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International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 2, Issue. 5, Sep.-Oct. 2012 pp-3641-3649 ISSN: 2249-6645 Supply Chain Production Inventory Model: Innovative Study for Shortages Allowed With Partial Backlogging Jasvinder Kaur1, Rajendra Sharma2 *(Department Of Mathematics, Graphic Era University Dehradun,India) ** (Department Of Mathematics Graphic Era University Dehradun, india) Abstract: In this paper, we have strived to combine all the Period before they settle the account with the supplier. This above mentioned factors into a single problem. We shall provides an advantage to the customers, due to the fact that undertake to explore a two echelon supply chain, they do not have to pay the supplier immediately after comprising of a vendor and a buyer. The whole environment receiving the product, but instead, can defer their payment of business dealings has been assumed to be progressive until the end of the allowed period. The customer pays no credit period, which conforms to the practical market interest during the fixed period they are supposed to settle situation. The whole combination is very unique and very the account; but if the payment is delayed beyond that much practical. The variable holding cost and variable period, interest will be charged. The customer can start to setup has been explored numerically as well; an optimal accumulate revenues on the sale or use of the product and solution has been reached. The final outcome shows that the earn interest on that revenue. So it is to the advantage of the model is not only economically feasible, but stable also. customer to offer the payment to the supplier until the end of the period. Keywords: Inventory model, partial backlogging, The two famous formulae of EOQ and EPQ are Progressive permissible delay, Supply chain, Shortages, treated separately for a buyer and a vendor respectively. EOQ (Economics Order Quantity). From the traditional point of view, the vendor and the buyer are two individual entities with different objectives and I. INTRODUCTION self-interest. Due to rising costs, the globalization trend, Inventory represents one of the most significant possessions shrinking resources, shortened product life cycle and that most businesses possess. It is in direct touch with the quicker response time, increasing attention has been placed user department in its day today activities. Inventory on the collaboration of the whole supply chain system. An management is playing a key role in setting up efficient effective supply chain network requires a cooperative closed loop supply chains. A supply chain is a network of relationship between the vendor and the buyer. It assumes facilities and distribution options that performs the that the buyer must pay off as soon as the items are functions of procurement of materials, transformation of received. Suppliers often offer trade credit as a marketing these materials into intermediate and finished products, and strategy to increase sales and reduce on-hand stock is the distribution of these finished products to customers. It reduced, and that leads to a reduction in the buyer’s holding consists of a network of companies which are dependent on cost of finance. In addition, during the time of the credit each other while making independent decisions. The supply period, buyers may earn interest on the money. In fact, chain not only includes the manufacturer and suppliers, but buyers, especially small businesses which tend to have a also transporters, warehouses, retailers, and customers limited number of financing opportunities rely on trade themselves. Therefore, supply chain analysis tools and credit as a source of short-term funds. The classical methodologies have become more and more important. It inventory models have considered demand rates which can be a source of great efficiency and cost-savings gains. were either constant or depended upon a single factor only, Supply chain speed and flexibility have become key levers like, stock, time etc. But changing market conditions have for competitive differentiation and increased profitability. rendered such a consideration quite unfruitful, since in real The faster the supply chain, the better a company can life situation, a demand cannot depend exclusively on a respond to changing market situation and the less it needs single parameter. A combination of two or more factors inventory which resulting in higher return on capital grants more authenticity to the formulation of the model. employed. Supply chain management offers a large Many delivery policies have been proposed in literature for potential this problem. Clark and Scarf (1960) presented the or organizations to reduce costs and improve customer concept of serial multi-echelon structures to determine the service performance. In the existing literature, most of the optimal policy. Goyal (1985) considered a mathematical inventory models studies only aimed at the determination of models with a permissible delay in payments to determine the optimum solutions that minimized cost or maximized the optimal order quantities. Ha and Kim (1997) used a profit from the vendor’s and vendor’s side. However, in the graphical method to analyze the integrated vendor-buyer modern global competitive market, the buyer and vendor inventory status to derive an optimal solution. Hwang and should be treated as strategic partners in the supply chain Shinn (1997) studied effects of permissible delay in with a long term cooperative relationship. Recently, many payments on retailer's pricing and lot sizing policy for researchers have considered the buyer and vendor as a unit exponentially deteriorating products. Yang and Wee to find the optimal EOQ in achieving the minimum total (2000) developed an integrated economic ordering policy of cost. In today's business transactions, it is more and more deteriorating items for a vendor and a buyer. Wang et al. common to see that the customers are allowed some grace (2000) analyzed supply chain models for perishable products under inflation and permissible delay in payment. www.ijmer.com 3641 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 2, Issue. 5, Sep.-Oct. 2012 pp-3641-3649 ISSN: 2249-6645 Teng (2002) modified Goyal (1985) model by considering 2.13 (Cvs 2 t ) = the setup cost for each production the selling price, instead of purchasing cost, as the base to calculate the interest. Abad and Jaggi (2003) studied a cycle for vendor. seller-buyer model with a permissible delay in payments by 2.14 (Cbs 1t ) = the setup cost per order for buyer. game theory to determine the optimal unit price and the 2.15 (Chv 2 t ) = holding cost per unit time for vendor. credit period, considering that the demand rate is a function of retail price. Huang, Y.F. et al. (2005) considered the 2.16 (Cbh 1t ) = holding cost per unit time for buyer. optimal inventory policies under permissible delay in 2.17 Cv = the unit cost for vendor. payments depending on the ordering quantity. Song and 2.18 Cb = the unit purchase cost for buyer. Cai (2006) has been taken on optimal payment time for a 2.19 Sb = shortage cost per unit time for buyer. retailer under permitted delay of payment by the 2.20 Lb= lost sale cost per unit time for buyer. wholesaler. Liao (2007) assumed on an EPQ model for 2.21 VC = the cost of vendor per unit time. deteriorating items under permissible delay in payments. 2.22 BC = the cost of buyer per unit time. In the present study, we have strived to combine 2.23 TC(T) = total cost of an inventory system / time unit. all the above mentioned factors into a single problem. We 2.24 B= Backlogging rate. shall undertake to explore a two echelon supply chain, 2.25 The deterioration function comprising of a vendor and a buyer. The whole environment of business dealings has been assumed to be (t , ) 0 ( )t , 0< 0 ( ) <<1, t>0 progressive credit period, which conforms to the practical This is a special form of the two parameter weibull function market situation. The whole combination is very unique and considered by Covert and Philip. The function is some very much practical. The variable holding cost and variable functions of the random variable which range over a setup has been explored numerically as well; an optimal space and in which a p.d.f. p( ) is defined such that solution has been reached. The final outcome shows that the model is not only economically feasible, but stable also. p( )d 1 II. PROPOSED ASSUMTIONS & NOTATIONS III. INDENTATIONS AND EQUATIONS MATHEMATICAL FORMULATION 1. ASSUMPTIONS The actual vendor’s average inventory level in the The following assumptions are used to develop aforesaid integrated two-echelon inventory model is difference model: between the vendor’s total average inventory level and the 1.1 The demand rate, D(t), is deterministic, the demand buyer’s average inventory level. Since the inventory level is function D(t) is given by D(t) = 0 e , a and b are t depleted due to a constant deterioration rate of the on-hand stock, the buyer’s inventory level is represented by the positive constants. following differential equation: 1.2 Shortages are allowed with partial backlogging. 1.3 If the retailer pays by M, then the supplier does not Ib (t ) 0 ( )tIb (t ) 0 e t ,0 t t1 ' (1) charge to the retailer. If the retailer pays after M and before N (N > M), he can keep the difference in the Ib (t ) B0 et , ' t1 t T (2) unit sale price and unit purchase price in an interest bearing account at the rate of Ie/unit/year. During [M, The vendor’s total inventory system consisting of N], the supplier charges the retailer an interest rate of production period and non-production period can be Ic1/unit/year on unpaid balance. If the retailer pays described as follows: after N, then supplier charges the retailer an interest rate of Ic2/unit/year (Ic1> Ic2) on unpaid balance. I v' 1 (t ) 0 ( )tI v1 (t ) ( K 1)0 et , 0 t T1 (3) 2. NOTATIONS: 2.1 P = the selling price / unit. I v' 2 (t ) 0 ( )tI v 2 (t ) 0 et , 0 t T2 (4) 2.2 KD = the production rate per year, where K>1 2.3 C = the unit purchase cost, with C < P. The boundary conditions are 2.4 M = the first offered credit period in settling the account without any charges. I v1 (t ) 0, t 0 (5) 2.5 N = the second permissible credit period in settling the account with interest charge Ic2 on unpaid balance and N > M. I v 2 (t ) 0, t T2 (6) 2.6 Ic1 = the interest charged per $ in stock per year by the supplier when retailer pays during [M, N]. Ib (t ) I 0 , t 0 (7) 2.7 Ic2 = the interest charged per $ in stock per year by the supplier when retailer pays during [N, T]. (Ic1 > Ic2) 2.8 Ie = the interest earned / $ / year. Ib (t ) 0, t t1 (8) 2.9 T = the replenishment cycle. 2.10 r = the discount rate (r > α) I v1 (T1 ) I v 2 (0) (9) 2.11 IE = the interest earned / time unit. 2.12 IC = the interest charged / time unit. www.ijmer.com 3642 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 2, Issue. 5, Sep.-Oct. 2012 pp-3641-3649 ISSN: 2249-6645 And, rt12 t3 t2 t3 t 4 r 2 t15 0 ( )t14 (r 2 0 ( )) 1 ] 0 [ 1 ( 2r ) 1 r (r ) 1 2 6 T 2 6 8 20 12 T2 0 ( )t15 (17) T (10) (3 2r ) 30 ] n Respectively. The solutions of the above differential equations obtained The annual total holding cost for the buyer and the vendor are are 1 1 t 0 ( ) t 2 ( ) t 2 t 2 0 ( )t 3 0 2 (11) HCb (Cbh 1t )e I b1 (t )dt rt Ib (t ) I 0 e 2 0 [t ]e , 0 t t1 T 0 2 6 B0 t1 t Ib (t ) e e , t1 t T 0 t 5 r 2 ( )t 4 ( )t 5 (3 2r ) (12) t2 t3 t4 Cbh [ 1 1 ( 2r ) 1 r (r ) 1 0 1 0 1 ] T 2 6 8 20 12 60 0 ( ) t 2 t2 0 ( )t 3 I v1 (t ) ( K 1)0 [t ]e 2 , 0 t T1 10 t13 t14 t5 ( )t15 Cbh I 0 rt 2 2 6 [ ( 2r ) 1 r (r ) 0 ] [t1 1 (13) T 3 8 10 15 T 2 ( ) I v 2 (t ) 0 [(T2 t ) (T22 t 2 ) 0 (T23 t 3 )], 0 t T2 (r 2 0 ( ))t13 I t 2 rt 3 (r 2 0 ( ))t14 ] 1 0 [ 1 1 ] 2 6 6 T 2 3 8 (14) (18) Using the condition that one can get, And t12 0 ( )t13 I 0 0 [t1 ] (15) 1 1 T T2 2 6 HCv (Chv 2 t )e rt I v1 (t )dt e rT1 (Chv 2t )e rt I v 2 (t )dt I b T2 0 0 If the product of the deterioration rate and the replenishment interval is much smaller than one, the 1 T2 T3 T4 T 5 r 2 ( )T 4 ( )T 5 (3 2r ) buyer’s and the vendor’s actual average inventory [( K 1)0Chv { 1 1 ( 2r ) 1 r (r ) 1 0 1 0 1 }] T2 2 6 8 20 12 60 level, I b and I v , are 2 ( K 1)0 T13 T14 T5 ( )T 5 C e rT1 T 2 T 3 ( 2r ) t1 [ ( 2r ) 1 r (r ) 0 1 ] bh 0 [ 2 2 1 rt T2 3 8 10 15 T 2 6 T Ib e I b (t )dt 0 r 2T24 0 ( )T24 0 ( )T25 (4 3r ) 0 e rT1 2 T23 T24 (3 2r ) I 0 ] [ ] [t1 I0 rt 2 t t t 3 2 3 24 24 60 T2 6 24 T [t1 (r 2 0 ( )) ] 0 [ ( 2r ) 1 1 1 1 T 2 6 T 2 6 rt12 t3 t2 t3 t 4 r 2 t15 0 ( )t14 (r 2 0 ( )) 1 ] 0 [ 1 ( 2r ) 1 r (r ) 1 2 6 T 2 6 8 20 12 t14 r 2 t15 0 ( )t14 ( )t15 0 ( )t1 5 (3 2r ) (19) r (r ) (3 2r ) 0 ] 30 ] 8 20 12 30 respectively. (16) The annual deterioration cost for the buyer and the vendor and are Cb t1 0 ( )te I b1 (t )dt = rt rtT1 T2 DCb 1 e I v1 (t )dt e 1 e I v 2 (t )dt I b 0 rT rt Iv T T2 0 0 I 00 ( ) t12 rt13 r 4 t14 00 ( ) t13 t14 ( 2r ) t15 1 T T 2 T 3 T r 0 ( )T 0 ( )T (3 2r ) 4 5 2 4 5 [ { } { r (r )}] [( K 1)0 ( ( 2r ) r (r ) 1 1 1 1 ) 1 1 T 2 6 8 T 3 8 10 T2 2 6 8 20 12 60 (20) T22 T23 r 2T 4 rT 4 ( )T 4 ( )T 5 (4 3r ) I 0 and e rT1 0 ( ( 2r ) 2 2 0 2 0 2 )] [t1 2 6 24 8 24 60 T www.ijmer.com 3643 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 2, Issue. 5, Sep.-Oct. 2012 pp-3641-3649 ISSN: 2249-6645 Cv 1 Regarding interest charged and interest earned based on the T T2 DCv 0 ( )te rt I v1 (t )dt e rT1 0 ( )te rt I v 2 (t )dt length of the cycle time t1, three cases arise: T2 0 0 IV. FIGURES AND TABLES = C [ ( K 1)00 ( ) {T1 T1 ( 2r ) T1 r (r )} e 00 ( ) {T2 3 4 5 rT1 3 Regarding interest charged and interest earned based on the v T2 3 8 10 T2 6 length of the cycle time t1, three cases arise: T2 4 (3 2r ) Case I: M ≥ t 1 }] (21) 24 Inventory level respectively. The annual set-up cost for the buyer and the vendor are t T 1 1 T OCb [ (Cbs 1t )dt (Cbs 1t )dt ] 0 t1 =C 1T (22) bs 2 and T T2 1 1 T2 (C OCv [ (Cvs 2 t )dt vs 2 t )dt ] 0 0 = [Cvs 2 2 (T1 T2 ) ] 2 2 1 (23) 2 respectively. t1 M Time The annual shortage cost for the buyer is 0 Fig 1: t1 ≥ M Sb e rt1 T 0 SCb T t1 et1 et e rt dt In the first case, retailer does not pay any interest to the supplier. Here, retailer sells I0 units during (0, t1) time Sb 0 e rt1 e( r )t1 e( t1 rT ) e( r )T (24) interval and paying for CI0 units in full to the supplier at [ ] time M ≥ t1, so interest charges are zero, i.e. T r ( r ) r ( r ) IC1 = 0 (28) The annual lost sale cost for the buyer is Retailers deposits the revenue in an interest bearing account Lb e rt1 T LCb T (1 B) t1 0 et e rt dt at the rate of Ie / $ / year. Therefore, interest earned IE1, per year is = Lb e 0 (1 B) [e( r )T e( r ) t ] rt 1 t t (25) PI e 1 rt 1 [ e D(t )tdt ( M t1 ) e rt D(t )dt ] 1 T ( r ) IE1 T2 0 0 The different costs associated with the system are set-up = PI e 0 [(M t )t {1 ( r )(M t )} t1 t1 ( r ) ] (29) 2 3 costs, holding costs, deterioration cost and shortage cost. 1 1 1 Our aim is to minimize the total cost. T2 2 3 From (9), one can derive the following condition: Total cost per unit time of an inventory system is T12 0 ( )T13 (2)T T 2 ( )T23 (2)T TCb (t1 , ) = OCb +HCb + DCb + SCb + IC1 – IE1 2 2 0 1 0 2 ( K 1)0 [T1 ]e 0 [T2 2 0 ]e 2 6 2 6 = [Cbs 1T 0 Cbh [ t1 t1 ( 2r ) t1 r (r ) t1 r 0 ( )t1 2 3 4 5 2 4 (26) 2 T 2 6 8 20 12 By Taylor’s series expansion, (4.26) is derived as 1 0 ( )t15 (3 2r ) 10 t13 t14 t5 ( )t15 T1 T2 1 2 T2 (27) ] [ ( 2r ) 1 r (r ) 0 ] K 1 60 T 3 8 10 15 www.ijmer.com 3644 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 2, Issue. 5, Sep.-Oct. 2012 pp-3641-3649 ISSN: 2249-6645 At15 ( K 1)0 A T13 T14 ( 2r ) T15 r (r ) (3 2r ) ] Cv [ { } Cbh I 0 rt (r 0 ( ))t 2 2 I t 3 rt (r 0 ( ))t 2 3 2 4 30 T2 3 8 10 [t1 1 ] 1 0 [ 1 1 1 ] 1 T 2 6 T 2 3 8 e rT1 0 A T23 T2 4 (3 2r ) (34) { }] T2 6 24 I 00 ( ) t12 rt13 r 4 t14 00 ( ) t13 t14 ( 2r ) t15 [ { } { r (r )}] T 2 6 8 T 3 8 10 To minimize the total cost per unit time, the optimum value of t1, T2 is the solution of following equation. Sb 0 e rt1 e(r )t1 e(t1 rT ) e(r )T Lb e rt1 0 (1 B) (r )T ( r )t1 [ ] [e e ] Case II: M < t1< N T r ( r ) r ( r ) T ( r ) Inventory level PI e 0 t 2 t 3 ( r ) [( M t1 )t1 {1 ( r )( M t1 )} 1 1 ]] T2 2 3 (30) Hence the mean cost Q < TCb >= TC b (t1 , ) p( )d (31) < TCb >= [Cbs 1T 0 Cbh [ t1 t1 ( 2r ) t1 r (r ) t1 r At1 2 3 4 5 2 4 2 T 2 6 8 20 12 At15 (3 2r ) 10 t13 t14 t5 At 5 ] [ ( 2r ) 1 r (r ) 1 ] 60 T 3 8 10 15 Cbh I 0 rt12 (r 2 A)t13 1 I 0 t12 rt13 (r 2 A)t14 [t1 ] [ ] T 2 6 T 2 3 8 I A t 2 rt 3 r 4 t 4 A t 3 t 4 ( 2r ) t15 [ 0 { 1 1 1 } 0 { 1 1 r (r )}] T 2 6 8 T 3 8 10 M t1 N Time Sb 0 e rt1 e(r )t1 e(t1 rT ) e(r )T Lb e rt1 0 (1 B) (r )T (r )t1 [ ] [e e ] Fig: 2 M < t1 < N T r ( r ) r ( r ) T ( r ) PI e 0 t 2 t 3 ( r ) In the second case, supplier charges interest at the rate Ic1 [( M t1 )t1 {1 ( r )( M t1 )} 1 1 ]] T2 2 3 on unpaid balance. Interest earned, IE2 during [0, M] is (32) M IE2 PI e e rt D(t )tdt Where A= 0 ( ) p( )d (33) 0 2 3 4 <TCv >= OCv +HCv + DCv - IC1 = PI e 0 [ M M ( r ) M ( r ) 2 ] (35) 2 6 8 Retailer pay for I0 units purchased at time t = 0 at the rate of 2 (T12 T22 ) 1 T2 T3 T4 = [Cvs 1 2 ] [( K 1)0Chv { 1 1 ( 2r ) 1 r (r ) C / $ / unit to the supplier during [0, M]. The retailer sells D 2 T2 2 6 8 (M).M units at selling price P/ unit. So, he has generated T15 r 2 AT14 AT15 (3 2r ) 2 ( K 1)0 T13 T14 revenue of P D(M).M + IE2. Then two sub cases may arise: }] [ ( 2r ) 20 12 60 T2 3 8 Sub Case: 2.1 T 5 AT 5 Cbh 0 e rT1 T2 2 T23 ( 2r ) r 2T2 4 AT2 4 Let P D(M).M + IE2 ≥ CI0, i.e. retailer has enough money to 1 r (r ) 1 ] [ 10 15 T 2 6 24 24 settle his account for all I0 units procured at time t = 0. Then interest charge will be IC2.1 = 0 (36) AT25 (4 3r ) 0 e rT1 2 T23 T2 4 (3 2r ) I 0 rt 2 ] [ ] [t1 1 and interest earned 60 T2 6 24 T 2 t13 0 t12 t3 t 4 r 2 t15 At14 (r 2 A) ] [ ( 2r ) 1 r (r ) 1 6 T 2 6 8 20 12 www.ijmer.com 3645 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 2, Issue. 5, Sep.-Oct. 2012 pp-3641-3649 ISSN: 2249-6645 IE2 To minimize the total cost per unit time, the optimum value IE2.1 of t1, T2 is the solution of following equation. T2 Sub Case: 2.2 = PI e 0 [ M M ( r ) M ( r ) 2 ] (37) 2 3 4 Let P D(M).M + IE2 < CI0. Here, retailer will have to pay T2 2 6 8 interest on unpaid balance U1 = CI0 – (P D(M).M + IE2) at the rate of Ic1 at time M to the supplier. Then interest paid So, total cost TC2.1 per unit time of inventory system is per unit time is given by t1 U12 Ic1 e rt <TCb >= OCb + HCb + DCb + SCb + LCb +IC2.1 – IE2.1 IC2.2 I (t )dt PI 0 M = U1 Ic1 [ (t1 M ) ( r )(t1 M ) ( r ) (t1 M ) ] 2 2 2 3 3 2 4 4 S= [Cbs 1T 0 Cbh [ t t ( 2r ) t r (r ) t r At 2 3 4 5 2 4 1 1 1 1 1 PI 0 2 2 8 2 T 2 6 8 20 12 (40) Where, At 5 (3 2r ) 10 t13 t14 t5 At 5 1 ] [ ( 2r ) 1 r (r ) 1 ] U1 = CI0 – (P D(M).M + IE2) 60 T 3 8 10 15 =CI0– Ie 2 I ( r ) 3 3 I e ( r ) 2 Cbh I 0 rt 2 (r 2 A)t13 1 I 0 t12 rt13 (r 2 A)t14 P0 [ M ( )M 2 ( e )M ( )M 4 ] [t1 1 ] [ ] 2 2 2 6 8 T 2 6 T 2 3 8 (41) And interest earned I 0 A t12 rt13 r 4 t14 A t 3 t 4 ( 2r ) t15 [ { } 0 { 1 1 r (r )}] T 2 6 8 T 3 8 10 IE2 IE2.2 T2 = PI e 0 [ M M ( r ) M ( r )2 ] 2 3 4 S e rt1 e(r )t1 e(t1 rT ) e(r )T Lb e rt1 0 (1 B) (r )T (r )t1 (42) b 0 [ ] [e e ] T2 2 6 8 T r ( r ) r ( r ) T ( r ) So, total cost TC2.2 per unit time of inventory system is PI e 0 M 2 M 3 M 4 (38) <TCb >= OCb + HCb + DCb + SCb + LCb + IC2.2 – IE2.2 [ ( r ) ( r ) 2 ] T2 2 6 8 = [Cbs 1T 0 Cbh [ t1 t1 ( 2r ) t1 r (r ) t1 r At1 <TCv >= OCv + HCv + DCv - IC2.1 2 3 4 5 2 4 2 T 2 6 8 20 12 = [C 2 (T1 T2 ) ] 1 [( K 1) C {T1 T1 ( 2r ) T1 r (r ) 2 2 2 3 4 vs 1 2 0 hv 2 T2 2 6 8 At15 (3 2r ) 10 t13 t14 t5 At 5 ] [ ( 2r ) 1 r (r ) 1 ] T r AT AT (3 2r ) 2 ( K 1)0 T T 5 2 4 5 3 4 60 T 3 8 10 15 1 1 }] 1 [ ( 2r ) 1 1 20 12 60 T2 3 8 Cbh I 0 rt 2 (r 2 A)t13 1 I 0 t12 rt13 (r 2 A)t14 T5 AT 5 C e rT1 T2 2 T23 ( 2r ) r 2T2 4 AT2 4 [t1 1 ] [ ] 1 r (r ) 1 ] bh 0 [ T 2 6 T 2 3 8 10 15 T 2 6 24 24 AT25 (4 3r ) 0 e rT1 2 T23 T2 4 (3 2r ) I 0 rt 2 I 0 A t12 rt13 r 4 t14 A t 3 t 4 ( 2r ) t15 ] [ ] [t1 1 [ { } 0 { 1 1 r (r )}] 60 T2 6 24 T 2 T 2 6 8 T 3 8 10 t13 0 t12 t3 t 4 r 2 t15 At14 (r 2 A) ] [ ( 2r ) 1 r (r ) 1 Sb 0 e rt1 e( r )t1 e( t1 rT ) e( r )T L e rt1 0 (1 B) ( r )T 6 T 2 6 8 20 12 [ ] b [e e( r )t1 ] T r ( r ) r ( r ) T ( r ) At15 ( K 1)0 A T13 T14 ( 2r ) T15 r (r ) (3 2r ) ] Cv [ { } PI e 0 M 2 M3 M4 (43) 30 T2 3 8 10 [ ( r ) ( r ) 2 ] T2 2 6 8 e rT1 0 A T23 T2 4 (3 2r ) - { }] <TCv >= OCv + HCv + DCv - IC2.2 T2 6 24 PI e 0 M 2 M 3 M4 ( r ) ( r ) 2 ] (39) =S [C 2 (T1 T2 ) ] 1 [( K 1) C {T1 T1 ( 2r ) T1 r (r ) [ 2 2 2 3 4 T2 2 6 8 vs 1 2 0 hv 2 T2 2 6 8 www.ijmer.com 3646 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 2, Issue. 5, Sep.-Oct. 2012 pp-3641-3649 ISSN: 2249-6645 T15 r 2 AT14 AT15 (3 2r ) ( K 1)0 T13 T14 in account at M and total money }] 2 [ ( 2r ) 20 12 60 T2 3 8 PD(N).N=P 0 e N N at N, there are three sub cases may T 5 AT C e 5 rT1 T2 T ( 2r ) r T2 3 2 4 AT 4 arise: r (r ) 1 ] bh 0 1 [ 2 2 2 10 15 T 2 6 24 24 Sub Case 3.1 Let P D(M).M + IE2 ≥ CI0 AT25 (4 3r ) 0 e rT1 2 T23 T2 4 (3 2r ) I 0 rt 2 This case is same as sub case 2.1, here 3.1 designate ] [ ] [t1 1 decision variables and objective function. 60 T2 6 24 T 2 Sub Case 3.2 Let P D(M).M + IE2 < CI0 and t3 t2 t3 t 4 r 2 t15 At14 (r A) 1 ] 0 [ 1 ( 2r ) 1 r (r ) 1 2 N 6 T 2 6 8 20 12 PD( N M ).( N M ) PI e D(t )dt CI 0 ( PD(M ).M IE2 ) M At 5 ( K 1)0 A T T ( 2r ) T r (r ) 3 4 5 (3 2r ) 1 ] Cv [ { 1 1 1 } 30 T2 3 8 10 P0 e ( N M ) ( N M ) PI e 0 [( N M ) ( N 2 M 2 )] CI 0 ( P0 eM M IE2 ) 2 This case similar to sub case 2.2. e rT1 0 A T23 T2 4 (3 2r ) U 2 Ic (t 2 M 2 ) Sub Case 3.3 Let P D(M).M + IE2 < CI0 and { }] 1 1 [ 1 T2 6 24 PI 0 2 P0 e ( N M ) ( N M ) PI e 0 [( N M ) ( N 2 M 2 )] CI 0 ( P0 eM M IE2 ) 2 ( r )(t13 M 3 ) ( r ) 2 (t14 M 4 ) Here, retailer does not have enough money to pay off total ] (44) 2 8 purchase cost at N. He will not pay money of P D(M).M + IE2 at M and PD( N M ).( N M ) PI [( N M ) ( N 2 M 2 )] at e To minimize the total cost per unit time, the optimum value 2 of t1, T2 is the solution of following equation. N. That’s why he has to pay interest on unpaid balance U1 = CI0 – (P D(M).M + IE2) with Ic1 interest rate during (M, N) Case III: t1 ≥ N N e and U U PD( N M ).( N M ) PI D(t )dt 2 1 M with interest rate Ic2 during (N, t1). Therefore, total interest charged on retailer, IC3.3 per unit time is U1 Ic1 ( N M ) U 2 Ic1 2 t1 e rt IC3.3 I b (t )dt T2 PI 0 N = U1 Ic1 ( N M ) U 22 Ic1 [ t12 N 2 (t13 N 3 )( 2r ) (t14 N 4 ) r (r ) T2 PI 0 2 6 8 (t15 N 5 ) 2 0 ( )(t14 N 4 ) 0 ( )(t15 N 5 )(3 2r ) U 22 Ic1 r ] [(t1 N ) 20 12 12 P r (t12 N 2 ) (r 2 0 ( ))(t13 N 3 ) ] (45) 2 6 Interest earned per unit time is IE2 IE3.3 T2 = PI e 0 [ M 2 M 3 ( r ) M 4 ( r ) 2 ] (46) Fig 3: t1 ≥ N T2 2 6 8 In the final case, retailer pays interest at the rate of Ic2 to the So, total cost TC3.3 per unit time of inventory system is supplier. Based on the total purchased cost, CI0, total money P D(M).M + IE2 <TCb >= OCb + HCb + DCb + SCb + LCb + IC3.3 – IE3.3 PI e 0 M 2 M 3 M4 = [ ( r ) ( r )2 ] T2 2 6 8 www.ijmer.com 3647 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 2, Issue. 5, Sep.-Oct. 2012 pp-3641-3649 ISSN: 2249-6645 NUMERICAL ILLUSTRATION: THE PRECEDING T t2 t3 t4 t 5 r 2 At 4 = [Cbs 1 0 Cbh [ 1 1 ( 2r ) 1 r (r ) 1 1 THEORY CAN BE ILLUSTRATED BY THE FOLLOWING NUMERICAL EXAMPLE WHERE 2 T 2 6 8 20 12 THE PARAMETERS ARE GIVEN AS FOLLOWS: Demand parameters, a = 500, b = 5, c = 2 At 5 (3 2r ) 10 t13 t14 t5 At 5 1 ] [ ( 2r ) 1 r (r ) 1 ] Selling price, P = 30 60 T 3 8 10 15 Buyer’s purchased cost, Cb = 35 C I rt 2 (r 2 A)t13 1 I 0 t12 rt13 (r 2 A)t14 bh 0 [t1 1 ] [ ] T 2 6 T 2 3 8 Buyer’s percentage holding cost per year per dollar, I A t 2 rt 3 r 4 t 4 A t 3 t 4 ( 2r ) t15 [ 0 { 1 1 1 } 0 { 1 1 r (r )}] Cbh = 0.2 T 2 6 8 T 3 8 10 Buyer’s ordering cost per order, Cbs = 500 S ert1 e(r )t1 e(t1 rT ) e(r )T Lb ert1 0 (1 B) (r )T (r )t1 b 0 [ ] [e e ] Buyer’s shortage cost, Sb = 50 T r ( r ) r ( r ) T ( r ) + U1 Ic1 ( N M ) U 2 Ic1 [ t1 N 2 (t1 N )( 2r ) (t1 N ) r (r ) 2 2 3 3 4 4 Vendor’s unit cost, Cv = 20 T2 PI 0 2 6 8 Vendor’s percentage holding cost per year per dollar, (t N ) 2 0 ( )(t N ) 0 ( )(t N )(3 2r ) U Ic 5 5 4 4 5 5 2 1 r 1 1 ] [(t1 N ) 2 1 Cvh = 0.2 20 12 12 P Vendor’s setup cost per order, Cvs = 1000 r (t 2 N 2 ) (r 2 0 ( ))(t13 N 3 ) 1 ] Vendor’s production rate per year, K = 5 2 6 PI M 2 M 3 M4 0 ( ) e 0[ ( r ) ( r )2 ] (47) Deterioration rate, = 0.01 T2 2 6 8 First delay period, M= 0.2 TCv = OCv + HCv + DCv - IC3.3 Second delay period, N= 0.4 = [C 2 (T1 T2 ) ] 1 [( K 1) C {T1 T1 ( 2r ) T1 r (r ) 2 2 2 3 4 vs 1 2 0 hv The interest earned, Ie = 0.05 2 T2 2 6 8 The interest charged, Ic1 = 0.10 T15 r 2 AT 4 AT 5 (3 2r ) ( K 1)0 T13 T14 1 1 }] 2 [ ( 2r ) The interest charged, Ic2 = 0.20 (Ic1 > Ic2) 20 12 60 T2 3 8 Backlogging rate, B=0 T5 AT 5 C e rT1 T2 2 T23 ( 2r ) r 2T2 4 AT2 4 1 r (r ) 1 ] bh 0 [ 10 15 T 2 6 24 24 Table 1: AT25 (4 3r ) 0 e rT1 2 T23 T2 4 (3 2r ) I 0 rt 2 ] [ ] [t1 1 N T2 t1 VC BC TC 60 T2 6 24 T 2 1 0.827183 0.800625 1757.09 1405.95 3163.03 t13 0 t12 t3 t 4 r 2 t15 At14 (r 2 A) ] [ ( 2r ) 1 r (r ) 1 2 0.942755 0.456282 2086.02 1517.28 3603.30 6 T 2 6 8 20 12 3 1.02889 0.331991 2274.14 1790.02 4064.16 At 5 ( K 1)0 A T13 T14 ( 2r ) T15 r (r ) (3 2r ) 1 ] Cv [ { } 30 T2 3 8 10 4 1.10067 0.266369 2425.17 2083.54 4508.71 5 1.16312 0.225188 2559.84 2374.64 4935.49 e rT1 0 A T23 T2 4 (3 2r ) { }] - T2 6 24 PI e 0 M 2 M3 M4 (48) [ ( r ) ( r ) 2 ] T2 2 6 8 To minimize the total cost per unit time, the optimum value of t1, T2 is the solution of following equation. www.ijmer.com 3648 | Page International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol. 2, Issue. 5, Sep.-Oct. 2012 pp-3641-3649 ISSN: 2249-6645 Table 2: AKNOWLEGEMENT We are thankful to Dr.A.P.Singh of SGRR (PG) College T2 t1 VC BC TC Dehradun for his constant support and review of our N research paper. We also thank Prof. Yudhveer Singh Moudgil of Uttranchal Institute of Technology for 1 0.792393 0.745431 1966.30 1774.98 3741.28 reviewing our research paper. 2 0.921355 0.433375 2435.64 1877.95 4313.59 OBSERVATION 3 1.01214 0.317385 2708.53 1969.85 4678.39 The data obtained clearly shows that individual optimal 4 1.08612 0.255438 2927.92 2215.75 5143.68 solutions are very different from each other. However, there exists a solution which ultimately provides the minimum 5 1.14978 0.216326 3121.89 2474.02 5595.9 operating cost to the whole supply chain. All the observations can be summed up as follows: 1. An increase in the interest charged, increases the buyer Table 3: cost BC and decrease the vendor cost VC of the commodity. N T2 t1 VC BC TC 2. Optimal solution for the buyer is n=1 in table first while 1 0.792393 0.745431 1780.22 1823.14 3603.36 for the vendor, it is n=5 in table 4. The overall optimal solution which ultimately minimizes the cost across the 2 0.921355 0.433375 1934.79 1957.21 3892.00 whole supply chain is n=5 in table 4 3 1.01214 0.317385 2265.29 2049.26 4314.55 REFERENCES 1) Hwang H. S. (1997). A Study on an inventory model 4 1.08612 0.255438 2315.26 2320.36 4635.62 for items with Weibull ameliorating. Com. & Ind. Eng. 33, 701-704. 5 1.14978 0.216326 2497.69 2546.68 5044.37 2) Hwang H. S. (1999). Inventory Model for Items for both Deteriorating and Ameliorating Items. Com. & Ind. Eng. 37, 257-260. Table 4: 3) Hwang, H. S. (2004): A Stochastic Set-covering Location Model for Both Ameliorating and N T2 t1 VC BC TC Deteriorating Items. Com. & Ind. Eng. 46, 313-319. 4) Law S. T. and Wee H. M. (2006). An integrated 1 1.43526 0.40970 1524.28 6918.29 8442.57 production-inventory model for ameliorating and deteriorating items taking account of time 2 2.22410 0.41285 1328.68 4504.47 5833.15 discounting. Math. & Comp. Mod., 43, 673-685. 5) Mondal B., Bhunia A. K. and Maiti M. (2003). An 3 2.69166 0.416398 1270.01 3076.3 4346.31 inventory system of ameliorating items for price 4 3.01918 0.420088 1172.2 2283.4 3455.6 dependent demand rate. Com. & Ind. Eng. 45, 443- 456. 5 3.27318 0.423883 1032.2 1856.66 2888.86 6) Moon I., Giri B. C. and Ko B. (2005). Economic order quantity models for ameliorating/ deteriorating items under inflation and time discounting. Euro. Jour. V. CONCLUSION Oper. Res. 162, 773-785. Here we have studied a two echelon supply chain with some very realistic assumptions. We studied our model in a progressive credit period. No doubt, this assumption imparts an economic viability to the whole study. In real world, it is noted that, as a result of progressive permissible delay in settling the replenishment account, the economic replenishment interval and order quantity generally increase marginally, although the annual cost decreases considerably. The saving in cost as a result of permissible delay in settling the replenishment account largely come the ability to delay payment without paying any interest. As a result of increasing order quantity under conditions or permissible delay in payments, we need to order less often. So this EOQ model is applicable when supplier gives the trade credit to the retailer. www.ijmer.com 3649 | Page