A Comparative Study Between Inventory Followed by Shortages and Shortages Followed by Inventory Under Trade-Credit Policy

Int. J. Appl. Comput. Math 05 :399 46 DOI 0.007/s4089-05-004-z ORIGINAL PAPER A Comparative Study Between Inventory Followed by Shortages Shortages Followed by Inventory Under Trade-Credit Policy S. Khanra Buddhadev Mal B. Sarkar Published online: January 05 Springer India Pvt. Ltd. 05 Abstract This paper deals with a comparison between inventory followed by shortages model shortages followed by inventory model with variable dem rate. It is assumed that the stock deteriorates over time which follows a two parameter Weibull distribution. Both the models are assumed fixed trade credit period to the retailer from the supplier. The model is solved analytically the results are illustrated with numerical examples. Keywords Comparative study Variable dem Probabilistic deterioration Permissible delay-in-payments Introduction The basic square root formula for economic order quantity EOQ was used in the inventory literature for a pretty long time under the assumption of constant dem. In real market the dem rate of any product is always in a dynamic state. The modification from constant dem to time varying dem was first studied by Silver Meal []. The first analytical model for the classical no-shortage inventory policy with a linear time-dependent dem was developedby Donaldson []. Deb Chaundhuri [3] were the first to incorporate shortages into the inventory lot-sizing problem with a linearly increasing time varying dem. Dr. Biswajit Sarkar is in leave on lien from Vidyasagar University. S. Khanra Department of Mathematics, Tamralipta Mahavidyalaya, Purba Medinipur, Tamluk 7 636, India B. Mal Department of Applied Mathematics with Oceanology Computer Programming, Vidyasagar University, Midnapore 7 0, West Bengal, India B. Sarkar B Department of Industrial & Management Engineering, Hanyang University, Ansan, Gyeonggi-do 46 79, South Korea e-mail: bsbiswajitsarkar@gmail.com 3

400 Int. J. Appl. Comput. Math 05 :399 46 In recent years, inventory problems for deteriorating items have been widely studied after Ghare Schrader [4]. They developed an inventory model for an exponentially deterioration. Later, Covert Philip [5] formulated the inventory model with variable deterioration rate with two-parameter Weibull distribution. In recent Sarkar Sarkar [6,7] improved an inventory model for variable deteriorating items. Philip [8] then developed the inventory model with a three-parameter Weibull distribution rate without shortage. Shah [9] extended Philip s [8] model with shortage was allowed. All these models deal with a replenishment policy that allows shortage in all cycles except the last one. Each of the cycles during which shortages permitted starts with replenishment ends with a shortage. These replenishment system is known as inventory followed by shortage IFS policy of replenishment, Goyal et al. [0] derived a new-replenishment policy in which shortages are permissible in every cycle. In this policy, every cycle starts with a shortage until replenishment is made followed by a period of positive inventory. It is known as shortage followed by inventory SFI policy. Several researchers started working on inventory models with time varying dem for items which undergo decay or deteriorating items over time. Some models for deteriorating items with trended dem IFS policy of replenishment are Goswami Chaudhuri [] Hariga []. The inventory models with shortage time varying dem Weibull distributed deterioration were studied by Ouyang et al. [3]. Using basic algebraic procedure Goyal Cárdenas-Barrón [4] derived several simple expressions for the total average cost of inventory items with shortages. Sarkar et al. [5] obtained an inventory model with trade-credit policy where deterioration was considered as variable for fixed lifetime products. In the conventional EOQ model, it was assumed that the customer must pay for the item as soon as it is received. In practice, however the supplier offers the retailer a certain trade credit period for paying the purchasing cost. During this delay period, the retailer can earn revenue by selling items earning interest. An inventory model with permissible delay in payments was first studied by Goyal [6]. Aggarwal Jaggi [7] determinedaneoq model with constant deterioration under permissible delay in payments. Sarkar [8], Chung Cárdenas-Barrón [9] studied two inventory models for deterioration under permissible delay in payments. Pal et al. [0] developed an integrated inventory model under three levels of trade credit policy. Chung Huang [] presented a mathematical model on EOQ for constant dem, shortages under permissible delay in payments. Pal et al. [] investigated an inventory model with price credit period dependent dem. Mahata [3]developed an EOQ model with considering time dependent linear dem shortages under permissible delay-in-payments. Singh Pattnayak [4], Singh Singh [5] considered two inventory models for both time-varying dem deterioration under permissible delay in payments. Amutha Chrasekaran [6] obtained an inventory model with constant dem, shortages, time varying deterioration rate under permissible delay in payments. An Bansal [7] developed an EOQ model for both time varying dem deterioration rate with shortages under permissible delay in payments. Several valuable contributions in this field were studied by Sarkar Sarkar [8], Sarkar Saren [9,30], Sana Chaudhuri [3], Ouyang et al. [3], Sana [33], Khanra et al. [34,35], Das Roy et al. [36], Sarkar [37,38]. Sarkar et al. [39] developed an economic order quantity EOQ model for various types of time-dependent dem where delay-in-payments price discount are permitted by suppliers to retailers. Sarkar Sarkar [6] formed an inventory model with partial backlogging time varying deterioration where dem is considered as stock-dependent. Wu et al. [40] described an inventory model for deteriorating items with expiration dates under two-level trade-credit financing. Chung et al. [4] addressed an integrated three layer supply chain system with non-instantaneous receipt exponentially deteriorating items under two levels of trade-credit policy. Chen et al. [4] obtained retailer s economic order quantity 3

Int. J. Appl. Comput. Math 05 :399 46 40 while the supplier offers conditionally permissible delay-in-payments link to order quantity. Sarkar et al. [43] produced an integrated inventory model by considering variable lead time, defective units, delay in payments. Sarkar Moon [44] developed the relationship between quality improvement, reorder point, lead time as affected by backorder rate in an imperfect production process. Sarkar et al. [45] considered an economic production quantity EPQ model with rework process at a single-stage manufacturing system with planned backorders. Three different inventory models are formulated for three different distribution density functions such as uniform, triangular, beta. Sarkar et al. [46] discussed an inventory model to improve the process quality backorder price discount with controllable lead time. Authors Linear dem Other dem IFS SFI Variable deterioration Trade credit Donalson [] Deb Chaudhuri [3] Covert Philip [5] Goyaletal.[0] Goswami Chaudhuri [] Hariga [] Aggarwal Jaggi [7] Chung Huang [] Mahata [3] Singh Pattnayak [4] Singh Singh [5] Amutha Chrasekaran [6] An Bansal [7] This Paper In this paper, a comparison for the effectiveness between inventory followed by shortages IFS shortages followed by inventory SFI have been discussed. The models are developed for deterioration items with linear trend in dem, shortages permissible delay-in-payments. When the dem rate for an item increases or decreases steadily with time, the dem rate is linear function of time of the form Dt = a bt, a > 0, b = 0. However, b = 0 gives the dem rate as constant. The deterioration rate follows a two parameter Weibull distribution. The model has been developed under two categories five circumstances as follows: Case IFS policy the credit period is less than the time of commencement of shortage. Case IFS policy the credit period is greater than the time of commencement of shortage period but less than cycle length. Case 3 IFS policy the credit period is greater than the total cycle time for settling the account. Case 4 SFI policy credit period is greater than commencement of shortages but less than cycle time Case 5 SFI policy credit period is greater than cycle time for settling the account. Finally, the model is illustrated with some numerical examples. Also justification of taking the SFI policy rather than IFS policy for managerial point of view is analyzed. Mathematical Model The following assumptions are made to develop this model. 3

40 Int. J. Appl. Comput. Math 05 :399 46 i The inventory system involves only single type of item. ii Replenishment occurs instantaneously. iii The dem rate for the item is represented by a linear continuous function of time. iv Shortages are considered completely backlogged. v Delay-in-payments are considered. vi No interest is to be charged after commencement of shortages. vii No interest is to be earned after permissible delay periods. viii The deterioration of products follows a Weibull distribution. ix The planning period is of infinite length. The following notation is used to develop this model. Decision variables T length of the replenishment cycle year T time when inventory level comes down to zero year Parameters m permissible delay in setting the account year I t inventory level at time t k ordering cost of inventory per order $/order h unit holding cost per unit time excluding interest charges $/unit/unit time c unit cost incurred from the deterioration of one item $/unit/unit time s unit shortage cost per item $/item unit short p unit purchase cost per item $/item Dt time dependent dem rate unit/year Dt = a bt, a > 0, b = 0, here a is initial rate of dem, b is the rate with which the dem rate increases. Zt two-parameter Weibull deterioration Zt = αt, 0 <α<<, >, here α is the scale parameter, is the shape parameter. I e interest earned per year $/year interest charges per year, I r I e $/year I r Shortages Followed by Inventory The inventory level I t at time t generally decreases from I 0 to meet market s dem product s deterioration reaches to zero at T. Thus shortages accumulate over [T, T ]. Hence, the variation of inventory with respect to time can be described by the governing differential equation dit ZtI t = a bt, 0 t T, with I 0 = I 0 dt dit = a bt, T t T, with I T = 0 dt The deterioration rate Zt = αt, 0 <α<<, >, t > 0 3 By virtue of 3, becomes dit I t αt = a bt, 0 t T 4 dt 3

Int. J. Appl. Comput. Math 05 :399 46 403 The solution of 4is I t = [See Appendix for value of I 0 ] The solution of is I 0 at b t aα t ξ bα t ξ exp αt, 0 t T 5 ξ ξ I t = at t b T t, T t T 6 The total deteriorated items are I 0 T 0 a btdt = I 0 at b T 7 The shortage cost SC over the time interval [T, T ] is T SC = s T a I tdt = s T T b 6 T 3 T 3 at T b T T at b T 3 The holding cost HC in [0, T ] is given by HC = h T 0 a I tdt = h T b 3 T 3 aσ T ξ bσ T ξ 3 aα T ξ 5 bα T ξ 6 ξ ξ ξ ξ 3 ξ ξ 5 ξ ξ 6 [See Appendix for all values of ξ i, i =,,...6] The total deterioration cost DC is given by 8 9 DC = c I 0 at b T 0 The permissible delay period m is settled by the whole-seller or distributor to the retailer or customer. Thus, three models can be developed depending on different values of m, T, T. Case Let m T < T Fig.. As the length of the period with positive stock is larger than credit period, the buyer can used sales revenue to earn interest at an annual rate I e in [0, m]. The interest earned E is given by E = pi e m 0 tdtdt = pi e a m b 3 m3 3

404 Int. J. Appl. Comput. Math 05 :399 46 Case. Let m T T. Inventory I 0 m T T Time I Fig. Inventory versus time for Model I m T < T Beyond the credit period, the unsold stock is supposed to be financed with an annual rate I r. L = pi r T m I tdt = pi r I 0 T m a T m b 6 T 3 m3 I 0α ξ T ξ m ξ aσ ξ ξ T ξ m ξ bσ ξ ξ 3 T ξ 3 m ξ 3 aα ξ ξ 5 T ξ 5 m ξ 5 bα ξ ξ 6 T ξ 6 m ξ 6 Therefore, the total average cost per unit time is given by Z T, T = T [k DC HC SC L E ] = [k c I 0 at b a T T h aα T ξ 5 bα T ξ 6 ξ ξ 5 ξ ξ 6 T b 3 T 3 aσ T ξ bσ T ξ 3 ξ ξ ξ ξ 3 sη pi r I 0 T m a T m b 6 T 3 m3 I 0α T ξ m ξ aσ T ξ m ξ bσ T ξ 3 m ξ 3 ξ ξ ξ ξ ξ 3 aα T ξ 5 m ξ 5 bα T ξ 6 m ξ 6 a pi e ξ ξ 5 ξ ξ m b3 ] m3 6 Our aim is to obtain the minimum average cost per unit time. We now take the following theorem which was established by [35]. Theorem If a function V T, T = T GT, T where GT, T admits continuous partial derivatives of second order then V T, T is minimum at T = T, T = T, if all principal minors are positive definite i.e., if GT, T GT,T GT,T > 0 GT,T GT,T > 0 [See Appendix for proof.] 3

Int. J. Appl. Comput. Math 05 :399 46 405 Case. Let T m T. Inventory I 0 T m T Time I Fig. Inventory versus time for Model II T < m < T Lemma Z T, T has the minimum value for those values of T T which satisfy following equations i c λ h at bt aσ h T ξ bσ T ξ aα ξ ξ sλ pλ 3 I r T m α T ξ m ξ ξ ii k c λ 4 hλ 5 sη pi r λ 6 pi e λ 7 Tsλ 8 = 0 provided the following conditions are satisfied bσξ iii c λ 9 hλ 0 h pi r b λ 4 T ξ aα ξ 4 ξ ξ T m αξ T ξ m ξ T bα ξ 5 T ξ 4 bα T ξ 5 ξ ξ I 0 αt τ = 0 sλ T ξ 4 ξ > pαi r λ 3 T I 0σ T a bt [ bσξ c λ 4 hλ 5 h T m αξ T ξ m ξ iv > sa bt [See Appendix for all values] [See Appendix 3 for proof.] T ξ aα ξ 4 ξ ξ T bα ξ 5 T ξ 4 sλ 6 pi r b λ 4 ξ ] pαi r λ 3 T I 0σ T a bt a bt Case Let T < m < T Fig.. In this case, the buyer pays no interest but earns interest at an annual rate I e. 3

406 Int. J. Appl. Comput. Math 05 :399 46 During the period [0, m] the interest earned E is given by E = pi e T 0 tdtdt m T pi e a = pi e T b 3 T 3 = pmi e at b T T pi e m T 0 Dtdt at b T a pi e T b 6 T 3 Therefore the total average cost per unit time is Z T, T = T [k DC HC SC E ] [ k c λ 4 h = T We now minimize Z T, T. a T b 3 T 3 aσ T ξ bσ T ξ 3 aα T ξ 5 ξ ξ ξ ξ 3 ξ ξ 5 ] bα T ξ 6 a sη pmi e τ pi e ξ ξ 6 T b 6 T 3 3 Lemma Z T, T has the minimum value for those values of T T which satisfy the following equations a i c λ h T b 3 T 3 aσ T ξ bσ T ξ 3 aα T ξ 5 bα T ξ 6 ξ ξ ξ ξ 3 ξ ξ 5 ξ ξ 6 sλ pi e τ pmi e a bt = 0 ii k c λ 4 hλ 5 sη pi e λ pmi e τ Tsλ 8 = 0 provided the following conditions are satisfied bσξ iii c λ 4 hλ 5 h T ξ aα ξ 4 ξ > pbmi e ξ T bα ξ 5 T ξ 4 ξ sλ 6 pi e a bt [ bσξ iv c λ 4 hλ 5 h T ξ aα ξ 4 T bα ξ 5 T ξ 4 sλ 6 pi e a bt ξ ξ ξ pbmi e ] a bt >sa bt [See Appendix 4 for proof.] Case 3 Let m T Fig. 3. In this case, the customer earns interest on the sales revenue upto the permissible delay period does not pay interest during the period for the time period [0, T ] is 3 pi e T 0 a tdtdt = pi e T b 3 T 3

Int. J. Appl. Comput. Math 05 :399 46 407 Case 3. Let m T. Inventory I 0 T T m Time I Fig. 3 Inventory versus time for Model III m T Interest earned for the permissible delay period [T, m] is m T pi e T 0 Dtdt = pi e m T at b T Therefore the total interest earned during the cycle is a E 3 = pi e T b 3 T 3 pi e m T at b T = pmi e τ pi e λ 4 Hence the total average cost per unit time is Z 3 T, T = T [k DC HC SC E 3] [ k c λ 4 h = T We minimize Z 3 T, T. a T b 3 T 3 aσ T ξ bσ T ξ 3 aα T ξ 5 ξ ξ ξ ξ 3 ξ ξ 5 bα T ξ 6 sη pi e λ pmi e τ] 5 ξ ξ 6 Lemma 3 Z 3 T, T has the minimum value for those values of T T which satisfy the following equations a i c λ h T b 3 T 3 aσ T ξ bσ T ξ 3 aα T ξ 5 bα T ξ 6 sλ pi e τ ξ ξ ξ ξ 3 ξ ξ 5 ξ ξ 6 pmi e a bt = 0 ii k c λ 4 hλ 5 sη pi e λ pmi e τ Tsλ 8 = 0 provided they satisfy the following conditions bσξ iii c λ 4 hλ 5 h T ξ aα ξ 4 T bα ξ 5 T ξ 4 sλ 6 pi e a bt ξ ξ ξ > pbmi e 3

408 Int. J. Appl. Comput. Math 05 :399 46 [ bσξ iv c λ 4 hλ 5 h T ξ aα ξ 4 T bα ξ 5 T ξ 4 sλ 6 pi e a bt ξ ξ ξ pbmi e ] a bt >sa bt [See Appendix 5 for proof.] Inventory Followed by Shortages The initial stock is zero. Shortages begin to accumulate over [0, T ].AtT, the replenishment takes place. The inventory gradually decreases during [T, T ] after adjusting existing dem deterioration it becomes zero at the end of cycle time T. The instantaneous state of the inventory level I t at any time t is governed by the differential equations dit dt dit dt The solution of 6is = a bt, 0 t T, with I 0 = 0I T = I 6 I tzt = a bt, T < t < T, with I T = 0 7 I t = at b t, 0 t T 8 The deterioration rate Zt follows a two-parameter Weibull distribution as By virtue of 0, 8 becomes Zt = αt,α >0, >0, t > 0 9 dit I tαt = a bt, T t < T 0 dt Hence, the solution of 5is I t = at t b T t aα T ξ t ξ bα T ξ t ξ ξ ξ [See Appendix for value of I ] The total deteriorated items are exp αt, T t < T T I a btdt = I at b T The shortage cost SCover the time interval [0, T ] is T 3 SC = s T 0 a I tdt = s T b 6 T 3 3

Int. J. Appl. Comput. Math 05 :399 46 409 Case 4. T m T. Inventory I 0 T m T Time Fig. 4 Inventory versus time for Model IV T < m < T The holding cost HC in [0, T ] is given by T HC = h T aα I tdt = h T ξ T ξ ξ bα T ξ 3 T ξ 3 ξ aα T ξ 5 T ξ 5 3 ξ ξ 5 bα ξ ξ 6 T ξ 6 T ξ 6 h bα T ξ 3 T ξ 3 ξ ξ 3 The total deterioration cost DC is given by DC = c I at b T = a T T b 6 T 3 T 3 aα ξ ξ T ξ T ξ hαμ T ζ T ζ 4 aα ξ T ξ bα αt aα T ξ 4 bα T ξ 5 ξ ξ T ξ ξ αt λ 8 5 Case 4 T < m < T Fig. 4. In this case, buyer pays no interest but earns interest at an annual rate I e during the period [T, m]. Interest earned E 4 is given by E 4 = pi e m T tdtdt = pi e a m T b 3 m3 T 3 Beyond the credit period, the interest payable L 4 is given by T L 4 = pi r I tdt = pi r T m at b T aα T ξ bα T ξ ξ ξ m a pi r T m b 6 T 3 m 3 aα T ξ m ξ bα T ξ 3 m ξ 3 ξ ξ ξ ξ 3 Therefore, the total average cost per unit time is given by 6 7 3

40 Int. J. Appl. Comput. Math 05 :399 46 Z 4 T, T = T [k DC HC SC L 4 E 4 ] = [k c αt μ c μ c αt aα T λ 8 c T ξ 4 bα T ξ 5 ξ ξ aα h T ξ T ξ ξ bα T ξ 3 T ξ 3 ξ aα T ξ 5 T ξ 5 3 ξ ξ bα T ξ 6 T ξ 6 5 ξ ξ 6 a h T T b 6 T 3 T 3 aα T ξ T ξ ξ ξ bα T ξ 3 T ξ 3 ξ ξ 3 hαμ T ζ T ζ a sλ pi r μ pi e m T b 3 m3 T 3 a pi r T m b 6 T 3 m 3 aα T ξ m ξ bα ] T ξ 3 m ξ 3 ξ ξ ξ ξ 3 Now we have minimize Z 4 T, T. For this purpose we consider the following lemma. Lemma 4 Z 4 T, T has the minimum value for those values of T T which satisfy the following equations i c μ 3 c σ T ζ λ 8 μ c μ 4 c αt a bt hμ 5 h τ μ 6 hαζ μ T ζ s a bt pi e at bt = 0 ii M Tc μ 5 αt T μ 8 pi r hα T ζ T ζ = 0 provided they satisfy the following sufficient conditions 8 Tc αt a bt hμ 7T T μ pi r h hαζ T ζ iii c μ 9 c αζ T ζ λ 8 μ c σ T ζ a bt c bαt hμ 0 hαζ ζ iv μ T ζ 3 sb pi e a bt >hμ c λ 9 [ c μ 9 sb pi e a bt c αζ T ζ c λ 9 hμ 0 hμ hαζ ζ b μ pi r hα T ζ T ζ μ 8 hαζ ζ μ T ζ 3 [See Appendix 6 for proof.] ] > μ T ζ 3 μ c σ T ζ a bt c bαt ] [ c μ αt c αbt hμ 3 [ c σμ T ζ c bσ T ζ hαζ b μ T ζ pi r h hαζ T ζ Case 5 m T Fig. 5. In this case the customer earns interest on the sales revenue upto the permissible delay period no interest to payable during the period for the time period [0, T ] is 3 ]

Int. J. Appl. Comput. Math 05 :399 46 4 Case 5. m T. Inventory I 0 T T m Time Fig. 5 Inventory versus time for Model V T < T m E 5 = pi e T T tdtdt pi e m T T pi e at T b T T Hence, the total average cost per unit time is T Dtdt = pi e a T T b 3 T 3 T 3 Z 5 T, T = T [k DC HC SC E 5] = [k c αt aα μ c μ c T ξ 4 bα T ξ 5 c αt T ξ ξ λ 8 aα h T ξ T ξ ξ bα T ξ 3 T ξ 3 ξ aα T ξ 5 T ξ 5 3 ξ ξ bα T ξ 6 T ξ 6 5 ξ ξ 6 a h T T b 6 T 3 T 3 aα T ξ T ξ ξ ξ bα T ξ 3 T ξ 3 ξ ξ 3 a pi e T T b 3 T 3 T 3 hα ] μ T ζ T ζ pm T λ 8I e sτ We have to minimize Z 5 T, T. We use the following lemma for this purpose. Lemma 5 Z 5 T, T has the minimum value for those values of T T which satisfy the following equations i c μ 3 c σ T ζ λ 8 μ c μ 4 c αt a bt hμ 5 h τ μ 6 hαζ T ζ μ s a bt pi e at bt pie m T a bt = 0 ii P Tc μ 4 αt Tc αt a bt hμ 7T T μ pi r h hαζ T ζ pi e at bt hα T ζ T ζ μ 8 pi e λ 8 pi e m T a bt = 0 9 30 3

4 Int. J. Appl. Comput. Math 05 :399 46 provided they satisfy the following sufficient conditions iii c μ 9 c αζ T ζ λ 8 μ 6 c σ T ζ a bt c bαt hμ 0 hαζ ζ iv μ T ζ 3 sb pi e a bt pb m T I e > hμ c λ 9 [ c μ 9 sb pi e a bt pb m T I e c αζ T ζ c bαt c μ hμ 0 hμ hαζ ζ hμ 3 h hαζ T ζ hαζ ] ζ μ T ζ 3 > [See Appendix for all values] [See Appendix 7 for proof.] μ T ζ 3 μ c σ T ζ a bt ] [ c μ αt c αbt μ 8 pi e a bt pbi e m T pi e a bt [ ] hαζ b μ T ζ c σ T ζ μ c bσ T ζ Numerical Examples Case Let us take the following parametric values of this inventory model as a = 500 units per year, b = 0.50 units per year, α = 0.00, =.5, k = $00 per order, I r = $0.5 per year, I e = $0.3 per year, h = $0.9 per unit per year, s = $0.80 per year, c = 0. per unit per year, p= $0 per unit m = 0.30 year. Solving Eq., we have T = 0.50 year T = 0.7 year the minimum average cost is Z T, T = $56.90. Case Let us take the following parametric values of this inventory model as a = 500 units per year, b = 0.50 units per year, α = 0.00, =.5, k = $00 per order, I r = $0.5 per year, I e = $0.3 per year, h = $0.9 per unit per year, s = $0.80 per year, c = 0. per unit per year, p = $0 per unit m = 0.3 year. Solving Eq. 3, we have T = 0.4 year T = 0.60 year the minimum average cost is Z T, T = $34.3. Case 3 Let us take the following parametric values of this inventory model as a = 500 units per year, b = 0.50 units per year, α = 0.00, =.5, k = $00 per order, I r = $0.5 per year, I e = $0.3 per year, h = $0.9 per unit per year, s = $0.80 per year, c = 0. per unit per year, p = $0 per unit m = 0.30 year. Solving Eq. 5, we have T = 0.3 year T = 0. year the minimum average cost is Z 3 T, T = $375.08. Case 4 Let us take the following parametric values of this inventory model as a = 500 units per year, b = 0.50 units per year, α = 0.00, =.5, k = $00 per order, I r = $0.5 per year, I e = $0.3 per year, h = $0.9 per unit per year, s = $0.80 per year, c = 0. per unit per year, p = $0 per unit m = 0.30 year. Solving Eq. 8, we have T = 0.5 year T = 0.4 year the minimum average cost is Z 4 T, T = $55.97. Case 5 Let us take the following parametric values of this inventory model as a = 500 units per year, b = 0.50 units per year, α = 0.00, =.5, k = $00 per order, I r = $0.5 per year, I e = $0.3 per year, h = $0.9 per unit per year, s = $0.80 per year, c = 0. per unit per year, p = $0 per unit m = 0.30 year. 3

Int. J. Appl. Comput. Math 05 :399 46 43 Solving Eq. 30, we have T = 0. year T = 0.7 year the minimum average cost is Z 5 T, T = $83.8. Conclusions Many physical goods are deteriorated over time. Electronics goods, radio-active substances, photographic film, grain etc. deteriorate through a gradual loss of potential with passage of time. Fruits, vegetables, foodstuff etc. suffer depletion by direct spoilage when they stored for a long time. Gasoline, petroleum, alcohol, etc. are highly perishable items. Thus decay or deterioration of physical goods held in stock is a very realistic area of researchers. This model assumed that the deterioration rate was a two-parameter Weibull distribution. This type of deterioration was justified for increasing rate, decreasing rate, or constant rate of deterioration for different choice of parameters of the distribution function. The dem rate was taken to be time-dependent in contract to constant dem rate in other models. A linear trend in dem is of the form D t = a bt it represents steady increase or decrease of dem rate over time. Thus it is certainly more realistic than constant dem in the real market. Shortages are very important for a managerial view. Some retailer allows shortages at the beginning of the planning period to avoid large holding cost or maintenance cost. Many retailers are compelled to allow shortages at the end of planning period. The allocation of cost savings is very important for the success of the joint relationship between the buyer vendor in supply chain management. Facing a competitive commercial environment, many vendors buyers would like to establish long-term cooperative relationships to obtain stable sources of supply dem to gain the optimum profit from each other. Traditional inventory models usually hypothesize that the buyer pays the vendor immediately when the items purchased are received. However, in practice, the vendor often gives the buyer a delay payment period to promote the buyer increasing the order quantity. Therefore, Delay-in-payments to the supplier is alternative ways of price discount for retailers, especially small businesses which tend to have a limited number of financing opportunities, rely on trade-credit as a source of short-term funds. Also, the managers of big suppliers are to take appropriate decisions for the different delay-period under different situation to encourage the retailer to run smooth business. Also, they have to face some difficulties like shortages, deterioration of products, inflation, etc. This model studied a comparative study between IFS policy SFI policy with the presence of stochastic deterioration of products. The deterioration rate followed a two-parameter Weibull deterioration. An attempt was made by assuming trade-credit financing in this comparative study. This study found that the total cost of the model starting with shortages is less than the total cost of the model starting with no shortages. The proposed model can be extended in several ways. For example, we may assume probabilistic dem a finite rate of replenishment. Also quantity discounts, time value of money inflation, etc. may be added in this paper for further study. Appendix Proof We have V T, T = T GT, T. For the minimum value of V T, T at T = T, T = T, necessary conditions are V T,T = 0, V T,T = 0. 3

44 Int. J. Appl. Comput. Math 05 :399 46 Now Therefore, V T,T V T, T = T GT,T GT, T T V T, T = GT, T T V T, T = 0givesT GT, T = 0gives GT,T = 0. Equations are satisfied for T = T *T = T. Again, V T, T T T GT,T GT,T = Thus, at T = T *T = T,wehave GT,T T 4 GT, T = 0 T T GT,T GT, T Similarly, at T = T * V T, T = T GT, T T = T, V T, T = GT, T T Hence, V T,T V T,T V T,T V T,T = V T, T GT,T T GT,T T = T GT, T. GT,T T GT,T T = T GT,T GT,T GT,T GT,T then Thus, if GT,T GT,T V T,T V T,T GT,T GT,T V T,T V T,T > GT, T 0 > 0 > V T, T 0 > 0 which indicates V T, T is minimum. 3

Int. J. Appl. Comput. Math 05 :399 46 45 Appendix ξ =, ξ 4 =, ξ =, ξ 5 =, ξ 3 = 3, ξ 6 = 3, ζ =, ζ =, ζ 3 = 3, α = σ, τ = at b T, I 0 = at b T aα T ξ bα T ξ ξ ξ, λ = aαt bαt, λ = at bt at bt T, λ 3 = a bt aαt λ 4 = I 0 at b T λ 5 = λ 6 =, bαt, a T b 3 T 3 aα T aα T I 0 T m a T m b 6 aα aα λ 7 = a m b 3 m3, λ 8 = at T b λ 9 = bα 3 T 3 bα 3 T 3, T 3 m3 I 0α T m T T, aαt bα T λ 0 = a bt aαt, λ = a bt bt, T m, bα 3 bα 3 T m T 3 m 3 T 3 m 3, 3

46 Int. J. Appl. Comput. Math 05 :399 46 λ = a T b 6 T 3, a η = T T b 6 T 3 T 3 T T at b T, I = at T b T T aα T ξ T ξ ξ bα T ξ T ξ ξ exp μ = at b T aα T bα T, μ = aα T bα T, aα μ 3 = T bα T, μ 4 = aαt bαt, aαt bα T aα μ 5 = T bα T μ 6 = aα T bα T, μ 7 = aαt bα T aα T bα T μ 8 = a bt aαt bαt, aα μ 9 = T bα T, μ 0 = at bt aαt bαt, μ = aα T μ = b aαt bα T, μ 3 = aα T bα T aα bα T aα T,, bα αt, T, T bα T, Appendix 3 Proof To minimize Z T, T = T F T, T, we have necessary conditions as Z T,T = 0, Z T,T = 0 from the above theorem the sufficient conditions are F T,T > 0 F T,T F T,T Now Z T,T i c aαt 3 F T,T F T,T = 0gives > 0 bαt h at bt aα T bα T aα T

Int. J. Appl. Comput. Math 05 :399 46 47 bα T T m s at bt at bt T pi r a bt aαt bαt α T α m I 0 αt at b T = 0 Also, Z T,T = 0gives ii k c I 0 at b T bα 3 T 3 a s T T b 6 T 3 T 3 T T a h T b 3 T 3 aα T aα T pi r I 0 T m a T m b 6 T 3 m3 I 0α aα T m bα 3 T 3 at b T bα T 3 m 3 3 bα T 3 m 3 a pi e 3 m b 3 m3 Ts at T b T T = 0 T m aα T m To obtain optimal values for T T, we have to solve the above two equations provided they satisfy the following sufficient conditions. Using the above theorem, F T, T > 0, F T,T F T,T F T,T F T,T > 0 we obtain the following results iii c aαt bα T h a bt aαt aα T bα T m α T > pαi r a bt aαt α m bα T T pi r b aαt bα T s a bt bt bαt T I 0αT a bt. 3

48 Int. J. Appl. Comput. Math 05 :399 46 iv [c aαt bα T h a bt aαt aα T T m bα α T pαi r a bt aαt > sa bt α m bα T T pi r b aαt bα T s a bt bt bαt T I 0αT a bt ] a bt Appendix 4 Proof To minimize Z T, T = T F T, T, we have necessary conditions as Z T,T = 0, Z T,T = 0 from the above theorem the sufficient conditions are F T,T > 0 F T,T F T,T Now Z T,T i c aαt F T,T F T,T = 0gives bα T pmi e a bt = 0 > 0 bαt h at bt aα T bα T aα T s at bt at bt T pi e at b T Also, Z T,T = 0gives ii k c I 0 at b a T h T b 3 T 3 aα T 3 bα 3 T 3 a s T T b 6 pi r I 0 T m a bα 3 T 3 T T at b T aα T T 3 T 3 T m b T 3 6 m 3 I 0α T m aα T m bα T 3 m 3 3 aα T m bα 3 a pi e m b 3 m3 Ts a T T b T T = 0 T 3 m 3

Int. J. Appl. Comput. Math 05 :399 46 49 To obtain optimal values for T T, we have to solve the above two equations provided they satisfy the following sufficient conditions. Using the above theorem, F T, T we obtain the following results > 0, iii c aαt bα T h aα T bα > pbmi e iv F T,T F T,T F T,T F T,T a bt aαt > 0 bα T T s a bt bt pi e a bt [c aαt bα T h a bt aαt aα T pbmi e ] a bt >sa bt bα bα T T s a bt bt pi e a bt Appendix 5 Proof To minimize Z 3 T, T = T F 3T, T, wehavenecessaryconditionsas Z 3 T,T = 0, Z 3T,T = 0 from the above theorem the sufficient conditions are F 3 T,T F 3 T,T F 3 T,T > 0 > 0 Now Z 3T,T i c aαt F 3 T,T = 0gives bαt bα T pmi e a bt = 0 Also, Z 3T,T ii k c h F 3 T,T at bt aα T bα T aα T s at bt at bt T pi e at b T = 0gives I 0 at b T h bα 3 T 3 a T T b 6 T 3 T 3 T T s a T b 3 T 3 aα T aα T bα 3 T 3 at b T 3

40 Int. J. Appl. Comput. Math 05 :399 46 pi r I 0 T m a T m b 6 T 3 m3 I 0α aα T m bα 3 aα T m bα 3 pi e a m b 3 m3 Ts at T b T T = 0 T m T 3 m 3 T 3 m 3 To obtain optimal values for T T, we have to solve the above two equations provided they satisfy the following sufficient conditions. Using the above theorem, F 3 T, T > 0, we obtain the following results iii c aαt bα T h aα T bα > pbmi e iv F 3 T,T F 3 T,T F 3 T,T F 3 T,T a bt aαt > 0 bα T T s a bt bt pi e a bt [c aαt bα T h a bt aαt aα T pbmi e ] a bt >sa bt bα bα T T s a bt bt pi e a bt Appendix 6 Proof To minimize Z 4 T, T = T F 4T, T, we have necessary conditions as Z 4T,T = 0, Z 4T,T = 0 from the above theorem the sufficient conditions are F 4 T,T > 0 F 4 T,T F 4 T,T Now Z 4T,T F 4 T,T F 4 T,T = 0gives aα i c T > 0 bα b T T aα T bα T 3 T c αt c aαt at T bαt

Int. J. Appl. Comput. Math 05 :399 46 4 c αt a bt h h at b T aα aα T bα T Also, Z 4T,T = 0gives aαt bα T aα T T bα T M Tc aαt bαt αt s a bt pi e at bt Tc αt bα T aα T bα T at b T aα T bα T a bt aαt bαt = 0 T bα T hα T = 0 at b T a bt ht aαt pi r h hα T T pi r hα T T Where M is given by ii M = [k c αt aα T bα T aα c T bα T aα c T bα T c αt at T b aα T T h T T bα T 3 T 3 aα T T 3 bα T 3 T 3 a h 3 T T b 6 T 3 T 3 aα T T bα 3 T 3 T 3 pi r hα T T at b T aα T bα T a pi r T m b 6 T 3 m 3 aα T m s at b T bα 3 T 3 m 3 a pi e m T b ] 3 m3 T 3 To obtain optimal values for T T, we have to solve the above two equations provided they satisfy the following sufficient conditions. Using the above theorem, F 4 T, T > 0, F 4 T,T F 4 T,T F 4 T,T F 4 T,T > 0 3

4 Int. J. Appl. Comput. Math 05 :399 46 we obtain the following results aα iii c T bα T c α T at T b T T aα T bα T c αt a bt c bαt h at bt aαt bαt hα T 3 sb pi e a bt >h c aαt bα T at b T aαt bα T aα T bα T aα T bα T [ aα iv c T bα c α T at T b T T aα c αt a bt c bαt c h at bt aαt bαt bα T αt b aαt bα T h h T sb pi e a bt T T bα aαt bα T aα T bα T T aα T bα hα at b T aα T bα ] [c T aαt c αbt aα T pi r h hα [ > c αt 3 hα bα T aα pi r hα T 3 T T hα a T bt aαt bαt T 3 at b T T bα T aα T bα aαt bα T c bαt T b aαt bα T ] ] T

Int. J. Appl. Comput. Math 05 :399 46 43 Appendix 7 Proof To minimize Z 5 T, T = T F 5T, T, we have necessary conditions as Z 5T,T = 0, Z 5T,T = 0 from the above theorem the sufficient conditions are F 5 T,T > 0 F 5 T,T F 5 T,T Now Z 5T,T F 5 T,T F 5 T,T = 0gives > 0 aα i c T bα aα T bα T h aαt bα T aα c aαt T c αt at T b T T bαt c αt T bα T h at b T aα T bα T at b T aα T bα T s a bt pi e at bt pie m T a bt = 0 Also, Z 5T,T = 0gives ii P Tc aαt bαt αt bα T aα T bα T at b T aα T bα T Tc αt T a bt hα T a bt ht aαt hα h T pi e at bt hα T T a bt aαt bαt pi e at T b T T pi e m T a bt = 0 Where P is given by P = [k c αt aα T bα aα T c T bα T aα c T bα T b aα T T h T T c αt at T bα 3 T 3 T 3 aα T T bα 3 T 3 T 3 a h T T b 6 T 3 T 3 aα T T 3

44 Int. J. Appl. Comput. Math 05 :399 46 bα 3 T 3 T 3 hα T T s at b T aα T bα T b 3 T 3 T 3 pi e m T at T b T T ] at b T pi e a T T To obtain optimal values for T T, we have to solve the above two equations provided they satisfy the following sufficient conditions. Using the above theorem, F 5 T, T > 0, we obtain the following results aα iii c T F 5 T,T F 5 T,T F 5 T,T F 5 T,T bα T > 0 c α T at T b T T aα T bα T c αt a bt c bαt h at bt aαt bαt hα T 3 at b T pi e a bt pbm T I e > h aα T bα [ aα iv c T pbm T I e c α T aα T bα T aα T bα T sb T c aαt bα T T sb pi e a bt bα at T b T T aα T bα T c αt a bt c bαt c aαt bα T hα T T b aαt bα T h at bt aαt h aα T bα T aα T bα T hα T 3 at b T aα T bα ] T [c aαt bα T αt c αbt h aα T bα T aα T bα T h a bt aαt bαt pi e a bt pbi e m T pi e a bt 3 bαt hα T

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