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Hierarchical Overlapping Coordination for Large Scale Systems
Copyright © IFAC Control Science and Tec hnology (8th Triennial World Congress) Kyoto. Japan . 1981
HIERARCHICAL OVERLAPPING COORDINATION FOR LARGE SCALE SYSTEMS T. Shima, K. Tarvainen and Y. Y. Haimes Systems Engineering Department, Case Institute of Technology, Case Western Reserve University, Cleveland, Ohio, USA
Abstract. The basic hierarchical overlapping coordination method is briefly reviewed. Three generalizations to this method with respect to convergence, problem formulation, and multiple objectives are presented. It is explained how a potential convergence problem in the basic scheme can be overcome by additional optimizations with respect to some interaction variables . In the problem formulation extension, the different decompositions are allowed to have different variables. Classical hierarchical methods are used for the overlapping coordination scheme resulting in a three-level structure. The single-objective hierarchical overlapping scheme is extended to multiple objectives using the Surrogate Worth Trade-off (SWT) method. Overlapping coordination and the Oantzig-Wolfe algorithm are compared. A simple example is used to articulate the respective attributes. Finally, the concept of hierarchical holographic modeling, obtained as an extension of the concept of hierarchical overlapping coordination, is presented with some preliminary mathematical results and illuminating examples. Keywords. Overlapping coordination; multiobjectives; hierarchical systems; SWT method; decomposition. 1.
INTRODUCTION
Critics attribute the present and pervasive skepticism of systems modeling and its relatively limited use in decision making to a welter of reasons known collectively as the lack of model credibility . Credible models connote believable, plausible and reliable models. As modelers, it is instructive to investigate the sources leading to the development of not too credible models. These include such issues as (a) the worth of available data; (b) model verification, calibration and testing; (c) model scope; (d) the time and resources allocated to model development--both in funds and in man power; (e) institutional support to model development, testing and maintenance; and, of course, (f) the assumptions and simplifications made in model development, and the respective optimization approach used to model solving. Focusing on large scale systems, this paper addresses issue (f) with some implication to the availability of data.
extent to which these simplifications can be justified is germane to the level of model credibility. Furthermore, the simplifying assumptions are central to a three-way compromise and trade-off among (a) the modeling process, (b) the optimization process, and (c) the availability and worth of data. For example, a very complex model might neither have the supportive data base nor be computationally tractable. The hierarchical approach, through decomposition and multi level optimization, is aimed at resolving some of the inherent conflicts between modeling and optimization. The hierarchical overlapping coordination extends this resolution to the data base as well (Haimes and Macko [1973], Macko and Haimes [1978], Sung [1978], and t1endu, Ha i mes and Macko [1980]). The a ttributes of the hierarchical approach are well documented in the literature and will not be repeated here (see, for example, Haimes [1977] and Titli and Singh [1979]). Hierarchical Overlapping Coordination (HOC) recognizes that in decomposing system's model into subsystems and/or levels within a hierarchical structure, more than one decomposition is possible, and most importantly-desirable. These different decompositions, each of which might be responsive to a different aspect of the system and/or data base, are coordinated through the respective couplings of the various decompositions. The
By necessity, mathematical models mimic real systems through simplified assumptions. The Invited paper submitted for presentation at the 8th Triennial World Congress/International Federation of Automatic Control, Kyoto, Japan, August 24-28, 1981. 1421
1422
T. Shima, K. Tarvainen and Y. Y. Haimes
coordination ultimately yields to an overall system's optimum in single-objective models and to preferred Pareto optimal solutions in multiple-objective models. Prior to articulating the attributes of HOC, it is instructive to consider the following generic problem drawn from the water resources area. The planning and management of water resources systems transcend hydrological as well as political-geographical boundaries, yielding to two natural respective model decompositions--hydrological and po 1 iti ca 1 • Without HOC, modelers are implicitly forced to choose between a hydrologically or politically based model/decomposition. Each decomposition represents and uncovers important aspects not amenable through the other. I~ost critical of all concerns is associated with the availability of the data base. The manipulation of data bases to serve and suit artificially (through the modeling process) imposed demands and constraints necessitates a compromise--an ultimate deterioration in model credibility . For example, data concerni ng s treamfl ow, wa ter qua 1i ty, and fl oods are available on a hydrological basis and collected by the U.S. Geological Survey, U.S. Environmental Protection Agency, and the U.S. Army Corps of Engineers, respectively. On the other hand, data concerning population dynamics, employment, and other economic activities are available on political/geographical bases and are collected by different agencies such as the U.S. Departments of Commerce, Labor, or Treasury. HOC enables the utmost utilization of these data bases with their minimum manipulation or misuse. This can be achieved by resorting to two simultaneous decompositions--hydrological and political/geographical--each of which might be with different number of subsystems. The Maumee River Basin, for example, lends itself to eight hydrological subsystems and to five political/geographical subsystems (Haimes, et al. [1979]). In general, water resources systems, as well as many other large scale systems, lend themselves to more than one decomposition/description, such as functional (water supply and demand for the various sectors--agriculture, industry, municipality, aquatic life, etc.), temporal (long, intermediate and short term and on line) as well as the hydrological and political/geographical decompositions discussed previously . The attributes of HOC might be summarized as follows (Macko and Haimes [1978]): 1) 2)
3)
Adds more fle xibility in modeling Yields more realistic models by increasing the responsiveness of the model structures to the organizational, political, technical, financial, and historical real ities. Increases the responsiveness of the model structures to the availability of data.
4) 5)
Models and coordinates system's transition from steady state to emergency state . Provides holographic view of modeled system.
In the following sections, theoretical aswell as methodological extensions of hierarchical overlapping coordination are presented. BASIC HIERARCHICAL OVERLAPPING COORDINATION (HOC)
2.
The basic algorithm and solution procedures are described in Haimes and Macko [1973], Macko and Haimes [1978], Haimes [1979], and Mendu et al. [1980]. A brief summary of the basic procedure is given in this section. The problem under consideration is represented as follows: ( 1)
max f(x) subject to Ax = c
( 2)
where f is a real-valued concave function and x is a decision vector. Suppose the system has two decompositions a and S. The original problem is transformed into the following problems by the rearrangement of the vectors x and c. The a decomposition problem is max f a (x a )
(3)
subject to
The
Aa Xa = ca S decomposition problem is
(4)
(5)
max fS(x S) subject to ASxS
Cs
(6)
where xa Aa
p x c = Qa c Xs = PSx, a a -1 Q AP- l A = QSAPS a a S
Cs
QSc
P and Q are nonsingular matrices representing the rearrangement of the vectors x and c. Suppose the a problem is decomposed into n subsystems and the S problem is decoma posed into nS subsystems. Here we assume the matrices Aa and AS are decomposable as follows: (7)
14 23
Hierarchical Overlapping Coordination
(8)
where Band H are block diagonal matrices compatible with the partitioning of the vectors by subsystems and G denotes the set of matrices of elements of A which are not on the block diagonal. The a and S problems are rewritten as follows: n a
max f a (xa ,ya ) = i=l L fa 1· (x a 1· ,ya 1'··· ,ya na ) (9)
subject to
na
B .x . +
a1 al
LG
• .y .
j=l alJ aJ
=c
.
a1
(10)
Haixai = Yai
(i = 1 , ••• ,na)
nS max fS(xS'YS) = i~l fSi (x Si 'YS1 , ... 'Ysns) (11 )
subject to (12 ) ( 13)
Ya and YS are vectors of the interaction variables. The above problems are equivalent to each other and to the original problem (1) and (2) •
The interaction variables Ya and Y are S determined by the decisions in the other decomposition as follows: Ya
Ha xa
(14 )
YS
HSxS
(15 )
Fi rs t, Y is fi xed and the a problem is a solved in each subsystem. Then Y is given S by the xa as represented in Eq. (15). The S problem is solved in each subsystem, and the solution Xs gives a new Ya ' and the a problem is solved again.
criterion. But it may happen that the result of the convergence does not reach the C~ C '(lld' solution. A method to overcome this possible difficulty is discussed below. First, the case where the problem of convergence is directly due to common interaction variables in different decompositions is addressed. This is the case in example problem 3 in Mendu et al. [1978] where the variables r 12 , r 13 , and r 23 act as coupling variables in both decompositions. Common interaction variables are never changed by the basi c scheme; the values remain as initial guesses. Assume that the problem is only due to the common interaction variables; that is, if they are guessed right at the beginning, the basic scheme converges to the right result, as in the example mentioned. A straightforward approach to this situation is simply to let one of the decompositions, or both, change the common coupling variables. In the basic scheme, when the algorithm is seen, say, from the point of the a decomposition, the procedure is the same to the a decomposition as a usual feasible decomposition scheme. That is, the a decomposition uses some initial values for the interaction variables, optimizes the subsystems, gets new values for the interaction variables (from the S decomposition which acts as an upper level to the a decomposition), and so on. In the case of common interaction variables, the 8 decomposition does not change these values. Hence, the a decomposition has to change these common interaction variables using methods developed for feasible hierarchical coordination. The basic scheme is, thus, modified slightly: before the a decompos it i on sends its resu lts to the S decomposition (Eq. (15)), the a decomposition improves, using standard methods, the values of the common interaction variables. Example 1. min
Consider the following problem: 8
L
i =1
2
x,
(16)
1
subject to xl x2
This procedure is repeated until convergence is achieved.
0
0
0
0
0
0
0
0
3.
3.1
EXTENSIONS OF BASIC HIERARCHICAL OVERLAPPING COORDINATION
Improvement of Convergence
In the basic scheme, the different decompositions alternatively improve the performance
[:
0
0
x3 x4
:] x5
x6 x7 x8
~]
( 17)
T. Shima, K. Tarvainen and Y. Y. Haimes
1424
Let x4 and x5 be interaction variables in the 0. decompos i ti on and x5 and x6 in the 8 decomposition'T That is, Ya = (x 4 ,x 5 )T and Y8 = (x 5 ,x 6 ) , and let the decompositions be as follows: a-decomposition: Subproblem 1(0.) . 2 2 2 2 2 2 mln (xl +X 2 +x 3 +x 6 ) + x4 + x5 X ,X
l
2
,X
3
,X
xl + x2 -x5 x3 + x6 = -x 4 - x5 Subproblem 2(0.) min (X~ + X~)
x7 ,x 8 subj ect to =
4
4
n x5
-n-l x5
Correspondingly, the values of the interaction variables of the 8 decomposition at the n-th iteration are determined by the values of the 0. decomposition at the n-th iteration: -n no x5 x5 -n n x5 x5
6
subject to
x7 + x8
xo5 ' where some guessed values are given to the interaction variables of the 0. decomposition. At the (n+l)-th iteration, the 0. decomposition uses values x4n and x5n for the interaction variables. These values are given by the results of the 8 decomposition at the n-th iteration: xn x-n-l
-x5
8-decomposition: Subproblem 1(8) -2 + x-2) + x5 -2 + -2 ml. n (-2 xl + x-22 + x3 x6 4
Where x~o denotes the optimal value of x5 at the n-th iteration to differentiate from x~ (= x~-l )--the initial value for x5 at the n-th iteration in the 0. decomposition. n -n-l . n -n-l Glven x4 = x4 and x5 = x5 at the beginning of the n-th iteration, it is easily seen that the corresponding solutions of the 0. decomposition problems denoted by -n-l ) are the following: xi ( x5 Subsystem 1 (0.) :
subject to xl + x2 x3 + x 4
-n-l ) xi ( x5 -x 5 -x 5
( 18)
(19)
Subsystem 2(0.): -n-l) _- -x-n-l /2 , k = 7,8 . (20) xk ( x5 5 The corresponding value for the performance -n-l ), is criteria (16), denoted by fan( x5
min ( x-27 + -2) x8 x7 ,x 8 subject to Let
i = 1,2
-n-l) /2, - ( x-n-l 4 + x5 j = 3,6,
x6
Subprobl em 2( 8)
x7 + x8 -x 5
-n-l -x5 /2
fn(x~-l) = (1/2)(x~-1 + x~-1)2 T
xa = [xl"" ,x 8] ,
The optimal solution of the problem given by Eqs. (16), (17) i s x~ = 0, i = 1 , ... ,8. The basic overlapping algorithm does not yield this optimal solution (unless the initial guess for x5 is zero) because x5 is a common interaction variable in the 0. and 8 decompositions. Consider the extension explained above, whereby x5 is optimized in the 0. decomposition. Let the superscript n denote the iteration number. Denote the initial values, by x~,
+ ( -n-l)2 x4 x5 • + 2 (-n-l)2
(21 )
Now, instead of using the value -n-l x5 for x5 ' an optimization is performed with respect to x5 • The optimal value is obtained from (21) as no -n-l x5 = -x 4 /5 The corresponding values for xl ,x 2 ,x 3 ,x 6 ' ... , x8 (given by Eqs. (18), ... ,(20)) are:
14 25
Hierar chical Overla pping Coordi nation
n xi x.n J
x~-l/lO
i = 1 ,2
(22)
_(2/5)x n- l 4
j = 3,6
( 23)
n (24) k = 7,8 xk x~-l/lO Next, consid er the S decomposition. At the n-th iterati on, it receive s x5 = x~o and x6 = x~ from the a decomposition. Optimization of the subproblems is then easily performed with the following results : -n-l -n (25) -x 5n/ 2 = x4 /10, i =1,2 xi 1 -(x~o+X~)/2 = (-3/5)x 2x1 j
= 3,4
(26) ( 27 )
-n /10,k= 7,8 xk =-x 5no /2=x-n-l 4 where the second equali ties are obtaine d using Eqs. (22), ... ,(24). Specif ically, for j = 4, -n x x = ( -3/5 )-n-l 4
Eq. (26) yields
4
From indica ting that x2 ~ 0 as n ~ all that shown be can Eqs. (23), ..• ,(27), it variab les in both decompositions approach zero--t he optimal solutio n. 00 .
In a general case, a convergence problem is not directl y due to some common interac tion variab les in the decompositions. Instead , there is a mathematical relatio nship in the equatio ns that binds the interac tion variab les of differe nt decompositions in such a way that they do not get varied over their origina l feasibl e set. In fact, exactly the same procedure as in the case of common interac tion variab les under optimi zation are release d. This will enlarge the areas of the feasib le sets of the decompositio ns to cover the origina l feasibl e set. As an extreme case, if all interac tion variables of the a decomposition are undero ptimizatio n, the optimum is guaranteed to be found. But this is genera lly not necess ary, because the S decomposition will cover a subspace of the feasibl e set. In practic e, a diffic ult problem is determining how many interac tion variab les in the a decomposition should be optimi zed. Mathematically , this relatio n is given below (cf. Eq. (14) in Macko and Haimes [1978] ):
N(A) .
N[H~Apal
+
N[HeApel
(28)
where H'a is a part of Ha correspondi ng those interac tion variab les that are not optimized . In an extreme case, when all variables are optimi zed, then H' = 0, and Eq. a
(28) holds trivial ly. Note also that if common interac tion variab les exist, there is, in genera l, no quarantee that the case does not imbed also more implic itly binding relatio ns hi ps. If the number of optimized interac tion variables cannot be determined, then, after the scheme has converged, the result should be tested by checking if the change of the interacti on variab les in one decomposition improves the result. 3.2
Extension of Problem Formulation
An import ant object for extending the basic scheme corresponds to problem formul ation. In it, is assumed that the differe nt decompositio ns comprise the same variab les (in a one-to- one corresp ondenc e). This is not usually the case: differe nt decompositions deal with differe nt aspects of the system, and, hence, with differe nt respec tive variables. Consider the following formulation as an immediate extensi on of the basic problen formul ation. The a decomposition deals and the S with a variab le vector (x,y) a a decomposition with a variab le vector (x s ,yS). Here, xa and Xs are the variab le vectors of the a and S decomposition that are directl y bound by equatio ns that specify the relatio nship between the decompositions. For example, (29)
xa
PaSx S'
Xs
PSax a ,
(29)
plx + PSxS = 0 , a a
(29)
or or ' Pas' PSa' Pa' h an d pI13 are cons t an t were matric es, not necess arily invert ible. The vectors Ya and ys comprise the otherv ariables in the a and 13 decompositions, respec tively.
Let the cost functio n be a sum of two cost functio ns relatin g to the decompositions. That is, the object ive is to maximize fa(xa'Y a) + fS(xS'YS).
(30)
In additio n, let the system and constr aint equatio ns in the decompositions be expressed by 9a (X a 'Ya)
0
gS(xs'YS)
0
(31)
(32)
Note that the problem given by (29a or b), (30), (31), (32) reduces to the basic case, when Ya and YS are vacuous and PaS or
1426
T. Shima, K. Tarvainen and Y. Y. Haimes
Psa is invertible. Mathematically, this problem (Z9a or b or c), (31), (31), and (3Z) is exactly of the same form as the problems in classical hierarchical systems theory. Hence, all the techniques developed there can be used in HOC. In effect, a three level scheme results when applying basic hierarchical schemes. At the highest level, the coordination variable relating to the coupling equation (Z9a or b or c) is changed. The coordination variable is the vector (xa 'x ) S in the feasible method, and a Lagrange multiplier vector in the non-feasible method. The two other levels are related to the a and S decompositions (which are solved, in parallel). Fig. 1 depicts a scheme where the non-feasible method is used for a case with the relationship between the two decompositions given by Eq. (Z9a). In Fig. 1, which depicts a non-feasible method, the lowest (1st) level optimizations have not been given. They can be formed using standard hierarchical methods for the criteria indicated at the second level. Overla
in
Let the original multiobjective hierarchical overlapping coordination problem be the following: (33) subject to Ax = c
(34 )
The a decomposition problem is ma~ [~(xa),~(xa) , .•• ,~(xa)]
( 36)
The
S decomposition problem is max [f~(XS ) ,f~(XS ) , ... ,f~(xS)]
In the above context of several decompositions and system aspects, different objectives related to different aspects arise in a natural way. Hence, the generalization of the basic overlapping coordination scheme to the multiobjective case is important.
( 37)
xS
subject to ASx S
cS.
(38)
Here, xa = pax,
Most planning of large-scale systems includes the optimization of several objectives. Methodologies to deal with multiobjective cases have been substantially developed during the past decade. Applications of multiobjective optimization in several fields of technology have produced much more realistic results than otherwise with classical singleobjective optimization.
(35)
x subject to
Aa = QaApa-l
cS
QSc
AS
QSApS-l
All f(·)'s tions.
xS
Ca = Qa c and
pSx,
and
are assumed to be concave func-
P and Q are non-singular matrices representing the rearrangement of the vectors x and c as described previously. Consider, first, the case with two linear objectives and inequality constraints: (39)
max [fl(x),fZ(x)]
The basic scheme can be generalized immediately to the case where the different decompositions have the same preferences for a common vector of objectives. Namely, if an underlying utility function for these preferences is assumed, then the problemisactually reduced to a single objective case, and the same basic Overlapping Coordination scher,le works under the same condition. Of course, the critical assumption is the existence of an underlying utility function.
subject to Ax ~ c, where
Another promising approach is to solve t~OC by the Surrogate Worth Trade-off (S~JT) method (Haimes, Hall and Freedman [1975]) which can generate the set of all needed non-inferior solutions and their corresponding trade-offs to the decision-makers and analysts.
A
The original problem (Eqs. (39) and (40)) is modified as follows:
The procedure can be briefly described as follows:
To solve this multiobjective problem, the SWT method is used. This problem is equivalent
x
~
(40)
0
fl (x)
dllx l + dlZx Z + ••• + dlnx n
(41 )
fZ(x)
dZlx l + dZZx Z + ••• + dZnx n
(4Z)
cl all·····a ln
C
C aml···· .ar.m
z
cm
Hierarchical Overlapping Coordination
to the following problem (cf. Haimes [1977J, Haimes, Hall and Freedman [1975J). n
L
Max fl (x)
j=l
(43)
d1 ·x . J J
subject to f 2 (x) ~ E2
Ax
~
c, x
(44)
~
0
(45) l 2 The trade-off A12 (= - af /af ) as the Lagrange multipller corresponding to the constraint (44). The trade-off, A , is a 12 function of the given E • 2 A1Z > 0 guarantees the non-inferiorsolution (cf. Haimes [1977J). Constraint (44) is modified by Eq. (42) as follows:: n
f 2 (x) =
L
j=l
dZ·x. ~ J J
E
(46)
Z
Therefore, the problem is modified again as follows: max fl = d xl + dlZx + ••• + dlnx ll Z n (47) subject to Ax ~ C , x ~ 0 (48) 0 0 where dZl all
dZZ a lZ
dZn a ln
A0
E
Z cl C0
aml
am2
...
amn
cm
The above problem (Eqs. (47) and (48)) is a linear programming problem. In the next section, the Dantzig-Wolfe decomposition is compared with the single-objective hierarchical overlapping coordination. Due to page limitations in the preprint several sections of this paper have been omitted. A complete version of the paper can be obtained from the authors. REFERENCES Haimes, Y. Y. (1977). Hierarchical Analysis of Water Resources Systems: Modelin~ and Optimization of Large-Scale Systems. McGraw-Hill, New York. Haimes, Y. Y., W. A. Hall, and H. T. Freedman. (1975). Multiobjective Optimization in Water Resources Systems: The Surrogate Worth Trade-off Method. Elsevier Scientific Publishing Company, The Netherlands.
14 27
Haimes, Y. Y., and D. Macko. (1973). Hierarchical structures in water resources systems management. IEEE Systems, ~an and Cybernetics, Vol. SMC-3, no. 4, pp. 396-40Z. Haimes, Y. Y. (1979). Large-scale water resources systems in large scale systems engineering applications. In: 1'1. Singh and A. Titli (Eds.). Handbook of LargeScale Systems Engineering Applications. North-Holland, pp. 227-273. Haimes, Y. Y. Hierarchical holographicmodeling. Technical Memo #SED-WRP-80. Case Institute of Technology, Cleveland, Ohio. Haimes, Y. Y., and V. Chankong. (1979). Kuhn-Tucker multipliers as trade-offs in multiobjective decision-making analysis. Automatica, Vol. 15. Hall, W. A., and Y. Y. Haimes. (1975). The surrogate worth trade-off method with multiple decision makers in multiple criteria decision making. Kyoto. Macko, D., and Y. Y. Haimes. (1978). Overlapping coordination of hierarchical structures. IEEE Transactions on Systems, Man and C~bernetics, Vol. SMC-8, no. 10 (October, pp. 745-751. Mendu, S. R., Y. Y. Haimes, and D. Macko. (1980). Computational aspects of overlapping coordination methodology for 1inear hierarchical systems, IEEE Trans., SMC-10, no. 2 (February). Schoeffler, J. D. (1971) . On -lin e multilevel systems. In: D. A. Wismer (Ed.) . Optimization Methods for Large-Scale Systems with Applications. McGraw-Hill. Singh, G., and A. Titli (editors). (1979). Handbook of Large-Scale Systems Engineering Applications. North-Holland, Amsterdam. Sung, K. (1978). Coordination of Overlap ping Hierarchical Water Resources Systems. Ph.D. Dissertation. Case Western Reserve University, Cleveland, Ohio. Sung, K., and Y. Y. Haimes. (1980). Hierarchical overlapping coordination and extensions. Technical Paper No. SEDWRP-10-80. Case Western Reserve University, Cleveland, Ohio. Tarvainen, K., and Y. Y. Haimes. (1980a). Coordination of hierarchical multiobjective systems theory and methodology . Tech. memo #SED-WRR-2-80. Submitted for publication. Tarvainen, K., and Y. Y. Haimes. (1980b). Hierarchical-multiobjective framework for energy storage systems. Fourth Multiple Criteria Decision Making Conference, Newark, Delaware.
T. Shirna, K. Tarvainen and Y. Y. Hairnes
1428
If yes, stop;
Fig. 1.
othe~lse
A non-feasible hierarchical scheme for the overlapping decompositions given by Eqs. (29a),(30), ... ,(32). Superscript k denotes iteration.
Discussion to Paper 47.1 Y.Y. Haimes (USA): Is it possible to use your very interesting approach - coordination of human information processing - to evaluate the efficacy and value of nel", data (i . e. the value of new information)? A.P. Sage (USA): Yes, there exist available approaches to determine the value of information. With the approach suggested here, it is possible to determine the value of information to a number of individuals in the hierarchy who have differing needs for a given set of data. Y. Kumata (Japan): An interpretation bias must be avoided and minimized, and I agree if you consider public planning such as city planning or regional planning, who is entitled to define who is biased and what interpretation is biased? Many participants, who are decision makers in different organizations with different social influence are likely to have different opinions over effects caused (or expected) by a proposed plan. Minimizing "so cial interpretation variance" among different interests groups is a vital problem for social systems planning. I would appreciate it if you would illuminate this problem further.
A.P . Sage (USA): Your comments are well taken and appropriate. One way of approaching the issues you discuss is to attempt to separate facts from values . The thought beh ind this is that we should agree on "facts" but there should be no attempt to "force" agreement on "value". Information processing and judgment may be flawed due to the use of any of a number of heuristics , many of which have been identified. Some of these, such as the failure to seek disconfirming information, the neglect of base rates or prior statistics when new information is received, and others are discussed in a forthcoming survey paper "Behavioral and Organizational Considerationss in the Design of Large Scale Systems for Planning and Decision Support", IEEE Transactions on Systems Man and Cybernetics, Vol. SML 11, No. 9, September 1981. Discussion to Paper 47.2 B. Sridhar (USA): How is the analysis affected when the polluters interact other than through the social cost function? i.e. when one polluter is downstream of another polluter, as in plants located at different points on a river. H. Tamura (Japan): When interactions exist among the polluters , as in the case of water pollution in rivers, we must take into account these interactions as additional constraints to the problem. I am not sure if the problem can be solved by taking this kind of constraint into account. Y.Y. Haimes (USA): Could you relate your approach of a system of emission charges to the existing literature on the subject? H. Tamura (Japan): In my paper I raised two new points compared with the existing literature on taxation and emission charges. First, I took into account "total emission regulation" in a prescribed region which was not taken into account before . Second , I proposed a new criterion for "impartiality" as to equalize the individual satisfaction of each player of the game.
HIERARCHICAL OVERLAPPING COORDINATION FOR LARGE SCALE SYSTEMS T. Shima, K. Tarvainen and Y. Y. Haimes Systems Engineering Department, Case Institute of Technology, Case Western Reserve University, Cleveland, Ohio, USA
Abstract. The basic hierarchical overlapping coordination method is briefly reviewed. Three generalizations to this method with respect to convergence, problem formulation, and multiple objectives are presented. It is explained how a potential convergence problem in the basic scheme can be overcome by additional optimizations with respect to some interaction variables . In the problem formulation extension, the different decompositions are allowed to have different variables. Classical hierarchical methods are used for the overlapping coordination scheme resulting in a three-level structure. The single-objective hierarchical overlapping scheme is extended to multiple objectives using the Surrogate Worth Trade-off (SWT) method. Overlapping coordination and the Oantzig-Wolfe algorithm are compared. A simple example is used to articulate the respective attributes. Finally, the concept of hierarchical holographic modeling, obtained as an extension of the concept of hierarchical overlapping coordination, is presented with some preliminary mathematical results and illuminating examples. Keywords. Overlapping coordination; multiobjectives; hierarchical systems; SWT method; decomposition. 1.
INTRODUCTION
Critics attribute the present and pervasive skepticism of systems modeling and its relatively limited use in decision making to a welter of reasons known collectively as the lack of model credibility . Credible models connote believable, plausible and reliable models. As modelers, it is instructive to investigate the sources leading to the development of not too credible models. These include such issues as (a) the worth of available data; (b) model verification, calibration and testing; (c) model scope; (d) the time and resources allocated to model development--both in funds and in man power; (e) institutional support to model development, testing and maintenance; and, of course, (f) the assumptions and simplifications made in model development, and the respective optimization approach used to model solving. Focusing on large scale systems, this paper addresses issue (f) with some implication to the availability of data.
extent to which these simplifications can be justified is germane to the level of model credibility. Furthermore, the simplifying assumptions are central to a three-way compromise and trade-off among (a) the modeling process, (b) the optimization process, and (c) the availability and worth of data. For example, a very complex model might neither have the supportive data base nor be computationally tractable. The hierarchical approach, through decomposition and multi level optimization, is aimed at resolving some of the inherent conflicts between modeling and optimization. The hierarchical overlapping coordination extends this resolution to the data base as well (Haimes and Macko [1973], Macko and Haimes [1978], Sung [1978], and t1endu, Ha i mes and Macko [1980]). The a ttributes of the hierarchical approach are well documented in the literature and will not be repeated here (see, for example, Haimes [1977] and Titli and Singh [1979]). Hierarchical Overlapping Coordination (HOC) recognizes that in decomposing system's model into subsystems and/or levels within a hierarchical structure, more than one decomposition is possible, and most importantly-desirable. These different decompositions, each of which might be responsive to a different aspect of the system and/or data base, are coordinated through the respective couplings of the various decompositions. The
By necessity, mathematical models mimic real systems through simplified assumptions. The Invited paper submitted for presentation at the 8th Triennial World Congress/International Federation of Automatic Control, Kyoto, Japan, August 24-28, 1981. 1421
1422
T. Shima, K. Tarvainen and Y. Y. Haimes
coordination ultimately yields to an overall system's optimum in single-objective models and to preferred Pareto optimal solutions in multiple-objective models. Prior to articulating the attributes of HOC, it is instructive to consider the following generic problem drawn from the water resources area. The planning and management of water resources systems transcend hydrological as well as political-geographical boundaries, yielding to two natural respective model decompositions--hydrological and po 1 iti ca 1 • Without HOC, modelers are implicitly forced to choose between a hydrologically or politically based model/decomposition. Each decomposition represents and uncovers important aspects not amenable through the other. I~ost critical of all concerns is associated with the availability of the data base. The manipulation of data bases to serve and suit artificially (through the modeling process) imposed demands and constraints necessitates a compromise--an ultimate deterioration in model credibility . For example, data concerni ng s treamfl ow, wa ter qua 1i ty, and fl oods are available on a hydrological basis and collected by the U.S. Geological Survey, U.S. Environmental Protection Agency, and the U.S. Army Corps of Engineers, respectively. On the other hand, data concerning population dynamics, employment, and other economic activities are available on political/geographical bases and are collected by different agencies such as the U.S. Departments of Commerce, Labor, or Treasury. HOC enables the utmost utilization of these data bases with their minimum manipulation or misuse. This can be achieved by resorting to two simultaneous decompositions--hydrological and political/geographical--each of which might be with different number of subsystems. The Maumee River Basin, for example, lends itself to eight hydrological subsystems and to five political/geographical subsystems (Haimes, et al. [1979]). In general, water resources systems, as well as many other large scale systems, lend themselves to more than one decomposition/description, such as functional (water supply and demand for the various sectors--agriculture, industry, municipality, aquatic life, etc.), temporal (long, intermediate and short term and on line) as well as the hydrological and political/geographical decompositions discussed previously . The attributes of HOC might be summarized as follows (Macko and Haimes [1978]): 1) 2)
3)
Adds more fle xibility in modeling Yields more realistic models by increasing the responsiveness of the model structures to the organizational, political, technical, financial, and historical real ities. Increases the responsiveness of the model structures to the availability of data.
4) 5)
Models and coordinates system's transition from steady state to emergency state . Provides holographic view of modeled system.
In the following sections, theoretical aswell as methodological extensions of hierarchical overlapping coordination are presented. BASIC HIERARCHICAL OVERLAPPING COORDINATION (HOC)
2.
The basic algorithm and solution procedures are described in Haimes and Macko [1973], Macko and Haimes [1978], Haimes [1979], and Mendu et al. [1980]. A brief summary of the basic procedure is given in this section. The problem under consideration is represented as follows: ( 1)
max f(x) subject to Ax = c
( 2)
where f is a real-valued concave function and x is a decision vector. Suppose the system has two decompositions a and S. The original problem is transformed into the following problems by the rearrangement of the vectors x and c. The a decomposition problem is max f a (x a )
(3)
subject to
The
Aa Xa = ca S decomposition problem is
(4)
(5)
max fS(x S) subject to ASxS
Cs
(6)
where xa Aa
p x c = Qa c Xs = PSx, a a -1 Q AP- l A = QSAPS a a S
Cs
QSc
P and Q are nonsingular matrices representing the rearrangement of the vectors x and c. Suppose the a problem is decomposed into n subsystems and the S problem is decoma posed into nS subsystems. Here we assume the matrices Aa and AS are decomposable as follows: (7)
14 23
Hierarchical Overlapping Coordination
(8)
where Band H are block diagonal matrices compatible with the partitioning of the vectors by subsystems and G denotes the set of matrices of elements of A which are not on the block diagonal. The a and S problems are rewritten as follows: n a
max f a (xa ,ya ) = i=l L fa 1· (x a 1· ,ya 1'··· ,ya na ) (9)
subject to
na
B .x . +
a1 al
LG
• .y .
j=l alJ aJ
=c
.
a1
(10)
Haixai = Yai
(i = 1 , ••• ,na)
nS max fS(xS'YS) = i~l fSi (x Si 'YS1 , ... 'Ysns) (11 )
subject to (12 ) ( 13)
Ya and YS are vectors of the interaction variables. The above problems are equivalent to each other and to the original problem (1) and (2) •
The interaction variables Ya and Y are S determined by the decisions in the other decomposition as follows: Ya
Ha xa
(14 )
YS
HSxS
(15 )
Fi rs t, Y is fi xed and the a problem is a solved in each subsystem. Then Y is given S by the xa as represented in Eq. (15). The S problem is solved in each subsystem, and the solution Xs gives a new Ya ' and the a problem is solved again.
criterion. But it may happen that the result of the convergence does not reach the C~ C '(lld' solution. A method to overcome this possible difficulty is discussed below. First, the case where the problem of convergence is directly due to common interaction variables in different decompositions is addressed. This is the case in example problem 3 in Mendu et al. [1978] where the variables r 12 , r 13 , and r 23 act as coupling variables in both decompositions. Common interaction variables are never changed by the basi c scheme; the values remain as initial guesses. Assume that the problem is only due to the common interaction variables; that is, if they are guessed right at the beginning, the basic scheme converges to the right result, as in the example mentioned. A straightforward approach to this situation is simply to let one of the decompositions, or both, change the common coupling variables. In the basic scheme, when the algorithm is seen, say, from the point of the a decomposition, the procedure is the same to the a decomposition as a usual feasible decomposition scheme. That is, the a decomposition uses some initial values for the interaction variables, optimizes the subsystems, gets new values for the interaction variables (from the S decomposition which acts as an upper level to the a decomposition), and so on. In the case of common interaction variables, the 8 decomposition does not change these values. Hence, the a decomposition has to change these common interaction variables using methods developed for feasible hierarchical coordination. The basic scheme is, thus, modified slightly: before the a decompos it i on sends its resu lts to the S decomposition (Eq. (15)), the a decomposition improves, using standard methods, the values of the common interaction variables. Example 1. min
Consider the following problem: 8
L
i =1
2
x,
(16)
1
subject to xl x2
This procedure is repeated until convergence is achieved.
0
0
0
0
0
0
0
0
3.
3.1
EXTENSIONS OF BASIC HIERARCHICAL OVERLAPPING COORDINATION
Improvement of Convergence
In the basic scheme, the different decompositions alternatively improve the performance
[:
0
0
x3 x4
:] x5
x6 x7 x8
~]
( 17)
T. Shima, K. Tarvainen and Y. Y. Haimes
1424
Let x4 and x5 be interaction variables in the 0. decompos i ti on and x5 and x6 in the 8 decomposition'T That is, Ya = (x 4 ,x 5 )T and Y8 = (x 5 ,x 6 ) , and let the decompositions be as follows: a-decomposition: Subproblem 1(0.) . 2 2 2 2 2 2 mln (xl +X 2 +x 3 +x 6 ) + x4 + x5 X ,X
l
2
,X
3
,X
xl + x2 -x5 x3 + x6 = -x 4 - x5 Subproblem 2(0.) min (X~ + X~)
x7 ,x 8 subj ect to =
4
4
n x5
-n-l x5
Correspondingly, the values of the interaction variables of the 8 decomposition at the n-th iteration are determined by the values of the 0. decomposition at the n-th iteration: -n no x5 x5 -n n x5 x5
6
subject to
x7 + x8
xo5 ' where some guessed values are given to the interaction variables of the 0. decomposition. At the (n+l)-th iteration, the 0. decomposition uses values x4n and x5n for the interaction variables. These values are given by the results of the 8 decomposition at the n-th iteration: xn x-n-l
-x5
8-decomposition: Subproblem 1(8) -2 + x-2) + x5 -2 + -2 ml. n (-2 xl + x-22 + x3 x6 4
Where x~o denotes the optimal value of x5 at the n-th iteration to differentiate from x~ (= x~-l )--the initial value for x5 at the n-th iteration in the 0. decomposition. n -n-l . n -n-l Glven x4 = x4 and x5 = x5 at the beginning of the n-th iteration, it is easily seen that the corresponding solutions of the 0. decomposition problems denoted by -n-l ) are the following: xi ( x5 Subsystem 1 (0.) :
subject to xl + x2 x3 + x 4
-n-l ) xi ( x5 -x 5 -x 5
( 18)
(19)
Subsystem 2(0.): -n-l) _- -x-n-l /2 , k = 7,8 . (20) xk ( x5 5 The corresponding value for the performance -n-l ), is criteria (16), denoted by fan( x5
min ( x-27 + -2) x8 x7 ,x 8 subject to Let
i = 1,2
-n-l) /2, - ( x-n-l 4 + x5 j = 3,6,
x6
Subprobl em 2( 8)
x7 + x8 -x 5
-n-l -x5 /2
fn(x~-l) = (1/2)(x~-1 + x~-1)2 T
xa = [xl"" ,x 8] ,
The optimal solution of the problem given by Eqs. (16), (17) i s x~ = 0, i = 1 , ... ,8. The basic overlapping algorithm does not yield this optimal solution (unless the initial guess for x5 is zero) because x5 is a common interaction variable in the 0. and 8 decompositions. Consider the extension explained above, whereby x5 is optimized in the 0. decomposition. Let the superscript n denote the iteration number. Denote the initial values, by x~,
+ ( -n-l)2 x4 x5 • + 2 (-n-l)2
(21 )
Now, instead of using the value -n-l x5 for x5 ' an optimization is performed with respect to x5 • The optimal value is obtained from (21) as no -n-l x5 = -x 4 /5 The corresponding values for xl ,x 2 ,x 3 ,x 6 ' ... , x8 (given by Eqs. (18), ... ,(20)) are:
14 25
Hierar chical Overla pping Coordi nation
n xi x.n J
x~-l/lO
i = 1 ,2
(22)
_(2/5)x n- l 4
j = 3,6
( 23)
n (24) k = 7,8 xk x~-l/lO Next, consid er the S decomposition. At the n-th iterati on, it receive s x5 = x~o and x6 = x~ from the a decomposition. Optimization of the subproblems is then easily performed with the following results : -n-l -n (25) -x 5n/ 2 = x4 /10, i =1,2 xi 1 -(x~o+X~)/2 = (-3/5)x 2x1 j
= 3,4
(26) ( 27 )
-n /10,k= 7,8 xk =-x 5no /2=x-n-l 4 where the second equali ties are obtaine d using Eqs. (22), ... ,(24). Specif ically, for j = 4, -n x x = ( -3/5 )-n-l 4
Eq. (26) yields
4
From indica ting that x2 ~ 0 as n ~ all that shown be can Eqs. (23), ..• ,(27), it variab les in both decompositions approach zero--t he optimal solutio n. 00 .
In a general case, a convergence problem is not directl y due to some common interac tion variab les in the decompositions. Instead , there is a mathematical relatio nship in the equatio ns that binds the interac tion variab les of differe nt decompositions in such a way that they do not get varied over their origina l feasibl e set. In fact, exactly the same procedure as in the case of common interac tion variab les under optimi zation are release d. This will enlarge the areas of the feasib le sets of the decompositio ns to cover the origina l feasibl e set. As an extreme case, if all interac tion variables of the a decomposition are undero ptimizatio n, the optimum is guaranteed to be found. But this is genera lly not necess ary, because the S decomposition will cover a subspace of the feasibl e set. In practic e, a diffic ult problem is determining how many interac tion variab les in the a decomposition should be optimi zed. Mathematically , this relatio n is given below (cf. Eq. (14) in Macko and Haimes [1978] ):
N(A) .
N[H~Apal
+
N[HeApel
(28)
where H'a is a part of Ha correspondi ng those interac tion variab les that are not optimized . In an extreme case, when all variables are optimi zed, then H' = 0, and Eq. a
(28) holds trivial ly. Note also that if common interac tion variab les exist, there is, in genera l, no quarantee that the case does not imbed also more implic itly binding relatio ns hi ps. If the number of optimized interac tion variables cannot be determined, then, after the scheme has converged, the result should be tested by checking if the change of the interacti on variab les in one decomposition improves the result. 3.2
Extension of Problem Formulation
An import ant object for extending the basic scheme corresponds to problem formul ation. In it, is assumed that the differe nt decompositio ns comprise the same variab les (in a one-to- one corresp ondenc e). This is not usually the case: differe nt decompositions deal with differe nt aspects of the system, and, hence, with differe nt respec tive variables. Consider the following formulation as an immediate extensi on of the basic problen formul ation. The a decomposition deals and the S with a variab le vector (x,y) a a decomposition with a variab le vector (x s ,yS). Here, xa and Xs are the variab le vectors of the a and S decomposition that are directl y bound by equatio ns that specify the relatio nship between the decompositions. For example, (29)
xa
PaSx S'
Xs
PSax a ,
(29)
plx + PSxS = 0 , a a
(29)
or or ' Pas' PSa' Pa' h an d pI13 are cons t an t were matric es, not necess arily invert ible. The vectors Ya and ys comprise the otherv ariables in the a and 13 decompositions, respec tively.
Let the cost functio n be a sum of two cost functio ns relatin g to the decompositions. That is, the object ive is to maximize fa(xa'Y a) + fS(xS'YS).
(30)
In additio n, let the system and constr aint equatio ns in the decompositions be expressed by 9a (X a 'Ya)
0
gS(xs'YS)
0
(31)
(32)
Note that the problem given by (29a or b), (30), (31), (32) reduces to the basic case, when Ya and YS are vacuous and PaS or
1426
T. Shima, K. Tarvainen and Y. Y. Haimes
Psa is invertible. Mathematically, this problem (Z9a or b or c), (31), (31), and (3Z) is exactly of the same form as the problems in classical hierarchical systems theory. Hence, all the techniques developed there can be used in HOC. In effect, a three level scheme results when applying basic hierarchical schemes. At the highest level, the coordination variable relating to the coupling equation (Z9a or b or c) is changed. The coordination variable is the vector (xa 'x ) S in the feasible method, and a Lagrange multiplier vector in the non-feasible method. The two other levels are related to the a and S decompositions (which are solved, in parallel). Fig. 1 depicts a scheme where the non-feasible method is used for a case with the relationship between the two decompositions given by Eq. (Z9a). In Fig. 1, which depicts a non-feasible method, the lowest (1st) level optimizations have not been given. They can be formed using standard hierarchical methods for the criteria indicated at the second level. Overla
in
Let the original multiobjective hierarchical overlapping coordination problem be the following: (33) subject to Ax = c
(34 )
The a decomposition problem is ma~ [~(xa),~(xa) , .•• ,~(xa)]
( 36)
The
S decomposition problem is max [f~(XS ) ,f~(XS ) , ... ,f~(xS)]
In the above context of several decompositions and system aspects, different objectives related to different aspects arise in a natural way. Hence, the generalization of the basic overlapping coordination scheme to the multiobjective case is important.
( 37)
xS
subject to ASx S
cS.
(38)
Here, xa = pax,
Most planning of large-scale systems includes the optimization of several objectives. Methodologies to deal with multiobjective cases have been substantially developed during the past decade. Applications of multiobjective optimization in several fields of technology have produced much more realistic results than otherwise with classical singleobjective optimization.
(35)
x subject to
Aa = QaApa-l
cS
QSc
AS
QSApS-l
All f(·)'s tions.
xS
Ca = Qa c and
pSx,
and
are assumed to be concave func-
P and Q are non-singular matrices representing the rearrangement of the vectors x and c as described previously. Consider, first, the case with two linear objectives and inequality constraints: (39)
max [fl(x),fZ(x)]
The basic scheme can be generalized immediately to the case where the different decompositions have the same preferences for a common vector of objectives. Namely, if an underlying utility function for these preferences is assumed, then the problemisactually reduced to a single objective case, and the same basic Overlapping Coordination scher,le works under the same condition. Of course, the critical assumption is the existence of an underlying utility function.
subject to Ax ~ c, where
Another promising approach is to solve t~OC by the Surrogate Worth Trade-off (S~JT) method (Haimes, Hall and Freedman [1975]) which can generate the set of all needed non-inferior solutions and their corresponding trade-offs to the decision-makers and analysts.
A
The original problem (Eqs. (39) and (40)) is modified as follows:
The procedure can be briefly described as follows:
To solve this multiobjective problem, the SWT method is used. This problem is equivalent
x
~
(40)
0
fl (x)
dllx l + dlZx Z + ••• + dlnx n
(41 )
fZ(x)
dZlx l + dZZx Z + ••• + dZnx n
(4Z)
cl all·····a ln
C
C aml···· .ar.m
z
cm
Hierarchical Overlapping Coordination
to the following problem (cf. Haimes [1977J, Haimes, Hall and Freedman [1975J). n
L
Max fl (x)
j=l
(43)
d1 ·x . J J
subject to f 2 (x) ~ E2
Ax
~
c, x
(44)
~
0
(45) l 2 The trade-off A12 (= - af /af ) as the Lagrange multipller corresponding to the constraint (44). The trade-off, A , is a 12 function of the given E • 2 A1Z > 0 guarantees the non-inferiorsolution (cf. Haimes [1977J). Constraint (44) is modified by Eq. (42) as follows:: n
f 2 (x) =
L
j=l
dZ·x. ~ J J
E
(46)
Z
Therefore, the problem is modified again as follows: max fl = d xl + dlZx + ••• + dlnx ll Z n (47) subject to Ax ~ C , x ~ 0 (48) 0 0 where dZl all
dZZ a lZ
dZn a ln
A0
E
Z cl C0
aml
am2
...
amn
cm
The above problem (Eqs. (47) and (48)) is a linear programming problem. In the next section, the Dantzig-Wolfe decomposition is compared with the single-objective hierarchical overlapping coordination. Due to page limitations in the preprint several sections of this paper have been omitted. A complete version of the paper can be obtained from the authors. REFERENCES Haimes, Y. Y. (1977). Hierarchical Analysis of Water Resources Systems: Modelin~ and Optimization of Large-Scale Systems. McGraw-Hill, New York. Haimes, Y. Y., W. A. Hall, and H. T. Freedman. (1975). Multiobjective Optimization in Water Resources Systems: The Surrogate Worth Trade-off Method. Elsevier Scientific Publishing Company, The Netherlands.
14 27
Haimes, Y. Y., and D. Macko. (1973). Hierarchical structures in water resources systems management. IEEE Systems, ~an and Cybernetics, Vol. SMC-3, no. 4, pp. 396-40Z. Haimes, Y. Y. (1979). Large-scale water resources systems in large scale systems engineering applications. In: 1'1. Singh and A. Titli (Eds.). Handbook of LargeScale Systems Engineering Applications. North-Holland, pp. 227-273. Haimes, Y. Y. Hierarchical holographicmodeling. Technical Memo #SED-WRP-80. Case Institute of Technology, Cleveland, Ohio. Haimes, Y. Y., and V. Chankong. (1979). Kuhn-Tucker multipliers as trade-offs in multiobjective decision-making analysis. Automatica, Vol. 15. Hall, W. A., and Y. Y. Haimes. (1975). The surrogate worth trade-off method with multiple decision makers in multiple criteria decision making. Kyoto. Macko, D., and Y. Y. Haimes. (1978). Overlapping coordination of hierarchical structures. IEEE Transactions on Systems, Man and C~bernetics, Vol. SMC-8, no. 10 (October, pp. 745-751. Mendu, S. R., Y. Y. Haimes, and D. Macko. (1980). Computational aspects of overlapping coordination methodology for 1inear hierarchical systems, IEEE Trans., SMC-10, no. 2 (February). Schoeffler, J. D. (1971) . On -lin e multilevel systems. In: D. A. Wismer (Ed.) . Optimization Methods for Large-Scale Systems with Applications. McGraw-Hill. Singh, G., and A. Titli (editors). (1979). Handbook of Large-Scale Systems Engineering Applications. North-Holland, Amsterdam. Sung, K. (1978). Coordination of Overlap ping Hierarchical Water Resources Systems. Ph.D. Dissertation. Case Western Reserve University, Cleveland, Ohio. Sung, K., and Y. Y. Haimes. (1980). Hierarchical overlapping coordination and extensions. Technical Paper No. SEDWRP-10-80. Case Western Reserve University, Cleveland, Ohio. Tarvainen, K., and Y. Y. Haimes. (1980a). Coordination of hierarchical multiobjective systems theory and methodology . Tech. memo #SED-WRR-2-80. Submitted for publication. Tarvainen, K., and Y. Y. Haimes. (1980b). Hierarchical-multiobjective framework for energy storage systems. Fourth Multiple Criteria Decision Making Conference, Newark, Delaware.
T. Shirna, K. Tarvainen and Y. Y. Hairnes
1428
If yes, stop;
Fig. 1.
othe~lse
A non-feasible hierarchical scheme for the overlapping decompositions given by Eqs. (29a),(30), ... ,(32). Superscript k denotes iteration.
Discussion to Paper 47.1 Y.Y. Haimes (USA): Is it possible to use your very interesting approach - coordination of human information processing - to evaluate the efficacy and value of nel", data (i . e. the value of new information)? A.P. Sage (USA): Yes, there exist available approaches to determine the value of information. With the approach suggested here, it is possible to determine the value of information to a number of individuals in the hierarchy who have differing needs for a given set of data. Y. Kumata (Japan): An interpretation bias must be avoided and minimized, and I agree if you consider public planning such as city planning or regional planning, who is entitled to define who is biased and what interpretation is biased? Many participants, who are decision makers in different organizations with different social influence are likely to have different opinions over effects caused (or expected) by a proposed plan. Minimizing "so cial interpretation variance" among different interests groups is a vital problem for social systems planning. I would appreciate it if you would illuminate this problem further.
A.P . Sage (USA): Your comments are well taken and appropriate. One way of approaching the issues you discuss is to attempt to separate facts from values . The thought beh ind this is that we should agree on "facts" but there should be no attempt to "force" agreement on "value". Information processing and judgment may be flawed due to the use of any of a number of heuristics , many of which have been identified. Some of these, such as the failure to seek disconfirming information, the neglect of base rates or prior statistics when new information is received, and others are discussed in a forthcoming survey paper "Behavioral and Organizational Considerationss in the Design of Large Scale Systems for Planning and Decision Support", IEEE Transactions on Systems Man and Cybernetics, Vol. SML 11, No. 9, September 1981. Discussion to Paper 47.2 B. Sridhar (USA): How is the analysis affected when the polluters interact other than through the social cost function? i.e. when one polluter is downstream of another polluter, as in plants located at different points on a river. H. Tamura (Japan): When interactions exist among the polluters , as in the case of water pollution in rivers, we must take into account these interactions as additional constraints to the problem. I am not sure if the problem can be solved by taking this kind of constraint into account. Y.Y. Haimes (USA): Could you relate your approach of a system of emission charges to the existing literature on the subject? H. Tamura (Japan): In my paper I raised two new points compared with the existing literature on taxation and emission charges. First, I took into account "total emission regulation" in a prescribed region which was not taken into account before . Second , I proposed a new criterion for "impartiality" as to equalize the individual satisfaction of each player of the game.