Cybernetics and Systems Analysis, Vol. 34, No. 2, I998

A L G O R I T H M S G E N E R A T I N G A L T E R N A T I V E S O L U T I O N S F O R

A M U L T I C R I T E R I O N L I N E A R P R O G R A M M I N G M O D E L

V. E. Plish 519.852

Modern support systems for group decisions in situational centers assume the availability of collective discussion and

decision-making subsystems. One of the main functions of such subsystems is to generate alternative decisions for the given

topic, which are then ranked in a certain way or from which one decision is chosen by consensus. The efficiency of the

system largely depends on the availability of a procedure that generates"reasonable" (nearly optimal) alternative decisions in

real time.

The most widespread model that describes economic, military, and social process is the multicriterion linear

programming model (MCLP):

find { max c l(x), max c2(x ) . . . . . max q(x)} (1) .1r . t . I

subject to Ax < b, x >__ 0, (2)

where q(x), i = 1, k are linear objective functions, A is an m • n matrix, x = (x 1, x 2 . . . . . x n) E R n, b = (b 1, b 2 . . . . . b m) E R m.

For a wide range of important applied problems, such as allocation of state orders and scarce resources, budget

negotiations, determination of strategic stability, or preparation of military operations, Eqs. (1), (2) reduce to the form

f'md (max x 1 , max x 2 . . . . . max x n) (3)

subject to constraints (2) with the additional condition that the elements of the matrix A and the vector b are nonnegative.

The objective is naturally set by the human decision maker in the category of decision criteria. Thus, the decision

maker specifies the vector

,,0,_ (x,0, ,:o, . . . . . .

If constraints (2) are satisfied for the vector x (~ then there is no problem: the objective can be met. Otherwise, we

have a choice problem, because the initial objective cannot be achieved subject to the given constraints (2). The difficulty can

be resolved in two ways:

1) first, we can alter (relax) the objective so that it remains optimally close (in a given sense) to the original objective

x (~ and yet satisfies the given constraints;

2) second, we can alter the constraints, i.e., the elements of the matrix A (the technologies) or (and) the components

of the right-hand side vector b (the resources) so as to achieve the sought objective.

Problems of the first class constitute the traditional "concentrated" MCLP problem [1], whereas problems of the

second class are classified as system optimization problems [2, 3].

Let us consider efficient interactive procedures that generate alternative solutions for these classes of problems.

Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 170-174, March-April, 1998. Original article submitted

January 29, 1997.

1060-0396/98/3402-0301520.00 �9 Plenum Publishing Corporation 301

We make a change of variables, introducing relative values of the unknowns, x i = a i r i (~ 0 <__ ot i <_ 1, i = 1 n .

Equations (2), (3) thus reduce to the form:

find {max c~ t, max c, 2 . . . . . max c%} (4)

subject to

~ / x i (~ ~'i -< b~ 0 < " i < t . j = i = l

(5)

Denote by I the index set (1, 2 . . . . . n), by J the index set (1, 2 . . . . . m), by / j the subset of indices i, i E I, of

constraint j in Eq. (5) for which a[ ;~ 0.

Equations (4), (5) have a whole set of Pareto-optimal (efficient) solutions.

Solving Eqs. (4), (5) with the aid of supplementary constraints, we obtain the set of alternative solutions.

The traditional methods [1] of ranking or scalarizing the criteria are ineffective, because they allow the decision

maker to express the attitude toward objectives only on a qualitative level, with very limited quantitative implications. The algorithms proposed here for the class of MCLP problems remove this shortcoming and, most importantly, are

applicable to large problems. The algorithm f'mds one Pareto-optimal solution in time O ( m • n), compared with O(n m) for

the simplex method, i.e., the proposed algorithm is computationally optimal.

GOAI~DIRECTED GENERATION OF ALTERNATIVE SOLUTIONS

1. Determination of an achievable goal x (l) (i.e., a goal that satisfies the constraints (5)) maximally close to x (~ by

all components. This is the traditional MCLP problem with equally important goal components [1] and is solvable

analytically:

b / _

~ J 'E ~ I E I

Given the constraints 0 < o~ i ___ 1, we f'mally obtain

(6)

If the set/Jl ;~ I, then

w

a l j l

a l ] l , i f aI j l

1 " i f ~t[ ' ./l

< _ 1 .

> 1 .

(6')

= ra i n

at j2 i E j \ j l

- X o/.,.i(o) b] - a l jl i E QI

,,/.,-i (~ i E I \ l.i I

(7)

m a I .

12

12

1 ,

i f a l . < 1 , 12

i f a t . > 1 , "/2

and so on, until we have determined the entire set of ~i, i E I, i.e., 0 S

J s = I .

(7 ')

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We thus obtain the solutions

(t)(xtO ~ ( t ) ) ,.,(o) x " x ( t ) ) r,(! a i i ~ L (8) . . . . . , X ! ! . . �9 --. . �9 .

of Eqs. (4), (5) that satisfy the condition of equal importance of the components of the goal vector x (~

2. Determination of an achievable goal x (2) maximally close to the given goal ~(2). Analyzing the original goal x (~

and the new achievable goal x (1) that is closest to x (~ by ali components, we can define a new goal for the subset of priority - ( 2 ) . - ( l ) _ ~ (2 ) - ( o ) .

components I l = (i l, i 2 . . . . . i s) which imposes the quantitative relationships xit txi t < ik <_ Xik ), i t E /(l). All other

components remain equally important.

The problem is to find the vector x (2) that satisfies the constraints (5) and is maximally close to x --(2).

The problem is solved by the following algorithm.

Check the achievability of the selected priority components, i.e., verify that the constraints

~] a/ .~.~7-) <_ b? /~j. (9)

~ t ( t )

are satisfied. If conditions (9) are satisfied, then set

r (2) = .~i (2).~ i~ ~ t (l) - i k

If conditions (9) are not satisfied, then f'md ~(2) , /(1) it i t E /(1) that are maximally close to ~.(2) ' ~t i t E i.e., solve the problem

{max c~ii, max c~i2 . . . . . . . . max C~is } subject to the constraint

~{k~ ~ ~b t j ~ z o~ ~, , k ~ ~ ~ r i k a i k �9 .

i k E l ( l )

using the previously described analytical method.

Find the nonpriority components Xi (2), i E /UO), that are maximally close to the corresponding components of the

original goal x (~ i.e., solve the problem

m a x r i . m a x or . . . . inax a .

subject to

,~/.,.(o) ~,..~ bj- ~ a/.,,(2)

t e l \ i i E l i

0 < O r i _ < 1 , i E l \ l ( 1 1 , j ~ J ,

by the same analytical method.

As a result, we obtain an alternative solution x (2) that corresponds to the imposed priorities.

Altering the priorities both qualitatively (the set/(1)) and quantitatively (the values of the goal components), we can

use the algorithm to generate any goal-directed solution in a perfectly acceptable time.

G E N E R A T I N G A L T E R N A T I V E SOLUTIONS BY STRUCTURAL CHANGES

Consider an algorithm that achieves the goal x (~ by altering the right-hand side vector of the constraints (5).

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Find the set J(t) of negative discrepancies

I !

A i hj ~ a !x (0) ai<0. i ~J(I) t �9 I

i.e., the set of shortages Aj for each resource bj, j E j~t) Thus, by altering bj to bj + Aj for all j E j(1), we simultaneously achieve the required goal.

A stepwise approximation to the goal is obtained through step-by-step, modification of bj, j E .ill) For simplicity, we assume that the matrix A is full, i.e., a/ # 0 for all i E I and j E J.

= ~ aix i ), j E j(l), and order {czj} in an ascending sequence o~jl < oej2 <_ ... < Otjm , i.e., in Compute (xj (bj/ j (o). i = 1

the order of occurrence of the shortages of bj, j E j(1), that restrict the approach of x (k) = x(~ k = 1, m, to the goal

x(0).

Now to satisfy the constraints by any vector x (~:), k = 1, m, we need to adjust the right-hand side vectors by the

amounts Ajs = (etjk -- ~js ) ~ a/sxiO, s = 1,2 . . . . . k -- 1. i=l

In conclusion note that the proposed algorithm has been incorporated in the interactive integrated decision support

system for the allocation of state orders in the machine-building industries. Currently these methods are implemented in the

FORA system for generation of alternative decisions using the MCLP model [4]. They are also used in the B YUDZHET

software package whose functions include budget balancing, allocation of state orders, and other problems.

REFERENCES

.

3.

~

G. S. Pospelov, V. A. Irikov, and A. E. Kurilov, Procedures and Algorithms for Software Package Construction [in

Russian], Nauka, Moscow (1985). V. M. Glushkov, "System optimization," Kibernetika, No. 5, 89-90 (1980). V. M. Glushkov, V. S. Mikhalevich, V. L. Volkovich, and G. A. Dolenko, "System optimization in multicriterion

linear programming problems with interval preferences," Kibernetika, No. 3, 1-8 (1983). V. E. Plish and A. E. Truten', "FORA -- a system for generation of alternative decisions using multicriterion linear

programming model," Kibern. Sist. Anal., No. 2, 150-155 (1997).

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