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Chapter 29. Vertex Generation Methods for Problems with Logical Constraints

Chapter 29. Vertex Generation Methods for Problems with Logical Constraints

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545



Partial orders related to Boolean optimization



(3.5)



C xi



x =



e,



t=1



with 0 . e , : = 0 and 1 * e, = e, for i E N.



I



- 0) (if the set is

(3.6) Definition. Let 5 be a preorder on B", k := min { j E N e, <_

empty put k : = n + 1). Then

x+:=



C xi * e i



ick



Let T ( x )= { j l ,j 2 , . . ., j,} with jl < j z < * . . < j,. Then x(') is given



(3.7) Definition.



by

for i E N.



T(x"')= { j l , j Z , . . .,jmt"(r,,)}



The special subvectors defined by (3.6) respectively (3.7) have some useful

properties in preordered semigroups. Let be 11 x I/:=/ T ( x ) ( .

(3.8) Proposition. Let (B", + , 5

- ) be a preordered semigroup. r = IIx 1). Then

(3.8.1)

o
(3.8.2)

y L'X

yzx+

(3.8.3)

y - XI')

(1 y I( = i A y L I X

(3.8.4) there exists t E N such that x(') = x+,



-a



X(')
=



=



=



x'''< X ( r - l ) < .

=

--



<



.
z



=



. . < x('+l)s

x+.



-



=



(3.8.1)-(3.8.4) follow by the monotonicity property (M) and (3.2'). Let us consider

for example (3.8.2). Let cp correspond to the definition of YL'X. Then, by repeated

application of (M)



y

for c p ( i )



=



C y,



*



C y, . e , , , )=: yf



ezs



i implies e, 5

- e,+,(,)by (3.2)'. Analogously



with k as defined in (3.6).

The application of the greedy algorithm to S yields step by step the sequence

(3.9)



x('),x(2) x ( 3,..-,

)

x(')= x ( S )

9



thus by (3.8.4) it is possibl(3.8.2) and (2.6) is



&-



determine x(S)'. An immediate consequence of



U. Zimmermnnn



546



c



(3.10) Theorem. Let 5 be a preorder on B" and S B". If

(3.10.1) (B", + , 5 ) a preordered semigroup,

(3.10.2) there exists the maximum x of S with regard to L',

then y 5 x + , V y E 9.



15-



The two assumptions describe a class of problems which can be solved by the

application of the greedy algorithm. After the determination of X + one has to check

whether X + E S or not. If x t E 9, S, then it is only an upper bound. If S = 9, then

x + is a solution of (P2). The following theorem implies by (3.8.4) Theorem (3.10).

Let usdenote S , : = { x E S [ I l x I ( = i } .



(3.11) Theorem.

max{llx Ib E S )



y <, X(I),



The assumptions of



(3.10) yield for ail



1 S i s I( S



/I:=



vy E (S),.



The theorem follows from (3.8.3) and (2.6). The two theorems refer to different

combinatorial structures.



(3.12) Corollary. Let B C B". If M

defined by (3.4), then



=



M ( N , T ( B ) )is a matroid by (2.7) and



5

- is



x ( B ) E max s ( B ) .



(3.13) Corollary. Let S

defined by (3.4) then



c B". If M = M ( N , T ( S ) )is a matroid by



(2.8) and



- is

5



[x(S)]+ E max z ( S ) .



The two corollaries follow from (3.11) respectively (3.10) and (2.10) respectively

(2.11). We consider the following class of functions in view of (Pl):



(3.14) Definition. Let F denote the set of all functions f : B" -+ H with

(3.14.1) ( H , c )is a n ordered set,

(3.14.2) f(e,) s f(e,-,) S . . . c f ( e l ) ,

(3.14.3) (B", +

) is a preordered semigroup with regard to

the preorder induced by f .



,s

-



c



(3.15) Corollary. Let S B" with S = 9. If (3.10.2) holds, then regardless of the

choice of the objective f E F, [x(S)]+ is a solution of the problem maxxEsf(x).

This follows from (3.10). Clearly there is an analogous corollary corresponding to

(3.11).



(3.16) Corollary. Let B C B" with 11 y 11 = IIB 1) V y E B. If (3.10.2) holds then

regardless of the choice of the objective f E F x ( B ) is a solution of the problem

maxxEsf(x).

These corollaries reflect t h e fact that the greedy algorithm only considers the

values of the objective function for the unit vectors.



Partial orders related to Boolean optimization



547



At the end of Section 2 we introduced regular sets with regard to L'. As shown by

(2.16) in this case the assumption (3.10.2) implies that M = M ( N , T ( S ) ) is a

matroid. Results for more general regular sets with regard to C and L' are given

by Hammer, Johnson and Peled in [7]. If the objective "agrees" with the partial

order R , that is



+ f(x)



xR y



(3.17)



f(y),



vx, Y E B",



then

max 5 S



(3.18)



max 5 SR



clearly holds for S CB", S R : = {x E B" I 3 y E S : x R y } .

If distinct vectors in (3.17) imply distinct function values, then equality holds in

(3.18). In this case the BOP (Pl) is equivalent to

(P3)



max f(x).

XCSR



As shown in [7] Sc can be described by the restrictions of a covering problem, that

means all restrictions are of the form

(l-xj)zl,



with J C N .



j€J



In the case R = L' a further simplification is possible and developed in [7].

In connection with covering problems the partial order i ' h a s been considered by

Bowman and Starr [l]. They present an enumerative algorithm for the problem of

maximizing a partial order on B", which fulfills (3.2)' and (M) in (3.3). If in this

section 5 denotes only a partial preorder, then under the additional assumption to

(3.2')



-



e , < 0 or there exists k E N,(1) such that e k z O <

=

+=

all results hold which refer to 5.



( 3 . 2 ) 0 5 en



or



-



ek-]



4. Dual partial orders

(4.1) Definition.



Let R be a partial order on B". Then the dual partial order of R is



R ' , defined by



x R' y :



G(y)RG(x)



with a E P n , a ( i ) : = n - i + l for I E N .



-



Partial orders and their duals may coincide more or less

(4.2) Proposition. (1) x C ' y C y C x,

(2) x L b ' y

x LbY.



U. Zimmermann



548



In view of proposition (2.4) the dual partial orders of those partial orders defined

by (2.1) and (2.3) have analogous properties.



*

*



(4.3) Proposition. (1) y C x

(2) X L b Y

XLI'Y,

(3) x L I ' y

x <'y.



=+-



x



L~

y,



In connection with dual partial orders we consider a modified greedy algorithm

(A')

(1) x:=O; j : = n ;

(2) if x + e, E S, set x:= x + ei;

(3) if j = 1, stop,

otherwise set j : = j - 1 and return to (2).

The output vector of this algorithm applied to S C B" shall be denoted by x ' ( S ) .

The application of (A') to S * := (1 - x x E S} is called dual greedy algorithm.



I



(4.4) Proposition. x'(S) is the minimum of S with regard to



<'.



Proof. The application of (A') to S is equivalent to the application of (A) to & ( S ) .

Hence x'(S) = x(&(S)). (4.4) follows by (4.1).



For an arbitrary set S the four vectors representing the maxima respectively the

minima of S with regard t o < respectively to <' may be pairwise distinct. For

example, take S = {x, y, u, v } with

x = (1 0 0 lo),



maximum with regard to



= (0 1100),



maximum with regard to



y



<,

<',



u = (0 1 0 0 l), minimum with regard to =S',



minimum with regard to



u = (00 1 lo),



<.



, L", 6 ,< '}. Then

(4.5) Proposition. Let R E { C , _> ,L ~L',



XRY



(l-y)R(l-X).



Let us show this for example in the case of R = L'. Equivalent to the left side

there is



I Tk(x)( I Tk(y)(,



v1



k



n



and this is equivalent to



1



Tk



(1 - x ) I 3 I



Tk



(1 - y ) 1,



v1



k s n.



The rest follows analogously to the first equivalence. An immediate consequence of

(4.5) is the next proposition.



549



Partial orders related to Boolean optimization



(4.6) Proposition. 1- x'(S*) is the maximum of S with regard to

the minimum of S with regard to < .



<'. 1- x ( S * ) is



The lexicographical maximum or minimum of S as well as the dual lexicographical maximum or minimum of S can be computed by the application of (A) or (A') to

S or S*.



B". The following statements are equivalent :

(4.7.1) there exists the maximum x E B with regard to L~



(4.7) Theorem. Let B



(4.7.2)



there exists the common maximum x E B with regard to L~and L'.



An implication of (4.7.1) or (4.7.2) is

(4.7.3) x ( S ) = 1- x'(S*).

Proof. (4.7.1) implies (4.7.2) by (2.4) and (4.3). Reversely, if y L " X

then follows by definition I(x 11 6 1) y 11 < 1) x (1 and therefore y L~

x.

(4.7.2) implies (4.7.3) by (2.6), (4.3) and (4.6).



and y



L'X,



If (4.7.1) or (4.7.2) hold, the dual greedy algorithm yields the complement of the

lexicographical maximum of S. This may be impdrtant in view of problem (Pl). The

crucial point

in the application of the greedy algorithm is the test whether x E or

not. If x E ( S * ) is easier t o check, then one will prefer the dual greedy algorithm.



5. Remarks

The combinatorial structure of problems for which the greedy algorithm is valid

is closely related to matroids. The corresponding algorithm for the intersection of

two matroids, namely the weighted intersection algorithm of Lawler [9], has not yet

been considered in this way, but similar studies have been published by Burkard,

Hahn, and Zimmermann [3] as well as Burkard [2] about the assignment problem

which is a special example of the intersection of two matroids. Already in this

special case it turned out that similar results as in (3.15) cannot be attained, yet an

algorithm is stated in [3] which solves the assignment problem with generalized

objectives.



References

[l] V.J. Bowman, and J.H. Starr, Set covering by ordinal cuts 1/11, Management Sciences Research

Reports No 321/322, 1973 Carnegie-Mellon University Pittsburgh, Pennsylvania.

[Z] R. Burkard, Kombinatorische Optimierung in Halbgruppen in: R. Bulirsch, W. Oettli, J. Stoer,

eds., Optimization and Optimal Control, Lecture notes in mathematics, 477 (Springer, Berlin, 1975)

pp. 1-17.



550



U. Zimmermann



[3] R. Burkard, W. Hahn and U. Zimmermann, An algebraic approach to assignment problems,

Report 1974-1, Mathematisches Institut der Universitat Koln 1974.

[4] F.D.J. Dunstan and D.J.A. Welsh, A greedy algorithm for solving a certain class of linear

programmes, Math. Programming 5 (1973) 338-353.

[5] J. Edmonds, Matroids and the greedy algorithm, Math. Programming 1 (1971) 127-136.

[6] D. Gale, Optimal assignments in an ordered set: an application of matroid theory, J. Comb. Theory

4 (1968) 176180.

[7] P.L. Hammer, E.L. Johnson and U.N. Peled, Regular 0-1-programs, Research Report CORR

73-18, University of Waterloo.

[8] J. B. Kruskal, On the shortest spanning subtree of a graph and the travelling salesman problem,

Proc. Am. Math. Soc. 7 (1956) 48-50.

[9] E.L. Lawler, Matroid intersection algorithms, Math. Programming 9 (1975) 31-56.

[lo] M.J. Magazine, G.L. Nemhauser and L.E. Trotter, When the greedy solution solves a class of

knapsack problems, MRC Technical Summary Report No. 1421 (1974), Mathematics Research

Center University of Wisconsin, Madison, Wisconsin, U.S.A.

[ll] L.A. Wolsey, Faces for a linear inequality in 0-1-Variables, Math. Programming 8 (1975) 165-178.



Annals of Discrete Mathematics 1 (1977) 551-562

@ North-Holland Publishing Company



INTEGER LINEAR PROGRAMMING WITH

MULTIPLE OBJECTIVES*

Stanley ZIONTS

School of Management, State University of New York ar Buffalo, Buffalo, N Y , U.S.A.



Although it may seem counterintuitive, a method for solving multiple criteria integer linear

programming problems is not an obvious extension of methods that solve multiple criteria linear

programming problems. The main difficulty is illustrated by means of an example. Then a way of

extending the Zionts-Wallenius algorithm [6] for solving integer problems is given, and two types

of algorithms for extending it are briefly presented. An example is presented for one of the two

types. Computational considerations are also discussed.



1. Introduction

In [6] a method was presented for solving multiple criteria linear programming

problems. Because integer programming is a generalization of linear programming

in that a subset of variables may be required to take on integer values, it is

reasonable to ask if multicriteria integer problems can be solved by an obvious

extension to the method: solving the multicriteria linear programming problem

using that method and then using the associated multipliers to solve the integer

problem. In general, unfortunately, such a procedure is not valid. Assuming that

the implicit utility function of the decision maker is a linear additive function of

objectives, the general idea can be modified into a workable algorithm for solving

mixed or all integer programming problems involving multiple objectives.

Numerous approaches to various problems involving multiple objective functions

have been proposed. B. Roy [3] discusses a number of them. He also develops a

typology of methods [3, p. 2401:

“1. aggregation of multiple objective functions in a single function defining a

complete preference order;

2. progressive definition of preferences together with exploration of the

feasible set;

3. definition of a partial order stronger than the product of the n complete

orders associated with the n objective functions;

4. maximum reduction of uncertainty and incomparability.”

To put things into perspective, the approach of [6] is a combination of 1 and 2 in

that an aggregation of the functions is accomplished by an interactive process in

* An earlier version of this paper has also been issued as Working Paper 75-32 of the European

Institute for Advanced Studies in Management in Brussels.

551



552



S. Zionts



which preferences are expressed. The use of multiple criteria in an integer

framework has been mentioned in [6] and more recently in [1] and [4].

The plan of this paper is to first indicate why noninteger methods cannot be

extended in an obvious way to solve multiple criteria integer problems. Then two

extensions of the method of [6] for solving integer problems are developed, an

example is solved, and some considerations for implementation are given. In an

appendix the method of [6] is briefly overviewed.



2. Some considerations for solving multiple criteria integer problems

The problem to be considered is a mixed integer linear programming problem.

Let the decision variables be a vector x of appropriate order where some or all of

the variables are required to take on integer values. Denote the set of integer

variables as J. The constraint set is then



Ax



=



b



x30

x,, j E J integer,



where A and b are, respectively, a matrix and vector of appropriate order. In

addition we have a matrix of objective functions C where row i of C gives the ith

objective C,. Each objective of u is to be maximized and we may thus write



Iu - c x



6 0.



(2)



The formulation (1) (2) is the most general formulation of the multiple criteria

integer programming problem if one grants that any nonlinearities are already

represented in the constraints (1) using piecewise linearizations and integer

variables as necessary. If we accept that the implicit utility function is a linear

function (as was done originally in [6]) of the objectives u, we may therefore say

that our objective is to maximize Au where A is an unknown vector of appropriate

order. Were A known, the problem of maximizing Au subject to (1) and (2) would

be an ordinary integer programming problem. Such a problem could be solved

using any method for solving integer linear programming problems. The problem is

that h is not known.

In an earlier paper [6] Wallenius and I developed a method for solving linear

programming problems having multiple objectives. That method is briefly summarized in the appendix. The method has been extensively tested and seems to work in

practice. A natural extension of that method would appear to be an extension for

solving problems involving integer variables:

1. Solve the continuous multiple criteria problem according to the method of [6];

2. Using the multipliers obtained in step 1, solve the associated integer linear

programming problem.



Integer linear programming with multiple objectives



553



Unfortunately as the following simple example shows, that extension does not

necessarily work.

Given the constraints:

XI



+ 4x2 c 3;



xl,xz 5 0 and integer

with objectives u1= xl, and u z = x2 then provided that the true multipliers A, and A 2

( >0) satisfy the following relationships

h 1> fh2

hl



< 3Az



then the continuous solution x1 = 2.34, xz = 2.34 is optimal. However, for this

problem there are three optimal integer solutions corresponding to the same

continuous optimum depending on the true weights:

If 3Az> A l > 2Az, then x1 = 3, xz = 0 is optimal;

If 2Az> h l > O S A , , then xI = x2 = 2 is optimal;

If 0.5Az> A l > !A2, then x1 = 0, xz = 3 is optimal.

The example could readily be made more complicated, but it serves to show that

further precision may be required in the specification of the multipliers than only to

identify the multiplier valid at a noninteger optimal solution. (Further precision is

not always required; change the constraint value of the problem from 3.125 to 2.99.)



3. Adapting the Zionts-Wallenius method for solving integer programming

problems



To further specify the multipliers A to find the optimal integer solution, it is

necessary to ask additional questions of the decision maker. There are numerous

ways in which this may be done, and we shall explore two cf them. Both of these

proposals represent untested procedures.

3.1. A branch and bound approach

We first consider branch and bound algorithms. The multiple criteria method can

be altered to work in a branch and bound integer framework. T o do this we first

present a flow chart of a simple branch-and-bound algorithm, [5, p. 4161 in Fig. 1.

As usual, [y] is the largest integer not exceeding y. The idea is to solve a sequence

of linear programming problems thereby implicitly enumerating all of the possible

integer solutions. The best one found is optimal. The procedure of Fig. 1 cannot be



554



Tc



Halt

optimum



S. Zionts



Choose an integer

variable Xk whose

solution value Vk

is not an integer.



Select solution with the maximum

objective function value from list,

I f l i s t is empty halt: there is no

feasible integer solution t o the problem.



FIG.1. Flow Chart ofa Simple Branch a n d Bound A l g o r i t h m

Taken f r o m [S,page 4161.



555



Integer linear programming with multiple objectives



used directly, but must be modified. The modifications wh'ich are to be made are

based on the following theorem.

Theorem. A solution can be excluded from further consideration (not added to the

list) provided the following two conditions hold :

(1) the decision maker prefers an integer solution to it,



Solve multicriteria linear programming

problem obtained by relaxing integer

constraints. If solution satisfies

integer constraints, stop.



Yes

the conditions of



-



Discard

the

Solution

I



Choose an integer variable xk whose

solution value y k is not integer.



Solve two problems, each having adjoined

one of the following constraints:

Xk 5 [Ykl

Xk 2 [Ykl -+ 1

Exclude any infeasible solutions

from further consideration.



FIG.2.

Flow Chart of a Branch and Bound Multicriteria

Integer Linear Programming Method



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