Some methods for solving interval linear programming (ILP) problems have
been reviewed based on feasibility and optimality conditions. Feasibility condition
is used to avoid violation of the constraints of the best problem, which has the
largest feasible space, and optimality condition to ensure that each point of the
solution space is optimal for at least one characteristic problem.
In some methods, the best and worst cases (BWC) and two-step method (TSM),
the feasibility condition was not been considered. Therefore, solution spaces given
by the BWC and TSM may be infeasible. Solutions obtained through improved
TSM (ITSM), robust TSM (RTSM), modified ILP (MILP), and three-step method
(ThSM) are completely feasible, but are not completely optimal. In these methods,
the optimality condition has not been considered. The solutions of improved MILP
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Yugoslav Journal of Operations Research
28 (2018), Number 4, 435-451
DOI: https://doi.org/10.2298/YJOR180117020A
AN IMPROVED THREE-STEP METHOD FOR
SOLVING THE INTERVAL LINEAR
PROGRAMMING PROBLEMS
Mehdi ALLAHDADI
Mathematics Faculty, University of Sistan and Baluchestan, Zahedan, Iran
m allahdadi@math.usb.ac.ir
Chongyang DENG
School of Science, Hangzhou Dianzi University, Hangzhou, PR China
dcy@hdu.edu.cn
Received: January 2018 / Accepted: July 2018
Abstract: Feasibility condition, which ensures that the solution space does not violate
any constraints, and optimality condition, which guarantees that all points of the solu-
tion space are optimal, are very significant conditions for the solution space of interval
linear programming (ILP) problems. Among the existing methods for ILP problems,
the best-worst cases (BWC) method and two-step method (TSM) do not ensure feasi-
bility condition, while the modified ILP (MILP), robust TSM (RTSM), improved TSM
(ITSM), and three-step method (ThSM) guarantee feasibility condition, whose solution
spaces may not be completely optimal. We propose an improved ThSM (IThSM) for ILP
problems, which ensures both feasibility and optimality conditions, i.e., we introduce an
extra step to optimality.
Keywords: Feasibility, Interval Linear Programming, Optimality, Robust two-step Method,
Three-step Method.
MSC: 65G40, 80M50, 90C05.
1. INTRODUCTION
Interval linear programming (ILP) is used to deal with uncertainties of many
real-world problems, where parameters may be specified as lying between lower
and upper bounds. There are several methods for solving ILP models, which are
divided into two sub-models to obtain the solution set [2, 3, 4, 10, 12, 14, 24, 26,
27, 28].
436 M. Allahdadi and C. Deng / An improved ThSM for solving the ILP problems
Also, there are some methods for ILP models. Tong [26] proposed the best and
worst cases (BWC) method by converting the ILP model into two sub-models,
the best and worst sub-models which have the largest and the smallest feasible
regions, respectively. This method has been extended by Chinneck and Ramadan
to ILP models with equality constraints [6]. A novel ILP method was proposed by
Huang and Moore [14]. Huang et al. proposed a two-step method (TSM) through
analyzing relations between variables and coefficients [11].
For ILP model, a given point is feasible if it satisfies the constraints of the
best problem. The solutions given by the BWC, ILP, and TSM may stray into
infeasible spaces. The solutions of BWC may appear in an infeasible area though
it introduces exact bounds for the values of the objective function. To achieve the
feasibility, the modified ILP method (MILP) [28], three-step method (ThSM) [13],
robust TSM (RTSM) [7] and improved TSM (ITSM) [27] were proposed. However,
the solutions resulting from the above methods may not be optimal. To our best
knowledge, only the solutions of IILP and IMILP methods proposed by Allahdadi
et al. are completely feasible and optimal [3].
Some researchers also focused on basis stability [9, 17, 20, 25], with which
it is possible to obtain the optimal solution set of ILP. Some have used interval
arithmetic and extensions of the simplex algorithm [5, 15, 16, 19].
In the ThSM, only the feasibility condition is considered. In such a way, the
solution might not be optimal in a part of the solution space. So, although the
basis stability condition can be disregarded, it is crucial to obtain a feasible and
optimal solution set of ILP problems for the whole space of solution.
1.1. Background knowledge
An interval number x± is represented as [x−, x+], where x− ≤ x+. If x− = x+,
then x± is degenerate. x± ≥ 0 if and only if x− ≥ 0. In addition, x± ≤ 0 if and
only if x+ ≤ 0. If A− and A+ are two matrices in Rm×n such that A− ≤ A+, then
the set of matrices
A± = [A−, A+] = {A| A− ≤ A ≤ A+}
is called an interval matrix, and the matrices A− and A+ are called its bounds.
Center and radius matrices are defined as
Ac =
1
2
(A+ +A−) , ∆A± =
1
2
(A+ −A−).
A square interval matrix A± is said to be regular if each A ∈ A± is non-singular.
A special case of an interval matrix is an interval vector x± = {x| x− ≤ x ≤ x+},
where x−,x+ ∈ Rn. Interval arithmetic has been studied in [1].
Consider the following ILP:
max z± =
∑n
j=1 c
±
j x
±
j
s.t.
∑n
j=1 a
±
ijx
±
j ≤ b±i , i = 1, 2, ...,m
x±j ≥ 0, j = 1, 2, ..., n.
(1)
M. Allahdadi and C. Deng / An improved ThSM for solving the ILP problems 437
A characteristic model of ILP problem (1) is
max z =
∑n
j=1 cjxj
s.t.
∑n
j=1 aijxj ≤ bi, i = 1, 2, ...,m
xj ≥ 0, j = 1, 2, ..., n,
(2)
where aij ∈ a±ij , cj ∈ c±j , and bi ∈ b±i .
The magnitude and mignitude of an interval number x± are defined as follows[8]:
mig(x±) =
{
x− x± ≥ 0
−x+ x± < 0 , mag(x
±) =
{
x+ x± ≥ 0
−x− x± < 0,
and sign(x±) is defined as follows:
sign(x±) =
{
1 x± ≥ 0
−1 x± < 0.
We use the notation |x±|− and |x±|+ to denote mig(x±) and mag(x±), respec-
tively.
The feasible solution set of the ILP is defined as {x ∈ Rn : ∑nj=1 a−ijxj ≤ b+i , xj ≥
0, i = 1, ...,m, j = 1, ..., n}.
Moreover, the optimal solution set of the ILP is defined as the set of all optimal
solutions over all characteristic models.
We first review some methods for solving ILP problems, and then discuss the
concept of stability. In the following, by considering the basis stability, we propose
a new method to obtain solution space that is both feasible and optimal.
1.2. Related works
There are only a few solution methods for solving the general ILP problems.
The fundamental idea of these methods is to divide an ILP model into two extreme
LP sub-models to obtain interval solutions. One LP sub-model is solved to obtain
the best optimal value, and the other is for the worst optimal value of the objective
function.
In this section, we review some methods for solving ILP model (1), see [3, 7, 11,
26, 27]. Suppose that c±j ≥ 0 for j = 1, ..., k, and c±j ≤ 0 for j = k+ 1, ..., n. Also,
a±ij ≥ 0 for j = 1, ..., li1; j = li2 + 1, ..., n and a±ij ≤ 0 for j = li1 + 1, ..., li2, where
li1 ≤ k, and li2 ≥ k. In the ITSM, we have
x′j =
x−j sign(a
±
ij) 6= sign(c±j ), j = 1, 2, ..., k
x+j opt sign(a
±
ij) = sign(c
±
j ), j = 1, 2, ..., k
x+j sign(a
±
ij) 6= sign(c±j ), j = k + 1, ..., n
x−j opt sign(a
±
ij) = sign(c
±
j ), j = k + 1, ..., n
The results are given in Table 1. Note that in each method, x+j opt for j = 1, 2, ..., k
and x−j opt for j = k + 1, ..., n are the optimal solutions of sub-model 1 of that
method.
438 M. Allahdadi and C. Deng / An improved ThSM for solving the ILP problems
In the IILP and IMILP methods, δ is the number of active constraints in sub-
model 1 for the optimal solution, as well as a±δj ≤ 0 for j = k − p + 1, ..., k and
a±δj ≥ 0 for j = n− q + 1, ..., n.
Since we want to improve the three-step method (ThSM), then let us first review
the ThSM. According to [13], the ThSM is based on the TSM solutions. The steps
of this method are as follows [13]:
Step 1. Obtain the solutions by the TSM to be x±opt.
Step 2. Test feasibility condition to analyze whether the TSM solutions can satisfy
the constraints of the best problem in the BWC method.
Step 3. Constrict the solutions to eliminate the infeasible solutions.
Let y±j = [x
c
j − q∆x±j , x
c
j + q∆x±j
] for j = 1, ..., n be the ThSM solutions. Two
constricting approaches are as follows:
• ThSM-I
max q
s.t.
∑
j∈B1 a
−
ijq∆x±j
−∑j∈B2 a−ijq∆x±j ≤ b+i −∑nj=1 a−ijxcj , i = 1, ...,m
0 ≤ q ≤ 1
(3)
• ThSM-II
max q1 × q2 × ...× qn0
s.t.
∑
j∈B1 a
−
ijqj∆x±j
−∑j∈B2 a−ijqj∆x±j ≤ b+i −∑nj=1 a−ijxcj , i = 1, ...,m
0 ≤ q ≤ 1,
(4)
where n0 is the number of interval solutions obtained by the sub-models of
the TSM and B1 = {j : a±ij ≥ 0} and B2 = {j : a±ij < 0}.
y±j is the shrunk interval of x
±
j opt, and the constricting rate depends on the value
of qj .
2. BASIS STABILITY (B-STABILITY)
In this section, we define the concept of basis stability (B-stability), which can
be used to obtain the optimal solution set of the ILP model [9, 17].
Definition 1. The problem max{cTx : Ax = b , x ≥ 0}, where c ∈ c± ⊆ Rn,
A ∈ A± ⊆ Rm×n, and b ∈ b± ⊆ Rm, is called B-stable with basis B if B is an
optimal basis for each characteristic problem. The ILP problem is called [unique]
non-degenerate B-stable if each characteristic model has a [unique] non-degenerate
optimal basic solution with the basis B.
M. Allahdadi and C. Deng / An improved ThSM for solving the ILP problems 439
Table 1: Sub-models of the methods for ILP model (1)
Method Sub-models
Sub-model 1 (the best problem)
max z+ =
∑n
j=1 c
+
j xj
s.t.
∑n
j=1 a
−
ijxj ≤ b+i , i = 1, ...,m, xj ≥ 0, j = 1, ..., n.
BWC Sub-model 2 (the worst problem)
max z− =
∑n
j=1 c
−
j xj
s.t.
∑n
j=1 a
+
ijxj ≤ b−i , i = 1, ...,m, xj ≥ 0, j = 1, ..., n.
Sub-model 1
max z+ =
∑k
j=1 c
+
j x
+
j +
∑n
j=k+1 c
+
j x
−
j
s.t.
∑k
j=1 sign(a
±
ij)|a±ij |−x+j +
∑n
j=k+1 sign(a
±
ij)|a±ij |+x−j ≤ b+i , i = 1, ...,m,
x±j ≥ 0, j = 1, ..., n.
TSM Sub-model 2
max z− =
∑k
j=1 c
−
j x
−
j +
∑n
j=k+1 c
−
j x
+
j
s.t.
∑k
j=1 sign(a
±
ij)|a±ij |+x−j +
∑n
j=k+1 sign(a
±
ij)|a±ij |−x+j ≤ b−i , i = 1, ...,m,
x−j ≤ x+j opt, j = 1, ..., k, x+j ≥ x−j opt, j = k + 1, ..., n, x±j ≥ 0, j = 1, ..., n.
Sub-model 1 same as that of the TSM
Sub-model 2
max z− =
∑k
j=1 c
−
j x
−
j +
∑n
j=k+1 c
−
j x
+
j
s.t.
∑k
j=1 sign(a
±
ij)|a±ij |+x−j +
∑n
j=k+1 sign(a
±
ij)|a±ij |−x+j ≤ b−i , i = 1, ...,m,
ITSM
∑n
j=1 a
−
ijx
′
j ≤ b+i
x−j ≤ x+j opt, j = 1, ..., k, x+j ≥ x−j opt, j = k + 1, ..., n, x±j ≥ 0, j = 1, ..., n.
Sub-model 1
max z− =
∑k
j=1 c
−
j x
−
j +
∑n
j=k+1 c
−
j x
+
j
s.t.
∑k
j=1 sign(a
±
ij)|a±ij |+x−j +
∑n
j=k+1 sign(a
±
ij)|a±ij |−x+j ≤ b−i , i = 1, ...,m,
x±j ≥ 0, j = 1, ..., n.
RTSM Sub-model 2
max z+ =
∑k
j=1 c
+
j x
+
j +
∑n
j=k+1 c
+
j x
−
j
s.t.
∑k
j=1 sign(a
±
ij)|a±ij |−x+j +
∑n
j=k+1 sign(a
±
ij)|a±ij |+x−j ≤ b+i , i = 1, ...,m,∑li1
j=1 a
−
ijx
+
j +
∑k
j=li1+1
a−ijx
−
j opt +
∑li2
j=k+1 a
−
ijx
−
j +
∑n
j=li2+1
a−ijx
+
j opt ≤ b+i
x+j ≥ x−j opt, j = 1, .., k, x−j ≤ x+j opt, j = k + 1, ..., n, x±j ≥ 0, j = 1, ..., n.
Sub-model 1
max z− =
∑k
j=1 c
−
j x
−
j +
∑n
j=k+1 c
−
j x
+
j
s.t.
∑k
j=1 sign(a
±
ij)|a±ij |+x−j +
∑n
j=k+1 sign(a
±
ij)|a±ij |−x+j ≤ b−i , i = 1, ...,m,
x±j ≥ 0, j = 1, ..., n.
IILP Sub-model 2
max z+ =
∑k
j=1 c
+
j x
+
j +
∑n
j=k+1 c
+
j x
−
j
s.t.
∑k
j=1 sign(a
±
ij)|a±ij |−x+j +
∑n
j=k+1 sign(a
±
ij)|a±ij |+x−j ≤ b+i , i = 1, ...,m,
x+j ≥ x−j opt, j = 1, ..., k, x−j ≤ x+j opt, j = k + 1, ..., n, x±j ≥ 0, j = 1, ..., n,∑k
j=k−p+1−(|a±δj |−x+j − |a±δj |+x−j opt) +
∑n
j=n−q+1(|a±δj |+x−j − |a±δj |−x+j opt) ≥ 0.
Sub-model 1 same as that of the TSM
Sub-model 2
min z− =
∑k
j=1 c
−
j x
−
j +
∑n
j=k+1 c
−
j x
+
j
IMILP s.t.
∑k
j=1 sign(a
±
ij)|a±ij |+x−j +
∑n
j=k+1 sign(a
±
ij)|a±ij |−x+j ≥ b−i , i = 1, ...,m,
x−j ≤ x+j opt, j = 1, ..., k, x+j ≥ x−j opt, j = k + 1, ..., n, x±j ≥ 0, j = 1, ..., n,∑k
j=k−p+1−(|a±δj |+x−j − |a±δj |−x+j opt) +
∑n
j=n−q+1(|a±δj |−x+j − |a±δj |+x−j opt) ≤ 0.
440 M. Allahdadi and C. Deng / An improved ThSM for solving the ILP problems
Let B ⊆ {1, 2, ..., n} be an index set such that AB (the restriction of A to the
columns indexed by B) is regular, (i.e., each real matrix in AB is non-singular).
Similarly, N = {1, 2, ..., n} \ B denotes the set of non-basic variables and AN is
its restriction to non-basic indices. B can be computed by solving an arbitrary
characteristic model of the ILP. We now review the conditions for B-stability[9].
• Regularity: AB is regular.
• Feasibility: The solution set of the interval system ABxB = b is non-
negative.
• Optimality: AB is optimal, i.e., cTBA−1B AN − cTN ≥ 0T .
Theorem 2. [21] If ρ(|(AcB)−1|∆AB ) < 1, then AB is regular, where ρ(.) denotes
the spectral radius.
Theorem 3. [21] If max1≤i≤n(|(AcB)−1|∆AB )ii ≥ 1, then AB is not regular.
Some conditions for the regularity of interval matrices have been proposed in[23].
Theorem 4. [22] The interval vector r is an enclosure to the solution set of a
system ABxB = b, where:
r−i = min{−x∗i + (xci + |xci |)Mii,
1
2Mii − 1(−x
∗
i + (x
c
i + |xci |)Mii)},
r+i = max{x∗i + (xci − |xci |)Mii,
1
2Mii − 1(x
∗
i + (x
c
i − |xci |)Mii)},
M = (I − |(AcB)−1|∆AB )−1, xc = (AcB)−1bc,
x∗ = M(|xc|+ |(AcB)−1|∆b),
AcB is non-singular, and ρ(|(AcB)−1|∆AB ) < 1.
Theorem 5. [9] Let y be an enclosure to the solution set of ATBy = cB. If
((ATN )y)
− ≥ c+N , then the optimality condition is valid.
Theorem 6. [9] Let diag(q) denote the diagonal matrix with entries q1, ..., qm.
For each q ∈ {±1}m, if the solution set of the system ((A
c
B)
T − (∆AB )T diag(q))y ≤ c+B
−((AcB)T + (∆AB )T diag(q))y ≤ −c−B
diag(q)y ≥ 0
lies in the solution set of the system{
((AcN )
T − (∆AN )T diag(q))y ≥ c+N
diag(q)y ≥ 0,
then, the optimality condition is valid.
M. Allahdadi and C. Deng / An improved ThSM for solving the ILP problems 441
Remark 6.1. Since there are no dependencies, the set of constraints Ax ≤ b,x ≥
0 is equivalent to Ax + Iy = b,x,y ≥ 0.
Theorem 7. [9] Let ILP model (1) be unique non-degenerate B-stable where B =
(1, 2, ...,m). The optimal solution set of the ILP model is equal to the solution
set of the interval system ABxB = b,xB ≥ 0,xN = 0, or A−BxB ≤ b+, A+BxB ≥
b−,xB ≥ 0,xN = 0.
Consider the model, i.e. an example given by [13]:
max z± = [2, 2.4]x±1 − [1, 1.3]x±2 + [1.5, 1.8]x±3
s.t. [2.6, 3.5]x±1 + [2, 2.4]x
±
2 + [3.2, 3.8]x
±
3 ≤ [18, 22]
[4.6, 5.5]x±1 + [3, 3.6]x
±
2 − [1.3, 1.6]x±3 ≤ [8, 9]
[1, 1.3]x±1 − [6, 6.5]x±2 + [2, 2.5]x±3 ≤ [2.2, 2.6]
x±1 , x
±
2 , x
±
3 ≥ 0.
(5)
To obtain the optimal solution set, we first check the basis stability conditions.
Based on Remark 6.1, model (5) is equivalent to the following model:
max z± = [2, 2.4]x±1 − [1, 1.3]x±2 + [1.5, 1.8]x±3
s.t. [2.6, 3.5]x±1 + [2, 2.4]x
±
2 + [3.2, 3.8]x
±
3 + x
±
4 = [18, 22]
[4.6, 5.5]x±1 + [3, 3.6]x
±
2 − [1.3, 1.6]x±3 + x±5 = [8, 9]
[1, 1.3]x±1 − [6, 6.5]x±2 + [2, 2.5]x±3 + x±6 = [2.2, 2.6]
x±1 , x
±
2 , x
±
3 , x
±
4 , x
±
5 , x
±
6 ≥ 0.
A candidate basis for B-stability is B = (1, 2, 3), since this is optimal for the
characteristic model with center values.
1. According to Theorem 2, the spectral radius is equal to 0.24, so AB is regular:
AB =
[2.6, 3.5] [2, 2.4] [3.2, 3.8][4.6, 5.5] [3, 3.6] −[1.3, 1.6]
[1, 1.3] −[6, 6.5] [2, 2.5]
.
2. According to Theorem 4, we can compute the enclosure of the solution set
of ABxB = b to be xB =
(
[1.32, 2.57], [0.94, 1.40], [2.72, 4.03]
)t
, which is non-
negative.
3. Using Theorem 4 again, we can compute the enclosure of the solution set
of ATBy = cB to be y =
(
[0.19, 0.43], [0.13, 0.21], [0.36, 0.39]
)t
. According to
Theorem 5, since ((ATN )y)
− ≥ c+N , the optimality condition holds.
Thus the problem is B-stable. Therefore in view of Theorem 7, the optimal solution
set of ILP model (5) is the solution set of the following inequalities:
2.6x1 + 2x2 + 3.2x3 ≤ 22, (6)
4.6x1 + 3x2 − 1.6x3 ≤ 9, (7)
x1 − 6.5x2 + 2x3 ≤ 2.6, (8)
3.5x1 + 2.4x2 + 3.8x3 ≥ 18, (9)
5.5x1 + 3.6x2 − 1.3x3 ≥ 8, (10)
1.3x1 − 6x2 + 2.5x3 ≥ 2.2 (11)
442 M. Allahdadi and C. Deng / An improved ThSM for solving the ILP problems
Note that the inequalities (6)-(8) and (9)-(11) are the feasibility and optimality
conditions, respectively.
3. THE IThSM
Suppose that x±opt is the solution obtained by the TSM. To guarantee that the
given solutions are completely feasible, we use the ThSM approach proposed by
Huang et al. [13] (Models (3) and (4)). To ensure that all points of the ThSM
solutions are optimal, the following approach of optimality test is developed. In
view of Theorem 7, if
∑n
j=1 a
+
ijxj opt ≥ b−i is tenable, then all solutions are optimal,
which means that x±opt pass the optimality test. Otherwise, we eliminate the non-
optimal solutions by the following approach.
Assume that y±opt is the IThSM solution, where y
±
j = [x
c
j opt − qj∆x±j , x
c
j opt +
qj∆x±j
] for j = 1, ..., n. To optimality, Theorem 7 implies that it is sufficient that
n∑
j=1
a+ijyj ≥ b−i ,
or ∑
j∈B1
a+ijy
−
j +
∑
j∈B2
a+ijy
+
j ≥ b−i ,
where B1 = {j : a±ij ≥ 0} and B2 = {j : a±ij < 0}. Therefore, we have:∑
j∈B1
a+ij(x
c
j opt − qj∆x±j ) +
∑
j∈B2
a+ij(x
c
j opt + qj∆x±j
) ≥ b−i .
Thus, to optimality, it is sufficient that
−
∑
j∈B1
a+ijqj∆x±j opt
+
∑
j∈B2
a+ijqj∆x±j opt
≥ b−i −
n∑
j=1
a+ijx
c
j opt.
Suppose that for each j = 1, 2, ..., n0, qj = q. Then, we have:
−
∑
j∈B1
a+ijq∆x±j opt
+
∑
j∈B2
a+ijq∆x±j opt
≥ b−i −
n∑
j=1
a+ijx
c
j opt.
Hence, the models for solving q and qj(j = 1, 2, ..., n), which are named IThSM-I
and IThSM-II, can be presented as follows:
• IThSM-I
max q
s.t.∑pi
j=1 a
−
ijq∆x±j opt
−∑n0j=pi+1 a−ijq∆x±j opt ≤ b+i −∑nj=1 a−ijxcj opt, i = 1, ...,m
−∑j∈B1 a+ijq∆x±j opt +∑j∈B2 a+ijq∆x±j opt ≥ b−i −∑nj=1 a+ijxcj opt, i = 1, ...,m
0 ≤ q ≤ 1,
(12)
M. Allahdadi and C. Deng / An improved ThSM for solving the ILP problems 443
• IThSM-II
max q1 × q2 × ...× qn0
s.t.∑pi
j=1 a
−
ijqj∆x±j opt
−∑n0j=pi+1 a−ijqj∆x±j opt ≤ b+i −∑nj=1 a−ijxcj opt, i = 1, ...,m
−∑j∈B1 a+ijqj∆x±j opt +∑j∈B2 a+ijqj∆x±j opt ≥ b−i −∑nj=1 a+ijxcj opt, i = 1, ...,m
0 ≤ q ≤ 1.
(13)
Remark 8. The IThSM consisted of the following steps:
Step 1. Obtain solutions of ILP model (1) by the TSM to be x±opt.
Step 2. If
∑n
j=1 a
−
ijxj opt ≤ b+i is tenable, then x±opt pass the feasibility test; oth-
erwise, it means that a part of solutions will be infeasible. Suppose that the ILP
model is B-stable. If
∑n
j=1 a
+
ijxj opt ≥ b−i is tenable, then x±opt pass the optimality
test; otherwise, it means that a part of solutions will not be optimal.
Step 3. Eliminate the infeasible and non-optimal solutions by models (12) or (13).
Therefore, the IThSM-I and IThSM-II solutions are y±j opt = [x
c
j opt−q∆x±j opt , x
c
j opt+
q∆x±j opt
] and y±j opt = [x
c
j opt − qj∆x±j opt , x
c
j opt + qj∆x±j opt
], respectively.
The optimal value of the objective function can be obtained as follows:
z−opt =
k∑
j=1
c−j y
−
j opt +
n∑
j=k+1
c−j y
+
j opt,
z+opt =
k∑
j=1
c+j y
+
j opt +
n∑
j=k+1
c+j y
−
j opt.
Note that similar to the IILP and IMILP, the IThSM obtains a solution space that
is both feasible and optimal. In fact, the IThSM gives a new feasible and optimal
solution space. Therefore, the union of the solution spaces obtained by which of
the mentioned methods will be closer to the exact optimal solution space of the
ILP model.
4. PROPERTIES OF IThSM
Let us first analyze the solving methods based on feasibility and optimality
conditions.
Consider ILP model (5). Since the ILP model is B-stable, in view of Theorem 7,
then the optimal solution set of the ILP model is equal to the solution set of the
system A−BxB ≤ b+, A+BxB ≥ b−,xB ≥ 0,xN = 0, where
AB =
[2.6, 3.5] [2, 2.4] [3.2, 3.8][4.6, 5.5] [3, 3.6] −[1.3, 1.6]
[1, 1.3] −[6, 6.5] [2, 2.5]
,
444 M. Allahdadi and C. Deng / An improved ThSM for solving the ILP problems
Table 2: Results for ILP model (5)
Method x±opt z
±
BWC ([1.40, 2.55], [1.09, 1.23], [2.76, 4.03])t [5.52, 12.15]
TSM ([1.56, 2.18], 1.22, [2.66, 4.18])t [5.51, 11.55]
ITSM ([1.26, 2.18], 1.22, [2.94, 4.18])t [5.33, 11.55]
RTSM ([1.63, 2.17], 1.09, [2.66, 3.76])t [5.83, 10.88]
ThSM − I ([1.61, 2.13], 1.22, [2.78, 4.06])t [5.80, 11.20]
ThSM − II ([1.63, 2.11], 1.22, [2.73, 4.11])t [5.77, 11.24]
IILP ([1.63, 2.17], 1.09, [2.66, 3.76])t [5.83, 10.88]
IMILP ([1.65, 2.18], 1.22, [2.95, 4.18])t [6.14, 11.55]
Table 3: Feasibility and optimality of the methods for ILP model (5)
Method Feasibility Optimality
BWC × ×
TSM × ×
ITSM X ×
RTSM X X
ThSM − I X ×
ThSM − II X ×
IILP X X
IMILP X X
or the solution set of inequalities (6)-(11).
We now solve ILP model (5). The results are given in Table 2.
Some points given by the BWC are either infeasible or non-optimal. For exam-
ple, point (2.55, 1.23, 2.76)t which does not satisfy (7), is not feasible, and point
(1.4, 1.09, 4.03)t which does not satisfy (10), is not optimal. In the TSM, since
points (2.18, 1.22, 4.18)t and (1.56, 1.22, 4.18)t do not satisfy (8) and (10), respec-
tively, then these points are infeasible and non-optimal, respectively. The solutions
obtained by ITSM are completely feasible, but are not completely optimal. For
example, point (1.26, 1.22, 4.18)t is non-optimal because it does not satisfy (10).
The solutions given by the RTSM, IILP, and IMILP are both feasible and optimal.
Some solutions obtained through ThSM-I and ThSM-II are non-optimal, too. For
example, point (1.61, 1.22, 2.78)t of ThSM-I and (1.63, 1.22, 2.73)t of ThSM-II do
not satisfy (11), therefore these points are non-optimal. The feasibility and opti-
mality for the methods are summarized in Table 3. In the following, we propose
an improved ThSM by considering the feasibility and optimality conditions. In
the ThSM, only the feasibility condition is considered [13]. In such a way, a part
of the solution space may not be optimal. To remove this non-optimal part, by
considering an optimality test, we propose an improved ThSM, namely IThSM.
M. Allahdadi and C. Deng / An improved ThSM for solving the ILP problems 445
5. NUMERICAL ANALYSIS AND COMPARISONS
In this section, two ILP models are solved by using the mentioned methods
and the results are then compared.
Example 9. Consider ILP model (5), again. We solve the ILP model by the
IThSM.
Step 1. The TSM solution is x±opt = ([1.56, 2.18], 1.22, [2.66, 4.18])
t.
Step 2. The ILP model is B-stable, therefore based on inequalities (6)-(11) the
following inequalities should be tested:
2.6x+1 opt + 2x
+
2 opt + 3.2x
+
3 opt ≤ 22, (14)
4.6x1 opt + 3x2 opt − 1.6x−3 opt ≤ 9, (15)
x+1 opt − 6.5x−2 opt + 2x+3 opt ≤ 2.6, (16)
3.5x−1 opt + 2.4x
−
2 opt + 3.8x
−
3 opt ≥ 18, (17)
5.5x−1 opt + 3.6x
−
2 opt − 1.3x+3 opt ≥ 8, (18)
1.3x−1 opt − 6x+2 opt + 2.5x−3 opt ≥ 2.2. (19)
Since the TSM solutions do not satisfy inequalities (15) and (18), then a part of
the solution space obtained by the TSM is not feasible and optimal.
Step 3. For the TSM solutions, we have: ∆x±opt
= (1.87, 1.22, 3.42)t and x±
c
opt =
(0.31, 0, 0.76)t. Suppose that y± is the solution of the IThSM, where
y± = ([1.87− 0.31q1, 1.87 + 0.31q1], 1.22, [3.42− 0.76q3, 3.42 + 0.76q3])t.
According to model (12), we have:
max q
s.t.
2.6× 0.31q + 3.2× 0.76q ≤ 22− (2.6× 1.87 + 2× 1.22 + 3.2× 3.42)
4.6× 0.31q − (−1.6)× 0.76q ≤ 9− (4.6× 1.87 + 3× 1.22− 1.6× 3.42)
1× 0.31q + 2× 0.76q ≤ 2.6− (1× 1.87− 6.5× 1.22 + 2× 3.42)
−3.5× 0.31q − 3.8× 0.76q ≥ 18− (3.5× 1.87 + 2.4× 3.8 + 3.8× 3.42)
−5.5× 0.31q − 1.3× 0.76q ≥ 8− (5.5× 1.87 + 3.6× 3.8− 1.3× 3.42)
−1.3× 0.31q − 2.5× 0.76q ≥ 2.2− (1.3× 1.87− 6× 2.5 + 3.8× 3.42)
0 ≤ q ≤ 1
The solution is q = 0.63, and hence y±opt = ([1.67, 2.07], 1.22, [2.94, 3.90])
t and
z±opt = [6.16, 10.77].
446 M. Allahdadi and C. Deng / An improved ThSM for solving the ILP problems
According to model (13), we have:
max q1q3
s.t.
2.6× 0.31q1 + 3.2× 0.76q3 ≤ 22− (2.6× 1.87 + 2× 1.22 + 3.2× 3.42)
4.6× 0.31q1 − (−1.6)× 0.76q3 ≤ 9− (4.6× 1.87 + 3× 1.22− 1.6× 3.42)
1× 0.31q1 + 2× 0.76q3 ≤ 2.6− (1× 1.87− 6.5× 1.22 + 2× 3.42)
−3.5× 0.31q1 − 3.8× 0.76q3 ≥ 18− (3.5× 1.87 + 2.4× 3.8 + 3.8× 3.42)
−5.5× 0.31q1 − 1.3× 0.76q3 ≥ 8− (5.5× 1.87 + 3.6× 3.8− 1.3× 3.42)
−1.3× 0.31q1 − 2.5× 0.76q3 ≥ 2.2− (1.3× 1.87− 6× 2.5 + 3.8× 3.42)
0 ≤ q ≤ 1
The solutions are q1 = 0.98, q3 = 0.56, and hence
y±opt = ([1.57, 2.17], 1.22, [2.99, 3.85])
t,
and z±opt = [6.04, 10.92].
The interval value of the objective function obtained by IThSM-II (i.e. [6.04, 10.92])
is wider than that of IThSM-I (i.e. [6.16, 10.77]). Since the value of z+opt given by
IThSM-II is larger than that of IThSM-I, then IThSM-II is preferred (Note that the
objective is to maximize the objective function). Based on the constraints of models
(12) and (13), the solutions obtained by IThSM are both feasible and optimal. The
interval width of x±1 opt given by IThSM-II is larger than that of IThSM-I, whereas
that of x±3 opt is converse.
Example 10. Consider the model, i.e., an example given by [13]:
max z± = [3, 3.5]x±1 − [1, 1.2]x±2
s.t. [1, 1.1]x±1 + [1.6, 1.8]x
±
2 ≤ [11.6, 12]
[3, 4]x±1 − [2, 3]x±2 ≤ [5, 7]
x±1 , x
±
2 ≥ 0.
(20)
Before using the IThSM, we first check the basis stability of ILP model (20).
Based on Remark 6.1, model (20) is equivalent to the following model:
max z± = [3, 3.5]x±1 − [1, 1.2]x±2
s.t. [1, 1.1]x±1 + [1.6, 1.8]x
±
2 + x
±
3 = [11.6, 12]
[3, 4]x±1 − [2, 3]x±2 + x±4 = [5, 7]
x±1 , x
±
2 , x
±
3 , x
±
4 ≥ 0.
A candidate basis for B-stability is B = (1, 2), since this is optimal for the char-
acteristic model with center values.
1. According to Theorem 2, the spectral radius is equal to 0.21, so AB is regular:
AB =
(
[1, 1.1] [1.6, 1.8]
[3, 4] [−3,−2]
)
.
M. Allahdadi and C. Deng / An improved ThSM for solving the ILP problems 447
Figure 1: Exact solution space of ILP model (20)
Figure 2: Solution spaces obtained by BWC, TSM, ITSM, RTSM, IILP, and IMILP for ILP
model (20)
2. According to Theorem 4, we can compute the enclosure of the solution set of
ABxB = b to be xB =
(
[3.42, 6.16], [3.72, 4.46]
)T
, which is non-negative.
3. Using Theorem 4 again, we can compute the enclosure of the solution set of
ATBy = cB to be y =
(
[0.19, 1.16], [0.81, 0.95]
)T
. According to Theorem 5, since
((ATN )y)
− ≥ c+N , the optimality condition holds.
Thus the problem is B-stable, and hence in view of Theorem 7, the optimal solution
set of ILP model (20) is the solution set of the following inequalities system, which
has been shown in Fig. 1.
{
x1 + 1.6x2 ≤ 12 , 1.1x1 + 1.8x2 ≥ 11.6
3x1 − 3x2 ≤ 7 , 4x1 − 2x2 ≥ 5.
We solve model (20) by using the BWC, TSM, ITSM, RTSM, IILP, IMILP,
ThSM, and IThSM. The results are given in Table 4 and Fig. 2. The feasibility and
optimality for the methods are summarized in Table 5. The solution space obtained
through BWC and TSM is not completely feasible and optimal. The ITSM and
RTSM solutions are completely feasible, but a part of solution spaces is non-optimal
448 M. Allahdadi and C. Deng / An improved ThSM for solving the ILP problems
Figure 3: Non-optimal space of ITSM for ILP model (20)
Figure 4: Non-optimal space of RTSM for ILP model (20)
Figure 5: Solution space obtained by ThSM for ILP model (20)
Figure 6: Solution space obtained by IThSM for ILP model (20)
M. Allahdadi and C. Deng / An improved ThSM for solving the ILP problems 449
Table 4: Results for ILP model (20)
Method z±opt x
±
opt
BWC [5.06, 17.46]
(
[3.43, 6.05], [3.72, 4.35]
)t
TSM [5.18, 16.80]
(
[3.63, 5.79], [3.45, 4.76]
)t
ITSM [4.91, 16.80]
(
[3.19, 5.79], [3.45, 3.88]
)t
RTSM [5.18, 13.31]
(
[3.63, 4.39], [2.06, 4.76]
)t
ThSM-I [7.86, 13.86]
(
[4.35, 5.07], [3.89, 4.32]
)t
ThSM-II [7.87, 13.85]
(
[4.35, 5.07], [3.88, 4.33]
)t
IILP [5.18, 11.11]
(
[3.63, 4.38], [4.23, 4.76]
)t
IMILP [10.04, 16.80]
(
[4.90, 5.79], [3.45, 3.88]
)t
IThSM-I [7.84, 13.89]
(
[4.34, 5.08], [3.88, 4.33]
)t
IThSM-II [7.88, 13.85]
(
[4.35, 5.07], [3.88, 4.33]
)t
Table 5: Feasibility and optimality of the methods for ILP model (20)
Method Feasibility Optimality
BWC × ×
TSM × ×
ITSM X ×
RTSM X ×
ThSM-I X X
ThSM-II X X
IILP X X
IMILP X X
IThSM-I X X
IThSM-II X X
as shown in Fig. 3 and Fig. 4. The solution spaces obtained through the ThSM
and IThSM are completely feasible and optimal as shown in Fig. 5 and Fig. 6.
6. CONCLUSIONS
Some methods for solving interval linear programming (ILP) problems have
been reviewed based on feasibility and optimality conditions. Feasibility condition
is used to avoid violation of the constraints of the best problem, which has the
largest feasible space, and optimality condition to ensure that each point of the
solution space is optimal for at least one characteristic problem.
In some methods, the best and worst cases (BWC) and two-step method (TSM),
the feasibility condition was not been considered. Therefore, solution spaces given
by the BWC and TSM may be infeasible. Solutions obtained through improved
TSM (ITSM), robust TSM (RTSM), modified ILP (MILP), and three-step method
(ThSM) are completely feasible, but are not completely optimal. In these methods,
the optimality condition has not been considered. The solutions of improved MILP
450 M. Allahdadi and C. Deng / An improved ThSM for solving the ILP problems
(IILP and IMILP) are feasible and optimal.
To ensure that solutions given by ThSM are optimal, optimality test is needed. By
considering basis stability, and hence adding the optimality condition to ThSM,
we propose an improved ThSM (IThSM) which gives both feasible and optimal
solution space.
Acknowledgments: The authors would like to thank to the Editor-in-Chief and
the anonymous referees for their constructive comments and suggestions that have
helped to improve this paper.
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