MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C3D82B.5957B760" This document is a Web archive file. If you are seeing this message, this means your browser or editor doesn't support Web archive files. For more information on the Web archive format, go to http://officeupdate.microsoft.com/office/webarchive.htm ------=_NextPart_01C3D82B.5957B760 Content-Location: file:///C:/2C839E43/TheoryandMethodologyreviseddoc.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" European Journal of Operational Research 81 (1995) 543-552 - = =20 543

Copyright © European Journal of Operati= onal Research 81 (1995) 543-552           =             &nb= sp;            =    

North-Holland http://www.ScienceDirect.com/s= cience/journal/03772217

Posted with permission from Elsevier.

Theo= ry and Methodology

 

        =        

        =         Some problems with the design  

 

        =           =  of self-learning Open-loop control systems

 

 

Ziny Flikop *

 

NYNEX Science and Technology, Inc., 500 Westchester Avenue, White Plains, NY 10= 604 US= A

 

Received October 1992; revised September 199= 3

 

Abst= ract: In this paper some problems in the design of open-loop control systems for complex objects are= discussed. Considering the absence of adequate models and the fact that human expertis= e in the control of non-stationary objects becomes obsolete quickly, the use of self-learning together with a two-step optimization for the development of on-line open-loop control rules is suggested. To prepare for the object analysis, a set of definitions has been proposed. Traditional and fuzzy sets (see Zadeh and Sugeno) approaches are used in the analysis.

 

Keyw= ords: Control processe= s; Decision theory; Fuzzy sets; Optimization

 

 

1. Introduction

 

  Automatic and semi-automatic control systems for the complex objects usually are base= d on sets of control rules. The development of such rules requires either comprehensive human expertise or an adequate object model or both. However, human expertise in the control of the complex non-stationary objects becomes obsolete with time. In addition, commonly used closed-loop control has a prolonged reaction time. These problems can be partially avoided if open-lo= op control together with a self-learning can be used. In this paper, which is = but another drop in the sea of control literature, we too are proposing and studying some variant of such an approach.

  The methodology for synthesizing control systems depends on the complexity of t= he controlled object. In this paper we discuss the control of complex objects = with multiple inputs and outputs. As an example of a controlled object we use a telecommunication network proposed in [3] (Fig. 1) and its model developed = in correspondence with [l].

  The numbers displayed in Fig. 1 represent line capacities  in packets-per-second, i =3D 1,. .= . ,16, is a line number. In this network, traffic transmitted between any source a= nd destination nodes can be split among different paths.

  In general, the .object transformation function (mapping) is defined by an obj= ect organization (structure) and by the values of the object element set-up parameters (control variables). In our example, network nodes, their connectivity and link capacities represent network structure. The traffic routing tables defined for each network node are control variables. Often object performance is evaluated by multiple criteria (via=

 

* Corresponding author.

 

        =             &nb= sp;            =             &nb= sp;            =             &nb= sp;        

 

multiple controlled variables). We evaluate performance of our network via average delay that packets undergo during traversing a network. In other words, delay is our controlled parameter (controlled output). Control system performance can be evaluated by the con= trol system's ability to maintain outputs at the predefined level (the simple control task) and by its ability to minimize the `cost' required for the control (the optimized control task). In our example, we evaluate performan= ce of the network control system by its ability to maintain an average delay below predefined level (the simple control task) and execute this control upon minimal network cost (the optimized control task). Control can be executed = by varying either control variables and/or object structure. In the developmen= t of control systems, one should consider that control system reaction time must= be much shorter than input drift, and processes of environmental and structural changes. This requirement is particularly important for the telecommunicati= ons network control systems.

 The ability of control systems to control and optimize objects depends on the efficiency of the algorithms used for these purposes. In turn, selection of these algorithms depends on continuity, separability, and monotonicity of controlled object mapping functions. In general, the behavior of mapping functions depends on the nature of controlled objects. If, for example, sim= ple physical devices often have mapping functions that are continuous, separabl= e, and monotonic, this is not always the case for more complex controlled obje= cts, such as heterogeneous telecommunication networks, for example. When mapping = functions are continuous, separable, and monotonic, relatively simple contr= ol and optimization algorithms can be applied. However, when mapping functions= are not continuous, separable, and monotonic, then significantly more powerful algorithms are needed. Because the behavior of mapping functions imposes limitations on the selection of control and optimization algorithms, and si= nce proposed self-learning methodology is based on such algorithms, we define what we me= an by continuity, separability, and monotonicity. We also define several other terms used in this paper. =

 

2. Definitions

 

From a control point-of-view, a general obje= ct can be defined via the following mapping functions:

 

  F(V, H): XY,  F(X, H):= V Y and F(X, H= ): V Q,        =              =             &nb= sp;            =             &nb= sp;       (1)

 

where:

is an input vector,  m =3D |M|, M is a set of input variables;

is an output vector,  n =3D |N|, N is a set of output variables;

 is a vector= of control variables,  p =3D |P|, P is a set of control variables= ;

is a vector of controlled variables,  b=3D |B|, B is a set of controlled variables;

 is a contro= lled object structure,  is an object element (link),  In our exam= ple L is a set of network elements and= links connecting such elements. We define some properties of these functions:

 

a. Let's consider a function  with any fi= xed V and H as continuous into space C if  when  , then  for .        =                   =             &nb= sp;            =         (2)

 

Z. F= likop / European Journal of Operational Research 81 (1995) 543-552        =         545

 

b. Let's consider a function  with any fi= xed X and H as continuous into

space   A  if when , then  for .        =             &nb= sp;                    =          (3)=

 

c. Let's consider a function with any fixed X<= /i> and H as continuous into space = D if  when , then  for .        =                   =             &nb= sp;            =           (4)=

 

d. Let's consider a function  with any fi= xed V and H, and as monotone if when X is changing in one direction along same monotone trajectory into C, then Y is also changing in one direction along a monotone trajectory into output space.        =             =             &nb= sp;            =             &nb= sp;            =              (5)

 

e. Let's consider a function with any fixed X<= /i> and H and  as monotone= if when V is changing in one direction = along same monotone trajectory into A= , then Y is also changing in one direc= tion along a monotone trajectory into output space.        =              =             &nb= sp;            =             &nb= sp;            =            (6= )

 

f. Let's consider a function  with any fi= xed X and H and  as monotone= if when V is changing in one direction = along same monotone trajectory into L= , then Q is also changing in one direc= tion along monotone trajectory into controlled variable space.               =             &nb= sp;            =             &nb= sp;            =    (7)

 

g. Let's consider a function as separable if   andthen         =             &nb= sp;            =             &nb= sp;            =    (8)

 

h. Let's consider a function as separable if and then         =             &nb= sp;            =                  =             &nb= sp;  

        =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;           =             &nb= sp;       (9)

 

i. Let's consider a function  as separabl= e if and then        =             &nb= sp;            

        =             &nb= sp;            =                   =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =  (10)

 

We can define the fluctuation range of = input variable by an ordered  set that co= nsists of real numbers representing possible measured values of this variable. T= he whole input space is:

  .        =             &nb= sp;                  =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =    (11)

 

We can define the reference (desired) value = for every output variable and the reference output vector for the whole object. As a particular case, we also can define the permissible output space:

 

,        =             &nb= sp;            =          (12)        =             &nb= sp;        

 

where  is an accur= acy of tracking i variable; ]  is a permissible interval of i outpu= t variable.

 

Each permissible interval also can be represented by a normalized fuzzy set  with a memb= ership function. In this set,  has a maxim= al possible grade . In this case, the permissible output space  can be defi= ned as:

 

   =3D supp supp supp .        =                   =             &nb= sp;            =                     =             &nb= sp;       (13)

 

The actual output vector , at moment t usually differs from . This difference is the result of either , drift or mapping function changes caused by V instability and environmental an= d H changes. Deviation of  from <= !--[if gte vml 1]> , is a control error for which the control system must compensate. Compensation can be done either by varying only V or only H or by simultaneous changes V and H.

 

The quality of control is evaluated either b= y an output error vector that at moment = t is:

 

 ), where         =             &nb= sp;            =             &nb= sp;            =                   (14)

 

or by  F =3D  that can be= used for estimating a degree that .              =             &nb= sp;       (15)<= /p>

 

 

We will consider that the controlled object = is working within a required accuracy if either of the following conditions are satisfied:

 

 or  supp          =             &nb= sp;            =             &nb= sp;            =         (16)

 

  or  or .        =           =                     =             &nb= sp;            =          =             &nb= sp;            =             &nb= sp;               (17)

All = X, for which conditions (16, 17) are satisfied for some combination of VH, are permissible input vectors = for this combination. We can propose a definition of permissible input subspace= for z-th comb= ination of VH:  or  for z-th combination of VH} .=

 

 We can define for each output variable i at any time t a. distance of from the border of a permissible output interval either v= ia:

 

 or via memb= ership grades as .              =              (18)

 

For the output vector we can use either:

 

,        =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;                (19)

 

or (15). The efficiency with which the contr= ol system executes its controlling functions can be different for different combinations of V and H. We can introduce a, multivalued= goal function G that can be used for evaluating the efficiency of the control system and optimizing the object:<= o:p>

 

        =             &nb= sp;            =             &nb= sp;             =             &nb= sp;            =             &nb= sp;            =             &nb= sp;              =       (20)

 

where   is a weight coefficient of a controlled variable f, f  B.

 

In our telecommunication network example con= trol can be presented via:

 

  ,        =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =              =     (21)

 

where  is the cont= rol vector that is implemented in the network via node routing tables and defin= es sets of paths for every source-destination pair of nodes. It also defines in what proportion traffic must be= split between paths. For example, is the portion of the traffic transmitted from source nod= e D to destination node C via path j. (Control vectors are presented in Table 1, columns "Cur= ve A", "Curve B", and "Curve C".)

 

  H =3D {H} is a set of network node= s and links.

 

 is an input vector, where for example, is traffic that is entering the

network via node D and destined for node C. When traff;c between nodes D an= d C is split, then

 

 is a set of= paths between which traffic from D to= C is split;  is a traffi= c in the path j.

 

One can see examples of splitting in Table 1, `Path' column.  is a

vector representing network output traffic (= it is a network performance constrain, not a controlled output from our network performance control point of view).  is, for exa= mple, traffic that is entering the network via node A, is transmitted by the network to node B, and is leaving successfully (without packet loss) the network via node B.

 

 

 

Table 1

Netw= ork traffic routing

 

Source

Destination

Traffic

(pkts/sec)

Path

Curve A

Traffic (%)

Curve B

Traffic (%)

Curve C

Traffic (%)

A

B

9.0

AB

100.0

100.0

100.0

A

C

4.0

ABC

AEC

40.0

60.0

47.5

52.5

16.25

83.75

A

D

1.0

ABCD

AECD

ABFD

90.0

10.0

100.0

10

20

70

A

E

7.0

AE

100.0

100.0

100.0

A

F

4.0

AEF

ABF

100.0

100.0

65.0

35.0

B

A

9.0

BA

100.0

100.0

100.0

B

C

8.0

BC

100.0

100.0

100.0

B

D

3.0

BCD

BFD

50.3

49.7

100.0

89.7

10.3

B

E

2.0

BFE

100.0

100.0

100.0

B

F

4.0 and vary

BF

100.0

100.0

100.0

C

A

4.0

CBA

CEA

51.25

48..75

32.5

67.5

71.25

28.75

C

B

8.0

CB

100.0

100.0

100.0

C

D

3.0

CD

100.0

100.0

100.0

C

E

3.0 and vary

CE

100.0

100.0

100.0

C

F

2.0

CEF

100.0

100.0

100.0

D

A

1.0

DCBA

DCEA

DFBA

DFEA

30.0

60.0

10.0

60.0

5.0

 

35.0

5.0

60.0

35.0

D

B

3.0

DCB

DFB

56.7

43.3

46.7

53.3

90.0

10.0

D

C

3.0

DC

100.0

100.0

100.0

D

E

3.0

DFE

DCE

60.0

40.0

40.0

60.0

93.3

6.7

D

F

4.0

DF

100.0

100.0

100.0

E

A

7.0

EA

100.0

100.0

100.0

E

B

2.0

EFB

100.0

100.0

100.0

E

D

3.0

EFD

ECD

57.7

42.3

72.3

27.7

75.0

25.0

E

C

3.0

EC

100.0

100.0

100.0

E

F

5.0

EF

100.0

100.0

100.0

F

A

4.0

FEA

FBA

100.0

90.0

10.0

100.0

F

B

4.0

FB

100.0

100.0

100.0

F

C

2.0

FEC

100.0

100.0

100.0

F

D

4.0

FD

100.0

100.0

100.0

F

E

5.0

FE

100.0

100.0

100.0

 

 (When traffic approaches link capac= ity, then the network can drop traffic to avoid congestion. In this case

 

.   is a network performance;        =             &nb= sp;            =             &nb= sp;            =             &nb= sp;          (22)

  

is an average traffic delay in the link i, is a value of traffic in link i ,        =        

 

 is a capaci= ty of link i.        =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;                 =          (23)

 

 

Since task of the network control consists in maintaining value of R below so= me predefined level and without packet loss (that is a constrain), R is a controlled output of our ne= twork performance control system. In this case we will use  mapping for= the control system development. For simplicity, let's evaluate an efficiency G of the network according to the following:

 

        =             &nb= sp;            =             &nb= sp;            =                  =             &nb= sp;            =             &nb= sp;            =             &nb= sp;        (24)

 

3. C= ontrol

 

Control systems designed in correspondence w= ith a proposed terminology should be able to work in two interrelated modes:

 

1.   &n= bsp;   Simple control mod= e.

This is either a process of minimizing an ou= tput error vector:

 

   )    or     &nb= sp;          =             &nb= sp;            =             &nb= sp;   (25)

 

or a process of confining, to permissible output space (12, 13), which is executed= by varying either only V or V together with H. For successful control, conditions (2) and (3) must be satisfied. Controlling algorithms can be relatively simple if conditions (5= , 6, 8, 9) are also satisfied.

 

2.   &n= bsp;   Optimized control = mode.

 

This is also a process of the object control that corresponds (25). However, here the object performance is optimized by varying either only V, or V and H:=

 

        =           =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;(26)

 

Relatively simple algorithms can be used for this optimization if conditions (4, 7, and 10) are satisfied. If conditions (5-10) are not satisfied, then the algorithm proposed in [l] can be recommended. The optimization (25, 26) that is executed by varying V is based, in general, on nonline= ar programming. During such an optimization, conditions (16, 17) can be preser= ved relatively easily. However, the optimization that is executed via controlled object structural changes often is based on the combinatorial approach. Dur= ing combinatorial optimization, conditions (16, 17) can be unexpectedly violate= d, since any changes of H create a= significant destabilization effect on the object mapping function. To decrease the poss= ibility of violations of (16, 17), the control system should, before changes of H are made, try to drive into the center of . This can be done, for example, by thorough optimization made via V variations. The occu= rrence of   near the ce= nter of  is an indic= ation that the object has an excess of stability. As a result, the optimization e= xecuted via changes of H becomes possib= le.

 

When, in our network example,  provides tr= affic in each network link which is below link capacitie, then with high level of probability =3D , i.e. network quality constrains are satisfied. If a rou= ting optimization algorithm allows for the continuous changing of , then mapping is a continuous. However, this mapping function is non-separable because traffic in every link consists of traffic flows originated by different sources and destined to different destinations and = also because (23) `is a non-linear function.

 

Different control system modes require the u= se of different models. Namely, the simple control mode requires the input-con= trol variables-object structure-output mapping model. This model reflects <= !--[if gte vml 1]> (Fig. 2a) and  (Fig. 2b). = For the optimized control mode, a control variables-controlled variables-mapping model that reflects  (Fig. 2c) s= hould also be used.

 

For the simple control mode two approaches b= ased either on or  can be used= . The first approach uses a feedback, i.e., the control system constantly monitors conditions (16, 17)

 

and corrects output, if necessary, by varyin= g V and H. This approach is relatively accurate. However, it is slow, s= ince control decisions are delayed by an object transformation time and by a decision process that requir= es CPU time. The second approach (open-loop control) does not use a feedback all = the time. Instead, the control system continuously monitors condition  and makes decisions either that the current  combination= has to be changed and thus what has to be done to satisfy (16, 17), or that no change has to be made. In other words, the second approach uses rules  of this kind: If then do nothing. If , then find other  to which , belongs and change the object in correspondence with . This approach is faster than the feedback approach, but requires that VH=3Df(X) (reacti= ons) for will be prepared in advance. We will study the possibilit= y of using a second (open-loop) controlling approach in combination with self-learning and adaptation.

 

4. Self-learning and adaptation

 

The purpose of self-learning is the developm= ent and modification of control rules based on cause- and-effect information received via trials. Trials can be made as on analytical models as on the r= eal objects.

 

Self-learning consists of three phases. Name= ly:

 

4.1.= The preliminary cause-and-effect trials phase

 

This phase is dedicated to the analysis of linearity of , and  = and studies where conditions (2-7) are satisfied. Analysi= s of conditions (8-10) must also be considered.

For the analysis of with fixed V = and H, we will either observe natural = X fluctuations on the real object,= or actively change X on the model = or on the real object. For each X, a = value of Y is defined. This process is repeated for different V and H. Similarly, for the analysis of = with fixed X = (if it is possible) and H, we will var= y V and define for each V a value of Y. This process is repeated for different X and H. The purpos= e of this is to check (2, 3, 5, 6, 8, 9).

 

For the analysis of  with fixed X = (if it is possible) and H, we will var= y V and define for each V value of Q. This process is repeated for different X and H. The purpos= e of this is to check (4, 7, and 10).

 

The number of such trials is dictated by the desired accuracy of verification of (2-10) and it should be held to the minimum. The results of the first phase are needed for the selection of optimization algorithms used in the control rule development phase.

 

4.2.= The development of the control rules phase

 

This phase is implemented via a two-step obj= ect optimization. During the first step, an VH combination (feasible solution) is received for the analyzed X in correspondence with (25) (Fig= . 3). The second step is the selection of the optimal (in correspondence with (26= ) VH combination for the same X.)

 

Control rule development starts from some X. When the optimal VH combination is received for this X, then a permissible input space  for this combination is defined by varying either model or real object inputs (Fig 4= )

 

 

        =    

        =           Fig 3        =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =        Fig 4

 

 

Consequently, we will formulate a rule: &quo= t;If , then  combination= has to be chosen." Then the two-stage optimization process is repeated for some other  . As a resu= lt, we will find other combinations and other input subspaces   (Fig. 5).

 

After this, the next X (neither  nor ) is selected and the process is repeated. Input vectors = are selected for the analysis until either the whole input space A becomes decomposed or a feasible solution for the same X is impo= ssible to find. As a result of decomposition, some input subspaces can intersect; i.e., more than one feasible solution exists for some X; i.e., , where represent a g= -th intersection, W is a number of intersected subspaces. If and if conditions (3) and (6), or (9) are satisfied, then= a combined weighted rule [41 can be executed to provide .

 

When the controlled object is non-linear, th= en it is possible that . As a rule, we should try to avoid intersections because= the more input space belongs to the intersections, the more VH combinations have to be analyzed. If  represents = an object stability input space for the z-th combination of VH, then represents a total object stability input space (Z is a number of subspaces received during input space decomposition).

 

When =3DA, then th= e object is stable. If , then the object is only partially stable. Results consi= sting of optimal input subspace-object structure-control vector rules should be stored in the input-reaction table. This table becomes a basis for the open-loop control.

 

Now, for the illustration purposes, let's tr= y to apply proposed above methodology to the development of

control rules for our network control system. Let's also define the network control task as one that provides

14.0 and min = G when traffic between nodes BF and CE varies and is fixed for all othe source -destination pairs of nodes. For the developing network control rules, we use the model an optimi= zation algorithm proposed in [l]. This model and optimization algorithm allow for = the continuous changing of , and they do not require monotonicity or separabilit= y of the optimized function. Since we have already

assumed the continuity of the  mappings, w= e can turn directly to the second phase of the control rules development.

 

We start the development with traffic values= for B-F and. C-E  source-destin= ation pairs which are proposed in [3]; namely 4.0 pkts/sec for B-F and 3.0 pkts/sec for C-E. During the first step of the optimization (which is based = on varying  upon fi= xed network structure H), the proce= dure described in [l] allows us to find a control vector that provides condition= s  14.0. Such = vector `A' is presented in Table 1 in = the column, `Curve A traffic (%). T= he value of R is 13.56 for this ve= ctor, analyzed network structure, traffic and corresponding G =3D 425. Since R < 14.0, the network has an ex= cess of capacity and its structure can be optimized.

 

During the second step of optimization, the network structure (link capacities) was changing. However, control vector was fixed. As a result of this optimization, the capacities of the EF and FE lines were decreased from 62.5 = to 42.63 pkts/sec. That corresponds to G =3D 385.26 and R =3D 14.0.=

 

 

 

        =             &nb= sp;            =             &nb= sp;    

 

        =             &nb= sp;            =             &nb= sp;            =             &nb= sp;          Fig. 5

 

Then permissible input spaces were defined f= or the original network structure and for the optimized one. The first was don= e on a fixed network structure by varying B-F and C-E traffic, while monitori= ng 14.0 conditions. We started with the control vector that = is optimal for B-F traffic that is equal t= o 4.0 pkts/sec and C-E traffic that is equal t= o 3.0 pkts/sec. Modelling allows us to plot curve "A" on Figure 6. The zone under this curve is a permissible input space and it defines possible combinations of B-F and C-E traffic= for which the analyzed control vector provides 14.0. In other words, a control rule `Until B-F and C-E traffic is in the zone below curve `A', the set of routing tables, corresponding to `Curve A traffic (%I' of the Table 1, shou= ld be used' can be applied.

 

We can see from curve `A' that the maximal B-F= traffic is limited to 9.66 pkts/sec= and C-E traffic is limited to 8.12 pkts/sec. To analyze the netw= ork's ability to absorb more B-F traf= fic, we can choose a value of B-F tr= affic which is above 9.66 pkts/sec. T= hen we can find the other control vector that provides R s 14.0 condition. This vector is presented in Table 1, column= `Curve B traffic (%)'. Analysis of= the network with a new control vector allows us to plot curve `B' in Figure 6. We repeated similar procedures for the C-E traffic that exceeds 8.12 pkts/sec. This give us one mo= re control vector (Table 1, column `C) and another curve `C'.

 

As a result of these studies, the following rules can be created: `Until B-F an= d C-E traffic is in the zone below curve `A', the set of routing tables correspon= ding to `Curve A traffic (70)' should be used. If traffic is in zone I, then use= the set of routing tables corresponding to `Curve B traffic (o/o)`. If traffic = is in zone II, then use the set of routing tables corresponding to `Curve C traffic (%I'. These rules apply to the initial H that corresponds to G= =3D 425.

 

A similar study was done with control vector= `A' and the network in which the capacities of lines EF and FE were decr= eased to 42.63 pkts/sec. As a result a curve was plotted. The previous rule can be modified by adding = the following: `If B-F and C-E traffic = is in the zone below the curve Aopt, then routing tables corresponding to `Curve A traffic (%)' should be used and the capacity of lines EF and FE can be decr= eased to 42.63 pkts/sec'.<= /i>

 

4.3.= The adaptation phase

 

Because an object and/or object environment usually are non-stationary, the control system's ability to control and its performance efficiency degrade with time. To maintain controlled object performance at the predefined level, the control system should constantly monitor the validity of the developed rules and the values of G. The purpose of monitoring is to detect a moment when control becomes inefficient. When control system inefficiency is detected, another self-learning process can be repeated.

 

 

 

 

        =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =     Fig. 6

 

 

5. Conclusion

 

Usage of proposed approach to the telecommunications network model and received results demonstrate feasibili= ty of design of the open-loop control systems for objects with multiple inputs, outputs and complex structures. Development of such systems can significant= ly decrease control reaction time, i.e. improve quality of the control.

 

References

 

[l] Flikop, Z., "Routing optimization in packet switching communication networks," European Journal of Operatio= nal Research 19 (1985) 262-267.

[2] Sugeno, M., "An introductory survey= of fuzzy control", .Inforrnation Sciences (1985) 59-83.=

[3] Tanenbaum, A.S., Computer Networks, Seco= nd edition, Prentice Hall, New Jersey, 1988.=

[4] Yager, R.R. and Filev, D.P., "Identification of nonlinear systems by fuzzy models" (forthcomin= g).

[5] Zadeh, L.A. "The concept= of a linguistic variable and its application to approximate reasoning", Information Sciences 8

 

 

 

 

        =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =     

 

 

 

 

 

 

        

        =    =

 

 

 

 

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