An exact extended formulation for the unrelated parallel machine total weighted completion time problem

Abstract

The plethora of research on $\mathcal {NP}$-hard parallel machine scheduling problems is focused on heuristics due to the theoretically and practically challenging nature of these problems. Only a handful of exact approaches are available in the literature, and most of these suffer from scalability issues. Moreover, the majority of the papers on the subject are restricted to the identical parallel machine scheduling environment. In this context, the main contribution of this work is to recognize and prove that a particular preemptive relaxation for the problem of minimizing the total weighted completion time (TWCT) on a set of unrelated parallel machines naturally admits a non-preemptive optimal solution and gives rise to an exact mixed integer linear programming formulation of the problem. Furthermore, we exploit the structural properties of TWCT and attain a very fast and scalable exact Benders decomposition-based algorithm for solving this formulation. Computationally, our approach holds great promise and may even be embedded into iterative algorithms for more complex shop scheduling problems as instances with up to 1000 jobs and 8 machines are solved to optimality within a few seconds.

Appendices

Appendix: Proof of Proposition 3.1

Proof

The proof consists of two main steps. First, we show that $(\overline{\varvec{u}}{}_k,\overline{\varvec{v}}{}_k)$ is a feasible solution for $\left( \mathbf {DS_{\varvec{k}}-R}\right)$, i.e., $\overline{v}{}_{kt} \le {} 0$ for all $t=1,\ldots {},H_k$ and $\overline{u}{}_{jk} + \overline{v}{}_{kt} \le {} c_{jkt}$ for all $j\in {}J_k$ and $t=1,\ldots {},H_k$. Then, we demonstrate that the objective function value associated with this feasible solution of the dual slave problem is equal to that of the optimal solution of the corresponding primal problem. Our focus in this proof is entirely on $\left( \mathbf {DS_{\varvec{k}}-R}\right) {}$ for a given machine k, and therefore, every specific job or set of jobs referred to in the following belongs to $J_k$.

The non-positivity of $\overline{v}{}_{kt}$ for all $t=1,\ldots {},H_k$ follows from

$$\begin{aligned} \overline{v}{}_{kt}= & {} \frac{w_{l(t){}}}{p_{l(t){}k}}\left( t - \sum _{i\le l(t){}} p_{ik} \right) - \sum _{i>l(t){}} w_i\\= & {} - r(t){} \frac{w_{l(t){}}}{p_{l(t){}k}} - \sum _{i>l(t){}} w_i \le {} 0, \end{aligned}$$

where

$$\begin{aligned} 0\le r(t){} = \sum _{i\le {}l(t){}} p_{ik} - t \le p_{l(t){}k}-1, \end{aligned}$$

(30)

and the weights and the processing times are strictly positive.

To show that $\overline{u}{}_{jk} + \overline{v}{}_{kt} \le {} c_{jkt}$ is satisfied for all $j\in {}J_k$ and $t=1,\ldots {},H_k$, we substitute the values of $\overline{u}_{jk}$ and $\overline{v}_{kt}$ from (27) and $c_{jkt}$ from (10). The constraint $\overline{u}{}_{jk} + \overline{v}{}_{kt} \le {} c_{jkt}$ then reduces to

$$\begin{aligned}&\frac{w_j}{p_{jk}}\left( \sum _{i\le j} p_{ik} + \frac{p_{jk}}{2}-\frac{1}{2} \right) \nonumber \\&\qquad + \sum _{i>j} w_i+ \frac{w_{l(t){}}}{p_{l(t){}k}}\left( t - \sum _{i\le l(t){}} p_{ik} \right) - \sum _{i>l(t){}} w_i\nonumber \\&\quad \le {}\frac{w_j}{p_{jk}} \left( t+\frac{p_{jk}}{2}-\frac{1}{2}\right) \nonumber \\ \iff {}&\frac{w_j}{p_{jk}}\left( \sum _{i\le {}j} p_{ik} +\frac{p_{jk}}{2}-\frac{1}{2}\right) \nonumber \\&\qquad + \sum _{i>j} w_i - \frac{w_{l(t){}}}{p_{l(t){}k}}r(t){} - \sum _{i>l(t){}} w_i\nonumber \\&\quad \le {}\frac{w_j}{p_{jk}}\left( \sum _{i\le {}l(t){}} p_{ik} - r(t){} +\frac{p_{jk}}{2}-\frac{1}{2}\right) \end{aligned}$$

(31)

by replacing $t - \sum _{i\le l(t){}} p_{ik}$ by $-r(t){}$ and t by $\sum _{i\le {}l(t){}} p_{ik} - r(t){}$ based on (30). In order to establish the validity of (31), we consider the cases $j \le {} l(t){}$ and $j > l(t){}$ separately.

If $j \le {} l(t){}$, (31) simplifies to

$$\begin{aligned} \frac{w_j}{p_{jk}}\left( - \sum _{j<i\le {}l(t){}} p_{ik} \right) + \sum _{j<i\le {}l(t){}} w_i \le {} r(t){} \left( \frac{w_{l(t){}}}{p_{l(t){}k}} - \frac{w_j}{p_{jk}} \right) . \end{aligned}$$

(32)

Note that $j \le {} l(t){}$ implies $\frac{w_j}{p_{jk}} \ge \frac{w_{l(t){}}}{p_{l(t){}k}}$ and leads to $\left( p_{l(t){}k} - 1\right) \left( \frac{w_{l(t){}}}{p_{l(t){}k}} - \frac{w_j}{p_{jk}} \right) \le {} r(t){} \left( \frac{w_{l(t){}}}{p_{l(t){}k}} - \frac{w_j}{p_{jk}} \right)$ based on (30). Therefore, (32) holds if

$$\begin{aligned}&- \sum _{j<i\le {}l(t){}} p_{ik}\frac{w_j}{p_{jk}} + \sum _{j<i\le {}l(t){}} p_{ik}\frac{w_i}{p_{ik}}\nonumber \\&\quad \le {} \left( p_{l(t){}k} - 1\right) \left( \frac{w_{l(t){}}}{p_{l(t){}k}} - \frac{w_j}{p_{jk}} \right) \end{aligned}$$

(33)

$$\begin{aligned} \iff {}&\sum _{j<i<l(t){}} p_{ik} \left( \frac{w_i}{p_{ik}} - \frac{w_j}{p_{jk}} \right) \nonumber \\&\quad \le {} \frac{w_j}{p_{jk}} - \frac{w_{l(t){}}}{p_{l(t){}k}} \end{aligned}$$

(34)

is satisfied, where the transition from (33) to (34) requires adding $p_{l(t){}k}\frac{w_j}{p_{jk}}-w_{l(t){}}$ to both sides of (33). The inequality (34) is clearly correct since $\left( \frac{w_i}{p_{ik}} - \frac{w_j}{p_{jk}} \right) \le {} 0$ for $i \ge {} j$ and $\left( \frac{w_j}{p_{jk}} - \frac{w_{l(t){}}}{p_{l(t){}k}} \right) \ge {} 0$, and this completes the argument for the first case with $j \le {} l(t){}$.

If $j > l(t){}$, re-arranging the terms of (31) leads to

$$\begin{aligned} \frac{w_j}{p_{jk}}\left( \sum _{l(t){}<i\le {}j} p_{ik} \right) - \sum _{l(t){}<i\le {}j} w_i \le {} r(t){} \left( \frac{w_{l(t){}}}{p_{l(t){}k}} - \frac{w_j}{p_{jk}} \right) .\qquad \end{aligned}$$

(35)

The inequality $\frac{w_j}{p_{jk}} \le \frac{w_{l(t){}}}{p_{l(t){}k}}$ follows from $j > l(t){}$, and we conclude that the right hand side of (35) is non-negative. Therefore, in order to prove that (35) is satisfied it is sufficient to demonstrate the correctness of this relation:

$$\begin{aligned}&\sum _{l(t){}<i\le {}j} p_{ik}\frac{w_j}{p_{jk}} - \sum _{l(t){}<i\le {}j} p_{ik}\frac{w_i}{p_{ik}}\nonumber \\&\quad =\sum _{l(t){}<i\le {}j} p_{ik} \left( \frac{w_j}{p_{jk}} - \frac{w_i}{p_{ik}} \right) \le {} 0. \end{aligned}$$

(36)

The validity of inequality (36) derives from $\left( \frac{w_j}{p_{jk}} - \frac{w_i}{p_{ik}} \right) \le {} 0$ for $i \le {} j$. This yields the correctness of (31) for the second case with $j > l(t){}$, and $(\overline{\varvec{u}}{}_k,\overline{\varvec{v}}{}_k)$ is certified as a feasible solution of $\left( \mathbf {DS_{\varvec{k}}-R}\right)$.

As noted before, the optimal schedule of the restricted cut generation subproblem $\left( \mathbf {TR_{\varvec{k}}-R}\right)$—the primal problem associated with $\left( \mathbf {DS_{\varvec{k}}-R}\right)$—follows the WSPT order for the jobs in $J_k$. Therefore, the associated optimal objective function value is calculated as $\sum _{j\in {}J_k} w_j \left( \sum _{i\le {}j} p_{ik} \right)$, where the completion time of job j in the non-preemptive WSPT schedule is equal to the sum of the processing times of the jobs placed earlier in the WSPT sequence. Thus, in order to complete the proof, we must argue that the objective function value associated with $(\overline{\varvec{u}}{}_k,\overline{\varvec{v}}{}_k)$ in $\left( \mathbf {DS_{\varvec{k}}-R}\right)$ is equal to $\sum _{j\in {}J_k} w_j \left( \sum _{i\le {}j} p_{ik} \right)$.

$$\begin{aligned}&\sum _{j\in {}J_k} p_{jk} \overline{u}{}_{jk} + \sum _{t=1}^{H_k} \overline{v}{}_{kt}\nonumber \\&\quad = \sum _{j\in {}J_k} p_{jk} \left( \frac{w_j}{p_{jk}}\left( \sum _{i\le {}j} p_{ik} + \frac{p_{jk}}{2}-\frac{1}{2} \right) + \sum _{i>j} w_i \right) \nonumber \\&\qquad + \sum _{t=1}^{H_k} \left( \frac{w_{l(t){}}}{p_{l(t){}k}}\left( t - \sum _{i\le {}l(t){}} p_{ik} \right) - \sum _{i>l(t){}} w_i \right) \end{aligned}$$

(37)

$$\begin{aligned}&\quad = \sum _{j\in {}J_k} p_{jk} \left( \frac{w_j}{p_{jk}}\left( \sum _{i\le {}j} p_{ik} + \frac{p_{jk}}{2}-\frac{1}{2} \right) + \sum _{i>j} w_i \right) \nonumber \\&\qquad + \sum _{j\in {}J_k} \left( \frac{w_j}{p_{jk}}\left( -\sum _{i=0}^{p_{jk}-1} i \right) - p_{jk}\sum _{i>j} w_i \right) \nonumber \\&\quad = \sum _{j\in {}J_k} p_{jk} \left( \frac{w_j}{p_{jk}}\left( \sum _{i\le {}j} p_{ik} + \frac{p_{jk}}{2}-\frac{1}{2} \right) + \sum _{i>j} w_i \right) \nonumber \\&\qquad - \sum _{j\in {}J_k} p_{jk} \left( \frac{w_j}{p_{jk}} \frac{(p_{jk}-1)}{2} + \sum _{i>j} w_i \right) \nonumber \\&\quad = \sum _{j\in {}J_k} p_{jk} \left( \frac{w_j}{p_{jk}}\left( \sum _{i\le {}j} p_{ik} + \frac{p_{jk}}{2}-\frac{1}{2} - \frac{p_{jk}-1 }{2} \right) \right) \nonumber \\&\quad = \sum _{j\in {}J_k} w_j \left( \sum _{i\le {}j} p_{ik} \right) . \end{aligned}$$

(38)

For the transition from (37) to (38), observe that each job $j\in J_k$ becomes the job $l(t){}$ for $p_{jk}$ consecutive time periods, and the difference $\left( t - \sum _{i\le {}l(t){}} p_{ik} \right)$ runs from $-(p_{jk}-1)$ to zero during these time periods. Therefore, we have $\sum _{t=1}^{H_k} \frac{w_{l(t){}}}{p_{l(t){}k}}\left( t - \sum _{i\le {}l(t){}} p_{ik} \right) =\sum _{j\in {}J_k} \frac{w_j}{p_{jk}}\left( -\sum _{i=0}^{p_{jk}-1} i \right)$. A similar argument yields $-\sum _{t=1}^{H_k} \sum _{i>l(t){}} w_i =-\sum _{j\in {}J_k} p_{jk}\sum _{i>j}w_i$.

Benders decomposition algorithm with cut strengthening for $\left( \mathbf {TR-A}\right)$