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Improved solution for inventory model with defective goods
Applied Mathematical Modelling 37 (2013) 5574–5579
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
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Improved solution for inventory model with defective goods Cheng-Tan Tung ⇑, Peter Shaohua Deng Department of Information Management, Central Police University, Taiwan
a r t i c l e
i n f o
Article history: Received 23 November 2011 Received in revised form 18 September 2012 Accepted 12 October 2012 Available online 30 October 2012 Keywords: Minimax distribution free procedure Analytical approach Crashable lead time Partial backorder
a b s t r a c t This paper is a consequence for a paper of Lin et al. [S.W. Lin, Y.W. Wou, P. Julian, Note on minimax distribution free procedure for integrated inventory model with defective goods and stochastic lead time demand, Appl. Math. Model. 35 (2011) 2087–2093]. We simplified their complicated solution procedure and then presented a revision to patch their negligence for the boundary minimums. Numerical examples are provided to demonstrate our findings. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Lin et al. [1] tried to improve the solution procedure of an integrated inventory model, with defective goods and stochastic lead time demand proposed by Ho [2], under the minimax distribution free procedure of Gallego and Moon [3]. From the first partial derivative system, Lin et al. [1] merged two equations into one equation to cancel the variable of safety factor, to express the results in the variable of shipping quantity. They found two upper bounds and one lower bound for the shipping quantity, and then used numerical evidence to reduce two upper bounds into one upper bound. Under this upper bound, and the lower bound, Lin et al. [1] proved that there cannot exist more than one positive solution for the shipping quantity. We must point out that the proof of existing more than one positive solution cannot guarantee the existence of the positive solution. We will find the condition to ensure the existence of the positive solution and then discuss the minimum solution on the boundary. Our improvement will provide a complete solution structure for the inventory model. 2. Notation and assumptions To be comparable with Ho [2] and Lin et al. [1], we used the same notation and assumptions as theirs in the following. Notation: D P Ab
Demand rate on the buyer (for non-defective goods) Production rate on the vendor Buyer’s ordering cost per order
⇑ Corresponding author. Address: 56 Su-Jen Rd., Ta-Kan Village, Kwei-San, Tao-Yuan County, Tao-Yuan, Taiwan. Tel.: +886 3 3282312 4491; fax: +886 3 3282312 4329. E-mail addresses: [email protected] (C.-T. Tung), [email protected] (P.S. Deng). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.10.022
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Av F hv hb1 hb2 w s
p p0 b k gðkÞ L Q q r m X xþ
Vendor’s set-up cost per set-up Transportation cost per delivery Vendor’s holding cost per item per unit time Buyer’s holding cost per non-defective item per unit time Buyer’s treatment cost (include holding cost) per defective item per unit time Vendor’s unit treatment cost (include warranty cost) of defective goods Buyer’s unit inspection cost Buyer’s unit shortage cost per unit short Buyer’s marginal profit per unit Fraction of the demand during the stock-out period will be backordered, b 2 ½0; 1 Defective rate in an order lot, k 2 ½0; 1Þ, a random variable The probability density function (p.d.f.) of k with finite mean M k and variance V k , where M k ¼ EðkÞ ¼
R1 0
kgðkÞdk
and V k ¼ Eðk2 Þ ðM k Þ2 Length of lead time for the buyer Order quantity (for non-defective goods) of the buyer (decision variable) Shipping quantity from vendor to buyer per shipment (decision variable) Reorder point of the buyer for non-defective goods (decision variable) The number of deliveries from vendor to buyer in one production cycle, a positive integer (decision variable) pffiffiffi The lead time demand which has a p.d.f. f with finite mean DL and standard derivation r Lð> 0Þ Maximum value of x and 0, i.e., xþ ¼ maxfx; 0g
Assumptions: 1. Single-vendor and single-buyer for a single product. 2. Inventory is continuously reviewed. The buyer places an order or requests successive shipments when on hand inventory (based on the number of non-defective goods) reaches the reorder point r. 3. The reorder point r = the expected demand pffiffiffi during lead time + safety stock (SS), and SS ¼ k (standard derivation of lead time demand), that is, r ¼ DL þ kr L where k is the safety factor. 4. The buyer orders a lot of size Q (for non-defective goods) and will receive the batch quantity in m equal sized shipments of size q. 5. An arrival lot, q, may contain some defective goods and the proportion defective, k, is a random variable. Upon arrival, all goods are quickly inspected and defective goods in each lot will be discovered and returned to the vendor at the time the next lot is delivered. Thus, the expected quantity of non-defective goods is reduced to ð1 M k Þq in each shipment and the order quantity, Q, is the sum of non-defective units in m shipments, i.e., Q ¼ mð1 M k Þq. 6. Inspection is non-destructive and error-free. The inspection is processed quickly and gives the buyer the ability to skim an entire lot efficiently and effectively. From this perspective, length of inspection period is neglected here. 7. The vendor’s production rate is finite. The expected production rate of non-defective goods is greater than the buyer’s ðA þAv Þ demand rate, i.e., Pð1 M k Þ > D. pffiffi 2D b m þF p r L ðm1Þð1M Þ , a2 ¼ Dðm1Þð1M , 8. To simplify the expression for the interior minimum, Lin et al. [1] used a1 ¼ 2 m 2 k k Þm Yþhv D Pþ P Yþhv D P þ D D P a3 ¼ hb1 ð1 M k Þð1 þ bÞ, and a4 ¼ hb1 ð1 M k Þð1 bÞ. h pffiffii Ab þAv hv D 1 D Y m kÞ 9. For the minimum on the boundary, we set a1 ¼ ð1M þ F þ p r2 L , a2 ¼ 2ð1M þ 1M ½ þ ðm1Þð1M 2P , and m 2D kÞ kÞ k P p ffiffi ffi DðsþwMk Þ 1b a3 ¼ 1Mk þhb1 r L½ 2 to further simplify the expression. pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 10. We assume FðkÞ ¼ a3 1 þ k þ a4 k 1 þ k þ k a1 ð 1 þ k þ kÞ þ a2 as an auxiliary function to set the condition to divide our minimum problem into two cases. pffiffiffi 11. To compare the interior minimum and the boundary minimum, we assume that a4 ¼ hb1 r L, h iqffiffiffiffi q ffiffiffiffi pffiffi pffiffiffiffiffiffiffiffiffiffi Ab þAv DðsþwMk Þ a2 a2 1b r L D Dp a5 ¼ ð1M þF m a1 þ 1Mk þ a1 a2 ; and a6 ¼ 2ð1Mk Þ a1 þ a4 2 . kÞ 3. Review of previous results We directly quoted the average total expected cost of Ho [2] by applying the minimax distribution-free approach ([3,4])
" pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi # D Ab þ Av p r L Dðs þ wMk Þ Yq 2 þ 1þk k JTECðq; k; mÞ ¼ þ þFþ qð1 M k Þ 1 Mk 2ð1 Mk Þ m 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #
2 pffiffiffi 1þk k hv Dq 1 ðm 1Þð1 Mk Þ m ; þ þ þ hb1 r L k þ ð1 bÞ 2 1 Mk P 2D 2P
ð1Þ
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¼ p þ p0 ð1 bÞ, and Y ¼ hb1 þ 2ðhb2 hb1 ÞM k þ ðhb1 2hb2 Þ M 2k þ V k . Ho [2] derived where p
@2 2DðAb þ Av Þ JTECðq; k; mÞ ¼ > 0: qð1 M k Þm3 @m2
ð2Þ
Hence, JTEC(q, k, m) is a convex function of m, when q and k are fixed. Moreover, in the Appendix of Ho [2], she showed that for a fixed positive integer m, JTEC(q, k, m) is a convex function in variables q and k. Lin et al. [1] claimed that when m is fixed, the minimum of JTEC(q, k, m) will occur at the point (q, k) that satisfies the first @ @ partial derivative system @q JTECðq; k; mÞ ¼ 0 and @k JTECðq; k; mÞ ¼ 0, simultaneously. In the next example, we will point out that is a questionable assertion. 1 2 Based on the assertion of Lin et al. [1], for the following convex function, say f ðxÞ ¼ xþ1 for x P 0 with f 00 ðxÞ ¼ ðxþ1Þ 3 for x > 0 0 1 0 to find the minimum that will occur at the solution of f ðxÞ ¼ 0 that is ðxþ1Þ2 ¼ 0. We agreed that if f ðxÞ ¼ 0 has solution, say x# , then f ðx# Þ will be the minimum. However, f 0 ðxÞ ¼ 0 did not always have solution. Hence, the convexity property only implied the uniqueness of the optimal solution, but it cannot ensure the existence of the optimal solution. Solving the system of first partial derivatives, Ho [2] found that
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi offi pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi u n uD 2½ðAb þ Av Þ=m þ F þ p L 1 þ k k r u h i ; q¼t m kÞ Y þ 2hv D 1P þ ðm1Þð1M 2P 2D
ð3Þ
and
k 2hb1 qð1 M k Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 : 2 D p þ hb1 qð1 M k Þð1 bÞ 1þk
ð4Þ
Ho [2] could not discuss the solution structure of Eqs. (3) and (4), and her solution approach applied by a graphical technique of Hadley and Whitin [5] that cannot derive the optimal shipping quantity and the safety factor that had been pointed out by numerical examples in Lin et al. [1]. When m is fixed, the solution approach of Lin et al. [1] was to merge Eqs. (3) and (4) to eliminate variable k to derive a function in one variable of q. They obtained three constraints of two upper bounds and a lower bound for the shipping quantity q. Based on two reasonable observations, they showed that the system of Eqs. (3) and (4) had at most a solution. Lin et al. [1] rewrote Eqs. (3) and (4) as follows,
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1þk k ; q ¼ a1 þ a2
ð5Þ
and
a3 q k Dp pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ : 2 þ a4 q D p 1þk
ð6Þ
4. Our improvement We will combine Eqs. (3) and (4) to cancel out the shipping quantity to express the results in the safety factor, k. We still accept the following observation to ensure the positivity of a1 and a2. We accept that the following assertion:
P m2 ð1 Mk Þ > 1 > : D m1
ð7Þ
From Eq. (6), we found that
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1þk k Dp pffiffiffiffiffiffiffiffiffiffiffiffiffiffi q¼ : 2 a3 1 þ k þ a4 k Using ð
ð8Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1 þ k þ kÞð 1 þ k kÞ ¼ 1, we compute q2 by combining Eqs. (5) and (8) to derive that
Þ2 ; FðkÞ ¼ ðDp
ð9Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 where FðkÞ ¼ ða3 1 þ k þ a4 kÞ2 ð 1 þ k þ kÞða1 ð 1 þ k þ kÞ þ a2 Þ. In Lin et al. [1], they have discussed the positivity of a1, a2, a3 and a4. From the definition of F(k), since all coefficients are all positive, it is a strictly increasing function such that Eq. (9) has at most one positive solution.
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Up to now, we finish the proof of Theorem 1 of Lin et al. [1] that the first partial derivative system has at most one solution. It is the uniqueness part for the optimal solution. Next, we must discuss the existence part for the optimal solution. We would divide our problem into two cases: (C1) Þ2 , and (C2) Fð0Þ P ðDp Þ2 . Fð0Þ < ðDp 2 Þ , F(k) is an strictly increasing function, and lim FðkÞ ¼ 1 > ðDp Þ2 , we knew that there is a unFor case (C1), Fð0Þ < ðDp k!1 2 ique point, say k that satisfies Fðk Þ ¼ ðDpÞ and then by Eq. (8),
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1þk k Dp qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ : 2 a3 1 þ k þ a4 k
ð10Þ
We know that ðq ; k Þ is the unique solution for the first partial derivation system of a convex function, JTEC(q, k, m) when m is fixed for the moment so ðq ; k Þ is the optimal solution for that m. 2 Þ has no positive solution so the minimum will occur at the boundary with k = 0 to reduce the For case (C2), FðkÞ ¼ ðDp inventory model as
JTECðq; k ¼ 0; mÞ ¼
" pffiffiffi# pffiffiffi 1 b
D Ab þ Av p r L Dðs þ wMk Þ Yq þ þ þFþ þ hb1 r L qð1 M k Þ 1 Mk 2ð1 M k Þ 2 m 2
hv Dq 1 ðm 1Þð1 M k Þ m þ þ ; 1 Mk P 2D 2P
ð11Þ
¼ p þ p0 ð1 bÞ, and Y ¼ hb1 þ 2ðhb2 hb1 ÞM k þ ðhb1 2hb2 ÞðM 2k þ V k Þ. where p We could simplify the expression of Eq. (11) as follows
JTECðq; k ¼ 0; mÞ ¼
a1 q
þ a2 q þ a3 ; pffiffi p r L, 2
ð12Þ
pffiffiffi hv D 1 Y m kÞ kÞ a3 ¼ DðsþwM þ hb1 r L½1b , and a2 ¼ 2ð1M þ 1M ½ þ ðm1Þð1M 2P . We know that a1, a2 1M k 2D 2 kÞ k P
þAv D where a1 ¼ ð1M ½A b m þF þ kÞ and a3 are all positive. By algebraic method, we rewrite Eq. (12) as
JTECðq; k ¼ 0; mÞ ¼
rffiffiffiffiffi
a1 q
2 pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi a2 q þ 2 a1 a2 þ a3 :
We obtained that the minimum would occur at q ¼
qffiffiffiffi a1 a2
and the minimum value is JTECðq ¼
ð13Þ qffiffiffiffi pffiffiffiffiffiffiffiffiffiffi a1 a2 ; k ¼ 0; mÞ ¼ 2 a1 a2 þ a3 .
Next, we begin qffiffiffiffito compare the interior minimum and the boundary minimum. When m is fixed for the moment, we rewrite JTECðq ¼ aa12 ; k; mÞ as a function of k
rffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 2 1þk k ; JTEC q ¼ ; k; m ¼ a4 k þ a5 þ a6
a2
ð14Þ
qffiffiffiffi pffiffi qffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffi þAv a2 1b r L D Dp kÞ where a4 ¼ hb1 r L, a5 ¼ ð1M ½A b m þ F aa21 þ DðsþwM þ a1 a2 , and a6 ¼ 2ð1M 1M k a1 þ a4 2 . We know that a4, a5 and a6 are kÞ kÞ all positive. It yields that
d JTEC dk
! rffiffiffiffiffi a1 k ; k; m ¼ a4 þ a6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ; 2 a2 1þk
ð15Þ
and 2
d
JTEC 2 dk
rffiffiffiffiffi a1 a6 ; k; m ¼ 3=2 > 0; a2 2 1þk
ð16Þ
qffiffiffiffi to imply that JTECðq ¼ aa12 ; k; mÞ is a convex function in variable k. qffiffiffiffi d JTECð aa12 ; k; mÞ ¼ 0 that is To solve dk
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi a6 2 2 1þk þk 1þk ¼ :
a4
ð17Þ
We know that the right hand side of Eq. (17) is an increasing function of k. For the solution for Eq. (17), we divide the problem into two cases: (C3) aa64 > 1, and (C4) aa64 6 1. # For case (C3) with aa64 > 1, there is a unique point, say k that satisfies
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1 þ ðk Þ2 þ k #
#
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a6 # 1 þ ðk Þ2 ¼ ;
ð18Þ
a4
qffiffiffiffi # such that k is the minimum of JTECðq ¼ aa12 ; k; mÞ for 0 6 k < 1, since it is a convex function. It follows that
rffiffiffiffiffi rffiffiffiffiffi a1 a1 # JTEC q ¼ ; 0; m > JTEC q ¼ ;k ;m :
a2
ð19Þ
a2
Let us recall that JTECðq ; k ; mÞ is the interior minimum so that
rffiffiffiffiffi a1 # JTEC q ¼ ; k ; m > JTECðq ; k ; mÞ:
ð20Þ
a2
We combine the results of Eqs. (19) and (20) to imply that the interior minimum is small than the boundary minimum as
rffiffiffiffiffi a1 JTEC q ¼ ; 0; m > JTECðq ; k ; mÞ:
ð21Þ
a2
For case (C4) with
a6 a4
6 1, it yields that there is no k with 0 < k < 1 that satisfies Eq. (17). We rewrite Eq. (15) as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rffiffiffiffiffi d a1 a4 a 1 þ k2 þ k 1 þ k2 6 > 0; JTEC ; k; m ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 dk a2 a4 1þk 1þk þk
ð22Þ
qffiffiffiffi qffiffiffiffi for k > 0 such that ð aa12 ; 0; mÞ is the minimum for JTECðq ¼ aa12 ; k; mÞ with 0 6 k < 1. We summarize our results in the following theorem. Theorem 1. When m is fixed for the moment, we find that
Þ2 , and a6 > a4 , then the interior minimum JTECðq ; k ; mÞ is the optimal solution. (a) If Fð0Þ < ðDp (b) If Fð0Þ < ðDpÞ2 , and a6 6 a4 , then the optimal solution is
rffiffiffiffiffi a1 min JTECðq ; k ; mÞ; JTEC ; 0; m :
ð23Þ
a2
qffiffiffiffi Þ2 , then the optimal solution is JTECðq ¼ aa1 ; 0; mÞ. (c) If Fð0Þ P ðDp 2 Þ2 that is equivalent to We examine the expression of Fð0Þ < ðDp
Þ2 ; a23 ða1 þ a2 Þ < ðDp
ð24Þ
that can be expressed as follows
h i pffiffiffi þAv Þ r L 2D ðAb m þ F þ Dp Þ2 : h i < ðDp ðhb1 ð1 M k Þð1 þ bÞÞ m kÞ Y þ hv D 2P þ ðm1Þð1M P D 2
ð25Þ
On the other hand, we study the expression of a6 > a4 that is equivalent to
pffiffiffi rffiffiffiffiffi r L Dp a2 1b > a4 : þ a4 2ð1 M k Þ a1 2
ð26Þ
Eq. (26) can be rewritten as
pffiffiffi rffiffiffiffiffi r L Dp a2 1þb ; > a4 2 2ð1 M k Þ a1
ð27Þ
and then it yields
11=2 0 kÞ Y þ hv D 2P þ ðm1Þð1M mP D Dp @ pffiffiffi A > hb1 ð1 þ bÞ: 1 Mk r L 2D Ab þAv þ F þ Dp
ð28Þ
m
Now, we compare Eqs. (25) and (28) to discover that they are equivalent such that we can further simplify our Theorem 1 as follows. Theorem 2. When m is fixed for the moment, we find that
Þ2 then the interior minimum JTECðq ; k ; q mÞffiffiffiffiis the optimal solution. (a) If Fð0Þ < ðDp Þ2 , then the optimal solution is JTECðq ¼ aa1 ; 0; mÞ. (b) If Fð0Þ P ðDp 2
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5. Numerical example We consider the same numerical example in Ho [2] and Lin et al. [1] with the following data: D = 1000 units/year, P = 3200 units/year, Ab = $25/order, Ab = $400/set-up, F = $30/shipment, hb1 = $5/unit/year, hb2 = $3/unit/year, hv = $2/unit/ year, s = $0.5/unit, w = $5/unit, p ¼ $10=unit, p0 ¼ $30=unit, L = 6 weeks, r = 7 units/week, the defective rate k has a Beta distribution with parameters a = 1 and b = 4 to imply the probability density function is gðkÞ ¼ 4ð1 kÞ3 for 0 < k < 1 to imply a 2 that Mk ¼ aþb ¼ 15 and V k ¼ ðaþbÞ2ab ¼ 75 . ðaþbþ1Þ Þ2 ¼ 1 108 , For the optimal solution is the interior minimum, we assume that b = 1 to find that Fð0Þ ¼ 7:328 106 < ðDp m ¼ 2, k ¼ 1:142032, q ¼ 309:574 and JTECðq ; k ; m Þ ¼ 4191:523 that is consistent with the finding of Lin et al. [1]. For the optimal solution is the boundary minimum, we change the buyer’s unit shortage cost per unit short, p, from p ¼ 10 to p ¼ 1. Þ2 ¼ 1 106 , the result of Theorem 2, (b) will happen to derive the boundary optimal Owing to Fð0Þ ¼ 5:605 106 > ðDp solution with m ¼ 2, k ¼ 0, q ¼ 295:945 and JTECðq ; k ; m Þ ¼ 3995:942.
6. Conclusion We derived a complete solution for the production inventory with defective items such that the boundary minimum was included in our derivation. The sophistic but complicated approach in Lin et al. [1] became unnecessary. Moreover, we pointed out that the convexity property only ensured the uniqueness of the interior minimum solution that could not guarantee the existence of the interior minimum solution. Up to now, we treated the number of deliveries from vendor to buyer in one production cycle, m, as a discrete variable and then examined for m = 1, m = 2, m ¼ 3;q ffiffiffiffi to locate the optimal solution of m. The possible direction for future research is to show that JTECðq ; k ; mÞ or JTECðq ¼ aa12 ; 0; mÞ is a convex function in variable m. Acknowledgments This research is partially supported by NSC of Taiwan, ROC with grant NSC 101-2410-H-015 -001. The authors want to express their gratitude to Sophia Liu ([email protected]) for her English revisions of our paper. References [1] S.W. Lin, Y.W. Wou, P. Julian, Note on minimax distribution free procedure for integrated inventory model with defective goods and stochastic lead time demand, Appl. Math. Model. 35 (2011) 2087–2093. [2] C.H. Ho, A minimax distribution free procedure for an integrated inventory model with defective goods and stochastic lead time demand, Int. J. Info. Manag. Sci. 20 (2009) 161–171. [3] G. Gallego, I. Moon, The distribution free newsboy problem: review and extensions, J. Oper. Res. Soc. 44 (8) (1993) 825–834. [4] I. Moon, G. Gallego, Distribution free procedures for some inventory models, J. Oper. Res. Soc. 45 (6) (1994) 651–658. [5] G. Hadley, T. Whitin, Analysis of Inventory Systems, Prentice-Hall, New Jersey, 1963.
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Short communication
Improved solution for inventory model with defective goods Cheng-Tan Tung ⇑, Peter Shaohua Deng Department of Information Management, Central Police University, Taiwan
a r t i c l e
i n f o
Article history: Received 23 November 2011 Received in revised form 18 September 2012 Accepted 12 October 2012 Available online 30 October 2012 Keywords: Minimax distribution free procedure Analytical approach Crashable lead time Partial backorder
a b s t r a c t This paper is a consequence for a paper of Lin et al. [S.W. Lin, Y.W. Wou, P. Julian, Note on minimax distribution free procedure for integrated inventory model with defective goods and stochastic lead time demand, Appl. Math. Model. 35 (2011) 2087–2093]. We simplified their complicated solution procedure and then presented a revision to patch their negligence for the boundary minimums. Numerical examples are provided to demonstrate our findings. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Lin et al. [1] tried to improve the solution procedure of an integrated inventory model, with defective goods and stochastic lead time demand proposed by Ho [2], under the minimax distribution free procedure of Gallego and Moon [3]. From the first partial derivative system, Lin et al. [1] merged two equations into one equation to cancel the variable of safety factor, to express the results in the variable of shipping quantity. They found two upper bounds and one lower bound for the shipping quantity, and then used numerical evidence to reduce two upper bounds into one upper bound. Under this upper bound, and the lower bound, Lin et al. [1] proved that there cannot exist more than one positive solution for the shipping quantity. We must point out that the proof of existing more than one positive solution cannot guarantee the existence of the positive solution. We will find the condition to ensure the existence of the positive solution and then discuss the minimum solution on the boundary. Our improvement will provide a complete solution structure for the inventory model. 2. Notation and assumptions To be comparable with Ho [2] and Lin et al. [1], we used the same notation and assumptions as theirs in the following. Notation: D P Ab
Demand rate on the buyer (for non-defective goods) Production rate on the vendor Buyer’s ordering cost per order
⇑ Corresponding author. Address: 56 Su-Jen Rd., Ta-Kan Village, Kwei-San, Tao-Yuan County, Tao-Yuan, Taiwan. Tel.: +886 3 3282312 4491; fax: +886 3 3282312 4329. E-mail addresses: [email protected] (C.-T. Tung), [email protected] (P.S. Deng). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.10.022
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Av F hv hb1 hb2 w s
p p0 b k gðkÞ L Q q r m X xþ
Vendor’s set-up cost per set-up Transportation cost per delivery Vendor’s holding cost per item per unit time Buyer’s holding cost per non-defective item per unit time Buyer’s treatment cost (include holding cost) per defective item per unit time Vendor’s unit treatment cost (include warranty cost) of defective goods Buyer’s unit inspection cost Buyer’s unit shortage cost per unit short Buyer’s marginal profit per unit Fraction of the demand during the stock-out period will be backordered, b 2 ½0; 1 Defective rate in an order lot, k 2 ½0; 1Þ, a random variable The probability density function (p.d.f.) of k with finite mean M k and variance V k , where M k ¼ EðkÞ ¼
R1 0
kgðkÞdk
and V k ¼ Eðk2 Þ ðM k Þ2 Length of lead time for the buyer Order quantity (for non-defective goods) of the buyer (decision variable) Shipping quantity from vendor to buyer per shipment (decision variable) Reorder point of the buyer for non-defective goods (decision variable) The number of deliveries from vendor to buyer in one production cycle, a positive integer (decision variable) pffiffiffi The lead time demand which has a p.d.f. f with finite mean DL and standard derivation r Lð> 0Þ Maximum value of x and 0, i.e., xþ ¼ maxfx; 0g
Assumptions: 1. Single-vendor and single-buyer for a single product. 2. Inventory is continuously reviewed. The buyer places an order or requests successive shipments when on hand inventory (based on the number of non-defective goods) reaches the reorder point r. 3. The reorder point r = the expected demand pffiffiffi during lead time + safety stock (SS), and SS ¼ k (standard derivation of lead time demand), that is, r ¼ DL þ kr L where k is the safety factor. 4. The buyer orders a lot of size Q (for non-defective goods) and will receive the batch quantity in m equal sized shipments of size q. 5. An arrival lot, q, may contain some defective goods and the proportion defective, k, is a random variable. Upon arrival, all goods are quickly inspected and defective goods in each lot will be discovered and returned to the vendor at the time the next lot is delivered. Thus, the expected quantity of non-defective goods is reduced to ð1 M k Þq in each shipment and the order quantity, Q, is the sum of non-defective units in m shipments, i.e., Q ¼ mð1 M k Þq. 6. Inspection is non-destructive and error-free. The inspection is processed quickly and gives the buyer the ability to skim an entire lot efficiently and effectively. From this perspective, length of inspection period is neglected here. 7. The vendor’s production rate is finite. The expected production rate of non-defective goods is greater than the buyer’s ðA þAv Þ demand rate, i.e., Pð1 M k Þ > D. pffiffi 2D b m þF p r L ðm1Þð1M Þ , a2 ¼ Dðm1Þð1M , 8. To simplify the expression for the interior minimum, Lin et al. [1] used a1 ¼ 2 m 2 k k Þm Yþhv D Pþ P Yþhv D P þ D D P a3 ¼ hb1 ð1 M k Þð1 þ bÞ, and a4 ¼ hb1 ð1 M k Þð1 bÞ. h pffiffii Ab þAv hv D 1 D Y m kÞ 9. For the minimum on the boundary, we set a1 ¼ ð1M þ F þ p r2 L , a2 ¼ 2ð1M þ 1M ½ þ ðm1Þð1M 2P , and m 2D kÞ kÞ k P p ffiffi ffi DðsþwMk Þ 1b a3 ¼ 1Mk þhb1 r L½ 2 to further simplify the expression. pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 10. We assume FðkÞ ¼ a3 1 þ k þ a4 k 1 þ k þ k a1 ð 1 þ k þ kÞ þ a2 as an auxiliary function to set the condition to divide our minimum problem into two cases. pffiffiffi 11. To compare the interior minimum and the boundary minimum, we assume that a4 ¼ hb1 r L, h iqffiffiffiffi q ffiffiffiffi pffiffi pffiffiffiffiffiffiffiffiffiffi Ab þAv DðsþwMk Þ a2 a2 1b r L D Dp a5 ¼ ð1M þF m a1 þ 1Mk þ a1 a2 ; and a6 ¼ 2ð1Mk Þ a1 þ a4 2 . kÞ 3. Review of previous results We directly quoted the average total expected cost of Ho [2] by applying the minimax distribution-free approach ([3,4])
" pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi # D Ab þ Av p r L Dðs þ wMk Þ Yq 2 þ 1þk k JTECðq; k; mÞ ¼ þ þFþ qð1 M k Þ 1 Mk 2ð1 Mk Þ m 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #
2 pffiffiffi 1þk k hv Dq 1 ðm 1Þð1 Mk Þ m ; þ þ þ hb1 r L k þ ð1 bÞ 2 1 Mk P 2D 2P
ð1Þ
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¼ p þ p0 ð1 bÞ, and Y ¼ hb1 þ 2ðhb2 hb1 ÞM k þ ðhb1 2hb2 Þ M 2k þ V k . Ho [2] derived where p
@2 2DðAb þ Av Þ JTECðq; k; mÞ ¼ > 0: qð1 M k Þm3 @m2
ð2Þ
Hence, JTEC(q, k, m) is a convex function of m, when q and k are fixed. Moreover, in the Appendix of Ho [2], she showed that for a fixed positive integer m, JTEC(q, k, m) is a convex function in variables q and k. Lin et al. [1] claimed that when m is fixed, the minimum of JTEC(q, k, m) will occur at the point (q, k) that satisfies the first @ @ partial derivative system @q JTECðq; k; mÞ ¼ 0 and @k JTECðq; k; mÞ ¼ 0, simultaneously. In the next example, we will point out that is a questionable assertion. 1 2 Based on the assertion of Lin et al. [1], for the following convex function, say f ðxÞ ¼ xþ1 for x P 0 with f 00 ðxÞ ¼ ðxþ1Þ 3 for x > 0 0 1 0 to find the minimum that will occur at the solution of f ðxÞ ¼ 0 that is ðxþ1Þ2 ¼ 0. We agreed that if f ðxÞ ¼ 0 has solution, say x# , then f ðx# Þ will be the minimum. However, f 0 ðxÞ ¼ 0 did not always have solution. Hence, the convexity property only implied the uniqueness of the optimal solution, but it cannot ensure the existence of the optimal solution. Solving the system of first partial derivatives, Ho [2] found that
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi offi pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi u n uD 2½ðAb þ Av Þ=m þ F þ p L 1 þ k k r u h i ; q¼t m kÞ Y þ 2hv D 1P þ ðm1Þð1M 2P 2D
ð3Þ
and
k 2hb1 qð1 M k Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 : 2 D p þ hb1 qð1 M k Þð1 bÞ 1þk
ð4Þ
Ho [2] could not discuss the solution structure of Eqs. (3) and (4), and her solution approach applied by a graphical technique of Hadley and Whitin [5] that cannot derive the optimal shipping quantity and the safety factor that had been pointed out by numerical examples in Lin et al. [1]. When m is fixed, the solution approach of Lin et al. [1] was to merge Eqs. (3) and (4) to eliminate variable k to derive a function in one variable of q. They obtained three constraints of two upper bounds and a lower bound for the shipping quantity q. Based on two reasonable observations, they showed that the system of Eqs. (3) and (4) had at most a solution. Lin et al. [1] rewrote Eqs. (3) and (4) as follows,
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1þk k ; q ¼ a1 þ a2
ð5Þ
and
a3 q k Dp pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ : 2 þ a4 q D p 1þk
ð6Þ
4. Our improvement We will combine Eqs. (3) and (4) to cancel out the shipping quantity to express the results in the safety factor, k. We still accept the following observation to ensure the positivity of a1 and a2. We accept that the following assertion:
P m2 ð1 Mk Þ > 1 > : D m1
ð7Þ
From Eq. (6), we found that
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1þk k Dp pffiffiffiffiffiffiffiffiffiffiffiffiffiffi q¼ : 2 a3 1 þ k þ a4 k Using ð
ð8Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1 þ k þ kÞð 1 þ k kÞ ¼ 1, we compute q2 by combining Eqs. (5) and (8) to derive that
Þ2 ; FðkÞ ¼ ðDp
ð9Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 where FðkÞ ¼ ða3 1 þ k þ a4 kÞ2 ð 1 þ k þ kÞða1 ð 1 þ k þ kÞ þ a2 Þ. In Lin et al. [1], they have discussed the positivity of a1, a2, a3 and a4. From the definition of F(k), since all coefficients are all positive, it is a strictly increasing function such that Eq. (9) has at most one positive solution.
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Up to now, we finish the proof of Theorem 1 of Lin et al. [1] that the first partial derivative system has at most one solution. It is the uniqueness part for the optimal solution. Next, we must discuss the existence part for the optimal solution. We would divide our problem into two cases: (C1) Þ2 , and (C2) Fð0Þ P ðDp Þ2 . Fð0Þ < ðDp 2 Þ , F(k) is an strictly increasing function, and lim FðkÞ ¼ 1 > ðDp Þ2 , we knew that there is a unFor case (C1), Fð0Þ < ðDp k!1 2 ique point, say k that satisfies Fðk Þ ¼ ðDpÞ and then by Eq. (8),
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1þk k Dp qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ : 2 a3 1 þ k þ a4 k
ð10Þ
We know that ðq ; k Þ is the unique solution for the first partial derivation system of a convex function, JTEC(q, k, m) when m is fixed for the moment so ðq ; k Þ is the optimal solution for that m. 2 Þ has no positive solution so the minimum will occur at the boundary with k = 0 to reduce the For case (C2), FðkÞ ¼ ðDp inventory model as
JTECðq; k ¼ 0; mÞ ¼
" pffiffiffi# pffiffiffi 1 b
D Ab þ Av p r L Dðs þ wMk Þ Yq þ þ þFþ þ hb1 r L qð1 M k Þ 1 Mk 2ð1 M k Þ 2 m 2
hv Dq 1 ðm 1Þð1 M k Þ m þ þ ; 1 Mk P 2D 2P
ð11Þ
¼ p þ p0 ð1 bÞ, and Y ¼ hb1 þ 2ðhb2 hb1 ÞM k þ ðhb1 2hb2 ÞðM 2k þ V k Þ. where p We could simplify the expression of Eq. (11) as follows
JTECðq; k ¼ 0; mÞ ¼
a1 q
þ a2 q þ a3 ; pffiffi p r L, 2
ð12Þ
pffiffiffi hv D 1 Y m kÞ kÞ a3 ¼ DðsþwM þ hb1 r L½1b , and a2 ¼ 2ð1M þ 1M ½ þ ðm1Þð1M 2P . We know that a1, a2 1M k 2D 2 kÞ k P
þAv D where a1 ¼ ð1M ½A b m þF þ kÞ and a3 are all positive. By algebraic method, we rewrite Eq. (12) as
JTECðq; k ¼ 0; mÞ ¼
rffiffiffiffiffi
a1 q
2 pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi a2 q þ 2 a1 a2 þ a3 :
We obtained that the minimum would occur at q ¼
qffiffiffiffi a1 a2
and the minimum value is JTECðq ¼
ð13Þ qffiffiffiffi pffiffiffiffiffiffiffiffiffiffi a1 a2 ; k ¼ 0; mÞ ¼ 2 a1 a2 þ a3 .
Next, we begin qffiffiffiffito compare the interior minimum and the boundary minimum. When m is fixed for the moment, we rewrite JTECðq ¼ aa12 ; k; mÞ as a function of k
rffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 2 1þk k ; JTEC q ¼ ; k; m ¼ a4 k þ a5 þ a6
a2
ð14Þ
qffiffiffiffi pffiffi qffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffi þAv a2 1b r L D Dp kÞ where a4 ¼ hb1 r L, a5 ¼ ð1M ½A b m þ F aa21 þ DðsþwM þ a1 a2 , and a6 ¼ 2ð1M 1M k a1 þ a4 2 . We know that a4, a5 and a6 are kÞ kÞ all positive. It yields that
d JTEC dk
! rffiffiffiffiffi a1 k ; k; m ¼ a4 þ a6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ; 2 a2 1þk
ð15Þ
and 2
d
JTEC 2 dk
rffiffiffiffiffi a1 a6 ; k; m ¼ 3=2 > 0; a2 2 1þk
ð16Þ
qffiffiffiffi to imply that JTECðq ¼ aa12 ; k; mÞ is a convex function in variable k. qffiffiffiffi d JTECð aa12 ; k; mÞ ¼ 0 that is To solve dk
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi a6 2 2 1þk þk 1þk ¼ :
a4
ð17Þ
We know that the right hand side of Eq. (17) is an increasing function of k. For the solution for Eq. (17), we divide the problem into two cases: (C3) aa64 > 1, and (C4) aa64 6 1. # For case (C3) with aa64 > 1, there is a unique point, say k that satisfies
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1 þ ðk Þ2 þ k #
#
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a6 # 1 þ ðk Þ2 ¼ ;
ð18Þ
a4
qffiffiffiffi # such that k is the minimum of JTECðq ¼ aa12 ; k; mÞ for 0 6 k < 1, since it is a convex function. It follows that
rffiffiffiffiffi rffiffiffiffiffi a1 a1 # JTEC q ¼ ; 0; m > JTEC q ¼ ;k ;m :
a2
ð19Þ
a2
Let us recall that JTECðq ; k ; mÞ is the interior minimum so that
rffiffiffiffiffi a1 # JTEC q ¼ ; k ; m > JTECðq ; k ; mÞ:
ð20Þ
a2
We combine the results of Eqs. (19) and (20) to imply that the interior minimum is small than the boundary minimum as
rffiffiffiffiffi a1 JTEC q ¼ ; 0; m > JTECðq ; k ; mÞ:
ð21Þ
a2
For case (C4) with
a6 a4
6 1, it yields that there is no k with 0 < k < 1 that satisfies Eq. (17). We rewrite Eq. (15) as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rffiffiffiffiffi d a1 a4 a 1 þ k2 þ k 1 þ k2 6 > 0; JTEC ; k; m ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 dk a2 a4 1þk 1þk þk
ð22Þ
qffiffiffiffi qffiffiffiffi for k > 0 such that ð aa12 ; 0; mÞ is the minimum for JTECðq ¼ aa12 ; k; mÞ with 0 6 k < 1. We summarize our results in the following theorem. Theorem 1. When m is fixed for the moment, we find that
Þ2 , and a6 > a4 , then the interior minimum JTECðq ; k ; mÞ is the optimal solution. (a) If Fð0Þ < ðDp (b) If Fð0Þ < ðDpÞ2 , and a6 6 a4 , then the optimal solution is
rffiffiffiffiffi a1 min JTECðq ; k ; mÞ; JTEC ; 0; m :
ð23Þ
a2
qffiffiffiffi Þ2 , then the optimal solution is JTECðq ¼ aa1 ; 0; mÞ. (c) If Fð0Þ P ðDp 2 Þ2 that is equivalent to We examine the expression of Fð0Þ < ðDp
Þ2 ; a23 ða1 þ a2 Þ < ðDp
ð24Þ
that can be expressed as follows
h i pffiffiffi þAv Þ r L 2D ðAb m þ F þ Dp Þ2 : h i < ðDp ðhb1 ð1 M k Þð1 þ bÞÞ m kÞ Y þ hv D 2P þ ðm1Þð1M P D 2
ð25Þ
On the other hand, we study the expression of a6 > a4 that is equivalent to
pffiffiffi rffiffiffiffiffi r L Dp a2 1b > a4 : þ a4 2ð1 M k Þ a1 2
ð26Þ
Eq. (26) can be rewritten as
pffiffiffi rffiffiffiffiffi r L Dp a2 1þb ; > a4 2 2ð1 M k Þ a1
ð27Þ
and then it yields
11=2 0 kÞ Y þ hv D 2P þ ðm1Þð1M mP D Dp @ pffiffiffi A > hb1 ð1 þ bÞ: 1 Mk r L 2D Ab þAv þ F þ Dp
ð28Þ
m
Now, we compare Eqs. (25) and (28) to discover that they are equivalent such that we can further simplify our Theorem 1 as follows. Theorem 2. When m is fixed for the moment, we find that
Þ2 then the interior minimum JTECðq ; k ; q mÞffiffiffiffiis the optimal solution. (a) If Fð0Þ < ðDp Þ2 , then the optimal solution is JTECðq ¼ aa1 ; 0; mÞ. (b) If Fð0Þ P ðDp 2
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5579
5. Numerical example We consider the same numerical example in Ho [2] and Lin et al. [1] with the following data: D = 1000 units/year, P = 3200 units/year, Ab = $25/order, Ab = $400/set-up, F = $30/shipment, hb1 = $5/unit/year, hb2 = $3/unit/year, hv = $2/unit/ year, s = $0.5/unit, w = $5/unit, p ¼ $10=unit, p0 ¼ $30=unit, L = 6 weeks, r = 7 units/week, the defective rate k has a Beta distribution with parameters a = 1 and b = 4 to imply the probability density function is gðkÞ ¼ 4ð1 kÞ3 for 0 < k < 1 to imply a 2 that Mk ¼ aþb ¼ 15 and V k ¼ ðaþbÞ2ab ¼ 75 . ðaþbþ1Þ Þ2 ¼ 1 108 , For the optimal solution is the interior minimum, we assume that b = 1 to find that Fð0Þ ¼ 7:328 106 < ðDp m ¼ 2, k ¼ 1:142032, q ¼ 309:574 and JTECðq ; k ; m Þ ¼ 4191:523 that is consistent with the finding of Lin et al. [1]. For the optimal solution is the boundary minimum, we change the buyer’s unit shortage cost per unit short, p, from p ¼ 10 to p ¼ 1. Þ2 ¼ 1 106 , the result of Theorem 2, (b) will happen to derive the boundary optimal Owing to Fð0Þ ¼ 5:605 106 > ðDp solution with m ¼ 2, k ¼ 0, q ¼ 295:945 and JTECðq ; k ; m Þ ¼ 3995:942.
6. Conclusion We derived a complete solution for the production inventory with defective items such that the boundary minimum was included in our derivation. The sophistic but complicated approach in Lin et al. [1] became unnecessary. Moreover, we pointed out that the convexity property only ensured the uniqueness of the interior minimum solution that could not guarantee the existence of the interior minimum solution. Up to now, we treated the number of deliveries from vendor to buyer in one production cycle, m, as a discrete variable and then examined for m = 1, m = 2, m ¼ 3;q ffiffiffiffi to locate the optimal solution of m. The possible direction for future research is to show that JTECðq ; k ; mÞ or JTECðq ¼ aa12 ; 0; mÞ is a convex function in variable m. Acknowledgments This research is partially supported by NSC of Taiwan, ROC with grant NSC 101-2410-H-015 -001. The authors want to express their gratitude to Sophia Liu ([email protected]) for her English revisions of our paper. References [1] S.W. Lin, Y.W. Wou, P. Julian, Note on minimax distribution free procedure for integrated inventory model with defective goods and stochastic lead time demand, Appl. Math. Model. 35 (2011) 2087–2093. [2] C.H. Ho, A minimax distribution free procedure for an integrated inventory model with defective goods and stochastic lead time demand, Int. J. Info. Manag. Sci. 20 (2009) 161–171. [3] G. Gallego, I. Moon, The distribution free newsboy problem: review and extensions, J. Oper. Res. Soc. 44 (8) (1993) 825–834. [4] I. Moon, G. Gallego, Distribution free procedures for some inventory models, J. Oper. Res. Soc. 45 (6) (1994) 651–658. [5] G. Hadley, T. Whitin, Analysis of Inventory Systems, Prentice-Hall, New Jersey, 1963.