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A novel approach for FNLP with piecewise linear membership functions
Chemometrics and Intelligent Laboratory Systems 191 (2019) 88–95
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Chemometrics and Intelligent Laboratory Systems journal homepage: www.elsevier.com/locate/chemometrics
A novel approach for FNLP with piecewise linear membership functions Bo Wen a, Wen Gu c, Bo Yang a, *, Hongguang Li a , Xiaochun Chen b a
College of Information Science & Technology, Beijing University of Chemical Technology, China College of Chemical Engineering, Beijing University of Chemical Technology, China c Changzheng Engineering Co., Limited, China b
A R T I C L E I N F O
A B S T R A C T
Keywords: Fuzzy nonlinear programming Piecewise linear membership functions Error analysis Iterative algorithm
In the fuzzy nonlinear programming (FNLP) domain, piecewise linear membership functions (PLMFs) are often employed to linearize the nonlinear membership functions. However, the linearization error will reduce the accuracy of the solution, which has become an obstacle to the application of PLMFs. In this paper, we present an iterative algorithm based on geometrical means to solve the problems mentioned above. Firstly, the linearization error can be reduced by gradually modifying the programming model. Subsequently, compared with traditional methods, the linearization model is determined only by weighing the linearization degree of membership function and the accuracy of programming results, the proposed method is more directly and effectively. Furthermore, indepth discussions are also given along with the algorithm for demonstrating the rationality. Finally, the proposed method is validated and compared with other methods. A real example is examined to demonstrate applicability of the proposed method in this paper.
1. Introduction Even though mathematical programming has become a well-known optimization method based on crisp models, it still remains less popular in industrial applications due to the uncertainty which occurs frequently in programming problems. For dealing with the uncertainty factors, fuzzy theory has been integrated into programming method, fuzzy decision and fuzzy linear programming theories were founded in 1970 and 1978 by Bellman & Zadeh [1] and Zimmermann [2] respectively. Recently, studies on fuzzy theory and fuzzy mathematical programming have been widely circulated in literature [3–7]. Since fuzzy models in according programming problems can be changed into crisp ones by means of suitable membership functions, the diverse types of membership function will lead to different properties of programming problem. In fuzzy nonlinear programming, nonlinear membership functions are employed to represent decision-makers’ (DM) subjective preferences. Taking advantage of the accurate description for fuzzy degree, nonlinear membership functions such as Gaussian membership functions are usually involved in practical problems. However, the involvement will inevitably leads to heavy calculation burden due to the severe nonlinear degree. Aiming at this problem, piecewise linear membership functions are introduced to decrease the nonlinearity by approximating the
nonlinear membership functions. The methodology that employs piecewise linear functions to approximate arbitrary shape nonlinear ones was firstly proposed by Stone [8], where the approximation accuracy is proportional to the number of segments. After that, many researchers have launched studies on this area [9–11]. In practical applications, piecewise linear functions also have been used extensively, such as stochastic sampling, pricing problem, real long-length noisy signals dealing, oblique survival tree and so on [12–15]. The functions have also been employed to discretize of occurring nonlinearities in mixed-integer nonlinear programs (MINLP), and solve the field of gas network optimization problem [16]. Rebennack and Kallrath also provided delta-approximations for solving MINLP, and his approach is approximating the nonlinear function from both sides [17]. Piecewise linear functions were firstly introduced to fuzzy programming by Narasimhan [18], which the linearization concept was used to deal with the complex nonlinear membership functions. However, although the solving difficulty of fuzzy programming with complex nonlinear membership functions can be simplified by PLMFs, the linear approximation would lead to a certain error between optimization results and the actual optimum values, which becomes a bottleneck of PLMF applications. Researchers have launched in-depth studies on the relations between segment numbers and approximation accuracy [19,20], where
* Corresponding author. E-mail address: [email protected] (B. Yang). https://doi.org/10.1016/j.chemolab.2019.06.007 Received 27 November 2018; Received in revised form 21 May 2019; Accepted 24 June 2019 Available online 25 June 2019 0169-7439/© 2019 Elsevier B.V. All rights reserved.
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the current research achievements still rest on decreasing the error by adding sub-segments rather than eliminating it authentically. Motivated by this challenge, this study proposes a novel iterative algorithm to eliminate the error which caused by PLMFs, and so as to make optimization solutions closer to the actual values or even equal with them completely. The rest of the paper is organized as follows. In section 2, the structure of PLMFs and causes of error have been discussed, based on that an iteration algorithm responsible for eliminate the error has been proposed. To analyze effectiveness of the proposed methodology, a plant case study is provided in Section 3. Finally, conclusion remarks are given in Section 4. 2. Error elimination 2.1. Approximation error A piecewise linear function is composed of several linear segments and knots. Thus, a nonlinear membership function with arbitrary shape can be approximated by changing quantity and position of the knots included in a PLMF. For instant, a nonlinear membership function gðxÞ is approximated by a 4 segments and 5 knots PLMF as shown in Fig. 1. A general replacement produces typically better errors (except for the quadratic function, where the equidistant placement is optimal, in terms of absolute error [21]). For minimizing the area between the nonlinear function and the piecewise linear function, for a given number of segments, by adjusting the placements of the knots [22]. According to the approximation property of PLMFs, it can be clearly seen that the replacing process will result in error (the gap between solid and dashed line). As shown in Fig. 2, gray areas represent the error between PLMFs and the original membership function, where the size of area is in proportion to the error. The error at any arbitrary point on linear segments is specified as the distance from this point to the original functions along the vertical direction. Therefore, each point on these linear segments is according to different errors. Taking the forth segment for example, which the amplified graph is shown in Fig. 3, the curve between points A and B is approximated with straight lineAB. The area enclosed by the straight line and the curve between A and B is according error area. Taking arbitrary point C on AB as an example, we draw a straight line perpendicular toAB through it, which intersects with curve y at pointC'. Thus, distance e from point C to C' signifies the error value corresponding to pointC. Thus, it is convincible that the optimal solutions obtained from fuzzy programming with PLMFs are somewhat different from those with original nonlinear membership functions. The error between a piecewise linear membership function and the original nonlinear one is proportional to the number of knots, which means the fitting degree between the two functions will increase with the
Fig. 2. Error areas in PLMFs.
Fig. 3. The error at arbitrary point.
knot adding. However, big amount of sub-segments would complicate the optimization model and ascend the calculation burden. In order to tradeoff these situations, it is crucial to appropriately choose the knot positions, which depend on the curvature, error and other factors. Even though, the error is only expected to be decreased rather than eliminated.
2.2. Iterative algorithms For the purpose of increasing the accuracy degree of the optimal solutions get from the programming problem with PLMFs, here suggests a novel iterative algorithm which can decrease or even eliminate the error. This algorithm is mainly based on geometrical means to achieve the error elimination purpose, Fig. 4 shows a flow chart of the algorithm along with a detailed implementing procedure as follow. Furthermore, the algorithm is based on the precondition which the inaccurate optimal satisfaction degree (λ) of programming problem has already been obtained. (1) Place the PLMFs and its original nonlinear membership function curve in a same rectangular coordinate system, and mark point λ on its according segment of the PLMFs. Specify this segment as the basis segment and the curve which approximated by this segment as the basis curve;
Fig. 1. The linear approximation of nonlinear function. 89
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(2) Construct a line A'B' parallel to line AB through point C', then the area surrounded by the original membership function along with below line A'B' is the new feasible region for λ, as shown in Fig. 6;
2.3. Additional remarks for the algorithm There are two supplementary remarks for the proposed iterative algorithm. For the convenience, we do a little modification to Fig. 5. Through point C' draw an auxiliary line parallel to the lateral axis, which intersects with A'B' at point D. Besides, label the abscissa as gðxÞ and ordinate as μðgðxÞÞ, as shown in Fig. 7. Firstly, the remark about the un-necessity of adding the constraint λ fCC' to additional constraints to prevent the λ point moving trend contrary to its optimal direction (moving to the left side) during iteration process. In programming problems, the role of constraints is to restrict the objective function via limiting decision variables. However, the real effective constraints are the active constraints. Therefore, in a standard Max programming, the objective function z will increase along with the adding of left-hand side function gðxÞin constraints (Min programming has the opposite situation). Such as point C and D in Fig. 7, although they have the same membership degree, but point D is corresponds to a larger objective function value, because the value of gðxÞ at D is bigger than C. Thus, if μðgðxÞÞ remains constant and gðxÞ gets bigger, μz will also increase accordingly. So the changing direction of the optimal point is from left to right, which means the location of the optimal point after iteration must in the right side area of last optimal one under the same constraint. According to the analysis above, the new optimal point should be located at the right side of C which is the current optimal point. Owing to the corresponding point of C is C'in original function, the new optimal area is shown as the dark area in Fig. 8. Thereupon, the constraint λ fCC' can be omitted. Secondly, the remark about the new optimal point location. By a large number of experiments, we found that the new optimal point is commonly located in the error area and getting closer to the actual one with the gradually increasing of λ. However, the optimal point will also occasionally locate outside the error area which beyond the feasible range of original function. Under this condition, the proposed methodology is also applicable. The only difference between the two conditions is the optimal point will approximate to the true value from the inside and outside of the error area respectively. It should be noted that in the iteration process of second condition, the value of λ is getting smaller which is opposite to the situation that the optimal point is inside of error area. However, both situations will make the final optimal value approximate to the true one.
Fig. 4. Algorithm flow chart.
(2) Erect a perpendicular, which will intersect with the basis curve, from point λ on the line. Construct a line parallel to the basis segment through the intersect point. Specify the line as a new constraint line, the area which below it becomes a new feasible region for λ; (3) Modify the feasible region by replacing the function of PLMFs with the one of the new constraint line; (4) Solve the new model and get the revised optimal solutions. Mark the revised point of λ on the new constraint line, and construct a line parallel to the basis segment through point λ as the new basis segment for the next iteration; (5) One-time iterative modification of the PLMFs is finished. If there exist other PLMFs, they can be modified by the above steps seriatim; (6) Circulate the above steps until the optimal solution is not changed anymore (represents the error is almost eliminated) or the required accuracy is achieved. Take a fuzzy programming problem with PLMFs which shown as follows for instance. Max
λ
s:t: λ μ1 ðg1 ðxi ÞÞ λ μ2 ðg2 ðxi ÞÞ λ μj ðgj ðxi ÞÞ
(1)
where g1 gj are original membership functions, μ1 μj are according PLMFs. We assume that the optimal solutions of this problem have been obtained. In order to eliminate the error brought by the approximation process of PLMFs, the iterative algorithm can be carried out by the following 5 steps. Since the PLMFs include several segments, we can fix the PLMFs from the first one. (1) Start the PLMFs error elimination from the first segmentμ1 ðxi Þ. Place the PLMFs and its original nonlinear membership function curve in one rectangular coordinate system. For convenience, we amplify the basis segment AB which optimal satisfaction degree point C locate in, as shown in Fig. 5;
Fig. 5. Amplification of segment AB 90
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Fig. 8. New optimal area.
Fig. 6. New feasible region. (3) Modify the model by replacing the constraint λ μ1 ðg1 ðxi ÞÞ with λ fA'B' ðg1 ðxi ÞÞ, and the model is converted into: Max
and MC is 98.42 C and 39.6 C respectively. Thus, the MC is collected in the top of the column, and HE is recovered in downstream unit. In order to save steam consumption and reaching the quality standard, the process engineer has built the fuzzy nonlinear programming model to obtain the optimal control strategy. The parameters in the model contain reflux ratio (calculated by feed flow and reflux flow), steam flow and bottom flow rate which listed in Table 1. The model is shown as follows. We initially obtained the optimum solutions of the corresponding crisp optimization problem as: x1 ¼ 0:431x2 ¼ 0:923x3 ¼ 2:276,z ¼ 43:164. For the comparison, this problem is solved with unilateral membership function, normal PLMFs and proposed methodology, respectively. Here, for building the object membership function of z, we set its expected value as 50.
λ
s:t: λ fA'B' ðg1 ðxi ÞÞ λ μ2 ðg2 ðxi ÞÞ λ μj ðgj ðxi ÞÞ
(2)
(4) Solve model (2) and obtain the new optimal solutions. Construct a line AB parallel to the basis segment through the new point. Line AB is the new basis segment of the next iteration; (5) Implement the steps on the remaining PLMFs, and circulate several times until the optimal solution is not changed anymore or the required accuracy is achieved.
(1) Construct bilateral membership function as constraint [23], then transform the fuzzy model into the crisp one. Max s:t:
λ λ μz ¼ z=50
arctanðg1 8Þ arctanðg1 8Þ þ 1Þμg1 ð½ þ 1Þ λ ð½ π =2 π =2 arctanðg2 7:5Þ arctanðg2 7:5Þ λ ð½ þ 1Þμg2 ð½ þ 1Þ π =2 π =2
μg1 ¼ e
ðg1 8Þ2 2p2 1
μg2 ¼ e
ðg2 7:5Þ2 2p2 2
(4)
where, p1 ¼ p2 ¼ 2 and the membership functions are both bilateral form, as shown in Fig. 10 and Fig. 11. The optimum solutions can be obtained by solving this model with GAMS (The General Algebraic Modeling System, a high-level modeling system for mathematical optimization): x1 ¼ 0:492, x2 ¼ 0:969, x3 ¼ 2:408, g1 ¼ 8:909, g2 ¼ 8:409, λ ¼ 0:902, z ¼ 45:092, μg1 ¼ μg2 ¼ μz ¼ 0:902. In the situation of μg1 ¼ μg2 ¼ 0:902, (4) can be simplified and rearranged as follows.
Fig. 7. Supplementary figure.
3. An illustrative example Consider a study case of Heptane (HE) & Methylene Chloride (MC) separation column, as shown in Fig. 9. The feed flow of this column is the mixed solvent of HE and MC, they are separated by the heating of 6 Bar steam (S6). The boiling point of HE
Max
91
f ¼λ
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Fig. 9. HE & MC separation column. Max
z ¼ x21 x22 2x23 þ 5x1 þ 5x2 þ 21x3
s:t:
g1 ¼ 1:03x21 þ 0:95x22 þ x23 þ x1 0:98x2 þ 1:07x3 ~8
g2 ¼ 0:97x21 þ 1:95x22 þ x23 þ 1:02x1 7:~5
(3)
Table 1 Corresponding relationship. Controlled Variable
Formula
Parameter
Unit
Reflux Ratio Steam Flow Bottom Flow Rate
¼ Reflux Flow/Feed Flow – –
x1 x2 x3
– t/h t/h
Fig. 11. Bilateral membership function of g2 ðg2 7:5Þ2 2p2 2
ω3 ¼ λ e
0
(5)
At the optimal point, the active constraints must satisfy Karush-KuhnTucker (KKT) conditions which are also known as the first order necessary conditions of optimality. Regarding (5), its KKT conditions are expressed by: Fig. 10. Bilateral membership function of g1
s:t:
3 ∂f X ∂ω η i¼0 ∂λ i¼1 i ∂λ
ω1 ¼ λ ð x21 x22 2x23 þ 5x1 þ 5x2 þ 21x3 Þ 50 0
ω2 ¼ λ e
ðg1 8Þ2 2p2 1
3 ∂f X ∂ω η i¼0 ∂xj i¼1 i ∂xj
0
92
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η1 ðω1 0Þ ¼ 0 η2 ðω2 0Þ ¼ 0 η3 ðω3 0Þ ¼ 0
(6)
where η1 η2 ,η3 are the vectors of KKT multipliers corresponding to each constraint. By solving equation (6), the solutions can be obtained as η1 ¼ 0:7938, η2 ¼ 0:1906, η3 ¼ 0:0156, which proves the optimality of the solutions. (2) Construct PLMFs on the basis of the bilateral form as constraints [19]. Choose the key knots in the membership function μg1 as (8, 1) (10, 0.607) (11.5, 0.2163) (15, 0) and μg2 as (7.5, 1) (9.52, 0.6) (11, 0.2163) (14.5, 0). Thus, construct PLMFs based on the key knots as shown in Fig. 12 and Fig. 13 respectively. According to the methodology proposed in paper [24], modeling the fuzzy programming problem as: Max s:t:
Fig. 13. PLMFs of g2
λ
Max
λ μz ¼ z=50
f ¼λ
ω1 ¼ λ 1
s:t:
λ1
ω2 ¼ λ þ 0:2g1 2:6
λ 0:2g1 þ 2:6 þ δ1 M
ω3 ¼ λ þ 0:2g2 2:5
λ 0:256g1 þ 3:165 þ δ1 M
ω4 ¼ λ ð x21 x22 2x23 þ 5x1 þ 5x2 þ 21x3 Þ 50 0
(8)
The KKT conditions of (8) are expressed by:
λ 0:062g1 þ 0:927 þ ð1 δ1 ÞM
4 ∂f X ∂ω η i¼0 ∂λ i¼1 i ∂λ
λ 0:2g2 þ 2:5 þ δ2 M λ 0:921g2 þ 10:25 þ δ2 M λ 0:0618g2 þ 0:8961 þ ð1 δ2 ÞM
4 ∂f X ∂ω η i¼0 ∂xj i¼1 i ∂xj
(7)
where δ1 and δ2 are binary variables. This problem is solved by GAMS, resulting in optimum solutions as: x1 ¼ 0:469x2 ¼ 0:952x3 ¼ 2:358g1 ¼ 8:562g2 ¼ 8:062λ ¼ 0:888z ¼ 44:378, δ1 ¼ δ2 ¼ 0. In the situation of δ1 ¼ δ2 ¼ 0 and according with the optimum solution, (7) can be simplified and re-arranged as follows.
η1 ðω1 1Þ ¼ 0 η2 ðω2 2:6Þ ¼ 0 η3 ðω3 2:5Þ ¼ 0 η4 ðω4 0Þ ¼ 0
(9)
where η1 , η2 , η3 , η4 are the vectors of KKT multipliers corresponding to each constraint. By solving equation (9), the solutions can be obtained as: η1 ¼ 0, η21 ¼ 0:1396, η3 ¼ 0:0336, η4 ¼ 0:8268, which proves the optimality of the solutions. (3) Optimize the solutions obtained from equation (7) by the proposed iterative algorithm. Firstly, do the first iteration to μg1 ¼ 0:2g1 þ 2:6. Through the optimum point (8.562,0.888) draw a straight line M1 perpendicular to this segment, the according linear equation is μM1 g1 ¼ 5g1 41:922. Gaussian membership function intersects with M1 at point A1 which the coordinate is (8.576, 0.958). Through point A1 draw a straight line K1 parallel to μg1 , the according linear equation is μK1 g1 ¼ 0:2g1 þ 2:6732, as shown in Fig. 14. According to the proposed methodology, replace the constraint of membership function μg1 for λ to the constraint of line K1 , the model is changed into: Max
Fig. 12. PLMFs of g1 93
λ
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Table 2 The solutions of FNLP. Bilateral style
PLMFs
First iteration forμg1
First iteration forμg2
Second iteration for
Second iteration for
μg1
μg2
x1 ¼ 0:492 x2 ¼ 0:969 x3 ¼ 2:408 g1 ¼ 8:909 g2 ¼ 8:409 λ ¼ 0:902
x1 ¼ 0:469 x2 ¼ 0:952 x3 ¼ 2:358 g1 ¼ 8:562 g2 ¼ 8:062 λ ¼ 0:888 z ¼ 44:378
x1 ¼ 0:482
x1 ¼ 0:489
x1 ¼ 0:491
x1 ¼ 0:492
x2 ¼ 0:845
x2 ¼ 0:958
x2 ¼ 0:943
x2 ¼ 0:968
x3 ¼ 2:424
x3 ¼ 2:403
x3 ¼ 2:413
x3 ¼ 2:408
g1 ¼ 8:882
g1 ¼ 8:866
g1 ¼ 8:908
g1 ¼ 8:909
g2 ¼ 8:016
g2 ¼ 8:34
g2 ¼ 8:333
g2 ¼ 8:408
λ ¼ 0:897
λ ¼ 0:9
λ ¼ 0:901
λ ¼ 0:902
z ¼ 44:838
z ¼ 44:995
z ¼ 44:065
z ¼ 45:092
z ¼ 45:092
Fig. 14. Analysis of μg1
s:t:
s:t:
λ 0:2g2 þ 2:568
λ 0:2g1 þ 2:6732
λ μz
λ 0:2g2 þ 2:5 λ μz
(11)
Solved by GAMS, the optimum solutions are: x1 ¼ 0:489, x2 ¼ 0:958, x3 ¼ 2:403, g1 ¼ 8:866, g2 ¼ 8:34, λ ¼ 0:9, z ¼ 44:995. Similarly, the optimal point is locate on line K2 . The first iteration for μg2 is finished. Do the second iteration to μg1 and μg2 , respectively. After that, as the optimum solution is almost unchanged, the iteration is over. The solutions of three modeling methodologies are presented in Table 2. By comparison, the solution gets from the model constructed by PLMF is smaller than the one get from the model constructed by bilateral membership function, because the certain errors existing between PLMF and the original one. Yet with the iteration algorithm proposed by this paper, the error caused by the PLMF is nearly eliminated after 4 time iterations. The changing of optimal point position of g1 and g2 in the 4 iterations is shown in Fig. 16 and Fig. 17, where point 1–4 are the optimal points after the each iteration and point O and point F are the optimal points get from the model constructed by PLMF and bilateral membership function respectively. Because the optimal solution is very close to the true value after 4 iterations, so the position of optimal point 4 is almost coincide with point F. From point positions in the figures, it can be seen that if one constraint is iterated then another will be affected according to the changes of xi , and another constraint's optimal point
(10)
Solved by GAMS, the optimum solutions are: x1 ¼ 0:482, x2 ¼ 0:845, x3 ¼ 2:424, g1 ¼ 8:882, g2 ¼ 8:016, λ ¼ 0:897, z ¼ 44:838. Through the coordinate, we know the optimal point is locate on line K1 . The first iteration for μg1 is finished. Secondly, do the first iteration to μg2 base on the above results. Similarly, through the optimal point (8.016, 0.897) in basis segment μg2 ¼ 0:2g2 þ 2:5 draw a straight line M2 perpendicular to this segment, the linear equation is μM2 g2 ¼ 5g2 39:183. Gaussian membership function intersects with M2 at point A2 which coordinate is (8.03, 0.962). Through point A2 draw a straight line K2 parallel to μg2 , the linear equation is μA2 g2 ¼ 0:2g2 þ 2:568, as shown in Fig. 15. According to the proposed methodology, replace the constraint of membership function μg2 for λ to the constraint of line K2 , the model is changed into: Max
λ 0:2g1 þ 2:6732
λ
Fig. 15. Analysis of μg2
Fig. 16. The location changing of optimum point in.g1 94
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Science Foundation of China No. 11702016. References [1] R.E. Bellman, L.A. Zadeh, Decision-making in a fuzzy envrionment, Manag. Sci. 17 (1970) 141–164. [2] H.J. Zimmermann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets Syst. 1 (1978) 45–55. [3] Bo Yang, Hongguang Li, A nove dynamic timed fuzzy Petri nets modelling method with applications to industrial process, Expert Syst. Appl. 97 (1) (2018) 276–289. [4] Bo Yang, Hongguang Li, A similarity elastic window based approach to process dynamic time delay analysis, Chemonetr. Intell. Lab. Sys. 170 (2017) 13–24. [5] Zhao Dong, Steven X. Ding, Hamid Reza Karimi, Yueyang Li, Youqing Wang, On robust Kalman filter for two-dimensional uncertain linear discrete time-varying systems: a least squares method, Automatica 99 (2019) 203–212. [6] Yu-Chung Tsao, Vo-Van Thanh, Jye-Chyi Lu, Yu Vincent, Designing sustainable supply chain networks under uncertain environments: fuzzy multi-objective programming, J. Clean. Prod. 174 (2018) 1550–1565. [7] Sohrab Moradi, Soleiman Mohammadi Limaei. Multi-objective game theory model and fuzzy programming approach for systainable watershed management, Land Use Policy 71 (2018) 363–371. [8] Stone Henry, Approximation of curves by line segments, Comput. Informat. Sci. 15 (1961) 40–47. [9] Luis Ibarra, Mario Rojas, Pedro Ponce, Arturo Molina, Type-2 fuzzy membership function design method through a piecewise-linear approach, Expert Syst. Appl. 42 (2015) 7530–7540. [10] Takashi Hasuike, Hideki Katagiri, Hiroe Tsubaki, A constructing algorithm for appropriate piecewise linear membership function based on statistics and information theory, Proc. Comput. Sci. 60 (2015) 994–1003. [11] Bo Wen, Hongguang Li, An approach to formulation of FNLP with complex piecewise linear membership functions, Chin. J. Chem. Eng. 22 (2014) 411–417. [12] Junbin Li, Renhong Wang, Min Xu, Fang Qin, Piecewise linear approximation methods with stochastic sampling sites, J. Comput. Appl. Math. 328 (2018) 173–178. [13] Qian Hu, Wenbin Zhu, Qin Hu, Andrew Lim, A branch-and-price algorithm for the two-dimensional vector packing problem with piecewise linear cost function, Eur. J. Oper. Res. 260 (2017) 70–80. [14] Matteo Rucco, ect, A new topological entropy-based approach for measuring similarities among piecewise linear functions, Signal Process. 134 (2017) 130–138. [15] Malgorzata Kretowska, Piecewise-linear criterion functons in oblique survival tree induction, Artifical Intell. Med. 75 (2017) 32–39. [16] Robert Burlacu, Solving mixed-integer nonlinear programmes using adaptively refined mixed-integer linear programme, Optim. Methods Software (2019) 1–28. [17] S. Rebennack, J. Kallrath, Continuous piecewise linear delta-approximations for univariate functions: computing minimal breakpoint systems, J. Optim. Theory Appl. 167 (2) (2015) 617–643. [18] R. Narasimhan, Goal programming in a fuzzy environment, Decision science 11 (1980) 325–336. [19] Elber Gershon, Error bounded piecewise linear approximation of freeform surfaces, Comput. Aided Des. 28 (1996) 51–57. [20] A. Imamoto, B. Tang, Optimal piecewise linear approxiamtion of convex functions, Proc. World Cong. Eng. Comput. Sci. 2008 (2008) 22–25. [21] S. Rebennack, Computing tight bounds via piecewise linear functions through the example of circle cutting problems, Math. Methods Oper. Res. 84 (1) (2016) 3–57. [22] Josef Kallrath, R. Steffen, Computing area-tight piecewise linear overestimators, underestimators and tubes for univariate funtions. Optimization in Science and Engineering, in: Honor of the 60th Birthday of Panos M. Pardalos., 2014, pp. 273–292. [23] Bo Wen Hongguang Li, Approaches to building piecewise linear membership functions in nonlinear fuzzy programming, CIE J. 62 (8) (2011) 2258–2264. [24] Bo Wen Hongguang Li, Modeling methodology for fuzzy programming with piecewise linear membership functions, CIE J. 61 (8) (2010) 2149–2153.
Fig. 17. The location changing of optimum point in g2
position will move slightly to left. The happening of this situation is caused by the coupling relationship between the two constraints, which is not contrary to the additional remark mentioned in Chapter 2.3, and the position of optimal point will keep moving to the right side and getting closer to the true value during each iteration process. Furthermore, this practical example is relatively simple, which its model with bilateral form membership functions can be solved easily, so as to illustrating the effectiveness of the given algorithm. However, when facing some complex practical problems which can only be solved with PLMFs the algorithm will play a bigger role. 4. Conclusion In this paper, the structure of PLMF is analyzed in detail, and the linearization error of the non-linear membership function is also explained firstly. Then, a novel iteration algorithm has been proposed to reduce the linearization error based on geometrical means. As the programming model is iteratively modified, the optimal solution will gradually approach the true value. Compared with the other conventional modeling methodologies through a practical case, it has been proved that the given contribution can provide a more accurate solution. Then, the programming method can be applied in the area which requires more precise solutions. At last, the effectiveness of the proposed method also been verified by a case of Heptane (HE) & Methylene Chloride (MC) separation column. Acknowledgment This research is supported by the Fundamental Research Funds for the Central Universities with Grant No. ZY1929 and the National Natural
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Chemometrics and Intelligent Laboratory Systems journal homepage: www.elsevier.com/locate/chemometrics
A novel approach for FNLP with piecewise linear membership functions Bo Wen a, Wen Gu c, Bo Yang a, *, Hongguang Li a , Xiaochun Chen b a
College of Information Science & Technology, Beijing University of Chemical Technology, China College of Chemical Engineering, Beijing University of Chemical Technology, China c Changzheng Engineering Co., Limited, China b
A R T I C L E I N F O
A B S T R A C T
Keywords: Fuzzy nonlinear programming Piecewise linear membership functions Error analysis Iterative algorithm
In the fuzzy nonlinear programming (FNLP) domain, piecewise linear membership functions (PLMFs) are often employed to linearize the nonlinear membership functions. However, the linearization error will reduce the accuracy of the solution, which has become an obstacle to the application of PLMFs. In this paper, we present an iterative algorithm based on geometrical means to solve the problems mentioned above. Firstly, the linearization error can be reduced by gradually modifying the programming model. Subsequently, compared with traditional methods, the linearization model is determined only by weighing the linearization degree of membership function and the accuracy of programming results, the proposed method is more directly and effectively. Furthermore, indepth discussions are also given along with the algorithm for demonstrating the rationality. Finally, the proposed method is validated and compared with other methods. A real example is examined to demonstrate applicability of the proposed method in this paper.
1. Introduction Even though mathematical programming has become a well-known optimization method based on crisp models, it still remains less popular in industrial applications due to the uncertainty which occurs frequently in programming problems. For dealing with the uncertainty factors, fuzzy theory has been integrated into programming method, fuzzy decision and fuzzy linear programming theories were founded in 1970 and 1978 by Bellman & Zadeh [1] and Zimmermann [2] respectively. Recently, studies on fuzzy theory and fuzzy mathematical programming have been widely circulated in literature [3–7]. Since fuzzy models in according programming problems can be changed into crisp ones by means of suitable membership functions, the diverse types of membership function will lead to different properties of programming problem. In fuzzy nonlinear programming, nonlinear membership functions are employed to represent decision-makers’ (DM) subjective preferences. Taking advantage of the accurate description for fuzzy degree, nonlinear membership functions such as Gaussian membership functions are usually involved in practical problems. However, the involvement will inevitably leads to heavy calculation burden due to the severe nonlinear degree. Aiming at this problem, piecewise linear membership functions are introduced to decrease the nonlinearity by approximating the
nonlinear membership functions. The methodology that employs piecewise linear functions to approximate arbitrary shape nonlinear ones was firstly proposed by Stone [8], where the approximation accuracy is proportional to the number of segments. After that, many researchers have launched studies on this area [9–11]. In practical applications, piecewise linear functions also have been used extensively, such as stochastic sampling, pricing problem, real long-length noisy signals dealing, oblique survival tree and so on [12–15]. The functions have also been employed to discretize of occurring nonlinearities in mixed-integer nonlinear programs (MINLP), and solve the field of gas network optimization problem [16]. Rebennack and Kallrath also provided delta-approximations for solving MINLP, and his approach is approximating the nonlinear function from both sides [17]. Piecewise linear functions were firstly introduced to fuzzy programming by Narasimhan [18], which the linearization concept was used to deal with the complex nonlinear membership functions. However, although the solving difficulty of fuzzy programming with complex nonlinear membership functions can be simplified by PLMFs, the linear approximation would lead to a certain error between optimization results and the actual optimum values, which becomes a bottleneck of PLMF applications. Researchers have launched in-depth studies on the relations between segment numbers and approximation accuracy [19,20], where
* Corresponding author. E-mail address: [email protected] (B. Yang). https://doi.org/10.1016/j.chemolab.2019.06.007 Received 27 November 2018; Received in revised form 21 May 2019; Accepted 24 June 2019 Available online 25 June 2019 0169-7439/© 2019 Elsevier B.V. All rights reserved.
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the current research achievements still rest on decreasing the error by adding sub-segments rather than eliminating it authentically. Motivated by this challenge, this study proposes a novel iterative algorithm to eliminate the error which caused by PLMFs, and so as to make optimization solutions closer to the actual values or even equal with them completely. The rest of the paper is organized as follows. In section 2, the structure of PLMFs and causes of error have been discussed, based on that an iteration algorithm responsible for eliminate the error has been proposed. To analyze effectiveness of the proposed methodology, a plant case study is provided in Section 3. Finally, conclusion remarks are given in Section 4. 2. Error elimination 2.1. Approximation error A piecewise linear function is composed of several linear segments and knots. Thus, a nonlinear membership function with arbitrary shape can be approximated by changing quantity and position of the knots included in a PLMF. For instant, a nonlinear membership function gðxÞ is approximated by a 4 segments and 5 knots PLMF as shown in Fig. 1. A general replacement produces typically better errors (except for the quadratic function, where the equidistant placement is optimal, in terms of absolute error [21]). For minimizing the area between the nonlinear function and the piecewise linear function, for a given number of segments, by adjusting the placements of the knots [22]. According to the approximation property of PLMFs, it can be clearly seen that the replacing process will result in error (the gap between solid and dashed line). As shown in Fig. 2, gray areas represent the error between PLMFs and the original membership function, where the size of area is in proportion to the error. The error at any arbitrary point on linear segments is specified as the distance from this point to the original functions along the vertical direction. Therefore, each point on these linear segments is according to different errors. Taking the forth segment for example, which the amplified graph is shown in Fig. 3, the curve between points A and B is approximated with straight lineAB. The area enclosed by the straight line and the curve between A and B is according error area. Taking arbitrary point C on AB as an example, we draw a straight line perpendicular toAB through it, which intersects with curve y at pointC'. Thus, distance e from point C to C' signifies the error value corresponding to pointC. Thus, it is convincible that the optimal solutions obtained from fuzzy programming with PLMFs are somewhat different from those with original nonlinear membership functions. The error between a piecewise linear membership function and the original nonlinear one is proportional to the number of knots, which means the fitting degree between the two functions will increase with the
Fig. 2. Error areas in PLMFs.
Fig. 3. The error at arbitrary point.
knot adding. However, big amount of sub-segments would complicate the optimization model and ascend the calculation burden. In order to tradeoff these situations, it is crucial to appropriately choose the knot positions, which depend on the curvature, error and other factors. Even though, the error is only expected to be decreased rather than eliminated.
2.2. Iterative algorithms For the purpose of increasing the accuracy degree of the optimal solutions get from the programming problem with PLMFs, here suggests a novel iterative algorithm which can decrease or even eliminate the error. This algorithm is mainly based on geometrical means to achieve the error elimination purpose, Fig. 4 shows a flow chart of the algorithm along with a detailed implementing procedure as follow. Furthermore, the algorithm is based on the precondition which the inaccurate optimal satisfaction degree (λ) of programming problem has already been obtained. (1) Place the PLMFs and its original nonlinear membership function curve in a same rectangular coordinate system, and mark point λ on its according segment of the PLMFs. Specify this segment as the basis segment and the curve which approximated by this segment as the basis curve;
Fig. 1. The linear approximation of nonlinear function. 89
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(2) Construct a line A'B' parallel to line AB through point C', then the area surrounded by the original membership function along with below line A'B' is the new feasible region for λ, as shown in Fig. 6;
2.3. Additional remarks for the algorithm There are two supplementary remarks for the proposed iterative algorithm. For the convenience, we do a little modification to Fig. 5. Through point C' draw an auxiliary line parallel to the lateral axis, which intersects with A'B' at point D. Besides, label the abscissa as gðxÞ and ordinate as μðgðxÞÞ, as shown in Fig. 7. Firstly, the remark about the un-necessity of adding the constraint λ fCC' to additional constraints to prevent the λ point moving trend contrary to its optimal direction (moving to the left side) during iteration process. In programming problems, the role of constraints is to restrict the objective function via limiting decision variables. However, the real effective constraints are the active constraints. Therefore, in a standard Max programming, the objective function z will increase along with the adding of left-hand side function gðxÞin constraints (Min programming has the opposite situation). Such as point C and D in Fig. 7, although they have the same membership degree, but point D is corresponds to a larger objective function value, because the value of gðxÞ at D is bigger than C. Thus, if μðgðxÞÞ remains constant and gðxÞ gets bigger, μz will also increase accordingly. So the changing direction of the optimal point is from left to right, which means the location of the optimal point after iteration must in the right side area of last optimal one under the same constraint. According to the analysis above, the new optimal point should be located at the right side of C which is the current optimal point. Owing to the corresponding point of C is C'in original function, the new optimal area is shown as the dark area in Fig. 8. Thereupon, the constraint λ fCC' can be omitted. Secondly, the remark about the new optimal point location. By a large number of experiments, we found that the new optimal point is commonly located in the error area and getting closer to the actual one with the gradually increasing of λ. However, the optimal point will also occasionally locate outside the error area which beyond the feasible range of original function. Under this condition, the proposed methodology is also applicable. The only difference between the two conditions is the optimal point will approximate to the true value from the inside and outside of the error area respectively. It should be noted that in the iteration process of second condition, the value of λ is getting smaller which is opposite to the situation that the optimal point is inside of error area. However, both situations will make the final optimal value approximate to the true one.
Fig. 4. Algorithm flow chart.
(2) Erect a perpendicular, which will intersect with the basis curve, from point λ on the line. Construct a line parallel to the basis segment through the intersect point. Specify the line as a new constraint line, the area which below it becomes a new feasible region for λ; (3) Modify the feasible region by replacing the function of PLMFs with the one of the new constraint line; (4) Solve the new model and get the revised optimal solutions. Mark the revised point of λ on the new constraint line, and construct a line parallel to the basis segment through point λ as the new basis segment for the next iteration; (5) One-time iterative modification of the PLMFs is finished. If there exist other PLMFs, they can be modified by the above steps seriatim; (6) Circulate the above steps until the optimal solution is not changed anymore (represents the error is almost eliminated) or the required accuracy is achieved. Take a fuzzy programming problem with PLMFs which shown as follows for instance. Max
λ
s:t: λ μ1 ðg1 ðxi ÞÞ λ μ2 ðg2 ðxi ÞÞ λ μj ðgj ðxi ÞÞ
(1)
where g1 gj are original membership functions, μ1 μj are according PLMFs. We assume that the optimal solutions of this problem have been obtained. In order to eliminate the error brought by the approximation process of PLMFs, the iterative algorithm can be carried out by the following 5 steps. Since the PLMFs include several segments, we can fix the PLMFs from the first one. (1) Start the PLMFs error elimination from the first segmentμ1 ðxi Þ. Place the PLMFs and its original nonlinear membership function curve in one rectangular coordinate system. For convenience, we amplify the basis segment AB which optimal satisfaction degree point C locate in, as shown in Fig. 5;
Fig. 5. Amplification of segment AB 90
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Fig. 8. New optimal area.
Fig. 6. New feasible region. (3) Modify the model by replacing the constraint λ μ1 ðg1 ðxi ÞÞ with λ fA'B' ðg1 ðxi ÞÞ, and the model is converted into: Max
and MC is 98.42 C and 39.6 C respectively. Thus, the MC is collected in the top of the column, and HE is recovered in downstream unit. In order to save steam consumption and reaching the quality standard, the process engineer has built the fuzzy nonlinear programming model to obtain the optimal control strategy. The parameters in the model contain reflux ratio (calculated by feed flow and reflux flow), steam flow and bottom flow rate which listed in Table 1. The model is shown as follows. We initially obtained the optimum solutions of the corresponding crisp optimization problem as: x1 ¼ 0:431x2 ¼ 0:923x3 ¼ 2:276,z ¼ 43:164. For the comparison, this problem is solved with unilateral membership function, normal PLMFs and proposed methodology, respectively. Here, for building the object membership function of z, we set its expected value as 50.
λ
s:t: λ fA'B' ðg1 ðxi ÞÞ λ μ2 ðg2 ðxi ÞÞ λ μj ðgj ðxi ÞÞ
(2)
(4) Solve model (2) and obtain the new optimal solutions. Construct a line AB parallel to the basis segment through the new point. Line AB is the new basis segment of the next iteration; (5) Implement the steps on the remaining PLMFs, and circulate several times until the optimal solution is not changed anymore or the required accuracy is achieved.
(1) Construct bilateral membership function as constraint [23], then transform the fuzzy model into the crisp one. Max s:t:
λ λ μz ¼ z=50
arctanðg1 8Þ arctanðg1 8Þ þ 1Þμg1 ð½ þ 1Þ λ ð½ π =2 π =2 arctanðg2 7:5Þ arctanðg2 7:5Þ λ ð½ þ 1Þμg2 ð½ þ 1Þ π =2 π =2
μg1 ¼ e
ðg1 8Þ2 2p2 1
μg2 ¼ e
ðg2 7:5Þ2 2p2 2
(4)
where, p1 ¼ p2 ¼ 2 and the membership functions are both bilateral form, as shown in Fig. 10 and Fig. 11. The optimum solutions can be obtained by solving this model with GAMS (The General Algebraic Modeling System, a high-level modeling system for mathematical optimization): x1 ¼ 0:492, x2 ¼ 0:969, x3 ¼ 2:408, g1 ¼ 8:909, g2 ¼ 8:409, λ ¼ 0:902, z ¼ 45:092, μg1 ¼ μg2 ¼ μz ¼ 0:902. In the situation of μg1 ¼ μg2 ¼ 0:902, (4) can be simplified and rearranged as follows.
Fig. 7. Supplementary figure.
3. An illustrative example Consider a study case of Heptane (HE) & Methylene Chloride (MC) separation column, as shown in Fig. 9. The feed flow of this column is the mixed solvent of HE and MC, they are separated by the heating of 6 Bar steam (S6). The boiling point of HE
Max
91
f ¼λ
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Fig. 9. HE & MC separation column. Max
z ¼ x21 x22 2x23 þ 5x1 þ 5x2 þ 21x3
s:t:
g1 ¼ 1:03x21 þ 0:95x22 þ x23 þ x1 0:98x2 þ 1:07x3 ~8
g2 ¼ 0:97x21 þ 1:95x22 þ x23 þ 1:02x1 7:~5
(3)
Table 1 Corresponding relationship. Controlled Variable
Formula
Parameter
Unit
Reflux Ratio Steam Flow Bottom Flow Rate
¼ Reflux Flow/Feed Flow – –
x1 x2 x3
– t/h t/h
Fig. 11. Bilateral membership function of g2 ðg2 7:5Þ2 2p2 2
ω3 ¼ λ e
0
(5)
At the optimal point, the active constraints must satisfy Karush-KuhnTucker (KKT) conditions which are also known as the first order necessary conditions of optimality. Regarding (5), its KKT conditions are expressed by: Fig. 10. Bilateral membership function of g1
s:t:
3 ∂f X ∂ω η i¼0 ∂λ i¼1 i ∂λ
ω1 ¼ λ ð x21 x22 2x23 þ 5x1 þ 5x2 þ 21x3 Þ 50 0
ω2 ¼ λ e
ðg1 8Þ2 2p2 1
3 ∂f X ∂ω η i¼0 ∂xj i¼1 i ∂xj
0
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η1 ðω1 0Þ ¼ 0 η2 ðω2 0Þ ¼ 0 η3 ðω3 0Þ ¼ 0
(6)
where η1 η2 ,η3 are the vectors of KKT multipliers corresponding to each constraint. By solving equation (6), the solutions can be obtained as η1 ¼ 0:7938, η2 ¼ 0:1906, η3 ¼ 0:0156, which proves the optimality of the solutions. (2) Construct PLMFs on the basis of the bilateral form as constraints [19]. Choose the key knots in the membership function μg1 as (8, 1) (10, 0.607) (11.5, 0.2163) (15, 0) and μg2 as (7.5, 1) (9.52, 0.6) (11, 0.2163) (14.5, 0). Thus, construct PLMFs based on the key knots as shown in Fig. 12 and Fig. 13 respectively. According to the methodology proposed in paper [24], modeling the fuzzy programming problem as: Max s:t:
Fig. 13. PLMFs of g2
λ
Max
λ μz ¼ z=50
f ¼λ
ω1 ¼ λ 1
s:t:
λ1
ω2 ¼ λ þ 0:2g1 2:6
λ 0:2g1 þ 2:6 þ δ1 M
ω3 ¼ λ þ 0:2g2 2:5
λ 0:256g1 þ 3:165 þ δ1 M
ω4 ¼ λ ð x21 x22 2x23 þ 5x1 þ 5x2 þ 21x3 Þ 50 0
(8)
The KKT conditions of (8) are expressed by:
λ 0:062g1 þ 0:927 þ ð1 δ1 ÞM
4 ∂f X ∂ω η i¼0 ∂λ i¼1 i ∂λ
λ 0:2g2 þ 2:5 þ δ2 M λ 0:921g2 þ 10:25 þ δ2 M λ 0:0618g2 þ 0:8961 þ ð1 δ2 ÞM
4 ∂f X ∂ω η i¼0 ∂xj i¼1 i ∂xj
(7)
where δ1 and δ2 are binary variables. This problem is solved by GAMS, resulting in optimum solutions as: x1 ¼ 0:469x2 ¼ 0:952x3 ¼ 2:358g1 ¼ 8:562g2 ¼ 8:062λ ¼ 0:888z ¼ 44:378, δ1 ¼ δ2 ¼ 0. In the situation of δ1 ¼ δ2 ¼ 0 and according with the optimum solution, (7) can be simplified and re-arranged as follows.
η1 ðω1 1Þ ¼ 0 η2 ðω2 2:6Þ ¼ 0 η3 ðω3 2:5Þ ¼ 0 η4 ðω4 0Þ ¼ 0
(9)
where η1 , η2 , η3 , η4 are the vectors of KKT multipliers corresponding to each constraint. By solving equation (9), the solutions can be obtained as: η1 ¼ 0, η21 ¼ 0:1396, η3 ¼ 0:0336, η4 ¼ 0:8268, which proves the optimality of the solutions. (3) Optimize the solutions obtained from equation (7) by the proposed iterative algorithm. Firstly, do the first iteration to μg1 ¼ 0:2g1 þ 2:6. Through the optimum point (8.562,0.888) draw a straight line M1 perpendicular to this segment, the according linear equation is μM1 g1 ¼ 5g1 41:922. Gaussian membership function intersects with M1 at point A1 which the coordinate is (8.576, 0.958). Through point A1 draw a straight line K1 parallel to μg1 , the according linear equation is μK1 g1 ¼ 0:2g1 þ 2:6732, as shown in Fig. 14. According to the proposed methodology, replace the constraint of membership function μg1 for λ to the constraint of line K1 , the model is changed into: Max
Fig. 12. PLMFs of g1 93
λ
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Table 2 The solutions of FNLP. Bilateral style
PLMFs
First iteration forμg1
First iteration forμg2
Second iteration for
Second iteration for
μg1
μg2
x1 ¼ 0:492 x2 ¼ 0:969 x3 ¼ 2:408 g1 ¼ 8:909 g2 ¼ 8:409 λ ¼ 0:902
x1 ¼ 0:469 x2 ¼ 0:952 x3 ¼ 2:358 g1 ¼ 8:562 g2 ¼ 8:062 λ ¼ 0:888 z ¼ 44:378
x1 ¼ 0:482
x1 ¼ 0:489
x1 ¼ 0:491
x1 ¼ 0:492
x2 ¼ 0:845
x2 ¼ 0:958
x2 ¼ 0:943
x2 ¼ 0:968
x3 ¼ 2:424
x3 ¼ 2:403
x3 ¼ 2:413
x3 ¼ 2:408
g1 ¼ 8:882
g1 ¼ 8:866
g1 ¼ 8:908
g1 ¼ 8:909
g2 ¼ 8:016
g2 ¼ 8:34
g2 ¼ 8:333
g2 ¼ 8:408
λ ¼ 0:897
λ ¼ 0:9
λ ¼ 0:901
λ ¼ 0:902
z ¼ 44:838
z ¼ 44:995
z ¼ 44:065
z ¼ 45:092
z ¼ 45:092
Fig. 14. Analysis of μg1
s:t:
s:t:
λ 0:2g2 þ 2:568
λ 0:2g1 þ 2:6732
λ μz
λ 0:2g2 þ 2:5 λ μz
(11)
Solved by GAMS, the optimum solutions are: x1 ¼ 0:489, x2 ¼ 0:958, x3 ¼ 2:403, g1 ¼ 8:866, g2 ¼ 8:34, λ ¼ 0:9, z ¼ 44:995. Similarly, the optimal point is locate on line K2 . The first iteration for μg2 is finished. Do the second iteration to μg1 and μg2 , respectively. After that, as the optimum solution is almost unchanged, the iteration is over. The solutions of three modeling methodologies are presented in Table 2. By comparison, the solution gets from the model constructed by PLMF is smaller than the one get from the model constructed by bilateral membership function, because the certain errors existing between PLMF and the original one. Yet with the iteration algorithm proposed by this paper, the error caused by the PLMF is nearly eliminated after 4 time iterations. The changing of optimal point position of g1 and g2 in the 4 iterations is shown in Fig. 16 and Fig. 17, where point 1–4 are the optimal points after the each iteration and point O and point F are the optimal points get from the model constructed by PLMF and bilateral membership function respectively. Because the optimal solution is very close to the true value after 4 iterations, so the position of optimal point 4 is almost coincide with point F. From point positions in the figures, it can be seen that if one constraint is iterated then another will be affected according to the changes of xi , and another constraint's optimal point
(10)
Solved by GAMS, the optimum solutions are: x1 ¼ 0:482, x2 ¼ 0:845, x3 ¼ 2:424, g1 ¼ 8:882, g2 ¼ 8:016, λ ¼ 0:897, z ¼ 44:838. Through the coordinate, we know the optimal point is locate on line K1 . The first iteration for μg1 is finished. Secondly, do the first iteration to μg2 base on the above results. Similarly, through the optimal point (8.016, 0.897) in basis segment μg2 ¼ 0:2g2 þ 2:5 draw a straight line M2 perpendicular to this segment, the linear equation is μM2 g2 ¼ 5g2 39:183. Gaussian membership function intersects with M2 at point A2 which coordinate is (8.03, 0.962). Through point A2 draw a straight line K2 parallel to μg2 , the linear equation is μA2 g2 ¼ 0:2g2 þ 2:568, as shown in Fig. 15. According to the proposed methodology, replace the constraint of membership function μg2 for λ to the constraint of line K2 , the model is changed into: Max
λ 0:2g1 þ 2:6732
λ
Fig. 15. Analysis of μg2
Fig. 16. The location changing of optimum point in.g1 94
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Science Foundation of China No. 11702016. References [1] R.E. Bellman, L.A. Zadeh, Decision-making in a fuzzy envrionment, Manag. Sci. 17 (1970) 141–164. [2] H.J. Zimmermann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets Syst. 1 (1978) 45–55. [3] Bo Yang, Hongguang Li, A nove dynamic timed fuzzy Petri nets modelling method with applications to industrial process, Expert Syst. Appl. 97 (1) (2018) 276–289. [4] Bo Yang, Hongguang Li, A similarity elastic window based approach to process dynamic time delay analysis, Chemonetr. Intell. Lab. Sys. 170 (2017) 13–24. [5] Zhao Dong, Steven X. Ding, Hamid Reza Karimi, Yueyang Li, Youqing Wang, On robust Kalman filter for two-dimensional uncertain linear discrete time-varying systems: a least squares method, Automatica 99 (2019) 203–212. [6] Yu-Chung Tsao, Vo-Van Thanh, Jye-Chyi Lu, Yu Vincent, Designing sustainable supply chain networks under uncertain environments: fuzzy multi-objective programming, J. Clean. Prod. 174 (2018) 1550–1565. [7] Sohrab Moradi, Soleiman Mohammadi Limaei. Multi-objective game theory model and fuzzy programming approach for systainable watershed management, Land Use Policy 71 (2018) 363–371. [8] Stone Henry, Approximation of curves by line segments, Comput. Informat. Sci. 15 (1961) 40–47. [9] Luis Ibarra, Mario Rojas, Pedro Ponce, Arturo Molina, Type-2 fuzzy membership function design method through a piecewise-linear approach, Expert Syst. Appl. 42 (2015) 7530–7540. [10] Takashi Hasuike, Hideki Katagiri, Hiroe Tsubaki, A constructing algorithm for appropriate piecewise linear membership function based on statistics and information theory, Proc. Comput. Sci. 60 (2015) 994–1003. [11] Bo Wen, Hongguang Li, An approach to formulation of FNLP with complex piecewise linear membership functions, Chin. J. Chem. Eng. 22 (2014) 411–417. [12] Junbin Li, Renhong Wang, Min Xu, Fang Qin, Piecewise linear approximation methods with stochastic sampling sites, J. Comput. Appl. Math. 328 (2018) 173–178. [13] Qian Hu, Wenbin Zhu, Qin Hu, Andrew Lim, A branch-and-price algorithm for the two-dimensional vector packing problem with piecewise linear cost function, Eur. J. Oper. Res. 260 (2017) 70–80. [14] Matteo Rucco, ect, A new topological entropy-based approach for measuring similarities among piecewise linear functions, Signal Process. 134 (2017) 130–138. [15] Malgorzata Kretowska, Piecewise-linear criterion functons in oblique survival tree induction, Artifical Intell. Med. 75 (2017) 32–39. [16] Robert Burlacu, Solving mixed-integer nonlinear programmes using adaptively refined mixed-integer linear programme, Optim. Methods Software (2019) 1–28. [17] S. Rebennack, J. Kallrath, Continuous piecewise linear delta-approximations for univariate functions: computing minimal breakpoint systems, J. Optim. Theory Appl. 167 (2) (2015) 617–643. [18] R. Narasimhan, Goal programming in a fuzzy environment, Decision science 11 (1980) 325–336. [19] Elber Gershon, Error bounded piecewise linear approximation of freeform surfaces, Comput. Aided Des. 28 (1996) 51–57. [20] A. Imamoto, B. Tang, Optimal piecewise linear approxiamtion of convex functions, Proc. World Cong. Eng. Comput. Sci. 2008 (2008) 22–25. [21] S. Rebennack, Computing tight bounds via piecewise linear functions through the example of circle cutting problems, Math. Methods Oper. Res. 84 (1) (2016) 3–57. [22] Josef Kallrath, R. Steffen, Computing area-tight piecewise linear overestimators, underestimators and tubes for univariate funtions. Optimization in Science and Engineering, in: Honor of the 60th Birthday of Panos M. Pardalos., 2014, pp. 273–292. [23] Bo Wen Hongguang Li, Approaches to building piecewise linear membership functions in nonlinear fuzzy programming, CIE J. 62 (8) (2011) 2258–2264. [24] Bo Wen Hongguang Li, Modeling methodology for fuzzy programming with piecewise linear membership functions, CIE J. 61 (8) (2010) 2149–2153.
Fig. 17. The location changing of optimum point in g2
position will move slightly to left. The happening of this situation is caused by the coupling relationship between the two constraints, which is not contrary to the additional remark mentioned in Chapter 2.3, and the position of optimal point will keep moving to the right side and getting closer to the true value during each iteration process. Furthermore, this practical example is relatively simple, which its model with bilateral form membership functions can be solved easily, so as to illustrating the effectiveness of the given algorithm. However, when facing some complex practical problems which can only be solved with PLMFs the algorithm will play a bigger role. 4. Conclusion In this paper, the structure of PLMF is analyzed in detail, and the linearization error of the non-linear membership function is also explained firstly. Then, a novel iteration algorithm has been proposed to reduce the linearization error based on geometrical means. As the programming model is iteratively modified, the optimal solution will gradually approach the true value. Compared with the other conventional modeling methodologies through a practical case, it has been proved that the given contribution can provide a more accurate solution. Then, the programming method can be applied in the area which requires more precise solutions. At last, the effectiveness of the proposed method also been verified by a case of Heptane (HE) & Methylene Chloride (MC) separation column. Acknowledgment This research is supported by the Fundamental Research Funds for the Central Universities with Grant No. ZY1929 and the National Natural
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