Single-machine scheduling with multi-agents to minimize total weighted late work

Abstract

We consider the competitive multi-agent scheduling problem on a single machine, where each agent’s cost function is to minimize its total weighted late work. The aim is to find the Pareto-optimal frontier, i.e., the set of all Pareto-optimal points. When the number of agents is arbitrary, the decision problem is shown to be unary $\mathcal {NP}$-complete even if all jobs have the unit weights. When the number of agents is two, the decision problems are shown to be binary $\mathcal {NP}$-complete for the case in which all jobs have the common due date and the case in which all jobs have the unit processing times. When the number of agents is a fixed constant, a pseudo-polynomial dynamic programming algorithm and a $(1+\epsilon )$-approximate Pareto-optimal frontier are designed to solve it.

Appendix

Example

Consider a two-agent scheduling problem $1 | CO | \mathcal {P}(\sum w_j^{(1)}Y_j^{(1)}, \sum w_j^{(2)}Y_j^{(2)})$, in which $\mathcal {J}^1=\{J_1^{(1)}, J_2^{(1)}\}$, $\mathcal {J}^2=\{J_1^{(2)}, J_2^{(2)}, J_3^{(2)}\}$, and the corresponding job parameters are given in Table 1.

Table 1 Job data of the problem set

Before executing algorithm $\mathcal {H}_1$, the jobs are first indexed in the EDD order, see Table 2.

Table 2 The indexed job data of the problem set

Table 3 Generation of $\mathcal {L}_1$ from $\mathcal {L}_0$

Table 4 Generation of $\mathcal {L}_2$ from $\mathcal {L}_1$

Table 5 Generation of $\mathcal {L}_3$ from $\mathcal {L}_2$

Table 6 Generation of $\mathcal {L}_4$ from $\mathcal {L}_3$

In the sequel, we apply algorithm $\mathcal {H}_1$ to solve this numerical example.

Step 1 [Initialization] Set $\mathcal {L}_0=\{(0, 0, 0, 0, 0)\}$, ${\mathcal L}_j=\emptyset$ for $j=1, 2, \ldots , 5$ and $j:=0$.

Step 2 [Generation] For $j=1$, $\mathcal {L}_1=\{(1, 0, 2, 0, 0)$, $(1, 0, 0, 0, 1), (1, 2, 0, 0, 0)\}$, see Table 3 for the generation process.

For $j=2$, $\mathcal {L}_2=\{(2, 0, 2, 6, 0), (2, 0, 2, 0, 2), (2, 3, 2, 0, 0)$, $(2, 0, 0, 6, 1), (2, 2, 0, 6, 0), (2, 2, 0, 0, 2), (2, 5, 0, 2, 0)\}$, see Table 4 for the generation process.

For $j=3$, $\mathcal {L}_3=\{(3, 0, 14, 6, 0), (3, 0, 2, 6, 3),$$(3, 4, 2, 6, 0), (3, 0, 14, 0, 2),$ (3, 3, 14, 0, 0), (3, 3, 2, 0, 3),  $(3, 7, 2, 0, 0),$ (3, 0, 12, 6, 1), (3, 2, 12, 6, 0),  $(3, 2, 0, 6, 3), (3, 6, 0, 6, 0),$ (3, 2, 12, 0, 2),  $(3, 5, 12, 2, 0), (3, 5, 0, 2, 3), (3, 9, 6, 2, 0)\}$, see Table 5 for the generation process.

Table 7 Generation of $\mathcal {L}_5$ from $\mathcal {L}_4$

For $j=4$, $\mathcal {L}_4=\{(4, 0, 14, 14, 0), (4, 0, 14, 6, 4),$ (4, 2, 14, 6, 0), (4, 0, 2, 14, 3), (4, 2, 2, 6, 3),  $(4, 6, 2, 6, 0), (4, 4, 2, 14, 0), (4, 4, 2, 6, 4),$$(4, 0, 14, 8, 2), (4, 2, 14, 0, 2), (4, 3, 14, 0, 4),$ (4, 5, 14, 0, 0), (4, 5, 2, 0, 3), (4, 9, 8, 0, 0),  (4, 0, 12, 14, 1), (4, 2, 12, 6, 1),  $(4, 2, 12, 14, 0), (4, 2, 12, 6, 4), (4, 4, 12, 6, 0)$, $(4, 2, 0, 14, 3), (4, 4, 0, 6, 3), (4, 6, 0, 14, 0),$ (4, 6, 0, 6, 4), (4, 8, 0, 10, 0), $(4, 2, 12, 8, 2), (4, 5, 12, 2, 4), (4, 7, 12, 2, 0)\}$, see Table 6 for the generation process, where the underlined states are eliminated in Step 2.2.

For $j=5$, $\mathcal {L}_5=\{(5, 0, 14, 23, 0), (5, 3, 14, 14, 0),$ (5, 2, 14, 15, 0),  (5, 5, 14, 6, 0), (5, 9, 8, 6, 0),  $(5, 4, 2, 23, 0), (5, 7, 2, 14, 0),$ (5, 6, 2, 15, 0), (5, 9, 2, 12, 0),  $(5, 8, 14, 3, 0), (5, 7, 2, 14, 0),$ (5, 2, 12, 23, 0), (5, 5, 12, 14, 0),  (5, 4, 12, 15, 0),  (5, 7, 12, 6, 0), (5, 6, 0, 23, 0),  $(5, 8, 0, 19, 0)\}$, see Table 7 for the generation process (note that we do not need to consider the case where job $J_5$ is deferred as it is the last job), where the underlined states are eliminated in Step 2.2.

Step 3 [Refinement] $\mathcal {L}_5=\{(5, 8, 0, 19, 0), (5, 9, 2, 12, 0),$$(5, 9, 8, 6, 0), (5, 8, 14, 3, 0)\}$.

Step 4 [Result] The Pareto-optimal frontier is $\mathcal {R}=\{(0, 19), (2, 9),$$(8, 6), (14, 3)\}$.

• The Pareto-optimal schedule corresponding to (0, 19) is $J_1^{(1)} \prec J_2^{(1)} \prec J_2^{(2)} \prec \underline{J_1^{(2)}} \prec \underline{J_3^{(2)}}$;

• The Pareto-optimal schedule corresponding to (2, 12) is $J_2^{(1)} \prec J_2^{(2)} \prec J_3^{(2)} \prec \underline{J_1^{(1)}} \prec \underline{J_1^{(2)}}$;

• The Pareto-optimal schedule corresponding to (8, 6) is $J_2^{(2)} \prec J_3^{(2)} \prec J_2^{(1)} \prec \underline{J_1^{(1)}} \prec \underline{J_1^{(2)}}$;

• The Pareto-optimal schedule corresponding to (14, 3) is $J_1^{(2)} \prec J_2^{(2)} \prec J_3^{(2)} \prec \underline{J_1^{(1)}} \prec \underline{J_2^{(1)}}$;

where the underlined jobs are late in the corresponding Pareto-schedule.