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An application of fuzzy sets theory to the EOQ model with imperfect quality items
Computers & Operations Research 31 (2004) 2079 – 2092
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An application of fuzzy sets theory to the EOQ model with imperfect quality items Hung-Chi Chang Department of Logistics Engineering and Management, National Taichung Institute of Technology, 129 Sec 3, San-Min Road, Taichung City 404, Taiwan, ROC
Abstract This article investigates the inventory problem for items received with imperfect quality, where, upon the arrival of order lot, 100% screening process is performed and the items of imperfect quality are sold as a single batch at a discounted price, prior to receiving the next shipment. The objective is to determine the optimal order lot size to maximize the total pro2t. We 2rst propose a model with fuzzy defective rate. Then, the model with fuzzy defective rate and fuzzy annual demand is presented. For each case, we employ the signed distance, a ranking method for fuzzy numbers, to 2nd the estimate of total pro2t per unit time in the fuzzy sense, and then derive the corresponding optimal lot size. Numerical examples are provided to illustrate the results of proposed models. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Inventory; Imperfect quality; Fuzzy set; Signed distance
1. Introduction In the classical economic production/order quantity (EPQ/EOQ) models, the items produced/ received are implicitly assumed to be with perfect quality. However, it may not always be the case. Due to imperfect production process, natural disasters, damage or breakage in transit, or for many other reasons, the lot sizes produced/received may contain some defective items. To capture the real situations better, several researchers considered the above scenarios in formulating the production/inventory models and studied the e=ect of imperfect quality on lot sizing policy. Speci2cally, we note that in Rosenblatt and Lee’s [1] study, they assumed that the defective items could be
E-mail address: [email protected] (H.-C. Chang). 0305-0548/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0305-0548(03)00166-7
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reworked instantaneously at a cost and found that the presence of defective products motivates smaller lot sizes. While in a recent article, Salameh and Jaber [2] assumed that the defective items could be sold as a single batch at a discounted price prior to receiving the next shipment, and found that the economic lot size quantity tends to increase as the average percentage of imperfect quality items increase. Later, Goyal and CIardenas-BarrIon [3] reconsidered the task done in [2] and presented a simple approach for determining the optimal lot size. Besides, more survey related to the studies of imperfect quality can be found in [2]. On the other hand, from [2] and the literature survey in them, we note that some of the previous researchers assumed that the defective rate in lot sizes produced/received is a 2xed constant, while others assumed it as a random variable with a known probability distribution to reKect the uncertainty of imperfect quality. In real environments, although we know that the defective rate may have a little change from one lot to another, however, it may lack historical data to estimate the probability distribution, especially, for the items o=ered by the new producers/suppliers. Thus, the traditional probability theory and statistical method cannot be used properly. As stated in Karsak and Tolga [4], an e=ective way to express factors including Kexibility, quality of the products, enhanced response to market demand, and reduction in inventory, which can neither be assessed by crisp values nor random processes, is using linguistic variables or fuzzy numbers. In this article, we shall then apply the fuzzy sets concept 2rst introduced by Zadeh [5] to deal with quality-related problem, as well as the imprecise demand. In recent years, several researchers have applied the fuzzy sets theory and technique to develop and solve the production/inventory problems. For example, Park [6] and VujoOseviIc et al. [7] extended the classical EOQ model by introducing the fuzziness of ordering cost and holding cost. Chen and Wang [8] fuzzi2ed the demand, ordering cost, inventory cost, and backorder cost into trapezoidal fuzzy numbers in EOQ model with backorder. Roy and Maiti [9] presented a fuzzy EOQ model with demand-dependent unit cost under limited storage capacity. Chang et al. [10] presented a fuzzy model for inventory with backorder, where the backorder quantity was fuzzi2ed as the triangular fuzzy number. Lee and Yao [11] and Lin and Yao [12] discussed the production inventory problems, where Lee and Yao [11] fuzzi2ed the demand quantity and production quantity per day, and Lin and Yao [12] fuzzi2ed the production quantity per cycle, treating all as the triangular fuzzy numbers. Yao et al. [13] proposed the EOQ model in the fuzzy sense, where both order quantity and total demand were fuzzi2ed as the triangular fuzzy numbers. Ouyang and Yao [14] presented a mixture inventory model with variable lead-time, where the annual demand was fuzzi2ed as the triangular fuzzy number and as the statistic fuzzy number. In this article, we investigate the inventory problem for items received with imperfect quality. Building upon the work of Salameh and Jaber [2], we propose two fuzzy models. The 2rst model incorporates the fuzziness of defective rate. In the second model, not only the defective rate but also the annual demand is considered as a fuzzy number. For each fuzzy case, we employ Yao and Wu’s [15] ranking method for fuzzy numbers to 2nd the estimate of total pro2t per unit time in the fuzzy sense, and then derive the corresponding optimal lot size. This paper is organized as follows. In Section 2, a brief review of Salameh and Jaber’s [2] model is given. In Section 3, some de2nitions and properties about fuzzy sets related to this study are introduced. Section 4 presents two fuzzy models as described earlier. Section 5 provides two numerical examples to illustrate the results of the proposed models. Section 6 summarizes the work done in this paper.
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2. Brief review of Salameh and Jaber’s model Recently, Salameh and Jaber [2] developed an EOQ model for items received with imperfect quality. They considered the following situations. (i) A lot of size y is delivered instantaneously with a purchasing price c per unit and an ordering cost K. (ii) Each lot received contains percentage defectives p. A 100% screening process of the lot is conducted at a rate x units per unit time with unit screening cost w, upon receiving the order. (iii) Items of poor quality are kept in stock and sold prior to receiving the next shipment as a single batch at a discounted price v per unit, whereas the good-quality item is sold at the regular price s per unit. (iv) Shortages are not allowed. The number of good items in each order (i.e., lot size less defective items), denoted by N (y; p), is at least equal to the demand during screening time t; that is, N (y; p) = y − py ¿ Dt, where D is the demand per year. The behavior of the inventory level is as that shown in Fig. 1, where T is the cycle length, py is the number of defectives withdrawn from inventory, and t is the total screening time of y units ordered per cycle. For a given percentage of defective items in each lot of size y, the total pro2t per cycle TP(y) is established as TP(y) = (sales of good-quality items + sales of imperfect quality items) − (procurement cost + screening cost + holding cost) y(1 − p)T py2 = [sy(1 − p) + vyp] − (K + cy) + wy + h + : 2 x
Inventory level
py y t
Time T
Fig. 1. Behavior of the inventory level over time.
(1)
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Dividing the total pro2t per cycle TP(y) by the cycle length T (=(1 − p)y=D), the total pro2t per unit time (we denote it by W (y)) of the model is obtained as follows: hy hy 1 hy(1 − p) K W (y) = D s − v + +D v− − −c−w− : (2) x x y 1−p 2 Furthermore, Salameh and Jaber considered the uncertainty of defectives and assumed that the defective rate p is a random variable with a known probability density function (pdf) f(p). They then derived the optimal order lot size by maximizing the expected value of W (y), i.e., E[W (y)] in which the terms 1=(1 − p) and p in (2) are replaced by their expected values E[1=(1 − p)] and E[p], respectively. As mentioned earlier, although the statistical method can be used to deal with the uncertainty, however, in some cases, it may lack historical data to estimate the above pdf f(p). Since the defective rate may have a slight change from one lot to another, it can be described by fuzzy language as “p˜ = the real defective rate is around p”. In this article, we shall represent the defective rate as a fuzzy number. Moreover, in (2), the demand per year D is a 2xed value, but in a practical situation it is diWcult for the decision-makers to assess it by a crisp value. Hence, we shall consider it as a fuzzy number also.
3. Preliminaries Before presenting the fuzzy inventory models, we introduce some de2nitions and properties about fuzzy numbers with relevant operations. Much of these can be found in [15–17]. De nition 1. For 0 6 6 1, the fuzzy set a˜ de2ned on R = (−∞; ∞) is called an -level fuzzy point if the membership function of a˜ is given by ; x = a; a˜ (x) = (3) 0; x = a: De nition 2. The fuzzy set A˜ = (a; b; c), where a ¡ b ¡ c and de2ned on R, is called the triangular fuzzy number, if the membership function of A˜ is given by (x − a)=(b − a); a 6 x 6 b; A˜(x) = (c − x)=(c − b); b 6 x 6 c; (4) 0; otherwise: Remark 1. (i) When = 1, the membership function of the 1-level fuzzy point a˜1 becomes the characteristic function, i.e., a˜1 (x) = 1 if x = a and a˜1 (x) = 0 if x = a. In this case, the real number a ∈ R is the same as the fuzzy point a˜1 except for their representations. (ii) If c = b = a, then the triangular fuzzy number A˜ = (a; b; c) is identical to the 1-level fuzzy point a˜1 .
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De nition 3. For 0 6 6 1, the fuzzy set [a ; b ] de2ned on R is called an -level fuzzy interval if the membership function of [a ; b ] is given by ; a 6 x 6 b; [a ;b ] (x) = (5) 0; otherwise: De nition 4. Let B˜ be a fuzzy set on R, and 0 6 6 1. The -cut B() of B˜ consists of points x such that B˜ (x) ¿ , that is, B() = {x|B˜ (x) ¿ }. Decomposition Principle. Let B˜ be a fuzzy set on R and 0 6 6 1. Suppose the -cut of B˜ to be a closed interval [BL (); BU ()], that is, B() = [BL (); BU ()]. Then, we have (see, e.g., [16]) B˜ = B() (6) 0661
or B˜ (x) =
CB() (x);
(7)
0661
where (i) B() is a fuzzy set with membership function ; x ∈ B(); B() (x) = 0; otherwise: (ii) CB() (x) is a characteristic function of B(), that is, 1; x ∈ B(); CB() (x) = 0; x ∈ B(): Remark 2. From the Decomposition Principle and (5), we obtain B() = [BL () ; BU () ] B˜ = 0661
or B˜ (x) =
0661
(8)
0 6 61
CB() (x) =
[BL () ;BU () ] (x):
(9)
0 6 61
For any a, b, c, d, k ∈ R, a ¡ b, and c ¡ d, the interval operations are as follows [16]: (i) (ii)
[a; b](+)[c; d] = [a + c; b + d]:
[a; b](−)[c; d] = [a − d; b − c]: [ka; kb]; k ¿ 0; (iii) k(·)[a; b] = [kb; ka]; k ¡ 0:
(10)
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Further, for a ¿ 0 and c ¿ 0, (iv) (v)
[a; b](·)[c; d] = [ac; bd]: a b ; : [a; b](÷)[c; d] = d c
Next, as in Yao and Wu [15], we introduce the concept of the signed distance of fuzzy set. We 2rst consider the signed distance on R. De nition 5. For any a and 0 ∈ R, de2ne the signed distance from a to 0 as d0 (a; 0) = a. If a ¿ 0, the distance from a to 0 is a = d0 (a; 0); if a ¡ 0, the distance from a to 0 is −a = −d0 (a; 0). Hence, d0 (a; 0) = a is called the signed distance from a to 0. Let ! be the family of all fuzzy sets B˜ de2ned on R with which the -cut B() = [BL (); BU ()] exists for every ∈ [0; 1], and both BL () and BU () are continuous functions on ∈ [0; 1]. Then, for any B˜ ∈ !, from (8) we have [BL () ; BU () ]: (11) B˜ = 0661
From De2nition 5, the signed distance of two end points, BL () and BU (), of the -cut B() = [BL (); BU ()] of B˜ to the origin 0 is d0 (BL (); 0) = BL () and d0 (BU (); 0) = BU (), respectively. Their average, (BL () + BU ())=2, is taken as the signed distance of -cut [BL (); BU ()] to 0. That is, the signed distance of interval [BL (); BU ()] to 0 is de2ned as d0 ([BL (); BU ()]; 0) = [d0 (BL (); 0) + d0 (BU (); 0)]=2 = (BL () + BU ())=2. In addition, for every ∈ [0; 1], there is a one-to-one mapping between the -level fuzzy interval [BL () ; BU () ] and the real interval [BL (); BU ()], that is, the following correspondence is one-to-one mapping: [BL () ; BU () ] ↔ [BL (); BU ()]:
(12)
Also, the 1-level fuzzy point 0˜ 1 is mapping to the real number 0. Hence, the signed distance of [BL () ; BU () ] to 0˜ 1 can be de2ned as d([BL () ; BU () ]; 0˜ 1 ) = d0 ([BL (); BU ()]; 0) = (BL () + BU ())=2. Moreover, for B˜ ∈ !, since the above function is continuous on 0 6 6 1, we can use the integration to obtain the mean value of the signed distance as follows: 1 1 1 d([BL () ; BU () ]; 0˜ 1 ) d = (BL () + BU ()) d: (13) 2 0 0 Then, from (11) and (13), we have the following de2nition. De nition 6. For B˜ ∈ !, de2ne the signed distance of B˜ to 0˜ 1 (i.e., y-axis) as 1 1 1 ˜ ˜ ˜ 01 ) = d([BL () ; BU () ]; 01 ) d = (BL () + BU ()) d: d(B; 2 0 0 According to De2nition 6, we obtain the following property.
(14)
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Property 1. For the triangular fuzzy number A˜ = (a; b; c), the -cut of A˜ is A() = [AL (); AU ()], ∈ [0; 1], where AL () = a + (b − a) and AU () = c − (c − b). The signed distance of A˜ to 0˜ 1 is ˜ 0˜ 1 ) = 1 (a + 2b + c): d(A; 4
(15)
˜ G˜ ∈ !, where B˜ = 0661 [BL () ; BU () ] and G˜ = 0661 Furthermore, for two fuzzy sets B; [GL () ; GU () ], and k ∈ R, using (10) and (12), we have ˜ [(BL () + GL ()) ; (BU () + GU ()) ]: (i) B(+) G˜ = 0661
(ii)
˜ B(−) G˜ =
(iii)
[(kBL ()) ; (kBU ()) ]; 61 06 ˜ k 1 (·)B˜ = [(kBU ()) ; (kBL ()) ]; 0661 ˜ 01 ;
[(BL () − GU ()) ; (BU () − GL ()) ]:
0661
k ¿ 0;
(16)
k ¡ 0; k = 0:
From the above and De2nition 6, we obtain the following property. ˜ G˜ ∈ ! and k ∈ R, Property 2. For two fuzzy sets B; (i)
˜ 0˜ 1 ): ˜ 0˜ 1 ) = d(B; ˜ 0˜ 1 ) + d(G; ˜ d(B(+) G;
(ii)
˜ 0˜ 1 ) = d(B; ˜ 0˜ 1 ): ˜ ˜ 0˜ 1 ) − d(G; d(B(−) G;
(iii)
˜ 0˜ 1 ) = kd(B; ˜ 0˜ 1 ): d(k˜1 (·)B;
(17)
4. Fuzzy EOQ models for items with imperfect quality 4.1. Model with fuzzy defective rate In this subsection, we modify the crisp defective rate model shown in (2) by incorporating the fuzziness of the defective rate p (or equivalently, the fuzziness of the good-quality rate, 1 − p). For convenience, we let q = 1 − p and rewrite (2) as D hy K hyq hy − + +c+w−v + : (18) W (y) = D s − v + x q x y 2 Then, we fuzzify q to be a triangular fuzzy number, q˜ = (q − $1 ; q; q + $2 ), where 0 ¡ $1 ¡ q and 0 ¡ $2 6 1 − q, $1 and $2 are determined by the decision-makers. In this case, the total pro2t per unit time is a fuzzy value also, which is expressed as hy K 1 hy hy − D + +c+w−v + q˜ : (19) W˜ ≡ W˜ (y) = D s − v + x x y q˜ 2
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Now, we defuzzify W˜ using the signed distance method. From Property 2, the signed distance of ˜ W to 0˜ 1 is given by hy K 1 ˜ hy hy − D + +c+w−v d ; 01 + d(q; ˜ 0˜ 1 ) ; (20) d(W˜ ; 0˜ 1 ) = D s − v + x x y q˜ 2 where d(q; ˜ 0˜ 1 ) and d(1= q; ˜ 0˜ 1 ) are measured as follows. From Property 1, the signed distance of fuzzy number q˜ to 0˜ 1 is d(q; ˜ 0˜ 1 ) = 14 [(q − $1 ) + 2q + (q + $2 )] = q + 14 ($2 − $1 ):
(21)
Also, the left and right end points of the -cut (0 6 6 1) of q˜ are qL () = (q − $1 ) + $1 ¿ 0
and
qU () = (q + $2 ) − $2 ¿ 0;
(22)
respectively. Since 0 ¡ qL () ¡ qU (), from (10), the left and right end points of the -cut (0 6 6 1) of 1= q˜ are 1 1 1 1 1 1 = and = ; (23) () = () = q L qU () (q + $2 ) − $2 q U qL () (q − $1 ) + $1 respectively. Then, from De2nition 6, the signed distance of 1= q˜ to 0˜ 1 is 1 1 1 1 1 1 q 1 q 1 ˜ ; () + () d = ln − ln d ; 01 = q˜ 2 0 q L q U 2 $1 q − $ 1 $2 q + $ 2 which is positive since $1 ¿ 0, $2 ¿ 0, ln(q=q − $1 ) ¿ 0, and ln(q=q + $2 ) ¡ 0. Substituting the results of (21) and (24) into (20), we have D hy K hy − + +c+w−v W ∗ (y) ≡ d(W˜ ; 0˜ 1 ) = D s − v + x 2 x y $2 − $1 hy q 1 q 1 q+ : + ln − ln × $1 q − $ 1 $2 q + $ 2 2 4
(24)
(25)
W ∗ (y) is regarded as the estimate of the total pro2t per unit time in the fuzzy sense. The objective of the problem is to determine the optimal order lot size, y∗ , such that W ∗ (y) has a maximum value. Utilizing the classical optimization technique, we take the 2rst and second derivatives of W ∗ (y) with respect to y, and obtain K $ 2 − $1 1 h q 1 q dW ∗ (y) Dh D h − − q+ (26) − ln − ln = dy x 2 x y2 $1 q − $ 1 $ 2 q + $ 2 2 4 and d 2 W ∗ (y) DK =− 3 dy2 y
1 q 1 q ln − ln $1 q − $ 1 $2 q + $ 2
:
(27)
Because d 2 W ∗ (y)=dy2 ¡ 0, i.e., W ∗ (y) is concave in y, and hence the maximum value of W ∗ (y) will occur at the point that satis2es dW ∗ (y)=dy = 0. Setting (26) equal to zero and solving for y, we obtain the optimal lot size: q DK( $11 ln q−q$1 − $12 ln q+$ ) 2 ∗ y = : (28) q q $2 − $1 D 1 1 h[(q + 4 ) + x ( $1 ln q−$1 − $2 ln q+$ − 2)] 2
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Remark 3. If $1 = $2 = $, then (21) and (24) reduce to d(q; ˜ 0˜ 1 ) = q and d(1= q; ˜ 0˜ 1 ) = (1=2$) ln (q + $)=(q − $), respectively. Furthermore, by the L’Hospital’s rule, we know that lim$→0 (1=2$) ln (q + $)=(q − $) = 1=q. As a result, when $1 = $2 = $ → 0, the estimate of the total pro2t per unit time in the fuzzy sense (25) is identical to the crisp case (18). Hence, the crisp defective rate model is a special case of the fuzzy model presented here. Besides, for the optimal order lot size (28), when $1 = $2 = $, it reduces to D 1 q+$ 1 q+$ ∗ =h q + ln −2 : DK ln y = $ q−$ x $ q−$ By taking q = 1 (i.e., the defective rate p = 0) and $ → 0, we then have y∗ = 2DK=h, which is the classical EOQ formula. 4.2. Model with fuzzy defective rate and fuzzy annual demand This subsection further incorporates the fuzziness of demand per year into the previous fuzzy model. The crisp annual demand D in (19) is fuzzi2ed as the triangular fuzzy number, D˜ = (D − $3 ; D; D + $4 ), where $3 and $4 are determined by the decision-makers and should satisfy the conditions 0 ¡ $3 ¡ D and 0 ¡ $4 . For this case, we express the fuzzy total pro2t per unit time as hy K D˜ hy hy ˜ ˜ ˜ − + +c+w−v + q˜ : (29) Z ≡ Z(y) = D s − v + x x y q˜ 2 Again, the approach employed in the previous subsection is utilized to 2nd the estimate of the total pro2t per unit time in the fuzzy sense. The signed distance of Z˜ to 0˜ 1 is given by D˜ ˜ hy hy hy K ˜ ˜ ˜ ˜ ˜ d(D; 01 ) − + +c+w−v d ; 01 + d(q; ˜ 01 ) ;(30) d(Z; 01 ) = s − v + x x y q˜ 2 ˜ 0˜ 1 ) = D + ($4 − $3 )=4 follows in the same where d(q; ˜ 0˜ 1 ) = q + ($2 − $1 )=4 (from (21)), and d(D; way. ˜ q; Next, we calculate the signed distance d(D= ˜ 0˜ 1 ). The left and right end points of the -cut (0 6 6 1) of D˜ are DL () = (D − $3 ) + $3 ¿ 0
and
DU () = (D + $4 ) − $4 ¿ 0;
(31)
respectively. The left and right end points of the -cut (0 6 6 1) of q˜ are as shown in (22). Since ˜ q˜ are 0 ¡ qL () ¡ qU (), from (10), the left and right end points of the -cut (0 6 6 1) of D= D D DL () (D − $3 ) + $3 DU () (D + $4 ) − $4 = and = ; (32) () = () = q L qU () (q + $2 ) − $2 q U qL () (q − $1 ) + $1 ˜ q˜ to 0˜ 1 can be derived as follows: respectively. Thus, the signed distance of D= D˜ ˜ 1 1 D D ; 01 = d () + () d q˜ 2 0 q L q U q $4 (q$3 + D$2 ) q + $2 $3 1 (q$4 + D$1 ) − ; ln − + ln = 2 q − $ 1 $1 q $2 $21 $22
(33)
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˜ q˜ as stated in (32) are positive which is positive, since the left and right end points of the -cut of D= continuous functions on 0 6 6 1; hence the above result of de2nite integral should be positive. ˜ 0˜ 1 ), and d(D= ˜ q; Using the results of d(q; ˜ 0˜ 1 ), d(D; ˜ 0˜ 1 ) in (30), we obtain ˜ 0˜ 1 ) Z ∗ (y) ≡ d(Z; $ 4 − $3 hy $2 − $1 hy K hy D+ − q+ + + +c+w−v = s−v+ x 4 2 4 x y q $4 (q$3 + D$2 ) q + $2 $3 1 (q$4 + D$1 ) − : (34) ln − + ln × 2 q − $ 1 $1 q $2 $21 $22 Z ∗ (y) is regarded as the estimate of total pro2t per unit time in the fuzzy sense. Now, we determine the optimal order lot size y∗∗ to maximize the total pro2t function Z ∗ (y) of (34). It can be shown that Z ∗ (y) is concave in y. Then, by solving the 2rst-order necessary condition, i.e., dZ ∗ (y)=dy = 0, we obtain the optimal lot size Kx& ; (35) y∗∗ = $2 − $ 1 h[& + x(q + 4 ) − 2(D + $4 −4 $3 )] where &=
(q$4 + D$1 ) q $4 (q$3 + D$2 ) q + $2 $3 − : ln − + ln q − $ 1 $1 q $2 $21 $22
Remark 4. When $3 = $4 = 0, it is clear the fuzzy annual demand reduces to D˜ = (D; D; D). In this ˜ 0˜ 1 ) = D and (33) reduces to case, the signed distance d(D; D˜ ˜ D 1 1 ˜ q 1 q ; 01 = ; 01 ; d = Dd ln − ln q˜ 2 $ 1 q − $ 1 $2 q + $ 2 q˜ which implies that the estimate of total pro2t per unit time in the fuzzy sense (34) reduces to (25), and the optimal order lot size (35) reduces to (28). That is, the fuzzy model proposed in Section 4.1 is a special case of the fuzzy model presented in this subsection. Furthermore, from Remark 3, we note that the optimal order lot size (35) can also be reduced to the classical EOQ formula.
5. Numerical examples To illustrate the results of the proposed models, we consider an inventory system with the data in Salameh and Jaber [2]: demand rate D = 50 000 units=year, ordering cost K = $100=cycle, holding cost h = $5=unit=year, screening rate x = 1 unit/min (equivalently, x = 175 200 units/year), screening cost w=$0.5/unit, purchase cost c = $25=unit, selling price of good-quality items s = $50=unit, and selling price of imperfect-quality items v = $20=unit. But instead of assuming that the defective rate p is uniformly distributed with mean 0.02, we consider that the defective rate is around p = 0.02 (i.e., the good-quality rate is around q = 1 − p = 0:98).
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Table 1 Optimal solution for the model with fuzzy defective rate $1
$2
q˜
0.0005 0.0010 0.0025 0.0050 0.0075 0.0150 0.0200
0.0200 0.0150 0.0075 0.0050 0.0025 0.0010 0.0005
(0.9795, (0.9790, (0.9775, (0.9750, (0.9725, (0.9650, (0.9600,
0.98, 0.98, 0.98, 0.98, 0.98, 0.98, 0.98,
1.0000) 0.9950) 0.9875) 0.9850) 0.9825) 0.9810) 0.9805)
d(q; ˜ 0˜ 1 )
d(1= q; ˜ 0˜ 1 )
y∗
W ∗ (y∗ )
Rel y (%)
Rel W (%)
0.9849 0.9835 0.9813 0.9800 0.9788 0.9765 0.9751
1.01540 1.01680 1.01912 1.02042 1.02172 1.02409 1.02556
1429.6 1431.0 1433.3 1434.6 1435.9 1438.2 1439.6
$1; 213; 661:75 $1; 213; 273:51 $1; 212; 632:34 $1; 212; 272:30 $1; 211; 911:03 $1; 211; 253:66 $1; 210; 848:14
−0:349 −0:251 −0:091 0.000 0.091 0.251 0.349
0.114 0.082 0.029 0.000 −0:030 −0:084 −0:118
Example 1. For the model proposed in Section 4.1, where q is fuzzi2ed as a triangular fuzzy number q˜ = (q − $1 ; q; q + $2 ) with 0 ¡ $1 ¡ q and 0 ¡ $2 6 1 − q, we solve the optimal order lot size y∗ and the maximum total pro2t per year W ∗ (y∗ ) in the fuzzy sense for various given sets of ($1 ; $2 ). Note that in practical situations, $1 and $2 are determined by the decision-makers due to the uncertainty of the problem. The results are summarized in Table 1. Furthermore, we compare the results of fuzzy case with those of crisp case. The optimal order lot size yc∗ and the maximum total pro2t per year W (yc∗ ) of crisp case can be derived easily from (18) using the classical optimization technique as performed in Section 4. After manipulation, we obtain yc∗ = 1434:6 units and W (yc∗ ) = $1 212 274:79. Then, the relative variation between fuzzy case and crisp case for the optimal order lot size and the maximum total pro2t per year can be measured by Rel y = [(y∗ − yc∗ )=yc∗ ] × 100% and Rel W = {[W ∗ (y∗ ) − W (yc∗ )]=W (yc∗ )} × 100%, respectively. The numerical results are listed in the last two columns of Table 1. From Table 1, we observe that ˜ 0˜ 1 ) ¿ q(= 0:98) and (i) When $1 ¡ $2 , the estimate of good-quality rate in the fuzzy sense d(q; d(1= q; ˜ 0˜ 1 ) ¡ 1=q(=1:02041). In this case, y∗ ¡ yc∗ and W ∗ (y∗ ) ¿ W (yc∗ ), which result in Rel y ¡ 0 and Rel W ¿ 0. Further, as the absolute value |$1 − $2 | decreases, both |Rel y| and |Rel W | decrease, which means the smaller the di=erence between $1 and $2 (i.e., the less the uncertainty of good-quality rate), the smaller the variation of the solutions between fuzzy case and crisp case. (ii) When $1 ¿ $2 , d(q; ˜ 0˜ 1 ) ¡ q and d(1= q; ˜ 0˜ 1 ) ¿ 1=q. In this case, y∗ ¿ yc∗ and W ∗ (y∗ ) ¡ W (yc∗ ), which result in Rel y ¿ 0 and Rel W ¡ 0. Also, as |$1 −$2 | increases, both |Rel y| and |Rel W | increase. (iii) When $1 = $2 (=0:005); d(q; ˜ 0˜ 1 ) = q and d(1= q; ˜ 0˜ 1 ) ∼ = 1=q. In this case, the solutions of fuzzy case are almost equal to those of the crisp case, and hence Rel y = 0 and Rel W = 0. This is consistent with that shown in Remark 3 i.e., when $1 = $2 → 0, the fuzzy case becomes the crisp case. Example 2. For the model proposed in Section 4.2, where q and D are fuzzi2ed as the triangular fuzzy numbers q˜ = (q − $1 ; q; q + $2 ) and D˜ = (D − $3 ; D; D + $4 ), respectively, we solve the optimal order lot size y∗∗ and the maximum total pro2t per year Z ∗ (y∗∗ ) in the fuzzy sense for various given
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Table 2 Optimal solution for the model with fuzzy defective rate and fuzzy annual demand $1
$2
0.0010
0.0150
$3 500 1000 1500
0.005
0.005
500 1000 1500
0.0150
0.0010
500 1000 1500
$4
d(q; ˜ 0˜ 1 )
˜ 0˜ 1 ) d(D;
˜ q; d(D= ˜ 0˜ 1 )
y∗∗
Z ∗ (y∗∗ )
Rel y (%)
Rel W (%)
350 500 650 750 1000 1250 1000 1500 2000
0.9835 0.9835 0.9835 0.9835 0.9835 0.9835 0.9835 0.9835 0.9835
49 963 50 000 50 038 49 938 50 000 50 063 49 875 50 000 50 125
50 803.25 50 841.54 50 879.84 50 779.10 50 842.92 50 906.74 50 716.65 50 844.29 50 971.93
1430.5 1431.0 1431.5 1430.1 1431.0 1431.9 1429.2 1431.0 1432.8
$1 212 354:11 $1 213 265:80 $1 214 177:49 $1 211 738:61 $1 213 258:09 $1 214 777:57 $1 210 211:42 $1 213 250:38 $1 216 289:35
−0:286 −0:251 −0:216 −0:314 −0:251 −0:188 −0:376 −0:251 −0:125
0.007 0.082 0.157 −0:044 0.081 0.206 −0:170 0.080 0.331
350 500 650 750 1000 1250 1000 1500 2000
0.9800 0.9800 0.9800 0.9800 0.9800 0.9800 0.9800 0.9800 0.9800
49 963 50 000 50 038 49 938 50 000 50 063 49 875 50 000 50 125
50 983.32 51 021.72 51 060.11 50 958.59 51 022.59 51 086.58 50 895.47 51 023.45 51 151.44
1434.0 1434.6 1435.1 1433.7 1434.6 1435.5 1432.8 1434.6 1436.4
$1 211 356:33 $1 212 267:44 $1 213 178:54 $1 210 744:05 $1 212 262:57 $1 213 781:08 $1 209 220:67 $1 212 257:70 $1 215 294:73
−0:042 0.000 0.035 −0:063 0.000 0.063 −0:125 0.000 0.125
−0:076 −0:001 0.075 −0:126 −0:001 0.124 −0:252 −0:001 0.249
350 500 650 750 1000 1250 1000 1500 2000
0.9765 0.9765 0.9765 0.9765 0.9765 0.9765 0.9765 0.9765 0.9765
49963 50 000 50 038 49 938 50 000 50 063 49 875 50 000 50 125
51 167.39 51 206.05 51 244.71 51 143.02 51 207.45 51 271.89 51 079.99 51 208.86 51 337.73
1437.7 1438.2 1438.7 1437.3 1438.2 1439.1 1436.4 1438.2 1440.0
$1; 210 336:15 $1; 211 245:78 $1 212 155:42 $1 209 721:86 $1 211 237:91 $1 212 753:97 $1 208 197:93 $1 211 230:04 $1 214 262:15
0.216 0.251 0.286 0.188 0.251 0.314 0.125 0.251 0.376
−0:160 −0:085 −0:010 −0:211 −0:086 0.040 −0:336 −0:086 0.164
Note: Rel y = [(y∗∗ − yc∗ )=yc∗ ] × 100% and Rel W = {[Z ∗∗ (y∗∗ ) − W (yc∗ )]=W (yc∗ )} × 100%.
sets of ($1 ; $2 ) and ($3 ; $4 ) that satisfy the conditions 0 ¡ $1 ¡ q, 0 ¡ $2 6 1 − q, 0 ¡ $3 ¡ D, and 0 ¡ $4 . Besides, as performed in Example 1, we compare the solutions of fuzzy case with those of crisp case by calculating the relative variation between them. The results are summarized in Table 2. From Table 2, we observe that ˜ 0˜ 1 ) increases (i) for 2xed ($1 ; $2 ), as the estimate of annual demand in the fuzzy sense d(D; ∗∗ by varying ($3 ; $4 ), the optimal lot size y and the maximum total pro2t per year Z ∗ (y∗∗ ) increase, and hence Rel y and Rel W increase, while their values could be positive, negative, or zero;
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(ii) for 2xed ($3 ; $4 ), as the estimate of good-quality rate in the fuzzy sense d(q; ˜ 0˜ 1 ) decreases ∗∗ by varying ($1 ; $2 ), the optimal lot size y increases but the maximum total pro2t per year Z ∗ (y∗∗ ) decreases, and hence Rel y increases but Rel W decreases; (iii) for various sets of ($1 ; $2 ) and ($3 ; $4 ), their joint e=ects on the optimal lot sizes are as follows: ˜ q; when the signed distance d(D= ˜ 0˜ 1 ) ¡ D=q(= 51020:41), we have Rel y ¡ 0 (i.e., y∗∗ ¡ yc∗ ) ˜ ˜ and, conversely, when d(D= q; ˜ 01 ) ¿ D=q, we have Rel y ¿ 0 (i.e., y∗∗ ¿ yc∗ ). Besides, when ˜ q; $1 = $2 (= 0:005) and $3 = $4 (= 500; 1000; 1500), d(D= ˜ 0˜ 1 ) ∼ = D=q, we then have Rel y = 0 ∗ ∗∗ (i.e., y = yc ), and in this case Rel W → 0. Note that from the above examples although we cannot ascertain which of the solutions (obtained from the fuzzy models or crisp model) are better, the advantages of fuzzy models themselves are that they keep the uncertainties, which can capture real situations better than the crisp model does. In addition, the decision-makers can use the solutions derived from the fuzzy models to perform the sensitivity analysis, so as to examine the e=ects of uncertainties.
6. Conclusions This paper proposed two fuzzy models for an inventory problem with imperfect-quality items. In the 2rst model, the good-quality rate (defective rate) is represented by a fuzzy number, while the annual demand is treated as a 2xed constant. In the second model, not only the good-quality rate but also the annual demand is represented by a fuzzy number. For each fuzzy model, a method of defuzzi2cation, namely the signed distance, is employed to 2nd the estimate of total pro2t per unit time in the fuzzy sense, and then the corresponding optimal order lot size is derived to maximize the total pro2t. Besides, we show that in some cases, the proposed fuzzy models can be reduced to the crisp model and the optimal order lot size in the fuzzy sense can be reduced to the classical EOQ formula. Two numerical examples are carried out to investigate the behavior of our proposed models, and the results are compared with those obtained from the crisp model. Uncertainties of product quality and demand are inherent in real inventory problems. The inventory models that include these uncertainties are often based on the concept of randomness and thus handled by probability theory. However, in practice, there may be a lack of historical data to estimate the probability distributions for the uncertain factors that are modeled by random variables. For such a situation, using probability theory is not appropriate. With these perspectives, it is worthwhile to reconsider the inventory with quality-related problem presented by Salameh and Jaber [2] and provide an alternative approach. Finally, we would like to point out the performance of using signed distance method for defuzzi2cation in this study. The previous researches on fuzzy production/inventory problems (e.g., [10–13]) often employed the centroid method to obtain the estimate of total cost in the fuzzy sense. To achieve this task, the membership function of fuzzy total cost has to be found 2rst using the extension principle, while the derivations are very complex and cumbersome, especially, for the case where the fuzzy number is located in the denominator (as the good-quality rate q˜ appears in Eqs. (19) and (29)). In this study, we showed that using the decomposition principle with the signed distance method can solve similar problems easier.
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Acknowledgements The author greatly appreciates the anonymous referees for their very valuable and helpful suggestions on an earlier version of this paper. References [1] Rosenblatt MJ, Lee HL. Economic production cycles with imperfect production processes. IIE Transactions 1986;18:48–55. [2] Salameh MK, Jaber MY. Economic production quantity model for items with imperfect quality. International Journal of Production Economics 2000;64:59–64. [3] Goyal SK, CIardenas-BarrIon LE. Note on: economic production quantity model for items with imperfect quality—a practical approach. International Journal of Production Economics 2002;77:85–7. [4] Karsak EE, Tolga E. Fuzzy mutli-criteria decision-making procedure for evaluating advanced manufacturing system investments. International Journal of Production Economics 2001;69:49–64. [5] Zadeh L. Fuzzy sets. Information and Control 1965;8:338–53. [6] Park KS. Fuzzy-set theoretic interpretation of economic order quantity. IEEE Transactions on Systems, Man, and Cybernetics 1987;SMC-17:1082–4. [7] VujoOseviIc M, PetroviIc D, PetroviIc R. EOQ formula when inventory cost is fuzzy. International Journal of Production Economics 1996;45:499–504. [8] Chen SH, Wang CC. Backorder fuzzy inventory model under functional principle. Information Sciences 1996;95: 71–9. [9] Roy TK, Maiti M. A fuzzy EOQ model with demand-dependent unit cost under limited storage capacity. European Journal of Operational Research 1997;99:425–32. [10] Chang SC, Yao JS, Lee HM. Economic reorder point for fuzzy backorder quantity. European Journal of Operational Research 1998;109:183–202. [11] Lee HM, Yao JS. Economic production quantity for fuzzy demand quantity and fuzzy production quantity. European Journal of Operational Research 1998;109:203–11. [12] Lin DC, Yao JS. Fuzzy economic production for production inventory. Fuzzy Sets and Systems 2000;111:465–95. [13] Yao JS, Chang SC, Su JS. Fuzzy inventory without backorder for fuzzy order quantity and fuzzy total demand quantity. Computers and Operations Research 2000;27:935–62. [14] Ouyang LY, Yao JS. A minimax distribution free procedure for mixed inventory model involving variable lead time with fuzzy demand. Computers and Operations Research 2002;29:471–87. [15] Yao JS, Wu K. Ranking fuzzy numbers based on decomposition principle and signed distance. Fuzzy Sets and Systems 2000;116:275–88. [16] Kaufmann A, Gupta MM. Introduction to fuzzy arithmetic: theory and applications. New York: Van Nostrand Reinhold, 1991. [17] Zimmermann HJ. Fuzzy set theory and its application, 3rd ed. Dordrecht: Kluwer Academic Publishers, 1996.
www.elsevier.com/locate/dsw
An application of fuzzy sets theory to the EOQ model with imperfect quality items Hung-Chi Chang Department of Logistics Engineering and Management, National Taichung Institute of Technology, 129 Sec 3, San-Min Road, Taichung City 404, Taiwan, ROC
Abstract This article investigates the inventory problem for items received with imperfect quality, where, upon the arrival of order lot, 100% screening process is performed and the items of imperfect quality are sold as a single batch at a discounted price, prior to receiving the next shipment. The objective is to determine the optimal order lot size to maximize the total pro2t. We 2rst propose a model with fuzzy defective rate. Then, the model with fuzzy defective rate and fuzzy annual demand is presented. For each case, we employ the signed distance, a ranking method for fuzzy numbers, to 2nd the estimate of total pro2t per unit time in the fuzzy sense, and then derive the corresponding optimal lot size. Numerical examples are provided to illustrate the results of proposed models. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Inventory; Imperfect quality; Fuzzy set; Signed distance
1. Introduction In the classical economic production/order quantity (EPQ/EOQ) models, the items produced/ received are implicitly assumed to be with perfect quality. However, it may not always be the case. Due to imperfect production process, natural disasters, damage or breakage in transit, or for many other reasons, the lot sizes produced/received may contain some defective items. To capture the real situations better, several researchers considered the above scenarios in formulating the production/inventory models and studied the e=ect of imperfect quality on lot sizing policy. Speci2cally, we note that in Rosenblatt and Lee’s [1] study, they assumed that the defective items could be
E-mail address: [email protected] (H.-C. Chang). 0305-0548/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0305-0548(03)00166-7
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reworked instantaneously at a cost and found that the presence of defective products motivates smaller lot sizes. While in a recent article, Salameh and Jaber [2] assumed that the defective items could be sold as a single batch at a discounted price prior to receiving the next shipment, and found that the economic lot size quantity tends to increase as the average percentage of imperfect quality items increase. Later, Goyal and CIardenas-BarrIon [3] reconsidered the task done in [2] and presented a simple approach for determining the optimal lot size. Besides, more survey related to the studies of imperfect quality can be found in [2]. On the other hand, from [2] and the literature survey in them, we note that some of the previous researchers assumed that the defective rate in lot sizes produced/received is a 2xed constant, while others assumed it as a random variable with a known probability distribution to reKect the uncertainty of imperfect quality. In real environments, although we know that the defective rate may have a little change from one lot to another, however, it may lack historical data to estimate the probability distribution, especially, for the items o=ered by the new producers/suppliers. Thus, the traditional probability theory and statistical method cannot be used properly. As stated in Karsak and Tolga [4], an e=ective way to express factors including Kexibility, quality of the products, enhanced response to market demand, and reduction in inventory, which can neither be assessed by crisp values nor random processes, is using linguistic variables or fuzzy numbers. In this article, we shall then apply the fuzzy sets concept 2rst introduced by Zadeh [5] to deal with quality-related problem, as well as the imprecise demand. In recent years, several researchers have applied the fuzzy sets theory and technique to develop and solve the production/inventory problems. For example, Park [6] and VujoOseviIc et al. [7] extended the classical EOQ model by introducing the fuzziness of ordering cost and holding cost. Chen and Wang [8] fuzzi2ed the demand, ordering cost, inventory cost, and backorder cost into trapezoidal fuzzy numbers in EOQ model with backorder. Roy and Maiti [9] presented a fuzzy EOQ model with demand-dependent unit cost under limited storage capacity. Chang et al. [10] presented a fuzzy model for inventory with backorder, where the backorder quantity was fuzzi2ed as the triangular fuzzy number. Lee and Yao [11] and Lin and Yao [12] discussed the production inventory problems, where Lee and Yao [11] fuzzi2ed the demand quantity and production quantity per day, and Lin and Yao [12] fuzzi2ed the production quantity per cycle, treating all as the triangular fuzzy numbers. Yao et al. [13] proposed the EOQ model in the fuzzy sense, where both order quantity and total demand were fuzzi2ed as the triangular fuzzy numbers. Ouyang and Yao [14] presented a mixture inventory model with variable lead-time, where the annual demand was fuzzi2ed as the triangular fuzzy number and as the statistic fuzzy number. In this article, we investigate the inventory problem for items received with imperfect quality. Building upon the work of Salameh and Jaber [2], we propose two fuzzy models. The 2rst model incorporates the fuzziness of defective rate. In the second model, not only the defective rate but also the annual demand is considered as a fuzzy number. For each fuzzy case, we employ Yao and Wu’s [15] ranking method for fuzzy numbers to 2nd the estimate of total pro2t per unit time in the fuzzy sense, and then derive the corresponding optimal lot size. This paper is organized as follows. In Section 2, a brief review of Salameh and Jaber’s [2] model is given. In Section 3, some de2nitions and properties about fuzzy sets related to this study are introduced. Section 4 presents two fuzzy models as described earlier. Section 5 provides two numerical examples to illustrate the results of the proposed models. Section 6 summarizes the work done in this paper.
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2. Brief review of Salameh and Jaber’s model Recently, Salameh and Jaber [2] developed an EOQ model for items received with imperfect quality. They considered the following situations. (i) A lot of size y is delivered instantaneously with a purchasing price c per unit and an ordering cost K. (ii) Each lot received contains percentage defectives p. A 100% screening process of the lot is conducted at a rate x units per unit time with unit screening cost w, upon receiving the order. (iii) Items of poor quality are kept in stock and sold prior to receiving the next shipment as a single batch at a discounted price v per unit, whereas the good-quality item is sold at the regular price s per unit. (iv) Shortages are not allowed. The number of good items in each order (i.e., lot size less defective items), denoted by N (y; p), is at least equal to the demand during screening time t; that is, N (y; p) = y − py ¿ Dt, where D is the demand per year. The behavior of the inventory level is as that shown in Fig. 1, where T is the cycle length, py is the number of defectives withdrawn from inventory, and t is the total screening time of y units ordered per cycle. For a given percentage of defective items in each lot of size y, the total pro2t per cycle TP(y) is established as TP(y) = (sales of good-quality items + sales of imperfect quality items) − (procurement cost + screening cost + holding cost) y(1 − p)T py2 = [sy(1 − p) + vyp] − (K + cy) + wy + h + : 2 x
Inventory level
py y t
Time T
Fig. 1. Behavior of the inventory level over time.
(1)
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Dividing the total pro2t per cycle TP(y) by the cycle length T (=(1 − p)y=D), the total pro2t per unit time (we denote it by W (y)) of the model is obtained as follows: hy hy 1 hy(1 − p) K W (y) = D s − v + +D v− − −c−w− : (2) x x y 1−p 2 Furthermore, Salameh and Jaber considered the uncertainty of defectives and assumed that the defective rate p is a random variable with a known probability density function (pdf) f(p). They then derived the optimal order lot size by maximizing the expected value of W (y), i.e., E[W (y)] in which the terms 1=(1 − p) and p in (2) are replaced by their expected values E[1=(1 − p)] and E[p], respectively. As mentioned earlier, although the statistical method can be used to deal with the uncertainty, however, in some cases, it may lack historical data to estimate the above pdf f(p). Since the defective rate may have a slight change from one lot to another, it can be described by fuzzy language as “p˜ = the real defective rate is around p”. In this article, we shall represent the defective rate as a fuzzy number. Moreover, in (2), the demand per year D is a 2xed value, but in a practical situation it is diWcult for the decision-makers to assess it by a crisp value. Hence, we shall consider it as a fuzzy number also.
3. Preliminaries Before presenting the fuzzy inventory models, we introduce some de2nitions and properties about fuzzy numbers with relevant operations. Much of these can be found in [15–17]. De nition 1. For 0 6 6 1, the fuzzy set a˜ de2ned on R = (−∞; ∞) is called an -level fuzzy point if the membership function of a˜ is given by ; x = a; a˜ (x) = (3) 0; x = a: De nition 2. The fuzzy set A˜ = (a; b; c), where a ¡ b ¡ c and de2ned on R, is called the triangular fuzzy number, if the membership function of A˜ is given by (x − a)=(b − a); a 6 x 6 b; A˜(x) = (c − x)=(c − b); b 6 x 6 c; (4) 0; otherwise: Remark 1. (i) When = 1, the membership function of the 1-level fuzzy point a˜1 becomes the characteristic function, i.e., a˜1 (x) = 1 if x = a and a˜1 (x) = 0 if x = a. In this case, the real number a ∈ R is the same as the fuzzy point a˜1 except for their representations. (ii) If c = b = a, then the triangular fuzzy number A˜ = (a; b; c) is identical to the 1-level fuzzy point a˜1 .
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De nition 3. For 0 6 6 1, the fuzzy set [a ; b ] de2ned on R is called an -level fuzzy interval if the membership function of [a ; b ] is given by ; a 6 x 6 b; [a ;b ] (x) = (5) 0; otherwise: De nition 4. Let B˜ be a fuzzy set on R, and 0 6 6 1. The -cut B() of B˜ consists of points x such that B˜ (x) ¿ , that is, B() = {x|B˜ (x) ¿ }. Decomposition Principle. Let B˜ be a fuzzy set on R and 0 6 6 1. Suppose the -cut of B˜ to be a closed interval [BL (); BU ()], that is, B() = [BL (); BU ()]. Then, we have (see, e.g., [16]) B˜ = B() (6) 0661
or B˜ (x) =
CB() (x);
(7)
0661
where (i) B() is a fuzzy set with membership function ; x ∈ B(); B() (x) = 0; otherwise: (ii) CB() (x) is a characteristic function of B(), that is, 1; x ∈ B(); CB() (x) = 0; x ∈ B(): Remark 2. From the Decomposition Principle and (5), we obtain B() = [BL () ; BU () ] B˜ = 0661
or B˜ (x) =
0661
(8)
0 6 61
CB() (x) =
[BL () ;BU () ] (x):
(9)
0 6 61
For any a, b, c, d, k ∈ R, a ¡ b, and c ¡ d, the interval operations are as follows [16]: (i) (ii)
[a; b](+)[c; d] = [a + c; b + d]:
[a; b](−)[c; d] = [a − d; b − c]: [ka; kb]; k ¿ 0; (iii) k(·)[a; b] = [kb; ka]; k ¡ 0:
(10)
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Further, for a ¿ 0 and c ¿ 0, (iv) (v)
[a; b](·)[c; d] = [ac; bd]: a b ; : [a; b](÷)[c; d] = d c
Next, as in Yao and Wu [15], we introduce the concept of the signed distance of fuzzy set. We 2rst consider the signed distance on R. De nition 5. For any a and 0 ∈ R, de2ne the signed distance from a to 0 as d0 (a; 0) = a. If a ¿ 0, the distance from a to 0 is a = d0 (a; 0); if a ¡ 0, the distance from a to 0 is −a = −d0 (a; 0). Hence, d0 (a; 0) = a is called the signed distance from a to 0. Let ! be the family of all fuzzy sets B˜ de2ned on R with which the -cut B() = [BL (); BU ()] exists for every ∈ [0; 1], and both BL () and BU () are continuous functions on ∈ [0; 1]. Then, for any B˜ ∈ !, from (8) we have [BL () ; BU () ]: (11) B˜ = 0661
From De2nition 5, the signed distance of two end points, BL () and BU (), of the -cut B() = [BL (); BU ()] of B˜ to the origin 0 is d0 (BL (); 0) = BL () and d0 (BU (); 0) = BU (), respectively. Their average, (BL () + BU ())=2, is taken as the signed distance of -cut [BL (); BU ()] to 0. That is, the signed distance of interval [BL (); BU ()] to 0 is de2ned as d0 ([BL (); BU ()]; 0) = [d0 (BL (); 0) + d0 (BU (); 0)]=2 = (BL () + BU ())=2. In addition, for every ∈ [0; 1], there is a one-to-one mapping between the -level fuzzy interval [BL () ; BU () ] and the real interval [BL (); BU ()], that is, the following correspondence is one-to-one mapping: [BL () ; BU () ] ↔ [BL (); BU ()]:
(12)
Also, the 1-level fuzzy point 0˜ 1 is mapping to the real number 0. Hence, the signed distance of [BL () ; BU () ] to 0˜ 1 can be de2ned as d([BL () ; BU () ]; 0˜ 1 ) = d0 ([BL (); BU ()]; 0) = (BL () + BU ())=2. Moreover, for B˜ ∈ !, since the above function is continuous on 0 6 6 1, we can use the integration to obtain the mean value of the signed distance as follows: 1 1 1 d([BL () ; BU () ]; 0˜ 1 ) d = (BL () + BU ()) d: (13) 2 0 0 Then, from (11) and (13), we have the following de2nition. De nition 6. For B˜ ∈ !, de2ne the signed distance of B˜ to 0˜ 1 (i.e., y-axis) as 1 1 1 ˜ ˜ ˜ 01 ) = d([BL () ; BU () ]; 01 ) d = (BL () + BU ()) d: d(B; 2 0 0 According to De2nition 6, we obtain the following property.
(14)
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Property 1. For the triangular fuzzy number A˜ = (a; b; c), the -cut of A˜ is A() = [AL (); AU ()], ∈ [0; 1], where AL () = a + (b − a) and AU () = c − (c − b). The signed distance of A˜ to 0˜ 1 is ˜ 0˜ 1 ) = 1 (a + 2b + c): d(A; 4
(15)
˜ G˜ ∈ !, where B˜ = 0661 [BL () ; BU () ] and G˜ = 0661 Furthermore, for two fuzzy sets B; [GL () ; GU () ], and k ∈ R, using (10) and (12), we have ˜ [(BL () + GL ()) ; (BU () + GU ()) ]: (i) B(+) G˜ = 0661
(ii)
˜ B(−) G˜ =
(iii)
[(kBL ()) ; (kBU ()) ]; 61 06 ˜ k 1 (·)B˜ = [(kBU ()) ; (kBL ()) ]; 0661 ˜ 01 ;
[(BL () − GU ()) ; (BU () − GL ()) ]:
0661
k ¿ 0;
(16)
k ¡ 0; k = 0:
From the above and De2nition 6, we obtain the following property. ˜ G˜ ∈ ! and k ∈ R, Property 2. For two fuzzy sets B; (i)
˜ 0˜ 1 ): ˜ 0˜ 1 ) = d(B; ˜ 0˜ 1 ) + d(G; ˜ d(B(+) G;
(ii)
˜ 0˜ 1 ) = d(B; ˜ 0˜ 1 ): ˜ ˜ 0˜ 1 ) − d(G; d(B(−) G;
(iii)
˜ 0˜ 1 ) = kd(B; ˜ 0˜ 1 ): d(k˜1 (·)B;
(17)
4. Fuzzy EOQ models for items with imperfect quality 4.1. Model with fuzzy defective rate In this subsection, we modify the crisp defective rate model shown in (2) by incorporating the fuzziness of the defective rate p (or equivalently, the fuzziness of the good-quality rate, 1 − p). For convenience, we let q = 1 − p and rewrite (2) as D hy K hyq hy − + +c+w−v + : (18) W (y) = D s − v + x q x y 2 Then, we fuzzify q to be a triangular fuzzy number, q˜ = (q − $1 ; q; q + $2 ), where 0 ¡ $1 ¡ q and 0 ¡ $2 6 1 − q, $1 and $2 are determined by the decision-makers. In this case, the total pro2t per unit time is a fuzzy value also, which is expressed as hy K 1 hy hy − D + +c+w−v + q˜ : (19) W˜ ≡ W˜ (y) = D s − v + x x y q˜ 2
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Now, we defuzzify W˜ using the signed distance method. From Property 2, the signed distance of ˜ W to 0˜ 1 is given by hy K 1 ˜ hy hy − D + +c+w−v d ; 01 + d(q; ˜ 0˜ 1 ) ; (20) d(W˜ ; 0˜ 1 ) = D s − v + x x y q˜ 2 where d(q; ˜ 0˜ 1 ) and d(1= q; ˜ 0˜ 1 ) are measured as follows. From Property 1, the signed distance of fuzzy number q˜ to 0˜ 1 is d(q; ˜ 0˜ 1 ) = 14 [(q − $1 ) + 2q + (q + $2 )] = q + 14 ($2 − $1 ):
(21)
Also, the left and right end points of the -cut (0 6 6 1) of q˜ are qL () = (q − $1 ) + $1 ¿ 0
and
qU () = (q + $2 ) − $2 ¿ 0;
(22)
respectively. Since 0 ¡ qL () ¡ qU (), from (10), the left and right end points of the -cut (0 6 6 1) of 1= q˜ are 1 1 1 1 1 1 = and = ; (23) () = () = q L qU () (q + $2 ) − $2 q U qL () (q − $1 ) + $1 respectively. Then, from De2nition 6, the signed distance of 1= q˜ to 0˜ 1 is 1 1 1 1 1 1 q 1 q 1 ˜ ; () + () d = ln − ln d ; 01 = q˜ 2 0 q L q U 2 $1 q − $ 1 $2 q + $ 2 which is positive since $1 ¿ 0, $2 ¿ 0, ln(q=q − $1 ) ¿ 0, and ln(q=q + $2 ) ¡ 0. Substituting the results of (21) and (24) into (20), we have D hy K hy − + +c+w−v W ∗ (y) ≡ d(W˜ ; 0˜ 1 ) = D s − v + x 2 x y $2 − $1 hy q 1 q 1 q+ : + ln − ln × $1 q − $ 1 $2 q + $ 2 2 4
(24)
(25)
W ∗ (y) is regarded as the estimate of the total pro2t per unit time in the fuzzy sense. The objective of the problem is to determine the optimal order lot size, y∗ , such that W ∗ (y) has a maximum value. Utilizing the classical optimization technique, we take the 2rst and second derivatives of W ∗ (y) with respect to y, and obtain K $ 2 − $1 1 h q 1 q dW ∗ (y) Dh D h − − q+ (26) − ln − ln = dy x 2 x y2 $1 q − $ 1 $ 2 q + $ 2 2 4 and d 2 W ∗ (y) DK =− 3 dy2 y
1 q 1 q ln − ln $1 q − $ 1 $2 q + $ 2
:
(27)
Because d 2 W ∗ (y)=dy2 ¡ 0, i.e., W ∗ (y) is concave in y, and hence the maximum value of W ∗ (y) will occur at the point that satis2es dW ∗ (y)=dy = 0. Setting (26) equal to zero and solving for y, we obtain the optimal lot size: q DK( $11 ln q−q$1 − $12 ln q+$ ) 2 ∗ y = : (28) q q $2 − $1 D 1 1 h[(q + 4 ) + x ( $1 ln q−$1 − $2 ln q+$ − 2)] 2
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Remark 3. If $1 = $2 = $, then (21) and (24) reduce to d(q; ˜ 0˜ 1 ) = q and d(1= q; ˜ 0˜ 1 ) = (1=2$) ln (q + $)=(q − $), respectively. Furthermore, by the L’Hospital’s rule, we know that lim$→0 (1=2$) ln (q + $)=(q − $) = 1=q. As a result, when $1 = $2 = $ → 0, the estimate of the total pro2t per unit time in the fuzzy sense (25) is identical to the crisp case (18). Hence, the crisp defective rate model is a special case of the fuzzy model presented here. Besides, for the optimal order lot size (28), when $1 = $2 = $, it reduces to D 1 q+$ 1 q+$ ∗ =h q + ln −2 : DK ln y = $ q−$ x $ q−$ By taking q = 1 (i.e., the defective rate p = 0) and $ → 0, we then have y∗ = 2DK=h, which is the classical EOQ formula. 4.2. Model with fuzzy defective rate and fuzzy annual demand This subsection further incorporates the fuzziness of demand per year into the previous fuzzy model. The crisp annual demand D in (19) is fuzzi2ed as the triangular fuzzy number, D˜ = (D − $3 ; D; D + $4 ), where $3 and $4 are determined by the decision-makers and should satisfy the conditions 0 ¡ $3 ¡ D and 0 ¡ $4 . For this case, we express the fuzzy total pro2t per unit time as hy K D˜ hy hy ˜ ˜ ˜ − + +c+w−v + q˜ : (29) Z ≡ Z(y) = D s − v + x x y q˜ 2 Again, the approach employed in the previous subsection is utilized to 2nd the estimate of the total pro2t per unit time in the fuzzy sense. The signed distance of Z˜ to 0˜ 1 is given by D˜ ˜ hy hy hy K ˜ ˜ ˜ ˜ ˜ d(D; 01 ) − + +c+w−v d ; 01 + d(q; ˜ 01 ) ;(30) d(Z; 01 ) = s − v + x x y q˜ 2 ˜ 0˜ 1 ) = D + ($4 − $3 )=4 follows in the same where d(q; ˜ 0˜ 1 ) = q + ($2 − $1 )=4 (from (21)), and d(D; way. ˜ q; Next, we calculate the signed distance d(D= ˜ 0˜ 1 ). The left and right end points of the -cut (0 6 6 1) of D˜ are DL () = (D − $3 ) + $3 ¿ 0
and
DU () = (D + $4 ) − $4 ¿ 0;
(31)
respectively. The left and right end points of the -cut (0 6 6 1) of q˜ are as shown in (22). Since ˜ q˜ are 0 ¡ qL () ¡ qU (), from (10), the left and right end points of the -cut (0 6 6 1) of D= D D DL () (D − $3 ) + $3 DU () (D + $4 ) − $4 = and = ; (32) () = () = q L qU () (q + $2 ) − $2 q U qL () (q − $1 ) + $1 ˜ q˜ to 0˜ 1 can be derived as follows: respectively. Thus, the signed distance of D= D˜ ˜ 1 1 D D ; 01 = d () + () d q˜ 2 0 q L q U q $4 (q$3 + D$2 ) q + $2 $3 1 (q$4 + D$1 ) − ; ln − + ln = 2 q − $ 1 $1 q $2 $21 $22
(33)
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˜ q˜ as stated in (32) are positive which is positive, since the left and right end points of the -cut of D= continuous functions on 0 6 6 1; hence the above result of de2nite integral should be positive. ˜ 0˜ 1 ), and d(D= ˜ q; Using the results of d(q; ˜ 0˜ 1 ), d(D; ˜ 0˜ 1 ) in (30), we obtain ˜ 0˜ 1 ) Z ∗ (y) ≡ d(Z; $ 4 − $3 hy $2 − $1 hy K hy D+ − q+ + + +c+w−v = s−v+ x 4 2 4 x y q $4 (q$3 + D$2 ) q + $2 $3 1 (q$4 + D$1 ) − : (34) ln − + ln × 2 q − $ 1 $1 q $2 $21 $22 Z ∗ (y) is regarded as the estimate of total pro2t per unit time in the fuzzy sense. Now, we determine the optimal order lot size y∗∗ to maximize the total pro2t function Z ∗ (y) of (34). It can be shown that Z ∗ (y) is concave in y. Then, by solving the 2rst-order necessary condition, i.e., dZ ∗ (y)=dy = 0, we obtain the optimal lot size Kx& ; (35) y∗∗ = $2 − $ 1 h[& + x(q + 4 ) − 2(D + $4 −4 $3 )] where &=
(q$4 + D$1 ) q $4 (q$3 + D$2 ) q + $2 $3 − : ln − + ln q − $ 1 $1 q $2 $21 $22
Remark 4. When $3 = $4 = 0, it is clear the fuzzy annual demand reduces to D˜ = (D; D; D). In this ˜ 0˜ 1 ) = D and (33) reduces to case, the signed distance d(D; D˜ ˜ D 1 1 ˜ q 1 q ; 01 = ; 01 ; d = Dd ln − ln q˜ 2 $ 1 q − $ 1 $2 q + $ 2 q˜ which implies that the estimate of total pro2t per unit time in the fuzzy sense (34) reduces to (25), and the optimal order lot size (35) reduces to (28). That is, the fuzzy model proposed in Section 4.1 is a special case of the fuzzy model presented in this subsection. Furthermore, from Remark 3, we note that the optimal order lot size (35) can also be reduced to the classical EOQ formula.
5. Numerical examples To illustrate the results of the proposed models, we consider an inventory system with the data in Salameh and Jaber [2]: demand rate D = 50 000 units=year, ordering cost K = $100=cycle, holding cost h = $5=unit=year, screening rate x = 1 unit/min (equivalently, x = 175 200 units/year), screening cost w=$0.5/unit, purchase cost c = $25=unit, selling price of good-quality items s = $50=unit, and selling price of imperfect-quality items v = $20=unit. But instead of assuming that the defective rate p is uniformly distributed with mean 0.02, we consider that the defective rate is around p = 0.02 (i.e., the good-quality rate is around q = 1 − p = 0:98).
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Table 1 Optimal solution for the model with fuzzy defective rate $1
$2
q˜
0.0005 0.0010 0.0025 0.0050 0.0075 0.0150 0.0200
0.0200 0.0150 0.0075 0.0050 0.0025 0.0010 0.0005
(0.9795, (0.9790, (0.9775, (0.9750, (0.9725, (0.9650, (0.9600,
0.98, 0.98, 0.98, 0.98, 0.98, 0.98, 0.98,
1.0000) 0.9950) 0.9875) 0.9850) 0.9825) 0.9810) 0.9805)
d(q; ˜ 0˜ 1 )
d(1= q; ˜ 0˜ 1 )
y∗
W ∗ (y∗ )
Rel y (%)
Rel W (%)
0.9849 0.9835 0.9813 0.9800 0.9788 0.9765 0.9751
1.01540 1.01680 1.01912 1.02042 1.02172 1.02409 1.02556
1429.6 1431.0 1433.3 1434.6 1435.9 1438.2 1439.6
$1; 213; 661:75 $1; 213; 273:51 $1; 212; 632:34 $1; 212; 272:30 $1; 211; 911:03 $1; 211; 253:66 $1; 210; 848:14
−0:349 −0:251 −0:091 0.000 0.091 0.251 0.349
0.114 0.082 0.029 0.000 −0:030 −0:084 −0:118
Example 1. For the model proposed in Section 4.1, where q is fuzzi2ed as a triangular fuzzy number q˜ = (q − $1 ; q; q + $2 ) with 0 ¡ $1 ¡ q and 0 ¡ $2 6 1 − q, we solve the optimal order lot size y∗ and the maximum total pro2t per year W ∗ (y∗ ) in the fuzzy sense for various given sets of ($1 ; $2 ). Note that in practical situations, $1 and $2 are determined by the decision-makers due to the uncertainty of the problem. The results are summarized in Table 1. Furthermore, we compare the results of fuzzy case with those of crisp case. The optimal order lot size yc∗ and the maximum total pro2t per year W (yc∗ ) of crisp case can be derived easily from (18) using the classical optimization technique as performed in Section 4. After manipulation, we obtain yc∗ = 1434:6 units and W (yc∗ ) = $1 212 274:79. Then, the relative variation between fuzzy case and crisp case for the optimal order lot size and the maximum total pro2t per year can be measured by Rel y = [(y∗ − yc∗ )=yc∗ ] × 100% and Rel W = {[W ∗ (y∗ ) − W (yc∗ )]=W (yc∗ )} × 100%, respectively. The numerical results are listed in the last two columns of Table 1. From Table 1, we observe that ˜ 0˜ 1 ) ¿ q(= 0:98) and (i) When $1 ¡ $2 , the estimate of good-quality rate in the fuzzy sense d(q; d(1= q; ˜ 0˜ 1 ) ¡ 1=q(=1:02041). In this case, y∗ ¡ yc∗ and W ∗ (y∗ ) ¿ W (yc∗ ), which result in Rel y ¡ 0 and Rel W ¿ 0. Further, as the absolute value |$1 − $2 | decreases, both |Rel y| and |Rel W | decrease, which means the smaller the di=erence between $1 and $2 (i.e., the less the uncertainty of good-quality rate), the smaller the variation of the solutions between fuzzy case and crisp case. (ii) When $1 ¿ $2 , d(q; ˜ 0˜ 1 ) ¡ q and d(1= q; ˜ 0˜ 1 ) ¿ 1=q. In this case, y∗ ¿ yc∗ and W ∗ (y∗ ) ¡ W (yc∗ ), which result in Rel y ¿ 0 and Rel W ¡ 0. Also, as |$1 −$2 | increases, both |Rel y| and |Rel W | increase. (iii) When $1 = $2 (=0:005); d(q; ˜ 0˜ 1 ) = q and d(1= q; ˜ 0˜ 1 ) ∼ = 1=q. In this case, the solutions of fuzzy case are almost equal to those of the crisp case, and hence Rel y = 0 and Rel W = 0. This is consistent with that shown in Remark 3 i.e., when $1 = $2 → 0, the fuzzy case becomes the crisp case. Example 2. For the model proposed in Section 4.2, where q and D are fuzzi2ed as the triangular fuzzy numbers q˜ = (q − $1 ; q; q + $2 ) and D˜ = (D − $3 ; D; D + $4 ), respectively, we solve the optimal order lot size y∗∗ and the maximum total pro2t per year Z ∗ (y∗∗ ) in the fuzzy sense for various given
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Table 2 Optimal solution for the model with fuzzy defective rate and fuzzy annual demand $1
$2
0.0010
0.0150
$3 500 1000 1500
0.005
0.005
500 1000 1500
0.0150
0.0010
500 1000 1500
$4
d(q; ˜ 0˜ 1 )
˜ 0˜ 1 ) d(D;
˜ q; d(D= ˜ 0˜ 1 )
y∗∗
Z ∗ (y∗∗ )
Rel y (%)
Rel W (%)
350 500 650 750 1000 1250 1000 1500 2000
0.9835 0.9835 0.9835 0.9835 0.9835 0.9835 0.9835 0.9835 0.9835
49 963 50 000 50 038 49 938 50 000 50 063 49 875 50 000 50 125
50 803.25 50 841.54 50 879.84 50 779.10 50 842.92 50 906.74 50 716.65 50 844.29 50 971.93
1430.5 1431.0 1431.5 1430.1 1431.0 1431.9 1429.2 1431.0 1432.8
$1 212 354:11 $1 213 265:80 $1 214 177:49 $1 211 738:61 $1 213 258:09 $1 214 777:57 $1 210 211:42 $1 213 250:38 $1 216 289:35
−0:286 −0:251 −0:216 −0:314 −0:251 −0:188 −0:376 −0:251 −0:125
0.007 0.082 0.157 −0:044 0.081 0.206 −0:170 0.080 0.331
350 500 650 750 1000 1250 1000 1500 2000
0.9800 0.9800 0.9800 0.9800 0.9800 0.9800 0.9800 0.9800 0.9800
49 963 50 000 50 038 49 938 50 000 50 063 49 875 50 000 50 125
50 983.32 51 021.72 51 060.11 50 958.59 51 022.59 51 086.58 50 895.47 51 023.45 51 151.44
1434.0 1434.6 1435.1 1433.7 1434.6 1435.5 1432.8 1434.6 1436.4
$1 211 356:33 $1 212 267:44 $1 213 178:54 $1 210 744:05 $1 212 262:57 $1 213 781:08 $1 209 220:67 $1 212 257:70 $1 215 294:73
−0:042 0.000 0.035 −0:063 0.000 0.063 −0:125 0.000 0.125
−0:076 −0:001 0.075 −0:126 −0:001 0.124 −0:252 −0:001 0.249
350 500 650 750 1000 1250 1000 1500 2000
0.9765 0.9765 0.9765 0.9765 0.9765 0.9765 0.9765 0.9765 0.9765
49963 50 000 50 038 49 938 50 000 50 063 49 875 50 000 50 125
51 167.39 51 206.05 51 244.71 51 143.02 51 207.45 51 271.89 51 079.99 51 208.86 51 337.73
1437.7 1438.2 1438.7 1437.3 1438.2 1439.1 1436.4 1438.2 1440.0
$1; 210 336:15 $1; 211 245:78 $1 212 155:42 $1 209 721:86 $1 211 237:91 $1 212 753:97 $1 208 197:93 $1 211 230:04 $1 214 262:15
0.216 0.251 0.286 0.188 0.251 0.314 0.125 0.251 0.376
−0:160 −0:085 −0:010 −0:211 −0:086 0.040 −0:336 −0:086 0.164
Note: Rel y = [(y∗∗ − yc∗ )=yc∗ ] × 100% and Rel W = {[Z ∗∗ (y∗∗ ) − W (yc∗ )]=W (yc∗ )} × 100%.
sets of ($1 ; $2 ) and ($3 ; $4 ) that satisfy the conditions 0 ¡ $1 ¡ q, 0 ¡ $2 6 1 − q, 0 ¡ $3 ¡ D, and 0 ¡ $4 . Besides, as performed in Example 1, we compare the solutions of fuzzy case with those of crisp case by calculating the relative variation between them. The results are summarized in Table 2. From Table 2, we observe that ˜ 0˜ 1 ) increases (i) for 2xed ($1 ; $2 ), as the estimate of annual demand in the fuzzy sense d(D; ∗∗ by varying ($3 ; $4 ), the optimal lot size y and the maximum total pro2t per year Z ∗ (y∗∗ ) increase, and hence Rel y and Rel W increase, while their values could be positive, negative, or zero;
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(ii) for 2xed ($3 ; $4 ), as the estimate of good-quality rate in the fuzzy sense d(q; ˜ 0˜ 1 ) decreases ∗∗ by varying ($1 ; $2 ), the optimal lot size y increases but the maximum total pro2t per year Z ∗ (y∗∗ ) decreases, and hence Rel y increases but Rel W decreases; (iii) for various sets of ($1 ; $2 ) and ($3 ; $4 ), their joint e=ects on the optimal lot sizes are as follows: ˜ q; when the signed distance d(D= ˜ 0˜ 1 ) ¡ D=q(= 51020:41), we have Rel y ¡ 0 (i.e., y∗∗ ¡ yc∗ ) ˜ ˜ and, conversely, when d(D= q; ˜ 01 ) ¿ D=q, we have Rel y ¿ 0 (i.e., y∗∗ ¿ yc∗ ). Besides, when ˜ q; $1 = $2 (= 0:005) and $3 = $4 (= 500; 1000; 1500), d(D= ˜ 0˜ 1 ) ∼ = D=q, we then have Rel y = 0 ∗ ∗∗ (i.e., y = yc ), and in this case Rel W → 0. Note that from the above examples although we cannot ascertain which of the solutions (obtained from the fuzzy models or crisp model) are better, the advantages of fuzzy models themselves are that they keep the uncertainties, which can capture real situations better than the crisp model does. In addition, the decision-makers can use the solutions derived from the fuzzy models to perform the sensitivity analysis, so as to examine the e=ects of uncertainties.
6. Conclusions This paper proposed two fuzzy models for an inventory problem with imperfect-quality items. In the 2rst model, the good-quality rate (defective rate) is represented by a fuzzy number, while the annual demand is treated as a 2xed constant. In the second model, not only the good-quality rate but also the annual demand is represented by a fuzzy number. For each fuzzy model, a method of defuzzi2cation, namely the signed distance, is employed to 2nd the estimate of total pro2t per unit time in the fuzzy sense, and then the corresponding optimal order lot size is derived to maximize the total pro2t. Besides, we show that in some cases, the proposed fuzzy models can be reduced to the crisp model and the optimal order lot size in the fuzzy sense can be reduced to the classical EOQ formula. Two numerical examples are carried out to investigate the behavior of our proposed models, and the results are compared with those obtained from the crisp model. Uncertainties of product quality and demand are inherent in real inventory problems. The inventory models that include these uncertainties are often based on the concept of randomness and thus handled by probability theory. However, in practice, there may be a lack of historical data to estimate the probability distributions for the uncertain factors that are modeled by random variables. For such a situation, using probability theory is not appropriate. With these perspectives, it is worthwhile to reconsider the inventory with quality-related problem presented by Salameh and Jaber [2] and provide an alternative approach. Finally, we would like to point out the performance of using signed distance method for defuzzi2cation in this study. The previous researches on fuzzy production/inventory problems (e.g., [10–13]) often employed the centroid method to obtain the estimate of total cost in the fuzzy sense. To achieve this task, the membership function of fuzzy total cost has to be found 2rst using the extension principle, while the derivations are very complex and cumbersome, especially, for the case where the fuzzy number is located in the denominator (as the good-quality rate q˜ appears in Eqs. (19) and (29)). In this study, we showed that using the decomposition principle with the signed distance method can solve similar problems easier.
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