This paper studies deterministic constraint optimization problem with two competitive agents in which the following objective functions on a single machine: the total weighted late work and the total completion time. We show that the constraint optimization problem is the binary NP-hard by Knapsack problem reduction. Furthermore, we present a pseudo-polynomial time algorithm by early due date maximum not-late sequence, and an approximation Pareto curve by dynamic programming algorithm and two eliminated states, which time complexity of the two approximation algorithms are $O(n_A^2{n}_{B}Q\sum (p_j^A+p_j^B))$ and $O\left(\frac{n^4}{{\theta }^2}\log U{B}_A\log U{B}_B\right),$ where $p_j,$ $\theta$are processing time of job $J_j,$ a given positive constant, and $U{B}_x$ an upper bound of the objective function of agent $x,$ $x\in \{A,B\}$. Finally, we present a simple approximation algorithm by the earliest due date (EDD) rule, which jobs of agent B are assigned an dummy due date.

本文研究了具有两个竞争代理的确定性约束优化问题，其中在单个机器上具有以下目标功能：总加权后期工作和总完成时间。我们证明了约束优化问题是通过背包问题归纳法解决的二元NP-hard问题。此外，我们提出了一个提前到期日期最大非延迟序列的伪多项式时间算法，以及一个动态规划算法和两个消除状态的逼近帕累托曲线，这两个逼近算法的时间复杂度分别为$Ø（{ñ}_{\mathrm{一种}}^{\mathrm{2个}}{ñ}_{乙}问\sum （p_{Ĵ}^{\mathrm{一种}}+p_{Ĵ}^{乙}））$ 和 $Ø（\frac{{ñ}^4}{{\theta }^{\mathrm{2个}}}\mathrm{日志}ü{乙}_{\mathrm{一种}}\mathrm{日志}ü{乙}_{乙}），$ 在哪里 $p_{Ĵ}，$ $\theta$正在处理工作时间 ${Ĵ}_{Ĵ}，$ 给定的正常数，以及 $ü{乙}_X$ 主体目标函数的上限 $X，$ $X\in \{\mathrm{一种}，乙\}$。最后，我们通过最早的到期日（EDD）规则提出了一种简单的近似算法，其中为代理B的工作分配了虚拟的到期日。