(PDF) Two-warehouse inventory model for deteriorating items with imperfect quality under the conditions of permissible delay in

Scientia Iranica E (2017) 24(1), 390{412 Sharif University of Technology Scientia Iranica Transactions E: Industrial Engineering www.scientiairanica.com Two-warehouse inventory model for deteriorating items with imperfect quality under the conditions of permissible delay in payments C.K. Jaggia , L.E. Cardenas-Barronb , S. Tiwaria and A.A. Sha ; a a. Department of Operational Research, Faculty of Mathematical Sciences, New Academic Block University of Delhi, Delhi-110007, India. b. School of Engineering and Sciences, Tecnol ogico de Monterrey, E. Garza Sada 2501 Sur, C.P. 64849, Monterrey, Nuevo Leon, Mexico. Received 5 August 2015; received in revised form 19 September 2015; accepted 14 December 2015 KEYWORDS Inventory; Deterioration; Imperfect items; Two-warehouse; Inspection; Trade credit. Regularly, manufacturing systems produce perfect and imperfect quality items. The perfect items start deteriorating as soon as they enter inventory. On the other hand, the suppliers make a delay in payment in order to motivate their buyers to purchase more products. This paper develops a two-warehouse inventory model that jointly considers the imperfect quality items, deterioration, and one level of trade credit. The proposed inventory model optimizes the order quantity to maximize the total pro t per unit time. Finally, the proposed inventory model and its solution procedure are validated with numerical examples and a sensitivity analysis is done to show how inventory model reacts to changes in parameters. 2017 Sharif University of Technology. All rights reserved. Abstract. © 1. Introduction Nowadays, production managers apply and implement ecacious manufacturing planning and complex production and control systems in order to have 100% quality items. However, the manufacturing process may still produce defective items. The defective items reduce the pro t for the retailer and their inadvertent supply to customer may cost the retailer to lose goodwill. Thus, to sustain the supply of good quality items, it is vital for the whole lot to be screened as soon as it comes into the inventory and the defective items identi ed must be removed from the lot. With this in mind, a great amount of research has been made in the direction of the development EOQ/EPQ models *. Corresponding author. Tel.: +52 81 83284235; Fax: +52 81 83284153 E-mail address:

(L.E. Cardenas-Barron) for defective items. Porteus [1] found a signi cant relationship between the fraction of defective items and the lot. Rosenblatt and Lee [2] and Lee and Rosenblatt [3] showed through their papers that the presence of defective items forces the lot size to be smaller. Salameh and Jaber [4] expanded the research made for imperfect quality items considering random yield. They developed an EOQ model that refuted the results of Rosenblatt and Lee [2]. Salameh and Jaber [4] concluded that the batch quantity increased as the average percentage of imperfect quality items rose. Cardenas-Barron [5] corrected an error in the model of Salameh and Jaber [4] without a ecting its main idea. Goyal and Cardenas-Barron [6] proposed an easy to apply method to determine the lot size in the model of Salameh and Jaber [4]. Papachristos and Konstantaras [7] examined the issue of non-shortages in the model of Salameh and Jaber [4] with proportional imperfect quality when the proportion of the imperfect was a random variable. Moussawi-Haidar et al. [8] C.K. Jaggi et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 390{412 studied the e ect of deterioration on the instantaneous replenishment of the lot with imperfect quality items. The items in inventory are under numerous risks like breakage, obsolescence, and evaporation. Another phenomenon that signi cantly a ects items in inventory, e.g. gasoline, chemicals, and food items, which deteriorate fast through time, is deterioration. Thus, the loss by deterioration cannot be disregarded. Researchers have been progressively modifying the existing inventory models for deteriorating items in order to make them more realistic. Goyal and Giri [9] presented a complete review of research on deteriorating items published up to 2001. Bakker et al. [10] provided an extensive and comprehensive review on advancements in the eld of inventory control of deteriorating items. In order to avoid the losses related to deterioration, a retailer may rent another warehouse with better preserving facilities than those of his/her own warehouse. Moreover, the trade credit period provided by supplier motivates the retailer to purchase a quantity that exceeds the owned warehouse capacity and, consequently, the retailer needs to rent another warehouse for storage. The two-warehouse inventory system was rst proposed by Hartley [11]. Das et al. [12] studied joint performance of a supply chain with two warehouse facilities. Hsieh et al. [13] proposed a deterministic inventory model for deteriorating items with two warehouses that minimized the net present value of the total cost. Several research papers in this interesting area were published by researchers in the last few decades [14-18]. The EOQ assumes that when the retailers receive the order, they immediately pay their suppliers. However, this is not always true. To facilitate the business for retailers, the suppliers sometimes make a delay in payment to retailers for a xed time period to settle the payment of the order without charging any interest on the retailers during this credit period. But, an interest is charged at pre-determined rate if payment is not made by the end of the established credit period. Both the retailer and the supplier bene t from credit period. During the period before the account has to be paid, the buyers can vend products and accumulate revenue, and earn interest by placing revenue within an interest bearing account. The delay in payment period motivates the retailers to order more products and, thus, turns out to be bene cial for the supplier. It is important to remark that large orders increase holding cost and losses due to deterioration. Naturally, the retailer must consider all pros and cons while ordering into bulk in order to earn a maximum pro t. This paper proposes a two-warehouse inventory model that considers imperfect quality items under deteriorating conditions and permissible delay in payments. The lot is screened as soon as it enters the inventory system. The screening rate is assumed to be 391 greater than the demand rate so that the demand can be satis ed along with the screening running in parallel, out of the items which are perfect in quality. Shortages are not allowed. The inventory model optimizes retailer's order quantity by maximizing his/her total pro t. 2. Assumptions and notation The mathematical model for the two-warehouse inventory problem is based on the following assumptions: 1. The Owned Warehouse (OW) has a xed capacity of w units; the Rented Warehouse (RW) has unlimited capacity; 2. The initial inventory level and lead time are zero; 3. The deterioration rate of RW ( ) is less than the deterioration rate of OW ( ); 4. The screening process and demand occur simultaneously, but the screening rate (x) is greater than the demand rate (D), x > D; 5. The supplier provides a xed credit period to the retailer to settle the account; 6. Shortages are not allowed; 7. The defective items exist in lot size y . The percentage of defective items is a random variable (p) with: Zl2 E (p) = pf (p)dp; 0 < l1 < l2 < 1: l1 The following notations are used: y Order size per cycle (units)-decision variable w Storage capacity of OW (units) D Demand rate per unit time (unit/time unit) p Percentage of defective items in y (%) f ( p) Probability function of p ts Screening time of RW (time unit) tw Screening time of OW (time unit) tr Time point when the stock level of RW reaches zero (time unit) T The replenishment cycle (time unit) M The retailer's trade credit period provided by the supplier (time unit) Io (t) Inventory level of OW at time t (units) Ir (t) Inventory level of RW at time t (units) X Screening rate (unit/time unit) Deterioration rate of OW Deterioration rate of RW c Unit purchasing cost per item ($/unit) 392 k s v d Ie Ip hr ho HCr HCo TPU(y ) C.K. Jaggi et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 390{412 Fixed cost of placing an order ($/order) Unit selling price per item of good items ($/unit) Unit selling price per item of defective items ($/unit) Unit screening cost per item ($/unit) Interest earned (%/unit time) Interest paid (%/unit time) Holding cost per unit item per unit time in RW, excluding interest charges ($/unit/time unit) Holding cost per unit item per unit time in OW, excluding interest charges ($/unit/time unit) Inventory holding cost of RW ($/time unit) Inventory holding cost of OW ($/time unit) The total pro t per unit time ($/time unit) of RW reaches zero. The inventory in RW depletes due to demand and deterioration from [0; tr ] and reaches zero. The inventory in OW depletes only due to deterioration during [0; tr ] and, then, by both demand and deterioration during [tr ; T ]. The behavior of the inventory model through the whole cycle [0; T ] is shown graphically in Figures 1 and 2. The di erential equations that model the inventory level in both warehouses RW and OW at any time, t, over the period (0; T ) are: dIr (t) + Ir (t) = D d(t) 0 t tr ; (1) dIo (t) = dt 0 t tr ; (2) Io (t) 3. Model formulation We consider that a lot of size y enters the inventory system, out of which w units are placed in OW and (y w) units are stored in RW. The RW is considered to have better preserving facilities than OW and, hence, deterioration rate of RW ( ) is less than the deterioration rate of OW ( ). The holding cost at RW is greater than the holding cost at OW (hr > ho ). In any production process, due to certain reasons such as improper transport, low labor skills, low quality of raw materials, among others, the production process may shift to an imperfect production process in which not all the items manufactured are of good quality. Due to this, a screening process must be conducted at screening rate of (x) units per unit time when the whole lot enters the inventory. It is assumed that each received lot y contains p percent of defective items, where p is a random variable with a known probability density function, f (p), and mean, E (p) = p. Thus, lot y has py defective items and (1 p)y non-defective items. The defective items found are kept in stock and sold at the end of the screening period at a salvage value of (v ) per unit, v < c. The screening process takes place in OW and RW, simultaneously, and gets completed at tw = w=x and ts = (y w)=x, respectively. Depending on the values of tw , ts , and tr , the following cases are discussed. Case I: When Figure 1. Inventory level for the two-warehouse system when tw < ts < tr . tw < t r This case considers that the screening period of OW (tw ) is less than the time point (tr ) when the stock level Figure 2. Inventory level for the two-warehouse system when ts < tw < tr . C.K. Jaggi et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 390{412 dIo (t) + Io (t) = D dt tr t T: 393 (3) Solving the above di erential equations with the boundary conditions Ir (0) = y w, Ir (t+ s ) = Ir (ts ) p(y w), Io (0) = w, Io (t+w ) = Io (tw ) pw, and Io (T ) = 0, the solutions are: D Ir (t)= D Ir (t)= + y w+ D e t 0 t ts ; (4) p(y w)e ts e t + (y w)+ D= ts < t tr ; (5) Io (t) = we t 0 t tw ; (6) Io (t) = w pwe tw e t tw < t tr ; (7) Io (t) = D e (T t) 1 tr t T: (8) Applying the boundary condition Ir (tr ) = 0, the value of tr is: tr = 1 ln 1 + D w) 1 pe ts (y : (9) Considering the continuity of Io (t) at t = tr , the value for T is: w wpe tw e 1 T = tr + ln tr = i h Case II: When D n (T tr ) o e 1 ; D w pwe tw e tr + 1 : (10) dIo (t) + Io (t) = D dt ts < t r < t w . and continuity of Io (t) at t = tr , the solutions are: 0 t tr ; Io (t) = we t Io (t) = D + we tr + D e ( (13) t tr ) tr t tw ; D (14) + we tr + D pwe t ( w tr ) e tw < t T: ( t tr ) (15) Io (T ) = 0 implies: tw > tr Io (t) when Io (t)= Figure 3 shows the behavior of the inventory model over the time interval [0; T ]. It is assumed that the screening period of OW (tw ) is greater than the time point (tr ) when the stock level of RW reaches zero. The inventory in RW diminishes due to demand and deterioration from [0; tr ] and reaches zero. The inventory in OW diminishes only due to deterioration during [0; tr ] and, then, by both demand and deterioration during [tr ; T ]. Notice that the equations for the inventory level of RW are identical to those in Case I. Thus, the di erential equations and their solutions for OW are: dIo (t) = dt Figure 3. Inventory level for the two-warehouse system 0 t tr ; tr t T: (11) (12) Solving the above di erential equations with the boundary conditions, Io (0) = w, Io (tw+ ) = Io (tw ) pw , 1 T = tr + ln i h D w pwe tw e tr + 1 : (16) It is easy to see that the total time cycle T is identical to that in Case I. Now, let Nr (y; p) and No (y; p) be the totals of good items in each lot at time t with respect to RW and OW, respectively. These items are obtained by removing the defective and deteriorated items from the inventory. Let w1r and w1o be the total numbers of deteriorated items during time intervals [0; tr ] and [0; T ] in RW and OW, respectively: Nr (y; p) = (y w)(1 p) w1r ; (17) No (y; p) = w(1 p) w1o : (18) Let I01r (t) be the inventory level of RW at time t when both e ects of lot quality and deterioration are ignored; thus, I01r (t) = Dt+y w. Let I02r (t) be the inventory level of RW when only the e ect of deterioration is 394 C.K. Jaggi et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 390{412 (where i = Case 1 - Case 3 and j = Sub-case 1 to Sub-case 5) is comprised of the following terms: T Pi:j (y) =Sales revenue + Interest earned Ordering cost Purchasing cost Screening cost Holding cost Interest paid: Figure 4. Inventory level for rented warehouse showing the deteriorated items at time tr . Thus: T Pi:j (y) =Sales revenue + Interest earned k cy dy Holding cost Interest paid: (24) Sales revenue, T R, is the sum of the total sale volumes of good quality and imperfect quality items from RW and OW: T R =s((y w)(1 p) w1r + w(1 p) w1o ) + v (y By replacing w1r and w1o , T R becomes: Figure 5. Inventory level for the owned warehouse showing the deteriorated items at time T. T R = sDT + vyp: ignored; hence, I02r (t) = Dt + (y According to Figure 4, w1r is: w1r = I02r (tr ): w)(1 p). (19) Let I01o (t) be the inventory level of OW at time t when both e ects of lot quality and deterioration are ignored; thus, I01o (t) = w. Let I02o (t) and I03o (t) be the inventory levels of OW when only the e ect of deterioration is ignored; hence, I02o (t) = w pw and I03o (t) = D(t tr ) + w pw. According to Figure 5, w1o is: w1o = I03o (T ): (20) To avoid shortages, it is assumed that the totals of good items Nr (y; p) and No (y; p) are at least equal to the demands during screening times, i.e. ts and tw : Nr (y; p) Dts ; (21) No (y; p) Dtw : (22) From Eqs. (21) and (22), the percent of defective items (p) must satisfy: p min 1 (y D w) D e tw w + 1+ (y D w) e ts ; (25) The inventory holding cost per cycle in RW is: HCr =hr 8 <Zts : ( =hr Ir (t)d(t) + 0 y w D (1 ln 2 D (23) The retailer's total pro t during a cycle T Pi:j (y ), Ztr ts Ir (t)d(t) 9 = ; p) (y w) 1 pe ts + 1 ) : (26) The inventory holding cost per cycle in OW for the case with tw < tr is: HCo =ho 8 <Ztw : ( =h o Io (t)dt + 0 w (1 p) ) o ! 1 a tr e tw : w)p + vwp: +1 : Ztr tw D 2 Io (t)dt + n ln D ZT tr Io (t)dt 9 = ; w wpe tw e tr (27) The inventory holding cost per cycle in OW for the case with tr < tw is: C.K. Jaggi et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 390{412 Table 1. Di erent sub-cases in each main case. HCo =ho : ( =ho Io (t)dt + 0 w (1 p) ) o +1 D Ztw Io (t)dt + tr 2 n ln D ts < t r < t w ts < t w < t r tw < t s < t r 8 <Ztr Case 3: Case 2: Case 1: Sub-case 1.1: 0 < M tw < ts < tr Sub-case 1.2: tw < M ts < tr Sub-case 1.3: tw < ts < M tr Sub-case 1.4: tw < ts < tr < M T Sub-case 1.5: tw < ts < tr < T < M Sub-case 2.1: 0 < M ts < tw < tr Sub-case 2.2: ts < M tw < tr Sub-case 2.3: ts < tw < M tr Sub-case 2.4: ts < tw < tr < M T Sub-case 2.5: ts < tw < tr < T < M ZT tw Io (t)dt w wpe : 9 = Sub-case 3.1: 0 < M ts < tr < tw Sub-case 3.2: ts < M tr < tw Sub-case 3.3: ts < tr < M tw Sub-case 3.4: ts < tr < tw < M T Sub-case 3.5: ts < tr < tw < T < M The interest payable per cycle for the inventory not sold after the due period M from RW and OW is given by: ; Interest paid from RW = tw e tr cIp 8 <Zts : Ir (t)d(t) + M ( (28) Hence, from Eqs. (27) and (28), we see that the holding costs in two cases with tw < tr and tr < tw are the same. The interest earned, interest paid, and pro t functions are calculated for di erent cases, which are shown in Table 1. The solution procedure is as follows. tw < t s < t r Sub-case 1.1: 0 < M tw < ts < tr Case 1: The retailer earns interest on revenue generated from the sale of good quality items up to M . Although, the account must be paid at M and for that, the money has to be arranged at some speci ed rate of interest in order to get remaining stocks nanced for the period M to T . The interest earned per cycle is equal to the area of triangle OAM in Figure 6. The interest earned is: 1 sI DM 2 (OM MA) = e : (29) 2 2 = cIp + 1 D Graphical representation of interest earned and interest charged for 0 < M tw < ts < tr . (M Ztr Ir (t)d(t) tr ) y w+ p 9 = ; ts D Interest paid from OW = cIp 8 <Ztw : Io (t)dt + M ( = cIp D 2 w ln Ztr tw Io (t)dt + e M n D p e tr e M w)e ts e tr + (y ) ts : e ZT tr Io (t)dt (30) 9 = ; w wpe tw e ) o tr + 1 : (31) Substituting the values of Eqs. (25)-(27) and (29)(31) into Eq. (24), the total pro t for Sub-case 1.1 is obtained by Eq. (32) as shown in Box I. Sub-case 1.2: Figure 6. 395 tw < M ts < tr The retailer receives interest on revenue created from the sale of good quality items up to M and also from the defective items sold as a single lot from OW for tw < t M . The account has to be paid at M and, hence, the retailer must arrange the money at a rate of interest in order to obtain the remaining stock nanced from M to T . Therefore, the interest earned on good items per cycle is equal to the area of triangle OAM . Thus, the 2 . interest earned on good items is determined as sIe DM 2 In addition, the retailer can earn interest on the sale 396 C.K. Jaggi et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 390{412 sI T P1:1 (y)= sDT + vyp + e 8 > > > > > > > > > > > > > > > > > > > < 2 DM 2 > > > > > > > > > > > > > > > > > > > : ( k + cy + dy + hr y w (1 ( +ho w (1 +cIp D (M + p (y +cIp tr ) + w)e ts ( w e p) e M p 1 ) o ts +1 )) ( ln D (y w) 1 pe 2 D ln 2 D (w p) ( n D wpe y w+ D tr +1 tw ) e e M ) e tr e D ln (w 2 D ) wpe tr +1 tw ) e : > > (32) > > > > > > > > > > > > > > > > > ; ts tr 9 > > > > > > > > > > > > > > > > > > > = Box I cIp 8 <Ztr : Io (t)d(t) + M ( = cIp Figure 7. Graphical representation of interest earned and interest charged for tw < M ts < tr . w 1 ZT tr 9 = Io (t)d(t) ; pe tw e M n ) o w 1 pe tw e tr + 1 : 2 ln D D (35) of defective items from OW, which is equal to the area of the rectangle CDMtw , as shown in Figure 7. Therefore, the interest earned from the sale of defective items from OW is given by vIe pw(M tw ) hence: Substituting the values from Eqs. (25)-(27) and (33)(35) in Eq. (24), the total pro t for Sub-case 1.2 is obtained by Eq. (36) as shown in Box II. The total interest earned = Sub-case 1.3: sIe DM 2 + vIe pw(M tw ): (33) 2 The interest payable per cycle for the inventory not sold after the due period M from RW and OW is given by: Interest paid from RW = cIp 8 <Zts : Ir (t)d(t) + M ( = cIp + 1 p D + (y (M Ztr 9 = Ir (t)d(t) ; ts tr ) y w+ D w)e ts e Interest paid from OW = e M e tr tr ts ) e : (34) tw < ts < M tr In this sub-case, the retailer wins interest on revenue produced from the sale of good quality items up to M and, likewise, from the sale of defective items wholesaled as one lot from OW for tw < t M and from RW for ts < t M . The retailer has to arrange the money at a speci ed rate of interest to settle the account at M in order to nance the remaining stocks for the period M to T . Therefore, the interest earned on good items per cycle is determined with the area of triangle OAM . Thus, the interest earned on good items is given by sIe DM 2 . Furthermore, the retailer can gain interest on 2 the sale of defective items from OW and RW, which is equal to the sum of the areas of rectangles CDMtw and EF Mts that are depicted in Figure 8. Consequently, the interest earned from the sale of defective items from OW is vIe pw(M tw ) and the interest earned from the sale of defective items from RW is vIe p(y w)(M ts ). Thus: The total interest earned = C.K. Jaggi et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 390{412 T P1:2 (y)= sDT + vyp + 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : sIe DM 2 2 + vIe pw(M tw ) ( k + cy + dy + hr y w (1 ( +h o +cIp w (1 ( D (M tr ) + w (1 +cIp D p) pe 1 tw ) e M ln D (y w) 1 pe o ts +1 ) ) wpe w+ D y n 2 D ln (w 2 D p) ( 397 tw ) e e M D ln w (1 2 D tr + 1 tr + p (y e pe tw ) e w)e ts e tr ) e ) ts tr + 1 9 > > > > > > > > > > > > > > = > > > > > > > > > > > > > > ; : (36) Box II Figure 8. Graphical representation of interest earned and interest charged for sIe DM 2 + vIe p(y w)(M ts )+ vIe pw(M tw ): (37) 2 The interest payable per cycle for the inventory not sold after the due period M from RW and OW is given by: Interest paid from RW = cIp 8 <Ztr 9 = Ir (t)d(t) = cIp ; : M + 1 ( Interest paid from OW = cIp : Io (t)d(t) + M ( = cIp D 2 ln (M D w)(1 p)e ts + (y 8 <Ztr D w n 1 ZT tr tr ) e ) M tr e : (38) 9 = Io (t)d(t) ; tw < ts < M tr . (37)-(39) into Eq. (24), the total pro t for Sub-case 1.3 is obtained by Eq. (40) as shown in Box III. Sub-case 1.4: tw < ts < tr < M T The interest earned in this sub-case is calculated in the similar way to that in Sub-case 1.3. The interest earned on good items per cycle is obtained with the area of triangle OAM . Thus, the interest earned on 2 good items is calculated with sIe DM . 2 Additionally, the retailer obtains interest on the sale of defective items from both OW and RW, which is computed with the sum of the areas of rectangles CDMtw and EF Mts that are displayed in Figure 9. Hence, the interest earned from the sale of defective items from OW is determined as vIe pw(M tw ) and the interest earned from the sale of defective items from RW is obtained by vIe p(y w)(M ts ). Then: The total interest earned = pe tw e M sIe DM 2 2 ) o w 1 pe tw e tr + 1 D : (39) Substituting the values given in Eqs. (25)-(27) and + vIe p(y w)(M ts ) + vIe pw(M tw ): (41) The interest paid in this case will result only from OW, since RW is exhausted, and it is given by: Interest paid from OW = 398 C.K. Jaggi et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 390{412 sIe DM 2 T P1:3 (y) = sDT + vyp + 8 > > > > > > > > > > > > > > < 2 + vIe p(y w)(M ts )+ vIe pw(M tw ) ( k + cy + dy + hr y w (1 ( w (1 +h o > > > > > > > > > > > > > > : +cIp D (M ( +cIp D ln (w 2 D p) ( w (1 p) tr )+ (y pe n ln D (y w) 1 pe 2 ) wpe D ln w (1 2 D M o ts +1 tr + 1 tw ) e ! w)(1 p)e ts + D 1 tw ) e D e M pe tw ) e tr e ) ) 9 > > > > > > > > > > > > > > = > > > > > > > > > > > > > > ; ) tr + 1 : (40) Box III Figure 9. Graphical representation of interest earned and interest charged for cIp 8 <ZT : 9 = ; = cIp D 2 e ( T M) D 1 (T M ) : (42) Substituting Eqs. (25)-(27) and (40)-(42) into Eq. (24), the total pro t for Sub-case 1.4 is obtained by Eq. (43) as shown in Box IV. Sub-case 1.5: sIe DT 2 + sIe DT (M T ): 2 Besides, the retailer can obtain interest on the sale of defective items from OW and RW, which is determined with the sum of the areas of rectangles CDMtw and EF Mts as exposed in Figure 10. tw < t s < t r < T < M In this sub-case, the retailer obtains interest on revenue generated from the sale of good quality items up to T sI DM 2 + vIe p(y w)(M ts ) + vIe pw(M T P1:4 (y) = sDT + vyp + e 2 8 > > > > > > < > > > > > > : ( k + cy + dy + hr y w (1 ( +ho w (1 +cIp D2 e p) D ln 2 D (w (T M ) 1 D (T p) . for time T < t M . Moreover, the retailer earns interest from the defective items vended as a sole lot from OW for tw < t M and from RW for ts < t M . The interest earned on good items per cycle is calculated with the area of triangle ABT . Also, the interest earned on good items for the period [T; M ] is computed with the area of the rectangle CDMT . Thus, the interest earned on good items is given by: I3o (t)d(t) M tw < t s < t r < M T n D ln 2 D (y w) 1 pe wpe tw ) e tr + 1 M) Box IV ) tw ) ts +1 o ) 9 > > > > > > = > > > > > > ; : (43) C.K. Jaggi et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 390{412 Figure 10. Graphical representation of interest earned and interest charged for 399 tw < t s < t r < T < M . As a result, the interest earned from the sale of defective items from OW is calculated with vIe pw(M tw ) and the interest earned from the sale of defective items from RW is computed with vIe p(y w)(M ts ). Hence: Total interest earned = sIe DT 2 2 + sIe DT (M T ) + vIe p(y w)(M + vIe pw(M tw ): ts ) (44) Since no inventory is left after time T , the interest paid is equal to zero. Substituting the values from Eqs. (25)-(27) and (44) into Eq. (24), the total pro t for Sub-case 1.5 is obtained by Eq. (45) as shown in Box V. Figure 11. Graphical representation of interest earned and interest charged for 0 Interest paid from RW = ( t s < tw < t r Sub-case 2.1: 0 < M ts < tw < tr cIp Case 2: ( cIp : (46) 2 The interest payable per cycle for the inventory not sold after the due period M from RW and OW is given by: 8 > > > > < > > > > : 2 + sIe DT (M ( +h o w (1 p) p) D ln 2 D (w w)e ts D y w+ e D e M e tr ) tr e w e M D 2 ln p ts : (47) n D w wpe tw e tr + 1 ) o : (48) Substituting the values given in Eqs. (25)-(27) T ) + vIe p(y w)(M ts )+ vIe pw(M tw ) ( k + cy + dy + hr y w (1 1 . Interest paid from OW = sIe DM 2 T P1:5 (y) = sDT + vyp + (M tr ) + + (y The total interest earned = sIe DT 2 D p Here, the interest earned on good items is equal to the area of triangle OAM , which is shown in Figure 11. Thus: < M ts < t w < t r n 2 ln D (y w) 1 pe ) wpe tw ) e Box V tr + 1 ts +1 o ) 9 > > > > = > > > > ; : (45) 400 C.K. Jaggi et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 390{412 T P2:1 (y) = sDT + vyp + 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : sIe DM 2 2 ( k + cy + dy + hr y w (1 ( +ho +cIp w (1 ( +cIp D (M w tr ) + M e p) 2 p 1 wpe w+ D y ln D (y w) 1 pe o ts +1 ) ) D ln (w 2 D p) ( n D D ln (w 2 D tr + 1 tw ) e e M wpe e tw ) e tr + p (y ) w)e ts e tr ) ts e tr + 1 9 > > > > > > > > > > > > > > = : > > (49) > > > > > > > > > > > > ; Box VI and (46)-(48) into Eq. (24), the total pro t for Subcase 2.1 is obtained by Eq. (49) as shown in Box VI. ts < M tw < t r Sub-case 2.2: Here, the interest earned on good items per cycle is determined by the area of triangle OAM . Therefore, the 2 . Furthermore, interest earned on good items is sIe DM 2 the retailer can gain interest on the sale of defective items from RW, which is given by the area of rectangle CDMts as shown in Figure 12. Hence, the interest earned from the sale of defective items from RW is vIe p(y w)(M ts ). Thus: Total interest earned = sIe DM 2 + vIe p(y w)(M ts ): (50) 2 Similarly, the interest payable per cycle for the inventory not sold after the due period M from RW and OW is given by: Interest paid from RW = cIp D + (M 1 tr ) D w)(1 p)e ts + (y e M tr e : (51) Interest paid from OW =%pagebreak[3] ( cIp w e D 2 ln M n D p w wpe tr + 1 and interest charged for ts < M tw < tr ) o : (52) . Substituting Eqs. (25)-(27) and (50)-(52) in Eq. (24), the total pro t for Sub-case 2.2 is obtained by Eq. (53) as shown in Box VII. Sub-case 2.3: ts < tw < M tr Here, the interest earned on good items per cycle is computed by the area of triangle OAM . Thus, the 2 . Besides, the interest earned on good items is sIe DM 2 retailer earns interest on the sale of defective items from RW and OW, which is equal to the sum of the areas of rectangles EF Mtw and CDMts as shown in Figure 13. Consequently, the interest earned from the sale of defective items from OW and RW is calculated with vIe pw(M tw ) and vIe p(y w)(M ts ), respectively. Thus: The total interest earned = sIe DM 2 tw e Figure 12. Graphical representation of interest earned + vIe p(y w)(M ts )+ vIe pw(M tw ): (54) 2 The interest payable per cycle for the inventory not sold after the due period M from RW and OW is given by: Interest paid from RW = C.K. Jaggi et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 390{412 sIe DM 2 T P2:2 (y) = sDT + vyp + 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : 2 + vIe p(y k + cy + dy + hr y w (1 ( +h o w (1 cIp D (M p) ( tr ) + ( w +cIp M e p) (y p ts ) n D 2 ln D (y w) 1 pe o ts +1 ) D ln (w 2 D 1 w)(M ( wpe w)(1 tr + 1 tw ) e ! p)e ts + D D ln (w 2 D 401 wpe e M tw ) e e tr ) ) tr + 1 ) 9 > > > > > > > > > > > > > > = > > > > > > > > > > > > > > ; : (53) Box VII Figure 13. Graphical representation of interest earned and interest charged for Figure 14. Graphical representation of interest earned and interest charged for ( cIp D + 1 (M (y Sub-case tr ) D w)(1 p)e ts + ) tr :(55) e M e cIp w 1 D 2 ln pe n tw e . ts < t w < t r < M T . ts < tw < tr < M T The interest earned in this sub-case is calculated in an analogous manner like that in Sub-case 2.3, which is shown in Figure 14. Hence: Total interest earned = Interest paid from OW = ( 2.4: ts < t w < M tr sIe DM 2 M 2 ) o w 1 pe tw e tr + 1 D : (56) Substituting the values from Eqs. (25)-(27) and (54)(56) in Eq. (24), the total pro t for Sub-case 2.3 is obtained by Eq. (57) as shown in Box VIII. + vIe p(y w)(M ts )+ vIe pw(M tw ): (58) Since RW is exhausted at tr , the interest paid occurs by the inventory not sold from OW and is given as: Interest paid from OW = cIp D 2 e ( T M) 1 D (T M) : (59) 402 C.K. Jaggi et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 390{412 : ( )= T P2 3 y sDT 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : + vyp + e sI DM 2 2 + vIe p(y w )(M ( y w (1 k + cy + dy + hr ( w (1 +h o p ) +cIp r )+ t ( w (1 +cIp pe 1 ) s )+ vIe pw(M ln D (y 2 wpe ) 1 w) pe o ts +1 ) ! D ln w (1 2 D M w t tr + 1 tw ) e D t w )(1 p)e s + (y tw ) e n D D ln (w 2 D ( D (M p t M e tr e ) > > > > > > > > > > > > > > ; ) pe tw ) e ) 9 > > > > > > > > > > > > > > = tr + 1 : (57) : (60) Box VIII : ( )= T P2 4 y sDT 8 > > > > > > > > > < > > > > > > > > > : + vyp + e sI DM 2 2 + vIe p(y w )(M ( k + cy + dy + hr y w (1 ( +ho w (1 p ( +cIp D2 e ( ) p ) D ln (w 2 D T M ) 1 D ( T t n D s )+ vIe pw(M ln D (y 2 wpe tw ) e w ) 1 t w) pe o ts +1 ) tr +1 > > > > > > > > > ; ) M ) 9 > > > > > > > > > = ) Box IX Substituting Eqs. (25)-(27) and (58)-(59) into Eq. (24), the total pro t for Sub-case 2.4 is obtained by Eq. (60) as shown in Box IX. Sub-case 2.5: ts < t w < t r < T < M Here, the interest obtained on good items per cycle is computed by the area of triangle OBT . Also, the interest gained on good items from the period [T ; M ] is calculated by the area of the rectangle BCMT. Hence, the interest gained on good items is determined with: e sI DT 2 + sIe DT (M T ): 2 In addition, the retailer can get interest on the sale of defective items from OW and RW, which is equal to sum of the areas of rectangles C DM tw and E F M ts that are displayed in Figure 15. Therefore, the interest earned from the sale of defective items from OW is vIe pw (M tw ) and the interest earned from the sale of Figure 15. Graphical representation of interest earned and interest charged for ts < t w < t r < T < M . C.K. Jaggi et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 390{412 defective items from RW is vIe p(y w)(M 403 ts ). Thus: Total interest earned = sIe DT 2 2 + sIe DT (M T ) + vIe p(y w)(M + vIe pw(M tw ): ts ) (61) As the inventory gets exhausted at time T , the interest paid is zero. Substituting the values given in Eqs. (25)-(27) and (61) in Eq. (24), the total pro t for Sub-case 2.5 is obtained by Eq. (62) as shown in Box X. t s < tr < tw Sub-case 3.1: 0 < M ts < tr < tw Case 3: Figure 16. Graphical representation of interest earned and interest charged for 0 ( cIp The retailer gains interest on the revenue caused by the sale of good quality items up to M , which is presented in Figure 16. Although, the account must be paid at M and for that, the money must be arranged at a speci ed rate of interest in order to obtain a nancing for the remaining stocks for the period M to T . Therefore: 2 : D (63) (M tr )+ p 1 y w+ w)e ts + (y tr e D Sub-case 3.2: e ) : T P2:5 (y) = sDT + vyp + 8 > > > > < > > > > : sIe DT 2 2 k + cy + dy + hr y w (1 ( +ho w (1 p) ts < M tr < tw sIe DM 2 + vIe p(y w)(M ts ): (67) 2 Also, the interest payable per cycle for the inventory + sIe DT (M T ) + vIe p(y w)(M ts )+ vIe pw(M tw ) ( p) D ln 2 D (w D (65) Interest earned = (64) Interest paid from OW = ) n The interest gained on good items per cycle is obtained with the area of triangle OAM . Thus, the interest 2 . Additionally, the gained on good items is sIe DM 2 retailer can win interest on the sale of defective items from RW, which is determined with the area of rectangle CDMts that is illustrated in Figure 17. Thus, the interest earned from the sale of defective items from RW is vIe p(y w)(M ts ). e M e tr ts Substituting the values from Eqs. (25)-(27) and (63)(65) in Eq. (24), the total pro t for Sub-case 3.1 is obtained by Eq. (66) as shown in Box XI. Interest paid from RW = cIp p . o w 1 pe tw e tr + 1 : 2 ln D Similarly, the interest payable per cycle for the inventory not sold after the due period M from RW and OW is given by: ( e M D The interest earned on good items = sIe DM 2 w < M ts < t r < t w n 2 ln D (y w) 1 pe ) wpe tw ) e Box X tr + 1 ts +1 o ) 9 > > > > = > > > > ; : (62) 404 C.K. Jaggi et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 390{412 T P3:1 (y) = sDT + vyp + 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : sIe DM 2 2 ( k + cy + dy + hr y w (1 ( +h o +cIp w (1 ( +cIp D (M w e p) 2 tr ) + M p 1 y wpe w+ D D ln w (1 2 D ln D (y w) 1 pe o ts +1 ) ) D ln (w 2 D p) ( n D tr + 1 tw ) e e M pe tr + p (y e w)e ts e tr ) tw ) e ) ts e tr + 1 9 > > > > > > > > > > > > > > = : > > (66) > > > > > > > > > > > > ; Box XI Figure 17. Graphical representation of interest earned Figure 18. Graphical representation of interest earned and interest charged for and interest charged for ts < M tr < t w . not sold after the due period M from RW and OW is given by: Interest paid from RW = ( cIp D + 1 (M (y tr ) w)(1 p)e ts + D e M e ) tr : (68) Interest paid from OW = ( cIp w e M p ) n (69) Substituting the values from Eqs. (25)-(27) and (67)(69) in Eq. (24), the total pro t for Sub-case 3.2 is obtained by Eq. (70) as shown in Box XII. Sub-case 3.3: ts < tr < M tw The interest gained in this sub-case is calculated in the same way as that in Sub-case 3.2 (see Figure 18). Thaus: . The total interest earned = sIe DM 2 + vIe p(y w)(M ts ): (71) 2 The RW is exhausted at time tr ; therefore, the interest to be paid occurs only due to the inventory left from OW and is given by: Interest paid from OW = ( cIp D + o w tw e t r + 1 : ln 1 pe 2 D D ts < tr < M tw (M D 2 T) + e ( w M tr ) e M 1 ) p : (72) Substituting the values from Eqs. (25)-(27) and (71)(72) in Eq. (24), the total pro t for Sub-case 3.3 is obtained by Eq. (73) as shown in Box XIII. Sub-case 3.4: ts < tr < tw < M T The interest earned on good items per cycle is calculated with the area of triangle OAM . As a result, 2 the interest earned on good items is sIe DM . As well, 2 the retailer can get interest on the sale of defective items from RW and OW, which is determined with the sum of the areas of rectangles CDMtw and EF Mts C.K. Jaggi et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 390{412 T P3:2 (y) = sDT + vyp + sIe DM 2 2 405 + vIe p(y w)(M ts ) 8 ) ( o n > > y w D > k + cy + dy + hr (1 p) 2 ln D (y w) 1 pe ts +1 > > > > > ) ( > > > > w D t t r w > +1 (1 p) 2 ln D (w wpe ) e > < +h o ( ) ! > > D D 1 > +cIp (M tr )+ e M e tr (y w )(1 p)e ts + > > > > > ( ) > > > w > w D t t M w r > +1 e p > 2 ln D (1 pe ) e : +cIp 9 > > > > > > > > > > > > > > = > > > > > > > > > > > > > > ; : (70) : (73) Box XII sI DM 2 T P3:3 (y) = sDT + vyp + e + vIe p(y w )(M ts ) 2 8 > > > > > > > > > < ( k + cy + dy + hr y w (1 ( + ho w (1 p) > > > ( > > > > D (M > > : +cIp D 2 ln p) D (w T) + w e M D 2 n ln D (y w) 1 ) pe wpe tw ) e tr + 1 p + D 2 e ( M tr ) ts +1 o ) 9 > > > > > > > > > = > > > > > > > > > ; ) 1 Box XIII that are portrayed in Figure 19. Hence, the interest sold after the due period M from OW is given by: received from the sale of defective items from OW is vIe pw(M tw ) and the interest gotten from the sale of defective items from RW = vIe p(y w)(M ts ). As a result: Interest paid from OW = ( cIp D ( M T) + The interest earned = sIe DM 2 2 + vIe p(y w)(M ts )+ vIe pw(M tw ): (74) The interest payable per cycle for the inventory not pwe (tw tr ) D we tr + 1 ! e ( M tr ) e ( T tr ) ) : (75) Substituting the values from Eqs. (25)-(27) and (74)- Figure 19. Graphical representation of interest earned and interest charged for ts < t r < t w < M T . 406 C.K. Jaggi et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 390{412 : ( )= T P3 4 y sDT 8 > > > > > > > > > < > > > > > > > > > : + vyp + e sI DM 2 2 + vIe p(y w )(M ( y w (1 k + cy + dy + hr ( +h o w (1 p ) D ln (w 2 D ) ( +cIp D (M p T )+ 1 s )+ vIe pw(M n D ln (y 2 D wpe w t ) 1 w) o ts +1 pe 9 > > > > > > > > > = ) ) tr + 1 tw ) e ! tr + D we t pwe t ( w tr ) e ( M tr ) e ) (T tr ) > > > > > > > > > ; (76) : Box XIV (75) in Eq. (24), the total pro t for the Sub-case 3.4 is obtained by Eq. (76) as shown in Box XIV. Sub-case 3.5: ts < t r < t w < T < M Here, the interest acquired on good items per cycle is obtained with the area of triangle OBT . Likewise, the interest gained on good items for the period [T ; M ] is given by the area of rectangle BC M T . Consequently, the interest received on good items is: e sI DT 2 + sIe DT (M T ): 2 In addition, the retailer can gain interest on the sale of defective items from OW and RW, which is equal to the sum of the areas of rectangles DC M tw and E F M ts as shown in Figure 20. Therefore, the interest earned from the sale of defective items from OW is determined with vIe pw (M tw ) and the interest received from the sale of defective items from RW is obtained with vIe p(y w )(M ts ). Thus: Total interest earned = e sI DT 2 2 + sIe DT (M T + vIe pw(M t ) + vIe p(y w ): w )(M t s) Substituting the values from Eqs. (25)-(27) and (77) in Eq. (24), the total pro t for Sub-case 3.5 is obtained by Eq. (78) as shown in Box XV. From all di erent equations of pro t functions, it is found that: From Eqs. (32), (50), and (68): : ( ) = T P2:1 (y ) = T P3:1 (y ); T P1 1 y From Eqs. (40) and (58): : ( ) = T P2:3 (y ); T P1 3 y From Eqs. (43) and (61): : ( ) = T P2:4 (y ); T P1 4 y From Eqs. (46), (64), and (81): : ( ) = T P2:5 (y ) = T P3:5 (y ); T P1 5 y From Eqs. (54) and (72): : ( ) = T P3:2 (y ): T P2 2 y Hence, eight di erent cases exist for the retailer's pro t per cycle, which can be expressed as: TPU(y ) = TPU1 (y )=TPU1:1 (y )=TPU2:1 (y )=TPU3:1 (y ); (79a) (77) As the inventory gets exhausted at T , the interest paid is equal to zero. TPU2 (y ) = TPU1:2 (y ); (79b) TPU3 (y ) = TPU1:3 (y ) = TPU2:3 (y ); (79c) Figure 20. Graphical representation of interest earned and interest charged for ts < t r < t w < T < M . C.K. Jaggi et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 390{412 T P3:5 (y) = sDT + vyp + 8 > > > > < > > > > : sIe DT 2 + sIe DT (M T ) + vIe p(y 2 ( k + cy + dy + hr y w (1 ( +h o w (1 p) D p) 2 n w)(M ln D (y w) 1 pe ts ) + vIe pw(M tw ) o ts +1 ) D ln (w 2 D wpe tw ) e ) 9 > > > > = > > > > ; tr +1 407 : (78) Box XV TPU4 (y ) = TPU1:4 (y ) = TPU2:4 (y ); (79d) TPU5 (y )=TPU1:5 (y )=TPU2:5 (y )=TPU3:5 (y ); (79e) TPU6 (y ) = TPU2:2 (y ) = TPU3:2 (y ); (79f) TPU7 (y ) = TPU3:3 (y ); (79g) TPU8 (y ) = TPU3:4 (y ): (79h) The main objective is to obtain the optimal value of y, which maximizes the total pro t function TPUi (y). In order to determine the optimal value of y , which maximizes the total pro t per unit time, the necessary and sucient conditions for optimality are: d(TPUi (y)) = 0; (80) dy and: d2 (TPUi (y)) dy2 0: which is the same pro t function as that in the model of Jaggi et al. [19]; b. When = = 0, M = 0, Ie = 0, and Ip = 0, i.e. there is no deterioration and no trade credit, the total pro t per unit time becomes: hr (y w)2 ho w2 TPU(y ) = D s v + + xy h (y w)2 ho w2 k c d r v + (1 p) y xy xy D 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 sDT + vyp 8 > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > : 7 97 7 y w > > >7 k + cy + dy + hr (1 p) > 7 > > 7 > > 7 > > ) >7 > 7 > >7 D ts +1 > > 7 ln ( y w ) 1 pe > 2 >7 ; D = 7 7 ( 7 > > 7 > w > 7 > +h o (1 p) >7 > > 7 > >7 > > 7 )> >7 o > > 5 D n > ln w wpe tw e tr +1 > ; ( 2 D which is the same pro t function as that given by Chung et al. [18]. c. When hr = hw , M = 0, Ie = 0, and Ip = 0, i.e. there is no trade credit and storage capacity of OW is unlimited, the total pro t per unit time becomes: 4. Special cases T ! hr (y w)2 (1 p) w2 (1 p) ho w(1 p) ; 2y 2y TPU(y ) = sD + a. When M = 0, Ie = 0, and Ip = 0, i.e. there is no trade credit, the total pro t per unit time becomes: 1 T P (y) = xy hD y vp c d h (1 p) k ; + ln D ln y + D pye ts which is the same pro t function as that obtained by Moussawi-Haidar et al. [8]. d. When hr = hw , = = 0, M = 0, Ie = 0, and Ip = 0, i.e. there is no deterioration and no trade credit, and storage capacity of OW is unlimited, the total pro t per unit time becomes, TPU(y ) = D s + D v v+ hy x k hy c d y x hy(1 p) ; (1 p) 2 which is the same pro t function as that obtained by Salameh and Jaber [4]. e. When hr = hw , s = c, = 0, = 0, p = 0, and storage capacity of OW is unlimited, the proposed model is same as that of Goyal [9]. 408 C.K. Jaggi et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 390{412 5. Solution procedure In order to nd the optimal value of y , which maximizes the total pro t function, the following algorithm is proposed: Determine y = y1 from Eq. (80). Now, using the value of y1 , calculate the values of tw , ts , tr , and T . If 0 < M tw < ts < tr or 0 < M ts < tw < tr or 0 < M ts < tr < tw , then the optimal value of total pro t is derived from Eq. (79a); Step 2. Determine y = y2 from Eq. (80). Now, using the value of y2 , calculate the values of tw , ts , tr , and T . If tw < M ts < tr , then the optimal value of total pro t is obtained from Eq. (79b); Step 3. Determine y = y3 from Eq. (80). Now, using the value of y3 , calculate the values of tw , ts , tr , and T . If tw < ts < M tr or ts < tw < M tr , then the optimal value of total pro t is determined from Eq. (79c); Step 4. Determine y = y4 from Eq. (80). Now, using the value of y4 , calculate the values of tw , ts , tr , and T . If tw < ts < tr < M T or ts < tw < tr < M T , then the optimal value of total pro t is calculated from Eq. (79d); Step 5. Determine y = y5 from Eq. (80). Now, using the value of y5 , calculate the values of tw , ts , tr , and T . If tw < ts < tr < T < M or ts < tw < tr < T < M or ts < tr < tw < T < M , then the optimal value of total pro t is computed from Eq. (79e); Step 6. Determine y = y6 from Eq. (80). Now, using the value of y6 , calculate the values of tw , ts , tr , and T . If ts < M tw < tr or ts < M tr < tw , then the optimal value of total pro t is derived from Eq. (79f); Step 7. Determine y = y7 from Eq. (80). Now, using the value of y7 , calculate the values of tw , ts , tr , and T . If ts < tr < M tw , then the optimal value of total pro t is obtained from Eq. (79g); Step 8. Determine y = y8 from Eq. (80). Now, using the value of y8 , calculate the values of tw , ts , tr , and T . If ts < tr < tw < M T , then the optimal value of total pro t is calculated from Eq. (79h). Step 1. 6. Numerical examples This section presents three numerical examples in order to illustrate the proposed inventory model. The values of parameters are: w = 500 units (thus, tw = 0:008 year), D = 15000 units/year, = 20%, = 12:5%, k = $1000/cycle, hr = $7/unit /year, ho = $5/unit/year, x = 60000 unit/year, c = $45/unit, s = $70/unit, v = $30/unit, d = $1:0/unit, Example 1. M = 20 days, and the percentage of defective random variable p with p.d.f is: ( 10 0 p 0:1 E (p) = 0:05: f ( p) = 0 otherwise Two cases are considered: (a) Let Ie = 0:10/year and Ip = 0:12/year (sIe > cIp ). Results are obtained, using the proposed algorithm, as y = 1311. Substituting the optimal value of y in the expressions of ts , tr , T , and TPU (y ), we get ts = 0:0135 year, tr = 0:051 year, T = 0:082 year, and TPU (y ) = $328198; (b) Let Ie = 0:05/year and Ip = 0:08/year (sIe < cIp ). Results are obtained, using the proposed algorithm, as y = 1408. Substituting the optimal value of y in the expressions of ts , tr , T , and TPU (y ), we get ts = 0:0151 year, tr = 0:057 year, T = 0:088 year, and TPU (y ) = $327362. Here, the values of parameters are: w = 800 units (thus tw = 0:013 year), D = 15000 units/ year, = 20%, = 12:5%, k = $1000/cycle, hr = $6 /unit/year, ho = $6/unit/year, x = 60000 unit/year, c = $35/unit, s = $60/unit, v = $25/unit, d = $1:0/ unit, M = 18 days, and the percentage of defective random variable p with p.d.f is: Example 2. f ( p) = ( 10 0 p 0:1 0 otherwise E (p) = 0:05: Now, we consider two cases: (a) Let Ie = 0:08/year and Ip = 0:10/year (sIe > cIp ). Results are obtained, using the proposed algorithm, as y = 1478. Substituting the optimal value of y in the expressions of ts , tr , T , and TPU (y ), we get ts = 0:0113 year, tr = 0:043 year, T = 0:093 year, and TPU (y) = $331970; (b) Let Ie = 0:04/year and Ip = 0:07/year (sIe < cIp ). Results are obtained, using the proposed algorithm, as y = 1555. Substituting the optimal value of y in the expressions of ts , tr , T , and TPU (y ), we get ts = 0:0126 year, tr = 0:048 year, T = 0:098 year, and TPU (y) = $331655. Here, the values of parameters are: w = 1200 units (thus tw = 0:02 year), D = 15000 units /year, = 20%, = 12:5%, k = $1000/cycle, hr = $6/unit/year, ho = $6/unit/year, x = 60000 unit/year, c = $35=unit, s = $60/unit, v = $25/unit, d = $1:0/ unit, M = 20 days, and the percentage of defective random variable p with p.d.f is: Example 3. f ( p) = ( 10 0 p 0:1 0 otherwise E (p) = 0:05: C.K. Jaggi et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 390{412 409 Now, we consider two cases: 7. Sensitivity analysis (a) Let Ie = 0:10/year and Ip = 0:12/year (sIe > cIp ). Results are obtained, using the proposed algorithm, as y = 1394. Substituting the optimal value of y in the expressions of ts , tr , T , and TPU (y ), we get ts = 0:0032 year, tr = 0:012 year, T = 0:087 year, and TPU (y) = $332178; (b) Let Ie = 0:05/year and Ip = 0:08/year (sIe < cIp ). Results are obtained, using the proposed algorithm, as y = 1492. Substituting the optimal value of y in the expressions of ts , tr , T , and TPU (y ), we get ts = 0:0049 year, tr = 0:018 year, T = 0:094 year, and TPU (y) = $331542. Sensitivity analysis was performed to study the e ects of permissible delay (M ), interest earned (Ie ), interest paid (Ip ), deterioration ( and ), percentage of defective items (p), and change in the capacity of OW (w) on the optimal lot size (y ) and the total pro t per unit time TPU (y ). The observations are shown in Tables 2 to 5. From Table 2, it is observed that when sIe > cIp , cycle lengths of RW and OW as well as the optimal order quantity decrease as the permissible delay period increases along with the increase in annual pro t. This insinuates that trade credit turns bene cial for Table 2. E ect of change in capacity of owned warehouse and trade credit period on the model (sIe > cIp ). W M (days) ts tr T y TPU (y ) 400 (tw = 0:007) 10 20 30 0.0166 0.0152 0.0147 0.063 0.058 0.056 0.088 0.082 0.081 1394 1312 1283 325628 328272 331110 900 (tw = 0:015) 10 20 30 0.0070 0.0068 0.0063 0.027 0.026 0.024 0.083 0.082 0.080 1322 1305 1276 325280 327897 330737 1200 (tw = 0:020) 10 20 30 0.0019 0.0016 0.0012 0.007 0.006 0.004 0.083 0.081 0.080 1315 1298 1270 325107 327725 330569 Table 3. Case tw < t s < M < t r < T tw < t s < M < t r < T tw < t s < t r < T < M ts < t w < M < t r < T ts < t w < t r < M < T ts < t w < t r < T < M ts < t r < t w < M < T ts < t r < t w < M < T ts < t r < t w < T < M E ect of change in capacity of owned warehouse and trade credit period on the model (sIe < cIp ). W M (days) ts tr T y TPU (y ) 400 (tw = 0:007) 10 20 30 0.0168 0.0168 0.0169 0.063 0.064 0.064 0.088 0.089 0.089 1406 1409 1412 325950 327450 328931 900 (tw = 0:015) 10 20 30 0.0083 0.0083 0.0084 0.031 0.032 0.032 0.088 0.088 0.088 1398 1400 1403 325565 327062 328544 1200 (tw = 0:02) 10 20 30 0.0032 0.0032 0.0032 0.012 0.012 0.012 0.087 0.087 0.088 1390 1391 1394 325391 326887 328368 Table 4. Case tw < t s < M < t r < T tw < t s < t r < M < T tw < t s < t r < T < M ts < t w < M < t r < T ts < t w < t r < M < T ts < t w < t r < T < M ts < t r < t w < M < T ts < t r < t w < M < T ts < t r < t w < T < M E ect of change of Ie and Ip on optimal replenishment policy (M = 10 days). Ie Ip y 0.1 0.15 0.2 928 888 854 0.03 TPU (y ) y 338260 338056 337876 920 880 847 0.05 TPU (y ) y 338382 338184 338008 898 872 840 0.07 TPU (y ) 338454 338312 338141 410 Table 5. C.K. Jaggi et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 390{412 E ect of change of percentage of defective items on the model (w = 900, Ip = 0:12/year). Percentage of defective items 0.025 0.05 0.075 p = 0:2, = 0:125, Ie = 0:1/year, ts tw tr T y TPU (y ) 0.0128 0.0133 0.0139 0.015 0.015 0.015 0.030 0.030 0.031 0.065 0.064 0.064 1666 1700 1734 562408 551454 539900 economic ordering policy. Thus, the retailer should procure less quantity and take the advantage of permissible delay in payments more often. Now, from Table 3, it is observed that if sIe < cIp , then, as the permissible delay period increases, the optimal order quantity and, thus, the total annual pro t increase. This implies that the retailer should procure more quantity to avoid higher interest charges on the inventory left after the credit period, which eventually results in higher pro ts. Table 4 shows that as interest earned (Ie ) by retailer increases, the optimal order quantity (y ) decreases; but, expected pro t increases, implying that when the interest earned per dollar is high, the expected total cost is low, which results in higher expected pro t. Also, increase in interest paid (Ip ) by retailer results in decrease in optimal order quantity, as well as the expected pro t, because the expected total cost increases when the interest payable rate for the items stocked is high. Thus, a retailer should order less but more frequently when the interest payable rate per dollar is high. Table 5 clearly shows that as the percentage of imperfect quality items (p) increases, the optimal order quantity (y ) increases to meet the demand out of perfect quality items; but, the retailer's total pro t TPU (y ) decreases signi cantly. Thus, the retailer should be more vigilant when ordering and should carefully select the suppliers. 8. Conclusion This paper amalgamates the concepts of two warehouses and the e ect of deterioration on the retailer's lot when the items are of imperfect quality under the permissible delay of payments. The screening rate is assumed to be more than the demand rate so that the demand can be ful lled out of the products that are found to be of perfect quality while the screening is in process. The numerical examples followed by the sensitivity analysis of various model parameters indicate that in case of highly deteriorating products, order should be made more frequently to reduce the losses due to deterioration. Also, as the defective items increase, the total pro t decreases; in such a situation, the corrective measures need to be taken in order to procure good quality products. The results of sensitivity analysis also show that the presence of trade credit period is bene cent for retailer ordering policy. The retailer should order more to avoid higher interest charged after the grace period that eventually increases his/her total pro t under the situation sIe < cIp ; whereas, in other situation, i.e. sIe > cIp , the retailer should order less to avail the bene t of permissible delay more frequently. Also, as the rate of interest to be paid increases, the retailer should order less but more frequently. The proposed inventory model is a general framework as it includes numerous previous models. References 1. Porteus, E.L. \Optimal lot sizing, process quality improvement and setup cost reduction", Oper. Res., 34(1), pp. 137-144 (1986). 2. Rosenblatt, M. and Lee, H. \Economic production cycles with imperfect production processes", IIE. Trans., 18(1), pp. 48-55 (1986). 3. Lee, H.L. and Rosenblatt, M.J. \Simultaneous determination of production cycles and inspection schedules in a production system", Manage. Sci., 33(9), pp. 1125-1137 (1987). 4. Salameh, M.K. and Jaber, M.Y. \Economic production quantity model for items with imperfect quality", Int. J. Prod. Econ., 64(3), pp. 59-64 (2000). 5. Cardenas-Barron, L.E. \Observation on: Economic production quantity model for items with imperfect quality", Int. J. Prod. Econ., 64, pp. 59-64 (2000), Int. J. Prod. Econ., 67(2), p. 201 (2000). 6. Goyal, S.K. and Cardenas-Barron, L.E. \Note on: Economic production quantity model for items with imperfect quality-a practical approach", Int. J. Prod. Econ., 77(1), pp. 85-87 (2002). 7. Papachristos, S. and Konstantaras, I. \Economic ordering quantity models for items with imperfect quality", Int. J. Prod. Econ., 100(1), pp. 148-154 (2006). 8. Moussawi-Haidar, L., Salameh, M. and Nasr, W. \E ect of deterioration on the instantaneous replenishment model with imperfect quality items", Appl. Math. Model., 38(24), pp. 5956-5966 (2014). 9. Goyal, S.K. and Giri, B.C. \Recent trends in modeling of deteriorating inventory", Eur. J. Oper. Res., 134(1), pp. 1-16 (2001). C.K. Jaggi et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 390{412 10. Bakker, M., Riezebos, J. and Teunter, R.H. \Review of inventory systems with deterioration since 2001", Eur. J. Oper. Res., 221(2), pp. 275-284 (2012). 11. Hartley, V.R., Operations Research - A Managerial Emphasis, Good Year Publishing Company, California, pp. 315-317 (1976). 12. Das, B., Maity, K. and Maiti, M. \A two warehouse supply-chain model under possibility/necessity/credibility measures", Math. Comput. Model., 46(3), pp. 398-409 (2007). 13. Hsieh, T.P., Dye, C.Y. and Ouyang, L.Y. \Determining optimal lot size for a two-warehouse system with deterioration and shortages using net present value", Eur. J. Oper. Res., 191(1), pp. 182-192 (2008). 14. Lee, C.C. \Two-warehouse inventory model with deterioration under FIFO dispatching policy", Eur. J. Oper. Res., 174(2), pp. 861-873 (2006). 15. Bhunia, A.K. and Maiti, M. \A two warehouses inventory model for deteriorating items with a linear trend in demand and shortages", J. Oper. Res. Soc., 49, pp. 287-292 (1998). 16. Niu, B. and Xie, J. \A note on two-warehouse inventory model with deterioration under FIFO dispatch policy", Eur. J. Oper. Res., 190(2), pp. 571-577 (2008). 17. Bhunia, A.K., Jaggi, C.K., Sharma, A. and Sharma, R. \A two-warehouse inventory model for deteriorating items under permissible delay in payment with partial backlogging", Appl. Math. Comput., 232, pp. 11251137 (2014). 18. Chung, K.J., Her, C.C. and Lin, S.D. \A twowarehouse inventory model with imperfect quality production process", Comput. Ind. Eng., 56(1), pp. 193-197 (2009). 19. Jaggi, C.K., Tiwari, S. and Sha , A. \E ect of deterioration on two-warehouse inventory model with imperfect quality", Comput. Ind. Eng., 88, pp. 378-385 (2015). Biographies Chandra Kant Jaggi is Professor and Head of the Department of Operational Research at University of Delhi, India. He has been Fellow Member of International Science Congress Association since 2012. He was awarded the Certi cate of Excellence in 2nd Academic Brilliance Award 2014 by EET CRS, Research wing for excellence in Professional Education & Industry, Noida. In 2010, he was awarded Certi cate for his Exceptional Contributions in the eld of Inventory Management by Lingaya's University, Faridabad, and in 2009, was awarded Certi cate for his Signi cant Contributions in Operation Management by the Society of Reliability Engineering, Quality and Operations Management, New Delhi. Also, he is a recipient of Shiksha Rattan Puraskar (for Meritorious Services, Outstanding 411 Performance and Remarkable Role) in 2007 by India International Friendship Society. He is Life Member of Operational Research Society of India, Indian Science Congress Association, and Fellow Member of International Science Congress Association, Computer Society of India, The Society of Mathematical Sciences, University of Delhi, India, Society for Reliability Engineering, Quality and Operations Management, Indian Society for Probability and Statistics. His research interest lies in the eld of supply chain and inventory management. He has guided 12 PhD and 21 MPhil candidates in Operations Research. He has published 7 book chapters and also papers in several national and international journals. He is also a reviewer of many international/national journals. He is Ex-Editor-inChief of International Journal of Inventory Control and Management; Associate Editor of International Journal of System Assurance Engineering and Management, Springer; Co-Editor/Reviewer-In-Charge of The Gstf Journal of Mathematics, Statistics and Operations Research; and a member of the Editorial Board of the IJSS: Operations & Logistics, International Journal of Services Operations and Informatics, American Journal of Operational Research, International Journal of Enterprise Computing and Business Systems, Journal of Applied Sciences Research, and Australian Journal of Basic and Applied Sciences. He has traveled to Canada, Philippines, Macau, Iran, and di erent parts of India to deliver keynote addresses and invited talks. Leopoldo Eduardo Cardenas-Barr on is currently a Professor in the School of Engineering and Sciences at Tecnologico de Monterrey, Campus Monterrey, Mexico. He is also a faculty member in the Department of Industrial and Systems Engineering at Tecnologico de Monterrey. He was the Associate Director of the Industrial and Systems Engineering programme from 1999 to 2005. Moreover, he was the Associate Director of the Department of Industrial and Systems Engineering from 2005 to 2009. His research areas are primarily related to inventory planning and control, logistics, and supply chain. He has published papers and technical notes in many international journals. He has co-authored one book in the eld of simulation in Spanish. He is also editorial board member in several international journals. Sunil Tiwari is an Assistant Professor in Department of Mathematics, Ambedkar University, Delhi. He received his PhD degree (Inventory Management) in 2016, M. Phil. (Inventory Management) in 2013 and MSc degree (Operational Research) in 2011 from Department of Operational Research, Faculty of Mathematical Sciences, University of Delhi. His research areas include primarily related to inventory planning and control, logistics, and supply chain. He has 412 C.K. Jaggi et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 390{412 published research papers in International Journal of Production Economics, Computers and Industrial Engineering, Annals of Operations Research, Applied Mathematics & Information Sciences, International Journal of Industrial Engineering Computations, International Journal of Operational Research, Iranian Journal of Fuzzy Systems, and one book chapter in this area. is an associate application developer at Oracle Financial Software Services. He completed his BSc degree in Electrical and Electronics Engineering at Birla Institute of Technology, Mesra, in 2014. He is primarily interested in research on optimization, logistics, and supply chain management. He has published a paper in Computers and Industrial Engineering. Ali Akbar Sha