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Using fuzzy numbers in knapsack problems
European Journal of Operational Research 135 (2001) 158±176
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Theory and Methodology
Using fuzzy numbers in knapsack problems Feng-Tse Lin a
a,*
, Jing-Shing Yao
b
Department of Applied Mathematics, Chinese Culture University, 55, Hwakang Rd., Yangminshan, Taipei 111, Taiwan, ROC b Department of Mathematics, National Taiwan University, Taipei, Taiwan, ROC Received 13 September 1999; accepted 6 November 2000
Abstract This paper investigates knapsack problems in which all of the weight coecients are fuzzy numbers. This work is based on the assumption that each weight coecient is imprecise due to the use of decimal truncation or rough estimation of the coecients by the decision-maker. To deal with this kind of imprecise data, fuzzy sets provide a powerful tool to model and solve this problem. Our work intends to extend the original knapsack problem into a more generalized problem that would be useful in practical situations. As a result, our study shows that the fuzzy knapsack problem is an extension of the crisp knapsack problem, and that the crisp knapsack problem is a special case of the fuzzy knapsack problem. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy sets; Optimization; Knapsack problem; Multiconstraint 0/1 knapsack problem; Signed distance ranking
1. Introduction In its original meaning, the knapsack problem involves ®nding a combination of dierent objects, that a hitchhiker chooses for his knapsack in which the total value of all of the objects he chooses is maximized [9]. The knapsack may correspond to a truck, a ship, a silicon chip, or a resource. The problem is to package these items so that there are a variety of applications for cargo loading, cutting stock, various packing problems, or economic planning. For example, this method could describe the problem of making investment decisions in which the size of an investment is based on the amount of money required, the knapsack capacity is the amount of available money to invest, and the investment pro®t is the expected return. The multi-constraint 0/1 knapsack problem (MCKP) is a very important knapsack problem, which arises in the areas of capital budgeting and resource allocation [3]. The reason for this interest is based on the fact that it is a special case of the general 0/1 integer programming and can be used as a subproblem for
*
Corresponding author. Tel.: +886-2-286-10511 ext. 741; Fax: +886-2-286-19509. E-mail address: ftlin@sta.pccu.edu.tw (F.-T. Lin).
0377-2217/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 0 ) 0 0 3 1 0 - 6
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solving many more complex problems. Since the MCKP has been proved to be an NP-hard problem [9], several heuristics [3,4,7,8,10,14,15] have been proposed for solving this problem approximately over the past few years. In our study, however, we are not proposing a new heuristic for solving the MCKP approximately. Instead, we are interested in studying how to deal with the imprecise data that would occur in practical situations. There are three motivations for our study. First, it is sometimes dicult for the decision-maker (DM) to determine the precise values of the weight coecients because the object information is vague or imprecise [1]. Second, as noted in [13], ¯oating-point arithmetic in the MCKP may lead to round-o errors, since a computer only works with approximations to the real numbers. Third, when using truncating functions, dramatic errors may be produced, even for small deviations, and thus lead to incorrect results. Although the knapsack problem has often been investigated, little of the related research involves resolving the uncertainties [1,2,11]. In our study, we assumed that the weight of each object in the knapsack problem is imprecise, which may be in the vicinity of a ®xed value, or substantially less than or greater than a ®xed value. To deal with this kind of imprecise data, fuzzy sets provide a powerful tool to model and solve the problem. Our approach can be depicted brie¯y as follows. Since each weight, wi , i 1; . . . ; n, is imprecise, the DM should determine an interval wi Di1 ; wi Di2 , 0 6 Di1 < wi and 0 < Di2 , to represent an acceptable range for the weight of each object. The interpretation of this range is as follows. If an estimate of the weight is exactly wi , then the acceptable grade for that weight will be 1; otherwise, the acceptable grade will get smaller when an estimate is approaching one of the ends of the interval, i.e., wi Di1 or wi Di2 . Accordingly, the DM needs to determine an appropriate estimate for each weight from the interval ~ i wi Di1 ; wi ; wi Di2 , in the knapsack wi Di1 ; wi Di2 . This leads to the use of fuzzy numbers, w problems. Obviously, the membership grade of a fuzzy number in the fuzzy set corresponds to the ac~ i using the ceptable grade of an estimate in a given interval. Thus, after defuzzifying the fuzzy number w proposed ranking method, we obtain an estimate for each object weight in the fuzzy sense, for instance wi , which is in the interval wi Di1 ; wi Di2 . Then we use wi as the weight of object i and use the approach for solving the crisp problem to solve the fuzzy model. The advantage of this fuzzy model is that it is much easier for the DM to specify a range value than to give an exact value for each object weight. Consequently, the proposed approach is a sound and a ¯exible way for specifying the weight coecient while we solve the knapsack problems in practical situations. The paper is organized as follows. Section 2 introduces the preliminaries and the method for ranking fuzzy numbers. The major contribution of this work is presented in Sections 3 and 4. In Section 3, we propose an approach to construct fuzzy knapsack models. After the defuzzi®cation of the fuzzy models, we obtain Propositions 1 and 2, the knapsack and the 0/1 knapsack problems in the fuzzy sense, which can be used for solving imprecise data. Two examples will be employed here to illustrate these problems. Section 4 presents Proposition 3, the MCKP in the fuzzy sense (i.e. the defuzzi®ed fuzzy MCKP), where an illustrative example is shown. Finally, the concluding remarks are given in Section 5. 2. Preliminaries This section reviews some basic de®nitions of fuzzy sets [6,17] and introduces the signed distance method for ranking fuzzy numbers. De®nition 1. Let b~1 be a fuzzy set on R 1; 1. It is called a level 1 fuzzy point if its membership function is as given below. 1; x b; 1 lb~1 x 0; x 6 b:
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Let FP 1 fb~1 j 8b 2 Rg denote the family of all level 1 fuzzy points. De®nition 2. Let pa ; qa be a fuzzy set on R 1; 1, p, q 2 R and 0 6 a 6 1. It is called a level a fuzzy interval, if its membership function is as given below. a; p 6 x 6 q; 2 lpa ;qa x 0; otherwise: De®nition 3. Let A~ be a fuzzy set on R 1; 1. It is called a triangular fuzzy number, if its membership function is as given below, and is denoted by A~ a; b; c. 8 < x a= b a; a 6 x 6 b; lA~ x c x= c b; b 6 x 6 c; a < b < c; 3 : 0; otherwise: Let FN f a; b; c j 8a < b < c; a; b; c 2 Rg be the family of all triangular fuzzy numbers. When (3) changes to lB~ x x a= b a if a 6 x 6 b, and lB~ x 0 if x < a or b < x, we can rewrite A~ a; b; c as B~ a; b; b and then call it a left triangular fuzzy number. The family of all left triangular fuzzy numbers can be denoted by FL f a; b; b j 8a < b; a; b 2 Rg. Similarly, we can de®ne a right triangular fuzzy number, C~ b; b; c. Let lC~ x c x= c b if b 6 x 6 c, and let lC~ x 0, if x < b or c < x. Then FR f b; b; c j 8b < c; b; c 2 Rg denotes the family of all right triangular fuzzy numbers. Note that A~ a; b; c b; b; b b~1 if c a b. It is clear that FP 1, FL , and FR are all special cases of FN . Therefore we have that F FN [ FL [ FR [ FP 1 f a; b; c j 8a 6 b 6 c; a; b; c 2 Rg. Now, consider the problem of ranking fuzzy numbers. We will use the signed distance ranking method, which was de®ned in [16], for ranking the fuzzy numbers on F in this paper. De®nition 4. The signed distance of b is de®ned by d b; 0 b, b; 0 2 R. The interpretation of the signed distance is as follows. Geometrically, 0 < b means that b lies to the right of the origin 0 and the distance between b and 0 is denoted by b d b; 0. Similarly, b < 0 means that b lies to the left of 0 and the distance between b and 0 is denoted by b d b; 0. In summary, d b; 0 stands for the signed distance of b measured from the origin 0. From Fig. 1, we can see that a a-cut of the fuzzy number A~ a; b; c is an interval AL a; AR a; 0 6 a 6 1, where AL a and AR a are the left endpoint and the right endpoint of the a-cut, respectively. From (3) we have that AL a a b aa and AR a c c ba, where AL a and AR a are the signed distances measured from 0. From De®nition 4, we have d AL a; 0 AL a and d AR a; 0 AR a, 0 6 a 6 1. Hence, the signed distance of the interval AL a; AR a is de®ned by d AL a; AR a; 0 12d AL a; 0 d AR a; 0 12AL a AR a 12a c 2b a ca. Since
~ Fig. 1. A a-cut of fuzzy number A.
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the function aRis continuous over the interval, R 1we can use the integration to obtain the mean of the signed 1 distance, i.e. 0 d AL a; AR a; 0 da 12 0 a c 2b a ca da 14 2b a c. In addition, the intervals AL a; AR a and AL aa ; AR aa , for each a 2 0; 1, have a one-to-one mapping as shown in Fig. 1. Thus we have the following de®nition. De®nition 5. Let A~ a; b; c 2 FN . The signed distance of A~ measured from ~01 (y-axis) is de®ned by Z 1 ~ ~ d AL aa ; AR aa ; ~ 01 da d A; 01 0
Z 1 1 AL a AR a da 2 0 1 4 2b a c: 4 As we mentioned earlier in this section, if c a b then A~ b~1 . We have AL a AR a b. From De®nitions 5 and 4, we also have d b~1 ; ~ 01 b and d b~1 ; ~01 b d b; 0, respectively. Since b~1 b; b; b ~ is a special case of A a; b; c, we can see that the signed distance on FN is an extension of the signed distance on R. Similarly, we can de®ne the signed distance on FL and FR using the same concept. Let B~ a; b; b 2 FL . Then we have that BL a a b aa and BR a b. From De®nition 5, we can see ~ ~01 1 3b a. Let C~ b; b; c 2 FR . that the signed distance of B~ measured from ~ 01 is de®ned by d B; 4 ~ ~ ~ ~01 1 3b c. As a result, we can Then the signed distance of C measured from 01 is de®ned by d C; 4 de®ne the signed distance of the fuzzy set A~ a; b; c 2 F , in the following property. ~ ~01 1 2b a c. Property 1. Let A~ a; b; c 2 F . The signed distance of A~ is given by d A; 4 ~ a; b; c and E~ p; q; r 2 F . We de®ne the rankings of the signed distance on F as De®nition 6. Let D follows: ~1 ; ~ E~ iff d D; ~ ~ ~ 0 D 01 < d E; ~ E~ iff d D; ~ ~ ~ ~ D 01 d E; 01 :
5
~ a; b; c and E~ p; q; r 2 F . We have the following binary operations from [6]: Property 2. Let D ~ E~ a p; b q; c r 2 F , 1. D ~ ka; kb; kc 2 F if k > 0, 2. k~1 D ~ 0; 0; 0 ~ 3. k~1 D 01 2 FP 1 if k 0. ~ a; b; c and E~ p; q; r 2 F . We derive two operations of the signed distance from Property 3. Let D Properties 1 and 2, as follows: ~ E; ~ ~ ~ ~ ~ ~ 1. d D 01 d D; 01 d E; 01 , ~ ~ ~ ~ 2. d k~1 D; 01 kd D; 01 if k P 0. Proof. ~ E~ a p; b q; c r 2 F , we obtain 1. Since D 1 ~ E; ~ ~ d D 01 2 b q a p c r 4 1 1 ~ ~01 d E; ~ ~01 : 2b a c 2q p r d D; 4 4 2. Similar to (1) .
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~ 2 F: ~ E; ~ Q Property 4. The order relations and satisfy the following three axioms in F. For any D; ~ - D. ~ (a) Re¯exive law: D ~ - E, ~ E~ - D ~)D ~ E. ~ (b) Antisymmetric law: D ~)D ~ ~ - E, ~ E~ - Q ~ Q. (c) Transitive law: D ~ 2 F ; either ~ E; ~ Q Property 5. The order relations and satisfy the law of trichotomy in F; i.e., for any D; ~ ~ ~ ~ ~ ~ D E; D E; or E D holds. ~ S~ 2 F . ~ E; ~ Q; Property 6. Let D; ~ ~ ~ ~ ~ ~ Q. 1. If E D; then E Q D ~ ~ ~ ~ ~ ~ E. ~ 2. If Q D and S E; then Q S~ D Properties 4±6 have the same characteristics as the rankings on R, i.e., for a; b 2 R; a < b i d a; 0 < d b; 0 and a b i d a; 0 d b; 0. Therefore, the ranking on F in De®nition 6 has the same feature as that on R. In conclusion, the signed distance ranking method in De®nitions 5 and 6 will be useful when we consider constructing the fuzzy knapsack models.
3. The knapsack problems with fuzzy numbers In this section, ®rst we consider the crisp knapsack problem. Next, we present an approach to construct fuzzy knapsack models. After defuzzifying the fuzzy models, we obtain Propositions 1 and 2, the knapsack and the 0/1 knapsack problems in the fuzzy sense (i.e. the defuzzi®ed fuzzy problems), which are used for solving imprecise data. The fuzzy MCKP model and the MCKP in the fuzzy sense (i.e. the defuzzi®ed fuzzy MCKP) will be stated in Section 4. 3.1. The crisp knapsack problem The knapsack problem is formulated by numbering the objects from 1 to n and introducing a vector of binary variables xi (i 1; . . . ; n) having the following meaning. Object i has a weight wi and the knapsack has a capacity M. If a fraction xi , 0 6 xi 6 1, of object i is placed into the knapsack then a pro®t, pi xi , is earned. The objective is to ®nd the combination of objects for the knapsack that maximizes the total pro®t from all of the objects chosen. Since the knapsack capacity is M, we require the total weight of all of the chosen objects to be at most M [5]. The knapsack problem is stated mathematically as follows: maximize
n X
p i xi
6
i1
subject to n X
wi xi 6 M;
7
i1
0 6 xi 6 1; 1 6 i 6 n:
8
Without loss of generality all weights and pro®ts are assumed to be positive numbers. The optimal solution is obtained by using the greedy strategy where the objects are considered in the order of non-increasing density pi =wi and objects are added to the knapsack if they ®t [5].
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3.2. The fuzzy knapsack models The essential problem we consider here is that the values of the coecient for each weight wi , i 1; . . . ; n, is imprecise because there is only vague knowledge about the objects or decimal truncation is used for the weights in the problem. Thus, the DM should determine an acceptable range of values for each wi , which is an interval wi Di1 ; wi Di2 , 0 6 Di1 < wi and 0 < Di2 . Then the DM chooses a value from the interval wi Di1 ; wi Di2 as an estimate of each weight. We say that the acceptable grade is 1 if the estimate is exactly wi ; otherwise, the acceptable grade will get smaller when the estimate approaches either wi Di1 or wi Di2 . It is clear that the acceptable grade for an estimate in an interval corresponds to the membership grade of a fuzzy number in the fuzzy sets. Thus this leads to the use of fuzzy numbers in the knapsack problem. We believe that the fuzzy tolerance interval of weight is the most valid information for the DM when he manages the knapsack problem in practical situations. The main advantage, compared to the crisp model, is that the DM is not forced into precise data, but into a range of values. This seems an easier task for the DM to determine the coecient of the weights in the knapsack problem. Besides, the membership functions, which monotonically increase or decrease in the interval wi Di1 ; wi Di2 , can also be handled quite easily. Nevertheless, a critical question may arise: how to guarantee an appropriate estimate value for each weight chosen in the decision-making process? Our approach is then stated as follows: ~ i be the fuzzy number denoted by Let w ~ i wi Di1 ; wi ; wi Di2 ; w 0 6 Di1 < wi ; 0 6 Di2 ; 1 6 i 6 n:
9
~ i is as shown below: The membership function of w 8 x wi Di1 > > ; wi Di1 6 x 6 wi ; > > < Di1 lw~i x wi Di2 x ; wi 6 x 6 wi Di2 ; > > > Di2 > : 0; otherwise: ~ i is 1. However, the more Fig. 2 shows that when an estimate x equals wi , the membership grade of x in w ~ i is obtained. The repaway from the position of wi an estimate x is, the less membership grade of x in w resentation of imprecise data as fuzzy numbers is useful when those data are used in fuzzy systems. Thus, ~ i by De®nition 5, we obtain an estimate of the weight in the fuzzy sense after defuzzifying the fuzzy number w ~ i ; ~01 wi 14 Di2 Di1 . The DM can then make use of from the interval wi Di1 ; wi Di2 , i.e., wi d w this equation to obtain a value as an estimate of the weight for solving the imprecise data problem. ~ i satis®es the following conditions. Furthermore, the fuzzy number w ~ i 2 FN . 1. If 0 < Di1 < wi and 0 < wi2 , then w ~ i 2 FL . 2. If 0 < Di1 < wi and Di2 0, then w
~i . Fig. 2. The fuzzy number w
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~ i 2 FR . 3. If Di1 0 and 0 < Di2 , then w ~ i 2 FP 1. 4. If Di1 Di2 0, then w ~ M; M; M 2 FP 1 and When fuzzifying the knapsack capacity, we obtain M ~ ~ d M; 01 M:
10
Let 1 ~i ; ~ wi d w 01 wi Di2 Di1 ; i 1; 2; . . . ; n: 11 4 ~ i measured from ~01 . Since From Property 1, we can see that (11) is the signed distance of w ~i ; ~ d w 01 34wi 14Di2 14 wi Di1 > 0, wi is a positive number measured from 0. Therefore, wi in (11) represents an estimate of the weight i in the fuzzy sense, which is equal to wi plus some fuzzy quantity 1 Di2 Di1 , wi 2 wi Di1 ; wi Di2 . When Di1 Di2 , we obtain wi wi . In particular, if Di1 Di2 0, 4 then the fuzzy model is equivalent to the crisp problem. From here, we can see that the fuzzy knapsack problem is an extension of the crisp knapsack problem. Thus we fuzzify (7) to obtain (12), where - is the ranking de®ned on F (see De®nition 6). n X ~ ~ i xi - M: 12 w i1
P ~ 1 ~x2 1 w ~ 2 ~xn 1 w ~ n . From (6), (8), (12), De®nition 6, ~ i xi equals ~x1 1 w Note that ni1 w and Properties 1 and 3, we derive the defuzzi®ed fuzzy knapsack problem in the following Proposition 1. Proposition 1. The knapsack problem in the fuzzy sense is formulated as follows: n X pi xi maximize
13
i1
subject to n X i1
wi xi 6 M;
0 6 xi 6 1;
14
i 1; . . . ; n;
15
where 1 wi wi Di2 4
Di1 ;
0 6 Di1 < wi ; 0 6 Di2 :
Proof. From (9), (12), and Properties 1 and 3, we have that ! X n n n n X X X 1 ~1 ~i ; ~ ~ i xi ; 0 d xi d w 01 xi wi Di2 Di1 wi xi : w 4 i1 i1 i1 i1 Pn ~ ~01 . Since d M; ~ ~01 M (see (10)), we obtain ~ i xi ; ~01 6 d M; From (12) and De®nition 6, we obtain d i1 w (14). Clearly, (13) and (15) are obtained directly from (6) and (8), respectively. Example 1. Consider an instance of knapsack problem given in Table 1. From (6)±(8), the optimal solution of the crisp problem obtained using the greedy strategy, of value 78.244, is xi 1; 1; 1; 0; 1; 25=41. Now consider the fuzzy knapsack model. The imprecise weight value
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Table 1 A knapsack problem with n 6 and M 80 i
1
2
3
4
5
6
wi pi
8 10
12 15
13 20
64 12
22 18
41 25
Table 2 Determine each coecient of weights in the fuzzy sense i
1
2
3
4
5
6
(Di1 ; Di2 ) wi
(0.2, 1) 8.2
(0.4, 1.6) 12.3
(0.6, 0.8) 13.05
(1, 0.6) 63.9
(0.7, 1.3) 22.15
(1, 0.6) 40.9
~i wi Di1 ; wi ; wi Di2 . Assume that the DM has determined the range of values for Di1 de®ned in (9) is w and Di2 , i 1; . . . ; 6, and has calculated wi wi 14 Di2 Di1 for each i, which are shown in Table 2. From (13)±(15) of Proposition 1, the optimal solution of the fuzzy model is xi 1; 1; 1; 0; 1; 24:3=40:9 with the optimal pro®t 77.8533. We compare the result obtained from Proposition 1 with that of the crisp case, as follows: 77:8533 78:244 100 78:244
0:499%:
Assume that the DM determines another set of values for Di1 and Di2 , i 1; . . . ; 6, and then another set of weight coecients in the fuzzy sense are obtained in Table 3. The optimal solution for this case is xi 1; 1; 1; 0; 1; 24:3=40:9 with a pro®t of 78.434. A comparison of this result with that for the crisp problem above is 78:434 78:244 100 0:243%: 78:244 Although the solution obtained from Proposition 1 may be slightly worse or better than that in the crisp case, the advantage of using the fuzzy model is that a range of weights is allowed in the problem. Remark 1. The interpretation of Fig. 2 using (11), wi wi 14 Di2 Di1 , i 1; . . . ; n, for comparing the dierence between Proposition 1 and the knapsack problem is as follows. When Di2 Di1 8i, Fig. 2 shows an isosceles triangle in which wi wi 8i. In this case, (13)±(15) are equivalent to (6)±(8). This means that the crisp knapsack problem is a special case of Proposition 1, the knapsack problem in the fuzzy sense. In particular, if Di2 Di1 0 8i, then Proposition 1 is equivalent to the crisp knapsack problem. Hence, Proposition 1 is an extension of the crisp knapsack problem. On the other hand, when Di2 > Di1 8i, the triangle in Fig. 2 is skewed to the right-hand side obtaining wi > wi 8i. From (7) and (14), we obtain n X i1
wi xi <
n X i1
wi xi 6 M:
Table 3 Another set of weights in the fuzzy sense i
1
2
3
4
5
6
(Di1 ; Di2 ) wi
(0.2, 0.6) 8.1
(0.8, 0.4) 11.9
(1, 0.4) 12.85
(0.3, 0.7) 64.1
(1.2, 0.4) 21.8
(1, 0.6) 40.9
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Let
( K1
) X n x1 ; . . . ; xn w x 6 M; 0 6 xi 6 1 8i i1 i i
denote the knapsack problem and let ( ) X n w x 6 M; 0 6 xi 6 1 8i K2 x1 ; . . . ; xn i1 i i denote Proposition 1. Since K2 K1 , we have n n X X max pi xi 6 max pi xi : x1 ;...;xn 2K2
x1 ;:::;xn 2K1
i1
i1
Similarly, if Di2 < Di1 , the triangle is skewed to the left-hand side obtaining wi < wi . Since K1 K2 , we have n n X X max pi xi 6 max pi xi : x1 ;...;xn 2K1
x1 ;...;xn 2K2
i1
i1
3.3. 0/1 knapsack problem If we add the requirement that xi 1 or xi 0, 1 6 i 6 n, i.e., an object is either included or not included into the knapsack. Then the problem is known as 0/1 knapsack problem. In other words, we wish to solve as follows: maximize
n X
p i xi
16
i1
subject to n X
wi xi 6 M;
17
i1
xi 0 or 1;
1 6 i 6 n:
18
Since the greedy strategy does not necessarily yield an optimal solution for the 0/1 knapsack problem, instead, we can use the dynamic programming approach for ®nding the optimal solution [5]. After fuzzifying (17), we obtain (12). From (16), (12) and (18), we derive Proposition 2, the defuzzi®ed fuzzy 0/1 knapsack problem, as follows. Proposition 2. The 0=1 knapsack problem in the fuzzy sense is formulated as follows: maximize
n X
pi xi
19
wi xi 6 M;
20
i1
subject to n X i1
xi 0 or 1;
1 6 i 6 n;
21
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167
where 1 wi wi Di2 4
Di1 ;
0 6 Di1 < wi ; 0 6 Di2 :
Example 2. We use the same instance as in Example 1. The 0/1 knapsack problem is maximize 10x1 15x2 20x3 12x4 18x5 25x6 subject to 8x1 12x2 13x3 64x4 22x5 41x6 6 80;
xi 0 or 1; 1 6 i 6 6:
The optimal solution for the 0/1 knapsack problem obtained using the dynamic programming technique is xi 1; 1; 1; 0; 0; 1 and the optimal pro®t is 70. Next, we consider the fuzzy 0/1 knapsack model. Case 1: From (19)±(21) of Proposition 2 and Table 2, the 0/1 knapsack problem in the fuzzy sense can be formulated as follows: maximize
10x1 15x2 20x3 12x4 18x5 25x6
subject to
8:2x1 12:3x2 13:05x3 63:9x4 22:15x5 40:9x6 6 80;
xi 0 or 1; 1 6 i 6 6:
The optimal solution obtained using the dynamic programming technique for this problem is still xi 1; 1; 1; 0; 0; 1 and the optimal pro®t is 70. Case 2: From Table 3, the 0/1 knapsack problem in the fuzzy sense is as follows: maximize
10x1 15x2 20x3 12x4 18x5 25x6
subject to
8:1x1 11:9x2 12:85x3 64:1x4 21:8x5 40:9x6 6 80;
xi 0 or 1; 1 6 i 6 6:
Again, the optimal solution is xi 1; 1; 1; 0; 0; 1 with the optimal pro®t of 70 that is obtained by using the dynamic programming technique. Example 3. Consider an instance of knapsack problem, which is given in Table 4. From (6)±(8), the optimal solution is xi 0; 1; 0; 1; 1; 1 and the optimal pro®t is 68 that can be obtained based on the greedy strategy. Consider the fuzzy model. Assume that the DM takes the values of Di1 and Di2 , i 1; . . . ; 6, and calculates the coecients of the weights in the fuzzy sense, as shown in Table 5. Table 4 An instance of knapsack problem with n 6 and M 60 i
1
2
3
4
5
6
wi pi
10 9
15 18
60 22
13 10
11 15
20 25
Table 5 Determine the coecients of weights in the fuzzy sense i
1
2
3
4
5
6
(Di1 ; Di2 ) wi
(1, 0.2) 9.8
(0.2, 1) 15.2
(1, 0.4) 59.85
(0.2, 1.8) 13.4
(0.3, 1.5) 11.3
(0.1, 0.9) 20.2
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From (19)±(21) of Proposition 2, the 0/1 knapsack problem in the fuzzy sense is formulated as follows: maximize
9x1 18x2 22x3 10x4 15x5 25x6
subject to
9:8x1 15:2x2 59:85x3 13:4x4 11:3x5 20:2x6 6 60;
xi 0 or 1; 1 6 i 6 6:
The optimal solution is xi 1; 1; 0; 0; 1; 1 and the optimal pro®t is 67 obtained by using the greedy strategy. Note that this solution is dierent from the crisp problem above. After comparing the result obtained from Proposition 2 with that of the crisp problem, we have 67
68 68
100
1:47%:
Remark 2. Let ( K3
) X n x1 ; . . . ; xn w x 6 M; xi 0 or 1 8i i1 i i
denote the 0/1 knapsack problem and let ( ) X n K4 x1 ; . . . ; xn w x 6 M; xi 0 or 1 8i i1 i i denote Proposition 2. If Di1 < Di2 8i, the triangle in Fig. 2 is skewed to the right-hand side obtaining wi < wi 8i. Thus we have n X
wi xi <
i1
n X i1
wi xi 6 M:
Since K4 K3 , we obtain max
x1 ;...;xn 2K4
n X
p i xi 6
i1
max
x1 ;...;xn 2K3
n X
pi xi :
i1
Similarly, ifPDi2 < Di1 8i, the triangle is skewed to the left-hand side obtaining wi < wi 8i. Then Pn n i1 wi xi < i1 wi xi 6 M. Since K3 K4 , we obtain max
x1 ;...;xn 2K3
n X i1
p i xi 6
max
x1 ;...;xn 2K4
n X
pi xi :
i1
4. The MCKP with fuzzy numbers The MCKP is a generalization of the 0/1 knapsack problem and a special case of general 0/1 integer programming. The problem is stated as follows: We are given n objects and a knapsack. The object i weighs wij when it is considered using the jth measure (1 6 j 6 m) and the knapsack also has m capacity because of the m dierent measures for objects. In other words, m is the number for the knapsack's constraint, each with a capacity Mj , 1 6 j 6 m, which is called the capacity of the knapsack using the jth measure. If the object i is placed into the knapsack, satisfying all of the m dierent measures then a pro®t pi is earned,
F.-T. Lin, J.-S. Yao / European Journal of Operational Research 135 (2001) 158±176
169
1 6 i 6 n. Since the capacity of the knapsack is limited, the total weight of all of the chosen objects can be at most Mj using each measure j, 1 6 j 6 m. We are interested in ®lling the knapsack with those objects that yield the maximum pro®t. Formally, the MCKP can be stated mathematically as follows: maximize
n X
pi xi
22
i1
subject to n X
wij xi 6 Mj
for j 1; . . . ; m;
23
i1
xi 0 or 1 for i 1; . . . ; n:
24
Without a loss of generality, all pi ; wij ; and Mj are assumed to be nonnegative. While considering an imprecise value for each coecient of weight, we let an acceptable range for each estimate of weights be denoted by wij hij1 ; wij hij2 , where 0 6 hij1 < wij and 0 < hij2 . A corresponding triangular fuzzy number for each weight wij is given by ~ ij wij w
hij1 ; wij ; wij hij2 :
25
~ ij is 1. Otherwise, the more Fig. 3 shows that when an estimate x equals wij , the membership grade of x in w ~ ij is obtained. Besides, the away from the position of wij an estimate x is, the less membership grade of x in w ~ ij satis®es the following conditions. fuzzy number w ~ ij 2 FN . 1. If 0 < hij1 < wij and 0 < hij2 , then w ~ ij 2 FL . 2. If 0 < hij1 < wij and hij2 0, then w ~ ij 2 FR . 3. If hij1 0 and 0 < hij2 , then w ~ij 2 FP 1. 4. If hij1 hij2 0, then w After fuzzifying the knapsack capacity Mj , we have that ~ j Mj ; Mj ; Mj 2 FP 1 and M
~j; ~ d M 01 Mj ; 1 6 j 6 m:
26
Let 1 ~ ij ; ~ wij d w 01 wij hij2 4
hij1 ;
1 6 i 6 n; 1 6 j 6 m:
27
~ ij measured from ~01 is From Property 1, we can see that the signed distance of w 1 1 1 ~ ~ ~ ij ; 01 wij 4 hij2 hij1 . Since d w ~ ij ; 01 4 3wij hij2 4 wij hij1 > 0, wij is a positive distance d w measured from 0. Hence, wij represents an estimate of the jth measure of object i in the fuzzy sense, which is
~ ij . Fig. 3. The fuzzy number w
170
F.-T. Lin, J.-S. Yao / European Journal of Operational Research 135 (2001) 158±176
wij plus some fuzzy quantity 14 hij2 hij1 . It is clear that the fuzzy MCKP is an extension of the crisp MCKP. Then (23) is fuzzi®ed to obtain (28), where - is the ranking de®ned on F, as follows: n X
~j ~ ij xi - M w
for j 1; . . . ; m:
28
i1
From (22), (24), (28), De®nition 6, and Properties 1 and 3, we can derive Proposition 3, the defuzzi®ed fuzzy MCKP, as follows: Proposition 3. The MCKP in the fuzzy sense is formulated as follows: maximize
n X
pi xi
29
i1
subject to n X i1
wij xi 6 Mj for j 1; . . . ; m;
30
xi 0 or 1 for i 1; . . . ; n;
31
where 1 wij wij hij2 4
hij1 ;
0 6 hij1 < wij ;
0 < hij2 :
Proof. From (25), (28), and Properties 1 and 3, we have ! n n n X X X 1 ~ ~ ~ ij ; 01 ~ ij xi ; 01 d xi d w xi wij hij2 w 4 i1 i1 i1
hij1
n X i1
wij xi :
Pn ~j; ~ ~ j ; ~01 Mj , we have (30). Clearly, (29) and (31) ~ ij xi ; ~ 01 6 d M 01 . Since d M From (28), we obtain d i1 w are obtained directly from (22) and (24). Example 4. Consider the instance of the MCKP from [12], which is shown in Table 6. The problem is n 10 and m 10, n is the number of objects and m is the number of constraints of capacity Mj , which can be formulated as follows: maximize
10 X
p i xi
i1
subject to
10 X
wij xi 6 Mj
for j 1; . . . ; 10;
i1
xi 1 or 0: The optimal solution for this problem instance is xi 0; 1; 0; 1; 1; 0; 0; 1; 0; 1, with the optimal value 8706.1, can be obtained using either simulated annealing or genetic algorithms [7]. Now, consider the fuzzy model. Let wij hij1 ; wij hij2 be the acceptable range of weight for wij . The corresponding fuzzy number ~ ij wij hij1 ; wij ; wij hij2 . Assume that the values of hij1 and hij2 , 1 6 i; j 6 10, are for that range is w already de®ned in Table 7.
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171
Table 6 The instance of MCKP with n 10 and m 10 wij
j1
2
3
4
5
6
7
8
9
10
pi
i1 2 3 4 5 6 7 8 9 10
20 5 100 200 2 4 60 150 80 40
20 7 130 280 2 8 110 210 100 40
60 3 50 100 4 2 20 40 6 12
60 8 70 200 4 6 40 70 16 20
60 13 70 250 4 10 60 90 20 24
60 13 70 280 4 10 70 105 22 28
5 2 20 100 2 5 10 60 0 0
45 14 80 180 6 10 40 100 20 0
55 14 80 200 6 10 50 140 30 40
65 14 80 220 6 10 50 180 30 50
600.1 310.5 1800 3850 18.6 198.7 882 4200 402.5 327
Mj
450
540
200
360
440
480
200
360
440
480
Table 7 The values of each hij1 and hij2 (hij1 ; hij2 )
j1
2
i1 2 3 4 5 6 7 8 9 10
(1, (2, (2, (1, (1, (2, (3, (3, (2, (3,
(1, (2, (1, (1, (2, (2, (1, (2, (3, (1,
2) 4) 3) 2) 3) 4) 1) 2) 3) 1)
3 1) 3) 2) 4) 3) 4) 3) 1) 4) 2)
(1, (1, (1, (1, (2, (1, (1, (2, (3, (1,
4 2) 2) 2) 2) 4) 3) 2) 3) 2) 4)
(1, (2, (1, (1, (1, (2, (3, (2, (3, (2,
5 2) 4) 3) 4) 2) 3) 1) 3) 4) 4)
(1, (1, (1, (1, (1, (1, (2, (4, (2, (2,
6 1) 3) 2) 3) 2) 3) 4) 2) 3) 1)
(1, (2, (1, (2, (1, (2, (3, (2, (1, (2,
7 2) 3) 2) 3) 2) 4) 2) 4) 3) 3)
(0, (0, (0, (0, (1, (0, (2, (1, (2, (1,
8 1) 1) 1) 1) 2) 1) 3) 4) 3) 3)
(1, (0, (1, (0, (1, (0, (1, (3, (1, (2,
9 2) 1) 1) 1) 3) 1) 2) 1) 4) 3)
(1, (1, (1, (0, (1, (1, (2, (4, (1, (2,
10 2) 1) 2) 1) 2) 2) 3) 2) 3) 4)
(1, (1, (1, (2, (3, (1, (3, (2, (1, (2,
2) 1) 2) 3) 2) 3) 2) 4) 3) 3)
Then the DM can obtain each weight in the fuzzy sense, using wij wij 14 hij2 hij1 , which are shown in Table 8. The optimal solution of this problem is still xi 0; 1; 0; 1; 1; 0; 0; 1; 0; 1 with the optimal pro®t 8706.1. If we change the capacity Mj in Tables 6 and 8 to M1 400, M2 539, M3 160, M4 340, M5 400, M6 450, and M8 320 (M7 ; M9 , and M10 are unchanged), then the optimal solution is xi 0; 1; 0; 1; 0; 0; 0; 1; 0; 1 with the optimal pro®t 8687.5. Table 8 Determine the weights in the fuzzy sense wij
j1
2
3
4
5
6
7
8
9
10
i1 2 3 4 5 6 7 8 9 10
20.25 5.5 100.25 200.25 2.5 4.5 59.5 149.75 80.25 39.5
20 7.25 130.25 280.75 2.25 8.5 110.5 209.75 100.25 40.25
60.25 3.25 50.25 100.25 4.5 2.25 20.25 40.25 5.75 12.75
60.25 8.5 70.5 200.75 4.25 6.25 39.5 70.25 16.25 20.5
60 13.5 70.25 250.5 4.25 10.5 60.5 89.5 20.25 23.75
60.25 13.25 70.25 280.25 4.25 10.5 69.75 104.5 22.5 28.25
5.25 2.25 20.25 100.25 2.25 5.25 10.25 60.75 0.25 0.75
45.25 14.25 80 180.25 6.5 10.25 40.25 99.5 20.75 0.25
55.25 14 80.25 200.25 6.25 10.25 50.25 139.5 30.5 40.5
65.25 14 80.25 220.25 5.75 10.5 49.75 180.5 30.5 30.25
Mj
450
540
200
360
440
480
200
360
440
480
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F.-T. Lin, J.-S. Yao / European Journal of Operational Research 135 (2001) 158±176
Remark 3. Similar to Remark 2. From (27), wij wij 14 hji2 hji1 , the interpretation of Fig. 3 is as follows. When hij2 hij1 8i; j, Fig. 3 shows an isosceles triangle obtaining wij wij 8i; j. This case shows that (29)±(31) are equivalent to (22)±(24). The crisp MCKP is a special case of Proposition 3, the MCKP in the fuzzy sense. However, if hij2 hij1 0 8i; j, then Proposition 3 is equivalent to the crisp knapsack problem. From above, Proposition 3 is an extension of the crisp knapsack problem. On the other hand, if hij2 > hij1 8i; j, the triangle in Fig. 3 is skewed to the right-hand side obtaining wij > wij 8i; j. From (23) and (30), we can see that n X
wij xi <
i1
n X i1
wij xj 6 Mj ;
j 1; . . . ; m:
Let ( L1
X n x1 ; x2 ; . . . ; xn w x 6 Mj ; i1 ij i
) j 1; . . . ; m; xi 0 or 1 8i
denote the MCKP. Let ( L2
X n x1 ; x2 ; . . . ; xn w x 6 Mj ; i1 ij i
) j 1; . . . ; m; xi 0 or 1 8i
denote Proposition 3. Since L2 L1 , max
x1 ;...;xn 2L2
n X
pi xi 6
i1
max
x1 ;...;xn 2L1
n X
pi xi :
i1
When hij2 6 hij1 8i; j, the triangle is skewed to the left-hand side obtaining wij < wij 8i; j, and n X i1
wij xi <
n X
wij xi 6 Mj ;
1 6 j 6 m:
i1
Since L1 L2 , we have max
x1 ;...;xn 2L1
n X
pi xi 6
i1
max
x1 ;...;xn 2L2
n X
pi xi :
i1
5. Concluding remarks In this paper, we have investigated the knapsack problem with imprecise object weights. We have proposed an approach for constructing fuzzy knapsack models. By using the signed distance ranking method to defuzzify the fuzzy models, we obtain Propositions 1, 2, and 3 used for solving the problem of imprecise weight coecients. In our study, our fuzzy knapsack models are an extension of the crisp knapsack problems, and the crisp knapsack problem is a special case of our fuzzy models. In conclusion, the generalization of our fuzzy knapsack model is as follows.
F.-T. Lin, J.-S. Yao / European Journal of Operational Research 135 (2001) 158±176
173
5.1. A generalized fuzzy knapsack model Let A~ be a fuzzy set on R. For each a-cut, 0 6 a 6 1, we have that A a AL a; AR a, where AL a and AR a are the left-end point and the right-end point for each a-cut. Assume that the a-cut A a holds and AL a and AR a are continuous functions in 0 6 a 6 1. Let G denote the family of those fuzzy sets, and let FP 1 G. Similar to De®nition 5, we have De®nition 7, which is the signed distance of fuzzy set A~ in G. De®nition 7. Let A~ 2 G. The signed distance of A~ measured from ~01 is de®ned by Z 1 1 ~~ dG A; 01 AL a AR a da: 2 0 Note that De®nition 5 is a special case of De®nition 7 and F G. ~ B~ G, we de®ne the signed distance rankings on G as follows: De®nition 8. For A; ~~ ~~ A~ B~ iff dG A; 01 < dG B; 01 ; ~ ~ ~ ~ ~ A B iff dG A; 01 dG B; ~ 01 : ~ B~ 2 G, we obtain the following equations Similarly, De®nition 6 is a special case of De®nition 8. For A; from De®nition 7. Z 1 1 ~ ~ dG ~ a1 A; 01 aAL a aAR ada 2 0 ~1 ; ~0 adG A; 32 ~~ dG A~ B; 01
1 2
Z
1 0
AL a BL a AR a BR ada
~~ ~~ dG A; 01 dG B; 01 :
33
~ i on R, the interpretation of the fuzzy number When the weight wi in (7) transforms into the fuzzy number w ~ i is 1 when an estimate equals wi . Otherwise, in the left-hand is then as follows. The membership grade in w side of wi is a monotonically increasing continuous function fL x and in the right-hand side of wi is a monotonically decreasing continuous function fR x, where x is an estimate for wi . The functions fL x and ~ M; M; M 2 FP 1 F G. When ~ i 2 G. Let M fR x are determined by the DM and we can see that w fuzzifying (7) using De®nition 8, we can obtain n X
~ ~ i xi - M: w
i1
From De®nitions 7 and 8 and by using (32)±(34) becomes n X
~i ; ~ dG w 01 xi 6 M:
i1
Finally, we derive a generalized knapsack problem in the fuzzy sense from (6)±(8) as follows.
34
174
F.-T. Lin, J.-S. Yao / European Journal of Operational Research 135 (2001) 158±176
Proposition 4. The generalized knapsack problem in the fuzzy sense is formulated as follows: n X maximize pi xi
35
i1
subject to
n X
~i ; ~ dG w 01 xi 6 M;
36
i1
0 6 xi 6 1
for i 1; . . . ; n:
37
~i ; ~ ~i ; ~ ~i 2 F , Proposition 4 becomes ProposiObviously, when dG w 01 d w 01 wi 14 Di2 Di1 for w tion 1. Similarly, Propositions 2 and 3 can also be generalized to the fuzzy set on G with the similar way. 5.2. Comparing the ranking method used in [11] with that of our work (1) The crisp MCKP that proposed by Okada and Gen [11] is as follows: minimize
z x
p X nt X t1
subject to
ctj xtj
38
j1
p X nt X atj xtj 6 b; t1 j1 nt X
xtj 1; t 1; 2; . . . ; p;
39 xtj 2 f0; 1g; j 1; 2; . . . ; nt ; t 1; 2; . . . ; p:
40
j1
The fuzzy MCKP is de®ned by p X nt X minimize ~z x c~tj xtj t1
subject to
p X nt X
41
j1
~ a~tj xtj K b;
t1 j1 nt X
xtj 1; t 1; 2; . . . ; p;
42 xtj 2 f0; 1g; j 1; 2; . . . ; nt ; t 1; 2; . . . ; p;
43
j1
where
R c~tj cLtj ; cM tj ; ctj ;
R a~tj aLtj ; aM tj ; atj ;
j 1; 2; . . . ; nt ; t 1; 2; . . . ; p;
and b~ bL ; bM ; bR are fuzzy numbers. Finally, the fuzzy MCKP is transformed into the following multiobjective programming problem as follows: minimize minimize
zL x zM x
p X nt X
cLtj xtj ;
44
cM tj xtj ;
45
t1 j1 p X nt X t1
j1
F.-T. Lin, J.-S. Yao / European Journal of Operational Research 135 (2001) 158±176
minimize
p X nt X
zR x
t1
subject to p X nt X
P
! a~tj xtj 6 b~ P a;
46
a 2 0; 1;
47
xtj 2 f0; 1g; j 1; 2; . . . ; nt ; t 1; 2; . . . ; p;
48
j1
t1 nt X
j1
cRtj xtj ;
175
xtj 1; t 1; 2; . . . ; p;
j1
where ~ P ~ a K b
R aR aL
minfla~ x; lb~ xg dx ; R aR la~ x dx aL
a~ and b~ are fuzzy numbers. (2) In our approach, when applying the proposed signed-distance method for ranking the fuzzy numbers in the fuzzy MCKP, (41)±(43), we obtain the following MCKP in the fuzzy sense, (49)±(51). Note that the ranking K in (42) is de®ned in De®nition 6. minimize
z0 x
p X nt X t1
j1
c0tj xtj
49
subject to p X nt X t1 nt X
j1
a0tj xtj 6 b0 ;
50
xtj 1; t 1; 2; . . . ; p;
xtj 2 f0; 1g; j 1; 2; . . . ; nt ; t 1; 2; . . . ; p;
51
j1
where L R c0tj d ~ ctj ; ~ 01 14 2cM tj ctj ctj ; L R a0tj d ~ atj ; ~ 01 14 2aM tj atj atj ;
~~ b0 d b; 01 14 2bM bL bR : (3) The major dierences between [11] and this paper are stated as follows: 1. When comparing the crisp MCKP of [11], (38)±(40), with the crisp MCKP of this paper, (22)±(24), we can see that the underlying problem is dierent. ~ is dierent from the signed distance rank2. The ranking K in (42) of the fuzzy MCKP of [11], P ~ a K b, ing method, De®nition 6, used in this paper. Pp Pnt 3. Consider the fuzzy objective function of (41), ~z x t1 j1 c~tj xij , which is de®ned in [11]. After applying the operation of fuzzy numbers, we obtain ! p X p X p X nt nt nt X X X L M R L M R ~z x z x; z x; z x ctj xtj ; ctj xtj ; ctj xtj : t1
j1
t1
j1
t1
j1
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F.-T. Lin, J.-S. Yao / European Journal of Operational Research 135 (2001) 158±176
Obviously, the objective functions of (44)±(46), which are de®ned in [11], are not obtained from the defuzzi®cation of the fuzzy numbers. On the contrary, if we use the signed distance of Property 1 to defuzzify the fuzzy objective function of (41), we will obtain " # p X p X p X p X nt nt nt nt X X X X 1 2 z0 x d ~z x; ~ 01 cM cLtj xtj cRtj xtj c0tj xtj : tj xtj 4 t1 j1 t1 j1 t1 j1 t1 j1 Thus the objective function of the MCKP de®ned in [11] and the objective function of the MCKP de®ned in this paper is dierent. R L M R L M R ~tj ; a~tj ; b~ will become level 1 4. If we let cLtj cM tj ctj ctj ; atj atj aij atj , and b b b b, then c fuzzy points, i.e., c~tj ~ ctj 1 ; a~tj ~ atj 1 and b~ b~1 . Hence, we have that ctj d ~ ctj 1 ; ~ 01 ctj ; atj atj , and b b. We can see that (49)±(51) are equivalent to (38)±(40). Obviously, in this paper, the crisp MCKP is a special case of the MCKP in the fuzzy sense. However, in [11], the crisp MCKP in (38)±(40) is not a special case of the multi-objective programming problem in (44)±(48). Acknowledgements The authors are grateful to anonymous referees whose valuable comments helped to improve the content of this paper. References [1] J.M. Cadenas, J.L. Verdegay, Using fuzzy numbers in linear programming, IEEE Transactions on Systems, Man and Cybernetics ± Part B: Cybernetics 27 (6) (1997) 1016±1022. [2] M. Delgado, J.L. Verdegay, M.A. Vila, A general model for fuzzy linear programming, Fuzzy Sets and Systems 29 (1997) 21±29. [3] A. Freville, G. Plateau, Heuristics and reduction methods for multiple constraints 0±1 linear programming problems, European Journal of Operational Research 24 (1986) 206±215. [4] J. Gavish, H. Pirkul, Ecient algorithms for solving multi-constraint zero±one knapsack problems to optimality, Mathematical Programming 31 (1985) 78±105. [5] E. Horowitz, S. Sahni, S. Rajasekaran, Computer Algorithms, Freeman, New York, 1998. [6] A. Kaufmann, M.M. Gupta, Introduction to Fuzzy Arithmetic Theory and Applications, van Nostrand Reinhold, New York, 1991. [7] F.T. Lin, C.Y. Kao, C.C. Hsu, Applying the genetic approach to simulated annealing in solving some NP-hard problems, IEEE Transactions on Systems, Man and Cybernetics 23 (6) (1993) 1752±1767. [8] M. Magazine, O. Oguz, A heuristic algorithm for the multi-dimensional zero±one knapsack problem, European Journal of Operational Research 16 (1984) 319±326. [9] S. Martello, P. Toth, Knapsack Problems: Algorithms and Computer Implementations, Wiley, Chichester, 1990. [10] S. Martello, P. Toth, Heuristic algorithms for the multiple knapsack problem, Computing 27 (1981) 93±112. [11] S. Okada, M. Gen, Fuzzy multiple choice knapsack problem, Fuzzy Sets and Systems 67 (1994) 71±80. [12] C.C. Peterson, Computational experience with variants of the Balas algorithm applied to the selection of R&D projects, Management Science 13 (1967) 736±750. [13] D. Pisinger, An expanding-core algorithm for the exact 0±1 knapsack problem, European Journal of Operational Research 87 (1995) 175±187. [14] S. Senyu, Y. Toyoda, An approach to linear programming with 0±1 variables, Management Science 15 (1968) 196±207. [15] W. Shih, A branch and bound method for the multi-constraint zero±one knapsack problem, Journal of the Operational Research Society 30 (1979) 369±378. [16] J.S. Yao, K.M. Wu, Ranking fuzzy numbers based on decomposition principle and signed distance, Fuzzy Sets and Systems 116 (2000) 275±288. [17] H.-J. Zimmermann, Fuzzy Set Theory and its Applications, second ed., Kluwer Academic Publishers, Boston, MA, 1991.
www.elsevier.com/locate/dsw
Theory and Methodology
Using fuzzy numbers in knapsack problems Feng-Tse Lin a
a,*
, Jing-Shing Yao
b
Department of Applied Mathematics, Chinese Culture University, 55, Hwakang Rd., Yangminshan, Taipei 111, Taiwan, ROC b Department of Mathematics, National Taiwan University, Taipei, Taiwan, ROC Received 13 September 1999; accepted 6 November 2000
Abstract This paper investigates knapsack problems in which all of the weight coecients are fuzzy numbers. This work is based on the assumption that each weight coecient is imprecise due to the use of decimal truncation or rough estimation of the coecients by the decision-maker. To deal with this kind of imprecise data, fuzzy sets provide a powerful tool to model and solve this problem. Our work intends to extend the original knapsack problem into a more generalized problem that would be useful in practical situations. As a result, our study shows that the fuzzy knapsack problem is an extension of the crisp knapsack problem, and that the crisp knapsack problem is a special case of the fuzzy knapsack problem. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy sets; Optimization; Knapsack problem; Multiconstraint 0/1 knapsack problem; Signed distance ranking
1. Introduction In its original meaning, the knapsack problem involves ®nding a combination of dierent objects, that a hitchhiker chooses for his knapsack in which the total value of all of the objects he chooses is maximized [9]. The knapsack may correspond to a truck, a ship, a silicon chip, or a resource. The problem is to package these items so that there are a variety of applications for cargo loading, cutting stock, various packing problems, or economic planning. For example, this method could describe the problem of making investment decisions in which the size of an investment is based on the amount of money required, the knapsack capacity is the amount of available money to invest, and the investment pro®t is the expected return. The multi-constraint 0/1 knapsack problem (MCKP) is a very important knapsack problem, which arises in the areas of capital budgeting and resource allocation [3]. The reason for this interest is based on the fact that it is a special case of the general 0/1 integer programming and can be used as a subproblem for
*
Corresponding author. Tel.: +886-2-286-10511 ext. 741; Fax: +886-2-286-19509. E-mail address: ftlin@sta.pccu.edu.tw (F.-T. Lin).
0377-2217/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 0 ) 0 0 3 1 0 - 6
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solving many more complex problems. Since the MCKP has been proved to be an NP-hard problem [9], several heuristics [3,4,7,8,10,14,15] have been proposed for solving this problem approximately over the past few years. In our study, however, we are not proposing a new heuristic for solving the MCKP approximately. Instead, we are interested in studying how to deal with the imprecise data that would occur in practical situations. There are three motivations for our study. First, it is sometimes dicult for the decision-maker (DM) to determine the precise values of the weight coecients because the object information is vague or imprecise [1]. Second, as noted in [13], ¯oating-point arithmetic in the MCKP may lead to round-o errors, since a computer only works with approximations to the real numbers. Third, when using truncating functions, dramatic errors may be produced, even for small deviations, and thus lead to incorrect results. Although the knapsack problem has often been investigated, little of the related research involves resolving the uncertainties [1,2,11]. In our study, we assumed that the weight of each object in the knapsack problem is imprecise, which may be in the vicinity of a ®xed value, or substantially less than or greater than a ®xed value. To deal with this kind of imprecise data, fuzzy sets provide a powerful tool to model and solve the problem. Our approach can be depicted brie¯y as follows. Since each weight, wi , i 1; . . . ; n, is imprecise, the DM should determine an interval wi Di1 ; wi Di2 , 0 6 Di1 < wi and 0 < Di2 , to represent an acceptable range for the weight of each object. The interpretation of this range is as follows. If an estimate of the weight is exactly wi , then the acceptable grade for that weight will be 1; otherwise, the acceptable grade will get smaller when an estimate is approaching one of the ends of the interval, i.e., wi Di1 or wi Di2 . Accordingly, the DM needs to determine an appropriate estimate for each weight from the interval ~ i wi Di1 ; wi ; wi Di2 , in the knapsack wi Di1 ; wi Di2 . This leads to the use of fuzzy numbers, w problems. Obviously, the membership grade of a fuzzy number in the fuzzy set corresponds to the ac~ i using the ceptable grade of an estimate in a given interval. Thus, after defuzzifying the fuzzy number w proposed ranking method, we obtain an estimate for each object weight in the fuzzy sense, for instance wi , which is in the interval wi Di1 ; wi Di2 . Then we use wi as the weight of object i and use the approach for solving the crisp problem to solve the fuzzy model. The advantage of this fuzzy model is that it is much easier for the DM to specify a range value than to give an exact value for each object weight. Consequently, the proposed approach is a sound and a ¯exible way for specifying the weight coecient while we solve the knapsack problems in practical situations. The paper is organized as follows. Section 2 introduces the preliminaries and the method for ranking fuzzy numbers. The major contribution of this work is presented in Sections 3 and 4. In Section 3, we propose an approach to construct fuzzy knapsack models. After the defuzzi®cation of the fuzzy models, we obtain Propositions 1 and 2, the knapsack and the 0/1 knapsack problems in the fuzzy sense, which can be used for solving imprecise data. Two examples will be employed here to illustrate these problems. Section 4 presents Proposition 3, the MCKP in the fuzzy sense (i.e. the defuzzi®ed fuzzy MCKP), where an illustrative example is shown. Finally, the concluding remarks are given in Section 5. 2. Preliminaries This section reviews some basic de®nitions of fuzzy sets [6,17] and introduces the signed distance method for ranking fuzzy numbers. De®nition 1. Let b~1 be a fuzzy set on R 1; 1. It is called a level 1 fuzzy point if its membership function is as given below. 1; x b; 1 lb~1 x 0; x 6 b:
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Let FP 1 fb~1 j 8b 2 Rg denote the family of all level 1 fuzzy points. De®nition 2. Let pa ; qa be a fuzzy set on R 1; 1, p, q 2 R and 0 6 a 6 1. It is called a level a fuzzy interval, if its membership function is as given below. a; p 6 x 6 q; 2 lpa ;qa x 0; otherwise: De®nition 3. Let A~ be a fuzzy set on R 1; 1. It is called a triangular fuzzy number, if its membership function is as given below, and is denoted by A~ a; b; c. 8 < x a= b a; a 6 x 6 b; lA~ x c x= c b; b 6 x 6 c; a < b < c; 3 : 0; otherwise: Let FN f a; b; c j 8a < b < c; a; b; c 2 Rg be the family of all triangular fuzzy numbers. When (3) changes to lB~ x x a= b a if a 6 x 6 b, and lB~ x 0 if x < a or b < x, we can rewrite A~ a; b; c as B~ a; b; b and then call it a left triangular fuzzy number. The family of all left triangular fuzzy numbers can be denoted by FL f a; b; b j 8a < b; a; b 2 Rg. Similarly, we can de®ne a right triangular fuzzy number, C~ b; b; c. Let lC~ x c x= c b if b 6 x 6 c, and let lC~ x 0, if x < b or c < x. Then FR f b; b; c j 8b < c; b; c 2 Rg denotes the family of all right triangular fuzzy numbers. Note that A~ a; b; c b; b; b b~1 if c a b. It is clear that FP 1, FL , and FR are all special cases of FN . Therefore we have that F FN [ FL [ FR [ FP 1 f a; b; c j 8a 6 b 6 c; a; b; c 2 Rg. Now, consider the problem of ranking fuzzy numbers. We will use the signed distance ranking method, which was de®ned in [16], for ranking the fuzzy numbers on F in this paper. De®nition 4. The signed distance of b is de®ned by d b; 0 b, b; 0 2 R. The interpretation of the signed distance is as follows. Geometrically, 0 < b means that b lies to the right of the origin 0 and the distance between b and 0 is denoted by b d b; 0. Similarly, b < 0 means that b lies to the left of 0 and the distance between b and 0 is denoted by b d b; 0. In summary, d b; 0 stands for the signed distance of b measured from the origin 0. From Fig. 1, we can see that a a-cut of the fuzzy number A~ a; b; c is an interval AL a; AR a; 0 6 a 6 1, where AL a and AR a are the left endpoint and the right endpoint of the a-cut, respectively. From (3) we have that AL a a b aa and AR a c c ba, where AL a and AR a are the signed distances measured from 0. From De®nition 4, we have d AL a; 0 AL a and d AR a; 0 AR a, 0 6 a 6 1. Hence, the signed distance of the interval AL a; AR a is de®ned by d AL a; AR a; 0 12d AL a; 0 d AR a; 0 12AL a AR a 12a c 2b a ca. Since
~ Fig. 1. A a-cut of fuzzy number A.
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the function aRis continuous over the interval, R 1we can use the integration to obtain the mean of the signed 1 distance, i.e. 0 d AL a; AR a; 0 da 12 0 a c 2b a ca da 14 2b a c. In addition, the intervals AL a; AR a and AL aa ; AR aa , for each a 2 0; 1, have a one-to-one mapping as shown in Fig. 1. Thus we have the following de®nition. De®nition 5. Let A~ a; b; c 2 FN . The signed distance of A~ measured from ~01 (y-axis) is de®ned by Z 1 ~ ~ d AL aa ; AR aa ; ~ 01 da d A; 01 0
Z 1 1 AL a AR a da 2 0 1 4 2b a c: 4 As we mentioned earlier in this section, if c a b then A~ b~1 . We have AL a AR a b. From De®nitions 5 and 4, we also have d b~1 ; ~ 01 b and d b~1 ; ~01 b d b; 0, respectively. Since b~1 b; b; b ~ is a special case of A a; b; c, we can see that the signed distance on FN is an extension of the signed distance on R. Similarly, we can de®ne the signed distance on FL and FR using the same concept. Let B~ a; b; b 2 FL . Then we have that BL a a b aa and BR a b. From De®nition 5, we can see ~ ~01 1 3b a. Let C~ b; b; c 2 FR . that the signed distance of B~ measured from ~ 01 is de®ned by d B; 4 ~ ~ ~ ~01 1 3b c. As a result, we can Then the signed distance of C measured from 01 is de®ned by d C; 4 de®ne the signed distance of the fuzzy set A~ a; b; c 2 F , in the following property. ~ ~01 1 2b a c. Property 1. Let A~ a; b; c 2 F . The signed distance of A~ is given by d A; 4 ~ a; b; c and E~ p; q; r 2 F . We de®ne the rankings of the signed distance on F as De®nition 6. Let D follows: ~1 ; ~ E~ iff d D; ~ ~ ~ 0 D 01 < d E; ~ E~ iff d D; ~ ~ ~ ~ D 01 d E; 01 :
5
~ a; b; c and E~ p; q; r 2 F . We have the following binary operations from [6]: Property 2. Let D ~ E~ a p; b q; c r 2 F , 1. D ~ ka; kb; kc 2 F if k > 0, 2. k~1 D ~ 0; 0; 0 ~ 3. k~1 D 01 2 FP 1 if k 0. ~ a; b; c and E~ p; q; r 2 F . We derive two operations of the signed distance from Property 3. Let D Properties 1 and 2, as follows: ~ E; ~ ~ ~ ~ ~ ~ 1. d D 01 d D; 01 d E; 01 , ~ ~ ~ ~ 2. d k~1 D; 01 kd D; 01 if k P 0. Proof. ~ E~ a p; b q; c r 2 F , we obtain 1. Since D 1 ~ E; ~ ~ d D 01 2 b q a p c r 4 1 1 ~ ~01 d E; ~ ~01 : 2b a c 2q p r d D; 4 4 2. Similar to (1) .
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~ 2 F: ~ E; ~ Q Property 4. The order relations and satisfy the following three axioms in F. For any D; ~ - D. ~ (a) Re¯exive law: D ~ - E, ~ E~ - D ~)D ~ E. ~ (b) Antisymmetric law: D ~)D ~ ~ - E, ~ E~ - Q ~ Q. (c) Transitive law: D ~ 2 F ; either ~ E; ~ Q Property 5. The order relations and satisfy the law of trichotomy in F; i.e., for any D; ~ ~ ~ ~ ~ ~ D E; D E; or E D holds. ~ S~ 2 F . ~ E; ~ Q; Property 6. Let D; ~ ~ ~ ~ ~ ~ Q. 1. If E D; then E Q D ~ ~ ~ ~ ~ ~ E. ~ 2. If Q D and S E; then Q S~ D Properties 4±6 have the same characteristics as the rankings on R, i.e., for a; b 2 R; a < b i d a; 0 < d b; 0 and a b i d a; 0 d b; 0. Therefore, the ranking on F in De®nition 6 has the same feature as that on R. In conclusion, the signed distance ranking method in De®nitions 5 and 6 will be useful when we consider constructing the fuzzy knapsack models.
3. The knapsack problems with fuzzy numbers In this section, ®rst we consider the crisp knapsack problem. Next, we present an approach to construct fuzzy knapsack models. After defuzzifying the fuzzy models, we obtain Propositions 1 and 2, the knapsack and the 0/1 knapsack problems in the fuzzy sense (i.e. the defuzzi®ed fuzzy problems), which are used for solving imprecise data. The fuzzy MCKP model and the MCKP in the fuzzy sense (i.e. the defuzzi®ed fuzzy MCKP) will be stated in Section 4. 3.1. The crisp knapsack problem The knapsack problem is formulated by numbering the objects from 1 to n and introducing a vector of binary variables xi (i 1; . . . ; n) having the following meaning. Object i has a weight wi and the knapsack has a capacity M. If a fraction xi , 0 6 xi 6 1, of object i is placed into the knapsack then a pro®t, pi xi , is earned. The objective is to ®nd the combination of objects for the knapsack that maximizes the total pro®t from all of the objects chosen. Since the knapsack capacity is M, we require the total weight of all of the chosen objects to be at most M [5]. The knapsack problem is stated mathematically as follows: maximize
n X
p i xi
6
i1
subject to n X
wi xi 6 M;
7
i1
0 6 xi 6 1; 1 6 i 6 n:
8
Without loss of generality all weights and pro®ts are assumed to be positive numbers. The optimal solution is obtained by using the greedy strategy where the objects are considered in the order of non-increasing density pi =wi and objects are added to the knapsack if they ®t [5].
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3.2. The fuzzy knapsack models The essential problem we consider here is that the values of the coecient for each weight wi , i 1; . . . ; n, is imprecise because there is only vague knowledge about the objects or decimal truncation is used for the weights in the problem. Thus, the DM should determine an acceptable range of values for each wi , which is an interval wi Di1 ; wi Di2 , 0 6 Di1 < wi and 0 < Di2 . Then the DM chooses a value from the interval wi Di1 ; wi Di2 as an estimate of each weight. We say that the acceptable grade is 1 if the estimate is exactly wi ; otherwise, the acceptable grade will get smaller when the estimate approaches either wi Di1 or wi Di2 . It is clear that the acceptable grade for an estimate in an interval corresponds to the membership grade of a fuzzy number in the fuzzy sets. Thus this leads to the use of fuzzy numbers in the knapsack problem. We believe that the fuzzy tolerance interval of weight is the most valid information for the DM when he manages the knapsack problem in practical situations. The main advantage, compared to the crisp model, is that the DM is not forced into precise data, but into a range of values. This seems an easier task for the DM to determine the coecient of the weights in the knapsack problem. Besides, the membership functions, which monotonically increase or decrease in the interval wi Di1 ; wi Di2 , can also be handled quite easily. Nevertheless, a critical question may arise: how to guarantee an appropriate estimate value for each weight chosen in the decision-making process? Our approach is then stated as follows: ~ i be the fuzzy number denoted by Let w ~ i wi Di1 ; wi ; wi Di2 ; w 0 6 Di1 < wi ; 0 6 Di2 ; 1 6 i 6 n:
9
~ i is as shown below: The membership function of w 8 x wi Di1 > > ; wi Di1 6 x 6 wi ; > > < Di1 lw~i x wi Di2 x ; wi 6 x 6 wi Di2 ; > > > Di2 > : 0; otherwise: ~ i is 1. However, the more Fig. 2 shows that when an estimate x equals wi , the membership grade of x in w ~ i is obtained. The repaway from the position of wi an estimate x is, the less membership grade of x in w resentation of imprecise data as fuzzy numbers is useful when those data are used in fuzzy systems. Thus, ~ i by De®nition 5, we obtain an estimate of the weight in the fuzzy sense after defuzzifying the fuzzy number w ~ i ; ~01 wi 14 Di2 Di1 . The DM can then make use of from the interval wi Di1 ; wi Di2 , i.e., wi d w this equation to obtain a value as an estimate of the weight for solving the imprecise data problem. ~ i satis®es the following conditions. Furthermore, the fuzzy number w ~ i 2 FN . 1. If 0 < Di1 < wi and 0 < wi2 , then w ~ i 2 FL . 2. If 0 < Di1 < wi and Di2 0, then w
~i . Fig. 2. The fuzzy number w
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~ i 2 FR . 3. If Di1 0 and 0 < Di2 , then w ~ i 2 FP 1. 4. If Di1 Di2 0, then w ~ M; M; M 2 FP 1 and When fuzzifying the knapsack capacity, we obtain M ~ ~ d M; 01 M:
10
Let 1 ~i ; ~ wi d w 01 wi Di2 Di1 ; i 1; 2; . . . ; n: 11 4 ~ i measured from ~01 . Since From Property 1, we can see that (11) is the signed distance of w ~i ; ~ d w 01 34wi 14Di2 14 wi Di1 > 0, wi is a positive number measured from 0. Therefore, wi in (11) represents an estimate of the weight i in the fuzzy sense, which is equal to wi plus some fuzzy quantity 1 Di2 Di1 , wi 2 wi Di1 ; wi Di2 . When Di1 Di2 , we obtain wi wi . In particular, if Di1 Di2 0, 4 then the fuzzy model is equivalent to the crisp problem. From here, we can see that the fuzzy knapsack problem is an extension of the crisp knapsack problem. Thus we fuzzify (7) to obtain (12), where - is the ranking de®ned on F (see De®nition 6). n X ~ ~ i xi - M: 12 w i1
P ~ 1 ~x2 1 w ~ 2 ~xn 1 w ~ n . From (6), (8), (12), De®nition 6, ~ i xi equals ~x1 1 w Note that ni1 w and Properties 1 and 3, we derive the defuzzi®ed fuzzy knapsack problem in the following Proposition 1. Proposition 1. The knapsack problem in the fuzzy sense is formulated as follows: n X pi xi maximize
13
i1
subject to n X i1
wi xi 6 M;
0 6 xi 6 1;
14
i 1; . . . ; n;
15
where 1 wi wi Di2 4
Di1 ;
0 6 Di1 < wi ; 0 6 Di2 :
Proof. From (9), (12), and Properties 1 and 3, we have that ! X n n n n X X X 1 ~1 ~i ; ~ ~ i xi ; 0 d xi d w 01 xi wi Di2 Di1 wi xi : w 4 i1 i1 i1 i1 Pn ~ ~01 . Since d M; ~ ~01 M (see (10)), we obtain ~ i xi ; ~01 6 d M; From (12) and De®nition 6, we obtain d i1 w (14). Clearly, (13) and (15) are obtained directly from (6) and (8), respectively. Example 1. Consider an instance of knapsack problem given in Table 1. From (6)±(8), the optimal solution of the crisp problem obtained using the greedy strategy, of value 78.244, is xi 1; 1; 1; 0; 1; 25=41. Now consider the fuzzy knapsack model. The imprecise weight value
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Table 1 A knapsack problem with n 6 and M 80 i
1
2
3
4
5
6
wi pi
8 10
12 15
13 20
64 12
22 18
41 25
Table 2 Determine each coecient of weights in the fuzzy sense i
1
2
3
4
5
6
(Di1 ; Di2 ) wi
(0.2, 1) 8.2
(0.4, 1.6) 12.3
(0.6, 0.8) 13.05
(1, 0.6) 63.9
(0.7, 1.3) 22.15
(1, 0.6) 40.9
~i wi Di1 ; wi ; wi Di2 . Assume that the DM has determined the range of values for Di1 de®ned in (9) is w and Di2 , i 1; . . . ; 6, and has calculated wi wi 14 Di2 Di1 for each i, which are shown in Table 2. From (13)±(15) of Proposition 1, the optimal solution of the fuzzy model is xi 1; 1; 1; 0; 1; 24:3=40:9 with the optimal pro®t 77.8533. We compare the result obtained from Proposition 1 with that of the crisp case, as follows: 77:8533 78:244 100 78:244
0:499%:
Assume that the DM determines another set of values for Di1 and Di2 , i 1; . . . ; 6, and then another set of weight coecients in the fuzzy sense are obtained in Table 3. The optimal solution for this case is xi 1; 1; 1; 0; 1; 24:3=40:9 with a pro®t of 78.434. A comparison of this result with that for the crisp problem above is 78:434 78:244 100 0:243%: 78:244 Although the solution obtained from Proposition 1 may be slightly worse or better than that in the crisp case, the advantage of using the fuzzy model is that a range of weights is allowed in the problem. Remark 1. The interpretation of Fig. 2 using (11), wi wi 14 Di2 Di1 , i 1; . . . ; n, for comparing the dierence between Proposition 1 and the knapsack problem is as follows. When Di2 Di1 8i, Fig. 2 shows an isosceles triangle in which wi wi 8i. In this case, (13)±(15) are equivalent to (6)±(8). This means that the crisp knapsack problem is a special case of Proposition 1, the knapsack problem in the fuzzy sense. In particular, if Di2 Di1 0 8i, then Proposition 1 is equivalent to the crisp knapsack problem. Hence, Proposition 1 is an extension of the crisp knapsack problem. On the other hand, when Di2 > Di1 8i, the triangle in Fig. 2 is skewed to the right-hand side obtaining wi > wi 8i. From (7) and (14), we obtain n X i1
wi xi <
n X i1
wi xi 6 M:
Table 3 Another set of weights in the fuzzy sense i
1
2
3
4
5
6
(Di1 ; Di2 ) wi
(0.2, 0.6) 8.1
(0.8, 0.4) 11.9
(1, 0.4) 12.85
(0.3, 0.7) 64.1
(1.2, 0.4) 21.8
(1, 0.6) 40.9
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Let
( K1
) X n x1 ; . . . ; xn w x 6 M; 0 6 xi 6 1 8i i1 i i
denote the knapsack problem and let ( ) X n w x 6 M; 0 6 xi 6 1 8i K2 x1 ; . . . ; xn i1 i i denote Proposition 1. Since K2 K1 , we have n n X X max pi xi 6 max pi xi : x1 ;...;xn 2K2
x1 ;:::;xn 2K1
i1
i1
Similarly, if Di2 < Di1 , the triangle is skewed to the left-hand side obtaining wi < wi . Since K1 K2 , we have n n X X max pi xi 6 max pi xi : x1 ;...;xn 2K1
x1 ;...;xn 2K2
i1
i1
3.3. 0/1 knapsack problem If we add the requirement that xi 1 or xi 0, 1 6 i 6 n, i.e., an object is either included or not included into the knapsack. Then the problem is known as 0/1 knapsack problem. In other words, we wish to solve as follows: maximize
n X
p i xi
16
i1
subject to n X
wi xi 6 M;
17
i1
xi 0 or 1;
1 6 i 6 n:
18
Since the greedy strategy does not necessarily yield an optimal solution for the 0/1 knapsack problem, instead, we can use the dynamic programming approach for ®nding the optimal solution [5]. After fuzzifying (17), we obtain (12). From (16), (12) and (18), we derive Proposition 2, the defuzzi®ed fuzzy 0/1 knapsack problem, as follows. Proposition 2. The 0=1 knapsack problem in the fuzzy sense is formulated as follows: maximize
n X
pi xi
19
wi xi 6 M;
20
i1
subject to n X i1
xi 0 or 1;
1 6 i 6 n;
21
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where 1 wi wi Di2 4
Di1 ;
0 6 Di1 < wi ; 0 6 Di2 :
Example 2. We use the same instance as in Example 1. The 0/1 knapsack problem is maximize 10x1 15x2 20x3 12x4 18x5 25x6 subject to 8x1 12x2 13x3 64x4 22x5 41x6 6 80;
xi 0 or 1; 1 6 i 6 6:
The optimal solution for the 0/1 knapsack problem obtained using the dynamic programming technique is xi 1; 1; 1; 0; 0; 1 and the optimal pro®t is 70. Next, we consider the fuzzy 0/1 knapsack model. Case 1: From (19)±(21) of Proposition 2 and Table 2, the 0/1 knapsack problem in the fuzzy sense can be formulated as follows: maximize
10x1 15x2 20x3 12x4 18x5 25x6
subject to
8:2x1 12:3x2 13:05x3 63:9x4 22:15x5 40:9x6 6 80;
xi 0 or 1; 1 6 i 6 6:
The optimal solution obtained using the dynamic programming technique for this problem is still xi 1; 1; 1; 0; 0; 1 and the optimal pro®t is 70. Case 2: From Table 3, the 0/1 knapsack problem in the fuzzy sense is as follows: maximize
10x1 15x2 20x3 12x4 18x5 25x6
subject to
8:1x1 11:9x2 12:85x3 64:1x4 21:8x5 40:9x6 6 80;
xi 0 or 1; 1 6 i 6 6:
Again, the optimal solution is xi 1; 1; 1; 0; 0; 1 with the optimal pro®t of 70 that is obtained by using the dynamic programming technique. Example 3. Consider an instance of knapsack problem, which is given in Table 4. From (6)±(8), the optimal solution is xi 0; 1; 0; 1; 1; 1 and the optimal pro®t is 68 that can be obtained based on the greedy strategy. Consider the fuzzy model. Assume that the DM takes the values of Di1 and Di2 , i 1; . . . ; 6, and calculates the coecients of the weights in the fuzzy sense, as shown in Table 5. Table 4 An instance of knapsack problem with n 6 and M 60 i
1
2
3
4
5
6
wi pi
10 9
15 18
60 22
13 10
11 15
20 25
Table 5 Determine the coecients of weights in the fuzzy sense i
1
2
3
4
5
6
(Di1 ; Di2 ) wi
(1, 0.2) 9.8
(0.2, 1) 15.2
(1, 0.4) 59.85
(0.2, 1.8) 13.4
(0.3, 1.5) 11.3
(0.1, 0.9) 20.2
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From (19)±(21) of Proposition 2, the 0/1 knapsack problem in the fuzzy sense is formulated as follows: maximize
9x1 18x2 22x3 10x4 15x5 25x6
subject to
9:8x1 15:2x2 59:85x3 13:4x4 11:3x5 20:2x6 6 60;
xi 0 or 1; 1 6 i 6 6:
The optimal solution is xi 1; 1; 0; 0; 1; 1 and the optimal pro®t is 67 obtained by using the greedy strategy. Note that this solution is dierent from the crisp problem above. After comparing the result obtained from Proposition 2 with that of the crisp problem, we have 67
68 68
100
1:47%:
Remark 2. Let ( K3
) X n x1 ; . . . ; xn w x 6 M; xi 0 or 1 8i i1 i i
denote the 0/1 knapsack problem and let ( ) X n K4 x1 ; . . . ; xn w x 6 M; xi 0 or 1 8i i1 i i denote Proposition 2. If Di1 < Di2 8i, the triangle in Fig. 2 is skewed to the right-hand side obtaining wi < wi 8i. Thus we have n X
wi xi <
i1
n X i1
wi xi 6 M:
Since K4 K3 , we obtain max
x1 ;...;xn 2K4
n X
p i xi 6
i1
max
x1 ;...;xn 2K3
n X
pi xi :
i1
Similarly, ifPDi2 < Di1 8i, the triangle is skewed to the left-hand side obtaining wi < wi 8i. Then Pn n i1 wi xi < i1 wi xi 6 M. Since K3 K4 , we obtain max
x1 ;...;xn 2K3
n X i1
p i xi 6
max
x1 ;...;xn 2K4
n X
pi xi :
i1
4. The MCKP with fuzzy numbers The MCKP is a generalization of the 0/1 knapsack problem and a special case of general 0/1 integer programming. The problem is stated as follows: We are given n objects and a knapsack. The object i weighs wij when it is considered using the jth measure (1 6 j 6 m) and the knapsack also has m capacity because of the m dierent measures for objects. In other words, m is the number for the knapsack's constraint, each with a capacity Mj , 1 6 j 6 m, which is called the capacity of the knapsack using the jth measure. If the object i is placed into the knapsack, satisfying all of the m dierent measures then a pro®t pi is earned,
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1 6 i 6 n. Since the capacity of the knapsack is limited, the total weight of all of the chosen objects can be at most Mj using each measure j, 1 6 j 6 m. We are interested in ®lling the knapsack with those objects that yield the maximum pro®t. Formally, the MCKP can be stated mathematically as follows: maximize
n X
pi xi
22
i1
subject to n X
wij xi 6 Mj
for j 1; . . . ; m;
23
i1
xi 0 or 1 for i 1; . . . ; n:
24
Without a loss of generality, all pi ; wij ; and Mj are assumed to be nonnegative. While considering an imprecise value for each coecient of weight, we let an acceptable range for each estimate of weights be denoted by wij hij1 ; wij hij2 , where 0 6 hij1 < wij and 0 < hij2 . A corresponding triangular fuzzy number for each weight wij is given by ~ ij wij w
hij1 ; wij ; wij hij2 :
25
~ ij is 1. Otherwise, the more Fig. 3 shows that when an estimate x equals wij , the membership grade of x in w ~ ij is obtained. Besides, the away from the position of wij an estimate x is, the less membership grade of x in w ~ ij satis®es the following conditions. fuzzy number w ~ ij 2 FN . 1. If 0 < hij1 < wij and 0 < hij2 , then w ~ ij 2 FL . 2. If 0 < hij1 < wij and hij2 0, then w ~ ij 2 FR . 3. If hij1 0 and 0 < hij2 , then w ~ij 2 FP 1. 4. If hij1 hij2 0, then w After fuzzifying the knapsack capacity Mj , we have that ~ j Mj ; Mj ; Mj 2 FP 1 and M
~j; ~ d M 01 Mj ; 1 6 j 6 m:
26
Let 1 ~ ij ; ~ wij d w 01 wij hij2 4
hij1 ;
1 6 i 6 n; 1 6 j 6 m:
27
~ ij measured from ~01 is From Property 1, we can see that the signed distance of w 1 1 1 ~ ~ ~ ij ; 01 wij 4 hij2 hij1 . Since d w ~ ij ; 01 4 3wij hij2 4 wij hij1 > 0, wij is a positive distance d w measured from 0. Hence, wij represents an estimate of the jth measure of object i in the fuzzy sense, which is
~ ij . Fig. 3. The fuzzy number w
170
F.-T. Lin, J.-S. Yao / European Journal of Operational Research 135 (2001) 158±176
wij plus some fuzzy quantity 14 hij2 hij1 . It is clear that the fuzzy MCKP is an extension of the crisp MCKP. Then (23) is fuzzi®ed to obtain (28), where - is the ranking de®ned on F, as follows: n X
~j ~ ij xi - M w
for j 1; . . . ; m:
28
i1
From (22), (24), (28), De®nition 6, and Properties 1 and 3, we can derive Proposition 3, the defuzzi®ed fuzzy MCKP, as follows: Proposition 3. The MCKP in the fuzzy sense is formulated as follows: maximize
n X
pi xi
29
i1
subject to n X i1
wij xi 6 Mj for j 1; . . . ; m;
30
xi 0 or 1 for i 1; . . . ; n;
31
where 1 wij wij hij2 4
hij1 ;
0 6 hij1 < wij ;
0 < hij2 :
Proof. From (25), (28), and Properties 1 and 3, we have ! n n n X X X 1 ~ ~ ~ ij ; 01 ~ ij xi ; 01 d xi d w xi wij hij2 w 4 i1 i1 i1
hij1
n X i1
wij xi :
Pn ~j; ~ ~ j ; ~01 Mj , we have (30). Clearly, (29) and (31) ~ ij xi ; ~ 01 6 d M 01 . Since d M From (28), we obtain d i1 w are obtained directly from (22) and (24). Example 4. Consider the instance of the MCKP from [12], which is shown in Table 6. The problem is n 10 and m 10, n is the number of objects and m is the number of constraints of capacity Mj , which can be formulated as follows: maximize
10 X
p i xi
i1
subject to
10 X
wij xi 6 Mj
for j 1; . . . ; 10;
i1
xi 1 or 0: The optimal solution for this problem instance is xi 0; 1; 0; 1; 1; 0; 0; 1; 0; 1, with the optimal value 8706.1, can be obtained using either simulated annealing or genetic algorithms [7]. Now, consider the fuzzy model. Let wij hij1 ; wij hij2 be the acceptable range of weight for wij . The corresponding fuzzy number ~ ij wij hij1 ; wij ; wij hij2 . Assume that the values of hij1 and hij2 , 1 6 i; j 6 10, are for that range is w already de®ned in Table 7.
F.-T. Lin, J.-S. Yao / European Journal of Operational Research 135 (2001) 158±176
171
Table 6 The instance of MCKP with n 10 and m 10 wij
j1
2
3
4
5
6
7
8
9
10
pi
i1 2 3 4 5 6 7 8 9 10
20 5 100 200 2 4 60 150 80 40
20 7 130 280 2 8 110 210 100 40
60 3 50 100 4 2 20 40 6 12
60 8 70 200 4 6 40 70 16 20
60 13 70 250 4 10 60 90 20 24
60 13 70 280 4 10 70 105 22 28
5 2 20 100 2 5 10 60 0 0
45 14 80 180 6 10 40 100 20 0
55 14 80 200 6 10 50 140 30 40
65 14 80 220 6 10 50 180 30 50
600.1 310.5 1800 3850 18.6 198.7 882 4200 402.5 327
Mj
450
540
200
360
440
480
200
360
440
480
Table 7 The values of each hij1 and hij2 (hij1 ; hij2 )
j1
2
i1 2 3 4 5 6 7 8 9 10
(1, (2, (2, (1, (1, (2, (3, (3, (2, (3,
(1, (2, (1, (1, (2, (2, (1, (2, (3, (1,
2) 4) 3) 2) 3) 4) 1) 2) 3) 1)
3 1) 3) 2) 4) 3) 4) 3) 1) 4) 2)
(1, (1, (1, (1, (2, (1, (1, (2, (3, (1,
4 2) 2) 2) 2) 4) 3) 2) 3) 2) 4)
(1, (2, (1, (1, (1, (2, (3, (2, (3, (2,
5 2) 4) 3) 4) 2) 3) 1) 3) 4) 4)
(1, (1, (1, (1, (1, (1, (2, (4, (2, (2,
6 1) 3) 2) 3) 2) 3) 4) 2) 3) 1)
(1, (2, (1, (2, (1, (2, (3, (2, (1, (2,
7 2) 3) 2) 3) 2) 4) 2) 4) 3) 3)
(0, (0, (0, (0, (1, (0, (2, (1, (2, (1,
8 1) 1) 1) 1) 2) 1) 3) 4) 3) 3)
(1, (0, (1, (0, (1, (0, (1, (3, (1, (2,
9 2) 1) 1) 1) 3) 1) 2) 1) 4) 3)
(1, (1, (1, (0, (1, (1, (2, (4, (1, (2,
10 2) 1) 2) 1) 2) 2) 3) 2) 3) 4)
(1, (1, (1, (2, (3, (1, (3, (2, (1, (2,
2) 1) 2) 3) 2) 3) 2) 4) 3) 3)
Then the DM can obtain each weight in the fuzzy sense, using wij wij 14 hij2 hij1 , which are shown in Table 8. The optimal solution of this problem is still xi 0; 1; 0; 1; 1; 0; 0; 1; 0; 1 with the optimal pro®t 8706.1. If we change the capacity Mj in Tables 6 and 8 to M1 400, M2 539, M3 160, M4 340, M5 400, M6 450, and M8 320 (M7 ; M9 , and M10 are unchanged), then the optimal solution is xi 0; 1; 0; 1; 0; 0; 0; 1; 0; 1 with the optimal pro®t 8687.5. Table 8 Determine the weights in the fuzzy sense wij
j1
2
3
4
5
6
7
8
9
10
i1 2 3 4 5 6 7 8 9 10
20.25 5.5 100.25 200.25 2.5 4.5 59.5 149.75 80.25 39.5
20 7.25 130.25 280.75 2.25 8.5 110.5 209.75 100.25 40.25
60.25 3.25 50.25 100.25 4.5 2.25 20.25 40.25 5.75 12.75
60.25 8.5 70.5 200.75 4.25 6.25 39.5 70.25 16.25 20.5
60 13.5 70.25 250.5 4.25 10.5 60.5 89.5 20.25 23.75
60.25 13.25 70.25 280.25 4.25 10.5 69.75 104.5 22.5 28.25
5.25 2.25 20.25 100.25 2.25 5.25 10.25 60.75 0.25 0.75
45.25 14.25 80 180.25 6.5 10.25 40.25 99.5 20.75 0.25
55.25 14 80.25 200.25 6.25 10.25 50.25 139.5 30.5 40.5
65.25 14 80.25 220.25 5.75 10.5 49.75 180.5 30.5 30.25
Mj
450
540
200
360
440
480
200
360
440
480
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F.-T. Lin, J.-S. Yao / European Journal of Operational Research 135 (2001) 158±176
Remark 3. Similar to Remark 2. From (27), wij wij 14 hji2 hji1 , the interpretation of Fig. 3 is as follows. When hij2 hij1 8i; j, Fig. 3 shows an isosceles triangle obtaining wij wij 8i; j. This case shows that (29)±(31) are equivalent to (22)±(24). The crisp MCKP is a special case of Proposition 3, the MCKP in the fuzzy sense. However, if hij2 hij1 0 8i; j, then Proposition 3 is equivalent to the crisp knapsack problem. From above, Proposition 3 is an extension of the crisp knapsack problem. On the other hand, if hij2 > hij1 8i; j, the triangle in Fig. 3 is skewed to the right-hand side obtaining wij > wij 8i; j. From (23) and (30), we can see that n X
wij xi <
i1
n X i1
wij xj 6 Mj ;
j 1; . . . ; m:
Let ( L1
X n x1 ; x2 ; . . . ; xn w x 6 Mj ; i1 ij i
) j 1; . . . ; m; xi 0 or 1 8i
denote the MCKP. Let ( L2
X n x1 ; x2 ; . . . ; xn w x 6 Mj ; i1 ij i
) j 1; . . . ; m; xi 0 or 1 8i
denote Proposition 3. Since L2 L1 , max
x1 ;...;xn 2L2
n X
pi xi 6
i1
max
x1 ;...;xn 2L1
n X
pi xi :
i1
When hij2 6 hij1 8i; j, the triangle is skewed to the left-hand side obtaining wij < wij 8i; j, and n X i1
wij xi <
n X
wij xi 6 Mj ;
1 6 j 6 m:
i1
Since L1 L2 , we have max
x1 ;...;xn 2L1
n X
pi xi 6
i1
max
x1 ;...;xn 2L2
n X
pi xi :
i1
5. Concluding remarks In this paper, we have investigated the knapsack problem with imprecise object weights. We have proposed an approach for constructing fuzzy knapsack models. By using the signed distance ranking method to defuzzify the fuzzy models, we obtain Propositions 1, 2, and 3 used for solving the problem of imprecise weight coecients. In our study, our fuzzy knapsack models are an extension of the crisp knapsack problems, and the crisp knapsack problem is a special case of our fuzzy models. In conclusion, the generalization of our fuzzy knapsack model is as follows.
F.-T. Lin, J.-S. Yao / European Journal of Operational Research 135 (2001) 158±176
173
5.1. A generalized fuzzy knapsack model Let A~ be a fuzzy set on R. For each a-cut, 0 6 a 6 1, we have that A a AL a; AR a, where AL a and AR a are the left-end point and the right-end point for each a-cut. Assume that the a-cut A a holds and AL a and AR a are continuous functions in 0 6 a 6 1. Let G denote the family of those fuzzy sets, and let FP 1 G. Similar to De®nition 5, we have De®nition 7, which is the signed distance of fuzzy set A~ in G. De®nition 7. Let A~ 2 G. The signed distance of A~ measured from ~01 is de®ned by Z 1 1 ~~ dG A; 01 AL a AR a da: 2 0 Note that De®nition 5 is a special case of De®nition 7 and F G. ~ B~ G, we de®ne the signed distance rankings on G as follows: De®nition 8. For A; ~~ ~~ A~ B~ iff dG A; 01 < dG B; 01 ; ~ ~ ~ ~ ~ A B iff dG A; 01 dG B; ~ 01 : ~ B~ 2 G, we obtain the following equations Similarly, De®nition 6 is a special case of De®nition 8. For A; from De®nition 7. Z 1 1 ~ ~ dG ~ a1 A; 01 aAL a aAR ada 2 0 ~1 ; ~0 adG A; 32 ~~ dG A~ B; 01
1 2
Z
1 0
AL a BL a AR a BR ada
~~ ~~ dG A; 01 dG B; 01 :
33
~ i on R, the interpretation of the fuzzy number When the weight wi in (7) transforms into the fuzzy number w ~ i is 1 when an estimate equals wi . Otherwise, in the left-hand is then as follows. The membership grade in w side of wi is a monotonically increasing continuous function fL x and in the right-hand side of wi is a monotonically decreasing continuous function fR x, where x is an estimate for wi . The functions fL x and ~ M; M; M 2 FP 1 F G. When ~ i 2 G. Let M fR x are determined by the DM and we can see that w fuzzifying (7) using De®nition 8, we can obtain n X
~ ~ i xi - M: w
i1
From De®nitions 7 and 8 and by using (32)±(34) becomes n X
~i ; ~ dG w 01 xi 6 M:
i1
Finally, we derive a generalized knapsack problem in the fuzzy sense from (6)±(8) as follows.
34
174
F.-T. Lin, J.-S. Yao / European Journal of Operational Research 135 (2001) 158±176
Proposition 4. The generalized knapsack problem in the fuzzy sense is formulated as follows: n X maximize pi xi
35
i1
subject to
n X
~i ; ~ dG w 01 xi 6 M;
36
i1
0 6 xi 6 1
for i 1; . . . ; n:
37
~i ; ~ ~i ; ~ ~i 2 F , Proposition 4 becomes ProposiObviously, when dG w 01 d w 01 wi 14 Di2 Di1 for w tion 1. Similarly, Propositions 2 and 3 can also be generalized to the fuzzy set on G with the similar way. 5.2. Comparing the ranking method used in [11] with that of our work (1) The crisp MCKP that proposed by Okada and Gen [11] is as follows: minimize
z x
p X nt X t1
subject to
ctj xtj
38
j1
p X nt X atj xtj 6 b; t1 j1 nt X
xtj 1; t 1; 2; . . . ; p;
39 xtj 2 f0; 1g; j 1; 2; . . . ; nt ; t 1; 2; . . . ; p:
40
j1
The fuzzy MCKP is de®ned by p X nt X minimize ~z x c~tj xtj t1
subject to
p X nt X
41
j1
~ a~tj xtj K b;
t1 j1 nt X
xtj 1; t 1; 2; . . . ; p;
42 xtj 2 f0; 1g; j 1; 2; . . . ; nt ; t 1; 2; . . . ; p;
43
j1
where
R c~tj cLtj ; cM tj ; ctj ;
R a~tj aLtj ; aM tj ; atj ;
j 1; 2; . . . ; nt ; t 1; 2; . . . ; p;
and b~ bL ; bM ; bR are fuzzy numbers. Finally, the fuzzy MCKP is transformed into the following multiobjective programming problem as follows: minimize minimize
zL x zM x
p X nt X
cLtj xtj ;
44
cM tj xtj ;
45
t1 j1 p X nt X t1
j1
F.-T. Lin, J.-S. Yao / European Journal of Operational Research 135 (2001) 158±176
minimize
p X nt X
zR x
t1
subject to p X nt X
P
! a~tj xtj 6 b~ P a;
46
a 2 0; 1;
47
xtj 2 f0; 1g; j 1; 2; . . . ; nt ; t 1; 2; . . . ; p;
48
j1
t1 nt X
j1
cRtj xtj ;
175
xtj 1; t 1; 2; . . . ; p;
j1
where ~ P ~ a K b
R aR aL
minfla~ x; lb~ xg dx ; R aR la~ x dx aL
a~ and b~ are fuzzy numbers. (2) In our approach, when applying the proposed signed-distance method for ranking the fuzzy numbers in the fuzzy MCKP, (41)±(43), we obtain the following MCKP in the fuzzy sense, (49)±(51). Note that the ranking K in (42) is de®ned in De®nition 6. minimize
z0 x
p X nt X t1
j1
c0tj xtj
49
subject to p X nt X t1 nt X
j1
a0tj xtj 6 b0 ;
50
xtj 1; t 1; 2; . . . ; p;
xtj 2 f0; 1g; j 1; 2; . . . ; nt ; t 1; 2; . . . ; p;
51
j1
where L R c0tj d ~ ctj ; ~ 01 14 2cM tj ctj ctj ; L R a0tj d ~ atj ; ~ 01 14 2aM tj atj atj ;
~~ b0 d b; 01 14 2bM bL bR : (3) The major dierences between [11] and this paper are stated as follows: 1. When comparing the crisp MCKP of [11], (38)±(40), with the crisp MCKP of this paper, (22)±(24), we can see that the underlying problem is dierent. ~ is dierent from the signed distance rank2. The ranking K in (42) of the fuzzy MCKP of [11], P ~ a K b, ing method, De®nition 6, used in this paper. Pp Pnt 3. Consider the fuzzy objective function of (41), ~z x t1 j1 c~tj xij , which is de®ned in [11]. After applying the operation of fuzzy numbers, we obtain ! p X p X p X nt nt nt X X X L M R L M R ~z x z x; z x; z x ctj xtj ; ctj xtj ; ctj xtj : t1
j1
t1
j1
t1
j1
176
F.-T. Lin, J.-S. Yao / European Journal of Operational Research 135 (2001) 158±176
Obviously, the objective functions of (44)±(46), which are de®ned in [11], are not obtained from the defuzzi®cation of the fuzzy numbers. On the contrary, if we use the signed distance of Property 1 to defuzzify the fuzzy objective function of (41), we will obtain " # p X p X p X p X nt nt nt nt X X X X 1 2 z0 x d ~z x; ~ 01 cM cLtj xtj cRtj xtj c0tj xtj : tj xtj 4 t1 j1 t1 j1 t1 j1 t1 j1 Thus the objective function of the MCKP de®ned in [11] and the objective function of the MCKP de®ned in this paper is dierent. R L M R L M R ~tj ; a~tj ; b~ will become level 1 4. If we let cLtj cM tj ctj ctj ; atj atj aij atj , and b b b b, then c fuzzy points, i.e., c~tj ~ ctj 1 ; a~tj ~ atj 1 and b~ b~1 . Hence, we have that ctj d ~ ctj 1 ; ~ 01 ctj ; atj atj , and b b. We can see that (49)±(51) are equivalent to (38)±(40). Obviously, in this paper, the crisp MCKP is a special case of the MCKP in the fuzzy sense. However, in [11], the crisp MCKP in (38)±(40) is not a special case of the multi-objective programming problem in (44)±(48). Acknowledgements The authors are grateful to anonymous referees whose valuable comments helped to improve the content of this paper. References [1] J.M. Cadenas, J.L. Verdegay, Using fuzzy numbers in linear programming, IEEE Transactions on Systems, Man and Cybernetics ± Part B: Cybernetics 27 (6) (1997) 1016±1022. [2] M. Delgado, J.L. Verdegay, M.A. Vila, A general model for fuzzy linear programming, Fuzzy Sets and Systems 29 (1997) 21±29. [3] A. Freville, G. Plateau, Heuristics and reduction methods for multiple constraints 0±1 linear programming problems, European Journal of Operational Research 24 (1986) 206±215. [4] J. Gavish, H. Pirkul, Ecient algorithms for solving multi-constraint zero±one knapsack problems to optimality, Mathematical Programming 31 (1985) 78±105. [5] E. Horowitz, S. Sahni, S. Rajasekaran, Computer Algorithms, Freeman, New York, 1998. [6] A. Kaufmann, M.M. Gupta, Introduction to Fuzzy Arithmetic Theory and Applications, van Nostrand Reinhold, New York, 1991. [7] F.T. Lin, C.Y. Kao, C.C. Hsu, Applying the genetic approach to simulated annealing in solving some NP-hard problems, IEEE Transactions on Systems, Man and Cybernetics 23 (6) (1993) 1752±1767. [8] M. Magazine, O. Oguz, A heuristic algorithm for the multi-dimensional zero±one knapsack problem, European Journal of Operational Research 16 (1984) 319±326. [9] S. Martello, P. Toth, Knapsack Problems: Algorithms and Computer Implementations, Wiley, Chichester, 1990. [10] S. Martello, P. Toth, Heuristic algorithms for the multiple knapsack problem, Computing 27 (1981) 93±112. [11] S. Okada, M. Gen, Fuzzy multiple choice knapsack problem, Fuzzy Sets and Systems 67 (1994) 71±80. [12] C.C. Peterson, Computational experience with variants of the Balas algorithm applied to the selection of R&D projects, Management Science 13 (1967) 736±750. [13] D. Pisinger, An expanding-core algorithm for the exact 0±1 knapsack problem, European Journal of Operational Research 87 (1995) 175±187. [14] S. Senyu, Y. Toyoda, An approach to linear programming with 0±1 variables, Management Science 15 (1968) 196±207. [15] W. Shih, A branch and bound method for the multi-constraint zero±one knapsack problem, Journal of the Operational Research Society 30 (1979) 369±378. [16] J.S. Yao, K.M. Wu, Ranking fuzzy numbers based on decomposition principle and signed distance, Fuzzy Sets and Systems 116 (2000) 275±288. [17] H.-J. Zimmermann, Fuzzy Set Theory and its Applications, second ed., Kluwer Academic Publishers, Boston, MA, 1991.