H. O. Stekler

S. S. L. Chang

English

Research Papers in Economics

This PDF is a selection from an out-of-print volume from the National Bureau
of Economic Research
Volume Title: Annals of Economic and Social Measurement, Volume 6, number 2
Volume Author/Editor: NBER
Volume Publisher:
Volume URL: http://www.nber.org/books/aesm77-2
Publication Date: April 1977
Chapter Title: Fuzziness in Economic Systems, Its Modeling and Control
Chapter Author: S. S. L. Chang, H. O. Stekler
Chapter URL: http://www.nber.org/chapters/c10514
Chapter pages in book: (p. 165 - 174)AnnaLc of Economic and Social Measurement/2. /977
FUZZINESS IN ECONOMIC SYSTEMS,
ITS MODELING AND CONTROL
BS. S. L. CHANC AND H. 0. SIFKI.FR
Control engineers have Just begun to investigate the properties ofj:zvsystems. These
systems contain stochastic variah!ec whose distributions are no: known cmnpleu'ly.On/v some
of the properties of these variables ?nay be known, the houndsfor instance. In thucave, the
theory of fuzzy dynamic programming has shown that thereis no unique control solution.
Rather there is a set of optimal policies dependingon the risk one is willing to take, The
least risky policy minimizes the 'naximmu loss,
This paper introduces the concept of fuzzysystems and fuzzy dynamic programming to
economic systems. There are a tiumber of possibilities for Jizzines.s ineconomic cisterns. We
present a general concept and illustrate it with one case,' stochastic coefficients whichare
bounded but where the entire distribution iv not known, .45an example, the control laity for a
'fuzzv'firsz.order 'nult:plier.accelerator system are developed, and thesolutions are anali':ed.
I. INTROI)UCTION
In recent years, both economists and control theorists havebeen working
with systems which contain a variety of uncertainties. Thesimplest ap-
proach for introducing uncertainty was to add random variablesto the
linear equations of the model. The solution to this problemis well known
and in economics involved the introduction of the certaintyequivalence
Concept.
The next step was to assume that the coefficients of the modelwere
stochastic. Chow [I] developed the macroeconomic stabilizationpolicy for
the case of stochastic coefficients. He showed that the policy involveda
linear feedback rule which was dependentupon the joint density of the
coefficients in the initial period. Chow presented two methods forcomput-
ing the means and covariances. The firstwas Bayesian; the second was an
approximation utilizing the asymptotic distribution of the structuralpa-
rameters from which the reduced form parameters are then derived.
However, Chow noted that in making the control laws dependent
upon the joint density of the coefficients in the ipiftial period, information
about the coefficients obtained during the control periodwas not being
utilized. Methods of efficiently obtaining this information and then using
the new information optimally for controlcome under the heading of
adaptive control problems.
However, these techniques do not take into accountsome of the un-
certainties that decision makers face in actuality. First, itis known that
there are alternative specifications of economic systems which yield widely
different results about policy multipliers. Second, it is a fact that economic
165data arc revised so frequentlythat the state of the system (even if that
could be precisely specified) lastperiod isnot known with certainty.
Third, when data revisions do occur,they may be substantial. 1 hen the
original coefficients of a regression andthe regression referring to the
same period hut usingthe revised data may differsubstantially.
Given the vast uncertainties, it isunlikely that the error should he
represented by a probability distributionwhere the mean and covariances
can be computed at agiven instant. However, if the probability (liStri-
bution is now known or cannot beused, none of the aforementioned
techniques are applicable.
A new set of techniques haverecently been developed to handle
stochastic control problenis where theprobability distribution of the co-
efficients or errors need not hespecified. All that is required is that the
error or parameter bebounded. This new technique is known as fuzzy
dynamic programming, and the analysisshow: that there is not a unique
optimal policy, but a set of policies depending onthe risk that is accept-
able. The most risky policy gives the least costunder favorable condi-
tions, but the largest loss when the uncertain parametersdisplay the worst
possible outcome. The least risky policy minimizesthe maximum loss
(mini-max). However, this policy frequentlyyields a non-linear compli-
cated control law, but there is an approximate hutsimple solution to this
least risky policy which is called guaranteed cost control. Itis also possible
to obtain the entire set of optimal controls,with each depending on the
risk preference of the decision maker.
In this paper, we shall first develop and explain sonic of the tech-
niques of fuzzy dynamic programming. We shall then apply this technique
to controlling a stochastic multiplier-accelerator system where itis as-
sumed that the distribution of the coetlIcients is unknown but bounded.
Since we arc analyzing a simple first-order system, it is feasible to derive
the exact minimax solution, and we need not examine the guaranteed cost
approximation.
II. Fuzzy DYNAMIC PROGRAMMING
A. An Economic Mode!
Assume that our economic system can be represented by the first
order system:
(1) Yr + i= A V, 4B G
where both A and B are stochastic coefficients. There are alternative as-
For an example, see the inventory equations iii H. 0. Sick !ers. Icononiic Forecasiing
(Praeger, 1970) Appendix A.
166sumptions which might he made about these coefficients.First, we could
estimate A and B from past data using appropriate statisticaltechniques.
However, this would not he an appropriate approach forcontrolling the
economy if thestructureof the economy were changing and if the coeffi-
cients varied with time. An alternative assumption thathas been made is
that the coefficients A and Bare composed of a known value plus a
random disturbance, i.e.
(2) A=A+ea and B=+e3
where e0 and e8 are assumed to be distributedas normal and independent
e or white. However, given the gross uncertainties inherent in the worldwe
can not realistically assume that we knowA,and the distributions of
e e0, e precisely.
We, therefore., propose another approach, whichwe feel more realis-
c tically reflects the real world in which economic decisionsare made. Ve
shall assume that the decision maker must controlan economy about
1- which there is some-but imprecise information. For instance,while there
St may be a consensus about the possible range of the A and B coefficients.
ss we explicitly assume that there is no knowledge about the exact disiribu-
110,1 of the random variables.
is Suppose equation (I) had been derived from the multiplier-accelera-
,lc tor model:
he (3) Vt +i+ G1
= '(.' i- .v,)
ue Then the A and B coefficients of(l) would then he
ed. (4) A=C',and B
I
lye
ost Some information about c is obviously available. We know thatc cannot
be greater than I, nor is it likely to be as low as .7. However,we are not
sure which of the values c is likely to take in the interval 0.7 < e < 1.0.
Similar information would be available aboutv,but our discussion will
concentrate on c.
hrst B. The Theory of FuzzySets[21andProgramming [3]:
This state of incomplete knowledge can be represented by describing
c as a fuzzy set: a membership functionon the real line as shown in
Figure 1. For any given x, mn(x) represents the likelihood ofc=x given
as- our imperfect knowledge. This membership function may be considered
a sting akin to a subjective probability distribution, where these prior probabili-
ties are based on the decision maker's own experience, knowledge and
1670
168
hgure IThe parameter c as afuizyset.
perceptions. If a value between 0.8 and 0.9 is highly likely, then p is close
to 1 for 0.8x0.9. Outside this range p falls gradually to zero.The
fuzzyset can also be described by its level set S=k:p(x) The
level set becomes smaller and smaller as v is increased, It is possible to pro-
vide an interpretation of a. This parameter determines the cut off between
those values of the coefficients which the decision maker would consider
and those which are considered irrelevant or extremely unlikely to occur.
For each value of a there isa setof optimal policies, and, in practice, it
would he necessary to undertake computer simulations to determine the
sensitivity of the sets of policies to the value of this parameter! In general.
the state equation can be written as
where q1 is a r-dimensional vector consisting ofruncertain parameters
(corresponding to the numhei of uncertain coefficients in our model) and
is represented as a fuzzy set in E' space.
While (5) is representative of our imprecise knowledge of the system,
we must now explain how a decision maker would proceed. We assume
that he selects the values of the coefficients which after careful considera-
tion of all the factors, seem subjectively most appropriate to him. Inter-
preted in the fuzzy set model, it means the selection of a ttireshold mem-
bership a, and only the possible values of q with a membershipa or
higher will be considered from this point on:
qES,,
Equations (5), (6) and the usual cost function equation, whichmay or may
not include the parameter q explicitly, now represents the control problem
21n Seciwn Ill below, we show how the policies forour simple model would vary with
changes in uncertainty. However, as a caveat, it should be noted that when thereare many
coefficients in the system, it may be difficult to characteri,e the boundary ofthe fuziy set.
As yet, no generally valid solution to this problem has beendeveloped.
I00 Zs
11.7)1 11.85
= f(.v1, u,q,)e
C
a-
r-
0
or
ay
ith
'fly
sct.
to be solved. It is intuitively clearthat any control law wouldgive a range of values for the cost,corresponding to the various values ofq in S. If a
control law is such thatno other control law can do betterforboththe maximum and minimumcosts (qSj,it is said to be one of theset of
optimum control laws. Theone which minimizes the maximumcost is
said to be least risky, and its riskparameterp= 0.
C. Soluiiopto the EconomicModel
a. Problem restatement
Equation (1) can be written inthe form:
(7) x,1 = (A0 + a,)x,- + (B + b,)u,
where (A0, B0) is the midpointof Sa and dependenton a, anda andb,j specify the range of Sa. Thevariables a, and b, arc not
known at the time of choosing U,. Thecost function is of the form
(8) J,= (x + Qu) + Px
The problem is to selectu,t = 0, I, 2 ...N - 1 to give the lowest J0. The
control law has the form
u,=
with (9), J can be expressedas
J,=C,x
whereC,satisfies the recurrent equation
C,=I +Qk+ C,,(A, -
C=P
b. Risk factors and minmim and Minmaxsolutions
In the above we have not yet specifiedthe exact values of A, andB,.If
A,andB,are chosen to minimize (maximize) the RHS (righthand side)
of(l I), the resultingC,gives the lowest (highest) value off,or Jmfl, (",na,)-
Given any sequence k,, t= 0, 1,2 ... N - 1, there is a corresponding
mjn andfl,a,.There is no unique optimum solutionas it depends on how much
relative weight we placeon theJ,and 'ma LetJbe defined as
'p(I- P)ma + pJ,,,,,,
and p is called the riskparameter. For any given p. 0 < p < I, there isan
optimum sequence k,, a'= 0, 1... iV - 1, which minimizesJ.
169I
A48 C,4,
Q+
We note that if k, >AO/BO,then a =?flah=1b would result in a higher
cost. Therefore the mininiax k21 is 40/B0. This is the policy which would
be used in a deterministic case if the cost of using policy is ignored.
The coefficients C', are calculated from (11) and (12) using A, B,
and kfor the minimum solution and using A4, 8, andk2for the
minimax solution.
c. Steady-state solution
If N approaches infinity
C,= c,4, =
We note (8) and (10) are of thesame form
(16) kABC
-Q+ CB2
Substituting (16) into (1 l)gives
C= I ±42C-- A282C2(Q + CB2Y'
Equation (16) can be solved for C
C= Qk
B(A -Bk)
Eliminating C between (17) and (18)gives an equation for k:
170
Wenow present the minirninandminimaxsolutions:
(1) p =I (minirnin solution)
A, =- =A
B, =B0 -i- h
A
= B,
=
Q+ C,IB
(2)p =0 (minimax solution)
A,=A0 + tn=A4
B, =B0 - nib=
(14)
L(2)= mm
"S
wherek2 +(I-A2
+-k -
\AR QAJ
where
l('l-42 B
2\AB QA
Equation(18) can be used only jfk satisfies(19). For other values of k, C
is obtained directly from (ii)
=i+Qk2
1(A -Bk)2
The minimum and maximum valacs of Ccorrespond to the minimum and
maximum values ofA - BK Iin the allowed ranges of A and B, if
A - BK< I.Otherwise the maximum cost is forA - BK=I
andC=
Ill. Cor'ci.usioAND INTERPRETATIONS: her
tild In the previous sections we developed therationale and some of the
methodology of fuzzy dynamic programmingand illustrated it with a sim- B, pie first-order example. Starting witha membership function, it was pos-
the sible to define the range of the stochastic coefficients.After a particular
value of that function is selected,a set of possible policies can he deter-
mined, with each policy dependentupon the risk the decision maker is
willing to assume. The procedurewas illustrated for the riskiest (minirnin)
and the least risky policies (minimax).
It should be noted that fuzzy dynamic solutionsare unlike the con-
ventional stochastic control problems. In the lattercase, with an exact
known distribution of the random variable, thereis only one solution
which minimizes the expected vtlue of the loss functions.With fuzzy dy-
narnic programming when the range of the coefficients is known,there is
a set of solutions.
Given the set of solutions, it is desirable to focuson the properties of
the extreme cases. We thereforeassume reasonable values for A and B,
and calculated how the vigor of policy (the gain)was related to the cost
associated with changing policy, Q. It can beseen from Figure 2, that
policy was used more vigorously in the minimax solutionthan in the
niinimin case.
Equation (19) shows that the minimax policymay he even more vig-
orous than the solution obtained from a deterministic model in whichthe
171
Q
0I.simun Cost
(I
Figure 3Minimum and maximum cost versus Q.
172
1) 0.5 1,0 .5 20
Q
Figure 2Minimin (p = 1) and minimax (p = 0) gain versus policy cost Q.
p= I
p=0
p=I
llnlmum Cost
0.5 1.0 .5 2.00 1 I
0 02 0.4 0.6
Figure4Minimum and maximum cost vcrsus gain.
cost of using policy, Q, was considered. However, thepolicy never exceeds
the full "gap", which would be the deterministicsolution when the cost of
using policy was ignored.
Figure 3 demonstrates thata minimin policy yields the lowest costs
when the most favorable eventsoccur, but an extremely high cost under
the most unfavorable circumstances. Thetwo outcomes for the minimax
policies lie within these bounds.
In Figure 4, we consider the cost and gain forthe entire set of solu-
tions. In our example, the optimumrange of the gain5 lies between .328
and .667. If the gain were larger than .667BOTHthe minimum and the
3The result presented here holds only for the first ordercase. In general there is a set of
optimal solutions which have the characteristic thatany other solution which gives a lower
minimum cost would also give a higher maximumcost, and vise versa. Moreover, in gerl-
oral the derivations of solutions for higher ordersystems are more complicated than are
presented here and the optimum set cannot be represented by thc gain.p
miximum cost are larger than the corresponding value at .667. Similar re-
sults hold br gain tess than .327. If we choose a gain within this range, an
of/Icr .vo!utwn it/lid! gites a lower mInI,nim?(I tsiiithl (lu'fl glee (1Jliç'/ls'r
inoxiniu?fleosi, and vice'versa.
Since all these results have been determined lr a given value of s, it
would be appropriate to determine how the policies would vary if the un-
certainty increased. If ec were decreased, the range of possible ValUes OF the
coefficients would be increased, i.e. the uncertainty is increased. Using
Equation (19), it can be shown that regardless of which coellicient displays
an increase in uncertainty the vigor of the minimin policy is reduced.4
For the minimax policy, equation ( 19) shows that an increase in un-
certainty in the A coefilcient. would increase the vigor of policy. An in-
crease in the uncertainty about the policy multipter. the B coellicient, may
either increase or decrease the vigor of policy. H owever with (2I. (i.e
the costs of using policy are no greater than those associated with system
deviations), an increase in uncertainty would increase the gain, provided
that it was not previously at the maximum, A0/B0. i\gain, the result differs
from that of stochastic programming. for then art increase in policy sin-
certainty in the first-order model reduces the vigorof policy.
These results even for a system as simple as a multiplier accelerator
model should serve to illustrate both the richness of fuziy programming
and its dilferences with the conventional stochastic programming solution.
SUN V. St ant' Brook
Revised December /976
4This leads to the result that with o = 0, no policy at all would he undertaken.
5This is the deterministic solution when the costs of using pol;cvare ignored.
REFER ENCES
II Chow, Gregory C. "Eficets of Uncertainty on Optimal Control Problems,"International
Economics Remjew, Vol. 14. No. 3 Oct. 1973.
121 L. A. Zadeh, "Fuizy Sets," Information and Control Vol.8. pp. 338 353, 1965
131 There are the following references on fuziySets and programmingS.S.L.. Chang.
"Fuzzy Dynamic Programming and the DecisionMaking Process",Third Annual
Pr,nceion Conference on information Sciences andSystems, 1969; S. S L Chung and
T. K. C. Pcng, "Adaptive Guaranteed CostControl of Systems with Uncertain
Parameters", IEEE Transon Automatic Control,Vol AC-17 No. 4, 1972,pp. 474 483.
S. S. 1. Chang and E. A. Zadeh, "Fuziymapping and control," IEEE irons. Si's:.,
Man. Cybern., Vol SMC-2,pp. 30 34, Jan. 1972. S. S. L. Chang and P. E. Barry, "Opti'
ma! Control of Systems with uncertainParameters", Fifth IFA(' ('ongrecs. Paris, 1972
S S. L. Chang "On Risk and DecisionMaking in a L'uziy Environment", Euz:v Sets and
Their Applications to Cognitii'e. and DecisionProces,sc.s Academic Press, 1975pp 219 226
C. V. Negoita and D. Ralescu,"Applications of Fuizy Sets to System Analysis," Wiley,
New York, 1975.
174

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