Large-scale energy-conscious bi-objective single-machine batch scheduling under time-of-use electricity tariffs via effective iterative heuristics

Abstract

Time-of-use (TOU) electricity pricing policy is widely encountered in the world, which provides new opportunities for power-intensive enterprises to save their energy cost. A good trade-off between the total electricity cost and production efficiency is desired by decision makers. This work addresses an energy-conscious bi-objective single-machine batch scheduling problem under TOU electricity tariffs, in which electricity price varies with time. The objective of the problem is to simultaneously minimize total electricity cost and makespan. Due to its strong NP-hard nature, two fast new $\epsilon$-constraint-based constructive heuristic algorithms are developed to solve it. The core idea is to transform the bi-objective problem into a series of single-objective problems that are fast and heuristically solved to obtain an approximate Pareto front. Especially, for each transformed single-objective problem, two novel constructive heuristic algorithms are proposed by solving a series of multiple knapsack problems and 0–1 knapsack problems, respectively. Computational results on 145 benchmark and 80 newly generated larger-scale instances show that the proposed algorithms are quite efficient and are able to find high-quality Pareto solutions for large-scale problems with up to 200 batches.

Appendices

Appendix A

The notations used to formulate SMBS–TOU are defined as follows:

Indices and parameters:

j::

Index of a job;

J::

Set of jobs, $j\in J$;

|J|::

Number of jobs;

$p_j$::

Processing time of job $j, j\in J$;

W::

Capacity of a batch;

B::

Set of batches, $B=\{1, 2,\ldots , |B|\}$;

b::

Index of a batch, $b \in B$;

I::

Set of periods;

|I|::

Number of periods;

i::

Index of a period, $i \in I$;

$D_i$::

Duration of period $i,~i\in I$;

$t_i$::

Starting time of period $i,~1 \le i \le |I|+1$;

$u_i$::

Unit electricity cost of period $i,~ i \in I$.

Decision variables:

|B|::

Number of the formed batches;

$x_{jbi}$::

$x_{jbi}=1$ if job j is assigned to batch b and the batch is processed in period i; and $x_{jbi}=0$ otherwise; $\forall j\in J, b\in B, i\in I$;

$y_{bi}$::

$y_{bi}=1$ if batch $b\in B$ is processed in period i; and $y_{bi}=0$ otherwise; $\forall b\in B, i\in I$;

$z_i$::

$z_i=1$ if at least one batch is processed in period i; and $z_i=0$ otherwise; $\forall i\in I$;

$P_{bi}$::

$P_{bi}=\max {p_j|j\in b}$, if batch $b\in B$ is processed in period i; and $P_{bi}=0$ otherwise; $\forall b\in B, i\in I$;

${\textit{T}EC}$::

Total electricity cost;

$C_{max}$::

Makespan, i.e., maximum completion time.

With the notations defined above, the studied SMBS–TOU can be formulated as the following integer program (Cheng et al. 2014).

Model ${\mathcal {P}}_{0}:$

$$\begin{aligned}&\text {Minimize} ~{\textit{T}EC} \end{aligned}$$

(29)

$$\begin{aligned}&\text {Minimize} ~C_{max} \end{aligned}$$

(30)

$$\begin{aligned} \text {subject to:} \nonumber \\&\sum ^{|I|}_{i=1}\sum ^{|B|}_{b=1}x_{jbi}=1, \quad \forall j\in J \end{aligned}$$

(31)

$$\begin{aligned}&\sum ^{|I|}_{i=1}y_{bi}=1,\quad \forall b\in B \end{aligned}$$

(32)

$$\begin{aligned}&\sum ^{|J|}_{j=1}x_{jbi}\le Wy_{bi},\quad \forall b\in B,\;\; \forall i\in I \end{aligned}$$

(33)

$$\begin{aligned}&p_jx_{jbi}\le P_{bi},\quad \forall j\in J,\;\; \forall b\in B, \;\; \forall i\in I \end{aligned}$$

(34)

$$\begin{aligned}&\sum ^{|B|}_{b=1}P_{bi}\le D_iz_i,\quad \forall i\in I \end{aligned}$$

(35)

$$\begin{aligned}&t_iz_i+\sum ^{|B|}_{b=1} P_{bi} \le C_{max},\quad \forall i\in I \end{aligned}$$

(36)

$$\begin{aligned}&\sum ^{|I|}_{i=1}u_i\sum ^{|B|}_{b=1}P_{bi} \le {\textit{T}EC}\end{aligned}$$

(37)

$$\begin{aligned}&z_i \in \{0, 1\},\quad \forall i\in I \end{aligned}$$

(38)

$$\begin{aligned}&y_{bi} \in \{0, 1\},\quad \forall b\in B,\;\; \forall i\in I \end{aligned}$$

(39)

$$\begin{aligned}&x_{jbi} \in \{0, 1\}, \quad \forall j\in J,\;\; \forall b\in B, \;\; \forall i\in I \end{aligned}$$

(40)

$$\begin{aligned}&P_{bi}\ge 0,\quad \forall b\in B, \;\; \forall i\in I \end{aligned}$$

(41)

$$\begin{aligned}&{\textit{T}EC} \ge 0 \end{aligned}$$

(42)

$$\begin{aligned}&C_{max} \ge 0 \end{aligned}$$

(43)

The objective functions (29) and (30) are to minimize the total electricity cost TEC and the makespan, respectively. (31) ensures that a job is processed in a batch within a period. (32) denotes that a batch is processed within a period. The batch capacity constraint is imposed by (33). (34) defines the processing time of a batch. (35) denotes the total batch processing time in a period should not exceed its duration. (36) and (37) define the makespan and the total electricity cost, respectively. (38)–(43) impose the restrictions on decision variables. Note that the number of batches |B| can be initially considered as its upper bound |J| to obtain an integer linear model.

Appendix B

The LPT batching rule is illustrated by the following example. Consider a small-scale problem, there are five jobs with processing time 5, 4, 3, 2, and 1 that need to be processed, then with the LPT rule, three bathes with processing time 5, 3, and 1 can be formed in which the first batch involves the jobs of the first two longest processing time while the last one involves the job of the shortest processing time, which is shown in Fig. 5.

Fig. 5

An example of the LPT rule