An EOQ model for weibull distribution deterioration with time-dependent cubic demand and backlogging

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1 IOP Conference Series: Materials Science and Engineering PAPER OPEN ACCESS An EOQ model for weibull distribution deterioration with time-dependent cubic demand and backlogging To cite this article: G Santhi and K Karthikeyan 07 IOP Conf. Ser.: Mater. Sci. Eng Related content - EOQ model for perishable products with price-dependent demand, pre and post discounted selling price G Santhi and K Karthikeyan - A Multiple Items EPQ/EOQ Model for a Vendor and Multiple Buyers System with Considering Continuous and Discrete Demand Simultaneously Jonrinaldi, T Rahman, Henmaidi et al. - The Weibull functional form for SEP event spectra M Laurenza, G Consolini, M Storini et al. View the article online for updates and enhancements. This content was downloaded from IP address on 3/03/08 at :33

2 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 043 doi:0.088/ x/63/4/043 An EOQ model for weibull distribution deterioration with time-dependent cubic demand and backlogging G Santhi and K Karthikeyan Department of Mathematics, School of Advanced Sciences, VIT University, Vellore- 6304, India kkarthikeyan67@yahoo.co.in Abstract. In this article we introduce an economic order quantity model with weibull deterioration and time dependent cubic demand rate where holding costs as a linear function of time. Shortages are allowed in the inventory system are partially and fully backlogging. The objective of this model is to minimize the total inventory cost by using the optimal order quantity and the cycle length. The proposed model is illustrated by numerical examples and the sensitivity analysis is performed to study the effect of changes in parameters on the optimum solutions.. Introduction It is important to maintain and control the inventories of deteriorating items for the modern industries. Deterioration is defined as decay, dryness, spoilage, obsolescence, damage, pilferage, evaporation. Ghare and Schrader [5] initiated an EOQ model for deterioration taking fixed rate. Covert and Philip [] derived an inventory system under conditions of constant demand and instantaneous delivery, the distribution of the time to deteriorating items is represented by weibull distribution. Jalan et al. [7] studied an inventory model for deteriorating items with linearly increasing demand rate and shortages. Wu et al. [5] derived an inventory of items that deteriorate at a weibull distribution and demand rate with a continuous function of time. Giri et al. [6] generalized this model with the ramp-type demand and weibull deterioration by taking single-period EOQ model. Dye [3] derived an economic order quantity model for deteriorating items which follows the weibull distribution. Mukhopadhyay et al. [9] formulated the replenishment a policy with demand rate is used price-dependent. Srichandan et al. [] obtained an inventory system for perishable items under the inflationary conditions and two parameter weibull distributions for deteriorating items. These models consider that, when there is a shortage, all the customers will be waiting for the arrival of the next replenishment (complete backlogging case) or all customers who are not served leave the system (lost sales case). Commonly, in many real-life situations, there are customers willing to wait for the next replenishment until they satisfy their demands, while others do not want to or cannot wait and leave the system. In such situations, the partial backlogging was used to formulate the model. In this line, there are many researchers contributing numerous papers. Abad [] studied the optimal pricing problem of the perishability and shortages. Wee [4] developed an inventory model with a quantity discount, pricing and partial backordering. Wu [6] generalized an EOQ model with Weibull distribution deterioration and shortages are allowed partial backlogging. Skouri and Papachristos [0] formulated an inventory model for deteriorating items, time dependent demand, Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by Ltd

3 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 043 doi:0.088/ x/63/4/043 backlogging rate is an exponentially decreasing. Wu [7] derived this model by taking Weibull distribution deterioration and shortages. Dye and Ouyang [4] presented an EOQ model for perishable with goods stock-level dependent selling rate and shortages are considered partial backlogging rate. Yan [8] established an inventory model for perishable goods with freshness-dependent demand rate and partial backlogging. Valliathal and Uthayakumar [3] described an EOQ model with general ramp type demand, deteriorating of the items and backlogging of unfulfilled demand. Karthikeyan and Santhi [8] presented an inventory model for deteriorating items in which shortages are not allowed and demand rate is a cubic function of time and salvage cost. Recently, Umakanta [] introduced an EOQ model for weibull deterioration with demand rate as quadratic and partially backlogging. They have not considered the effect of cubic demand, complete backlogging and without shortage. In this paper, we developed an economic order quantity model for weibull deteriorating items and partial backlogging. In addition, the effect of cubic demand, complete backlogging and without shortage are considered. The two parameter weibull distribution is used to represent the time to deterioration. Here in this model, the total inventory cost is minimized. Numerical examples are provided for with and without shortage model and sensitivity analysis is performed to show the effect of changes in the parameters of the optimum solution.. Assumptions and notation of the model. Assumptions () The replenishment rate is infinite and the lead time is zero. 3 () The demand rate is time dependent cubic function, i.e., D() t a bt ct dt a 0, b 0, c 0, d 0. (3) T is the length of the replenishment cycle, Q is the order quantity per cycle. t, t andq are decision variables. (4) () t 0 t t in which the product has demand and deterioration. I t : Inventory level at time I t : Inventory level at time t t t t t (5) () in which the product has shortage. (6) The rate of deterioration t t, 0 follows two parameter Weibull distribution, where is the scale parameter, is the shape parameter. The assumed that the deterioration increases with time t 0. t t t t t t is the (7) The unsatisfied demand is backlogged at a rate exp where, time up to the next replenishment and is the backlogging parameter 0.,. Notations To develop this mathematical model, the following notations are used throughout this paper: A : Order cost per unit, C : Purchase cost per unit, HC : Holding cost per unit time, H( t) g h( t), g 0, h 0 C : Shortage cost per unit per unit time, C : Lost sales per unit, C d : Deterioration cost per unit of deteriorated item, : Backlogging parameter is which lies in0 t : Length of time interval with positive or zero inventory level in a replenishment cycle, wheret 0 t : Length of time interval with negative inventory (or shortages) in a cycle, where t 0

4 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 043 doi:0.088/ x/63/4/043 Q : Order quantity per cycle, i.e. Q Q0 BI T : Length of the inventory cycle, It (): Inventory level at time t, I () t : Inventory level at time t, 0 t t I () t : Inventory level at time t, t t t t Q 0 : The maximum inventory level, BI : The maximum inventory level during the stockout period, TC : Total relevant inventory cost, 3. Model formulation 3. The model with partial backlogging Figure.The graphical presentation of inventory level The instantaneous inventory level It () at time t 0 t t t can be modelled by the following differential equations. During the interval 0, t, the inventory is depleted due to the combined effects of demand as well as the deterioration and Shortages occur during the period t, t t. The behavior of inventory in a cycle is depicted in Figure. Hence, the differential equation below represents the inventory status is given by di t 3 t I t a bt ct dt, 0 t t () dt With boundary conditions I (0) Q 0 and It 0, The solution of the equation () is b c 3 3 d 4 4 a t a t a( t t) ( t t ) ( t t ) ( t t ) b t b t c t c t dt dt I() t a tt, 0 t t () b t t b t c t t c t dt t dt a t

5 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 043 doi:0.088/ x/63/4/043 We get the maximum inventory level is bt ct dt at b t ct dt Q0 at (3) Due to shortage during [ t, t t ], the demand at time ' t ' is partially backlogged at rate ( t t t) e. The differential equation governing the amount of demand is backlogged. di () t 3 ( tt t) ( a bt ct dt ) e t t t t (4) dt with boundary condition I( t) 0, the solution of Equation (4) is b c 3 3 d 4 4 a( t t) ( t t ) ( t t ) ( t t ) 3 4 a t ( t t) t( t t) ( t t ) 3 3 I( t) b t( t t ) t( t t ) ( t t ) c t( t t ) t( t t ) ( t t ) d t( t t ) t( t t ) ( ) t t t t t t (5) The maximum backlogged inventory BI is obtained at t t t then from equation (5) 3 t t at btt ct t tt t t t t tt t BI I( t t) d t t tt a b t t tt t t t t t tt t c d From equations (3) and (6), we have Q Q BI bt ct dt a t b t c t dt at at t t 3 3 3t t t t Q b tt c t t tt d t t tt a t t t t t tt t t t t t tt t b c d (6) (7) Cost components: The cost components per cycle of the total inventory cost are follows Ordering cost ( OC) A 4

6 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 043 doi:0.088/ x/63/4/043 t Holding cost ( HC) ( g ht) I( t) dt agt bgt cgt dgt agt bgt cgt ( ) ( 3) ( 4) dgt agt bgt cgt dgt ( 5) ( )( ) ( )( 3) ( )( 4) ( )( 5) HC aht bht cht dht a ht bht cht ( 3) ( 4) ( 5) dht a ht b ht c ht dht ( 6) ( )( 3) ( )( 4) ( )( 5) ( )( 6) The cost of deterioration per cycle DC is t 3 DC C d Q0 a bt ct dt dt a t b t c t dt DC Cd 3 4 The cost of shortage per cycle due to backlogged SC is t t SC C I( t) dt, t t t t t t tt t t t tt t a b c t t t t tt t t tt t C d a b t t tt t t t t t tt t c d The cost due to lost sales LS is given by tt 3 tt t LS C a bt ct dt e dt t t t t t a t b t t b t c t t c t dt t dt LS C a tt The purchase cost per cycle ( PC) is given by PC CQ bt ct dt a t b t c t dt at at t t 3 3 3t t t t PC C b tt c t t tt d t t tt a t t t t t tt t t t t t tt t b c d (8) (9) (0) () () 5

7 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 043 doi:0.088/ x/63/4/043 The total annual cost per unit time is given by TC OC HC DC SC LS PC T agt bgt cgt dgt agt bgt cgt ( ) ( 3) ( 4) dgt agt bgt cgt dgt ( 5) ( )( ) ( )( 3) ( )( 4) ( )( 5) A aht bht cht dht a ht b ht c ht ( 3) ( 4) ( 5) dht a ht b ht c ht dht ( 6) ( )( 3) ( )( 4) ( )( 5) ( )( 6) 3 4 a t b t c t dt Cd t tt t t t tt t a b c t t t t tt t t tt t TC C d a b t t t t tt t t t t t tt t c d a t b tt b t c tt c t d tt d t C a tt bt ct dt a t b t c t dt at at t t 3 3 3t t t t C btt c t t tt d t t tt a tt t t t tt t t t t t tt t (3) b c d The conditions for minimization of the total cost ( TC) per unit time are TC TC =0 and =0 t t Providing that the optimal values for t and tobtained from equation (4) satisfies the sufficient condition TC TC TC 0 tt t t (4) 3.. The special case of completely backlogging 6

8 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 043 doi:0.088/ x/63/4/043 Due to shortage during [ t, t t ], the demand at time ' t ' is completely backlogged. Hence, the inventory level can be represented by the following differential equation: di () t 3 ( a bt ct dt ), t t t t dt with the boundary condition I( t) 0 b c 3 3 d 4 4 I( t) a( t t) ( t t ) ( t t ) ( t t ) 3 4 The maximum backlogged inventory BI is obtained at t t t then from equation (5) 3 4 t t 3 3 3t t t BI I( t t) at btt ct t tt d t t tt 3 4 Hence the order quantity during 0, t t Q Q BI bt ct dt a t b t c t dt at Q 3 4 t t 3 3 3t t t at btt ct t tt d t t tt 3 4 (5) (6) (7) The total shortage cost during interval [ t, t t ] is tt SC C I( t) dt, t t t t t t tt t t t tt t t t t t tt t C a b c d Purchase cost per cycle ( PC) CQ (8) b c 3 3 d 4 4 a t a( t t ) ( t t ) ( t t ) ( t t ) a t d b t b t c t d ct d 3 3 PC CQ 4 4 dt dt b t d t at d t a t 4 4 d d b t d c t t d c t d dt t d dt d (9) Hence, the total relevant cost per unit time is given by TC OC HC DC SC PC T 7

9 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 043 doi:0.088/ x/63/4/ agt bgt cgt dgt agt bgt cgt ( ) ( 3) ( 4) dgt agt bgt cgt dgt ( 5) ( )( ) ( )( 3) ( )( 4) ( )( 5) A aht bht cht dht a ht b ht c ht ( 3) ( 4) ( 5) dht a ht b ht c ht dht ( 6) ( )( 3) ( )( 4) ( )( 5) ( )( 6) 3 4 a t b t c t dt TC Cd t t t tt t t t tt t a b c 6 3 C t t t t tt t d bt ct dt a t b t c t dt at at C 3 4 t t 3 3 3t t t b tt c t t tt d t t tt 3 4 The conditions for minimization of the total cost ( TC) per unit time are TC TC =0 and =0 () t t Providing that the optimal values for t and tobtained from equation () satisfies the sufficient condition TC TC TC 0 tt t t (0) 3.3. Without shortage model Figure. The graphical presentation of inventory level 8

10 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 043 doi:0.088/ x/63/4/043 The decreases of the inventory level occurs due to demand and deterioration during the time 0, T. Hence, the differential equation below represents the inventory status is given by di t I t D( t), 0 t T dt Now putting the value of in above equation we get, di t 3 t I t a bt ct dt, 0 t T () dt with boundary conditions I (0) Q and I T 0, The solution of the equation () is b c 3 3 d 4 4 at at a( T t) ( T t ) ( T t ) ( T t ) bt b t ct ct dt dt I() t att, 0 t T (3) bt t bt ct t ct dt t dt a t We get the optimum order quantity is given by bt ct dt at bt ct dt Q at (4) Cost components The cost components per cycle of the total inventory cost are follows Ordering cost ( OC) A Holding cost per cycle ( HC) ( g ht) I( t) dt T agt bgt cgt dgt agt bgt cgt ( ) ( 3) ( 4) dgt agt bgt cgt dgt ( 5) ( )( ) ( )( 3) ( )( 4) ( )( 5) HC aht bht cht dht aht bht c ht ( 3) ( 4) ( 5) dht a ht b ht c ht dht ( 6) ( )( 3) ( )( 4) ( )( 5) ( )( 6) The cost of deterioration per cycle DC is T 3 DC C d Q a bt ct dt dt at bt ct dt DC Cd 3 4 The purchase cost per cycle ( PC) is PC CQ (6) (5) 9

11 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 043 doi:0.088/ x/63/4/ bt ct dt at bt ct dt C PC at TC OC HC DC PC T (7) agt bgt cgt dgt agt bgt cgt ( ) ( 3) ( 4) dgt agt bgt cgt dgt ( 5) ( )( ) ( )( 3) ( )( 4) ( )( 5) A HC aht bht cht dht aht bht c ht ( 3) ( 4) ( 5) dht a ht b ht c ht dht ( 6) ( )( 3) ( )( 4) ( )( 5) ( )( 6) 3 4 at bt ct dt TC Cd (8) T bt ct dt at bt ct dt C at The condition for minimization of the total cost ( TC) per unit time is dtc =0 dt The condition is also satisfied for the valuet from equation (9) d TC > 0 for all T 0 dt (9) 4. Numerical illustrations Example.Partially backlogging model Let A 500, a 5, b 9, c 5, d 0, g 0.9, h 0.4, 0.5,, C 0.8, C 0.5, C 0.6, Cd 0., 0.5. We get the optimal values aret 0.053, t.4538, Q and TC Example. Completely backlogging model Let A 500, a 5, b 9, c 5, d 0, g 0.9, h 0.4, 0.5,, C 0.8, C 0.5, C 0.6, Cd 0.. We get the optimal values are t 0.383, t.843, Q.5 and TC Example 3. Without shortage model 0

12 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 043 doi:0.088/ x/63/4/043 Let A 500, a 5, b 9, c 5, d 0, g 0.9, h 0.4, 0.5,, C 0.6, Cd 0.. The optimal solutions aret.63, Q and TC Sensitivity analysis This section, we now study the effect of changes in the major parameters such as A, a, b, c, d,,, h, g, C, C, Cd, C and on T, Q, andtc in the EOQ model of the Examples, and 3. We change one parameter at a time and keep the other parameter remains unchanged to examine the sensitivity analysis. The results are summarized in Tables, and 3 respectively. Table. Sensitivity analysis of the partially backlogging model Parameters Optimum values t Q TC t A a b c d g h

13 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 043 doi:0.088/ x/63/4/ C C C d C Observations: The following observations are based on the results of Table- ) When parameters A increases, t, Q and TC increases. ) When parameters a, b, c and d are increases, t is decreases while t,q and TC increases. 3) When parameters and are increases, t, t and Q decreases while TC increases. 4) When the value of the model parameter C and C increases, t and TC increases while t and Q decreases. 5) When parameter g, h, C, and decreases. C d are increases, t andtc increases while t and Q Table. Sensitivity analysis of the completely backlogging model Parameters Optimum values t Q TC t

14 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 043 doi:0.088/ x/63/4/043 A a b c d g h C

15 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 043 doi:0.088/ x/63/4/043 Cd C Observations: The following observations are based on the results of Table- ) When the value of the model parameters A increases, t, Q and TC increases. ) When the value of the model parameters a, b, c and d are increases, t is decreases while t,q and TC increases. 3) When parameters and are increases, t, t and Q decreases while TC increases. 4) When parameter C increases, t and TC increases while t and Q decreases. 5) When parameter g, h, C and C d are increases, t andtc increases while t and Q decreases. Table 3. Sensitivity analysis of the without shortage model Parameters Optimum values T Q TC A a b c d

16 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 043 doi:0.088/ x/63/4/043 g h Cd C The following observations are based on the results of Table Observations ) When the value of the model parameters A increases, T, Q and TC increases. ) When parameters a, b, c and d are increases, T is decreases while Q and TC increases. 3) When parameters and are increases, T and Q decreases while TC increases. 4) When parameter g, h, C and C are increases, T andtc increases whileq decreases. Three dimensional graphs are shown in the following Figures 3, 4 and 5 d Figure 3 shows that in partially backlogging model, while increasing the Backlogging rate obtain that the Order quantity Q decreases and the total cost increases. we Figure 4 shows that in completely backlogging model, while increasing Ordering cost A we obtain that the Order quantity Q and the total cost are increases. Figure 5 shows that in without shortage model, while increasing Ordering cost A we obtain that the Order quantity Q and the total cost are increases. 5

17 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 043 doi:0.088/ x/63/4/043 Figure 3. Total cost versusq and Figure 4. Total cost versusq and A 6

18 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 043 doi:0.088/ x/63/4/043 Figure 5. Total cost versusq and A 6. Conclusion In this article, we developed an economic order quantity (EOQ) model for deteriorating items with time-dependent demand and two-parameter weibull distribution for deteriorating items. Shortages are allowed in the inventory system are complete and partial backlogging model. The numerical examples have been given to illustrate it. Finally, the sensitivity analysis shows the changes in the values of different parameters. Our research result implies that, total cost per unit time of partial backlogging is more than the complete backlogging which is clear from the numerical examples and sensitivity analysis. In future we extend the study further the proposed model can be incorporate into more realistic assumptions, such as stock-dependent demand, price-dependent demand, price discount, quantity discount. References [] Abad P L 996Management Science [] Covert R P and Philip G C 973 Aiie Transactions [3] Dye C Y 004 Information and Management Sciences [4] Dye C Y and Ouyang L Y005European Journal of Operational Research [5] Ghare P MandSchrader G H 963Int. J. Production Research [6] Giri B C, Jalan A K and Chaudhuri K S 003Int. J. Syst. Sci [7] Jalan A K, GiriR R and ChaudhuriK S996Int. J. Syst. Sci [8] Karthikeyan KandSanthi G 05Int. J. Applied Engineering Research [9] Mukhopadhyay S, Mukherjee R N and Chaudhuri K S 005Int. J. Mathematical Education in Science and Technology [0] Skouri K and PapachristosS 00Applied Mathematical Modelling [] Srichandan M, Umakanta M, Gopabandhu M, BarikS and Paikray S K 0Int.J. Advances in Engineering and Technology [] Umakanta M05Int. J. Applied and Computational Mathematics [3] Valliathal MandUthayakumar R03Yugoslav Journal of Operations Research [4] Wee H M 999Int. J. Production Economics

19 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 043 doi:0.088/ x/63/4/043 [5] Wu J W, Lin J W C, Tan B and Lee W C 000Int. J. Syst. Sci [6] Wu K S 00 Production Planning and Control [7] Wu K S 00 Int. J. Systems Science [8] Yan X 0 Int. J. Control and Automation