Depth-first heuristic search for the job shop scheduling problem

Abstract

We evaluate two variants of depth-first search algorithms and consider the classic job shop scheduling problem as a test bed. The first one is the well-known branch-and-bound algorithm proposed by P. Brucker et al. which uses a single chronological backtracking strategy. The second is a variant that uses partially informed depth-first search strategy instead. Both algorithms use the same heuristic estimation; in the first case, it is only used for pruning states that cannot improve the incumbent solution, whereas in the second it is also used to sort the successors of an expanded state. We also propose and analyze a new heuristic estimation which is more informed and more time consuming than that used by Brucker’s algorithm. We conducted an experimental study over well-known instances showing that the proposed partially informed depth-first search algorithm outperforms the original Brucker’s algorithm.

Appendices

Appendix A: Details of the expansion mechanism

Let us consider a feasible schedule S being compatible with the disjunctive arcs fixed in state n. Of course, S might be a schedule calculated by Algorithm 2. The solution graph G _S has a critical path with critical blocks B ₁,…,B _k . For block $B_{j} = (u_{1}^{j}, \dots, u_{m_{j}}^{j})$ the sets of operations

$$E_{j}^B = B_j \setminus \bigl\{u_1^j\bigr\} \quad \mbox{and} \quad E_{j}^A = B_j \setminus\bigl\{u_{m_j}^j\bigr\} $$

are called before-candidates and after-candidates respectively. For each before-candidate (after-candidate) a successor s of n is generated by moving the candidate before (after) its corresponding block. An operation $l \in E_{j}^{B}$ is moved before B _j by fixing the arcs {l→i; i∈B _j ∖{l}}. Similarly, $l \in E_{j}^{A}$ is moved after B _j by fixing the arcs {i→l; i∈B _j ∖{l}}.

This expansion strategy is complete as it guarantees that at least one optimal solution is contained in the search graph. However, it can be improved by fixing additional arcs in order to define a complete search tree. Let us consider a permutation (E ₁,…,E _2k ) of all the sets $E_{j}^{B}$ and $E_{j}^{A}$. This permutation defines an ordering for successors generation. When a successor is created from a candidate E _t , we can assume that all the solutions reachable from n by fixing the arcs corresponding to the candidates E ₁,…,E _t−1 will be explored from the successors associated to these candidates. So, for the successor state s generated from E _t the following sets of disjunctive arcs can be fixed: $F_{j} = \{u_{1}^{j} \rightarrow i;\ i=u_{2}^{j}, \dots, u_{m_{j}}^{j}\}$, for each $E_{j}^{B} < E_{t}$ and $L_{j} = \{i \rightarrow u_{m_{j}}^{j};\ i=u_{1}^{j}, \dots, u_{m_{j}-1}^{j}\}$, for each $E_{j}^{A} < E_{t}$ in the permutation above. So the successors of a search tree node n generated from the permutation (E ₁,…,E _2k ) are defined as follows. For each operation $l \in E_{j}^{B}$, j=1…k, generate a candidate successor s by fixing the arcs $\mathit{FD}_{s} = \mathit{FD}_{n} \cup S_{l}^{B}$, where

$$ S_l^B = \bigcup_{E_i^B < E_j^B}F_i \cup\bigcup_{E_i^A < E_j^B}L_i \cup \bigl \{l\rightarrow i: i \in B_j\backslash\{l\}\bigr\}. $$

(11)

And for each operation $l \in E_{j}^{A}$ generate a search tree node s by fixing the arcs $\mathit{FD}_{s} = \mathit{FD}_{n} \cup S_{l}^{A}$ with

$$ S_l^A = \bigcup_{E_i^B < E_j^A}F_i \cup\bigcup_{E_i^A < E_j^A}L_i \cup \bigl \{i\rightarrow l: i \in B_j\backslash\{l\}\bigr\}. $$

(12)

After fixing one of these sets of arcs to build a new successor n′ from a state n, heads and tails are calculated by the algorithm Computing new Heads and Tails as it is indicated in Sect. 4.1. To compute heads, the algorithm is applied from the start node following a topological ordering in G _n′ (analogously for tails starting from node end backwards). If heads and tails get fixed for all operations, then n′ is a feasible search state, otherwise G _n′ contains some cycle and so n′ is unfeasible.

For example, if we consider the search state of Fig. 2(a) the heuristic solution might be that of Fig. 1(b). Then, the first successor state would be that represented in Fig. 2(b).

Appendix B: Improving heads and tails

The algorithms to improve heads and tails proposed in Brucker et al. (1994a) are based on the following ideas. If (J,c) is primal pair, the operation c cannot start at a time lower than

$$ r_J = \max_{J' \subseteq J} \biggl\{ \min_{j \in J'}r_j + \sum_{j \in J'}p_j \biggr\}. $$

(13)

So, if r _c <r _J , we can set r _c =r _J and then the procedure Select fixes all the primal arcs {j→c; j∈J}. Analogously, if (c,J) is dual pair, q _c cannot be lower than

$$ q_J = \max_{J' \subseteq J} \biggl\{ \sum _{j \in J'}p_j + \min_{j \in J'}q_j \biggr\}. $$

(14)

So, if q _c <q _J , we can set q _c =q _J and then the procedure Select fixes all the dual arcs {c→j; j∈J}.

In Brucker et al. (1994a) and Brucker (2004) an algorithm is given that computes all primal and dual arcs in polynomial time (O(|I|²)). This algorithm is based on other two algorithms for improving heads and tails from the current upper bound UB and then fixing additional arcs with the procedure Select.

To improve the head of an operation c, the idea is to compute a lower bound, s _c , of the completion time of c in any solution that improves UB, under the assumption that preemption is allowed. Clearly, s _c is also a lower bound for the completion of c in a non preemptive schedule, so r _c may be improved to $r'_{c} = s_{c} - p_{c}$. The algorithm for improving the head of an operation c is given in Algorithm 5. Analogously for improving tails.

Algorithm 5

Improving Heads (upper bound UB)

Figure 8 shows the schedule built by the Algorithm 5 to improve the head of operation c=4 with UB=52 for the same instance as in Fig. 3. As we can observe, the lower bound obtained for the completion time of operation 4 is s ₄=39, so $r'_{4} = s_{4} - p_{4} = 39 - 5 = 34$. Therefore, after improving heads, the arc (5→4) gets fixed by the procedure Select, due to $r'_{4} + p_{4} + p_{5} + q_{5} = 60 \geq52$. However, before improving heads that arc would not be fixed as r ₄+p ₄+p ₅+q ₅=42<52.

Fig. 8

Calculation of earliest completion times s _c to c=4

In (Brucker 2004) the following results on Algorithm Improving Heads are given.

Theorem 7

• (i) The schedule S calculated by Algorithm Improving Heads is a feasible preemptive schedule.

• (ii) Algorithm Improving Heads provides a lower bound for the completion time of operation c in a preemptive schedule under the assumption that preemption at integer times is allowed.

Theorem 8

Procedure Select calculates all primal and dual arcs associated with I if we replace the original r _j -values and q _j -values by the modified values obtained with the algorithms for improving heads and improving tails respectively.

Appendix C: Proof of Lemma 1

Lemma 1

Let n ₁ and n ₂ be two states fulfilling conditions (6) and (7). If the procedure Improving Heads is applied to n ₁ and n ₂ with tentative upper bounds P ₁ and P ₂≤P ₁ respectively, and then new arcs are fixed by the procedure Select, previous conditions (6) and (7) hold.

Proof

Let S(n ₁,P ₁,r _c (n ₁)) be the preemptive schedule built by the algorithm Improving Heads when applied to the set of operations I that require the same machine as operation c, considering heads and tails in state n ₁ and tentative upper bound P ₁ (analogously S(n ₂,P ₂,r _c (n ₂))). And let $s_{c}^{11}$ and $s_{c}^{22}$ be the completion times of operation c in S(n ₁,P ₁,r _c (n ₁)) and S(n ₂,P ₂,r _c (n ₂)) respectively.

First of all, we have to clarify that if S(n ₂,P ₂,r _c (n ₂)) is feasible, then S(n ₁,P ₁,r _c (n ₁)) is feasible as well. This is true as condition (6) guarantees that any feasible preemptive schedule for the operations in I with heads and tails as they are in n ₂ and tentative upper bound P ₂ is a feasible preemptive schedule for the set I with heads and tails as they are in n ₁ and tentative upper bound P ₁. If S(n ₂,P ₂,r _c (n ₂)) is not feasible, then $s_{c}^{22} = \infty$.

From now on, we consider that both S(n ₁,P ₁,r _c (n ₁)) and S(n ₂,P ₂,r _c (n ₂)) are feasible. In order to prove that $s_{c}^{11} \leq s_{c}^{22}$ we can do the following reasoning. Let us consider the preemptive schedule S(n ₂,P ₁,r _c (n ₁)), i.e., a schedule built by Algorithm Immediate Selection with heads and tails as they are in n ₂ and P ₁, but considering the head of operation c as it is in n ₁. Let $s_{c}^{21}$ be the completion time of operation c in S(n ₂,P ₁,r _c (n ₁)).

On one hand, due to r _i (n ₁)≤r _i (n ₂) and q _i (n ₁)≤q _i (n ₂) it is clear that S(n ₂,P ₁,r _c (n ₁)) is a feasible preemptive schedule for the set I considering P ₁ and the heads and tails as they are in n ₁. So, from Theorem 7 it follows that $s_{c}^{11} \leq s_{c}^{21}$ due to the fact that $s_{c}^{11}$ is a lower bound for the completion time of operation c in a preemptive schedule for I in that context.

At the same time, if we compare the preemptive schedules S(n ₂,P ₁,r _c (n ₁)) and S(n ₂,P ₂,r _c (n ₂)) we can observe that they are the same up to the time r _c (n ₁). So, the remaining time units for all the operations after r _c (n ₁) are the same in both schedules, even though each one of these unit intervals is scheduled at a different time in each preemptive schedule.

Now, in order to prove that $s_{c}^{21} \leq s_{c}^{22}$ we can use the following arguments: let x be a unit time interval of an operation u scheduled after r _c (n ₁), and tx ₁ and tx ₂ the times at which this unit interval is scheduled in S(n ₂,P ₁,r _c (n ₁)) and S(n ₂,P ₂,r _c (n ₂)) respectively. It is clear that tx ₂≤tx ₁ if u is an operation other than c, but tx ₂≥tx ₁ if u is c. In particular, this holds for the last unit interval of c, so $s_{c}^{21} \leq s_{c}^{22}$.

Then $s_{c}^{11} \leq s_{c}^{22}$ and $r'_{c}(n_{1}) = s_{c}^{11} - p_{c} \leq s_{c}^{22} - p_{c} = r'_{c}(n_{2})$. So, after replacing all heads by the improved values we have r _i (n ₁)≤r _i (n ₂) for all i. As tails remain unchanged, condition (6) holds after the procedure Improving Heads. Condition (7) trivially holds as none of the graphs is modified by this procedure.

Now it is easy to see that both conditions hold after Procedure Select. Let (i→j) be a disjunctive arc fixed in n ₁, then r _j (n ₁)+p _j +p _i +q _i (n ₁)≥P ₁ and trivially r _j (n ₂)+p _j +p _i +q _i (n ₂)≥P ₂ due to r _j (n ₁)≤r _j (n ₂), q _i (n ₁)≤q _i (n ₂) and P ₂≤P ₁, so the arc (i→j) is fixed in n ₂ as well. □