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Method for solving nonlinear goal programming with interval coefficients using genetic algorithm
Computers ind. Engng Vol. 33, Nos 3--4, pp. 597-600, 1997 0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0360-8352/97 $17.00 + 0.00
Pergamon PH: S0360-8352(97)00201-5
Method for Solving Nonlinear Goal Programming with Interval Coefficients using Genetic Algorithm Takeaki Taguchi
KenichiIda
Mitsuo Gen
Department of Industrial & Systems Engineering Ashikaga Institute of Technology 268 Ohmae-cho, Ashikaga, 326 JAPAN
Abstract Traditional formulations on reliabilityoptimization problems have assumed that the coefficients of models are known as fixed quantities and reliabilitydesign problem is treated as deterministic optimization problems. Because that the optimal design of system reliabilityis resolved in the same stage of overall system design, model coefficientsare highly uncertainty and imprecision during design phase and it is usually very difficultto determine the precise values for them. However, these coefficients can be roughly given as the intervals of confidence. In this paper, we formulated reliabihty optimization problem as nonlinear goal programming with interval coefficients and develop a genetic algorithm to solve it. The key point is how to evaluate each solution with interval data. We give a new definition on deviation variables which take interval relation into account. Numerical example is given to demonstrate the efficiency of the proposed approach. © 1997 Elsevier Science Ltd K e y W o r d s : nonlinear goal programming, genetic algorithm, interval coefficients.
1
Introduction
2
In this paper, the technique to solve the nonlinear integer goal programming with interval coefficients using genetic algorithm more efficiently is proposed. Numerical example is given to demonstrate tile efficiency of the proposed approach.
Model
A typical formulation of Interval Nonlinear Integer Goal Programming (I-NIGP) is given as follows:
As one of solution methods for a muttiobjective decision making (MODM) problem, there is a goal programming[4]. If a way of thinking of this goal programming was applied to a system reliability optimization problem, it is frequently formulated as a nonlinear goal programming problem. Formerly, though they dealt with reliability as definite value, when realistic optimal design of system reliability to value this as a reliable interval, it is one of an important element[I, 5, 6]. Genetic algorithm (GA) - a kind of stochastic search method - is regarded as the powerful solution method in searching best solutions for complexes function optimization problems and large scale combinatorial optimization problems[2, 3].
Interval NIGP
min
f i f i Pt(w'dd ~- + w,+d+)
(1)
/ = 1 i----1
s.t.
Gi(z) = Ai(z) + d~- - d + = Bi, i= 1,2,...,m (2)
z=[xl z~ x.] li < z i < uI :integer d~,d + > 0 , i = 1 , 2 , . - . , m ...
xi
""
(3) (4)
where,Pi is the I-th preemptive priority (Pa >> Pt+l for all l), d+ is the positive deviation variable representing the over-achievement of goal i, d~" is the negative deviation variable representing the under-achievement of goal i, wti+ is the positive weight assigned to d + at priority PI, w~ is the negative weight assigned to d~" at priority/~, Ai(z) = [ai(:~)L,ai(z) R] is the nonlinear function with interval coefficients of goal constraints i, Bi = [biL,bi R] is the interval target value ac-
597
Proceedings of 19961CC&IC
598
cording to goal i, ljanduy are the lower and upper bound for xy. We sometimes rewrite the objective function (1) as follows:
lexmin
{ fi zx =
(w~d? + w + d +) ,...,
where ai(z) c is center of interval A i ( z ) , bt ¢ is center of interbal Bi, and p is the weighting factor of order relation between interval numbers in range [0,1]. With p more close to 0, more pessirnistic cases axe considered. Similarly, we can consider the expected cases with p more close to 1.
i=1
-
}
3.3
3.1
¢
G A for I - N I G P
~t(vk) = ~ P , zz, k--1,2,...,pop_~ize (9) I--1
Representation
For a given problem with n decision variables, we use a vector x = [xtz2... x~] as a chromosome to represent a solution to the problem. A chromosome, denoted as v~(k = 1,2,..-,pop.size) is an ordered list of n genes as follows: vk =
3.2
[xklx k 2
Deviation
...
However, zt is given by equation (1) and equation (6) - (8) corresponding to a type of a goal. Moreover, Pt is the weight which considered a priority level, and it is defined as follows: Pz =
x~.]
Variables
There are three types of decision making in goal programming: 1)which is to be optimized under goals, 2)which is to be optimized over goals, and 3)which is to be optimized to be close to goals. For the I-NIGP problems, we define the deviation variables in the following way: Moreover, the definition and the order relation for interval numbers presented by Ishibuchi and Tanaka[l], interval maximum/minimum problems correspond to maximizing/minimizing the center and left bound / right bound of the interval number.
¢k
3.4
(ii) selection
(6) Algorithm
The solving procedure of GA for interval NIGP problems is illustrated as follows:
:
(7)
3) which is to be optimized to be close to goals:
0
+ (1 - p) (max{0, a i ( x ) R -- bi R} + max{O, b, L - ai(=)z})
Operations
Crossover is implemented with uniform crossover operator and mutation is performed as random perturbation with in the permissive range of integer variables[4].
3.5
+d,+ =
Genetic
(i) crossover and mutation
Deterministic selection is used, that is, delete all duplicate among parents and offspring and them in ascending order and select the first pop.size chromosomes as the new population.
2) which is to be optimized over goals:
+ (1 - p) (max{0, blL - ai(z)L})
(10)
v* = argrtun{eval(vk)]k = 1, 2,...,pop.size} (11)
b,c )
+ (1 - p) (max{O, ai(:e)/~ - biR))
10~*(q-0, 1 = 1 , 2 , . . . , q
A chromosome whose eval(vk) is the smallest has the best fitness in the problem. The best chromasome v* is derived from:
1) which is to be optimized under goals:
t/_- p
Function
The evaluation function is defined in consideration of a priority level as follows:
i=l
3
Evaluation
(8)
Step 1: Set population size pop_size, crossover probability pc, mutation probability pro, maximum generation mazgen, weight p of orderly valuation of interval Step 2: Set initial generation 9en = 0. Generate initial population vk(k = 1,2,-. • ,pop~ize) with n genes randomly. However, the scope which each decision variable zj can take is tj
599
Proceedings of 1996 I C C & I C
Table 1: Interval coefficnts of optimal design problem for system reliability De= pa
Subsystem t
2
1
[1.0,1.2] [1.9,2.3] (1.4, 2.31 [2.5,3.01
[3.0, 3.5]
1
[0.87, 0.90]
2 3 4 5 6 `7
[0.95,0.97]
8
[0.81,0.87]
[2.4, 3.0]
9 10 11 12 13 14
[1.'7, 2.1]
[7.4, 8.1]
[0.83,0.87'] [0.90, 0.96] [0.'/9, 0.86] [0.97, 0.99]
[3.8, 4.21 [2.5, 3.8] [2.0, 2.5] [I.8, 2.2] [3.0,4.8]
[4.8, 6.7] [5.0, 5.5] [4.0, 5.0] [4.8, 5.31 [5.5,6.0]
Subsystem t 1 2 3 4 5 6 7 8 9 10 11 12 13 14
[0.84,0.87] [0.83,0.84]
[0.90, 0.96]
[0.95,0.99] [0.90, 0.921 [0.93, 0.9'7]
[0.90,0.921
[I.7, 2.2] [2.5,3.41
[4.0, 4.11
[7.1,9.6]
[5.2,7.01
[4.1,5.81
[3.5, 4.5]
[3.8,5.2]
[7.0, '7.5]
[4.0, 6.0]
92, 0.93] (0~,0.981 [0.90,0.931
[0.8,1.31 [0.7,1.01 [2.2,3.01
[3.2, 4.01 [r.5,10.01 [5.0,6.81
[0.86,0.89] [0.92, 0.931 [0.97, 0.98] [0.92,0.94] [0.88,0.90]
[3.2,4.4] [1.5, 2.4] [2.8, 3.0] [3.8, 4.5] [4.5, 5.81
[0.83, 0.85]
[3.4,4.5] [3.4, 4.0]
[5.3,6.1] [2.9, 4.01 [3.8, 4.2] [7.5, 8.01 [6.2, 7.0] [8.5, 9.01 [4.7, 5.8]
[0.97, 0.991
[0.94, 0.95] [O.80, 0.82] [0.94, 0.99] [0.88, 0.96]
3
t't4Ltrt4 ~
[0.91,0.93]
[1.7, 2.0]
[1.0, 1.5] [1.o, 1.61
[8.2,9.o1
[0.85, 0.88] [0.95, 0.971
[4.8, 5.1]
[4.0, 4.9]
[0.96, 0.98]
[0.90, 0.96]
[0.86, 0.93] [0.95, 0.98] [0.87, 0.91] [0.93, 0.98] [0.80, 0.87] [0.95, 0.981 [0.90, 0.95]
[2.3, 3.0]
[2.0, 2.6] [4.3, 5.51 [5.4, 6.1] [3.1, 4.0] [4.6, 5.01 [4.8, 5.2] [3.5, 4.0]
[1.4, 3.0]
[4.7, 5.5]
[4.9, 6.81 [e.o, 6.o1
[2.6, 3.8] [2.7, 3.6]
[5.0, 5.8]
[6.0, 7.0]
[3.8, 4.4] 4
[~,3~,,,3~1 to,3
[2.4, 3.01
[5.4, 6.21 [4.1,5.0] [4.8, 5.4]
[8.1,9.ol [5.6,6.7"] [87,761 [5.8,6.ol [5.4,6.2] [6.0,8.5]
Is 7,6.o1
[5.6,7.1]
.92, 0.97]
[1.9, 2.3]
[3.8, 5.5]
[0.89, 0.95]
[3.0, 4.2]
[4.0, 5.0]
[0.95, 0.97]
[1.8, 2.1]
[3.0, 4.9]
[0.90, 0.931
[2.4, 3.81
[7.0, 8.6]
{0.89,0.911
[41, s 21
[8.s, 7.01
[0.94,0.99]
[5.4,6.0]
[7.9, 9.0]
RQ = [0.95, 0.9'7],G'Q == [110, 150], WQ = [150,190], -
S t e p 3: Set g e n = g e n + 1. Calculate Ai(=) by interval arithmetics['?] and calculate deviation variables by equation (6) - (8). In addition, they ask for evaluation function eval(vk) by equation (9). Moreover, equation (11) is used and the best chromosome v* is preserved.
: No design
thinks about the following three goals concerning this system. (I) System reliability [0.95,0.971.
is
established
over
(II) Unused quantity of system resource 1 is lost. S t e p 4: Reproduce new chromosomes during uniform crossover and mutation process and perform the Elitist Selection[4]. S t e p 5: If g e n < m a z g e n , return to S t e p 3. If g e n = m a z g e n , output v* and terminate.
(III) Excessive use of system resource 2 is prevented. An I-NIGP of this case is formulated as follows:
lexmin
4
Numerical
Example
The optimal design for system reliability problem with interval coefficients that chooses the best alternative design and the best corresponding red u n d a n t units for subsystems. The system consists of 14 subsystems and their alternative designs are shown in Table 1. Now, a decision maker
{ z l = d ? , 32 = d ~ , 33 = d + }
14 s.t. t----1
+ d r - dt = Rq
14 O2(lrn, o~) = E Cta, rnt t--1 +a~ - 4 = cq
600
Proceedings of 1996 ICC&IC 14
G ~ ( , , , ~ ) = ~ W,,,mt t=l
+d~-~=wq 0 < m t < 6 :integer, t = 1 , . . . , 1 4
Table 2: Results with various p p 0.0
1 < at
0.1 0.3 0.4 0.9 1.0
i 1 2 3 1 2 3 ] 2 3 1
g, o" 122.1000 179.2000 0.966614 130.2000 184.4500 0.9657]5 130.1000 184.1600 0.961774
[g, = ,g, ~] [0.950862,0.981036] [111.5000,132.7000] [166.0000,192.4000] [0.952145,0.979083] [118.0000,142.4000] [171.6000,197.4000] [0.9,51266,0.980164] [120.0000,140.2000] [170.7000,197.6000] [0.943871,0.979677]
2 3
130.1500 182.5500
[120.3000,140.0000] [167.6000,197.5000]
0.965949
RQ, C o and Wo are interval target values of each goal constraint, and denoted as R~ =
[~oL,~oR],cq = [cqL, cqR],Wq = [~qL, ~qR].
However, a gene denoted as v~t consists of two kinds of variables: alternative design v ~ ) and redundant unit v ~ ) for subsystem t. A chromosome, denoted as Vk(k= 1,... ,pop_size), is an ordered list of T genes. We solve the optimal design for system reliability problem with our proposed algorithm. The parameters are set as follows: population size: pop.size = 20 crossover probability: pC = 0.4 mutation probability: p m = 0.1 maximum generation: maxgen = 500 weight: p = 0.5 We apply the proposed method to this problem and repeat the process for 500 generations. At the 251-th generation of the chromosome and goal function variables the following results were obtained: y*
:1"(3,3)(1,2)(4,3)(3,3)(2,2)(2,2)(1,2) (3,3) (3,2) (3,3)(3,2) (4,3) (1,2) (3,2)] eval(V*) = 10.874992 * goal functions * zl
O. 000000
z2 z3
0. 000000 10. 874992
Table 2 shows that various sizes of weight p(0 < p < 1) have influences on wlole system concerned in this numerical example.
5
Conclusions
In this paper, the technique to solve nonlinear integer goal programming with interval coefficients using genetic algorithm more efficiently was proposed. The proposed method makes the optimal design of system reliability done more flexible and
more practical, since it considered the vagueness of designer's measurement and the vagueness of data resulted from the situations changes by introducing the interval of goals and interval coefficients. The subjective preferences of decision makers were shown in problem solving by defining deviation variables with the weighting factor of order relation between interval numbers for INIGP problems. Also the method is easy to apply nonlinear optimization problems with holding their nonlinearity, without any transformations of nonlinear model to linear model.
References [1] Ishibuchi, H. & H. Tanaka: Formulation and analysis of liner programming problem with interval coefficients, Journal o] Japan Industrial Management Association, Vol. 40, No.5, pp. 320-329, 1989 (in Japanese). [2] Goldberg, D. : Genetic Algorithms in Search, Optimization, and Machine Learning, AddisonWesley, 1989. [3] Michalewicz, Z. : Genetic Algorithms + Data Structures = Evolution Programs. SpringerVerlag, 1992. [4] Gen, M. & R. Cheng : Genetic Algorithms and Engineering Design, John Wiley & Sons, New York, 1996. [5] Gun, M. & R. Cheng : Interval Programming Using Genetic Algorithms, Proc. o] l~th Fuzzy System Symposium, pp.861-864, 1996. [6] Yokota, T., M. Gun, K. Ida & T. Taguchi: Optimal design of system reliability by an improved genetic algorithm, Transactions of Institute o] Electronics, In]ormation and Computer Engineering, Vol.J78-A,No.6,pp.702-709, 1995(in Japanese). [7] Hansen, E. :Global Optimization Using lnterval Analysis, Marcel dekker Inc., 1992.
Pergamon PH: S0360-8352(97)00201-5
Method for Solving Nonlinear Goal Programming with Interval Coefficients using Genetic Algorithm Takeaki Taguchi
KenichiIda
Mitsuo Gen
Department of Industrial & Systems Engineering Ashikaga Institute of Technology 268 Ohmae-cho, Ashikaga, 326 JAPAN
Abstract Traditional formulations on reliabilityoptimization problems have assumed that the coefficients of models are known as fixed quantities and reliabilitydesign problem is treated as deterministic optimization problems. Because that the optimal design of system reliabilityis resolved in the same stage of overall system design, model coefficientsare highly uncertainty and imprecision during design phase and it is usually very difficultto determine the precise values for them. However, these coefficients can be roughly given as the intervals of confidence. In this paper, we formulated reliabihty optimization problem as nonlinear goal programming with interval coefficients and develop a genetic algorithm to solve it. The key point is how to evaluate each solution with interval data. We give a new definition on deviation variables which take interval relation into account. Numerical example is given to demonstrate the efficiency of the proposed approach. © 1997 Elsevier Science Ltd K e y W o r d s : nonlinear goal programming, genetic algorithm, interval coefficients.
1
Introduction
2
In this paper, the technique to solve the nonlinear integer goal programming with interval coefficients using genetic algorithm more efficiently is proposed. Numerical example is given to demonstrate tile efficiency of the proposed approach.
Model
A typical formulation of Interval Nonlinear Integer Goal Programming (I-NIGP) is given as follows:
As one of solution methods for a muttiobjective decision making (MODM) problem, there is a goal programming[4]. If a way of thinking of this goal programming was applied to a system reliability optimization problem, it is frequently formulated as a nonlinear goal programming problem. Formerly, though they dealt with reliability as definite value, when realistic optimal design of system reliability to value this as a reliable interval, it is one of an important element[I, 5, 6]. Genetic algorithm (GA) - a kind of stochastic search method - is regarded as the powerful solution method in searching best solutions for complexes function optimization problems and large scale combinatorial optimization problems[2, 3].
Interval NIGP
min
f i f i Pt(w'dd ~- + w,+d+)
(1)
/ = 1 i----1
s.t.
Gi(z) = Ai(z) + d~- - d + = Bi, i= 1,2,...,m (2)
z=[xl z~ x.] li < z i < uI :integer d~,d + > 0 , i = 1 , 2 , . - . , m ...
xi
""
(3) (4)
where,Pi is the I-th preemptive priority (Pa >> Pt+l for all l), d+ is the positive deviation variable representing the over-achievement of goal i, d~" is the negative deviation variable representing the under-achievement of goal i, wti+ is the positive weight assigned to d + at priority PI, w~ is the negative weight assigned to d~" at priority/~, Ai(z) = [ai(:~)L,ai(z) R] is the nonlinear function with interval coefficients of goal constraints i, Bi = [biL,bi R] is the interval target value ac-
597
Proceedings of 19961CC&IC
598
cording to goal i, ljanduy are the lower and upper bound for xy. We sometimes rewrite the objective function (1) as follows:
lexmin
{ fi zx =
(w~d? + w + d +) ,...,
where ai(z) c is center of interval A i ( z ) , bt ¢ is center of interbal Bi, and p is the weighting factor of order relation between interval numbers in range [0,1]. With p more close to 0, more pessirnistic cases axe considered. Similarly, we can consider the expected cases with p more close to 1.
i=1
-
}
3.3
3.1
¢
G A for I - N I G P
~t(vk) = ~ P , zz, k--1,2,...,pop_~ize (9) I--1
Representation
For a given problem with n decision variables, we use a vector x = [xtz2... x~] as a chromosome to represent a solution to the problem. A chromosome, denoted as v~(k = 1,2,..-,pop.size) is an ordered list of n genes as follows: vk =
3.2
[xklx k 2
Deviation
...
However, zt is given by equation (1) and equation (6) - (8) corresponding to a type of a goal. Moreover, Pt is the weight which considered a priority level, and it is defined as follows: Pz =
x~.]
Variables
There are three types of decision making in goal programming: 1)which is to be optimized under goals, 2)which is to be optimized over goals, and 3)which is to be optimized to be close to goals. For the I-NIGP problems, we define the deviation variables in the following way: Moreover, the definition and the order relation for interval numbers presented by Ishibuchi and Tanaka[l], interval maximum/minimum problems correspond to maximizing/minimizing the center and left bound / right bound of the interval number.
¢k
3.4
(ii) selection
(6) Algorithm
The solving procedure of GA for interval NIGP problems is illustrated as follows:
:
(7)
3) which is to be optimized to be close to goals:
0
+ (1 - p) (max{0, a i ( x ) R -- bi R} + max{O, b, L - ai(=)z})
Operations
Crossover is implemented with uniform crossover operator and mutation is performed as random perturbation with in the permissive range of integer variables[4].
3.5
+d,+ =
Genetic
(i) crossover and mutation
Deterministic selection is used, that is, delete all duplicate among parents and offspring and them in ascending order and select the first pop.size chromosomes as the new population.
2) which is to be optimized over goals:
+ (1 - p) (max{0, blL - ai(z)L})
(10)
v* = argrtun{eval(vk)]k = 1, 2,...,pop.size} (11)
b,c )
+ (1 - p) (max{O, ai(:e)/~ - biR))
10~*(q-0, 1 = 1 , 2 , . . . , q
A chromosome whose eval(vk) is the smallest has the best fitness in the problem. The best chromasome v* is derived from:
1) which is to be optimized under goals:
t/_- p
Function
The evaluation function is defined in consideration of a priority level as follows:
i=l
3
Evaluation
(8)
Step 1: Set population size pop_size, crossover probability pc, mutation probability pro, maximum generation mazgen, weight p of orderly valuation of interval Step 2: Set initial generation 9en = 0. Generate initial population vk(k = 1,2,-. • ,pop~ize) with n genes randomly. However, the scope which each decision variable zj can take is tj
599
Proceedings of 1996 I C C & I C
Table 1: Interval coefficnts of optimal design problem for system reliability De= pa
Subsystem t
2
1
[1.0,1.2] [1.9,2.3] (1.4, 2.31 [2.5,3.01
[3.0, 3.5]
1
[0.87, 0.90]
2 3 4 5 6 `7
[0.95,0.97]
8
[0.81,0.87]
[2.4, 3.0]
9 10 11 12 13 14
[1.'7, 2.1]
[7.4, 8.1]
[0.83,0.87'] [0.90, 0.96] [0.'/9, 0.86] [0.97, 0.99]
[3.8, 4.21 [2.5, 3.8] [2.0, 2.5] [I.8, 2.2] [3.0,4.8]
[4.8, 6.7] [5.0, 5.5] [4.0, 5.0] [4.8, 5.31 [5.5,6.0]
Subsystem t 1 2 3 4 5 6 7 8 9 10 11 12 13 14
[0.84,0.87] [0.83,0.84]
[0.90, 0.96]
[0.95,0.99] [0.90, 0.921 [0.93, 0.9'7]
[0.90,0.921
[I.7, 2.2] [2.5,3.41
[4.0, 4.11
[7.1,9.6]
[5.2,7.01
[4.1,5.81
[3.5, 4.5]
[3.8,5.2]
[7.0, '7.5]
[4.0, 6.0]
92, 0.93] (0~,0.981 [0.90,0.931
[0.8,1.31 [0.7,1.01 [2.2,3.01
[3.2, 4.01 [r.5,10.01 [5.0,6.81
[0.86,0.89] [0.92, 0.931 [0.97, 0.98] [0.92,0.94] [0.88,0.90]
[3.2,4.4] [1.5, 2.4] [2.8, 3.0] [3.8, 4.5] [4.5, 5.81
[0.83, 0.85]
[3.4,4.5] [3.4, 4.0]
[5.3,6.1] [2.9, 4.01 [3.8, 4.2] [7.5, 8.01 [6.2, 7.0] [8.5, 9.01 [4.7, 5.8]
[0.97, 0.991
[0.94, 0.95] [O.80, 0.82] [0.94, 0.99] [0.88, 0.96]
3
t't4Ltrt4 ~
[0.91,0.93]
[1.7, 2.0]
[1.0, 1.5] [1.o, 1.61
[8.2,9.o1
[0.85, 0.88] [0.95, 0.971
[4.8, 5.1]
[4.0, 4.9]
[0.96, 0.98]
[0.90, 0.96]
[0.86, 0.93] [0.95, 0.98] [0.87, 0.91] [0.93, 0.98] [0.80, 0.87] [0.95, 0.981 [0.90, 0.95]
[2.3, 3.0]
[2.0, 2.6] [4.3, 5.51 [5.4, 6.1] [3.1, 4.0] [4.6, 5.01 [4.8, 5.2] [3.5, 4.0]
[1.4, 3.0]
[4.7, 5.5]
[4.9, 6.81 [e.o, 6.o1
[2.6, 3.8] [2.7, 3.6]
[5.0, 5.8]
[6.0, 7.0]
[3.8, 4.4] 4
[~,3~,,,3~1 to,3
[2.4, 3.01
[5.4, 6.21 [4.1,5.0] [4.8, 5.4]
[8.1,9.ol [5.6,6.7"] [87,761 [5.8,6.ol [5.4,6.2] [6.0,8.5]
Is 7,6.o1
[5.6,7.1]
.92, 0.97]
[1.9, 2.3]
[3.8, 5.5]
[0.89, 0.95]
[3.0, 4.2]
[4.0, 5.0]
[0.95, 0.97]
[1.8, 2.1]
[3.0, 4.9]
[0.90, 0.931
[2.4, 3.81
[7.0, 8.6]
{0.89,0.911
[41, s 21
[8.s, 7.01
[0.94,0.99]
[5.4,6.0]
[7.9, 9.0]
RQ = [0.95, 0.9'7],G'Q == [110, 150], WQ = [150,190], -
S t e p 3: Set g e n = g e n + 1. Calculate Ai(=) by interval arithmetics['?] and calculate deviation variables by equation (6) - (8). In addition, they ask for evaluation function eval(vk) by equation (9). Moreover, equation (11) is used and the best chromosome v* is preserved.
: No design
thinks about the following three goals concerning this system. (I) System reliability [0.95,0.971.
is
established
over
(II) Unused quantity of system resource 1 is lost. S t e p 4: Reproduce new chromosomes during uniform crossover and mutation process and perform the Elitist Selection[4]. S t e p 5: If g e n < m a z g e n , return to S t e p 3. If g e n = m a z g e n , output v* and terminate.
(III) Excessive use of system resource 2 is prevented. An I-NIGP of this case is formulated as follows:
lexmin
4
Numerical
Example
The optimal design for system reliability problem with interval coefficients that chooses the best alternative design and the best corresponding red u n d a n t units for subsystems. The system consists of 14 subsystems and their alternative designs are shown in Table 1. Now, a decision maker
{ z l = d ? , 32 = d ~ , 33 = d + }
14 s.t. t----1
+ d r - dt = Rq
14 O2(lrn, o~) = E Cta, rnt t--1 +a~ - 4 = cq
600
Proceedings of 1996 ICC&IC 14
G ~ ( , , , ~ ) = ~ W,,,mt t=l
+d~-~=wq 0 < m t < 6 :integer, t = 1 , . . . , 1 4
Table 2: Results with various p p 0.0
1 < at
0.1 0.3 0.4 0.9 1.0
i 1 2 3 1 2 3 ] 2 3 1
g, o" 122.1000 179.2000 0.966614 130.2000 184.4500 0.9657]5 130.1000 184.1600 0.961774
[g, = ,g, ~] [0.950862,0.981036] [111.5000,132.7000] [166.0000,192.4000] [0.952145,0.979083] [118.0000,142.4000] [171.6000,197.4000] [0.9,51266,0.980164] [120.0000,140.2000] [170.7000,197.6000] [0.943871,0.979677]
2 3
130.1500 182.5500
[120.3000,140.0000] [167.6000,197.5000]
0.965949
RQ, C o and Wo are interval target values of each goal constraint, and denoted as R~ =
[~oL,~oR],cq = [cqL, cqR],Wq = [~qL, ~qR].
However, a gene denoted as v~t consists of two kinds of variables: alternative design v ~ ) and redundant unit v ~ ) for subsystem t. A chromosome, denoted as Vk(k= 1,... ,pop_size), is an ordered list of T genes. We solve the optimal design for system reliability problem with our proposed algorithm. The parameters are set as follows: population size: pop.size = 20 crossover probability: pC = 0.4 mutation probability: p m = 0.1 maximum generation: maxgen = 500 weight: p = 0.5 We apply the proposed method to this problem and repeat the process for 500 generations. At the 251-th generation of the chromosome and goal function variables the following results were obtained: y*
:1"(3,3)(1,2)(4,3)(3,3)(2,2)(2,2)(1,2) (3,3) (3,2) (3,3)(3,2) (4,3) (1,2) (3,2)] eval(V*) = 10.874992 * goal functions * zl
O. 000000
z2 z3
0. 000000 10. 874992
Table 2 shows that various sizes of weight p(0 < p < 1) have influences on wlole system concerned in this numerical example.
5
Conclusions
In this paper, the technique to solve nonlinear integer goal programming with interval coefficients using genetic algorithm more efficiently was proposed. The proposed method makes the optimal design of system reliability done more flexible and
more practical, since it considered the vagueness of designer's measurement and the vagueness of data resulted from the situations changes by introducing the interval of goals and interval coefficients. The subjective preferences of decision makers were shown in problem solving by defining deviation variables with the weighting factor of order relation between interval numbers for INIGP problems. Also the method is easy to apply nonlinear optimization problems with holding their nonlinearity, without any transformations of nonlinear model to linear model.
References [1] Ishibuchi, H. & H. Tanaka: Formulation and analysis of liner programming problem with interval coefficients, Journal o] Japan Industrial Management Association, Vol. 40, No.5, pp. 320-329, 1989 (in Japanese). [2] Goldberg, D. : Genetic Algorithms in Search, Optimization, and Machine Learning, AddisonWesley, 1989. [3] Michalewicz, Z. : Genetic Algorithms + Data Structures = Evolution Programs. SpringerVerlag, 1992. [4] Gen, M. & R. Cheng : Genetic Algorithms and Engineering Design, John Wiley & Sons, New York, 1996. [5] Gun, M. & R. Cheng : Interval Programming Using Genetic Algorithms, Proc. o] l~th Fuzzy System Symposium, pp.861-864, 1996. [6] Yokota, T., M. Gun, K. Ida & T. Taguchi: Optimal design of system reliability by an improved genetic algorithm, Transactions of Institute o] Electronics, In]ormation and Computer Engineering, Vol.J78-A,No.6,pp.702-709, 1995(in Japanese). [7] Hansen, E. :Global Optimization Using lnterval Analysis, Marcel dekker Inc., 1992.