mathematical modeling of dynamic decisions; priorities and hierarchies with time dependence

Mathematics and Computers in Simulation XXI (1979) 352-358 0 North-Holland Publishing Company

MATHEMATICAL MODELING OF DYNAMIC DECISIONS;

PRIORITIES AND HIERARCHIES WITH TIME DEPENDENCE *

Thomas L. SAATY

Graduate School of Business, University ofPittsburgh, Pittsburgh, PA, OXA.

1. Introduction

The question often arises in regard to the use of Analytic Hierarchy Process: What would one do if the judgments where to change? A simple answer to that problem is that one should solve the new problem. But this is not what people usually have in mind. Presumably what they would like is a parameter- ized eigenvector solution as a function of time in order to make the implementation compatible, not just with what people think now, but what they are likely to think later on. Thus one would like an analytic solution of the eigenvalue problem A(t) w(t) = x,,(t)

w(t). See [2,3,4]. Judgments by their very nature may be expected

to vary according to different situations. If they follow a known trend corresponding to a particular parameter, then one could adjust the judgments to follow the changes in the parameter. For example, a combat pilot may have a number of strategies to choose from depending upon the speed of his aircraft, his distance from an enemy aircraft or on the amount of fuel in the tanks. The importance of one strategy over another would then be a function of speed or distance or amount of fuel. One way to solve this problem is to repeatedly fix the value of the time parameter and then use curve fitting for the different values obtained for each of the eigenvector components.

An elegant approach would be to decompose the hierarchy into clusters whose number does not ex- ceed four for any set of pairwise comparisons, obtain the solution for X,, as a function of the coef-

* This research was partially supported by AFOSR, Grant No.

77-3366, to the Wharton School, University of Pennsylvania,

ficients by solving a quadratic, a cubic, or a quartic as the need may be and then solve the eigenvalue problem explicitly in terms of the coefficients and also in terms of X,,,. One could then apply the hierarchial composition principle to obtain the overall weighting as a time dependent function.

It is well known that by using simple quadrature according to Galois theory n = 4 is the highest order matrix for which we can obtain solutions for h,, in closed form. As we said before, if one insists on using higher order matrices one should enter static numerical judgments provided for different periods

of time and solve the corresponding problem. For the pairwise comparison judgments one may

attempt to fit one of the functions given in Table 1 to the changing judgments. These functions have been left in parametric form so that the parameter may be set for the particular comparison; hopefully adhering to the l-9 scale we have been using in the discrete case as a limit on the range of values (or any other convenient scale used in the discrete case). These functions reflect our intuitive feeling about change in trend: constant, linear, logarithmic and exponential, rising to a maximum and declining or falling to a minimum and rising, oscillating, and finally allowing for catastrophic change.

We give the results for the dynamic case for n = 2, 3, 4, solving for the principal eigenvalue and its eigenvector for a reciprocal matrix, in Section 2,3,4, resp.

2. The quadratic case

For this case h,,(t) = 2 and the time dependent eigenvalue problem is given by

T.L. Saaty /Mathematical modeling 353

Table 1

Dynamic judgments

Time-dependent Description Explanation

importance intensity

a

alt + a2

bllog(t + 1) + b2

cle C2t + c3

dlt2 + d2t + d3

eltn sin@ + e2) + e3

Catastrophies

Constant for aII t, No change in

lGolG9an relative standing

integer

Linear relation in Steady increase in

t, increasing or value of one

decreasing to a activity over an-

point and then a other

constant value

thereafter. Note

that the reciprocal

is a hyperbola

Logarithmic growth Rapid increase

up to a certain

point and con-

stant thereafter

Exponential

growth (or decay

if c2 is negative)

to a certain point

and constant there-

after (note reci-

procal of case c2

is negative is the

logistic S-curve)

A parabola giving

a maximum or

minimum (depending on dl being

negative or posi-

tive) with a con-

stant value there-

after (may be

modified for skew-

ness to the right or

left)

Oscillatory

(decrease follow-

ed by slow in-

crease (decrease)

Slow increase (decrease) fol-

lowed by rapid

increase

(decrease)

Increase (decrease)

to maximum

(minimum) and

then decrease (increase)

Oscillates depend-

ing on n > 0

(n < 0) with

decreasing (in-

creasing) ampli-

tude

Violent change

in intensity

Discontinuities

indicated

From which we have

WI(t) + a(t) wz(t) = 2w,(t) ,

The first equation yields

Wl@) = 4) w2W 1

which is also what we can obtain from the second equation. These two equations cannot be indepen- dent, otherwise the determinant of A(t) would not be zero and we would not have a nonzero solution. Thus, we can fix w2(t) arbitrarily, e.g., put w,(t) = 1 from which we have wr(t) = a(t). The normalized right eigenvector has the form {a(t)/[a(t) +l], l/ [a(t) + 11). The normalized left eigenvector is the componentwise reciprocal of this given by

Ul4~)[4~) + 11, Ub(f) + 11).

3. The cubic case

In a straightforward fashion PC. Morris [l] by solving a cubic equation, showed that h,,, for the 3 by 3 case with aii = l/aii is given by

Note that h,, is always >3 (we have proved that in general X,, 2 n).

The system of equations corresponding to this problem is given by

wr(t) +a12w2(f) +a13w3(r) = imax WI(f) )

wl(t)/42 + w2(t) +a23W3tt)=h,,,(t)W2(t),

Put WI = 1. The first equation becomes

ar2w2 +u13w3 = -(l -A) )

and the second

(1 - V w2 + a23w3 = -l/al2 .

We now solve the above equations for w2 and w3. We have

w2 = (A - l) a23 + al3/al2)/A ,

w3 = (-1 + (1 - h)2)/A ,

354 T.L. Saaty /Mathematical modeling

the components we form:

WI + w2 + w3 =

= alza23 +a,,(h-1) + (h-l)023 +a13b12- 1 +(1-h>2

A

D =_ -A’

Thus, finally,

w1 = A/D ,

w2 = ((A - l)a23 +a13/a12)/D a

w3 = (-1 + (1 - h)2)/D .

For the left eigenvector which is the elementwise reciprocal of the above (the reciprocal property between left and right eigenvectors of a reciprocal matrix holds only for n = 2 and n = 3):

u1 = (-1 + (1 - Q2)/E )

u2 = (ad’ - l) + a13/a23)/E >

v3 = A/E ,

where

E = -1 t (I - h)2 +a12(h - 1) + a13/u23 + a12a23

+a13(h - 1) .

4. The quartic case

Consider the 4 by 4 matrix with reciprocal entries

A=

Note that all the coefficients may be functions of a parameter t. The characteristic equation of this matrix is:

h4-44X3-(B-8)h+(B+C-S)=O,

where

B=($+-g+(fy+;)+($+-g+(y+g,

We consider reduction of the quartic as follows: We write

(h2-2h)2=(B-8)h-(B+C-5)+4h2;

adding (71’ - 2h) I + br2 (where r is a parameter) to both sides we have

(A2 - 2h + ;,’ = (r t 4) X2 t (B - 8 - 2r) h

t $2 - (B + c - 5) .

The right side is a perfect square of a linear function in X if and only if its discriminant is zero, i.e.,

A = [(B - 8) - 2r]’ - 4[;r2 - (B + C ~ 5)](r + 4)

=-r3t4(Ct3)rt(-16tB2+16C)=0

which is called the resolvent cubic equation of the quark. Whenever r is a root of this equation we

have A = 0. We have

(A2 ;,2 = B - +

- 2h + 2(r 4) 1 2

2(rt4)

=(r+4)b-1 t& 1 2

from which using the largest value of I we get

h 2++7

max= 2 ty&Cg

Now let us look at the solution of the cubic. The resolvant cubic may be put in the form

r3+pr+q=0,

where p = -4(C t 3) and 4 = 16 - B2 - 16C. Using the transformation r = z - p/32 the resolvent is trans.


formed to = - f(Tl + T2) - $fi(T1 - T2).

z3 - p3/27z3 + 9 = 0 ,

or

Zb + 923 - p3/27 = 0,

and this equation is quadratic in z3. So the solutions are

z3 = -;qaa

where R = ($.J)~ + (;9)*.

Let

When T1 = T2, R = 0 and r2 = r3 = (- i9)‘13 is real. Also, rl = 2(- $9) ‘I3 In addition, since com- . plex roots occur in conjugate pairs, rl is always a real root of the resolvent cubic. This follows from the fact that p > 0. To see this we note that C has three terms of the form x + l/x. The minimum value of such a term is 2. Hence, C S -3 and p =

-4(C + 3) Z 0. Thus r = rl is always real. In addition, r Z 0 since 9 < 0. This follows from

B* •t 16C> 16, B* > 16(1 - C) .

T1 = (- ;9 + @)1’3, T, = (- ;9 - fi)‘13 .

Now the cube roots of unity are

1, w=-i+iid, w*=-i-$ia.

We obtain the following six solutions:

Ti, wTr> w*Tr, T2, wT2, w2T2

to the equation

ze + 923 -p3/27=0.

It is known that the roots of the reduced cubic equa- tionr3+pr+9=Oaregivenby:

rl = T1 + T2 ,

r2=wT1+w2T2,

r3=w2T1+wT2.

Therefore,

Now the minimum of 16( 1 - C) is 64. Similarly the minimum value of B* is 64. Thus 9 < 0.

Now the first term in rl is positive. The second term is always dominated by the first term. Thus, r E rl > 0

is what we use in the expression h,, above. I am grateful to my student Luis Vargas for his help

in working out the details of this case. The solution of the system AW = Xmaxw which in

expanded form is given by

(1 -X)wr+aw* +bWa+CW4=0,

1 -w,+(1-X)w,+dWJ+ew~=O, a

‘,,+A, +(1-X)wa+fwq=O, b d *

rl=[-8+iB2+8C+

([- !(C+ 3)] 3 + (8 - ;B* - 8C)2)1’2]1’3

+ [(-8 + ;B* + 8C) -

([- ;(C+ 3)] 3 + (8 - ;B* - 8C)2)“2]1’3 ,

r2=(-~+~ifi)T1+(-~-~ifi)T2

= - i(T, + T2) + iifi(T, - T,),

1 -wl+~w2+~w3+(1-A)W~=0 c e f

is, when normalized, given by

Wl w2 23 w4 w1=- ,w*=-, w3=--, w4=-,

Q Q Q Q where

Q = (X - 1)3 t (c +f+ e)(h - l)*

+[(ae-3)+(b+d)f+(i+k)c+i](X-I)

+

r3 = (- i - kid) T1 + (- i + fi&) T2

+cd+ae+c-b

b -1 ad ’


WI = C(X - 1)2 + (ae + bfl(X - I) + ( d )’ &f+ E- c

w3 =f(h-1)2+ (;+f-)(A-1,t(;+;-f);

wq=(x-1)3- a A-l)- f+-$ . ( )

5. Example of dynamic judgments and priorities

Let us consider the simple case of a family consist- ing of a father, a mother and a child. Obviously the amount of time the child spends at home will depend on his age. The infant would spend the same amount of time as the mother and then as he grows older he will progressively spend less time at home as com- pared to the time spent by the mother. We assume that the mother does not go to work.

If we were to compare the length of time spent at home by mother and child and plot this relation as a

function of time (i.e. as the child grows older) we would get the type of curve as shown in Fig. 1.

Thus the curve begins with mother and child spending the same amount of time, then the ratio of mother’s to child’s time increases until it levels off by the time the child is in in his midteens.

Comparison of father to child times yields a relation- ship which is a mirror image of the above - reflected about a horizontal axis halfway up the curve. This is illustrated in Fig. 2. The relative length of time spent by father and mother would not vary too much and could be expected to be fairly constant.

t,me

Fig. 1.

Fig. 2.

If we were to make a pairwise comparison of the different lengths of time spent at home by the different members of the family we would get a sequence of comparison matrices each corresponding to parti-

cular period of time. Consider the time period corresponding to the

child’s age O-4 years. If we were to exclude, say, eight

hours of the night, we would expect the mother and child to spend about two to three times the length of time the father spends at home. The mother and child would of course spend the same amount of

time. This would give rise to the following matrix:

FM C

F 1 l/2.5 l/2.5

M 2.5 1 1

: 1

.

c 2.5 1 1

This yields the following eigenvector for their relative

times at home:

F: 0.167, M: 0.417, C: 0.417,

which is a reasonable reflection of the proportions of time they each spend at home.

M’s ttme j

--t C’S time

2-

l-

I I I

2 6 10 IL '8 t,me

Fig. 3. Mother and child; age O-16 years.


2-

l-

I I I I I I

2 6 10 Ii 18 time

Fig. 4. Father and mother; age O-16 years.

Around the age of four the child begins school

so there is a sudden change in the relative propor-

tions of time spent at home by mother and child and by father and child.

We can express the varying proportions in a single matrix using a time dependent expression for these proportions. We have

F M C

1 1 z 3 -In t/2

M 2 1 0.4tlnt/21 ,

C 3-lnt/2 i

1

0.4+ln t/2 ’ J

where t denotes the time ranging between 4 and 16 years.

This matrix along with the previous one give rise to the curves in Figs. 3,4 and 5 depicting the corresponding pairwise comparisons as time varies from zero to 16 years.

M’s t,me Ca3

t ----- 1

2- i_____-_________

I 1 I I I I

2 6 10 1L 18 ,,me

Fig. 5. Father and child; age O-16 years.

mother

father

child

Fig. 6. The solution.

The solution of the maximum eigenvalue problem corresponding to these pairwise comparison curves for (4 < t < 16) yields:

h=lt [

2 1 113

(3 -In t/2)(0.4 + In t/2)

t (3 -In t/2)(0.4 + In t/2)

[

U3

2 1 . The corresponding eigenvector is given by

a/D 7

- 1)(0.4 + In t/2) + 3 _2n t,2]/D ,

i-1 + (1 - h)2]/D J where

A = 0.5(0.4 + In t/2) t h-l

3 - In t/2 ’

D = (X - 0.5)(0.4 + In t/2) + x+ 1

3 - In t/2

-;I + (1 - A)2 .

As the child finishes school he begins spending even less time at home than the father. The proportions once again become fairly constant and are reflected in the following (consistent) pairwise com-


parison matrix

F M c

relative time (with respect to all others) which each spends at home (see Fig. 6).

References

[ 1] C. Morris, Weighting inconsistent judgments, Pi Mu Epsi-

lon J. (1979).

whose eigenvector solution is given by

F: 0.263, M: 0.526, C: 0.211 .

[2] T.L. Saaty, Modeling unstructured decision problems:

theory of analytical hieratchies, Math. Comput. Simula-

tion 20 (3) (1978) 147-157.

[3] T.L. Saaty, Applications of analytical hierarchies, Math.

Plotting these results together for 0 < t < 4,4 < t < 16, and 16 < t gives a realistic representation of the

Comput. Simulation 21 (1) (1979) l-20.

]4] T.L. Saaty, A scaling method for priorities in hierarchical

structures, J. Mathematical Psychology 15 (3) (1977)

234-281.

mathematical modeling of dynamic decisions; priorities and hierarchies with time dependence

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