Dominance Relations

In this section, the job shop problem with the objective of minimizing the sum of job completion times under uncertain numerical input data is modeled in terms of a mixed graph. As far as scheduling is concerned, the mean flow time seems to be more important than makespan. While C_max aims for minimizing the schedule duration, in industry this duration is often defined by the periodicity of the process, say, a working day or a work-ing week. So, the criterion ^PC_i is more suitable for a periodical scheduling environment.

Theoretically speaking, the problem with ^PC_i is more difficult to solve than that with C_max as evidenced by the facts that the two-machine open shop problem O2//C_max and the flow shop problem F 2//C_max are polynomially solvable while problems O2//^PC_i and F 2//^PC_i are NP-hard (see [BEP⁺96, LLRKS93]). (Obviously, the NP-hardness of the latter problem implies the NP-hardness of problem J /a_i≤ p_i≤ b_i/^PC_i under consider-ation). In academic research, criterion C_max has mainly been considered for multi-stage systems (open, flow and job shops) while criterion ^PC_i for single-stage systems, i.e., systems with one machine or with a set of parallel (usually identical) machines. One explanation could be that makespan is easier to attack than mean flow time.

In the problem J /ai≤ pi≤ bi/^PCi, it is assumed that only the structural input data (i.e., precedence and capacity constraints) are fixed while for the operation processing times only their lower and upper bounds are known before scheduling and the probability distribution functions of the random processing times are unknown. The structural input data are defined by the technological routes of the jobs, e.g., for a flow or open shop fixing the structural input data simply means to fix the number of jobs and the number of machines.

In this and next sections, two variants of a branch-and-bound method are developed.

The first one constructs a set of k schedules which are the best with respect to the mean flow time criterion for a fixed feasible vector of the processing times. The second variant constructs a set of potentially optimal schedules for all perturbations of the processing times within the given lower and upper bounds. To exclude redundant schedules, we use a stability analysis based on the pairwise comparison of schedules. Along with implicit enumerations based on a branch-and-bound method, we realize an explicit enumeration of all feasible schedules.

We assume that n jobs J = {J₁, J₂, . . . , J_n} have to be processed on m machines M = {M₁, M₂, . . . , M_m} when only the technological routes of the jobs are given before scheduling. At the stage j ∈ {1, 2, . . . , ni} of processing job Ji, operation Oij ∈ Q^J_k has to be processed by machine M_k ∈ M. The distribution of the operations to the machines M is fixed via the given technological routes of the jobs J . Each machine can process at most one operation at a time (see Condition 1 at page 2) and preemptions of an operation are forbidden (see Condition 4 at page 3).

We consider the job shop problem J /a_i≤ p_i≤ b_i/^PC_i with fixed technological routes and uncertain processing times p_ij, which have to satisfy only the inequalities a_ij ≤ p_ij ≤ b_ij, J_i ∈ J; j = 1, 2, . . . , n_i (see Condition 5 at page 6). The sum of the job completion times (mean flow time) is considered as the objective function Φ = Φ(C₁, C₂, . . . , C_n) =

Pn

i=1C_i =^PC_i, where C_i = c_ini is the completion time of job J_i ∈ J.

To present the structural input data for problem J /a_i ≤ p_i ≤ b_i/^PC_i, we use the mixed graph (Q^J, A^J, E^J) introduced at page 13. Such a mixed graph G = (Q^J, A^J, E^J)

defines the structural input data (precedence and capacity constraints) which are supposed to be known before scheduling. A schedule is defined as a circuit-free digraph G_s = (Q^J, A^J ∪ E_s^J, ∅) generated from the mixed graph (Q^J, A^J, E^J) by replacing each edge [Oij, Ouv] ∈ E^J by one of the arcs (Oij, Ouv) or (Ouv, Oij). In the rest of this chapter, we use the terms of an optimal schedule (digraph), a better and a best schedule (digraph) with respect to the mean flow time criterion^PC_i. However, the makespan criterion C_max and a regular criterion Φ are considered in the rest of this chapter as well.

Due to Definition 3.1 given at page 110, a solution Λ^∗(G) of the problem J /a_i≤ p_i≤ b_i/^PC_i is a set of digraphs containing at least one optimal digraph for each feasible vector p = (p_1,1, p_1,2, . . . , p_nnn) of the processing times, i.e., for each vector p ∈ T , where

T = {x = (x_1,1, x_1,2, . . . , x_nnn) : a_ij ≤ x_ij ≤ b_ij; i = 1, 2, . . . , n; j = 1, 2, . . . , n_i} is the polytope of all feasible vectors of the processing times in the space R^q₊, with q =

|Q^J| = ^Pn_i=1n_i = ^Pm_k=1|Q^J_k|. It is practically important to look for a minimal solution Λ^T(G) of problem J /ai≤ pi≤ bi/^PCi, i.e., for a minimal subset of the set Λ(G) containing at least one optimal digraph for each fixed vector p ∈ T of the processing times such that any proper subset of the set Λ^T(G) is not a solution (see Definition 3.1).

If the processing times p_ij of all operations O_ij ∈ Q^J are fixed, one can calculate the value of the objective function for a digraph Gs ∈ Λ(G) using the critical path method.

As it follows from Section 1.2, to solve problem J //Φ we must find a digraph G_s such that Φ^p_s = min{Φ^p_k : k = 1, 2, . . . , λ} (see formula (4.17) at page 196), where

Φ^p_k= Φ(max

ν∈H_k¹

l^p(ν), max

ν∈H²_k

l^p(ν), . . . , max

ν∈H_kⁿl^p(ν))

is the value of the objective function of the job completion times for the digraph G_k∈ Λ(G) with fixed processing times p ∈ R^q₊, and l^p(µ) is the weight of path µ: l^p(µ) =^P_Oij∈[µ]p_ij. Remind that Φ^p_s = l^p_s for criterion C_max and Φ^p_s = L^p_s for criterion ^PC_i.

As it has been proven in Chapter 1 (see Theorem 1.3 at page 22), there exists problem J m/n = n⁰/C_max with any given number of machines m and number of jobs n = n⁰, for which the optimality of digraph Gs ∈ Λ(G) does not depend on the numerical input data.

In other words, %_bs(p) = ∞ which means that schedule s minimizes the makespan for all non-negative processing times. However, such a schedule cannot exist for criterion ^PC_i. In other words, each optimal digraph (for mean flow time criterion) loses its optimality for some vectors p ∈ R^q₊ of the processing times, i.e., %_s(p) < ∞ (see Theorem 1.7 at page 36 and Remark 1.1). As it will be shown in the proof of Theorem 3.7 in the case of the relative stability radius %^B_s(p ∈ T ) (see Definition 3.6 below) when T ⊂ R^q₊ and B ⊂ Λ(G), an unrestricted value of %^B_s(p ∈ T ) is still possible.

For the problem J /a_i≤ p_i≤ b_i/Φ, we introduce the following two transitive dominance relations which define partial orderings on the set of feasible digraphs Λ(G).

Definition 3.5 Feasible digraph Gs (strongly) dominates feasible digraph Gk in set D ⊆ R^q₊ if inequality Φ^p_s ≤ Φ^p_k (inequality Φ^p_s < Φ^p_k, respectively) holds for any vector p ∈ D of the processing times. We denote the dominance relation by G_s _D G_k, and the strong dominance relation by Gs≺D Gk.

Let equality aij = bij hold for each operation Oij ∈ Q^J. In other words, set T turns into one point: T = {a} where a = (a_1,1, a_1,2, . . . , a_nnn). Therefore, problem J /a_i≤ p_i≤ b_i/Φ becomes a deterministic problem J //Φ, and the dominance relation _T defines a total ordering on the set of digraphs Λ(G). Consequently, minimal solution Λ^T(G) consists of a single digraph: Λ^T(G) = {G_s}, where G_s is any optimal digraph for problem J //Φ with processing times p_ij being equal to a_ij = b_ij for each operation O_ij ∈ Q^J. In other words, digraph Gs dominates all digraphs Gk ∈ Λ(G) at the point a ∈ R^q₊: Gs a Gk. Moreover, if the strong dominance relation holds for each digraph G_k∈ Λ(G) at the point a, i.e., if G_s ≺_a G_k, then digraph G_s is the unique optimal digraph for the processing times pij equal to aij = bij. As it was shown in the computational results (see Section 5.2 below), an optimal digraph for problem J //^PC_i was mainly uniquely determined. In such cases, if the dominance relation G_s _aG_k is valid for each digraph G_k ∈ Λ(G), then generally the strong dominance relation Gs ≺a Gk is valid for each digraph Gk ∈ Λ(G) with k 6= s. (Note that this is not the case for the makespan criterion: For the most job shop instances which have been randomly generated makespan optimal digraphs were not uniquely determined.)

In the general case of the problem J /a_i≤ p_i≤ b_i/^PC_i, the operation processing times may vary between given lower and upper bounds and therefore it is a priori unknown which path from the set H_kⁱ will have the largest weight in a practical realization of schedule k corresponding to digraph G_k. Thus, we have to consider the whole set Ω^u_k of representatives of the family of sets (H_kⁱ)_Ji∈J in a similar way to the approach considered for the problem J //^PC_i (see Section 1.4).

Each of these sets Ω^u_k includes exactly one path from each set H_kⁱ, J_i ∈ J. Since H_kⁱ ∩ H_k^j = ∅ for any pair of different jobs J_i and J_j, we have the equality |Ω^u_k| = n and so there exist ω_k =^Qn_i=1|H_kⁱ| different sets of representatives for digraph G_k, namely:

Ω¹_k, Ω²_k, . . . , Ω^ω_k^k. Next, we show how to restrict the number of sets of representatives which have to be considered while solving problem J /a_i≤ p_i≤ b_i/^PC_i.

Similarly to the notion of a critical path which is particularly important for criterion C_max, we use the notion of a critical set Ω^u_k^∗ for criterion ^PC_i. The set Ω^u_k^∗, u^∗ ∈ {1, 2, . . . , ω_k}, is a critical set in G_k ∈ Λ(G) if the objective function value L^p_k is reached on this set of paths, i.e., if equality ^P_ν∈Ωu∗

k l^p(ν) = L^p_k holds.

For different vectors p ∈ R₊^q of the processing times, different sets Ω^u_k, u ∈ {1, 2, . . . , ω_k}, may be critical, however a path ν ∈ H_kⁱ, J_i ∈ J, may belong to a critical set only if l^p(ν) = max_µ∈Hⁱ

kl^p(µ). Therefore, while solving problem J /ai≤ pi≤ bi/^PCi, it is sufficient to consider only paths from the set H_kⁱ which may have the largest weight for at least one vector p ∈ T of the processing times. Moreover, if there are two or more paths in H_kⁱ which have the largest weight at the same vector p ∈ T , it is sufficient to consider only one of them. Thus, it is sufficient to consider only dominant paths which were defined in Section 3.2 (see Definition 3.6 at page 119).

Using Corollary 3.2, one can simplify digraph G_s while solving problem J /a_i≤ p_i≤ b_i/^PC_i or problem J /a_i≤ p_i≤ b_i/C_max. First, we delete all transitive arcs, then we delete some arcs on the basic of domination of path sets (see Definition 3.4).

Let H_sⁱ(T ) denote the set of all dominant paths in H_sⁱ with respect to the polytope T . Since H_s⊆ ∪ⁿ_i=1H_sⁱ for problem J /a_i≤ p_i≤ b_i/C_max, one can construct a set H_sⁱ(T ) as a subset of the set of all dominant paths Hs by selecting all paths ending in vertex Oini

(if they exist). Let G^T_s = (Q^T_s, E_s^T, ∅) be a minimal subgraph of digraph Gs such that, if µ ∈^Sn_i=1H_sⁱ(T ), then digraph G^T_s contains path µ. To construct the digraph G^T_s, one can use the following straightforward modification of the critical path method [Dij59].

Assume that the path µ has the maximal weight among all paths in digraph Gsending in vertex O_ij when the processing times are defined by the vector p ∈ R^q₊. As usual, the weight of path µ minus p_ij is called earliest start time of operation O_ij and we denote it by l_s^p(Oij) :

Starting with a vertex in digraph G_s which has a zero in-degree and following the critical path method, we define values l^a_s(O_ij) and l_s^b(O_ij) for each vertex O_ij ∈ Q^J. Then, using backtracking, we define vertices Q^T_s and arcs E_s^T of digraph G^T_s as follows. Initially, we set Q^T_s = {O_ini : J_i ∈ J} for problem J /a_i≤ p_i≤ b_i/^PC_i (if H_sⁱ(T ) 6= ∅, H_sⁱ(T ) ⊆ H_s, for problem J /a_i≤ p_i≤ b_i/C_max) and we set E_s^T = ∅. Then we add vertex O_uv to the set Q^T_s, and we add arc (O_uv, O_ini) to the set E_s^T:

Q^T_s := Q^T_s ∪ {O_uv}, E_s^T := E_s^T ∪ {(O_uv, O_ini)}

if and only if the following two conditions hold:

1) there is no arc (O_u1v1, O_ini) such that l^b_s(O_u1v1) < l^a_s(O_uv), and 2) inequality l^b_s(Ouv) + bini ≥ l^a_s(Oini) holds.

Continuing in a similar way for each vertex which is already included in the set Q^T_s, we construct the digraph G^T_s (see Lemma 3.1).

Thus, instead of digraphs Gk, k = 1, 2, . . . , λ, one can consider digraphs G^T_k which contain all dominant paths ^Sn_i=1H_kⁱ(T ) and which are often essentially simpler than the corresponding digraphs G_k. The transformation of digraph G_kinto digraph G^T_k by testing inequality (3.7)

(see page 119) takes O(q²) elementary steps (q is the number of operations).

Let for criterion^PC_i the superscripts of the sets Ω¹_k, Ω²_k, . . . , Ω^ω_k^kT, . . . , Ω^ω_k^k be such that

Example 3.2 To illustrate the above notions and definitions, we introduce a job shop problem J 3/n = 3, ai≤ pi≤ bi/^PCi with Q^J₁ = {O1,1, O1,3, O3,2}, Q^J₂ = {O1,2, O2,1, O3,3}

and Q^J₃ = {O2,2, O3,1}. The mixed graph G = (Q^J, A^J, E^J) represented in Figure 3.6 defines the structural input data. The numerical input data are defined by polytope T ∈ R₊⁸ (see Table 3.7) within which the actual vector p of the processing times has to be contained.

For this small example, one can explicitly enumerate all feasible digraphs of set Λ(G).

(the cardinality λ of the set Λ(G) is equal to 22). Since not all digraphs may be optimal for the given segments [a_ij, b_ij] of possible variations of the processing times p_ij, we construct a subset B of the set Λ(G) of possible candidates of competitive digraphs (optimal digraphs) using the algorithms from Section 3.6 below. The cardinality of the set B is equal to 12, while λ = 22.

Table 3.7: Numerical data for problem J 3/n = 3, a_i≤ p_i≤ b_i/^PC_i

i 1 1 1 2 2 3 3 3

j 1 2 3 1 2 1 2 3

a_ij 60 20 45 10 50 60 30 30 bij 80 40 60 30 70 80 50 40

Before finding a minimal solution Λ^T(G) of this problem J 3/n = 3, a_i≤ p_i≤ b_i/^PC_i, we consider its deterministic version J 3/n = 3/^PCi by setting the vector of the processing times to be equal to p⁰ = (p⁰_1,1, p⁰_1,2, . . . , p⁰_3,3) ∈ T with p⁰_1,1 = 70, p⁰_1,2 = 30, p⁰_1,3 = 60, p⁰_2,1 = 20, p⁰_2,2 = 60, p⁰_3,1 = 70, p⁰_3,2 = 40, and p⁰_3,3 = 30. (Note that this vector can be chosen arbitrarily from the polytope T .) We number the digraphs G1, G2, . . . , G12 in accordance with non-decreasing values of the function^PC_i calculated for the vector p⁰of the processing times: L^p₁⁰ = 440, L^p₂⁰ = 470, L^p₃⁰ = 500, L^p₄⁰ = 500, L^p₅⁰ = 520, L^p₆⁰ = 530, L^p₇⁰ = 540, L^p₈⁰ = 550, L^p₉⁰ = 570, L^p₁₀⁰ = 610, L^p₁₁⁰ = 610, L^p₁₂⁰ = 620.

For the vector p⁰ ∈ T , the digraph G₁ = (Q^J, A^J ∪ E₁^J, ∅) with the signature E₁^J = {(O_1,1, O_3,2), (O_3,2, O_1,3), (O_2,1, O_1,2), (O_1,2, O_3,3), (O_2,1, O_3,3), (O_3,1, O_2,2)} is the only optimal digraph. Therefore, for the initial problem J 3/n = 3, a_i≤ p_i≤ b_i/^PC_i, we have to include the digraph G₁ in the desired minimal solution Λ^T(G). Using the above modifica-tion of critical path method, we simplify the digraphs G₁, G₂, . . . , G₁₂. Then we compare

































-

--

-O_1,1 O_1,2 O_1,3

O_2,1 O_2,2

O_3,1 O_3,2 O_3,3

Figure 3.6: Mixed graph G = (Q^J, A^J, E^J) for problem J 3/n = 3, ai≤ p_i≤ b_i/^PC_i

 comparison of these sets, we find that only two digraphs may be better than the digraph G1 (provided that the vector of the processing times belongs to the polytope T defined in Table 3.7). These two digraphs are as follows: Digraph G₂ = (Q^J, A^J ∪ E₂^J, ∅) with the signature E₂^J = {(O_1,1, O_3,2), (O_1,3, O_3,2), (O_2,1, O_1,2), (O_1,2, O_3,3), (O_2,1, O_3,3), (O_2,2, O_3,1)}

and digraph G5 = (Q^J, A^J ∪ E₅^J, ∅) with the signature E₅^J = {(O1,1, O3,2), (O1,3, O3,2),

(O2,1, O1,2), (O1,2, O3,3), (O2,1, O3,3), (O3,1, O2,2)}. Moreover, the digraph G2 is the only optimal one, for example, for the vector p⁰ = (60, 20, 60, 10, 60, 80, 40, 30) ∈ T , and di-graph G₅ is the only optimal one for the vector p⁰⁰ = (60, 20, 45, 30, 70, 80, 50, 30) ∈ T . Consequently, a minimal solution of problem J 3/n = 3, ai≤ pi≤ bi/^PCi consists of three digraphs, namely: Λ^T(G) = {G₁, G₂, G₅}. The corresponding digraphs G^T₁, G^T₂ and G^T₅ are represented in Figure 3.7 a), b) and c), respectively. Note that while digraph G₁ has ω1 = 4 · 2 · 5 = 40 sets of representatives, digraph G^T₁ has only ω^T₁ = 3 · 1 · 3 = 9 sets of representatives. For the digraphs G₂ and G^T₂, these numbers are ω₂ = 16 and ω₂^T = 2, and for the digraphs G₅ and G^T₅, these numbers are ω₅ = 28 and ω^T₅ = 1.

The above full enumeration of digraphs Λ(G) is only possible for a small number of edges in the mixed graph G, and for a practical use one has to reduce the number of digraphs which have to be constructed. E.g., for the example under consideration, it is sufficient to construct only k = 5 digraphs, which are the best for the initial vector p⁰ of the processing times. Further, in Section 3.6 such a calculation will be developed on the basis of a branch-and-bound method for constructing k best digraphs. Moreover, the digraphs G₃ and G₄ in the set of the k = 5 best digraphs are also redundant. In Section 3.6, we present a branch-and-bound method for constructing all digraphs which are the only ones that may be optimal for feasible vectors of the processing times. We also show how to calculate the stability radius of an optimal digraph on the basis of an explicit enumeration of the digraphs Λ(G). The calculation of the stability radius will be used in Sections 3.6 and 5.4 as the main procedure for finding a minimal solution of problem J /a_i≤ p_i≤ b_i/^PC_i.

The remainder of this chapter is organized as follows. Necessary and sufficient con-ditions for a set of digraphs to be a solution respectively a minimal solution of the job shop problem with uncertain processing times are proven in Section 3.5. Three exact and four heuristic algorithms for problems J /a_i ≤ p_i ≤ b_i/C_max and J /a_i ≤ p_i ≤ b_i/^PC_i are given in Section 3.6. Section 3.6 contains a branch-and-bound method (B&B1) for constructing k schedules which are the best for problem J /ai ≤ pi ≤ bi/Φ with some fixed processing times. As it has been shown for the traveling salesman prob-lem [Lib99, LvdPSvdV96, vdP97] and for linear binary programming [PZ76, WJ88], the running time of such a branch-and-bound variant grows relatively slowly with k. We develop also a branch-and-bound method (B&B2) for constructing all schedules which may be optimal if the processing times vary between given lower and upper bounds. Un-fortunately, both algorithms B&B1 and B&B2 may construct some redundant schedules, which are not necessarily contained in a minimal solution of problem J /a_i≤ p_i≤ b_i/Φ.

To reject such redundant schedules, we use a deeper (but more time-consuming) stability analysis of an optimal schedule.