fully backlogged; 6. Lead time is assumed as negligible and the relenishment rate is innite. 3. Model formulation he model considers an inventory model for deteriorating items with ram-tye demand and stockdeendent selling rate. he relenishment at the beginning of the cycle brings the inventory level u to I max. he inventory level decreases during the time interval [0; t ] due to combined eects of the demand and deterioration, and falls to zero at t = t. Shortages occur during the eriod (t ; ), which are fully backlogged. During the relenishment cycle [0; ], the inventory level, I(t), satises the following dierential equations = R(t) (t)i(t); 0 t t with I(0) = I max ; () = R(t); t t ; with I(t ) = 0 () o solve the above dierential equations, two cases are considered (a) t, and (b) t. he uctuation of the inventory level for the two cases is deicted in Figures and, resectively.

4 B. Kumar Sett et al./scientia Iranica, ransactions E Industrial Engineering 3 (06) 38{39 3 Author (s) Wee [] Sana [] Sarkar [3] Sett, Sarkar, and Goswami [4] Sarkar, Saren, and Cardenas-Barron [5] Wu, Ouyang, Cardenas-Barron, and Goyal [6] Sarkar, Sarkar, and Yun [7] Sarkar [8] Ghare and Schrader [9] Covert and Phili [0] Phili [] Dave and Patel [] Wee [3] Goyal and Giri [4] Yang and Wee [5] Law and Wee [6] Chung and Wee [7] Hsu, Wee, and eng [8] Sarkar and Sarkar [9] Sarkar, Saren, and Wee [0] Sarkar [] Sarkar and Sarkar [] Shah and Cardenas-Barron [3] Deb and Chaudhuri [4] Abad [5] Wee, Law, Yu, and Chen [6] Cardenas-Barron [7] aleizadeh, Mohammadi, Crdenas-Barrn, and Samimi [8] Cardenas-Barron [9] Sarkar and Sarkar [30] Sarkar, Cardenas-Barron, Sarkar, and Singgih [3] Vishkaei, Pasandideh, and Farhangi [3] Sarkar, Mandal, and Sarkar [33] Sarkar, Sett, Goswami, and Sarkar [34] Mandal and Pal [35] Giri, Jalan, and Chaudhuri [36] Manna and Chaudhuri [37] Skouri, Konstantaras, Paachristos, and Ganas [38] Skouri, Konstantaras, Manna, and Chaudhuri [39] Mahaatra, and Samanta [4] his aer able. Comarison between the contributions of dierent authors. Ram-tye demand Deterioration Maximum ime-deendent Shortage lifetime holding cost

5 3 B. Kumar Sett et al./scientia Iranica, ransactions E Industrial Engineering 3 (06) 38{39 = R 0 t with I( ) = S (5) Solution of Eq. (3) is of the form t I(t) =R 0 ( t) t + ( ) ln + I max ; 0 t t (6) R 0 ( ) Figure. Grahical resentation of the inventory system (Model t ). Considering the boundary condition I(t ) = 0 and Eq. (6), the maximum inventory level for each cycle is I max = R 0 ( ) ( ) ln t t (7) After substituting I max into Eq. (6), it becomes t I(t) =R 0 ( t) ( ) ln t + (t t ) ; 0 t t (8) Solving Eqs. (4) and (5) with the boundary conditions, one has I(t) = R 0 (t t ); t t ; (9) I(t) = R 0 ( t) S; t (0) Figure. Grahical resentation of the inventory system (Model t ). 3.. Model when t, in this case, the demand rate R(t) is R(t) = 8 >< > R 0 t; 0 t t R 0 t; R 0 ; t t t Due to deterioration of items and ram-tye demand rate, the inventory level gradually decreases during the eriod [0; t ] and ultimately falls to zero at time t. herefore, from Eqs. () and (), one has = R 0 t (t)i(t) 0 t t with I(0) = I max ; (3) = R 0 t t t with I(t ) = 0; (4) Considering the continuity of I(t) at t =, it follows from Eqs. (9) and (0) that the maximum amount of demand backlogged er cycle is S = R 0 ( + t ) () Substituting the value of S, Eq. (0) becomes I(t) = R 0 ( + t ) t hus the order quantity Q is Q =I max + S = R 0 ( ) ln t t + ; t () ( ) ( + t ) Now, the total cost er cycle time consists of the following values Ordering cost er cycle OC = A.

6 B. Kumar Sett et al./scientia Iranica, ransactions E Industrial Engineering 3 (06) 38{39 33 Purchase cost er cycle P C = C Q = C R 0 ( ) ( ) ln t + ( + t ) Holding cost er cycle HC = Z t 0 C h (t)i(t) = = R 0 ( ) 3 ln Z t 3t ( ) 6( ) h 0 + (h 0 + h t)i(t) 0 + t t t ( ) 4 6 ln( ) + 3t 4 6t ( ) 3 t 36 3t ( ) t 3 ( ) h Deterioration cost er cycle DC Z t =C d I max R(t) ( ) ln t 0 t = C d R 0 t Shortage cost er cycle SC Z =C s I(t) = C s R 0 t t3 3 ( ) t t herefore, the average total cost er unit time under the condition t is C (t ) = [OC + P C + HC + DC + SC] = R 0 A ( ) 3 + h 0 ln R 0 t + t t 3t ( ) 6( ) ( ) 4 + ln( ) 6 t + 3t 4 6t ( ) 3 3t 36 ( ) t 3 ( ) h (C + C d )( ) ( ) ln( ) t t + C t Cd t + C s 6 + t3 3 t Model when t In this case, the demand R(t) is R(t) = 8 >< > R 0 t; 0 t ; R 0 ; = t t ; R 0 ; t t (3) Hence, Eqs. () and () are reduced to the following equations = (t)i(t) R 0 t; 0 t with I (0) = I max ; (4) = (t)i(t) R 0 ; t t with I (t ) = 0; (5) = R 0 ; t t ; with I 3 (t ) = 0 (6) Utilizing the boundary conditions, the solutions of Eqs. (4)-(6) are as follows t I(t) =R 0 ( t) t + ( ) ln + I max ; 0 t ; (7) R 0 ( ) t I(t) = R 0 ( t) ln ; t t t ; (8) I(t) = R 0 (t t); t t (9) Considering the continuity of I(t) at t =, the maximum inventory level I max is obtained from Eqs. (7) and (8) as I max =R 0 ( ) ln t ( ) ln (0)

7 34 B. Kumar Sett et al./scientia Iranica, ransactions E Industrial Engineering 3 (06) 38{39 Using the value of I max, Eq. (7) is reduced to t I(t) =R 0 ( t) ( ) ln + ln + (t ) () t Putting t = in Eq. (9), the maximum amount of demand backlogged er cycle can be obtained as S I( ) = R 0 ( t ) hus, the order quantity Q is Q =I max + S = R 0 ( ) ln t ( ) ln + R 0 ( t ) Now, the total cost er cycle time consists of the following Ordering cost er cycle OC = A. Purchase cost er cycle P C = C Q =C R 0 ( ) ln t ( ) ln Holding cost er cycle HC = + R 0 ( t ) Z t Z C h (t)i(t) = (h 0 + h t)i(t) 0 0 Z t + (h 0 + h t)i(t) = R 0 h 0 ( ) 4 3t ( ) 4 ( )3 + 6t ( ) + h ( ) + 3h 0 ( ) 6 + ( ) ln ln t ( ) + ( 8 3t )( ) 6( 3 ) 3 + 4t 3 + h (3 6t ) 36 Deterioration cost er cycle DC Z t = C d I max R(t) 0 = C d R 0 ( ) ( ) ln Shortage cost er cycle SC = C s Z t I(t) ln t + (t ) = C sr 0 ( t ) herefore, the average total cost er unit time under the condition t is C (t ) = [OC + P C + HC + DC + SC] = R 0 A + h 0 ( )( ) R ( )3 + ( ) + 6 t + h 36 3t 6t ( ) [h ( ) + 3h 0 ] ln + ( ) ln (3 6t )(+) +( 8 3t ) ( ) 6( 3 ) 3 + 4t (C +C d )(+) ln t ( ) ln C d +(t ) + C ( t ) + C s ( t ) () From the above analysis, one can obtain the total average cost of the system over the interval [0; ] of the form ( C (t ) if t C(t ) = (3) C (t ) if t It is easy to check if this function is continuous at.

8 B. Kumar Sett et al./scientia Iranica, ransactions E Industrial Engineering 3 (06) 38{ Solution rocedure his section rovides the results which can ensure the existence of a unique t to minimize the total average cost for the model. aking rst- and second-order derivatives of C (t ) with resect to t, one has and d C (t ) = (t )f(t ); (4) d C (t ) where f(t ) = On the other hand = (t ) df(t ) + d(t ) f(t ); (5) ( ) 3h 0 + ( )h 6( t ) 6 h ( )( t ) t h 0( t ) + t (C d + C ) t C s ( t ) (6) f(0) = C s < 0; (7) and ( ) f( ) = 6 Further 3h0 + h + 6 (3h 0 + h ) + (C d + C ) > 0 (8) df(t ) =C s + ( ) [3h 0 + ( )h ] 6( t ) + ( )(C d + C ) ( t ) + 3 h t h 0 6 h ( ) > 0 (9) From Eqs. (7)-(9), one can conclude that f(t ) is strictly increasing function of t. herefore, the equation f(t ) = 0; (30) has a unique root t (0; ) for which d C (t ) j t =t = (t ) df(t ) = (t ) df(t ) + d(t ) f(t ) > 0; (3) so that t corresonds to the unconstrained global minimum of C(t ). aking rst- and second-order derivatives of C (t ) with resect to t, one has and d C (t ) = ()f(t ); (3) d C (t ) = () df(t ) > 0; (33) where f(t ) is given by Eq. (6). From the inequality in Eq. (33), following from Eq. (9), one can ensure the strict convexity of C (t ). Based on the roerties of f(t ), it can be concluded that the rst-order derivative of C (t ) with resect to t vanishes at the oint t, which is a unique solution of f(t ) = 0. his t corresonds to the unconstrained global minimum of C (t ). 4. Numerical examles his section considers some numerical examles to check the uniqueness of our solution. Some inut arametric values are considered to obtain the otimum results. 4.. Examle o derive the otimal solution, two examles are given, which consist of dierent situations of the ram-tye demand and deteriorated rates. Let us consider the following arametric values D 0 = 400, A = $50/order, h 0 = $0/unit/unit time, h = $0/unit/unit time, C s = $/unit, C = $5/unit, C d = $5/unit, = 08, = week, and = 5 months. he otimal value of t is t = 0438(< ) week and the minimum cost er unit of time is Z (t ) = $06333/week (see Figure 3). 4.. Examle All the arametric values are identical to Examle, excet for = 0. he otimal value of t is t = 0438(> ) week and the minimum cost er unit of time is Z (t ) = $4374/week (see Figure 4) Sensitivity analysis In this section, the eects of changes in arameters such as A; C s ; C ; C d, h ; h, and L on the total cost are studied. he sensitivity analysis is erformed by changing each of the arameters by -50%, -5%, +5%, and +50%, taking one arameter at a time

9 36 B. Kumar Sett et al./scientia Iranica, ransactions E Industrial Engineering 3 (06) 38{39 able. Eects of changes in the arameters on the total costs for Models and. Parameters Changes (in %) Z(t) Z(t) A -50% % % % Figure 3. Variation of total cost versus time (Examle ). C s C C d -50% % % % % % % % % % % % Figure 4. Variation of total cost versus time (Examle ). while keeing the remaining arameters unchanged. he results are resented in able. On the basis of able, the following features are observed ˆ Increasing value of ordering cost A increases the material cost, shiing cost, and lacing order's cost; as a result, the total relevant cost increases. From able, it is observed that this arameter is a highly a sensitive cost arameter for Model comared to Model. For the ositive and negative changes in the total cost, it follows symmetrical change; ˆ Increasing value of shortage cost indicates more shortages in the system, imlying increase in total cost. From able, one can notice that this arameter is slightly sensitive to total cost for both models; ˆ From able, it is noticed that slight change in urchasing cost results in larger change in total cost for both models; ˆ Increasing values of deterioration cost and holding cost increase the total cost. In both models, these two arameters are slightly sensitive to the total cost; h h L -50% % % % % % % % % % % % ˆ Higher maximum lifetime indicates less deterioration, signifying decrease in the total cost. 5. Conclusions In modern marketing environment, a ractical roblem is to control the deterioration of items. Some roducts (e.g., fruits, vegetables, harmaceutical, volatile liquids, etc.) not only deteriorate continuously due to evaoration, obsolescence, soilage, etc., but also deteriorate with increase in time (i.e., a deteriorating item has a maximum lifetime). In existing literature, very few researchers have considered maximum lifetime of deteriorating items in their model. In this

10 B. Kumar Sett et al./scientia Iranica, ransactions E Industrial Engineering 3 (06) 38{39 37 regard, the research aer considered an inventory model for roducts with maximum lifetime, timevarying deterioration rate, and ram-tye demand. A simle solution rocedure was given, and existence and uniqueness of the otimal solutions were obtained analytically. his model minimized the associated cost function at the otimal values of the decision variable. Finally, the sensitivity analysis on the otimal solution with resect to key arameters was erformed to illustrate the model and some managerial insights were rovided. his model used the concet of xed lifetime of roducts as a time-varying deterioration rate. he industry managers easily use the suggested olicy for their roducts with xed lifetime and timevarying deterioration rate. his model can be extended for items having increasing linear demand, rice, and advertising-deendent demand or ower-demand. his study can be extended further by considering the reservation technology cost for the deteriorating roducts. One another extension can be considered by assuming multi-item inventory model with inations. Acknowledgment he authors are thankful to the reviewers for their helful comments and suggestions to imrove the revious version of this aer. his study was nancially suorted by 0 Post-Doctoral Develoment Program, Pusan National University, Korea. Conict of interest and suort for this research here is no conict of interest regarding the research area by all authors. he third author has received his nancial suort from Post-Doctoral Develoment Program, Pusan National University, Korea, 0. he rest of the authors have not received any fund for this research. References. Wee, H.M. \Economic roduction lot size model for deteriorating items with artialback-ordering", Com. Ind. Eng., 4, (993).. Sana, S.S. \Otimal selling rice and lotsize with timevarying deterioration and artial backlogging", Al. Math. Com., 7, (00). 3. Sarkar, B. \An EOQ model with delay in ayments and time-varying demand", Math. Com. Model., 55, (0). 4. Sett, B.K., Sarkar, B. and Goswami, A. \A twowarehouse inventory model with increasing demand and time varying deterioration", Sci. Iran., 9, (0). 5. Sarkar, B., Saren, S. and Cardenas-Barron, L.E. \An inventory model with trade-credit olicy and variable deterioration for xed lifetime roducts", Anna. Oer. Res., 9, (05). 6. Wu, J., Ouyang, L.Y., Cardenas-Barron, L.E. and Goyal, S.K. \Otimal credit eriod and lot size for deteriorating items with exiration dates under twolevel trade credit nancing", Eur. J. Oer. Res., 37, (04). 7. Sarkar, B., Sarkar, S. and Yun, W.Y. \Retailers otimal strategy for xed lifetime roducts", Int. J. Mach. Learn. Cyber., 7,. -33 (06). 8. Sarkar, B. \Suly chain coordination with variable backorder, insections, and discount olicy for xed lifetime roducts", 06, Article ID , 4 ages (06). 9. Ghare, P.M. and Schrader, G.F. \A model for exonentially decaying inventory system", Int. J. Prod. Res.,, (963). 0. Covert, R.P. and Phili, G.C. \An EOQ model for items with weibull distribution deterioration", AIIE rans., 5, (973).. Phili, G.C. \A generalized EOQ model for items with Weibull distribution deterioration", AIIE rans., 6,. 596 (974).. Dave, U. and Patel, L.K. \(, Si) olicy inventory model for deteriorating items with time roortional demand", J. Oer. Res. Soc., 3,. 374 (98). 3. Wee, H.M. \A relenishment olicy for items with a rice-deendent demand and a varying rate of deteriorating", Prod. Plan. Cont., 8, (997). 4. Goyal, S.K. and Giri, B.C. \Recent trends in modeling of deteriorating inventory", Eur. J. Oer. Res., 34,. -6 (00). 5. Yang, P.C. and Wee, H.M. \A collaborative inventory system with ermissible delay in ayment for deteriorating items", Math. Com. Model., 43,. 09- (006). 6. Law, S. and Wee, H.M. \An integrated roductioninventory model for amelio-rating and deteriorating items taking account of time discounting", Math. Com. Model., 43, (006). 7. Chung, C.J. and Wee, H.M. \An integrated roduction-inventory deteriorating model for ricing olicy considering imerfect roduction, insection lanning and warranty-eriod and stock-leveldeendent demand", Int. J. Syst. Sci., 39, (008). 8. Hsu, P., Wee, H.M. and eng, H. \Preservation technology investment for deterio-rating inventory", Int. J. Prod. Econ., 4, (00). 9. Sarkar, B. and Sarkar, S. \Variable deterioration and demandan inventory model", Econ. Model., 3, (03). 0. Sarkar, B., Saren, S. and Wee, H.M. \An inventory model with variable demand, comonent cost and selling rice for deteriorating items", Econ. Model., 30, (03).

11 38 B. Kumar Sett et al./scientia Iranica, ransactions E Industrial Engineering 3 (06) 38{39. Sarkar, B. \A roduction-inventory model with robabilistic deterioration in two-echelon suly chain management", Al. Math. Model., 37, (03).. Sarkar, M. and Sarkar, B. \An economic manufacturing quantity model with robabilistic deterioration in a roduction system", Econ. Model., 3, (03). 3. Shah, N.H. and Cardenas-Barron, L.E. \Retailers decision for ordering and credit olicies for deteriorating items when a sulier oers order-linked credit eriod or cash discount", Al. Math. Com., 59, (05). 4. Deb, M. and Chaudhuri, K.S. \A note on the heuristic for relenishment of trended inventories considering shortages", J. Oer. Res. Soc., 38, (987). 5. Abad, P.L. \Otimal ricing and lot sizing under conditions of erishability and artial backordering", Manag. Sci., 4, (996). 6. Wee, H.M., Law, S.., Yu, J. and Chen, H.C. \An inventory model for ameliorating and deteriorating items with artial backordering under ination", Int. J. Syst. Sci., 39, (008). 7. Cardenas-Barron, L.E. \Otimal order size to take advantage of a one-time discount oer with allowed backorders", Al. Math. Model., 34, (00). 8. aleizadeh, A., Mohammadi, B., Cardenas-Barron, L.E. and Samimi, H. \An EOQ model for erishable roduct with secial sale and shortage", Int. J. Prod. Econ., 45, (03). 9. Cardenas-Barron, L.E. \Economic roduction quantity with rework rocess at a single-stage manufacturing system with lanned backorders", Com. Indu. Eng., 57, (009). 30. Sarkar, B. and Sarkar, S. \An imroved inventory model with artial backlogging, time-varying deterioration and stock-deendent demand", Econ. Model., 30, (03). 3. Sarkar, B., Cardenas-Barron, L.E., Sarkar, M. and Singgih, M.L. \An economic roduction quantity model with random defective rate, rework rocess and back-orders for a single stage roduction system", J. Manuf. Syst., 33, (04). 3. Vishkaei, B.M., Pasandideh, S.H.R. and Farhangi, M. \00% screening economic order quantity model under shortage and delay in ayment", Scien. Iran. E,, (04). 33. Sarkar, B., Mandal, B. and Sarkar, S. \Quality imrovement and backorder rice discount under controllable lead time in an inventory model", J. Manuf. Syst., 35, (05). 34. Sarkar, B., Sett, B.K., Goswami, A. and Sarkar, S.A. \Mitigation of high-tech roducts with robabilistic deterioration and ination", Amer. J. Indus. Busi. Manag., 5, (05). 35. Mandal, B. and Pal, A.K. \Order level inventory system with ram tye demand rate for deteriorating items", J. Inter. Math.,, (998). 36. Giri, B.C., Jalan, A.K. and Chaudhuri, K.S. \Economic order quantity model with weibull deterioration distribution, shortage and ram-tye demand", Int. J. Syst. Sci., 34, (003). 37. Manna, S.K. and Chaudhuri, K.S. \An EOQ model with ram-tye demand rate, time-deendent deterioration rate, unit roduction cost and shortages", Eur. J. Oer. Res., 7, (006). 38. Skouri, K., Konstantaras, I., Paachristos, S. and Ganas, I.S. \Inventory models with ram-tye demand rate, artial backlogging and Weibull deterioration rate", Eur. J. Oer. Res., 9, (009). 39. Skouri, K., Konstantaras, I., Manna, S.K. and Chaudhuri, K.S. \Inventory models with ram-tye demand rate, time-deendent deterioration rate, unit roduction cost and shortages", Anna. Oer. Res., 9, (0). 40. Sarkar, B., Mandal, P. and Sarkar, S. \An EMQ model with rice and time- deendent demand under the eect of reliability and ination", Al. Math. Com., 3, (04). 4. Pal, S., Mahaatra, G.S. and Samanta, G.P. \An EPQ model of ram-tye demand with Weibull deterioration under in environment", Int. J. Prod. Econ., 56, (04). Biograhies Bimal Kumar Sett is an Assistant Professor of Mathematics at Hooghly Mohsin College, India. He received his BSc and MSc degrees from University of Calcutta. Currently, he is ursuing his PhD in Alied Mathematics from Vidyasagar University, India. He has ublished his research aers in Scientia Iranica, American Journal of Industrial and Business Management, International Journal of Alied and Comutational Mathematics. Sumon Sarkar is a Research Assistant in the Deartment of Industrial & Management Engineering, Hanyang University, Korea. He received his BSc degree in Mathematics from University of Calcutta, India, and MSc degree in Alied Mathematics from Vidyasagar University, India. He has ublished several aers in the Journal of Manufacturing System, Alied Mathematics and Comutation, Economic Modelling, Journal of Industrial and Management Otimization, International Journal of Machine Learning and Cybernetics, and others. Biswajit Sarkar is currently an Assistant Professor in the Deartment of Industrial and Management Engineering of Hanyang University, Korea. He was an

12 B. Kumar Sett et al./scientia Iranica, ransactions E Industrial Engineering 3 (06) 38{39 39 Assistant Professor of Oceanology and Comuter Programming in the Deartment of Alied Mathematics, Vidyasagar University, India, during 00 to 04. He served as an Assistant Professor in the Deartment of Mathematica at Darjeeling Government College, India, during 009 to 00. He received his BSc, MSc, and PhD degrees from Jadavur University, India, in 00, 004, and 00, resectively. He was awarded M.Phil in 008 from Annamalai University, India. His research includes otimization in inventory management, suly chain management, reliability, transortations, and roduction system. He has ublished several research aers in international journals of reute, including Euroean Journal of Oerational Research, International Journal of Production Economics, Alied Mathematical Modelling, Alied Mathematics and Comutation, Annals of Oeration Research, Exert Systems with Alications, Economic Modelling, and others. Presently, he is serving as the Editorial Board Member of several journals. Recently, he became the Editor-in-Chief of the Journal of Engineering and Alied Mathematics, DJ Publications. Won Young Yun is a Professor in the Deartment of Industrial Engineering, Pusan National University, Busan, Korea. He received his ME and PhD from Korea Advanced Institute of Science and echnology (KAIS) in 984 and 988, resectively. His research interests include otimization roblems in system reliability, maintenance, warranty, and inland reositioning in which simulation and meta-heuristics are used to nd otimal solutions.