In Chapter 1, stability radius%bs(p) of an optimal digraph Gs has been investigated which denotes the largest quantity of independent variations of the processing times pi of oper-ations i ∈ Q within the interval [0, ∞) such that digraph Gs remains ‘the best’ (i.e., the weighted digraph Gs(p) has the minimal critical weight) among all feasible digraphs Λ(G) (see Definition 1.2 at page 16).
From Example 3.1 it follows that for solving problem G/ai≤ pi≤ bi/Cmax, we need a more general notion of a stability radius since the processing time of operation i ∈ Q falls within the given closed interval [ai, bi], 0 ≤ ai ≤ bi, and competitive digraphs have to belong to a subset B of set Λ(G). The following generalization of stability radius%bs(p) (we call it relative stability radius) is defined by considering the closed interval [ai, bi] instead of [0, ∞) and considering set B ⊆ Λ(G) instead of the whole set Λ(G). In Definition 3.2, lps denotes the critical weight of the weighted digraph Gs(p), p ∈ T , see equality (1.15) at page 19.
Definition 3.2 Let digraph Gs ∈ B ⊆ Λ(G) have the minimal critical weight lps0 for each vector p0 ∈ O%(p) ∩ T among all digraphs from the given set B
lsp0 = min{lpk0 : Gk ∈ B}. (3.4) The maximal value of radius % of such a ball O%(p) is denoted by %bBs(p ∈ T ) and is called relative stability radius of digraph Gs with respect to polytope T (for criterion Cmax).
Note that relativity of%bBs(p ∈ T ) is defined not only by polytope T of feasible vectors, but also by set B of feasible digraphs. From Definition 1.2 and Definition 3.2, if follows:
%bs(p) =%bΛ(G)s (p ∈ Rq+).
Thus, relative stability radius is equal to the maximal error of the given processing times pi (ai ≤ pi ≤ bi, i ∈ Q) within which the ‘superiority’ of digraph Gs is still preserved over the given subset B of feasible digraphs.
The following two extreme cases of relative stability radius are of particular importance for solving problem G/ai≤ pi≤ bi/Cmax. On the one hand, if for any positive real number
> 0 which may be as small as desired, there exist vector p0 ∈ O(p) ∩ T and digraph Gk∈ B such that lsp0 > lpk0, we obtain zero relative stability radius:
%bBs(p ∈ T ) = 0.
On the other hand, if lps0 ≤ lkp0 for any vector p0 ∈ T and for any digraph Gk ∈ B, we obtain infinitely large relative stability radius:
%bBs(p ∈ T ) = ∞.
Note that even in the case of finite upper bonds (bi < ∞, i ∈ Q), i.e., when the maximal error of processing time pi for each operation i ∈ Q is restricted by
max= max{{pi− ai, bi− pi} : i ∈ Q}, (3.5) value of%bBs(p ∈ T ) may be infinitely large as it follows from Definition 3.2. Deterministic problem G//Cmax with vector p of the processing times and optimal digraph Gs provides such a trivial example with an infinitely large relative stability radius%bBs(p ∈ T ). Indeed, if ai = pi = bi for each operation i ∈ Q, then polytope T degenerates into a single point:
T = {p}
and therefore from inclusion p0 ∈ O%(p) ∩ T it follows that vector p0 mentioned in Defini-tion 3.2 is definitely equal to vector p, for which digraph Gs is optimal.
To characterize the extreme values of%bBs(p ∈ T ), we define the following binary relation which generalizes the dominance relation used in Chapter 1.
Definition 3.3 Path ν dominates path µ in set T if and only if for any vector x = (x1, x2, . . . , xq) ∈ T the following inequality holds:
lx(µ) ≤ lx(ν). (3.6)
Note that binary relation introduced in Definition 3.3 is an extension of the dominance relation introduced in Definition 1.3 (see page 17) in the sense that path ν dominates path µ in any set T ⊆ Rq+ (i.e., due to Definition 3.3) if path ν dominates path µ (due to Definition 1.3). Indeed, if [µ] ⊂ [ν], then inequality lx(µ) ≤ lx(ν) holds for any vector x ∈ Rq+. Note also that both dominance relations coincide at least when ai = 0 and bi = ∞ for each operation i ∈ Q (it is easy to see that inclusion [µ] ⊂ [ν] holds if and only if inequality (3.6) holds for ai = 0 and bi = ∞, i ∈ Q). Moreover, in this case equality lx(µ) = lx(ν) is achieved only if xi = ai = 0 for any operation i ∈ [ν] \ [µ].
Thus, we conclude that the dominance relation introduced in Definition 1.3 is a special case of the dominance relation defined by inequality (3.6) when T is equal to the whole space Rq+ (i.e., ai = 0 and bi = ∞ for each operation i ∈ Q). Hence, the phrase “path ν dominates path µ” is identical to the phrase “path ν dominates path µ in Rq+”.
The following lemma gives a simple criterion for the dominance relation defined by inequality (3.6) in Definition 3.3.
Lemma 3.1 Path ν dominates path µ in set T if and only if the following inequality
Proof. By subtracting all common variables from the left-hand side and the right-hand side of inequality (3.6) and taking into account that ai ≤ bi for each operation i ∈ Q, we obtain that inequality (3.6) is equivalent to the following ones:
X
i∈[µ]\[ν]
xi ≤ X
j∈[ν]\[µ]
xj for any xi with ai ≤ xi ≤ bi, i ∈ [ν] ∪ [µ]. (3.8)
Vector x ∈ T satisfies inequalities (3.8) if and only if inequality (3.7) holds since we have: On the basis of the above path domination, we can introduce the following dominance relation of path sets.
Definition 3.4 Set of paths Hk dominates set of paths Hs in T if and only if for any path µ ∈ Hs, there exists a path ν ∈ Hk, which dominates path µ in set T.
The following statement gives a simple sufficient condition, when domination of sets of paths does not hold.
Lemma 3.2 Set of paths Hk does not dominate set of path Hs in T if there exists a path µ ∈ Hs such that system
P
i∈[ν]\[µ]ai <Pj∈[µ]\[ν]bj,
ai ≤ xi ≤ bi, i ∈ Q, (3.9)
has a solution for any path ν ∈ Hk.
Proof. From Definition 3.4 it follows that set of paths Hkdoes not dominate set of paths Hs in T if there exists a path µ∗ ∈ Hs such that there is no path ν ∈ Hk which dominates path µ∗ in set T . This means that inequality (3.6) is violated for path µ∗ ∈ Hs for some vector x0 ∈ T , i.e., system
lx(ν) < lx(µ),
ai ≤ xi ≤ bi, i ∈ Q, (3.10)
has a solution for any path ν ∈ Hk. Furthermore, system (3.10) is consistent if and only if it has the following solution:
xi = x0i =
ai, if i ∈ [µ∗] \ [ν],
bi, if i ∈ [ν] \ [µ∗]. (3.11) It is easy to see that vector x according to (3.11) is a solution of system (3.10) if and only if condition (3.7) does not hold for any vertex i ∈ [ν] ∪ [µ∗]. In other words, vector x0 = (x01, x02, . . . , x0q) ∈ T and path µ∗ ∈ T provide a solution of the equivalent system (3.9).
3
Obviously, if Hk = Hk(p), we have Hk(p0) ⊆ Hk = Hk(p) for any vector p0 ∈ R+q of the processing times. The following lemma shows that set of the critical paths is not expanded for small variations of the processing times.
Lemma 3.3 If Hk 6= Hk(p), the inclusion Hk(p0) ⊆ Hk(p) holds for any vector p0 ∈ O(p) ∩ Rq+ with real number k> > 0 defined as follows:
k= 1 q
hlpk− max{lp(ν) : ν ∈ Hk\ Hk(p)}i. (3.12)
Proof. Since Hk\ Hk(p) 6= ∅, we can consider any path ν? ∈ Hk with lp(ν?) = max{lp(ν) : ν ∈ Hk\ Hk(p)}.
From (3.12) it follows that lpk−lp(ν?) = q·k, and therefore, to make the difference lpk−lp(ν?) equal to zero, we need a vector p0 with a distance from the vector p greater than or equal to k : d(p, p0) ≥ k. But due to condition of Lemma 3.3, we have d(p, p0) ≤ < k. Consequently, ν? ∈ H/ k(p0).
Since for any path ν ∈ Hk\ Hk(p) with lp(ν) < lp(ν?) the difference lkp− lp(ν) is still greater than the product q · k, such a path ν cannot belong to set Hk(p0).
3 Next, we present a generalization of Theorem 1.1 with the necessary and sufficient conditions for a zero stability radius to the case of a zero relative stability radius.
Theorem 3.1 Let digraph Gs have the minimal critical weight lsp, p ∈ T, within the given subset B ⊆ Λ(G) of feasible digraphs. Then equality %bBs(p ∈ T ) = 0 holds if and only if there exists a digraph Gk ∈ B such that lsp = lpk, k 6= s, and set of paths Hk(p) does not dominate set of paths Hs(p) in T .
Proof. Sufficiency (if ). Let conditions of Theorem 3.1 hold: There exists a digraph Gk∈ B such that lps = lpk, k 6= s, and Hk(p) does not dominate set Hs(p) in T. We show that %bBs(p ∈ T ) < for any given > 0 which may be as small as desired.
Since set Hk(p) does not dominate set Hs(p) in T, there exists a path µ∗ ∈ Hs(p) such that no path ν ∈ Hk(p) dominates path µ∗ in set T , i.e., system (3.10) has a solution for any path ν ∈ Hk(p). First, we make the following remark.
Remark 3.1 From the consistency of system (3.10), it follows that for the considered problem G/ai ≤ pi ≤ bi/Cmax, the trivial case with ai = bi for each i ∈ Q does not hold. Indeed, in this case the first inequality in (3.10) is transformed into inequality lp(ν) < lp(µ∗) which is certainly wrong: lp(ν) = lpk= lsp = lp(µ∗).
We construct a vector p0 = (p01, p02, . . . , p0q) with the following components:
p0i =
pi + 0, if i ∈ [µ∗], pi 6= bi,
pi − 0, if i ∈ {∪ν∈Hk(p)[ν]}\[µ∗], pi 6= ai, pi, otherwise,
(3.13)
where 0 is chosen as a strictly positive real number less than both value and value
min = max{0, min{min{pi − ai : pi > ai, i ∈ Q}, min{bi− pi : bi > pi, i ∈ Q}}}.
We can also choose 0 less than k> 0 defined in (3.12). More precisely, if Hk 6= Hk(p), then k > 0, and we can choose 0 such that 0 < 0 < min{, k, min}. Otherwise, if Hk = Hk(p), we choose 0 such that 0 < 0 < min{, min}. Such choices are possible since in both cases, inequality min > 0 holds due to Remark 3.1.
The following arguments are the same for both cases of the choice of 0 except the ‘last step’ since Hk\ Hk(p) = ∅ in the latter case.
Since system (3.10) has a solution for each path ν ∈ Hk, the first inequality in (3.10) lx(ν) < lx(µ∗)
has a solution for x ∈ T which implies that inclusion [µ∗] ⊂ [ν] does not hold for any path ν ∈ Hk(p). Therefore, from the equalities lp(ν) = lpk = lsp = lp(µ∗) and (3.13), we can conclude that vector p0 is a solution of system (3.10) for each path ν ∈ Hk(p). In other words, vector p0 is a solution of the following system of inequalities:
lx(ν) < lx(µ∗), ν ∈ Hk(p), ai ≤ xi ≤ bi, i ∈ Q.
Thus, we obtain inequality lp0(ν) < lp0(µ∗) for each path ν ∈ Hk(p), and therefore max{lp0(ν) : ν ∈ Hk(p)} < lp0(µ∗). (3.14) The ‘last step’ in the proof of sufficiency is as follows. Since p0 ∈ O0(p) ∩ Rq+ with 0 < 0 < k, due to Lemma 3.3 we obtain Hk(p0) ⊆ Hk(p) and, as a result,
lp0(τ ) < lpk0 = max{lp0(ν) : ν ∈ Hk(p)} (3.15) for each path τ ∈ Hk\ Hk(p).
From inequalities (3.14) and (3.15), it follows that lpk0 < lps0. Taking into account that d(p0, p) = 0 < , we conclude that %bBs(p ∈ T ) < .
Necessary (only if ). We prove necessity by contradiction. Let us suppose that%bBs(p ∈ T ) = 0 but condition of Theorem 3.1 does not hold. The following two cases (i) and (ii) of violating condition of Theorem 3.1 may hold.
(i) There does not exist a digraph Gk∈ B such that lsp = lpk, k 6= s.
In the trivial case when B = {Gs}, we have %bBs(p ∈ T ) = ∞ due to Definition 3.2.
Let B \ {Gs} 6= ∅. Then we calculate the following real number:
∗ = 1
qmin{lpt − lsp : Gt ∈ B, t 6= s} (3.16) which is strictly positive since inequality lps < lpt holds for each digraph Gt ∈ B, t 6= s.
Arguing in a similar way as in the proof of Lemma 3.3, we can show that the difference lpt − lps cannot become negative when vector p is replaced by an arbitrary vector p0 ∈ O∗(p) ∩ Rq+. Next, we show that the difference lpt − lsp cannot become negative when vector p is replaced by an arbitrary vector p0 ∈ O∗(p) ∩ T ⊆ Rq+ with 0 < ∗ < k.
Since Hk\Hk(p) 6= ∅, we can consider any path ν? ∈ Hk with lp(ν?) = max{lp(ν) : ν ∈ Hk\Hk(p)}.
From (3.16) it follows that lkp0− lps0 ≥ q · ∗, and therefore, to make the difference lpk0− lsp0 equal to zero, we need a vector p0 with a distance from the vector p greater than or equal to k : d(p0, p0) ≥ k. However, due to conditions of Lemma 3.3, we have d(p, p0) ≤ ∗ < k. Consequently, ν? ∈ H/ k(p0).
Since for any digraph Gt∈ B, with lp0(ν) < lp0(ν?) the difference lpk0−lsp0 is still greater than the product q · ∗, such a path ν cannot belong to the set Hk(p0). So we conclude that digraph Gs remains ‘the best’ (perhaps one of the ‘best’) within the set B for any vector p0 of the processing times. Due to Definition 3.2, we have %bBs(p ∈ T ) ≥ ∗ > 0 which contradicts the above assumption of%bBs(p ∈ T ) = 0.
(ii) There exists a digraph Gk∈ B such that lsp = lpk, k 6= s, and for any such digraph Gk, set of paths Hk(p) dominates set of paths Hs(p) in T .
In this case, we can take any that satisfies the following inequalities:
0 < < minnmin{k : lpk= lps, Gk ∈ B}, 1
qmin{ltp− lsp : ltp > lps, Gt∈ B}o. Due to inequality > s, we get from Lemma 3.3 that equalities
lps0 = max
µ∈Hs(p0)lp0(µ) = max
µ∈Hs(p)lp0(µ) (3.17) hold for any vector p0 ∈ O(p) ∩ Rq+. The statement that for any digraph Gk ∈ B, k 6= s, with lsp = lpk set of paths Hk(p) dominates set of paths Hs(p) in T means that for any path µ ∈ Hs(p), there exists a path ν∗ ∈ Hk(p) such that system
lx(ν∗) < lx(µ), ai ≤ xi ≤ bi, i ∈ Q, has no solution. Therefore, inequality
lx(µ) ≤ lx(ν∗) (3.18)
holds for any vector x ∈ T. Due to inequality (3.18) and taking into account that < k and < s, we obtain the following inequality using Lemma 3.3:
µ∈Hmaxs(p)lp0(µ) ≤ max
ν∈Hk(p)lp0(ν). (3.19)
Thus, due to (3.17) and (3.19), we obtain inequality lsp0 ≤ max
ν∈Hk(p)lp0(ν) (3.20)
for any digraph Gk ∈ B, lps = lpk, k 6= s. Since
< 1
q min{ltp− lps : ltp > lps, Gt∈ B},
inequality lpt > lps implies inequality lpt0 > lps0. Taking into account (3.20), we conclude that lps0 ≤ lpk0 for any digraph Gk ∈ B and for any vector p0 ∈ T with d(p, p0) ≤ .
Consequently, %bBs(p ∈ T ) ≥ > 0, which contradicts the assumption of %bBs(p ∈ T ) = 0.
3
Theorem 3.1 directly implies the following statement.
Corollary 3.1 If Gs ∈ B is unique optimal digraph for vector p ∈ T of the processing times, then %bBs(p ∈ T ) > 0.
From Theorem 3.1 we obtain the following lower bound for relative stability radius.
Corollary 3.2 If Gs ∈ B and lps = min{lpk : Gk ∈ B}, then %bBs(p ∈ T ) ≥ ∗ with ∗ calculated according to (3.16).
Proof. If there exists a digraph Gk ∈ B such that lps = lkp, k 6= s, the equality %bBs(p ∈ T ) ≥ ∗ = 0 holds due to Definition 3.2. Otherwise, inequality %bBs(p ∈ T ) ≥ ∗ follows from the above proof of necessity of Theorem 3.1 (see case (i)).
3 Theorem 3.1 identifies a digraph Gs ∈ Λ(G) whose ‘superiority’ within the set B is unstable: Even a very small change in the processing times can make another digraph from the set B to be ‘better’ than Gs. The following theorem identifies a digraph Gs
whose ‘superiority’ within the set B in the polytope T is ‘absolute’: Any changes of the processing times within the polytope T cannot make another digraph from the set B to be ‘better’ than digraph Gs.
Theorem 3.2 For digraph Gs ∈ B, we have %bBs(p ∈ T ) = ∞ if and only if for any digraph Gt∈ B, t 6= s, set of paths Ht dominates set of paths Hs\ H in T.
Proof. Sufficiency. Let % be any positive number (as large as desired). We take any vector p ∈ O%(p) ∩ T ⊆ Rq+ and consider a path µ ∈ Hs such that lsp = lp(µ).
(j) If µ ∈ H, then inequality lps = lp(µ) ≤ ltp holds for any digraph Gt∈ Λ(G).
(jj) If µ ∈ Hs\H, then due to condition of Theorem 3.2, it follows that for any digraph Gt∈ B, t 6= s, there exists a path ν∗ ∈ Ht such that inequality
lx(µ) ≤ lx(ν∗)
holds for any vector x ∈ T (and for the vector p as well). Therefore, we have lsp = lp(µ) <
lp(ν∗) ≤ ltp. Thus, in both above cases (j) and (jj) we have lps = min{ltp : Gt ∈ B}.
Necessity. We prove necessity by contradiction. Let us suppose that %bBs(p ∈ T ) = ∞, but there exists a digraph Gt ∈ B, t 6= s, such that set of paths Ht does not dominate set of paths Hs\ H in T . Thus, there exists a path µ0 ∈ Hs\ H such that for any path ν ∈ Ht, system of inequalities
lx(ν) < lx(µ0),
ai ≤ xi ≤ bi, i ∈ Q, (3.21)
has a solution. Therefore, due to Lemma 3.2, inequality
X
i∈[ν]\[µ0]
ai < X
j∈[µ0]\[ν]
bj (3.22)
holds. We consider the vector p∗ = (p∗1, p∗2, . . . , p∗q) ∈ T with
p∗i =
ai, if i ∈ {S[ν]∈Ht[ν]} \ [µ0], bi, if i ∈ [µ0],
pi otherwise.
Adding to the left-hand side and to the right-hand side of (3.22) the value Pj∈[ν]∩[µ0]bj, we obtain that inequality
X
i∈[ν]\[µ0]
ai+ X
j∈[ν]∩[µ0]
bj < X
j∈[µ0]
bj
holds. Thus, we can conclude that vector p∗is a solution of the system of linear inequalities obtained by joining systems (3.21) for all paths ν ∈ Ht, i.e., we obtain
(lp∗(ν) < lp∗(µ0), ν ∈ Ht, ai ≤ xi ≤ bi, i ∈ Q.
Therefore, lpt∗ < lp∗(µ0) ≤ lps∗, and hence, we get a contradiction to the above assumption:
%bBs(p ∈ T ) < d(p∗, p) ≤ max < ∞.