CaROMaD, Volume 01 (2016) : 39 − 52
http://www.laromad.usthb.dz
Cahiers de Recherche Opérationnelle
et Mathématiques de la Décision
Optimizing a nonlinear function in multi-objective integer fractional Programming
Abdelhak Mezghiche(1) & Ouiza Zerdani(2) & Mustapha Moulaı̈(3)
(1,2)
LaROMad, Fac. Maths, USTHB, PO 32, 16111 Bab Ezzouar, Algeria
(3)
LAROMAD, Fac. Sciences, UMMT, Tizi Ouzou, Algeria
(1)
[email protected],
(2)
[email protected],
(3)
[email protected]
Abstract: An extension of optimizing over the integer efficient set of a multi-objective integer
linear programming problem to a nonlinear case which maximizes a ratio of two linear functions
over the integer set of a multi-objective integer linear fractional programming problem is considered in this paper. An exact method is presented for optimizing a main linear fractional function
ψ over the set of integer efficient solutions of a MOILFP problem without explicitly having to
enumerate all of them. We Combine two strategies: the first one reduces the feasible region
using a cutting plane technique while eliminating dominated points; the second one explores
incident edges to a current solution in order to find a new efficient solution increasing the function ψ. The proposed algorithm, able to give an optimal value of the linear fractional function
ψ, converges in a finite number of steps. An numerical example is presented for illustrating the
method.
Key Words: multi-objective optimization, fractional programming, integer programming, efficient set, level sets.
MSC(2010): 90C29, 90C32, 90C26
Résumé : Une extension de l’optimisation sur l’ensemble efficient discret d’un problème de
programmation linéaire multi-objectif au cas non linéaire où l’on maximise un rapport de deux
fonctions linéaires sur l’ensemble efficient discret d’un problème de programmation hyperbolique
multi-objectif (MOILFP) est considéré dans cet article. Pour optimiser la fonction hyperbolique
ψ représentant les préférences du décideurs sur l’ensemble efficient discret du problème MOILFP
sans avoir explicitement à les énumérer toutes, une méthode exacte est présentée. Nous combinons deux stratégies: la première réduit le domaine admissible en utilisant une technique
de coupe tout en éliminant les points dominés; La seconde explore les arêtes incidentes à une
solution courante afin de trouver une nouvelle solution efficace augmentant la fonction psi.
L’algorithme proposé converge vers la valeur optimale de la fonction hyperbolique ψ en un
nombre fini d’étapes. Un exemple numérique est présenté pour illustrer le procédé.
Mots clés : optimisation multi-objectif , programmation fractionnaire, programmation en
nombres entiers, ensemble efficient, ensemble niveau.
40
1
A. Mezghiche & O. Zerdani & M. Moulaı̈
Introduction
The multiobjective integer linear fractional programming , say (MOILFP), is an important
class of problems arising in multicriteria decisionmaking when some or all the model variables
represent discrete decisions. Several methods have been developed to solve MOILFP problems
see for instance [?, ?]. Mathematically, this problem can be classified as a global optimization
problem. One of its main difficulties arises from the fact that the efficient set E(P ) is generally
non connected. In continuous linear case, the problem was first considered by Philip [?], in which
an algorithm based on moving to adjacent efficient vertices is outlined. In Isermann and Steuer
[?], the main idea of the algorithm is based on the use of a cutting plane procedure. Benson [?, ?]
has given two relaxation algorithms for solving (PE ). The survey of Yamamoto [?] proposes a
classification of the existing algorithms for optimization over the efficient set. Thi, Pham and
Thoai [?] propose a branch and bound procedure based on some properties in Lagrange duality.
Yamada, Tanino, Inuiguchi [?] propose a method for approximate minimization of a convex
function over the weakly efficient set.
In [?], Benson has proposed a bisecting search algorithm and he has suggested in [?] a linear
programming procedure for detecting and solving the problem (PE ) in four special cases and
many others references see for instance [?, ?, ?]. When the decision variables are integers in
linear case, few methods exist in the literature and cuts or branch and bound techniques are
unavoidable. However, It is rare to find an article in the literature addressing the problem of
optimization of a non linear function over the integer efficient of multi-objective integer linear
fractional problem.
As an application for our method is the following multiple objective fractional minimal cost flow
problem on network [?, ?]. Let G = (V, E) be a connected and directed graph with vertex set
V = {v1 , v2 , ..., vm } and
P arc set E. To each kvertex vki ∈ V (i = 1, 2, ..., m) is assigned a real
number bi such that m
i=1 bi = 0. Constants pij and qij are given cost coefficients for every arc
(i, j) ∈ E, and uij ≥ 0 is the upper bound of flow volume on arc (i, j). Given constant costs αk ,
β k , the multi-objective linear fractional minimum cost flow problem is following

P
k
k

(i,j)∈E pij xij + α

k

P
min
z
=

k
k


(i,j)∈E qij xij + β


s.t.
X
X


x
−
xki = bi ,
ij




(i,j)∈E
(k,i)∈E


0 ≤ xij ≤ uij
k = 1, . . . , r
i = 1, . . . , m
(1)
Since the profitability is an index of operation efficiency expressed as a fraction, the utility
function of the decision-maker can be written as
P
(i,j)∈E cij xij + λ
ψ(x) = P
(i,j)∈E dij xij + µ
where cij , dij are cost coefficients for every arc (i, j) ∈ E and λ, µ are fixed costs.
The decision maker, say DM, optimizes this function over the efficient solutions in order to
choose a robust solution. In this paper we focus on the problem of optimizing a linear fractional
function, denoted by ψ, over the efficient set of a M OILF P problem. Based on the Ehrgott’s
Optimizing over an integer efficient set
41
idea [?] for reducing progressively the feasible set, the proposed algorithm avoids searching for
all efficient solutions but guarantees finding one that optimizes ψ and an exploration process of
the edges incident to the current optimal solution is performed [?].
Consider the M OILF P problem

pk x + αk

k

, k = 1, ..., r
 ”max ” Z (x) = k
q x + βk
(P )
s.t



x∈S
where αi , β i are scalars; pi , q i ∈ Rn for each i ∈ {1, 2, ..., r}; b ∈ Zm ; A ∈ Zm×n ; D = {x ∈
Rn /Ax ≤ b, x ≥ 0}; S = D ∩ Zn and Z is the set of integers. It is assumed that S is not empty
and D is a bounded convex polyhedron in Rn and q i x + β i > 0 over D for all i ∈ {1, 2, ..., r}.
The set of all integer efficient solutions of (P ) is denoted by E(P ).
As in multiple objective integer linear programming see [?], the solution to the problem (P ) is
to find all solutions that are efficient in the sense of the following definition.
Definition 1 [?, ?] A point x0 ∈ S is said to be efficient if and only if there does not exist
another point x1 ∈ S such that: Zi (x1 ) ≥ Zi (x0 ), i ∈ {1, 2, ..., r} and Zi (x1 ) > Zi (x0 ) for at
least one i ∈ {1, 2, ..., r}.
Otherwise, x0 is called a dominated solution and the vector (Z1 (x0 ),Z2 (x0 ),...,Zr (x0 )) is said to
be a dominated r−tuple.
In practical applications of multiple criteria decision making, the decision makers often have to
choose some preferred point from the efficient set E(P ). This involves the problem of finding
efficient solutions and describing the structure of E(P ) see [?].
Since in many cases the criteria are in conflict, the decision maker try to optimize a linear
fractional function over the efficient set E(P ). This leads to finding a method for solving the
optimization problem (PE )
(PE )
where c, d ∈ Rn and λ, µ ∈ R.


cx + λ
dx + µ
x ∈ E(P )
max ψ(x) =
 s.t.
(2)
Although the discrete case has by no means seen a similar development as continuous case, linear
function optimization on an integer efficient set is considered only by:
− N.C. Nguyen [?] who gives an upper bound for the optimal objective value of ψ.
− Chaabane et al. [?] where different types of cuts are imposed in such a way that the
improvement of the objective value is guaranteed at each iteration.
− Jesus M. Jorge [?] who proposes an algorithm based on the analysis of an appropriate
sequence of scalar linear integer problems.
42
A. Mezghiche & O. Zerdani & M. Moulaı̈
In this paper, a method for optimizing a linear fractional function over an integer efficient set of
a multi-objective fractional problem without having to determine all integer efficient solutions
is presented. The proposed algorithm is based on a cutting plane technique and on a simple
selection procedure that improves ψ at each iteration.
Let the relaxed problem be:
(PR )


max ψ(x) =
 s.t.
cx + λ
dx + µ
(3)
x∈S
The problem (Pi (S)), i ∈ {1, 2, ..., r}, is the following integer linear fractional programming
problem [?, ?]:

pi x + αi

max Zi (x) = i
(Pi (S))
(4)
q x + βi

n
s.t. x ∈ S = D ∩ Z
2
Necessary theoretical tools
We introduce the following notations:
− D1 = {x ∈ Rn1 : A1 x ≤ b1 ; A1 ∈ Rm1 ×n1 ; b1 ∈ Rm1 ; x ≥ 0}. D1 is a current truncated
region of D obtained by successive Gomory cuts introduced when optimizing problem
(P1 (S)). Note that S1 = S = D1 ∩ Zn , because Gomory cuts do not eliminate integer
solutions from D.
− (Z11 , Z21 , ..., Zr1 ) is the first non-dominated r−tuple corresponding to the optimal integer
pi x + αi
for i = 1, 2, ..., r.
solution x11 obtained in D1 , where Zi1 = i
q x + βi
For k ≥ 2 ,we have:
×nk , b ∈ Rmk , x ≥ 0}. D is the current truncated
− Dk = {x ∈ Rnk \ Ak x ≤ bk , Ak ∈ RmkP
k
k
region obtained after applying the cut j∈Nk−1{jk−1 } xj ≥ 1 where jk−1 ∈ Γk−1 (see below)
and successive Gomory cuts eventually.
− x1k = (x1k,j ) the kth optimal integer solution of problem P1 (S) obtained on Dk at step k.
( Note that in place of (P1 (S)),one can similarly consider the problem (Pi (S)), i ∈ {2, ..., r}).
− Bk1 is a basis associated with solution x1k .
− a1k,j ∈ Rmk ×1 is the activity vector of x1k,j with respect to Dk .
1 = (y 1 ) = (B 1 )−1 × a1 where y 1 ∈ Rmk×1
− yk,j
k,j
k,j
k
k,ij
- Ik = {i/a1k,i ∈ Bk1 } (indices of basic variables)
− Nk = {j/a1k,j ∈
/ Bk1 } (indices of non-basic variables)
− p1j = the j th component of vector p1
Optimizing over an integer efficient set
43
− qj1 = the j th component of vector q 1
P
1
− p1k,j = i∈Ik p1i · yk,ij
P
1
1
1 =
− qk,j
i∈Ik qi · yk,ij
- Z1 (x1k ) =
− ψ(x1k ) =
1
Zk,1
1
Zk,2
=
p1 x1k + α1
q 1 x1k + β 1
ψk,1
cx1 + λ
= k1
ψk,2
dxk + µ
1 (p1 − p1 ) − Z 1 (q 1 − q 1 ) , the updated value of the j th component of the
− γ 1k,j = Zk,2
j
k,j
k,1 j
k,j
reduced gradient vector γ 1k .
− xuk = (xuk,j ) are the (tk − 1) alternate integer solutions to x1k , if they exist, where tk is an
integer number and u ∈ {2, ..., tk }.
ψ
− Γk = {j ∈ Nk /γ 1k,j ≤ 0 and γ ψ
k,j ≥ 0}, where γ k,j = ψk,2 (cj − ck,j ) − ψk,1 (dj − dk,j ) , the
updated value of the j th component of the reduced gradient vector γ ψ
k,j .
Theorem 1 ([?]) The point x1k of S is an optimal solution of problem (P1 (S)) if and only if
the reduce gradient vector γ 1k is such that γ 1k,j ≤ 0 for all j ∈ Nk .
Remark 2 Recall that a sufficient condition for the uniqueness of the optimal solution x1k of
P1 (S) is that the set Jk = {j ∈ Nk /γ 1k,j = 0} be empty.
In this case, there does not exist any other integer feasiblesolution x in S, such that Z1 (x) =
Z1 (x1k ). Werefer to x as an alternate optimal solution to x1k .
Corollary 3 [?] A point x0 that is unique solution of the integer linear fractional programming
problem (P1 (S)), is efficient for (P ).
3
Theoretical results
Since the problem MOILFP is to determine the set of integer efficient solutions, we scan all
integer points of the feasible region D by a cutting plane technique which is described in the
present section.
3.1
Exploring edges
Definition 2 Assume that jk ∈ Nk . An edge Ejk incident to a solution x1k is defined as the set


1
xi = x1k,i − θjk .yk,ij
for
i
∈
I


k
k
(5)
Ejk =
x = (xi ) ∈ Dk : x
=
θ
jk
jk


xr = 0
for all v ∈ Nk \{jk }
)
(
x1k,i 1
1
are integers for
/yk,ijk > 0 , θjk is a positive integer and θjk .yk,ij
where 0 ≤ θjk ≤ min
1
k
i∈Ik
yk,ij
k
all i ∈ Ik if such integer values exist.
44
A. Mezghiche & O. Zerdani & M. Moulaı̈
Remark 4 Note that equation (??) enables us to compute the integer feasible alternate solutions
when the optimal solution obtained by solving (P1 (S)) is not unique.
The following theorem addresses the case in which the optimal solution of (P1 (S)) is not unique.
Theorem 5 [?] All integer feasible solutions xuk , u ∈ {2, ..., tk } of problem (P1 (S)) alternate to
x1k on an edge Ejk of region S (or truncated region Sk ) emanating from it, in the direction of a
vector a1k,jk , jk ∈ Jk , exist in the open half space.
X
xj < 1
(6)
j∈Nk \{jk }
The following theorem suggests a cut that can be viewed as a generalization of Dantzig’s cut.
Theorem 6 [?] An integer feasible solution of problem (P1 (S)) that is distinct from x1k and not
on an edge Ejk , jk ∈ Jk of the truncated region Sk (or S) through an integer feasible point x1k
of problem (P1 (S)) exists in the closed half space
X
xj ≥ 1
(7)
j∈Nk \{jk }
P
Remark 7 The cut j∈Nk \{jk } xj ≥ 1 truncating the whole edge Ejk with respect to jk ∈ Jk , is
a modified form of the Dantzig-cut, whereas the Dantzig-cut, for which Jk = ∅ defined by
X
xj ≥ 1
(8)
j∈Nk
truncated a point.
We now calculate the value ψk′ of the linear function ψ at any solution xuk = (xu1 , xu2 , ..., xun ) lying
on Ejk .
ψk′ = ψ(xuj ) =
Pn
cj xuj
Pnj=1
u
j=1 dj xj
P
i∈I
+λ
+µ
P
i∈Ik
= P
i∈Ik
1
) + cjk θjk + λ
cj (x1k,i − θjk yk,ij
k
1
dj (x1k,i − θjk yk,ij
) + djk θjk + µ
k
ci x1k,i + λ + θjk (cjk −
P
i∈I
1
)
ci yk,ij
k
Pk 1
= Pk 1
di yk,ijk )
di xk,i + µ + θjk (djk −
i∈Ik
i∈Ik
=
ψk′ =
(cx1k + λ) + θjk (cjk − ck,jk )
(dx1k + µ) + θjk (djk − dk,jk )
θj k γ ψ
k,jk
ψk,2 [ψk,2 + θjk (djk − dk,jk )]
+ ψ(x1k )
Optimizing over an integer efficient set
where θ is an integer verifying 0 < θ ≤
45
θj0k
and
θj0k
is the integer part of min
i∈Ik
We put :
υk =
xk,i
/yk,ijk > 0 .
yk,ijk
θj k γ ψ
k,jk
ψk,2 [ψk,2 + θjk (djk − dk,jk )]
(9)
Then along an edge Ejk , jk ∈ Γk , we have υk ≥ 0. Therefore, the values of ψk′ are increasing
and ψk′ reaches its maximum for θ = θj0k .
Definition 3 [?] Let f : S ⊂ Rn −→ R and x ∈ S. Then
L≥ f (x) = {x ∈ S : f (x) ≥ f (x)} is called the level set of x for f and
L= f (x) = {x ∈ S : f (x) = f (x)} is called the level curve of x for f .
The following theorem is used in various steps of our algorithm to test the efficiency of a given
integer feasible solution of multi-objectiveinteger linear fractional programming (P ).
Theorem 8 [?]
i=r
T
x ∈ S is Pareto optimal if and only if
L≥ Zi (x) =
i=1
i=r
T
L= Zi (x).
i=1
Preuve. x is Pareto optimal of (P)
⇐⇒ x ∈ S such that [ Zi (x) ≥ Zi (x) ∀i = 1...r and Zj (x) > Zj (x) for some j]
i=r
T
⇐⇒ x ∈ S such that x ∈
L≥ Zi (x) and j : x ∈ L> Zj (x)
i=1
⇐⇒
i=r
T
L≥ Zi (x) =
i=1
i=r
T
L= Zi (x)
i=1
Before starting the description of the algorithm we introduce the following inequality (≥ ψopt )
which eliminates only solutions that are strictly worse than the current optimal solution.
4
Development of the algorithm
The algorithm that we propose here is proved to provide an optimal solution of (PE ) without
having to compute the set of all efficient solutions of the underlying problem (P ).
In doing this, the procedure starts by solving the relaxed problem (PR ).
Step 0: Initialization let
ψopt = −∞
Solve the relaxed problem (PR ) : max
cx + λ
/x∈S
dx + µ
− If (PR ) is unfeasible ⇒ stop, (PE ) is unfeasible.
− Otherwise, let x0 be an optimal solution of (PR ).
Step 1: This solution is tested for efficiency by applying Theorem (??).
46
A. Mezghiche & O. Zerdani & M. Moulaı̈
− If x0 ∈ E(P ) ⇒ stop, xopt = x0 is an optimal solution of (PE ) and the algorithm terminates.
− Otherwise, go to step 2.
Step 2: Solve the problem (P1 (S)) [one may alternatively consider any of the problems (Pi (S))
i = 2, 3, ..., r instead of (P1 (S))].
1 = 0} = ∅ then the optimal solution found x1 is unique and it is
2.1 If J1 = {j ∈ N1 /γ1,j
1
cx11 + λ
1
1
, set ψopt to ψ , set xopt to x11 and go to step 3.
efficient (Corollary ??). Let ψ = 1
dx1 + µ
2.2 If J1 6= ∅, then x11 may not be unique, test the efficiency of x11 .
− If it is not efficient go to step 3.
− Otherwise let ψ 1 =
cx11 + λ
, set ψopt to ψ 1 , set xopt to x11 and go to step 3.
dx11 + µ
Step 3: Let k = 1 and perform the following sub-steps:
k ≤ 0 and γ ψ
3.1 Construct the set Γk = {j ∈ Nk /γk,j
k,jk ≥ 0}.
− If
PΓk = ∅, then go to step 3.3 and the cut in that step becomes the Dantzig cut
j∈Nk xj ≥ 1.
− Otherwise, let γ = Γk . go to (a).
a − If γ = ∅, then let jk ∈ Γk and go to 3.3.
− Otherwise, select jk ∈ γ and calculate θj0k the integer part of mini∈Ik {
x1k,i
1
yk,ij
k
1
>
/yk,ij
k
0}
1 If θj0k = 0 then there is no integer feasible solution on the edge Ejk , put
γ = γ\{jk } and go to (a).
2 Otherwise, if θj0k ≥ 1, then go to (b).
cx1k + λ
≥ ψopt , then calculate the value of the parameter
dx1k + µ
υk defined in equation (??).
If υk 6= 0, then go to (c).
Otherwise, put γ = γ\{jk } and go to (a).
cx1k + λ
If x1k is not efficient or
< ψopt , then go to (c).
dx1k + µ
Explore the edge Ejk , searching for a feasible solutions of (P1 (S)) corresponding to θ (θ is an integer verifying 0 < θ ≤ θj0k ) and test for efficiency starting
from θ = θj0k until θ = 1.
cxuk + λ
>
Once a first integer efficient solution is found, say xuk such that
dxuk + µ
u
cxk + λ
ψopt , set xopt to xuk and ψopt to
and go to sub-step 3.2.
dxuk + µ
b − If x1k is efficient and
−
−
−
c −
Optimizing over an integer efficient set
47
− If there is no integer efficient solution on this edge, then put γ = γ\{jk } and
go to (a).
3.2 Let k = k + 1. Define the new truncated region Dk as the subset of Dk−1 obtained by
cx1 + λ
≥ ψ(x1k−1 ) and using the dual simplex method and Gomory cuts
applying the cut k1
dxk + µ
(whenever they are needed) to find a new optimal solution x1k .
Set xopt to x1k and ψopt to ψ(x1k ) and go to (3.1).
3.3 Let k = k + 1. The new truncated region Dk is obtained as a subset of Dk−1 by applying
the specifiedP
cut
P
Dantzig cut j∈Nk xj ≥ 1 or cut j∈Nk \{jk } xj ≥ 1
and using the dual simplex method and Gomory cuts, if necessary, to find a new optimal
solution x1k . Set ψ k = ψ(x1k ).
− If the solution x1k is efficient and ψ(x1k ) > ψopt , set xopt to x1k and ψopt to ψ k go to
(3.2)
− Otherwise, go to (3.1).
Terminal step: The procedure terminates either at the first step when the solution x0 is efficient
or the impossibility of pivot operations appears indicating that the current region contains no
integer feasible point.
The optimal solution of problem (PE ) is then xopt and its value on criterion ψ is ψopt .
Proposition 9 Under the hypothesis that S is not empty and D is bounded, the algorithm ends
up with an efficient solution of problem (P ).
P
Preuve. SincePD is bounded, S is non-empty and finite. Each cut of Dantzig j∈Nk xj ≥ 1 or
a cut of type j∈Nk \{Jk } xj ≥ 1 reduces strictly the domain. Hence the procedure terminales
with an efficient solution of (P ) because at least one such solution exists in S.
Theorem 10 If S is non-empty and D is bounded, then
1. The algorithm terminates in a finite number of iterations.
2. The solution xopt is an optimal solution of problem (PE ).
Preuve. Proposition (??) guarantees that we can obtain an initial efficient solution of (P ), at
iteration p, p ≥ 1. WeP
see also from the description P
of the algorithm that, during iteration k,
either a cut of Dantzig j∈Nk xj ≥ 1 or a cut of type j∈Nk \{Jk } xj ≥ 1 is applied which strictly
reduces the domain or a new efficient solution is found that improves ψopt . Obviously, since the
domain S is finite, it may not be strictly reduced an infinite number of times. For the same
cx + λ
may be observed when x moves
reason, only a finite number of improvements of ψ(x) =
dx + µ
in the finite set S. This proves that the algorithm stops after a finite number of iterations.
Provided S is non-empty and D is bounded, the algorithm stops at iteration k > p if and only if
the problem (P1 (Sk )) is unfeasible, this is seen from the fact that, the dual simplex algorithm, at
some stage, possibly after the adjunction of Gomory cuts, can not perform any pivot operation.
The current value of ψopt at that iteration is optimal and xopt is an optimal solution of problem
(PE ).
48
5
A. Mezghiche & O. Zerdani & M. Moulaı̈
Numerical illustration
To illustrate the use of this algorithm, we consider the following integer linear fractional program
with three objectives.

−x1 + 4


max Z1 (x) =


x2 + 1



 max Z (x) = x1 − 4
2
−x2 + 3
(P )

max
Z
(x)
=
−x

3
1 + x2




Subject
to


S = {x ∈ R2 / − x1 + 4x2 ≤ 0, x1 − 1/2x2 ≤ 4, x1 , x2 ≥ 0, x ≥ 0 and integers}
Let the main problem be

5x1 + x2 + 1

 max ψ =
2x1 + x2 + 1
s.t.
x
,
x
∈
E(P )
1 2


(PE )
Step 0: Initialization, let ψopt = −∞

5x1 + x2 + 1

 max
2x1 + x2 + 1
We solve the relaxed problem (PR )
x
∈
S


The optimal solution is x0 = (4, 0)′ and ψ 0 = 21/9.
Step 1: This solution x0 is tested for efficiency and we obtain:
3
\
L≥ Zi (4, 0) = {(4, 0)′ ; (4, 1)′ } =
6
3
\
L= Zi (4, 0) = {(4, 0)′ }
i=1
i=1
Thus x0 is not efficient, go to step 2.

−x1 + 4

 max
x2 + 1
Step 2: We solve the problem (P1 (S))
x
∈
S


The results of solving problem (P1 (S)) are summarized in table I.
Table I
B
x3
x4
−p1
−q 1
γ 11,j
−cψ
1
−dψ
1
γψ
1,j
xB
0
4
-4
-1
4
-1
-1
1
x1
-1
1
-1
0
-1
5
2
3
x2
4
− 12
0
1
-4
1
1
0
Optimizing over an integer efficient set
49
The optimal solution x11 = (0, 0)′ is unique, thus it is efficient. Let it be a first efficient solution
that corresponds to ψ 1 = 1. We have ψ 1 = 1 > −∞ then ψopt = 1 and xopt = (0, 0)′ .
Step 3:
3.1 k = 1
6 ∅ . We put
I1 = {3, 4}; N1 = {1, 2}, Γ1 = {j ∈ N1 /γ 11,j ≤ 0 and γ ψ
1,j ≥ 0} = {1, 2} =
γ = Γ1 = {1, 2}.
Let j1 = 2 ∈ γ. We start exploring the edge E2 by calculating θ20 = min{0/4} = 0. No
integer efficient solution exists in this direction. Let γ = γ\{2} and consider the second
index j2 = 1 ∈ γ ,θ10 = min{ 14 } = 4. For θ = 4 , x21 (4) = (4, 0)′ which is not efficient.
For θ = 3, the corresponding solution on the edge E1 is x31 (3) = (3, 0)′ .
The solution x31 (3) is being tested for efficiency and we obtain:
3
\
L≥ Zi (3, 0) =
3
\
L= Zi (3, 0) = {(3, 0)′ }
i=1
i=1
Thus x31 (3) is efficient. We calculate ψ11 = 16/7.
As ψ11 > ψopt = 1, then ψopt = 16/7 and xopt = (3, 0)′ .
3.2 k = k + 1 = 2
5x1 + x2 + 1
≥ 16/7 ⇔ x1 − 3x2 ≥ 3
We cut by
2x1 + x2 + 1
After adjusting table I for the reduced feasible region and applying the dual simplex
method, the optimal feasible solution is x12 = (3, 0)′ which is efficient. It corresponds to
ψ 2 = 16/7; ψopt = 16/7 and xopt = (3, 0)′ (see table II)
Table II
B
x3
x4
x1
−p2
−q 2
γ 12,j
−cψ
2
−dψ
2
γψ
2,j
xB
3
1
3
-1
-1
1
-16
-7
16
7
x2
1
5/2
-3
-3
1
-4
16
7
0
x5
-1
1
-1
-1
0
-1
5
2
3
I2 = {1, 3, 4}, N2 = {2, 5}, Γ2 = {2, 5} =
6 ∅
Let γ = Γ2 and j2 = 2. We start exploring the edge E2 by calculating θ20 = min{3, 2/5} =
2/5. Then there is no integer feasible solution on the edge E2 .
Let γ = γ\{2} and consider the second index j2 = 5 ∈ γ, θ50 = min{1/1} = 1.
The corresponding solution on the edge E5 is: x22 (2) = (4, 0) which is not efficient. We
have γ = γ\{5} = ∅.
3.3 k = k + 1 = 3
P
and we cut the current feasible region by j∈N2 \{5} xj ≥ 1 ⇔ x2 ≥ 1.
50
A. Mezghiche & O. Zerdani & M. Moulaı̈
We add this constraint at the bottom of table II and apply dual simplex method to obtain
table III
Table III
B
x3
x4
x1
x2
−p3
−q 3
γ 13,j
−cψ
3
−dψ
3
γψ
2,j
xB
2
− 32
6
1
-2
-2
1
-32
-14
16
7
x6
-1
1
-1
0
-1
0
-2
5
2
6
x7
1
5/2
-3
-1
-3
1
-8
16
7
0
The dual is not feasible then the algorithm terminates. The optimal solution of problem
(PE ) is then xopt = (3, 0)′ and ψopt = 16/7.
6
Conclusion
An exact method for optimizing a linear fractional function over an integer efficient solutions
set of a MOILFP problem is presented in this paper and it can be viewed as an extension of
chaabane’s method [?] to a nonlinear case where two ideas are combined to achieve the objective :
one uses Ehrgott’s efficiency test to reduce the admissible region and the second explores incident
edges to a current solution in order to find a new efficient solution increasing the linear function
φ(x). The studied problem is also interesting when the DM’s criterion becomes quadratic.
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