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A periodic review inventory model involving fuzzy expected demand short and fuzzy backorder rate
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Computers & Industrial Engineering 54 (2008) 666–676 www.elsevier.com/locate/dsw
A periodic review inventory model involving fuzzy expected demand short and fuzzy backorder rate Yu-Jen Lin
*
General Education Center, St. John’s University, 499, Sec. 4 Tam King Road, Tamsui, Taipei Hsien 251, Taiwan Received 18 December 2006; received in revised form 2 October 2007; accepted 2 October 2007 Available online 9 October 2007
Abstract The purpose of this paper is to extend [Ouyang, L. Y., Chuang, B. R. (2001). A periodic review inventory-control system with variable lead time. International Journal of Information and Management Sciences, 12, 1–13] periodic review inventory model with variable lead time by considering the fuzziness of expected demand shortage and backorder rate. We fuzzify the expected shortage quantity at the end of cycle and the backorder (or lost sales) rate, and then obtain the fuzzy total expected annual cost. Using the signed distance method to defuzzify, we derive the estimate of total expected annual cost in the fuzzy sense. For the proposed model, we provide a solution procedure to find the optimal review period and optimal lead time in the fuzzy sense so that the total expected annual cost in the fuzzy sense has a minimum value. Furthermore, a numerical example is provided and the results of fuzzy and crisp models are compared. 2007 Elsevier Ltd. All rights reserved. Keywords: Inventory; Periodic review; Fuzzy membership function; Signed distance
1. Introduction In recent years, Yao and others (Chang, Yao, & Ouyang, 2006; Chang, Yao, & Lee, 1998; Lee & Yao, 1998, 1999; Lin & Yao, 2000; Ouyang & Yao, 2002; Yao & Lee, 1996, 1999; Yao, Chang, & Su, 2000; Yao & Su, 2000; Yao & Chiang, 2003; Yao, Ouyang, & Chang, 2003) have contributed several articles by applying the fuzzy sets theory to deal with the production/inventory problems. In Lee and Yao (1999), Yao and Lee (1999) and Yao et al. (2000), they presented the fuzzy EOQ models for the inventory with backorders. Chang et al. (1998) fuzzfied the backorder quantity to be the triangular fuzzy number, while the order quantity is an ordinary variable. By contrast, Yao and Lee (1996) fuzzified the order quantity to be the triangular fuzzy number, while the backorder quantity is an ordinary variable. In Yao and Su (2000), the total demand was fuzzified to be the interval-value fuzzy set. In Lee and Yao (1998) and Lin and Yao (2000), the authors discussed the production inventory problems and proposed the fuzzy EPQ models, where Lee and Yao (1998) fuzzified the demand quantity and production quantity per day and Lin and Yao (2000) fuzzified the production quantity *
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0360-8352/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2007.10.002
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per cycle, all to be the triangular fuzzy numbers, and the fuzzy total costs were derived using the extension principle. Besides, there are several authors presented various fuzzy inventory models. For example, Petrovic and Sweeney (1994) fuzzified the demand, lead time and inventory level into triangular fuzzy numbers in an inventory control model, and then decided the order quantity by the method of fuzzy propositions. Chen and Wang (1996) fuzzified the demand, ordering cost, inventory cost, and backorder cost into trapezoid fuzzy numbers and used the functional principle to obtain the estimate of total cost in the fuzzy sense. Vujosevic, Petrovic, and Petrovic (1996) fuzzified the ordering cost and holding cost into trapezoid fuzzy numbers in the total cost of an inventory without backorder model and obtained the fuzzy total cost. Then they did the defuzzification by using centroid and gained the total cost in the fuzzy sense. In this paper, we recast the Ouyang and Chuang’s (2001) model by introducing the fuzziness of expected demand shortage and backorder rate, and then using the signed distance method to defuzzify, we derive the estimate of total expected annual cost in the fuzzy sense. The same signed distance method has also been used in Yao and Wu (2000), Yao and Chiang (2003), Lin and Yao (2003), Chiang, Yao, and Lee (2005), and others. This article is organized as follows. In Section 2, some definitions and properties about fuzzy sets, which will be needed later, are introduced. Section 3, explores the inventory problem for a periodic review model with variable lead time. Specifically, in Section 3.1, we consider the crisp case, and given the basis of the notation and assumptions. In Sections 3.2 and 3.3, we consider the fuzzy case, so we fuzzify the expected demand shortage and backorder rate. In Section 4, we derive the optimal review period and optimal lead time by minimizing the estimate of total cost in the fuzzy sense. A numerical example is provided to illustrate the results. In Section 5, we discuss some problems for the proposed model. Finally, some concluding remarks are given in Section 6. 2. Preliminaries In order to consider the fuzziness of an inventory problem, we need some definitions and property relative to this study. We state them in the following. e be a fuzzy set on R = (1,1) and 0 6 a 6 1, the a cut D(a) of D e consists of the points Definition 2.1. Let D x such that le ðxÞ P a, that is, DðaÞ ¼ fxjl ðxÞ P ag. D D e
e on R with which the a cut DðaÞ ¼ fxjl ðxÞ P ag ¼ ½DL ðaÞ; DU ðaÞ Let R be the family of the fuzzy sets D D e exists for every a 2 [0, 1], where DL(a) and DU(a) are continuous functions on a 2 [0, 1], then by the Decomposition Principle (see, e.g. Kaufmann & Gupta, 1991), we have e ¼ [ aDðaÞ ¼ [ ½DL ðaÞ; DU ðaÞ; a: D 06a61
06a61
ð2:1Þ
A new ranking method for fuzzy numbers, namely the signed distance has been first introduced by Yao and Wu (2000), and it has been utilized in many studies (e.g. Yao & Chiang, 2003; Chiang et al., 2005). Here we describe the concept of the signed distance on R which will be needed later. We first consider the signed distance on R. Definition 2.2. For any a and 0 2 R, define the signed distance of a to 0 as d0(a, 0) = a. If a > 0, implies that a is on the right-hand side of origin 0 with distance d0(a, 0) = a; and if a < 0, implies that a is on the left-hand side of origin 0 with distance d0(a, 0) = a. So, we called d0(a, 0) = a is the signed distance of a to 0. e 2 R, from Eq. (2.1), we have For any D e ¼ [ ½DL ðaÞ; DU ðaÞ; a: D 06a61
ð2:2Þ
And for every a 2 [0, 1], there is an one-to-one mapping between the a -level fuzzy interval [DL(a), DU (a);a] and real interval [DL(a), DU(a)], that is, the following correspondence is one-to-one mapping:
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½DL ðaÞ; DU ðaÞ $ ½DL ðaÞ; DU ðaÞ; a:
ð2:3Þ
We shall use this relation later. e to the From Definition 2.2, the signed distance of the left end point DL(a) of the a cut [DL(a), DU(a)] of D origin 0 is DL(a), and the signed distance of the right end point DU(a) to the origin 0 is DU(a). Their average, 1 ½DL ðaÞ þ DU ðaÞ, is defined as the signed distance of a cut [DL(a), DU(a)] to 0, that is, we define the signed 2 distance of the interval [DL(a), DU(a)] to 0 as: 1 1 d 0 ð½DL ðaÞ; DU ðaÞ; 0Þ ¼ ½d 0 ðDL ðaÞ; 0Þ þ d 0 ðDU ðaÞ; 0Þ ¼ ½DL ðaÞ þ DU ðaÞ: ð2:4Þ 2 2 Further, from Eqs. (2.3) and (2.4), the signed distance of a-level fuzzy interval [DL(a), DU(a);a] to the fuzzy point ~ 0 can be defined as: 1 dð½DL ðaÞ; DU ðaÞ; a; ~ 0Þ ¼ d 0 ð½DL ðaÞ; DU ðaÞ; 0Þ ¼ ½DL ðaÞ þ DU ðaÞ: 2 e 2 Thus, from Eqs. (2.2) and (2.5), we can define the signed distance of a fuzzy set D follows.
ð2:5Þ P
to ~0 as
e 2 R, define the signed distance of D e to ~0 as Definition 2.3. For D
e ~ dð D; 0Þ ¼
Z
1
dð½DL ðaÞ; DU ðaÞ; a; ~ 0Þda ¼ 0
1 2
Z
1
½DL ðaÞ þ DU ðaÞda: 0
e ¼ ða; b; cÞ to ~0 is dð D; e ~0Þ ¼ 1 ða þ 2b þ cÞ. Property 1. The signed distance of the fuzzy number D 4 e is D(a) = [DL(a), DU(a)], where DL (a) = a + (b a)a and Proof. For a 2 [0, 1], the a cut of D R e ~0Þ ¼ 1 1 ½DL ðaÞ þ DU ðaÞda ¼ 1 ða þ 2b þ cÞ. h DU(a) = c (c b)a. By Definition 2.3, we have dð D; 2 0 4 3. A fuzzy periodic review inventory model 3.1. The crisp inventory model We first review the crisp period review inventory model which was asserted by Ouyang and Chuang (2001). Under the assumptions that: 1. The inventory level is reviewed every T units of time. A sufficient quantity is ordered up to the target inventory level r, and the ordering quantity is arrived after L units of time. 2. The length of the lead time L does not exceed an inventory cycle time T so that there is never more than a single order outstanding in any cycle. 3. The target inventory level r = expected demand during the protection interval + safety stock (SS), and pffiffiffiffiffiffiffiffiffiffiffiffi SS = k· (standard deviation of protection interval demand), i.e., r ¼ DðT þ LÞ þ kr T þ L, where D is the average demand per year, r is the standard deviation of demand per unit time, k is the safety factor and satisfies P(X > r) = q, q is given to represent the allowable stockout probability during the protection interval. 4. The lead time L consists of n mutually independent components. The ith component has a minimum duration ai and normal duration bi, and a crashing cost per unit time ci, where c1 6 c2 6 6 cn. Then, it is clear that the reduction of lead time should be first on component 1 because it has the minimum unit crashing cost, and P then component 2, and so on. n 5. Let L0 ¼ j¼1 bj and Li be the length of lead time components 1, 2, . . . , i crashed to their minimum Pn with P i duration, then Li can be expressed as Li ¼ j¼1 bj j¼1 ðbj aj Þ; i ¼ 1; 2; . . . ; n; andPthe lead time crashing cost C(L) per cycle for a given L 2 [Li, Li1] is given by CðLÞ ¼ ci ðLi1 LÞ þ i1 j¼1 cj ðbj aj Þ.
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Ouyang and Chuang (2001) established the following total expected annual cost function: A DT pEðX rÞþ CðLÞ þ þ h r DL þ ð1 bÞEðX rÞ þ þ T 2 T T h i p ffiffiffiffiffiffiffiffiffiffiffiffi A þ CðLÞ DT p þ ¼ þh þ kr T þ L þ þ hð1 bÞ EðX rÞ ; T 2 T
EACðT ; LÞ ¼
ð3:1Þ
where A = fixed ordering cost per order h = inventory holding cost per item per year b = backorder rate, b 2 [0, 1] p = stockout cost per unit short X = the protection interval, T + L, demand which has a normal probability density function (p.d.f.) fX with pffiffiffiffiffiffiffiffiffiffiffiffi finite mean D(T + L) and standard deviation r T þ L E(X r)+ = the expected shortage quantity at the end of the cycle, and is given by R1 pffiffiffiffiffiffiffiffiffiffiffiffi þ EðX rÞ ¼ r ðx rÞfX ðxÞdx ¼ r T þ LwðkÞ > 0, where w(k) / (k) k[1 U(k)], /(k) and U(k) denote the standard normal p.d.f. and distribution function, respectively. Note that if (R, B, P) be the probability space, then the protection interval demand X is a random variable on (R, B, P). 3.2. The fuzzy inventory model involving variable lead time with fuzzy random variable and fuzzy expected demand short In this subsection, we use the concepts of fuzzy set to extend the Ouyang and Chuang (2001) model for a periodic review inventory control system with variable lead time. In real business transactions, owing to the unstable environments, it would not be easily determined the exact values of the expected shortage quantity at the end of the cycle by the decision-maker. Thus the decision-maker determines the shortage quantity at the end of the cycle is always near x r as X = x. Corresponde here. From Kwakernaak ing to the crisp random variable X in (3.1), we consider the fuzzy random variable X (1978) and Puri and Ralescu (1986), the fuzzy random variable can be defined as a mapping from R to a family of triangular fuzzy numbers, i.e., e :x2R!X e ðxÞ ¼ ðx D1 ; x; x þ D2 Þ; X
if
X ¼ x;
ð3:2Þ
where D1, D2 are determined by the decision-maker and should satisfy the conditions: 0 < D1 < x r, and 0 < D2, where r < x. Corresponding to the crisp random variable Y = X r, we set the fuzzy random variable as e ðxÞðÞ~r ¼ ðx r D1 ; x r; x r þ D2 Þ; if X r ¼ x r Ye : x 2 R ! Ye ðxÞ ¼ X
ð3:3Þ
where ~r is a fuzzy point. Thus, we have the membership function of Ye ðxÞ as follows: 8 tðxrD1 Þ ; x r D1 6 t 6 x r; > D1 < ðxrþD Þt 2 le ðtÞ ¼ ; x r 6 t 6 x r þ D2 ; Y ðxÞ D2 > : 0; otherwise:
ð3:4Þ
And the a cut of Ye ðxÞ is ½ Ye ðxÞL ðaÞ; Ye ðxÞU ðaÞ ¼ ½x r D1 þ aD1 ; x r þ ð1 aÞD2 ;
for
a 2 ½0; 1:
ð3:5Þ
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Since Ye ðxÞ 2 R, the estimate of expectation of fuzzy random variable Ye in the fuzzy sense as follows: Z Z i 1 1 1 he þ e e þ E ð X ðÞ~rÞ ¼ E ð Y Þ ¼ Y ðxÞL ðaÞ þ Ye ðxÞU ðaÞ fX ðxÞdxda 2 0 rþD1 Z 1 1 1 2ðx rÞ þ ðD2 D1 Þ fX ðxÞdx ¼ 2 rþD1 2 Z Z 1 1 1 ðx rÞfX ðxÞdx þ ðD2 D1 ÞfX ðxÞdx ¼ 4 rþD1 rþD1 Z rþD1 Z 1 1 ¼ EðX rÞþ ðx rÞfX ðxÞdx ðD1 D2 Þ fX ðxÞdx: ð3:6Þ 4 r rþD1 Theorem 1. In the Eq. (3.1), using the fuzzy random variable of Eqs. (3.2) and (3.3), then the crisp random e ðÞ~r. Thus, we obtained the total expected annual variable Y = X r, is changed to fuzzy random variable Ye ¼ X cost in the fuzzy sense h i pffiffiffiffiffiffiffiffiffiffiffiffi A þ CðLÞ DT p þh þ kr T þ L þ þ hð1 bÞ EAC ðT ; L; D1 ; D2 Þ ¼ T 2 T Z rþD1 Z 1 1 þ EðX rÞ ðx rÞfX ðxÞdx ðD1 D2 Þ fX ðxÞdx 4 r rþD1 hp i ¼ EACðT ; LÞ þ hð1 bÞ T Z rþD1 Z 1 1 ðx rÞfX ðxÞdx þ ðD1 D2 Þ fX ðxÞdx ; ð3:7Þ 4 r rþD1 for T > 0,
L 2 [Li, Li1], i = 1,2, ,n.
3.3. The fuzzy inventory model involving variable lead time with fuzzy random variable and fuzzy backorder rate In the real situation, due to various uncertainties, it is also difficult for the decision-maker to determine a single value for the backorder (or lost sales), we attempt to modify Eq. (3.7) in Theorem 1 by introducing the fuzziness of the backorder rate (or equivalently, the fuzziness of the lost sales rate). For convenience, we let d 1 b denote the lost sales rate. Thus, we now replace the lost sales rate, d, by the fuzzy number ~d, and consider ~ d as a triangular fuzzy number, i:e:; ~ d ¼ ðd D3 ; d; d þ D4 Þ; where0 < D3 < d and 0 < D4 6 1 d: Then the signed distance of ~ d to ~ 0 is given by Z 1 1 1 d ¼ dð~ d; ~ 0Þ ¼ ½dL ðaÞ þ dU ðaÞda ¼ d þ ðD4 D3 Þ > 0; 2 0 4
ð3:8Þ
ð3:9Þ
where d* is estimate of the lost sales rate in the fuzzy sense. Now, we rewrite Eq. (3.7) in Theorem 1 as follows: GT ;L ðd; D1 ; D2 Þ EAC ðT ; L; D1 ; D2 Þ pffiffiffiffiffiffiffiffiffiffiffiffi A þ CðLÞ DT þh þ kr T þ L ¼ T 2 Z rþD1 Z 1 p 1 þ þ hd EðX rÞþ ðx rÞfX ðxÞdx ðD1 D2 Þ fX ðxÞdx T 4 rþD1 r
ð3:10Þ
We use a to represent the fuzzy point ~ a and use +, , Æ, to represent the operations of fuzzy set: (+), (), (Æ), and then utilizing the triangular fuzzy number ~ d in Eq. (3.8) to fuzzify Eq. (3.10), we obtain
Y.-J. Lin / Computers & Industrial Engineering 54 (2008) 666–676
pffiffiffiffiffiffiffiffiffiffiffiffi A þ CðLÞ DT p ~ þh þ kr T þ L þ þ h~d GT ;L ðd; D1 ; D2 Þ ¼ T 2 T Z rþD1 Z 1 1 þ EðX rÞ ðx rÞfX ðxÞdx ðD1 D2 Þ fX ðxÞdx : 4 r rþD1
671
ð3:11Þ
Eq. (3.11) is triangular fuzzy number. For any triangular fuzzy number, utilizing the signed distance to defuzzify is better than utilizing the centroid to defuzzify, in the viewpoint of membership grade (see, e.g. Yao & Chiang, 2003, (property 2)). Therefore, from Definition 2.3 of Section 2, we utilize the signed distance to defuzzify Eq. (3.11), and then we have EAC ðT ; L; D1 ; D2 ; D3 ; D4 Þ dðGT ;L ð~ d; D1 ; D2 Þ; ~0Þ pffiffiffiffiffiffiffiffiffiffiffiffi A þ CðLÞ DT þh þ kr T þ L ¼ T 2 Z rþD1 Z 1 p 1 þ þ EðX rÞ ðx rÞfX ðxÞdx ðD1 D2 Þ fX ðxÞdx T 4 r rþD1 Z rþD1 Z 1 1 þ þ EðX rÞ ðx rÞfX ðxÞdx ðD1 D2 Þ fX ðxÞdx hdð~d; ~0Þ 4 r rþD1 pffiffiffiffiffiffiffiffiffiffiffiffi A þ CðLÞ DT p h þh þ kr T þ L þ þ hd þ ðD4 D3 Þ ¼ T 2 T 4 Z rþD1 Z 1 1 EðX rÞþ ðx rÞfX ðxÞdx ðD1 D2 Þ fX ðxÞdx 4 r rþD1 h ¼ EAC ðT ; L; D1 ; D2 Þ þ ðD4 D3 Þ 4 Z rþD1 Z 1 1 þ ðx rÞfX ðxÞdx ðD1 D2 Þ fX ðxÞdx : EðX rÞ 4 r rþD1 ð3:12Þ Hence, we have the following theorem. Theorem 2. The estimate of the total expected annual cost in the fuzzy sense by signed distance is given by pffiffiffiffiffiffiffiffiffiffiffiffi A þ CðLÞ DT p h EAC ðT ; L; D1 ; D2 ; D3 ; D4 Þ ¼ þh þ kr T þ L þ þ hd þ ðD4 D3 Þ T 2 T 4 Z rþD1 Z 1 1 þ EðX rÞ ðx rÞfX ðxÞdx ðD1 D2 Þ fX ðxÞdx : ð3:13Þ 4 r rþD1 4. The optimal solution This section provides the solution procedure for the problem of determining the optimal review periodic and the optimal lead time such that the total expected annual cost in the fuzzy sense has a minimum value, while the decision-maker takes D1, D2, D3, D4 satisfying the conditions: 0 < D1 < x r, 0 < D2, 0 < D3 < d and 0 < D4 6 1 d. Given that the protection interval demand X has a normal p.d.f. fX with finite mean D(T + L) and standard pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi deviation r T þ L, and r ¼ DðT þ LÞ þ kr T þ L, we have Z 1 Z rþD1 pffiffiffiffiffiffiffiffiffiffiffiffi EðX rÞþ ¼ ðx rÞfX ðxÞdx ¼ r T þ LwðkÞ > 0; ðx rÞfX ðxÞdx r r
pffiffiffiffiffiffiffiffiffiffiffiffi D1 D1 ; ¼ r T þ L ½/ðkÞ þ kUðkÞ / k þ pffiffiffiffiffiffiffiffiffiffiffiffi þ kU k þ pffiffiffiffiffiffiffiffiffiffiffiffi r T þL r T þL
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Y.-J. Lin / Computers & Industrial Engineering 54 (2008) 666–676
and Z
D1 fX ðxÞdx ¼ 1 U k þ pffiffiffiffiffiffiffiffiffiffiffiffi ; r T þL rþD1 1
where fZ(z) is the p.d.f. ofR the standard normal random variable Z, and wðkÞ /ðkÞ k½1 UðkÞ; 2 k ffi ek2 and UðkÞ ¼ 1 /ðkÞ ¼ p1ffiffiffi /ðxÞdx. 2p Therefore, the Eq. (3.13) becomes to pffiffiffiffiffiffiffiffiffiffiffiffi A þ CðLÞ DT p h þh þ kr T þ L þ þ hd þ ðD4 D3 Þ EAC ðT ; L; D1 ; D2 ; D3 ; D4 Þ ¼ T 2 T 4 i 9 8 pffiffiffiffiffiffiffiffiffiffiffiffih D1 ffi 1 ffi > > pDffiffiffiffiffiffi þ kU k þ = < r T þ L k þ / k þ rpffiffiffiffiffiffi T þL r T þL : ð4:1Þ h i > > 1 ffi ; : 1 ðD1 D2 Þ 1 U k þ pDffiffiffiffiffiffi 4 r T þL
For any given safety factor k and fixed T, it can be chosen suitable values of D1, D2, D3, and D4 by the decision-maker such that # " p h r 5D21 þ 3D1 D2 D1 p ffiffiffiffiffiffiffiffiffiffiffiffi / k þ þ hd þ ðD4 D3 Þ þ þ T 4 r T þL 4ðT þ LÞ3=2 16rðT þ LÞ5=2 8 9 D21 ð3D1 þD2 Þ kr 1 ffi > > pDffiffiffiffiffiffi k þ < = 3 3=2 þ 2 r T þL ðT þLÞ 16r 4ðT þLÞ kr D1 p h p ffiffiffiffiffiffiffiffiffiffiffiffi > U k þ D Þ þ hd þ ðD þ : 4 3 3=2 > > T 4 r T þL 4ðT þ LÞ 1 ffi : / k þ pDffiffiffiffiffiffi ;
hkr 4ðT þ LÞ3=2
r T þL
2
1 ;D2 ;D3 ;D4 Þ In this case, it can be shown that o EAC ðT ;L;D < 0 (see Appendix for a detail proof). And hence, oL2 ** EAC (T, L; D1, D2, D3, D4) is concave in L 2 [Li, Li1]. Therefore, for fixed T, the minimum total expected annual cost of the fuzzfy sense will occur at the end points of the interval [Li, Li1]. By examining the second order sufficient condition for the minimization problem, the minimum value of EAC**(T, L; D1, D2, D3, D4) oEAC ðT ; L;D1 ; D2 ; D3 ;D4 Þ for each L 2 [Li, Li1] will occur at the point T which satisfies ¼ 0, i.e., oT
i 9 8 pffiffiffiffiffiffiffiffiffiffiffiffih 1 ffi 1 ffi > > r T þ L k þ / k þ rpDffiffiffiffiffiffi þ kU k þ rpDffiffiffiffiffiffi = < T þL T þL A þ CðLÞ hD hkr p p ffiffiffiffiffiffiffiffiffiffiffiffi þ ¼ h i 2 2> > 2 T 2 T þ L T : 1 ðD D Þ 1 U k þ pDffiffiffiffiffiffi 1 ffi ; 1 2 4 r T þL h i 9 8 r ffi 1 ffi 1 ffi > > pffiffiffiffiffiffi pDffiffiffiffiffiffi pDffiffiffiffiffiffi k þ / k þ þ kU k þ = < r T þL r T þL 2 T þL p h : þ þ hd þ ðD4 D3 Þ > > T 4 1 ffi ; : þ D1 ð3D1 þD3=22 Þ / k þ pDffiffiffiffiffiffi 8rðT þLÞ
ð4:2Þ
r T þL
Consequently, for given A, D, h, p, b, r, D1, D2, D3, D4, k and each Li(i = 0, 1, 2, . . . , n), we can establish the following algorithm to find the optimal review period T* and optimal lead time L*. Algorithm Step 1. Step 2. Step 3.
For each Li (i = 0, 1, 2, . . . , n), use a numerical search technique to obtain Ti which satisfies Eq. (4.2). For each pair (Ti, Li), compute the corresponding total expected annual cost of Eq. (4.1) ði:e:; EAC ðT i ; Li ; D1 ; D2 ; D3 ; D4 Þ; i ¼ 0; 1; 2; . . . ; nÞ. Find mini¼0;1;2;n EAC ðT i ; Li ; D1 ; D2 ; D3 ; D4 Þ. If QEAC ðT ; L ; D1 ; D2 ; D3 ; D4 Þ ¼ mini¼0;1;2;n EAC ðT i ; Li ; D1 ; D2 ; D3 ; D4 Þ, then (T**,L**ffi ) is the optimal solution. And the optimal target inventory level pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ DðT þ L Þ þ kr T þ L follows.
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673
4.1. Numerical example In order to illustrate the above solution procedure and compare them with those obtained from the crisp model, let us consider an inventory system with the following data used in Ouyang and Chuang (2001): D = 600 units/year, A = $ 200 per order, h = $ 20/units/year, r = 7 units/week, p = $ 50, and the lead time has three components with data shown in Table 1. In the fuzzy case, decision-maker chooses suitable value of D1, D2, D3, D4, which satisfies 0 < D1 < x r, 0 < D2, 0 < D3 < d, 0 < D4 6 1 d and Appendix (A.1), we solve the cases when b = 0.01, 0.2, 0.4, 0.6, 0.99 and q = 0.3 (in this situation, the value of safety factor, k, can be found directly from the normal table, and is given by 0.524). Applying the Algorithm procedure, the results are shown in Table 2. Furthermore, from Table 2, the optimal inventory policy can be found by comparing EAC**(Ti, Li; D1, D2, D3, D4), i = 0, 1, 2, 3; we summarize the results in Table 3. Moreover, in order to compare the results with those obtained from crisp model, we list the optimal solution of crisp model in Table 4. Then, the relative error of review period, target inventory level, lead time and minimum total expected annual cost in the fuzzy sense can be measured by:
Table 1 Lead time data Lead time component i
Normal duration bi (days)
Minimum duration ai (days)
Unit crashing cost ci ($/day)
1 2 3
20 20 16
6 6 9
0.4 1.2 5.0
Table 2 Solution procedure of the algorithm (Ti,Li in weeks) i
Li
C(Li)
Ti
ri
EAC**(Ti, Li,D1,D2,D3,D4)
b = 0.01 0 1 2 3
8 6 4 3
D1 = 0.3 0 5.6 22.4 57.4
D2 = 0.4 15.41 14.95 14.59 14.80
D3 = 0.001 287.96 258.55 230.37 220.86
D4 = 0.009 4968.14 4865.41 4794.23 4851.98
b = 0.2 0 1 2 3
8 6 4 3
D1 = 0.3 0 5.6 22.4 57.4
D2 = 0.4 15.47 15.01 14.66 14.86
D3 = 0.009 288.69 259.32 231.17 221.69
D4 = 0.001 4923.91 4823.81 4755.26 4813.93
b = 0.4 0 1 2 3
8 6 4 3
D1 = 0.3 0 5.6 22.4 57.4
D2 = 0.3 15.54 15.07 14.72 14.93
D3 = 0.001 289.41 260.05 231.94 222.49
D4 = 0.001 4877.64 4780.16 4714.26 4773.87
b = 0.6 0 1 2 3
8 6 4 3
D1 = 0.4 0 5.6 22.4 57.4
D2 = 0.3 15.78 15.30 14.93 15.13
D3 = 0.001 292.36 262.80 234.48 224.92
D4 = 0.009 4900.62 4801.44 4733.33 4790.93
b = 0.99 0 1 2 3
8 6 4 3
D1 = 0.4 0 5.6 22.4 57.4
D2 = 0.3 15.92 15.44 15.08 15.29
D3 = 0.009 293.96 264.48 236.23 226.73
D4 = 0.001 4807.01 4713.46 4650.96 4710.52
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Table 3 Summary of the optimal procedure solution (T**, L** in weeks) b
D1
D2
D3
D4
T**
r**
L**
EAC**(T**, L**, D1, D2, D3, D4)
0.01 0.20 0.40 0.60 0.99
0.3 0.3 0.3 0.4 0.4
0.4 0.4 0.3 0.3 0.3
0.001 0.009 0.001 0.001 0.009
0.009 0.001 0.001 0.009 0.001
14.59 14.66 14.72 14.93 15.08
230.37 231.17 231.94 234.48 236.23
4 4 4 4 4
4794.23 4755.26 4714.26 4733.33 4650.96
Table 4 The optimal solution of crisp model (T*, L* in weeks) b
(T*,L*)
0.01 0.20 0.40 0.60 0.99
(12.62, (12.65, (12.69, (12.72, (12.79,
4) 4) 4) 4) 4)
r*
EAC(T*, L*)
206.75 207.15 207.57 207.99 208.82
3899.21 3878.54 3856.76 3834.96 3792.38
Table 5 The relative error (%) of review period, target inventory level, lead time and minimum total expected annual cost in the fuzzy sense b
D1
D2
D3
D4
R1 (%)
R2 (%)
R3 (%)
R4 (%)
0.01 0.20 0.40 0.60 0.99
0.3 0.3 0.3 0.4 0.4
0.4 0.4 0.3 0.3 0.3
0.001 0.009 0.001 0.001 0.009
0.009 0.001 0.001 0.009 0.001
15.61 15.88 15.99 17.37 17.90
11.42 11.59 11.74 12.73 13.12
0 0 0 0 0
22.95 22.60 22.23 23.42 22.63
1 ;D2 ;D3 ;D4 ÞEACðT ;L Þ R1 ¼ T TT 100%; R2 ¼ r rr 100%; R3 ¼ L LL 100% and R4 ¼ EAC ðT ;L ;DEACðT 100%, ;L Þ respectively. Using the values in the Tables 3 and 4, and these formulas, we obtained the results as summarized in Table 5.
5. Discussions (1) Compare the fuzzy case (3.7) and crisp case (3.1). From (3.1) and (3.7), we have the following results: (i) If 0 < D2 6 D1, then EAC*(T, L, D1, D2) < EAC(T, L). R rþD1 4 ðxrÞfX ðxÞdx (ii) If 0 < D2 D1 < rR 1 , thenEAC*(T, L, D1, D2) < EAC(T, L). rþD
fX ðxÞdx
1 R rþD1 4 ðxrÞfX ðxÞdx > 0, thenEAC*(T, L, D1, D2) > EAC(T, L). (iii) If D2 D1 > rR 1 rþD
fX ðxÞdx
R rþD1 1 4 ðxrÞfX ðxÞdx > 0, then EAC*(T, L, D1, D2) = EAC(T, L), this implies that the fuzzy (iv) If D2 D1 ¼ rR 1 rþD1
fX ðxÞdx
case becomes the crisp case, i.e., the crisp case is a special case of the fuzzy case and crisp random variable is a special case of fuzzy random variable. (2) Compare the Theorems 1 and 2. From Eqs. (3.7) and (3.13), we see that if D3 = D4, EAC**(T, L, D1, D2, D3, D4) = EAC*(T, L, D1, D2) and Theorem 1 is a special case of Theorem 2.
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675
6. Conclusions For a periodic review inventory control system with variable lead time, Ouyang and Chuang (2001) have treated the review period and lead time as decision variables in crisp set. This paper explores a similar inventory model in which not only the expected demand shortage E(X r)+ is fuzzified as a fuzzy random variable in the fuzzy sense, but the lost sales rate, d = 1 b, is also considered to be a fuzzy number ~d. We then obtain the total expected annual cost in the fuzzy sense. For this fuzzy model, we use signed distance method to defuzzify the fuzzy total expected annual cost and obtain an estimate of the total expected annual cost in the fuzzy sense. Furthermore, we provide the solution procedure for the problem of determining the optimal review period and the optimal lead time so that the expected annual cost in the fuzzy sense has a minimum value. In addition, a numerical example is provided and the results of fuzzy and crisp models are compared. Acknowledgements The author greatly appreciates the anonymous referees for their very valuable and helpful suggestions on an earlier version of the paper. Appendix From (4.1), we have
oEAC ðT ; L; D1 ; D2 ; D3 ; D4 Þ ci hkr p h ¼ þ pffiffiffiffiffiffiffiffiffiffiffiffi þ þ hd þ ðD4 D3 Þ oL T 4 T 2 T þL 8 h i 9 r ffi 1 ffi 1 ffi > > k þ / k þ rpDffiffiffiffiffiffi þ kU k þ rpDffiffiffiffiffiffi = < 2pffiffiffiffiffiffi T þL T þL T þL : > > 1 ffi ; : þ D1 ð3D1 þD3=22 Þ / k þ pDffiffiffiffiffiffi 8rðT þLÞ
r T þL
o2 EAC ðT ; L; D1 ; D2 ; D3 ; D4 Þ oL2 hkr p h þ hd þ ðD4 D3 Þ ¼ þ T 4 4ðT þ LÞ3=2 h i 9 8 r 1 ffi 1 ffi pDffiffiffiffiffiffi pDffiffiffiffiffiffi > > þ kU k þ 3=2 k þ / k þ r T þL > > r T þL 4ðT þLÞ > > > > > > 2 3 > > > > > > D D D 1 1 1 > > pffiffiffiffiffiffiffi / k þ pffiffiffiffiffiffiffi k þ þ > > 3=2 r T þL r T þL > > 2rðT þLÞ 6 7 > > r > > ffi4 = < þ 2pffiffiffiffiffiffi 5 T þL D D 1 1 ffi k/ k þ rpffiffiffiffiffiffi 3=2 T þL 2rðT þLÞ > > > > > > > > 3D ð3D þD Þ > > D1 ð3D1 þD2 Þ D D 1 1 2 > > 1 1 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi > > / k þ k þ 5=2 3=2 > > 16rðT þLÞ r T þL r T þL 8rðT þLÞ > > > > > > > > > > D1 ; : / k þ pDffiffiffiffiffiffi 1 ffi r T þL 2rðT þLÞ3=2 hkr p h ¼ þ hd þ ðD4 D3 Þ þ T 4 4ðT þ LÞ3=2 h i 9 8 Dffiffiffiffiffiffi Dffiffiffiffiffiffi r 1 ffi 1 ffi > > p p k þ / k þ þ kU k þ > > r T þL r T þL > > 4ðT þLÞ3=2 > > > > = < 2 2 5D1 þ3D1 D2 D1 ð3D1 þD2 Þ D D 1 1 p ffi þ 2 ffiffiffiffiffiffi ffi : 16rðT þLÞ5=2 / k þ rpffiffiffiffiffiffi k þ T þL r T þL 16r ðT þLÞ3 > > > > > > > > > > ; : / k þ pDffiffiffiffiffiffi 1 ffi r T þL
While the decision-maker can choose the values of D1, D2, D3, and D4 such that
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4ðT þ LÞ
# " p h r 5D21 þ 3D1 D2 D1 p ffiffiffiffiffiffiffiffiffiffiffiffi þ hd þ ðD4 D3 Þ / k þ þ þ 3=2 5=2 T 4 r T þL 4ðT þ LÞ 16rðT þ LÞ kr D1 p h þ hd þ ðD4 D3 Þ þ U k þ pffiffiffiffiffiffiffiffiffiffiffiffi > 3=2 T 4 r T þL 4ðT þ LÞ 8 9 2 ð3D þD Þ D > > 1 ffi = < kr 3=2 þ 1 2 1 23 k þ rpDffiffiffiffiffiffi T þL 16r ðT þLÞ 4ðT þLÞ ; > > 1 ffi ; : / k þ pDffiffiffiffiffiffi
hkr 3=2
ðA:1Þ
r T þL
then we can obtain
o2 EAC ðT ;L;D1 ;D2 ;D3 ;D4 Þ oL2
< 0.
References Chang, H. C., Yao, J. S., & Ouyang, L. Y. (2006). Fuzzy mixture inventory model involving fuzzy random variable lead time demand and fuzzy total demand. European Journal of Operational Research, 169, 65–80. Chang, S. C., Yao, J. S., & Lee, H. M. (1998). Economic reorder point for fuzzy backorder quantity. European Journal of Operational Research, 109, 183–202. Chen, S. H., & Wang, C. C. (1996). Backorder fuzzy inventory model under functional principle. Information Sciences, 95, 71–79. Chiang, J., Yao, J. S., & Lee, H. M. (2005). Fuzzy inventory with backorder defuzzification by signed distance method. Journal of Information Sciences and Engineering, 21, 673–694. Kwakernaak, H. (1978). Fuzzy random variable definitions and theorems. Information Science, 15, 1–29. Kaufmann, A., & Gupta, M. M. (1991). Introduction to fuzzy arithmetic: Theory and applications. New York: Van Nostrand Reinhold. Lee, H. M., & Yao, J. S. (1998). Economic production quantity for fuzzy demand quantity and fuzzy production quantity. European Journal of Operational Research, 109, 203–211. Lee, H. M., & Yao, J. S. (1999). Economic order quantity in fuzzy sense for inventory without backorder model. Fuzzy Sets and Systems, 105, 13–31. Lin, D. C., & Yao, J. S. (2000). Fuzzy economic production for production inventory. Fuzzy Sets and Systems, 111, 465–495. Lin, F. T., & Yao, J. S. (2003). Fuzzy critical path method based on signed distance ranking and statistical confidence-interval estimates. Journal of Supercomputing, 24(3), 305–325. Ouyang, L. Y., & Chuang, B. R. (2001). A periodic review inventory-control system with variable lead time. International Journal of Information and Management Sciences, 12, 1–13. Ouyang, L. Y., & Yao, J. S. (2002). A minimax distribution free for mixed inventory model involving variable lead time with fuzzy demand. Computers and Operations Research, 29, 471–487. Puri, M. L., & Ralescu, D. A. (1986). Fuzzy random variable. Journal of Mathematical Analysis and Applications, 114, 409–422. Petrovic, D., & Sweeney, E. (1994). Fuzzy knowledge-based approach to treating uncertainty in inventory control. Computer Integrated Manufacturing Systems, 7(3), 147–152. Vujosevic, M., Petrovic, D., & Petrovic, R. (1996). EOQ formula when inventory cost is fuzzy. International Journal of Production Economics, 45, 499–504. Yao, J. S., & Lee, H. M. (1996). Fuzzy inventory with or without backorder for fuzzy order quantity. Information Sciences, 93, 283–319. Yao, J. S., & Lee, H. M. (1999). Fuzzy inventory with or without backorder for fuzzy order quantity with trapezoid fuzzy number. Fuzzy Sets and Systems, 105, 311–337. Yao, J. S., Chang, S. C., & Su, J. S. (2000). Fuzzy inventory without backorder for fuzzy quantity and fuzzy total demand quantity. Computers and Operations Research, 27, 935–962. Yao, J. S., & Su, J. S. (2000). Fuzzy inventory with backorder for fuzzy total demand based on interval-value fuzzy sets. European Journal of Operational Research, 124, 390–408. Yao, J. S., & Wu, K. (2000). Ranking fuzzy numbers based on decomposition principle and signed distance. Fuzzy Sets and Systems, 116, 275–288. Yao, J. S., & Chiang, J. (2003). Inventory without backorder with fuzzy total cost and fuzzy storing cost defuzzified by centroid and signed distance. European Journal of Operational Research, 148, 401–409. Yao, J. S., Ouyang, L. Y., & Chang, H. C. (2003). Models for a fuzzy inventory of two replaceable merchandises without backorder based on the signed distance of fuzzy sets. European Journal of Operational Research, 150, 601–616.
Computers & Industrial Engineering 54 (2008) 666–676 www.elsevier.com/locate/dsw
A periodic review inventory model involving fuzzy expected demand short and fuzzy backorder rate Yu-Jen Lin
*
General Education Center, St. John’s University, 499, Sec. 4 Tam King Road, Tamsui, Taipei Hsien 251, Taiwan Received 18 December 2006; received in revised form 2 October 2007; accepted 2 October 2007 Available online 9 October 2007
Abstract The purpose of this paper is to extend [Ouyang, L. Y., Chuang, B. R. (2001). A periodic review inventory-control system with variable lead time. International Journal of Information and Management Sciences, 12, 1–13] periodic review inventory model with variable lead time by considering the fuzziness of expected demand shortage and backorder rate. We fuzzify the expected shortage quantity at the end of cycle and the backorder (or lost sales) rate, and then obtain the fuzzy total expected annual cost. Using the signed distance method to defuzzify, we derive the estimate of total expected annual cost in the fuzzy sense. For the proposed model, we provide a solution procedure to find the optimal review period and optimal lead time in the fuzzy sense so that the total expected annual cost in the fuzzy sense has a minimum value. Furthermore, a numerical example is provided and the results of fuzzy and crisp models are compared. 2007 Elsevier Ltd. All rights reserved. Keywords: Inventory; Periodic review; Fuzzy membership function; Signed distance
1. Introduction In recent years, Yao and others (Chang, Yao, & Ouyang, 2006; Chang, Yao, & Lee, 1998; Lee & Yao, 1998, 1999; Lin & Yao, 2000; Ouyang & Yao, 2002; Yao & Lee, 1996, 1999; Yao, Chang, & Su, 2000; Yao & Su, 2000; Yao & Chiang, 2003; Yao, Ouyang, & Chang, 2003) have contributed several articles by applying the fuzzy sets theory to deal with the production/inventory problems. In Lee and Yao (1999), Yao and Lee (1999) and Yao et al. (2000), they presented the fuzzy EOQ models for the inventory with backorders. Chang et al. (1998) fuzzfied the backorder quantity to be the triangular fuzzy number, while the order quantity is an ordinary variable. By contrast, Yao and Lee (1996) fuzzified the order quantity to be the triangular fuzzy number, while the backorder quantity is an ordinary variable. In Yao and Su (2000), the total demand was fuzzified to be the interval-value fuzzy set. In Lee and Yao (1998) and Lin and Yao (2000), the authors discussed the production inventory problems and proposed the fuzzy EPQ models, where Lee and Yao (1998) fuzzified the demand quantity and production quantity per day and Lin and Yao (2000) fuzzified the production quantity *
Tel.: +886 2 2801 3131x6936; fax: +886 2 2801 3127. E-mail address: [email protected]
0360-8352/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2007.10.002
Y.-J. Lin / Computers & Industrial Engineering 54 (2008) 666–676
667
per cycle, all to be the triangular fuzzy numbers, and the fuzzy total costs were derived using the extension principle. Besides, there are several authors presented various fuzzy inventory models. For example, Petrovic and Sweeney (1994) fuzzified the demand, lead time and inventory level into triangular fuzzy numbers in an inventory control model, and then decided the order quantity by the method of fuzzy propositions. Chen and Wang (1996) fuzzified the demand, ordering cost, inventory cost, and backorder cost into trapezoid fuzzy numbers and used the functional principle to obtain the estimate of total cost in the fuzzy sense. Vujosevic, Petrovic, and Petrovic (1996) fuzzified the ordering cost and holding cost into trapezoid fuzzy numbers in the total cost of an inventory without backorder model and obtained the fuzzy total cost. Then they did the defuzzification by using centroid and gained the total cost in the fuzzy sense. In this paper, we recast the Ouyang and Chuang’s (2001) model by introducing the fuzziness of expected demand shortage and backorder rate, and then using the signed distance method to defuzzify, we derive the estimate of total expected annual cost in the fuzzy sense. The same signed distance method has also been used in Yao and Wu (2000), Yao and Chiang (2003), Lin and Yao (2003), Chiang, Yao, and Lee (2005), and others. This article is organized as follows. In Section 2, some definitions and properties about fuzzy sets, which will be needed later, are introduced. Section 3, explores the inventory problem for a periodic review model with variable lead time. Specifically, in Section 3.1, we consider the crisp case, and given the basis of the notation and assumptions. In Sections 3.2 and 3.3, we consider the fuzzy case, so we fuzzify the expected demand shortage and backorder rate. In Section 4, we derive the optimal review period and optimal lead time by minimizing the estimate of total cost in the fuzzy sense. A numerical example is provided to illustrate the results. In Section 5, we discuss some problems for the proposed model. Finally, some concluding remarks are given in Section 6. 2. Preliminaries In order to consider the fuzziness of an inventory problem, we need some definitions and property relative to this study. We state them in the following. e be a fuzzy set on R = (1,1) and 0 6 a 6 1, the a cut D(a) of D e consists of the points Definition 2.1. Let D x such that le ðxÞ P a, that is, DðaÞ ¼ fxjl ðxÞ P ag. D D e
e on R with which the a cut DðaÞ ¼ fxjl ðxÞ P ag ¼ ½DL ðaÞ; DU ðaÞ Let R be the family of the fuzzy sets D D e exists for every a 2 [0, 1], where DL(a) and DU(a) are continuous functions on a 2 [0, 1], then by the Decomposition Principle (see, e.g. Kaufmann & Gupta, 1991), we have e ¼ [ aDðaÞ ¼ [ ½DL ðaÞ; DU ðaÞ; a: D 06a61
06a61
ð2:1Þ
A new ranking method for fuzzy numbers, namely the signed distance has been first introduced by Yao and Wu (2000), and it has been utilized in many studies (e.g. Yao & Chiang, 2003; Chiang et al., 2005). Here we describe the concept of the signed distance on R which will be needed later. We first consider the signed distance on R. Definition 2.2. For any a and 0 2 R, define the signed distance of a to 0 as d0(a, 0) = a. If a > 0, implies that a is on the right-hand side of origin 0 with distance d0(a, 0) = a; and if a < 0, implies that a is on the left-hand side of origin 0 with distance d0(a, 0) = a. So, we called d0(a, 0) = a is the signed distance of a to 0. e 2 R, from Eq. (2.1), we have For any D e ¼ [ ½DL ðaÞ; DU ðaÞ; a: D 06a61
ð2:2Þ
And for every a 2 [0, 1], there is an one-to-one mapping between the a -level fuzzy interval [DL(a), DU (a);a] and real interval [DL(a), DU(a)], that is, the following correspondence is one-to-one mapping:
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Y.-J. Lin / Computers & Industrial Engineering 54 (2008) 666–676
½DL ðaÞ; DU ðaÞ $ ½DL ðaÞ; DU ðaÞ; a:
ð2:3Þ
We shall use this relation later. e to the From Definition 2.2, the signed distance of the left end point DL(a) of the a cut [DL(a), DU(a)] of D origin 0 is DL(a), and the signed distance of the right end point DU(a) to the origin 0 is DU(a). Their average, 1 ½DL ðaÞ þ DU ðaÞ, is defined as the signed distance of a cut [DL(a), DU(a)] to 0, that is, we define the signed 2 distance of the interval [DL(a), DU(a)] to 0 as: 1 1 d 0 ð½DL ðaÞ; DU ðaÞ; 0Þ ¼ ½d 0 ðDL ðaÞ; 0Þ þ d 0 ðDU ðaÞ; 0Þ ¼ ½DL ðaÞ þ DU ðaÞ: ð2:4Þ 2 2 Further, from Eqs. (2.3) and (2.4), the signed distance of a-level fuzzy interval [DL(a), DU(a);a] to the fuzzy point ~ 0 can be defined as: 1 dð½DL ðaÞ; DU ðaÞ; a; ~ 0Þ ¼ d 0 ð½DL ðaÞ; DU ðaÞ; 0Þ ¼ ½DL ðaÞ þ DU ðaÞ: 2 e 2 Thus, from Eqs. (2.2) and (2.5), we can define the signed distance of a fuzzy set D follows.
ð2:5Þ P
to ~0 as
e 2 R, define the signed distance of D e to ~0 as Definition 2.3. For D
e ~ dð D; 0Þ ¼
Z
1
dð½DL ðaÞ; DU ðaÞ; a; ~ 0Þda ¼ 0
1 2
Z
1
½DL ðaÞ þ DU ðaÞda: 0
e ¼ ða; b; cÞ to ~0 is dð D; e ~0Þ ¼ 1 ða þ 2b þ cÞ. Property 1. The signed distance of the fuzzy number D 4 e is D(a) = [DL(a), DU(a)], where DL (a) = a + (b a)a and Proof. For a 2 [0, 1], the a cut of D R e ~0Þ ¼ 1 1 ½DL ðaÞ þ DU ðaÞda ¼ 1 ða þ 2b þ cÞ. h DU(a) = c (c b)a. By Definition 2.3, we have dð D; 2 0 4 3. A fuzzy periodic review inventory model 3.1. The crisp inventory model We first review the crisp period review inventory model which was asserted by Ouyang and Chuang (2001). Under the assumptions that: 1. The inventory level is reviewed every T units of time. A sufficient quantity is ordered up to the target inventory level r, and the ordering quantity is arrived after L units of time. 2. The length of the lead time L does not exceed an inventory cycle time T so that there is never more than a single order outstanding in any cycle. 3. The target inventory level r = expected demand during the protection interval + safety stock (SS), and pffiffiffiffiffiffiffiffiffiffiffiffi SS = k· (standard deviation of protection interval demand), i.e., r ¼ DðT þ LÞ þ kr T þ L, where D is the average demand per year, r is the standard deviation of demand per unit time, k is the safety factor and satisfies P(X > r) = q, q is given to represent the allowable stockout probability during the protection interval. 4. The lead time L consists of n mutually independent components. The ith component has a minimum duration ai and normal duration bi, and a crashing cost per unit time ci, where c1 6 c2 6 6 cn. Then, it is clear that the reduction of lead time should be first on component 1 because it has the minimum unit crashing cost, and P then component 2, and so on. n 5. Let L0 ¼ j¼1 bj and Li be the length of lead time components 1, 2, . . . , i crashed to their minimum Pn with P i duration, then Li can be expressed as Li ¼ j¼1 bj j¼1 ðbj aj Þ; i ¼ 1; 2; . . . ; n; andPthe lead time crashing cost C(L) per cycle for a given L 2 [Li, Li1] is given by CðLÞ ¼ ci ðLi1 LÞ þ i1 j¼1 cj ðbj aj Þ.
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Ouyang and Chuang (2001) established the following total expected annual cost function: A DT pEðX rÞþ CðLÞ þ þ h r DL þ ð1 bÞEðX rÞ þ þ T 2 T T h i p ffiffiffiffiffiffiffiffiffiffiffiffi A þ CðLÞ DT p þ ¼ þh þ kr T þ L þ þ hð1 bÞ EðX rÞ ; T 2 T
EACðT ; LÞ ¼
ð3:1Þ
where A = fixed ordering cost per order h = inventory holding cost per item per year b = backorder rate, b 2 [0, 1] p = stockout cost per unit short X = the protection interval, T + L, demand which has a normal probability density function (p.d.f.) fX with pffiffiffiffiffiffiffiffiffiffiffiffi finite mean D(T + L) and standard deviation r T þ L E(X r)+ = the expected shortage quantity at the end of the cycle, and is given by R1 pffiffiffiffiffiffiffiffiffiffiffiffi þ EðX rÞ ¼ r ðx rÞfX ðxÞdx ¼ r T þ LwðkÞ > 0, where w(k) / (k) k[1 U(k)], /(k) and U(k) denote the standard normal p.d.f. and distribution function, respectively. Note that if (R, B, P) be the probability space, then the protection interval demand X is a random variable on (R, B, P). 3.2. The fuzzy inventory model involving variable lead time with fuzzy random variable and fuzzy expected demand short In this subsection, we use the concepts of fuzzy set to extend the Ouyang and Chuang (2001) model for a periodic review inventory control system with variable lead time. In real business transactions, owing to the unstable environments, it would not be easily determined the exact values of the expected shortage quantity at the end of the cycle by the decision-maker. Thus the decision-maker determines the shortage quantity at the end of the cycle is always near x r as X = x. Corresponde here. From Kwakernaak ing to the crisp random variable X in (3.1), we consider the fuzzy random variable X (1978) and Puri and Ralescu (1986), the fuzzy random variable can be defined as a mapping from R to a family of triangular fuzzy numbers, i.e., e :x2R!X e ðxÞ ¼ ðx D1 ; x; x þ D2 Þ; X
if
X ¼ x;
ð3:2Þ
where D1, D2 are determined by the decision-maker and should satisfy the conditions: 0 < D1 < x r, and 0 < D2, where r < x. Corresponding to the crisp random variable Y = X r, we set the fuzzy random variable as e ðxÞðÞ~r ¼ ðx r D1 ; x r; x r þ D2 Þ; if X r ¼ x r Ye : x 2 R ! Ye ðxÞ ¼ X
ð3:3Þ
where ~r is a fuzzy point. Thus, we have the membership function of Ye ðxÞ as follows: 8 tðxrD1 Þ ; x r D1 6 t 6 x r; > D1 < ðxrþD Þt 2 le ðtÞ ¼ ; x r 6 t 6 x r þ D2 ; Y ðxÞ D2 > : 0; otherwise:
ð3:4Þ
And the a cut of Ye ðxÞ is ½ Ye ðxÞL ðaÞ; Ye ðxÞU ðaÞ ¼ ½x r D1 þ aD1 ; x r þ ð1 aÞD2 ;
for
a 2 ½0; 1:
ð3:5Þ
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Since Ye ðxÞ 2 R, the estimate of expectation of fuzzy random variable Ye in the fuzzy sense as follows: Z Z i 1 1 1 he þ e e þ E ð X ðÞ~rÞ ¼ E ð Y Þ ¼ Y ðxÞL ðaÞ þ Ye ðxÞU ðaÞ fX ðxÞdxda 2 0 rþD1 Z 1 1 1 2ðx rÞ þ ðD2 D1 Þ fX ðxÞdx ¼ 2 rþD1 2 Z Z 1 1 1 ðx rÞfX ðxÞdx þ ðD2 D1 ÞfX ðxÞdx ¼ 4 rþD1 rþD1 Z rþD1 Z 1 1 ¼ EðX rÞþ ðx rÞfX ðxÞdx ðD1 D2 Þ fX ðxÞdx: ð3:6Þ 4 r rþD1 Theorem 1. In the Eq. (3.1), using the fuzzy random variable of Eqs. (3.2) and (3.3), then the crisp random e ðÞ~r. Thus, we obtained the total expected annual variable Y = X r, is changed to fuzzy random variable Ye ¼ X cost in the fuzzy sense h i pffiffiffiffiffiffiffiffiffiffiffiffi A þ CðLÞ DT p þh þ kr T þ L þ þ hð1 bÞ EAC ðT ; L; D1 ; D2 Þ ¼ T 2 T Z rþD1 Z 1 1 þ EðX rÞ ðx rÞfX ðxÞdx ðD1 D2 Þ fX ðxÞdx 4 r rþD1 hp i ¼ EACðT ; LÞ þ hð1 bÞ T Z rþD1 Z 1 1 ðx rÞfX ðxÞdx þ ðD1 D2 Þ fX ðxÞdx ; ð3:7Þ 4 r rþD1 for T > 0,
L 2 [Li, Li1], i = 1,2, ,n.
3.3. The fuzzy inventory model involving variable lead time with fuzzy random variable and fuzzy backorder rate In the real situation, due to various uncertainties, it is also difficult for the decision-maker to determine a single value for the backorder (or lost sales), we attempt to modify Eq. (3.7) in Theorem 1 by introducing the fuzziness of the backorder rate (or equivalently, the fuzziness of the lost sales rate). For convenience, we let d 1 b denote the lost sales rate. Thus, we now replace the lost sales rate, d, by the fuzzy number ~d, and consider ~ d as a triangular fuzzy number, i:e:; ~ d ¼ ðd D3 ; d; d þ D4 Þ; where0 < D3 < d and 0 < D4 6 1 d: Then the signed distance of ~ d to ~ 0 is given by Z 1 1 1 d ¼ dð~ d; ~ 0Þ ¼ ½dL ðaÞ þ dU ðaÞda ¼ d þ ðD4 D3 Þ > 0; 2 0 4
ð3:8Þ
ð3:9Þ
where d* is estimate of the lost sales rate in the fuzzy sense. Now, we rewrite Eq. (3.7) in Theorem 1 as follows: GT ;L ðd; D1 ; D2 Þ EAC ðT ; L; D1 ; D2 Þ pffiffiffiffiffiffiffiffiffiffiffiffi A þ CðLÞ DT þh þ kr T þ L ¼ T 2 Z rþD1 Z 1 p 1 þ þ hd EðX rÞþ ðx rÞfX ðxÞdx ðD1 D2 Þ fX ðxÞdx T 4 rþD1 r
ð3:10Þ
We use a to represent the fuzzy point ~ a and use +, , Æ, to represent the operations of fuzzy set: (+), (), (Æ), and then utilizing the triangular fuzzy number ~ d in Eq. (3.8) to fuzzify Eq. (3.10), we obtain
Y.-J. Lin / Computers & Industrial Engineering 54 (2008) 666–676
pffiffiffiffiffiffiffiffiffiffiffiffi A þ CðLÞ DT p ~ þh þ kr T þ L þ þ h~d GT ;L ðd; D1 ; D2 Þ ¼ T 2 T Z rþD1 Z 1 1 þ EðX rÞ ðx rÞfX ðxÞdx ðD1 D2 Þ fX ðxÞdx : 4 r rþD1
671
ð3:11Þ
Eq. (3.11) is triangular fuzzy number. For any triangular fuzzy number, utilizing the signed distance to defuzzify is better than utilizing the centroid to defuzzify, in the viewpoint of membership grade (see, e.g. Yao & Chiang, 2003, (property 2)). Therefore, from Definition 2.3 of Section 2, we utilize the signed distance to defuzzify Eq. (3.11), and then we have EAC ðT ; L; D1 ; D2 ; D3 ; D4 Þ dðGT ;L ð~ d; D1 ; D2 Þ; ~0Þ pffiffiffiffiffiffiffiffiffiffiffiffi A þ CðLÞ DT þh þ kr T þ L ¼ T 2 Z rþD1 Z 1 p 1 þ þ EðX rÞ ðx rÞfX ðxÞdx ðD1 D2 Þ fX ðxÞdx T 4 r rþD1 Z rþD1 Z 1 1 þ þ EðX rÞ ðx rÞfX ðxÞdx ðD1 D2 Þ fX ðxÞdx hdð~d; ~0Þ 4 r rþD1 pffiffiffiffiffiffiffiffiffiffiffiffi A þ CðLÞ DT p h þh þ kr T þ L þ þ hd þ ðD4 D3 Þ ¼ T 2 T 4 Z rþD1 Z 1 1 EðX rÞþ ðx rÞfX ðxÞdx ðD1 D2 Þ fX ðxÞdx 4 r rþD1 h ¼ EAC ðT ; L; D1 ; D2 Þ þ ðD4 D3 Þ 4 Z rþD1 Z 1 1 þ ðx rÞfX ðxÞdx ðD1 D2 Þ fX ðxÞdx : EðX rÞ 4 r rþD1 ð3:12Þ Hence, we have the following theorem. Theorem 2. The estimate of the total expected annual cost in the fuzzy sense by signed distance is given by pffiffiffiffiffiffiffiffiffiffiffiffi A þ CðLÞ DT p h EAC ðT ; L; D1 ; D2 ; D3 ; D4 Þ ¼ þh þ kr T þ L þ þ hd þ ðD4 D3 Þ T 2 T 4 Z rþD1 Z 1 1 þ EðX rÞ ðx rÞfX ðxÞdx ðD1 D2 Þ fX ðxÞdx : ð3:13Þ 4 r rþD1 4. The optimal solution This section provides the solution procedure for the problem of determining the optimal review periodic and the optimal lead time such that the total expected annual cost in the fuzzy sense has a minimum value, while the decision-maker takes D1, D2, D3, D4 satisfying the conditions: 0 < D1 < x r, 0 < D2, 0 < D3 < d and 0 < D4 6 1 d. Given that the protection interval demand X has a normal p.d.f. fX with finite mean D(T + L) and standard pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi deviation r T þ L, and r ¼ DðT þ LÞ þ kr T þ L, we have Z 1 Z rþD1 pffiffiffiffiffiffiffiffiffiffiffiffi EðX rÞþ ¼ ðx rÞfX ðxÞdx ¼ r T þ LwðkÞ > 0; ðx rÞfX ðxÞdx r r
pffiffiffiffiffiffiffiffiffiffiffiffi D1 D1 ; ¼ r T þ L ½/ðkÞ þ kUðkÞ / k þ pffiffiffiffiffiffiffiffiffiffiffiffi þ kU k þ pffiffiffiffiffiffiffiffiffiffiffiffi r T þL r T þL
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Y.-J. Lin / Computers & Industrial Engineering 54 (2008) 666–676
and Z
D1 fX ðxÞdx ¼ 1 U k þ pffiffiffiffiffiffiffiffiffiffiffiffi ; r T þL rþD1 1
where fZ(z) is the p.d.f. ofR the standard normal random variable Z, and wðkÞ /ðkÞ k½1 UðkÞ; 2 k ffi ek2 and UðkÞ ¼ 1 /ðkÞ ¼ p1ffiffiffi /ðxÞdx. 2p Therefore, the Eq. (3.13) becomes to pffiffiffiffiffiffiffiffiffiffiffiffi A þ CðLÞ DT p h þh þ kr T þ L þ þ hd þ ðD4 D3 Þ EAC ðT ; L; D1 ; D2 ; D3 ; D4 Þ ¼ T 2 T 4 i 9 8 pffiffiffiffiffiffiffiffiffiffiffiffih D1 ffi 1 ffi > > pDffiffiffiffiffiffi þ kU k þ = < r T þ L k þ / k þ rpffiffiffiffiffiffi T þL r T þL : ð4:1Þ h i > > 1 ffi ; : 1 ðD1 D2 Þ 1 U k þ pDffiffiffiffiffiffi 4 r T þL
For any given safety factor k and fixed T, it can be chosen suitable values of D1, D2, D3, and D4 by the decision-maker such that # " p h r 5D21 þ 3D1 D2 D1 p ffiffiffiffiffiffiffiffiffiffiffiffi / k þ þ hd þ ðD4 D3 Þ þ þ T 4 r T þL 4ðT þ LÞ3=2 16rðT þ LÞ5=2 8 9 D21 ð3D1 þD2 Þ kr 1 ffi > > pDffiffiffiffiffiffi k þ < = 3 3=2 þ 2 r T þL ðT þLÞ 16r 4ðT þLÞ kr D1 p h p ffiffiffiffiffiffiffiffiffiffiffiffi > U k þ D Þ þ hd þ ðD þ : 4 3 3=2 > > T 4 r T þL 4ðT þ LÞ 1 ffi : / k þ pDffiffiffiffiffiffi ;
hkr 4ðT þ LÞ3=2
r T þL
2
1 ;D2 ;D3 ;D4 Þ In this case, it can be shown that o EAC ðT ;L;D < 0 (see Appendix for a detail proof). And hence, oL2 ** EAC (T, L; D1, D2, D3, D4) is concave in L 2 [Li, Li1]. Therefore, for fixed T, the minimum total expected annual cost of the fuzzfy sense will occur at the end points of the interval [Li, Li1]. By examining the second order sufficient condition for the minimization problem, the minimum value of EAC**(T, L; D1, D2, D3, D4) oEAC ðT ; L;D1 ; D2 ; D3 ;D4 Þ for each L 2 [Li, Li1] will occur at the point T which satisfies ¼ 0, i.e., oT
i 9 8 pffiffiffiffiffiffiffiffiffiffiffiffih 1 ffi 1 ffi > > r T þ L k þ / k þ rpDffiffiffiffiffiffi þ kU k þ rpDffiffiffiffiffiffi = < T þL T þL A þ CðLÞ hD hkr p p ffiffiffiffiffiffiffiffiffiffiffiffi þ ¼ h i 2 2> > 2 T 2 T þ L T : 1 ðD D Þ 1 U k þ pDffiffiffiffiffiffi 1 ffi ; 1 2 4 r T þL h i 9 8 r ffi 1 ffi 1 ffi > > pffiffiffiffiffiffi pDffiffiffiffiffiffi pDffiffiffiffiffiffi k þ / k þ þ kU k þ = < r T þL r T þL 2 T þL p h : þ þ hd þ ðD4 D3 Þ > > T 4 1 ffi ; : þ D1 ð3D1 þD3=22 Þ / k þ pDffiffiffiffiffiffi 8rðT þLÞ
ð4:2Þ
r T þL
Consequently, for given A, D, h, p, b, r, D1, D2, D3, D4, k and each Li(i = 0, 1, 2, . . . , n), we can establish the following algorithm to find the optimal review period T* and optimal lead time L*. Algorithm Step 1. Step 2. Step 3.
For each Li (i = 0, 1, 2, . . . , n), use a numerical search technique to obtain Ti which satisfies Eq. (4.2). For each pair (Ti, Li), compute the corresponding total expected annual cost of Eq. (4.1) ði:e:; EAC ðT i ; Li ; D1 ; D2 ; D3 ; D4 Þ; i ¼ 0; 1; 2; . . . ; nÞ. Find mini¼0;1;2;n EAC ðT i ; Li ; D1 ; D2 ; D3 ; D4 Þ. If QEAC ðT ; L ; D1 ; D2 ; D3 ; D4 Þ ¼ mini¼0;1;2;n EAC ðT i ; Li ; D1 ; D2 ; D3 ; D4 Þ, then (T**,L**ffi ) is the optimal solution. And the optimal target inventory level pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ DðT þ L Þ þ kr T þ L follows.
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673
4.1. Numerical example In order to illustrate the above solution procedure and compare them with those obtained from the crisp model, let us consider an inventory system with the following data used in Ouyang and Chuang (2001): D = 600 units/year, A = $ 200 per order, h = $ 20/units/year, r = 7 units/week, p = $ 50, and the lead time has three components with data shown in Table 1. In the fuzzy case, decision-maker chooses suitable value of D1, D2, D3, D4, which satisfies 0 < D1 < x r, 0 < D2, 0 < D3 < d, 0 < D4 6 1 d and Appendix (A.1), we solve the cases when b = 0.01, 0.2, 0.4, 0.6, 0.99 and q = 0.3 (in this situation, the value of safety factor, k, can be found directly from the normal table, and is given by 0.524). Applying the Algorithm procedure, the results are shown in Table 2. Furthermore, from Table 2, the optimal inventory policy can be found by comparing EAC**(Ti, Li; D1, D2, D3, D4), i = 0, 1, 2, 3; we summarize the results in Table 3. Moreover, in order to compare the results with those obtained from crisp model, we list the optimal solution of crisp model in Table 4. Then, the relative error of review period, target inventory level, lead time and minimum total expected annual cost in the fuzzy sense can be measured by:
Table 1 Lead time data Lead time component i
Normal duration bi (days)
Minimum duration ai (days)
Unit crashing cost ci ($/day)
1 2 3
20 20 16
6 6 9
0.4 1.2 5.0
Table 2 Solution procedure of the algorithm (Ti,Li in weeks) i
Li
C(Li)
Ti
ri
EAC**(Ti, Li,D1,D2,D3,D4)
b = 0.01 0 1 2 3
8 6 4 3
D1 = 0.3 0 5.6 22.4 57.4
D2 = 0.4 15.41 14.95 14.59 14.80
D3 = 0.001 287.96 258.55 230.37 220.86
D4 = 0.009 4968.14 4865.41 4794.23 4851.98
b = 0.2 0 1 2 3
8 6 4 3
D1 = 0.3 0 5.6 22.4 57.4
D2 = 0.4 15.47 15.01 14.66 14.86
D3 = 0.009 288.69 259.32 231.17 221.69
D4 = 0.001 4923.91 4823.81 4755.26 4813.93
b = 0.4 0 1 2 3
8 6 4 3
D1 = 0.3 0 5.6 22.4 57.4
D2 = 0.3 15.54 15.07 14.72 14.93
D3 = 0.001 289.41 260.05 231.94 222.49
D4 = 0.001 4877.64 4780.16 4714.26 4773.87
b = 0.6 0 1 2 3
8 6 4 3
D1 = 0.4 0 5.6 22.4 57.4
D2 = 0.3 15.78 15.30 14.93 15.13
D3 = 0.001 292.36 262.80 234.48 224.92
D4 = 0.009 4900.62 4801.44 4733.33 4790.93
b = 0.99 0 1 2 3
8 6 4 3
D1 = 0.4 0 5.6 22.4 57.4
D2 = 0.3 15.92 15.44 15.08 15.29
D3 = 0.009 293.96 264.48 236.23 226.73
D4 = 0.001 4807.01 4713.46 4650.96 4710.52
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Table 3 Summary of the optimal procedure solution (T**, L** in weeks) b
D1
D2
D3
D4
T**
r**
L**
EAC**(T**, L**, D1, D2, D3, D4)
0.01 0.20 0.40 0.60 0.99
0.3 0.3 0.3 0.4 0.4
0.4 0.4 0.3 0.3 0.3
0.001 0.009 0.001 0.001 0.009
0.009 0.001 0.001 0.009 0.001
14.59 14.66 14.72 14.93 15.08
230.37 231.17 231.94 234.48 236.23
4 4 4 4 4
4794.23 4755.26 4714.26 4733.33 4650.96
Table 4 The optimal solution of crisp model (T*, L* in weeks) b
(T*,L*)
0.01 0.20 0.40 0.60 0.99
(12.62, (12.65, (12.69, (12.72, (12.79,
4) 4) 4) 4) 4)
r*
EAC(T*, L*)
206.75 207.15 207.57 207.99 208.82
3899.21 3878.54 3856.76 3834.96 3792.38
Table 5 The relative error (%) of review period, target inventory level, lead time and minimum total expected annual cost in the fuzzy sense b
D1
D2
D3
D4
R1 (%)
R2 (%)
R3 (%)
R4 (%)
0.01 0.20 0.40 0.60 0.99
0.3 0.3 0.3 0.4 0.4
0.4 0.4 0.3 0.3 0.3
0.001 0.009 0.001 0.001 0.009
0.009 0.001 0.001 0.009 0.001
15.61 15.88 15.99 17.37 17.90
11.42 11.59 11.74 12.73 13.12
0 0 0 0 0
22.95 22.60 22.23 23.42 22.63
1 ;D2 ;D3 ;D4 ÞEACðT ;L Þ R1 ¼ T TT 100%; R2 ¼ r rr 100%; R3 ¼ L LL 100% and R4 ¼ EAC ðT ;L ;DEACðT 100%, ;L Þ respectively. Using the values in the Tables 3 and 4, and these formulas, we obtained the results as summarized in Table 5.
5. Discussions (1) Compare the fuzzy case (3.7) and crisp case (3.1). From (3.1) and (3.7), we have the following results: (i) If 0 < D2 6 D1, then EAC*(T, L, D1, D2) < EAC(T, L). R rþD1 4 ðxrÞfX ðxÞdx (ii) If 0 < D2 D1 < rR 1 , thenEAC*(T, L, D1, D2) < EAC(T, L). rþD
fX ðxÞdx
1 R rþD1 4 ðxrÞfX ðxÞdx > 0, thenEAC*(T, L, D1, D2) > EAC(T, L). (iii) If D2 D1 > rR 1 rþD
fX ðxÞdx
R rþD1 1 4 ðxrÞfX ðxÞdx > 0, then EAC*(T, L, D1, D2) = EAC(T, L), this implies that the fuzzy (iv) If D2 D1 ¼ rR 1 rþD1
fX ðxÞdx
case becomes the crisp case, i.e., the crisp case is a special case of the fuzzy case and crisp random variable is a special case of fuzzy random variable. (2) Compare the Theorems 1 and 2. From Eqs. (3.7) and (3.13), we see that if D3 = D4, EAC**(T, L, D1, D2, D3, D4) = EAC*(T, L, D1, D2) and Theorem 1 is a special case of Theorem 2.
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6. Conclusions For a periodic review inventory control system with variable lead time, Ouyang and Chuang (2001) have treated the review period and lead time as decision variables in crisp set. This paper explores a similar inventory model in which not only the expected demand shortage E(X r)+ is fuzzified as a fuzzy random variable in the fuzzy sense, but the lost sales rate, d = 1 b, is also considered to be a fuzzy number ~d. We then obtain the total expected annual cost in the fuzzy sense. For this fuzzy model, we use signed distance method to defuzzify the fuzzy total expected annual cost and obtain an estimate of the total expected annual cost in the fuzzy sense. Furthermore, we provide the solution procedure for the problem of determining the optimal review period and the optimal lead time so that the expected annual cost in the fuzzy sense has a minimum value. In addition, a numerical example is provided and the results of fuzzy and crisp models are compared. Acknowledgements The author greatly appreciates the anonymous referees for their very valuable and helpful suggestions on an earlier version of the paper. Appendix From (4.1), we have
oEAC ðT ; L; D1 ; D2 ; D3 ; D4 Þ ci hkr p h ¼ þ pffiffiffiffiffiffiffiffiffiffiffiffi þ þ hd þ ðD4 D3 Þ oL T 4 T 2 T þL 8 h i 9 r ffi 1 ffi 1 ffi > > k þ / k þ rpDffiffiffiffiffiffi þ kU k þ rpDffiffiffiffiffiffi = < 2pffiffiffiffiffiffi T þL T þL T þL : > > 1 ffi ; : þ D1 ð3D1 þD3=22 Þ / k þ pDffiffiffiffiffiffi 8rðT þLÞ
r T þL
o2 EAC ðT ; L; D1 ; D2 ; D3 ; D4 Þ oL2 hkr p h þ hd þ ðD4 D3 Þ ¼ þ T 4 4ðT þ LÞ3=2 h i 9 8 r 1 ffi 1 ffi pDffiffiffiffiffiffi pDffiffiffiffiffiffi > > þ kU k þ 3=2 k þ / k þ r T þL > > r T þL 4ðT þLÞ > > > > > > 2 3 > > > > > > D D D 1 1 1 > > pffiffiffiffiffiffiffi / k þ pffiffiffiffiffiffiffi k þ þ > > 3=2 r T þL r T þL > > 2rðT þLÞ 6 7 > > r > > ffi4 = < þ 2pffiffiffiffiffiffi 5 T þL D D 1 1 ffi k/ k þ rpffiffiffiffiffiffi 3=2 T þL 2rðT þLÞ > > > > > > > > 3D ð3D þD Þ > > D1 ð3D1 þD2 Þ D D 1 1 2 > > 1 1 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi > > / k þ k þ 5=2 3=2 > > 16rðT þLÞ r T þL r T þL 8rðT þLÞ > > > > > > > > > > D1 ; : / k þ pDffiffiffiffiffiffi 1 ffi r T þL 2rðT þLÞ3=2 hkr p h ¼ þ hd þ ðD4 D3 Þ þ T 4 4ðT þ LÞ3=2 h i 9 8 Dffiffiffiffiffiffi Dffiffiffiffiffiffi r 1 ffi 1 ffi > > p p k þ / k þ þ kU k þ > > r T þL r T þL > > 4ðT þLÞ3=2 > > > > = < 2 2 5D1 þ3D1 D2 D1 ð3D1 þD2 Þ D D 1 1 p ffi þ 2 ffiffiffiffiffiffi ffi : 16rðT þLÞ5=2 / k þ rpffiffiffiffiffiffi k þ T þL r T þL 16r ðT þLÞ3 > > > > > > > > > > ; : / k þ pDffiffiffiffiffiffi 1 ffi r T þL
While the decision-maker can choose the values of D1, D2, D3, and D4 such that
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Y.-J. Lin / Computers & Industrial Engineering 54 (2008) 666–676
4ðT þ LÞ
# " p h r 5D21 þ 3D1 D2 D1 p ffiffiffiffiffiffiffiffiffiffiffiffi þ hd þ ðD4 D3 Þ / k þ þ þ 3=2 5=2 T 4 r T þL 4ðT þ LÞ 16rðT þ LÞ kr D1 p h þ hd þ ðD4 D3 Þ þ U k þ pffiffiffiffiffiffiffiffiffiffiffiffi > 3=2 T 4 r T þL 4ðT þ LÞ 8 9 2 ð3D þD Þ D > > 1 ffi = < kr 3=2 þ 1 2 1 23 k þ rpDffiffiffiffiffiffi T þL 16r ðT þLÞ 4ðT þLÞ ; > > 1 ffi ; : / k þ pDffiffiffiffiffiffi
hkr 3=2
ðA:1Þ
r T þL
then we can obtain
o2 EAC ðT ;L;D1 ;D2 ;D3 ;D4 Þ oL2
< 0.
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