Tamkang Journal of Science and Engineering, Vol. 7, No 4, pp. 259-263 (2004)
259
Linear Optimization over Efficiency Set and Weakly
Efficiency Set of a Multiple Objective Linear Program
Chien-Hsiung Lin
Industry and Business Management Department
The Open University of Kaohsiung
Kaohsiung, Taiwan 812, R.O.C.
E-mail: axel@ms2.ouk.edu.tw
Abstract
This paper develops an algorithm for finding an optimal solution of
maximizing a linear function over the weakly efficiency set of a multiple
objective line program. This optimization problem is solved by solving a
bilevel linear program.
Key Words: Multiple Objective Linear Program (Molp), Bilevel Linear
Program, Weak Efficient Solution, Optimal Solution, Optimization, Post Optimality Analysis
1. Introduction
A Multiple Objective Linear Program (MOLP) is defined as
max Cx
xÎ X
where X = {x Î Rn : Ax = b, x ³ 0} is the feasible region
and the matrix of objective functions C = (cij)kxn with
objective row vectors c1,c2, ck, and the matrix of constraints A = (aij)mxn. To solve an (MOLP) problem, one
maximizes k objective functions simultaneously, an optimal solution of the problem is called an efficient solution. The set of all efficient solutions is denoted by E
[3,4,6,10].
The well known scalarization theorem of vector optimization is stated as, x* Î X is an efficient solution of
(MOLP) if and only if there is a l Î Rk, l = (l1,l2, lk), li
> 0 for all i, such that x* solves
( Pl )
max l Cx.
xÎ X
x* is an efficient solution of (MOLP) if there exists
no x Î X such that cix ³ cix*"i and cjx > cjx* for some j Î
{1,2,…k}.
It can be seen that the case if one considers the case
when l ³ 0 instead of l ³ 0. Then, one (or more) of the
components of l is allowed to be zero. This leads to the
concept of weak efficient solution. The set of all weak efficient solutions, Ew is defined as x*Î Ew if and only if x*
solve (Pl) for l ³ 0 [10].
Similarly one can easily see that Ew = {x* Î X : {x Î
X : Cx > Cx*} = f} That is to say, if x* is a weak efficient
solution, then there is no x Î X such that Cx > Cx*.
In summary:
E = {x* Î X : Cx ³ Cx* Þ Cx = Cx*}
E = {x* Î X : $l Î Rk, l > 0, lCx* ³ lCx, "x Î X}
Ew = {x Î X : {x Î X : Cx > Cx} = f}
Ew = {x Î X : $l Î Rk, l ³ 0, s.t. lCx ³ lCx, "x Î X}
Obviously, E Ì Ew.
Let f : Rn ® R be a real-valued function. Then the
problem max f ( x ) is called the optimization over the effixÎ E
ciency set (OE) and max f ( x ) is called the optimization
over the weakly efficiency set (OWE).
Since E = {x* Î X : $l > 0, Cx* ³ Cx, "x Î X} and
Ew = {x Î X : $l ³ 0, Cx ³ Cx, "x Î X}
xÎ E w
260
Kuei-Hsien Chen
Then
max ax where x solves
(OE) max f ( x )
l
xÎE
(BL)
and
s.t. x Î X
(OEW) max f
This is a linear price control problem if let lC = y which
is one of the well known form of bilevel program.
xÎEw
can be written as
max f ( x)
l >0
(BOE)
max l Cx
where x solves
max ax where x solves
y
max l Cx
max y T x
x
s.t. x Î X
(BLP)
and
s.t. Ax £ b
T
max y x
x
max f ( x)
l ³0
(BOEW)
max l Cx
s.t. x Î X
respectively. While, l > 0 (BOE) is not easy to find an
optimal solution. Let first consider
max f ( x)
l ³0
(BOEW)
s.t . Ax £ b
where x solves
To develop a solution procedure for this bilevel linear
program, the duality theorem and post optimality analysis related theorem, such as complementary slackness
conditions are applied.
The dual linear program of the following linear program
max y T x
where x solves
x
max l Cx
(PLP)
y = CT l
s.t. x Î X
This (BOEW) is a bilevel problem. When f(x) is a linear function, f(x) = ax, (BOEW) is a bilevel linear program, which can be formulated as
s.t. Ax £ b
x, l ³ 0
is
min ub
x
max ax where x solves
l ³0
(BL)
s.t. uA ³ y
max l Cx
y = CT l
s.t. x Î X
u, l ³ 0
Let y = (lC)T
Then (BL) can be written as
max ax where x solves
y
max y t x
(BO)
(DLP)
s.t. Ax £ b,
Theorem. Let x* and u* be optimal solutions of
(PLP) and (DLP), respectively, then
1. yx* = u*b
2. u*(b – Ax*) = 0
3. (u*A - y)x* = 0
Proof. 1,2,3 follow immediately from the duality
theorem and the complimentary conditions. Consider
max ax where x solves
y = lC ,
x, l ³ 0
y
(BLP)
max y T x
x
This is a bilevel linear programming, since Ew is the set
of optimal weakly efficient solutions of (MOLP). Next
we reformulate this model:
s.t. Ax £ b
y = lC
x, l ³ 0
Generally Applicable Self-masking Technique for Nanotips Array Fabrication
by above theorem we have the follows:
max ax
( x , l ,u , v , z )
s.t. Ax + z = b
(OEP)
AT u - v = l C
vT x = 0
uT z = 0
x, l , u , v, z ³ 0
Note that the program (OEP) is not a linear program, because the non linear complementary conditions. In order to develop the algorithm for this problem let us consider the follow linear program first.
max ax.
( x , l ,u , v , z )
(OELP)
Ax + z = b
s.t. A u - v = l C
T
x, l , u , v, z ³ 0
Next theorem is fundamental to the algorithm.
Theorem. (x, l, u, n, z) is a feasible solution of (OEP)
if and only if (x, l, u, n, z) is a feasible solution of (OELP)
with nx = 0 and uz = 0.
Proof. It is obvious that (OEP) is equivalent to
(OELP) with nx = 0 and uz = 0.
Now we are ready for the algorithm. It is easy to see
now, if we keep one of n or x, and one of u or z as non basic variable in each of the iteration. We can then, by theorem, to search an optimal solution using the following algorithm.
2. Rule of Changing Basic Variable
If xi (or ni) and zi (or ui) are the basic variables with
respect to the k-th vertex of the modified simplex method,
then in the (k + 1)-th iteration, the corresponding ni (or xi)
and ui (or zi) must not enter the basis.
Algorithm
In the algorithm, we use p(i(j),j) to denote the j-th iteration point derived from the i(j)-th iteration point. The
following is the steps of the algorithm.
Step 0. Get an initial iteration point p(0,1) Î S, set j =
1, i(j) = 0, Mj = {p(0,1)} and k = 1.
Step 1. If all the reduced cost of p(i(j),j) are negative,
let x* = p(i(j),j), goto step 5.
Step 2. If p(i(k),k) is not a live node (a live node is a
261
point which has at least an adjacent vertex in S that has
not been checked yet), go to step 3. Otherwise, according
to the rule of changing variable, move to an adjacent
point p(i(j + 1),j + 1) Ï Mj, i(j + 1) = k, Mj+1 = MjÈ{p(i(j +
1),j + 1)}. Set j = j + 1, k = j + 1, goto step 1.
Step 3. If no live node exists, goto step 4. Otherwise,
backtrack from the current point to p(i(i(k)), (i(k)), set k:
= i(k), goto step 2.
Step 4. Get x* from {max aT x | x Î M}.
Step 5. x* is the optimal solution to (P), Stop.
Remark. By introducing slack or artificial variables,
it is easy to obtain the initial point p(0,1).
3. Conclusion
The following example shows how the algorithm applies to a numerical problem concludes this work.
Example
max 2 x1 + x2
s.t. x Î EW
where Ew is the weakly efficient set of the following
problem
æ -1 -1 ö æ x1 ö
÷ç ÷
è 0 1 ø è x2 ø
max ç
s.t. x1 + x2 £ 2
2 x1 - x2 £ 2
- x1 + 2 x2 £ 2
x1 , x2 ³ 0.
After transforming the problem into a price control
problem, replacing the inner program of the price control problem by its corresponding duality theorem and
post optimality conditions and removing the complementary condition, we obtain the following linear programming problem:
max 2 x1 + x2
s.t. x1 + x2 + z1 = 2
2 x1 - x2 + z2 = 2
- x1 + 2 x2 + z3 = 2
- u1 - 2u2 + u3 - l1 + v1 = 1
- u1 + u2 - 2u3 - l1 + l2 + v2 = 0
xi , ui , li , vi ³ 0.
Let basis (z1,z2,z3,n1,n2) be initial.
262
Kuei-Hsien Chen
Table 1. Iteration (1) searching for the basic variable
z1
z2
z3
n1
n2
x1
x2
u1
u2
u3
l1
l2
n1
n2
z1
z2
z3
1
2
-1
0
0
2
1
-1
2
0
0
1
0
0
0
-1
-1
0
0
0
0
-2
1
0
0
0
0
1
-2
0
0
0
0
-1
-1
0
0
0
0
0
1*
0
0
0
0
1
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
2
2
2
1
0
According to the rule of changing basic variable, only l2 can enter the base.
Table 2. Iteration (2) searching for the basic variable
z1
z2
z3
n1
l2
x1
x2
u1
u2
u3
l1
l2
n1
n2
z1
z2
z3
1
2
-1
0
0
2
1
-1
2*
0
0
1
0
0
0
-1
-1
0
0
0
0
-2
1
0
0
0
0
1
-2
0
0
0
0
-1
-1
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
2
2
2
1
0
Only x2 can enter the base.
Table 3. Iteration (3) searching for the basic variable
z1
z2
x2
n1
l2
x1
x2
u1
u2
u3
l1
l2
n1
n2
z1
z2
z3
3/2
3/2
-1/2
0
0
5/2
0
0
1
0
0
0
0
0
0
-1
-1
0
0
0
0
-2
1
0
0
0
0
1*
-2
0
0
0
0
-1
-1
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
1
0
0
0
0
-1/2
1/2
1/2
0
0
-1/2
1
3
1
1
0
Only u3 satisfies the rule of changing base.
Table 4. Iteration (4) searching for the basic variable
z1
z2
x2
u3
l2
x1
x2
u1
u2
u3
l1
l2
n1
n2
z1
z2
z3
3/2*
3/2
-1/2
0
0
5/2
0
0
1
0
0
0
0
0
0
-1
-3
0
0
0
0
-2
-3
0
0
0
0
1
0
0
0
0
0
-1
-3
0
0
0
0
0
1
0
0
0
0
1
2
0
0
0
0
0
1
0
1
0
0
0
0
0
0
1
0
0
0
0
-1/2
1/2
1/2
0
0
-1/2
1
3
1
1
2
x1 is the only variable which can enter the base.
Table 5. Iteration (5) searching for the basic variable
x1
z2
x2
u3
l2
x1
x2
u1
u2
u3
l1
l2
n1
n2
z1
z2
z3
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
-1
-3
0
0
0
0
-2
-3
0
0
0
0
1
0
0
0
0
0
-1
-3
0
0
0
0
0
1
0
0
0
0
1
2
0
0
0
0
0
1
0
2/3
-1
1/3
0
0
5/3
0
1
0
0
0
0
-1/3
1
1/3
0
0
3/4
No variable can enter the base and there is no live node left. The optimal solution is (2/3, 4/3).
2/3
2
4/3
1
2
Generally Applicable Self-masking Technique for Nanotips Array Fabrication
References
[1] Bard, J. F., “Coordination of Multidivisional Organization through Two Levels Management,” Omege,
Vol. 11, pp. 457-468 (1983).
[2] Bialas, W. and Karwan, M., “Two-level Linear Programming,” Management Science, Vol. 30, pp. 10041020 (1984).
[3] Lin, C. H., Siddiqui, K. J. and Liu, Y. H., Chapter 20:
“Multiple Objective Programming to Analyze and
Classify Business Information,” New Frontier of Decision Making for the Information Technology Era,
World Scientific Publishing, Singapore (2000).
[4] Liu, Y. H. and Dauer, J. P., “Bicriteria Programming
with Several Applications,” Computer and Industrial
Engineering, Vol. 37, pp. 563-580 (1999).
[5] Liu, Y. H. and Siddiqui, K. J., “Using Bicriteria
263
Boolean Linear Programming Model for Parameter
Selection in Large Multicategory Classification Problem,” Mathematical and Computer Modeling, Vol. 24,
pp. 75-81 (1995).
[6] Luenberger, D. G., Linear and Nonlinear Programming.
Addison Wesly, Reading, MA, U.S.A. (1969).
[7] Rockafellar, R. T., Convex Analysis, Princeton Press,
NJ, U.S.A. (1970).
[8] Stadler, W., Multicriterica Optimization in Engineering
and in the Sciences, Plenum Press, NY, U.S.A. (1988).
[9] Schrage, L., LINDO User Manual, Release 5.0, The
Scientific Press, South San Francisco, CA, U.S.A.
(1989).
Manuscript Received: May 28, 2004
Accepted: June 20, 2004