- Email: [email protected]
Solving fuzzy shortest path problems by neural networks
Compatcrs ind. Engng, VoL 31, No. 3/4, pp. 861 - 865,1996
Pergamon
Cop~ghtO 1995~ Mad~e Prm PublishedbyElsevierSeie~eLtd.Printedin GreatBritain S0360-8352(96)00278-1 0360-8352J96 $15.00+ 0.00
Solving Fuzzy Shortest Path Problems by Neural Networks Yinzhen Li, Mitsuo Gen a n d Kenichi I d a D e p a r t m e n t of Industrial and S y s t e m s Engineering Ashikaga I n s t i t u t e of Technology, Ashikaga, 326 J a p a n Email: {gen ida i i } ~ g e n l a b . a s h i t e c h . a c . j p
Abstract In this paper, we introduce the neural networks for solving fuzzy shortest path problems. The penalization of the neural networks is realized after transforming into crisp shortest path model. The procedure and efficiency of this approach are shown with numerical simulations. Keywords: Shortest Path Problem, Fuzzy Sets Theory, Neural Network.
1
Introduction
Shortest path problems (SPP) lie at the heart of network flows, and its application has the practical importance as an essential methodology in transportation, routing and communication problems area. Having been formulated as a linear network programming problem, its particularly simple structure allowed researchers to develop several intuitively appealing algorithms for solving SPP. The network flow literature typically classifies algorithmic approaches for solving SPP into two groups: label setting and label correcting. Both approaches are iterative. They assign tentative distance labels to nodes at each step; the distance labels are estimates of the shortest path distances, for example, snch traditional methods as the network simplex method, Bellman-Ford algorithm, Dijkstra algorithm, Floyd-Warshall algorithm, and so on [1]. llowever, for more complex and large scale problems, the increase of compnting time proved to be one of important problems when applying the conventional methods. Also, when we deal with real-world problems, it might be considered more appropriate that represent imprecise conditions in fuzzy coefficients, because of the nature of measurement. Traditional mathematical programming techniques cannot solve the all fuzzy programming problem. On the other hand, the most optimization problems are solved numerically by using iterative methods. Many methods of these iterative schemes can be considered as discrete-time realizations of continuous-time dynamic systems. And the certain optimization problems are expressed and/or can be transformed as the minimization problems. Therefore, neural networks with minimizing energy functions can be applied to solve these problen~s. In this paper, we present, a procedure for solving fuzzy SPP by using the neural networks. The penalization of the neural network is realized after transforming into crisp shortest path model. The procedure and efficiency of this approach are shown with nnmerical simulations.
2
Mathematical M o d e l
For the filzzy SPP we wish to find a path of minimum length (or cost) from a specified source node s to another specified sink node I, a~suming that each arc (i,j) E N has an associated length (or cost) cij. Some of tile simplest SPP applications are to determine a path between two specified nodes of a network that has minimum cost, or a path that takes least time to traverse, or a path that has tile minimum reliability. Suppose that we are given a directed network G = (S, N) with m nodes, n ares, and an arc length (or arc cost) ~ij is fuzzy associated with each arc ( i , j ) E N. This viewpoint gives rise to the following linear programming formulation of the SPP.
i=1 j = l
861
862
18th International Conference on Computers and Industrial Engineering
s.t.
EziJ
-
ztl =
j=l zij > 0 ,
k=l i, j =
O;
i = 2,3,...,m
(2)
- 1
-1; i = m 1,2,--.,m
(3)
In the study of SPP, we will impose several assumptions. A s s u m p t i o n I : All arc lengths are integers. A s s u m p t i o n 2: It contains a directed path from node s to every other node in the network. A s s u m p t i o n 3: The network is directed. Generally, the practical problems are represented imprecise mathematical models because of the real-world situation are often not. so deterministic. Hence, fuzzy mathematical theory is developed by many researchers and several techniques are appeals. These approaches take the way that transform from the fuzzy mathematical model into crisp model and then solve it [3]. From these technique, we interest the simple method which is transform the fuzzy maximizing mathematical model into crisp model developed by Okada et aL [2]. Also, we extend the method for minimizing cases will be discussed in follows. In the objective function of the model (1), we use the triangular fuzzy number to represent the fuzzy length (or cost) cij which are denoted as ~q = (cql,cij2, cq3), and 0 _< ci~t ~ cij2 <_ cq3. After the L-R fuzzy number scalar times and addition operations, the z also is a triangular fuzzy number represented as £ = (zl, z2, za), like the following equation: rB
Wt
• , = ~__~__~cu,x,~,
t =
1,2,3.
i=1 j---I
By introducing the decision making threshold value ~ for possibility, we have
01- z-z.........~l>_~ Z 2 -- Z 1
from its membership function. T h e o r e m : If 01 >_ ~ and zl _< z < z2, V a E [0,1], t h e n (1 - a ) z l + ~ z 2 < z ~ z2. Therefore, the objective value z can be take as z = (1 - e)Zl + oz2 as in optimistic case. So, the minimizing fuzzy SPP (1) - (3) can be converted with following objective function under same constraints. z=
rain
- e)cijl + acij2}z,)
E{(I
(4)
i=I j=l
s. t
3
Neural Network
constraints (2),(3)
Approach
The SPP model (4) can be represented as given by m
rain
m
(5)
z = E E {(1 - c~)cijl + ac,j2}zq i----1 ~=1 L
m
s.t. j=l =
k=l -
~=I m j=l
=
o,
i =
2,3,...,m
-
I
k=l m k--1
rlj >_0, i, j - - 1 , 2 , . . . , m
(9)
18th International Conference on Computers and Indu,Ttrial Engineering
863
For this SPP, we construct a neural network for which global minimum z* is simultaneously a solution of the above problem. The penalty melhod approach belongs to a class of indirect method for solving a constrained, nonlinear programming problem via a sequence of one or more unconstrained minimization problems. This approach is to transform the constrained problem into an unconstrained one by adding penalty function terms to the objective function. Some or all constraints are eliminated by increasing the objective function by a quantity which depends on the amount by which the constraints are violated. With penalty function term it. is converted into one of the following unconstrained minimization problems rn
El.,
¢n
zlz)+
in
+ i=l
(10) i=1 j=l
where ~1, t¢2 > 0.are control parameters and [xij]- = min{O, zij }. Thus, original problem equivalent to the following unconstrained optimization problem. min
z = E(z)
s. t.
z 6 Rn
(11)
Then, the equivalent unconstrained optimization problem is solved with the following dynamic system of differential equations: dz d--/" = - ~ V . E ( z , ~ )
(12)
where the gradient of objective function is as follows:
where that
a[zii]_ = f - I 0zij I, 0
if z i j < O if zij > 0
which can be written in the scalar form dzlj dt . . . .
I,(eij + ~1 ( h i ( z ) - hi(z)) + Kz[xij]-) i,j = 1 , 2 , . . . , m
zi~(0) = x[o), i,j = 1 , 2 , . . . , m
and (i,j) ~ N
(14)
(15)
where penalty parameter Kl, K2 > 0 and p is a learning parameter, typically, p = ~, where r is the time-integrated parameter and e is a positive constant. So, the constructed neural network is globally stable and that only are the solutions to the corresponding SPP problem which from arbitrary initial state. We show an algorithm for carrying out numerical simulation as follows: Procedure: f-spp/nn Step 1: Input coefficients of the objective function and constraints, and set initial states z(0) randomly such that near to zero. Step 2: Convert the filzzy SPP (1) into crisp SPP (4). Step 3: Transform tlw problem (4) into an unconstrained optimization problem by using (10). Step 4: Apply the gradient method to obtain the system of ordinary differential equations (14), (15). Step 5: Select the penalty COemcients that Xl > 0 for the constraints (6) - (8), and ~2 > 0 for the negative constraints (9). Also set the learning parameter p (or time-integrated r). Step 6: Use the simula!ion tools of M a t h e m a t i c a to solve the globally stability points. Step 7: If the problem converge to stability state, then stop, else goto Step 5 and adjust the learning parameters t A.
864
18th International Conference on Computers and Industrial Engineering
Figure I: : A directed network in example Table 1: The fuzzy associated length of arcs arc I-2 1-3 1-4 2-5 3-5
4
~ij 14,7,7) (1,2,4) (2,2,5) (3,3,4) (2,3,4)
arc 3-6 4-6 5-7 6-7 -
~j (4,5,6)
(6,8,9) 14,6,7) (6,7,9) -
Numerical Example
Consider a network with seven nodes and nine arcs in the Figure 1. The fuzzy associated length of arcs ~j are given in Table 1. This problem is formulated as follows: min
7 7 z ~ E E cij~ij i--1 j--i 7
S. t.
hi(z)--
.,(.)
=
-
j=l 7 j=l
xo>O,
7
E x l j - E xkl ~- 1, j=l i'=1 7 7 = o,
i = 2 ,
.6
k--I 7 k--I
i,j=1,2,...,7
and (i,j) E N
For the neural network approach we use the developed Mathematicu program to simulate in three cases: crO, 0.5 and 1 for the problem. 1) For a = O, the converted objective function of tile problem is as: z = 4ah2 + xz3 + 2z14 + 3x25 + 2z35 + 4z36 + 6x46 + 4xs7 + 6x67; 2) For a = 0.5, tile converted objective function of the problem is as: z = 5.5z12 + 1.5x13 + 2x14 + 3x2s + 2.5x35 + 4.5x36 + 7x46 + 5xs7 + 6.5xs7; 3) For cr = 1, the converted objective function of tile problem is as: z = 7x12 + 2z13 + 2x14 + 3x2s + 3zas + 5zs6 + 8z4~ + 6zs7 + 7~T; From Table 2 that shown the simulation results of three cases, we can see the obtained solution is closed enough to the optimal solution =* = [0, 1,0,0, 1,0, 0, 1,0] in three cases which is corresponding to the optimal routing on the problem is 1 --* 3 - , 5 --* 7. And that constraints are closed to satisfactory level.
I 8th International Conference on Computers and Industrial Engineering
865
Table 2: Simulation results for example
Z12 ~'13 Z14 Z25 Z35 Z36
Z46 Zr,7
:(=) hi(=) h2(:.) h3(=) h4(=.) hs(~.) h6(~.) hz(z)
~=0 -0.00019 1.00039 -0.0~19 -0.0~20 1.0~48 -0.0~10 -0.0~20 1.0~27 -0.00030 6.99722 -7.00000x 10 -6 -4.97512x 1O -6 -6.0OO00xl0 -~ -6.95234x 10 -e -4.00000x10 -* -2.97420xI0 -6
3.19017x 10-5
~=0.5 • -0.00022 1.0~4 -0.0~18 -0.0~23 1.0~47 -0.0~08 -0.0~19 1.0~23 -0.00027 8.99718 -9.0000Ox 10-° -5.72139xI0 -s -7.5000Oxl0 -6 -8.83943x I0 -~ -5.00000xI0 -e -3.76725xI0 -6 3.98281 x I0 -~
o=l
-0.00025 1.00041 -0.00017 -0.00025 1.00046 -0.00006 -0.00018
1.0002
-0.00024 10.9971 -O.O00OI -6.46766x I0 -~ -9.00000 x 10- ~ -0.0000 l -6.00000 x l0- 6 -4.56031 x 1O -*~ 4.77545x I0 -~
if we choose largor p it can make, the neural system stabilize fast, and if we choose larger i¢i it can get more accuracy solution. The two capacitive parameters p and icj can be used scale of the neural networks.
5
Conclusion
In this paper, we presented the neural network approach to the fuzzy SPP. For the fuzzy SPP, we used the penalty method to construct the energy function of the neural network after converted into crisp model. The mapped dynamic system can be find out the optimal solution. Finally, a part of this research was supported with University-to-University Cooperative Research, the International Scientific Research Program (No. 07045032), Grant-in-Aid for Scientific Research by the Ministry of Japan, the Japanese Government.
References [1] Ahuja, R. K., T. L. Magnanti and J. B. Orlin: Network Flows: Theory , Algorithms, and Applications, Prentice-]loll, Inc. 1993. [2] Okada, S., M. Gen and K. Ida: "A Method for Solving Multiple Objective Linear Programming Problems with 'IYiangular Fuzzy Coefficients", Joarnai of EICA A, Vol. J75-A, No.6, pp.l109-1112, 1992(!n Japanese). [3] Lai, Y.- J. and C.- L. llwang: Fizzy Mathematical Programming: Methods and Applications, Springer-Verlag, 1994. [4] Ida, K. and M. Gon: "An Algorithm for Solving Bicriteria Shortest Path Problems with Fuzzy Coefficients", be appeared in Journal of Japan Society for Fuzzy Theory and Systems, Vol. 7, No. 2, 1995. [5] Loot, Chee-Kit: "Neural Network Methods in Combinatorial Optimization", Computers Ops. Res. Vol. 19, No. 3/4, pp.191-208, 1992. [6] Wang J. and V. Chankong: "l~current Neural Network for Linear Programming: Analysis and l)esign Principles", Computer.* Ops fits. Vol. 19, pp.297-31 I, 1992. [7] Zhu, X.Z., S.W. Zhang and A.G. Constantinides : " l,agrange Neural Networks for Linear Programnling", Journal of Parallel and Distributed Computing 1~, pp.354-360, 1992. [8] Cichocki, A. and R. Unbehsuen : Neural Networks for Optimization ~ Signal Processing, Wiley,1993 [9] Zak, S. II., V. Upatising, and S. llui: "Solving Linear Programming Problems with Neural Networks: A Comparative Study", IEEff 7)'ans. on Neural Networks, Vol.6, No.l, 1995.
CAIE 31/3-4~ M
Pergamon
Cop~ghtO 1995~ Mad~e Prm PublishedbyElsevierSeie~eLtd.Printedin GreatBritain S0360-8352(96)00278-1 0360-8352J96 $15.00+ 0.00
Solving Fuzzy Shortest Path Problems by Neural Networks Yinzhen Li, Mitsuo Gen a n d Kenichi I d a D e p a r t m e n t of Industrial and S y s t e m s Engineering Ashikaga I n s t i t u t e of Technology, Ashikaga, 326 J a p a n Email: {gen ida i i } ~ g e n l a b . a s h i t e c h . a c . j p
Abstract In this paper, we introduce the neural networks for solving fuzzy shortest path problems. The penalization of the neural networks is realized after transforming into crisp shortest path model. The procedure and efficiency of this approach are shown with numerical simulations. Keywords: Shortest Path Problem, Fuzzy Sets Theory, Neural Network.
1
Introduction
Shortest path problems (SPP) lie at the heart of network flows, and its application has the practical importance as an essential methodology in transportation, routing and communication problems area. Having been formulated as a linear network programming problem, its particularly simple structure allowed researchers to develop several intuitively appealing algorithms for solving SPP. The network flow literature typically classifies algorithmic approaches for solving SPP into two groups: label setting and label correcting. Both approaches are iterative. They assign tentative distance labels to nodes at each step; the distance labels are estimates of the shortest path distances, for example, snch traditional methods as the network simplex method, Bellman-Ford algorithm, Dijkstra algorithm, Floyd-Warshall algorithm, and so on [1]. llowever, for more complex and large scale problems, the increase of compnting time proved to be one of important problems when applying the conventional methods. Also, when we deal with real-world problems, it might be considered more appropriate that represent imprecise conditions in fuzzy coefficients, because of the nature of measurement. Traditional mathematical programming techniques cannot solve the all fuzzy programming problem. On the other hand, the most optimization problems are solved numerically by using iterative methods. Many methods of these iterative schemes can be considered as discrete-time realizations of continuous-time dynamic systems. And the certain optimization problems are expressed and/or can be transformed as the minimization problems. Therefore, neural networks with minimizing energy functions can be applied to solve these problen~s. In this paper, we present, a procedure for solving fuzzy SPP by using the neural networks. The penalization of the neural network is realized after transforming into crisp shortest path model. The procedure and efficiency of this approach are shown with nnmerical simulations.
2
Mathematical M o d e l
For the filzzy SPP we wish to find a path of minimum length (or cost) from a specified source node s to another specified sink node I, a~suming that each arc (i,j) E N has an associated length (or cost) cij. Some of tile simplest SPP applications are to determine a path between two specified nodes of a network that has minimum cost, or a path that takes least time to traverse, or a path that has tile minimum reliability. Suppose that we are given a directed network G = (S, N) with m nodes, n ares, and an arc length (or arc cost) ~ij is fuzzy associated with each arc ( i , j ) E N. This viewpoint gives rise to the following linear programming formulation of the SPP.
i=1 j = l
861
862
18th International Conference on Computers and Industrial Engineering
s.t.
EziJ
-
ztl =
j=l zij > 0 ,
k=l i, j =
O;
i = 2,3,...,m
(2)
- 1
-1; i = m 1,2,--.,m
(3)
In the study of SPP, we will impose several assumptions. A s s u m p t i o n I : All arc lengths are integers. A s s u m p t i o n 2: It contains a directed path from node s to every other node in the network. A s s u m p t i o n 3: The network is directed. Generally, the practical problems are represented imprecise mathematical models because of the real-world situation are often not. so deterministic. Hence, fuzzy mathematical theory is developed by many researchers and several techniques are appeals. These approaches take the way that transform from the fuzzy mathematical model into crisp model and then solve it [3]. From these technique, we interest the simple method which is transform the fuzzy maximizing mathematical model into crisp model developed by Okada et aL [2]. Also, we extend the method for minimizing cases will be discussed in follows. In the objective function of the model (1), we use the triangular fuzzy number to represent the fuzzy length (or cost) cij which are denoted as ~q = (cql,cij2, cq3), and 0 _< ci~t ~ cij2 <_ cq3. After the L-R fuzzy number scalar times and addition operations, the z also is a triangular fuzzy number represented as £ = (zl, z2, za), like the following equation: rB
Wt
• , = ~__~__~cu,x,~,
t =
1,2,3.
i=1 j---I
By introducing the decision making threshold value ~ for possibility, we have
01- z-z.........~l>_~ Z 2 -- Z 1
from its membership function. T h e o r e m : If 01 >_ ~ and zl _< z < z2, V a E [0,1], t h e n (1 - a ) z l + ~ z 2 < z ~ z2. Therefore, the objective value z can be take as z = (1 - e)Zl + oz2 as in optimistic case. So, the minimizing fuzzy SPP (1) - (3) can be converted with following objective function under same constraints. z=
rain
- e)cijl + acij2}z,)
E{(I
(4)
i=I j=l
s. t
3
Neural Network
constraints (2),(3)
Approach
The SPP model (4) can be represented as given by m
rain
m
(5)
z = E E {(1 - c~)cijl + ac,j2}zq i----1 ~=1 L
m
s.t. j=l =
k=l -
~=I m j=l
=
o,
i =
2,3,...,m
-
I
k=l m k--1
rlj >_0, i, j - - 1 , 2 , . . . , m
(9)
18th International Conference on Computers and Indu,Ttrial Engineering
863
For this SPP, we construct a neural network for which global minimum z* is simultaneously a solution of the above problem. The penalty melhod approach belongs to a class of indirect method for solving a constrained, nonlinear programming problem via a sequence of one or more unconstrained minimization problems. This approach is to transform the constrained problem into an unconstrained one by adding penalty function terms to the objective function. Some or all constraints are eliminated by increasing the objective function by a quantity which depends on the amount by which the constraints are violated. With penalty function term it. is converted into one of the following unconstrained minimization problems rn
El.,
¢n
zlz)+
in
+ i=l
(10) i=1 j=l
where ~1, t¢2 > 0.are control parameters and [xij]- = min{O, zij }. Thus, original problem equivalent to the following unconstrained optimization problem. min
z = E(z)
s. t.
z 6 Rn
(11)
Then, the equivalent unconstrained optimization problem is solved with the following dynamic system of differential equations: dz d--/" = - ~ V . E ( z , ~ )
(12)
where the gradient of objective function is as follows:
where that
a[zii]_ = f - I 0zij I, 0
if z i j < O if zij > 0
which can be written in the scalar form dzlj dt . . . .
I,(eij + ~1 ( h i ( z ) - hi(z)) + Kz[xij]-) i,j = 1 , 2 , . . . , m
zi~(0) = x[o), i,j = 1 , 2 , . . . , m
and (i,j) ~ N
(14)
(15)
where penalty parameter Kl, K2 > 0 and p is a learning parameter, typically, p = ~, where r is the time-integrated parameter and e is a positive constant. So, the constructed neural network is globally stable and that only are the solutions to the corresponding SPP problem which from arbitrary initial state. We show an algorithm for carrying out numerical simulation as follows: Procedure: f-spp/nn Step 1: Input coefficients of the objective function and constraints, and set initial states z(0) randomly such that near to zero. Step 2: Convert the filzzy SPP (1) into crisp SPP (4). Step 3: Transform tlw problem (4) into an unconstrained optimization problem by using (10). Step 4: Apply the gradient method to obtain the system of ordinary differential equations (14), (15). Step 5: Select the penalty COemcients that Xl > 0 for the constraints (6) - (8), and ~2 > 0 for the negative constraints (9). Also set the learning parameter p (or time-integrated r). Step 6: Use the simula!ion tools of M a t h e m a t i c a to solve the globally stability points. Step 7: If the problem converge to stability state, then stop, else goto Step 5 and adjust the learning parameters t A.
864
18th International Conference on Computers and Industrial Engineering
Figure I: : A directed network in example Table 1: The fuzzy associated length of arcs arc I-2 1-3 1-4 2-5 3-5
4
~ij 14,7,7) (1,2,4) (2,2,5) (3,3,4) (2,3,4)
arc 3-6 4-6 5-7 6-7 -
~j (4,5,6)
(6,8,9) 14,6,7) (6,7,9) -
Numerical Example
Consider a network with seven nodes and nine arcs in the Figure 1. The fuzzy associated length of arcs ~j are given in Table 1. This problem is formulated as follows: min
7 7 z ~ E E cij~ij i--1 j--i 7
S. t.
hi(z)--
.,(.)
=
-
j=l 7 j=l
xo>O,
7
E x l j - E xkl ~- 1, j=l i'=1 7 7 = o,
i = 2 ,
.6
k--I 7 k--I
i,j=1,2,...,7
and (i,j) E N
For the neural network approach we use the developed Mathematicu program to simulate in three cases: crO, 0.5 and 1 for the problem. 1) For a = O, the converted objective function of tile problem is as: z = 4ah2 + xz3 + 2z14 + 3x25 + 2z35 + 4z36 + 6x46 + 4xs7 + 6x67; 2) For a = 0.5, tile converted objective function of the problem is as: z = 5.5z12 + 1.5x13 + 2x14 + 3x2s + 2.5x35 + 4.5x36 + 7x46 + 5xs7 + 6.5xs7; 3) For cr = 1, the converted objective function of tile problem is as: z = 7x12 + 2z13 + 2x14 + 3x2s + 3zas + 5zs6 + 8z4~ + 6zs7 + 7~T; From Table 2 that shown the simulation results of three cases, we can see the obtained solution is closed enough to the optimal solution =* = [0, 1,0,0, 1,0, 0, 1,0] in three cases which is corresponding to the optimal routing on the problem is 1 --* 3 - , 5 --* 7. And that constraints are closed to satisfactory level.
I 8th International Conference on Computers and Industrial Engineering
865
Table 2: Simulation results for example
Z12 ~'13 Z14 Z25 Z35 Z36
Z46 Zr,7
:(=) hi(=) h2(:.) h3(=) h4(=.) hs(~.) h6(~.) hz(z)
~=0 -0.00019 1.00039 -0.0~19 -0.0~20 1.0~48 -0.0~10 -0.0~20 1.0~27 -0.00030 6.99722 -7.00000x 10 -6 -4.97512x 1O -6 -6.0OO00xl0 -~ -6.95234x 10 -e -4.00000x10 -* -2.97420xI0 -6
3.19017x 10-5
~=0.5 • -0.00022 1.0~4 -0.0~18 -0.0~23 1.0~47 -0.0~08 -0.0~19 1.0~23 -0.00027 8.99718 -9.0000Ox 10-° -5.72139xI0 -s -7.5000Oxl0 -6 -8.83943x I0 -~ -5.00000xI0 -e -3.76725xI0 -6 3.98281 x I0 -~
o=l
-0.00025 1.00041 -0.00017 -0.00025 1.00046 -0.00006 -0.00018
1.0002
-0.00024 10.9971 -O.O00OI -6.46766x I0 -~ -9.00000 x 10- ~ -0.0000 l -6.00000 x l0- 6 -4.56031 x 1O -*~ 4.77545x I0 -~
if we choose largor p it can make, the neural system stabilize fast, and if we choose larger i¢i it can get more accuracy solution. The two capacitive parameters p and icj can be used scale of the neural networks.
5
Conclusion
In this paper, we presented the neural network approach to the fuzzy SPP. For the fuzzy SPP, we used the penalty method to construct the energy function of the neural network after converted into crisp model. The mapped dynamic system can be find out the optimal solution. Finally, a part of this research was supported with University-to-University Cooperative Research, the International Scientific Research Program (No. 07045032), Grant-in-Aid for Scientific Research by the Ministry of Japan, the Japanese Government.
References [1] Ahuja, R. K., T. L. Magnanti and J. B. Orlin: Network Flows: Theory , Algorithms, and Applications, Prentice-]loll, Inc. 1993. [2] Okada, S., M. Gen and K. Ida: "A Method for Solving Multiple Objective Linear Programming Problems with 'IYiangular Fuzzy Coefficients", Joarnai of EICA A, Vol. J75-A, No.6, pp.l109-1112, 1992(!n Japanese). [3] Lai, Y.- J. and C.- L. llwang: Fizzy Mathematical Programming: Methods and Applications, Springer-Verlag, 1994. [4] Ida, K. and M. Gon: "An Algorithm for Solving Bicriteria Shortest Path Problems with Fuzzy Coefficients", be appeared in Journal of Japan Society for Fuzzy Theory and Systems, Vol. 7, No. 2, 1995. [5] Loot, Chee-Kit: "Neural Network Methods in Combinatorial Optimization", Computers Ops. Res. Vol. 19, No. 3/4, pp.191-208, 1992. [6] Wang J. and V. Chankong: "l~current Neural Network for Linear Programming: Analysis and l)esign Principles", Computer.* Ops fits. Vol. 19, pp.297-31 I, 1992. [7] Zhu, X.Z., S.W. Zhang and A.G. Constantinides : " l,agrange Neural Networks for Linear Programnling", Journal of Parallel and Distributed Computing 1~, pp.354-360, 1992. [8] Cichocki, A. and R. Unbehsuen : Neural Networks for Optimization ~ Signal Processing, Wiley,1993 [9] Zak, S. II., V. Upatising, and S. llui: "Solving Linear Programming Problems with Neural Networks: A Comparative Study", IEEff 7)'ans. on Neural Networks, Vol.6, No.l, 1995.
CAIE 31/3-4~ M