Decomposition of satisficing decision problems

I N F O R M A T I O N S C I E N C E S 22, 1 3 9 - 1 4 8 (1980)

139

Decomposition of Satisficing Decision P r o b l e m s SHINZO TAKATSU

Department of Management Engineering, Hiroshima University, Hiroshima, Japan Communicated by M. D. Mesarovic

ABSTRACT This paper intends to clarify the decomposition of satisficing (decision) problems. Many researchers have studied the several aspects of satisficing problems verbal-logically (e.g. Cyert and March [2]; Simon [12]; Simon, Smithburgh, and Thompson [13D or mathematically (e.g. Charnes and Cooper [1]; ljiri [4D, and a formulation of satisficing problems under true uncertainty was done about one decade ago by M. D. Mesaxovic and his group (e.g. Mesarovic, Macko, and Takahara [9]; Mesarovic and Takahara [10D. Its fundamental properties have been studied qualitatively by Matsuda and Takatsu [6-8]. However, the decomposition of satisficing problems has not been explicitly mentioned anywhere, even though implicit discussions have been givenl Without the decomposition principles, we cannot solve large-scale satisficing problems, and cannot analyze or design coordination processes of hierarchical systems with satisficing problems as overall decision problems. Our study is mainly concentrated on the decomposition. Our results are somewhat restricted, because we deal with satisficing problems of relatively manageable forms. However, we can develop more sophisticated decompositions by combining our decomposition principles or by adding the other assumptions to our formulations. The results will be a starting point of a more complete study of satisficing decision problem solving.

1.

INTRODUCTION

In this paper, we c o n s t r u c t d e c o m p o s i t i o n principles for satisficing (decision) problems. T h e word "satisficing" is a d o p t e d from the works of H. A. S i m o n [12, 13], a n d o u r f o r m u l a t i o n is d u e to M. D. Mesarovic [10]. W e believe that the satisficing (decision) criterion is one of the most i m p o r t a n t decision criteria u n d e r u n c e r t a i n t y . This does n o t m e a n that the other criteria are n o t i m p o r t a n t . Rather, we consider that the i m p l i c a t i o n a n d practicability of a decision criterion do n o t d e p e n d on its axiomatic or m a t h e m a t i c a l structure b u t on the d e c i s i o n - m a k e r ' s value j u d g m e n t of the u n c e r t a i n t y in his decision e n v i r o n m e n t a n d o n his responsibility for decision consequences. T h a t is, the satisficing criterion is i m p o r t a n t for some d e c i s i o n - m a k e r s in some decision e n v i r o n m e n t s , especially, w h e n the cognitive limitations of h u m a n beings m u s t @Elsevier North Holland, Inc., 1980 52 Vanderbilt Ave., New York, NY 1 0 0 1 7

0020-0255/80/08139-10501.75

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be taken into account; see e . g . R . M . Cyert and J. G. March [2] or H. A. Simon [12]. A satisficing problem is defined as follows: Find r~ E M such that

(v,o)(,o ~ ~ g ( , ~ ,

,o)

>

~(,~)),

where M is the set of decision alternatives, f~ is the set of uncertainties, g: M × f~--.R is the performance function, and r : f~--~R is the aspiration-level function We have already shown an axiomatic characterization of the satisficing criterion and its basic properties in [6], studied its algebraic properties in [7], and investigated the existence of satisficing solutions and the treatment of constrained satisficing decision problems in [8]. We are now at a stage where decomposition principles for satisficing problems must be studied, since such principles have not been explicitly developed. In order to enhance the practical applicability of the satisficing criterion to large-scale problems, it is a fundamental task to develop them. The organization of this paper is as follows. In Sec. 2, we show a decomposition of a satisficing problem that is completely separable. Notice that the decomposition of a satisficing problem is not trivial even if it is completely separable. In Sec. 3, we study the decomposition of a satisficing problem with separable constraints whose performance function is separable. We give two types of decompositions: by resource pricing and by resource allocation. In Sec. 4, we investigate a decomposition of a satisficing problem with process interactions, so that the performance function is inseparable. In Sec. 5, we summarize our results and suggest further research. 2.

C O M P L E T E L Y S E P A R A B L E CASE

In this section, we show a decomposition of the following completely separable satisficing problem: PROBLEM 2.1. Find th E M such that

(v,o)(,o ~ a ~ g ( , ~ , ,o)

>

,(oo)),

where

m = m , x . . , xm.,

g(m, w)= ~gi(m,, w), •(~)= ~,~).

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141

In an optimizing problem with a completely separable performance function with no constraints, it is unnecessary to derive its decomposition, because the decomposition is trivial from the definition. However, in satisficing decision problems of the above type, the decomposition is not trivial, because we have no guarantee of the existence of solutions of each satisficing subproblem even if the overall satisficmg problem has a solution. Here, we assume the following: ASSUMPTION 2. I. (I) (2) (3) (4)

~2 is a compact convex subset of a linear topological space. 3,/,. is a compact convex subset of a linear topological space. g~ is concave in m~ and upper semicontinuous in mi for each 60. g l - h is convex in 60 and lower semicontinuous in to for each m.

Now we can show the foUowing proposition. PROPOSITION 2. l. Suppose that Assumption 2. I is satisfied. Then there exists an overaYl satisficing solution da in M if and only if there exists a vector function 8 = ( 8 j ..... ~ ) (with 8i:~2---~R concave and Y.8i(60)>0 for each 60)1 and a solution vector ~ = ( ml ..... ran) of the satisficing subproblems such that

g,( ~ i , 60) > ,ri( 60) + 8i( 60)

for each 60.

Proof. Let rh E M be an overall solution; that is, g(rh, 60)~ ~'(60) for each 60. Then

Y. [g,(,~,. 60)- ~-,(to)] > o. From the assumption, each function gi(r~i, 60 ) -- "r/ ( to ) takes a finite value in each 60 at rh i. Define 8 , ( 6 0 ) = g i ( r h , 6 0 ) - h ( 6 0 ) for each 60. This function evidently satisfies the requirements. The converse is self-evident. This proposition shows that in a satisficing problem, even without any constraint, solutions of the subproblems must be coordinated in order to find an overall solution for the system. In this case each mfimal problem is as follows: PROBLEM 2.2. For a given concave function 8i, find rh, which satisfies

( v ~ ) ( ~ E a.-,~,(m,, to) > ~-~(60)+ a,(~)). All the propositions in this paper can be easily transformed into ones for reference decision problems in which a uniformaUy better decision ~ E M than a reference decision m * ~ M is sought. 2 Therefore, we do not show the

t ln this paper, we assume that a vector function 8 always satisfies these conditions. 2For ~ to b,¢ uniforr~allv better than m* m ~ that g(~, w)>g(m*, oJ) for each ~0.

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transformed results in this paper. However, notice that the converse transformation is not so easy, even though Mesarovic, Macko, a n d T a k a h a r a [9] believed that it might be.

3.

SEPARABLE CASE (TREATMENT OF SYSTEM INTERACTION) Here, we study decomposition of the following problem: PROBLEM 3.1. F i n d rfi E M which satisfies

(Wo)(~ ~ - ~ g ( r ~ , w) > ~(~o)) subject to H ( r f i ) < 0, where

M = M I × . . . ×M,, g(m, w ) = ~g,(rn,, w),

H: M--~R ~

[ H ( m ) r = ( H ' ( m ) ..... H ' ( m ) ) ] ,

and

Hg(m)= ~ H/(mi). i

In this paper, we do not deal with constraints with uncertainty in their arguement variables, because we cannot introduce a concept of "certainty equivalence" in a decision p r o b l e m under true uncertainty. 3 Now, let us consider the following Lagrangian problem: PROBLEM 3.2. F o r a given X=(;k I. . . . . M) ~ 0, find thEM such that g(rfi, w) - X - H ( r h ) ~ ' r ( w ) for any w. D E F ~ m O N 3.1. W h e n Problem 3.2 is solved at ( ~ , ~ ) a n d ~ , . H ( r h ) = 0 [ H i ( r h ) = 0 if M > 0 and H i ( r h ) < 0 if h-' = 0], 2~/> 0 is called a Kuhn-Tucker vector for the satisficing problem. On the existence of K u h n - T u c k e r vectors, see [8]. ASSUMPTION 3.1. (1) ~2 is a compact convex subset of a linear topological space. (2) M is a compact convex subset of a linear topological space.

3For example, see A. Charnes and W. W. Cooper [1].

SATISFICING DECISION PROBLEMS

143

(3) g is concave in m and upper semicontinuous in m for each to. (4) g-~" is convex in to and lower semicontinuous in ~o for each m. (5) For any given m * E M , g ( m , t o ) - g ( m * , t o ) is convex in to a n d lower semicontinuous in to for each m. (6) H j is a convex function for each 1 < j < s. Let show the following proposition. PROPOSITION 3.1. Suppose that Assumption 3.1 is satisfied. I f Fn(X) solves Problem 3.2 for a given ~ >10 and ~ is eventually a Kuhn-Tucker vector, then Yh(~ ) is a solution of Problem 3.2.

Proof. Let rh(X) be a solution of Problem 3.2 and X be a K u h n - T u c k e r vector. It is apparent that g(rfi()~),co)=g(rfi()t),to)-)t.H(Fn()t))>~'c(to)

for any ~ ;

at the same time, H ( m ( h ) ) < 0 , because h is a K u h n - T u c k e r vector. Hence, rfi(X) is a satisficing solution of Problem 3,1. Though we will not use the converse result of Proposition 3.1 in the main part of this paper, we show it in the following proposition. First we define the following sets: M °p= { r h l r h ~ M & (3~)(~5 E~2 &

( V m ) ( mGM--~g( ,h, ~o) ) g( m, ~ ) ) ) ) and

f'i={mlH(m)
}.

PROPOSITION 3.2. Suppose that Assumption 3.1 is satisfied. Let assume M °p N 121v~q~. Then if rh is a solution of Problem 3.1, there exists a Kuhn-Tucker vector ~k which makes Problem 3.2 solvable at (~, ~ ( ~ )), where ~ ( ~ ) ~ M °p is uniformally better than r~ if rh is not in M °p.

Proof. See T. Matsuda and S. Takatsu [8]. Now let us study the satisficing subproblems. PROBLEM 3.3. For a given concave function 6 , : f ~ R r f i , ~ M such that

and given X, find

gi(r~,, w) - X. H~(r~)/> ~,(~) + 8 , ( ~ ) . ASSUMPTION 3.2. (1) f~ is a compact convex subset of a linear topological space. (2) M i is a compact convex subset of a linear topological space.

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SHINZO TAKATSU

(3) g~ is concave in mr and upper semicontinuous in m~ for each to. (4) gr-~'r is convex in to and lower semicontinuous in to for each mr. (5) For any given m~EM~, gr(mi, ~)-g~(m~, to) is convex in to and lower semicontinuous in ~0 for each mr. (6) Hr(mr) is a convex vector function. PROPOSITIOta 3.3. Suppose that Assumption 3.2 is satisfied. Suppose also that for each i, rhr()~) solves Problem 3.3 for given 8r=(81 ..... 8,)(~,8i~ 0, 8 concave) and ~ • O, and that rh()t)= (rhl()t) ..... ~ , ( X ) ) satisfies (a) h-H(tli(;t)) = O,

Co) HJ(eh(X))=O if hi>O, HY( fft()t )) < 0 if )tJ=O. Then the combined solution ~(X ) is an overall satisficing solution of Problem 3.1. Proof. F r o m P r o p o s i t i o n 3.1, it is evident that tfi()t) is an overall solution because the assumption in that proposition is satisfied. We investigate another decomposition of Problem 3.1 by a different approach which is equivalent to so-called "decomposition by resource allocation" when fl is a singleton set. Let define a set Y as

r={yiy=(y,

..... y . ) & y Z = ( y ' ..... y : ) a

yyt=0}. i

Now, define each satisficing subproblem directly from the Problem 3.1 as follows: PROBLEM 3.4. For given concave function 8i and vectory~, find mr ~ Mr such that gr(tfir, ~0)~ ¢i(t~)+Sr(to) for each to subject to H~(tfi~) 0, find g t , E M i such that

gr(thr,~o)+)%'(yl-Hl(rh~))>%(to)+8i(to)

for each to

subject to (a) Xr.(y,-Hr(,~r))=O Co) H/(tfir)-~y / if h~>O,

st/(,ttr) ,:y/ if X{ o. =

Hence, we can show the following proposition. PROPOSlTIOIq 3.4. Suppose that Assumption 3.2 is satisfied. For given 8r, Yr, and 7%, if each satisficing subproblem as given in Problem 3.5 is solved at rhr, then • =(tfi I..... ~ , ) is an overall satisficing solution for Problem 3.1.

S A T I S F I C I N G DECISION PROBLEMS

145

Proof. From the assumptions, gi(r?*,,to)+Xi.[y,-Hi(d*i)]>,l(to)+8,(to)

for each o~

and

x,.(y,-n,(m,)) =0; hence, ~,[g,(th,,to)+)~i.(y,-Hi(~i))l>~,[rl(to)+8,(to)] for each to. Therefore, g(rh, to)~, ~(to) for each to; at the same time, Y.HA6tl)• Y.y~=0. That is, t~ is an overall solution of Problem 3.1, which was to be proved. Notice that in Proposition 3.4 it is unnecessary for the system to have equal Kuhn-Tucker vectors; that is, A~:A{ ( i ~ k ) in general. This means that the system adopts a multiple pricing mechanism. (For a definition see L. S. Lasdon [5].) As mentioned above, we have studied the decompositions of satisficing problems with systems interaction. 4.

INSEPARABLE CASE ( T R E A T M E N T OF PROCESS I N T E R A C T I O N )

In this section, we study a decomposition of a satisficing problem with an inseparable performance function, as follows: PROBLEM 4.1. Find d ~ M

such that g ( ~ , to) > ,(to) for each to, where

g ( m , to)= ~ g i ( r n , to), M=M t×...

XMn,

,(,o)= Y~,,(to). Following Mesarovic, Macko, and Takahara [9], introduce a process interaction function K and a modified performance function ~ of the ith subproblem for each i: U=U,x

...

xU,,

K: M×~2--~U, K = ( K I ..... K . ) ,

Ki: M× fi~U~, gi: M, X Ui x~--*R. The modified performance function gi must satisfy

gi( m, to) = g , ( m i , Ki( m, to), ~o)

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SHINZO TAKATSU

for each rn and to. F o r simplicity, we investigate only linear process interactions; i.e.,

Ki( m ) = A , m + Bito, where Ai and B~ are appropriate matrices and A , is a zero vector. To deal with process interactions, we need another decomposition of Problem 4.1, different from those in Sec. 3, because we cannot expect that a = K ( r f i , to) in general for each to. PROBLEM 4.2. F o r given / ~ • 0 [ # = ( # ~ . . . . . ~t")], concave vector function 8 r = ( 6 t . . . . . ~.) (~6i • 0), and O~c_ U~, find tfi~ such that gi(rfi~, u . to)+#i.u~Y.~#~'Aor~-Ia~'B~to•r~(to)+~i(to ) for any u i ~ O i and t o ~ ; i.e.,

~ ( ~ , u~, t o ) - ~ l~-i.A~jgh~ • "r~(to ) + d~( to ) + l.d. B~to-l,t~.ui J

As will be apparent, the estimation principle of Mesarovic et al. [9] is appropriate for this problem. ASSUMPTION 4.1. (1) fi is a c o m p a c t convex subset of a linear topological space. (2) M~ is a c o m p a c t convex subset of a linear topological space. (3) U~ is a convex c o m p a c t subset of a linear topological space. (4) ~l is concave in m~ and upper semicontinuous in rn~ for each to and u~. (5) ~ is concave in u i a n d upper semicontinuous in us for each m, and to. (6) ~ j - ~'~is convex in to and lower semicontinuous in co for each m~ a n d u~. (7) F o r any given m ~ E M i , ~ ( r n , , ui, t o ) - ~ i ( m ~ , ui, to) is convex in to and lower semicontinuous in to for each m~ and u,. (8) F o r any given u~ ~ Ui, gi(mi, ui, to)-g~(mi, u*, to) is convex in to and ower semicontinuous in to for each m~ and u,. Define X = { U' IU' C_K ( M, f~ ) C_U& U' compact & convex }. PROPOSITION 4.1. Suppose that AsSUMP'nON 4.1 is satisfied. For given # • O, t;, and O E X , if Fai solves Problem 4.2 over ~ and if a combined decision r h = ( r h I ..... rh,) satisfies K( r~, ~2) C O, then rfi is an overall satisficing solution .for Problem 4.1.

Proof. F r o m the assumption, for each i, ¢~(rh~, u~, to)-EltJ.A~Jh~ • z~(to) +ri(to)+tz~'B~to-~i.u~ for any u i ~ 0 / a n d to; hence

u,, t o ) -

•

(,,(to) + 8,(to)

~.. ~i(rfii, u,, to) --/~.A .rh • v(to) +/z. B t o - ~. u;

Y. g,(,~, u,, t o ) + ~ . ( u - A .r~- B.to) • ~-(to)

SATISFICING DECISION PROBLEMS

147

for any ~o and uE/9. Because K(rh, f~)C 0, we can replace u with K ( ~ , ~0); ~', g,(,-fi,, K~(tfi, o:), ¢ o ) + g . [ K(rfi, w ) - A . r f i - B . ~ ]

~>~-(o~); Y~ £(,~,, K,(m, ,o),~) > ~(,~); ~. g,(r?t, to) > r(~0) ; g ( ~ , to) > r(lo), which was to be proved. In this section, we used a somewhat different decomposition method from those in the proceeding two sections because of the existence of the equality constraints due to process interactions. This method is equivalent to the estimation principle studied by Mesarovic et al. [9] when the latter is applied to a decision problem under uncertainty. 5.

DISCUSSION

In this paper we have shown several decomposition principles for satisficing problems. In the real world, there simultaneously exist system interactions and process interactions in large-scale satisficing problems. However, as shown in this paper, the two interactions can be dealt with separately; this means that our problems can be solved when each interaction is separated and coordinated effectively. The author believes that the decomposition principles described in this paper will give us foundations for the multilevel coordination of large-scale satisficing problems. Now, let us mention possibilities for further research. First, we must clarify the coordination processes more precisely. To do this, we must find algorithms for solving satisficing problems. As mentioned in our paper [8], we must study the duality in satisficing problems because this may reduce the problem-solving load. Finally, we firmly believe that the satisficing criterion is one of the most meaningful decision criteria with which we can compromise conflicting interests in systems or find a solution which is at least second best for each decision participant. In our next paper, we will show the coordination processes and coordination principles in hierarchical decision systems to solve large-scale satisficing decision problems. REFERENCES 1. A. Charnes and W. W. Cooper, Deterministic equivalents for optimizing and satisficing under chance-constraints, Operations Res. 11(I):18-39 (1963).

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TAKATSU

2. R. M. Cyert and J. G. March, Beha~'oral Theory of the Firm, Prentice-HaU, Englewood Cliffs, NJ., 1963. 3. H. Everette, Generalized Lagrange multiplier method for solving problems of optimum allocation of resources, Operations Res. ! 1:399-417 (1963). 4. Y. ljiri, Management Goals and Accountingfar Control, North-Holland, Amsterdam, 1965. 5. L. S. Lasdon, Optimization Theoryfor Large @stems, Macmillan; New York, 1970. 6. T. Matsuda and S. Takatsu, Characterization of satisficing decision criterion, Information $ci. 17:131-151 (1979). 7. T. Matsuda and S. Takatsu, Algebraic properties of safisficing decision criterion, Information $ci. 17:221-237 (1979). 8. T. Matsuda and S. Takatsu, Algebraic treatment of satisficing decision problems, J. Operations Rea. Soc. Japan, to appear. 9. M. D. Mcsarovic, E. Macko, and Y. Tak~hara, Theory of Hierarchical, Multi-Leoel Systems, Academic, New York, 1970. 10. M. D. Mesarovic and Y. Takahara, On a qualitative theory of sati~actory control, Information Sci. 4(4):291-314 (1972). 11. R. T. Rockafeller, Convex Ana/ys/s, Princeton U.P., Princeton, 1970. 12. H. A. Simon, Models of Man, Wiley, New York, 1967. 13. H. A. Simon, D. W. Smithburg, and V. A. Thompson, Public Administration, Knopf, New York, 1968.

Received February 1980