Due date assignment single-machine scheduling with delivery times, position-dependent weights and deteriorating jobs

Abstract

This article studies the single-machine scheduling problem with due date assignments, deteriorating jobs, and past-sequence-dependent delivery times. Under three assignments (i.e., common, slack, and different due dates), the goal is to determine a feasible sequence and due dates of all jobs in order to minimize the weighted sum of earliness, tardiness, and due date costs of all jobs, where the weight is not related to the job but to the position in which some job is scheduled. Through a series of optimal properties, efficient and fast polynomial time algorithms are designed for solving the studied scheduling problem with three due date assignments.

1 Introduction

In classical production scheduling problems, it is typically assumed that each job’s processing time on a specific machine is a given constant. But under many practical cases, the processing time may be related to the start time of each job; that is, the processing time of the job increases with the increase of its start time. More specifically, the later the job starts to process, the longer the processing times on machines and vice versa. This problem is called a deteriorating job (time-dependent) scheduling (Li et al. 2019; Gawiejnowicz 2020). For example, Cheng et al. (2020) proved that the due date assignment scheduling problem with deteriorating jobs and minimal the total completion time is NP-hard, and thus proposed a pseudo-polynomial time algorithm for the considered problem, in which the processing time of a job on a specific machine is a step function of its start time.

In practical problems, it is also necessary to consider the extra time for delivering the items to their customers, which is known as the past-sequence-dependent delivery time in the literature (Koulamas and Kyparisis 2010). For example, Zhao and Tang (2014) studied the scheduling problem with the minimization of the makespan, the total completion time, and the absolute difference between the completion time and the past-sequence-dependent delivery time. Their results verified that two types of scheduling problems, namely including and excluding the due date, can be solved in polynomial time. Ji et al. (2015) investigated the single-machine scheduling problem with simultaneous considering the slack due window assignment, the past-sequence-dependent delivery time and the controllable processing time. Note that in their work, each job’s processing time was supposed to be a linear or convex function of learning effect and resource allocation, and the authors proposed an efficient polynomial algorithm with the time complexity of $O({\bar{N}}^3)$, where ${\bar{N}}$ denotes the number of jobs. Recently, Qian and Han (2022) investigated a problem considering the due date assignment, the delivery time, and deterioration effects at the same time, and proposed a polynomial algorithm, which runs in $O({\bar{N}}\log {\bar{N}})$ time.

Similarly, as an important research direction, due date assignment problems can be divided into three cases (see Gordon et al. 2002a, b; Yin et al. 2014, 2018). Among them, the first case is common due date, abbreviated as CON (see Yin et al. 2016; Xiong et al. 2018; T’kindt V, Shang L, Croce FD, 2020; Lv and Wang 2021; Cheng et al. 2020). That is, each job has the same due date, which can be represented by $d_{opt}$. Toksari and Güner Toksari (2010) simultaneously worked on the common due date earliness/tardiness scheduling problem with time-dependent learning effects and linear or non-linear deterioration effects under parallel machines, and verified that the optimal schedule follows the V-shaped sequence. Koulamas (2017) proposed a dynamic programming (DP) algorithm for the CON problem with general earliness/tardiness penalties under the single-machine scenario and proposed a faster algorithm within $O({\bar{N}})$ for the special situation without taking into account the earliness/tardiness of jobs. The second case of the due date assignment problem is slack due date, abbreviated as SLK (see Yin et al. 2016; Chen et al. 2023); that is, there is the same common flow allowance for each job, which is usually represented by $q_{opt}$, and the due date of job $J_i$ can be denoted as $d_i=p_i+q_{opt}$, in which $p_i$ is the processing time of $J_i$. Liu et al. (2017) studied the earliness-tardiness scheduling problem with only considering the single-machine and proposed effective polynomial time algorithms for general processing time and position-dependent processing time functions, respectively, with corresponding time complexities of $O({\bar{N}}\log {\bar{N}})$ and $O({\bar{N}}^3)$. Recently, Liu et al. (2020) scrutinized the single-machine resource allocation problem with deterioration effects and position-dependent weights. Under the CON and SLK cases, they gave a corresponding optimal algorithm and optimal resource allocation. For the third case of due date assignments, that is, an unrestricted due date assignment scheduling problem, abbreviated as DIF (see Yang et al. 2022), namely the due date of each job $J_i$ can be expressed as $d_i$, where $i=1,2, \ldots ,{{\bar{N}}}$. Very recently, Yin et al. (2021) examined the serial-batch delivery scheduling problem with only considering the single-machine and two competing agents. Under the CON and DIF due date models, they proved that the total cost (comprising the earliness, tardiness (weighted number of tardy jobs), job holding, due date assignment, and batch delivery costs) minimization is NP-hard and presented a pseudo-polynomial DP algorithm for each of the explored problems.

Furthermore, Qian and Han (2022) researched the single-machine scheduling problem with simple linear deterioration and past-sequence-dependent delivery times. Under three due date assignment methods, the objective was to simultaneously minimize the weighted sum of earliness, tardiness, and the due date of all jobs. They have proved that the optimal schedule to the minimization problem can be obtained in $O({\bar{N}}\log {\bar{N}})$ time. However, the scheduling problems with position-dependent weights exist in many practical production services environments, such as in Didi taxi dispatching, orders placed in the morning offer a higher bonus to the driver, which can effectively improve customer satisfaction in these locations by better meeting the needs of customers going to work in the morning (see Sun et al. 2020). Hence, in this article, we mainly investigate the three due date assignment problems with position-dependent weights (see Wang et al. 2020, 2021), deteriorating jobs, and past-sequence-dependent delivery times. The research objective is to minimize the weighted sum of earliness, tardiness and due date of all jobs, where the weights depend on the position in which a job is scheduled. The contributions of this study are given as follows: (1) we examine the single-machine due date assignment scheduling problem along with deteriorating jobs and past-sequence-dependent delivery times; (2) under CON, SLK and DIF, our goal is to minimize the weighted sum of earliness, tardiness, and due date assignment cost, where the weights are position-dependent weights (i.e., the weight is not related to the job but to the position in which some job is scheduled); (3) we present the structural properties of the optimal solutions and demonstrate that the problem is polynomially solvable. The structure of the paper is organized as follows: Sect. 2 describes the studied problem. Sections 3, 4 and 5 give the solution algorithms for the researched problem under CON, SLK and DIF, respectively. Section 6 gives the conclusion.

2 Problem description

Consider that there exist ${\bar{N}}$ jobs in total to be processed continuously on a single-machine, and all jobs are processed at time instant $s_0$, where $s_0>0$. Under the simple linear deterioration, the processing time of job $J_i$ can be defined as $p_i=\chi _is_i$, in which $s_i$ and $\chi _i$ represent the start time and the deterioration rate of job $J_i$, respectively. The past-sequence-dependent delivery time (denoted by psddt) of job $J_i$ can be defined as $q_i=v s_i$, where v is the delivery rate. For the case of common due date assignment, i.e., CON, the earliness and tardiness of job $J_{[i]}$ at the i-th position in the sequence can be expressed as ${\widehat{E}}_{[i]}=\max \{0,d_{opt}-C_{[i]}\}$ and ${\widehat{T}}_{[i]}=\max \{0,C_{[i]}-d_{opt}\}$, respectively, where $d_{opt}$ is the optimal common due date of all jobs, and $C_{[i]}$ is the completion time of job $J_{[i]}$. For the slack due date assignment problem, i.e., SLK, the earliness and tardiness of job $J_{[i]}$ can be written as ${\widehat{E}}_{[i]}=\max \{0,p_{[i]}+q_{opt}-C_{[i]}\}$ and ${\widehat{T}}_{[i]}=\max \{0,C_{[i]}-p_{[i]}-q_{opt}\}$, in which $q_{opt}$ denotes the optimal common flow allowance of all jobs. For the different (unrestricted) due dates assignment, i.e., DIF, the earliness and tardiness of job $J_{[i]}$ are ${\widehat{E}}_{[i]}=\max \{0,d_{[i]}-C_{[i]}\}$ and ${\widehat{T}}_{[i]}=\max \{0,C_{[i]}-d_{[i]}\}$ respectively, where $d_{[i]}$ is the due date of job $J_{[i]}$ ($i=1,2, \ldots ,n$).

Brucker (2001) has demonstrated that the problem $1|CON|\sum _{i=1}^{{\bar{N}}}({\widehat{\xi }}_i |{\widehat{L}}_{[i]}|+{\widehat{\gamma }} d_{opt})$ is polynomially solvable, where ${\widehat{L}}_{[i]}=C_{[i]}-d_{opt}$ is the lateness of job $J_{[i]}$, ${\widehat{\xi }}_i$ is the position-dependent weight for lateness of ith position and ${\widehat{\gamma }}$ is the weight for common due date (obviously, $|{\widehat{L}}_{[i]}|={\widehat{E}}_{[i]}+ {\widehat{T}}_{[i]}$). Liu et al. (2017) have showed that the problem $1|SLK|\sum _{i=1}^{{\bar{N}}}({\widehat{\xi }}_i |{\widehat{L}}_{[i]}|+{\widehat{\gamma }} q_{opt})$ is can be solved efficiently. Qian and Han (2022) have proved that the problems $1|p_{i}=\chi _{i}s_{i},CON,psddt|\sum _{i=1}^{{\bar{N}}}({\widehat{\alpha }} {\widehat{E}}_{[i]}+{\widehat{\beta }} {\widehat{T}}_{[i]}$ $+{\widehat{\gamma }} d_{opt}),$ $1|p_{i}=\chi _{i}s_{i},SLK,psddt|\sum _{i=1}^{{\bar{N}}}\left( {\widehat{\alpha }} {\widehat{E}}_{[i]}+{\widehat{\beta }} {\widehat{T}}_{[i]}+{\widehat{\gamma }} q_{opt}\right)$ and $1|p_{i}=\chi _{i}s_{i},DIF,psddt|\sum _{i=1}^{{\bar{N}}}$ $\left( {\widehat{\alpha }} {\widehat{E}}_{[i]}+{\widehat{\beta }} {\widehat{T}}_{[i]}+{\widehat{\gamma }} d_{[i]}\right)$ can be solved in polynomial time, respectively, where ${\widehat{\alpha }}$, ${\widehat{\beta }}$ and ${\widehat{\gamma }}$ are the weights for earliness, tardiness, and due date, respectively. In this paper, we consider the problem of minimizing the weighted sum of earliness, tardiness and due date of all jobs, in which the weights we considered refer to the position-dependent weights, i.e., the weight is related to position of the job and not simply the job itself (Jiang et al. 2020). Under three due date assignment cases, the problems can be expressed as follows:

$$\begin{aligned} 1|p_{i}=\chi _{i}s_{i},CON,psddt|\sum _{i=1}^{\bar{N}}\left( \alpha _i \widehat{E}_{[i]}+\beta _i \widehat{T}_{[i]}+\gamma _i d_{opt}\right) ;\\ 1|p_{i}=\chi _{i}s_{i},SLK,psddt|\sum _{i=1}^{\bar{N}}\left( \alpha _i \widehat{E}_{[i]}+\beta _i \widehat{T}_{[i]}+\gamma _i q_{opt}\right) ;\\ 1|p_{i}=\chi _{i}s_{i},DIF,psddt|\sum _{i=1}^{\bar{N}}\left( \alpha _i \widehat{E}_{[i]}+\beta _i \widehat{T}_{[i]}+\gamma _i d_{[i]}\right) , \end{aligned}$$

where 1 denotes the single-machine, the middle (resp. third) field denotes the job characteristics (resp. the objective function), $\alpha _i$, $\beta _i$ and $\gamma _i$ are the position-dependent weights for earliness, tardiness, and due date, respectively, in which a position-dependent weight does not change with a change in some job.

Obviously, there exists an optimal schedule or sequence in which all jobs are processed consecutively without any idle time from the start time $s_0$. For a given job schedule or sequence, by the mathematical induction, we have $C_{[i]}=s_0\prod _{h=1}^i(1+v+\chi _{[h]})$. From this equation, if the jobs at any two positions are exchanged with the others positions remaining unchanged, then the completion time and start time of the jobs before and after the two jobs will essentially remain unchanged.

3 CON case

In this section, we focus on the problem $1|p_{i}=\chi _{i}s_{i},CON,psddt|\sum _{i=1}^{{\bar{N}}}\left( \alpha _i {\widehat{E}}_{[i]}\right.$$\left. +\beta _i {\widehat{T}}_{[i]}+\gamma _i d_{opt}\right)$. First, we present some properties that will be useful for later analysis.

Property 3.1

For a given job schedule or sequence, suppose that there exists an optimal common due date, then we can know that $d_{opt}$ equals either the completion of a job (i.e., $d_{opt}=C_{[l]}$) or $s_0$ (i.e., $d_{opt}=s_0$).

Proof

First, suppose that the $d_{opt}$ does not represent the job’s the completion time, namely $C_{[l]}<d_{opt}<C_{[l+1]}$, where $1\le l <{{\bar{N}}}$. Then, based on the above assumption, the objective function could be written as the following:

$$\begin{aligned} f= & {} \sum _{i=1}^{{\bar{N}}}\left( \alpha _i {\widehat{E}}_{[i]}+\beta _i {\widehat{T}}_{[i]}+\gamma _i d_{opt}\right) \\= & {} \sum _{i=1}^l \alpha _i(d_{opt}-C_{[i]})+\sum _{i=l+1}^{{\bar{N}}} \beta _i(C_{[i]}-d_{opt})+\sum _{i=1}^{{\bar{N}}}\gamma _i d_{opt}. \end{aligned}$$

Next, move the common due date $d_{opt}$ to the left to $C_{[l]}$ and set $x=d_{opt}-C_{[l]}$; that is, $d_{opt}=x+C_{[l]}$, and $x>0$. Then,

$$\begin{aligned} f_1=\sum _{i=1}^l \alpha _i(C_{[l]}-C_{[i]})+\sum _{i=l+1}^{{\bar{N}}} \beta _i(C_{[i]}-C_{[l]})+\sum _{i=1}^{{\bar{N}}} \gamma _i C_{[l]}, \end{aligned}$$

and

$$\begin{aligned} f-f_1=\left( \sum _{i=1}^l \alpha _i-\sum _{i=l+1}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i\right) x. \end{aligned}$$

In the next, shift the common due date $d_{opt}$ to the right to $C_{[l+1]}$, and let $y=C_{[l+1]}-d_{opt}$, i.e., $d_{opt}=C_{[l+1]}-y$, and $y>0$. Then,

$$\begin{aligned} f_2=\sum _{i=1}^l \alpha _i(C_{[l+1]}-C_{[i]})+\sum _{i=l+1}^{{\bar{N}}} \beta _i(C_{[i]}-C_{[l+1]})+\sum _{i=1}^{{\bar{N}}} \gamma _i C_{[l+1]}, \end{aligned}$$

and

$$\begin{aligned} f-f_2=-\left( \sum _{i=1}^l \alpha _i-\sum _{i=l+1}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i\right) y. \end{aligned}$$

When $\left( \sum _{i=1}^l \alpha _i-\sum _{i=l+1}^{{\bar{N}}}\beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i\right) \ge 0$, then we can easily know that $f\ge f_1$ holds; If $\left( \sum _{i=1}^l \alpha _i-\sum _{i=l+1}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i\right) <0$, then $f>f_2$ holds. That is, either $d_opt$ equals the completion of a job or $s_0$. $\square$

Property 3.2

For a given job schedule or sequence, its common due date $d_{opt}$ is equal to the completion time of the l-th position, i.e., $d_{opt}=C_{[l]}$, where l satisfies $\left( \sum _{i=1}^{l-1}\alpha _i-\sum _{i=l}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i\right) \le 0$ and $\left( \sum _{i=1}^{l}\alpha _i-\sum _{i=l+1}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i\right) \ge 0$.

Proof

The conclusion can be easily proved by the classical small perturbation technique. On the basis of Property 3.1, assuming that $d_{opt}=C_{[l]}$ and the corresponding objective function f can be obtained as follows:

$$\begin{aligned} f=\sum _{i=1}^{l-1}\alpha _i(C_{[l]}-C_{[i]})+\sum _{i=l}^{{\bar{N}}} \beta _i(C_{[i]}-C_{[l]})+\sum _{i=1}^{{\bar{N}}} \gamma _i C_{[l]}. \end{aligned}$$

Moving the common due date $d_{opt}$ to the left by u units, it follows that

$$\begin{aligned} f_1=\sum _{i=1}^{l-1}\alpha _i(C_{[l]}-u-C_{[i]})+\sum _{i=l}^{{\bar{N}}} \beta _i(C_{[i]}-C_{[l]}+u)+\sum _{i=1}^{{\bar{N}}} \gamma _i (C_{[l]}-u), \end{aligned}$$

then

$$\begin{aligned} f_1=f-\left( \sum _{i=1}^{l-1} \alpha _i-\sum _{i=l}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i\right) u. \end{aligned}$$

When f is optimal, then $\left( \sum _{i=1}^{l-1} \alpha _i-\sum _{i=l}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}}\gamma _i\right) \le 0$ is satisfied.

Shifting the common due date $d_{opt}$ to the right by u units yields

$$\begin{aligned} f_2=\sum _{i=1}^{l}\alpha _i(C_{[l]}+u-C_{[i]})+\sum _{i=l+1}^{{\bar{N}}} \beta _i(C_{[i]}-C_{[l]}-u)+\sum _{i=1}^{{\bar{N}}}\gamma _i (C_{[l]}+u), \end{aligned}$$

then

$$\begin{aligned} f_2=f+\left( \sum _{i=1}^{l} \alpha _i-\sum _{i=l+1}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i\right) u. \end{aligned}$$

Similarly, $\left( \sum _{i=1}^{l} \alpha _i-\sum _{i=l+1}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i\right) \ge 0$ can be satisfied.

Hence, the optimal common due date is $d_{opt}=C_{[l]}$, where l satisfies the following inequalities: $\left( \sum _{i=1}^{l-1}\alpha _i-\sum _{i=l}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i\right) \le 0$ and $\left( \sum _{i=1}^{l}\alpha _i-\sum _{i=l+1}^{{\bar{N}}}\beta _i\right.$$\left. +\sum _{i=1}^{{\bar{N}}} \gamma _i\right) \ge 0$. $\square$

For convenience, let the sets be $S=\{J_i\in \sigma |C_i\le d_{opt}\}$ and $R=\{J_i\in \sigma |C_i>d_{opt}\}$, where $\sigma$ is a given sequence.

Property 3.3

In the optimal schedule or sequence, all jobs in set S are sorted according to the non-increasing order of $\chi _i$, i.e., the LDR (Largest Deterioration Rate first) order.

Proof

Suppose that there exist two successive jobs $J_j$ and $J_k$, where $J_j$ is at the u-th position in a sequence $A_1$, in which $A_1=\{J_1, \ldots ,J_j,J_k, \ldots ,J_{{\bar{N}}}\}$, the start time of job $J_{[1]}$ is $s_0$, and $d_{opt}=C_{[l]}$, where $1\le u<l \le {{\bar{N}}}$. Now, by exchanging the positions of jobs $J_j$ and $J_k$, the sequence $A_2=\{J_1, \ldots ,J_k,J_j, \ldots ,J_{{\bar{N}}}\}$ can be obtained. For the objective function $f_1$ and $f_2$ of sequences $A_1$ and $A_2$, it follows that

$$\begin{aligned} f_1-f_2=-\alpha _u s_0 (\chi _j-\chi _k)\prod _{i=1}^{u-1}(1+\chi _{[i]}+v). \end{aligned}$$

If $\chi _j\ge \chi _k$, we have $f_1\le f_2$, that is, the jobs in set S are sorted in non-increasing order of $\chi _i$. $\square$

Property 3.4

In the optimal schedule or sequence, all jobs in set R are sorted according to the non-decreasing order of $\chi _i$, i.e., the SDR (Smallest Deterioration Rate first) order.

Proof

As in Property 3.3, suppose that there exist two successive jobs $J_j$ and $J_k$ in set R, and $J_j$ is at the u-th position in sequence $A_1$, in which $A_1=\{J_1, \ldots ,J_j,J_k, \ldots ,J_{{\bar{N}}}\}$. Similarly, $C_{[0]}=s_0$ and $d_{opt}=C_{[l]}$, where $l+1\le u <{{\bar{N}}}$. Now the sequence $A_2=\{J_1, \ldots ,J_k,J_j, \ldots ,J_{{\bar{N}}}\}$ can be obtained by exchanging jobs $J_j$ and $J_k$, then, for the objective function $f_1$ and $f_2$ of sequences $A_1$ and $A_2$, it follows that

$$\begin{aligned} f_1-f_2=\beta _u s_0(\chi _j-\chi _k)\prod _{i=1}^{u-1}(1+\chi _{[i]}+v). \end{aligned}$$

If $\chi _j\le \chi _k$, we have $f_1\le f_2$, that is, the jobs in set R are sorted in non-decreasing order of $\chi _i$. $\square$

Now, the following notation is defined:

$$\begin{aligned} Q_{x,y}= & {} \left[ \begin{array}{l} -\sum _{i=x}^{l-1}\alpha _i \prod _{g=x+1}^i(1+\chi _{[g]}+v)+\sum _{i=l+1}^{y-1}\beta _i \prod _{g=x+1}^i (1+\chi _{[g]}+v)\\ +\left( \sum _{i=1}^{l-1}\alpha _i-\sum _{i=l+1}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i\right) \prod _{i=x+1}^l(1+\chi _{[i]}+v) \end{array} \right] \end{aligned}$$

where $1\le x \le l$ and $l+1\le y \le {{\bar{N}}}$.

Property 3.5

Supposing there are two different jobs $J_j$ and $J_k$ in which job $J_j$ is supposed to be at the x-th position and job $J_k$ is supposed to be at the y-th position ($1\le x \le l$ and $l+1\le y \le {{\bar{N}}}$), consider the following cases:

• 1. If $Q_{x,y}\ge 0$, then $\chi _j < \chi _k$;

• 2. If $Q_{x,y}<0$, then $\chi _j \ge \chi _k$.

Proof

First, we assume that there are two different jobs $J_j$ and $J_k$ in an optimal sequence $A_1=\{J_1, \ldots ,J_j, \ldots ,J_k, \ldots ,J_{{\bar{N}}}\}$ and the former is supposed to be processed at time $s_0$, in which $J_j$ is at the x-th position and $J_k$ is supposed to be at the y-th position, $d_{opt}=C_{[l]}$, $1\le x \le l$, and $l+1\le y \le {{\bar{N}}}$. Based on the above assumptions, the sequence $A_2=\{J_1, \ldots ,J_k, \ldots ,J_j, \ldots ,J_{{\bar{N}}}\}$ can be obtained by swapping jobs $J_j$ and $J_k$. As a result, it follows that

$$\begin{aligned} f_1-f_2&{=}&s_0(\chi _j-\chi _k)\prod _{i=1}^{x-1}(1+\chi _{[i]}+v) \\{} & {} {\times } \left[ \begin{array}{l} {-}\sum _{i=x}^{l-1}\alpha _i \prod _{g=x{+}1}^i(1{+}\chi _{[g]}{+}v){+}\sum _{i=l{+}1}^{y{-}1}\beta _i \prod _{g=x{+}1}^i (1{+}\chi _{[g]}{+}v)\\ {+}\left( \sum _{i=1}^{l{-}1}\alpha _i{-}\sum _{i=l{+}1}^{{\bar{N}}} \beta _i{+}\sum _{i=1}^{{\bar{N}}} \gamma _i\right) \prod _{i=x{+}1}^l(1{+}\chi _{[i]}{+}v) \end{array} \right] \end{aligned}$$

Because sequence $A_1$ is an optimal sequence, it follows that $f_1-f_2\le 0$. Therefore, if $Q_{x,y}\ge 0$, then $\chi _j < \chi _k$ definitely holds; if $Q_{x,y}< 0$, then $\chi _j\ge \chi _k$ must hold. $\square$

Based on the above analysis, the following algorithm holds:

Algorithm 3.1

Input: ${{\bar{N}}}$, v, $\chi _i, \alpha _i, \beta _i, \gamma _i$ ($1\le i\le {{\bar{N}}}$).

Output: An optimal sequence A, and an optimal $d_{opt}$.

Step 1. Arrange jobs in non-decreasing order of $\chi _i$, i.e., $\chi _{[1]}\le \chi _{[2]} \le \ldots \le \chi _{[{{\bar{N}}}]}$;

Step 2. Calculate the value of l based on Property 3.2;

Step 3. Determine the optimal sequence based on Property 3.5;

Step 4. Calculate $d_{opt}=C_{[l]}$.

Theorem 3.1

The scheduling problem $1|p_{i}=\chi _{i}s_i,CON,psddt|\sum _{i=1}^{{\bar{N}}}\left( \alpha _i {\widehat{E}}_{[i]}\right.$$\left. +\beta _i {\widehat{T}}_{[i]}+\gamma _i d_{opt}\right)$ can be solved by Algorithm 3.1 in $O({{\bar{N}}}\log {{\bar{N}}})$ time.

Proof

First, it is not hard to find that the time required for Step 1 is $O({{\bar{N}}}\log {{\bar{N}}})$. Besides, it is easy to know that Step 2 can be solved in a constant time. Steps 3 and 4 require $O({{\bar{N}}})$ time. In summary, the total time of Algorithm 3.1 is $O({{\bar{N}}}\log {{\bar{N}}})$. $\square$

4 SLK case

In this section, we focus on the problem $1|p_{i}=\chi _{i}s_i,SLK,psddt|\sum _{i=1}^{{\bar{N}}} \left( \alpha _i {\widehat{E}}_{[i]}\right.$$\left. +\beta _i {\widehat{T}}_{[i]}+\gamma _i q_{opt}\right)$. First, the following properties to obtain the optimal solution are presented.

Property 4.1

For a given sequence, the optimal common flow allowance $q_{opt}$ is equal to $(1+v)$ times the completion time of a job or $(1+v)s_0$.

Proof

Proof by contradiction. That is, assuming that $(1+v)C_{[l-1]}<q_{opt}<(1+v)C_{[l]}$, $1 \le l \le {{\bar{N}}}$, and the start time of the job $J_1$ is $s_0$, then

$$\begin{aligned} f= & {} \sum _{i=1}^{{\bar{N}}}\left( \alpha _i {\widehat{E}}_{[i]}+\beta _i {\widehat{T}}_{[i]}+\gamma _i q\right) \\= & {} \sum _{i=1}^l \alpha _i (p_{[i]}+q_{opt}-C_{[i]})s+\sum _{i=l+1}^{{\bar{N}}} \beta _i(C_{[i]}-p_{[i]}-q_{opt})+\sum _{i=1}^{{\bar{N}}} \gamma _i q_{opt}\\= & {} -\sum _{i=1}^l \alpha _i(1+v)C_{[i-1]}+\sum _{i=l+1}^{{\bar{N}}} \beta _i (1+v) C_{[i-1]}\\{} & {} \quad +\left( \sum _{i=1}^l \alpha _i-\sum _{i=l+1}^{{\bar{N}}}\beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i\right) q_{opt}. \end{aligned}$$

First, move the $q_{opt}$ to the left side of the $(1+v)C_{[l-1]}$ and set $q_{opt}-(1+v)C_{[l-1]}=s$, where $s>0$;, then,

$$\begin{aligned} f_1= & {} \sum _{i=1}^{l-1} \alpha _i (p_{[i]}+q_{opt}-C_{[i]})+\sum _{i=l+1}^{{\bar{N}}} \beta _i(C_{[i]}-p_{[i]}-q_{opt})+\sum _{i=1}^{{\bar{N}}} \gamma _i q_{opt}\\= & {} -\sum _{i=1}^{l-1} \alpha _i(1+v)C_{[i-1]}+\sum _{i=l+1}^{{\bar{N}}} \beta _i (1+v) C_{[i-1]}\\{} & {} \quad +\left( \sum _{i=1}^{l-1} \alpha _i-\sum _{i=l+1}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i\right) (1+v)C_{[l-1]}. \end{aligned}$$

Then, shift the $q_{opt}$ to the right side of the $(1+v)C_{[l]}$ and set $(1+v)C_{[l]}-q_{opt}=t$, where $t>0$; it holds that

$$\begin{aligned} f_2= & {} \sum _{i=1}^{l} \alpha _i (p_{[i]}+q_{opt}-C_{[i]})+\sum _{i=l+1}^{{\bar{N}}} \beta _i(C_{[i]}-p_{[i]}-q_{opt})+\sum _{i=1}^{{\bar{N}}} \gamma _i q_{opt}\\= & {} -\sum _{i=1}^{l} \alpha _i(1+v)C_{[i-1]}+\sum _{i=l+2}^{{\bar{N}}} \beta _i (1+v) C_{[i-1]}\\{} & {} \quad +\left( \sum _{i=1}^{l} \alpha _i-\sum _{i=l+2}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i\right) (1+v)C_{[l]}. \end{aligned}$$

Hence

$$\begin{aligned} f-f_1=\left( \sum _{i=1}^l \alpha _i-\sum _{i=l+1}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i\right) s \end{aligned}$$

and

$$\begin{aligned} f-f_2=-\left( \sum _{i=1}^l \alpha _i-\sum _{i=l+1}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i\right) t. \end{aligned}$$

When $\left( \sum _{i=1}^l \alpha _i-\sum _{i=l+1}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i\right) \ge 0$, it is easy to find that $f\ge f_1$ holds, otherwise $f> f_2$ must hold. That is to say, $q_{opt}$ is equal to $(1+v)$ times the completion of a job or $(1+v)s_0$. $\square$

Property 4.2

For a given sequence, $q_{opt}=(1+v)C_{[l-1]}$, where l satisfies

$$\begin{aligned} \left( \sum _{i=1}^{l-1}\alpha _i-\sum _{i=l}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i\right) \le 0 \end{aligned}$$

and

$$\begin{aligned} \left( \sum _{i=1}^{l}\alpha _i-\sum _{i=l+1}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i\right) \ge 0. \end{aligned}$$

Proof

It is similar to the proof of Property 3.3. $\square$

Now we define two sets: $W=\{J_j\in \sigma | C_j\le q_{opt}\}$ and $U=\{J_j\in \sigma | C_j>q_{opt}\}$, where $q_{opt}=(1+v)C_{[l-1]}$, and $\sigma$ is the given sequence.

Property 4.3

In an optimal schedule or sequence, all jobs in the set W are sorted by the LDR order of $\chi _i$.

Proof

First, suppose there are two consecutive jobs, $J_j$ and $J_k$ in W, where $J_j$ is at the u-th position in sequence $A_1=\{J_1, \ldots ,J_j,J_k, \ldots ,J_{{\bar{N}}} \}$. Also, suppose that $C_{[0]}=s_0$, and $q_{opt}=(1+v)C_{[l-1]}$, where $1 \le u \le l-2$. Then the sequence $A_2=\{J_1, \ldots ,J_k,J_j, \ldots ,J_{{\bar{N}}} \}$ can be obtained by exchanging the two jobs, then the objective function $f_1$ of $A_1$ is subtracted from $f_2$ of $A_2$:

$$\begin{aligned} f_1-f_2=-\alpha _{u+1}s_0 (1+v)(\chi _j-\chi _k)\prod _{i=1}^{u-1}(1+\chi _{[i]}+v). \end{aligned}$$

If $f_1 \le f_2$, $\chi _j\ge \chi _k$, that means the jobs in W are sorted in descending order of $\chi _i$. $\square$

Property 4.4

In an optimal schedule or sequence, all jobs in the set U are sorted by the SDR order of $\chi _i$.

Proof

Also, suppose that there are two successive jobs, $J_j$ and $J_k$, in set U, where $J_j$ is at the u-th position in sequence $A_1=\{J_1, \ldots ,J_j,J_k, \ldots ,J_{{\bar{N}}} \}$. Assume that $q_{opt}=(1+v)C_{[l-1]}$, $l \le u < {{\bar{N}}}$. Now exchange the job $J_j$ and $J_k$, and the sequence $A_2=\{J_1, \ldots ,J_k,J_j, \ldots ,J_{{\bar{N}}} \}$ can be obtained. Then, the objective function $f_1$ of $A_1$ is subtracted from $f_2$ of $A_2$:

$$\begin{aligned} f_1-f_2=\beta _{u+1}s_0 (1+v)(\chi _j-\chi _k)\prod _{i=1}^{u-1}(1+\chi _{[i]}+v). \end{aligned}$$

If $f_1 \le f_2$, then $\chi _j\le \chi _k$ holds, which indicates that all jobs in U are sorted in ascending order of $\chi _i$. $\square$

Similarly, we can define:

$$\begin{aligned} Q_{x,y}&{=}&\left[ \begin{array}{l} {-}\sum _{i=x+1}^{l-1}\alpha _i \prod _{g=x+1}^{i-1}(1+\chi _{[g]}+v){+}\sum _{i=l+1}^{y}\beta _i \prod _{g=x+1}^i (1+\chi _{[g]}{+}v)\\ {+}\left( \sum _{i=1}^{l-1}\alpha _i{-}\sum _{i=l+1}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i\right) \prod _{i=x+1}^l(1+\chi _{[i]}+v) \end{array} \right] \end{aligned}$$

If $Q_{xy}\ge 0$, it is known that the job $J_j$ should be at the x-th position; otherwise, the job $J_j$ should be at y-th position.

Property 4.5

Assuming there are two different jobs, $J_j$ and $J_k$, in which $J_j$ is at the x-th position and $J_k$ is at the y-th position ($1\le x \le l-1$ and $l\le y \le {{\bar{N}}}$), consider the following two cases:

• 1. If $Q_{x,y}\ge 0$, then $\chi _j < \chi _k$;

• 2. If $Q_{x,y}<0$, then $\chi _j \ge \chi _k$.

Proof

Similar to the proofing process of Property 3.6. $\square$

In summary, the following scheduling algorithm and theorem can be obtained:

Algorithm 4.1

Input: ${{\bar{N}}}$, v, $\chi _i, \alpha _i, \beta _i, \gamma _i$ ($1\le i\le {{\bar{N}}}$).

Output: An optimal sequence A, and an optimal $q_{opt}$.

Step 1. First, arrange jobs in non-decreasing order of $\chi _i$, i.e., $\chi _{[1]}\le \chi _{[2]} \le \ldots \le \chi _{[{{\bar{N}}} ]}$;

Step 2. Then, calculate the value of l based on Property 4.2;

Step 3. Next, determine the optimal sequence based on Property 4.5;

Step 4. Finally, calculate $q_{opt}=(1+v)C_{[l-1]}$.

Theorem 4.1

Algorithm 4.1 solves $1|p_{i}=\chi _{i}s_i,SLK,psddt|\sum _{i=1}^{{\bar{N}}} \left( \alpha _i {\widehat{E}}_{[i]}+\beta _i {\widehat{T}}_{[i]}\right.$$\left. +\gamma _i q_{opt}\right)$ in $O(n\log n)$ time.

5 DIF case

In this section, we first analyze the optimal properties for the problem $1|p_{[i]}=\chi _{[i]}s_i,DIF,psddt|$ $\sum _{i=1}^{{\bar{N}}} \left( \alpha _i {\widehat{E}}_{[i]}+\beta _i {\widehat{T}}_{[i]}+\gamma _i d_{[i]}\right)$.

Property 5.1

For a given schedule or sequence, when $\gamma _i \ge \beta _i$, the due date of job $J_i$ is equal to 0; that is, $\min \{\beta _i,\gamma _i\}=\beta _i$, $d_{[i]}=0$. Otherwise, its due date time is equal to the completion time of job $J_i$; that is, $\min \{\beta _i,\gamma _i\}=\gamma _i$, $d_{[i]}=C_{[i]}$.

Proof

For job $J_i$, it follows that

$$\begin{aligned} f_i= & {} \alpha _i {\widehat{E}}_{[i]}+\beta _i {\widehat{T}}_{[i]}+\gamma _i d_{[i]}\\= & {} \alpha _i\max \{0,d_{[i]}-C_{[i]}\}+\beta _i \max \{0,C_{[i]}-d_{[i]}\}+\gamma _i d_{[i]}. \end{aligned}$$

Then, it has the following two cases:

Case I When $J_i$ is a tardy job, that is, $d_{[i]} \le C_{[i]}$, the corresponding objective function can be rewritten as the following:

$$\begin{aligned} f_i=\beta _i C_{[i]}+(\gamma _i-\beta _i)d_{[i]}. \end{aligned}$$

If $\gamma _i \ge \beta _i$, in order to minimize the objective function, there exists $d_{[i]}=0$, and the objective function is $f_i=\beta _i C_{[i]}$; If $\gamma _i < \beta _i$, then it follows that $d_{[i]}=C_{[i]}$, and $f_i=\gamma _i C_{[i]}$.

Case II If $J_i$ is an early job, that is, $d_{[i]} \ge C_{[i]}$, then the objective function of job $J_i$ can be rewritten as follows:

$$\begin{aligned} f_i=(\alpha _i+\gamma _i)d_{[i]}-\alpha _i C_{[i]}. \end{aligned}$$

Then, if there exists $d_{[i]}=C_{[i]}$, it follows that $f_i=\gamma _i C_{[i]}$. $\square$

From Lemma 5.1, it is easily known that, if $\min \{\beta _i,\gamma _i\}=\beta _i$, then $d_{[i]}=0$ and $f=\sum _{i=1}^n \beta _i C_{[i]}$ hold; If $\min \{\beta _i,\gamma _i\}=\gamma _i$, then $d_{[i]}=C_{[i]}$, and $f=\sum _{i=1}^n \gamma _i C_{[i]}$ must hold. Based on the above works, it follows that

$$\begin{aligned} f=\sum _{i=1}^n \min \{\beta _i,\gamma _i\} C_{[i]}. \end{aligned}$$

Property 5.2

If an optimal sequence exists, then it can be obtained by the SDR order of $\chi _{i}$, i.e., $\chi _{[1]} \le \chi _{[2]} \le \ldots \le \chi _{[{{\bar{N}}}]}$.

Proof

For sequence $A_1=\{J_1, \ldots ,J_j,J_k, \ldots ,J_{{\bar{N}}} \}$, $J_j$ and $J_k$ are two consecutive jobs in the sequence, job $J_j$ is supposed to be at the x-th position, and job $J_k$ is supposed to be at the $(x+1)$-th position. Now swap the two jobs to obtain sequence $A_2=\{J_1, \ldots ,J_k,J_j, \ldots ,J_{{\bar{N}}} \}$. Subsequently, the objective function $f_1$ of $A_1$ is subtracted from the objective function $f_2$ of $A_2$:

$$\begin{aligned} f_1-f_2=\min \{\beta _x,\gamma _x\}(\chi _j-\chi _k)s_0\prod _{i=1}^{x-1}(1+\chi _{[i]}+v), \end{aligned}$$

and the optimal sequence is that the jobs are sorted in the increasing order of $\chi _i$.

In summary, the following scheduling algorithm and theorem can be obtained: $\square$

Algorithm 5.1

Input: ${{\bar{N}}}$, v, $\chi _i, \alpha _i, \beta _i, \gamma _i$ ($1\le i\le {{\bar{N}}}$).

Output: An optimal sequence A, and an optimal $d_{i}$.

Step 1. Determine the optimal sequence by using the SDR order of $\chi _i$, i.e., $\chi _{[1]}\le \chi _{[2]} \le \ldots \le \chi _{[{{\bar{N}}} ]}$;

Step 2. If $\gamma _i \ge \beta _i$, the optimal due date is $d_{[i]}=0$. Else, the optimal due date is $d_{[i]}=C_{[i]}$.

Theorem 5.1

Algorithm 5.1 can solve the $1|p_{[i]}=\chi _{[i]}s_i,DIF,psddt|\sum _{i=1}^{{\bar{N}}} \left( \alpha _i {\widehat{E}}_{[i]}\right.$$\left. +\beta _i {\widehat{T}}_{[i]}+\gamma _i d_{[i]}\right)$ problem within $O({{\bar{N}}} \log {{\bar{N}}} )$ time.

6 Example

Example 6.1

There are four jobs, that is, ${{\bar{N}}}=4$, in which $s_0=2$, $v=0.3$, $\{\chi _1,\chi _2,\chi _3,\chi _4\}$$=\{0.2,0.5,0.4,0.3\}$, $\{\alpha _1,\alpha _2,\alpha _3,\alpha _4\}=\{1,1,2,3\}$, $\{\beta _1,\beta _2,\beta _3,\beta _4\}=\{1,1,3,5\}$, and $\{\gamma _1,\gamma _2,\gamma _3,\gamma _4\}=\{1,2,1,3\}$.

For the problem $1|p_{i}=\chi _{i}s_i,CON,psddt|\sum _{i=1}^{{\bar{N}}}\left( \alpha _i {\widehat{E}}_{[i]}+\beta _i {\widehat{T}}_{[i]}+\gamma _i d_{opt}\right)$, according to Algorithm 3.1, the solution steps are given as follows:

Step 1. $J_1\rightarrow J_4\rightarrow J_3 \rightarrow J_2$ can be obtained by $b_1< b_4< b_3< b_2$.

Step 2. Calculate the value of l according to Property 3.2:

When $l=1$, $(\sum _{i=1}^{l-1}\alpha _i-\sum _{i=l}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i)=-10+7=-3<0$, and $(\sum _{i=1}^{l}\alpha _i-\sum _{i=l+1}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}}\gamma _i)=1-9+7=-1<0$;

When $l=2$, $(\sum _{i=1}^{l-1}\alpha _i-\sum _{i=l}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i)=-1<0$, and $(\sum _{i=1}^{l}\alpha _i-\sum _{i=l+1}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i)=2-8+7=1>0$;

When $l=3$, $(\sum _{i=1}^{l-1}\alpha _i-\sum _{i=l}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i)=1>0$, and $(\sum _{i=1}^{l}\alpha _i-\sum _{i=l+1}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i)=4-5+7=6>0$;

When $l=4$, $(\sum _{i=1}^{l-1}\alpha _i-\sum _{i=l}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i)=6>0$, and $(\sum _{i=1}^{l}\alpha _i-\sum _{i=l+1}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i)=7+7=14>0$;

Then $l=2$ can be obtained.

Step 3. From Step 2, $l=2$, and the solution steps are given as follows:

• (1) $x=2, y=3$, and $Q_{2,3}=0$; then, job $J_1$ is at the 2-th position;

• (2) $x=1, y=3$, and $Q_{1,3}=-1<0$; then, job $J_4$ is at the 3-th position;

• (3) $x=2, y=4$, and $Q_{2,4}=4.8>0$; then, job $J_2$ is at the 4-th position;

• (4) $x=1, y=4$, and $Q_{1,4}=6.2>0$; then, job $J_3$ is at the 1-th position.

Based on the above works, the optimal sequence is $J_3\rightarrow J_1\rightarrow J_4 \rightarrow J_2$, and the processing times, delivery times, and completion times can be seen as follows (see Table 1):

Table 1 Results for CON

It can be seen that $d_{opt}=C_{[2]}=5.1000$, and we can easily know that the corresponding optimal value is $f=\sum _{i=1}^{{\bar{N}}}\left( \alpha _i {\widehat{E}}_{[i]}+\beta _i {\widehat{T}}_{[i]}+\gamma _i d_{opt}\right) =94.5200$.

For the problem $1|p_{i}=\chi _{i}s_i,SLK,psddt|\sum _{i=1}^{{\bar{N}}} \left( \alpha _i {\widehat{E}}_{[i]}+\beta _i {\widehat{T}}_{[i]}+\gamma _i q_{opt}\right)$, according to Algorithm 4.1, the solution steps are given as follows:

Step 1. $J_1\rightarrow J_4\rightarrow J_3 \rightarrow J_2$ can be obtained by $\chi _1< \chi _4< \chi _3< \chi _2$.

Step 2. Calculate the value of l according to Property 4.2:

When $l=2$, $(\sum _{i=1}^{l-1}\alpha _i-\sum _{i=l}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i)=1-9+7=-1<0$, and $(\sum _{i=1}^{l}\alpha _i-\sum _{i=l+1}^{{\bar{N}}} \beta _i+\sum _{i=1}^{{\bar{N}}} \gamma _i)=2-8+7=1>0$, that is $l=2$.

Step 3. From Step 2, $l=2$, and the solution steps are given as follows:

• (1) $x=2, y=3$, and $Q_{2,3}=5.1>0$; then, job $J_4$ is at the 2-th position;

• (2) $x=1, y=3$, and $Q_{1,3}=8.16>0$; then, job $J_1$ is at the 1-th position;

• (3) $x=2, y=4$, and $Q_{2,4}=20.4>0$; then, job $J_2$ is at the 4-th position;

• (4) $x=1, y=4$, and $Q_{1,4}=32.64>0$; then, job $J_3$ is at the 3-th position.

Based on the above works, the optimal sequence is $J_1\rightarrow J_4\rightarrow J_3 \rightarrow J_2$, and the processing times, delivery times, completion times and due dates can be seen in Table 2.

Table 2 Results for SLK

It can be known that $q_{opt}=(1+v)C_{[2]}=6.2400$, and the corresponding objective function value is $f=\sum _{i=1}^{{\bar{N}}}\left( \alpha _i {\widehat{E}}_{[i]}+\beta _i {\widehat{T}}_{[i]}+\gamma _i q_{opt}\right) =71.2000$.

For the problem $1|p_{[i]}=\chi _{[i]}s_i,DIF,psddt|\sum _{i=1}^{{\bar{N}}} \left( \alpha _i {\widehat{E}}_{[i]}+\beta _i {\widehat{T}}_{[i]}+\gamma _i d_{[i]}\right)$, according to Algorithm 5.1, the solution steps are given as follows:

Step 1. The optimal sequence: $J_1\rightarrow J_4\rightarrow J_3 \rightarrow J_2$ can be obtained by $\chi _1< \chi _4< \chi _3< \chi _2$.

Step 2. The processing times, delivery times, completion times and due dates (see Property 5.1) are shown in Table 3.

Table 3 Results for DIF

The corresponding objective function value is $\sum _{i=1}^{{\bar{N}}} \left( \alpha _i {\widehat{E}}_{[i]}+\beta _i {\widehat{T}}_{[i]}+\gamma _i d_{[i]}\right) =60.0240$.

7 Conclusion

We extended the results of scheduling with position-dependent weights and due date assignment to a setting of scheduling with psddt and deterioration effects. Under three kinds of due date assignments, our objective is to minimize weighted sum of earliness, tardiness, and due date cost, where the weights are position-dependent weights. Through a series of optimal properties, it can be obtained that the CON, SLK, and DIF assignment scheduling problems can be solved in $O({\bar{N}}\log {\bar{N}})$ time. Future research could delve into a general linear deterioration, such as $p_i=a_i+\chi _is_i$ (where $a_i$ is the normal processing time of job $J_i$), consider the model under a flow shop setting, or study the model with job-dependent weights (i.e., the objective function would be to minimize $\sum _{i=1}^{{\bar{N}}} \left( \alpha _i {\widehat{E}}_{i}+\beta _i {\widehat{T}}_{i}+\gamma _i d_{i}\right)$, where $\alpha _i$, $\beta _i$ and $\gamma _i$ are the weights of job $J_i$ for earliness, tardiness and due date).