Mean Flow Time

In this section, we consider stability radius %_s(p) of an optimal schedule for problem G//^PC_i with criterion^PC_i. If Φ =^PC_i, conditions (4.17) and (1.6) for the general shop problem G//Φ are converted to the following conditions (1.30) and (1.31), respectively.

n

Obviously, the value Ci for a digraph Gs(p) is equal to the largest weight of a path from the set H_sⁱ, and hence, to solve problem G//^PC_i, it is sufficient to find a digraph G_s(p) such that equality (1.30) holds.

Due to equality (1.31) to find the stability radius %_s(p), it is sufficient to construct a vector x ∈ R^q₊ that satisfies the following three conditions.

(a’) There exists a digraph G_k(p) ∈ Λ(G), k 6= s, such that

(b’) For any given real  > 0, which may be as small as desired, there exists a vector

(c’) The distance d(p, x) achieves its minimal value among the distances between the vector p and the other vectors in the space R^q₊ which satisfy both above conditions (a’) and (b’).

Similarly as in the previous section (see conditions (a), (b) and (c)), after having constructed such a vector x ∈ R^q₊ one can define the stability radius of digraph G_s(p):

%_s(p) = d(p, x). Thus, due to (1.32) and (1.33) the calculation of the stability radius may again be reduced to an extremal problem on the set of weighted digraphs Λ(G).

However, in this case we are forced to consider sets of representatives of the family of sets H_kⁱ, 1 ≤ i ≤ n, which may be defined as follows.

Similarly to the proof of Theorem 1.4, one can find vector x satisfying conditions (a’) and (b’) in the form x(r) = (x1(r), x2(r), . . ., xq(r)) with the components xi(r) from the set {p_i, p_i + r, p_i − r} on the basis of a direct comparison of the set Ω^u_s ∈ Ω_s,k of representatives of the family of sets (H_sⁱ)_1≤i≤n, and the set Ω^u_k of representatives of the family of sets (H_kⁱ)1≤i≤n, where k = 1, 2, . . . , λ, k 6= s, and

Ω_s,k = {Ω^v_s : There does not exist a set Ω^u_k such that n_i(Ω^v_s) ≤ n_i(Ω^u_k) for each i = 1, 2, . . . , q}.

As a result, the following lower bound of the stability radius has been obtained:

%_s(p) ≥ r = min

It is easy to see that it may happen that the above vector x(r) ∈ R^q does not belong to set R₊^q since substraction value r from some component of vector p may result a negative number. To obtain the exact value of %_s(p), one can use the vector x^∗(r) =

In contrast to vector x(r), vector x^∗(r) necessary has no negative components.

Let the set of operations Q be ordered in the following way:

i₁, i₂, . . . , i_m, i_m+1, . . . , i_q, (1.35) where n_iα(Ω^u_k) ≤ n_iα(Ω^v_s) for each α = 1, 2, . . . , m and n_iα(Ω^u_k) > n_iα(Ω^v_s) for each α = m + 1, m + 2, . . . , q. Moreover, for the sequence (1.35) the inequalities

p_im+1 ≤ p_im+2 ≤ . . . ≤ p_iq

have to be satisfied. Using sequence of operations 1.35 it is easy to derive the following formula for calculating %_s(p). If only a subset of the processing times can be changed but the other ones cannot be changed, a formulas similar to (1.36) and (1.37) can be derived (see the remark after Theorem 1.4).

Zero Stability Radius

We consider the case of %_s(p) = 0. Similarly to the notion of a critical path and the critical weight, which is important for problem G//Cmax (see Section 1.3), we introduce the notion of a critical set of paths Ω^u_k^∗ and the critical sum of weights of digraph G_k(p) ∈ Λ(G) for

for the weighted digraph G_k(p) is reached on this set:

X

The value L^p_k defined in (1.38) is called critical sum of weights for digraph Gk(p).

Obviously, a critical set Ω^u_k^∗ may include a path ν ∈ H_kⁱ, i = 1, 2, . . . , n, if and only if

Lemma 1.4 There exists a real  > 0 such that the set Ωk\Ωk(p) contains no critical set of digraph G_k(p) ∈ Λ(G) for any vector p⁰ ∈ O_(p) ∩ R^q₊ of the processing times, i.e.

Ω_k(p⁰) ⊆ Ω_k(p).

Proof. After having calculated the value

_k = 1

2qnminⁿL^p_k− ^X

ν∈Ω^u_k

l^p(ν) : Ω^u_k ∈ Ω_k\Ω_k(p)^o, (1.39)

one can verify that for any real , which satisfies the inequalities 0 <  < _k, the difference in the right side of equality (1.39) remains positive when vector p is replaced by any vector p⁰ ∈ O_(p) ∩ R₊^q. Indeed, for any u ∈ {1, 2, . . . , ω_k}, the cardinality of set Ω^u_k may be at most equal to qn. Thus, the difference L^p_k−^P_ν∈Ω^u

k l^p(ν) may not be ’overcome’ by a vector p⁰ if d(p, p⁰) < k.

3 Next, we prove the following necessary and sufficient conditions for equality %_s(p) = 0.

Theorem 1.6 Let G_s be an optimal digraph for problem G//^PC_i with positive processing times p_i > 0 of all operations i ∈ Q. The equality %_s(p) = 0 holds if and only if the following three conditions hold:

1) there exists another optimal schedule k ∈ S^Φ(p), Φ =^PC_i, k 6= s,

2) there exists a set Ω^v_s^∗ ∈ Ω_s(p) such that for any set Ω^u_k^∗ ∈ Ω_k(p), there exists an operation i ∈ Q for which the condition

n_i(Ω^v_s^∗) ≥ n_i(Ω^u_k), Ω^u_k ∈ Ω_k(p), (1.40) holds (or the condition

n_i(Ω^v_s^∗) ≤ n_i(Ω^u_k), Ω^u_k ∈ Ω_k(p), (1.41) holds) and

3) inequality (1.40) (or inequality (1.41), respectively) is satisfied as a strict one for the set Ω^u_k^∗.

Proof. We prove necessity by contradiction. Assume that %_s(p) = 0 but the conditions of the theorem are not satisfied. We consider three cases (j), (jj) and (jjj) of violating these conditions.

(j) Assume that there does not exist another optimal schedule, i.e., we have S^Φ(p) = {s} with Φ =^PC_i. Then we consider a real  such that

0 <  < 1 2qnmin

t6=s(L^p_t − L^p_s)

holds. Similarly to the proof of Lemma 1.4, we can show that digraph G_sremains optimal for any vector p⁰ = (p⁰₁, p⁰₂, . . . , p⁰_q) ∈ R^q₊of the processing times provided that d(p, p⁰) ≤ .

Therefore, we have %_s(p) ≥  > 0 which contradicts the assumption %_s(p) = 0.

(jj) Assume that |S^Φ(p)| > 1 and for any optimal schedule k ∈ S^Φ(p) with k 6= s, and for any set Ω^v_s ∈ Ω_s(p), there exists a set Ω^u_k^∗ ∈ Ω_k(p) such that n_i(Ω^v_s) = n_i(Ω^u_k^∗) for any operation i ∈ Q.

In this case, we can take any  that satisfies the inequalities 0 <  < minⁿ_s, _k, 1

2qn min

t6∈φ^Σ(p)(L^p_t − L^p_s)^o. (1.42) From Lemma 1.4, due to inequality  < _s, we get that equality

L^p_s⁰ = max

Sufficiency. We show that, if the conditions of Theorem 1.6 are satisfied, then %_s(p) <  for any given  > 0.

We construct a vector p^∗ = (p^∗₁, p^∗₂, . . . , p^∗_q) ∈ R^q₊with components p^∗_i ∈ {p_i, p_i+ ^∗, p_i−

^∗}, where ^∗ = min{_k, , min_i∈Qp_i}, using the following rule: For each Ω^u_k^∗ ∈ Ω_k(p), mentioned in Theorem 1.6, we set p^∗_i = p_i + ^∗, if inequalities (1.40) hold, or we set p^∗_i = p_i− ^∗, if inequalities (1.41) hold. Note that ^∗ > 0 since p_i > 0, i ∈ Q.

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

X

Thus, we conclude that s /∈ S^Φ(p^∗) with d(p, p^∗) = ^∗ which implies %_s(p) < ^∗ ≤ .

3 Theorem 1.6 directly implies the following assertion.

Corollary 1.5 If s ∈ S is a unique optimal schedule for problem G//^PCi, then %_s(p) > 0.

It is easy to prove the following upper bound for the stability radius of an optimal schedule for problem G//^PCi.

Theorem 1.7 If s ∈ S is an optimal schedule for problem G//^PC_i with λ > 1 and p_i > 0 for at least one operation i ∈ Q, then

%_s(p) ≤ max

i∈Q p_i.

Proof. We consider vector p⁰ ∈ R^q₊ with zero components: p⁰_i = 0 for each i ∈ Q. For this vector of processing times, each feasible digraph G_t ∈ Λ(G) is optimal and each set of representatives Ω^u_t is critical. We can take a schedule k ∈ S^Φ(p⁰) which has only one arc (j, i) ∈ E_k different from the arcs in E_s, i.e. (i, j) ∈ E_s and E_s\{(i, j)} = E_k\{(j, i)}.

It is easy to see that there exist sets Ω^v_s ∈ Ω_s(p) and Ω^u_k ∈ Ω_k(p) such that n_i(Ω^v_s) > n_i(Ω^u_k).

Setting p^_i =  > 0 and p^_l = 0 for each l ∈ Q\{i}, we obtain s /∈ S^Φ(p^) and d(p^, p) <

max{p_i : i ∈ Q}.

3 Remark 1.1 As it follows from Theorem 1.7, problem J //^PC_i with λ > 1 cannot have an optimal schedule with an infinitely large stability radius in contrast to problem J //Cmax and problem J //Lmax (see Section 1.3).

Note that all of the results in this section and in Section ?? are valid for any general shop scheduling problem. However, we use the partition of the set of operations Q into n chains Q⁽ⁱ⁾, i = 1, 2, . . . , n, (which is necessary for the job shop and flow shop but is not necessary for the general shop) for a better presentation of the results. Table 1.1 collects special cases of the shop scheduling problem which are characterized by the machine service and the technological routes of the jobs. In scheduling theory often the classical job shop is considered for which each job has to be processed exactly once on each machine (see problem in the third row in Table 1.1). For our consideration this restriction is not important. We will consider the job shop problem J m//Φ with recirculation (see [Pin95a]), which may occur when a job may visit a machine more than once.

To illustrate the above notations and some results, we consider in Section 6 an example of a job shop scheduling problem with two jobs and two machines.

Example 1.5 The job shop problem is specified by the mixed graph G = (Q, A, E) given in Figure 1.5. The first job consists of operations 1 and 2, and the second job consists of operations 3 and 4. So we have the precedence constraints 1 → 2 and 3 → 4. The

assignment of the operations to the machines is as follows: Q1 = {1, 4}, Q2 = {2, 3}. The vector p = (10, 30, 20, 40) defines the processing times of the operations Q = {1, 2, 3, 4}.

For this problem we get Λ(G) = {G1, G2, G3} with the following signatures of semi-active schedules: E₁ = {(1, 4), (3, 2)}, E₂ = {(1, 4), (2, 3)} and E₃ = {(4, 1), (3, 2)}.

The corresponding sets of dominant paths are as follows: H₁¹ = {(1, 2), (3, 2)}, H₁² = {(1, 4), (3, 4)}, H₂¹ = {(1, 2)}, H₂² = {(1, 2, 3, 4)}, H₃¹ = {(3, 4, 1, 2)} and H₃² = {(3, 4)}.

The optimal digraph G₁ = (Q, A ∪ E₁, ∅) is represented in Figure 1.5 and it defines the unique optimal semiactive schedule (10, 50, 20, 60) for both criteria C_max and^PC_i. There-fore, due to Corollary 1.4 and Corollary 1.5 we have %b_s(p) > 0 and %_s(p) > 0.

















-l

l l

l l

l

1 2

3 4

p₁ = 10 p₂ = 30

p₃ = 20 p₄ = 40

Figure 1: Mixed graph G = (Q, A, E) for example 1.5

We can calculate the exact value of %b₁(p) on the basis of Theorem 1.5. First, we compare the digraphs G1 and G2. We have four sets of representatives for di-graph G₁, namely: Ω¹₁ = {(1, 2), (1, 4)}, Ω²₁ = {(1, 2), (3, 4)}, Ω³₁ = {(3, 2), (1, 4)} and Ω⁴₁ = {(3, 2), (3, 4)}. We can calculate vectors n(Ω¹₁) = (2, 1, 0, 1), n(Ω²₁) = (1, 1, 1, 1), n(Ω³₁) = (1, 1, 1, 1) and n(Ω⁴₁) = (0, 1, 2, 1). Digraph G2 has only one set of representatives:

Ω¹₂ = {(1, 2), (1, 2, 3, 4)} with n(Ω¹₂) = (2, 2, 1, 1). Obviously, we have n_i(Ω^u₁) ≤ n_i(Ω¹₂) for u ∈ {1, 2, 3} and for all i ∈ {1, 2, 3, 4}. Thus, we have to calculate only

r_Ω¹

2,Ω⁴₁ = (40 + 100) − (50 + 60)

| 2 − 0 | + | 2 − 1 | + | 1 − 2 | + | 1 − 1 | = 30

4 = 7.5.

Next, we compare the digraphs G₁ and G₃. Digraph G₃ has only one set of representatives:

Ω¹₃ = {(3, 4, 1, 2), (3, 4)} with n(Ω¹₃) = (1, 1, 2, 2). Obviously, we have n_i(Ω^u₁) ≤ n(Ω¹₃) for u ∈ {2, 3, 4} and for all i ∈ {1, 2, 3, 4}. Thus, we have to calculate only

r_Ω¹

3,Ω¹₁ = (100 + 60) − (40 + 50)

| 1 − 2 | + | 1 − 1 | + | 2 − 0 | + | 2 − 1 | = 70

4 = 17.5.

Since 7.5 < 10 = min{p₁, p₂, p₃, p₄}, the bound (1.34) is reached and we obtain %^Σ₁(p) = min{7.5; 17.5} = 7.5 and G₂ is a competitive digraph for digraph G₁, which is optimal for p = (10, 30, 20, 40).

















-1 2

3 4

c₁(1) = 10 c₂(1) = 50

c₃(1) = 20 c₄(1) = 60

1

@

@

@

@

@ R

Figure 2: The optimal digraph G₁ = (Q, A ∪ E₁, ∅)

Next we consider the stability radius of the optimal digraph G₁ with respect to crite-rion C_max. We have the following sets of dominant paths: H = {(1, 2), (3, 4)}, H₁ = {(1, 2), (1, 4), (3, 2), (3, 4)}, H₂ = {(1, 2, 3, 4)} and H₃ = {(3, 4, 1, 2)}. It is clear that any path µ ∈ H₁\H is dominated by the path (1, 2, 3, 4) ∈ H₂ and by the path (3, 4, 1, 2) ∈ H₃. Thus, due to Theorem 1.1 we have %_b1(p) = ∞.

In scheduling theory open shop problem is also considered when technological routes for processing jobs are not fixed before scheduling, i.e., for each job J_i ∈ J only set of operations Q^Ji is given but order of these operations is not fixed. While solving open shop problem O//Φ it is necessary to find optimal orders of operations Q^Ji for each job J_i ∈ J along with optimal orders of operations on each machine M_k ∈ M . Similarly to flow shop problem, each job J_i ∈ J has to be processed by each machine exactly once.

The same property of the technological routes are assumed for so-called classical job shop. Classification of shop scheduling problems is given in Table 1.1.

Table 1.1: Different shop scheduling problems

Characterization Shop scheduling Technological routes

of machine service problem of the jobs

Open shop Different jobs may have O//Φ different routes, which are Each job J_i ∈ J has not fixed before scheduling to be processed by Flow shop Jobs have the same route, each machine M_k ∈ M F //Φ which is fixed before

exactly once scheduling

Classical Different jobs may have job shop different routes, which A job may be processed Job shop are fixed before

by a machine more J //Φ scheduling

than once

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