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Improving the modified interval linear programming method by new techniques
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Improving the modified interval linear programming method by new techniques M. Allahdadi∗, H. Mishmast Nehi, H.A. Ashayerinasab, M. Javanmard
Q1
Mathematics Faculty, University of Sistan and Baluchestan, Zahedan, Iran
a r t i c l e
i n f o
Article history: Received 19 January 2015 Revised 1 December 2015 Accepted 25 December 2015 Available online xxx Keywords: Feasibility Interval linear programming MILP method Optimality Uncertainty
a b s t r a c t In this study, we consider interval linear programming (ILP) problems, which are used to deal with uncertainties resulting from the range of admissible values in problem coefficients. In most existing methods for solving ILP problems, a part of the solution region is not feasible. The solution set obtained through the modified ILP (MILP) method is completely feasible (i.e., it does not violate any constraints), but is not completely optimal (i.e., some points of the region are not optimal). In this paper, two new ILP methods and their sub-models are presented. These techniques improve the MILP method, giving a solution region that is not only completely feasible, but also completely optimal. © 2016 Published by Elsevier Inc.
1
1. Introduction
2
Uncertainties in many real-world problems mean that their parameters may be specified as lying between lower and upper bounds. To deal with such uncertainties, interval linear programming (ILP) is used. Researchers working on ILP problems have proposed several methods for solving ILP models [1–34]. Some have used interval arithmetic and extensions of the simplex algorithm [3,16,17,23], whereas others have focused on basis stability [9,18,25,30]. Under the assumption of basis stability, it is possible to obtain the optimal solution set of ILP. The ILP model has sometimes been divided into two sub-models to obtain the solution set [10,13,15,29,32–34]. In [2], the authors obtained the optimal solution set to an ILP problem by using the best and worst problem constraints when all components of the optimal solutions to the ILP model are positive. This assumption can be derived by solving the best problem. If all components of the feasible solution to the best problem are positive, then the feasible solution components (and hence the optimal solution components) in all of the characteristic models (and thus the ILP model) are positive. In the best and worst cases (BWC) method proposed by Tong [32], the ILP model was converted into two sub-models, the best and worst sub-models, which have the largest and smallest feasible regions, respectively. A given point is feasible for the ILP model if it satisfies the constraints of the best problem, and it is optimal for the ILP model if it is optimal for at least one characteristic model. The BWC method has been extended by Chinneck and Ramadan to ILP models with equality constraints [4]. A novel ILP method was proposed by Huang and Moore [15].
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
∗
Corresponding author. Tel.: +98 5431132517. E-mail addresses: [email protected], [email protected] (M. Allahdadi), [email protected] (H. Mishmast Nehi), hassan.ali. [email protected] (H.A. Ashayerinasab), [email protected] (M. Javanmard). http://dx.doi.org/10.1016/j.ins.2015.12.037 0020-0255/© 2016 Published by Elsevier Inc.
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The solutions given by the BWC and ILP methods may stray into infeasible regions. Although the BWC method introduces exact bounds for the values of the objective function, the solution could exist in an area that is infeasible. To guarantee that the given solution is completely feasible, Zhou et al. [34] proposed the modified ILP method (MILP), in which an extra constraint is added to the second sub-model. However, the solutions resulting from MILP may not be optimal. In this paper, we propose improved ILP (IILP) and improved MILP (IMILP) methods for solving ILP problems. The solutions to these methods are guaranteed to be completely feasible and completely optimal. The feasibility and optimality are illustrated by some numerical examples. In the following, 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 and x− ≥ 0. In addition, x± ≤ 0 if and only if x+ ≤ 0 and 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+ } 28
is called an interval matrix, and the matrices A− and A+ are called its bounds. Center and radius matrices are defined as
Ac =
1 + ( A + A− ) , 2
A ± =
1 + ( A − A− ) . 2
31
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].
32
2. Overview of ILP model solving methods
33
In this section, we review some methods for solving ILP models with inequality constraints. Models with equality constraints have also been investigated [11,29]. Consider the following ILP:
29 30
34
max s.t.
z± = n
n
c±j x±j
j=1
a± x± ij j
≤ b± , i
i = 1, 2, . . . , m
(1)
j=1
x±j ≥ 0, 35
j = 1, 2, . . ., n.
The characteristic model of the ILP model (1) is
max s.t.
z= n
n
c jx j
j=1
ai j x j ≤ bi ,
i = 1, 2, . . ., m
(2)
j=1
x j ≥ 0, 36 37 38 39 40
where ai j ∈
a± , cj ij
∈
j = 1, 2, . . ., n,
c±j ,
and bi ∈ b± . i
The feasible solution set of the ILP is defined as {x ∈ Rn : nj=1 a− x ≤ b+ , x j ≥ 0, i = 1, . . ., m, j = 1, . . ., n}. ij j i Moreover, the optimal solution set of the ILP is defined as the set of all optimal solutions over all characteristic models. According to [32], the BWC method can be used to solve model (1). The two sub-models are as follows: Sub-model 1 (the best problem).
max s.t.
z+ = n
n
c+j x j
j=1
a− x ij j
≤ b+ , i
i = 1, 2, . . ., m,
(3)
j=1
x j ≥ 0, 41
j = 1, 2, . . ., n.
Sub-model 2 (the worst problem).
max s.t.
z− = n
n
c−j x j
j=1
a+ x ij j
≤ b− , i
i = 1, 2, . . ., m,
(4)
j=1
x j ≥ 0, 42 43
j = 1, 2, . . ., n. + zopt
− and zopt are the best and worst optimal values of the objective function, respectively, then all of the Theorem 2.1 [32]. If − + optimal values of the objective function lie in [zopt , zopt ].
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44 45 46
z+ =
k
c+j x+j +
j=1
s.t.
k
sign(
n
c+j x−j
j=k+1
a± ij
)| |
− + a± xj ij
+
j=1
s.t.
z− = k
k
c−j x−j +
sign(a± )|a±i j |− x+j ≤ b−i , ij
i = 1, 2, . . ., m,
c−j x+j n j=k+1
j = 1, . . ., k, j = k + 1, . . ., n,
where c±j ≥ 0 for j = 1, . . ., k, c±j ≤ 0 for j = k + 1, . . ., n, and
sign(a± )= ij
a− ij
a± ≥0 ij
−a+ ij
a± <0 ij
1
a± ≥0 ij
−1
< 0. a± ij
|a±i j |+ =
,
a+ ij
a± ≥0 ij
−a− ij
a± <0 ij
x+j opt for j = 1, . . ., k and x−j opt for j = k + 1, . . ., n are the optimal solutions of sub-model 1. A part of the solution region obtained by the ILP method may be infeasible. To guarantee that the solution given by the ILP method is completely feasible, Zhou et al. proposed the following MILP method [34]: The first sub-model of this method is the same as that of the ILP method. The second sub-model is as follows.
max
z− =
k
c−j x−j +
j=1
s.t.
k
n
c−j x+j
j=k+1
sign(a± )|a±i j |+ x−j + ij
j=1
n
sign(a± )|a±i j |− x+j ≤ b−i , i = 1, 2, . . ., m, ij
j=k+1
x−j ≤ x+j opt ,
j = 1, . . ., k,
x+j
j = k + 1, . . ., n,
≥
x−j opt ,
k
(7)
+ − ± − + − |a± δ j | x j − |aδ j | x j opt +
n
|a±δ j |− x+j − |a±δ j |+ x−j opt ≤ 0,
j=n−q+1
j=k−p+1
x±j ≥ 0, 54
(6)
j = 1, 2, . . ., n,
0,
53
(5)
j=k+1
x+j opt , x−j opt ,
|a±i j |− =
51
n
sign(a± )|a±i j |+ x−j + ij
j=1 x−j ≤ x+j ≥ x±j ≥
52
i = 1, 2, . . ., m,
j = 1, 2, . . ., n.
j=1
50
sign(a± )|a±i j |+ x−j ≤ b+i , ij
Sub-model 2.
max
49
n j=k+1
x±j ≥ 0,
48
3
Suppose that the interval coefficients of the ILP model are either all non-negative or all non-positive. The two sub-models of the ILP method are then defined as follows [15]: Sub-model 1.
max
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j = 1, 2, . . ., n,
± where δ is the number of active constraints in model (5), and a± δ j ≥ 0 for j = 1, . . ., k − p, aδ j ≤ 0 for j = k − p + 1, . . ., k, k + ± 1, . . ., n − q, and aδ j ≥ 0 for j = n − q + 1, . . ., n.
56
Though the MILP method guarantees that the solution region is completely feasible, the following example shows that a part of the solution region may be non-optimal.
57
Example 2.1. Consider the example given by [34]:
55
max
z± = [26, 30]x± − [5.5, 6]x± 1 2
s.t.
[8, 10]x± − [12, 14]x± ≤ [3.8, 4.2] 1 2 [1, 1.1]x± + [0.19, 0.2]x± ≤ [6.5, 7] 1 2
(8)
x± , x± ≥ 0. 1 2 58 59
To determine the exact optimal solution set of ILP model (8), we solve the characteristic models for most values of ai j ∈ a± , c j ∈ c±j , and bi ∈ b± for i, j = 1, 2. This gives their optimal solutions. The region obtained through the optimal solution ij i Please cite this article as: M. Allahdadi et al., Improving the modified interval linear programming method by new techniques, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2015.12.037
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Fig. 1. Approximation of the optimal solution region for ILP model (8).
Fig. 2. Optimal solution set resulting from the BWC, ILP, and MILP methods for ILP model (8). Table 1 Solutions of BWC, ILP, and MILP methods for ILP model (8).
60 61 62
Method
x± 1
x± 2
z±
BWC ILP MILP
[5.182, 6.366] [5.213, 6.336] [4.574, 6.336]
[3.338, 4.001] [3.320, 4.028] [3.320, 3.495]
[110.71, 172.62] [111.38, 171.81] [97.96, 171.81]
points is shown in Fig. 1. The results obtained from the BWC, ILP, and MILP methods are given in Table 1 and Fig. 2. The value of z− pertaining to MILP (97.96) is less than that obtained by BWC (110.71). In other words, the optimal value of the − objective function in the second sub-model of MILP is less than zopt , and thus this result is not optimal. However, according Please cite this article as: M. Allahdadi et al., Improving the modified interval linear programming method by new techniques, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2015.12.037
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5
66
− to Theorem 2.1, zopt given by BWC is the worst value of the objective function. Obviously, many points resulting from the MILP method are not optimal. In the following, we first discuss the concept of stability, and then propose two improvements to the MILP method to obtain solution regions that are both feasible and optimal.
67
3. Basis stability
68
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 [9,18].
63 64 65
69 70 71 72 73 74 75
Definition 3.1. The problem max{cT x : Ax = b, x ≥ 0}, where c ∈ c± ⊆ Rn , A ∈ A± ⊆ Rm×n , and b ∈ b± ⊆ Rm , is called Bstable 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. Let B ⊆ {1, 2, . . . , n} be an index set such that AB (the restriction of A to the columns indexed by B) 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. We now review the conditions for B-stability [9].
76
•
77
•
78
•
Regularity: AB is regular. Feasibility: The solution set of the interval system AB xB = b is non-negative. Optimality: AB is optimal, i.e., cTB A−1 AN − cTN ≥ 0T . B
79
Theorem 3.1 [26]. If ρ (|(Ac )−1 |AB ) < 1, then AB is regular, where ρ (.) denotes the spectral radius. B
80
Theorem 3.2 [26]. If max1≤i≤n (|(Ac )−1 |AB )ii ≥ 1, then AB is not regular. B
81 82
Some conditions for the regularity of interval matrices have been proposed in [28]. Theorem 3.3 [27]. The interval vector r is an enclosure to the solution set of a system AB xB = b, where:
1 (−x∗i + (xci + |xci | )Mii ) , 2Mii − 1 1 ri+ = max x∗i + (xci − |xci | )Mii , (x∗i + (xci − |xci | )Mii ) , 2Mii − 1
ri− = min −x∗i + (xci + |xci | )Mii ,
M = (I − |(AcB )−1 |AB )−1 ,
xc = (AcB )−1 bc ,
x∗ = M (|xc | + |(AcB )−1 |b ), 83
AcB is non-singular, and ρ (|(AcB )−1 |A ) < 1.
84
, then the optimality condition holds. Theorem 3.4 [9]. Let y be an enclosure to the solution set of ATB y = cB . If ((ATN )y )− ≥ c+ N
86
Theorem 3.5 [9]. Let diag(q) denote the diagonal matrix with entries q1 , . . ., qm . For each q ∈ { ± 1}m , if the solution set of the system
87
lies in the solution set of the system
85
⎧ c T + T ⎨((AB ) − (AB ) diag(q ))y ≤ cB −((AcB )T + (AB )T diag(q ))y ≤ −c− B ⎩ diag(q )y ≥ 0
((AcN )T − (AN )T diag(q ))y ≥ c+N diag(q )y ≥ 0,
88
then the optimality condition holds.
89
Remark 3.1. Since there are no dependencies, the set of constraints Ax ≤ b, x ≥ 0 is equivalent to Ax + Iy = b, x, y ≥ 0.
90
4. Our methods
91 92
We now propose two improved methods, namely IILP and IMILP, which guarantee that the obtained solution region is completely optimal. Moreover, the obtained solution region lies in the exact optimal solution region.
93
Definition 4.1. For each 1 ≤ i ≤ m, we define Si and Ti as follows:
Si =
x:
n j=1
a+ x ij j
≥
b− i
,
Ti =
x:
n
a− x ij j
≤
b+ i
.
j=1
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Note that Ti and Si are the feasible solution sets of the best problem and the worst problem with the inverse sign, respectively.
98
Theorem 4.1 [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 the intersection of the region generated by the best problem constraints and the worst problem constraints with the inverse sign.
99
Theorem 4.2 [34]. For ILP model (1), |a± |− sign(a±i j ) is the coefficient of x+j for j = 1, 2, . . ., k and |a±i j |+ sign(a±i j ) is the coefficient ij
96 97
100 101
of x−j for j = k + 1, k + 2, . . ., n for the first sub-model z+ . In addition, |a± |+ sign(a±i j ) is the coefficient of x−j for j = 1, 2, . . ., k, ij and |a± |− sign(a±i j ) is the coefficient of x+j for j = k + 1, k + 2, . . ., n for the second sub-model z− . ij
103
Theorem 4.3 [34]. For ILP model (1), b+ and b− are the right-hand sides of the constraints for the first and second sub-models, i i respectively.
104
4.1. IILP method
105
In this subsection, we introduce two sub-models of the IILP method. First, we solve the model corresponding to z− , which is the second sub-model of the MILP method without the extra constraints. In the sub-model corresponding to z+ , we add an extra constraint that guarantees that the resulting solution region is completely optimal. According to Theorems 4.2 and 4.3, for ILP model (1), we can write the IILP constraints corresponding to z− as
102
106 107 108 109
k
sign(a± )|a±i j |+ x−j + ij
j=1
110
111
113
114
sign(a± )|a±i j |− x+j + ij
n
119
sign(a± )|a±i j |+ x−j ≤ b+i , ij
ti = 1, 2, . . ., m,
(10a)
x+j ≥ x−j opt ,
j = 1, . . ., k,
(10b)
x−j ≤ x+j opt ,
j = k + 1, . . ., n,
(10c)
x±j ≥ 0,
j = 1, 2, . . ., n.
(10d)
Theorem 4.4. Let ILP model (1) be unique non-degenerate B-stable, where B = (1, 2, . . ., m ). To guarantee that the feasible solution x± opt is optimal, the following additional constraints must be added to (10) (for each δ ). − + ± + − −(|a± δ j | x j − |aδ j | x j opt ) +
n
(|a±δ j |+ x−j − |a±δ j |− x+j opt ) ≥ 0,
(11)
j=n−q+1
j=k−p+1
120
(9b)
j=k+1
k
118
(9a)
and those corresponding to z+ as
112
117
i = 1, 2, . . ., m,
j = 1, 2, . . ., n,
j=1
116
sign(a± )|a±i j |− x+j ≤ b−i , ij
j=k+1
x±j ≥ 0,
k
115
n
where δ is the number of active constraints in constraint (9a) for the optimal solution, as well as a± δ j ≤ 0 for j = k − p + 1, . . ., k and a± ≥ 0 for j = n − q + 1, . . ., n. δj Proof. ILP model (1) is unique non-degenerate B-stable with B = (1, 2, . . ., m ). Thus, for each i = δ = 1, . . ., n, the δ th constraint in (9a) is active. To prove the optimality, Theorem 4.1 implies that it is sufficient to show that, for each x j ∈ x±j , n
− a+ δ j x j ≥ bδ ,
j=1
121
or k j=1
122
a+ δ jx j +
n
− a+ δ j x j ≥ bδ .
(12)
j=k+1
± ± Suppose a± δ j ≥ 0 for j = 1, . . ., k − p, aδ j ≤ 0 for j = k − p + 1, . . ., k, k + 1, . . ., n − q, and aδ j ≥ 0 for j = n − q + 1, . . ., n.
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Then, from (12), we have:
123
k−p
k
± + sign(a± δ j )|aδ j | x j +
j=1
± − sign(a± δ j )|aδ j | x j +
j=k−p+1
≥
k
n−q
n
± − sign(a± δ j )|aδ j | x j +
± + sign(a± δ j )|aδ j | x j
j=n−q+1
j=k+1 n
± + − sign(a± δ j )|aδ j | x jopt +
j=1
124
7
± − + sign(a± δ j )|aδ j | x jopt ,
j=k+1
or k−p
k
± + − sign(a± δ j )|aδ j | (x j − x jopt ) +
j=1
± − ± + − sign(a± δ j )(|aδ j | x j − |aδ j | x jopt )
j=k−p+1 n−q
+
n
± − + sign(a± δ j )|aδ j | (x j − x jopt ) +
± + ± − + sign(a± δ j )(|aδ j | x j − |aδ j | x jopt )
j=n−q+1
j=k+1
≥ 0.
(13) ± + sign(a± δ j )|aδ j | (x j
125
Note that, for j = 1, . . ., k − p,
126
is the left-hand side of (3), then we have k
≥
n
− ± + − −(|a± δ j | x j − |aδ j | x j opt ) +
≥ 0 and, for j = k + 1, . . ., n − q,
− + ± + − −(|a± δ j | x j − |aδ j | x j opt ) +
n
(|a±δ j |+ x−j − |a±δ j |− x+j opt ) ≥ 0 ∀δ.
j=n−q+1
j=k−p+1
Remark 4.1. The two sub-models of the IILP method are formulated as follows: Sub-model 1.
z− =
max
k
c−j x−j +
j=1
s.t.
k
n
c−j x+j
j=k+1
sign(a± )|a±i j |+ x−j + ij
j=1
n
sign(a± )|a±i j |− x+j ≤ b−i , ij
i = 1, 2, . . ., m,
sign(a± )|a±i j |+ x−j ≤ b+i , ij
i = 1, 2, . . ., m,
j=k+1
x±j ≥ 0, 131
≥ 0. If
(|a±δ j |+ x j − |a±δ j |− x+j opt ) ≥ 0.
128
130
− x+jopt )
Therefore, to prove the optimality, it is sufficient that k
129
± − sign(a± δ j )|aδ j | (x j
j=n−q+1
j=k−p+1
127
− x−jopt )
j = 1, 2, . . ., n.
Sub-model 2.
max
z+ =
k
c+j x+j +
j=1
s.t.
k
n
c+j x−j
j=k+1
sign(a± )|a±i j |− x+j + ij
j=1
n j=k+1
x+j ≥ x−j opt ,
j = 1, . . ., k,
x−j
j = k + 1, . . ., n,
≤
x+j opt ,
k
− + ± + − −(|a± δ j | x j − |aδ j | x j opt ) +
(|a±δ j |+ x−j − |a±δ j |− x+j opt ) ≥ 0,
j=n−q+1
j=k−p+1
x±j ≥ 0.
n
j = 1, 2, . . ., n.
132
4.2. IMILP method
133
In this subsection, we propose a pair of sub-models of the IMILP method. The first sub-model is the same as the first sub-model of MILP. For the second sub-model of the IMILP method, we consider the following MILP constraints corresponding to z− :
134 135
x−j ≤ x+j opt ,
j = 1, . . ., k,
(14a)
136
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x+j ≥ x−j opt , 137
k
j = k + 1, . . ., n,
(14b)
j = 1, 2, . . ., n,
± + − sign(a± δ j )|aδ j | x j +
j=1
143
(14d)
Theorem 4.5. Let ILP model (1) be unique non-degenerate B-stable, where B = (1, 2, . . ., m ). To guarantee that a feasible solution x± opt is optimal, the following constraints should be added to (14): k
144
(14c)
Note that constraint (14c) is the extra constraint of the MILP method that guarantees that solution x± opt is feasible. We now prove that the sign of the first constraint of the second sub-model of MILP should be reversed.
140 141
(|a±δ j |− x+j − |a±δ j |+ x−j opt ) ≤ 0, ∀δ,
j=n−q+1
x±j ≥ 0.
139
142
n
+ − ± − + −(|a± δ j | x j − |aδ j | x j opt ) +
j=k−p+1
138
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n
± − + − sign(a± δ j )|aδ j | x j ≥ bδ .
(15)
j=k+1
Proof. ILP model (1) is unique non-degenerate B-stable with B = (1, 2, . . ., m ). Thus, based on Theorem 4.1, the following condition is sufficient for optimality: n
− a+ δ j x j ≥ bδ ,
∀x j ∈ x±j
j=1
145
or k
a+ δ jx j +
j=1
146 147 148 149
n
− a+ δ j x j ≥ bδ .
(16)
j=k+1
Note that the inequality
n j=1
+ a− δ j x j ≤ bδ from Ti is used to prove the feasibility, leading to the extra constraint in the
MILP method. ± ± Suppose that a± δ j ≥ 0 for j = 1, . . ., k − p, aδ j ≤ 0 for j = k − p + 1, . . ., k, k + 1, . . ., n − q, and aδ j ≥ 0 for j = n − q + 1, . . ., n. Therefore, from (16), we have: k−p
± + sign(a± δ j )|aδ j | x j +
j=1
k
n−q
± − sign(a± δ j )|aδ j | x j +
j=k−p+1
± − sign(a± δ j )|aδ j | x j +
j=k+1
n
± + sign(a± δ j )|aδ j | x j
j=n−q+1
≥ b− δ. 150
To verify the above inequality, it is sufficient that k−p
k
± + − sign(a± δ j )|aδ j | x j +
j=1
± − + sign(a± δ j )|aδ j | x j opt
j=k−p+1 n−q
+
n
± − + sign(a± δ j )|aδ j | x j +
± + − sign(a± δ j )|aδ j | x j opt
j=n−q+1
j=k+1
≥ b− δ. 151
(17)
From (14c), we can conclude that k
k
− + −|a± δ j | x j opt +
k j=k−p+1
=
k j=k−p+1
152
n
|a±δ j |+ x−j opt
j=n−q+1
j=k−p+1
≥
± + − sign(a± δ j )|aδ j | x j opt
j=n−q+1
j=k−p+1
=
n
± − + sign(a± δ j )|aδ j | x j opt +
+ − −|a± δ j| x j +
n
|a±δ j |− x+j
j=n−q+1 ± + − sign(a± δ j )|aδ j | x j +
n
± − + sign(a± δ j )|aδ j | x j .
j=n−q+1
Therefore, by considering (17) for optimality, it is sufficient that Please cite this article as: M. Allahdadi et al., Improving the modified interval linear programming method by new techniques, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2015.12.037
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M. Allahdadi et al. / Information Sciences xxx (2016) xxx–xxx k−p
k
± + − sign(a± δ j )|aδ j | x j +
j=1
± + − sign(a± δ j )|aδ j | x j +
j=k−p+1
n−q
± − + sign(a± δ j )|aδ j | x j +
j=k+1
n
9
± − + sign(a± δ j )|aδ j | x j
j=n−q+1
≥ b− δ, 153
and so k
n
± + − sign(a± δ j )|aδ j | x j +
j=1
± − + − sign(a± δ j )|aδ j | x j ≥ bδ .
j=k+1
154 155
Remark 4.2. In the IMILP method, the first sub-model is the same as (5), and the second sub-model is as follows:
min
z− =
k
c−j x−j +
j=1
s.t.
k
n
c−j x+j
j=k+1 n
sign(a± )|a±i j |+ x−j + ij
j=1
sign(a± )|a±i j |− x+j ≥ b−i , i = 1, 2, . . ., m, ij
j=k+1
x−j ≤ x+j opt ,
j = 1, . . ., k,
x+j
j = k + 1, . . ., n,
≥
x−j opt ,
k
+ − ± − + −(|a± δ j | x j − |aδ j | x j opt ) +
156 157
(|a±δ j |− x+j − |a±δ j |+ x−j opt ) ≤ 0, ∀δ,
j=n−q+1
j=k−p+1
x±j ≥ 0
n
j = 1, 2, . . ., n.
Note that because the signs of the main constraints have been reversed, the problem of maximization changes to a problem of minimization.
159
Note 4.1. Since the solution regions obtained through the IILP and IMILP methods are completely optimal, the union of these regions will be closer to the exact optimal solution region of the ILP model.
160
5. Numerical results
158
161 162 163 164 165 166
In this section, we solve three ILP models by using the proposed methods. Example 5.1. Consider ILP model (8). According to Table 1, the interval optimal solutions given by the MILP method are x± = [4.574, 6.336] and x± = 1 2 [3.320, 3.495]. MILP leads to a completely feasible, but not completely optimal, solution. Before using the IILP and IMILP methods, we first check the basis stability of ILP model (8). Based on Remark 3.1, model (8) is equivalent to the following model:
max s.t.
± z± = [26, 30]x± 1 − [5.5, 6]x2 ± ± [8, 10]x± 1 − [12, 14]x2 + x3 = [3.8, 4.2] ± ± [1, 1.1]x± 1 + [0.19, 0.2]x2 + x4 = [6.5, 7] ± ± ± x± 1 , x2 , x3 , x4 ≥ 0.
167 168
A candidate basis for B-stability is B = (1, 2 ), since this is optimal for the characteristic model with center values. 1. According to Theorem 3.1, the spectral radius is equal to 0.104, so AB is regular:
AB = 169 170 171 172 173 174 175 176
[8, 10] [1, 1.1]
[−14, −12] . [0.19, 0.2]
2. According to Theorem 3.3, we can compute the enclosure of the solution set of AB xB = b to be xB =
([5.176, 6.382], [3.096, 4.342] )T , which is non-negative.
3. Using Theorem 3.3 again, we can compute the enclosure of the solution set of ATB y = cB to be y = ([0.693, 0.806], [16.541, 24.354] )T . According to Theorem 3.4, since ((ATN )y )− ≥ c+ , the optimality condition holds. Thus, the N problem is B-stable. We now use Corollaries 4.1 and 4.2 to avoid the non-optimal region. The results are given in Table 2 and Fig. 3. Obviously, not all of the solution regions of BWC and MILP are completely feasible and completely optimal, respectively, whereas those of IILP and IMILP are completely feasible and optimal. Please cite this article as: M. Allahdadi et al., Improving the modified interval linear programming method by new techniques, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2015.12.037
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M. Allahdadi et al. / Information Sciences xxx (2016) xxx–xxx Table 2 Sub-models and solutions of BWC, MILP, IILP, and IMILP for ILP model (8). Method
Sub-models
BWC
max z+ = 30x1 − 5.5x2
± zopt = [110.71, 172.62]
s.t. 8x1 − 14x2 ≤ 4.2
x± = [5.182, 6.366] 1 opt
x1 + 0.19x2 ≤ 7
Results
x± = [3.338, 4.001] 2 opt
x 1 , x2 ≥ 0 max z− = 26x1 − 6x2 s.t. 10x1 − 12x2 ≤ 3.8 1.1x1 + 0.2x2 ≤ 6.5 x1 , x2 ≥ 0 MILP
− 5.5x− max z+ = 30x+ 1 2 − 14x− ≤ 4.2 s.t. 8x+ 1 2 + 0.2x− ≤7 x+ 1 2
± zopt = [97.96, 171.81]
x± = [4.574, 6.336] 1 opt x± = [3.320, 3.495] 2 opt
, x− ≥0 x+ 1 2
− 6x+ max z− = 26x− 1 2 s.t.10x− − 12x+ ≤ 3.8 1 2
+ 0.19x+ ≤ 6.5 1.1x− 1 2
, x+ ≥ 0,x− ≤ 6.336 x− 1 2 1 x+ ≥ 3.3205,0.19x+ ≤ 0.2 × 3.3205 2 2 IILP
max z− = 26x− − 6x+ 1 2
s.t.10x− − 12x+ ≤ 3.8 1 2 + 0.19x+ ≤ 6.5 1.1x− 1 2
± zopt = [111.38, 165.99]
x± = [5.213, 6.235] 1 opt x± = [3.827, 4.028] 2 opt
, x+ ≥0 x− 1 2
− 5.5x− max z+ = 30x+ 1 2
s.t. 8x+ − 14x− ≤ 4.2 1 2 − x+ + 0.2x ≤7 1 2
x+ , x− ≥ 0,x+ ≥ 5.21 1 2 1
≤ 4.03,0.2x− ≥ 0.19 × 4.03 x− 2 2 IMILP
− 5.5x− max z+ = 30x+ 1 2 s.t. 8x+ − 14x− ≤ 4.2 1 2 + 0.2x− ≤7 x+ 1 2
± zopt = [116.97, 171.81]
x± = [5.305, 6.336] 1 opt x± = [3.320, 3.495] 2 opt
, x− ≥0 x+ 1 2
− 6x+ min z− = 26x− 1 2
s.t.10x− − 12x+ ≥ 3.8 1 2 1.1x− + 0.19x+ ≥ 6.5 1 2
, x+ ≥ 0,x− ≤ 6.336 x− 1 2 1
≥ 3.3205,0.19x+ ≤ 0.2 × 3.3205 x+ 2 2
177 178 179 180
In [34], the infeasible regions of MILP were avoided using an extra constraint of 0.19 × x+ ≤ 0.2 × 3.3205 in the second 2 sub-model. This produced optimal interval solutions of x± = [4.574, 6.336] and x± = [3.320, 3.495], and the optimal value of 1 2 the objective function was [97.96, 171.81]. In contrast, the worst value of the objective function given by the BWC method was 110.71. Therefore, the interval value of the objective function (i.e., [97.96, 171.81]), and hence the interval values of the
T
186
solutions (i.e., ) cannot be optimal. [4.574, 6.336], [3.320, 3.495] MILP outputs a completely feasible solution region that is not completely optimal. This is because the extra constraint 0.19 × x+ ≤ 0.2 × 3.3205 does not guarantee the optimality of the solutions. The non-optimal solution part of MILP is repre2 sented in Fig. 3. Note that the optimal interval values of the objective function in the IILP and IMILP sub-models lie within the optimal intervals of the BWC method.
187
Example 5.2. Consider the ILP model
181 182 183 184 185
max
± ± z± = [2, 2.5]x± 1 − [1, 1.5]x2 + [1.5, 2]x3
± ± s.t. [2, 3]x± 1 + [1.5, 2]x2 + [3, 3.4]x3 ≤ [15, 20] ± ± [4.5, 5.5]x± 1 + [2.5, 3.5]x2 − [1, 1.5]x3 ≤ [5, 10] ± ± [1, 1.6]x± 1 − [6, 7]x2 + [2, 3]x3 ≤ [4, 6] ± ± x± 1 , x2 , x3 ≥ 0.
188
(18)
Model (18) is equivalent to Please cite this article as: M. Allahdadi et al., Improving the modified interval linear programming method by new techniques, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2015.12.037
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11
Fig. 3. Graphical interpretation of the BWC, ILP, MILP, IILP, and IMILP methods for ILP model (8).
max
± ± z± = [2, 2.5]x± 1 − [1, 1.5]x2 + [1.5, 2]x3
± ± ± s.t. [2, 3]x± 1 + [1.5, 2]x2 + [3, 3.4]x3 + x4 = [15, 20] ± ± ± [4.5, 5.5]x± 1 + [2.5, 3.5]x2 − [1, 1.5]x3 + x5 = [5, 10] ± ± ± [1, 1.6]x± 1 − [6, 7]x2 + [2, 3]x3 + x6 = [4, 6] ± ± ± ± ± x± 1 , x2 , x3 , x4 , x5 , x6 ≥ 0.
189 190
A candidate basis for B-stability is B = (1, 2, 3 ), since this is optimal for the characteristic model taking the center values. 1. According to Theorem 3.1, the spectral radius is equal to 0.3377, so AB is regular:
AB = 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211
[2, 3] [4.5, 5.5] [1, 1.6]
[1.5, 2] [2.5, 3.5] [−7, −6]
[3, 3.4] [−1.5, −1] . [2, 3]
2. According to Theorem 3.3, we can compute the enclosure of the solution set of AB xB = b to be xB = [0.6849, 3.2513], [0.6264, 1.3119], [2.9358, 4.1662] )T , which is non-negative. 3. By Theorem 3.3, we can compute the enclosure of the solution set of ATB y = cB to be y = ([0.1198, 0.5903], [0.2155, 0.3441], [0.3716, 0.4513] )T . According to Theorem 3.4, since ((ATN )y )− ≥ c+ , the optimality condition is valid. Thus, N the model is B-stable. We solved model (18) by using the BWC, MILP, IILP, and IMILP methods. The results are given in Table 3. The solutions obtained from the BWC method are not completely feasible. For example, the extreme point (3.1728, 1.0842, 4.1592)T is infeasible, because it does not satisfy the inequalities 2x1 + 1.5x2 + 3x3 ≤ 20 and 4.5x1 + 2.5x2 − 1.5x3 ≤ 10, which are the first and second constraints of the best problem. The solution region obtained with the MILP method is completely feasible, but not completely optimal, because the extreme point (1.0217, 0.7760, 2.8578)T does not satisfy the inequality 3x1 + 2x2 + 3.4x3 ≥ 15, which is the first constraint of the worst problem with the inverse sign. The solution region obtained through the IILP and IMILP methods is completely feasible and optimal. Example 5.3. Suppose that the interval coefficients in ILP model (1) are random intervals. The entries of a± , b± , and c±j ij i for i = 1, 2, . . ., m and j = 1, 2, . . ., n are generated randomly with a uniform distribution in [−1, 5], [5, 10], and [−1, 2], respectively. The lower and upper bounds of the intervals are assumed to have the same sign. The computations were mainly carried out by using MATLAB R2009b, with some interval computations performed by the interval toolbox INTLAB v6 [31]. For given dimensions, the optimal values of the objective function are presented in Table 4. In all cases, the lower bound of the objective function obtained through MILP is less than the lower bound of the objective function given by the BWC method. Therefore, MILP does not provide the optimal solution region. The optimal interval values of the objective function obtained through IILP and IMILP lie in the optimal intervals given by the BWC method. Please cite this article as: M. Allahdadi et al., Improving the modified interval linear programming method by new techniques, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2015.12.037
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M. Allahdadi et al. / Information Sciences xxx (2016) xxx–xxx Table 3 Solutions obtained using BWC, MILP, IILP, and IMILP for ILP model (18). Method
BWC
MILP
± zopt
x±
[4.5625,15.4660]
[4.7780,14.5560]
⎛
⎞
⎛
⎞
[0.7802, 3.1728] ⎝[0.7845, 1.0842]⎠ [3.0856, 4.1592] [1.0217, 2.6000] ⎝[0.7760, 1.0347]⎠ [2.8578, 4.4160]
⎛ IILP
[5.0718,14.4208]
⎛ IMILP
[5.2084,14.5560]
[1.1976, 2.5667]
⎞
⎝[0.8173, 1.0898]⎠ [2.8743, 4.4107]
⎞
[1.2232, 2.6000] ⎝ [0.7760, 1.0347] ⎠ [2.8760, 4.4160]
Table 4 Optimal values of the objective function obtained using BWC, MILP, IILP, and IMILP for Example 5.3. m
n
BWC
MILP
IILP
IMILP
5 100 100 200 500
10 100 150 250 700
[2.13,9.22] [2.51,6.40] [2.85,7.19] [3.07,7.22] [2.98,7.11]
[2.02,8.73] [1.74,6.29] [2.17,7.07] [2.29,7.06] [2.25,6.97]
[2.14,5.80] [2.53,3.91] [2.87,3.53] [3.11,3.82] [3.01,3.33]
[2.79,8.73] [2.96,6.29] [3.78,7.07] [3.94,7.06] [3.20,6.97]
212
6. Conclusion
213
219
In this paper, we analyzed the MILP method of solving ILP problems. A part of the solution region obtained through the MILP method is not completely optimal. To remove this non-optimal part, under the assumption of basis stability, we proposed two improved MILP methods, namely IILP and IMILP. These methods guarantee that the solution regions are completely feasible and optimal. In fact, the solution regions obtained through these methods are not only feasible, but also completely optimal. In future studies, we will attempt to identify new methods for obtaining ILP solution sets without the assumption of basis stability such that the solution region is both feasible and optimal.
220
Acknowledgments
221 222
We would like to thank the anonymous referees for their constructive comments and suggestions that have helped to improve this paper.
223
References
214 215 216 217 218
224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
G. Alefeld, J. Herzberger, Introduction to Interval Computations, Academic Press, New York, 1983. M. Allahdadi, H. Mishmast Nehi, The optimal solutions set of the interval linear programming problems, Optim. Lett. 7 (2013) 893–1911. H. Beeck, Linear Programming with Inexact Data, Technical report tum-isu-7830, Technical University of Munich, Munich, 1978. J.W. Chinneck, K. Ramadan, Linear programming with interval coefficients, J. Oper. Res. Soc. 51 (2000) 209–220. P. Guo, G.H. Huang, L. He, H.L. Li, Interval-parameter fuzzy-stochastic semi-infinite mixed-integer linear programming for waste management under uncertainty, Environ. Model. Assess 14 (2009) 521–537. L. He, G. Huang, G. Zeng, H. Lu, Fuzzy inexact mixed-integer semiinfinite programming for municipal solid waste management planning, J. Environ. Eng. 134 (2008) 572–581. L. He, G.H. Huang, G.M. Zeng, H.W. Lu, Identifying optimal regional solid waste management strategies through an inexact integer programming model containing infinite objectives and constraints, Waste Manage. 29 (2009) 21–31. M. Hladik, AE solutions and AE solvability to general interval linear systems, Linear Algebra Appl. 465 (0) (2015) 221–238. M. Hladik, How to determine basis stability in interval linear programming, Optim. Lett. 8 (2014) 375–389. M. Hladik, Interval linear programming: A survey, in: Z.A. Mann (Ed.), Linear Programming, New Frontiers in Theory and Applications, chapter 2, Nova Science Publishers, New York, 2012, pp. 85–120.2012 M. Hladik, Optimal value range in interval linear programming, Fuzzy Optim. Decis. Mak. 8 (3) (2009) 283–294. G.H. Huang, B.W. Baetz, G.G. Patry, A grey linear programming approach for municipal solid waste management planning under uncertainty, Civ. Eng. Environ. Syst. 9 (1992) 319–335. G.H. Huang, B.W. Baetz, G.G. Patry, Grey integer programming: an application to waste management planning under uncertainty, Eur. J. Oper. Res. 83 (1995) 594–620. G.H. Huang, B.W. Baetz, G.G. Patry, Trash-flow allocation: planning under uncertainty, Interfaces 28 (1998) 36–55. G.H. Huang, R.D. Moore, Grey linear programming, its solving approach, and its application, Int. J. Syst. Sci. 24 (1993) 159–172.
Please cite this article as: M. Allahdadi et al., Improving the modified interval linear programming method by new techniques, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2015.12.037
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245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 Q2268 269 270 271 272
[m3Gsc;January 13, 2016;15:0] 13
[16] C. Jansson, A self-validating method for solving linear programming problems with interval input data, in: U. Kulisch, H.J. Stetter (Eds.), Scientific Computation with Automatic Result Verification. Computing Suppl., 6, 1988, pp. 33–45. [17] C. Jansson, S.M. Rump, Rigorous solution of linear programming problems with uncertain data, Z. Oper. Res. 35 (2) (1991) 87–111. [18] J. Konickova, Sufficient condition of basis stability of an interval linear programming problem, ZAMM. Z. Angew. Math. Mech. 81 (3) (2001) 677–678. [19] H.W. Lu, M.F. Cao, Y. Wang, X. Fan, L. He, Numerical solutions comparison for interval linear programming problems based on coverage and validity rates, Appl. Math. Modell. 38 (2014) 1092–1100. [20] H.W. Lu, G.H. Huang, L. He, A semi-infinite analysis-based inexact two-stage stochastic fuzzy linear programming approach for water resources management, Eng. Opt. 41 (2009) 73–85. [21] H.W. Lu, G.H. Huang, L. Liu, L. He, An interval-parameter fuzzy-stochastic programming approach for air quality management under uncertainty, Environ. Eng. Sci. 25 (2008) 895–909. [22] H.W. Lu, G.H. Huang, Z.F. Liu, L. He, Greenhouse gas mitigation-induced rough-interval programming for municipal solid waste management, J. Air. Waste. Manage. Assoc. 58 (2008) 1546–1559. [23] B. Machost, Numerische Behandlung des Simplexverfahrens mit Intervallanalytischen Methoden, Technical report 30, Berichte der Gesellschaft für Mathematik und Datenverarbeitung, 1970. 54 pages, Bonn [24] A. Mostafaee, M. Hladik, M. Cerny, Inverse linear programming with interval coefficients, J. Comput. Appl. Math. 292 (2016) 591–608. [25] F. Mraz, On infimum of Optimal Objective Function Values in Interval Linear Programming, Technical report kam series, Department of Applied Mathematics, Charles University, Prague, 1996. [26] J. Rex, J. Rohn, Sufficient conditions for regularity and singularity of interval matrices, SIAM J. Matrix Anal. Appl. 20 (2) (1998) 437–445. [27] J. Rohn, Cheap and tight bound: The recent result by e. hansen can be made more efficient, Interval Comput. 4 (1993) 13–21. [28] J. Rohn, Forty necessary and sufficient conditions for regularity of interval matrices: A survey, Electron. J. Linear Algebra 18 (2009) 500–512. [29] J. Rohn, Interval linear programming, in: M. Fiedler, et al. (Eds.), Linear Optimization Problems with Inexact Data, Chapter 3, Springer, New York, 2006, pp. 79–100. [30] J. Rohn, Stability of the optimal basis of a linear program under uncertainty, Oper. Res. Lett. 13 (1) (1993) 9–12. [31] S.M. Rump, INTLAB - INTerval LABoratory, in: T. Csendes (Ed.), Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, 1999, pp. 77–104. [32] S.C. Tong, Interval number, fuzzy number linear programming, Fuzzy Sets Syst. 66 (1994) 301–306. [33] X. Wang, G. Huang, Violation analysis on two-step method for interval linear programming, Inf. Sci. 281 (2014) 85–96. [34] F. Zhou, G.H. Huang, G. Chen, H. Guo, Enhanced-interval linear programming, Eur. J. Oper. Res. 199 (2009) 323–333.
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Information Sciences journal homepage: www.elsevier.com/locate/ins
Improving the modified interval linear programming method by new techniques M. Allahdadi∗, H. Mishmast Nehi, H.A. Ashayerinasab, M. Javanmard
Q1
Mathematics Faculty, University of Sistan and Baluchestan, Zahedan, Iran
a r t i c l e
i n f o
Article history: Received 19 January 2015 Revised 1 December 2015 Accepted 25 December 2015 Available online xxx Keywords: Feasibility Interval linear programming MILP method Optimality Uncertainty
a b s t r a c t In this study, we consider interval linear programming (ILP) problems, which are used to deal with uncertainties resulting from the range of admissible values in problem coefficients. In most existing methods for solving ILP problems, a part of the solution region is not feasible. The solution set obtained through the modified ILP (MILP) method is completely feasible (i.e., it does not violate any constraints), but is not completely optimal (i.e., some points of the region are not optimal). In this paper, two new ILP methods and their sub-models are presented. These techniques improve the MILP method, giving a solution region that is not only completely feasible, but also completely optimal. © 2016 Published by Elsevier Inc.
1
1. Introduction
2
Uncertainties in many real-world problems mean that their parameters may be specified as lying between lower and upper bounds. To deal with such uncertainties, interval linear programming (ILP) is used. Researchers working on ILP problems have proposed several methods for solving ILP models [1–34]. Some have used interval arithmetic and extensions of the simplex algorithm [3,16,17,23], whereas others have focused on basis stability [9,18,25,30]. Under the assumption of basis stability, it is possible to obtain the optimal solution set of ILP. The ILP model has sometimes been divided into two sub-models to obtain the solution set [10,13,15,29,32–34]. In [2], the authors obtained the optimal solution set to an ILP problem by using the best and worst problem constraints when all components of the optimal solutions to the ILP model are positive. This assumption can be derived by solving the best problem. If all components of the feasible solution to the best problem are positive, then the feasible solution components (and hence the optimal solution components) in all of the characteristic models (and thus the ILP model) are positive. In the best and worst cases (BWC) method proposed by Tong [32], the ILP model was converted into two sub-models, the best and worst sub-models, which have the largest and smallest feasible regions, respectively. A given point is feasible for the ILP model if it satisfies the constraints of the best problem, and it is optimal for the ILP model if it is optimal for at least one characteristic model. The BWC method has been extended by Chinneck and Ramadan to ILP models with equality constraints [4]. A novel ILP method was proposed by Huang and Moore [15].
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
∗
Corresponding author. Tel.: +98 5431132517. E-mail addresses: [email protected], [email protected] (M. Allahdadi), [email protected] (H. Mishmast Nehi), hassan.ali. [email protected] (H.A. Ashayerinasab), [email protected] (M. Javanmard). http://dx.doi.org/10.1016/j.ins.2015.12.037 0020-0255/© 2016 Published by Elsevier Inc.
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The solutions given by the BWC and ILP methods may stray into infeasible regions. Although the BWC method introduces exact bounds for the values of the objective function, the solution could exist in an area that is infeasible. To guarantee that the given solution is completely feasible, Zhou et al. [34] proposed the modified ILP method (MILP), in which an extra constraint is added to the second sub-model. However, the solutions resulting from MILP may not be optimal. In this paper, we propose improved ILP (IILP) and improved MILP (IMILP) methods for solving ILP problems. The solutions to these methods are guaranteed to be completely feasible and completely optimal. The feasibility and optimality are illustrated by some numerical examples. In the following, 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 and x− ≥ 0. In addition, x± ≤ 0 if and only if x+ ≤ 0 and 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+ } 28
is called an interval matrix, and the matrices A− and A+ are called its bounds. Center and radius matrices are defined as
Ac =
1 + ( A + A− ) , 2
A ± =
1 + ( A − A− ) . 2
31
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].
32
2. Overview of ILP model solving methods
33
In this section, we review some methods for solving ILP models with inequality constraints. Models with equality constraints have also been investigated [11,29]. Consider the following ILP:
29 30
34
max s.t.
z± = n
n
c±j x±j
j=1
a± x± ij j
≤ b± , i
i = 1, 2, . . . , m
(1)
j=1
x±j ≥ 0, 35
j = 1, 2, . . ., n.
The characteristic model of the ILP model (1) is
max s.t.
z= n
n
c jx j
j=1
ai j x j ≤ bi ,
i = 1, 2, . . ., m
(2)
j=1
x j ≥ 0, 36 37 38 39 40
where ai j ∈
a± , cj ij
∈
j = 1, 2, . . ., n,
c±j ,
and bi ∈ b± . i
The feasible solution set of the ILP is defined as {x ∈ Rn : nj=1 a− x ≤ b+ , x j ≥ 0, i = 1, . . ., m, j = 1, . . ., n}. ij j i Moreover, the optimal solution set of the ILP is defined as the set of all optimal solutions over all characteristic models. According to [32], the BWC method can be used to solve model (1). The two sub-models are as follows: Sub-model 1 (the best problem).
max s.t.
z+ = n
n
c+j x j
j=1
a− x ij j
≤ b+ , i
i = 1, 2, . . ., m,
(3)
j=1
x j ≥ 0, 41
j = 1, 2, . . ., n.
Sub-model 2 (the worst problem).
max s.t.
z− = n
n
c−j x j
j=1
a+ x ij j
≤ b− , i
i = 1, 2, . . ., m,
(4)
j=1
x j ≥ 0, 42 43
j = 1, 2, . . ., n. + zopt
− and zopt are the best and worst optimal values of the objective function, respectively, then all of the Theorem 2.1 [32]. If − + optimal values of the objective function lie in [zopt , zopt ].
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44 45 46
z+ =
k
c+j x+j +
j=1
s.t.
k
sign(
n
c+j x−j
j=k+1
a± ij
)| |
− + a± xj ij
+
j=1
s.t.
z− = k
k
c−j x−j +
sign(a± )|a±i j |− x+j ≤ b−i , ij
i = 1, 2, . . ., m,
c−j x+j n j=k+1
j = 1, . . ., k, j = k + 1, . . ., n,
where c±j ≥ 0 for j = 1, . . ., k, c±j ≤ 0 for j = k + 1, . . ., n, and
sign(a± )= ij
a− ij
a± ≥0 ij
−a+ ij
a± <0 ij
1
a± ≥0 ij
−1
< 0. a± ij
|a±i j |+ =
,
a+ ij
a± ≥0 ij
−a− ij
a± <0 ij
x+j opt for j = 1, . . ., k and x−j opt for j = k + 1, . . ., n are the optimal solutions of sub-model 1. A part of the solution region obtained by the ILP method may be infeasible. To guarantee that the solution given by the ILP method is completely feasible, Zhou et al. proposed the following MILP method [34]: The first sub-model of this method is the same as that of the ILP method. The second sub-model is as follows.
max
z− =
k
c−j x−j +
j=1
s.t.
k
n
c−j x+j
j=k+1
sign(a± )|a±i j |+ x−j + ij
j=1
n
sign(a± )|a±i j |− x+j ≤ b−i , i = 1, 2, . . ., m, ij
j=k+1
x−j ≤ x+j opt ,
j = 1, . . ., k,
x+j
j = k + 1, . . ., n,
≥
x−j opt ,
k
(7)
+ − ± − + − |a± δ j | x j − |aδ j | x j opt +
n
|a±δ j |− x+j − |a±δ j |+ x−j opt ≤ 0,
j=n−q+1
j=k−p+1
x±j ≥ 0, 54
(6)
j = 1, 2, . . ., n,
0,
53
(5)
j=k+1
x+j opt , x−j opt ,
|a±i j |− =
51
n
sign(a± )|a±i j |+ x−j + ij
j=1 x−j ≤ x+j ≥ x±j ≥
52
i = 1, 2, . . ., m,
j = 1, 2, . . ., n.
j=1
50
sign(a± )|a±i j |+ x−j ≤ b+i , ij
Sub-model 2.
max
49
n j=k+1
x±j ≥ 0,
48
3
Suppose that the interval coefficients of the ILP model are either all non-negative or all non-positive. The two sub-models of the ILP method are then defined as follows [15]: Sub-model 1.
max
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j = 1, 2, . . ., n,
± where δ is the number of active constraints in model (5), and a± δ j ≥ 0 for j = 1, . . ., k − p, aδ j ≤ 0 for j = k − p + 1, . . ., k, k + ± 1, . . ., n − q, and aδ j ≥ 0 for j = n − q + 1, . . ., n.
56
Though the MILP method guarantees that the solution region is completely feasible, the following example shows that a part of the solution region may be non-optimal.
57
Example 2.1. Consider the example given by [34]:
55
max
z± = [26, 30]x± − [5.5, 6]x± 1 2
s.t.
[8, 10]x± − [12, 14]x± ≤ [3.8, 4.2] 1 2 [1, 1.1]x± + [0.19, 0.2]x± ≤ [6.5, 7] 1 2
(8)
x± , x± ≥ 0. 1 2 58 59
To determine the exact optimal solution set of ILP model (8), we solve the characteristic models for most values of ai j ∈ a± , c j ∈ c±j , and bi ∈ b± for i, j = 1, 2. This gives their optimal solutions. The region obtained through the optimal solution ij i Please cite this article as: M. Allahdadi et al., Improving the modified interval linear programming method by new techniques, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2015.12.037
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Fig. 1. Approximation of the optimal solution region for ILP model (8).
Fig. 2. Optimal solution set resulting from the BWC, ILP, and MILP methods for ILP model (8). Table 1 Solutions of BWC, ILP, and MILP methods for ILP model (8).
60 61 62
Method
x± 1
x± 2
z±
BWC ILP MILP
[5.182, 6.366] [5.213, 6.336] [4.574, 6.336]
[3.338, 4.001] [3.320, 4.028] [3.320, 3.495]
[110.71, 172.62] [111.38, 171.81] [97.96, 171.81]
points is shown in Fig. 1. The results obtained from the BWC, ILP, and MILP methods are given in Table 1 and Fig. 2. The value of z− pertaining to MILP (97.96) is less than that obtained by BWC (110.71). In other words, the optimal value of the − objective function in the second sub-model of MILP is less than zopt , and thus this result is not optimal. However, according Please cite this article as: M. Allahdadi et al., Improving the modified interval linear programming method by new techniques, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2015.12.037
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5
66
− to Theorem 2.1, zopt given by BWC is the worst value of the objective function. Obviously, many points resulting from the MILP method are not optimal. In the following, we first discuss the concept of stability, and then propose two improvements to the MILP method to obtain solution regions that are both feasible and optimal.
67
3. Basis stability
68
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 [9,18].
63 64 65
69 70 71 72 73 74 75
Definition 3.1. The problem max{cT x : Ax = b, x ≥ 0}, where c ∈ c± ⊆ Rn , A ∈ A± ⊆ Rm×n , and b ∈ b± ⊆ Rm , is called Bstable 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. Let B ⊆ {1, 2, . . . , n} be an index set such that AB (the restriction of A to the columns indexed by B) 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. We now review the conditions for B-stability [9].
76
•
77
•
78
•
Regularity: AB is regular. Feasibility: The solution set of the interval system AB xB = b is non-negative. Optimality: AB is optimal, i.e., cTB A−1 AN − cTN ≥ 0T . B
79
Theorem 3.1 [26]. If ρ (|(Ac )−1 |AB ) < 1, then AB is regular, where ρ (.) denotes the spectral radius. B
80
Theorem 3.2 [26]. If max1≤i≤n (|(Ac )−1 |AB )ii ≥ 1, then AB is not regular. B
81 82
Some conditions for the regularity of interval matrices have been proposed in [28]. Theorem 3.3 [27]. The interval vector r is an enclosure to the solution set of a system AB xB = b, where:
1 (−x∗i + (xci + |xci | )Mii ) , 2Mii − 1 1 ri+ = max x∗i + (xci − |xci | )Mii , (x∗i + (xci − |xci | )Mii ) , 2Mii − 1
ri− = min −x∗i + (xci + |xci | )Mii ,
M = (I − |(AcB )−1 |AB )−1 ,
xc = (AcB )−1 bc ,
x∗ = M (|xc | + |(AcB )−1 |b ), 83
AcB is non-singular, and ρ (|(AcB )−1 |A ) < 1.
84
, then the optimality condition holds. Theorem 3.4 [9]. Let y be an enclosure to the solution set of ATB y = cB . If ((ATN )y )− ≥ c+ N
86
Theorem 3.5 [9]. Let diag(q) denote the diagonal matrix with entries q1 , . . ., qm . For each q ∈ { ± 1}m , if the solution set of the system
87
lies in the solution set of the system
85
⎧ c T + T ⎨((AB ) − (AB ) diag(q ))y ≤ cB −((AcB )T + (AB )T diag(q ))y ≤ −c− B ⎩ diag(q )y ≥ 0
((AcN )T − (AN )T diag(q ))y ≥ c+N diag(q )y ≥ 0,
88
then the optimality condition holds.
89
Remark 3.1. Since there are no dependencies, the set of constraints Ax ≤ b, x ≥ 0 is equivalent to Ax + Iy = b, x, y ≥ 0.
90
4. Our methods
91 92
We now propose two improved methods, namely IILP and IMILP, which guarantee that the obtained solution region is completely optimal. Moreover, the obtained solution region lies in the exact optimal solution region.
93
Definition 4.1. For each 1 ≤ i ≤ m, we define Si and Ti as follows:
Si =
x:
n j=1
a+ x ij j
≥
b− i
,
Ti =
x:
n
a− x ij j
≤
b+ i
.
j=1
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Note that Ti and Si are the feasible solution sets of the best problem and the worst problem with the inverse sign, respectively.
98
Theorem 4.1 [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 the intersection of the region generated by the best problem constraints and the worst problem constraints with the inverse sign.
99
Theorem 4.2 [34]. For ILP model (1), |a± |− sign(a±i j ) is the coefficient of x+j for j = 1, 2, . . ., k and |a±i j |+ sign(a±i j ) is the coefficient ij
96 97
100 101
of x−j for j = k + 1, k + 2, . . ., n for the first sub-model z+ . In addition, |a± |+ sign(a±i j ) is the coefficient of x−j for j = 1, 2, . . ., k, ij and |a± |− sign(a±i j ) is the coefficient of x+j for j = k + 1, k + 2, . . ., n for the second sub-model z− . ij
103
Theorem 4.3 [34]. For ILP model (1), b+ and b− are the right-hand sides of the constraints for the first and second sub-models, i i respectively.
104
4.1. IILP method
105
In this subsection, we introduce two sub-models of the IILP method. First, we solve the model corresponding to z− , which is the second sub-model of the MILP method without the extra constraints. In the sub-model corresponding to z+ , we add an extra constraint that guarantees that the resulting solution region is completely optimal. According to Theorems 4.2 and 4.3, for ILP model (1), we can write the IILP constraints corresponding to z− as
102
106 107 108 109
k
sign(a± )|a±i j |+ x−j + ij
j=1
110
111
113
114
sign(a± )|a±i j |− x+j + ij
n
119
sign(a± )|a±i j |+ x−j ≤ b+i , ij
ti = 1, 2, . . ., m,
(10a)
x+j ≥ x−j opt ,
j = 1, . . ., k,
(10b)
x−j ≤ x+j opt ,
j = k + 1, . . ., n,
(10c)
x±j ≥ 0,
j = 1, 2, . . ., n.
(10d)
Theorem 4.4. Let ILP model (1) be unique non-degenerate B-stable, where B = (1, 2, . . ., m ). To guarantee that the feasible solution x± opt is optimal, the following additional constraints must be added to (10) (for each δ ). − + ± + − −(|a± δ j | x j − |aδ j | x j opt ) +
n
(|a±δ j |+ x−j − |a±δ j |− x+j opt ) ≥ 0,
(11)
j=n−q+1
j=k−p+1
120
(9b)
j=k+1
k
118
(9a)
and those corresponding to z+ as
112
117
i = 1, 2, . . ., m,
j = 1, 2, . . ., n,
j=1
116
sign(a± )|a±i j |− x+j ≤ b−i , ij
j=k+1
x±j ≥ 0,
k
115
n
where δ is the number of active constraints in constraint (9a) for the optimal solution, as well as a± δ j ≤ 0 for j = k − p + 1, . . ., k and a± ≥ 0 for j = n − q + 1, . . ., n. δj Proof. ILP model (1) is unique non-degenerate B-stable with B = (1, 2, . . ., m ). Thus, for each i = δ = 1, . . ., n, the δ th constraint in (9a) is active. To prove the optimality, Theorem 4.1 implies that it is sufficient to show that, for each x j ∈ x±j , n
− a+ δ j x j ≥ bδ ,
j=1
121
or k j=1
122
a+ δ jx j +
n
− a+ δ j x j ≥ bδ .
(12)
j=k+1
± ± Suppose a± δ j ≥ 0 for j = 1, . . ., k − p, aδ j ≤ 0 for j = k − p + 1, . . ., k, k + 1, . . ., n − q, and aδ j ≥ 0 for j = n − q + 1, . . ., n.
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Then, from (12), we have:
123
k−p
k
± + sign(a± δ j )|aδ j | x j +
j=1
± − sign(a± δ j )|aδ j | x j +
j=k−p+1
≥
k
n−q
n
± − sign(a± δ j )|aδ j | x j +
± + sign(a± δ j )|aδ j | x j
j=n−q+1
j=k+1 n
± + − sign(a± δ j )|aδ j | x jopt +
j=1
124
7
± − + sign(a± δ j )|aδ j | x jopt ,
j=k+1
or k−p
k
± + − sign(a± δ j )|aδ j | (x j − x jopt ) +
j=1
± − ± + − sign(a± δ j )(|aδ j | x j − |aδ j | x jopt )
j=k−p+1 n−q
+
n
± − + sign(a± δ j )|aδ j | (x j − x jopt ) +
± + ± − + sign(a± δ j )(|aδ j | x j − |aδ j | x jopt )
j=n−q+1
j=k+1
≥ 0.
(13) ± + sign(a± δ j )|aδ j | (x j
125
Note that, for j = 1, . . ., k − p,
126
is the left-hand side of (3), then we have k
≥
n
− ± + − −(|a± δ j | x j − |aδ j | x j opt ) +
≥ 0 and, for j = k + 1, . . ., n − q,
− + ± + − −(|a± δ j | x j − |aδ j | x j opt ) +
n
(|a±δ j |+ x−j − |a±δ j |− x+j opt ) ≥ 0 ∀δ.
j=n−q+1
j=k−p+1
Remark 4.1. The two sub-models of the IILP method are formulated as follows: Sub-model 1.
z− =
max
k
c−j x−j +
j=1
s.t.
k
n
c−j x+j
j=k+1
sign(a± )|a±i j |+ x−j + ij
j=1
n
sign(a± )|a±i j |− x+j ≤ b−i , ij
i = 1, 2, . . ., m,
sign(a± )|a±i j |+ x−j ≤ b+i , ij
i = 1, 2, . . ., m,
j=k+1
x±j ≥ 0, 131
≥ 0. If
(|a±δ j |+ x j − |a±δ j |− x+j opt ) ≥ 0.
128
130
− x+jopt )
Therefore, to prove the optimality, it is sufficient that k
129
± − sign(a± δ j )|aδ j | (x j
j=n−q+1
j=k−p+1
127
− x−jopt )
j = 1, 2, . . ., n.
Sub-model 2.
max
z+ =
k
c+j x+j +
j=1
s.t.
k
n
c+j x−j
j=k+1
sign(a± )|a±i j |− x+j + ij
j=1
n j=k+1
x+j ≥ x−j opt ,
j = 1, . . ., k,
x−j
j = k + 1, . . ., n,
≤
x+j opt ,
k
− + ± + − −(|a± δ j | x j − |aδ j | x j opt ) +
(|a±δ j |+ x−j − |a±δ j |− x+j opt ) ≥ 0,
j=n−q+1
j=k−p+1
x±j ≥ 0.
n
j = 1, 2, . . ., n.
132
4.2. IMILP method
133
In this subsection, we propose a pair of sub-models of the IMILP method. The first sub-model is the same as the first sub-model of MILP. For the second sub-model of the IMILP method, we consider the following MILP constraints corresponding to z− :
134 135
x−j ≤ x+j opt ,
j = 1, . . ., k,
(14a)
136
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x+j ≥ x−j opt , 137
k
j = k + 1, . . ., n,
(14b)
j = 1, 2, . . ., n,
± + − sign(a± δ j )|aδ j | x j +
j=1
143
(14d)
Theorem 4.5. Let ILP model (1) be unique non-degenerate B-stable, where B = (1, 2, . . ., m ). To guarantee that a feasible solution x± opt is optimal, the following constraints should be added to (14): k
144
(14c)
Note that constraint (14c) is the extra constraint of the MILP method that guarantees that solution x± opt is feasible. We now prove that the sign of the first constraint of the second sub-model of MILP should be reversed.
140 141
(|a±δ j |− x+j − |a±δ j |+ x−j opt ) ≤ 0, ∀δ,
j=n−q+1
x±j ≥ 0.
139
142
n
+ − ± − + −(|a± δ j | x j − |aδ j | x j opt ) +
j=k−p+1
138
[m3Gsc;January 13, 2016;15:0]
n
± − + − sign(a± δ j )|aδ j | x j ≥ bδ .
(15)
j=k+1
Proof. ILP model (1) is unique non-degenerate B-stable with B = (1, 2, . . ., m ). Thus, based on Theorem 4.1, the following condition is sufficient for optimality: n
− a+ δ j x j ≥ bδ ,
∀x j ∈ x±j
j=1
145
or k
a+ δ jx j +
j=1
146 147 148 149
n
− a+ δ j x j ≥ bδ .
(16)
j=k+1
Note that the inequality
n j=1
+ a− δ j x j ≤ bδ from Ti is used to prove the feasibility, leading to the extra constraint in the
MILP method. ± ± Suppose that a± δ j ≥ 0 for j = 1, . . ., k − p, aδ j ≤ 0 for j = k − p + 1, . . ., k, k + 1, . . ., n − q, and aδ j ≥ 0 for j = n − q + 1, . . ., n. Therefore, from (16), we have: k−p
± + sign(a± δ j )|aδ j | x j +
j=1
k
n−q
± − sign(a± δ j )|aδ j | x j +
j=k−p+1
± − sign(a± δ j )|aδ j | x j +
j=k+1
n
± + sign(a± δ j )|aδ j | x j
j=n−q+1
≥ b− δ. 150
To verify the above inequality, it is sufficient that k−p
k
± + − sign(a± δ j )|aδ j | x j +
j=1
± − + sign(a± δ j )|aδ j | x j opt
j=k−p+1 n−q
+
n
± − + sign(a± δ j )|aδ j | x j +
± + − sign(a± δ j )|aδ j | x j opt
j=n−q+1
j=k+1
≥ b− δ. 151
(17)
From (14c), we can conclude that k
k
− + −|a± δ j | x j opt +
k j=k−p+1
=
k j=k−p+1
152
n
|a±δ j |+ x−j opt
j=n−q+1
j=k−p+1
≥
± + − sign(a± δ j )|aδ j | x j opt
j=n−q+1
j=k−p+1
=
n
± − + sign(a± δ j )|aδ j | x j opt +
+ − −|a± δ j| x j +
n
|a±δ j |− x+j
j=n−q+1 ± + − sign(a± δ j )|aδ j | x j +
n
± − + sign(a± δ j )|aδ j | x j .
j=n−q+1
Therefore, by considering (17) for optimality, it is sufficient that Please cite this article as: M. Allahdadi et al., Improving the modified interval linear programming method by new techniques, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2015.12.037
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M. Allahdadi et al. / Information Sciences xxx (2016) xxx–xxx k−p
k
± + − sign(a± δ j )|aδ j | x j +
j=1
± + − sign(a± δ j )|aδ j | x j +
j=k−p+1
n−q
± − + sign(a± δ j )|aδ j | x j +
j=k+1
n
9
± − + sign(a± δ j )|aδ j | x j
j=n−q+1
≥ b− δ, 153
and so k
n
± + − sign(a± δ j )|aδ j | x j +
j=1
± − + − sign(a± δ j )|aδ j | x j ≥ bδ .
j=k+1
154 155
Remark 4.2. In the IMILP method, the first sub-model is the same as (5), and the second sub-model is as follows:
min
z− =
k
c−j x−j +
j=1
s.t.
k
n
c−j x+j
j=k+1 n
sign(a± )|a±i j |+ x−j + ij
j=1
sign(a± )|a±i j |− x+j ≥ b−i , i = 1, 2, . . ., m, ij
j=k+1
x−j ≤ x+j opt ,
j = 1, . . ., k,
x+j
j = k + 1, . . ., n,
≥
x−j opt ,
k
+ − ± − + −(|a± δ j | x j − |aδ j | x j opt ) +
156 157
(|a±δ j |− x+j − |a±δ j |+ x−j opt ) ≤ 0, ∀δ,
j=n−q+1
j=k−p+1
x±j ≥ 0
n
j = 1, 2, . . ., n.
Note that because the signs of the main constraints have been reversed, the problem of maximization changes to a problem of minimization.
159
Note 4.1. Since the solution regions obtained through the IILP and IMILP methods are completely optimal, the union of these regions will be closer to the exact optimal solution region of the ILP model.
160
5. Numerical results
158
161 162 163 164 165 166
In this section, we solve three ILP models by using the proposed methods. Example 5.1. Consider ILP model (8). According to Table 1, the interval optimal solutions given by the MILP method are x± = [4.574, 6.336] and x± = 1 2 [3.320, 3.495]. MILP leads to a completely feasible, but not completely optimal, solution. Before using the IILP and IMILP methods, we first check the basis stability of ILP model (8). Based on Remark 3.1, model (8) is equivalent to the following model:
max s.t.
± z± = [26, 30]x± 1 − [5.5, 6]x2 ± ± [8, 10]x± 1 − [12, 14]x2 + x3 = [3.8, 4.2] ± ± [1, 1.1]x± 1 + [0.19, 0.2]x2 + x4 = [6.5, 7] ± ± ± x± 1 , x2 , x3 , x4 ≥ 0.
167 168
A candidate basis for B-stability is B = (1, 2 ), since this is optimal for the characteristic model with center values. 1. According to Theorem 3.1, the spectral radius is equal to 0.104, so AB is regular:
AB = 169 170 171 172 173 174 175 176
[8, 10] [1, 1.1]
[−14, −12] . [0.19, 0.2]
2. According to Theorem 3.3, we can compute the enclosure of the solution set of AB xB = b to be xB =
([5.176, 6.382], [3.096, 4.342] )T , which is non-negative.
3. Using Theorem 3.3 again, we can compute the enclosure of the solution set of ATB y = cB to be y = ([0.693, 0.806], [16.541, 24.354] )T . According to Theorem 3.4, since ((ATN )y )− ≥ c+ , the optimality condition holds. Thus, the N problem is B-stable. We now use Corollaries 4.1 and 4.2 to avoid the non-optimal region. The results are given in Table 2 and Fig. 3. Obviously, not all of the solution regions of BWC and MILP are completely feasible and completely optimal, respectively, whereas those of IILP and IMILP are completely feasible and optimal. Please cite this article as: M. Allahdadi et al., Improving the modified interval linear programming method by new techniques, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2015.12.037
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M. Allahdadi et al. / Information Sciences xxx (2016) xxx–xxx Table 2 Sub-models and solutions of BWC, MILP, IILP, and IMILP for ILP model (8). Method
Sub-models
BWC
max z+ = 30x1 − 5.5x2
± zopt = [110.71, 172.62]
s.t. 8x1 − 14x2 ≤ 4.2
x± = [5.182, 6.366] 1 opt
x1 + 0.19x2 ≤ 7
Results
x± = [3.338, 4.001] 2 opt
x 1 , x2 ≥ 0 max z− = 26x1 − 6x2 s.t. 10x1 − 12x2 ≤ 3.8 1.1x1 + 0.2x2 ≤ 6.5 x1 , x2 ≥ 0 MILP
− 5.5x− max z+ = 30x+ 1 2 − 14x− ≤ 4.2 s.t. 8x+ 1 2 + 0.2x− ≤7 x+ 1 2
± zopt = [97.96, 171.81]
x± = [4.574, 6.336] 1 opt x± = [3.320, 3.495] 2 opt
, x− ≥0 x+ 1 2
− 6x+ max z− = 26x− 1 2 s.t.10x− − 12x+ ≤ 3.8 1 2
+ 0.19x+ ≤ 6.5 1.1x− 1 2
, x+ ≥ 0,x− ≤ 6.336 x− 1 2 1 x+ ≥ 3.3205,0.19x+ ≤ 0.2 × 3.3205 2 2 IILP
max z− = 26x− − 6x+ 1 2
s.t.10x− − 12x+ ≤ 3.8 1 2 + 0.19x+ ≤ 6.5 1.1x− 1 2
± zopt = [111.38, 165.99]
x± = [5.213, 6.235] 1 opt x± = [3.827, 4.028] 2 opt
, x+ ≥0 x− 1 2
− 5.5x− max z+ = 30x+ 1 2
s.t. 8x+ − 14x− ≤ 4.2 1 2 − x+ + 0.2x ≤7 1 2
x+ , x− ≥ 0,x+ ≥ 5.21 1 2 1
≤ 4.03,0.2x− ≥ 0.19 × 4.03 x− 2 2 IMILP
− 5.5x− max z+ = 30x+ 1 2 s.t. 8x+ − 14x− ≤ 4.2 1 2 + 0.2x− ≤7 x+ 1 2
± zopt = [116.97, 171.81]
x± = [5.305, 6.336] 1 opt x± = [3.320, 3.495] 2 opt
, x− ≥0 x+ 1 2
− 6x+ min z− = 26x− 1 2
s.t.10x− − 12x+ ≥ 3.8 1 2 1.1x− + 0.19x+ ≥ 6.5 1 2
, x+ ≥ 0,x− ≤ 6.336 x− 1 2 1
≥ 3.3205,0.19x+ ≤ 0.2 × 3.3205 x+ 2 2
177 178 179 180
In [34], the infeasible regions of MILP were avoided using an extra constraint of 0.19 × x+ ≤ 0.2 × 3.3205 in the second 2 sub-model. This produced optimal interval solutions of x± = [4.574, 6.336] and x± = [3.320, 3.495], and the optimal value of 1 2 the objective function was [97.96, 171.81]. In contrast, the worst value of the objective function given by the BWC method was 110.71. Therefore, the interval value of the objective function (i.e., [97.96, 171.81]), and hence the interval values of the
T
186
solutions (i.e., ) cannot be optimal. [4.574, 6.336], [3.320, 3.495] MILP outputs a completely feasible solution region that is not completely optimal. This is because the extra constraint 0.19 × x+ ≤ 0.2 × 3.3205 does not guarantee the optimality of the solutions. The non-optimal solution part of MILP is repre2 sented in Fig. 3. Note that the optimal interval values of the objective function in the IILP and IMILP sub-models lie within the optimal intervals of the BWC method.
187
Example 5.2. Consider the ILP model
181 182 183 184 185
max
± ± z± = [2, 2.5]x± 1 − [1, 1.5]x2 + [1.5, 2]x3
± ± s.t. [2, 3]x± 1 + [1.5, 2]x2 + [3, 3.4]x3 ≤ [15, 20] ± ± [4.5, 5.5]x± 1 + [2.5, 3.5]x2 − [1, 1.5]x3 ≤ [5, 10] ± ± [1, 1.6]x± 1 − [6, 7]x2 + [2, 3]x3 ≤ [4, 6] ± ± x± 1 , x2 , x3 ≥ 0.
188
(18)
Model (18) is equivalent to Please cite this article as: M. Allahdadi et al., Improving the modified interval linear programming method by new techniques, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2015.12.037
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11
Fig. 3. Graphical interpretation of the BWC, ILP, MILP, IILP, and IMILP methods for ILP model (8).
max
± ± z± = [2, 2.5]x± 1 − [1, 1.5]x2 + [1.5, 2]x3
± ± ± s.t. [2, 3]x± 1 + [1.5, 2]x2 + [3, 3.4]x3 + x4 = [15, 20] ± ± ± [4.5, 5.5]x± 1 + [2.5, 3.5]x2 − [1, 1.5]x3 + x5 = [5, 10] ± ± ± [1, 1.6]x± 1 − [6, 7]x2 + [2, 3]x3 + x6 = [4, 6] ± ± ± ± ± x± 1 , x2 , x3 , x4 , x5 , x6 ≥ 0.
189 190
A candidate basis for B-stability is B = (1, 2, 3 ), since this is optimal for the characteristic model taking the center values. 1. According to Theorem 3.1, the spectral radius is equal to 0.3377, so AB is regular:
AB = 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211
[2, 3] [4.5, 5.5] [1, 1.6]
[1.5, 2] [2.5, 3.5] [−7, −6]
[3, 3.4] [−1.5, −1] . [2, 3]
2. According to Theorem 3.3, we can compute the enclosure of the solution set of AB xB = b to be xB = [0.6849, 3.2513], [0.6264, 1.3119], [2.9358, 4.1662] )T , which is non-negative. 3. By Theorem 3.3, we can compute the enclosure of the solution set of ATB y = cB to be y = ([0.1198, 0.5903], [0.2155, 0.3441], [0.3716, 0.4513] )T . According to Theorem 3.4, since ((ATN )y )− ≥ c+ , the optimality condition is valid. Thus, N the model is B-stable. We solved model (18) by using the BWC, MILP, IILP, and IMILP methods. The results are given in Table 3. The solutions obtained from the BWC method are not completely feasible. For example, the extreme point (3.1728, 1.0842, 4.1592)T is infeasible, because it does not satisfy the inequalities 2x1 + 1.5x2 + 3x3 ≤ 20 and 4.5x1 + 2.5x2 − 1.5x3 ≤ 10, which are the first and second constraints of the best problem. The solution region obtained with the MILP method is completely feasible, but not completely optimal, because the extreme point (1.0217, 0.7760, 2.8578)T does not satisfy the inequality 3x1 + 2x2 + 3.4x3 ≥ 15, which is the first constraint of the worst problem with the inverse sign. The solution region obtained through the IILP and IMILP methods is completely feasible and optimal. Example 5.3. Suppose that the interval coefficients in ILP model (1) are random intervals. The entries of a± , b± , and c±j ij i for i = 1, 2, . . ., m and j = 1, 2, . . ., n are generated randomly with a uniform distribution in [−1, 5], [5, 10], and [−1, 2], respectively. The lower and upper bounds of the intervals are assumed to have the same sign. The computations were mainly carried out by using MATLAB R2009b, with some interval computations performed by the interval toolbox INTLAB v6 [31]. For given dimensions, the optimal values of the objective function are presented in Table 4. In all cases, the lower bound of the objective function obtained through MILP is less than the lower bound of the objective function given by the BWC method. Therefore, MILP does not provide the optimal solution region. The optimal interval values of the objective function obtained through IILP and IMILP lie in the optimal intervals given by the BWC method. Please cite this article as: M. Allahdadi et al., Improving the modified interval linear programming method by new techniques, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2015.12.037
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M. Allahdadi et al. / Information Sciences xxx (2016) xxx–xxx Table 3 Solutions obtained using BWC, MILP, IILP, and IMILP for ILP model (18). Method
BWC
MILP
± zopt
x±
[4.5625,15.4660]
[4.7780,14.5560]
⎛
⎞
⎛
⎞
[0.7802, 3.1728] ⎝[0.7845, 1.0842]⎠ [3.0856, 4.1592] [1.0217, 2.6000] ⎝[0.7760, 1.0347]⎠ [2.8578, 4.4160]
⎛ IILP
[5.0718,14.4208]
⎛ IMILP
[5.2084,14.5560]
[1.1976, 2.5667]
⎞
⎝[0.8173, 1.0898]⎠ [2.8743, 4.4107]
⎞
[1.2232, 2.6000] ⎝ [0.7760, 1.0347] ⎠ [2.8760, 4.4160]
Table 4 Optimal values of the objective function obtained using BWC, MILP, IILP, and IMILP for Example 5.3. m
n
BWC
MILP
IILP
IMILP
5 100 100 200 500
10 100 150 250 700
[2.13,9.22] [2.51,6.40] [2.85,7.19] [3.07,7.22] [2.98,7.11]
[2.02,8.73] [1.74,6.29] [2.17,7.07] [2.29,7.06] [2.25,6.97]
[2.14,5.80] [2.53,3.91] [2.87,3.53] [3.11,3.82] [3.01,3.33]
[2.79,8.73] [2.96,6.29] [3.78,7.07] [3.94,7.06] [3.20,6.97]
212
6. Conclusion
213
219
In this paper, we analyzed the MILP method of solving ILP problems. A part of the solution region obtained through the MILP method is not completely optimal. To remove this non-optimal part, under the assumption of basis stability, we proposed two improved MILP methods, namely IILP and IMILP. These methods guarantee that the solution regions are completely feasible and optimal. In fact, the solution regions obtained through these methods are not only feasible, but also completely optimal. In future studies, we will attempt to identify new methods for obtaining ILP solution sets without the assumption of basis stability such that the solution region is both feasible and optimal.
220
Acknowledgments
221 222
We would like to thank the anonymous referees for their constructive comments and suggestions that have helped to improve this paper.
223
References
214 215 216 217 218
224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
G. Alefeld, J. Herzberger, Introduction to Interval Computations, Academic Press, New York, 1983. M. Allahdadi, H. Mishmast Nehi, The optimal solutions set of the interval linear programming problems, Optim. Lett. 7 (2013) 893–1911. H. Beeck, Linear Programming with Inexact Data, Technical report tum-isu-7830, Technical University of Munich, Munich, 1978. J.W. Chinneck, K. Ramadan, Linear programming with interval coefficients, J. Oper. Res. Soc. 51 (2000) 209–220. P. Guo, G.H. Huang, L. He, H.L. Li, Interval-parameter fuzzy-stochastic semi-infinite mixed-integer linear programming for waste management under uncertainty, Environ. Model. Assess 14 (2009) 521–537. L. He, G. Huang, G. Zeng, H. Lu, Fuzzy inexact mixed-integer semiinfinite programming for municipal solid waste management planning, J. Environ. Eng. 134 (2008) 572–581. L. He, G.H. Huang, G.M. Zeng, H.W. Lu, Identifying optimal regional solid waste management strategies through an inexact integer programming model containing infinite objectives and constraints, Waste Manage. 29 (2009) 21–31. M. Hladik, AE solutions and AE solvability to general interval linear systems, Linear Algebra Appl. 465 (0) (2015) 221–238. M. Hladik, How to determine basis stability in interval linear programming, Optim. Lett. 8 (2014) 375–389. M. Hladik, Interval linear programming: A survey, in: Z.A. Mann (Ed.), Linear Programming, New Frontiers in Theory and Applications, chapter 2, Nova Science Publishers, New York, 2012, pp. 85–120.2012 M. Hladik, Optimal value range in interval linear programming, Fuzzy Optim. Decis. Mak. 8 (3) (2009) 283–294. G.H. Huang, B.W. Baetz, G.G. Patry, A grey linear programming approach for municipal solid waste management planning under uncertainty, Civ. Eng. Environ. Syst. 9 (1992) 319–335. G.H. Huang, B.W. Baetz, G.G. Patry, Grey integer programming: an application to waste management planning under uncertainty, Eur. J. Oper. Res. 83 (1995) 594–620. G.H. Huang, B.W. Baetz, G.G. Patry, Trash-flow allocation: planning under uncertainty, Interfaces 28 (1998) 36–55. G.H. Huang, R.D. Moore, Grey linear programming, its solving approach, and its application, Int. J. Syst. Sci. 24 (1993) 159–172.
Please cite this article as: M. Allahdadi et al., Improving the modified interval linear programming method by new techniques, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2015.12.037
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ARTICLE IN PRESS M. Allahdadi et al. / Information Sciences xxx (2016) xxx–xxx
245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 Q2268 269 270 271 272
[m3Gsc;January 13, 2016;15:0] 13
[16] C. Jansson, A self-validating method for solving linear programming problems with interval input data, in: U. Kulisch, H.J. Stetter (Eds.), Scientific Computation with Automatic Result Verification. Computing Suppl., 6, 1988, pp. 33–45. [17] C. Jansson, S.M. Rump, Rigorous solution of linear programming problems with uncertain data, Z. Oper. Res. 35 (2) (1991) 87–111. [18] J. Konickova, Sufficient condition of basis stability of an interval linear programming problem, ZAMM. Z. Angew. Math. Mech. 81 (3) (2001) 677–678. [19] H.W. Lu, M.F. Cao, Y. Wang, X. Fan, L. He, Numerical solutions comparison for interval linear programming problems based on coverage and validity rates, Appl. Math. Modell. 38 (2014) 1092–1100. [20] H.W. Lu, G.H. Huang, L. He, A semi-infinite analysis-based inexact two-stage stochastic fuzzy linear programming approach for water resources management, Eng. Opt. 41 (2009) 73–85. [21] H.W. Lu, G.H. Huang, L. Liu, L. He, An interval-parameter fuzzy-stochastic programming approach for air quality management under uncertainty, Environ. Eng. Sci. 25 (2008) 895–909. [22] H.W. Lu, G.H. Huang, Z.F. Liu, L. He, Greenhouse gas mitigation-induced rough-interval programming for municipal solid waste management, J. Air. Waste. Manage. Assoc. 58 (2008) 1546–1559. [23] B. Machost, Numerische Behandlung des Simplexverfahrens mit Intervallanalytischen Methoden, Technical report 30, Berichte der Gesellschaft für Mathematik und Datenverarbeitung, 1970. 54 pages, Bonn [24] A. Mostafaee, M. Hladik, M. Cerny, Inverse linear programming with interval coefficients, J. Comput. Appl. Math. 292 (2016) 591–608. [25] F. Mraz, On infimum of Optimal Objective Function Values in Interval Linear Programming, Technical report kam series, Department of Applied Mathematics, Charles University, Prague, 1996. [26] J. Rex, J. Rohn, Sufficient conditions for regularity and singularity of interval matrices, SIAM J. Matrix Anal. Appl. 20 (2) (1998) 437–445. [27] J. Rohn, Cheap and tight bound: The recent result by e. hansen can be made more efficient, Interval Comput. 4 (1993) 13–21. [28] J. Rohn, Forty necessary and sufficient conditions for regularity of interval matrices: A survey, Electron. J. Linear Algebra 18 (2009) 500–512. [29] J. Rohn, Interval linear programming, in: M. Fiedler, et al. (Eds.), Linear Optimization Problems with Inexact Data, Chapter 3, Springer, New York, 2006, pp. 79–100. [30] J. Rohn, Stability of the optimal basis of a linear program under uncertainty, Oper. Res. Lett. 13 (1) (1993) 9–12. [31] S.M. Rump, INTLAB - INTerval LABoratory, in: T. Csendes (Ed.), Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, 1999, pp. 77–104. [32] S.C. Tong, Interval number, fuzzy number linear programming, Fuzzy Sets Syst. 66 (1994) 301–306. [33] X. Wang, G. Huang, Violation analysis on two-step method for interval linear programming, Inf. Sci. 281 (2014) 85–96. [34] F. Zhou, G.H. Huang, G. Chen, H. Guo, Enhanced-interval linear programming, Eur. J. Oper. Res. 199 (2009) 323–333.
Please cite this article as: M. Allahdadi et al., Improving the modified interval linear programming method by new techniques, Information Sciences (2016), http://dx.doi.org/10.1016/j.ins.2015.12.037