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A note on “Solving linear programming problems under fuzziness and randomness environment using attainment values”
Information Sciences 179 (2009) 4083–4088
Contents lists available at ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
A note on ‘‘Solving linear programming problems under fuzziness and randomness environment using attainment values” Shuo-Yan Chou a,*, Jennifer Shu-Jen Lin b, Peterson Julian c a
Department of Industrial Management, National Taiwan University of Science and Technology, Taiwan Institute of Technological and Vocational Education, National Taipei University of Technology, Taiwan c Department of Traffic Science, Central Police University, Taoyuan County, Taiwan b
a r t i c l e
i n f o
Article history: Received 1 February 2009 Received in revised form 6 July 2009 Accepted 8 August 2009
Keywords: Fuzzy linear programming Fuzzy stochastic linear programming Fuzzy number Attainment values Attainment index
a b s t r a c t This paper is an amendment to Hop’s paper [N.V. Hop, Solving linear programming problems under fuzziness and randomness environment using attainment values, Information Sciences 177 (2007) 2971–2984], in solving linear programming problems under fuzziness and randomness environments. Hop introduced a new characterization of relationship, attainment values, to enable the conversion of fuzzy (stochastic) linear programming models into corresponding deterministic linear programming models. The purpose of this paper is to provide a correction and an improvement of Hop’s analytical work through rationalization and simplification. More importantly, it is shown that Hop’s analysis does not support his demonstration or the solution-finding mechanism; the attainment values approach as he had proposed does not result in superior performance as compared to other existing approaches because it neglects some relevant and inevitable theoretical essentials. Two numerical examples from Hop’s paper are also employed to show that his approach, in the conversion of fuzzy (stochastic) linear programming problems to corresponding problems, is questionable and can neither find the maximum nor the minimum in the examples. The models of the examples are subsequently amended in order to derive the correct optimal solutions. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction In view of its properties that is applicable to real-life business applications, fuzzy linear programming has attracted a great deal of attention from researchers. Among these investigations [2,21,3,10,17] the most widely-adopted procedure to find the solution was through the conversion of fuzzy linear programming models into corresponding deterministic linear programming models [8]. For instance, Leung [11] tried to classify fuzzy mathematical programming models into four categories: a precise objective with fuzzy constraints, a fuzzy objective with precise constraints, a fuzzy objective with fuzzy constraints, and robust programming. In trying to find a solution to the problem, the procedure in each of the categories were also developed which includes the signed distance method [3], the area compensation method [4], the expected mid-point approach [10] and the grade of possibility and necessity measures [1]. Zimmermann [22] is viewed as the pioneer in solving linear programming problems with fuzzy resources and fuzzy objectives. He developed a max–min tolerance method with the criteria of the highest membership degree to convert the initial fuzzy linear programming models into corresponding crisp ones. A unique solution could then be found by using the Simplex method. A number of researchers have developed
* Corresponding author. Tel.: +886 227376327. E-mail address: [email protected] (S.-Y. Chou). 0020-0255/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2009.08.013
4084
S.-Y. Chou et al. / Information Sciences 179 (2009) 4083–4088
various approaches with different degrees of efficiency and effectiveness, but most were built upon Zimmermann’s max–min approach. As for extending the research of fuzzy optimization, a new direction in the investigation focuses on a fuzzy and stochastic environment. Solving fuzzy stochastic linear programming problems, therefore, becomes important. In essence, fuzzy numbers/variables and stochastic variables are considered to be a more suitable characterization for real-world problems where uncertain and imprecise information is inherent. However, the inclusion of those components creates a challenge for finding efficient methods to deal with such conditions. An effective way to handle the fuzzy stochastic optimization problems is to convert the problems by de-fuzzifying the fuzzy numbers/variables, de-randomizing the stochastic variables and to solve the resulting deterministic problems instead. Two main approaches were established to cope with the fuzzy stochastic problems. One is to perform the conversion of the de-fuzzification and de-randomization in a sequential manner [16,15,9]. The other is to perform both processes in a simultaneous manner [6,12,14]. As for the sequential approach, a major disadvantage is that it generates an excessive amount of new variables and constraints to the model. On the other hand, a major disadvantage of the simultaneous approach is having a cumbersome workload for calculating the expected value for removing fuzziness and randomness at the same time. In the pursuit of improving the performance of existing approaches, Hop has established a new approach that enables a reduction in the number of additional constraints and an achievement to a certain degree of computational efficiency. The primary feature of Hop’s proposed approach is that instead of using the absolute relationship, as applied in the signed distance method, he employed the relative relationship between fuzzy numbers and fuzzy stochastic variables. The relative relationship is obtained through the calculation of so-called ‘‘attainment values” or degrees such as lower-side attainment index, both-side attainment index and average index as mentioned in his study. The relative relationship and thus the attainment values, play the key role in the conversion of fuzzy and fuzzy stochastic linear programming problems into more simple, conventional crisp problems, while reducing the number of constraints and computational complexity. Hop emphasized that his converted deterministic LP with few additional constraints can be easily solved by using standard optimization packages such as LINGO. Two examples were provided to illustrate the procedure of his proposed approach. However, we can verify that the method of attainment values that Hop [7] proposed is flawed since it can neither find the maximum nor the minimum values of the desired objectives owing to negligence of some relevant and unavoidable theoretical essentials. To the best of our knowledge, at least four papers [5,18–20], thus far have referred to Hop [7] in their references. Of them, none have noticed the flawed results established and claimed in Hop’s work. Here we have proceeded with a thorough investigation of Hop’s paper and highlighted mistakes in his conversion and his solution-finding procedures in an effort to alert future adoption of this particular solution approach in fuzzy/fuzzy stochastic programming and related areas. The remainder of this paper is organized as follows. Section 2 reviews the mathematical formulation of Hop’s research. Section 3 provides our revision necessary in order to resolve the problems in Hop’s approach. Section 4 reconsiders the two numerical examples of Hop’s investigation and compares these results with those from our approach. Finally, conclusions are drawn and future work outlined.
2. Review of Hop’s results in mathematical formulation Hop [7] tried to provide a new approach to solve linear programming problems under fuzzy and random conditions, and that is without the complicated computation of expected values as appeared in Liu and Liu [13]. To achieve this objective, the e , with P e , the lower-side attainment index of e and Q e6Q attainment indices are defined and utilized. For two fuzzy numbers P e is defined by Hop [7] as e to Q P
eÞ ¼ e Q Dð P;
Z
1
n o e ðrÞ P ag da: e P ag inffr : Q max 0; supfs : PðsÞ
ð1Þ
0
e ¼ ðu; a; bÞ with the membership function For triangular fuzzy number P more compact expression.
(
leP ðxÞ ¼
; if x 6 u; max 0; 1 ux a ; if x > u: max 0; 1 xu b
leP ðxÞ, we rewrite the membership function in a
ð2Þ
e ¼ ðv ; c; dÞ with u 6 v , the average lower-side e ¼ ðu; a; bÞ and Q Proposition 1 of Hop [7]. For two triangular fuzzy numbers, P e Þ ¼ uv þbþc. e is defined as Dð P; e Q e to Q attainment index of P 2 Proposition 1 as proved by Hop is as follows. e is ½Q l ðaÞ; Q u ðaÞ ¼ ½v cð1 aÞ; e is defined as ½Pl ðaÞ; P u ðaÞ ¼ ½u að1 aÞ; u þ bð1 aÞ, and of Q The a-cut of P v þ dð1 aÞ. Hop [7] had the lower-side attainment index of fuzzy number Pe to Qe at a-level:
e Þ ¼ maxf0; Pu ðaÞ Q l ðaÞg ¼ maxf0; u v þ ðb þ cÞð1 aÞg: e Q Dð P; a
ð3Þ
S.-Y. Chou et al. / Information Sciences 179 (2009) 4083–4088
4085
He derived that
eÞ ¼ e Q Dð P;
Z
1
maxf0; u v þ ðb þ cÞð1 aÞgda ¼
Z
u k ¼ð1vbþc Þ
½u v þ ðb þ cÞð1 aÞda;
ð4Þ
0
0
e is defined e to Q where k denotes the maximum value of a with P u ðaÞ P Q l ðaÞ. The average lower-side attainment index of P as
e Þ ¼ 1 Dð P; e Þ: e Q e Q Dð P; k
ð5Þ
Detailed computation for expression (5) provided by Hop [7] is as follows:
" !# 1 e e 1 k2 k e e ¼ ðu v þ b þ cÞ ðb þ cÞ Dð P; Q Þ ¼ Dð P; Q Þ ¼ ðu v þ b þ cÞk ðb þ cÞ k k 2 2 ðb þ cÞ v u bþc v u uv þbþc 1 ¼v uþbþc þ ¼ : ¼ ðu v þ b þ cÞ 2 bþc 2 2 2
ð6Þ
3. Our revisions In this section, we show that the average lower side attainment index as defined in the previous section can, in fact, be computed much more easily. In addition, Hop assumed that the intersection of two membership functions always exists, as e. e and the left side of Q shown in Fig. 1, Case I, where k is the y-coordinate of the intersection between the right side of P However, such an intersection does not always exist. Under the condition where v P u, two cases can be established: (I) u þ b P v c and (II) u þ b < v c. For Case I, the intersection does exist with the coordinate at
bv þ cu v u ;1 ¼k : bþc bþc
ð7Þ
e Þ denotes the area of the shaded triangle shown in Fig. 1. In Eq. (5), the average lower-side attainment index e Q In Eq. (4), Dð P; e Þ, is expressed as the area divided by the height. Hence, it follows that e , Dð P; e Q e to Q of P
eÞ ¼ e Q Dð P;
1 2
ðthe length of the baseÞ:
ð8Þ
This observation leads to a much simpler way than Hop’s proof to derive
eÞ ¼ e Q Dð P;
1 ½ðu þ bÞ ðv cÞ; 2
ð9Þ
under the condition of u þ b P v c for Case I. For Case II, the intersection does not exist. The inequality
Pu ðaÞ < Q l ðaÞ for 0 6 a 6 1
ð10Þ
e Þ ¼ 0 for 0 6 a 6 1 and thus Dð P; e Þ ¼ 0. As the intersection k does not exist, it follows that Dð P; e Þ ¼ 0. e Q e Q e Q implies that Dð P; a Proposition 1 of Hop [7] can therefore be revised as follows. e ¼ ðv ; c; dÞ with u 6 v , the average e ¼ ðu; a; bÞ and Q Revised Proposition 1 of Hop [7]. For two triangular fuzzy numbers, P e Þ ¼ max 0; uv þbþc . e is defined as Dð P; e Q e to Q lower-side attainment index of P 2 Similar problems also appear in Proposition 3 of Hop [7] for the average lower-side attainment index of the fuzzy random variables.
e as P e ¼ ðv ; c; dÞ with u 6 v . e to Q e ¼ ðu; a; bÞ and Q Fig. 1. The average lower-side attainment index of P
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4. Review of Hop’s results in the numerical examples Hop provided two examples in his study to show the superiority of the attainment values approach. In Example 1, in a manufacturing context, he formulated the following fuzzy linear programming problem:
Maximize
7x1 þ 9x2 ~ 2 6 9; ~ ~ Subject to 2x1 þ 3x ~ 2 6 18; ~ ~ 1 þ 4x 6x
ð11Þ
x1 ; x2 P 0; where fuzzy coefficients are represented by fuzzy numbers in the form of ~t ¼ ðt; 1; 1Þ. Hop [7] converted this fuzzy linear programming problem into a deterministic linear programming problem before solving for its optimal solution:
Maximize
7x1 þ 9x2 k1 k2 1 Subject to ½ð9 2x1 3x2 Þ þ ð1 þ x1 þ x2 Þ ¼ k1 ; 2 1 ½ð18 6x1 4x2 Þ þ ð1 þ x1 þ x2 Þ ¼ k2 ; 2 x1 ; x2 P 0:
ð12Þ
Hop declared that the optimal solution to the model is ðx1 ; x2 Þ ¼ ð0:875; 4:875Þ and the objective value 50. In contrast between the solution ðx1 ; x2 Þ ¼ ð1:8; 1:8Þ and the objective value, 28.8, obtained by the signed distance method, he claimed that his new approach has a better performance. This claim in fact was based on an erroneous result and therefore cannot be substantiated. In Hop’s process for solving Example 1, the resulting deterministic linear program clearly disregarded the non-negativity constraints k1 P 0 and k2 P 0. Hop’s solution of ðx1 ; x2 Þ ¼ ð0:875; 4:875Þ would correspond to k1 ¼ 0:3125 and k2 ¼ 0, with the value of k1 a violation of the constraint k1 P 0, rendering the solution infeasible. In addition, in Hop’s conversion, both in the theory and in the example, the inequality conditions on pairs of fuzzy numbers were missing in the deterministic model. His optimal solution, ðx1 ; x2 Þ ¼ ð0:875; 4:875Þ, evidently violated the inequality constraints 2x1 þ 3x2 6 9 and 6x1 þ 4x2 6 18. With an incomplete model and subsequently a flawed result, the comparison is clearly invalid. Under our revised Proposition 1, the correct converted deterministic linear programming model for Example 1 should be as follows:
Maximize
7x1 þ 9x2 k1 k2
Subject to
2x1 þ 3x2 6 9;
ð13Þ
6x1 þ 4x2 6 18; 1 max 0; ½ð9 2x1 3x2 Þ þ ð1 þ x1 þ x2 Þ ¼ k1 ; 2 1 max 0; ½ð18 6x1 4x2 Þ þ ð1 þ x1 þ x2 Þ ¼ k2 ; 2 k1 ; k2 P 0;
x1 ; x2 P 0:
From 2x1 þ 3x2 6 9, we know that k1 ¼ 12 ð10 x1 2x2 Þ P 0. By 6x1 þ 4x2 6 18, it yields that k2 ¼ 12 ð19 5x1 3x2 Þ P 0. We can therefore further simplify linear programming model (13) as follows:
Maximize
10x1 þ 11:5x2 14:5
ð14Þ
Subject to 2x1 þ 3x2 6 9; 6x1 þ 4x2 6 18; x1 ; x2 P 0: It follows that the optimal solution for the simplified model (14) occurs at ðx1 ; x2 Þ ¼ ð1:8; 1:8Þ with the maximum objective value equal to 24.2, which is consistent with the optimal solution obtained by the signed distance method [3]. The difference in the optimal objective values, 28.8 and 24.2 for the signed distance method [3] and the revised Hop’s method, is due to different optimization objectives that bear no consequence to their comparison. Based on these observations, the illustration with Example 1 provides no evidence to show the superiority of the attainment values method. In Example 2 below, Hop provided a fuzzy stochastic linear programming problem intended to show that the solution to Luhandjula’s model [16] could be improved by the attainment values method.
S.-Y. Chou et al. / Information Sciences 179 (2009) 4083–4088
Minimize
3x1 þ 2x2
ð15Þ
e e Subject to Ax 6 b x P 0; w 2 X; 8 ( > > ~w1 Þ ¼ > e w1 ; b > ð A > < e ~ ( where ð A; bÞ ¼ > > > ðA ~ e > ; b Þ ¼ > w2 w2 : pðw1 Þ ¼ 0:25;
4087
pðw2 Þ ¼ 0:75;
! ~ 1 ~ 1 ; ~ 1 ~ 2 ! ~ 3 ~ 1 ; ~ 2 ~ 1
~ 3 ~ 4 ~ 5 ~ 4
X ¼ ðw1 ; w2 Þ;
!) ; !) ; ~t ¼ ðt; 0:5; 0:5Þ:
Based on the attainment values, Hop [7] first converted the fuzzy stochastic linear programming problem into the corresponding deterministic linear programming problem:
1 3x1 þ 2x2 þ ðk11 þ k21 þ 3k12 þ 3k22 Þ 4 ~ 1 þ 1x ~ 2 ; 3Þ ~ ¼ k11 ; Subject to Dð1x ~ ~ ~ Dð2x1 þ 1x2 ; 4Þ ¼ k21 ; Minimize
ð16Þ
~ 1 þ 3x ~ 2 ; 5Þ ~ ¼ k12 ; Dð1x ~ ~ ~ ¼ k22 ; Dð1x1 þ 2x2 ; 4Þ xj P 0;
kij P 0;
for i; j ¼ 1; 2:
By Hop’s Proposition 1, the constraints of the model can then be expressed explicitly as:
1 3x1 þ 2x2 þ ðk11 þ k21 þ 3k12 þ 3k22 Þ 4 1 ðx1 þ x2 3 þ 0:5x1 þ 0:5x2 þ 0:5Þ ¼ k11 ; Subject to 2 1 ð2x1 þ x2 4 þ 0:5x1 þ 0:5x2 þ 0:5Þ ¼ k21 ; 2 1 ðx1 þ 3x2 5 þ 0:5x1 þ 0:5x2 þ 0:5Þ ¼ k12 ; 2 1 ðx1 þ 2x2 4 þ 0:5x1 þ 0:5x2 þ 0:5Þ ¼ k22 ; 2 xj P 0; kij P 0; for i; j ¼ 1; 2: Minimize
ð17Þ
Hop claimed that his solution ðx1 ; x2 Þ ¼ ð0:85; 0:92Þ and the minimum value 4.9 are better than the results from Luhandjula [16] with ðx1 ; x2 Þ ¼ ð1:0; 1:0Þ and the minimum value 6.02, under the converted Eq. (17). Notice that the result in Eq. (17), that is based on his Proposition 1, has already been demonstrated flawed. The claim for superiority in the example would not be sustainable, and as a result, what he had obtained was incorrect. The correct model of Eq. (17) should be based on Revised Proposition 1, which yields the following:
Minimize
1 3x1 þ 2x2 þ ðk11 þ k21 þ 3k12 þ 3k22 Þ 4
Subject to x1 þ x2 6 3; 2x1 þ x2 6 4; x1 þ 3x2 6 5; x1 þ 2x2 6 4; 1 max 0; ðx1 þ x2 3 þ 0:5x1 þ 0:5x2 þ 0:5Þ ¼ k11 ; 2 1 max 0; ð2x1 þ x2 4 þ 0:5x1 þ 0:5x2 þ 0:5Þ ¼ k21 ; 2 1 max 0; ðx1 þ 3x2 5 þ 0:5x1 þ 0:5x2 þ 0:5Þ ¼ k12 ; 2 1 max 0; ðx1 þ 2x2 4 þ 0:5x1 þ 0:5x2 þ 0:5Þ ¼ k22 ; 2 xj P 0;
kij P 0;
for i; j ¼ 1; 2:
ð18Þ
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In addition to having an incorrect conversion from a fuzzy linear programming problem to a crisp linear programming problem, this example has its own deficiency in its entirety. From Eq. (18), it can be easily observed that choosing ðx1 ; x2 Þ ¼ ð0; 0Þ results in kij ¼ 0, for i; j ¼ 1; 2, and subsequently the minimum value zero. This example becomes a trivial problem. This problem, however, would make sense if it is a maximization rather than a minimization problem. In fact, in Luhandjula [16], he considered a maximization problem under the same objective function of fuzzy and of random conditions. 5. Conclusion In the study, we have described critical flaws in Hop’s conversion from fuzzy or fuzzy/stochastic linear programming problems to the corresponding deterministic linear programming problems. We have provided a revision with theoretical validity and efficiency for mathematical analysis based on his proposed model. The amended results of the two examples that Hop provided were also shown to disagree with his own claim on the superiority of his approach. Different distance as measured in metrics can be adopted under different circumstances to explore alternatives for problem formulation and problem solving. However, it is our contention that Hop’s approach, which is based on a proposed metric distance, does not out-perform any other existing approaches. References [1] L. Campos, J.L. Verdegay, Linear programming problems and ranking of fuzzy numbers, Fuzzy Sets and Systems 32 (1) (1989) 1–11. [2] S. Chanas, P. Zielinski, On the equivalence of two optimization methods for fuzzy linear programming problems, European Journal of Operational Research 121 (1) (2000) 56–63. [3] J. Chiang, Fuzzy linear programming based on statistical confidence interval and interval-valued fuzzy set, European Journal of Operational Research 129 (1) (2001) 65–86. [4] P. Fortemps, M. Roubens, Ranking and defuzzification methods based on area compensation, Fuzzy Sets and Systems 82 (3) (1996) 319–330. [5] Y. Gao, G. Zhang, J. Lu, A particle swarm optimization based algorithm for fuzzy bilevel decision making, IEEE International Conference on Fuzzy Systems 4630563 (2008) 1452–1457. [6] D. Garcia, M.A. Lubiano, M.C. Alonso, Estimating the expected value of fuzzy random variables in the stratified random sampling from finite populations, Information Sciences 138 (1–4) (2001) 165–184. [7] N.V. Hop, Solving linear programming problems under fuzziness and randomness environment using attainment values, Information Sciences 177 (14) (2007) 2971–2984. [8] M. Inuiguchi, J. Ramik, Possibilistic linear programming: a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem, Fuzzy Sets and Systems 111 (1) (2000) 3–28. [9] M.G. Iskander, A suggested approach for possibility and necessity dominance indices in stochastic fuzzy linear programming, Applied Mathematics Letters 18 (4) (2005) 395–399. [10] K.D. Jamison, W.A. Lodwick, Fuzzy linear programming using a penalty method, Fuzzy Sets and Systems 119 (1) (2001) 97–110. [11] Y. Leung, Spatial Analysis and Planning under Imprecision, North-Holland, Amsterdam, 1988. [12] B. Liu, Fuzzy random chance-constrained programming, IEEE Transactions on Fuzzy Systems 9 (5) (2001) 713–720. [13] B. Liu, Y.K. Liu, Expected value of fuzzy variable and fuzzy expected value models, IEEE Transactions on Fuzzy Systems 10 (4) (2002) 445–450. [14] Y.K. Liu, B. Liu, A class of fuzzy random optimization: expected value models, Information Sciences 155 (1–2) (2003) 89–102. [15] M.K. Luhandjula, Optimization under hybrid uncertainty, Fuzzy Sets and Systems 146 (2) (2004) 187–203. [16] M.K. Luhandjula, M.M. Gupta, On fuzzy stochastic optimization, Fuzzy Sets and Systems 81 (1) (1996) 47–55. [17] H.R. Maleki, M. Tata, M. Mashinchi, Linear programming with fuzzy variables, Fuzzy Sets and Systems 109 (1) (2000) 21–33. [18] D. Qiu, L. Shu, Supremum metric on the space of fuzzy sets and common fixed point theorems for fuzzy mappings, Information Sciences 178 (18) (2008) 3595–3604. [19] J. Xu, Q. Liu, R. Wang, A class of multi-objective supply chain networks optimal model under random fuzzy environment and its application to the industry of Chinese liquor, Information Sciences 178 (8) (2008) 2022–2043. [20] J. Xu, Y. Liu, Multi-objective decision making model under fuzzy random environment and its application to inventory problems, Information Sciences 178 (14) (2008) 2899–2914. [21] J.-S. Yao, K. Wu, Ranking fuzzy numbers based on decomposition principle and signed distance, Fuzzy Sets and Systems 116 (2) (2000) 275–288. [22] H.J. Zimmermann, Description and optimization of fuzzy systems, International Journal of General Systems 2 (4) (1976) 209–215.
Contents lists available at ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
A note on ‘‘Solving linear programming problems under fuzziness and randomness environment using attainment values” Shuo-Yan Chou a,*, Jennifer Shu-Jen Lin b, Peterson Julian c a
Department of Industrial Management, National Taiwan University of Science and Technology, Taiwan Institute of Technological and Vocational Education, National Taipei University of Technology, Taiwan c Department of Traffic Science, Central Police University, Taoyuan County, Taiwan b
a r t i c l e
i n f o
Article history: Received 1 February 2009 Received in revised form 6 July 2009 Accepted 8 August 2009
Keywords: Fuzzy linear programming Fuzzy stochastic linear programming Fuzzy number Attainment values Attainment index
a b s t r a c t This paper is an amendment to Hop’s paper [N.V. Hop, Solving linear programming problems under fuzziness and randomness environment using attainment values, Information Sciences 177 (2007) 2971–2984], in solving linear programming problems under fuzziness and randomness environments. Hop introduced a new characterization of relationship, attainment values, to enable the conversion of fuzzy (stochastic) linear programming models into corresponding deterministic linear programming models. The purpose of this paper is to provide a correction and an improvement of Hop’s analytical work through rationalization and simplification. More importantly, it is shown that Hop’s analysis does not support his demonstration or the solution-finding mechanism; the attainment values approach as he had proposed does not result in superior performance as compared to other existing approaches because it neglects some relevant and inevitable theoretical essentials. Two numerical examples from Hop’s paper are also employed to show that his approach, in the conversion of fuzzy (stochastic) linear programming problems to corresponding problems, is questionable and can neither find the maximum nor the minimum in the examples. The models of the examples are subsequently amended in order to derive the correct optimal solutions. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction In view of its properties that is applicable to real-life business applications, fuzzy linear programming has attracted a great deal of attention from researchers. Among these investigations [2,21,3,10,17] the most widely-adopted procedure to find the solution was through the conversion of fuzzy linear programming models into corresponding deterministic linear programming models [8]. For instance, Leung [11] tried to classify fuzzy mathematical programming models into four categories: a precise objective with fuzzy constraints, a fuzzy objective with precise constraints, a fuzzy objective with fuzzy constraints, and robust programming. In trying to find a solution to the problem, the procedure in each of the categories were also developed which includes the signed distance method [3], the area compensation method [4], the expected mid-point approach [10] and the grade of possibility and necessity measures [1]. Zimmermann [22] is viewed as the pioneer in solving linear programming problems with fuzzy resources and fuzzy objectives. He developed a max–min tolerance method with the criteria of the highest membership degree to convert the initial fuzzy linear programming models into corresponding crisp ones. A unique solution could then be found by using the Simplex method. A number of researchers have developed
* Corresponding author. Tel.: +886 227376327. E-mail address: [email protected] (S.-Y. Chou). 0020-0255/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2009.08.013
4084
S.-Y. Chou et al. / Information Sciences 179 (2009) 4083–4088
various approaches with different degrees of efficiency and effectiveness, but most were built upon Zimmermann’s max–min approach. As for extending the research of fuzzy optimization, a new direction in the investigation focuses on a fuzzy and stochastic environment. Solving fuzzy stochastic linear programming problems, therefore, becomes important. In essence, fuzzy numbers/variables and stochastic variables are considered to be a more suitable characterization for real-world problems where uncertain and imprecise information is inherent. However, the inclusion of those components creates a challenge for finding efficient methods to deal with such conditions. An effective way to handle the fuzzy stochastic optimization problems is to convert the problems by de-fuzzifying the fuzzy numbers/variables, de-randomizing the stochastic variables and to solve the resulting deterministic problems instead. Two main approaches were established to cope with the fuzzy stochastic problems. One is to perform the conversion of the de-fuzzification and de-randomization in a sequential manner [16,15,9]. The other is to perform both processes in a simultaneous manner [6,12,14]. As for the sequential approach, a major disadvantage is that it generates an excessive amount of new variables and constraints to the model. On the other hand, a major disadvantage of the simultaneous approach is having a cumbersome workload for calculating the expected value for removing fuzziness and randomness at the same time. In the pursuit of improving the performance of existing approaches, Hop has established a new approach that enables a reduction in the number of additional constraints and an achievement to a certain degree of computational efficiency. The primary feature of Hop’s proposed approach is that instead of using the absolute relationship, as applied in the signed distance method, he employed the relative relationship between fuzzy numbers and fuzzy stochastic variables. The relative relationship is obtained through the calculation of so-called ‘‘attainment values” or degrees such as lower-side attainment index, both-side attainment index and average index as mentioned in his study. The relative relationship and thus the attainment values, play the key role in the conversion of fuzzy and fuzzy stochastic linear programming problems into more simple, conventional crisp problems, while reducing the number of constraints and computational complexity. Hop emphasized that his converted deterministic LP with few additional constraints can be easily solved by using standard optimization packages such as LINGO. Two examples were provided to illustrate the procedure of his proposed approach. However, we can verify that the method of attainment values that Hop [7] proposed is flawed since it can neither find the maximum nor the minimum values of the desired objectives owing to negligence of some relevant and unavoidable theoretical essentials. To the best of our knowledge, at least four papers [5,18–20], thus far have referred to Hop [7] in their references. Of them, none have noticed the flawed results established and claimed in Hop’s work. Here we have proceeded with a thorough investigation of Hop’s paper and highlighted mistakes in his conversion and his solution-finding procedures in an effort to alert future adoption of this particular solution approach in fuzzy/fuzzy stochastic programming and related areas. The remainder of this paper is organized as follows. Section 2 reviews the mathematical formulation of Hop’s research. Section 3 provides our revision necessary in order to resolve the problems in Hop’s approach. Section 4 reconsiders the two numerical examples of Hop’s investigation and compares these results with those from our approach. Finally, conclusions are drawn and future work outlined.
2. Review of Hop’s results in mathematical formulation Hop [7] tried to provide a new approach to solve linear programming problems under fuzzy and random conditions, and that is without the complicated computation of expected values as appeared in Liu and Liu [13]. To achieve this objective, the e , with P e , the lower-side attainment index of e and Q e6Q attainment indices are defined and utilized. For two fuzzy numbers P e is defined by Hop [7] as e to Q P
eÞ ¼ e Q Dð P;
Z
1
n o e ðrÞ P ag da: e P ag inffr : Q max 0; supfs : PðsÞ
ð1Þ
0
e ¼ ðu; a; bÞ with the membership function For triangular fuzzy number P more compact expression.
(
leP ðxÞ ¼
; if x 6 u; max 0; 1 ux a ; if x > u: max 0; 1 xu b
leP ðxÞ, we rewrite the membership function in a
ð2Þ
e ¼ ðv ; c; dÞ with u 6 v , the average lower-side e ¼ ðu; a; bÞ and Q Proposition 1 of Hop [7]. For two triangular fuzzy numbers, P e Þ ¼ uv þbþc. e is defined as Dð P; e Q e to Q attainment index of P 2 Proposition 1 as proved by Hop is as follows. e is ½Q l ðaÞ; Q u ðaÞ ¼ ½v cð1 aÞ; e is defined as ½Pl ðaÞ; P u ðaÞ ¼ ½u að1 aÞ; u þ bð1 aÞ, and of Q The a-cut of P v þ dð1 aÞ. Hop [7] had the lower-side attainment index of fuzzy number Pe to Qe at a-level:
e Þ ¼ maxf0; Pu ðaÞ Q l ðaÞg ¼ maxf0; u v þ ðb þ cÞð1 aÞg: e Q Dð P; a
ð3Þ
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He derived that
eÞ ¼ e Q Dð P;
Z
1
maxf0; u v þ ðb þ cÞð1 aÞgda ¼
Z
u k ¼ð1vbþc Þ
½u v þ ðb þ cÞð1 aÞda;
ð4Þ
0
0
e is defined e to Q where k denotes the maximum value of a with P u ðaÞ P Q l ðaÞ. The average lower-side attainment index of P as
e Þ ¼ 1 Dð P; e Þ: e Q e Q Dð P; k
ð5Þ
Detailed computation for expression (5) provided by Hop [7] is as follows:
" !# 1 e e 1 k2 k e e ¼ ðu v þ b þ cÞ ðb þ cÞ Dð P; Q Þ ¼ Dð P; Q Þ ¼ ðu v þ b þ cÞk ðb þ cÞ k k 2 2 ðb þ cÞ v u bþc v u uv þbþc 1 ¼v uþbþc þ ¼ : ¼ ðu v þ b þ cÞ 2 bþc 2 2 2
ð6Þ
3. Our revisions In this section, we show that the average lower side attainment index as defined in the previous section can, in fact, be computed much more easily. In addition, Hop assumed that the intersection of two membership functions always exists, as e. e and the left side of Q shown in Fig. 1, Case I, where k is the y-coordinate of the intersection between the right side of P However, such an intersection does not always exist. Under the condition where v P u, two cases can be established: (I) u þ b P v c and (II) u þ b < v c. For Case I, the intersection does exist with the coordinate at
bv þ cu v u ;1 ¼k : bþc bþc
ð7Þ
e Þ denotes the area of the shaded triangle shown in Fig. 1. In Eq. (5), the average lower-side attainment index e Q In Eq. (4), Dð P; e Þ, is expressed as the area divided by the height. Hence, it follows that e , Dð P; e Q e to Q of P
eÞ ¼ e Q Dð P;
1 2
ðthe length of the baseÞ:
ð8Þ
This observation leads to a much simpler way than Hop’s proof to derive
eÞ ¼ e Q Dð P;
1 ½ðu þ bÞ ðv cÞ; 2
ð9Þ
under the condition of u þ b P v c for Case I. For Case II, the intersection does not exist. The inequality
Pu ðaÞ < Q l ðaÞ for 0 6 a 6 1
ð10Þ
e Þ ¼ 0 for 0 6 a 6 1 and thus Dð P; e Þ ¼ 0. As the intersection k does not exist, it follows that Dð P; e Þ ¼ 0. e Q e Q e Q implies that Dð P; a Proposition 1 of Hop [7] can therefore be revised as follows. e ¼ ðv ; c; dÞ with u 6 v , the average e ¼ ðu; a; bÞ and Q Revised Proposition 1 of Hop [7]. For two triangular fuzzy numbers, P e Þ ¼ max 0; uv þbþc . e is defined as Dð P; e Q e to Q lower-side attainment index of P 2 Similar problems also appear in Proposition 3 of Hop [7] for the average lower-side attainment index of the fuzzy random variables.
e as P e ¼ ðv ; c; dÞ with u 6 v . e to Q e ¼ ðu; a; bÞ and Q Fig. 1. The average lower-side attainment index of P
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4. Review of Hop’s results in the numerical examples Hop provided two examples in his study to show the superiority of the attainment values approach. In Example 1, in a manufacturing context, he formulated the following fuzzy linear programming problem:
Maximize
7x1 þ 9x2 ~ 2 6 9; ~ ~ Subject to 2x1 þ 3x ~ 2 6 18; ~ ~ 1 þ 4x 6x
ð11Þ
x1 ; x2 P 0; where fuzzy coefficients are represented by fuzzy numbers in the form of ~t ¼ ðt; 1; 1Þ. Hop [7] converted this fuzzy linear programming problem into a deterministic linear programming problem before solving for its optimal solution:
Maximize
7x1 þ 9x2 k1 k2 1 Subject to ½ð9 2x1 3x2 Þ þ ð1 þ x1 þ x2 Þ ¼ k1 ; 2 1 ½ð18 6x1 4x2 Þ þ ð1 þ x1 þ x2 Þ ¼ k2 ; 2 x1 ; x2 P 0:
ð12Þ
Hop declared that the optimal solution to the model is ðx1 ; x2 Þ ¼ ð0:875; 4:875Þ and the objective value 50. In contrast between the solution ðx1 ; x2 Þ ¼ ð1:8; 1:8Þ and the objective value, 28.8, obtained by the signed distance method, he claimed that his new approach has a better performance. This claim in fact was based on an erroneous result and therefore cannot be substantiated. In Hop’s process for solving Example 1, the resulting deterministic linear program clearly disregarded the non-negativity constraints k1 P 0 and k2 P 0. Hop’s solution of ðx1 ; x2 Þ ¼ ð0:875; 4:875Þ would correspond to k1 ¼ 0:3125 and k2 ¼ 0, with the value of k1 a violation of the constraint k1 P 0, rendering the solution infeasible. In addition, in Hop’s conversion, both in the theory and in the example, the inequality conditions on pairs of fuzzy numbers were missing in the deterministic model. His optimal solution, ðx1 ; x2 Þ ¼ ð0:875; 4:875Þ, evidently violated the inequality constraints 2x1 þ 3x2 6 9 and 6x1 þ 4x2 6 18. With an incomplete model and subsequently a flawed result, the comparison is clearly invalid. Under our revised Proposition 1, the correct converted deterministic linear programming model for Example 1 should be as follows:
Maximize
7x1 þ 9x2 k1 k2
Subject to
2x1 þ 3x2 6 9;
ð13Þ
6x1 þ 4x2 6 18; 1 max 0; ½ð9 2x1 3x2 Þ þ ð1 þ x1 þ x2 Þ ¼ k1 ; 2 1 max 0; ½ð18 6x1 4x2 Þ þ ð1 þ x1 þ x2 Þ ¼ k2 ; 2 k1 ; k2 P 0;
x1 ; x2 P 0:
From 2x1 þ 3x2 6 9, we know that k1 ¼ 12 ð10 x1 2x2 Þ P 0. By 6x1 þ 4x2 6 18, it yields that k2 ¼ 12 ð19 5x1 3x2 Þ P 0. We can therefore further simplify linear programming model (13) as follows:
Maximize
10x1 þ 11:5x2 14:5
ð14Þ
Subject to 2x1 þ 3x2 6 9; 6x1 þ 4x2 6 18; x1 ; x2 P 0: It follows that the optimal solution for the simplified model (14) occurs at ðx1 ; x2 Þ ¼ ð1:8; 1:8Þ with the maximum objective value equal to 24.2, which is consistent with the optimal solution obtained by the signed distance method [3]. The difference in the optimal objective values, 28.8 and 24.2 for the signed distance method [3] and the revised Hop’s method, is due to different optimization objectives that bear no consequence to their comparison. Based on these observations, the illustration with Example 1 provides no evidence to show the superiority of the attainment values method. In Example 2 below, Hop provided a fuzzy stochastic linear programming problem intended to show that the solution to Luhandjula’s model [16] could be improved by the attainment values method.
S.-Y. Chou et al. / Information Sciences 179 (2009) 4083–4088
Minimize
3x1 þ 2x2
ð15Þ
e e Subject to Ax 6 b x P 0; w 2 X; 8 ( > > ~w1 Þ ¼ > e w1 ; b > ð A > < e ~ ( where ð A; bÞ ¼ > > > ðA ~ e > ; b Þ ¼ > w2 w2 : pðw1 Þ ¼ 0:25;
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pðw2 Þ ¼ 0:75;
! ~ 1 ~ 1 ; ~ 1 ~ 2 ! ~ 3 ~ 1 ; ~ 2 ~ 1
~ 3 ~ 4 ~ 5 ~ 4
X ¼ ðw1 ; w2 Þ;
!) ; !) ; ~t ¼ ðt; 0:5; 0:5Þ:
Based on the attainment values, Hop [7] first converted the fuzzy stochastic linear programming problem into the corresponding deterministic linear programming problem:
1 3x1 þ 2x2 þ ðk11 þ k21 þ 3k12 þ 3k22 Þ 4 ~ 1 þ 1x ~ 2 ; 3Þ ~ ¼ k11 ; Subject to Dð1x ~ ~ ~ Dð2x1 þ 1x2 ; 4Þ ¼ k21 ; Minimize
ð16Þ
~ 1 þ 3x ~ 2 ; 5Þ ~ ¼ k12 ; Dð1x ~ ~ ~ ¼ k22 ; Dð1x1 þ 2x2 ; 4Þ xj P 0;
kij P 0;
for i; j ¼ 1; 2:
By Hop’s Proposition 1, the constraints of the model can then be expressed explicitly as:
1 3x1 þ 2x2 þ ðk11 þ k21 þ 3k12 þ 3k22 Þ 4 1 ðx1 þ x2 3 þ 0:5x1 þ 0:5x2 þ 0:5Þ ¼ k11 ; Subject to 2 1 ð2x1 þ x2 4 þ 0:5x1 þ 0:5x2 þ 0:5Þ ¼ k21 ; 2 1 ðx1 þ 3x2 5 þ 0:5x1 þ 0:5x2 þ 0:5Þ ¼ k12 ; 2 1 ðx1 þ 2x2 4 þ 0:5x1 þ 0:5x2 þ 0:5Þ ¼ k22 ; 2 xj P 0; kij P 0; for i; j ¼ 1; 2: Minimize
ð17Þ
Hop claimed that his solution ðx1 ; x2 Þ ¼ ð0:85; 0:92Þ and the minimum value 4.9 are better than the results from Luhandjula [16] with ðx1 ; x2 Þ ¼ ð1:0; 1:0Þ and the minimum value 6.02, under the converted Eq. (17). Notice that the result in Eq. (17), that is based on his Proposition 1, has already been demonstrated flawed. The claim for superiority in the example would not be sustainable, and as a result, what he had obtained was incorrect. The correct model of Eq. (17) should be based on Revised Proposition 1, which yields the following:
Minimize
1 3x1 þ 2x2 þ ðk11 þ k21 þ 3k12 þ 3k22 Þ 4
Subject to x1 þ x2 6 3; 2x1 þ x2 6 4; x1 þ 3x2 6 5; x1 þ 2x2 6 4; 1 max 0; ðx1 þ x2 3 þ 0:5x1 þ 0:5x2 þ 0:5Þ ¼ k11 ; 2 1 max 0; ð2x1 þ x2 4 þ 0:5x1 þ 0:5x2 þ 0:5Þ ¼ k21 ; 2 1 max 0; ðx1 þ 3x2 5 þ 0:5x1 þ 0:5x2 þ 0:5Þ ¼ k12 ; 2 1 max 0; ðx1 þ 2x2 4 þ 0:5x1 þ 0:5x2 þ 0:5Þ ¼ k22 ; 2 xj P 0;
kij P 0;
for i; j ¼ 1; 2:
ð18Þ
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In addition to having an incorrect conversion from a fuzzy linear programming problem to a crisp linear programming problem, this example has its own deficiency in its entirety. From Eq. (18), it can be easily observed that choosing ðx1 ; x2 Þ ¼ ð0; 0Þ results in kij ¼ 0, for i; j ¼ 1; 2, and subsequently the minimum value zero. This example becomes a trivial problem. This problem, however, would make sense if it is a maximization rather than a minimization problem. In fact, in Luhandjula [16], he considered a maximization problem under the same objective function of fuzzy and of random conditions. 5. Conclusion In the study, we have described critical flaws in Hop’s conversion from fuzzy or fuzzy/stochastic linear programming problems to the corresponding deterministic linear programming problems. We have provided a revision with theoretical validity and efficiency for mathematical analysis based on his proposed model. The amended results of the two examples that Hop provided were also shown to disagree with his own claim on the superiority of his approach. Different distance as measured in metrics can be adopted under different circumstances to explore alternatives for problem formulation and problem solving. However, it is our contention that Hop’s approach, which is based on a proposed metric distance, does not out-perform any other existing approaches. References [1] L. Campos, J.L. Verdegay, Linear programming problems and ranking of fuzzy numbers, Fuzzy Sets and Systems 32 (1) (1989) 1–11. [2] S. Chanas, P. Zielinski, On the equivalence of two optimization methods for fuzzy linear programming problems, European Journal of Operational Research 121 (1) (2000) 56–63. [3] J. Chiang, Fuzzy linear programming based on statistical confidence interval and interval-valued fuzzy set, European Journal of Operational Research 129 (1) (2001) 65–86. [4] P. Fortemps, M. Roubens, Ranking and defuzzification methods based on area compensation, Fuzzy Sets and Systems 82 (3) (1996) 319–330. [5] Y. Gao, G. Zhang, J. Lu, A particle swarm optimization based algorithm for fuzzy bilevel decision making, IEEE International Conference on Fuzzy Systems 4630563 (2008) 1452–1457. [6] D. Garcia, M.A. Lubiano, M.C. Alonso, Estimating the expected value of fuzzy random variables in the stratified random sampling from finite populations, Information Sciences 138 (1–4) (2001) 165–184. [7] N.V. Hop, Solving linear programming problems under fuzziness and randomness environment using attainment values, Information Sciences 177 (14) (2007) 2971–2984. [8] M. Inuiguchi, J. Ramik, Possibilistic linear programming: a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem, Fuzzy Sets and Systems 111 (1) (2000) 3–28. [9] M.G. Iskander, A suggested approach for possibility and necessity dominance indices in stochastic fuzzy linear programming, Applied Mathematics Letters 18 (4) (2005) 395–399. [10] K.D. Jamison, W.A. Lodwick, Fuzzy linear programming using a penalty method, Fuzzy Sets and Systems 119 (1) (2001) 97–110. [11] Y. Leung, Spatial Analysis and Planning under Imprecision, North-Holland, Amsterdam, 1988. [12] B. Liu, Fuzzy random chance-constrained programming, IEEE Transactions on Fuzzy Systems 9 (5) (2001) 713–720. [13] B. Liu, Y.K. Liu, Expected value of fuzzy variable and fuzzy expected value models, IEEE Transactions on Fuzzy Systems 10 (4) (2002) 445–450. [14] Y.K. Liu, B. Liu, A class of fuzzy random optimization: expected value models, Information Sciences 155 (1–2) (2003) 89–102. [15] M.K. Luhandjula, Optimization under hybrid uncertainty, Fuzzy Sets and Systems 146 (2) (2004) 187–203. [16] M.K. Luhandjula, M.M. Gupta, On fuzzy stochastic optimization, Fuzzy Sets and Systems 81 (1) (1996) 47–55. [17] H.R. Maleki, M. Tata, M. Mashinchi, Linear programming with fuzzy variables, Fuzzy Sets and Systems 109 (1) (2000) 21–33. [18] D. Qiu, L. Shu, Supremum metric on the space of fuzzy sets and common fixed point theorems for fuzzy mappings, Information Sciences 178 (18) (2008) 3595–3604. [19] J. Xu, Q. Liu, R. Wang, A class of multi-objective supply chain networks optimal model under random fuzzy environment and its application to the industry of Chinese liquor, Information Sciences 178 (8) (2008) 2022–2043. [20] J. Xu, Y. Liu, Multi-objective decision making model under fuzzy random environment and its application to inventory problems, Information Sciences 178 (14) (2008) 2899–2914. [21] J.-S. Yao, K. Wu, Ranking fuzzy numbers based on decomposition principle and signed distance, Fuzzy Sets and Systems 116 (2) (2000) 275–288. [22] H.J. Zimmermann, Description and optimization of fuzzy systems, International Journal of General Systems 2 (4) (1976) 209–215.