On the linear programming problems with interval coefficients

Computers and Industrial Engineering Voi. 23, Nos I-.4, pp. 301-304, 1992 Printed in Great Britain. All rights reserved

0360-8352/92 $5.00+0.00 Copyright © 1992 Pergamon Press Ltd

ON THE LINEAR PROGRAMMING PROBLEMS WITH INTERVAL COEFFICIENTS Y. NAkahara, M. Sasaki and M. Gen

Department of Industrial and Systems Engineering Ashikaga Instituteof Technology Ashikaga, 326 Japan

is more detailed or natural than the conventional concepts. Moreover, we show that the feasible region by this concept includes a corresponding simple region. And we suggest that we can use this simple region as the substitution of the feasible region, and that it is especially useful when DM hopes to seek solutions necessarily belonging to the feasible region.

ABSTRACT In this paper, we investigate a LP problem with interval coefficients and propose the new concept of constraints based on the probability. We demonstrate that this concept is more detailed or natural than the conventional concepts. And we show that the feasible region by this concept includes the corresponding simple region which is useful espedally in some case.

LP P R O B L E M S WITH INTERVAL COEFFICIENTS(I.C.)

INTRODUCTION In numerical programming theory, various methods have been proposed in trying to take the fuzzyness of the state into account. Recently, a LP problem with interval coefficie~ts(ILP problem) which express fuzzyness of the state is formulated. In this ILP, an interval coefficient(I.C.) is set by the DM(decision maker) as the region of the value which the coefficient possibly takes. This methods is excellent in the point that for it is easy to express the fuzzyness of the value by the interval, the ILP problem to be solved can be easily set. So far, the feasible region of the constraints with the inequalities between the intervals has been defined as follows: (a) Regarding the intervals as the fuzzy numbers which ~ support are the intervals, the region is defined as the set of x such as the possibility or necessity of inequality between fuzzy numbers which are introduced by Dubois and Prade is more than certain value decided by the DM. (b) By introducing degree of the inequality between the intervals, the region is defined as the set of x such as degree of the inequality is more than certain value decided by the DM. The degree is defined as it takes any real value between 0 and 1. By using the definition (a) to interval fuzzy numbers, the possibility and the necessity is either 0 or 1. The definition (b) was introduced in order to treat the relative position between the intervals in more detail than the treatment in the definition (a). In this point, this definition is highly evaluated. But the relationship between the definition of (b) and the meaning of the interval comes not dear and not natural. In this paper, we investigate a LP problem with I.C. and propose the new concept of constraints based on the probability. We demonstrate that this concept

301

In mathematical programming problems, the coefficients of objective functions and constraints have been put as real numbers according to experience or knowledge of specialists. But in many cases, the state has the fuzziness as does in prediction of economy and so on. In such cases, there is fuzziness in the coefficients when we formulate this as a mathematical programruing problem. It is easier to express this fuzziness in I.C. than in fuzzy theory. In this view point, we investigate the following LP problem with LC. consisting of ra system constraints. .2."

maxlmin

Z(x) = l~ CixJ

(I)

i=*

subj. to

a,(x) = ~

AOz ~ < B,,

j=l

i = 1,2,...,m

z# > 0,

where

Z =

[~. ~ ]. C~

(2)

j = 1,2,.-.,n

= [~j~. ~,~ ].

A,~

=

(3)

[.,,~..,j~]

and B, = [biZ,b~u] for i = 1,2,...,m,j = 1,2,...,n,

and C# : the objective I.C. of the j - t h decision variable, A o : the I.C. of the j-th decision variable in the i-th system constraint, Bi : Interval constant in the right-hand side of the i-th system constraint, , : the number of decision variables, z~ : the j-th decision variable. In the above, let addition of interval and multiplication with interval and real number be defined as usually. And we denote the left-hand side or the righthand side of each closed interval by the subscript or the superscript of L or U respectively. The various methods of treating the objective function with I.C. have been proposed. In this paper,

302

Proceedings of the 14th Annual Conference on Computers and Industrial Engineering

we concentrate our attention to constraints with I.C. without the loss of generality. CONSTRAINTS

W I T H I.C.

r(x) in (8)=

Various definitions of the feasible region r(x) of any constraint with I.C.

t

A i z j <_ B,

i=1 Ai :

P r o p o s i t i o n . For arbitrary intervals A i = [ai L, a1U ], j = 1 , . . . , n and B = [bL,bv], wehave

[ajL,aiU],

j =

r(x) in (5), ff q : 0 ; r(x) in (6), if q = l .

Moreover, we will show the reason why we consider the definition (8) is more natural than the definition (7), in Remark 1. At first, we define a probability and degree of any interval inequality. Definition 1. For any interval A = [aL,au] and B = [bz,bv], we define P(A (_ B) as

1. . . . . n,

B = [br, by ]

(4)

have been made so far. The famous definition based on the possibility by Dubois and Prade is the following: r(x) = {x • R+" l ~'~ aiL xj <_by}

(5)

j=l

where

R +" = { x = (= i ,.. . , =, ) • R" I=i _>0,...,=, >0}.

P(A < B) = P(a <_b) where a is the random variable uniformly distributed on [aL ,av], b is the random variable uniforrrdy distributed on [bL,bv], and P(a < b) represents the conventional probability of a < b. And we call P(A < B) a probability of A < B. Definition 2. For any interval A = [aL,av] and B = [bL,bu], we define P'(A ~_ B) as

The definition based on the necessity by Dubois and Prade is the followings:

P'(A ~_ B) = (bu - av) + (av - at.)

av --at. +by --br And we call P'(A ~ B) degree of A _< B.

r(x) = (x • R+" I ~ , / ~ i

< b~)

(6)

i=1

The definition proposed by Tanaka et al. is the following: for given q (0 < q < 1) by dicision maker, r(x) = { x • R +" [ q ~

aiuxi + ( l - q )

i=1

ai% i i=1

(I - q)bv + qbL }

(7)

In this paper, we propose the following definitionbaaed on probability:

R e m a r k 1. The definition (8) is based on Definitinn 1. And definition (7) is essentially based on Definition 2. We consider that "degree of A < B" is defined more naturally by Definition 1 than Definition 2, if an LC. represents the set of value which the coef~cient may have with equal possibility. And the mean/ng of I.C. such that [.C. represents such set is one of good approximation of the meaning of LC. which DM really set, we think. Definition 3. For any interval A = [ a L , a U ] and B = [bL,bv] , we define v ( A , B ) and to(A,B) as v ( A , B ) = m a z { m i n { a u ,bu },aL } w ( A , B ) = min{raaz{aL,bL },au }.

n

r(x) -- {x • R +" [ P ( ~

Aiz i _< B) _> q},

(8)

(10) (11)

1=1

where P ( A <_ B) for any interval A and B represents the degree of A _< B based on probability, and will be defined later using the conventional probability. And we will show that this region contains n

{ X • R + " [ q ~ aiUzi-l-(l-q)~ ai"cxi i=I 1=I _< bL }.

(9)

In other words, any element of the region (9) is the necessarily contained by r(x) in (8). We suggest that we can use the region (9) as the substitution of the feasible region (8) when we solve a real problem. When DM hopes to seek solutions in the region of which element necess~wily satisfy (8), if we use the region (9), it is assured. In the later of this section, we will prove that the region (8) includes region (9). And we will show that the definition (8) is more detailed than the definition (4) or (5). More precisely, we will show the following Proposition.

We may use v or to instead of v( A, B ) or to(A, B ) / / w e identify it clearly. R e m a r k 2. For any interval A = [aL,av] and B = [bL, by ], the following 1) or 2) or.... ,or 6) holds. 1) aL ~_ a u ~_ bL ~ bv ~) aL < bL ~_ au < bu 3) aL ~_bL ~ b u ~_au 4) bL ~_ a£ < a U ~ bu 5) bL < aL < bu < au bL ~_bu ~_aL ~_av. For each case, the v and w have the following valo ues.

Case

[

1) 2) 3) 4) 5) 6)

Value of v

Value of to

~U

au

au bv

bL bL

au bu

aL GL

aL

aL

Figure 1. Values of v and w in each case

NAKAHARAelf al.: Linear Programming By calculation, we have the following result. Theorem [bs , bt] ],

lary 1.

I. For any interval A = [as,au] and B =

P(A <_B)=

w--as -

G~T(i-~

-

bs ) (bt]

-

~ + w

2

""

P r o o f . B y Deftaition 1 and Remark 2, we have P(A <_B)= f~

1-4) W h e n b L <_ aL <_at] <_ bt], t ~ a s .

--at] 1 aL dz

+

at]

-

as bu

From Theorem 2 and Remark 1, we have the Proposition stated above. That is, the region (4) is the special case of the region (8), and the region (5) is also. T h e o r e m 3. Let A = [aL,at]] and B = [bs,bt]] be arbitrary intervals, and let us set the right-band side of (16) as t. Then, t is estimated as follows: I-i) Whenas <_at] <_bs <_bt],t=at]. 1-2) When as <_ bs <_ at] <_ bt], t > bs. 1 I-3) W h e n a s <_bs <_bu < a t ] , t = ~(bt] +bs).

at] -- a s v -- to (at]

303

dz.

(12)

- -

Set the first term as tt and the second term as t2. We

I-5) When bs <_ aL <_ bt] <_ at], t ~_ ](bt] + bs). I-6) When bs <_ bt] <_ as <_ at], t = as. S k e t c h of P r o o f . Note that for any interval A = [as,at]] and B = [bL,bt]]

have

1 W ~ as

t, = - - ,

(13)

at] -- a s

1

Jr 2 (bt] "~ bL )

and 1

t2 = - - [ b u z -

1 1 2 =bt] - bs (•bt] - wbs - 5(v - w ' ) ).

--;z~ "

~]"

at] -- as

=

1

bt] = bs (--2 (" - bt] )" "t" ~(w -- bs )')

v- w (bt] - v + w ~ (au - as )(bu - bs ) 2 ""

(14)

The result follows from (12),(13) and (14).

From this relation and Remark 2, we have the result using the following two relations: When

as <_bs <_at] <_bt],

When

bL ~ a L

The following Corollary is dearly obtained from Theorem 1: C o r o l l a r y 1. For any interval A = [as,at]], B = [bs, bt] ], and number q E [0,1], P ( A < B) > q

(15)

¢=~ qat] + (I - q)as 1 1 <-- bu --bs ( - . ( v - b t ] ) 2

1 + 2 (w-bL)

+ 2(bt] + b s ) ,

L e m m a 1. Let A = [aL,at]] and B = [bs,bu] bearbitrary intervals. And let us suppose that the following (1 - 1) or (i - 2) or .... ,or (1 - 6) holds: (1 - 1) :

aL <_ au <_ bL <_ bt],

(1-2):

qat] + (1 q)a L <_ art. aL <_bL <_au < _ b u , -

(16)

~3rom this Corollary 1, the following Theorem 2 is obtained. T h e o r e m 2. For arbitrary intervals A¢ = [a~r,a~t]], j = 1. . . . . n, B = [bs,bu], and number q E [0,1], we have n

{x e R +" IP(~-~ A~x~ < B) _> q}

qat] A- (1 - q)aL <_ bL. (1 -- 3) :

aL <_ bL <_ bt] < at],

(1--4):

qat] A- (1 - q)a L <_ bL. bL_
(1 -- 5) :

qat] + (1 - q)as < a s . bL <_ as <_ bu <_ at], qat] Jr (1 -- q)aL <_ bs.

(1 -- 6) :

bs <_ bt] <_ aL <_ at], qat] + (1 - q)as = as.

jffil It

tt

Then, the estimate (16) holds.

= { x e R +" Iq ~-"~ ajt]=: + (1 - q ) ~"~ a;%~ /=1

P r o o f . The result is obtained from Theorem 3, dearly.

j=l

1 ( _ l ( v _ b u ) l + ~ ( w _1b s ) <-- bt] - b¢ 1 -{- ~(bt] + bL)}.

2)

L e m m a 2. Let A = [as,at]] and B = [bs,bt] l be arbitrary intervals. Then, (1-1) or (1-2) or,... ,or (1-6) of the L e m m a 1 is equivalent to the following: qau -4- (1 - q)aL <_ bL.

P r o o f . ~ o m the interval calculation, we have

j----I

<_at] ~ b t ] ,

0 <_ bt] - at] <_ bt] - a t .

)

where ¢==~ means "is equivalent to".

t

0 <_ bt] - at] <_ bt] - bs.

n

tt

@ffil

j----I

A,,, = [~ .j%~,~

aj%A,

(17)

for any x E R +" • H e n ~ , the result follows from Corol-

(18)

P r o o f . There are no pair of intervals ( A , B ) such as satisfy the (1-4), and there are no pair of intervals ( A , B ) such as satisfy the (1-6) Hence, (1-1) or (12) or . . . . . or (1-6) of Theorem 3 is equivalent to (1-1) or (1-2) or (1-3) or (1-5) of Theorem 3. Now, the f o f i o w ~

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Proceedings of the 14th Annual Conference on Computers and Industrial Engineering

relations hold clearly: ( I - - I ) ==~(18), (1-3)==~(18),

(1-2)==~(18), (1-5)==~(18).

CONCLUSION

From the above relations, the following relation holds:

(1 - 1)or (1 - 2) or (1 - 3)or (1 - 5 ) ==¢, (18). Hence, if we can prove the following re/atioa:

(18) ==~ ( 1 - 1 ) or ( 1 - 2) or ( 1 - 3) or ( 1 - 5),

(19)

the result is obtained. For we have that (1-1) is equivaJent to av < bL , and that

(1-3) or(1-5)

In this paper, we investigate a LP problem with I.C. and propose the new concept of constraints based on the probability. We demonstrate that this concept is more detailed or natural than the conventional concepts. Moreover, we show that the feasible region by this concept includes a corresponding simple region. And we suggest that we can use this simple region as the substitution of the feasible region, and that it is especially useful when DM hopes to seek solutions necessarily belonging to the feasible region.

¢=~aL
the following re/ation is obtained:

REFERENCES (18), -~(1 - 1), -~{(1 - 3)or (1 - 5)} bu < aL, or av <_ bu,

(20) (21) (22) (23)

or qat] + (1 - q ) a £ > bL, and (10), bL < at].

Noting that (20) contradicts (18) and that (22) also, we have the following relation:

(20) or (21) or (22) and (23) (18), bt. < at], at] < bu, ==~(18), at. < bL, bL < at], at] < bt],

=,(1-2). By virtue of the following relations, we have (19) and the result.

(19) ¢=* (18), -,(1 - 1), -,(1 - 3), -1(1 - 5) :=~ (1 - 2) (18), -,(I - I), --,{(I - 3) o r (I - 5)} ~

(I - 2).

T h e o r e m 4. Let A = [aL,at]] and B = [bL,bt]] be arbitrary intervals and q be any number such as q E [0,1]. Then, (18) contains (15). Proof. Let (18) hold. Then, from Lemma 1 and Lemma 2, we have (16). Hence, from Corollary 1, (15) is obtained. T h e o r e m 5. For arbitrary intervals A t = [at L, at u ], j = 1 . . . . . n, B ---- [bL,bu], and number q e [0,1], we have n

{x e R +" Iq ~

aj t] zj + (1 - q) ~

~----1

a t L xj < bL }

j=l n

C: { x E R +" IP(~-'~ Aix~ <

B) _> q}.

,j=l

Proof. From the interval calculation, we have (1 7) for any x E R +" • Hence, the result follows from Theorem 4.

[1] C.V.Negoita: The Current Interest in Fuzzy Optimization, Fuzzy Sets Syst., 6, 3, pp.261-269 (1980). [2] G.Alefeld and J.Herzberger: Introduction to Interval Computations (translated by J.Rokne), pp.l-9, Academic Press, New York (1983). [3] M. Inuiguchi and Y. Kume: On Four Formulations of Goal Program Based on the Interval Concept, Journal of Japan Industrial Management Association, 39, 3, pp.146-152 (1988), in Japanese. [4] H. Ishibuchi and H. Tanaka: Formulation and Analysis of Linear Programming Problem with Interval Coefficients, Journal of Japan Industrial Management Association, 40, 5, pp.320-329 (1988), in Japanese. [5] H. Ishibuchi and H. Tanaka: Identification of Fuzzy Parameters by Interval Regression Models, Trans. IEICE, J72-A, 12, pp.1958-1964 (1989), in Japanese [6] H. Ishibuchi and H. Tanaka: Interval 0-1 Programming Problem and Produced-Mix Analysis, J. Oper. Res. Soc. Jap., 32, 3, pp.352-369 (1989), in Japanese. [7] H. Ishibuchi and H. Tanaka: Multiobjective Programming in Optimization of the Interval Objective Function, Euro. J. Oper. Res., 48, pp.219-225 (1990). [8] Y. Nakahara, M. Sasaki, K. Ida and, M. Gen: An Effective Method for Solving Large Scale 0-1 Linear Programming problems with Interval Coefficients, Proceeding in the Conference of ,lap. Ind. Mona. Association, pp.234-235 (1990.11), in Japanese. [9] Y. Nakahara, M. Sasaki, K. Ida and, M. Gem A Method for Solving 0-1 Linear Programming Problem with Interval Coefficients, J. Jap. Ind. Mana. Association, 42, 5, pp.345-351 (1991) , in Japanese. [10] Y. Nakahara, M. Sasaki, M. Gen and, K. Ida: An Effective Method for Large-Scale 0-1 Linear Programming Problem with Interval Coefficients, Proceeding in the Conference of IFIP, pp.ll0-111 (1991.9)