Characterization of a Solution

A characterization of a solution Λ of problem J /ai≤ pi≤ bi/Φ which is a proper subset of the set Λ(G), Λ ⊂ Λ(G), may be obtained on the basis of the dominance relation _D introduced in Section 3.4. Next, we prove necessary and sufficient conditions for a set of feasible digraphs to be a solution of the problem J /ai≤ pi≤ bi/^PCi.

Theorem 3.4 The set Λ ⊂ Λ(G) is a solution of problem J /ai≤ pi≤ bi/Φ if and only if there exists a finite covering of the polytope T by convex closed sets D_j ⊂ R^q₊:

T ⊆

d

[

j=1

D_j, d ≤ |Λ|,

such that for any digraph G_k ∈ Λ(G) and for any set D_j, j = 1, 2, . . . , d, there exists a digraph G_s∈ Λ for which dominance relation G_s _Dj G_k holds.

Proof. Sufficiency. For any fixed vector p ∈ T, one can find a set Dj, 1 ≤ j ≤ d, such that p ∈ D_j. From the condition of Theorem 3.4, it follows that for any digraph G_k ∈ Λ(G), there exists a digraph G_s such that dominance relation G_s _Dj G_k holds.

Hence, we have Φ^p_s ≤ Φ^p_k and so inequality min{Φ^p_s : Gs ∈ Λ ⊆ Λ(G)} ≤ Φ^p_k holds for each k = 1, 2, . . . , λ. Consequently, for any vector p ∈ T of the processing times, set Λ contains an optimal digraph.

Necessity. Let set Λ ⊆ Λ(G) be a solution of problem J /ai≤ pi≤ bi/Φ. We define a subset Λ⁰ of the set Λ such that each digraph G_s ∈ Λ⁰ is optimal for at least one vector p ∈ T of the processing times.

For each digraph G_s ∈ Λ, one can define its stability region, i.e., the set of all vectors p ∈ T ⊆ R₊^q for which digraph G_s is optimal. Let D_s be the intersection of the stability region of the digraph G_s with the polytope T :

Ds = {p ∈ R₊^q : Φ^p_s ≤ Φ^p_k, k = 1, 2, . . . , λ} ∩ T. (3.38) Since Λ⁰ is a solution, we have

T ⊆

|Λ⁰|

[

j=1

D_j ⊂ R^q₊

and for each digraph Gk ∈ Λ(G) and each set Ds, the dominance relation Gs Ds Gk

holds. The inclusion G_s ∈ Λ⁰ implies D_s6= ∅. From inequality (3.38) it follows that D_s is a closed set.

Note that, if digraph Gs is optimal for the vector p, it remains optimal for a feasible vector αp with any positive real number α > 0. Consequently, the stability region is convex set and so set D_s is convex as the intersection of convex sets.

3 Theorem 3.4 implies the following corollary which characterizes a single-element solu-tion of problem J /a_i≤ p_i≤ b_i/Φ, which is necessarily a minimal solution.

Corollary 3.6 The equality Λ^T(G) = {G_s} holds if and only if the dominance relation G_s_T G_k holds for any digraph G_k ∈ Λ(G).

A minimal solution which includes more than one digraph may be characterized on the basis of the strong dominance relation ≺_D as follows.

Theorem 3.5 Let set Λ^∗(G) be a solution to problem J /ai≤ pi≤ bi/Φ with |Λ^∗(G)| ≥ 2.

This solution is minimal if and only if for each digraph G_s ∈ Λ^∗(G), there exists a vector p^(s) ∈ T such that the strong dominance relation G_s ≺_p(s) G_k holds for each digraph Gk∈ Λ^∗(G) \ {Gs}.

Proof. Sufficiency. If the condition of Theorem 3.5 holds, then for any digraph Gs ∈ Λ^∗(G), the set Λ^∗(G) \ {G_s} is no longer a solution to problem J /a_i≤ p_i≤ b_i/Φ.

Indeed, for the above vector p^(s) ∈ T , inequality Φ^p_s^(s) < Φ^p_k^(s) holds for each digraph Gk ∈ Λ^∗(G) \ {Gs}. It follows that Gs is the unique optimal digraph to problem J //Φ with vector p^(s) of the processing times. Therefore, solution Λ^∗(G) is minimal.

Necessity. We assume that Λ^∗(G) is a minimal solution but the condition of Theo-rem 3.5 does not hold, i.e., there exists a digraph G_s ∈ Λ^∗(G) such that for each vector p^(s)∈ T , there exists a digraph G_k∈ Λ^∗(G)\{G_s} for which the strong dominance relation G_s ≺_p(s) G_k does not hold, i.e., we have Φ^p_s ≥ Φ^p_k. It follows that the set Λ^∗(G) \ {G_s} is also a solution of problem J /a_i≤ p_i≤ b_i/Φ (since the set Λ^∗(G) is supposed to be a solution). Thus, we get a contradiction to the assumption that solution Λ^∗(G) is minimal.

3 Section 3.6 deals with different algorithms for finding a solution and a minimal so-lution on the basis of an explicit or an implicit enumeration of feasible digraphs. All algorithms developed are based on the fact that a digraph G_s ∈ Λ(G) being optimal for the fixed vector p ∈ R₊^q of the processing times, generally remains optimal within some neighborhood of the point p in the space R^q₊. In other words, digraph Gs dominates all digraphs in a neighborhood of the point p. We consider the closed ball O_r(p) ⊂ R^q with the center p ∈ T and the radius r > 0 as the neighborhood of the point p ∈ T ⊂ R₊^q in the space R^q. Next, we rewrite some basis notions using dominance relation D.

The closed ball Or(p) is called a stability ball of the digraph Gs if this digraph dom-inates all digraphs G_k ∈ Λ(G) in the polytope T^∗ = O_r(p) ∩ T, i.e., if G_s _T^∗ G_k for each digraph G_k ∈ Λ(G) (in this case, from Corollary 3.6 it follows that Λ^T∗(G) = {G_s}).

As it was noted in Section 3.2, the radius r of a stability ball may be interpreted as the error of the given processing times p = (p_1,1, p_1,2, . . . , p_nnn) ∈ R₊^q such that for all variable processing times x = (x_1,1, x_1,2, . . . , x_nnn) ∈ R^q₊ with p_ij − r ≤ x_ij ≤ p_ij + r digraph Gs remains the best. The maximal value of such a radius is of particular importance for finding a minimal solution Λ^T(G). Similarly to Definition 3.2 of the relative stability radius for the makespan criterion, we give the definition of the relative stability radius for the mean flow time criterion.

Definition 3.6 Assume that for each vector p⁰ ∈ O_%(p) ∩ T digraph G_s ∈ B ⊆ Λ(G) with the vector p⁰ of weights has the minimal critical sum of weights L^p_s⁰ among all digraphs of the set B. The maximal value of the radius % of such a ball O_%(p) is denoted by %^B_s(p ∈ T ) and is called the relative stability radius of the digraph G_s with respect to the polytope T for criterion ^PC_i.

Remark 3.6 From Definition 3.5 and Definition 3.6, it follows that the relative stability radius %^B_s(p ∈ T ) of the digraph G_s∈ B is equal to the maximal value of the radius % of a ball O_%(p) such that for each digraph G_k ∈ B ⊆ Λ(G) dominance relation G_s _T^∗ G_k holds where T^∗ = O_%(p) ∩ T.

Similarly to Section 1.4 (which deals with the stability radius %_s(p) (see conditions (1.30) and (1.31) at page 31), to find the relative stability radius %^B_s(p ∈ T ) for problem J /ai≤ pi≤ bi/^PCi it is sufficient to construct a vector x = (x1,1, x1,2, . . . , xnnn) ∈ T ⊆ R₊^q

which satisfies the following three conditions.

(b^∗) For any given real  > 0, which may be as small as desired, there exists a vector p^∈ T such that d(x, p^) =  and L^p_s^ > L^p_k^, i.e., inequality

(c^∗) The distance d(p, x) achieves its minimal value among the distances between the vector p and the other vectors in the polytope T which satisfy both above conditions (a^∗) and (b^∗).

Next, we describe the calculation of the relative stability radius %^B_s(p ∈ T ) using the above notation of the dominance relation. To this end, we prove Lemma 3.5 below about the dominance relation _T, and then we derive a formula for the calculation of the relative stability radius %^B_s(p ∈ T ) which is presented in Theorem 3.6.

If Λ^T(G) = {Gs}, then digraph Gs dominates all digraphs in the polytope T (see Corollary 3.6). In such a case, we assume that %^Λ(G)_s (p ∈ T ) = ∞, since digraph G_s remains the best for all variable feasible vectors x ∈ T of the processing times. Otherwise, there exists a digraph Gk ∈ Λ(G) such that dominance relation GsT Gk does not hold, and from Corollary 3.6 and Remark 3.6, it follows that the stability radius %^Λ(G)_s (p ∈ T ) has to be finite, i.e., there exists a vector p^ ∈ T such that inequality (3.41)

L^p_s^ > L^p_k^ (3.41)

holds. To calculate the stability radius %^B_s(p ∈ T ), B ⊆ Λ(G), we will consider digraphs G_k ∈ B such that dominance relation G_s _T G_k does not hold, and for each of these digraphs Gk, we will look for the vector p^ ∈ T which is the closest to p, among all vectors for which inequality (3.41) holds (see condition (c^∗)). The following lemma allows to restrict the set of digraphs G_k∈ B which have to be considered for any regular criterion.

Lemma 3.5 Digraph G_s ∈ B dominates digraph G_k ∈ B in the polytope T if (only if ) the following inequality (3.42) holds (inequalities (3.43) hold, respectively):

L^b_s≤ L^a_k (3.42)

(L^a_s ≤ L^a_k, L^b_s≤ L^b_k). (3.43) Proof. Sufficiency. Since the objective function is non-decreasing, it follows from

in-equality (3.42) that

for any vector x ∈ T . Therefore, dominance relation Gs T Gk holds.

Necessity. Dominance relation G_s_T G_k means that inequality L^x_s ≤ L^x_k holds for any vector x ∈ T and thus for both vectors a ∈ T and b ∈ T , i.e., inequalities (3.43) hold.

3 The test of inequalities (3.42) and (3.43) takes O(q²) elementary steps, however, there is a ‘gap’ between the necessary and sufficient conditions of Lemma 3.5, if L^a_s 6= L^b_s. To overcome this gap for problem J /a_i ≤ p_i≤ b_i/^PC_i, we are forced to compare the sets in order to obtain vector x with

X

µ∈Ω^v_s

l^x(µ) = ^X

ν∈Ω^u_k

l^x(ν). (3.45)

It is easy to convince that equality (3.45) holds for the vector x obtained from the vector p by adding the value r calculated in (3.44) to all components p_ij with n_ij(Ω^u_k) < n_ij(Ω^v_s) and by subtracting the same value r from all components p_ij with n_ij(Ω^u_k) > n_ij(Ω^v_s). Note that for the above vector x, the inclusion x ∈ T may not hold. To guarantee this inclusion, we have to look for a vector x in the form x = p(r) = (p_1,1(r), p_1,2(r), . . . , p_nnn(r)), where

k,Ω^v_s denote the minimal distance between the given vector p ∈ T and the desired vector x = p(r) ∈ T for which equality (3.45) holds: r_Ω^u

Note that each value ∆^ij_α(Ω^v_s, Ω^u_k) is calculated according to (3.47) for all different opera-tions O_ij, and the subscript α = 1, 2, . . . , |N (Ω^v_s, Ω^u_k)| indicates the location of value (3.47) in the sequence (3.48). We define value

N_α(∆) = |n_ij(Ω^u_k) − n_ij(Ω^v_s)|

for each ∆^ij_α(Ω^v_s, Ω^u_k), α = 1, 2, . . . , |N (Ω^v_s, Ω^u_k)|, and we assume that ∆^ij₀(Ω^v_s, Ω^u_k) = 0 and N₀(∆) = 0.

From (3.46) and (3.48), it follows that equality (3.49) holds:

r_Ω^v_s,Ω^u take the minimum of the maximum obtained:

r^B_ks= min

Note that, if there exists a vector x ∈ T such that equality L^x_s = L^x_k holds (see condition (a^∗)), nevertheless it may be that there exists no vector p^ ∈ T defined in condition (b^∗) such that L^p_s^ > L^p_k^. However, as follows from Definition 3.5, only inequality (3.40) guaranties that digraph G_s does not dominate digraph G_k in the polytope T . Therefore, we look for a vector p^ ∈ T such that inequality (3.40) holds which may be rewritten in the following equivalent form:

does not influence the difference ^P_ν∈Ω^u Lemma 3.5 may be generalized as follows.

Lemma 3.6 Digraph G_s ∈ B dominates digraph G_k∈ B in the polytope T if Ω^∗_sk = ∅. Due to Lemma 3.6, we can rewrite equality (3.50) as follows:

r^B_ks = min

To obtain the desired vector p^ ∈ T , we have to calculate r^B_ks according to (3.55) for each digraph G_k ∈ B which is not dominated by digraph G_s(if G_s 6_T G_k) and to take the minimum over all such digraphs G_k. We summarize the above arguments in the following claim.

Theorem 3.6 If for digraph G_s ∈ B ⊆ Λ(G) dominance relation G_s _p G_k holds for each digraph G_k ∈ B and fixed vector p ∈ T of the processing times, then equalities

%^B_s(p ∈ T ) = min{ min

hold, where value rΩ^v_s,Ω^u_k is calculated according to (3.49).

The following corollary is used in the proof of Theorem 3.8.

Corollary 3.7 The value r_Ωv0

s ,Ω^u_k calculated according to (3.49) for the set Ω^v_s⁰ ∈ Ω^∗_sk \ Ωs(p) is strongly positive.

Proof. Due to formula (3.56), we have to repeat the calculation (3.49) for each set Ω^v_s ∈ Ω^∗_sk and each set Ω^u_k⁰, u⁰ ∈ {1, 2, . . . , ω_k^T}, such that ^P_ν∈Ωu0

hold. Therefore, due to the calculation of the value r_Ωv0

s ,Ω^u0_k , the numerator in (3.49) is strongly positive at least for β = 0. Since we have to take the maximum value among all values calculated for each β = 0, 1, . . . , |N (Ω^v_s⁰, Ω^u_k⁰)| − 1 (see formula (3.49)), we get r_Ωv0

s ,Ω^u0_k > 0.

3 Next, we present necessary and sufficient conditions for an infinitely large relative stability radius %^B_s(p ∈ T ) for problem J /a_i≤ p_i≤ b_i/^PC_iif B ⊂ Λ(G) and T ⊂ R^q₊. Note that deterministic problem J //^PC_i with λ > 1 cannot have an optimal digraph with an infinitely large stability radius %_s(p) (see Remark 1.1). Recall that %_s(p) = %^Λ(G)_s (p ∈ R^q₊).

Theorem 3.7 For digraph G_s ∈ B ⊆ Λ(G), we have %^B_s(p ∈ T ) = ∞ if and only if Ω^∗_sk = ∅ for each digraph G_k ∈ B.

Proof. Necessity. Following the contradiction method, we suppose that %^B_s(p ∈ T ) =

∞ but there exists a digraph G_k ∈ B such that the set of representatives Ω^v_s⁰, v⁰ ∈

holds for the vector p^∗ calculated according to formula (3.53) for the set of representatives Ω^u_k, u ∈ {1, 2, . . . , ω_k^T}. Thus, due to Remark 3.7 there exists a vector p⁰ ∈ T such that

and hence digraph Gs cannot be optimal for the processing times given by vector p⁰ ∈ T.

We get a contradiction:

%^B_s(p ∈ T ) < d(p, p⁰) ≤ max

Oij∈Q^J{b_ij − p_ij, p_ij − a_ij} < ∞.

Sufficiency. Due to Lemma 3.6, equality Ω^∗_sk = ∅ (valid for each digraph Gk ∈ B) implies that digraph G_s ∈ B dominates all digraphs G_k ∈ B in polytope T. Hence, inequality L^p_s⁰ ≤ L^p_k⁰ holds for each vector p⁰ ∈ T and so %^B_s(p ∈ T ) = ∞.

3 From the above proof of the necessity, we obtain an upper bound for the relative stability radius %^B_s(p ∈ T ).

Corollary 3.8 If %^B_s(p ∈ T ) < ∞, then %^B_s(p ∈ T ) ≤ max{{b_ij − p_ij, p_ij − a_ij} : O_ij ∈ Q^J}.

Moreover, we can strengthen Corollary 3.6 as follows.

Corollary 3.9 The following propositions are equivalent:

1) Λ^T(G) = {G_s}, 2) %^Λ(G)_s (p ∈ T ) = ∞,

3) G_k ∈ Λ(G) ⇒ G_s _T G_k, 4) G_k ∈ Λ(G) ⇒ Ω^∗_sk = ∅.

To present necessary and sufficient conditions for %^B_s(p ∈ T ) = 0, we need the following auxiliary lemma. Let Ωk denote the set {Ω^u_k : u = 1, 2, . . . , ωk}.

Lemma 3.7 If Ω_k 6= Ω_k(p), the inclusion Ω_k(p⁰) ⊆ Ω_k(p) holds for any vector p⁰ ∈ O_(p) ∩ R^q₊ with _k >  > 0 defined as follows:

_k= 1

qnminⁿL^p_k− ^X

ν∈Ω^u_k

l^p(ν) : Ω^u_k ∈ Ω_k\Ω_k(p)^o. (3.57) The following theorem is a generalization of Theorem 1.6.

Theorem 3.8 Let G_s be an optimal digraph of problem J /a_i ≤ p_i ≤ b_i/^PC_i with the minimal objective function value L^p_s, p ∈ T, within the set B ⊆ Λ(G) of feasible digraphs.

The equality %^B_s(p ∈ T ) = 0 holds if and only if the following three conditions hold:

1) there exists a digraph G_k ∈ B such that L^p_s = L^p_k, k 6= s,

2) the set Ω^v_s^∗ ∈ Ω^∗_sk∩ Ω_s(p) is such that for any set Ω^u_k^∗ ∈ Ω_k(p), there exists an operation O_ij ∈ Q^J for which condition

n_ij(Ω^v_s^∗) ≥ n_ij(Ω^u_k^∗) (3.58) holds (or condition

n_ij(Ω^v_s^∗) ≤ n_ij(Ω^u_k^∗) (3.59) holds) and

3) inequality (3.58) (or inequality (3.59), respectively) is satisfied as a strict one for at least one set Ω^u_k⁰ ∈ Ωk(p): nij(Ω^v_s^∗) > nij(Ω^u_k⁰) (or nij(Ω^v_s^∗) < nij(Ω^u_k⁰)).

Proof. Necessity. We prove necessity by contradiction. Assume that %^Bs(p ∈ T ) = 0 but the conditions of the theorem are not satisfied. We consider four cases i, ii, iii and iv of violating these conditions.

i) Assume that there does not exist another optimal digraph G_k ∈ B such that L^p_s = L^p_k, k 6= s. If B \ {G_s} 6= ∅, we can calculate the value

^∗ = 1 qnmin

t6=s(L^p_t − L^p_s), (3.60)

which is strictly positive since L^p_s < L^p_t for each G_t ∈ B, t 6= s. Using Lemma 3.7, one can verify that for any real , which satisfies the inequalities 0 <  < ^∗, the difference in the right-hand side of equality (3.60) remains positive when vector p is replaced by any vector p⁰ ∈ O_(p) ∩ T . Indeed, for any v ∈ {1, 2, . . . , ω_s^T}, the cardinality of the obtain a strongly positive numerator in formula (3.49) at least for β = 0 :

X

ν∈Ω^u_t

l^p(ν) − ^X

µ∈Ω^v0_s

l^p(µ) > 0.

Therefore, the maximum taken according to (3.49) is also strongly positive, i.e. r_Ωv0 s ,Ω^u_t >

 > 0, where we can choose any  such that the inequality

 < minⁿ_s, _k, 1

is satisfied. This means that only in the case of the calculation of the value r_Ω^v_s,Ω^u

k for the r^B_ts is calculated due to formula (3.55) using the value rΩ^v_s,Ω^u_t > 0. Therefore, the relative

stability radius satisfies the following inequalities: %^B_s(p ∈ T ) > min{, ⁰} > 0, which implication, we conclude that the digraph G_s becomes an optimal digraph for any vector p⁰ ∈ T, provided that d(p, p⁰) ≤ . Consequently, we have %^B_s(p ∈ T ) ≥  > 0, which contradicts the assumption %^B_s(p ∈ T ) = 0.

iv) Assume that conditions 1 and 2 of Theorem 3.8 hold. More exactly, there exists a digraph G_k ∈ B such that L^p_s = L^p_k, k 6= s, and one of the two cases of condition 2 and

Sufficiency. We show that, if the conditions of Theorem 3.8 are satisfied, then %^B_s(p ∈ T ) <  for any given  > 0. First, we make the following remark.

Remark 3.8 In the trivial case of a_ij = b_ij for each operation O_ij ∈ Q^J, the set Ω^∗_sk∩Ω_s(p) is empty, since in this case the vector p is equal to the vector p^∗ constructed according to (3.53), and the strong inequality (3.54) does not hold.

We construct a vector p⁰ = (p⁰_1,1, p⁰_1,2, . . . , p⁰_nnn) ∈ T with components p⁰_ij ∈ {pij, pij +

⁰, p_ij − ⁰}, where ⁰ = min{, _k, _min} with the value _k> 0 defined in (3.57), and

_min = max{0, min{min{p_ij−a_ij : p_ij > a_ij, O_ij ∈ Q^J}, min{b_ij−p_ij : b_ij > p_ij, O_ij ∈ Q^J}}}, using the following rule: For each Ω^u_k^∗ ∈ Ωk(p), mentioned in Theorem 3.8, we set p⁰_ij = p_ij+ ⁰, if inequalities (3.58) hold, or we set p⁰_ij = p_ij − ⁰, if inequalities (3.59) hold.

More precisely, we can choose ⁰ as follows: If Ω_k 6= Ω_k(p), then _k > 0, and we can choose ⁰ such that 0 < ⁰ < min{, k, min}. Otherwise, if Ωk = Ωk(p), we choose ⁰ such that 0 < ⁰ < min{, _min}. Such choices are possible since in both above cases, inequality

_min > 0 holds due to Remark 3.8. Note that ⁰ > 0 since p_ij > 0, O_ij ∈ Q^J. The following arguments are the same for both cases of the choice of ⁰.

After changing at most |Ω_k(p)| components of the vector p according to this rule, we obtain a vector p⁰ of the processing times for which inequality

X

µ∈Ω^v∗_s

l^p0(µ) > ^X

ν∈Ω^u∗_k

l^p0(ν)

holds for each set Ω^u_k^∗ ∈ Ω_k(p). Due to ⁰ ≤ _min, we have p⁰ ∈ T. Since ⁰ ≤ _k, we have L^p_k⁰ = max

u∈{1,2,...,ω^T_k}

X

ν∈Ω^u_k

l^p0(ν) = max

Ω^u_k∈Ω_k(p)

X

ν∈Ω^u_k

l^p0(ν)

= ^X

ν∈Ω^u∗_k

l^p0(ν) < ^X

µ∈Ω^v∗_s

l^p0(µ) ≤ L^p_s⁰.

Thus, we conclude that digraph G_s is not optimal for the vector p⁰ ∈ T with d(p, p⁰) = ⁰ which implies %^B_s(p ∈ T ) < ⁰ ≤ .

3 Corollary 3.10 If G_s∈ B is a unique optimal digraph for the vector p ∈ T , then %^B_s(p ∈ T ) > 0.

From Theorem 3.8 we obtain the following lower bound for the relative stability radius

%^B_s(p ∈ T ).

Corollary 3.11 If G_s ∈ B is an optimal digraph, then %^B_s(p ∈ T ) ≥ ^∗, where ^∗ is calculated according to (3.60).

Proof. If there exists a digraph Gk ∈ B such that L^p_s = L^p_k, k 6= s, the equality

%^B_s(p ∈ T ) ≥ ^∗ = 0 holds due to Definition 3.6. Otherwise, inequality %^B_s(p ∈ T ) ≥ ^∗ follows from the above proof of necessity (see case i).

3

Example 3.2 (continued). Returning to the Example 3.2 and using Theorem 3.6, we can calculate the relative stability radius of the digraph G₁ ∈ B ⊆ Λ(G), |B| = 12, for the vector p = p⁰ = (70, 30, 60, 20, 60, 70, 40, 30) of the processing times according to formula (3.56). After a pairwise comparison of the sets of representatives for digraph G^T₁ with

those for digraphs G^T₂, G^T₃, . . . , G^T₁₂, we obtain equality %^B₁(p⁰ ∈ T ) = 3, which means that digraph G₁ remains optimal at least for all vectors p ∈ O₃(p⁰) ∩ T of the processing times. Due to calculation of the relative stability radius, we show that only digraphs G₂ and G5 may be better than digraph G1 provided that vector p of the processing times belongs to polytope T , and for each digraph G_k ∈ Λ(G) with k 6= 2 and k 6= 5, dominance relation G₁ _T G_k holds. We also obtain the following equalities: %^B₁(p⁰ ∈ T ) = r^B_2,1 = 3, %^B\{G₁ ^2}(p⁰ ∈ T ) = r^B\{G_5,1 ^2} = 10, where the values r^B_k1 are calculated according to (3.55), and they give the minimum over all digraphs G_k ∈ B which are not dominated by digraph G₁. Next, it follows from Theorem 3.7 that %^B\{G₁ ^2,G5}(p⁰ ∈ T ) = ∞.

Due to Theorem 3.4, set Λ^∗(G) = {G1, G2, G5} is a solution to problem J 3/n = 3, a_i≤ p_i ≤ b_i/^PC_i, since there exists a covering of the polytope T by sets D_s = {p ∈ R⁸₊ : L^p_s ≤ L^p_k, k = 1, 2 . . . , λ} ∩ T with s ∈ {1, 2, 5}. More exactly, for any digraph Gk∈ Λ(G) and for any set Ds, s ∈ {1, 2, 5}, there exists a digraph Gs ∈ Λ^∗(G) for which dominance relation G_s _Ds G_k holds (since the dominance relation G₁ _T G_k holds for each digraph G_k ∈ Λ(G), k 6= 2, k 6= 5, it follows that set {D₁, D₂, D₅} is indeed a covering of the polytope T ). Moreover, since for each digraph Gs ∈ Λ^∗(G) there exists a point (see vectors p⁰, p^ and x, given in Section 3.4), for which this digraph is the unique optimal one, it follows from Theorem 3.5 that solution Λ^∗(G) = {G₁, G₂, G₅} is minimal.

Note that from a practical point of view, it is more useful to consider a covering of the polytope T by nested balls O3(p⁰), O10(p⁰) and Or^∗(p⁰), where r^∗ may be any real number no less than max{b_ij − p⁰_ij, p⁰_ij− a_ij : i = 1, 2, . . . , n; j = 1, 2, . . . , n_i}. Indeed, due to the calculation of the stability radius %^B₁(p⁰ ∈ T ), we know that for each vector p ∈ O₃(p⁰) digraph G1 is optimal. Moreover, for each vector p ∈ O10(p⁰) at least one digraph G1 or G₂ is optimal since %^B\{G₁ ^2}(p⁰ ∈ T ) = 10. Finally, for each vector p ∈ O_r^∗(p⁰) at least one digraph G₁, G₂ or G₅ is optimal since %^B\{G₁ ^2,G5}(p⁰ ∈ T ) = ∞.

Table 3.8: Solution of problem J 3/n = 3, ai≤ p_i≤ b_i/^PC_i with the initial vector p⁰ ∈ T

i Set B %^B₁(p⁰ ∈ T ) Set Γ_i of competitive digraphs of digraph G₁ 1 B = {G₁, G₂, . . . , G₁₂} 3 {G₂}

2 B \ {G₂} 10 {G₅}

3 B \ {G2, G5} ∞ ∅

Remark 3.9 Solving problem J 3/n = 3, ai≤ pi≤ bi/^PCi takes three iterations by the above algorithm (see Table 3.8). But similarly to the calculation of the relative stability radius and the construction of a solution of the scheduling problem with the makespan criterion (see Remark 3.4), we can construct a solution Λ^∗(G) for the mean flow time criterion in one scan as follows.

We union one of the optimal digraphs G_s with all digraphs G_k, k 6= s, for which a nonempty set Ω^∗_sk 6= ∅ exists, i.e., for which the dominance relation G_s _T G_k does not hold, and the union of these digraphs composes such a solution Λ^∗(G). In other words,

a solution of problem J /ai≤ pi≤ bi/^PCi is the union of an optimal digraph and of all its competitive digraphs Λ^∗(G) = {G_s} ∪ {G_k : G_s 6_T G_k} = {G_s} ∪ {∪^I_i=1Γ_i}, where Γ_i is the set of competitive digraphs of digraph G_s with respect to the set B in the iteration i = 1, 2, . . . , I.

Next, we consider a small problem J 3/n = 2/^PCi to illustrate the calculation of %₁(p) by formulas (1.36) and (1.37). Then we calculate the relative stability radius for problem J 3/n = 2, a_i≤ p_i≤ b_i/^PC_i. Notice that we use the same notations for different examples:

Example 3.1 and Example 3.2.

Example 3.1 (continued). Returning to the Example 3.1 from Section 3.1, we consider the job shop problem with the mean flow time criterion J 3/n = 2/^PC_i, whose input data are given by the weighted mixed graph G(p) with p = (75, 50, 40, 60, 55, 30), presented in Figure 3.1. Obviously, the set of all feasible digraphs Λ(G) is the same, but we number these digraphs in non-decreasing order of the objective function values: L^p₁ ≤ L^p₂ ≤ . . . ≤ L^p₅ (see Figure 3.8). Note that for criterion^PC_i we do not need to use dummy operations.

As we can see, digraph G₁(p) is optimal for both criteria C_maxand^PC_i. Next, we calculate stability radius %₁(p) for digraph G₁(p).



To this end, we construct an auxiliary Table 3.9, where for each feasible digraph G_k, k = 1, 2, . . . , 5, column 2 presents the sets Ω^u_k of representatives of the family of sets (H_kⁱ)_Ji∈J, column 3 presents the integer vector n(Ω^u_k) = (n_1,1(Ω^u_k), n_1,2(Ω^u_k), . . . , n_2,3(Ω^u_k)), where the value nij(Ω^u_k) is equal to the number of vertices Oij in the multiset {[ν] : ν ∈ Ω^u_k}

Table 3.9: Auxiliary information for problem J 3/n = 2/^PC_i

G_k Ω^u_k, u = 1, 2, . . . , ω_k n_1,1 n_1,2n_1,3n_2,1 n_2,2n_2,3 X

ν∈Ω^u_k

l^p(ν)

1 2 3 4

G1 Ω¹₁={O1,1, O1,2, O1,3; O1,1, O1,2, O2,3} 2 2 1 0 0 1 320 Ω²₁={O1,1, O1,2, O1,3; O2,1, O1,2, O2,3} 1 2 1 1 0 1 305 Ω³₁={O1,1, O1,2, O1,3; O1,1, O2,2, O2,3} 2 1 1 0 1 1 325 Ω⁴₁={O1,1, O1,2, O1,3; O2,1, O2,2, O2,3} 1 1 1 1 1 1 310 Ω⁵₁={O2,1, O1,2, O1,3; O1,1, O1,2, O2,3} 1 2 1 1 0 1 305 Ω⁶₁={O2,1, O1,2, O1,3; O2,1, O1,2, O2,3} 0 2 1 2 0 1 290 Ω⁷₁={O_2,1, O_1,2, O_1,3; O_1,1, O_2,2, O_2,3} 1 1 1 1 1 1 310 Ω⁸₁={O_2,1, O_1,2, O_1,3; O_2,1, O_2,2, O_2,3} 0 1 1 2 1 1 295 G2 Ω¹₂={O1,1, O2,2, O2,3, O1,2, O1,3; O1,1, O2,2, O2,3} 2 1 1 0 2 2 410 Ω²₂={O_1,1, O_2,2, O_2,3, O_1,2, O_1,3; O_2,1, O_2,2, O_2,3} 1 1 1 1 2 2 395 Ω³₂={O_2,1, O_2,2, O_2,3, O_1,2, O_1,3; O_1,1, O_2,2, O_2,3} 1 1 1 1 2 2 395 Ω⁴₂={O_2,1, O_2,2, O_2,3, O_1,2, O_1,3; O_2,1, O_2,2, O_2,3} 0 1 1 2 2 2 380 G₃ Ω¹₃={O_2,1, O_2,2, O_1,1, O_1,2, O_1,3; O_2,1, O_2,2, O_2,3} 1 1 1 2 2 1 425 Ω²₃={O_2,1, O_2,2, O_2,3, O_1,2, O_1,3; O_2,1, O_2,2, O_2,3} 0 1 1 2 2 2 380 G₄ Ω¹₄={O_1,1, O_1,2, O_1,3; O_1,1, O_1,2, O_2,1, O_2,2, O_2,3} 2 2 1 1 1 1 435 G5 Ω¹₅={O2,1, O2,2, O1,1, O1,2, O1,3; 2 2 1 2 2 1 550

O2,1, O2,2, O1,1, O1,2, O2,3}

(for simplicity we use notation n_ij instead of n_ij(Ω^u_k)), and column 4 presents the value

X

ν∈Ω^u_k

l^p(ν) = ^X

Oij∈[ν], ν∈Ω^u

k

p_ij · n_ij(Ω^u_k).

The calculation of %₁(p) by formula (1.37) is given in Table 3.10, which presents the results of the computations for each β = 1, 2, . . . , q − m, where m is the number of operations Oij ∈ Ω^v₁ ∪ Ω^u_k, Ω^v₁ ∈ Ω1k, for which nij(Ω^v₁) < nij(Ω^u_k). The cardinal-ity of the set Ω_1k, k = 1, 2, . . . , 5, and the elements Ω^ν₁ of this set are presented in col-umn 2 and colcol-umn 3, respectively. The elements of the set Ω^u_k, u = 1, 2, . . . , ω_k, for which

P

ν∈Ω^u_kl^p(ν) ≥ L^p₁ = 325 are presented in column 4.

Since the vector n(Ω^u_k) = (n1,1(Ω^u_k), n1,2(Ω^u_k), . . . , n2,3(Ω^u_k)) is the same for both sets Ω²₁ and Ω⁵₁, for both sets Ω⁴₁ and Ω⁷₁, and for both sets Ω²₂ and Ω³₂ (see Table 3.9), the results calculated by formula (1.37) are the same for these pairs of sets, too. Therefore, we combine these calculations in column 7 in Table 3.10. In column 6 we give the sequence of processing times of the operations O_ij ∈ Ω^v₁ ∪ Ω^u_k with n_ij(Ω^v₁) < n_ij(Ω^u_k) ordered in the following way: p_ij(m+1) ≥ p_ij(m+2) ≥ . . . ≥ p_ij(q). Note that in column 7 we do not write components with nij(Ω^v₁) = nij(Ω^u_k) in the fraction from formula (1.37). For the sets Ω¹₁ and Ω¹₂, we give a more detailed computation and for each other pair of the sets Ω^v₁ and Ω^u_k at each following iteration, we use the value of the fraction obtained at the previous iteration. From the derived values in column 7, we write their maximum for β = 1, 2, . . . , q −m, the maximum for Ω^u_k, u = 1, 2, . . . , ω_k, and the minimum for Ω^v₁ ∈ Ω_1k, respectively, in columns 8, 9 and 10. Using formula (1.36), we take the minimum value from column 10. Therefore, we obtain %₁(p) = 17.5.

Table 3.10: Calculation of the stability radius %₁(p) for problem J 3/n = 2/^PC_i

Table 3.10 (continuation): Calculation of the stability radius %₁(p) for problem J 3/n = 2/^PC_i the aim to illustrate the idea of constructing a solution to this problem mentioned in Remark 3.9. The structural input data are given by mixed graph G in Figure 3.1. The numerical input data are given in Table 3.3. Obviously, the set of all feasible digraphs Λ(G) is identical for Cmax and ^PCi, and here we number these digraphs in non-decreasing order of the values ^PC_i with the same initial vector p = (75, 50, 40, 60, 55, 30) as for problem J 3/n = 2, a_i≤ p_i≤ b_i/C_max considered in Chapter 3: L^p₁ ≤ L^p₂ ≤ . . . ≤ L^p₅ (see Figure 3.8). Using the modification of critical path method described at page 142, we can simplify digraphs G₁, G₂, . . . , G₅, but for these numerical input data (see Table 3.3) the corresponding digraphs G^T₁, G^T₂, . . . , G^T₅ remain the same. It means that the number of sets of representatives ω^T_k is equal to the number ωk for each digraph Gk, k = 1, 2, . . . , 5,

Table 3.10 (continuation): Calculation of the stability radius %₁(p) for problem J 3/n = 2/^PC_i the aim to illustrate the idea of constructing a solution to this problem mentioned in Remark 3.9. The structural input data are given by mixed graph G in Figure 3.1. The numerical input data are given in Table 3.3. Obviously, the set of all feasible digraphs Λ(G) is identical for Cmax and ^PCi, and here we number these digraphs in non-decreasing order of the values ^PC_i with the same initial vector p = (75, 50, 40, 60, 55, 30) as for problem J 3/n = 2, a_i≤ p_i≤ b_i/C_max considered in Chapter 3: L^p₁ ≤ L^p₂ ≤ . . . ≤ L^p₅ (see Figure 3.8). Using the modification of critical path method described at page 142, we can simplify digraphs G₁, G₂, . . . , G₅, but for these numerical input data (see Table 3.3) the corresponding digraphs G^T₁, G^T₂, . . . , G^T₅ remain the same. It means that the number of sets of representatives ω^T_k is equal to the number ωk for each digraph Gk, k = 1, 2, . . . , 5,

在文檔中 不確定數據下新排程理論研究專書 (頁 156-180)