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doi: 10.3934/jimo.2021008
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## A unified analysis for scheduling problems with variable processing times

 1 School of Science, Shenyang Aerospace University, Shenyang 110136, China 2 Post-doctoral Mobile Station, Dalian Commodity Exchange, Dalian, China

*Corresponding author: Ji-Bo Wang

Received  June 2020 Revised  October 2020 Early access January 2021

This paper considers single-machine scheduling problems with variable processing times, in which the actual processing time of a job is a function of its additional resources, starting time, and position in a sequence. Four problems arising from two criteria (a scheduling cost and a total resource consumption cost) are investigated. Under the linear and convex resource consumption functions, we provide unified approaches and consequently prove that these four problems are solvable in polynomial time.

Citation: Ji-Bo Wang, Bo Zhang, Hongyu He. A unified analysis for scheduling problems with variable processing times. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021008
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List of notations
 notations definitions $n$ total number of jobs $J_i$ index of job $\bar{p}_i$ normal processing time of job $J_i$ $C_i$ completion time of job $J_i$ $W_i$ waiting time of job $J_i$ $p_i$ actual processing time of job $J_i$ $d_i$ due date of job $J_i$ $E_i$ earliness of job $J_i$ $T_i$ tardiness of job $J_i$ $u_i$ amount of resources allocated to job $J_i$ $\bar{u}_i$ the upper bound of $u_i$ $\theta_i$ compression rate corresponding to job $J_i$ $v_i$ the cost allocated to job $J_i$ $\varphi_i$ positional weight of $ith$ position $[i]$ the job that is placed in $ith$ position $A$ a truncation parameter $t$ starting process time of some job $b$ common deterioration rate $\sigma$ sequence of all jobs $C_{\max}$ makespan $\sum_{i=1}^n C_{i}$ total completion time $\sum_{i=1}^n W_{i}$ total waiting time $\sum_{i=1}^n\sum_{j=i}^n|C_i-C_j|$ total absolute differences in completion times $\sum_{i=1}^n\sum_{j=i}^n|W_i-W_j|$ total absolute differences in waiting times
 notations definitions $n$ total number of jobs $J_i$ index of job $\bar{p}_i$ normal processing time of job $J_i$ $C_i$ completion time of job $J_i$ $W_i$ waiting time of job $J_i$ $p_i$ actual processing time of job $J_i$ $d_i$ due date of job $J_i$ $E_i$ earliness of job $J_i$ $T_i$ tardiness of job $J_i$ $u_i$ amount of resources allocated to job $J_i$ $\bar{u}_i$ the upper bound of $u_i$ $\theta_i$ compression rate corresponding to job $J_i$ $v_i$ the cost allocated to job $J_i$ $\varphi_i$ positional weight of $ith$ position $[i]$ the job that is placed in $ith$ position $A$ a truncation parameter $t$ starting process time of some job $b$ common deterioration rate $\sigma$ sequence of all jobs $C_{\max}$ makespan $\sum_{i=1}^n C_{i}$ total completion time $\sum_{i=1}^n W_{i}$ total waiting time $\sum_{i=1}^n\sum_{j=i}^n|C_i-C_j|$ total absolute differences in completion times $\sum_{i=1}^n\sum_{j=i}^n|W_i-W_j|$ total absolute differences in waiting times
Models studied
 Ref. Problem Relation with this article Due date methods Complexity Wang et al. [29] $1|p_i=\bar{p}_ir^a+bt-\theta_iu_i|\delta_1C_{\max}+\delta_2\sum_{i=1}^nC_{i}+\delta_3\sum_{i=1}^n\sum_{j=1}^i|C_i-C_j|+\delta_4\sum_{i=1}^nv_iu_i$ $1|p_i=\bar{p}_ir^a+bt-\theta_iu_i|\delta_1C_{\max}+\delta_2\sum_{i=1}^nW_{i}+\delta_3\sum_{i=1}^n\sum_{j=1}^i|W_i-W_j|+\delta_4\sum_{i=1}^nv_iu_i$ $p_i$ is type 1b $g(r)=r^a$, $A=0$ $p_i$ is type 1b $g(r)=r^a$, $A=0$ $O(n^3)$ $O(n^3)$ Li et al. [12] $1|p_i=\left(\left(\frac{\bar{p}_i}{u_i}\right)^k+bt\right)r^a|\delta_1C_{\max}+\delta_2\sum_{i=1}^nC_{i}+\delta_3\sum_{i=1}^n\sum_{j=1}^i|C_i-C_j|+\delta_4\sum_{i=1}^nv_iu_i$ $1|p_i=\left(\left(\frac{\bar{p}_i}{u_i}\right)^k+bt\right)r^a|\delta_1C_{\max}+\delta_2\sum_{i=1}^nW_{i}+\delta_3\sum_{i=1}^n\sum_{j=1}^i|W_i-W_j|+\delta_4\sum_{i=1}^nv_iu_i$ $p_i$ is type 2a $g(r)=r^a$, $A=0$ $p_i$ is type 2a $g(r)=r^a$, $A=0$ $O(n\log n)$ $O(n\log n)$ Sun et al. [23] $1|p_i=\left(\left(\frac{\bar{p}_i}{u_i}\right)^k+bt\right)r^a|\sum_{i=1}^n(\alpha E_i+\beta T_i+\gamma d_i+v_iu_i)$ $1|p_i=\left(\left(\frac{\bar{p}_i}{u_i}\right)^k+bt\right)r^a, \sum_{i=1}^nu_i\leq U|\sum_{i=1}^n(\alpha E_i+\beta T_i+\gamma d_i)$ $1|p_i=\left(\left(\frac{\bar{p}_i}{u_i}\right)^k+bt\right)r^a, \sum_{i=1}^n(\alpha E_i+\beta T_i+\gamma d_i)\leq V|\sum_{i=1}^nu_i$ $p_i$ is type 2a $g(r)=r^a, A=0$ $p_i$ is type 2a $g(r)=r^a$, $A=0$ $p_i$ is type 2a $g(r)=r^a$, $A=0$ CON/SLK/DIF CON/SLK/DIF CON/SLK/DIF $O(nlogn)$ $O(n\log n)$ $O(n\log n)$ Wang et al. [27] $1|p_i=\bar{p}_i\max\{r^{a_i}, A\}+bt-\theta_iu_i|\beta_1P+\beta_2\sum_{i=1}^nv_iu_i, P\in\left\{C_{\max}, \sum_{i=1}^nC_i, \sum_{i=1}^n\sum_{j=1}^i|C_i-C_j|\right\}$ $1|p_i=\left(\frac{\bar{p}_i\max\{r^{a_i}, A\}}{u_i}\right)^k+bt|\beta_1P+\beta_2\sum_{i=1}^nv_iu_i, P\in\left\{C_{\max}, \sum_{i=1}^nC_i, \sum_{i=1}^n\sum_{j=1}^i|C_i-C_j|\right\}$ $1|p_i=\left(\frac{\bar{p}_i\max\{r^{a_i}, A\}}{u_i}\right)^k+bt, \sum_{i=1}^nu_i\leq U|P, P\in\left\{C_{\max}, \sum_{i=1}^nC_i, \sum_{i=1}^n\sum_{j=1}^i|C_i-C_j|\right\}$ $1|p_i=\left(\frac{\bar{p}_i\max\{r^{a_i}, A\}}{u_i}\right)^k+bt, P\leq V|\sum_{i=1}^nu_i, P\in\left\{C_{\max}, \sum_{i=1}^nC_i, \sum_{i=1}^n\sum_{j=1}^i|C_i-C_j|\right\}$ $p_i$ is type 1b $g(r)=r^{a_i}$ $p_i$ is type 2b $g(r)=r^{a_i}$ $p_i$ is type 2b $g(r)=r^{a_i}$ $p_i$ is type 2b $g(r)=r^{a_i}$ $O(n\log n)$ $O(n\log n)$ $O(n\log n)$ $O(n\log n)$ Liu et al. [15] $1|p_i=(\bar{p}_i-bt)g(r)-\theta_iu_i|\sum(\alpha E_i+\beta T_i+\gamma d_i^1+\delta D)+\sigma C_{\max}+\eta\sum v_iu_i$ $1|\left(\left(\frac{\bar{p}_i}{u_i}\right)^k-bt\right)g(r)|\sum(\alpha E_i+\beta T_i+\gamma d_i^1+\delta D)+\sigma C_{\max}+\eta\sum v_iu_i$ $1|\left(\left(\frac{\bar{p}_i}{u_i}\right)^k-bt\right)g(r), \sum(\alpha E_i+\beta T_i+\gamma d_i^1+\delta D)+\sigma C_{\max}\leq V|\sum v_iu_i$ $1|\left(\left(\frac{\bar{p}_i}{u_i}\right)^k-bt\right)g(r), \sum u_i \leq U|\sum(\alpha E_i+\beta T_i+\gamma d_i^1+\delta D)+\sigma C_{\max}$ $p_i$ is type 1a $A=0$ $p_i$ is type 2a $A=0$ $p_i$ is type 2a $A=0$ $p_i$ is type 2a $A=0$ CON/SLK/DIF due window CON/SLK/DIF due window CON/SLK/DIF due window CON/SLK/DIF due window $O(n^3)$ $O(n\log n)$ $O(n\log n)$ h$O(n\log n)$ This paper P1: $1|p_i\in\{\mbox {type 1a, type 1b}\}|\sum_{i=1}^np_{[i]}\varphi_i+\eta\sum_{i=1}^nv_iu_i$ P1: $1|p_i\in\{\mbox{type 1a, type 1b}\}|P+\eta\sum_{i=1}^nv_iu_i$ $P\in\left\{C_{\max}, \sum_{i=1}^nC_i, \sum_{i=1}^nW_i, \sum_{i=1}^n\sum_{j=1}^i|C_i-C_j|, \sum_{i=1}^n\sum_{j=1}^i|W_i-W_j|, \sum_{i=1}^n(\alpha E_i +\beta T_i+ \gamma d_i)\right\}$ P2: $1|p_i\in\{\mbox {type 2a, type 2b}\}|P+\eta\sum_{i=1}^nv_iu_i$hP2: $1|p_i\in\{\mbox {type 2a, type 2b}\}|\sum_{i=1}^np_{[i]}\varphi_i+\eta\sum_{i=1}^nv_iu_i$ P2: $1|p_i\in\{\mbox {type 2a, type 2b}\}|P+\eta\sum_{i=1}^nv_iu_i$ $P\in\left\{C_{\max}, \sum_{i=1}^nC_i, \sum_{i=1}^nW_i, \sum_{i=1}^n\sum_{j=1}^i|C_i-C_j|, \sum_{i=1}^n\sum_{j=1}^i|W_i-W_j|, \sum_{i=1}^n(\alpha E_i +\beta T_i+ \gamma d_i)\right\}$ P3: $1|p_i\in\{\mbox{type 2a, type 2b}\}, \sum_{i=1}^np_{[i]}\varphi_i\leq \bar{Z}|\sum_{i=1}^nv_iu_i$ P3: $1|p_i\in\{\mbox{type 2a, type 2b}\}, P\leq \bar{Z}|\sum_{i=1}^nv_iu_i$ $P\in\left\{C_{\max}, \sum_{i=1}^nC_i, \sum_{i=1}^nW_i, \sum_{i=1}^n\sum_{j=1}^i|C_i-C_j|, \sum_{i=1}^n\sum_{j=1}^i|W_i-W_j|, \sum_{i=1}^n(\alpha E_i +\beta T_i+ \gamma d_i)\right\}$ P4: $1|p_i\in\{\mbox{type 2a, type 2b}\}, \sum_{i=1}^nv_iu_i\leq\bar{V}|\sum_{i=1}^np_{[i]}\psi_i$ P4: $1|p_i\in\{\mbox{type 2a, type 2b}\}, \sum_{i=1}^nv_iu_i\leq\bar{V}|P$ $P\in\left\{C_{\max}, \sum_{i=1}^nC_i, \sum_{i=1}^nW_i, \sum_{i=1}^n\sum_{j=1}^i|C_i-C_j|, \sum_{i=1}^n\sum_{j=1}^i|W_i-W_j|, \sum_{i=1}^n(\alpha E_i +\beta T_i+ \gamma d_i)\right\}$ $p_i$ is type 1a/1b $p_i$ is type 1a/1b $p_i$ is type 2a/2b $p_i$ is type 2a/2b $p_i$ is type 2a/2b $p_i$ is type 2a/2b $p_i$ is type 2a/2b $p_i$ is type 2a/2b $O(n^3)$ $O(n^3)$ $O(n\log n)$ $o(n\log n)$ $O(n\log n)$ $o(n\log n)$ $O(n\log n)$ $o(n\log n)$ $W_i$, $T_i, E_i$ is the waiting time, earliness, tardiness of job $J_i$, respectively; $C_{max}$ is the makespan of all jobs; $\delta_1, \delta_2, \delta_3, \delta_4, \alpha, \beta, \gamma, \delta, \sigma, \beta_1, \beta_2, U$ and $V$ are given constants
 Ref. Problem Relation with this article Due date methods Complexity Wang et al. [29] $1|p_i=\bar{p}_ir^a+bt-\theta_iu_i|\delta_1C_{\max}+\delta_2\sum_{i=1}^nC_{i}+\delta_3\sum_{i=1}^n\sum_{j=1}^i|C_i-C_j|+\delta_4\sum_{i=1}^nv_iu_i$ $1|p_i=\bar{p}_ir^a+bt-\theta_iu_i|\delta_1C_{\max}+\delta_2\sum_{i=1}^nW_{i}+\delta_3\sum_{i=1}^n\sum_{j=1}^i|W_i-W_j|+\delta_4\sum_{i=1}^nv_iu_i$ $p_i$ is type 1b $g(r)=r^a$, $A=0$ $p_i$ is type 1b $g(r)=r^a$, $A=0$ $O(n^3)$ $O(n^3)$ Li et al. [12] $1|p_i=\left(\left(\frac{\bar{p}_i}{u_i}\right)^k+bt\right)r^a|\delta_1C_{\max}+\delta_2\sum_{i=1}^nC_{i}+\delta_3\sum_{i=1}^n\sum_{j=1}^i|C_i-C_j|+\delta_4\sum_{i=1}^nv_iu_i$ $1|p_i=\left(\left(\frac{\bar{p}_i}{u_i}\right)^k+bt\right)r^a|\delta_1C_{\max}+\delta_2\sum_{i=1}^nW_{i}+\delta_3\sum_{i=1}^n\sum_{j=1}^i|W_i-W_j|+\delta_4\sum_{i=1}^nv_iu_i$ $p_i$ is type 2a $g(r)=r^a$, $A=0$ $p_i$ is type 2a $g(r)=r^a$, $A=0$ $O(n\log n)$ $O(n\log n)$ Sun et al. [23] $1|p_i=\left(\left(\frac{\bar{p}_i}{u_i}\right)^k+bt\right)r^a|\sum_{i=1}^n(\alpha E_i+\beta T_i+\gamma d_i+v_iu_i)$ $1|p_i=\left(\left(\frac{\bar{p}_i}{u_i}\right)^k+bt\right)r^a, \sum_{i=1}^nu_i\leq U|\sum_{i=1}^n(\alpha E_i+\beta T_i+\gamma d_i)$ $1|p_i=\left(\left(\frac{\bar{p}_i}{u_i}\right)^k+bt\right)r^a, \sum_{i=1}^n(\alpha E_i+\beta T_i+\gamma d_i)\leq V|\sum_{i=1}^nu_i$ $p_i$ is type 2a $g(r)=r^a, A=0$ $p_i$ is type 2a $g(r)=r^a$, $A=0$ $p_i$ is type 2a $g(r)=r^a$, $A=0$ CON/SLK/DIF CON/SLK/DIF CON/SLK/DIF $O(nlogn)$ $O(n\log n)$ $O(n\log n)$ Wang et al. [27] $1|p_i=\bar{p}_i\max\{r^{a_i}, A\}+bt-\theta_iu_i|\beta_1P+\beta_2\sum_{i=1}^nv_iu_i, P\in\left\{C_{\max}, \sum_{i=1}^nC_i, \sum_{i=1}^n\sum_{j=1}^i|C_i-C_j|\right\}$ $1|p_i=\left(\frac{\bar{p}_i\max\{r^{a_i}, A\}}{u_i}\right)^k+bt|\beta_1P+\beta_2\sum_{i=1}^nv_iu_i, P\in\left\{C_{\max}, \sum_{i=1}^nC_i, \sum_{i=1}^n\sum_{j=1}^i|C_i-C_j|\right\}$ $1|p_i=\left(\frac{\bar{p}_i\max\{r^{a_i}, A\}}{u_i}\right)^k+bt, \sum_{i=1}^nu_i\leq U|P, P\in\left\{C_{\max}, \sum_{i=1}^nC_i, \sum_{i=1}^n\sum_{j=1}^i|C_i-C_j|\right\}$ $1|p_i=\left(\frac{\bar{p}_i\max\{r^{a_i}, A\}}{u_i}\right)^k+bt, P\leq V|\sum_{i=1}^nu_i, P\in\left\{C_{\max}, \sum_{i=1}^nC_i, \sum_{i=1}^n\sum_{j=1}^i|C_i-C_j|\right\}$ $p_i$ is type 1b $g(r)=r^{a_i}$ $p_i$ is type 2b $g(r)=r^{a_i}$ $p_i$ is type 2b $g(r)=r^{a_i}$ $p_i$ is type 2b $g(r)=r^{a_i}$ $O(n\log n)$ $O(n\log n)$ $O(n\log n)$ $O(n\log n)$ Liu et al. [15] $1|p_i=(\bar{p}_i-bt)g(r)-\theta_iu_i|\sum(\alpha E_i+\beta T_i+\gamma d_i^1+\delta D)+\sigma C_{\max}+\eta\sum v_iu_i$ $1|\left(\left(\frac{\bar{p}_i}{u_i}\right)^k-bt\right)g(r)|\sum(\alpha E_i+\beta T_i+\gamma d_i^1+\delta D)+\sigma C_{\max}+\eta\sum v_iu_i$ $1|\left(\left(\frac{\bar{p}_i}{u_i}\right)^k-bt\right)g(r), \sum(\alpha E_i+\beta T_i+\gamma d_i^1+\delta D)+\sigma C_{\max}\leq V|\sum v_iu_i$ $1|\left(\left(\frac{\bar{p}_i}{u_i}\right)^k-bt\right)g(r), \sum u_i \leq U|\sum(\alpha E_i+\beta T_i+\gamma d_i^1+\delta D)+\sigma C_{\max}$ $p_i$ is type 1a $A=0$ $p_i$ is type 2a $A=0$ $p_i$ is type 2a $A=0$ $p_i$ is type 2a $A=0$ CON/SLK/DIF due window CON/SLK/DIF due window CON/SLK/DIF due window CON/SLK/DIF due window $O(n^3)$ $O(n\log n)$ $O(n\log n)$ h$O(n\log n)$ This paper P1: $1|p_i\in\{\mbox {type 1a, type 1b}\}|\sum_{i=1}^np_{[i]}\varphi_i+\eta\sum_{i=1}^nv_iu_i$ P1: $1|p_i\in\{\mbox{type 1a, type 1b}\}|P+\eta\sum_{i=1}^nv_iu_i$ $P\in\left\{C_{\max}, \sum_{i=1}^nC_i, \sum_{i=1}^nW_i, \sum_{i=1}^n\sum_{j=1}^i|C_i-C_j|, \sum_{i=1}^n\sum_{j=1}^i|W_i-W_j|, \sum_{i=1}^n(\alpha E_i +\beta T_i+ \gamma d_i)\right\}$ P2: $1|p_i\in\{\mbox {type 2a, type 2b}\}|P+\eta\sum_{i=1}^nv_iu_i$hP2: $1|p_i\in\{\mbox {type 2a, type 2b}\}|\sum_{i=1}^np_{[i]}\varphi_i+\eta\sum_{i=1}^nv_iu_i$ P2: $1|p_i\in\{\mbox {type 2a, type 2b}\}|P+\eta\sum_{i=1}^nv_iu_i$ $P\in\left\{C_{\max}, \sum_{i=1}^nC_i, \sum_{i=1}^nW_i, \sum_{i=1}^n\sum_{j=1}^i|C_i-C_j|, \sum_{i=1}^n\sum_{j=1}^i|W_i-W_j|, \sum_{i=1}^n(\alpha E_i +\beta T_i+ \gamma d_i)\right\}$ P3: $1|p_i\in\{\mbox{type 2a, type 2b}\}, \sum_{i=1}^np_{[i]}\varphi_i\leq \bar{Z}|\sum_{i=1}^nv_iu_i$ P3: $1|p_i\in\{\mbox{type 2a, type 2b}\}, P\leq \bar{Z}|\sum_{i=1}^nv_iu_i$ $P\in\left\{C_{\max}, \sum_{i=1}^nC_i, \sum_{i=1}^nW_i, \sum_{i=1}^n\sum_{j=1}^i|C_i-C_j|, \sum_{i=1}^n\sum_{j=1}^i|W_i-W_j|, \sum_{i=1}^n(\alpha E_i +\beta T_i+ \gamma d_i)\right\}$ P4: $1|p_i\in\{\mbox{type 2a, type 2b}\}, \sum_{i=1}^nv_iu_i\leq\bar{V}|\sum_{i=1}^np_{[i]}\psi_i$ P4: $1|p_i\in\{\mbox{type 2a, type 2b}\}, \sum_{i=1}^nv_iu_i\leq\bar{V}|P$ $P\in\left\{C_{\max}, \sum_{i=1}^nC_i, \sum_{i=1}^nW_i, \sum_{i=1}^n\sum_{j=1}^i|C_i-C_j|, \sum_{i=1}^n\sum_{j=1}^i|W_i-W_j|, \sum_{i=1}^n(\alpha E_i +\beta T_i+ \gamma d_i)\right\}$ $p_i$ is type 1a/1b $p_i$ is type 1a/1b $p_i$ is type 2a/2b $p_i$ is type 2a/2b $p_i$ is type 2a/2b $p_i$ is type 2a/2b $p_i$ is type 2a/2b $p_i$ is type 2a/2b $O(n^3)$ $O(n^3)$ $O(n\log n)$ $o(n\log n)$ $O(n\log n)$ $o(n\log n)$ $O(n\log n)$ $o(n\log n)$ $W_i$, $T_i, E_i$ is the waiting time, earliness, tardiness of job $J_i$, respectively; $C_{max}$ is the makespan of all jobs; $\delta_1, \delta_2, \delta_3, \delta_4, \alpha, \beta, \gamma, \delta, \sigma, \beta_1, \beta_2, U$ and $V$ are given constants
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