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An economic production quantity model with non-synchronized screening and rework
Applied Mathematics and Computation 233 (2014) 127–138
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
An economic production quantity model with non-synchronized screening and rework Hui-Ming Wee a,⇑, Wan-Tsu Wang a,b, Tsai-Chi Kuo a, Yung-Lung Cheng a,c, Yen-Deng Huang a a
Department of Industrial & Systems Engineering, Chung Yuan Christian University, Chungli 32023, Taiwan, ROC Department of Marketing and Distribution Management, Chien Hsin University of Science and Technology, Jungli 32097, Taiwan, ROC c Department of Industrial Management, Chien Hsin University of Science and Technology, Jungli 32097, Taiwan, ROC b
a r t i c l e
i n f o
Keywords: Economic production quantity Rework Backorder Non-synchronized screening
a b s t r a c t This study investigates an economic production quantity model for imperfect quality items with non-synchronized screening and rework. With time intervals as decision variables, our approach establishes a feasible optimal policy for the inventory system. We presented a solution procedure to derive the optimal policy, and generalized the proposed model to the classic EPQ model by taking the limiting parameter values for the optimal solution. A numerical example is provided to illustrate the theory and its application. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction In a classic economic order/production quantity (EOQ/EPQ) model, all items purchased/produced are assumed 100% good quality. Since defective items often exist in order/production lots, 100% perfect quality may not be practical in real-life situations. Salameh and Jaber [1] developed an extended EOQ model for a random fraction of imperfect quality items. The poor quality items are sold at the end of the 100% screening process. Maddah and Jaber [2] revisited the model of Salameh and Jaber [1] and rectified the optimal expected annual profit using the Renewal Reward Theorem. Other studies related to the model of Salameh and Jaber [1] include Papachristos and Konstantaras [3], Wee et al. [4], Eroglu and Ozdemir [5] and Yassine et al. [6]. The effect of defective items on the production lot sizing policy has also received considerable attention by many researchers. Rosenblatt and Lee [7] proposed an EPQ model to deal with an imperfect production process system. They assumed that defective items could be reworked instantaneously; their study showed that the presence of defective products motivates a smaller lot size. Hayek and Salameh [8] proposed an EPQ model in which all imperfect quality products were reworked and shortages were allowed. Chiu et al. [9] presented a finite production model that considered rework and backorders under a service level constraint. Jamal et al. [10] developed an EPQ model in which all defective products are reworked to fix the defects during the same production cycle. Related to this research were the papers by Sarker et al. [11] and Cárdenas-Barrón [12,13]. Taleizadeh et al. [14] introduced a multi-product single-machine production system with a stochastic scrapped production rate, partial backorders, and a service level constraint. Chiu et al. [15] provided an optimal inventory replenishment policy for the EPQ with rework and multiple shipments. Recently, Wee and Widyadana [16] considered a production system for deteriorating items with stochastic preventive maintenance time and rework. Taleizadeh et al. [17] developed an EPQ inventory model for multi products in a single machine with rework and interruption in process. In our literature review, there are very few EOQ/EPQ models considering two backorder costs (see for example, Sphicas [18], Omar et al. [19], Cárdenas-Barrón [20], Chung and Cárdenas-Barrón [21]). Cárdenas-Barrón [22] and Wee et al. [23] ⇑ Corresponding author. E-mail address: [email protected] (H.-M. Wee). http://dx.doi.org/10.1016/j.amc.2014.01.150 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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extended the model of Jamal et al. [10] to an EPQ model with planned backorders (linear and fixed costs). Most EPQ models with rework assumed that the production process implicitly includes an inspection/screening procedure, or that the production and screening procedures occur at the same rate. In considering two backorder costs, our proposed models allow the screening rate to be less than the production rate. If the screening rate is less than the production rate, the screening rate determines the rate to fulfill backorders and demand. To derive closed-form solutions, this study uses time intervals rather than production and backorder sizes as decision variables. The optimal production lot size and optimal backorder quantity are derived from the optimal intervals. By analyzing three conditions for the optimal inventory policy, our approach distinguishes between feasible and infeasible optimal policies. If the screening rate is less than the production rate, we have found that the optimal production time is not always greater than the optimal time to eliminate backorders. We provide an alternate model for which infeasible optimal intervals are derived. Moreover, we provide a solution procedure to derive the feasible optimal policy. We show that special cases of EPQ models can be derived by using the parameter limiting values for the proposed model. The remainder of this paper is organized as follows. Model formulation and notation are presented in Section 2. Special case models are developed in Section 3. In Section 4, we use a numerical example to illustrate our solution procedure. Section 5 concludes the paper. 2. Model formulation The following notation is used for this study: T, S Q, Q0 W, W0
u k x d c cR k h b bF p
cycle lengths for Models I and II, respectively production lot sizes in units per cycle for Models I and II, respectively maximum backorder levels in units for Models I and II, respectively production rate in units per unit time demand rate in units per unit time screening rate in units per unit time, x 6 u screening cost per unit production cost per unit rework cost per unit setup cost per production cycle holding cost per unit per unit time backorder cost per unit per unit time (linear backorder cost) backorder cost per unit (fixed backorder cost) percentage of defective items in Q; p is regarded as an expected defective percentage for items during the production; 0 6 p < 1.
In addition, we make the following assumptions: (1) (2) (3) (4) (5) (6)
The replenishment rate is dependent on the screening rate. The demand rate is known and constant. All items are screened 100% with a known screening rate. Defective items are proportional to the production lot. Screening and demand proceed simultaneously. The screening rate is less than or equal to the production rate. Shortages are completely backordered. Production and rework process occur at the same speed. All defective items are reworked to remove the defects.
2.1. Model I: the production time is greater than or equal to the time to eliminate backorders Fig. 1 depicts the production-inventory model with rework (during the same cycle) and backorders. We assume the production process contains p fraction of defective items. The items are screened to separate the perfect and defective items. At the beginning of interval t2, production and screening start simultaneously. The production interval is t2 + t3. If the production rate is greater than the screen rate, the interval t4 will be greater than zero. The rate of perfect items that are screened during t2 + t3 + t4 is 1 p. During the interval t2, with a known screening rate x, the fraction of the perfect items are used to meet demand and eliminate backorders, at a rate of (1 p)x k; the remaining perfect items are used to increase inventory, at a rate of u x. The defective inventory increases at a rate of px. To eliminate backorders and achieve a positive inventory, we assume that (1 p)x k > 0. The screening process terminates at the end of interval (t2 + t3 + t4) and the defective items of pu(t2 + t3) are reworked immediately. Referring to Fig. 1, the interval t2 builds inventory and eliminates backorders/shortages simultaneously: t 2 ¼ W=ðð1 pÞ x kÞ ¼ z1 =ðu xÞ. We have:
W ¼ ðð1 pÞx kÞt 2 ;
ð1Þ
H.-M. Wee et al. / Applied Mathematics and Computation 233 (2014) 127–138
129
z4 z3 z2 z1
W
Fig. 1. Inventory of good items for EPQ model I.
z1 ¼ ðu xÞt2 :
ð2Þ
The shortage interval without production is
t1 ¼ W=k ¼ ðð1 pÞx kÞt 2 =k:
ð3Þ
Since the interval t2 + t3 + t4 is consumed to screen production units per cycle, u(t2 + t3), we have xðt 2 þ t3 þ t4 Þ ¼ uðt2 þ t 3 Þ. It results in:
t4 ¼ ðu xÞðt 2 þ t 3 Þ=x:
ð4Þ
From Eq. (4), we can see that t4 > 0 if u > x. Referring to Fig. 2, the interval t5 is consumed to rework pu(t2 + t3) defective units with a rate of u. We now have:
t5 ¼ puðt 2 þ t 3 Þ=u ¼ pðt2 þ t3 Þ:
ð5Þ
Subsequently, referring to Fig. 1, we can derive the following expressions:
z4 ¼ z1 þ ðu px kÞt 3 ¼ ðu xÞt2 þ ðu px kÞt 3 ;
ð6Þ
z2 ¼ z4 ðpx þ kÞt4 ¼ ðð1 pÞx kÞðuðt 2 þ t 3 Þ xt 2 Þ=x;
ð7Þ
z3 ¼ z2 þ ðu kÞt 5 ¼ pðu kÞðt 2 þ t 3 Þ þ ðð1 pÞx kÞðuðt2 þ t3 Þ xt 2 Þ=x;
ð8Þ
t6 ¼ z3 =k ¼ pðu kÞðt 2 þ t 3 Þ=k þ ðð1 pÞx kÞðuðt2 þ t 3 Þ xt2 Þ=ðxkÞ:
ð9Þ
The total inventory cost per cycle, TC, is composed of the production cost, the screening cost, the rework cost, the backorder cost, and the holding cost. We have
Fig. 2. Inventory of defective items for EPQ model I.
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bðt1 þ t 2 ÞW TC ¼ ½cuðt2 þ t3 Þ þ k þ duðt2 þ t 3 Þ þ cR puðt2 þ t 3 Þ þ þ bF W 2 t 2 z1 t 3 ðz1 þ z4 Þ t4 ðz4 þ z2 Þ t 5 ðz2 þ z3 Þ t6 z3 ðt 2 þ t 3 þ t 4 þ t5 Þpuðt 2 þ t3 Þ þh þh : þ þ þ þ 2 2 2 2 2 2
ð10Þ
From Eqs. (3)–(5) and (9), the total cycle time can be rewritten as:
T ¼ t1 þ t2 þ t3 þ t4 þ t5 þ t6 ¼
ðt2 þ t3 Þu : k
ð11Þ
For ease of notation, we use t2 and t (=t 2 þ t 3 ) as decision variables in the total cost function. The total inventory cost per unit time is written as:
TCUðt 2 ; tÞ ¼
TC T 2
kk bt ð1 pÞx bF t 2 k þ dk þ cR pk þ 2 ðð1 pÞx kÞ þ ðð1 pÞx kÞ tu 2t u tu 2 t ð1 pÞx t ðð1 pÞx kÞ þ ðu kÞ t 2 ðð1 pÞx kÞ : þh 2 2t u 2
¼ ck þ
ð12Þ
The objective is to minimize TCUðt2 ; tÞ subject to t P t2 P 0. In Eq. (12), one can see that c, cR and d do not affect the optimal solutions. Lemma 1. For any given t > 0, there are the following cases to minimize TCUðt 2 jtÞ: Case A. For a given t 6 bhFuk ; t2 ¼ 0. Case B. If hu ðb þ hÞð1 pÞx P 0, for a given If hu ðb þ hÞð1 pÞx < 0, for a given
bF k hu bF k hu
bF k < t 6 huðbþhÞð1pÞx ; 0 < t 2 6 t.
< t; 0 < t2 < t.
bF k ; t 2 > t. Case C. If hu ðb þ hÞð1 pÞx > 0, for a given t > huðbþhÞð1pÞx
Proof. See Appendix A.
h
Lemma 1 shows that the optimal time to eliminate backorders, t2 , is dependent on the given production time, t. The interval t2 = 0 implies that the shortage is not allowed. If production time is greater than shortage. However, if production time is greater than
bF k , huðbþhÞð1pÞx
bF k , hu
the optimal policy is to allow for
the optimal time to eliminate backorders is greater than
hu , then the production time, which is an infeasible optimal policy for Model I. In Lemma 1, if x > ð1pÞðbþhÞ
hu ðb þ hÞð1 pÞx < 0, which means 0 6 t2 < t. In other words, if
hu ð1pÞðbþhÞ
< x 6 u, Model I always attains a feasible
optimal policy, i.e., Lemma 1 Case C does not exist. Consider the first-order-derivative condition for TCUðt 2 ; tÞ. Taking the first derivatives of TCUðt 2 ; tÞ with respect to t2 and t, we have:
@ ðð1 pÞx kÞððb þ hÞð1 pÞxt2 þ bF k hutÞ TCUðt2 ; tÞ ¼ ; @t2 ut
ð13Þ
@ 2kk t 2 ðð1 pÞx kÞððb þ hÞð1 pÞxt 2 þ 2bF kÞ þ hut2 ðu kÞ TCUðt 2 ; tÞ ¼ ; @t 2u t 2
ð14Þ
Letting
@ @ TCUðt 2 ; tÞ ¼ 0 and TCUðt2 ; tÞ ¼ 0, we have: @t 2 @t
t2 ðtÞ ¼
hut bF k ; ðb þ hÞð1 pÞx
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2kk þ t 2 ðð1 pÞx kÞððb þ hÞð1 pÞxt 2 þ 2bF kÞ tðt2 Þ ¼ : huðu kÞ
ð15Þ
ð16Þ
Substituting Eq. (15) into Eq. (16), we can derive the solutions for t and t2, as follows:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k 2kðb þ hÞð1 pÞx bF kðð1 pÞx kÞ ; t ¼ hu bxð1 pÞðu kÞ þ hkðu ð1 pÞxÞ
ð17Þ
H.-M. Wee et al. / Applied Mathematics and Computation 233 (2014) 127–138
t2 ¼
hut bF k : ðb þ hÞð1 pÞx
131
ð18Þ
In the denominator of Eq. (17), we see that bxð1 pÞðu kÞ þ hkðu ð1 pÞxÞ is greater than 0. If TCUðt 2 ; tÞ is convex in t and t2, Eqs. (17) and (18) are the optimal solutions to minimize TCUðt 2 ; tÞ. Theorem 1. 2
(i) If 2hku bF kðu kÞ > 0, then TCU is convex in t and t2. (a) If hu ðb þ hÞð1 pÞx < 0, the feasible optimal solution ðt ; t2 Þ of TCUðt2 ; tÞ is ðt ; t 2 Þ. (b) If hu ðb þ hÞð1 pÞx P 0 and t P t 2 , the feasible optimal solution ðt ; t 2 Þ of TCUðt 2 ; tÞ is ðt ; t 2 Þ. qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 (ii) If 2hku bF kðu kÞ 6 0, then t ¼ hu2kk and t 2 ¼ 0. ðukÞ
Proof. See Appendix A. h The optimal total cost per unit time can be derived by substituting ðt ; t2 Þ for ðt; t 2 Þ in Eq. (12). From t and t 2 , the optimal production size and maximal backorder level can be derived:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uk 2kðb þ hÞð1 pÞx b2F kðð1 pÞx kÞ Q ¼ ut ¼ ; h bxð1 pÞðu kÞ þ hkðu ð1 pÞxÞ
ð19Þ
W ¼ ðð1 pÞx kÞt2 :
ð20Þ
If p ! 0 and x ! u, Eqs. (19) and (20) are reduced to the optimal solutions for the classic EPQ model with linear and fixed backorder costs. Since p ! 0 and x ! u, we have hu ðb þ hÞð1 pÞx ¼ bu < 0. Therefore, for a given t > 0, this leads to two cases: t 2 ¼ 0 and 0 < t2 < t in Lemma 1. Subsequently, Theorem 1 reduces to the model without reworking defective items. Corollary 1. For the classic EPQ model with linear and fixed backorder costs: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 2kkðbþhÞ=ð1k=uÞb2F k2 F kÞð1k=uÞ (i) If 2hku bF kðu kÞ > 0; Q ¼ . and W ¼ ðhQ bhþb hb qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2kk (ii) If 2hku bF kðu kÞ 6 0; Q ¼ hð1k= uÞ and W ¼ 0. 2
If 2hku bF kðu kÞ ¼ 0; Q and W in Corollary 1 (i) are reduced to identical to Theorem 4 in Chung and Cárdenas-Barrón [21] if Q 2 in bF k In Lemma 1 Case C, we can see that, if t > huðbþhÞð1pÞx , then t 2 >
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2kk hð1k=uÞ
and 0, respectively. Therefore, Corollary 1 is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi 2KDðhþpÞ=ð1D=PÞp2 D . hp
their equation is amended by
t , i.e., Model I derives infeasible solutions with t < t 2.
Fig. 1 cannot represent the correct inventory behavior for Lemma 1 Case C. When the optimal production time (t ) is less than the optimal time to eliminate backorders (t 2 ), in Section 2.2, we show how to apply Model II to derive the feasible optimal policy. 2.2. Model II: the production time is less than the time to eliminate backorders Fig. 3 depicts the inventory behavior when the production run time is less than the time to eliminate backorders. The inventory behavior for the defective items is the same as that shown in Fig. 2. Similar to the analysis process in Section 2.1, the following equations can be expressed in terms of s2 and s3.
W 0 ¼ ðð1 pÞx kÞðs2 þ s3 Þ;
ð21Þ
s1 ¼ ðð1 pÞx kÞðs2 þ s3 Þ=k;
ð22Þ
s4 ¼ s2 ðu=x 1Þ s3 ;
ð23Þ
s5 ¼ pus2 =u ¼ ps2 ;
ð24Þ
s6 ¼ s2 ðx kÞðu ð1 pÞxÞ=ðxkÞ ðð1 pÞx kÞs3 =k;
ð25Þ
z01 ¼ ðð1 pÞx kÞðus2 xðs2 þ s3 ÞÞ=x;
ð26Þ
z02 ¼ us2 xðs2 þ s3 Þ;
ð27Þ
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z’4 z’3 z’2 z’1
-W’
Fig. 3. Inventory of good items for EPQ model II.
z03 ¼ s2 ðx kÞðu ð1 pÞxÞ=x s3 ðð1 pÞx kÞ;
ð28Þ
z04 ¼ ðu xÞs2 :
ð29Þ 0
The total inventory cost per cycle, TC , is shown as:
bðs1 þ s2 þ s3 ÞW 0 TC 0 ¼ ½cus2 þ k þ dus2 þ cR pus2 þ þ bF W 0 2 0 0 0 0 0 s4 z2 þ z1 s5 z01 þ z03 s2 z4 s3 z4 þ z2 s6 z0 ðs2 þ s3 þ s4 þ s5 Þpus2 : þh þ þ þ þ 3 þh 2 2 2 2 2 2
ð30Þ
From Eqs. (22)–(25), the total cycle time, S, is rewritten as
S ¼ s1 þ s2 þ s3 þ s4 þ s5 þ s6 ¼
s2 u : k
ð31Þ
For ease of notation, we use s2 and s (=s2 þ s3 ) as decision variables. The total inventory cost per unit time is written as:
TCU 0 ðs2 ; sÞ ¼
TC 0 S 2
kk bs ð1 pÞx bF sk þ dk þ cR pk þ ðð1 pÞx kÞ þ ðð1 pÞx kÞ s2 u 2s2 u s2 u 2 s ð1 pÞx s2 þh ðð1 pÞx kÞ þ ðu kÞ sðð1 pÞx kÞ : 2s2 u 2
¼ ck þ
ð32Þ
The objective is to minimize TCU 0 ðs2 ; sÞ, subject to s P s2 P 0. We can see that TCU 0 ðs2 ; sÞ is the same as TCUðt2 ; tÞ if t2 and t are replaced by s and s2, respectively. Similar to Lemma 1, we can now develop Lemma 2. Lemma 2. For any given s2 > 0, there are the following cases to minimize TCU 0 ðsjs2 Þ:
Case A. For a given s2 6 bhFuk ; s = 0. Case B. If hu ðb þ hÞð1 pÞx P 0, for a given If hu ðb þ hÞð1 pÞx < 0, for a given
bF k bF k < s2 6 huðbþhÞð1pÞx , hu bF k < s ; 0 < s < s . 2 2 hu
then 0 < s 6 s2 ;
bF k ; s > s2 . Case C. If hu ðb þ hÞð1 pÞx > 0, for a given s2 > huðbþhÞð1pÞx
Proof. Similar to Lemma 1. h @ The solutions for @s@2 TCU 0 ðs2 ; sÞ ¼ 0 and @s TCU 0 ðs2 ; sÞ ¼ 0 can be solved by substituting s and s2 for t2 and t in Eqs. (17) and (18), respectively:
H.-M. Wee et al. / Applied Mathematics and Computation 233 (2014) 127–138
s2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k 2kðb þ hÞð1 pÞx bF kðð1 pÞx kÞ ¼ ; hu bxð1 pÞðu kÞ þ hkðu ð1 pÞxÞ
s ¼
133
ð33Þ
hus2 bF k : ðb þ hÞð1 pÞx
ð34Þ
It is shown that s 2 ¼ t and s ¼ t 2 .
Theorem 2. 2
0 (i) If 2hku bF kðu kÞ > 0, then TCU0 is convex in s2 and s. If s P s 2 , the feasible optimal solution ðs2 ; s Þ of TCU ðs2 ; sÞ is ðs2 ; s Þ. qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 and s ¼ 0. (ii) If 2hku bF kðu kÞ 6 0, then s2 ¼ hu2kk ðukÞ
Proof. Similar to Theorem 1. h From s2 and s , the optimal production size and the optimal maximal backorder level can be derived as follows: 0
Q ¼u
s2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uk 2kðb þ hÞð1 pÞx b2F kðð1 pÞx kÞ ; ¼ h bxð1 pÞðu kÞ þ hkðu ð1 pÞxÞ
ð35Þ
W 0 ¼ ðð1 pÞx kÞs : 0
ð36Þ
0
0
ðs2 ; s Þ
TCUðt2 ;
We can show that Q ¼ Q ; W ¼ W and TCU ¼ t Þ. However, Model I (Fig. 1) and Model II (Fig. 2) represent different production schedules and inventory levels, respectively. Therefore, using production size and backorder level as decision variables may not reveal the infeasible optimal policy. Lemma 1 Case A and Lemma 2 Case A have the same optimal policy without shortage. In Lemma 2 Case B, we have that if bF k hu
bF k < s2 < huðbþhÞð1pÞx , then 0 < s < s2 , i.e., s3 < 0. This shows that the optimal time to eliminate backorders is less than the
optimal production time. If s < s2 , Model II cannot represent the correct production schedule. So, if s < s2 , the optimal policy is derived by Model I rather than Model II. We provide a solution procedure to determine the optimal policy. 2
Step 1: Start with Model I. Calculate the value of 2hku bF kðu kÞ. qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2kk (1) If 2hku bF kðu kÞ 6 0, then the optimal solution ðt ; t 2 Þ for TCUðt2 ; tÞ is ;0 . huðukÞ 2
(2) If 2hku bF kðu kÞ > 0 with t P t 2 , the optimal solution ðt ; t 2 Þ for TCUðt 2 ; tÞ is ðt ; t 2 Þ, as shown in Eqs. (17) and (18). The optimal total inventory cost per unit time is TCUðt 2 ; t Þ. 2 (3) If 2hku bF kðu kÞ > 0 with t < t 2 , go to Step 2. Step 2: Apply Model II to find the optimal policy. The optimal solution ðs2 ; s Þ for TCU 0 ðs2 ; sÞ is ðs 2 ; s Þ is shown in Eqs. (33)
and (34). The optimal total inventory cost per unit time is TCU 0 ðs2 ; s Þ. 3. Special cases Here we modify Model I in Section 2.1 by not considering the holding cost of the defective items. In other words, the total h i inventory cost does not include the last term of TC in Eq. (10), i.e., h ðt2 þt3 þt4 þt25 Þpuðt2 þt3 Þ . We therefore have the total inventory cost per cycle:
bðt1 þ t2 ÞW TCM ¼ ½cuðt 2 þ t 3 Þ þ k þ duðt2 þ t3 Þ þ cR puðt2 þ t3 Þ þ þ bF W 2 t 2 z1 t3 ðz1 þ z4 Þ t 4 ðz4 þ z2 Þ t5 ðz2 þ z3 Þ t 6 z3 þh : þ þ þ þ 2 2 2 2 2
ð37Þ
The total inventory cost per unit time is simplified as:
TCMUðt 2 ; tÞ ¼
TCM T 2
kk bt ð1 pÞx bF t 2 k þ dk þ cR pk þ 2 ðð1 pÞx kÞ þ ðð1 pÞx kÞ tu 2tu tu 2 t ð1 pÞx t kðx þ pu þ p2 xÞ t 2 ðð1 pÞx kÞ : ðð1 pÞx kÞ þ u þh 2 2tu 2 x
¼ ck þ
ð38Þ
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Similar to Lemma 1, for any given t > 0, we can obtain three cases for the optimal t2 values that minimize TCMUðt 2 jt Þ. Solving @t@2 TCMUðt2 ; tÞ ¼ 0 and @t@ TCMUðt 2 ; tÞ ¼ 0, we have:
t2 ðtÞ ¼
tðt2 Þ ¼
hut bF k ; ðb þ hÞð1 pÞx
ð39Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2kk þ t 2 ðð1 pÞx kÞððb þ hÞð1 pÞxt 2 þ 2bF kÞ : huðu kðx þ pu þ p2 xÞ=xÞ
ð40Þ
Substituting Eq. (39) into Eq. (40), we can derive the solutions for t and t2:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k 2kðb þ hÞð1 pÞx bF kðð1 pÞx kÞ t ¼ ; hu hku þ ð1 pÞðbxu ðb þ hÞkðð1 þ p2 Þx þ puÞÞ
t2 ¼
hut bF k : ðb þ hÞð1 pÞx
ð41Þ
ð42Þ
Theorem 3 2
(i) If 2hkxu bF kðxu kðx þ pu þ p2 xÞÞ P 0, then TCMU is convex in t and t2. (a) If hu ðb þ hÞð1 pÞx < 0, the feasible optimal solution ðt ; t 2 Þ of TCMUðt 2 ; tÞ is ðt ; t 2 Þ. (b) If hu ðb þ hÞð1 pÞx P 0 and t P t 2 , the feasible optimal solution ðt ; t 2 Þ of TCMUðt 2 ; tÞ is ðt ; t 2 Þ. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2kk (ii) If 2hkxu bF kðxu kðx þ pu þ p2 xÞÞ < 0, then t ¼ huðuxkðxþp uþp2 xÞÞ=x and t 2 ¼ 0.
Proof. See Appendix A.
h
From Eqs. (41) and (42), the optimal production size and the optimal maximal backorder level can be derived, as follows:
Q m
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ku 2kðb þ hÞð1 pÞx bF kðð1 pÞx kÞ ; ¼ ut ¼ h hku þ ð1 pÞðbxu ðb þ hÞkðð1 þ p2 Þx þ puÞÞ
W m ¼ ðð1 pÞx kÞt 2 :
ð43Þ ð44Þ
t 2
If Model I results in infeasible optimal solutions, i.e., > t , we can directly extend Model II to a model in which the holding cost for defective items is not considered. According to Theorem 3, when the screening rate approaches the production rate (i.e., x ! u), the condition that TCMU is 2 convex in t and t2 is transformed into 2hku bF kðu kð1 þ p þ p2 ÞÞ > 0. The optimal production size is derived by:
limQ m x!u
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k 2kðb þ hÞð1 pÞu bF kðð1 pÞu kÞ ¼ : h hk þ ð1 pÞbu ð1 p3 Þðb þ hÞk
ð45Þ
Eq. (45) is identical to Eq. (19) in Cárdenas-Barrón [22]. Furthermore, setting the backorder costs (b and bF) to infinity, Eq. (45) is reduced to
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2kku : hðu ð1 þ p þ p2 ÞkÞ
ð46Þ
Eq. (46) is identical to Eq. (11) in Jamal et al. [10]. Since a shortage is not allowed, Eq. (46) can also be obtained from limx!u ðut Þ, where this t is shown in Theorem 3 (ii). Taking the limiting values of the optimal solutions in our models with respect to different parameters, our models can be reduced to different special cases. For instance, letting x ! u; p ! 0, and bF ! 0, we can derive Q and W for the optimal solutions of Model I, the classic EPQ with linear backorder costs. 4. Numerical example Example. k = 300 unit per year, u = 550 units per year, k = $50 per lot size, h = $50 per unit per year, bF = $1 per unit backorder, b = $10 per unit backorder per year, c = cR = $7 per unit, d = $0 per unit and p = 0.1.
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H.-M. Wee et al. / Applied Mathematics and Computation 233 (2014) 127–138 Table 1 The results of the example for Model I with different screening rates. Screening rate (x)
350
400
450
500
550
TCU⁄ Q⁄ W⁄ z1 z2 z3 z4 t1 t2 t3 t4 t5 t6
3109.82 37.98 1.27 16.9230 0.3587 2.0853 13.5792 0.004231 0.084615 0.015552 0.039464 0.006906 0.006951
3036.90 43.42 5.20 12.9917 1.3157 3.2892 11.3804 0.017322 0.086611 0.007673 0.029602 0.007894 0.010964
2969.04 49.83 9.47 9.0188 2.1576 4.4226 9.1038 0.031566 0.090188 0.000415 0.020134 0.009060 0.014742
2903.95 57.77 14.38 4.7938 2.9505 5.5766 6.6270 0.047938 0.095876 0.009166 0.010504 0.010504 0.018589
2839.37 68.25 20.44 0.0000 3.7622 6.8646 3.7622 0.068121 0.104802 0.019293 0.000000 0.012410 0.022882
Table 2 The results of the example for Model II with different screening rates. Screening rate (x)
350
400
450
500
550
TCU 0 Q 0 W 0 z0 1 z0 2 z0 3 z0 4 s1 s2 s3 s4 s5 s6
3109.82 37.98 1.27 0.3587 8.3692 2.0853 13.8125 0.004231 0.069063 0.015552 0.023912 0.006906 0.006951
3036.90 43.42 5.20 1.3157 8.7716 3.2892 11.8408 0.017322 0.078938 0.007673 0.021929 0.007894 0.010964
2969.04 49.83 9.47 2.1576 9.2468 4.4226 9.0602 0.031566 0.090602 0.000415 0.020548 0.009060 0.014742
2903.95 57.77 14.38 2.9505 9.8350 5.5766 5.2521 0.047938 0.105042 0.009166 0.019670 0.010504 0.018589
2839.37 68.25 20.44 3.7622 10.6113 6.8646 0.0000 0.068121 0.124095 0.019293 0.019293 0.012410 0.022882
k 300 Since x is constrained by (1 p)x k > 0, we use the values of x > 1p ¼ 10:1 ¼ 333:33 in this example. Since 2
2hku bF kðu kÞ ¼ 2:675 106 > 0, we have t 2 > 0. In Table 1, if t ð¼ t 2 þ t 3 Þ < t 2 , i.e., t 3 < 0, the derived optimal policy of Model I is infeasible. It is shown that x = 350 and x = 400 derive infeasible optimal polices for Model I. When t < t 2 , Model II is applied to find the feasible optimal policy. Referring to Fig. 1, when x ¼ u = 550, all good items filtered from production are immediately used to eliminate backorders during t2, resulting in z1 ¼ 0. Moreover, since the production and screening are synchronized (at the same rate), we can see that t 4 ¼ 0. In Table 2, we can see that x = 450, x = 500, and x = 550 derive infeasible optimal polices for Model II. This example shows that Q 0 ¼ Q ; W 0 ¼ W and TCU 0 ðs2 ; s Þ ¼ TCUðt 2 ; t Þ. However, Models I and II result in different production schedules, respectively. This implies that using production and backorder sizes as decision variables may obtain an infeasible optimal production schedule. Tables 1 and 2 show the optimal total inventory cost decreases as the screening rate increases. Therefore, for a scheduled production rate, a better policy is to ensure the screening rate is the same as the production rate.
5. Conclusion remarks This paper investigates EPQ models with two backorder costs and all defective products are reworked in the same production cycle. In contrast to many related studies, the proposed models allow the screening rate to be less than the production rate. To derive the closed-form optimal solutions, we use time intervals rather than ordering size and backorder size as decision variables. Our approach ensures the feasibility of the optimal policy. For the case with infeasible optimal intervals, we have suggested an alternate solution procedure to derive the feasible optimal policy. Taking the limiting parameter values for the optimal solutions, our models can be generalized to the classic EPQ models. A numerical example is given to illustrate the solution procedure. Further research can investigate the case in which production rate is not the same as the reworking rate and shortages are partially backordered.
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Appendix A
Proof for Lemma 1. For given t > 0, taking the first derivative with respective to t2, we have
d ðð1 pÞx kÞððb þ hÞð1 pÞxt 2 þ bF k hutÞ TCUðt2 jtÞ ¼ : dt2 ut
ðA1Þ
From Eq. (A1), we have
d
lim
t2 !0 dt 2
lim t 2 !t
TCUðt 2 jt Þ ¼
ðð1 pÞx kÞðbF k hutÞ ; ut
ðA2Þ
d ðð1 pÞx kÞððb þ hÞð1 pÞxt þ bF k hutÞ TCUðt 2 jt Þ ¼ : dt 2 ut
ðA3Þ
Since 2
d
2
dt2
TCUðt2 jtÞ ¼
ðb þ hÞð1 pÞðð1 pÞx kÞx >0 ut
ðA4Þ
TCUðt 2 jt Þ is convex in t2 for given t > 0. If t < bhFuk, then lim
d t 2 !t dt 2
d t 2 !0 dt 2
TCUðt2 jtÞ > lim
TCUðt 2 jtÞ > 0. Thus,
d dt 2
TCUðt 2 jtÞ remains positive but never reaches zero. Because
TCUðt2 jtÞ is convex, it reaches a minimum at the point of t2 = 0. If t ¼ bhFuk, then lim
d t 2 !0 dt2
d dt 2
d t 2 !t dt 2
TCUðt 2 jt Þ ¼ 0 and lim
TCUðt2 jtÞ > 0. So,
TCUðt2 jt Þ reaches zero at t2 = 0, and the optimal solution is attained at t 2 ¼ 0. d t 2 !t dt 2
To analyze the value of lim
TCUðt 2 jt Þ, we consider the following conditions.
(i) If hu ðb þ hÞð1 pÞx > 0: If
bF k hu
bF k < t < huðbþhÞð1pÞx ; lim
d t 2 !0 dt 2
d t 2 !t dt 2
TCUðt 2 jtÞ < 0, and lim
TCUðt 2 jt Þ > 0, then
d dt 2
TCUðt2 jtÞ reaches zero at the interior of
the interval ð0; tÞ. bF k t ¼ huðbþhÞð1pÞx ,
If
t2 ¼ t ¼
then
lim d t 2 !0 dt 2
TCUðt 2 jt Þ < 0
and
d t 2 !t dt 2
lim
TCUðt2 jtÞ ¼ 0.
Here,
d dt 2
TCUðt 2 jt Þ
reaches
zero
at
bF k . huðbþhÞð1pÞx
bF k , then lim If t > huðbþhÞð1pÞx
d t 2 !0 dt 2
d t 2 !t dt 2
TCUðt 2 jt Þ < lim
TCUðt2 jtÞ < 0. Because TCUðt2 jt Þ is convex, it reaches a minimum at
the interior of interval ðt; 1Þ, which means t 2 > t. (ii) If hu ðb þ hÞð1 pÞx ¼ 0: If
bF k hu
bF k < t < huðbþhÞð1pÞx ! 1; lim
d t 2 !0 dt 2
d t 2 !t dt 2
TCUðt 2 jt Þ < 0, and lim
TCUðt2 jtÞ > 0;
d dt 2
TCUðt2 jtÞ reaches zero at the interior of
the interval ð0; 1Þ. If hu ðb þ hÞð1 pÞx ¼ 0, Case C does not exist. (iii) If hu ðb þ hÞð1 pÞx < 0: If hu ðb þ hÞð1 pÞx < 0, then bF k hu
bF k huðbþhÞð1pÞx
< 0. So, the condition for Case B,
bF k hu
bF k < t < huðbþhÞð1pÞx , is replaced by
< t, which shows that the optimal t2 is at the interior of ð0; tÞ. If hu ðb þ hÞð1 pÞx < 0, Case C does not exist. h
Proof for Theorem 1. The second derivatives of TCUðt2 ; tÞ with respect to t2 and t are as follows.
@2 ðb þ hÞð1 pÞðð1 pÞx kÞx TCUðt2 ; tÞ ¼ > 0; ut @t22
ðA5Þ
@2 2kk þ t 2 ðð1 pÞx kÞððb þ hÞð1 pÞxt 2 þ 2bF kÞ TCUðt2 ; tÞ ¼ > 0: 2 @t ut3
ðA6Þ
Since ð1 pÞx k > 0, we can see that Eqs. (A5) and (A6) are greater than zero. Subsequently, we have:
@2 ðð1 pÞx kÞððb þ hÞð1 pÞxt 2 þ bF kÞ : TCUðt2 ; tÞ ¼ @t2 @t ut 2 To verify the convexity of TCUðt 2 ; tÞ, we calculate the following condition:
ðA7Þ
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H.-M. Wee et al. / Applied Mathematics and Computation 233 (2014) 127–138
@2 TCUðt2 ; tÞ @t22
!
! !2 2 @2 @2 kðð1 pÞx kÞð2kðb þ hÞð1 pÞx bF kðð1 pÞx kÞÞ TCUðt ; tÞ ; tÞ ¼ : TCUðt 2 2 @t2 @t @t 2 u2 t4 ðA8Þ
2 If 2kðb þ hÞð1 pÞx bF kðð1 pÞx kÞ > 0, then Eq. (A8) is greater than and t2. The optimal solution ðt ; t 2 Þ is derived by Eqs. (17) and (18). 2 2 In Lemma 1, we can see that if h t bF k > 0, i.e., ðh tÞ ðbF kÞ > 0, 2 2 ðh tÞ ðbF kÞ , we have
u
u
zero, which means that TCUðt2 ; tÞ is convex in t then t2 > 0. Substituting t by t in Eq. (17) in
u
2
2
2
ðhut Þ ðbF kÞ ¼
ðb þ hÞð1 pÞxkð2hku bF kðu kÞÞ : bxð1 pÞðu kÞ þ hkðu ð1 pÞxÞ
ðA9Þ
2
If 2hku bF kðu kÞ > 0, Eq. (A9) > 0. Subsequently, we obtain 2
2
2kðb þ hÞð1 pÞx bF kðð1 pÞx kÞ ¼ 2bkð1 pÞx þ 2hkð1 pÞx bF kðð1 pÞx kÞ k 2 2 ð1 pÞx bF kðð1 pÞx kÞ > 2bkð1 pÞx þ bF k 1 u ! ! k2 2 2 ð1 pÞx þ k ¼ 2bkð1 pÞx þ bF
u
x 2 > 0: ð*u P xÞ: ¼ 2bkð1 pÞx þ bF k2 1 ð1 pÞ
ðA10Þ
u
2
So, if 2hku bF kðu kÞ > 0, Eq. (A10) > 0 implies that TCUðt2 ; tÞ is convex in t > 0 and t2 > 0. The optimal solution ðt ; t2 Þ of TCUðt 2 ; tÞ is derived from Eqs. (17) and (18). 2 If 2hku bF kðu kÞ 6 0, then t 2 is attained at zero. Therefore, the optimal t can be derived from Eq. (16):
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2kk þ t2 ðð1 pÞx kÞððb þ hÞð1 pÞxt 2 þ 2bF kÞ 2kk ¼ : t 2 !0 huðu kÞ huðu kÞ
t ¼ lim tðt 2 Þ ¼ lim t 2 !0
ðA11Þ
2
Proof for Theorem 3. Similar to Theorem 1, if 2kðb þ hÞð1 pÞx bF kðð1 pÞx kÞ > 0, then TCMUðt 2 ; tÞ is convex in t and t2. In the denominator of Eq. (43), we have
hku þ ð1 pÞ bxu ðb þ hÞk ð1 þ p2 Þx þ pu > hku þ bku ð1 pÞðb þ hÞk 1 þ p2 x þ pu ð*ð1 pÞx > kÞ ¼ ðb þ hÞ ku ð1 pÞk ð1 þ p2 Þx þ pu ¼ ðb þ hÞ ku ð1 pÞkpu ð1 pÞ 1 þ p2 kx ¼ ðb þ hÞ 1 p þ p2 ku 1 p þ p2 p3 kx P ðb þ hÞ 1 p þ p2 kx 1 p þ p2 p3 kx ð*u P xÞ ¼ ðb þ hÞkxp3 > 0:
ðA12Þ
Therefore, Eqs. (43) and (44) are real solutions for t and t2, respectively. 2 2 From Lemma 1, we see that, if hut bF k > 0, i.e., ðhutÞ ðbF kÞ > 0, then t2 > 0. Substituting t by t in Eq. (43) in 2 2 ðhutÞ ðbF kÞ , we have: 2
2
2
ðhut Þ ðbF kÞ ¼
ðb þ hÞð1 pÞð2hkxu bF kðxu kðx þ pu þ p2 xÞÞÞ : hku þ ð1 pÞðbxu ðb þ hÞkðð1 þ p2 Þx þ puÞÞ
ðA13Þ
2
If 2hkxu bF kðxu kðx þ pu þ p2 xÞÞ > 0, then Eq. (A13) > 0. Subsequently, we can obtain the following: 2
2
2kðb þ hÞð1 pÞx bF kðð1 pÞx kÞ ¼ 2bkð1 pÞx þ 2hkð1 pÞx bF kðð1 pÞx kÞ 2
2
> 2bkð1 pÞx þ bF kðxu kðx þ pu þ p2 xÞÞð1 pÞ=u bF kðð1 pÞx kÞ 2
¼ 2bkð1 pÞx þ bF k2 ðu ð1 pÞðx þ p2 x þ puÞÞ=u 2
P 2bkð1 pÞx þ bF k2 ðu ð1 pÞðu þ p2 u þ puÞÞ=u ð*u P xÞ 2
2
¼ 2bkð1 pÞx þ bF k2 ðu ð1 p3 ÞuÞ=u ¼ 2bkð1 pÞx þ bF k2 p3 > 0: 2 bF kðx
2
ðA14Þ
Therefore, if 2hkxu u kðx þ pu þ p xÞÞ > 0; TCMUðt2 ; tÞ is convex in t > 0 and t2 > 0. The optimal solution ðt ; t 2 Þ of TCMUðt 2 ; tÞ is derived from Eqs. (43) and (44).
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H.-M. Wee et al. / Applied Mathematics and Computation 233 (2014) 127–138 2
If 2hkxu bF kðxu kðx þ pu þ p2 xÞÞ 6 0; t2 is attained at zero. Therefore, the optimal t can be derived from Eq. (42), as follows:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2kk þ t2 ðð1 pÞx kÞððb þ hÞð1 pÞxt2 þ 2bF kÞ 2kk ¼ : t2 !0 huðu kðx þ pu þ p2 xÞ=xÞ huðux kðx þ pu þ p2 xÞÞ=x
t ¼ lim tðt2 Þ ¼ lim t 2 !0
ðA15Þ
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Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
An economic production quantity model with non-synchronized screening and rework Hui-Ming Wee a,⇑, Wan-Tsu Wang a,b, Tsai-Chi Kuo a, Yung-Lung Cheng a,c, Yen-Deng Huang a a
Department of Industrial & Systems Engineering, Chung Yuan Christian University, Chungli 32023, Taiwan, ROC Department of Marketing and Distribution Management, Chien Hsin University of Science and Technology, Jungli 32097, Taiwan, ROC c Department of Industrial Management, Chien Hsin University of Science and Technology, Jungli 32097, Taiwan, ROC b
a r t i c l e
i n f o
Keywords: Economic production quantity Rework Backorder Non-synchronized screening
a b s t r a c t This study investigates an economic production quantity model for imperfect quality items with non-synchronized screening and rework. With time intervals as decision variables, our approach establishes a feasible optimal policy for the inventory system. We presented a solution procedure to derive the optimal policy, and generalized the proposed model to the classic EPQ model by taking the limiting parameter values for the optimal solution. A numerical example is provided to illustrate the theory and its application. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction In a classic economic order/production quantity (EOQ/EPQ) model, all items purchased/produced are assumed 100% good quality. Since defective items often exist in order/production lots, 100% perfect quality may not be practical in real-life situations. Salameh and Jaber [1] developed an extended EOQ model for a random fraction of imperfect quality items. The poor quality items are sold at the end of the 100% screening process. Maddah and Jaber [2] revisited the model of Salameh and Jaber [1] and rectified the optimal expected annual profit using the Renewal Reward Theorem. Other studies related to the model of Salameh and Jaber [1] include Papachristos and Konstantaras [3], Wee et al. [4], Eroglu and Ozdemir [5] and Yassine et al. [6]. The effect of defective items on the production lot sizing policy has also received considerable attention by many researchers. Rosenblatt and Lee [7] proposed an EPQ model to deal with an imperfect production process system. They assumed that defective items could be reworked instantaneously; their study showed that the presence of defective products motivates a smaller lot size. Hayek and Salameh [8] proposed an EPQ model in which all imperfect quality products were reworked and shortages were allowed. Chiu et al. [9] presented a finite production model that considered rework and backorders under a service level constraint. Jamal et al. [10] developed an EPQ model in which all defective products are reworked to fix the defects during the same production cycle. Related to this research were the papers by Sarker et al. [11] and Cárdenas-Barrón [12,13]. Taleizadeh et al. [14] introduced a multi-product single-machine production system with a stochastic scrapped production rate, partial backorders, and a service level constraint. Chiu et al. [15] provided an optimal inventory replenishment policy for the EPQ with rework and multiple shipments. Recently, Wee and Widyadana [16] considered a production system for deteriorating items with stochastic preventive maintenance time and rework. Taleizadeh et al. [17] developed an EPQ inventory model for multi products in a single machine with rework and interruption in process. In our literature review, there are very few EOQ/EPQ models considering two backorder costs (see for example, Sphicas [18], Omar et al. [19], Cárdenas-Barrón [20], Chung and Cárdenas-Barrón [21]). Cárdenas-Barrón [22] and Wee et al. [23] ⇑ Corresponding author. E-mail address: [email protected] (H.-M. Wee). http://dx.doi.org/10.1016/j.amc.2014.01.150 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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extended the model of Jamal et al. [10] to an EPQ model with planned backorders (linear and fixed costs). Most EPQ models with rework assumed that the production process implicitly includes an inspection/screening procedure, or that the production and screening procedures occur at the same rate. In considering two backorder costs, our proposed models allow the screening rate to be less than the production rate. If the screening rate is less than the production rate, the screening rate determines the rate to fulfill backorders and demand. To derive closed-form solutions, this study uses time intervals rather than production and backorder sizes as decision variables. The optimal production lot size and optimal backorder quantity are derived from the optimal intervals. By analyzing three conditions for the optimal inventory policy, our approach distinguishes between feasible and infeasible optimal policies. If the screening rate is less than the production rate, we have found that the optimal production time is not always greater than the optimal time to eliminate backorders. We provide an alternate model for which infeasible optimal intervals are derived. Moreover, we provide a solution procedure to derive the feasible optimal policy. We show that special cases of EPQ models can be derived by using the parameter limiting values for the proposed model. The remainder of this paper is organized as follows. Model formulation and notation are presented in Section 2. Special case models are developed in Section 3. In Section 4, we use a numerical example to illustrate our solution procedure. Section 5 concludes the paper. 2. Model formulation The following notation is used for this study: T, S Q, Q0 W, W0
u k x d c cR k h b bF p
cycle lengths for Models I and II, respectively production lot sizes in units per cycle for Models I and II, respectively maximum backorder levels in units for Models I and II, respectively production rate in units per unit time demand rate in units per unit time screening rate in units per unit time, x 6 u screening cost per unit production cost per unit rework cost per unit setup cost per production cycle holding cost per unit per unit time backorder cost per unit per unit time (linear backorder cost) backorder cost per unit (fixed backorder cost) percentage of defective items in Q; p is regarded as an expected defective percentage for items during the production; 0 6 p < 1.
In addition, we make the following assumptions: (1) (2) (3) (4) (5) (6)
The replenishment rate is dependent on the screening rate. The demand rate is known and constant. All items are screened 100% with a known screening rate. Defective items are proportional to the production lot. Screening and demand proceed simultaneously. The screening rate is less than or equal to the production rate. Shortages are completely backordered. Production and rework process occur at the same speed. All defective items are reworked to remove the defects.
2.1. Model I: the production time is greater than or equal to the time to eliminate backorders Fig. 1 depicts the production-inventory model with rework (during the same cycle) and backorders. We assume the production process contains p fraction of defective items. The items are screened to separate the perfect and defective items. At the beginning of interval t2, production and screening start simultaneously. The production interval is t2 + t3. If the production rate is greater than the screen rate, the interval t4 will be greater than zero. The rate of perfect items that are screened during t2 + t3 + t4 is 1 p. During the interval t2, with a known screening rate x, the fraction of the perfect items are used to meet demand and eliminate backorders, at a rate of (1 p)x k; the remaining perfect items are used to increase inventory, at a rate of u x. The defective inventory increases at a rate of px. To eliminate backorders and achieve a positive inventory, we assume that (1 p)x k > 0. The screening process terminates at the end of interval (t2 + t3 + t4) and the defective items of pu(t2 + t3) are reworked immediately. Referring to Fig. 1, the interval t2 builds inventory and eliminates backorders/shortages simultaneously: t 2 ¼ W=ðð1 pÞ x kÞ ¼ z1 =ðu xÞ. We have:
W ¼ ðð1 pÞx kÞt 2 ;
ð1Þ
H.-M. Wee et al. / Applied Mathematics and Computation 233 (2014) 127–138
129
z4 z3 z2 z1
W
Fig. 1. Inventory of good items for EPQ model I.
z1 ¼ ðu xÞt2 :
ð2Þ
The shortage interval without production is
t1 ¼ W=k ¼ ðð1 pÞx kÞt 2 =k:
ð3Þ
Since the interval t2 + t3 + t4 is consumed to screen production units per cycle, u(t2 + t3), we have xðt 2 þ t3 þ t4 Þ ¼ uðt2 þ t 3 Þ. It results in:
t4 ¼ ðu xÞðt 2 þ t 3 Þ=x:
ð4Þ
From Eq. (4), we can see that t4 > 0 if u > x. Referring to Fig. 2, the interval t5 is consumed to rework pu(t2 + t3) defective units with a rate of u. We now have:
t5 ¼ puðt 2 þ t 3 Þ=u ¼ pðt2 þ t3 Þ:
ð5Þ
Subsequently, referring to Fig. 1, we can derive the following expressions:
z4 ¼ z1 þ ðu px kÞt 3 ¼ ðu xÞt2 þ ðu px kÞt 3 ;
ð6Þ
z2 ¼ z4 ðpx þ kÞt4 ¼ ðð1 pÞx kÞðuðt 2 þ t 3 Þ xt 2 Þ=x;
ð7Þ
z3 ¼ z2 þ ðu kÞt 5 ¼ pðu kÞðt 2 þ t 3 Þ þ ðð1 pÞx kÞðuðt2 þ t3 Þ xt 2 Þ=x;
ð8Þ
t6 ¼ z3 =k ¼ pðu kÞðt 2 þ t 3 Þ=k þ ðð1 pÞx kÞðuðt2 þ t 3 Þ xt2 Þ=ðxkÞ:
ð9Þ
The total inventory cost per cycle, TC, is composed of the production cost, the screening cost, the rework cost, the backorder cost, and the holding cost. We have
Fig. 2. Inventory of defective items for EPQ model I.
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bðt1 þ t 2 ÞW TC ¼ ½cuðt2 þ t3 Þ þ k þ duðt2 þ t 3 Þ þ cR puðt2 þ t 3 Þ þ þ bF W 2 t 2 z1 t 3 ðz1 þ z4 Þ t4 ðz4 þ z2 Þ t 5 ðz2 þ z3 Þ t6 z3 ðt 2 þ t 3 þ t 4 þ t5 Þpuðt 2 þ t3 Þ þh þh : þ þ þ þ 2 2 2 2 2 2
ð10Þ
From Eqs. (3)–(5) and (9), the total cycle time can be rewritten as:
T ¼ t1 þ t2 þ t3 þ t4 þ t5 þ t6 ¼
ðt2 þ t3 Þu : k
ð11Þ
For ease of notation, we use t2 and t (=t 2 þ t 3 ) as decision variables in the total cost function. The total inventory cost per unit time is written as:
TCUðt 2 ; tÞ ¼
TC T 2
kk bt ð1 pÞx bF t 2 k þ dk þ cR pk þ 2 ðð1 pÞx kÞ þ ðð1 pÞx kÞ tu 2t u tu 2 t ð1 pÞx t ðð1 pÞx kÞ þ ðu kÞ t 2 ðð1 pÞx kÞ : þh 2 2t u 2
¼ ck þ
ð12Þ
The objective is to minimize TCUðt2 ; tÞ subject to t P t2 P 0. In Eq. (12), one can see that c, cR and d do not affect the optimal solutions. Lemma 1. For any given t > 0, there are the following cases to minimize TCUðt 2 jtÞ: Case A. For a given t 6 bhFuk ; t2 ¼ 0. Case B. If hu ðb þ hÞð1 pÞx P 0, for a given If hu ðb þ hÞð1 pÞx < 0, for a given
bF k hu bF k hu
bF k < t 6 huðbþhÞð1pÞx ; 0 < t 2 6 t.
< t; 0 < t2 < t.
bF k ; t 2 > t. Case C. If hu ðb þ hÞð1 pÞx > 0, for a given t > huðbþhÞð1pÞx
Proof. See Appendix A.
h
Lemma 1 shows that the optimal time to eliminate backorders, t2 , is dependent on the given production time, t. The interval t2 = 0 implies that the shortage is not allowed. If production time is greater than shortage. However, if production time is greater than
bF k , huðbþhÞð1pÞx
bF k , hu
the optimal policy is to allow for
the optimal time to eliminate backorders is greater than
hu , then the production time, which is an infeasible optimal policy for Model I. In Lemma 1, if x > ð1pÞðbþhÞ
hu ðb þ hÞð1 pÞx < 0, which means 0 6 t2 < t. In other words, if
hu ð1pÞðbþhÞ
< x 6 u, Model I always attains a feasible
optimal policy, i.e., Lemma 1 Case C does not exist. Consider the first-order-derivative condition for TCUðt 2 ; tÞ. Taking the first derivatives of TCUðt 2 ; tÞ with respect to t2 and t, we have:
@ ðð1 pÞx kÞððb þ hÞð1 pÞxt2 þ bF k hutÞ TCUðt2 ; tÞ ¼ ; @t2 ut
ð13Þ
@ 2kk t 2 ðð1 pÞx kÞððb þ hÞð1 pÞxt 2 þ 2bF kÞ þ hut2 ðu kÞ TCUðt 2 ; tÞ ¼ ; @t 2u t 2
ð14Þ
Letting
@ @ TCUðt 2 ; tÞ ¼ 0 and TCUðt2 ; tÞ ¼ 0, we have: @t 2 @t
t2 ðtÞ ¼
hut bF k ; ðb þ hÞð1 pÞx
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2kk þ t 2 ðð1 pÞx kÞððb þ hÞð1 pÞxt 2 þ 2bF kÞ tðt2 Þ ¼ : huðu kÞ
ð15Þ
ð16Þ
Substituting Eq. (15) into Eq. (16), we can derive the solutions for t and t2, as follows:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k 2kðb þ hÞð1 pÞx bF kðð1 pÞx kÞ ; t ¼ hu bxð1 pÞðu kÞ þ hkðu ð1 pÞxÞ
ð17Þ
H.-M. Wee et al. / Applied Mathematics and Computation 233 (2014) 127–138
t2 ¼
hut bF k : ðb þ hÞð1 pÞx
131
ð18Þ
In the denominator of Eq. (17), we see that bxð1 pÞðu kÞ þ hkðu ð1 pÞxÞ is greater than 0. If TCUðt 2 ; tÞ is convex in t and t2, Eqs. (17) and (18) are the optimal solutions to minimize TCUðt 2 ; tÞ. Theorem 1. 2
(i) If 2hku bF kðu kÞ > 0, then TCU is convex in t and t2. (a) If hu ðb þ hÞð1 pÞx < 0, the feasible optimal solution ðt ; t2 Þ of TCUðt2 ; tÞ is ðt ; t 2 Þ. (b) If hu ðb þ hÞð1 pÞx P 0 and t P t 2 , the feasible optimal solution ðt ; t 2 Þ of TCUðt 2 ; tÞ is ðt ; t 2 Þ. qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 (ii) If 2hku bF kðu kÞ 6 0, then t ¼ hu2kk and t 2 ¼ 0. ðukÞ
Proof. See Appendix A. h The optimal total cost per unit time can be derived by substituting ðt ; t2 Þ for ðt; t 2 Þ in Eq. (12). From t and t 2 , the optimal production size and maximal backorder level can be derived:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uk 2kðb þ hÞð1 pÞx b2F kðð1 pÞx kÞ Q ¼ ut ¼ ; h bxð1 pÞðu kÞ þ hkðu ð1 pÞxÞ
ð19Þ
W ¼ ðð1 pÞx kÞt2 :
ð20Þ
If p ! 0 and x ! u, Eqs. (19) and (20) are reduced to the optimal solutions for the classic EPQ model with linear and fixed backorder costs. Since p ! 0 and x ! u, we have hu ðb þ hÞð1 pÞx ¼ bu < 0. Therefore, for a given t > 0, this leads to two cases: t 2 ¼ 0 and 0 < t2 < t in Lemma 1. Subsequently, Theorem 1 reduces to the model without reworking defective items. Corollary 1. For the classic EPQ model with linear and fixed backorder costs: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 2kkðbþhÞ=ð1k=uÞb2F k2 F kÞð1k=uÞ (i) If 2hku bF kðu kÞ > 0; Q ¼ . and W ¼ ðhQ bhþb hb qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2kk (ii) If 2hku bF kðu kÞ 6 0; Q ¼ hð1k= uÞ and W ¼ 0. 2
If 2hku bF kðu kÞ ¼ 0; Q and W in Corollary 1 (i) are reduced to identical to Theorem 4 in Chung and Cárdenas-Barrón [21] if Q 2 in bF k In Lemma 1 Case C, we can see that, if t > huðbþhÞð1pÞx , then t 2 >
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2kk hð1k=uÞ
and 0, respectively. Therefore, Corollary 1 is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi 2KDðhþpÞ=ð1D=PÞp2 D . hp
their equation is amended by
t , i.e., Model I derives infeasible solutions with t < t 2.
Fig. 1 cannot represent the correct inventory behavior for Lemma 1 Case C. When the optimal production time (t ) is less than the optimal time to eliminate backorders (t 2 ), in Section 2.2, we show how to apply Model II to derive the feasible optimal policy. 2.2. Model II: the production time is less than the time to eliminate backorders Fig. 3 depicts the inventory behavior when the production run time is less than the time to eliminate backorders. The inventory behavior for the defective items is the same as that shown in Fig. 2. Similar to the analysis process in Section 2.1, the following equations can be expressed in terms of s2 and s3.
W 0 ¼ ðð1 pÞx kÞðs2 þ s3 Þ;
ð21Þ
s1 ¼ ðð1 pÞx kÞðs2 þ s3 Þ=k;
ð22Þ
s4 ¼ s2 ðu=x 1Þ s3 ;
ð23Þ
s5 ¼ pus2 =u ¼ ps2 ;
ð24Þ
s6 ¼ s2 ðx kÞðu ð1 pÞxÞ=ðxkÞ ðð1 pÞx kÞs3 =k;
ð25Þ
z01 ¼ ðð1 pÞx kÞðus2 xðs2 þ s3 ÞÞ=x;
ð26Þ
z02 ¼ us2 xðs2 þ s3 Þ;
ð27Þ
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z’4 z’3 z’2 z’1
-W’
Fig. 3. Inventory of good items for EPQ model II.
z03 ¼ s2 ðx kÞðu ð1 pÞxÞ=x s3 ðð1 pÞx kÞ;
ð28Þ
z04 ¼ ðu xÞs2 :
ð29Þ 0
The total inventory cost per cycle, TC , is shown as:
bðs1 þ s2 þ s3 ÞW 0 TC 0 ¼ ½cus2 þ k þ dus2 þ cR pus2 þ þ bF W 0 2 0 0 0 0 0 s4 z2 þ z1 s5 z01 þ z03 s2 z4 s3 z4 þ z2 s6 z0 ðs2 þ s3 þ s4 þ s5 Þpus2 : þh þ þ þ þ 3 þh 2 2 2 2 2 2
ð30Þ
From Eqs. (22)–(25), the total cycle time, S, is rewritten as
S ¼ s1 þ s2 þ s3 þ s4 þ s5 þ s6 ¼
s2 u : k
ð31Þ
For ease of notation, we use s2 and s (=s2 þ s3 ) as decision variables. The total inventory cost per unit time is written as:
TCU 0 ðs2 ; sÞ ¼
TC 0 S 2
kk bs ð1 pÞx bF sk þ dk þ cR pk þ ðð1 pÞx kÞ þ ðð1 pÞx kÞ s2 u 2s2 u s2 u 2 s ð1 pÞx s2 þh ðð1 pÞx kÞ þ ðu kÞ sðð1 pÞx kÞ : 2s2 u 2
¼ ck þ
ð32Þ
The objective is to minimize TCU 0 ðs2 ; sÞ, subject to s P s2 P 0. We can see that TCU 0 ðs2 ; sÞ is the same as TCUðt2 ; tÞ if t2 and t are replaced by s and s2, respectively. Similar to Lemma 1, we can now develop Lemma 2. Lemma 2. For any given s2 > 0, there are the following cases to minimize TCU 0 ðsjs2 Þ:
Case A. For a given s2 6 bhFuk ; s = 0. Case B. If hu ðb þ hÞð1 pÞx P 0, for a given If hu ðb þ hÞð1 pÞx < 0, for a given
bF k bF k < s2 6 huðbþhÞð1pÞx , hu bF k < s ; 0 < s < s . 2 2 hu
then 0 < s 6 s2 ;
bF k ; s > s2 . Case C. If hu ðb þ hÞð1 pÞx > 0, for a given s2 > huðbþhÞð1pÞx
Proof. Similar to Lemma 1. h @ The solutions for @s@2 TCU 0 ðs2 ; sÞ ¼ 0 and @s TCU 0 ðs2 ; sÞ ¼ 0 can be solved by substituting s and s2 for t2 and t in Eqs. (17) and (18), respectively:
H.-M. Wee et al. / Applied Mathematics and Computation 233 (2014) 127–138
s2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k 2kðb þ hÞð1 pÞx bF kðð1 pÞx kÞ ¼ ; hu bxð1 pÞðu kÞ þ hkðu ð1 pÞxÞ
s ¼
133
ð33Þ
hus2 bF k : ðb þ hÞð1 pÞx
ð34Þ
It is shown that s 2 ¼ t and s ¼ t 2 .
Theorem 2. 2
0 (i) If 2hku bF kðu kÞ > 0, then TCU0 is convex in s2 and s. If s P s 2 , the feasible optimal solution ðs2 ; s Þ of TCU ðs2 ; sÞ is ðs2 ; s Þ. qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 and s ¼ 0. (ii) If 2hku bF kðu kÞ 6 0, then s2 ¼ hu2kk ðukÞ
Proof. Similar to Theorem 1. h From s2 and s , the optimal production size and the optimal maximal backorder level can be derived as follows: 0
Q ¼u
s2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uk 2kðb þ hÞð1 pÞx b2F kðð1 pÞx kÞ ; ¼ h bxð1 pÞðu kÞ þ hkðu ð1 pÞxÞ
ð35Þ
W 0 ¼ ðð1 pÞx kÞs : 0
ð36Þ
0
0
ðs2 ; s Þ
TCUðt2 ;
We can show that Q ¼ Q ; W ¼ W and TCU ¼ t Þ. However, Model I (Fig. 1) and Model II (Fig. 2) represent different production schedules and inventory levels, respectively. Therefore, using production size and backorder level as decision variables may not reveal the infeasible optimal policy. Lemma 1 Case A and Lemma 2 Case A have the same optimal policy without shortage. In Lemma 2 Case B, we have that if bF k hu
bF k < s2 < huðbþhÞð1pÞx , then 0 < s < s2 , i.e., s3 < 0. This shows that the optimal time to eliminate backorders is less than the
optimal production time. If s < s2 , Model II cannot represent the correct production schedule. So, if s < s2 , the optimal policy is derived by Model I rather than Model II. We provide a solution procedure to determine the optimal policy. 2
Step 1: Start with Model I. Calculate the value of 2hku bF kðu kÞ. qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2kk (1) If 2hku bF kðu kÞ 6 0, then the optimal solution ðt ; t 2 Þ for TCUðt2 ; tÞ is ;0 . huðukÞ 2
(2) If 2hku bF kðu kÞ > 0 with t P t 2 , the optimal solution ðt ; t 2 Þ for TCUðt 2 ; tÞ is ðt ; t 2 Þ, as shown in Eqs. (17) and (18). The optimal total inventory cost per unit time is TCUðt 2 ; t Þ. 2 (3) If 2hku bF kðu kÞ > 0 with t < t 2 , go to Step 2. Step 2: Apply Model II to find the optimal policy. The optimal solution ðs2 ; s Þ for TCU 0 ðs2 ; sÞ is ðs 2 ; s Þ is shown in Eqs. (33)
and (34). The optimal total inventory cost per unit time is TCU 0 ðs2 ; s Þ. 3. Special cases Here we modify Model I in Section 2.1 by not considering the holding cost of the defective items. In other words, the total h i inventory cost does not include the last term of TC in Eq. (10), i.e., h ðt2 þt3 þt4 þt25 Þpuðt2 þt3 Þ . We therefore have the total inventory cost per cycle:
bðt1 þ t2 ÞW TCM ¼ ½cuðt 2 þ t 3 Þ þ k þ duðt2 þ t3 Þ þ cR puðt2 þ t3 Þ þ þ bF W 2 t 2 z1 t3 ðz1 þ z4 Þ t 4 ðz4 þ z2 Þ t5 ðz2 þ z3 Þ t 6 z3 þh : þ þ þ þ 2 2 2 2 2
ð37Þ
The total inventory cost per unit time is simplified as:
TCMUðt 2 ; tÞ ¼
TCM T 2
kk bt ð1 pÞx bF t 2 k þ dk þ cR pk þ 2 ðð1 pÞx kÞ þ ðð1 pÞx kÞ tu 2tu tu 2 t ð1 pÞx t kðx þ pu þ p2 xÞ t 2 ðð1 pÞx kÞ : ðð1 pÞx kÞ þ u þh 2 2tu 2 x
¼ ck þ
ð38Þ
134
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Similar to Lemma 1, for any given t > 0, we can obtain three cases for the optimal t2 values that minimize TCMUðt 2 jt Þ. Solving @t@2 TCMUðt2 ; tÞ ¼ 0 and @t@ TCMUðt 2 ; tÞ ¼ 0, we have:
t2 ðtÞ ¼
tðt2 Þ ¼
hut bF k ; ðb þ hÞð1 pÞx
ð39Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2kk þ t 2 ðð1 pÞx kÞððb þ hÞð1 pÞxt 2 þ 2bF kÞ : huðu kðx þ pu þ p2 xÞ=xÞ
ð40Þ
Substituting Eq. (39) into Eq. (40), we can derive the solutions for t and t2:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k 2kðb þ hÞð1 pÞx bF kðð1 pÞx kÞ t ¼ ; hu hku þ ð1 pÞðbxu ðb þ hÞkðð1 þ p2 Þx þ puÞÞ
t2 ¼
hut bF k : ðb þ hÞð1 pÞx
ð41Þ
ð42Þ
Theorem 3 2
(i) If 2hkxu bF kðxu kðx þ pu þ p2 xÞÞ P 0, then TCMU is convex in t and t2. (a) If hu ðb þ hÞð1 pÞx < 0, the feasible optimal solution ðt ; t 2 Þ of TCMUðt 2 ; tÞ is ðt ; t 2 Þ. (b) If hu ðb þ hÞð1 pÞx P 0 and t P t 2 , the feasible optimal solution ðt ; t 2 Þ of TCMUðt 2 ; tÞ is ðt ; t 2 Þ. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2kk (ii) If 2hkxu bF kðxu kðx þ pu þ p2 xÞÞ < 0, then t ¼ huðuxkðxþp uþp2 xÞÞ=x and t 2 ¼ 0.
Proof. See Appendix A.
h
From Eqs. (41) and (42), the optimal production size and the optimal maximal backorder level can be derived, as follows:
Q m
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ku 2kðb þ hÞð1 pÞx bF kðð1 pÞx kÞ ; ¼ ut ¼ h hku þ ð1 pÞðbxu ðb þ hÞkðð1 þ p2 Þx þ puÞÞ
W m ¼ ðð1 pÞx kÞt 2 :
ð43Þ ð44Þ
t 2
If Model I results in infeasible optimal solutions, i.e., > t , we can directly extend Model II to a model in which the holding cost for defective items is not considered. According to Theorem 3, when the screening rate approaches the production rate (i.e., x ! u), the condition that TCMU is 2 convex in t and t2 is transformed into 2hku bF kðu kð1 þ p þ p2 ÞÞ > 0. The optimal production size is derived by:
limQ m x!u
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k 2kðb þ hÞð1 pÞu bF kðð1 pÞu kÞ ¼ : h hk þ ð1 pÞbu ð1 p3 Þðb þ hÞk
ð45Þ
Eq. (45) is identical to Eq. (19) in Cárdenas-Barrón [22]. Furthermore, setting the backorder costs (b and bF) to infinity, Eq. (45) is reduced to
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2kku : hðu ð1 þ p þ p2 ÞkÞ
ð46Þ
Eq. (46) is identical to Eq. (11) in Jamal et al. [10]. Since a shortage is not allowed, Eq. (46) can also be obtained from limx!u ðut Þ, where this t is shown in Theorem 3 (ii). Taking the limiting values of the optimal solutions in our models with respect to different parameters, our models can be reduced to different special cases. For instance, letting x ! u; p ! 0, and bF ! 0, we can derive Q and W for the optimal solutions of Model I, the classic EPQ with linear backorder costs. 4. Numerical example Example. k = 300 unit per year, u = 550 units per year, k = $50 per lot size, h = $50 per unit per year, bF = $1 per unit backorder, b = $10 per unit backorder per year, c = cR = $7 per unit, d = $0 per unit and p = 0.1.
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H.-M. Wee et al. / Applied Mathematics and Computation 233 (2014) 127–138 Table 1 The results of the example for Model I with different screening rates. Screening rate (x)
350
400
450
500
550
TCU⁄ Q⁄ W⁄ z1 z2 z3 z4 t1 t2 t3 t4 t5 t6
3109.82 37.98 1.27 16.9230 0.3587 2.0853 13.5792 0.004231 0.084615 0.015552 0.039464 0.006906 0.006951
3036.90 43.42 5.20 12.9917 1.3157 3.2892 11.3804 0.017322 0.086611 0.007673 0.029602 0.007894 0.010964
2969.04 49.83 9.47 9.0188 2.1576 4.4226 9.1038 0.031566 0.090188 0.000415 0.020134 0.009060 0.014742
2903.95 57.77 14.38 4.7938 2.9505 5.5766 6.6270 0.047938 0.095876 0.009166 0.010504 0.010504 0.018589
2839.37 68.25 20.44 0.0000 3.7622 6.8646 3.7622 0.068121 0.104802 0.019293 0.000000 0.012410 0.022882
Table 2 The results of the example for Model II with different screening rates. Screening rate (x)
350
400
450
500
550
TCU 0 Q 0 W 0 z0 1 z0 2 z0 3 z0 4 s1 s2 s3 s4 s5 s6
3109.82 37.98 1.27 0.3587 8.3692 2.0853 13.8125 0.004231 0.069063 0.015552 0.023912 0.006906 0.006951
3036.90 43.42 5.20 1.3157 8.7716 3.2892 11.8408 0.017322 0.078938 0.007673 0.021929 0.007894 0.010964
2969.04 49.83 9.47 2.1576 9.2468 4.4226 9.0602 0.031566 0.090602 0.000415 0.020548 0.009060 0.014742
2903.95 57.77 14.38 2.9505 9.8350 5.5766 5.2521 0.047938 0.105042 0.009166 0.019670 0.010504 0.018589
2839.37 68.25 20.44 3.7622 10.6113 6.8646 0.0000 0.068121 0.124095 0.019293 0.019293 0.012410 0.022882
k 300 Since x is constrained by (1 p)x k > 0, we use the values of x > 1p ¼ 10:1 ¼ 333:33 in this example. Since 2
2hku bF kðu kÞ ¼ 2:675 106 > 0, we have t 2 > 0. In Table 1, if t ð¼ t 2 þ t 3 Þ < t 2 , i.e., t 3 < 0, the derived optimal policy of Model I is infeasible. It is shown that x = 350 and x = 400 derive infeasible optimal polices for Model I. When t < t 2 , Model II is applied to find the feasible optimal policy. Referring to Fig. 1, when x ¼ u = 550, all good items filtered from production are immediately used to eliminate backorders during t2, resulting in z1 ¼ 0. Moreover, since the production and screening are synchronized (at the same rate), we can see that t 4 ¼ 0. In Table 2, we can see that x = 450, x = 500, and x = 550 derive infeasible optimal polices for Model II. This example shows that Q 0 ¼ Q ; W 0 ¼ W and TCU 0 ðs2 ; s Þ ¼ TCUðt 2 ; t Þ. However, Models I and II result in different production schedules, respectively. This implies that using production and backorder sizes as decision variables may obtain an infeasible optimal production schedule. Tables 1 and 2 show the optimal total inventory cost decreases as the screening rate increases. Therefore, for a scheduled production rate, a better policy is to ensure the screening rate is the same as the production rate.
5. Conclusion remarks This paper investigates EPQ models with two backorder costs and all defective products are reworked in the same production cycle. In contrast to many related studies, the proposed models allow the screening rate to be less than the production rate. To derive the closed-form optimal solutions, we use time intervals rather than ordering size and backorder size as decision variables. Our approach ensures the feasibility of the optimal policy. For the case with infeasible optimal intervals, we have suggested an alternate solution procedure to derive the feasible optimal policy. Taking the limiting parameter values for the optimal solutions, our models can be generalized to the classic EPQ models. A numerical example is given to illustrate the solution procedure. Further research can investigate the case in which production rate is not the same as the reworking rate and shortages are partially backordered.
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Appendix A
Proof for Lemma 1. For given t > 0, taking the first derivative with respective to t2, we have
d ðð1 pÞx kÞððb þ hÞð1 pÞxt 2 þ bF k hutÞ TCUðt2 jtÞ ¼ : dt2 ut
ðA1Þ
From Eq. (A1), we have
d
lim
t2 !0 dt 2
lim t 2 !t
TCUðt 2 jt Þ ¼
ðð1 pÞx kÞðbF k hutÞ ; ut
ðA2Þ
d ðð1 pÞx kÞððb þ hÞð1 pÞxt þ bF k hutÞ TCUðt 2 jt Þ ¼ : dt 2 ut
ðA3Þ
Since 2
d
2
dt2
TCUðt2 jtÞ ¼
ðb þ hÞð1 pÞðð1 pÞx kÞx >0 ut
ðA4Þ
TCUðt 2 jt Þ is convex in t2 for given t > 0. If t < bhFuk, then lim
d t 2 !t dt 2
d t 2 !0 dt 2
TCUðt2 jtÞ > lim
TCUðt 2 jtÞ > 0. Thus,
d dt 2
TCUðt 2 jtÞ remains positive but never reaches zero. Because
TCUðt2 jtÞ is convex, it reaches a minimum at the point of t2 = 0. If t ¼ bhFuk, then lim
d t 2 !0 dt2
d dt 2
d t 2 !t dt 2
TCUðt 2 jt Þ ¼ 0 and lim
TCUðt2 jtÞ > 0. So,
TCUðt2 jt Þ reaches zero at t2 = 0, and the optimal solution is attained at t 2 ¼ 0. d t 2 !t dt 2
To analyze the value of lim
TCUðt 2 jt Þ, we consider the following conditions.
(i) If hu ðb þ hÞð1 pÞx > 0: If
bF k hu
bF k < t < huðbþhÞð1pÞx ; lim
d t 2 !0 dt 2
d t 2 !t dt 2
TCUðt 2 jtÞ < 0, and lim
TCUðt 2 jt Þ > 0, then
d dt 2
TCUðt2 jtÞ reaches zero at the interior of
the interval ð0; tÞ. bF k t ¼ huðbþhÞð1pÞx ,
If
t2 ¼ t ¼
then
lim d t 2 !0 dt 2
TCUðt 2 jt Þ < 0
and
d t 2 !t dt 2
lim
TCUðt2 jtÞ ¼ 0.
Here,
d dt 2
TCUðt 2 jt Þ
reaches
zero
at
bF k . huðbþhÞð1pÞx
bF k , then lim If t > huðbþhÞð1pÞx
d t 2 !0 dt 2
d t 2 !t dt 2
TCUðt 2 jt Þ < lim
TCUðt2 jtÞ < 0. Because TCUðt2 jt Þ is convex, it reaches a minimum at
the interior of interval ðt; 1Þ, which means t 2 > t. (ii) If hu ðb þ hÞð1 pÞx ¼ 0: If
bF k hu
bF k < t < huðbþhÞð1pÞx ! 1; lim
d t 2 !0 dt 2
d t 2 !t dt 2
TCUðt 2 jt Þ < 0, and lim
TCUðt2 jtÞ > 0;
d dt 2
TCUðt2 jtÞ reaches zero at the interior of
the interval ð0; 1Þ. If hu ðb þ hÞð1 pÞx ¼ 0, Case C does not exist. (iii) If hu ðb þ hÞð1 pÞx < 0: If hu ðb þ hÞð1 pÞx < 0, then bF k hu
bF k huðbþhÞð1pÞx
< 0. So, the condition for Case B,
bF k hu
bF k < t < huðbþhÞð1pÞx , is replaced by
< t, which shows that the optimal t2 is at the interior of ð0; tÞ. If hu ðb þ hÞð1 pÞx < 0, Case C does not exist. h
Proof for Theorem 1. The second derivatives of TCUðt2 ; tÞ with respect to t2 and t are as follows.
@2 ðb þ hÞð1 pÞðð1 pÞx kÞx TCUðt2 ; tÞ ¼ > 0; ut @t22
ðA5Þ
@2 2kk þ t 2 ðð1 pÞx kÞððb þ hÞð1 pÞxt 2 þ 2bF kÞ TCUðt2 ; tÞ ¼ > 0: 2 @t ut3
ðA6Þ
Since ð1 pÞx k > 0, we can see that Eqs. (A5) and (A6) are greater than zero. Subsequently, we have:
@2 ðð1 pÞx kÞððb þ hÞð1 pÞxt 2 þ bF kÞ : TCUðt2 ; tÞ ¼ @t2 @t ut 2 To verify the convexity of TCUðt 2 ; tÞ, we calculate the following condition:
ðA7Þ
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H.-M. Wee et al. / Applied Mathematics and Computation 233 (2014) 127–138
@2 TCUðt2 ; tÞ @t22
!
! !2 2 @2 @2 kðð1 pÞx kÞð2kðb þ hÞð1 pÞx bF kðð1 pÞx kÞÞ TCUðt ; tÞ ; tÞ ¼ : TCUðt 2 2 @t2 @t @t 2 u2 t4 ðA8Þ
2 If 2kðb þ hÞð1 pÞx bF kðð1 pÞx kÞ > 0, then Eq. (A8) is greater than and t2. The optimal solution ðt ; t 2 Þ is derived by Eqs. (17) and (18). 2 2 In Lemma 1, we can see that if h t bF k > 0, i.e., ðh tÞ ðbF kÞ > 0, 2 2 ðh tÞ ðbF kÞ , we have
u
u
zero, which means that TCUðt2 ; tÞ is convex in t then t2 > 0. Substituting t by t in Eq. (17) in
u
2
2
2
ðhut Þ ðbF kÞ ¼
ðb þ hÞð1 pÞxkð2hku bF kðu kÞÞ : bxð1 pÞðu kÞ þ hkðu ð1 pÞxÞ
ðA9Þ
2
If 2hku bF kðu kÞ > 0, Eq. (A9) > 0. Subsequently, we obtain 2
2
2kðb þ hÞð1 pÞx bF kðð1 pÞx kÞ ¼ 2bkð1 pÞx þ 2hkð1 pÞx bF kðð1 pÞx kÞ k 2 2 ð1 pÞx bF kðð1 pÞx kÞ > 2bkð1 pÞx þ bF k 1 u ! ! k2 2 2 ð1 pÞx þ k ¼ 2bkð1 pÞx þ bF
u
x 2 > 0: ð*u P xÞ: ¼ 2bkð1 pÞx þ bF k2 1 ð1 pÞ
ðA10Þ
u
2
So, if 2hku bF kðu kÞ > 0, Eq. (A10) > 0 implies that TCUðt2 ; tÞ is convex in t > 0 and t2 > 0. The optimal solution ðt ; t2 Þ of TCUðt 2 ; tÞ is derived from Eqs. (17) and (18). 2 If 2hku bF kðu kÞ 6 0, then t 2 is attained at zero. Therefore, the optimal t can be derived from Eq. (16):
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2kk þ t2 ðð1 pÞx kÞððb þ hÞð1 pÞxt 2 þ 2bF kÞ 2kk ¼ : t 2 !0 huðu kÞ huðu kÞ
t ¼ lim tðt 2 Þ ¼ lim t 2 !0
ðA11Þ
2
Proof for Theorem 3. Similar to Theorem 1, if 2kðb þ hÞð1 pÞx bF kðð1 pÞx kÞ > 0, then TCMUðt 2 ; tÞ is convex in t and t2. In the denominator of Eq. (43), we have
hku þ ð1 pÞ bxu ðb þ hÞk ð1 þ p2 Þx þ pu > hku þ bku ð1 pÞðb þ hÞk 1 þ p2 x þ pu ð*ð1 pÞx > kÞ ¼ ðb þ hÞ ku ð1 pÞk ð1 þ p2 Þx þ pu ¼ ðb þ hÞ ku ð1 pÞkpu ð1 pÞ 1 þ p2 kx ¼ ðb þ hÞ 1 p þ p2 ku 1 p þ p2 p3 kx P ðb þ hÞ 1 p þ p2 kx 1 p þ p2 p3 kx ð*u P xÞ ¼ ðb þ hÞkxp3 > 0:
ðA12Þ
Therefore, Eqs. (43) and (44) are real solutions for t and t2, respectively. 2 2 From Lemma 1, we see that, if hut bF k > 0, i.e., ðhutÞ ðbF kÞ > 0, then t2 > 0. Substituting t by t in Eq. (43) in 2 2 ðhutÞ ðbF kÞ , we have: 2
2
2
ðhut Þ ðbF kÞ ¼
ðb þ hÞð1 pÞð2hkxu bF kðxu kðx þ pu þ p2 xÞÞÞ : hku þ ð1 pÞðbxu ðb þ hÞkðð1 þ p2 Þx þ puÞÞ
ðA13Þ
2
If 2hkxu bF kðxu kðx þ pu þ p2 xÞÞ > 0, then Eq. (A13) > 0. Subsequently, we can obtain the following: 2
2
2kðb þ hÞð1 pÞx bF kðð1 pÞx kÞ ¼ 2bkð1 pÞx þ 2hkð1 pÞx bF kðð1 pÞx kÞ 2
2
> 2bkð1 pÞx þ bF kðxu kðx þ pu þ p2 xÞÞð1 pÞ=u bF kðð1 pÞx kÞ 2
¼ 2bkð1 pÞx þ bF k2 ðu ð1 pÞðx þ p2 x þ puÞÞ=u 2
P 2bkð1 pÞx þ bF k2 ðu ð1 pÞðu þ p2 u þ puÞÞ=u ð*u P xÞ 2
2
¼ 2bkð1 pÞx þ bF k2 ðu ð1 p3 ÞuÞ=u ¼ 2bkð1 pÞx þ bF k2 p3 > 0: 2 bF kðx
2
ðA14Þ
Therefore, if 2hkxu u kðx þ pu þ p xÞÞ > 0; TCMUðt2 ; tÞ is convex in t > 0 and t2 > 0. The optimal solution ðt ; t 2 Þ of TCMUðt 2 ; tÞ is derived from Eqs. (43) and (44).
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H.-M. Wee et al. / Applied Mathematics and Computation 233 (2014) 127–138 2
If 2hkxu bF kðxu kðx þ pu þ p2 xÞÞ 6 0; t2 is attained at zero. Therefore, the optimal t can be derived from Eq. (42), as follows:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2kk þ t2 ðð1 pÞx kÞððb þ hÞð1 pÞxt2 þ 2bF kÞ 2kk ¼ : t2 !0 huðu kðx þ pu þ p2 xÞ=xÞ huðux kðx þ pu þ p2 xÞÞ=x
t ¼ lim tðt2 Þ ¼ lim t 2 !0
ðA15Þ
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