Two-agent preemptive Pareto-scheduling to minimize the number of tardy jobs and total late work

Abstract

We consider the single-machine preemptive Pareto-scheduling problem with two competing agents A and B, where agent A wants to minimize the number of its jobs (the A-jobs) that is tardy, while agent B wants to minimize the total late work of its jobs (the B-jobs). We provide an $O(nn_{A}\log n_{A}+n_B\log n_B)$-time algorithm that generates all the Pareto-optimal points, where $n_A$ is the number of the A-jobs, $n_B$ is the number of the B-jobs, and $n=n_A+n_B$.

Appendix

Now we give an instance to demonstrate the execution of Algorithm 5.1. Let ${\mathcal {J}}=\{J^A_{1}, J^A_{2}, \ldots ,J^A_{6}, J^B_{1}, \ldots , J^B_{4}\}$ be the instance displayed in Table 1.

Table 1 The job instance ${\mathcal {J}}$

Note that $P^*=1+\max \{d^A_{n_A}, d^B_{n_B}\}=1+\max \{25,26\}=27$. Let $\Omega =\Omega ({\mathcal {J}}^A, {\mathcal {J}}^B)$. The key steps in applying Algorithm 5.1 to solve the instance are as follows:

(i) Generate the schedule $\sigma _0^B=(J_1^B, J_2^B, J_3^B, J_4^B)$ and calculate the Y-value $Y^{(0)}=T_{\max }(\sigma _0^B)=1$. Then generate the schedule $\sigma ^{B(Y^{(0)})}$, and the forbidden intervals ${\mathcal {I}}^{B(Y^{(0)})}=\{h_1,h_2,h_3,h_4,h_5\}$ is determined, where $h_1=[0,4]$, $h_2=[5,10]$, $h_3=[12,18]$, $h_4=[22,25]$, and $h_5=[27,28]$. Then, for each $y\in (Y^{(0)},P_B]=(1, 19]$, $\sigma ^{B(y)}$ and ${\mathcal {I}}^{B(y)}$ can be easily generated. The forbidden intervals of ${\mathcal {I}}^{B(Y^{(0)})}={\mathcal {I}}^{B(1)}$ are displayed in Fig. 1.

Fig. 1

The forbidden intervals of ${\mathcal {I}}^{B(1)}$

(ii) Generate the schedule $\sigma ^{A(Y^{(0)})}=\sigma ^{A(1)}$ and calculate the U-value $U^{(0)}=\sum U_j^A(\sigma ^{A(1)})=4$. Then $\sigma ^{(1)}=(\sigma ^{A(1)}, \sigma ^{B(1)})$ is a Pareto-optimal schedule corresponding to $(4,1)\in \Omega$, as displayed in Fig. 2.

Fig. 2

Schedule $\sigma ^{(1)}$ corresponding to $(4,1)\in \Omega$

(iii) Calculate $Y^{(1)}= Y^{(0)}+|h_1|=5$, generate $\sigma ^{A(5)}$, and calculate the U-value $U^{(1)}=U(5)=3$. The schedule $\sigma ^{A(5)}$ is displayed in Fig. 3.

Fig. 3

Schedule $\sigma ^{A(5)}$ corresponding to $U^{(1)}=3$

(iv) Now $U^{(0)}=4>3=U^{(1)}$. Calculate $Y(3)=4$ and generate $\sigma ^{A(4)}$. Then $\sigma ^{(4)}=(\sigma ^{A(4)}, \sigma ^{B(4)})$ is a Pareto-optimal schedule corresponding to $(3,4)\in \Omega$, as displayed in Fig. 4.

Fig. 4

Schedule $\sigma ^{(4)}$ corresponding to $(3,4)\in \Omega$

(v) Calculate $Y^{(2)}= Y^{(1)}+|h_2|=10$, generate $\sigma ^{A(10)}$, and calculate the U-value $U^{(2)}=U(10)=2$. The schedule $\sigma ^{A(10)}$ is displayed in Fig. 5.

Fig. 5

Schedule $\sigma ^{A(10)}$ corresponding to $U^{(2)}=2$

(vi) Now $U^{(1)}=3>2=U^{(2)}$. Calculate $Y(2)=7$ and generate $\sigma ^{A(7)}$. Then $\sigma ^{(7)}=(\sigma ^{A(7)}, \sigma ^{B(7)})$ is a Pareto-optimal schedule corresponding to $(2,7)\in \Omega$, as displayed in Fig. 6.

Fig. 6

Schedule $\sigma ^{(7)}$ corresponding to $(2,7)\in \Omega$

(vii) Calculate $Y^{(3)}= Y^{(2)}+|h_3|=16$, generate $\sigma ^{A(16)}$, and calculate the U-value $U^{(3)}=U(16)=0$. The schedule $\sigma ^{A(16)}$ is displayed in Fig. 7.

Fig. 7

Schedule $\sigma ^{A(16)}$ corresponding to $U^{(3)}=0$

(viii) Now $U^{(2)}=2>0=U^{(3)}$. For $u=U^{(2)}-1=1$, we calculate $Y(u)=Y(1)=11$, and generate $\sigma ^{A(11)}$. Then $\sigma ^{(11)}=(\sigma ^{A(11)}, \sigma ^{B(11)})$ is a Pareto-optimal schedule corresponding to $(1,11)\in \Omega$, as displayed in Fig. 8.

Fig. 8

Schedule $\sigma ^{(11)}$ corresponding to $(1,11)\in \Omega$

(ix) For $u=U^{(3)}=0$, calculate $Y(u)=Y(0)=16$ and generate $\sigma ^{A(16)}$. Then $\sigma ^{(16)}=(\sigma ^{A(16)}, \sigma ^{B(16)})$ is a Pareto-optimal schedule corresponding to $(0,16)\in \Omega$, as displayed in Fig. 9.

Fig. 9

Schedule $\sigma ^{(16)}$ corresponding to $(0,16)\in \Omega$

(x) Finally, we conclude that $\Omega =\{(4,1), (3,4), (2,7), (1,11),(0,16)\}$ and $\sigma ^{(1)}, \sigma ^{(4)},\sigma ^{(7)},\sigma ^{(11)}$, and $\sigma ^{(16)}$ are the corresponding Pareto-optimal schedules.