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Fuzzy inventory without backorder for fuzzy order quantity and fuzzy total demand quantity
Computers & Operations Research 27 (2000) 935}962
Fuzzy inventory without backorder for fuzzy order quantity and fuzzy total demand quantity Jing-Shing Yao, San-Chyi Chang*, Jin-Shieh Su Department of Applied Mathematics, Chinese Culture University, Hwakang, Yangminshan, Taipei, Taiwan Received 1 January 1998; received in revised form 1 May 1999
Abstract In this paper, we consider the inventory problem without backorder such that both order and the total demand quantities are triangular fuzzy numbers QI "(q , q , q ), and RI "(r , r , r ), respectively, where q "q !* , q "q #* , r "r !* , r "r #* such that 0(* (q , 0(* , 0(* (r , 0(* , and r is a known positive number. Under conditions 0)q (q (q (r (r (r we "nd the member ship function k I I (z) of the total fuzzy cost function G(QI , RI ) and their centroid, then obtain order quantity %/ 0 qHH in the fuzzy sense and the estimate of the total demand quantity. Scope and purpose This paper deals with the inventory problem without backorder with total cost function F(q)" c¹q/2#ar/q, q'0. In the classical inventory (without backorder) model, both the total demand over the planning time period [0, ¹] and the period from ordering to arriving are "xed. In the real situation, the total demand r and order quantity q probably will be di!erent from the values used in the total cost function. Also, r in#uences the values of ¹. In view of this circumstances, we consider the inventory problem in which both order and total demand quantities are triangular fuzzy numbers QI "(q , q , q ), and RI "(r , r , r ), respec tively, where q "q !* , q "q #* , r "r !* , r "r #* such that 0(* (q , 0(* , 0(* (r , 0(* ; where r is a known number and q is unknown. Letting G(QI , RI )"c¹QI /2#aRI /QI , we use the extension principle to "nd the membership function k I I of the fuzzy total cost function G(QI , RI ) and %/ 0 their centroid (see Proposition 3). Therefore, given the value of q , q , q , r and r , we can "nd an estimate of the total cost in the fuzzy sense. Finally, we make a comparison between the crisp sense and fuzzy sense by some numerical result. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Inventory; Membership functions; Extension principle; Fuzzy inventory without backorder; Fuzzy economic order quantity; Fuzzy demand quantity
* Corresponding author. Tel.: 0088628610511;368/9; fax: 00-886-2-861-9509. E-mail address: [email protected] (J.-S. Su). 0305-0548/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 5 4 8 ( 9 9 ) 0 0 0 6 8 - 4
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1. Introduction When we discuss the classical inventory without backorder model, we get the total cost function F(q)"c¹q/2#ar/q, q'0, where c is the cost of storing one unit for one day, a is the cost of placing an order, q is the order quantity for one cycle, ¹ is the length of the planning time period, measure in days, and r is the total demand over the planning time period [0, ¹]. Inventory in fuzzy sense have been discussed recently in papers such as [1}6]. In [3], inventory without backorder model is discussed. They fuzzify the order quantity q to a fuzzy number. In [1,5], they use two approaches to discuss fuzzy sets concepts in the backorder model. In the "rst, the order quantity is fuzzi"ed and the shortage quantity is a real number [5]. In the second, shortage quantity is fuzzi"ed and the order quantity is a real variable [1]. In [4], they consider fuzzy demand quantity and fuzzy product quantity in which the inventory system is the factory warehouse. In [2], they consider the backorder fuzzy inventory model. They replaced the extension principle by the function principle to solve this problem. They fuzzify some variables in the numerator, but, the order quantity q, both in the numerator and denominator, is a crisp variable. In this paper, we consider the order quantity q and the total demand quantity r as fuzzy numbers in the inventory model without the backorder. Because the total cost function contains the demand quantity q in both the numerator and denominator, this problem is more complex than [2]. In the crisp case, the economic order quantity q is solved by min F(q)"F(q ), under the H O H condition that both the total demand over the planning time period [0, ¹] and the period from ordering to arrival are "xed. However, in the real situation, both of them will probably be di!erent from the values used in the total cost function. Also, the total demand r over the planning time period [0, ¹] could change due to some uncertain in#uence in the market. Thus, consider the total demand r as a fuzzy number near r is more reasonable than a "xed value r . Furthermore, given a, c, ¹ as constants, they e!ect q and r. It follows that q and r are subject to some kind of uncertainty. We will consider this problem under the condition that a, c, ¹ are "xed values. Hence, since q and r in#uenced F(q), we express them as the vague variables instead of crisp variables to treat this problem. Therefore, corresponding to crisp order quantity q '0 and crisp total demand r , we denote QI "(q , q , q ) and RI "(r , r , r ) where q "q !* , q "q #* (QI is just around the crisp order quantity q ) such that 0(* (q , 0(* . Similarly, r "r !* , r "r #* such that 0(* (r , 0(* . Based on the fuzzy behavior, the value of * , G i"1, 2, 3, 4 may be determined by the decision maker. In other words, we consider the order quantity and the total demand quantity as fuzzy numbers QI and RI with menbership function
q!q q !q k I (q)" q!q / q !q 0
q )q)q , q )q)q , elsewhere,
r!r r !r k I (r)" r!r 0 r !r 0
r )r)r , r )r)r , elsewhere,
where 0(q (q (q (r (r (r , and we suppose r is a known constant. We obtain the fuzzy total cost G(QI , RI )"c¹QI /2#aRI /QI . In Section 2, we use the extension principle to "nd the membership function k I I of the fuzzy total cost G(QI , RI ). In Section 3, we "nd the centroid of %/ 0
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k I I . This centroid is an estimate of the total cost. In Section 4, under conditions 0)q ( %/ 0 q (q , 0)r (r (r we use a computer program to "nd the total cost in fuzzy sense M (q , q , q , r , r ) for fuzzy order quantity (q , q , q ) and the fuzzy total demand quantity GHI (r , r , r ). In Section 5, if the centroid qHH of fuzzy order quantity is near the optimal crisp order quantity qH and the centroid rHH of fuzzy total demand quantity is near the crisp total demand quantity r , then the estimate total cost approaches the crisp total cost. 2. Membership function of the fuzzy total cost function The graph in Fig. 1 shows an inventory model without the backorder. Where q denotes the order quantity for one period, t denotes the length of one cycle and ¹ denotes the length of the planning O time period, both are measured in days. For convenience, we denote 1. c: the cost of storing one unit for one day. 2. a: the cost of placing an order. 3. r: the total demand over the planning time period [0, ¹]. Then q/t "r/¹ and the total storing cost is given by O q c¹q . ct " O2 2r Furthermore, r/q is the number of periods of the time interval [0, ¹] (r/q may be not an integer, for convenience we assume it is the number of periods). Thus the total cost of the planning time period [0, ¹] is given by
r c¹q ar q F(q)" ct #a " # , q'0. O2 q 2 q By the classical approach for inventory model without backorder, we obtain the crisp economic order quantity q "(2acr¹/c¹ and F(q )"(2acr¹ is the minimum total cost. If r"r (r is H H a known positive number), then q "(2acr ¹/c¹, F(q )"(2acr ¹. H H
Fig. 1. Inventory without backorder.
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Since the order quantity q may slightly change for some uncontrollable situations (as mentioned in the introduction) therefore, corresponding to the crisp order quantity q ('0) which is an unknown number, we consider the fuzzy order quantity QI "(q !* , q , q #* ) which may change around q , where 0(* (q , 0(* . Suppose the membership function of QI is given by q!q q )q)q , q !q k I (q)" q!q q )q)q , (1) / q !q 0 elsewhere,
where q "q !* , q "q #* , 0(* (q , 0(* , and * , * are determined by the deci sion maker based on the uncertainty of the problem. Then * !* M (q , q , q )"(q #q #q )"q # / 3 is the centroid of k I (q). Also, the total demand r is inherently extremely di$cult to estimate without / any error. Therefore, we consider the total demand quantity r is a fuzzy numbers RI near r with membership function RI given by
r!r r )r)r , r !r k I (r)" r!r r )r)r , 0 r !r 0 elsewhere, where 0(q (q (q )r (r (r (r is a known constant). Similarly, M (r , r , r )"(r #r #r ) 0 is the centroid of k I (r). 0 For the total cost function F(q), consider q, r are variables and denote c¹q ar # , 0(q(r, G(q, r)"F(q)" q 2 where a, c and ¹ are given positive numbers. Let G(q, r)"z, then c¹q!2zq#2ar"0 and hence 2zq!c¹q r" . 2a
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Therefore, for every q3[q , q ], r*0 if z*c¹q/2. Hence, z*max c¹q/2"c¹q /2 and O WOWO c¹q c¹q#2ar , z" , 0(q(r . G\(z)" (q, r) " ∀ z* 2 2q
By the extension principle, the membership function of the fuzzy cost function is given by k I I (z)" sup [k I (q)k I (r)] 0 %/ 0 / O PZ%\X 2zq!c¹q " S k I (q)k I . / 0 2a OOO In order to "nd k I I (z), if z*(2ac¹r , j"1, 0, 2 and z*c¹q /2, let %/ 0 H (z!2ac¹r #z H , j"1, 0, 2. u" H c¹
Then u )q)u if r )(2zq!c¹q)/2a)r . Thus, by (2) we have f (q) u )q)u , 2zq!c¹q kI " f (q) u )q)u , 0 2a 0 elsewhere,
where
c¹q!2zq#2ar . f (q)" 2a(r !r )
!c¹q#2zq!2ar , f (q)" 2a(r !r )
(3)
(4)
(5)
Since max (2ac¹r "(2ac¹r , H H so z*(2ac¹r for all j"1, 0, 2 if z*(2ac¹r , and hence z must satisfy z*c¹q /2 and H z*(2ac¹r . Therefore, we denote c¹q , (2ac¹r (*F(q )"(2acr ¹"G(q , r )). (6) a (q , r )"max H H 2
For convenience, under the conditions u )q and u *q the graph for Eqs. (1) and (4) are shown in Fig. 2. Also, Fig. 3 is the graph for Eqs. (1) and (4) under the conditions u *q and u )q . For z*a (q , r ) we consider k I I (z) in Eq. (3) under conditions (A) and (B) as follows: %/ 0 (A) u )q and u *q : From Fig. 2, by the "rst equation in (1) and the second equation in (4), we have q!q !c¹q#2zq!2ar " , q !q 2a(r !r )
u )q)u .
(7)
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Fig. 2. Graph under (1).
Fig. 3. Graph under (2).
Eq. (7) is equivalent to c¹(q !q )q!2[(q !q )z!a(r !r )]q#2a(q r !q r )"0 The discriminant for (8) is given by
(8)
D "[(q !q )z!a(r !r )]!2ac¹(q !q )(q r !q r ). (9) Let D "q r !q r and a (q , q , q , r , r )"max[z , a (q , r )], where H (2ac¹(q !q )D #a(r !r ) . z " H q !q From conditions (a) and (b) in Appendix A, if (D )0 and z*a (q , r )) or (D '0 and z*a (q , q , q , r , r )) we obtain PQ as follows: (q !q )z!a(r !r )!c¹q (q !q )#(D ,k (z) (say) PQ" (10) c¹(q !q )
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(B) u *q and u )q : From Fig. 3, by the second equation in (1) and the "rst equation in (4), we have q !q c¹q!2zq#2ar , u )q)u . " q !q 2a(r !r ) Eq. (11) is equivalent to
(11)
c¹(q !q )q!2[(q !q )z!a(r !r )]q#2a(q r !q r )"0. The discriminant for (12) is given by
(12)
D "[(q !q )z!a(r !r )]!2ac¹(q !q )(q r !q r ). Let D "q r !q r and a (q , q , r )"max[z , a (q , r )], where HH
(13)
(2ac¹(q !q )D #a(r !r ) . z " HH q !q Similar to (1), from conditions (c) and (d) in Appendix A, if D )0 and z*a (q , r ) or D '0 and z*a (q , q , r ), we obtain P Q as follows: !(q !q )z#a(r !r )#c¹q (q !q )!(D ,k (z) (say) P Q " c¹(q !q ) Thus, by Appendix A, we have the range of z as follows: D )0 or D )0: z*a (q , r ) (from (a) and (c)), D '0: z*a (q , q , q , r , r ) (from (b)), D '0: z*a (q , q , r ) (from (d)). Furthermore, c¹q#2ar H H ('0), j"1, 0, 2. Q *c¹q 0 (2ac¹r *c¹q , where Q " H H H H H 2q H Therefore, we can "nd the range of z for PQ"k (z) and P Q "k (z) in Section 2.1.
(14)
(15) (16) (17)
(18)
2.1. The range of z for PQ"k (z) and P Q "k (z) (i) u )q and u *q : (from (A)) The range of z for PQ"k (z) in (10) are given by the following: If u )q , then (z!2ac¹r )c¹q !z and hence z)c¹q and z)Q .
(19)
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If u *q , then (z!2ac¹r *c¹q !z and hence z(c¹q , z*c¹q , or (20) z*Q z*(2ac¹r . (ii) u *q and u )q : (from (B)) Similar to (i) we "nd the range of z for P Q "k (z) in (14) as follows: If u )q , then z)c¹q and z)Q . (21) If u *q , then we have z(c¹q , z*c¹q , or (22) z*Q z*(2ac¹r . Under conditions u )q , u *q and Eq. (15), (16), (18)}(20), since a (q , r )*(2acr ¹, H a (q , q , q , r , r )*(2acr ¹ and a (q , q , r )*(2acr ¹, ∀j"1, 0, 2; the range of z for H H PQ"k (z) in (10) can be divided into four cases (1 )}(4 ) as follows: (1 ) If Q )c¹q and Q )c¹q , then by (18) we have (2ac¹r )c¹q ; by (19) and Q )c¹q we have z)Q ; by (20) and Q )c¹q we have z*Q . Hence: (1 .1) If D )0 (implies z*a (q , r )) and max[a (q , r ), Q ])Q , since z)Q and z*Q we have (23) k I I (z)"k (z) for max[a (q , r ), Q ])z)Q . %/ 0 (1 .2) If D '0 (implies z*a (q , q , q , r , r )) and max[a (q , q , q , r , r ), Q ])Q , since z)Q and z*Q , we have (24) k I I (z)"k (z) for max[a (q , q , q , r , r ), Q ])z)Q . %/ 0 (2 ) If Q )c¹q and Q *c¹q , similar to (1 ), we have: (2 .1) If D )0 and a (q , r ))Q , since a (q , r )*(2ac¹r , z)Q and z*(2ac¹r , we have (25) k I I (z)"k (z) for a (q , r ))z)Q . %/ 0 (2 .2) If D '0 (implies z*a (q , q , q , r , r )) and a (q , q , q , r , r ))Q , (since a (q , q , q , r , r )*(2ac¹r ), z)Q and z*(2ac¹r , we have (26) k I I (z)"k (z) for a (q , q , q , r , r ))z)Q . %/ 0 (3 ) If Q *c¹q and Q )c¹q , similar to (1 ) we get: (3 .1) If D )0 and max[a (q , r ), Q ])c¹q , then k I I (z)"k (z) for max[a (q , r ), Q ])z)c¹q . (27) %/ 0
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(3 .2) If D '0 and max[a (q , q , q , r , r ), Q ])c¹q , then (28) k I I (z)"k (z) for max[a (q , q , q , r , r ), Q ])z)c¹q . %/ 0 (4 ) If Q *c¹q and Q *c¹q , then we have: (4 .1) If D )0 and a (q , r ))c¹q , then k I I (z)"k (z) for a (q , r ))z)c¹q . (29) %/ 0 (4 .2) If D '0 and a (q , q , q , r , r ))c¹q , then (30) k I I (z)"k (z) for a (q , q , q , r , r ))z)c¹q . %/ 0 Similarly, in cases (1 )}(4 ), we replace D with D (denoted by D PD ); furthermore, take a (q , q , q , r , r )Pa (q , q , r ), Q PQ , q Pq , Q PQ , q Pq . Thus, under conditions u *q and u )q , from Eqs. (15), (17), (18), (21) and (22), the range of z for P Q "k (z) in (14) can be divided into four cases (5 )}(8 ) as follows: (5 ) If Q )c¹q and Q )c¹q , then we have: (5 .1) If D )0 (implies z*a (q , r )) and max[a (q , r ), Q ])Q , we have k I I (z)"k (z) for max[a (q , r ), Q ])z)Q . (31) %/ 0 (5 .2) If D '0 (implies z*a (q , q , r )) and max[a (q , q , r ), Q ])Q , we have (32) k I I (z)"k (z) for max[a (q , q , r ), Q ])z)Q . %/ 0 (6 ) If Q )c¹q and Q *c¹q , we get: (6 .1) If D )0 and a (q , r ))Q , then (33) k I I (z)"k (z) for a (q , r ))z)Q . %/ 0 (6 .2) If D '0 and a (q , q , r ))Q , then k I I (z)"k (z) for a (q , q , r ))z)Q . (34) %/ 0 (7 ) If Q *c¹q and Q )c¹q , we get: (7 .1) If D )0 and max[a (q , r ), Q ])c¹q , then (35) k I I (z)"k (z) for max[a (q , r ), Q ])z)c¹q . %/ 0 (7 .2) If D '0 and max[a (q , q , r ), Q ])c¹q , then k I I (z)"k (z) for max[a (q , q , r ), Q ])z)c¹q . (36) %/ 0 (8 ) If Q *c¹q and Q *c¹q , we get: (8 .1) If D )0 and a (q , r ))c¹q , then k I I (z)"k (z) for a (q , r ))z)c¹q . (37) %/ 0 (8 .2) If D '0 and a (q , q , r ))c¹q , then k I I (z)"k (z) for a (q , q , r ))z)c¹q . (38) %/ 0
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2.2. The range of z in k
%/I 0I
(z)
In order to describe the range of z in k I I (z), we denote the following sets: %/ 0 S "+(q , q , q , r , r ) " Q )c¹q , Q )c¹q , Q )c¹q , case (1 ), (5 ), S "+(q , q , q , r , r ) " Q )c¹q , Q )c¹q , Q *c¹q , case (1 ), (7 ), S "+(q , q , q , r , r ) " Q *c¹q , Q )c¹q , Q )c¹q , case (2 ), (5 ), S "+(q , q , q , r , r ) " Q *c¹q , Q )c¹q , Q *c¹q , case (2 ), (7 ), S "+(q , q , q , r , r ) " Q )c¹q , Q *c¹q , Q )c¹q , case (3 ), (6 ), S "+(q , q , q , r , r ) " Q )c¹q , Q *c¹q , Q *c¹q , case (3 ), (8 ), S "+(q , q , q , r , r ) " Q *c¹q , Q *c¹q , Q )c¹q , case (4 ), (6 ), S "+(q , q , q , r , r ) " Q *c¹q , Q *c¹q , Q *c¹q , case (4 ), (8 ), ¹ "+(q , q , q , r , r ) " D )0 and D )0,, ¹ "+(q , q , q , r , r ) " D )0 and D '0,, ¹ "+(q , q , q , r , r ) " D '0 and D )0,, ¹ "+(q , q , q , r , r ) " D '0 and D '0,. Also, for j"1, 2,2, 8, and i"1, 2, 3, 4, we denote E "+E( j, i, l, 1))E( j, i, l, 2) l"1, 2 and E( j, i, 1, 2))E( j, i, 2, 1),. HG Combine (1 )}(4 ) and (5 )}(8 ), we obtain k I I (z) and the corresponding range of z as follows: %/ 0 (a) For i"1, 2, 3, 4, if (q , q , q , r , r )3S 5E 5¹ , then G G k (z) if E(1, i, 1, 1))z)E(1, i, 1, 2), k I I (z)" k (z) if E(1, i, 2, 1))z)E(1, i, 2, 2), (39) %/ 0 0 elsewhere,
where by (1 ) and (5 ): for i"1, by (1 .1), E(1, 1, 1, 1)"max[a (q , r ), Q ] and E(1, 1, 1, 2)"Q ; by (5 .1), E(1, 1, 2, 1)"max[a (q , r ), Q ] and E(1, 1, 2, 2)"Q ; for i"2, by (1 .1), we have E(1, 2, 1, k)"E(1, 1, 1, k), k"1, 2; by (5 .2), E(1, 2, 2, 1)" max[a (q , q , r ), Q ] and E(1, 2, 2, 2)"Q ; for i"3, by (1 .2) we have E(1, 3, 1, 1)"max[a (q , q , q , r , r ), Q ] and E(1, 3, 1, 2)"Q ; by (5 .1), E(1, 3, 2, k)"E(1, 1, 2, k), k"1, 2; for i"4, by (1 .2) and (5 .2), we have E(1, 4, 1, k)"E(1, 3, 1, k), k"1, 2 and E(1, 4, 2, k)" E(1, 2, 2, k), k"1, 2.
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(b) For i"1, 2, 3, 4, if (q , q , q , r , r )3S 5E 5¹ , then G G k (z) if E(2, i, 1, 1))z)E(2, i, 1, 2), k I I (z)" k (z) if E(2, i, 2, 1))z)E(2, i, 2, 2), %/ 0 0 elsewhere, where:
by (1 ), E(2, i, 1, k)"E(1, i, 1, k), i"1, 2, 3, 4 for k"1, 2; by (7 ), E(2, i, 2, 1)"E(1, i, 2, 1), E(2, i, 2, 2)"c¹q for i"1, 2, 3, 4. (c) For i"1, 2, 3, 4, if (q , q , q , r , r )3S 5E 5¹ , then G G k (z) if E(3, i, 1, 1))z)E(3, i, 1, 2), k I I (z)" k (z) if E(3, i, 2, 1))z)E(3, i, 2, 2), %/ 0 0 elsewhere, where:
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(41)
by (2 .1), E(3, 1, 1, 1)"a (q , r ), E(3, 1, 1, 2)"Q , E(3, 2, 1, k)"E(3, 1, 1, k) for k"1, 2; by (2 .2), E(3, 3, 1, 1)"a (q , q , q , r , r ), E(3, 3, 1, 2)"Q , E(3, 4, 1, k)"E(3, 3, 1, k) for k"1, 2; by (5 ), E(3, i, 2, k)"E(1, i, 2, k) for i"1, 2, 3, 4 and k"1, 2. (d) For i"1, 2, 3, 4, if (q , q , q , r , r )3S 5E 5¹ , then G G k (z) if E(4, i, 1, 1))z)E(4, i, 1, 2), (42) k I I (z)" k (z) if E(4, i, 2, 1))z)E(4, i, 2, 2), %/ 0 0 elsewhere, where:
by (2 ), E(4, i, 1, k)"E(3, i, 1, k) for i"1, 2, 3, 4 and k"1, 2; by (7 ), E(4, i, 2, k)"E(2, i, 2, k) for i"1, 2, 3, 4 and k"1, 2. (e) For i"1, 2, 3, 4, if (q , q , q , r , r )3S 5E 5¹ , then G G k (z) if E(5, i, 1, 1))z)E(5, i, 1, 2), k I I (z)" k (z) if E(5, i, 2, 1))z)E(5, i, 2, 2), %/ 0 0 elsewhere,
where similar to (a), we have: E(5, 1, 1, 1)"max[a (q , r ), Q ], E(5, 1, 1, 2)"c¹q , E(5, 1, 2, 1)"a (q , r ), E(5, 1, 2, 2)"Q . E(5, 2, 1, k)"E(5, 1, 1, k), k"1, 2, E(5, 2, 2, 1)"a (q , q , r ), E(5, 2, 2, 2)"Q , E(5, 3, 1, 1)"max[a (q , q , q , r , r ), Q ], E(5, 3, 1, 2)"c¹q , E(5, 3, 2, k)"E(5, 1, 2, k), k"1, 2, E(5, 4, 1, k)"E(5, 3, 1, k), k"1, 2, E(5, 4, 2, k)"E(5, 2, 2, k), k"1, 2.
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(f ) For i"1, 2, 3, 4, if (q , q , q , r , r )3S 5E 5¹ , then G G k (z) if E(6, i, 1, 1))z)E(6, i, 1, 2), k I I (z)" k (z) if E(6, i, 2, 1))z)E(6, i, 2, 2), %/ 0 0 elsewhere, where:
by (3 ), E(6, i, 1, k)"E(5, i, 1, k) for i"1, 2, 3, 4 and k"1, 2; by (8 ), E(6, i, 2, 1)"E(5, i, 2, 1), E(6, i, 2, 2)"c¹q for i"1, 2, 3, 4. (g) For i"1, 2, 3, 4, if (q , q , q , r , r )3S 5E 5¹ , then G G k (z) if E(7, i, 1, 1))z)E(7, i, 1, 2), k I I (z)" k (z) if E(7, i, 2, 1))z)E(7, i, 2, 2), %/ 0 0 elsewhere, where:
(44)
(45)
by (4 .1), E(7, 1, 1, 1)"E(3, 1, 1, 1), E(7, 1, 1, 2)"c¹q , E(7, 2, 1, k)"E(7, 1, 1, k) for k"1, 2; by (4 .2), E(7, 3, 1, 1)"E(3, 3, 1, 1), E(7, 3, 1, 2)"c¹q , E(7, 4, 1, k)"E(7, 3, 1, k) for k"1, 2; by (6 ), E(7, i, 2, k)"E(5, i, 2, k) for i"1, 2, 3, 4 and k"1, 2. (h) For i"1, 2, 3, 4, if (q , q , q , r , r )3S 5E 5¹ , then G G k (z) if E(8, i, 1, 1))z)E(8, i, 1, 2), (46) k I I (z)" k (z) if E(8, i, 2, 1))z)E(8, i, 2, 2), %/ 0 0 elsewhere, where:
by (4 ), E(8, i, 1, k)"E(7, i, 1, k) for i"1, 2, 3, 4 and k"1, 2; by (8 ), E(8, i, 2, k)"E(6, i, 2, k) for i"1, 2, 3, 4 and k"1, 2. Next, we will "nd the range of z to preserve 0)k (z))1, i"1, 2 in Sections 2.3 and 2.4. G 2.3. The range of z for 0)k (z))1 From Eq. (10), we have (q !q )z!a(r !r )!c¹q (q !q )#(D . k (z)" c¹(q !q ) Thus, 0)k (z))1 is equivalent to 0)(q !q )z!a(r !r )!c¹q (q !q )#(D )c¹(q !q ). From (a) and (b) in Appendix A, we note that If D )0 and z*a (q , r ), then D *0. If D '0, z*a (q , r ) and z*z , then D *0. H
(47)
(48) (49)
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947
Therefore, we will discuss inequality (47) as follows: (B1) When !(q !q )z#a(r !r )#c¹q (q !q ))(D , under (48) and (49), denote
(50)
2aq (r !r )#c¹q(q !q )#2aD , 2q (q !q ) 2aq (r !r )#c¹q(q !q )#2aD , A "max a (q , q , q , r , r ), 2q (q !q ) a(r !r )#c¹q (q !q ) , B "max a (q , r ), q !q a(r !r )#c¹q (q !q ) , B "max a (q , q , q , r , r ), q !q a(r !r )#c¹q (q !q ) (D )0) H " (q , q , q , r , r ) " A ) q !q A "max a (q , r ),
or
a(r !r )#c¹q (q !q ) (D '0) , A ) q !q 2aq (r !r )#c¹q(q !q )#2aD (D )0), H " (q , q , q , r , r ) " B ( 2q (q !q )
or
2aq (r !r )#c¹q(q !q )#2aD (D '0) . B ( 2q (q !q ) By (S1) and (S2) in Appendix B, since a (q , q , q , r , r )"max[z , a (q , r )], we have H If (q , q , q , r , r ,)3H , then a(r !r )#c¹q (q !q ) when D )0, A )z) q !q a(r !r )#c¹q (q !q ) when D '0. A )z) q !q If (q , q , q , r , r ,)3H , then B )z when D )0, B )z when D '0. (B2) When (D )!(q !q )z#a(r !r )#c¹q (q !q )#c¹(q !q ),
(51) (52)
(53) (54)
(55)
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let
a(r !r )#c¹q (q !q ) 2aq (r !r )#c¹q(q !q )#2aD , . (q !q ) 2q (q !q ) Then we have the following results: CH"min
If D )0 and a (q , r ))CH, then a (q , r ))z)CH. If D '0 and a (q , q , q , r , r ,))CH, then a (q , q , q , r , r ,))z)CH. Furthermore, denote
min
C "
a(r !r )#c¹q (q !q ) ,CH (q !q )
CH
(56) (57)
if (q , q , q , r , r ,)3H , if (q , q , q , r , r ,)3H .
By (51)}(54) and (56)}(57), we have the range of z in which 0)k (z))1 as follows: 1. If (q , q , q , r , r ,)3H , then A )z)C when D )0 and a (q , r ))C , A )z)C when D '0 and a (q , q , q , r , r ,))C . 2. If (q , q , q , r , r ,)3H , then B )z)C when D )0 and a (q , r ))C , B )z)C when D '0 and a (q , q , q , r , r ,))C . 2.4. The range of z for 0)k (z))1 From (14) we have !(q !q )z#a(r !r )#c¹q (q !q )!(D . k (z)" c¹(q !q ) Thus, 0)k (z))1 is equivalent to 0)!(q !q )z#a(r !r )#c¹q (q !q )!(D )c¹(q !q ). Similar to Section 2.3, let
2aq (r !r )#c¹q(q !q )#2aD , 2q (q !q ) 2aq (r !r )#c¹q(q !q )#2aD , A "max a (q , q , r ), 2q (q !q )
A "max a (q , r ),
(58) (59)
(60) (61)
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949
a(r !r )#c¹q (q !q ) , q !q a(r !r )#c¹q (q !q ) , B "max a (q , q , r ), q !q a(r !r )#c¹q (q !q ) 2aq (r !r )#c¹q(q !q )#2aD , , CH"min (q !q ) 2q (q !q ) a(r !r )#c¹q (q !q ) (D )0), H " (q , q , q , r , r ) A ) q !q B "max a (q , r ),
or
a(r !r )#c¹q (q !q ) (D '0) , A ) q !q 2aq (r !r )#c¹q(q !q )#2aD (D )0), H " (q , q , q , r , r ) " B ( 2q (q !q )
or
2aq (r !r )#c¹q(q !q )#2aD (D '0) , B ( 2q (q !q ) a(r !r )#c¹q (q !q ) , CH min if (q , q , q , r , r ,)3H , (q !q ) C " if (q , q , q , r , r ,)3H . CH Therefore, we have the range of z in which 0)k (z))1 as follows: 1. If (q , q , q , r , r ,)3H , then A )z)C when D )0 and a (q , r ))C , A )z)C when D '0 and a (q , q , r ,))C . 2. If (q , q , q , r , r ,)3H , then B )z)C when D )0 and a (q , r ))C , B )z)C when D '0 and a (q , q , r ,))C .
2.5. More about the range of z in k
(z) %/I 0I
Corresponding to ¹ , i"1, 2, 3, 4, and Eqs. (58)}(65), we let G < "+(q , q , q , r , r ) " a (q , r ))C , a (q , r ))C ,, < "+(q , q , q , r , r ) " a (q , r ))C , a (q , q , r ))C ,,
(62) (63)
(64) (65)
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< "+(q , q , q , r , r ) " a (q , q , q , r , r ))C , a (q , r ))C ,, < "+(q , q , q , r , r ) " a (q , q , q , r , r ))C , a (q , q , r ))C ,. Combining the range of z for 0)k (z))1 (in (58)}(61)) and 0)k (z))1 (in (62)}(65)), and based on the sign of D and D , we rewrite the range of z as follows: 1. In case (D )0)(a (q , r ))C ) and (D )0)(a (q , r ))C ), for every j"1, 2,2, 8, by (58), (60), (62) and (64), we denote EH (1, 1)"EH (1, 1)"max(A , E( j, 1, 1, 1)), H H H H E (1, 2)"E (1, 2)"min(C , E( j, 1, 1, 2)), H H EH (1, 1)"EH (1, 1)"max(B , E( j, 1, 1, 1)), H H EH (1, 2)"EH (1, 2)"min(C , E( j, 1, 1, 2)), H H EH (2, 1)"EH (2, 1)"max(A , E( j, 1, 2, 1)), H H EH (2, 2)"EH (2, 2)"min(C , E( j, 1, 2, 2)), H H EH (2, 1)"EH (2, 1)"max(B , E( j, 1, 2, 1)), H H EH (2, 2)"EH (2, 2)"min(C , E( j, 1, 2, 2)). H H 1. By (58) and (62), (q , q , q , r , r )3H 5H 5< . Let K "+(q , q , q , r , r ) " EH (1, 1))EH (1, 2))EH (2, 1))EH (2, 2),. H H H H H 1. By (58) and (64), (q , q , q , r , r )3H 5H 5< . Let K "+(q , q , q , r , r ) " EH (1, 1))EH (1, 2))EH (2, 1))EH (2, 2),. H H H H H 1. By (60) and (62), (q , q , q , r , r )3H 5H 5< . Let K "+(q , q , q , r , r ) " EH (1, 1))EH (1, 2))EH (2, 1))EH (2, 2),. H H H H H 1. By (60) and (64), (q , q , q , r , r )3H 5H 5< . Let K "+(q , q , q , r , r ) " EH (1, 1))EH (1, 2))EH (2, 1))EH (2, 2),. H H H H H Similarly: 2. In case (D )0)(a (q , r ))C ) and (D '0)(a (q , q , r ))C ), for every j"1, 2,2, 8, k"1, 2, 3, 4, by (58), (60), (63) and (65) we let K "+(q , q , q , r , r ) " EH (1, 1))EH (1, 2))EH (2, 1))EH (2, 2),, HI HI HI HI HI where EH (1, 1)"EH (1, 1)"max(A , E( j, 2, 1, 1)), H H EH (1, 2)"EH (1, 2)"min(C , E( j, 2, 1, 2)), H H
J.-S. Yao et al. / Computers & Operations Research 27 (2000) 935}962
951
EH (1, 1)"EH (1, 1)"max(B , E( j, 2, 1, 1)), H H EH (1, 2)"EH (1, 2)"min(C , E( j, 2, 1, 2)), H H EH (2, 1)"EH (2, 1)"max(A , E( j, 2, 2, 1)), H H EH (2, 2)"EH (2, 2)"min(C , E( j, 2, 2, 2)), H H EH (2, 1)"EH (2, 1)"max(B , E( j, 2, 2, 1)), H H H H E (2, 2)"E (2, 2)"min(C , E( j, 2, 2, 2)). H H 3. In case (D '0)(a (q , q , q , r , r ,))C ) and (D )0)(a (q , r ))C ), for every j"1, 2,2, 8, k"1, 2, 3, 4, by (59), (61), (62) and (64) we let K "+(q , q , q , r , r ) " EH (1, 1))EH (1, 2))EH (2, 1))EH (2, 2),, HI HI HI HI HI where EH (1, 1)"EH (1, 1)"max(A , E( j, 3, 1, 1)), H H H H E (1, 2)"E (1, 2)"min(C , E( j, 3, 1, 2)), H H EH (1, 1)"EH (1, 1)"max(B , E( j, 3, 1, 1)), H H H H E (1, 2)"E (1, 2)"min(C , E( j, 3, 1, 2)), H H EH (2, 1)"EH (2, 1)"max(A , E( j, 3, 2, 1)), H H H H E (2, 2)"E (2, 2)"min(C , E( j, 3, 2, 2)), H H EH (2, 1)"EH (2, 1)"max(B , E( j, 3, 2, 1)), H H EH (2, 2)"EH (2, 2)"min(C , E( j, 3, 2, 2)). H H 4. In case (D '0)(a (q , q , q , r , r ,))C ) and (D '0)(a (q , q , r ,))C ), for every j"1, 2,2, 8, k"1, 2, 3, 4 by (59), (61), (63) and (65) we let K "+(q , q , q , r , r ) " EH (1, 1))EH (1, 2))EH (2, 1))EH (2, 2),, HI HI HI HI HI where EH (1, 1)"EH (1, 1)"max(A , E( j, 4, 1, 1)), H H H H E (1, 2)"E (1, 2)"min(C , E( j, 4, 1, 2)), H H EH (1, 1)"EH (1, 1)"max(B , E( j, 4, 1, 1)), H H H H E (1, 2)"E (1, 2)"min(C , E( j, 4, 1, 2)), H H EH (2, 1)"EH (2, 1)"max(A , E( j, 4, 2, 1)), H H
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EH (2, 2)"EH (2, 2)"min(C , E( j, 4, 2, 2)), H H EH (2, 1)"EH (2, 1)"max(B , E( j, 4, 2, 1)), H H EH (2, 2)"EH (2, 2)"min(C , E( j, 4, 2, 2)). H H Then, for a summary of the above discussion, we have k I I (z) in Proposition 1. %/ 0 Proposition 1. For every j"1, 2,2, 8, i"1, 2, 3, 4 and (q , q , q , r , r )3S 5E 5¹ , H HG G the membership function of G(QI , RI ) is described as follows: (1 ) If (q , q , q , r , r )3H 5H 5< 5K , then G HG k (z) if EH (1, 1))z)EH (1, 2), HG HG k I I (z)" k (z) if EH (2, 1))z)EH (2, 2), HG HG %/ 0 0 elsewhere.
(2 ) If (q , q , q , r , r )3H 5H 5< 5K , then G HG
k (z) if EH (1, 1))z)EH (1, 2), HG HG k I I (z)" k (z) if EH (2, 1))z)EH (2, 2), HG HG %/ 0 0 elsewhere. (3 ) If (q , q , q , r , r )3H 5H 5< 5K , then G HG
k (z) if EH (1, 1))z)EH (1, 2), HG HG k I I (z)" k (z) if EH (2, 1))z)EH (2, 2), HG HG %/ 0 0 elsewhere. (4 ) If (q , q , q , r , r )3H 5H 5< 5K , then G HG
k (z) if EH (1, 1))z)EH (1, 2), HG HG k I I (z)" k (z) if EH (2, 1))z)EH (2, 2), HG HG %/ 0 0 elsewhere. Denote H "H 5H , H "H 5H , H "H 5H and H "H 5H then Proposi tion 1 can be rewritten as follows: Proposition 2. For every j"1, 2,2, 8, k, i"1, 2, 3, 4, if (q , q , q , r , r )3S 5E 5¹ 5H 5< 5K , H HG G I G HGI
J.-S. Yao et al. / Computers & Operations Research 27 (2000) 935}962
then
k (z) if EH (1, 1))z)EH (1, 2), HGI HGI k I I (z)" k (z) if EH (2, 1))z)EH (2, 2), HGI HGI %/ 0 0 elsewhere. 3. The centroid of k
%/I 0I
953
(66)
(z)
From Eqs. (C.1)}(C.10) in Appendix C, we can "nd the centroid for k I I (z) in Proposition 2 as %/ 0 follows: For every j"1, 2,2, 8, i, k"1, 2, 3, 4, if (q , q , q , r , r )3S 5E 5¹ 5H 5< 5K , H HG G I G HGI let
k I I (z) dz %/ 0 \ 1 1 #HHGI #HHGI " H (z) dz# H (z) dz, c¹(q !q ) HHGI c¹(q !q ) HHGI # # zk I I (z) dz Q( j, i, k)" %/ 0 \ 1 1 #HHGI #HHGI zH (z) dz# zH (z) dz, " c¹(q !q ) HHGI c¹(q !q ) HHGI # # where P( j, i, k)"
(67)
(68)
H (z)"(q !q )z!a(r !r )!c¹q (q !q )#(D , H (z)"!(q !q )z#a(r !r )#c¹q (q !q )!(D . For convenience, we give the following notations: 1. If i"1, 2, by ¹ we have D )0 and hence G 1. By (C.5) and (C.2) we denote
#HHGI H (z) dz #HHGI "R (EH (1, 1), EH (1, 2))#= (EH (1, 1), EH (1, 2)). HGI HGI HGI HGI 1. By (C.9) and (C.7) we denote P ( j, i, k)"
(69)
#HHGI zH (z) dz H #HGI "R (EH (1, 1), EH (1, 2))#= (EH (1, 1), EH (1, 2)). HGI HGI HGI HGI
Q ( j, i, k)"
(70)
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2. If i"1, 3, by ¹ we have D )0 and hence G 1. By (C.6) and (C.4) we denote
#HHGI H (z) dz H #HGI
P ( j, i, k)"
"R (EH (2, 1), EH (2, 2))!= (EH (2, 1), EH (2, 2)). HGI HGI HGI HGI
(71)
1. By (C.10) and (C.8) we denote
#HHGI zH (z) dz #HHGI
Q ( j, i, k)"
"R (EH (2, 1), EH (2, 2))!= (EH (2, 1), EH (2, 2)). HGI HGI HGI HGI
(72)
3. If i"3, 4, by ¹ we have D '0 and hence G 1. By (C.5) and (C.1) we denote
#HHGI H (z) dz #HHGI
P ( j, i, k)"
"R (EH (1, 1), EH (1, 2))#= (EH (1, 1), EH (1, 2)). HGI HGI HGI HGI
(73)
1. By (C.9) and (C.7) we denote
#HHGI zH (z) dz Q ( j, i, k)" #HHGI "R (EH (1, 1), EH (1, 2))#= (EH (1, 1), EH (1, 2)). HGI HGI HGI HGI
(74)
4. If i"2, 4, by ¹ we have D '0 and hence G 1. By (C.6) and (C.3) we denote
#HHGI H (z) dz H #HGI
P ( j, i, k)"
"R (EH (2, 1), EH (2, 2))!= (EH (2, 1), EH (2, 2)). HGI HGI HGI HGI
(75)
1. By (C.10) and (C.8) we denote
Q ( j, i, k)"
#HHGI zH (z) dz #HHGI
"R (EH (2, 1), EH (2, 2))!= (EH (2, 1), EH (2, 2)). HGI HGI HGI HGI
(76)
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955
Therefore, by (67)}(76) we obtain the following: P ( j, 1, k) P ( j, 1, k) # , P( j, 1, k)" c¹(q !q ) c¹(q !q ) P ( j, 2, k) P ( j, 2, k) # , P( j, 2, k)" c¹(q !q ) c¹(q !q ) P ( j, 3, k) P ( j, 3, k) P( j, 3, k)" # , c¹(q !q ) c¹(q !q ) P ( j, 4, k) P ( j, 4, k) P( j, 4, k)" # , c¹(q !q ) c¹(q !q )
Q ( j, 1, k) Q ( j, 1, k) Q( j, 1, k)" # , c¹(q !q ) c¹(q !q ) Q ( j, 2, k) Q ( j, 2, k) Q( j, 2, k)" # , c¹(q !q ) c¹(q !q ) Q ( j, 3, k) Q ( j, 3, k) Q( j, 3, k)" # , c¹(q !q ) c¹(q !q ) Q ( j, 4, k) Q ( j, 4, k) Q( j, 4, k)" # . c¹(q !q ) c¹(q !q )
Proposition 3. For every j"1, 2,2, 8 and i, k"1, 2, 3, 4, if (q , q , q , r , r )3S 5E 5¹ 5H 5< 5K , H HG G I G HGI then the centroid for k I I (z) is given by %/ 0 Q M (q , q , q , r , r )" H G I HGI P H G I which is an estimate of the total cost.
(77)
4. Numerical example Let a"3, c"4, r "30 and ¹"5, then q "3 is the crisp optimal order quantity and H G(q , r )"F(q )"60 is the crisp minimum total cost. In the fuzzy sense, we use a FORTRAN H H program to "nd MHH"M (q , q , q , r , r ), which is an estimate of the total cost under the fuzzy HGI order quantity (q , q , q ) and the fuzzy total demand quantity (r , r , r ). Also, let qHH" (q #q #q )/3 and rHH"(r #r #r )/3 be the centroid of (q , q , q ) and (r , r , r ), respectively. Then the relative error in the fuzzy sense for order quantity, total demand quantity and total cost are given by rHH!r MHH!G(q , r ) qHH!qH , Rel C" H , , Rel R" Rel Q" r G(q , r ) qH H respectively. For some sets of (q , q , q ) and (r , r , r ), we have the numerical results in Table 1 as follows: Roughly speaking, Table 1 displays that the value of MHH approaches the crisp minimum cost G(q , r ) as both qHH approaches the crisp order quantity q and rHH approaches the crisp total H H demand quantity r . That is, Rel CP0 as Rel QP0 and Rel RP0. Therefore, if both the relative error for order quantity and total demand quantity approaches zero, then the relative error for total cost approaches zero. Otherwise, if qHH are far from q and rHH is far from r , then the relative H error for the total cost is large.
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Table 1 Numerical Results for a"3, c"4, r "30, ¹"5, q "3, G(q , r )"60 H H Input data
Output data
q
q
q
r
r
qHH
rHH
MHH
Rel Q
Rel R
Rel C
1.0 2.2 2.2 2.2 1.0 2.8 2.8 2.8 1.0 2.8 2.9 2.8 2.2 2.8 2.9 4.9 4.9 4.9 4.9 4.9 6.5 6.5 7.1 7.1 7.1 7.1 6.5 6.5
4.1 4.1 4.1 4.1 4.1 4.1 4.7 4.7 4.7 4.7 4.9 4.7 4.7 4.7 4.9 6.9 6.9 6.9 6.9 6.9 7.9 7.9 7.9 7.9 8.7 8.7 8.7 9.5
4.7 4.7 5.3 5.3 5.0 5.9 5.3 5.3 5.9 5.9 5.9 6.5 6.5 6.5 6.7 7.9 7.9 7.9 7.9 7.9 8.1 8.1 8.1 8.9 8.9 8.9 9.7 9.7
29.0 23.0 26.0 29.0 25.0 29.0 20.0 29.0 29.0 29.0 21.0 29.0 20.0 20.0 27.0 25.0 28.0 19.0 19.0 28.0 28.8 23.4 27.0 28.8 23.4 23.4 28.8 23.4
31.0 31.0 31.0 31.0 33.0 31.0 34.0 34.0 34.0 34.0 34.0 31.0 34.0 34.0 34.0 32.0 33.0 37.0 40.0 41.0 35.9 44.0 38.6 35.9 37.7 44.0 36.8 39.5
3.27 3.67 3.87 3.87 3.37 4.27 4.27 4.27 3.87 4.47 4.57 4.67 4.47 4.67 4.83 6.57 6.57 6.57 6.57 6.57 7.50 7.50 7.70 7.97 8.23 8.23 8.30 8.57
30.00 28.00 29.00 30.00 29.33 30.00 28.00 31.00 31.00 31.00 28.33 30.00 28.00 28.00 30.33 29.00 30.33 28.67 29.67 33.00 31.57 32.47 31.87 31.57 30.37 32.47 31.87 30.97
63.12 63.12 64.29 64.32 65.20 65.69 66.89 66.93 68.16 68.21 68.40 69.74 70.21 70.22 72.03 83.36 83.49 83.98 84.36 84.51 87.94 88.64 89.37 93.16 95.00 95.66 100.81 102.29
0.0889 0.2222 0.2889 0.2889 0.1222 0.4222 0.4222 0.4222 0.2889 0.4889 0.5222 0.5556 0.4889 0.5556 0.6111 1.1889 1.1889 1.1889 1.1889 1.1889 1.5000 1.5000 1.5667 1.6556 1.7444 1.7444 1.7667 1.8556
0.0000 !0.0667 !0.0333 0.0000 !0.0222 0.0000 !0.0667 0.0333 0.0333 0.0333 !0.0556 0.0000 !0.0667 !0.0667 0.0111 !0.0333 0.0111 !0.0444 !0.0111 0.1000 0.0522 0.0822 0.0622 0.0522 0.0122 0.0822 0.0622 0.0322
0.0520 0.0520 0.0715 0.0719 0.0866 0.0948 0.1148 0.1154 0.1360 0.1369 0.1400 0.1624 0.1702 0.1703 0.2005 0.3893 0.3915 0.3996 0.4060 0.4085 0.4657 0.4773 0.4895 0.5526 0.5833 0.5943 0.6801 0.7049
5. Conclusions In the classical inventory without the backorder model, we consider the order quantity q and the total demand quantity r are fuzzy numbers. If the centroid qHH of fuzzy order quantity near the optimal crisp order quantity qH and the centroid rHH of fuzzy total demand quantity near the crisp total demand quantity r , then the estimate total cost in Proposition 3 approach the crisp optimal total cost F(q ). H In brief, when the fuzzy situation is known, we can consider this inventory model in the fuzzy sense. This method may also be used in comparison with the crisp case.
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Appendix A (a) If D )0 then D *0. Therefore for z*a (q , r ) the maximum positive root for (8) is given by (q !q )z!a(r !r )#(D ('0). (*) q" c¹(q !q ) When D )0 and z*a (q , r ), substituting this q (in (*)) in the left-hand side of (7), we obtain (q !q )z!a(r !r )!c¹q (q !q )#(D ,k (z) (say). PQ" c¹(q !q ) (b) If D '0 then z*a (q , r ) and z*z imply D *0, where H (2ac¹(q !q )D #a(r !r ) ('0). z " H q !q Also, z*z N(q !q )z!a(r !r )*(q !q )z !a(r !r )"(2ac¹(q !q )D '0. H H Similarly, the maximum positive root for (8) is given by (*) if z*a (q , r ) and z*z . Denote H a (q , q , q , r , r )"max+z , a (q , r )]. Therefore, H if D '0 and z*a (q , q , q , r , r ) then PQ"k (z). (c) If D )0 then D *0, for z*a (q , r ). The maximum positive root for (12) is given by (q !q )z!a(r !r )#(D ('0). q" c¹(q !q ) Substituting q in the left-hand side of (11), when D )0 and z*a (q , r ), we obtain !(q !q )z#a(r !r )#c¹q (q !q )!(D ,k (z) (say). P Q " c¹(q !q ) (d) If D '0 then z*a (q , r ) and z*z imply D *0, where HH (2ac¹(q !q )D #a(r !r ) ('0) z " HH q !q and z*z N(q !q )z!a(r !r )*(q !q )z !a(r !r )"(2ac¹(q !q )D '0. HH HH
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Thus, the maximum positive root for (12) is given by (q !q )z!a(r !r )#(D ('0). q" c¹(q !q ) Denote a (q , q , r )"max[z , a (q , r )]. Therefore, HH if D '0 and z*a (q , q , r ) then P Q "k (z). Appendix B (S1) If !(q !q )z#a(r !r )#c¹q (q !q )*0, then inequality (50) implies 2aq (r !r )#c¹q(q !q )#2aD )2q (q !q )z. Therefore, If D )0, then z*a (q , r ), a(r !r )#c¹q (q !q ) z) (q !q ) and 2aq (r !r )#c¹q(q !q )#2aD . z* 2q (q !q ) If D '0, then z*a (q , r ), z*z , H a(r !r )#c¹q (q !q ) z) (q !q ) and 2aq (r !r )#c¹q(q !q )#2aD . z* 2q (q !q ) (S2) If !(q !q )z#a(r !r )#c¹q (q !q )(0, under conditions (48) and (49), inequality (50) holds. Then we have If D )0, then z*a (q , r ) and a(r !r )#c¹q (q !q ) . z' (q !q ) If D '0, then z*a (q , r ), z*z and H a(r !r )#c¹q (q !q ) . z' (q !q )
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Appendix C In order to "nd the centroid for the membership function k I I (z) in Proposition 2, we consider %/ 0 the following integrals: 1. If D '0 denote
= (a , a )"
?
?
"
?
?
(D dz ([(q !q )z!a(r !r )]!2ac¹(q !q )D dz,
where D "(q r !q r ). Substituting y"(q !q )z!a(r !r ) and y"(2ac¹(q !q )D sec h in the above equa tion we get
sec h #tan h , = (a , a )"ac¹D sec h tan h !sec h tan h !ln sec h #tan h
(C.1)
where aH H sec h " and aH"(q !q )a !a(r !r ). H (2ac¹(q !q )D H H 2. If D (0 we have
=H (a , a )"
?
?
(D dz
sec hH#tan hH , "!ac¹D sec hH tan hH!sec hH tan hH#ln sec hH#tan hH where aH H tan hH" . H (!2ac¹(q !q )D 3. If D "0 let
=H (a , a )"
?
?
"(q !q )z!a(r !r )" dz
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then
1 ?H =H (a , a )" "y" dy q !q H ? [(aH)!(aH)] if 0)aH, O\O " [(aH)#(aH)] if aH(0)aH, O \O [(aH )!(aH )] if aH (aH )0. O\O For convenience, denote
=H (a , a ) if D (0, = (a , a )" =H (a , a ) if D "0. On the other hand, consider the signs of D , we obtain the following: 4. If D '0, then ? = (a , a )" (D dz ? ? " ([(q !q )z!a(r !r )]!2ac¹(q !q )D dz ? sec u #tan u , "ac¹D sec u tan u !sec u tan u !ln sec u #tan u where
(C.2)
(C.3)
cH H and cH"(q !q )a !a(r !r ). sec u " H H H (2ac¹(q !q )D 5. If D (0, then
=H (a , a )"
where
?
?
(D dz
cH H . tan uH" H (!2ac¹(q !q )D 6. If D "0, then
sec uH#tan uH , "!ac¹D sec uH tan uH!sec uH tan uH#ln sec uH#tan uH
AH 1 =H (a , a )" "y" dy q !q H A
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[(cH)!(cH)] if 0)cH, O\O " [(cH)#(cH)] if cH(0)cH, O \O [(cH )!(cH )] if cH (cH )0. O\O For convenience, denote
=H (a , a ) if D (0, = (a , a )" =H (a , a ) if D "0. 7.
(C.4)
R (a , a )"
? [(q !q )z!a(r !r )!c¹q (q !q )] dz ?
"(a !a ) 8.
q !q (a #a )!a(r !r )!c¹q (q !q ) . 2
(C.5)
R (a , a )"
? [!(q !q )z#a(r !r )#c¹q (q !q )] dz ?
q !q (a #a )#a(r !r )#c¹q (q !q ) . "(a !a ) ! 2
(C.6)
9. Substituting z"[y#a(r !r )]/(g !g ) in the following integration we get
= (a , a )" H
? z(D dz (D '0 for j"1 and D )0 for j"2) ?
?H 1 " [y#a(r !r )](y!2ac¹(q !q )D dy (q !q ) H ? a(r !r ) 1 = (a , a )# " H q !q 3(q !q ) ;+[(aH)!2ac¹(q !q )D ] ![(aH)!2ac¹(q !q )D ] ,. (C.7) Similarly, 10.
= (a , a )" H
?
?
z(D dz (D '0 for j"1 and D )0 for j"2)
a(r !r ) 1 = (a , a )# " H q !q 3(q !q ) ;+[(cH)!2ac¹(q !q )D ] ![(cH)!2ac¹(q !q )D ] ,. (C.8)
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11.
R (a , a )"
?
?
z[(q !q )z!a(r !r )!c¹q (q !q )] dz
(a!a) 1 [a(r !r )#c¹q (q !q )]. " (q !q )(a!a)! 2 3 12.
R (a , a )"
?
?
(C.9)
z[!(q !q )z#a(r !r )#c¹q (q !q )] dz
(a!a) 1 [a(r !r )#c¹q (q !q )]. "! (q !q )(a!a)# 2 3
(C.10)
References [1] San-Chyi Chang, Jing-Shing Yao, Huey-Ming Lee. Economic reorder point for fuzzy backorder quantity. European Journal of Operation Research 1998;109:183}202. [2] Shan-Huo Chen, Chien-Chung Wang, Arthur Ramer. Backorder fuzzy inventory model under function principle. Information Science 1996;95:71}9. [3] Huey-Ming Lee, Jing-Shing Yao. Economic order quantity in fuzzy sense for inventory without backorder model. Fuzzy Sets and Systems 1999;105:13}31. [4] Huey-Ming Lee, Jing-Shing Yao. Economic production quantity for fuzzy demand quantity and fuzzy production quantity. European Journal of Operation Research 1998;109:203}11. [5] Jing-Shing Yao, Huey-Ming Lee. Fuzzy inventory with backorder for fuzzy order quantity. Information Sciences 1996;93:283}319. [6] Jing-Shing Yao, Huey-Ming Lee. Fuzzy inventory with or without backorder for order quantity with trapezoid fuzzy number. Fuzzy Sets and Systems 1999;105:311}37.
Jing-Shing Yao is a professor in the department of applied mathematics at the Chinese Culture University. He earned his Ph.D from the department of mathematics at Kyushu University in Japan. His research interests are in the "eld of fuzzy set theory and its application, operation research and statistics. He has publication in Fuzzy Sets and Systems, European Journal of Operation Research and Information Sciences. San-Chyi Chang is an associate professor in the department of applied mathematics at the Chinese Culture University. He earned his Ph.D from the graduate institute of management science at Tamkang University in Taiwan. His research interests are in the "eld of statistics and the application of fuzzy mathematics. He has publication in the European Journal of Operation Research and some papers will appear in Fuzzy Sets and Systems. Jin-Shieh Su is an instructor in the department of applied mathematics at the Chinese Culture University. His research interests are in the "eld of computer science and the application of fuzzy mathematics. He has publication in Hwa Kang Journal of Sciences and some papers will appear in the European Journal of Operation Research.
Fuzzy inventory without backorder for fuzzy order quantity and fuzzy total demand quantity Jing-Shing Yao, San-Chyi Chang*, Jin-Shieh Su Department of Applied Mathematics, Chinese Culture University, Hwakang, Yangminshan, Taipei, Taiwan Received 1 January 1998; received in revised form 1 May 1999
Abstract In this paper, we consider the inventory problem without backorder such that both order and the total demand quantities are triangular fuzzy numbers QI "(q , q , q ), and RI "(r , r , r ), respectively, where q "q !* , q "q #* , r "r !* , r "r #* such that 0(* (q , 0(* , 0(* (r , 0(* , and r is a known positive number. Under conditions 0)q (q (q (r (r (r we "nd the member ship function k I I (z) of the total fuzzy cost function G(QI , RI ) and their centroid, then obtain order quantity %/ 0 qHH in the fuzzy sense and the estimate of the total demand quantity. Scope and purpose This paper deals with the inventory problem without backorder with total cost function F(q)" c¹q/2#ar/q, q'0. In the classical inventory (without backorder) model, both the total demand over the planning time period [0, ¹] and the period from ordering to arriving are "xed. In the real situation, the total demand r and order quantity q probably will be di!erent from the values used in the total cost function. Also, r in#uences the values of ¹. In view of this circumstances, we consider the inventory problem in which both order and total demand quantities are triangular fuzzy numbers QI "(q , q , q ), and RI "(r , r , r ), respec tively, where q "q !* , q "q #* , r "r !* , r "r #* such that 0(* (q , 0(* , 0(* (r , 0(* ; where r is a known number and q is unknown. Letting G(QI , RI )"c¹QI /2#aRI /QI , we use the extension principle to "nd the membership function k I I of the fuzzy total cost function G(QI , RI ) and %/ 0 their centroid (see Proposition 3). Therefore, given the value of q , q , q , r and r , we can "nd an estimate of the total cost in the fuzzy sense. Finally, we make a comparison between the crisp sense and fuzzy sense by some numerical result. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Inventory; Membership functions; Extension principle; Fuzzy inventory without backorder; Fuzzy economic order quantity; Fuzzy demand quantity
* Corresponding author. Tel.: 0088628610511;368/9; fax: 00-886-2-861-9509. E-mail address: [email protected] (J.-S. Su). 0305-0548/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 5 4 8 ( 9 9 ) 0 0 0 6 8 - 4
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1. Introduction When we discuss the classical inventory without backorder model, we get the total cost function F(q)"c¹q/2#ar/q, q'0, where c is the cost of storing one unit for one day, a is the cost of placing an order, q is the order quantity for one cycle, ¹ is the length of the planning time period, measure in days, and r is the total demand over the planning time period [0, ¹]. Inventory in fuzzy sense have been discussed recently in papers such as [1}6]. In [3], inventory without backorder model is discussed. They fuzzify the order quantity q to a fuzzy number. In [1,5], they use two approaches to discuss fuzzy sets concepts in the backorder model. In the "rst, the order quantity is fuzzi"ed and the shortage quantity is a real number [5]. In the second, shortage quantity is fuzzi"ed and the order quantity is a real variable [1]. In [4], they consider fuzzy demand quantity and fuzzy product quantity in which the inventory system is the factory warehouse. In [2], they consider the backorder fuzzy inventory model. They replaced the extension principle by the function principle to solve this problem. They fuzzify some variables in the numerator, but, the order quantity q, both in the numerator and denominator, is a crisp variable. In this paper, we consider the order quantity q and the total demand quantity r as fuzzy numbers in the inventory model without the backorder. Because the total cost function contains the demand quantity q in both the numerator and denominator, this problem is more complex than [2]. In the crisp case, the economic order quantity q is solved by min F(q)"F(q ), under the H O H condition that both the total demand over the planning time period [0, ¹] and the period from ordering to arrival are "xed. However, in the real situation, both of them will probably be di!erent from the values used in the total cost function. Also, the total demand r over the planning time period [0, ¹] could change due to some uncertain in#uence in the market. Thus, consider the total demand r as a fuzzy number near r is more reasonable than a "xed value r . Furthermore, given a, c, ¹ as constants, they e!ect q and r. It follows that q and r are subject to some kind of uncertainty. We will consider this problem under the condition that a, c, ¹ are "xed values. Hence, since q and r in#uenced F(q), we express them as the vague variables instead of crisp variables to treat this problem. Therefore, corresponding to crisp order quantity q '0 and crisp total demand r , we denote QI "(q , q , q ) and RI "(r , r , r ) where q "q !* , q "q #* (QI is just around the crisp order quantity q ) such that 0(* (q , 0(* . Similarly, r "r !* , r "r #* such that 0(* (r , 0(* . Based on the fuzzy behavior, the value of * , G i"1, 2, 3, 4 may be determined by the decision maker. In other words, we consider the order quantity and the total demand quantity as fuzzy numbers QI and RI with menbership function
q!q q !q k I (q)" q!q / q !q 0
q )q)q , q )q)q , elsewhere,
r!r r !r k I (r)" r!r 0 r !r 0
r )r)r , r )r)r , elsewhere,
where 0(q (q (q (r (r (r , and we suppose r is a known constant. We obtain the fuzzy total cost G(QI , RI )"c¹QI /2#aRI /QI . In Section 2, we use the extension principle to "nd the membership function k I I of the fuzzy total cost G(QI , RI ). In Section 3, we "nd the centroid of %/ 0
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k I I . This centroid is an estimate of the total cost. In Section 4, under conditions 0)q ( %/ 0 q (q , 0)r (r (r we use a computer program to "nd the total cost in fuzzy sense M (q , q , q , r , r ) for fuzzy order quantity (q , q , q ) and the fuzzy total demand quantity GHI (r , r , r ). In Section 5, if the centroid qHH of fuzzy order quantity is near the optimal crisp order quantity qH and the centroid rHH of fuzzy total demand quantity is near the crisp total demand quantity r , then the estimate total cost approaches the crisp total cost. 2. Membership function of the fuzzy total cost function The graph in Fig. 1 shows an inventory model without the backorder. Where q denotes the order quantity for one period, t denotes the length of one cycle and ¹ denotes the length of the planning O time period, both are measured in days. For convenience, we denote 1. c: the cost of storing one unit for one day. 2. a: the cost of placing an order. 3. r: the total demand over the planning time period [0, ¹]. Then q/t "r/¹ and the total storing cost is given by O q c¹q . ct " O2 2r Furthermore, r/q is the number of periods of the time interval [0, ¹] (r/q may be not an integer, for convenience we assume it is the number of periods). Thus the total cost of the planning time period [0, ¹] is given by
r c¹q ar q F(q)" ct #a " # , q'0. O2 q 2 q By the classical approach for inventory model without backorder, we obtain the crisp economic order quantity q "(2acr¹/c¹ and F(q )"(2acr¹ is the minimum total cost. If r"r (r is H H a known positive number), then q "(2acr ¹/c¹, F(q )"(2acr ¹. H H
Fig. 1. Inventory without backorder.
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Since the order quantity q may slightly change for some uncontrollable situations (as mentioned in the introduction) therefore, corresponding to the crisp order quantity q ('0) which is an unknown number, we consider the fuzzy order quantity QI "(q !* , q , q #* ) which may change around q , where 0(* (q , 0(* . Suppose the membership function of QI is given by q!q q )q)q , q !q k I (q)" q!q q )q)q , (1) / q !q 0 elsewhere,
where q "q !* , q "q #* , 0(* (q , 0(* , and * , * are determined by the deci sion maker based on the uncertainty of the problem. Then * !* M (q , q , q )"(q #q #q )"q # / 3 is the centroid of k I (q). Also, the total demand r is inherently extremely di$cult to estimate without / any error. Therefore, we consider the total demand quantity r is a fuzzy numbers RI near r with membership function RI given by
r!r r )r)r , r !r k I (r)" r!r r )r)r , 0 r !r 0 elsewhere, where 0(q (q (q )r (r (r (r is a known constant). Similarly, M (r , r , r )"(r #r #r ) 0 is the centroid of k I (r). 0 For the total cost function F(q), consider q, r are variables and denote c¹q ar # , 0(q(r, G(q, r)"F(q)" q 2 where a, c and ¹ are given positive numbers. Let G(q, r)"z, then c¹q!2zq#2ar"0 and hence 2zq!c¹q r" . 2a
(2)
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Therefore, for every q3[q , q ], r*0 if z*c¹q/2. Hence, z*max c¹q/2"c¹q /2 and O WOWO c¹q c¹q#2ar , z" , 0(q(r . G\(z)" (q, r) " ∀ z* 2 2q
By the extension principle, the membership function of the fuzzy cost function is given by k I I (z)" sup [k I (q)k I (r)] 0 %/ 0 / O PZ%\X 2zq!c¹q " S k I (q)k I . / 0 2a OOO In order to "nd k I I (z), if z*(2ac¹r , j"1, 0, 2 and z*c¹q /2, let %/ 0 H (z!2ac¹r #z H , j"1, 0, 2. u" H c¹
Then u )q)u if r )(2zq!c¹q)/2a)r . Thus, by (2) we have f (q) u )q)u , 2zq!c¹q kI " f (q) u )q)u , 0 2a 0 elsewhere,
where
c¹q!2zq#2ar . f (q)" 2a(r !r )
!c¹q#2zq!2ar , f (q)" 2a(r !r )
(3)
(4)
(5)
Since max (2ac¹r "(2ac¹r , H H so z*(2ac¹r for all j"1, 0, 2 if z*(2ac¹r , and hence z must satisfy z*c¹q /2 and H z*(2ac¹r . Therefore, we denote c¹q , (2ac¹r (*F(q )"(2acr ¹"G(q , r )). (6) a (q , r )"max H H 2
For convenience, under the conditions u )q and u *q the graph for Eqs. (1) and (4) are shown in Fig. 2. Also, Fig. 3 is the graph for Eqs. (1) and (4) under the conditions u *q and u )q . For z*a (q , r ) we consider k I I (z) in Eq. (3) under conditions (A) and (B) as follows: %/ 0 (A) u )q and u *q : From Fig. 2, by the "rst equation in (1) and the second equation in (4), we have q!q !c¹q#2zq!2ar " , q !q 2a(r !r )
u )q)u .
(7)
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Fig. 2. Graph under (1).
Fig. 3. Graph under (2).
Eq. (7) is equivalent to c¹(q !q )q!2[(q !q )z!a(r !r )]q#2a(q r !q r )"0 The discriminant for (8) is given by
(8)
D "[(q !q )z!a(r !r )]!2ac¹(q !q )(q r !q r ). (9) Let D "q r !q r and a (q , q , q , r , r )"max[z , a (q , r )], where H (2ac¹(q !q )D #a(r !r ) . z " H q !q From conditions (a) and (b) in Appendix A, if (D )0 and z*a (q , r )) or (D '0 and z*a (q , q , q , r , r )) we obtain PQ as follows: (q !q )z!a(r !r )!c¹q (q !q )#(D ,k (z) (say) PQ" (10) c¹(q !q )
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(B) u *q and u )q : From Fig. 3, by the second equation in (1) and the "rst equation in (4), we have q !q c¹q!2zq#2ar , u )q)u . " q !q 2a(r !r ) Eq. (11) is equivalent to
(11)
c¹(q !q )q!2[(q !q )z!a(r !r )]q#2a(q r !q r )"0. The discriminant for (12) is given by
(12)
D "[(q !q )z!a(r !r )]!2ac¹(q !q )(q r !q r ). Let D "q r !q r and a (q , q , r )"max[z , a (q , r )], where HH
(13)
(2ac¹(q !q )D #a(r !r ) . z " HH q !q Similar to (1), from conditions (c) and (d) in Appendix A, if D )0 and z*a (q , r ) or D '0 and z*a (q , q , r ), we obtain P Q as follows: !(q !q )z#a(r !r )#c¹q (q !q )!(D ,k (z) (say) P Q " c¹(q !q ) Thus, by Appendix A, we have the range of z as follows: D )0 or D )0: z*a (q , r ) (from (a) and (c)), D '0: z*a (q , q , q , r , r ) (from (b)), D '0: z*a (q , q , r ) (from (d)). Furthermore, c¹q#2ar H H ('0), j"1, 0, 2. Q *c¹q 0 (2ac¹r *c¹q , where Q " H H H H H 2q H Therefore, we can "nd the range of z for PQ"k (z) and P Q "k (z) in Section 2.1.
(14)
(15) (16) (17)
(18)
2.1. The range of z for PQ"k (z) and P Q "k (z) (i) u )q and u *q : (from (A)) The range of z for PQ"k (z) in (10) are given by the following: If u )q , then (z!2ac¹r )c¹q !z and hence z)c¹q and z)Q .
(19)
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If u *q , then (z!2ac¹r *c¹q !z and hence z(c¹q , z*c¹q , or (20) z*Q z*(2ac¹r . (ii) u *q and u )q : (from (B)) Similar to (i) we "nd the range of z for P Q "k (z) in (14) as follows: If u )q , then z)c¹q and z)Q . (21) If u *q , then we have z(c¹q , z*c¹q , or (22) z*Q z*(2ac¹r . Under conditions u )q , u *q and Eq. (15), (16), (18)}(20), since a (q , r )*(2acr ¹, H a (q , q , q , r , r )*(2acr ¹ and a (q , q , r )*(2acr ¹, ∀j"1, 0, 2; the range of z for H H PQ"k (z) in (10) can be divided into four cases (1 )}(4 ) as follows: (1 ) If Q )c¹q and Q )c¹q , then by (18) we have (2ac¹r )c¹q ; by (19) and Q )c¹q we have z)Q ; by (20) and Q )c¹q we have z*Q . Hence: (1 .1) If D )0 (implies z*a (q , r )) and max[a (q , r ), Q ])Q , since z)Q and z*Q we have (23) k I I (z)"k (z) for max[a (q , r ), Q ])z)Q . %/ 0 (1 .2) If D '0 (implies z*a (q , q , q , r , r )) and max[a (q , q , q , r , r ), Q ])Q , since z)Q and z*Q , we have (24) k I I (z)"k (z) for max[a (q , q , q , r , r ), Q ])z)Q . %/ 0 (2 ) If Q )c¹q and Q *c¹q , similar to (1 ), we have: (2 .1) If D )0 and a (q , r ))Q , since a (q , r )*(2ac¹r , z)Q and z*(2ac¹r , we have (25) k I I (z)"k (z) for a (q , r ))z)Q . %/ 0 (2 .2) If D '0 (implies z*a (q , q , q , r , r )) and a (q , q , q , r , r ))Q , (since a (q , q , q , r , r )*(2ac¹r ), z)Q and z*(2ac¹r , we have (26) k I I (z)"k (z) for a (q , q , q , r , r ))z)Q . %/ 0 (3 ) If Q *c¹q and Q )c¹q , similar to (1 ) we get: (3 .1) If D )0 and max[a (q , r ), Q ])c¹q , then k I I (z)"k (z) for max[a (q , r ), Q ])z)c¹q . (27) %/ 0
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(3 .2) If D '0 and max[a (q , q , q , r , r ), Q ])c¹q , then (28) k I I (z)"k (z) for max[a (q , q , q , r , r ), Q ])z)c¹q . %/ 0 (4 ) If Q *c¹q and Q *c¹q , then we have: (4 .1) If D )0 and a (q , r ))c¹q , then k I I (z)"k (z) for a (q , r ))z)c¹q . (29) %/ 0 (4 .2) If D '0 and a (q , q , q , r , r ))c¹q , then (30) k I I (z)"k (z) for a (q , q , q , r , r ))z)c¹q . %/ 0 Similarly, in cases (1 )}(4 ), we replace D with D (denoted by D PD ); furthermore, take a (q , q , q , r , r )Pa (q , q , r ), Q PQ , q Pq , Q PQ , q Pq . Thus, under conditions u *q and u )q , from Eqs. (15), (17), (18), (21) and (22), the range of z for P Q "k (z) in (14) can be divided into four cases (5 )}(8 ) as follows: (5 ) If Q )c¹q and Q )c¹q , then we have: (5 .1) If D )0 (implies z*a (q , r )) and max[a (q , r ), Q ])Q , we have k I I (z)"k (z) for max[a (q , r ), Q ])z)Q . (31) %/ 0 (5 .2) If D '0 (implies z*a (q , q , r )) and max[a (q , q , r ), Q ])Q , we have (32) k I I (z)"k (z) for max[a (q , q , r ), Q ])z)Q . %/ 0 (6 ) If Q )c¹q and Q *c¹q , we get: (6 .1) If D )0 and a (q , r ))Q , then (33) k I I (z)"k (z) for a (q , r ))z)Q . %/ 0 (6 .2) If D '0 and a (q , q , r ))Q , then k I I (z)"k (z) for a (q , q , r ))z)Q . (34) %/ 0 (7 ) If Q *c¹q and Q )c¹q , we get: (7 .1) If D )0 and max[a (q , r ), Q ])c¹q , then (35) k I I (z)"k (z) for max[a (q , r ), Q ])z)c¹q . %/ 0 (7 .2) If D '0 and max[a (q , q , r ), Q ])c¹q , then k I I (z)"k (z) for max[a (q , q , r ), Q ])z)c¹q . (36) %/ 0 (8 ) If Q *c¹q and Q *c¹q , we get: (8 .1) If D )0 and a (q , r ))c¹q , then k I I (z)"k (z) for a (q , r ))z)c¹q . (37) %/ 0 (8 .2) If D '0 and a (q , q , r ))c¹q , then k I I (z)"k (z) for a (q , q , r ))z)c¹q . (38) %/ 0
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2.2. The range of z in k
%/I 0I
(z)
In order to describe the range of z in k I I (z), we denote the following sets: %/ 0 S "+(q , q , q , r , r ) " Q )c¹q , Q )c¹q , Q )c¹q , case (1 ), (5 ), S "+(q , q , q , r , r ) " Q )c¹q , Q )c¹q , Q *c¹q , case (1 ), (7 ), S "+(q , q , q , r , r ) " Q *c¹q , Q )c¹q , Q )c¹q , case (2 ), (5 ), S "+(q , q , q , r , r ) " Q *c¹q , Q )c¹q , Q *c¹q , case (2 ), (7 ), S "+(q , q , q , r , r ) " Q )c¹q , Q *c¹q , Q )c¹q , case (3 ), (6 ), S "+(q , q , q , r , r ) " Q )c¹q , Q *c¹q , Q *c¹q , case (3 ), (8 ), S "+(q , q , q , r , r ) " Q *c¹q , Q *c¹q , Q )c¹q , case (4 ), (6 ), S "+(q , q , q , r , r ) " Q *c¹q , Q *c¹q , Q *c¹q , case (4 ), (8 ), ¹ "+(q , q , q , r , r ) " D )0 and D )0,, ¹ "+(q , q , q , r , r ) " D )0 and D '0,, ¹ "+(q , q , q , r , r ) " D '0 and D )0,, ¹ "+(q , q , q , r , r ) " D '0 and D '0,. Also, for j"1, 2,2, 8, and i"1, 2, 3, 4, we denote E "+E( j, i, l, 1))E( j, i, l, 2) l"1, 2 and E( j, i, 1, 2))E( j, i, 2, 1),. HG Combine (1 )}(4 ) and (5 )}(8 ), we obtain k I I (z) and the corresponding range of z as follows: %/ 0 (a) For i"1, 2, 3, 4, if (q , q , q , r , r )3S 5E 5¹ , then G G k (z) if E(1, i, 1, 1))z)E(1, i, 1, 2), k I I (z)" k (z) if E(1, i, 2, 1))z)E(1, i, 2, 2), (39) %/ 0 0 elsewhere,
where by (1 ) and (5 ): for i"1, by (1 .1), E(1, 1, 1, 1)"max[a (q , r ), Q ] and E(1, 1, 1, 2)"Q ; by (5 .1), E(1, 1, 2, 1)"max[a (q , r ), Q ] and E(1, 1, 2, 2)"Q ; for i"2, by (1 .1), we have E(1, 2, 1, k)"E(1, 1, 1, k), k"1, 2; by (5 .2), E(1, 2, 2, 1)" max[a (q , q , r ), Q ] and E(1, 2, 2, 2)"Q ; for i"3, by (1 .2) we have E(1, 3, 1, 1)"max[a (q , q , q , r , r ), Q ] and E(1, 3, 1, 2)"Q ; by (5 .1), E(1, 3, 2, k)"E(1, 1, 2, k), k"1, 2; for i"4, by (1 .2) and (5 .2), we have E(1, 4, 1, k)"E(1, 3, 1, k), k"1, 2 and E(1, 4, 2, k)" E(1, 2, 2, k), k"1, 2.
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(b) For i"1, 2, 3, 4, if (q , q , q , r , r )3S 5E 5¹ , then G G k (z) if E(2, i, 1, 1))z)E(2, i, 1, 2), k I I (z)" k (z) if E(2, i, 2, 1))z)E(2, i, 2, 2), %/ 0 0 elsewhere, where:
by (1 ), E(2, i, 1, k)"E(1, i, 1, k), i"1, 2, 3, 4 for k"1, 2; by (7 ), E(2, i, 2, 1)"E(1, i, 2, 1), E(2, i, 2, 2)"c¹q for i"1, 2, 3, 4. (c) For i"1, 2, 3, 4, if (q , q , q , r , r )3S 5E 5¹ , then G G k (z) if E(3, i, 1, 1))z)E(3, i, 1, 2), k I I (z)" k (z) if E(3, i, 2, 1))z)E(3, i, 2, 2), %/ 0 0 elsewhere, where:
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(40)
(41)
by (2 .1), E(3, 1, 1, 1)"a (q , r ), E(3, 1, 1, 2)"Q , E(3, 2, 1, k)"E(3, 1, 1, k) for k"1, 2; by (2 .2), E(3, 3, 1, 1)"a (q , q , q , r , r ), E(3, 3, 1, 2)"Q , E(3, 4, 1, k)"E(3, 3, 1, k) for k"1, 2; by (5 ), E(3, i, 2, k)"E(1, i, 2, k) for i"1, 2, 3, 4 and k"1, 2. (d) For i"1, 2, 3, 4, if (q , q , q , r , r )3S 5E 5¹ , then G G k (z) if E(4, i, 1, 1))z)E(4, i, 1, 2), (42) k I I (z)" k (z) if E(4, i, 2, 1))z)E(4, i, 2, 2), %/ 0 0 elsewhere, where:
by (2 ), E(4, i, 1, k)"E(3, i, 1, k) for i"1, 2, 3, 4 and k"1, 2; by (7 ), E(4, i, 2, k)"E(2, i, 2, k) for i"1, 2, 3, 4 and k"1, 2. (e) For i"1, 2, 3, 4, if (q , q , q , r , r )3S 5E 5¹ , then G G k (z) if E(5, i, 1, 1))z)E(5, i, 1, 2), k I I (z)" k (z) if E(5, i, 2, 1))z)E(5, i, 2, 2), %/ 0 0 elsewhere,
where similar to (a), we have: E(5, 1, 1, 1)"max[a (q , r ), Q ], E(5, 1, 1, 2)"c¹q , E(5, 1, 2, 1)"a (q , r ), E(5, 1, 2, 2)"Q . E(5, 2, 1, k)"E(5, 1, 1, k), k"1, 2, E(5, 2, 2, 1)"a (q , q , r ), E(5, 2, 2, 2)"Q , E(5, 3, 1, 1)"max[a (q , q , q , r , r ), Q ], E(5, 3, 1, 2)"c¹q , E(5, 3, 2, k)"E(5, 1, 2, k), k"1, 2, E(5, 4, 1, k)"E(5, 3, 1, k), k"1, 2, E(5, 4, 2, k)"E(5, 2, 2, k), k"1, 2.
(43)
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(f ) For i"1, 2, 3, 4, if (q , q , q , r , r )3S 5E 5¹ , then G G k (z) if E(6, i, 1, 1))z)E(6, i, 1, 2), k I I (z)" k (z) if E(6, i, 2, 1))z)E(6, i, 2, 2), %/ 0 0 elsewhere, where:
by (3 ), E(6, i, 1, k)"E(5, i, 1, k) for i"1, 2, 3, 4 and k"1, 2; by (8 ), E(6, i, 2, 1)"E(5, i, 2, 1), E(6, i, 2, 2)"c¹q for i"1, 2, 3, 4. (g) For i"1, 2, 3, 4, if (q , q , q , r , r )3S 5E 5¹ , then G G k (z) if E(7, i, 1, 1))z)E(7, i, 1, 2), k I I (z)" k (z) if E(7, i, 2, 1))z)E(7, i, 2, 2), %/ 0 0 elsewhere, where:
(44)
(45)
by (4 .1), E(7, 1, 1, 1)"E(3, 1, 1, 1), E(7, 1, 1, 2)"c¹q , E(7, 2, 1, k)"E(7, 1, 1, k) for k"1, 2; by (4 .2), E(7, 3, 1, 1)"E(3, 3, 1, 1), E(7, 3, 1, 2)"c¹q , E(7, 4, 1, k)"E(7, 3, 1, k) for k"1, 2; by (6 ), E(7, i, 2, k)"E(5, i, 2, k) for i"1, 2, 3, 4 and k"1, 2. (h) For i"1, 2, 3, 4, if (q , q , q , r , r )3S 5E 5¹ , then G G k (z) if E(8, i, 1, 1))z)E(8, i, 1, 2), (46) k I I (z)" k (z) if E(8, i, 2, 1))z)E(8, i, 2, 2), %/ 0 0 elsewhere, where:
by (4 ), E(8, i, 1, k)"E(7, i, 1, k) for i"1, 2, 3, 4 and k"1, 2; by (8 ), E(8, i, 2, k)"E(6, i, 2, k) for i"1, 2, 3, 4 and k"1, 2. Next, we will "nd the range of z to preserve 0)k (z))1, i"1, 2 in Sections 2.3 and 2.4. G 2.3. The range of z for 0)k (z))1 From Eq. (10), we have (q !q )z!a(r !r )!c¹q (q !q )#(D . k (z)" c¹(q !q ) Thus, 0)k (z))1 is equivalent to 0)(q !q )z!a(r !r )!c¹q (q !q )#(D )c¹(q !q ). From (a) and (b) in Appendix A, we note that If D )0 and z*a (q , r ), then D *0. If D '0, z*a (q , r ) and z*z , then D *0. H
(47)
(48) (49)
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Therefore, we will discuss inequality (47) as follows: (B1) When !(q !q )z#a(r !r )#c¹q (q !q ))(D , under (48) and (49), denote
(50)
2aq (r !r )#c¹q(q !q )#2aD , 2q (q !q ) 2aq (r !r )#c¹q(q !q )#2aD , A "max a (q , q , q , r , r ), 2q (q !q ) a(r !r )#c¹q (q !q ) , B "max a (q , r ), q !q a(r !r )#c¹q (q !q ) , B "max a (q , q , q , r , r ), q !q a(r !r )#c¹q (q !q ) (D )0) H " (q , q , q , r , r ) " A ) q !q A "max a (q , r ),
or
a(r !r )#c¹q (q !q ) (D '0) , A ) q !q 2aq (r !r )#c¹q(q !q )#2aD (D )0), H " (q , q , q , r , r ) " B ( 2q (q !q )
or
2aq (r !r )#c¹q(q !q )#2aD (D '0) . B ( 2q (q !q ) By (S1) and (S2) in Appendix B, since a (q , q , q , r , r )"max[z , a (q , r )], we have H If (q , q , q , r , r ,)3H , then a(r !r )#c¹q (q !q ) when D )0, A )z) q !q a(r !r )#c¹q (q !q ) when D '0. A )z) q !q If (q , q , q , r , r ,)3H , then B )z when D )0, B )z when D '0. (B2) When (D )!(q !q )z#a(r !r )#c¹q (q !q )#c¹(q !q ),
(51) (52)
(53) (54)
(55)
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let
a(r !r )#c¹q (q !q ) 2aq (r !r )#c¹q(q !q )#2aD , . (q !q ) 2q (q !q ) Then we have the following results: CH"min
If D )0 and a (q , r ))CH, then a (q , r ))z)CH. If D '0 and a (q , q , q , r , r ,))CH, then a (q , q , q , r , r ,))z)CH. Furthermore, denote
min
C "
a(r !r )#c¹q (q !q ) ,CH (q !q )
CH
(56) (57)
if (q , q , q , r , r ,)3H , if (q , q , q , r , r ,)3H .
By (51)}(54) and (56)}(57), we have the range of z in which 0)k (z))1 as follows: 1. If (q , q , q , r , r ,)3H , then A )z)C when D )0 and a (q , r ))C , A )z)C when D '0 and a (q , q , q , r , r ,))C . 2. If (q , q , q , r , r ,)3H , then B )z)C when D )0 and a (q , r ))C , B )z)C when D '0 and a (q , q , q , r , r ,))C . 2.4. The range of z for 0)k (z))1 From (14) we have !(q !q )z#a(r !r )#c¹q (q !q )!(D . k (z)" c¹(q !q ) Thus, 0)k (z))1 is equivalent to 0)!(q !q )z#a(r !r )#c¹q (q !q )!(D )c¹(q !q ). Similar to Section 2.3, let
2aq (r !r )#c¹q(q !q )#2aD , 2q (q !q ) 2aq (r !r )#c¹q(q !q )#2aD , A "max a (q , q , r ), 2q (q !q )
A "max a (q , r ),
(58) (59)
(60) (61)
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a(r !r )#c¹q (q !q ) , q !q a(r !r )#c¹q (q !q ) , B "max a (q , q , r ), q !q a(r !r )#c¹q (q !q ) 2aq (r !r )#c¹q(q !q )#2aD , , CH"min (q !q ) 2q (q !q ) a(r !r )#c¹q (q !q ) (D )0), H " (q , q , q , r , r ) A ) q !q B "max a (q , r ),
or
a(r !r )#c¹q (q !q ) (D '0) , A ) q !q 2aq (r !r )#c¹q(q !q )#2aD (D )0), H " (q , q , q , r , r ) " B ( 2q (q !q )
or
2aq (r !r )#c¹q(q !q )#2aD (D '0) , B ( 2q (q !q ) a(r !r )#c¹q (q !q ) , CH min if (q , q , q , r , r ,)3H , (q !q ) C " if (q , q , q , r , r ,)3H . CH Therefore, we have the range of z in which 0)k (z))1 as follows: 1. If (q , q , q , r , r ,)3H , then A )z)C when D )0 and a (q , r ))C , A )z)C when D '0 and a (q , q , r ,))C . 2. If (q , q , q , r , r ,)3H , then B )z)C when D )0 and a (q , r ))C , B )z)C when D '0 and a (q , q , r ,))C .
2.5. More about the range of z in k
(z) %/I 0I
Corresponding to ¹ , i"1, 2, 3, 4, and Eqs. (58)}(65), we let G < "+(q , q , q , r , r ) " a (q , r ))C , a (q , r ))C ,, < "+(q , q , q , r , r ) " a (q , r ))C , a (q , q , r ))C ,,
(62) (63)
(64) (65)
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< "+(q , q , q , r , r ) " a (q , q , q , r , r ))C , a (q , r ))C ,, < "+(q , q , q , r , r ) " a (q , q , q , r , r ))C , a (q , q , r ))C ,. Combining the range of z for 0)k (z))1 (in (58)}(61)) and 0)k (z))1 (in (62)}(65)), and based on the sign of D and D , we rewrite the range of z as follows: 1. In case (D )0)(a (q , r ))C ) and (D )0)(a (q , r ))C ), for every j"1, 2,2, 8, by (58), (60), (62) and (64), we denote EH (1, 1)"EH (1, 1)"max(A , E( j, 1, 1, 1)), H H H H E (1, 2)"E (1, 2)"min(C , E( j, 1, 1, 2)), H H EH (1, 1)"EH (1, 1)"max(B , E( j, 1, 1, 1)), H H EH (1, 2)"EH (1, 2)"min(C , E( j, 1, 1, 2)), H H EH (2, 1)"EH (2, 1)"max(A , E( j, 1, 2, 1)), H H EH (2, 2)"EH (2, 2)"min(C , E( j, 1, 2, 2)), H H EH (2, 1)"EH (2, 1)"max(B , E( j, 1, 2, 1)), H H EH (2, 2)"EH (2, 2)"min(C , E( j, 1, 2, 2)). H H 1. By (58) and (62), (q , q , q , r , r )3H 5H 5< . Let K "+(q , q , q , r , r ) " EH (1, 1))EH (1, 2))EH (2, 1))EH (2, 2),. H H H H H 1. By (58) and (64), (q , q , q , r , r )3H 5H 5< . Let K "+(q , q , q , r , r ) " EH (1, 1))EH (1, 2))EH (2, 1))EH (2, 2),. H H H H H 1. By (60) and (62), (q , q , q , r , r )3H 5H 5< . Let K "+(q , q , q , r , r ) " EH (1, 1))EH (1, 2))EH (2, 1))EH (2, 2),. H H H H H 1. By (60) and (64), (q , q , q , r , r )3H 5H 5< . Let K "+(q , q , q , r , r ) " EH (1, 1))EH (1, 2))EH (2, 1))EH (2, 2),. H H H H H Similarly: 2. In case (D )0)(a (q , r ))C ) and (D '0)(a (q , q , r ))C ), for every j"1, 2,2, 8, k"1, 2, 3, 4, by (58), (60), (63) and (65) we let K "+(q , q , q , r , r ) " EH (1, 1))EH (1, 2))EH (2, 1))EH (2, 2),, HI HI HI HI HI where EH (1, 1)"EH (1, 1)"max(A , E( j, 2, 1, 1)), H H EH (1, 2)"EH (1, 2)"min(C , E( j, 2, 1, 2)), H H
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EH (1, 1)"EH (1, 1)"max(B , E( j, 2, 1, 1)), H H EH (1, 2)"EH (1, 2)"min(C , E( j, 2, 1, 2)), H H EH (2, 1)"EH (2, 1)"max(A , E( j, 2, 2, 1)), H H EH (2, 2)"EH (2, 2)"min(C , E( j, 2, 2, 2)), H H EH (2, 1)"EH (2, 1)"max(B , E( j, 2, 2, 1)), H H H H E (2, 2)"E (2, 2)"min(C , E( j, 2, 2, 2)). H H 3. In case (D '0)(a (q , q , q , r , r ,))C ) and (D )0)(a (q , r ))C ), for every j"1, 2,2, 8, k"1, 2, 3, 4, by (59), (61), (62) and (64) we let K "+(q , q , q , r , r ) " EH (1, 1))EH (1, 2))EH (2, 1))EH (2, 2),, HI HI HI HI HI where EH (1, 1)"EH (1, 1)"max(A , E( j, 3, 1, 1)), H H H H E (1, 2)"E (1, 2)"min(C , E( j, 3, 1, 2)), H H EH (1, 1)"EH (1, 1)"max(B , E( j, 3, 1, 1)), H H H H E (1, 2)"E (1, 2)"min(C , E( j, 3, 1, 2)), H H EH (2, 1)"EH (2, 1)"max(A , E( j, 3, 2, 1)), H H H H E (2, 2)"E (2, 2)"min(C , E( j, 3, 2, 2)), H H EH (2, 1)"EH (2, 1)"max(B , E( j, 3, 2, 1)), H H EH (2, 2)"EH (2, 2)"min(C , E( j, 3, 2, 2)). H H 4. In case (D '0)(a (q , q , q , r , r ,))C ) and (D '0)(a (q , q , r ,))C ), for every j"1, 2,2, 8, k"1, 2, 3, 4 by (59), (61), (63) and (65) we let K "+(q , q , q , r , r ) " EH (1, 1))EH (1, 2))EH (2, 1))EH (2, 2),, HI HI HI HI HI where EH (1, 1)"EH (1, 1)"max(A , E( j, 4, 1, 1)), H H H H E (1, 2)"E (1, 2)"min(C , E( j, 4, 1, 2)), H H EH (1, 1)"EH (1, 1)"max(B , E( j, 4, 1, 1)), H H H H E (1, 2)"E (1, 2)"min(C , E( j, 4, 1, 2)), H H EH (2, 1)"EH (2, 1)"max(A , E( j, 4, 2, 1)), H H
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EH (2, 2)"EH (2, 2)"min(C , E( j, 4, 2, 2)), H H EH (2, 1)"EH (2, 1)"max(B , E( j, 4, 2, 1)), H H EH (2, 2)"EH (2, 2)"min(C , E( j, 4, 2, 2)). H H Then, for a summary of the above discussion, we have k I I (z) in Proposition 1. %/ 0 Proposition 1. For every j"1, 2,2, 8, i"1, 2, 3, 4 and (q , q , q , r , r )3S 5E 5¹ , H HG G the membership function of G(QI , RI ) is described as follows: (1 ) If (q , q , q , r , r )3H 5H 5< 5K , then G HG k (z) if EH (1, 1))z)EH (1, 2), HG HG k I I (z)" k (z) if EH (2, 1))z)EH (2, 2), HG HG %/ 0 0 elsewhere.
(2 ) If (q , q , q , r , r )3H 5H 5< 5K , then G HG
k (z) if EH (1, 1))z)EH (1, 2), HG HG k I I (z)" k (z) if EH (2, 1))z)EH (2, 2), HG HG %/ 0 0 elsewhere. (3 ) If (q , q , q , r , r )3H 5H 5< 5K , then G HG
k (z) if EH (1, 1))z)EH (1, 2), HG HG k I I (z)" k (z) if EH (2, 1))z)EH (2, 2), HG HG %/ 0 0 elsewhere. (4 ) If (q , q , q , r , r )3H 5H 5< 5K , then G HG
k (z) if EH (1, 1))z)EH (1, 2), HG HG k I I (z)" k (z) if EH (2, 1))z)EH (2, 2), HG HG %/ 0 0 elsewhere. Denote H "H 5H , H "H 5H , H "H 5H and H "H 5H then Proposi tion 1 can be rewritten as follows: Proposition 2. For every j"1, 2,2, 8, k, i"1, 2, 3, 4, if (q , q , q , r , r )3S 5E 5¹ 5H 5< 5K , H HG G I G HGI
J.-S. Yao et al. / Computers & Operations Research 27 (2000) 935}962
then
k (z) if EH (1, 1))z)EH (1, 2), HGI HGI k I I (z)" k (z) if EH (2, 1))z)EH (2, 2), HGI HGI %/ 0 0 elsewhere. 3. The centroid of k
%/I 0I
953
(66)
(z)
From Eqs. (C.1)}(C.10) in Appendix C, we can "nd the centroid for k I I (z) in Proposition 2 as %/ 0 follows: For every j"1, 2,2, 8, i, k"1, 2, 3, 4, if (q , q , q , r , r )3S 5E 5¹ 5H 5< 5K , H HG G I G HGI let
k I I (z) dz %/ 0 \ 1 1 #HHGI #HHGI " H (z) dz# H (z) dz, c¹(q !q ) HHGI c¹(q !q ) HHGI # # zk I I (z) dz Q( j, i, k)" %/ 0 \ 1 1 #HHGI #HHGI zH (z) dz# zH (z) dz, " c¹(q !q ) HHGI c¹(q !q ) HHGI # # where P( j, i, k)"
(67)
(68)
H (z)"(q !q )z!a(r !r )!c¹q (q !q )#(D , H (z)"!(q !q )z#a(r !r )#c¹q (q !q )!(D . For convenience, we give the following notations: 1. If i"1, 2, by ¹ we have D )0 and hence G 1. By (C.5) and (C.2) we denote
#HHGI H (z) dz #HHGI "R (EH (1, 1), EH (1, 2))#= (EH (1, 1), EH (1, 2)). HGI HGI HGI HGI 1. By (C.9) and (C.7) we denote P ( j, i, k)"
(69)
#HHGI zH (z) dz H #HGI "R (EH (1, 1), EH (1, 2))#= (EH (1, 1), EH (1, 2)). HGI HGI HGI HGI
Q ( j, i, k)"
(70)
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2. If i"1, 3, by ¹ we have D )0 and hence G 1. By (C.6) and (C.4) we denote
#HHGI H (z) dz H #HGI
P ( j, i, k)"
"R (EH (2, 1), EH (2, 2))!= (EH (2, 1), EH (2, 2)). HGI HGI HGI HGI
(71)
1. By (C.10) and (C.8) we denote
#HHGI zH (z) dz #HHGI
Q ( j, i, k)"
"R (EH (2, 1), EH (2, 2))!= (EH (2, 1), EH (2, 2)). HGI HGI HGI HGI
(72)
3. If i"3, 4, by ¹ we have D '0 and hence G 1. By (C.5) and (C.1) we denote
#HHGI H (z) dz #HHGI
P ( j, i, k)"
"R (EH (1, 1), EH (1, 2))#= (EH (1, 1), EH (1, 2)). HGI HGI HGI HGI
(73)
1. By (C.9) and (C.7) we denote
#HHGI zH (z) dz Q ( j, i, k)" #HHGI "R (EH (1, 1), EH (1, 2))#= (EH (1, 1), EH (1, 2)). HGI HGI HGI HGI
(74)
4. If i"2, 4, by ¹ we have D '0 and hence G 1. By (C.6) and (C.3) we denote
#HHGI H (z) dz H #HGI
P ( j, i, k)"
"R (EH (2, 1), EH (2, 2))!= (EH (2, 1), EH (2, 2)). HGI HGI HGI HGI
(75)
1. By (C.10) and (C.8) we denote
Q ( j, i, k)"
#HHGI zH (z) dz #HHGI
"R (EH (2, 1), EH (2, 2))!= (EH (2, 1), EH (2, 2)). HGI HGI HGI HGI
(76)
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955
Therefore, by (67)}(76) we obtain the following: P ( j, 1, k) P ( j, 1, k) # , P( j, 1, k)" c¹(q !q ) c¹(q !q ) P ( j, 2, k) P ( j, 2, k) # , P( j, 2, k)" c¹(q !q ) c¹(q !q ) P ( j, 3, k) P ( j, 3, k) P( j, 3, k)" # , c¹(q !q ) c¹(q !q ) P ( j, 4, k) P ( j, 4, k) P( j, 4, k)" # , c¹(q !q ) c¹(q !q )
Q ( j, 1, k) Q ( j, 1, k) Q( j, 1, k)" # , c¹(q !q ) c¹(q !q ) Q ( j, 2, k) Q ( j, 2, k) Q( j, 2, k)" # , c¹(q !q ) c¹(q !q ) Q ( j, 3, k) Q ( j, 3, k) Q( j, 3, k)" # , c¹(q !q ) c¹(q !q ) Q ( j, 4, k) Q ( j, 4, k) Q( j, 4, k)" # . c¹(q !q ) c¹(q !q )
Proposition 3. For every j"1, 2,2, 8 and i, k"1, 2, 3, 4, if (q , q , q , r , r )3S 5E 5¹ 5H 5< 5K , H HG G I G HGI then the centroid for k I I (z) is given by %/ 0 Q M (q , q , q , r , r )" H G I HGI P H G I which is an estimate of the total cost.
(77)
4. Numerical example Let a"3, c"4, r "30 and ¹"5, then q "3 is the crisp optimal order quantity and H G(q , r )"F(q )"60 is the crisp minimum total cost. In the fuzzy sense, we use a FORTRAN H H program to "nd MHH"M (q , q , q , r , r ), which is an estimate of the total cost under the fuzzy HGI order quantity (q , q , q ) and the fuzzy total demand quantity (r , r , r ). Also, let qHH" (q #q #q )/3 and rHH"(r #r #r )/3 be the centroid of (q , q , q ) and (r , r , r ), respectively. Then the relative error in the fuzzy sense for order quantity, total demand quantity and total cost are given by rHH!r MHH!G(q , r ) qHH!qH , Rel C" H , , Rel R" Rel Q" r G(q , r ) qH H respectively. For some sets of (q , q , q ) and (r , r , r ), we have the numerical results in Table 1 as follows: Roughly speaking, Table 1 displays that the value of MHH approaches the crisp minimum cost G(q , r ) as both qHH approaches the crisp order quantity q and rHH approaches the crisp total H H demand quantity r . That is, Rel CP0 as Rel QP0 and Rel RP0. Therefore, if both the relative error for order quantity and total demand quantity approaches zero, then the relative error for total cost approaches zero. Otherwise, if qHH are far from q and rHH is far from r , then the relative H error for the total cost is large.
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Table 1 Numerical Results for a"3, c"4, r "30, ¹"5, q "3, G(q , r )"60 H H Input data
Output data
q
q
q
r
r
qHH
rHH
MHH
Rel Q
Rel R
Rel C
1.0 2.2 2.2 2.2 1.0 2.8 2.8 2.8 1.0 2.8 2.9 2.8 2.2 2.8 2.9 4.9 4.9 4.9 4.9 4.9 6.5 6.5 7.1 7.1 7.1 7.1 6.5 6.5
4.1 4.1 4.1 4.1 4.1 4.1 4.7 4.7 4.7 4.7 4.9 4.7 4.7 4.7 4.9 6.9 6.9 6.9 6.9 6.9 7.9 7.9 7.9 7.9 8.7 8.7 8.7 9.5
4.7 4.7 5.3 5.3 5.0 5.9 5.3 5.3 5.9 5.9 5.9 6.5 6.5 6.5 6.7 7.9 7.9 7.9 7.9 7.9 8.1 8.1 8.1 8.9 8.9 8.9 9.7 9.7
29.0 23.0 26.0 29.0 25.0 29.0 20.0 29.0 29.0 29.0 21.0 29.0 20.0 20.0 27.0 25.0 28.0 19.0 19.0 28.0 28.8 23.4 27.0 28.8 23.4 23.4 28.8 23.4
31.0 31.0 31.0 31.0 33.0 31.0 34.0 34.0 34.0 34.0 34.0 31.0 34.0 34.0 34.0 32.0 33.0 37.0 40.0 41.0 35.9 44.0 38.6 35.9 37.7 44.0 36.8 39.5
3.27 3.67 3.87 3.87 3.37 4.27 4.27 4.27 3.87 4.47 4.57 4.67 4.47 4.67 4.83 6.57 6.57 6.57 6.57 6.57 7.50 7.50 7.70 7.97 8.23 8.23 8.30 8.57
30.00 28.00 29.00 30.00 29.33 30.00 28.00 31.00 31.00 31.00 28.33 30.00 28.00 28.00 30.33 29.00 30.33 28.67 29.67 33.00 31.57 32.47 31.87 31.57 30.37 32.47 31.87 30.97
63.12 63.12 64.29 64.32 65.20 65.69 66.89 66.93 68.16 68.21 68.40 69.74 70.21 70.22 72.03 83.36 83.49 83.98 84.36 84.51 87.94 88.64 89.37 93.16 95.00 95.66 100.81 102.29
0.0889 0.2222 0.2889 0.2889 0.1222 0.4222 0.4222 0.4222 0.2889 0.4889 0.5222 0.5556 0.4889 0.5556 0.6111 1.1889 1.1889 1.1889 1.1889 1.1889 1.5000 1.5000 1.5667 1.6556 1.7444 1.7444 1.7667 1.8556
0.0000 !0.0667 !0.0333 0.0000 !0.0222 0.0000 !0.0667 0.0333 0.0333 0.0333 !0.0556 0.0000 !0.0667 !0.0667 0.0111 !0.0333 0.0111 !0.0444 !0.0111 0.1000 0.0522 0.0822 0.0622 0.0522 0.0122 0.0822 0.0622 0.0322
0.0520 0.0520 0.0715 0.0719 0.0866 0.0948 0.1148 0.1154 0.1360 0.1369 0.1400 0.1624 0.1702 0.1703 0.2005 0.3893 0.3915 0.3996 0.4060 0.4085 0.4657 0.4773 0.4895 0.5526 0.5833 0.5943 0.6801 0.7049
5. Conclusions In the classical inventory without the backorder model, we consider the order quantity q and the total demand quantity r are fuzzy numbers. If the centroid qHH of fuzzy order quantity near the optimal crisp order quantity qH and the centroid rHH of fuzzy total demand quantity near the crisp total demand quantity r , then the estimate total cost in Proposition 3 approach the crisp optimal total cost F(q ). H In brief, when the fuzzy situation is known, we can consider this inventory model in the fuzzy sense. This method may also be used in comparison with the crisp case.
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957
Appendix A (a) If D )0 then D *0. Therefore for z*a (q , r ) the maximum positive root for (8) is given by (q !q )z!a(r !r )#(D ('0). (*) q" c¹(q !q ) When D )0 and z*a (q , r ), substituting this q (in (*)) in the left-hand side of (7), we obtain (q !q )z!a(r !r )!c¹q (q !q )#(D ,k (z) (say). PQ" c¹(q !q ) (b) If D '0 then z*a (q , r ) and z*z imply D *0, where H (2ac¹(q !q )D #a(r !r ) ('0). z " H q !q Also, z*z N(q !q )z!a(r !r )*(q !q )z !a(r !r )"(2ac¹(q !q )D '0. H H Similarly, the maximum positive root for (8) is given by (*) if z*a (q , r ) and z*z . Denote H a (q , q , q , r , r )"max+z , a (q , r )]. Therefore, H if D '0 and z*a (q , q , q , r , r ) then PQ"k (z). (c) If D )0 then D *0, for z*a (q , r ). The maximum positive root for (12) is given by (q !q )z!a(r !r )#(D ('0). q" c¹(q !q ) Substituting q in the left-hand side of (11), when D )0 and z*a (q , r ), we obtain !(q !q )z#a(r !r )#c¹q (q !q )!(D ,k (z) (say). P Q " c¹(q !q ) (d) If D '0 then z*a (q , r ) and z*z imply D *0, where HH (2ac¹(q !q )D #a(r !r ) ('0) z " HH q !q and z*z N(q !q )z!a(r !r )*(q !q )z !a(r !r )"(2ac¹(q !q )D '0. HH HH
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Thus, the maximum positive root for (12) is given by (q !q )z!a(r !r )#(D ('0). q" c¹(q !q ) Denote a (q , q , r )"max[z , a (q , r )]. Therefore, HH if D '0 and z*a (q , q , r ) then P Q "k (z). Appendix B (S1) If !(q !q )z#a(r !r )#c¹q (q !q )*0, then inequality (50) implies 2aq (r !r )#c¹q(q !q )#2aD )2q (q !q )z. Therefore, If D )0, then z*a (q , r ), a(r !r )#c¹q (q !q ) z) (q !q ) and 2aq (r !r )#c¹q(q !q )#2aD . z* 2q (q !q ) If D '0, then z*a (q , r ), z*z , H a(r !r )#c¹q (q !q ) z) (q !q ) and 2aq (r !r )#c¹q(q !q )#2aD . z* 2q (q !q ) (S2) If !(q !q )z#a(r !r )#c¹q (q !q )(0, under conditions (48) and (49), inequality (50) holds. Then we have If D )0, then z*a (q , r ) and a(r !r )#c¹q (q !q ) . z' (q !q ) If D '0, then z*a (q , r ), z*z and H a(r !r )#c¹q (q !q ) . z' (q !q )
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959
Appendix C In order to "nd the centroid for the membership function k I I (z) in Proposition 2, we consider %/ 0 the following integrals: 1. If D '0 denote
= (a , a )"
?
?
"
?
?
(D dz ([(q !q )z!a(r !r )]!2ac¹(q !q )D dz,
where D "(q r !q r ). Substituting y"(q !q )z!a(r !r ) and y"(2ac¹(q !q )D sec h in the above equa tion we get
sec h #tan h , = (a , a )"ac¹D sec h tan h !sec h tan h !ln sec h #tan h
(C.1)
where aH H sec h " and aH"(q !q )a !a(r !r ). H (2ac¹(q !q )D H H 2. If D (0 we have
=H (a , a )"
?
?
(D dz
sec hH#tan hH , "!ac¹D sec hH tan hH!sec hH tan hH#ln sec hH#tan hH where aH H tan hH" . H (!2ac¹(q !q )D 3. If D "0 let
=H (a , a )"
?
?
"(q !q )z!a(r !r )" dz
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then
1 ?H =H (a , a )" "y" dy q !q H ? [(aH)!(aH)] if 0)aH, O\O " [(aH)#(aH)] if aH(0)aH, O \O [(aH )!(aH )] if aH (aH )0. O\O For convenience, denote
=H (a , a ) if D (0, = (a , a )" =H (a , a ) if D "0. On the other hand, consider the signs of D , we obtain the following: 4. If D '0, then ? = (a , a )" (D dz ? ? " ([(q !q )z!a(r !r )]!2ac¹(q !q )D dz ? sec u #tan u , "ac¹D sec u tan u !sec u tan u !ln sec u #tan u where
(C.2)
(C.3)
cH H and cH"(q !q )a !a(r !r ). sec u " H H H (2ac¹(q !q )D 5. If D (0, then
=H (a , a )"
where
?
?
(D dz
cH H . tan uH" H (!2ac¹(q !q )D 6. If D "0, then
sec uH#tan uH , "!ac¹D sec uH tan uH!sec uH tan uH#ln sec uH#tan uH
AH 1 =H (a , a )" "y" dy q !q H A
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961
[(cH)!(cH)] if 0)cH, O\O " [(cH)#(cH)] if cH(0)cH, O \O [(cH )!(cH )] if cH (cH )0. O\O For convenience, denote
=H (a , a ) if D (0, = (a , a )" =H (a , a ) if D "0. 7.
(C.4)
R (a , a )"
? [(q !q )z!a(r !r )!c¹q (q !q )] dz ?
"(a !a ) 8.
q !q (a #a )!a(r !r )!c¹q (q !q ) . 2
(C.5)
R (a , a )"
? [!(q !q )z#a(r !r )#c¹q (q !q )] dz ?
q !q (a #a )#a(r !r )#c¹q (q !q ) . "(a !a ) ! 2
(C.6)
9. Substituting z"[y#a(r !r )]/(g !g ) in the following integration we get
= (a , a )" H
? z(D dz (D '0 for j"1 and D )0 for j"2) ?
?H 1 " [y#a(r !r )](y!2ac¹(q !q )D dy (q !q ) H ? a(r !r ) 1 = (a , a )# " H q !q 3(q !q ) ;+[(aH)!2ac¹(q !q )D ] ![(aH)!2ac¹(q !q )D ] ,. (C.7) Similarly, 10.
= (a , a )" H
?
?
z(D dz (D '0 for j"1 and D )0 for j"2)
a(r !r ) 1 = (a , a )# " H q !q 3(q !q ) ;+[(cH)!2ac¹(q !q )D ] ![(cH)!2ac¹(q !q )D ] ,. (C.8)
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11.
R (a , a )"
?
?
z[(q !q )z!a(r !r )!c¹q (q !q )] dz
(a!a) 1 [a(r !r )#c¹q (q !q )]. " (q !q )(a!a)! 2 3 12.
R (a , a )"
?
?
(C.9)
z[!(q !q )z#a(r !r )#c¹q (q !q )] dz
(a!a) 1 [a(r !r )#c¹q (q !q )]. "! (q !q )(a!a)# 2 3
(C.10)
References [1] San-Chyi Chang, Jing-Shing Yao, Huey-Ming Lee. Economic reorder point for fuzzy backorder quantity. European Journal of Operation Research 1998;109:183}202. [2] Shan-Huo Chen, Chien-Chung Wang, Arthur Ramer. Backorder fuzzy inventory model under function principle. Information Science 1996;95:71}9. [3] Huey-Ming Lee, Jing-Shing Yao. Economic order quantity in fuzzy sense for inventory without backorder model. Fuzzy Sets and Systems 1999;105:13}31. [4] Huey-Ming Lee, Jing-Shing Yao. Economic production quantity for fuzzy demand quantity and fuzzy production quantity. European Journal of Operation Research 1998;109:203}11. [5] Jing-Shing Yao, Huey-Ming Lee. Fuzzy inventory with backorder for fuzzy order quantity. Information Sciences 1996;93:283}319. [6] Jing-Shing Yao, Huey-Ming Lee. Fuzzy inventory with or without backorder for order quantity with trapezoid fuzzy number. Fuzzy Sets and Systems 1999;105:311}37.
Jing-Shing Yao is a professor in the department of applied mathematics at the Chinese Culture University. He earned his Ph.D from the department of mathematics at Kyushu University in Japan. His research interests are in the "eld of fuzzy set theory and its application, operation research and statistics. He has publication in Fuzzy Sets and Systems, European Journal of Operation Research and Information Sciences. San-Chyi Chang is an associate professor in the department of applied mathematics at the Chinese Culture University. He earned his Ph.D from the graduate institute of management science at Tamkang University in Taiwan. His research interests are in the "eld of statistics and the application of fuzzy mathematics. He has publication in the European Journal of Operation Research and some papers will appear in Fuzzy Sets and Systems. Jin-Shieh Su is an instructor in the department of applied mathematics at the Chinese Culture University. His research interests are in the "eld of computer science and the application of fuzzy mathematics. He has publication in Hwa Kang Journal of Sciences and some papers will appear in the European Journal of Operation Research.