Single-machine scheduling with resource-dependent processing times and multiple unavailability periods

Abstract

We consider a single-machine scheduling problem with multiple unavailability periods such that the processing time of each job is inversely proportional to the kth power of its own resource consumption amount. The objective is to minimize the sum of the makespan and the total resource consumption cost. For every $k>0$, we show that the problem with one unavailability period is NP-hard while it admits a fully polynomial-time approximation scheme. Furthermore, we analyze the (in)approximability for the case with multiple unavailability periods. For every $k>0$, we show that the problem with an arbitrary number of unavailability periods is strongly NP-hard. Finally, we extend the results to the problem of minimizing the makespan with a constraint on the total resource consumption cost.

Appendix A. Proof of Lemma 1

Proof

For $i<l$, the jobs in ${\mathcal {J}}_i$ contribute $\sum _{j \in {\mathcal {J}}_i} c_j u_j$ in $z_{1}(\sigma )$ by (4). Hence, the optimal resource allocation problem for jobs in ${\mathcal {J}}_i$ can be written as follows:

$$\begin{aligned} \begin{array}{lll} \min &{} \sum _{j \in {\mathcal {J}}_i} c_j u_j\\ \mathrm {s.t.} &{} \sum _{j \in {\mathcal {J}}_i} p_j(u_j) \le L_i ~~\text {and}~~u_j > 0 \quad \text {for each}~~ j \in {\mathcal {J}}_i. \end{array}\nonumber \\ \end{aligned}$$

(31)

Shabtay and Zofi (2018) showed that $\sum _{j \in {\mathcal {J}}_i} p_j(u_j) = L_i$ and (8) hold in problem (31).

The jobs in ${\mathcal {J}}_l$ contribute $\sum _{j \in {\mathcal {J}}_l} t_j(u_j)$ in $z_{1}(\sigma )$ by (4). Hence, the optimal resource allocation problem for jobs in ${\mathcal {J}}_l$ can be written as follows:

$$\begin{aligned} \begin{array}{lll} \min &{} \sum _{j \in {\mathcal {J}}_l} t_j(u_j)\\ \mathrm {s.t.} &{} \sum _{j \in {\mathcal {J}}_l} p_j(u_j) \le L_l ~~\text {and}~~u_j > 0 \quad \text {for each}~~ j \in {\mathcal {J}}_l. \end{array}\nonumber \\ \end{aligned}$$

(32)

If $s_l \le L_l$, then the optimal solution exists at $u_j = \tau _j$ for $j \in {\mathcal {J}}_l$ by (7). Henceforth, we focus on the case with $s_l > L_l$. If $\sum _{j \in {\mathcal {J}}_l} p_j(u_j) < L_l$, then a job $j \in {\mathcal {J}}_l$ exists with $u_j> \tau _j$. Since $t_j(u_j)$ is an increasing function for $u_j > \tau _j$, however, we can decrease the objective value without violating the feasibility by decreasing $u_j$ by a sufficiently small value. This implies that $\sum _{j \in {\mathcal {J}}_l} p_j(u_j) = L_l$ in the optimal solution for problem (32). Then, the objective function can be represented as $(L_l + \sum _{j \in {\mathcal {J}}_l} c_j u_j)$, which implies that problem (32) is reduced to problem (31). Thus, the optimal solution for problem (32) can be represented by (9). $\square$