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Credit financing in economic ordering policies for defective items with allowable shortages
Applied Mathematics and Computation 219 (2013) 5268–5282
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Credit financing in economic ordering policies for defective items with allowable shortages Chandra K. Jaggi a,⇑, Satish K. Goel a, Mandeep Mittal b a b
Department of Operational Research, Faculty of Mathematical Sciences, New Academic Block, University of Delhi, Delhi 110007, India Department of Computer Science Engineering, Amity School of Engineering and Technology, 580 Delhi Palam Vihar Road, Bijwasan, New Delhi 110061, India
a r t i c l e
i n f o
Keywords: Inventory Imperfect items Shortages Permissible delay
a b s t r a c t In the classical inventory models, the common unrealistic assumption is that all the items produced are of good quality in nature. However, in realistic environment, it can be observed that there may be some defective items in an ordered lot. These items are usually picked up during the screening process and are sold as a single lot at the end of screening process. Further, it is tacitly assumed that the supplier must be paid for the items as soon as the items are received. Whereas, in today business transaction, it is common to see that the retailer is allowed some grace period before they settle the account with the supplier. Under this scenario, a new inventory model for imperfect quality items has been developed under permissible delay in payments. Shortages are allowed and fully backlogged, which are eliminated during screening process as it has been assumed that screening rate is greater than the demand rate. This model jointly optimizes the order quantity and shortages by maximizing the expected total profit. Results have been validated with the help of numerical example using Matlab 7.0.1. Comprehensive sensitivity analysis has also been presented. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction In today’s technology driven world, despite of efficient planning of manufacturing system and emergence of sophisticated production methods and control systems; the items produced may have some fraction of defectives. By considering this fact, researchers devoted a great amount of effort to develop EPQ/EOQ models for defective items [1–5]. In 2000, Salameh and Jaber [6] extended the traditional EPQ/EOQ model for the imperfect quality items. They also considered that the imperfect – quality items are sold at a discounted price as a single batch by the end of the screening process. Cárdenas-Barrón [7] corrected the optimum order size formula obtained by Salameh and Jaber [6] by adding constant parameter which was missing in their optimum order size formula. Further, Goyal and Cárdenas-Barrón [8] presented a simple approach for determining economic production quantity for imperfect quality items and compare the results based on the simple approach with optimal method suggested by Salameh and Jaber [6], which results in almost zero penalty. Papachristos and Konstantaras [9] examined the Salameh and Jaber [6] paper closely and rectify the proposed conditions to ensure that shortages will not occur. They extended their model to the case in which withdrawing takes place at the end of the planning horizon. Further, Wee et al. [10] extended the model of Salameh and Jaber [6] for the case where shortages are back ordered in each cycle. They also studied the effect of varying backordering cost value and showed that as the backordering cost increases then the rate of change in their annual profit decreases as compared to that in Salameh and Jaber [6] model. Maddah and Jaber
⇑ Corresponding author. E-mail address: [email protected] (C.K. Jaggi). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.11.027
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[11] corrected Salameh and Jaber [6] work related to the method of evaluating the expected total profit per unit time. They applied renewal-reward theorem [12] which leads to simple expressions of the optimal order quantity and expected profit per unit time. In addition, they have extended their model by allowing for several batches of imperfect quality items to be consolidated and shipped in one lot. Further, Eroglu and Ozdemir [13] also extended the model of Salameh and Jaber [6] by allowing shortages to be backordered. They suggest that a fraction of good quality items in each cycle not only fulfills current demand but also fulfills backorders during screening process. They also examined effects of defective items on lot size and optimal total profit and showed that the optimal total profit per unit time decreases when percentage of defective items increases. Thereafter, several interesting papers for controlling imperfect quality items have appeared in different journals (e.g. [14–19]. In all the above mentioned papers, it is tacitly assumed that payment will be made to the supplier for the goods immediately after receiving the consignment. However, in day-to-day dealing, it is found that the supplier allows a certain fixed period to settle the account. During this period, no interest is charged by the supplier, but beyond this period interest is charged under certain terms and conditions agreed upon, since inventories are usually financed through debt or equity. Owing to this fact, during the past few years, a lot of research work has been done on inventory models with permissible delay in payments, which has been summarized by Soni et al. [20]. Further, Chung and Huang [21] incorporated the concept of inspection of imperfect items with trade credit. Jaggi et al. [22] formulated an inventory model with imperfect quality deteriorating items with the assumption that the screening rate is more than the demand rate. This assumption helps one to meet his demand parallel to the screening process, out of the items which are of perfect quality. Recently, Sarkar [23,24] discussed inventory models with delay in payments having timevarying deterioration rate and stock dependent demand. In this paper, an inventory model is developed for imperfect quality items under permissible delay in payments where shortages are allowed and fully backlogged. The screening rate is assumed to be more than the demand rate. This assumption helps one to meet his demand and backorders parallel to the screening process, out of the items which are of perfect quality. The proposed model jointly optimizes retailer’s order quantity and shortages by maximizing his expected total profit. A comprehensive sensitivity analysis has also been performed to study the impact of M, Ie, Ip and E[a] on Q, B and expected total profit. 2. Assumption and notations The following assumptions are used to develop the model: 1. 2. 3. 4. 5. 6.
Demand rate is known with certainty and is uniform. The replenishment is instantaneous. Shortages are allowed and are full backlogged. The lead-time is negligible. The supplier provides a fixed credit period to settle the accounts to the retailer. Both screening as well as demand proceeds simultaneously, but the screening rate is assumed to be greater than demand rate. 7. It is assumed that each lot contains percentage defectives of ‘a’ with known uniform probability function, f(a). The following notations are used in developing the model: D Q B K c p cs h b k t1 t2 t3 T1 T C2 Ie
demand rate in units per unit time order size for each cycle maximum backorder level allowed fixed cost of placing an order unit cost unit selling price of good quality items unit selling price of imperfect quality items, cs < p holding cost per unit time unit screening cost screening rate in units per unit time, k > D time to build up backorder level of ‘B’ units time to eliminate the backorder level of ‘B’ units time to screen Q units ordered per cycle time where inventory level will become zero cycle length shortage cost per unit per unit time interest earned per unit per unit time (continued on next page)
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Ip z z1
interest paid per unit per unit time, inventory level at time t2 inventory level at time t3 percentage of defective items in Q probability density function of a expected value operator
a f(a) E() EðaÞ
Rb expected value of a, which is equal to ¼ a af ðaÞda; 0 < a < b < 1 rate of good quality items during t2 rate of good quality items to eliminate backorder, ð1 aÞk D > 0
ð1 aÞk ð1 aÞk D
3. Mathematical model This paper develops a mathematical model for defective items under permissible delay in payments. Shortages are allowed and fully backlogged which are eliminated during the screening process as it has been assumed that screening rate is greater than the demand rate. The behavior of the inventory model is illustrated in Fig. 1. It is assumed that a lot of Q units enter in the inventory system at time t = 0 and contains a percent defective items with a probability density function, f(a), which can be estimated from the past data. Screening is done for the entire received lot at a rate of k units per unit time to separate good and defective quality items. Further, it is assumed that the good quality items are screened at the rate of ð1 aÞk during time t2 and a fraction of good quality items fulfill the demand at the rate D and the rest is used to eliminate backorders with a rate of ð1 aÞk D:. After the end of screening process at time t3, defective items i.e. aQ are sold immediately as a single batch at discounted price cs. Thereafter, inventory level gradually decreases due to demand and reaches zero at time T1. The cycle length T for the proposed inventory model is given by
T¼
ð1 aÞQ : D
ð1Þ
As the percent defective items (a), is a random variable, the expected value of cycle length is given by
E½T ¼
ð1 E½aÞQ : D
ð2Þ
The time to build up a backorder of ‘B’ units is,
t1 ¼
B D
ð3Þ
and the time to eliminate the ‘B’ units is
t2 ¼
B B ¼ ð1 aÞk D kA
where A ¼ 1 a D=k
Fig. 1. Inventory system with inspection for the cases: M 6 t2 6 T; t 2 6 M 6 t 3 6 T, M 6 T 1 6 T and T 1 6 M 6 T.
ð4Þ
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and also
t2 ¼
Q z : ð1 aÞk
ð5Þ
The value of z is obtained by using Eqs. (4) and (5) as follows:
z¼Q
ð1 aÞB : A
ð6Þ
The screening time, t3, is obtained as
t3 ¼
Q : k
ð7Þ
Using Fig. 1, (t3 t2) can be written as
t3 t2 ¼
ðz z1 aQ Þ D
ð8Þ
and z1 and T1 are obtained as
z1 ¼ AQ B
ð9Þ
and
T1 ¼
AQ B Q þ D k
ð10Þ
The present model has been developed under the condition of permissible delay in payments, therefore, depending upon the credit period, there would be five distinct possible cases for the retailer’s total profit, pj ðQ ; BÞ; j ¼ 1; 2; 3; 4; 5, viz. (i) (ii) (iii) (iv) (v)
M 6 t2 6 t3 t2 6 M 6 t3 t2 6 t3 6 M T1 6 M 6 T T 6 M.
6 T1, 6 T1, 6 T1, and,
Since, the retailer’s total profit consists of the following components:
pj ðQ; BÞ ¼ Sales revenue Ordering cost Purchasing cost Screening cost Holding cost þ Interest earned Interest paid:
ð11Þ
Therefore, various individual components are evaluated as follows:
ðaÞ Sales revenue is the sum of revenue generated by the demand meet during the time period ð0; TÞ and by the sale of imperfect quality items is ¼ pð1 aÞQ þ cs aQ ;
ð12Þ
ðbÞ Ordering Cost ¼ K;
ð13Þ
ðcÞ Purchase Cost ¼ cQ ;
ð14Þ
ðdÞ Screening Cost ¼ bQ ;
ð15Þ
ðeÞ Holding Cost during the time period 0 to t 2 ðQ þ zÞ ðt 3 t 2 Þðz þ z1 þ aQ Þ ðT 1 t 3 Þz1 T1; ¼ h þ þ ; 2 2 2
ð16Þ
B where t1 ¼ DB ; t2 ¼ kA ; t3 ¼ Qk , z1 = AQ B, z ¼ Q ð1AaÞB and T 1 ¼ AQDB þ Qk and
ðf Þ Shortage Cost ¼
C 2 ðt 1 þ t2 ÞB : 2
ð17Þ
The interest earned and paid is calculated for the five different cases as follows: Case (i): M 6 t2 6 t 3 6 T 1 The retailer earns interest on average sales revenue generated for the time period 0 to M. At M account is settled and the finances are to be arranged to make the payment to the supplier for the remaining stock at some specified rate of interest for the period M to T1. (Fig. 2).
Therefore; the interest earned for the cycle ¼
fð1 aÞk DgM 2 Ie p DM2 Ie p þ ; 2 2
ð18Þ
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Fig. 2. Inventory system with inspection for the case 1: M 6 t 2 6 T.
And the interest payable per cycle for the inventory not sold after M is given as ¼
½ð1 aÞQ DM fð1 aÞk DgMcIp ðT 1 MÞ þ cs Ip aQðt 3 MÞ: 2
ð19Þ
Substitute the values from Eqs. (12)–(19) in Eq. (11), the total profit for case (i), p1(Q, B) becomes
p1 ðQ ; BÞ ¼ pð1 aÞQ þ cs aQ K cQ bQ h
t2 ðQ þ zÞ ðt3 t2 Þðz þ z1 þ aQÞ ðT 1 t3 Þz1 C 2 ðt 1 þ t 2 ÞB þ þ 2 2 2 2
" # fð1 aÞk DgM2 Ie p DM2 Ie p ½ð1 aÞQ DM fð1 aÞk DgMcIp ðT 1 MÞ þ þ þ cs Ip aQðt 3 MÞ : 2 2 2 ð20Þ Case (ii): t 2 6 M 6 t3 6 T 1 In this case, the retailer can earn interest on revenue generated from the sales up to M. Although, he has to settle the account at M, for that, he has to arrange money at some specified rate of interest in order to get his remaining stocks financed for the period M to T1. Retailer will also earn interest due to shortages meet for the time period (M t2) (Fig. 3).
Interest earned ¼
½ð1 aÞk Dt 22 Ie p DM2 Ie p þ BpIe ðM t 2 Þ þ ; 2 2
Interest payable ¼
½ð1 aÞQ DM fð1 aÞk Dgt 2 cIp ðT 1 MÞ þ cs Ip aQðt 3 MÞ: 2
Fig. 3. Inventory system with inspection for the case 2: t2 6 M 6 t3 6 T.
ð21Þ
ð22Þ
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Substitute the values from Eqs. (12)–(17), (21) and (22) in Eq. (11), the total profit for case (ii), p2(Q, B) becomes
t2 ðQ þ zÞ ðt3 t2 Þðz þ z1 þ aQ Þ ðT 1 t 3 Þz1 C 2 ðt 1 þ t 2 ÞB þ þ 2 2 2 2 " # 2 2 ½ð1 aÞk Dt 2 Ie p DM Ie p þ þ BpIe ðM t 2 Þ þ 2 2 ½ð1 aÞQ DM fð1 aÞk Dgt2 cIp ðT 1 MÞ þ cs Ip aQ ðt 3 MÞ 2
p2 ðQ; BÞ ¼ pð1 aÞQ þ cs aQ K cQ bQ h
ð23Þ
Case (iii): t2 6 t3 6 M 6 T 1 Here, in addition to interest earned and paid as in case (i), retailer not only earn interest from the sales of defective items, aQ, after screening process for the time period (M t3) but also earn interest from shortages which are backlogged during (M t2) (Fig. 4).
Interest earned ¼
½ð1 aÞk Dt 22 Ie p DM2 Ie p þ BpIe ðM t 2 Þ þ þ Ie cs aQ ðM t3 Þ; 2 2
Interest payable ¼
½ð1 aÞQ DM fð1 aÞk Dgt 2 cIp ðT 1 MÞ : 2
ð24Þ
ð25Þ
Substitute the values from Eqs. (12)–(17), (24) and (25) in Eq. (11), the total profit for case (iii), p3(Q, B) becomes
t2 ðQ þ zÞ ðt3 t2 Þðz þ z1 þ aQ Þ ðT 1 t 3 Þz1 C 2 ðt 1 þ t 2 ÞB þ þ 2 2 2 2 " # 2 2 ½ð1 aÞk Dt 2 Ie p DM Ie p þ þ BpIe ðM t 2 Þ þ þ Ie cs aQ ðM t3 Þ 2 2 ½ð1 aÞQ DM fð1 aÞk Dgt2 cIp ðT 1 MÞ : 2
p3 ðQ; BÞ ¼ pð1 aÞQ þ cs aQ K cQ bQ h
ð26Þ
Case (iv): T 1 6 M 6 T This case discusses the situation when inventory cycle is less than or equal to permissible delay. Thus, in this scenario no interest is payable by the retailer, where as he earns interest on the revenue generated from the sales from time 0 to M. Further, not only retailer earns interest due to sales of defective items, aQ, for the time period (M t3) but also earn interest from the shortages which are backlogged for the time period (M t2) (Fig. 5).
Interest earned ¼
½ð1 aÞk Dt 22 Ie p DT 2 Ie p þ BpIe ðM t 2 Þ þ 1 þ DT 1 Ie pðM T 1 Þ þ Ie cs aQ ðM t3 Þ; 2 2
Interest payable = 0
Fig. 4. Inventory system with inspection for the case 3: t2 6 t3 6 M 6 T 1 .
ð27Þ
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Fig. 5. Inventory system with inspection for the case 4: T 1 6 M 6 T.
Substitute the values from Eqs. (12)–(17) and (27) in Eq. (11), the total profit for case (iv), p4(Q, B) becomes
t2 ðQ þ zÞ ðt3 t2 Þðz þ z1 þ aQÞ ðT 1 t3 Þz1 þ þ 2 2 2 " # ½ð1aÞkDt 22 Ie p DT 21 Ie p C 2 ðt 1 þ t2 ÞB þ BpIe ðM t 2 Þ þ 2 2 : þ 2 þDT 1 Ie pðM T 1 Þ þ Ie cs aQðM t 3 Þ
p4 ðQ ; BÞ ¼ pð1 aÞQ þ cs aQ K cQ bQ h
ð28Þ
Case (v): T 6 M In this case, all the expressions for interest earned and paid coincide with that of case (iv). Hence, effectively we have four different cases for the retailer’s total profit per cycle, p(Q, B), which can be expressed as
8 p1 ðQ ; BÞ; > > > < p2 ðQ ; BÞ; pðQ ; BÞ ¼ > > p3 ðQ ; BÞ; > : p4 ðQ ; BÞ;
M 6 t 2 6 t3 6 T 1 ; case 1; t2 6 M 6 t3 6 T 1 ; case 2;
ð29Þ
t2 6 t 3 6 M 6 T 1 ; case 3; M P T1;
case 4:
To determine the expected total profit per unit time, we apply renewal-reward theorem Ross [12] as a is a random variable with known probability density function, f(a) and get the expected total profit per unit time for different cases which follows:
pðQ ; BÞ E½pðQ ; BÞ ¼ E½pT ðQ; BÞ ¼ E ; T E½T 8 E½pT1 ðQ ; BÞ ¼ E½p1 ðQ ; BÞ=E½T; > > > < E½pT2 ðQ ; BÞ ¼ E½p2 ðQ ; BÞ=E½T; pðQ ; BÞ ¼ T > > E½p3 ðQ ; BÞ ¼ E½p3 ðQ ; BÞ=E½T; > : E½pT4 ðQ ; BÞ ¼ E½p4 ðQ ; BÞ=E½T;
ð30Þ M 6 t 2 6 t3 6 T 1 ; case 1 ð31aÞ t2 6 M 6 t3 6 T 1 ; case 2 ð31bÞ t2 6 t3 6 M 6 T 1 ; case 3 ð31cÞ M P T1;
case 4 ð31dÞ
where
t 2 ðQ þ zÞ ðt 3 t 2 Þðz þ z1 þ E½aQÞ ðT 1 t3 Þz1 þ þ E½p1 ðQ ; BÞ ¼ pð1 E½aÞQ þ cs E½aQ K cQ bQ h 2 2 2 " # 2 2 C 2 ðt1 þ t 2 ÞB fð1 E½aÞk DgM Ie p DM Ie p þ þ 2 2 2 ½ð1 E½aÞQ DM fð1 E½aÞk DgMcIp ðT 1 MÞ þ cs Ip E½aQ ðt3 MÞ ; 2
ð32Þ
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t2 ðQ þ zÞ ðt3 t 2 Þðz þ z1 þ E½aQÞ ðT 1 t 3 Þz1 E½p2 ðQ ; BÞ ¼ pð1 E½aÞQ þ cs E½aQ K cQ bQ h þ þ 2 2 2 " # 2 2 C 2 ðt1 þ t2 ÞB ½ð1 E½aÞk Dt2 Ie p DM Ie p þ þ BpIe ðM t 2 Þ þ 2 2 2 ½ð1 E½aÞQ DM fð1 E½aÞk Dgt 2 cIp ðT 1 MÞ þ cs Ip E½aQ ðt 3 MÞ ; 2
5275
ð33Þ
t2 ðQ þ zÞ ðt3 t2 Þðz þ z1 þ E½aQ Þ ðT 1 t 3 Þz1 E½p3 ðQ ; BÞ ¼ pð1 E½aÞQ þ cs E½aQ K cQ bQ h þ þ 2 2 2 2 3 ½ð1E½aÞkDt22 Ie p þ BpIe ðM t2 Þ C 2 ðt1 þ t2 ÞB 4 2 5 ½ð1 E½aÞQ DM fð1 E½aÞk Dgt2 cIp ðT 1 MÞ ; þ 2 2 2 þ DM Ie p þ I c E½aQðM t Þ
ð34Þ
t 2 ðQ þ zÞ ðt 3 t2 Þðz þ z1 þ E½aQ Þ ðT 1 t 3 Þz1 E½p4 ðQ ; BÞ ¼ pð1 E½aÞQ þ cs E½aQ K cQ bQ h þ þ 2 2 2 2 3 ½ð1E½aÞkDt 22 Ie p þ BpIe ðM t 2 Þ C 2 ðt1 þ t2 ÞB 4 2 5 þ DT 21 Ie p 2 þ þ DT I pðM T Þ þ I c E½aQðM t Þ
ð35Þ
e s
2
2
3
1 e
1
e s
3
and E½T ¼ ð1E½DaÞQ Also, at T = M, the retailer’s expected total profit, E[p(Q, B)] is given by
pðQ ; BÞ E½pðQ ; BÞ ¼ ; E½pT ðQ ; BÞ ¼ E M E½M
ð36Þ
where E½M ¼ ð1E½DaÞQ and
t2 ðQ þ zÞ ðt3 t 2 Þðz þ z1 þ E½aQ Þ ðT 1 t 3 Þz1 E½pðQ ; BÞ ¼ pð1 E½aÞQ þ cs E½aQ K cQ bQ h þ þ 2 2 2 2 3 ½ð1E½aÞkDt 22 Ie p þ BpIe ðM t 2 Þ C 2 ðt 1 þ t 2 ÞB 4 2 5: þ 2 DT I p 2 þ 1 e þ DT I pðM T Þ þ I c E½aQ ðM t Þ 2
1 e
1
e s
3
4. Solution procedure Our objective is to find the optimal values of Q and B which maximize the expected total profit function, E[pT(Q, B)], therefore, the necessary conditions for E[pT(Q, B)] to be optimal are
@E½pT ðQ ;BÞ @Q
¼ 0 and
@E½pT ðQ ;BÞ @B
¼ 0 which follows as case wise:
Case (i): M 6 t2 6 t 3 6 T 1 The optimal values of Q = Q1 (say) and B = B1 (say), which maximizes, E½pT1 ðQ ; BÞ can be obtained by solving the Eq. (31a),
Q 2 ðhX 1 þ X 8 Þ B2 Q 2 hX 2 þ B2 X 6 þ QX 9 þ ðX 4 þ h X 7 X 10 Þ ¼ 0;
ð37Þ
B ¼ QX 14 :
ð38Þ
All Xi (i = 1, 2, . . . , 14) are elaborated in Appendix A. Case (ii): t 2 6 M 6 t3 6 T 1 The optimal values of Q = Q2 (say) and B = B2 (say), which maximizes, E½pT2 ðQ ; BÞ, are
B2 Y 19 þ Y 20 BY 21 þ B2 Q 2 Y 10 þ Q 2 Y 22 ¼ 0;
ð39Þ
BQY 12 þ B2 Y 7 þ QY 13 þ Q 2 Y 14 ¼ 0:
ð40Þ
All Yi’s are elaborated in Appendix B. Case (iii): t2 6 t3 6 M 6 T 1 The optimal values of Q = Q3 (say) and B = B3 (say), which maximizes, E½pT3 ðQ ; BÞ, are
w1 B2 Y 19 þ Y 20 BY 21 þ B2 Q 2 Y 10 þ Q 2 Y 22 þ Q 2 Mw1 ¼ 0; k
ð41Þ
BQY 12 þ B2 Y 7 þ QY 13 þ Q 2 Y 14 ¼ 0:
ð42Þ
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Fig. 6. Concavity of expected total profit function (case 1).
Case (iv): M P T 1 The optimal values of Q = Q4 (say) and B = B4 (say), which maximizes, E½pT4 ðQ ; BÞ, are
3Q 4 L1 2Q 3 L3 B þ Q 2 B2 ðL2 þ L15 Þ þ Q 2 ðL14 þ L12 Þ þ B2 ðL17 L7 þ L11 Þ þ BðL10 þ L6 Þ L13 L4 1 ¼ 0;
ð43Þ
Q 3 L3 þ 2Q 2 BL2 QL16 þ Bð2L15 2L17 þ 2L11 þ L9 Þ þ L16 L6 ¼ 0:
ð44Þ
All Li’s are elaborated in Appendix C. Further, for the expected profit function to be concave, the following sufficient conditions must be satisfied:
@ 2 E½pT ðQ ; BÞ @Q @B
and
!2
@ 2 E½pT ðQ ; BÞ @Q 2
! @ 2 E½pT ðQ; BÞ @Q 2 ! 6 0;
! @ 2 E½pT ðQ ; BÞ @B2
! @ 2 E½pT ðQ ; BÞ @B2
6 0:
6 0;
ð45Þ
ð46Þ
Second order derivatives have been calculated and these are very complicated (Appendix D). Therefore, it is very difficult to prove the concavity mathematically; hence the concavity of all the expected total profit functions have been established graphically and graph for one of the case is shown below (Fig. 6). Now, in order to find the optimal values of Q⁄ and B⁄, which maximizes the retailer’s expected profit, we propose the following algorithm: Algorithm: Step 1: Determine Q⁄ = Q1 (say) and B⁄ = B1 (say) from Eqs. (37) and (38). Now, using the values of Q1 and B1, calculate the values of t2, t3 and T1 from Eqs. (5), (7) and (10). If M 6 t2 6 t3 6 T 1 then the optimal value of expected total profit is obtained from Eq. (31a). Step 2: Determine Q⁄ = Q2 (say) and B⁄ = B2 (say) from Eqs. (39) and (40). Now, using the values of Q2 and B2, calculate the values of t2, t3 and T1 from Eqs. (5), (7) and (10). If t 2 6 M 6 t3 6 T 1 then the optimal values of expected profit is obtained from Eq. (31b). Step 3: Determine Q⁄ = Q3 (say) and B⁄ = B3 (say) from Eqs. (41) and (42). Now, using the values of Q3 and B3, calculate the values of t2, t3 and T1 from Eqs. (5), (7) and (10). If t 2 6 t 3 6 M 6 T 1 then the optimal values of expected profit is obtained from Eq. (31c). Step 4: Determine Q⁄ = Q4 (say) and B⁄ = B4 (say) from Eqs. (43) and (44). Now, using the values of Q3 and B3, calculate the values of t2, t3 and T1 from Eqs. (5), (7) and (10). If M P T 1 then the optimal values of expected profit then the optimal values of expected profit is obtained from Eq. (31d).
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Step 5: Determine the profit at T = M from Eq. (36). Step 6: Compare the calculated expected profit for cases 1, 2, 3, 4 and at M and select the optimal values of Q and B associated with the maximum expected profit. 5. An illustrative example
Example 1. An example is devised to illustrate the effect of permissible delay in payments on the retailer’s ordering policy for the developed model using the following data. D = 15000 units/year, A = $400/cycle, h = $4/unit/year, k = 60000 unit/year, c = $35/unit, p = $60/unit, cs = $25/unit, 10; 0 6 a 6 0:1; E½a ¼ 0:05, d = $1.0/unit and C2 = $6/year, percentage defective random variable a with its pdf, f ðaÞ ¼ 0; otherwise; M = 18 days. Two cases are considered (a) Let Ie = 0.08/year and Ip = 0.10/year, ðpIe ¼ 4:8 > 3:5 ¼ cIp Þ Results are obtained using the proposed algorithm as: Q⁄ = 1642 units, B⁄ = 674. Substituting the optimal values of Q⁄ in Eqs. (2) and (7), we get, T⁄ = 0.104 year, t3⁄ = 0.0274 year and E½pT ðQ ; B Þ = $347086. (b) Let Ie = 0.04 and Ip = 0.07, ðpIe ¼ 2:4 < 2:45 ¼ cIp Þ Using the proposed algorithm, results are obtained as: Q⁄ = 1804 units, B⁄ = 653, T⁄ = 0.114 year and t3⁄ = 0.0301 year and E½pT ðQ ; B Þ = $346095. Now, if supplier offers no trade credit, i.e. M = 0, we get: Q⁄ = 2130 units, B⁄ = 596, T⁄ = 0.135 year, t3⁄ = 0.0355 year and E½pT ðQ ; B Þ = $345384. Example 2. An example is devised to illustrate the effect of permissible delay in payments on the retailer’s ordering policy for the developed model using the following data. D = 25000 units/year, A = $1000/cycle, h = $5/unit/year, k = 60000 unit/year, c = $45/unit, p = $70/unit, cs = $30/unit, 10; 0 6 a 6 0:1; E½a ¼ 0:05, d = $1.0/unit and C2 = $7/year, percentage defective random variable a with its pdf, f ðaÞ ¼ 0; otherwise; M = 20 days. Two cases are considered: (a) Let Ie = 0.10/year and Ip = 0.12/year, ðpIe ¼ 7 > 5:4 ¼ cIp Þ Results are obtained using the proposed algorithm as: Q⁄ = 2671 units, B⁄ = 886. Substituting the optimal values of Q⁄ in Eqs. (2) and (7), we get, T⁄ = 0.101/year, t3⁄ = 0.0445 year and E½pT ðQ ; B Þ = $568588. (b) Let Ie = 0.05 and Ip = 0.08, ðpIe ¼ 3:5 < 3:6 ¼ cIp Þ Using the proposed algorithm, results are obtained as: Q⁄ = 3004 units, B⁄ = 881, T⁄ = 0.114 year and t3⁄ = 0.0501 year and E½pT ðQ ; B Þ = $566251. Now, if supplier offers no trade credit, i.e. M = 0, we get: Q⁄ = 3693 units, B⁄ = 821, T⁄ = 0.140 year, t3⁄ = 0.0616 year and E½pT ðQ ; B Þ = $564697. Example 3. D = 40000 units/year, A = $1000/cycle, h = $5/unit/year, k = 60000 unit/year, c = $45/unit, p = $70/unit, cs = $30/ 10; 0 6 a 6 0:1; , unit, d = $1.0/unit and C2 = $7/year, percentage defective random variable a with its pdf, f ðaÞ ¼ 0; otherwise; E½a ¼ 0:05, M = 20 days. Two cases are considered: (a) Let Ie = 0.10/year and Ip = 0.12/year, ðpIe ¼ 7 > 5:4 ¼ cIp Þ Results are obtained using the proposed algorithm as: Q⁄ = 2941 units, B⁄ = 530. Substituting the optimal values of Q⁄ in Eqs. (2) and (7), we get, T⁄ = 0.070/year, t3⁄ = 0.049 year and E½pT ðQ ; B Þ = $914760. (b) Let Ie = 0.05 and Ip = 0.08, ðpIe ¼ 3:5 < 3:6 ¼ cIp Þ Using the proposed algorithm, results are obtained as: Q⁄ = 3439 units, B⁄ = 535, T⁄ = 0.082 year and t3⁄ = 0.0573 year and E½pT ðQ ; B Þ = $910292. Now, if supplier offers no trade credit, i.e. M = 0, we get: Q⁄ = 4321 units, B⁄ = 510, T⁄ = 0.103 year, t3⁄ = 0.072 year and E½pT ðQ ; B Þ = $906826. 6. Sensitivity analysis Sensitivity analysis has been performed to study the impact of permissible delay (M), Interest earned (Ie), interest paid (Ip) and expected number of imperfect quality items (E[a]) on the lot size (Q⁄), backorders (B⁄) and the retailer’s expected total profit (E½pT ðQ ; B Þ). Results are summarized in Tables 1–3. Based on the computational results as shown in the Tables 1–3, we obtain the following managerial phenomena:
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Table 1 Impact of M on the optimal replenishment policy. M (days)
pIe < cIp
10 20 35
cIp < pIe T
Q
B
T
1792 1804 1805
649 653 652
0.113 0.114 0.114
Case T
E½p ðQ ; BÞ
Q
B
T
E½p ðQ ; BÞ
345298 346293 347778
1668 1629 1621
661 675 675
0.105 0.103 0.103
341649 347473 350440
2 3 4
Table 2 Impact of Ie and Ip on the optimal replenishment policy (M = 20 days). Ie ? Ip ;
0.03
0.20 0.15 0.10
0.05
0.07
Q
B
E½pT ðQ ; BÞ
Q
B
E½pT ðQ ; BÞ
Q
B
E½pT ðQ ; BÞ
1686 1716 1758
671 669 667
353075 353094 353120
1643 1665 1695
683 682 681
353670 353679 353692
1596 1608 1625
692 691 690
354285 354288 354292
Table 3 Impact of E[a] on Q, B and E½pT ðQ ; BÞ (M = 18 days, Ie = 0.08 and Ip = 0.1). E[a]
Q
B
E½pT ðQ ; BÞ
0.01 0.02 0.03 0.04 0.05
1600 1610 1621 1631 1642
694 689 684 679 674
354217 352489 350725 348924 347086
(i) It is observed from Table 1, when pIe < cIp the cycle length increases with permissible delay period which implies that the retailer should procure more quantity to avoid higher interest charges after the credit period that helps him to increase his expected profit e.g. in case of credit cards, usually banks charges very high rate of interest after the grace period. Further, when pIe > cIp, then both cycle length as well as order quantity decreases with increase in permissible delay period which means that the trade credit offered to the retailer has a positive impact on the economic ordering policy for imperfect quality items. In this scenario, results reveal that the retailer should order less quantity and take the advantage of permissible delay in payments more frequently. (ii) Table 2 reveals as Ie increases then order quantity decreases but expected profit increases which implies that when the interest earned per dollar is high, the expected total cost is low, which results in higher expected profit. Further, increase in Ip results a decrease in order quantity as well as the expected profit because the expected total cost increases when the capital opportunity cost in stock per dollar is high. From retailer point of view, it implies that when the capital opportunity cost per dollar is high, the retailer should order less but more frequently. (iii) It is clearly evident from Table 3 that as the percentage of defective items increases the optimal order quantity increases but the backorders decreases significantly, which results in less expected profit. In this scenario, the retailer should allow less backorders and be more vigilant while ordering. 7. Conclusions In this paper, we have developed an EOQ-based inventory model for imperfect quality items to determine the optimal ordering policies of a retailer under permissible delay in payments with allowable shortages. Screening rate is assumed to be more than the demand rate. During the screening process, retailer not only fulfills his current demand but also fulfills the back orders. In the above scenario, the model jointly optimizes the order quantity and shortages with the help of proposed algorithm. A comprehensive sensitivity analysis is also conducted to explore the effects of the key parameters (viz. M, Ie, Ip and E[a]) on the optimal results. Findings clearly suggest that the presence of trade credit has got affirmative effect on retailer ordering policy. The retailer should order more to avoid higher interest charged after the grace period that eventually increases his expected profit under the situation, pIe < cIp, whereas in other situation, i.e. pIe > cIp, the retailer should order less to avail the benefit of permissible delay more frequently. Further, when the interest payable rate increases, the retailer should order less but more frequently. However, increase in percentage defective items alert the retailer to look into source of supply and take the corrective measures in order to improve the quality of supply. Further, the proposed model can be extended for more realistic situations such as deteriorating items, stock dependent and stochastic demand with partial- trade credit, inflation etc.
C.K. Jaggi et al. / Applied Mathematics and Computation 219 (2013) 5268–5282
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Acknowledgments The authors are thankful to anonymous referees and the editor for their constructive comments and valuable suggestions that improved the presentation of the paper. The first author would like to acknowledge the support of Research Grant No. Dean (R)/R&D/2012/917, provided by University of Delhi (India) for conducting this research. Appendix A
X1 ¼
Dð1 þ a þ A þ A2 Þ ; ð1 E½aÞ
X4 ¼
KD ; ð1 E½aÞ
X8 ¼
cIp D ; 2k ð1 E½aÞ
X 11 ¼
X 13 ¼
X5 ¼
D½3ð1 E½aÞ þ 2A þ A2 ; ð1 E½aÞ
X3 ¼
C 2 ½1=D þ 1=kA ; 2
½ð1 E½aÞk DM 2 Iep DM2 Iep þ ; 2 2
X6 ¼
X3D ; ð1 E½aÞ
X2 ¼
X9 ¼
cIp D ½DM þ fð1 E½aÞk DgM ; 2k ð1 E½aÞ
" # 2D½3ð1 E½aÞ þ 2A þ A2 ; ð1 E½aÞ
X 12 ¼
DC 2 1 1 þ ; 2ð1 E½aÞ D kA
X7 ¼
X 10 ¼
X 14 ¼
cs IpE½a D ; k ð1 E½aÞ
X 13 ; X 12 X 11
1 cIp cIp D ½DM þ fð1 E½aÞk DgM ; D ð1 E½aÞ 2k 2k
Appendix B
Y1 ¼
cIp D ; 2k ð1 E½aÞ
Y4 ¼
Y8 ¼
cIp 2
2k A
Y2 ¼
1þA ½fð1 E½aÞk Dg þ ð1 E½aÞ; kA
½ð1 E½aÞk D;
2D ; ð1 E½aÞ
Y9 ¼
Y5 ¼
pIeMD ; ð1 E½aÞ
C2D 1 1 þ ; 2ð1 E½aÞ D kA
Y 10 ¼
D ½3ð1 EðaÞ þ 2A þ A2 ; ð1 E½aÞ
Y 23 ¼
D ½ð1 þ EðaÞ þ A þ A2 ; ð1 E½aÞ
Y 11 ¼
Y 24 ¼
cIpDM ; 2k
½ð1 E½aÞk DDIep
Y7 ¼
Y9 ¼
Y3 ¼
2k2 A2 ð1 E½aÞ
DC 2 ½1=D þ 1=kA ; 2ð1 E½aÞ
D ½1 2EðaÞ þ A2 ; ð1 E½aÞ
KD ; ð1 E½aÞ
Y 12 ¼ 2Y 10 þ 2Y 9 Y 8 2Y 4 ;
Y 13 ¼ Y 5 Y 3 ;
Y 19 ¼ 2Y 10 þ 2Y 9 Y 8 2Y 4 ;
Y 20 ¼ Y 24 Y 16 1;
w1 ¼ Iecs E½a;
Y 14 ¼ Y 1 Y 2 þ Y 11 ; Y 21 ¼ Y 3 þ Y 5 ;
Y 22 ¼ Y 23 Y 1 Y 17 ð2Y 10 þ 2Y 9 Y 16 2Y 4 Þ: Appendix C
L1 ¼
L5 ¼
Dð1 þ AÞ2 2
ð1 E½aÞk
;
L2 ¼
D2 pIeMð1 þ AÞ ; kð1 E½aÞ
D ð1 E½aÞk
L6 ¼
2
;
D2 Iep ; ð1 E½aÞ
L3 ¼
2Dð1 þ AÞ ð1 E½aÞk
L7 ¼
2
;
L4 ¼
pIeMD2 ; ð1 E½aÞ
½ð1 E½aÞk DDIep 2k2 A2 ð1 E½aÞ
;
X5D ; ð1 E½aÞ
;
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½ð1 E½aÞk DDIep
L9 ¼
2
2
k A ð1 E½aÞ
L12 ¼
IeDcs E½a ; ð1 E½aÞ
L16 ¼
ð1 2a A2 ÞD ; ð1 E½aÞ
L14 ¼
L10 ¼
;
pIeMD ; ð1 E½aÞ
ð1 þ a þ A þ A2 ÞD ; ð1 E½aÞ L17 ¼
L11 ¼
L15 ¼
D ; kAð1 E½aÞ
ð3ð1 aÞ þ 2A þ A2 ÞD ; ð1 E½aÞ
C 2 ð1=D þ 1=kAÞD ; ð1 E½aÞ
Appendix D Find below second derivatives of expected total profit functions for each case: Case (i): M 6 t2 6 t 3 6 T 1
@ 2 E½pT1 ðQ ; BÞ @Q 2 @ 2 E½pT1 ðQ ; BÞ @B2
¼ 2Q ðhX 1 þ X 8 Þ 2B2 QhX 2 þ X 9 ;
ðD:1Þ
¼ 1;
ðD:2Þ
where
X1 ¼
Dð1 þ a þ A þ A2 Þ ; ð1 E½aÞ
X4 ¼
KD ; ð1 E½aÞ
X5 ¼
½ð1 E½aÞk DM 2 Iep DM2 Iep þ ; 2 2
X6 ¼
X3D ; ð1 E½aÞ
X7 ¼
X5D ; ð1 E½aÞ
X9 ¼
cIp D ½DM þ fð1 E½aÞk DgM ; 2k ð1 E½aÞ
X 11 ¼
X 13 ¼
X2 ¼
D½3ð1 E½aÞ þ 2A þ A2 ; ð1 E½aÞ
X8 ¼
" # 2D½3ð1 E½aÞ þ 2A þ A2 ; ð1 E½aÞ
X3 ¼
C 2 ½1=D þ 1=kA ; 2
cIp D ; 2k ð1 E½aÞ
X 12 ¼
X 10 ¼
cs IpE½a D ; k ð1 E½aÞ
DC 2 1 1 þ ; 2ð1 E½aÞ D kA
1 cIp cIp D ½DM þ fð1 E½aÞk DgM ; D ð1 E½aÞ 2k 2k
X 14 ¼
X 13 : X 12 X 11
Case (ii): t 2 6 M 6 t3 6 T 1
@ 2 E½pT2 ðQ ; BÞ @Q 2 @ 2 E½pT2 ðQ ; BÞ @B2
¼ 2B2 QY 10 þ 2QY 22 ;
ðD:3Þ
¼ QY 12 þ 2BY 7 ;
ðD:4Þ
where
Y1 ¼
cIp D ; 2k ð1 E½aÞ
Y4 ¼
Y8 ¼
cIp
2k2 A
Y2 ¼
1þA ½fð1 E½aÞk Dg þ ð1 aÞ; kA
½ð1 E½aÞk D;
2D ; ð1 E½aÞ
Y9 ¼
Y5 ¼
pIeMD ; ð1 E½aÞ
C2 D 1 1 ; þ 2ð1 E½aÞ D kA
Y7 ¼
Y9 ¼
Y3 ¼
cIpDM ; 2k
½ð1 E½aÞk DDIep 2k2 A2 ð1 aÞ
DC 2 ½1=D þ 1=kA ; 2ð1 aÞ
;
C.K. Jaggi et al. / Applied Mathematics and Computation 219 (2013) 5268–5282
Y 10 ¼
D ½3ð1 EðaÞ þ 2A þ A2 ; ð1 E½aÞ
Y 23 ¼
D ½ð1 þ EðaÞ þ A þ A2 ; ð1 E½aÞ
Y 11 ¼
Y 24 ¼
5281
D ½1 2EðaÞ þ A2 ; ð1 E½aÞ
KD ; ð1 E½aÞ
Y 12 ¼ 2Y 10 þ 2Y 9 Y 8 2Y 4 ;
Y 13 ¼ Y 5 Y 3 ;
Y 14 ¼ Y 1 Y 2 þ Y 11 ;
Y 19 ¼ 2Y 10 þ 2Y 9 Y 8 2Y 4 ;
Y 20 ¼ Y 24 Y 16 1;
Y 21 ¼ Y 3 þ Y 5 ;
Y 22 ¼ Y 23 Y 1 Y 17 ð2Y 10 þ 2Y 9 Y 16 2Y 4 Þ: Case (iii): t2 6 t3 6 M 6 T 1
@ 2 E½pT3 ðQ; BÞ @Q
2
@ 2 E½pT3 ðQ; BÞ @B2
w1 ¼ 2B2 QY 10 þ 2QY 22 þ 2Q Mw1 ; k
ðD:5Þ
¼ QY 12 þ 2BY 7 ;
ðD:6Þ
where
Y1 ¼
Y4 ¼
Y8 ¼
cIp D ; 2k ð1 E½aÞ
cIp
2k2 A
Y2 ¼
1þA ½fð1 E½aÞk Dg þ ð1 E½aÞ; kA
½ð1 E½aÞk D;
2D ; ð1 E½aÞ
Y9 ¼
Y5 ¼
pIeMD ; ð1 E½aÞ
C2D 1 1 ; þ 2ð1 E½aÞ D kA
Y 10 ¼
D ½3ð1 EðaÞ þ 2A þ A2 ; ð1 E½aÞ
Y 23 ¼
D ½ð1 þ EðaÞ þ A þ A2 ; ð1 E½aÞ
Y 11 ¼
Y 24 ¼
cIpDM ; 2k
½ð1 E½aÞk DDIep
Y7 ¼
Y9 ¼
Y3 ¼
2k2 A2 ð1 E½aÞ
;
DC 2 ½1=D þ 1=kA ; 2ð1 E½aÞ
D ½1 2EðaÞ þ A2 ; ð1 E½aÞ
KD ; ð1 E½aÞ
Y 12 ¼ 2Y 10 þ 2Y 9 Y 8 2Y 4 ;
Y 13 ¼ Y 5 Y 3 ;
Y 19 ¼ 2Y 10 þ 2Y 9 Y 8 2Y 4 ;
Y 20 ¼ Y 24 Y 16 1;
w1 ¼ Iecs E½a;
Y 14 ¼ Y 1 Y 2 þ Y 11 ; Y 21 ¼ Y 3 þ Y 5 ;
Y 22 ¼ Y 23 Y 1 Y 17 ð2Y 10 þ 2Y 9 Y 16 2Y 4 Þ: Case (iv): M P T 1
@ 2 E½pT4 ðQ; BÞ @Q 2 @ 2 E½pT4 ðQ; BÞ @B2
¼ 12Q 3 L1 6Q 2 L3 B þ 2QB2 ðL2 þ L15 Þ þ 2QðL14 þ L12 Þ;
ðD:7Þ
¼ 2Q 2 L2 þ ð2L15 2L17 þ 2L11 þ L9 Þ;
ðD:8Þ
where
L1 ¼
L5 ¼
L9 ¼
Dð1 þ AÞ2 2
ð1 E½aÞk
;
L2 ¼
D2 pIeMð1 þ AÞ ; kð1 E½aÞ
D ð1 E½aÞk
L6 ¼
½ð1 E½aÞk DDIep k2 A2 ð1 E½aÞ
;
2
;
D2 Iep ; ð1 E½aÞ L10 ¼
L3 ¼
2Dð1 þ AÞ ð1 E½aÞk
L7 ¼
pIeMD ; ð1 E½aÞ
2
;
L4 ¼
pIeMD2 ; ð1 E½aÞ
½ð1 E½aÞk DDIep 2k2 A2 ð1 E½aÞ L11 ¼
D ; kAð1 E½aÞ
;
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L12 ¼
IeDcs E½a ; ð1 E½aÞ
L16 ¼
ð1 2a A2 ÞD ; ð1 E½aÞ
L14 ¼
ð1 þ a þ A þ A2 ÞD ; ð1 E½aÞ L17 ¼
L15 ¼
ð3ð1 aÞ þ 2A þ A2 ÞD ; ð1 E½aÞ
C 2 ð1=D þ 1=kAÞD : ð1 E½aÞ
References [1] E.L. Porteus, Optimal lot sizing, process quality improvement and setup cost reduction, Oper. Res. 34 (1986) 137–144. [2] M. Rosenblatt, H. Lee, Economic production cycles with imperfect production processes, IIE Trans. 18 (1986) 48–55. [3] H.L. Lee, M.J. Rosenblatt, Simultaneous determination of production cycles and inspection schedules in a production system, Manage. Sci. 33 (1987) 1125–1137. [4] R.L. Schwaller, EOQ under inspection costs, Prod. Invest. Man. 29 (1988) 22–24. [5] X. Zhang, Y. Gerchak, Joint lot sizing and inspection policy in an EOQ model with random yield, IIE Trans. 22 (1990) 41–47. [6] M.K. Salameh, M.Y. Jaber, Economic production quantity model for items with imperfect quality, Int. J. Prod. Econ. 64 (2000) 59–64. [7] L.E. Cárdenas-Barrón, Observation on: economic production quantity model for items with imperfect quality, Int. J. Prod. Econ. 67 (2000) 201. [8] S.K. Goyal, L.E. Cárdenas-Barrón, Note on: economic production quantity model for items with imperfect quality – a practical approach, Int. J. Prod. Econ. 77 (2002) 85–87. [9] S. Papachristos, I. Konstantaras, Economic ordering quantity models for items with imperfect quality, Int. J. Prod. Econ. 100 (2006) 148–154. [10] H.M. Wee, J. Yu, M.C. Chen, Optimal inventory model for items with imperfect quality and shortage backordering, Omega 35 (2007) 7–11. [11] B. Maddah, M.Y. Jaber, Economic order quantity for items with imperfect quality: revisited, Int. J. Prod. Econ. 112 (2008) 808–815. [12] S.M. Ross, Stochastic Processes, second ed., Wiley, New York, 1996. [13] A. Eroglu, G. Ozdemir, An economic order quantity model with defective items and shortages, Int. J. Prod. Econ. 106 (2007) 544–549. [14] B. Sarkar, I.K. Moon, An EPQ model with inflation in an imperfect production system, Appl. Math. Comput. 217 (2011) 6159–6167. [15] B. Sarkar, S.S. Sana, K.S. Chaudhuri, An imperfect production process for time varying demand with inflation and time value of money – an EMQ model, Expert Syst. Appl. 38 (2011) 13543–13548. [16] B. Sarkar, S.S. Sana, K.S. Chaudhuri, An economic production quantity model with stochastic demand in an imperfect production system, Int. J. Serv. Oper. Manage. 9 (2011) 259–283. [17] M.D. Roy, S.S. Sana, K. Chaudhuri, An EOQ model for imperfect quality products with partial backlogging – a comparative study, Int. J. Serv. Oper. Manage. 9 (2011) 83–110. [18] M.D. Roy, S.S. Sana, K. Chaudhuri, An economic order quantity model of imperfect quality items with partial backlogging, Int. J. Syst. Sci. 42 (2011) 1409–1419. [19] S.S. Sana, A production-inventory model of imperfect quality products in a three-layer supply chain, Decis. Support Syst. 50 (2011) 539–547. [20] H. Soni, N.H. Shah, C.K. Jaggi, Inventory models and trade credit, J. Con. Cyb. 39 (2010) 867–882. [21] K.J. Chung, Y.F. Huang, Retailer’s optimal cycle times in the EOQ model with imperfect quality and permissible credit period, J. Qual. Quant. 40 (2006) 59–77. [22] C.K. Jaggi, S.K. Goel, M. Mittal, Economic order quantity model for deteriorating items with imperfect quality and permissible delay on payment, Int. J. Ind. Eng. Comput. 2 (2010) 237–248. [23] B. Sarkar, An EOQ model with delay-in-payments and time-varying deterioration rate, Math. Comput. Model. 55 (2012) 367–377. [24] B. Sarkar, An EOQ model with delay in payments and stock dependent demand in the presence of imperfect production, Appl. Math. Comput. 218 (2012) 8295–8308.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Credit financing in economic ordering policies for defective items with allowable shortages Chandra K. Jaggi a,⇑, Satish K. Goel a, Mandeep Mittal b a b
Department of Operational Research, Faculty of Mathematical Sciences, New Academic Block, University of Delhi, Delhi 110007, India Department of Computer Science Engineering, Amity School of Engineering and Technology, 580 Delhi Palam Vihar Road, Bijwasan, New Delhi 110061, India
a r t i c l e
i n f o
Keywords: Inventory Imperfect items Shortages Permissible delay
a b s t r a c t In the classical inventory models, the common unrealistic assumption is that all the items produced are of good quality in nature. However, in realistic environment, it can be observed that there may be some defective items in an ordered lot. These items are usually picked up during the screening process and are sold as a single lot at the end of screening process. Further, it is tacitly assumed that the supplier must be paid for the items as soon as the items are received. Whereas, in today business transaction, it is common to see that the retailer is allowed some grace period before they settle the account with the supplier. Under this scenario, a new inventory model for imperfect quality items has been developed under permissible delay in payments. Shortages are allowed and fully backlogged, which are eliminated during screening process as it has been assumed that screening rate is greater than the demand rate. This model jointly optimizes the order quantity and shortages by maximizing the expected total profit. Results have been validated with the help of numerical example using Matlab 7.0.1. Comprehensive sensitivity analysis has also been presented. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction In today’s technology driven world, despite of efficient planning of manufacturing system and emergence of sophisticated production methods and control systems; the items produced may have some fraction of defectives. By considering this fact, researchers devoted a great amount of effort to develop EPQ/EOQ models for defective items [1–5]. In 2000, Salameh and Jaber [6] extended the traditional EPQ/EOQ model for the imperfect quality items. They also considered that the imperfect – quality items are sold at a discounted price as a single batch by the end of the screening process. Cárdenas-Barrón [7] corrected the optimum order size formula obtained by Salameh and Jaber [6] by adding constant parameter which was missing in their optimum order size formula. Further, Goyal and Cárdenas-Barrón [8] presented a simple approach for determining economic production quantity for imperfect quality items and compare the results based on the simple approach with optimal method suggested by Salameh and Jaber [6], which results in almost zero penalty. Papachristos and Konstantaras [9] examined the Salameh and Jaber [6] paper closely and rectify the proposed conditions to ensure that shortages will not occur. They extended their model to the case in which withdrawing takes place at the end of the planning horizon. Further, Wee et al. [10] extended the model of Salameh and Jaber [6] for the case where shortages are back ordered in each cycle. They also studied the effect of varying backordering cost value and showed that as the backordering cost increases then the rate of change in their annual profit decreases as compared to that in Salameh and Jaber [6] model. Maddah and Jaber
⇑ Corresponding author. E-mail address: [email protected] (C.K. Jaggi). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.11.027
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[11] corrected Salameh and Jaber [6] work related to the method of evaluating the expected total profit per unit time. They applied renewal-reward theorem [12] which leads to simple expressions of the optimal order quantity and expected profit per unit time. In addition, they have extended their model by allowing for several batches of imperfect quality items to be consolidated and shipped in one lot. Further, Eroglu and Ozdemir [13] also extended the model of Salameh and Jaber [6] by allowing shortages to be backordered. They suggest that a fraction of good quality items in each cycle not only fulfills current demand but also fulfills backorders during screening process. They also examined effects of defective items on lot size and optimal total profit and showed that the optimal total profit per unit time decreases when percentage of defective items increases. Thereafter, several interesting papers for controlling imperfect quality items have appeared in different journals (e.g. [14–19]. In all the above mentioned papers, it is tacitly assumed that payment will be made to the supplier for the goods immediately after receiving the consignment. However, in day-to-day dealing, it is found that the supplier allows a certain fixed period to settle the account. During this period, no interest is charged by the supplier, but beyond this period interest is charged under certain terms and conditions agreed upon, since inventories are usually financed through debt or equity. Owing to this fact, during the past few years, a lot of research work has been done on inventory models with permissible delay in payments, which has been summarized by Soni et al. [20]. Further, Chung and Huang [21] incorporated the concept of inspection of imperfect items with trade credit. Jaggi et al. [22] formulated an inventory model with imperfect quality deteriorating items with the assumption that the screening rate is more than the demand rate. This assumption helps one to meet his demand parallel to the screening process, out of the items which are of perfect quality. Recently, Sarkar [23,24] discussed inventory models with delay in payments having timevarying deterioration rate and stock dependent demand. In this paper, an inventory model is developed for imperfect quality items under permissible delay in payments where shortages are allowed and fully backlogged. The screening rate is assumed to be more than the demand rate. This assumption helps one to meet his demand and backorders parallel to the screening process, out of the items which are of perfect quality. The proposed model jointly optimizes retailer’s order quantity and shortages by maximizing his expected total profit. A comprehensive sensitivity analysis has also been performed to study the impact of M, Ie, Ip and E[a] on Q, B and expected total profit. 2. Assumption and notations The following assumptions are used to develop the model: 1. 2. 3. 4. 5. 6.
Demand rate is known with certainty and is uniform. The replenishment is instantaneous. Shortages are allowed and are full backlogged. The lead-time is negligible. The supplier provides a fixed credit period to settle the accounts to the retailer. Both screening as well as demand proceeds simultaneously, but the screening rate is assumed to be greater than demand rate. 7. It is assumed that each lot contains percentage defectives of ‘a’ with known uniform probability function, f(a). The following notations are used in developing the model: D Q B K c p cs h b k t1 t2 t3 T1 T C2 Ie
demand rate in units per unit time order size for each cycle maximum backorder level allowed fixed cost of placing an order unit cost unit selling price of good quality items unit selling price of imperfect quality items, cs < p holding cost per unit time unit screening cost screening rate in units per unit time, k > D time to build up backorder level of ‘B’ units time to eliminate the backorder level of ‘B’ units time to screen Q units ordered per cycle time where inventory level will become zero cycle length shortage cost per unit per unit time interest earned per unit per unit time (continued on next page)
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Ip z z1
interest paid per unit per unit time, inventory level at time t2 inventory level at time t3 percentage of defective items in Q probability density function of a expected value operator
a f(a) E() EðaÞ
Rb expected value of a, which is equal to ¼ a af ðaÞda; 0 < a < b < 1 rate of good quality items during t2 rate of good quality items to eliminate backorder, ð1 aÞk D > 0
ð1 aÞk ð1 aÞk D
3. Mathematical model This paper develops a mathematical model for defective items under permissible delay in payments. Shortages are allowed and fully backlogged which are eliminated during the screening process as it has been assumed that screening rate is greater than the demand rate. The behavior of the inventory model is illustrated in Fig. 1. It is assumed that a lot of Q units enter in the inventory system at time t = 0 and contains a percent defective items with a probability density function, f(a), which can be estimated from the past data. Screening is done for the entire received lot at a rate of k units per unit time to separate good and defective quality items. Further, it is assumed that the good quality items are screened at the rate of ð1 aÞk during time t2 and a fraction of good quality items fulfill the demand at the rate D and the rest is used to eliminate backorders with a rate of ð1 aÞk D:. After the end of screening process at time t3, defective items i.e. aQ are sold immediately as a single batch at discounted price cs. Thereafter, inventory level gradually decreases due to demand and reaches zero at time T1. The cycle length T for the proposed inventory model is given by
T¼
ð1 aÞQ : D
ð1Þ
As the percent defective items (a), is a random variable, the expected value of cycle length is given by
E½T ¼
ð1 E½aÞQ : D
ð2Þ
The time to build up a backorder of ‘B’ units is,
t1 ¼
B D
ð3Þ
and the time to eliminate the ‘B’ units is
t2 ¼
B B ¼ ð1 aÞk D kA
where A ¼ 1 a D=k
Fig. 1. Inventory system with inspection for the cases: M 6 t2 6 T; t 2 6 M 6 t 3 6 T, M 6 T 1 6 T and T 1 6 M 6 T.
ð4Þ
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and also
t2 ¼
Q z : ð1 aÞk
ð5Þ
The value of z is obtained by using Eqs. (4) and (5) as follows:
z¼Q
ð1 aÞB : A
ð6Þ
The screening time, t3, is obtained as
t3 ¼
Q : k
ð7Þ
Using Fig. 1, (t3 t2) can be written as
t3 t2 ¼
ðz z1 aQ Þ D
ð8Þ
and z1 and T1 are obtained as
z1 ¼ AQ B
ð9Þ
and
T1 ¼
AQ B Q þ D k
ð10Þ
The present model has been developed under the condition of permissible delay in payments, therefore, depending upon the credit period, there would be five distinct possible cases for the retailer’s total profit, pj ðQ ; BÞ; j ¼ 1; 2; 3; 4; 5, viz. (i) (ii) (iii) (iv) (v)
M 6 t2 6 t3 t2 6 M 6 t3 t2 6 t3 6 M T1 6 M 6 T T 6 M.
6 T1, 6 T1, 6 T1, and,
Since, the retailer’s total profit consists of the following components:
pj ðQ; BÞ ¼ Sales revenue Ordering cost Purchasing cost Screening cost Holding cost þ Interest earned Interest paid:
ð11Þ
Therefore, various individual components are evaluated as follows:
ðaÞ Sales revenue is the sum of revenue generated by the demand meet during the time period ð0; TÞ and by the sale of imperfect quality items is ¼ pð1 aÞQ þ cs aQ ;
ð12Þ
ðbÞ Ordering Cost ¼ K;
ð13Þ
ðcÞ Purchase Cost ¼ cQ ;
ð14Þ
ðdÞ Screening Cost ¼ bQ ;
ð15Þ
ðeÞ Holding Cost during the time period 0 to t 2 ðQ þ zÞ ðt 3 t 2 Þðz þ z1 þ aQ Þ ðT 1 t 3 Þz1 T1; ¼ h þ þ ; 2 2 2
ð16Þ
B where t1 ¼ DB ; t2 ¼ kA ; t3 ¼ Qk , z1 = AQ B, z ¼ Q ð1AaÞB and T 1 ¼ AQDB þ Qk and
ðf Þ Shortage Cost ¼
C 2 ðt 1 þ t2 ÞB : 2
ð17Þ
The interest earned and paid is calculated for the five different cases as follows: Case (i): M 6 t2 6 t 3 6 T 1 The retailer earns interest on average sales revenue generated for the time period 0 to M. At M account is settled and the finances are to be arranged to make the payment to the supplier for the remaining stock at some specified rate of interest for the period M to T1. (Fig. 2).
Therefore; the interest earned for the cycle ¼
fð1 aÞk DgM 2 Ie p DM2 Ie p þ ; 2 2
ð18Þ
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Fig. 2. Inventory system with inspection for the case 1: M 6 t 2 6 T.
And the interest payable per cycle for the inventory not sold after M is given as ¼
½ð1 aÞQ DM fð1 aÞk DgMcIp ðT 1 MÞ þ cs Ip aQðt 3 MÞ: 2
ð19Þ
Substitute the values from Eqs. (12)–(19) in Eq. (11), the total profit for case (i), p1(Q, B) becomes
p1 ðQ ; BÞ ¼ pð1 aÞQ þ cs aQ K cQ bQ h
t2 ðQ þ zÞ ðt3 t2 Þðz þ z1 þ aQÞ ðT 1 t3 Þz1 C 2 ðt 1 þ t 2 ÞB þ þ 2 2 2 2
" # fð1 aÞk DgM2 Ie p DM2 Ie p ½ð1 aÞQ DM fð1 aÞk DgMcIp ðT 1 MÞ þ þ þ cs Ip aQðt 3 MÞ : 2 2 2 ð20Þ Case (ii): t 2 6 M 6 t3 6 T 1 In this case, the retailer can earn interest on revenue generated from the sales up to M. Although, he has to settle the account at M, for that, he has to arrange money at some specified rate of interest in order to get his remaining stocks financed for the period M to T1. Retailer will also earn interest due to shortages meet for the time period (M t2) (Fig. 3).
Interest earned ¼
½ð1 aÞk Dt 22 Ie p DM2 Ie p þ BpIe ðM t 2 Þ þ ; 2 2
Interest payable ¼
½ð1 aÞQ DM fð1 aÞk Dgt 2 cIp ðT 1 MÞ þ cs Ip aQðt 3 MÞ: 2
Fig. 3. Inventory system with inspection for the case 2: t2 6 M 6 t3 6 T.
ð21Þ
ð22Þ
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Substitute the values from Eqs. (12)–(17), (21) and (22) in Eq. (11), the total profit for case (ii), p2(Q, B) becomes
t2 ðQ þ zÞ ðt3 t2 Þðz þ z1 þ aQ Þ ðT 1 t 3 Þz1 C 2 ðt 1 þ t 2 ÞB þ þ 2 2 2 2 " # 2 2 ½ð1 aÞk Dt 2 Ie p DM Ie p þ þ BpIe ðM t 2 Þ þ 2 2 ½ð1 aÞQ DM fð1 aÞk Dgt2 cIp ðT 1 MÞ þ cs Ip aQ ðt 3 MÞ 2
p2 ðQ; BÞ ¼ pð1 aÞQ þ cs aQ K cQ bQ h
ð23Þ
Case (iii): t2 6 t3 6 M 6 T 1 Here, in addition to interest earned and paid as in case (i), retailer not only earn interest from the sales of defective items, aQ, after screening process for the time period (M t3) but also earn interest from shortages which are backlogged during (M t2) (Fig. 4).
Interest earned ¼
½ð1 aÞk Dt 22 Ie p DM2 Ie p þ BpIe ðM t 2 Þ þ þ Ie cs aQ ðM t3 Þ; 2 2
Interest payable ¼
½ð1 aÞQ DM fð1 aÞk Dgt 2 cIp ðT 1 MÞ : 2
ð24Þ
ð25Þ
Substitute the values from Eqs. (12)–(17), (24) and (25) in Eq. (11), the total profit for case (iii), p3(Q, B) becomes
t2 ðQ þ zÞ ðt3 t2 Þðz þ z1 þ aQ Þ ðT 1 t 3 Þz1 C 2 ðt 1 þ t 2 ÞB þ þ 2 2 2 2 " # 2 2 ½ð1 aÞk Dt 2 Ie p DM Ie p þ þ BpIe ðM t 2 Þ þ þ Ie cs aQ ðM t3 Þ 2 2 ½ð1 aÞQ DM fð1 aÞk Dgt2 cIp ðT 1 MÞ : 2
p3 ðQ; BÞ ¼ pð1 aÞQ þ cs aQ K cQ bQ h
ð26Þ
Case (iv): T 1 6 M 6 T This case discusses the situation when inventory cycle is less than or equal to permissible delay. Thus, in this scenario no interest is payable by the retailer, where as he earns interest on the revenue generated from the sales from time 0 to M. Further, not only retailer earns interest due to sales of defective items, aQ, for the time period (M t3) but also earn interest from the shortages which are backlogged for the time period (M t2) (Fig. 5).
Interest earned ¼
½ð1 aÞk Dt 22 Ie p DT 2 Ie p þ BpIe ðM t 2 Þ þ 1 þ DT 1 Ie pðM T 1 Þ þ Ie cs aQ ðM t3 Þ; 2 2
Interest payable = 0
Fig. 4. Inventory system with inspection for the case 3: t2 6 t3 6 M 6 T 1 .
ð27Þ
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Fig. 5. Inventory system with inspection for the case 4: T 1 6 M 6 T.
Substitute the values from Eqs. (12)–(17) and (27) in Eq. (11), the total profit for case (iv), p4(Q, B) becomes
t2 ðQ þ zÞ ðt3 t2 Þðz þ z1 þ aQÞ ðT 1 t3 Þz1 þ þ 2 2 2 " # ½ð1aÞkDt 22 Ie p DT 21 Ie p C 2 ðt 1 þ t2 ÞB þ BpIe ðM t 2 Þ þ 2 2 : þ 2 þDT 1 Ie pðM T 1 Þ þ Ie cs aQðM t 3 Þ
p4 ðQ ; BÞ ¼ pð1 aÞQ þ cs aQ K cQ bQ h
ð28Þ
Case (v): T 6 M In this case, all the expressions for interest earned and paid coincide with that of case (iv). Hence, effectively we have four different cases for the retailer’s total profit per cycle, p(Q, B), which can be expressed as
8 p1 ðQ ; BÞ; > > > < p2 ðQ ; BÞ; pðQ ; BÞ ¼ > > p3 ðQ ; BÞ; > : p4 ðQ ; BÞ;
M 6 t 2 6 t3 6 T 1 ; case 1; t2 6 M 6 t3 6 T 1 ; case 2;
ð29Þ
t2 6 t 3 6 M 6 T 1 ; case 3; M P T1;
case 4:
To determine the expected total profit per unit time, we apply renewal-reward theorem Ross [12] as a is a random variable with known probability density function, f(a) and get the expected total profit per unit time for different cases which follows:
pðQ ; BÞ E½pðQ ; BÞ ¼ E½pT ðQ; BÞ ¼ E ; T E½T 8 E½pT1 ðQ ; BÞ ¼ E½p1 ðQ ; BÞ=E½T; > > > < E½pT2 ðQ ; BÞ ¼ E½p2 ðQ ; BÞ=E½T; pðQ ; BÞ ¼ T > > E½p3 ðQ ; BÞ ¼ E½p3 ðQ ; BÞ=E½T; > : E½pT4 ðQ ; BÞ ¼ E½p4 ðQ ; BÞ=E½T;
ð30Þ M 6 t 2 6 t3 6 T 1 ; case 1 ð31aÞ t2 6 M 6 t3 6 T 1 ; case 2 ð31bÞ t2 6 t3 6 M 6 T 1 ; case 3 ð31cÞ M P T1;
case 4 ð31dÞ
where
t 2 ðQ þ zÞ ðt 3 t 2 Þðz þ z1 þ E½aQÞ ðT 1 t3 Þz1 þ þ E½p1 ðQ ; BÞ ¼ pð1 E½aÞQ þ cs E½aQ K cQ bQ h 2 2 2 " # 2 2 C 2 ðt1 þ t 2 ÞB fð1 E½aÞk DgM Ie p DM Ie p þ þ 2 2 2 ½ð1 E½aÞQ DM fð1 E½aÞk DgMcIp ðT 1 MÞ þ cs Ip E½aQ ðt3 MÞ ; 2
ð32Þ
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t2 ðQ þ zÞ ðt3 t 2 Þðz þ z1 þ E½aQÞ ðT 1 t 3 Þz1 E½p2 ðQ ; BÞ ¼ pð1 E½aÞQ þ cs E½aQ K cQ bQ h þ þ 2 2 2 " # 2 2 C 2 ðt1 þ t2 ÞB ½ð1 E½aÞk Dt2 Ie p DM Ie p þ þ BpIe ðM t 2 Þ þ 2 2 2 ½ð1 E½aÞQ DM fð1 E½aÞk Dgt 2 cIp ðT 1 MÞ þ cs Ip E½aQ ðt 3 MÞ ; 2
5275
ð33Þ
t2 ðQ þ zÞ ðt3 t2 Þðz þ z1 þ E½aQ Þ ðT 1 t 3 Þz1 E½p3 ðQ ; BÞ ¼ pð1 E½aÞQ þ cs E½aQ K cQ bQ h þ þ 2 2 2 2 3 ½ð1E½aÞkDt22 Ie p þ BpIe ðM t2 Þ C 2 ðt1 þ t2 ÞB 4 2 5 ½ð1 E½aÞQ DM fð1 E½aÞk Dgt2 cIp ðT 1 MÞ ; þ 2 2 2 þ DM Ie p þ I c E½aQðM t Þ
ð34Þ
t 2 ðQ þ zÞ ðt 3 t2 Þðz þ z1 þ E½aQ Þ ðT 1 t 3 Þz1 E½p4 ðQ ; BÞ ¼ pð1 E½aÞQ þ cs E½aQ K cQ bQ h þ þ 2 2 2 2 3 ½ð1E½aÞkDt 22 Ie p þ BpIe ðM t 2 Þ C 2 ðt1 þ t2 ÞB 4 2 5 þ DT 21 Ie p 2 þ þ DT I pðM T Þ þ I c E½aQðM t Þ
ð35Þ
e s
2
2
3
1 e
1
e s
3
and E½T ¼ ð1E½DaÞQ Also, at T = M, the retailer’s expected total profit, E[p(Q, B)] is given by
pðQ ; BÞ E½pðQ ; BÞ ¼ ; E½pT ðQ ; BÞ ¼ E M E½M
ð36Þ
where E½M ¼ ð1E½DaÞQ and
t2 ðQ þ zÞ ðt3 t 2 Þðz þ z1 þ E½aQ Þ ðT 1 t 3 Þz1 E½pðQ ; BÞ ¼ pð1 E½aÞQ þ cs E½aQ K cQ bQ h þ þ 2 2 2 2 3 ½ð1E½aÞkDt 22 Ie p þ BpIe ðM t 2 Þ C 2 ðt 1 þ t 2 ÞB 4 2 5: þ 2 DT I p 2 þ 1 e þ DT I pðM T Þ þ I c E½aQ ðM t Þ 2
1 e
1
e s
3
4. Solution procedure Our objective is to find the optimal values of Q and B which maximize the expected total profit function, E[pT(Q, B)], therefore, the necessary conditions for E[pT(Q, B)] to be optimal are
@E½pT ðQ ;BÞ @Q
¼ 0 and
@E½pT ðQ ;BÞ @B
¼ 0 which follows as case wise:
Case (i): M 6 t2 6 t 3 6 T 1 The optimal values of Q = Q1 (say) and B = B1 (say), which maximizes, E½pT1 ðQ ; BÞ can be obtained by solving the Eq. (31a),
Q 2 ðhX 1 þ X 8 Þ B2 Q 2 hX 2 þ B2 X 6 þ QX 9 þ ðX 4 þ h X 7 X 10 Þ ¼ 0;
ð37Þ
B ¼ QX 14 :
ð38Þ
All Xi (i = 1, 2, . . . , 14) are elaborated in Appendix A. Case (ii): t 2 6 M 6 t3 6 T 1 The optimal values of Q = Q2 (say) and B = B2 (say), which maximizes, E½pT2 ðQ ; BÞ, are
B2 Y 19 þ Y 20 BY 21 þ B2 Q 2 Y 10 þ Q 2 Y 22 ¼ 0;
ð39Þ
BQY 12 þ B2 Y 7 þ QY 13 þ Q 2 Y 14 ¼ 0:
ð40Þ
All Yi’s are elaborated in Appendix B. Case (iii): t2 6 t3 6 M 6 T 1 The optimal values of Q = Q3 (say) and B = B3 (say), which maximizes, E½pT3 ðQ ; BÞ, are
w1 B2 Y 19 þ Y 20 BY 21 þ B2 Q 2 Y 10 þ Q 2 Y 22 þ Q 2 Mw1 ¼ 0; k
ð41Þ
BQY 12 þ B2 Y 7 þ QY 13 þ Q 2 Y 14 ¼ 0:
ð42Þ
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Fig. 6. Concavity of expected total profit function (case 1).
Case (iv): M P T 1 The optimal values of Q = Q4 (say) and B = B4 (say), which maximizes, E½pT4 ðQ ; BÞ, are
3Q 4 L1 2Q 3 L3 B þ Q 2 B2 ðL2 þ L15 Þ þ Q 2 ðL14 þ L12 Þ þ B2 ðL17 L7 þ L11 Þ þ BðL10 þ L6 Þ L13 L4 1 ¼ 0;
ð43Þ
Q 3 L3 þ 2Q 2 BL2 QL16 þ Bð2L15 2L17 þ 2L11 þ L9 Þ þ L16 L6 ¼ 0:
ð44Þ
All Li’s are elaborated in Appendix C. Further, for the expected profit function to be concave, the following sufficient conditions must be satisfied:
@ 2 E½pT ðQ ; BÞ @Q @B
and
!2
@ 2 E½pT ðQ ; BÞ @Q 2
! @ 2 E½pT ðQ; BÞ @Q 2 ! 6 0;
! @ 2 E½pT ðQ ; BÞ @B2
! @ 2 E½pT ðQ ; BÞ @B2
6 0:
6 0;
ð45Þ
ð46Þ
Second order derivatives have been calculated and these are very complicated (Appendix D). Therefore, it is very difficult to prove the concavity mathematically; hence the concavity of all the expected total profit functions have been established graphically and graph for one of the case is shown below (Fig. 6). Now, in order to find the optimal values of Q⁄ and B⁄, which maximizes the retailer’s expected profit, we propose the following algorithm: Algorithm: Step 1: Determine Q⁄ = Q1 (say) and B⁄ = B1 (say) from Eqs. (37) and (38). Now, using the values of Q1 and B1, calculate the values of t2, t3 and T1 from Eqs. (5), (7) and (10). If M 6 t2 6 t3 6 T 1 then the optimal value of expected total profit is obtained from Eq. (31a). Step 2: Determine Q⁄ = Q2 (say) and B⁄ = B2 (say) from Eqs. (39) and (40). Now, using the values of Q2 and B2, calculate the values of t2, t3 and T1 from Eqs. (5), (7) and (10). If t 2 6 M 6 t3 6 T 1 then the optimal values of expected profit is obtained from Eq. (31b). Step 3: Determine Q⁄ = Q3 (say) and B⁄ = B3 (say) from Eqs. (41) and (42). Now, using the values of Q3 and B3, calculate the values of t2, t3 and T1 from Eqs. (5), (7) and (10). If t 2 6 t 3 6 M 6 T 1 then the optimal values of expected profit is obtained from Eq. (31c). Step 4: Determine Q⁄ = Q4 (say) and B⁄ = B4 (say) from Eqs. (43) and (44). Now, using the values of Q3 and B3, calculate the values of t2, t3 and T1 from Eqs. (5), (7) and (10). If M P T 1 then the optimal values of expected profit then the optimal values of expected profit is obtained from Eq. (31d).
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Step 5: Determine the profit at T = M from Eq. (36). Step 6: Compare the calculated expected profit for cases 1, 2, 3, 4 and at M and select the optimal values of Q and B associated with the maximum expected profit. 5. An illustrative example
Example 1. An example is devised to illustrate the effect of permissible delay in payments on the retailer’s ordering policy for the developed model using the following data. D = 15000 units/year, A = $400/cycle, h = $4/unit/year, k = 60000 unit/year, c = $35/unit, p = $60/unit, cs = $25/unit, 10; 0 6 a 6 0:1; E½a ¼ 0:05, d = $1.0/unit and C2 = $6/year, percentage defective random variable a with its pdf, f ðaÞ ¼ 0; otherwise; M = 18 days. Two cases are considered (a) Let Ie = 0.08/year and Ip = 0.10/year, ðpIe ¼ 4:8 > 3:5 ¼ cIp Þ Results are obtained using the proposed algorithm as: Q⁄ = 1642 units, B⁄ = 674. Substituting the optimal values of Q⁄ in Eqs. (2) and (7), we get, T⁄ = 0.104 year, t3⁄ = 0.0274 year and E½pT ðQ ; B Þ = $347086. (b) Let Ie = 0.04 and Ip = 0.07, ðpIe ¼ 2:4 < 2:45 ¼ cIp Þ Using the proposed algorithm, results are obtained as: Q⁄ = 1804 units, B⁄ = 653, T⁄ = 0.114 year and t3⁄ = 0.0301 year and E½pT ðQ ; B Þ = $346095. Now, if supplier offers no trade credit, i.e. M = 0, we get: Q⁄ = 2130 units, B⁄ = 596, T⁄ = 0.135 year, t3⁄ = 0.0355 year and E½pT ðQ ; B Þ = $345384. Example 2. An example is devised to illustrate the effect of permissible delay in payments on the retailer’s ordering policy for the developed model using the following data. D = 25000 units/year, A = $1000/cycle, h = $5/unit/year, k = 60000 unit/year, c = $45/unit, p = $70/unit, cs = $30/unit, 10; 0 6 a 6 0:1; E½a ¼ 0:05, d = $1.0/unit and C2 = $7/year, percentage defective random variable a with its pdf, f ðaÞ ¼ 0; otherwise; M = 20 days. Two cases are considered: (a) Let Ie = 0.10/year and Ip = 0.12/year, ðpIe ¼ 7 > 5:4 ¼ cIp Þ Results are obtained using the proposed algorithm as: Q⁄ = 2671 units, B⁄ = 886. Substituting the optimal values of Q⁄ in Eqs. (2) and (7), we get, T⁄ = 0.101/year, t3⁄ = 0.0445 year and E½pT ðQ ; B Þ = $568588. (b) Let Ie = 0.05 and Ip = 0.08, ðpIe ¼ 3:5 < 3:6 ¼ cIp Þ Using the proposed algorithm, results are obtained as: Q⁄ = 3004 units, B⁄ = 881, T⁄ = 0.114 year and t3⁄ = 0.0501 year and E½pT ðQ ; B Þ = $566251. Now, if supplier offers no trade credit, i.e. M = 0, we get: Q⁄ = 3693 units, B⁄ = 821, T⁄ = 0.140 year, t3⁄ = 0.0616 year and E½pT ðQ ; B Þ = $564697. Example 3. D = 40000 units/year, A = $1000/cycle, h = $5/unit/year, k = 60000 unit/year, c = $45/unit, p = $70/unit, cs = $30/ 10; 0 6 a 6 0:1; , unit, d = $1.0/unit and C2 = $7/year, percentage defective random variable a with its pdf, f ðaÞ ¼ 0; otherwise; E½a ¼ 0:05, M = 20 days. Two cases are considered: (a) Let Ie = 0.10/year and Ip = 0.12/year, ðpIe ¼ 7 > 5:4 ¼ cIp Þ Results are obtained using the proposed algorithm as: Q⁄ = 2941 units, B⁄ = 530. Substituting the optimal values of Q⁄ in Eqs. (2) and (7), we get, T⁄ = 0.070/year, t3⁄ = 0.049 year and E½pT ðQ ; B Þ = $914760. (b) Let Ie = 0.05 and Ip = 0.08, ðpIe ¼ 3:5 < 3:6 ¼ cIp Þ Using the proposed algorithm, results are obtained as: Q⁄ = 3439 units, B⁄ = 535, T⁄ = 0.082 year and t3⁄ = 0.0573 year and E½pT ðQ ; B Þ = $910292. Now, if supplier offers no trade credit, i.e. M = 0, we get: Q⁄ = 4321 units, B⁄ = 510, T⁄ = 0.103 year, t3⁄ = 0.072 year and E½pT ðQ ; B Þ = $906826. 6. Sensitivity analysis Sensitivity analysis has been performed to study the impact of permissible delay (M), Interest earned (Ie), interest paid (Ip) and expected number of imperfect quality items (E[a]) on the lot size (Q⁄), backorders (B⁄) and the retailer’s expected total profit (E½pT ðQ ; B Þ). Results are summarized in Tables 1–3. Based on the computational results as shown in the Tables 1–3, we obtain the following managerial phenomena:
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Table 1 Impact of M on the optimal replenishment policy. M (days)
pIe < cIp
10 20 35
cIp < pIe T
Q
B
T
1792 1804 1805
649 653 652
0.113 0.114 0.114
Case T
E½p ðQ ; BÞ
Q
B
T
E½p ðQ ; BÞ
345298 346293 347778
1668 1629 1621
661 675 675
0.105 0.103 0.103
341649 347473 350440
2 3 4
Table 2 Impact of Ie and Ip on the optimal replenishment policy (M = 20 days). Ie ? Ip ;
0.03
0.20 0.15 0.10
0.05
0.07
Q
B
E½pT ðQ ; BÞ
Q
B
E½pT ðQ ; BÞ
Q
B
E½pT ðQ ; BÞ
1686 1716 1758
671 669 667
353075 353094 353120
1643 1665 1695
683 682 681
353670 353679 353692
1596 1608 1625
692 691 690
354285 354288 354292
Table 3 Impact of E[a] on Q, B and E½pT ðQ ; BÞ (M = 18 days, Ie = 0.08 and Ip = 0.1). E[a]
Q
B
E½pT ðQ ; BÞ
0.01 0.02 0.03 0.04 0.05
1600 1610 1621 1631 1642
694 689 684 679 674
354217 352489 350725 348924 347086
(i) It is observed from Table 1, when pIe < cIp the cycle length increases with permissible delay period which implies that the retailer should procure more quantity to avoid higher interest charges after the credit period that helps him to increase his expected profit e.g. in case of credit cards, usually banks charges very high rate of interest after the grace period. Further, when pIe > cIp, then both cycle length as well as order quantity decreases with increase in permissible delay period which means that the trade credit offered to the retailer has a positive impact on the economic ordering policy for imperfect quality items. In this scenario, results reveal that the retailer should order less quantity and take the advantage of permissible delay in payments more frequently. (ii) Table 2 reveals as Ie increases then order quantity decreases but expected profit increases which implies that when the interest earned per dollar is high, the expected total cost is low, which results in higher expected profit. Further, increase in Ip results a decrease in order quantity as well as the expected profit because the expected total cost increases when the capital opportunity cost in stock per dollar is high. From retailer point of view, it implies that when the capital opportunity cost per dollar is high, the retailer should order less but more frequently. (iii) It is clearly evident from Table 3 that as the percentage of defective items increases the optimal order quantity increases but the backorders decreases significantly, which results in less expected profit. In this scenario, the retailer should allow less backorders and be more vigilant while ordering. 7. Conclusions In this paper, we have developed an EOQ-based inventory model for imperfect quality items to determine the optimal ordering policies of a retailer under permissible delay in payments with allowable shortages. Screening rate is assumed to be more than the demand rate. During the screening process, retailer not only fulfills his current demand but also fulfills the back orders. In the above scenario, the model jointly optimizes the order quantity and shortages with the help of proposed algorithm. A comprehensive sensitivity analysis is also conducted to explore the effects of the key parameters (viz. M, Ie, Ip and E[a]) on the optimal results. Findings clearly suggest that the presence of trade credit has got affirmative effect on retailer ordering policy. The retailer should order more to avoid higher interest charged after the grace period that eventually increases his expected profit under the situation, pIe < cIp, whereas in other situation, i.e. pIe > cIp, the retailer should order less to avail the benefit of permissible delay more frequently. Further, when the interest payable rate increases, the retailer should order less but more frequently. However, increase in percentage defective items alert the retailer to look into source of supply and take the corrective measures in order to improve the quality of supply. Further, the proposed model can be extended for more realistic situations such as deteriorating items, stock dependent and stochastic demand with partial- trade credit, inflation etc.
C.K. Jaggi et al. / Applied Mathematics and Computation 219 (2013) 5268–5282
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Acknowledgments The authors are thankful to anonymous referees and the editor for their constructive comments and valuable suggestions that improved the presentation of the paper. The first author would like to acknowledge the support of Research Grant No. Dean (R)/R&D/2012/917, provided by University of Delhi (India) for conducting this research. Appendix A
X1 ¼
Dð1 þ a þ A þ A2 Þ ; ð1 E½aÞ
X4 ¼
KD ; ð1 E½aÞ
X8 ¼
cIp D ; 2k ð1 E½aÞ
X 11 ¼
X 13 ¼
X5 ¼
D½3ð1 E½aÞ þ 2A þ A2 ; ð1 E½aÞ
X3 ¼
C 2 ½1=D þ 1=kA ; 2
½ð1 E½aÞk DM 2 Iep DM2 Iep þ ; 2 2
X6 ¼
X3D ; ð1 E½aÞ
X2 ¼
X9 ¼
cIp D ½DM þ fð1 E½aÞk DgM ; 2k ð1 E½aÞ
" # 2D½3ð1 E½aÞ þ 2A þ A2 ; ð1 E½aÞ
X 12 ¼
DC 2 1 1 þ ; 2ð1 E½aÞ D kA
X7 ¼
X 10 ¼
X 14 ¼
cs IpE½a D ; k ð1 E½aÞ
X 13 ; X 12 X 11
1 cIp cIp D ½DM þ fð1 E½aÞk DgM ; D ð1 E½aÞ 2k 2k
Appendix B
Y1 ¼
cIp D ; 2k ð1 E½aÞ
Y4 ¼
Y8 ¼
cIp 2
2k A
Y2 ¼
1þA ½fð1 E½aÞk Dg þ ð1 E½aÞ; kA
½ð1 E½aÞk D;
2D ; ð1 E½aÞ
Y9 ¼
Y5 ¼
pIeMD ; ð1 E½aÞ
C2D 1 1 þ ; 2ð1 E½aÞ D kA
Y 10 ¼
D ½3ð1 EðaÞ þ 2A þ A2 ; ð1 E½aÞ
Y 23 ¼
D ½ð1 þ EðaÞ þ A þ A2 ; ð1 E½aÞ
Y 11 ¼
Y 24 ¼
cIpDM ; 2k
½ð1 E½aÞk DDIep
Y7 ¼
Y9 ¼
Y3 ¼
2k2 A2 ð1 E½aÞ
DC 2 ½1=D þ 1=kA ; 2ð1 E½aÞ
D ½1 2EðaÞ þ A2 ; ð1 E½aÞ
KD ; ð1 E½aÞ
Y 12 ¼ 2Y 10 þ 2Y 9 Y 8 2Y 4 ;
Y 13 ¼ Y 5 Y 3 ;
Y 19 ¼ 2Y 10 þ 2Y 9 Y 8 2Y 4 ;
Y 20 ¼ Y 24 Y 16 1;
w1 ¼ Iecs E½a;
Y 14 ¼ Y 1 Y 2 þ Y 11 ; Y 21 ¼ Y 3 þ Y 5 ;
Y 22 ¼ Y 23 Y 1 Y 17 ð2Y 10 þ 2Y 9 Y 16 2Y 4 Þ: Appendix C
L1 ¼
L5 ¼
Dð1 þ AÞ2 2
ð1 E½aÞk
;
L2 ¼
D2 pIeMð1 þ AÞ ; kð1 E½aÞ
D ð1 E½aÞk
L6 ¼
2
;
D2 Iep ; ð1 E½aÞ
L3 ¼
2Dð1 þ AÞ ð1 E½aÞk
L7 ¼
2
;
L4 ¼
pIeMD2 ; ð1 E½aÞ
½ð1 E½aÞk DDIep 2k2 A2 ð1 E½aÞ
;
X5D ; ð1 E½aÞ
;
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½ð1 E½aÞk DDIep
L9 ¼
2
2
k A ð1 E½aÞ
L12 ¼
IeDcs E½a ; ð1 E½aÞ
L16 ¼
ð1 2a A2 ÞD ; ð1 E½aÞ
L14 ¼
L10 ¼
;
pIeMD ; ð1 E½aÞ
ð1 þ a þ A þ A2 ÞD ; ð1 E½aÞ L17 ¼
L11 ¼
L15 ¼
D ; kAð1 E½aÞ
ð3ð1 aÞ þ 2A þ A2 ÞD ; ð1 E½aÞ
C 2 ð1=D þ 1=kAÞD ; ð1 E½aÞ
Appendix D Find below second derivatives of expected total profit functions for each case: Case (i): M 6 t2 6 t 3 6 T 1
@ 2 E½pT1 ðQ ; BÞ @Q 2 @ 2 E½pT1 ðQ ; BÞ @B2
¼ 2Q ðhX 1 þ X 8 Þ 2B2 QhX 2 þ X 9 ;
ðD:1Þ
¼ 1;
ðD:2Þ
where
X1 ¼
Dð1 þ a þ A þ A2 Þ ; ð1 E½aÞ
X4 ¼
KD ; ð1 E½aÞ
X5 ¼
½ð1 E½aÞk DM 2 Iep DM2 Iep þ ; 2 2
X6 ¼
X3D ; ð1 E½aÞ
X7 ¼
X5D ; ð1 E½aÞ
X9 ¼
cIp D ½DM þ fð1 E½aÞk DgM ; 2k ð1 E½aÞ
X 11 ¼
X 13 ¼
X2 ¼
D½3ð1 E½aÞ þ 2A þ A2 ; ð1 E½aÞ
X8 ¼
" # 2D½3ð1 E½aÞ þ 2A þ A2 ; ð1 E½aÞ
X3 ¼
C 2 ½1=D þ 1=kA ; 2
cIp D ; 2k ð1 E½aÞ
X 12 ¼
X 10 ¼
cs IpE½a D ; k ð1 E½aÞ
DC 2 1 1 þ ; 2ð1 E½aÞ D kA
1 cIp cIp D ½DM þ fð1 E½aÞk DgM ; D ð1 E½aÞ 2k 2k
X 14 ¼
X 13 : X 12 X 11
Case (ii): t 2 6 M 6 t3 6 T 1
@ 2 E½pT2 ðQ ; BÞ @Q 2 @ 2 E½pT2 ðQ ; BÞ @B2
¼ 2B2 QY 10 þ 2QY 22 ;
ðD:3Þ
¼ QY 12 þ 2BY 7 ;
ðD:4Þ
where
Y1 ¼
cIp D ; 2k ð1 E½aÞ
Y4 ¼
Y8 ¼
cIp
2k2 A
Y2 ¼
1þA ½fð1 E½aÞk Dg þ ð1 aÞ; kA
½ð1 E½aÞk D;
2D ; ð1 E½aÞ
Y9 ¼
Y5 ¼
pIeMD ; ð1 E½aÞ
C2 D 1 1 ; þ 2ð1 E½aÞ D kA
Y7 ¼
Y9 ¼
Y3 ¼
cIpDM ; 2k
½ð1 E½aÞk DDIep 2k2 A2 ð1 aÞ
DC 2 ½1=D þ 1=kA ; 2ð1 aÞ
;
C.K. Jaggi et al. / Applied Mathematics and Computation 219 (2013) 5268–5282
Y 10 ¼
D ½3ð1 EðaÞ þ 2A þ A2 ; ð1 E½aÞ
Y 23 ¼
D ½ð1 þ EðaÞ þ A þ A2 ; ð1 E½aÞ
Y 11 ¼
Y 24 ¼
5281
D ½1 2EðaÞ þ A2 ; ð1 E½aÞ
KD ; ð1 E½aÞ
Y 12 ¼ 2Y 10 þ 2Y 9 Y 8 2Y 4 ;
Y 13 ¼ Y 5 Y 3 ;
Y 14 ¼ Y 1 Y 2 þ Y 11 ;
Y 19 ¼ 2Y 10 þ 2Y 9 Y 8 2Y 4 ;
Y 20 ¼ Y 24 Y 16 1;
Y 21 ¼ Y 3 þ Y 5 ;
Y 22 ¼ Y 23 Y 1 Y 17 ð2Y 10 þ 2Y 9 Y 16 2Y 4 Þ: Case (iii): t2 6 t3 6 M 6 T 1
@ 2 E½pT3 ðQ; BÞ @Q
2
@ 2 E½pT3 ðQ; BÞ @B2
w1 ¼ 2B2 QY 10 þ 2QY 22 þ 2Q Mw1 ; k
ðD:5Þ
¼ QY 12 þ 2BY 7 ;
ðD:6Þ
where
Y1 ¼
Y4 ¼
Y8 ¼
cIp D ; 2k ð1 E½aÞ
cIp
2k2 A
Y2 ¼
1þA ½fð1 E½aÞk Dg þ ð1 E½aÞ; kA
½ð1 E½aÞk D;
2D ; ð1 E½aÞ
Y9 ¼
Y5 ¼
pIeMD ; ð1 E½aÞ
C2D 1 1 ; þ 2ð1 E½aÞ D kA
Y 10 ¼
D ½3ð1 EðaÞ þ 2A þ A2 ; ð1 E½aÞ
Y 23 ¼
D ½ð1 þ EðaÞ þ A þ A2 ; ð1 E½aÞ
Y 11 ¼
Y 24 ¼
cIpDM ; 2k
½ð1 E½aÞk DDIep
Y7 ¼
Y9 ¼
Y3 ¼
2k2 A2 ð1 E½aÞ
;
DC 2 ½1=D þ 1=kA ; 2ð1 E½aÞ
D ½1 2EðaÞ þ A2 ; ð1 E½aÞ
KD ; ð1 E½aÞ
Y 12 ¼ 2Y 10 þ 2Y 9 Y 8 2Y 4 ;
Y 13 ¼ Y 5 Y 3 ;
Y 19 ¼ 2Y 10 þ 2Y 9 Y 8 2Y 4 ;
Y 20 ¼ Y 24 Y 16 1;
w1 ¼ Iecs E½a;
Y 14 ¼ Y 1 Y 2 þ Y 11 ; Y 21 ¼ Y 3 þ Y 5 ;
Y 22 ¼ Y 23 Y 1 Y 17 ð2Y 10 þ 2Y 9 Y 16 2Y 4 Þ: Case (iv): M P T 1
@ 2 E½pT4 ðQ; BÞ @Q 2 @ 2 E½pT4 ðQ; BÞ @B2
¼ 12Q 3 L1 6Q 2 L3 B þ 2QB2 ðL2 þ L15 Þ þ 2QðL14 þ L12 Þ;
ðD:7Þ
¼ 2Q 2 L2 þ ð2L15 2L17 þ 2L11 þ L9 Þ;
ðD:8Þ
where
L1 ¼
L5 ¼
L9 ¼
Dð1 þ AÞ2 2
ð1 E½aÞk
;
L2 ¼
D2 pIeMð1 þ AÞ ; kð1 E½aÞ
D ð1 E½aÞk
L6 ¼
½ð1 E½aÞk DDIep k2 A2 ð1 E½aÞ
;
2
;
D2 Iep ; ð1 E½aÞ L10 ¼
L3 ¼
2Dð1 þ AÞ ð1 E½aÞk
L7 ¼
pIeMD ; ð1 E½aÞ
2
;
L4 ¼
pIeMD2 ; ð1 E½aÞ
½ð1 E½aÞk DDIep 2k2 A2 ð1 E½aÞ L11 ¼
D ; kAð1 E½aÞ
;
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L12 ¼
IeDcs E½a ; ð1 E½aÞ
L16 ¼
ð1 2a A2 ÞD ; ð1 E½aÞ
L14 ¼
ð1 þ a þ A þ A2 ÞD ; ð1 E½aÞ L17 ¼
L15 ¼
ð3ð1 aÞ þ 2A þ A2 ÞD ; ð1 E½aÞ
C 2 ð1=D þ 1=kAÞD : ð1 E½aÞ
References [1] E.L. Porteus, Optimal lot sizing, process quality improvement and setup cost reduction, Oper. Res. 34 (1986) 137–144. [2] M. Rosenblatt, H. Lee, Economic production cycles with imperfect production processes, IIE Trans. 18 (1986) 48–55. [3] H.L. Lee, M.J. Rosenblatt, Simultaneous determination of production cycles and inspection schedules in a production system, Manage. Sci. 33 (1987) 1125–1137. [4] R.L. Schwaller, EOQ under inspection costs, Prod. Invest. Man. 29 (1988) 22–24. [5] X. Zhang, Y. Gerchak, Joint lot sizing and inspection policy in an EOQ model with random yield, IIE Trans. 22 (1990) 41–47. [6] M.K. Salameh, M.Y. Jaber, Economic production quantity model for items with imperfect quality, Int. J. Prod. Econ. 64 (2000) 59–64. [7] L.E. Cárdenas-Barrón, Observation on: economic production quantity model for items with imperfect quality, Int. J. Prod. Econ. 67 (2000) 201. [8] S.K. Goyal, L.E. Cárdenas-Barrón, Note on: economic production quantity model for items with imperfect quality – a practical approach, Int. J. Prod. Econ. 77 (2002) 85–87. [9] S. Papachristos, I. Konstantaras, Economic ordering quantity models for items with imperfect quality, Int. J. Prod. Econ. 100 (2006) 148–154. [10] H.M. Wee, J. Yu, M.C. Chen, Optimal inventory model for items with imperfect quality and shortage backordering, Omega 35 (2007) 7–11. [11] B. Maddah, M.Y. Jaber, Economic order quantity for items with imperfect quality: revisited, Int. J. Prod. Econ. 112 (2008) 808–815. [12] S.M. Ross, Stochastic Processes, second ed., Wiley, New York, 1996. [13] A. Eroglu, G. Ozdemir, An economic order quantity model with defective items and shortages, Int. J. Prod. Econ. 106 (2007) 544–549. [14] B. Sarkar, I.K. Moon, An EPQ model with inflation in an imperfect production system, Appl. Math. Comput. 217 (2011) 6159–6167. [15] B. Sarkar, S.S. Sana, K.S. Chaudhuri, An imperfect production process for time varying demand with inflation and time value of money – an EMQ model, Expert Syst. Appl. 38 (2011) 13543–13548. [16] B. Sarkar, S.S. Sana, K.S. Chaudhuri, An economic production quantity model with stochastic demand in an imperfect production system, Int. J. Serv. Oper. Manage. 9 (2011) 259–283. [17] M.D. Roy, S.S. Sana, K. Chaudhuri, An EOQ model for imperfect quality products with partial backlogging – a comparative study, Int. J. Serv. Oper. Manage. 9 (2011) 83–110. [18] M.D. Roy, S.S. Sana, K. Chaudhuri, An economic order quantity model of imperfect quality items with partial backlogging, Int. J. Syst. Sci. 42 (2011) 1409–1419. [19] S.S. Sana, A production-inventory model of imperfect quality products in a three-layer supply chain, Decis. Support Syst. 50 (2011) 539–547. [20] H. Soni, N.H. Shah, C.K. Jaggi, Inventory models and trade credit, J. Con. Cyb. 39 (2010) 867–882. [21] K.J. Chung, Y.F. Huang, Retailer’s optimal cycle times in the EOQ model with imperfect quality and permissible credit period, J. Qual. Quant. 40 (2006) 59–77. [22] C.K. Jaggi, S.K. Goel, M. Mittal, Economic order quantity model for deteriorating items with imperfect quality and permissible delay on payment, Int. J. Ind. Eng. Comput. 2 (2010) 237–248. [23] B. Sarkar, An EOQ model with delay-in-payments and time-varying deterioration rate, Math. Comput. Model. 55 (2012) 367–377. [24] B. Sarkar, An EOQ model with delay in payments and stock dependent demand in the presence of imperfect production, Appl. Math. Comput. 218 (2012) 8295–8308.