A cutting plane approach for the multi-machine precedence-constrained scheduling problem

Abstract

A cutting-plane approach is developed for the problem of optimally scheduling jobs with arbitrary precedence constraints on unrelated parallel machines to minimize weighted completion time. While the single machine version of this problem has attracted much research efforts, enabling solving problems with up to 100 jobs, not much has been done on the multiple machines case. A novel mixed-integer programming model is presented for the problem with multiple machines. For this model, many classes of valid inequalities that cut off fractional linear programming solutions are developed. This leads to an increase of the linear programming lower bound from 89.3 to 94.6% of the corresponding optimal solution, and a substantial reduction in the computational time of an optimal branch-and-bound algorithm for this problem. This enables us to report optimal solutions for problem instances with up to 25 jobs and 5 machines, which is more than twice the size of problems for which optimal solutions have been reported in the literature thus far. For a special case of the problem—that of minimizing makespan—application of our model helps solve 18 of 27 previously unsolved problem instances to optimality.

Appendix: More valid inequalities

In this section, nine different classes of valid inequalities are presented. While inclusion of these inequalities did not help improve $lb_1$, these inequalities play a significant role in fathoming the branch-and-bound tree quicker; refer to Sect. 5.1 for empirical evidence. Additional notation is now introduced that is needed to describe these inequalities.

1.1 Derived values

• $MinC^k(S)$: a lower bound on the sum of completion times of jobs in set S, all scheduled to be processed on k

• $fthr^{k}(i,j,i^{'})$: directed flowthrough (defined below) on k along $i\rightarrow j\rightarrow i^{'}$

• $InActivity^k_j$: $\sum \nolimits _{i=0,i\ne j}^{N}x^k_{ij}$, denoting the total flow into job j on k.

Flowthrough, $fthr^{k}(i,j,i^{'})$, denotes a lower bound on the “flow” (as defined by the x variables) originating at i and reaching $i^{'}$ via j. This is calculated as $fthr^{k}(i,j,i^{'})=x^k_{ij}-\sum \nolimits _{h=1, h\ne i, h\ne i^{'}}^{N+1}x^k_{jh}$. Intuitively, this captures the “incoming” flow at j from i, reduced by the amount of flow that leaves j but does not go to $i^{'}$.

1.2 Inequality class 1

Consider machine k and a subset $S\subseteq {\mathcal {N}}$ of s jobs that complete on k. Then, the following inequality holds:

$$\begin{aligned} \sum \limits _{i\in S}C_i \ge MinC^k(S)\left[ \sum \limits _{i\in S}\sum \limits _{h=0,h\ne i}^{N}x^k_{hi}-s+1\right] \end{aligned}$$

The validity of the inequality stems from the fact that in an integral solution, the right-hand side of the inequality equals $MinC^k(S)$ if the jobs in S are scheduled on machine k, and negative otherwise.

Computing $MinC^k(S)$ is based on the following observation. Given a set of jobs, $i_1,\ldots ,i_s$, re-indexed if necessary such that $p^k_{i_1}\le \ldots \le p^k_{i_s}$, a lower bound on the sum of their completion times on machine k is $p^k_{i_1}+(p^k_{i_1}+p^k_{i_2})+\ldots +(p^k_{i_1}+p^k_{i_2}+\ldots +p^k_{i_s-1}+p^k_{i_s})$.

1.3 Inequality class 2

If h succeeds j in G, then, for a machine k, the following inequality holds:

$$\begin{aligned} \sum \limits _{i=0,i\ne j}^{N}x^k_{ij} \le 1 - x^{k}_{0h} \end{aligned}$$

The validity of the inequality stems from the fact that if h is scheduled to be processed first on k, the right-hand side is 0. The left-hand side of the inequality, $InActivity^k_j$, cannot be positive as j cannot be scheduled on k as j should complete before h’s start.

1.4 Inequality class 3

Consider triplet $h, i, j\in {\mathcal {N}}$, each job distinct, and two distinct machines $k_1$ and $k_2$. Then, the following inequality holds:

$$\begin{aligned} x^{k_1}_{hi} + x^{k_1}_{ih} + x^{k_2}_{hj} + x^{k_2}_{jh}\le 1 \end{aligned}$$

The validity of the inequality follows from the fact that if $x^{k_1}_{hi}=1$, i is scheduled on $k_1$ immediately after h. This precludes the possibility of h being scheduled after i, or of j being scheduled after h on $k_2$ or of h being scheduled after j on $k_2$. Other cases are similarly proven.

1.5 Inequality class 4

Consider pair $i,j\in {\mathcal {N}}$, each job distinct, and two distinct machines $k_1$ and $k_2$. The following inequality holds:

$$\begin{aligned} x^{k_1}_{ij} + x^{k_1}_{ji} + \sum \limits _{h=0,h\ne j}^{N}x^{k_2}_{hj}\le 1 \end{aligned}$$

The validity of this inequality is proven along similar lines to that of Inequality 3.

1.6 Inequality class 5

Consider distinct jobs $i, j\in {\mathcal {N}}$. Then, the following inequality holds:

$$\begin{aligned} \sum _{k\in {\mathcal {K}}}x^{k}_{ij}+\sum _{k\in {\mathcal {K}}}x^{k}_{ji}\le 1 \end{aligned}$$

The validity of this inequality follows since if i immediately precedes j on a machine k, then neither can j immediately precede i on any machine, nor can i immediately precede j on machine other than k.

1.7 Inequality class 6

Consider $j\in {\mathcal {N}}$ and k. The following inequality holds:

$$\begin{aligned} \sum \limits _{i=0,i\ne j}^{N}V^{k}_{ij} + p^{k}_{j}\sum \limits _{i=0,i\ne j}^{N}x^{k}_{ij} \le \sum \limits _{i=1,i\ne j}^{N+1}V^{k}_{ji} \end{aligned}$$

In a feasible integer solution, the left-hand side is the sum of the completion time of job i that immediately precedes j in a schedule, both on k and the processing time of j. The right-hand side denotes the actual completion time of j on machine k. This proves the validity of the inequality.

1.8 Inequality class 7

Consider the case where $fthr^k(i,j,i^{'})=1$. In this case, the lower bound on the completion time of $i^{'}$ can be updated to $C=max(MinCT^{k}(i^{'}), p^k_{i} + p^{k}_{j} + p^k_{i^{'}})$. Then, for $h\notin \{i,j,i^{'}\}$, the following inequality holds:

$$\begin{aligned} V^k_{i^{'}h} \ge (C - MinCT^k(i^{'}))(fthr^k(i,j,i^{'}) - 1) + C x^k_{i^{'}h} \end{aligned}$$

To prove the validity of this inequality, first note that in a feasible solution, the possible values of $fthr^k(i,j,i^{'})$ can be 1, 0 or $-\,1$. If $fthr^k(i,j,i^{'})=1$, then $V^k_{i^{'}h}=C_{i^{'}} x^k_{i^{'}h}\ge C x^k_{i^{'}h}$. If $fthr^k(i,j,i^{'})=0$, the right-hand side of the inequality becomes $MinCT^k(i^{'})-C(1-x^k_{i^{'}h})$. If $x^k_{i^{'}h}=1$, the inequality simplifies to $C_{i^{'}}\ge MinCT^k(i^{'})$. If $x^k_{i^{'}h}=0$, the right-hand side is non-positive and the inequality is trivially true. The case where $fthr^k(i,j,i^{'})=-1$ follows along similar lines.

1.9 Inequality class 8

Consider i, and j that is either independent of i or succeeds i in G. If $x^k_{ij}=1$ for some k, then $V^k_{ij}\ge MinCT(i)$, by virtue of definition of MinCT(i). If, however, j does not immediately succeed i on any machine, $\sum \nolimits _{k\in {\mathcal {K}}}\sum \nolimits _{i^{'}=1,i^{'}\ne j}^{N+1}x^k_{ii^{'}}=1$. This proves the validity of the following inequality.

$$\begin{aligned} \sum \limits _{k\in {\mathcal {K}}}V^{k}_{ij} + MinCT(i)\left( \sum \limits _{k\in {\mathcal {K}}}\sum \limits _{i^{'}=1,i^{'}\ne j}^{N+1}x^k_{ii^{'}}\right) \ge MinCT(i) \end{aligned}$$

1.10 Inequality class 9

Consider distinct $i,j,i^{'}\in {\mathcal {N}}$ such that $(i,i^{'})\in E$ and $(j,i^{'})\in E$. For machine k, the following inequality holds:

$$\begin{aligned} C_{i^{'}} - C_i \ge P^{min}_{i^{'}} + p^k_j x^k_{ij} \end{aligned}$$

Note that if $x^k_{ij}=1$, earliest that $i^{'}$ can start is $C_i+p^k_j$, proving the validity of the inequality. In the computational tests, a generalized version of this inequality is used. The generalization is the case where $(i,h)\in E$, $(h,i^{'})\in E$, $(j,j^{'})\in E$ and $(j^{'},i^{'})\in E$. In this case, the inequality becomes $C_i^{'} - C_i \ge P^{min}(i^{'}) + Max\{p^k_j+P^{min}(j^{'}),P^{min}(h)\}x^k_{ij}$. Further generalization is possible by considering predecessors that are 3 or more levels above $i^{'}$, but computationally that did not provide significantly better improvement.