Fuzzy Inventory with Backorder Defuzzification by Signed Distance Method

Transcription

1 JOURNAL OF INFORMAION SCIENCE AND ENGINEERING, (5) Fuzzy Inventory with Backorder Defuzzification by Signed Distance Method JERSHAN CHIANG, JING-SHING YAO + AND HUEY-MING LEE * Department of Applied Mathematics * Department of Information Management Chinese Culture University aipei, aiwan hmlee@faculty.pccu.edu.tw + Department of Mathematics National aiwan University aipei, 6 aiwan In this paper, we consider fuzzy inventory with backorder. First, we fuzzify the storing cost a, backorder cost b, cost of placing an order c, total demand r, order quantity q, and shortage quantity s as the triangular fuzzy numbers in the total cost. From these, we can obtain the fuzzy total cost. Using the signed distance method to defuzzify, we get the estimate of the total cost in the fuzzy sense. wo special cases of the optimal solutions on fuzzifying the storage quantity and order quantity as triangular fuzzy numbers will be treated numerically by the Nedler-Mead algorithm. Keywords: fuzzy inventory, fuzzy sets, fuzzy total cost, signed distance, extension principle. INRODUCION In crisp inventory models, all the parameters in the total cost are known and have definite values without ambiguity; in addition the real variable of the total cost is positive. But, in reality, it is not so certain. Hence there is a need to consider the fuzzy inventory models. Due to the various fuzzy cases, one may consider different fuzzy inventory models as follows. Yao et al. [, 7, 3-6] discussed fuzzy inventory with and without backorder models. Papers [, 3, 5, 6] related to this paper treated fuzzy inventory with backorder. In [], they fuzzified the shortage quantity s as a triangular fuzzy number in the total cost of inventory with backorder and kept the order quantity q as a crisp real variable. In this way, they obtained a fuzzy total cost. In [3, 5], they fuzzified the order quantity q as triangular fuzzy numbers and trapezoid fuzzy numbers, and kept the shortage quantity s as a crisp real variable in the total cost of inventory with backorder. In [, 3, 5] the authors used the extension principle to find the membership functions of the fuzzy total cost. hen they defuzzified by the centroid to find the estimate of the total cost in the fuzzy sense. Such methods are very difficult and complicated. In [6], they fuzzified the total demand quantity to an interval-valued fuzzy set in the total cost of inventory with backorder. hen they used the extension principle to find the membership Received November 7, 3; revised July 9 & October 8, 4; accepted November, 4. Communicated by Pau-Choo Chung. 673

2 674 JERSHAN CHIANG, JING-SHING YAO AND HUEY-MING LEE function and defuzzified to get an estimate of the total cost in the fuzzy sense. Papers [7, 3, 4] discussed fuzzy inventory without backorder. In [7, 3], they fuzzified order quantity q in the total cost of inventory without backorder to a triangular fuzzy number and trapezoid fuzzy number to get the fuzzy total cost. hen they used the extension principle to find their membership function. In [4], they fuzzified order quantity q and total demand quantity r in the total cost of inventory without backorder to triangular fuzzy numbers. In this way, they could compute a fuzzy total cost. Similarly, they used the extension principle to find their membership function. All the articles [, 7, 3-6] used the extension principle and centroid to find the estimate of the total cost in the fuzzy sense. his treatment is difficult and very complicated. Petrovic and Sweeney [9] fuzzified the demand, lead time and inventory level into triangular fuzzy numbers in an inventory control model, then decided the order quantity by the method of fuzzy proposition. Vujosevic et al. [] fuzzified the ordering cost into a trapezoidal fuzzy number in the total cost of an inventory without backorder model and obtained the fuzzy total cost. hey did the defuzzification by using centroid and obtained the total cost in the fuzzy sense. Chen et al. [3] fuzzified the order cost, inventory cost, and backorder cost into trapezoidal fuzzy numbers and used the functional principle to obtain the estimate of the total cost in the fuzzy sense. Roy and Maiti [] rewrote the problem of classic economic order quantity into a form of nonlinear programming problem, and introduced fuzziness both in the objective function and storage area. It was solved by fuzzifying both nonlinear and geometric programming techniques for linear membership functions. Ishii and Konno [4] fuzzified the shortage cost L to a fuzzy number in a classical newsboy problem aimed at finding an optimal ordering quantity in the sense of fuzzy ordering. In this article we do not use the extension principle, centroid or other methods in the above papers. Instead, we will use the signed distance method to defuzzify the fuzzy total cost and obtain an estimate of the total cost in the fuzzy sense. We will discuss signed distance in section. Section 3 consists of three subsections. In section 3., we will fuzzify q, s, r, a, b, and c in the total cost to triangular fuzzy numbers, and get the fuzzy total cost of inventory with backorder. hrough defuzzification by signed distance, we have the estimate of the total cost in the fuzzy sense. We fuzzify s to a triangular fuzzy number in the total cost and defuzzify by signed distance in section 3.. In section 3.3, we fuzzify q as a triangular fuzzy number and defuzzify by the signed distance method in the total cost. We give an example in section 5 and make comparisons with [, 5].. PRELIMINARIES For the fuzzy total cost in inventory with backorder based on the signed distance method, all pertinent definitions of fuzzy sets are given below. Definition. Fuzzy Point (Definition. in Pu and Liu []): Let ã be a fuzzy set on R = (, ). It is called a fuzzy point if its membership function is, if x = a µ a ( x) =., if x a ()

3 FUZZY INVENORY WIH BACKORDER 675 Definition. Level α Fuzzy Interval: Let [a, b; α] be a fuzzy set on R = (, ). It is called a level α fuzzy interval, α, a < b, if its membership function is µ [ a, b; α ] α, if a x b ( x) =. (), otherwise Definition.3 riangular Fuzzy Numbers: Let à = (p, q, r), p < q < r, be a fuzzy set on R = (, ). It is called a triangular fuzzy number, if its membership function is x p, if p x q q p r x µ ( x), if q x r. A = (3) r q, otherwise If p = q = r then P = ( p, p, p). Let D be a fuzzy number on R. Denote by D(α) = [D L (α), D R (α)] the α-cut of D, where α. D L (α) and D R (α) are the left and right hand side of D(α); they exist and are integrable for α [, ]. In addition, we let F be the family of all these fuzzy numbers D on R. As in Yao and Wu [7], we consider the definition of the signed distance on F. Definition.4 he Signed Distance: We define d (a, ) = a, for a, R. Remark.: he meaning of Definition.4 is as the follows, if < a then the distance between a and is d (a, ) = a. If a < then the distance between a and is d (a, ) = a. herefore, we call d (a, ) = a is the signed distance between a and. For D F, from Definition.4, we have that the signed distance of D L (α) and D R (α) measured from are d (D L (α), ) = D L (α) and d (D R (α), ) = D R (α), respectively. herefore, we may define the signed distance of the interval [D L (α), D R (α)], which is measured from the origin, by d [[D L (α), D R (α)], ] = [d (D L (α), ) + d (D R (α), )] = [D L(α) + D R (α)]. For each α [, ], the crisp interval [D L (α), D R (α)] and the level α fuzzy interval [D L (α), D R (α); α] are in one to one correspondence. herefore, we may define the signed distance from [D L (α), D R (α); α] to as d([d L (α), D R (α); α], ) = d ([D L (α), D R (α)], ) = [D L(α) + D R (α)]. Since D F, D L (α) and D R (α) exist and are integrable for α [, ], we have the following definition. Definition.5 Let D F. We define the signed distance of D measured from as

4 676 JERSHAN CHIANG, JING-SHING YAO AND HUEY-MING LEE d( D, ) = [ DL( α) + DR( α)] dα. We have the ordering definition of F as follows: Definition.6 Let D, Ẽ F. he ordering of D, Ẽ is as follows D E iff dd (, ) < de (, ), D E iff dd (, ) = de (, ). Using Definitions.5 and.6, and the properties of ordering relation <, = on R, we have the following Proposition. Proposition : (a) D, Ẽ F, then one and only one of the following, D E, D E, or E D, is true. (b) C, D, Ẽ F, then the following three axioms of the ordering relations are true. (i) C C. (ii) C D and D C, then C D. (iii) C D and D E, then C E. From Proposition, we have that, are the linear order on F. Remark.: If C = (p, q, r), then the left endpoint and the right endpoint of the α-cut of C are C L (α) = p + (q p)α and C R (α) = r (r q)α, respectively. he centroid of C is CC ( ) = 3 (p + q + r), and the signed distance of C is dc (, ) = (q + p + r). he mid- 4 point of the interval [p, r] is M = (p + r). CC ( ) dc (, ) = ( M q). 6 dc (, ) q= ( M q). M C( C ) = ( M q). 3 (a) If M > q, then p < q < dc (, ) < CC ( ) < M < r. (b) If M < q, then p < M < CC ( ) < dc (, ) < q < r. (c) If M = q, then p < M = CC ( ) = dc (, ) = q < r. From (a) and (b), we know that dc (, ) is near q, and CC ( ) is near M. From Figs. and, we know that the membership grade of C at q is. he membership grade of C at dc (, ) is greater than that at CC ( ). hen, we have µ ( (, )) C dc > µ ( ( )). C CC herefore, from the membership grade viewpoint, it is better for us to defuzzify the fuzzy number C by dc (, ) than by CC ( ).

5 FUZZY INVENORY WIH BACKORDER 677 p q M r dc (, ) C( C ) Fig.. Case M > q. p M q r C( C ) d( C, ) Fig.. Case M < q. Let D, Ẽ F. For α [, ], we have four operations,,, fuzzy intervals. For more details, please refer to [6]. for the level α 3. DEFUZZIFICAION BY SIGNED DISANCE MEHOD For a crisp inventory model with backorder, we use the following notations and related parameters. Fig. 3 illustrates the role of all of the parameters where is the length of plan (days). a is the storing cost for one unit per day. b is the backorder cost for one unit per day. c is the cost of placing an order. r is the total demand over the planning time period [, ]. t q is length of a cycle. q is the order quantity per cycle.

6 678 JERSHAN CHIANG, JING-SHING YAO AND HUEY-MING LEE q q-s s t () t () t q Fig. 3. Inventory with backorder. s is the shortage quantity per cycle. q s q s r hen, we have = = =. t() tq t() he crisp total cost on the planning period [, ] is given by q s s r Fq (, s) = [ at() + bt() + c] q aq ( s) bs cr = + +,( < s < q). (4) q q q he crisp optimal solutions are optimal order quantity q ( a+ b) cr =. ab acr optimal backorder quantity s =. ba ( + b ) minimal total cost abcr Fq (, s ) =. a + b 3. Fuzzifying q, s, r, a, b, and c to riangular Fuzzy Numbers in the otal Cost Under the condition that the period of the ordering and arriving of the commodities per cycle is the same, we can find the total cost Eq. (4). In practical situations, it will fluctuate a little. It will influence the ordering quantity q and shortage quantity s. herefore, we consider fuzzifying q and s to the following fuzzy numbers q = ( q, q, q ), (5)

7 FUZZY INVENORY WIH BACKORDER 679 s = ( s, s, s ), (6) where < s < s < s < q < q < q and s, s, s, q, q, and q are unknown positive numbers. Remark 3.: Under the condition < s < q in Eq. (4), we may fuzzify as follows: Since < s < s < s < q < q < q, we have d(, ) = < d( s, ) = ( s+ s+ s) < d( q, 3 ) = ( ). 3 q + q + q hen, we have d (,) < d (,) s < d (,). q By definition.6, we have s q. It is difficult to determine a fixed value r for the total demand in the plan period. On the contrary, it is easier to set the total demand in the interval [r, r + ], where < < r, < and, are determined by the decision maker. Let r be a known number. he decision maker wants to choose a suitable value in the interval [r, r + ] as an appropriate estimate of the total demand. If it happens that the value coincides with r, then the error is. If the value deviates from r farther from both sides of r, the error is bigger. he error will attain a maximum at the two endpoints r, and r +. From the fuzzy point of view, we can transform the error to a confidence level. If the error is zero, then the confidence level is. he farther the value is from both sides of r, the less the confidence level is. At the two endpoints r, and r +, the confidence level will be the minimum. We set them to zero. From the arguments above and below, corresponding to the interval [r, r + ], we set the following fuzzy numbers r = (r, r, r + ), < < r, <. (7) he membership grade of r in r is. he farther the point in [r, r + ] is from both sides of r, the less the membership grade is. he membership grade shares the same property with the confidence level. If we make a correspondence between membership grade and confidence level, it is reasonable to set a fuzzy number in Eq. (7) corresponding to the interval [r, r + ]. In a perfectly competitive market, it will be perturbed a little for the inventory cost per unit per day in the plan period. We can set the inventory cost per unit to lie in the interval [a 3, a + 4 ] where < 3 < a, and < 4. With the same arguments as above, corresponding to the interval [a 3, a + 4 ], we set the following fuzzy number ã = (a 3, a, a + 4 ), < 3 < a, < 4, a is known. (8) Similarly, let the backorder cost per unit per day lie in the interval [b 5, b + 6 ]. Corresponding to the interval [b 5, b + 6 ], we set the following fuzzy number b = (b 5, b, b + 6 ), < 5 < b, < 6, and b is known. (9) Let the ordering cost each time lie in the interval [c 7, c + 8 ]. Corresponding to the interval [c 7, c + 8 ], we set the following fuzzy number

8 68 JERSHAN CHIANG, JING-SHING YAO AND HUEY-MING LEE c = (c 7, c, c + 8 ), < 7 < c, < 8, and c is known. () j, j =,,, 8 will be decided appropriately by the decision maker. From Eq. (4), let a q a s b s c r Kq (, s, r, a, b, c) = Fq (, s) = a s () q q q We fuzzify q, s, r, a, b, and c in Eq. () to Eqs. (5) - (), then we have the fuzzy total cost Eq. (). Let = (,, ),,, = be the fuzzy points, and Q = a q, Q = a s, Q 3 = a ( s s ) q, Q 4 = b ( s s ) q, Q 5 = c r s q. hen we have the fuzzy total cost Kq (, s, r, a, b, c ) = Q Q Q 3 Q 4 Q 5. () If we use the extension principle to find the membership functions of the fuzzy total cost (Eq. ()), and defuzzify by the centroid to find the estimate of the total cost in the fuzzy sense, then it will be difficult and complicated to derive the membership functions and centroid. In this article we don t use the extension principle or centroid. Instead, we will use the signed distance method (by Definition.5) to defuzzify the fuzzy total cost and obtain an estimate of the total cost in the fuzzy sense. he procedure is as follows: he left and right hand side of the α-cut, ( α ), of q, s, r, a, b, c are ql( α) = q+ ( q q) α >, qr( α) = q ( q q) α = q( α) + qα > sl( α) = s+ ( s s) α >, sr( α) = s ( s s) α > rl( α) = r + α >, rr( α) = r+ α >. al( α) = a 3 + α 3 >, ar( α) = a+ 4 α 4 > bl( α) = b 5 + α 5 >, br( α) = b+ 6 α 6 > cl ( α) = c 7 + α 7 >, cr ( α) = c+ 8 α 8 > (3) From Eq. (3), we have the left and right hand side of the α-cut, ( α ), of Q j, j =,,, 5 are QL( α) = al( α) ql( α), QR( α) = ar( α) qr( α) QL( α) = al( α) sl( α), QR( α) = ar( α) sr( α) Q3L( α) = al( α) sl( α) qr( α), Q3R( α) = ar( α) sr( α) ql( α). Q4L( α) = bl( α) sl( α) qr( α), Q4R( α) = br( α) sr( α) ql( α) Q5 L( α) = cl( α) rl( α) qr( α), Q5 R( α) = cr( α) rr( α) ql( α) (4)

9 FUZZY INVENORY WIH BACKORDER 68 he left and right hand side of the α-cut, ( α ), of Kq (, s, r, a, b, c ) are Kq (, s, r, a, b, c ) ( α) = Q ( α) Q ( α) + Q ( α) + Q ( α) + Q ( α), L L R 3L 4L 5L Kq (, s, r, a, b, c ) ( α) = Q ( α) Q ( α) + Q ( α) + Q ( α) + Q ( α). (5) R R L 3R 4R 5R rivially, we have Kq (, s, r, a, b, c ) F. By Definition.5 d( K( q, s, r, a, b, c ), ) = [ K( q, s, r, a, b, c) L( α) + K( q, s, r, a, b, c) R( α)] dα FC(q, q, q, s, s, s ; j, j =,,, 8). (6) Let G(e, f, g, h, k, w) = 3 eα + fα + gα+ h d α kα + w e fk ew gk fkw+ ew = k k k 3 3 hk gk w + fkw ew k + w ln. 4 k w (7) wα + vα + u H(w, v, u, t, p) = d α tα + p w vt wp ut vtp+ wp t+ p = + + ln. (8) 3 t t t p From Eqs. (5) to (8), we have the following Proposition. Proposition : If we fuzzify the q, s, r, a, b, and c in the crisp inventory with backorder in Eq. () as the fuzzy numbers q (in Eq. (5)), s (in Eq. (6)), r (in Eq. (7)), ã (in Eq. (8)), b (in Eq. (9)), and c (in Eq. ()), then we can obtain the estimate of the total cost in the fuzzy sense as FC(q, q, q, s, s, s ; j, j =,,, 8) = [3 aq ( + q+ q) + ( 4 3) q+ 4q 3q] [3 as ( + s+ s) + ( 4 3) s+ 4 4s 3s] + G( e, f, g, h, k, w) + G( e, f, g, h, k, w) Ge ( 3, f3, g3, h3, k3, w3) + Ge ( 4, f4, g4, h4, k 4, w4) H( w, v, u, t, p) (,,,, ), + H w v u t p

10 68 JERSHAN CHIANG, JING-SHING YAO AND HUEY-MING LEE where e = 3 (s s ), f = (a 3 )(s s ) + 3 s (s s ), g = 3 s + (a 3 )s (s s ), h = (a 3 )s, k = q + q, w = q, e = 4 (s s), f = (a + 4 )(s s) + 4 s (s s), g = 4 s (a + 4 )s (s s), h = (a + 4 )s, k = q q, w = q, e 3 = 5 (s s ), f 3 = (b 5 )(s s ) + 5 s (s s ), g 3 = 5 s + (b 5 )s (s s ), h 3 = (b 5 )s, k 3 = q + q, w 3 = q, e 4 = 6 (s s), f 4 = (b + 6 )(s s) + 6 s (s s), g 4 = 6 s (b + 6 )s (s s), h 4 = (b + 6 )s, k 4 = q q, w 4 = q, w = 7, v = 7 (r ) + (c 7 ), u = (c 7 )(r ), t = q + q, p = q, w = 8, v = 8 (r + ) (c + 8 ), u = (c + 8 )(r + ), t = q q, p = q. 3. Fuzzify s in the otal Cost as riangular Fuzzy Number We replace the fuzzy numbers in Eqs. (5) to () by q = (q, q, q), s = (s, s, s 3 ), r = (r, r, r), ã = (a, a, a), b = (b, b, b), c = (c, c, c), and = (,, ). Under this condition, k j =, j =,, 3, 4, and t j =, j =,, in Proposition, from Eqs. (7) and (8), we cannot solve G(e j, f j, g j, h j, k j, w j ) and H(w j, v j, u j, t j, p j ). herefore, from Eq. (3) we could solve it as follows. he left and right hand side of the α-cut, ( α ), of q, s, r, a, b, c are ql( α) = q ( α) = q s ( α) s ( s s ) α, s ( α) s ( s s) α L R = + > R = > rl( α) = rr( α) = r. al( α) = ar( α) = a bl( α) = br( α) = b cl( α) = cr( α) = c From Eq. (4), we have a QL( α) = QR( α) = q QL( α) = a[ s+ ( s s) α], QR( α) = a[ s ( s s) α)] a a Q3L( α) = [ s+ ( s s) α], Q3R( α) = [ s ( s s) α] q q. b b Q4L( α) = [ s+ ( s s) α], Q4R( α) = [ s ( s s) α] q q cr Q5L( α) = Q5R( α) = q

11 FUZZY INVENORY WIH BACKORDER 683 he left and right hand side of the α-cut, ( α ), of Kq (, s, r, a, b, c ) are aq a Kq (, s, r, a, b, c ) L ( α) = as [ ( s s) α] + [ s+ ( s s) α] + q b cr [ s+ ( s s) α] +, q q aq a Kq (, s, r, a, b, c ) R ( α) = as [ + ( s s) α] + [ s ( s s) α] + q b cr [ s ( s s) α] +. q q By Definition.5, we have d( K( q, s, r, a, b, c ), ) = [ K( q, s, r, a, b, c) L( α) + K( q, s, r, a, b, c) R( α)] dα F q (q, s, s, s ). (9) We have the following proposition. Proposition 3: If we fuzzify s in the crisp inventory with backorder in Eq. () as the fuzzy numbers s (in Eq. (6)), then we obtain an estimate of the total cost in the fuzzy sense as Fq ( q, s, s, aq cr a a s) = + [ s+ s+ s] + [ s + s + s + ss+ ss] + q 4 q b [ s + s + s + s s + s s ]. q Remark 3.: Since ds (,) = ( ), 4 s + s+ s ds (, ) = ( ), 6 s + s + s + s s+ s s aq cr a therefore, we have F q (q, s, s, s ) = + ad(,) s + ds (, ) + b ds (, ). q q q his formula is just the same as if we replaced s by ds (,) and s by ds (, ) in Eq. (). 3.3 Fuzzify q in the otal Cost as riangular Fuzzy Number We replace the fuzzy numbers in Eqs. (5) to () by q = (q, q, q ), s = (s, s, s), r = (r, r, r), ã = (a, a, a), b = (b, b, b), c = (c, c, c), and = (,, ). Let s = s = s and j =, j =,,, 8 in Proposition, then we have the following proposition. Proposition 4: If we fuzzify q in the crisp inventory with backorder in Eq. () as the fuzzy numbers q (in Eq. (5)), then we can obtain an estimate of the total cost in the fuzzy sense as

12 684 JERSHAN CHIANG, JING-SHING YAO AND HUEY-MING LEE a Fs (, s q, q, q) = [ q+ q+ q] as+ 8 as bs cr q q q q q q q q ln ln. Remark 3.3: Since dq (, ) = ( ), 4 q + q+ q ln q ln q ln q ln q d, = +, q q q q q a as bs we have F s (s, q, q, q ) = dq (, ) as+ + + cr d,. his formula q is just the same as if we replaced q by dq (, ) and q by d, in Eq. (). q 4. OPIMAL SOLUION For FC(q, q, q, s, s, s ; j, j =,,, 8), the estimated total cost in the fuzzy sense is shown in Proposition. For given j, j =,,, 8, there are six variables q, q, q, s, s, s which satisfy < s < s < s < q < q < q. We want to find q, q, q, s, s, s such that FC( q, q, q, s, s, s ; j, j =,,, 8) is the minimum, with the optimal order quantity q () = ( ) 4 q + q + q and the optimal backorder quantity s () = ( s + s + s ). For the estimated total cost in the fuzzy 4 sense, F q (q, s, s, s ) is shown in Proposition 3. here are four variables q, s, s, s which satisfy < s < s < s < q. We want to find q, s, s, s such that Fq ( q, s, s, s ) is minimum, the optimal order quantity q () = q, and the optimal backorder quantity s () = ( ). 4 s + s + s For the estimated total cost in the fuzzy sense, F s(s, q, q, q ) is shown in Proposition 4. here are four variables s, q, q, q which satisfy < s < q < q < q. We want to find s, q, q, q such that Fs ( s, q, q, q ) is the minimum, the optimal order quantity q() = ( ), 4 q + q + q and the optimal backorder quantity s () = s. We apply the Nelder-Mead method [8]. But, in our paper, the variables should have the ordering relation as shown above. For example, in Proposition, q, q, q, s, s, s should satisfy < s < s < s < q < q < q. herefore, when we apply the Nelder-Mead simplex algorithm [], the two transformations Eqs. () and () we use are shown in Figs. 4 and 5 instead of the two transformations of Algorithm 6.5 of the Nelder-Mead method [8]. R = X + e(x G) = ( + e)x eg, where < e, () E = X + d(r X) = ( d)x + dr, where d >. () For the Proposition 4 (similar arguments as Propositions and 3), we denote q for R(), X(), G(), and E(), q for R(), X(), G(), and E(), and q for R(3), X(3), G(3), and E(3), s for R(4), X(4), G(4), and E(4) instead of the symbols in Algorithm 6.5 of the Nelder-Mead method [8].

13 FUZZY INVENORY WIH BACKORDER 685 H Optimal point for the last time G X R S Fig. 4. Contraction step. H Optimal point for the last time G X R E S Fig. 5. Expansion step. Suppose X() < X() < X(3) < X(4) and G() < G() < G(3) < G(4). Step : Let H(k +, k) = X(k) X(k + ) G(k) + G(k + ), for k =,, 3 and, if H( k+, k) > I( H( k +, k)) =,, if H( k+, k) H X() X() X(3) X() I( H (, )), I( H(3, )), H(, ) H(3, ) = min, X(4) X(3) I( H (4, 3)), H (4, 3) for k =,, 3. If we take e in Eq. () satisfying < e < H, () then it is easy to show that R() < R() < R(3) < R(4). Step : Let L(k +, k) = X(k + ) X(k) + R(k) R(k + ), for k =,, 3 and

14 686 JERSHAN CHIANG, JING-SHING YAO AND HUEY-MING LEE, if Lk ( +, k), k=,, 3 L = X( k + ) X( k). min I ( L ( k +, k )), otherwise Lk ( +, k) If we take d in Eq. () satisfying < d < L, (3) then it is easy to show that E() < E() < E(3) < E(4). We modify R(k) = M(k) V(H i, k) in subroutine Newpoints of Algorithm 6.5 [8] to be R(k) = ( + e)m(k) ev(h i, k) where e satisfies Eq. (). Also, we modify E(k) = R(k) M(k) to be E(k) = dr(k) + ( d)m(k) where d satisfies Eq. (3). Applying the modi- fied Algorithm 6.5 [8], we can find s, q, q, q such that Fs ( s, q, q, q ) (in Proposition 4) is the local minimal value. hen the optimal order quantity is q () = ( ), 4 q + q + q and the optimal backorder quantity s() = s. 5. NUMERICAL EXAMPLE Example 5.: In [], Chang et al. fuzzified s as triangular fuzzy number s = (s, s, s ) and kept the q as crisp variable in Eq. (4), they obtained the fuzzy total cost Gq (). s hey derived the membership function by the extension principle, and then they obtained the estimate of the total cost in the fuzzy sense M = (s, s, s, q) by the centroid method. But, it is very complex and difficult to derive []. () Example 4. in []: given a =, b =, c =, r =, =, we get the optimal solution q =, s = and the minimal total cost F(q, s ) = 8 in the crisp case. From [], we have the results shown in able for each case of given sets of initial points s, s, s, q, where R q = q q %, R s = s s %, M = q s * * * ** M F( q, s) M( s, s, s, q ), R c = %. Fq (, s) () Using the data of the initial points s, s, s, q as shown in able for each case, and Proposition in this article, we have the optimal quantities s, s, s, q, and s () = () ( s + s + s ), q () = q q q, as shown in able, where r q = 4 q () s s F q F( q, s ) () q q %, r s = %, r c = %, t q = %, s Fq (, s) q () s s F q M () t s = %, t c = %, and Fq = Fq s M ( q, s, s, s ). Example 5.: he formula form of total cost in [5] and in this paper are not the same. If we interchange s with q s in Eq. (4) within this paper, then we obtain the same total cost form as in [5]: F a s b ( q s) c r (q, s) = + +. Let G s (q) = F (q, s). For s q q q

15 FUZZY INVENORY WIH BACKORDER 687 able. Results of Example 4. shown in [] and comparison with the crisp case. Case 3 4 Initial points (s, s, s, q) (3, 33, 35, ) (3, 34, 35, ) (33, 35, 36, ) (34, 35, 36, ) (33, 34, 36, ) (3, 33, 36, ) (3, 33, 35, ) (3, 35, 36, ) (3, 33, 34, ) (33, 34, 36, ) (3, 33, 35, ) (3, 3, 35, ) (33, 35, 36, ) (34, 35, 36, ) (33, 34, 36, ) (3, 33, 37, ) (3, 3, 35, ) (3, 35, 36, ) (3, 33, 34, ) (33, 34, 36, ) Optimal quantities M R q (%) R s (%) R c (%) s = 3. s = s = 35.3 s = q =.3 s = s = s = s = q =.6498 s = s = s = s = q =.9 s = 3.37 s = s = s = q = able. Computed result of Example 5. by Proposition using the initial points shown in able, comparison with the crisp case and []. Case 3 4 Initial points (s, s, s, q) (3, 33, 35, ) (3, 34, 35, ) (33, 35, 36, ) (34, 35, 36, ) (33, 34, 36, ) (3, 33, 36, ) (3, 33, 35, ) (3, 35, 36, ) (3, 33, 34, ) (33, 34, 36, ) (3, 33, 35, ) (3, 3, 35, ) (33, 35, 36, ) (34, 35, 36, ) (33, 34, 36, ) (3, 33, 37, ) (3, 3, 35, ) (3, 35, 36, ) (3, 33, 34, ) (33, 34, 36, ) Optimal quantities s = s = s = s = q =.3984 s = s = s = s = q =.7494 s = 3. s = 33. s = 35. s = 33.5 q =. s = s = s = s = q = F q r q(%) t q(%) r s(%) t s(%) r c(%) t c(%)

16 688 JERSHAN CHIANG, JING-SHING YAO AND HUEY-MING LEE >, Yao and Lee [5] fuzzified q as triangular fuzzy number q = (q, q, q ) and kept s as a crisp variable. hey obtained the fuzzy total cost Gs ( q ). hey derived the membership function by the extension principle, and then they obtained the estimate of the total cost in the fuzzy sense M = (q, q, q, s) by the centroid method. But, it is very complex and difficult to derive (see [4]). In order to compare F (q, s) in [4] with F(q, s) in Eq. (5), we interchange a and b. () Example 4. in [4], given a =, b =, c =, r =, =, we get the optimal order quantity q =, optimal backorder quantity s = , and the minimal total cost F(q, s ) = 8 in the crisp case. From [5], we have the results shown in able 3 for each case of given sets of initial points q, q, q, s, where R q = q q %, q = ( ), q 3 q q q + + M = M( q, q, q, s ), and R c = M F( q, s) %. Fq (, s) able 3. Result of Example 4. shown in [8] and comparison with the crisp case. Case 3 4 Initial points (q, q, q, s) (,, 3, ) (, 3, 4, ) (,, 4, ) (,,, ) (,, 4, ) (98, 99,, 55) (97, 98, 99, 5) (96, 98, 99, 5) (,, 3, 5) (96, 97, 98, 5) (,, 3, 67) (, 3, 4, 67) (,, 4, 67) (, 3, 5, 67) (,, 5, 67) (98, 99,, 9) (97, 98, 99, 85) (96, 98, 99, 85) (,, 3, 88) (96, 97, 98, 85) Optimal quantities M R q (%) R c (%) q =.3364 q =.385 q =.467 q =.3689 s = q = q = q = q = s = q =. q =. q = 3.43 q =.8 s = q = q = q = q = s = () As the discussion above, if we apply the Proposition 3 we should change the value of a and b in Example 4. in [5]. Given a =, b =, c =, r =, =, we get the optimal order quantity q =, optimal backorder quantity s = , and the minimal total cost F(q, s ) = 8 in the crisp case. Using the data of the

17 FUZZY INVENORY WIH BACKORDER 689 initial points q, q, q, s as shown in able 3 for each case, and Proposition 3 in this article, we obtain the optimal quantities ( q, q, q, s ), and q () = () ( ), 4 q + q + q s() = s q q, as shown in able 4. Let r q = %, q () () Fs F( q, s) q q Fs = Fs( q, q, q, s ), r c = %, t q = Fq (, s) q %, M = M( q, q, q, s ), Fs M t c = %. M able 4. Computed result of Example 5. using the initial points shown in able 3 and comparison with the crisp case. Case 3 4 Initial points (q, q, q, s) (,, 3, ) (, 3, 4, ) (,, 4, ) (,,, ) (,, 4, ) (98, 99,, 55) (97, 98, 99, 5) (96, 98, 99, 5) (,, 3, 5) (96, 97, 98, 5) (,, 3, 67) (, 3, 4, 67) (,, 4, 67) (, 3, 5, 67) (,, 5, 67) (98, 99,, 9) (97, 98, 99, 85) (96, 98, 99, 85) (,, 3, 88) (96, 97, 98, 85) Optimal quantities q =. q =. q =. q =. s = q =. q =. q = 3. q =. s = 6. q =. q =. q = 3. q =. s = 67. q = q = q = q = s = M r q (%) t q (%) r c (%) t c (%) Example 5.3: We apply Proposition of this article to solve the optimal solution q, q, q, s, s, s, the optimal order quantity q () = ( 4 q + q + q ), the optimal backorder quantity s () = ( ), 4 s + s + s and the minimal total cost () q q F c ( q, q, q, s, s, s ; j, j =,,, 8) F c. Let r q = %, q () s s Fc F( q, s) r s = %, r c = %. s Fq (, s)

18 69 JERSHAN CHIANG, JING-SHING YAO AND HUEY-MING LEE Given a =, b =, c =, r =, =, we get the crisp optimal order quantity q =, optimal backorder quantity s = , and the minimal total cost F(q, s ) = 8 in the crisp case. Since there are six variables < s < s < s < q < q < q, we should give a set of seven initial points. If we assign two sets of initial points as shown in ables 5 and 6, and the j, j =,,, 8, then we have the computation results shown in ables 7 and 8. able 5. A set of initial points for Proposition. s s s q q q able 6. A set of initial points for Proposition. s s s q q q DISCUSSIONS (A) he comparison of this article with [] and [5]. In this article, we use the signed distance to find the estimated total cost in the fuzzy sense in an easy way. he articles [] and [5] used the extension principle and the centroid method. It was difficult for them to get the estimated total cost in the fuzzy sense. From able, we know that the computation results from Proposition in this paper are very close to those of []. From able 4, we know that the computation results from Proposition 4 in this paper are very close to those of [5]. Moreover, from ables 7 and 8, we find that the computation results from Proposition (fuzzification of six variables) are very close to the crisp case if the fuzzy situation is insignificant.

19 FUZZY INVENORY WIH BACKORDER 69 able 7. Example 5.3 (Computed result with Proposition ) s s s q q q..... s () q () FC r q (%) r s (%) r c (%) able 8. Example 5.3 (Computed result with Proposition ) s s s q q q s () q () FC r q (%) r s (%) r c (%)

20 69 JERSHAN CHIANG, JING-SHING YAO AND HUEY-MING LEE (B) If we let j =, j =,,, 8 in Proposition, then we have the total cost in the fuzzy sense FC(q, q, q, s, s, s ; j =, j =,,, 8) a a + + = ( ) ( a b)( s s) ( a b) s( s s) ( q q q) ( s s s) q q q q + q q ln + + ln + ( ) ( a b) s ( a b)( s s) ( a b) s ( s s) ( a b) s q q q q q q q q q q cr q q ln + ln. q q q q q q If we let s = s = s then we obtain Proposition 4. (C) In section 3., we fuzzify the length of plan. We fuzzify q, s, r, a, b in the crisp total cost in Eq. (4) as the triangular fuzzy numbers. In Eqs. (5) to (9), we do the same way and fuzzify the length of plan as the following triangular fuzzy number = ( w,, + w ), < w <, < w where w, w are properly determined by the decision maker, L (α) = ( α)w >, R (α) = + ( α)w >, and fuzzify total cost in Eq. () with Q = a q, Q = a s, Q 3 = ( ) a s s q, Q 4 = b ( s s) q, Q 5 = c r q. Eq. (4) becomes QL( α) = L( α) al( α) ql( α), QR( α) = R( α) ar( α) qr( α) QL( α) = L( α) al( α) sl( α), QR( α) = R( α) ar( α) sr( α) Q3L( α) = L( α) al( α) sl( α) qr( α), Q3R( α) = R( α) ar( α) sr( α) ql( α). Q4L( α) = L( α) bl( α) sl( α) qr( α), Q4R( α) = R( α) br( α) sr( α) ql( α) Q5L( α) = cl( α) rl( α) qr( α), Q5R( α) = cr( α) rr( α) ql( α). Eq. (5) becomes K (, q, s, r, a, b, c ) L( α) = QL( α) QR( α) + Q3L( α) + Q4L( α) + Q5L( α), K(, q, s, r, a, b, c ) ( α) = Q ( α) Q ( α) + Q ( α) + Q ( α) + Q ( α). By the same way as section 3., we have R R L 3R 4R 5R

21 FUZZY INVENORY WIH BACKORDER 693 d( K( q, s, r, a, b, c ), ) = [ K(, q, s, r, a, b, c ) L ( ) α + K(, q, s, r, a, b, c ) ( α)] dα. R ACKNOWLEDGMENS he authors would like to express their sincere gratitude toward the anonymous referees for their great comments. REFERENCES. J. L. Buchanan and P. R. urner, Numerical Methods and Analysis, Mc Graw-Hill Inc., New York, 99.. S. C. Chang, J. S. Yao, and H. M. Lee, Economic reorder point for fuzzy backorder quantity, European Journal of Operation Research, Vol. 9, 998, pp S. H. Chen, C. C. Wang, and A. Ramer, Backorder fuzzy inventory model under functional principle, Information Sciences, Vol. 95, 996, pp H. Ishii and. Konno, A stochastic inventory problem with fuzzy shortage cost, European Journal of Operational Research, Vol. 6, 998, pp A. Kaufmann and M. M. Gupta, Introduction to Fuzzy Arithematic heory and Applications, Van Nostrand Reinhold, New York, G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic heory and Applications, Prentice Hall, H. M. Lee and J. S. Yao, Economic order quantity in fuzzy sense for inventory without backorder model, Fuzzy Sets and Systems, Vol. 5, 999, pp J. H. Mathews, Numerical Methods for Mathematics, Science, and Engineering, Prentice-Hall International, Inc., London, D. Petrovic and E. Sweeney, Fuzzy knowledge-based approach to treating uncertainty in inventory control, Computer Integrated Manufacturing Systems, Vol. 7, 994, pp P. M. Pu and Y. M. Liu, Fuzzy topology, neighborhood structure of a fuzzy point and moore-smith convergence, Journal of Mathematical Analysis and Applications, Vol. 76, 98, pp K. Roy and M. Maiti, A fuzzy EOQ model with demand-dependent unit cost under limited storage capacity, European Journal of Operational Research, Vol. 99, 997, pp M. Vujosevic, D. Petrovic, and R. Petrovic, EOQ formula when inventory cost is fuzzy, International Journal of Production Economics, Vol. 45, 996, pp J. S. Yao and H. M. Lee, Fuzzy inventory with or without backorder for fuzzy order quantity with trapezoid fuzzy number, Fuzzy Sets and Systems, Vol. 5, 999, pp J. S. Yao, S. C. Chang, and J. S. Su, Fuzzy inventory without backorder for fuzzy order quantity and fuzzy total demand quantity, Computer and Operation Research, Vol. 7,, pp J. S. Yao and H. M. Lee, Fuzzy inventory with backorder for fuzzy order quantity,

22 694 JERSHAN CHIANG, JING-SHING YAO AND HUEY-MING LEE Information Sciences, Vol. 93, 996, pp J. S. Yao and J. S. Su, Fuzzy inventory with backorder for fuzzy total demand based on interval-valued fuzzy set, European Journal of Operation Research, Vol. 4,, pp J. S. Yao and K. Wu, Ranking fuzzy numbers based on decomposition principle and signed distance, Fuzzy Sets and Systems, Vol. 6,, pp H. J. Zimmermann, Fuzzy Set heory and Its Application, nd Revised Edition, Kluwer Academic Publishers, BostonDordrechtLondon, 99. Jershan Chiang ( 江哲賢 ) is a Professor in the Department of Applied Mathematics at the Chinese Culture University. He earned his Ph.D. from the Department of Mathematical Sciences (now Department of Computational and Applied Mathematics) at Rice University in U.S.A. His research interests are in the field of numerical optimization, fuzzy sets theory and its applications, and operation research. His papers appeared in Fuzzy Sets and Systems, and European Journal of Operational Research. He is a member of Mathematics Society of China, and the Society for Industrial and Applied Mathematics (SIAM), U.S.A. Jing-Shing Yao ( 姚景星 ) is an Emeritus Professor of Department of Mathematics at the National aiwan University. He earned his Ph.D. from the Department of Mathematics at Kyushu University in Japan. His research interests are in the field of fuzzy sets theory and its applications, operation research and statistics. He has publications in Biometrika, Communication in Statistics, Fuzzy Sets and Systems, European Journal of Operational Research, Information Sciences, Computers and Operations Research, and IEEE ransactions on Systems, Man and Cybernetics, and Journal of Information Science and Engineering. Huey-Ming Lee ( 李惠明 ) is a Professor in the Department of Information Management at the Chinese Culture University. He earned his Ph.D. from the School of Computer Science and Engineering at the University of New South Wales in Australia. His research interests are in the field of fuzzy sets theory and its applications, operation research, software engineering, and information systems. His papers appeared in European Journal of Operational Research, Fuzzy Sets and Systems, Information Sciences, and International Journal of Reliability, Quality and Safety Engineering. He is a member of aiwanese Association for Artificial Intelligence (AAI), and Chinese Fuzzy Systems Association aiwan (CFSA).