Research article Special Issues

Exploring brain activity and transforming knowledge in visual and textual programming using neuroeducation approaches

  • Received: 05 May 2019 Accepted: 12 August 2019 Published: 02 September 2019
  • Eight (8) computer science students, novice programmers, who were in the first semester of their studies, participated in a field study in order to explore potential differences in their brain activity during programming with a visual programming language versus a textual programming language. The eight students were asked to develop two specific programs in both programming languages (a total of four tasks). The order of these programs was determined, while the order of languages in which they worked differed between the students. Measurement of cerebral activity was performed by the electroencephalography (EEG) imaging method. According to the analysis of the data it appears that the type of programming language did not affect the students' brain activity. Also, six students needed more time to successfully develop the programs they were asked with the first programming language versus the second one, regardless of the type of programming language that was first. In addition, it appears that six students did not show reducing or increasing brain activity as they spent their time on tasks and at the same time did not show a reduction or increase in the time they needed to develop the programs. Finally, the students showed higher average brain activity in the development of the fourth task than the third, and six of them showed higher average brain activity when developing the first versus the second program, regardless of the programming language. The results can contribute to: a) highlighting the need for a diverse educational approach for students when engaging in program development and b) identifying appropriate learning paths to enhance student education in programming.

    Citation: Spyridon Doukakis. Exploring brain activity and transforming knowledge in visual and textual programming using neuroeducation approaches[J]. AIMS Neuroscience, 2019, 6(3): 175-190. doi: 10.3934/Neuroscience.2019.3.175

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  • Eight (8) computer science students, novice programmers, who were in the first semester of their studies, participated in a field study in order to explore potential differences in their brain activity during programming with a visual programming language versus a textual programming language. The eight students were asked to develop two specific programs in both programming languages (a total of four tasks). The order of these programs was determined, while the order of languages in which they worked differed between the students. Measurement of cerebral activity was performed by the electroencephalography (EEG) imaging method. According to the analysis of the data it appears that the type of programming language did not affect the students' brain activity. Also, six students needed more time to successfully develop the programs they were asked with the first programming language versus the second one, regardless of the type of programming language that was first. In addition, it appears that six students did not show reducing or increasing brain activity as they spent their time on tasks and at the same time did not show a reduction or increase in the time they needed to develop the programs. Finally, the students showed higher average brain activity in the development of the fourth task than the third, and six of them showed higher average brain activity when developing the first versus the second program, regardless of the programming language. The results can contribute to: a) highlighting the need for a diverse educational approach for students when engaging in program development and b) identifying appropriate learning paths to enhance student education in programming.


    In actual industrial production, the processing times of jobs have deterioration (aging) effects, i.e., the later of a job starts, the longer it takes to process it (see Wang and Wang [1]). Wang and Wang [2] considered the single-machine makespan minimization problem with time dependent processing times and group technology. Under ready times of the jobs, they proved that the problem can be solved in polynomial time. Wang and Wang [3] investigated the single-machine weighted sum of the $ h $th power of waiting times problems with simple linear deterioration and precedence constraints. Under some precedence constraints, they proved that these problems remain polynomially solvable. Cheng et al. [4] studied single-machine problems with an accelerating deterioration effect. They proved that some regular and non-regular objective functions can be solved in polynomial time. Yin et al. [5] addressed some two-agent scheduling problems with deterioration effects on a single-machine. Zhang et al. [6] considered single-machine problems with time-dependent processing times. Under the common and slack due window assignments, they proved that two non-regular objective function minimizations can be solved in polynomial time. Liu et al. [7] considered single-machine group scheduling with deteriorating jobs. For the makespan minimization with ready times, they proposed branch-and-bound algorithm and heuristic algorithm. Wang and Liang [8] studied single-machine group scheduling with deterioration effects and resource allocation. For the makespan minimization under limited resource availability constraint, they proposed branch-and-bound and heuristic algorithms. Gawiejnowicz [9] reviewed scheduling problems with deteriorations effects (time-dependent processing times). In addition, with the deterioration (aging) effects, the machine can perform the maintenance activities and increase work efficiency, so, the scheduling problems with maintenance activities have also been considered (see Ma et al. [10] and Wang and Wei [11]). Hsu et al. [12] considered unrelated parallel-machine problem with rate-modifying activities. For the total completion time minimization, they proposed a more efficient algorithm. Ji et al. [13] studied single-machine slack due date assignment problem with job-dependent aging effects. Under a deteriorating maintenance activity, that proved that a non-regular objective minimization can be solved in polynomial time. Rustogi and Strusevich [14,15] considered single-machine scheduling problems with rate modifying activities. Liu et al. [16] studied single-machine multiple common due-date assignments scheduling with deterioration effects. Under an maintenance activity, they proved that the linear weighted sum of earliness, tardiness, and the due dates minimization can be solved in polynomial time. Xiong et al. [17] considered the single-machine common due date assignment problem potential machine disruption. Zhu et al. [18] investigated multitasking scheduling problems with multiple rate-modifying activities. For the single-criterion and multi-criteria minimizations, they proposed optimal algorithms.

    Recently, Zhang et al. [19] studied machine scheduling problems with deteriorating effects. Under the deteriorating rate-modifying activities, they proved that some objective function minimizations can be solved in polynomial time. Wang and Li [20] studied the unrelated parallel processors problem with deterioration effects and deteriorating maintenance activities. For some regular objective function minimizations, they proved that the problem can be solved in polynomial time. Zhang et al. [21] considered parallel machines scheduling problems with linear deteriorating jobs and maintenance activities. Under the resumable and non-resumable cases, the goal is to minimize the expected sum of completion times. They presented the pseudo-polynomial time algorithms to solve the problems. Zhang et al. [22] addressed single-machine scheduling problems with multiple maintenance activities and position-and-resource-dependent processing times. For some regular and non-regular objective function minimizations, they proposed polynomial time and pseudo-polynomial time algorithms. Sun and Geng [23] investigated the single-machine maintenance activity scheduling with deteriorating effects. The goal is to minimize the makespan and total completion time, they showed that the problem can be solved in polynomial time, respectively. Wang et al. [24] examined the single-machine common due-window assignment problem with a maintenance activity. Under constant and time-dependent processing times, they proved that a non-regular general earliness and tardiness minimization can be solved in polynomial time. Jia et al. [25] considered the single-machine scheduling problem with deterioration effects. Under the deterioration maintenance activity and slack due-window assignment, they showed that a non-regular objective function minimization can be solved in polynomial time. He et al. [26] discussed unrelated parallel processor scheduling with deterioration effects and maintenance activities. The objective is to minimize the total completion time and total machine load, they showed that these problems remain polynomially solvable.

    The phenomena of deteriorating jobs and machine maintenance activity occurring simultaneously can be found in real-life situations. For example, consider a set of tasks (jobs) available for processing by a human operator. Perhaps the most distinguishing factor that differentiates human operators from machines within the context of task-sequencing is the notion of fatigue and its effect on task processing. More specifically, the rate at which a human operator performs a given task is known to be a decreasing function of the operator's level of fatigue, which manifests as the task processing time taking longer than expected. Another distinguishing characteristic is that human operators regularly engage in rest breaks during work shifts, which allows them to recover and mitigate some of the negative effects of fatigue (Sun and Geng [23]; Eilon [27]; Lodree and Geiger [28]). In this paper, we extend the results of Sun and Geng [23], by studying a more general processing times that includes the one given in Sun and Geng [23] as a special case. For the makespan minimization, we prove that the problem is polynomial-time solvable.

    The organization of this article is as follows. Section 2 gives a description of the problem. Section 3 presents a polynomial-time solution for the problem. Conclusions are presented in Section 4.

    There are $ \ddot{n} $ independent jobs $ T_1, T_2, \ldots, T_{{n}} $ to be processed on a single-machine, and before processing, the machine needs $ \tilde{t} $ $ (\tilde{t} > 0) $ preparation time. Due to the time-and-position dependent deteriorating effects (denoted by $ tpdde $), the machine's production efficiency is reduced, the machine will perform a maintenance activity (denoted by $ ma $), and when the repair is completed, the machine will return to the initial setting and the $ tpdde $ will start again.

    It is assumed that the machine performs a $ ma $ after the $ k $th job is completed, and the maintenance time is $ D $. In the case of $ tpdde $, the actual processing time of job $ T_j $ if it is scheduled in $ r $th position in a sequence is given by

    $ pj={(aj+bjsj)rc,if rk,(θjaj+bj(sjC[k]D))(rk)c,if r>k,
    $
    (1)

    where $ a_j $ (resp. $ b_j $) is the normal processing time (resp. Deteriorating rate) of $ T_j $ $ (j = 1, 2, \ldots, n) $, $ s_j $ (resp. $ \theta_j $ $ (\theta_j > 0) $) is the starting processing time (resp. Maintenance rate) of $ T_j $, $ c $ ($ c\geq0 $) is the common aging rate for all the jobs, and $ [k] $ is some job scheduled in $ k $th position. Ji et al. [29] studied the following (i.e., simple linear deterioration and aging) model $ p_{j} = \left\{ bjsjrc,if rk,bj(sjC[k]D)(rk)c,if r>k.

    \right. $ Ji et al. [29] proved that the makespan minimization under simple linear deterioration and aging model is NP-hard in the strong sense. Sun and Geng [23] considered the following (i.e., linear deterioration) model $ p_{j} = \left\{ aj+bsj,if rk,θaj+b(sjC[k]D),if r>k,
    \right. $ where $ b $ (resp. $ 0 < \theta\leq1 $) is the common deterioration rate (resp. Maintenance rate) for all the jobs. Sun and Geng [23] proved that the makespan and total completion time minimizations can be solved in polynomial time, respectively.

    In this paper, we mainly concentrate on the following model:

    $ pj={(aj+bsj)rc,if rk,(θjaj+b(sjC[k]D))(rk)c,if r>k,
    $
    (2)

    Let $ C_{j} $ denote the completion time of job $ T_{j} $, the objective is to determine the location of the maintenance activity and the job sequence $ \varphi $ such that the makespan $ C_{\max } = \max\{C_1, C_2, \ldots, C_{{n}}\} $ is to be minimized. From Gawiejnowicz [9], this problem can be written as:

    $ 1|ma,tpdde|Cmax
    $
    (3)

    The machine is repaired after the $ k $th job is processed, the machine can be repaired in any position from $ 1 $ to $ n-1 $, and the maintenance time is $ D $, hence, from Figure 1, we have

    $ Cmax=˜t+kj=1p[j]+D+˜t+nj=k+1p[j]=D+2˜t+kj=1p[j]+nj=k+1p[j]
    $
    (4)
    Figure 1.  A sample of $ 1|ma, tpdde|C_{\max } $.

    According to the location of $ ma $, we are divided into the following three cases, that is, case ⅰ: $ k = 0 $, case ⅱ: $ 0 < k < n $, and case ⅲ: $ k = n $.

    If $ k = 0 $, the actual machining time $ p_{[j]} $ of the job $ T_{[j]} $ is

    $ p[j]=(θ[j]a[j]+b(s[j]D))jc,j=1,2,,n.
    $
    (5)

    If $ j = 1 $, the machine needs $ \tilde{t} $ time to prepare and maintenance time $ D $,

    $ s_{[1]} = \tilde{t}+D $, $ p_{[1]} = \left(\theta_{[1]}{a_{[1]}}+b{(s_{[1]}-D)}\right)1^c = \theta_{[1]}{a_{[1]}}+b\tilde{t} $.

    If $ j = 2 $, $ s_{[2]} = C_{[1]} = p_{[1]}+s_{[1]} = \theta_{[1]}{a_{[1]}}+(1+b)\tilde{t}+D $,

    $ p[2]=(θ[2]a[2]+b(s[2]D))2c=θ[2]a[2]2c+b2c(θ[1]a[1]+(1+b)˜t)=θ[2]a[2]2c+b2cθ[1]a[1]+b2c(1+b)˜t.
    $

    If $ j = 3 $, $ s_{[3]} = C_{[2]} = p_{[2]}+s_{[2]} = \theta_{[2]}{a_{[2]}}2^c+\theta_{[1]}a_{[1]}(1+b2^c) +\tilde{t}(1+b)(1+b2^c)+D $,

    $ p_{[3]} = \left(\theta_{[3]}{a_{[3]}}+b{(s_{[3]}-D)}\right)3^c = \theta_{[3]}a_{[3]}3^c+b3^c (\theta_{[2]}a_{[2]}2^c+\theta_{[1]}a_{[1]}{(1+b2^c)}+\tilde{t}(1+b)(1+b2^c)) $.

    ......

    If $ j = i $,

    $ s_{[i]} = C_{[i-1]} = \sum_{h = 1}^{i-1} \theta_{[h]}a_{[h]}h^c\prod_{l = h+1}^{i-1}(1+bl^c) +\tilde{t}\prod_{l = 1}^{i-1}(1+bl^c)+D $,

    $ p_{[i]} = \theta_{[i]}a_{[i]}i^c+bi^c\left(\sum_{h = 1}^{i-1} a_{[h]}h^c\prod_{l = h+1}^{i-1}(1+bl^c)+\tilde{t}\prod_{l = 1}^{i-1}(1+bl^c)\right) $.

    ......

    If $ j = n $,

    $ s_{[n]} = C_{[n-1]} = \sum_{h = 1}^{{n}-1} \theta_{[h]}a_{[h]}h^c\prod_{l = h+1}^{{n}-1}(1+bl^c) +\tilde{t}\prod_{l = 1}^{n-1}(1+bl^c)+D $,

    $ p_{[{n}]} = \theta_{[{n}]}a_{[{n}]}{n^c}+b{n}^c\left(\sum_{h = 1}^{{n}-1} a_{[h]}h^c\prod_{l = h+1}^{{n}-1}(1+bl^c)+\tilde{t}\prod_{l = 1}^{{n}-1}(1+bl^c)\right). $

    By the simple algebraic calculation, we have

    $ Cmax=C[n]=s[n]+p[n]=n1h=1θ[h]a[h]hcn1l=h+1(1+blc)+˜tn1l=1(1+blc)+D+θ[n]a[n]ic+bnc(n1h=1a[h]hcn1l=h+1(1+blc)+˜tn1l=1(1+blc))=(1+b2c+b3c(1+b2c)+b4c(1+b2c)(1+b3c)++bncn1h=2(1+bhc))θ[1]a[1]+(2c+b3c2c+b4c2c(1+b3c)++bnc2cn1h=3(1+bhc))θ[2]a[2]+...+((n1)c+bnc(n1)c)θ[n1]a[n1]+ncθ[n]a[n]+(b+b2c(1+b)+b3c(1+b)(1+b2c)+...+bncn1l=1(1+blc))˜t+D=nj=1Δjθ[j]a[j]+E,
    $
    (6)

    where

    $ Δ1=1+b2c+b3c(1+b2c)+b4c(1+b2c)(1+b3c)++bncn1h=2(1+bhc)Δ2=2c+b3c2c+b4c2c(1+b3c)++bnc2cn1h=3(1+bhc)......Δn1=(n1)c+bnc(n1)cΔn=nc,
    $
    (7)

    and

    $ E=D+(b+b2c(1+b)+b3c(1+b)(1+b2c)+...+bncn1l=1(1+blc))˜t.
    $
    (8)

    If $ 0 < k < {n} $, the actual processing time of job $ T_j $ is:

    $ p[j]={(a[j]+bs[j])jc,if jk(θ[j]a[j]+b(s[j]C[k]D))(jk)c,if j>k
    $
    (9)

    If $ j = 1, 2, \ldots, k $, we have $ s_{[1]} = \tilde{t} $, $ p_{[1]} = (a_{[1]}+bs_{[1]})1^c = a_{[1]}+b\tilde{t} $,

    $ s_{[2]} = C_{[1]} = p_{[1]}+s_{[1]} = a_{[1]}+(1+b)\tilde{t} $,

    $ p_{[2]} = (a_{[2]}+bs_{[2]})2^c = a_{[2]}2^c+b2^c{(a_{[1]}1^c+(1+b)\tilde{t})} $,

    $ s_{[3]} = C_{[2]} = p_{[2]}+s_{[2]} = a_{[2]}2^c+a_{[1]}1^c{(1+b2^c)}+(1+b)(1+b2^c)\tilde{t} $,

    $ p_{[3]} = (a_{[3]}+bs_{[3]})3^c = a_{[3]}3^c+b3^c (a_{[2]}2^c+a_{[1]}{(1+b2^c)}+(1+b)(1+b2^c)\tilde{t}) $,

    ......

    $ s_{[k]} = C_{[k-1]} = \sum_{h = 1}^{k-1} a_{[h]}h^c\prod_{l = h+1}^{k-1}(1+bl^c)+\tilde{t}\prod_{l = 1}^{k-1}(1+bl^c) $,

    $ p_{[k]} = (a_{[k]}+bs_{[k-1]})i^c = a_{[k]}k^c+bk^c\left(\sum_{h = 1}^{k-1} a_{[h]}h^c\prod_{l = h+1}^{k-1}(1+bl^c)+\tilde{t}\prod_{l = 1}^{k-1}(1+bl^c)\right) $,

    $ C_{[k]} = \sum_{h = 1}^{k} a_{[h]}h^c\prod_{l = h+1}^{k}(1+bl^c)+\tilde{t}\prod_{l = 1}^{k}(1+bl^c) $.

    If $ j = k+1 $, the machine need to be repaired after the job $ T_{[k]} $ is completed, we have

    $ s_{[k+1]} = C_{[k]}+D+\tilde{t}, $
    $ p_{[k+1]} = (\theta_{[k+1]}{a_{[k+1]}}+b{(s_{[k+1]}-C_{[k]}-D)})1^c = \theta_{[k+1]}{a_{[k+1]}}+b\tilde{t}. $

    If $ j = k+2 $, $ j = k+3 $, $ \ldots $, $ j = {n} $, we have

    $ s_{[k+2]} = C_{[k+1]} = \theta_{[k+1]}{a_{[k+1]}}+(1+b)\tilde{t}+C_{[k]}+D $,

    $ p_{[k+2]} = (\theta_{[k+2]}{a_{[k+2]}}+b{(s_{[k+2]}-C_{[k]}-D)})2^c = \theta_{[k+2]}{a_{[k+2]}}2^c+b2^c{\theta_{[k+1]}{a_{[k+1]}}+b2^c(1+b)\tilde{t}} $.

    $ s_{[k+3]} = C_{[k+2]} = \theta_{[k+2]}{a_{[k+2]}}2^c+{\theta_{[k+1]}{a_{[k+1]}}(1+b2^c) +(1+b)(1+b2^c)\tilde{t}}+C_{[k]}+D $,

    $ p[k+3]=(θ[k+3]a[k+3]+b(s[k+3]C[k]d))3c=θ[k+3]a[k+3]3c+b3c(θ[k+2]a[k+2]2c+θ[k+1]a[k+1](1+b2c)+(1+b)(1+b2c)˜t)
    $

    ......

    $ s_{[{n}]} = C_{[{n}-1]} = \sum_{h = 1}^{{n}-k-1} \theta_{[k+h]}a_{[k+h]}h^c\prod_{l = h+1}^{{n}-k-1}(1+bl^c)+\tilde{t}\prod_{l = 1}^{{n}-k-1}(1+bl^c) +C_{[k]}+D $,

    $ p_{[{n}]} = \theta_{[{n}]}{a_{[{n}]}}({n}-k)^c+b({n}-k)^c \left(\sum\limits_{h = 1}^{{n}-k-1} \theta_{[k+h]}a_{[k+h]}h^c\prod\limits_{l = h+1}^{{n}-k-1}(1+bl^c) +\tilde{t}\prod\limits_{l = 1}^{{n}-k-1}(1+bl^c)\right). $

    Hence,

    $ Cmax=s[n]+p[n]=nk1h=1θ[k+h]a[k+h]hcnk1l=h+1(1+blc)+˜tnk1l=1(1+blc)+kh=1a[h]hckl=h+1(1+blc)+˜tkl=1(1+blc)+D+θ[n]a[n](nk)c+b(nk)c(nk1h=1θ[k+h]a[k+h]hcnk1l=h+1(1+blc)+˜tnk1l=1(1+blc))=(1+b2c+b3c(1+b2c)+b4c(1+b2c)(1+b3c)++bkck1h=2(1+bhc))a[1]+(2c+b3c2c+b4c2c(1+b3c)++bkc2ck1h=3(1+bhc))a[2]+...+((k1)c+bkc(k1)c)a[k1]+kca[k]+(+b2c+b3c(1+b2c)+b4c(1+b2c)(1+b3c)++b(nk)cnk1h=2(1+bhc))θ[k+1]a[k+1]+(2c+b3c2c+b4c2c(1+b3c)++b(nk)c2cnk1h=3(1+bhc))θ[k+2]a[k+2]+...+((nk1)c+b(nk1)c(nk)cθ[n1]a[n1]+(nk)cθ[n]a[n]+(b+b2c(1+b)+b3c(1+b)(1+b2c)+...+bkck1l=1(1+blc)+b+b2c(1+b)+...+b(nk)cnk1l=1(1+blc))˜t+D=kj=1Δja[j]+nj=k+1Δjθ[j]a[j]+E
    $
    (10)

    where,

    $ Δ1=1+b2c+b3c(1+b2c)+b4c(1+b2c)(1+b3c)++bkck1h=2(1+bhc)Δ2=2c+b3c2c+b4c2c(1+b3c)++bkc2ck1h=3(1+bhc)......Δk1=(k1)c+bkc(k1)cΔk=kcΔk+1=1+b2c+b3c(1+b2c)+b4c(1+b2c)(1+b3c)          ++b(nk)cnk1h=2(1+bhc)Δk+2=2c+b3c2c+b4c2c(1+b3c)++b(nk)c2cnk1h=3(1+bhc)......Δn1=(nk1)c+b(nk1)c(nk)cΔn=(nk)c,
    $
    (11)

    and

    $ E=D+˜t(b+b2c(1+b)+b3c(1+b)(1+b2c)+...+bkck1l=1(1+blc)+b+b2c(1+b)+...+b(nk)cnk1l=1(1+blc)).
    $
    (12)

    Similarly, if $ k = {n} $, the actual processing time of $ T_{[j]} $ is

    $ p[j]=(a[j]+bs[j])jc,j=1,2,,n,
    $
    (13)

    we have

    $ Cmax=(a[1]+b˜t)+(a[2]2c+b2ca[1]+b2c(1+b)˜t)+...+(a[n]nc+bnc(n1h=1a[h]hcn1l=h+1(1+blc)+˜tn1l=1(1+blc)))+D=(1+b2c+b3c(1+b2c)+b4c(1+b2c)(1+b3c)++bncn1h=2(1+bhc))a[1]+(2c+b3c2c+b4c2c(1+b3c)++bnc2cn1h=3(1+bhc))a[2]+...+((n1)c+bnc(n1)c)a[n1]+nca[n]+(b+b2c(1+b)+b3c(1+b)(1+b2c)+...+bncn1l=1(1+blc))˜t+D=nj=1Δja[j]+E
    $
    (14)

    where

    $ Δ1=1+b2c+b3c(1+b2c)+b4c(1+b2c)(1+b3c)++bncn1h=2(1+bhc)Δ2=2c+b3c2c+b4c2c(1+b3c)++bnc2cn1h=3(1+bhc)......Δn1=(n1)c+bnc(n1)cΔn=nc,
    $
    (15)

    and

    $ E=D+(b+b2c(1+b)+b3c(1+b)(1+b2c)+...+bncn1l=1(1+blc))˜t.
    $
    (16)

    Lemma 1. (Hardy et al. [30]) $ \sum_{i = j}^{{n}}{\mu_j}\nu_j $ get the minimum, when sequence $ \mu_j $ and sequence $ \nu_j $ is arranged in opposite monotonous order.

    From Eqs (8), (12) and (16), we have that $ E $ is a constant.

    For case ⅰ (i.e., $ k = 0 $), the problem $ 1|ma, tpdde|C_{\max } $ can be easily solved by Lemma 1 in $ O({n}\log {n}) $ time, i.e., $ \mu_j = \Delta_{j} $ (see Eqs (6) and (7)), $ \nu_j = \theta_{j}{a_{j}} $.

    For case ⅲ (i.e., $ k = {n} $), the problem $ 1|ma, tpdde|C_{\max } $ can be easily solved by Lemma 1 in $ O({n}\log {n}) $ time, i.e., $ \mu_j = \Delta_{j} $ (see Eqs (14) and (15)), $ \nu_j = {a_{j}} $.

    For case ⅱ (i.e., $ 0 < k < {n} $), let $ Z_{j, r} = 1 $ if job $ T_j $ is scheduled in $ r $th position, and $ Z_{j, r} = 0 $, otherwise. Form Eq (10), the optimal solution of the problem $ 1|ma, tpdde|C_{\max } $ can be solved by the following assignment problem:

    $ Min(nj=1kr=1ΔrajZj,r+nj=1nr=k+1ΔrθjajZj,r)
    $
    (17)
    $ Subject to nj=1Zj,r=1,r=1,2,,n,
    $
    (18)
    $  nr=1Zj,r=1,j=1,2,,n,
    $
    (19)
    $   Zj,r=0or1,j,r=1,2,,n,
    $
    (20)

    where $ \Delta_{r} $ ($ r = 1, 2, \ldots, {n} $) is given by Eq (11).

    Let $ C_{\max }(k) $ be the makespan under a given $ k $ ($ k = 0, 1, 2, \ldots, {n} $), from the above analysis, we propose the following algorithm to solve the problem $ 1|ma, tpdde|C_{\max } $.

    Algorithm 1

    Step 1. Set $ k = 0 $, calculate $ \mu_j = \Delta_{j} $ (see Eqs (6) and (7)) and $ \nu_j = \theta_{j}{a_{j}} $, determine a local optimal job sequence by Lemma 1, and record the objective function $ C_{\max }(k = 0) $.

    Step 2. Set $ k = 1, 2, \ldots, {n}-1 $, calculate $ \Delta_{r} $ (see Eq (11)) and determine a local optimal job sequence by an assignment problem Eqs (17)–(20), and record the objective function $ C_{\max }(k) $ ($ k = 1, 2, \ldots, {n}-1 $).

    Step 3. Set $ k = {n} $, calculate $ \mu_j = \Delta_{j} $ (see Eqs (14) and (15)) and $ \nu_j = {a_{j}} $, determine a local optimal job sequence by Lemma 1, and record the objective function $ C_{\max }(k = {n}) $.

    Step 4. The optimal job sequence is the one with the minimum objective function value $ C_{\max }^* = \min\{C_{\max }(k)|k = 0, 1, 2, \ldots, {n}\} $.

    Theorem 1. The problem $ 1|ma, tpdde|C_{\max } $ can be solved by Algorithm 1 in $ O({n}^4) $ time.

    Proof. Steps 1 and 3 need $ O({n}\log {n}) $ time respectively. For each $ k $ ($ k = 1, 2, \ldots, {n}-1 $), the running time for solving each assignment problem needs $ O(n^3) $ time. Step 4 needs $ O(n) $ time. Hence, the overall time complexity of Algorithm 1 is $ O(n^4) $ time.

    Example 1. Consider a $ 6 $-job problem $ 1|ma, tpdde|C_{\max } $, where $ a_1 = 3, a_2 = 4, a_3 = 5, a_4 = 8, a_5 = 9, a_6 = 7 $, $ \theta_1 = 0.7 $, $ \theta_2 = 0.6 $, $ \theta_3 = 0.7 $, $ \theta_4 = 0.8 $, $ \theta_5 = 0.6 $, $ \theta_6 = 0.9 $, $ \tilde{t} = 1 $, $ b = 0.15 $, $ c = 0.3 $, and $ D = 3 $.

    Solution:

    For $ k = 0 $, from Eqs (7) and (8), we have $ \Delta_{1} = 2.7453 $, $ \Delta_{2} = 2.8530 $, $ \Delta_{3} = 2.6660 $, $ \Delta_{4} = 2.3680 $, $ \Delta_{5} = 2.0368 $, $ \Delta_{6} = 1.7118 $, and $ E = 5.1571 $. The local optimal job sequence is $ \varphi = \{J_2\rightarrow J_1\rightarrow J_3\rightarrow J_5\rightarrow J_6\rightarrow J_4\} $, and $ C_{\max }(0) = 63.6427 $.

    For $ k = 1 $, from Eqs (11) and (12), we have $ \Delta_{1} = 1 $, $ \Delta_{2} = 2.1845 $, $ \Delta_{3} = 2.2701 $, $ \Delta_{4} = 2.1214 $, $ \Delta_{5} = 1.8842 $, $ \Delta_{6} = 1.6207 $, and $ E = 4.6621 $. The local optimal job sequence is $ \varphi = \{J_6\rightarrow J_2\rightarrow J_1\rightarrow J_3\rightarrow J_5\rightarrow J_4\} $, and $ C_{\max }(1) = 49.6442 $.

    For $ k = 2 $, from Eqs (11) and (12), we have $ \Delta_{1} = 1.1847 $, $ \Delta_{2} = 1.2311 $, $ \Delta_{3} = 1.7573 $, $ \Delta_{4} = 1.8262 $, $ \Delta_{5} = 1.7065 $, $ \Delta_{6} = 1.5157 $, and $ E = 4.3832 $. The local optimal job sequence is $ \varphi = \{J_4\rightarrow J_6\rightarrow J_2\rightarrow J_1\rightarrow J_3\rightarrow J_5\} $, and $ C_{\max }(2) = 44.6886 $.

    For $ k = 3 $, from Eqs (11) and (12), we have $ \Delta_{1} = 1.4317 $, $ \Delta_{2} = 1.4879 $, $ \Delta_{3} = 1.3904 $, $ \Delta_{4} = 1.4317 $, $ \Delta_{5} = 1.4879 $, $ \Delta_{6} = 1.3904 $, and $ E = 4.2930 $. The local optimal job sequence is $ \varphi = \{J_3\rightarrow J_1\rightarrow J_6\rightarrow J_5\rightarrow J_2\rightarrow J_4\} $, and $ C_{\max }(3) = 45.8487 $.

    For $ k = 4 $, from Eqs (11) and (12), we have $ \Delta_{1} = 1.7573 $, $ \Delta_{2} = 1.8262 $, $ \Delta_{3} = 1.7065 $, $ \Delta_{4} = 1.5157 $, $ \Delta_{5} = 1.1847 $, $ \Delta_{6} = 1.2311 $, and $ E = 4.3832 $. The local optimal job sequence is $ \varphi = \{J_2\rightarrow J_1\rightarrow J_3\rightarrow J_6\rightarrow J_4\rightarrow J_5\} $, and $ C_{\max }(4) = 50.2634 $.

    For $ k = 5 $, from Eqs (11) and (12), we have $ \Delta_{1} = 2.1845 $, $ \Delta_{2} = 2.2701 $, $ \Delta_{3} = 2.1214 $, $ \Delta_{4} = 1.8842 $, $ \Delta_{5} = 1.6207 $, $ \Delta_{6} = 1 $, and $ E = 4.6621 $. The local optimal job sequence is $ \varphi = \{J_2\rightarrow J_1\rightarrow J_3\rightarrow J_6\rightarrow J_4\rightarrow J_5\} $, and $ C_{\max }(5) = 62.3724 $.

    For $ k = 6 $, from Eqs (15) and (16), we have $ \Delta_{1} = 2.7453 $, $ \Delta_{2} = 2.8530 $, $ \Delta_{3} = 2.6660 $, $ \Delta_{4} = 2.3680 $, $ \Delta_{5} = 2.0368 $, $ \Delta_{6} = 1.7118 $, and $ E = 5.1571 $. The local optimal job sequence is $ \varphi = \{J_2\rightarrow J_1\rightarrow J_3\rightarrow J_6\rightarrow J_4\rightarrow J_5\} $, and $ C_{\max }(6) = 86.3039 $.

    From above analysis, the optimal value is $ k = 2 $, the optimal job sequence is $ \varphi^* = \{J_4\rightarrow J_6\rightarrow J_2\rightarrow J_1\rightarrow J_3\rightarrow J_5\} $, and the optimal value is $ C_{\max }^* = 44.6886 $.

    We studied the single-machine problem with $ tpdde $ and $ ma $, where the objective function is to minimize the makespan. It is showed that the problem $ 1|ma, tpdde|C_{\max } $ can be solved in $ O({n}^4) $ time. Future research may focus on the problems with general time-and-position dependent deteriorating effects $ p_{[j]} = \left\{ (a[j]+b[j]s[j])jc,if jk,(θ[j]a[j]+b[j](s[j]C[k]D))(jk)c,if j>k.

    \right. $ Another possible challenging is to consider the problems under parallel machines. The model assumptions can also be extended to for several special cases of processing set restrictions like as Scenario-Dependent Component Processing Times or Release Dates (see Wu et al. [31]; Wu et al. [32]; Wu et al. [33]).

    This research was supported by the National Natural Science Regional Foundation of China (71861031 and 72061029).

    The authors declare that they have no conflicts of interest.


    Acknowledgments



    This research was partially supported by Fulbright Foundation–Greece, The American College of Greece, The Institute of Educational Policy, Greece, BiHeLab, Ionian University, Greece and Villanova University, USA. I would like to thank Dr Plerou for her support in EEG analysis, Professor Papalaskari and Mrs Giannopoulou for their comments.

    Conflict of interest



    The authors declare no conflicts of interest.

    [1] Giraffa LMM, Moraes MC, Uden L (2014) Teaching Object-Oriented Programming in First-Year Undergraduate Courses Supported by Virtual Classrooms. In Uden L. (Ed.), The 2nd International Workshop on Learning Technology for Education in Cloud (pp. 15–26). Springer Proceedings in Complexity.
    [2] Santos Á, Gomes A, Mendes AJ (2010) Integrating new technologies and existing tools to promote programming learning. Algorithms 3: 183–196. doi: 10.3390/a3020183
    [3] Price TW, Barnes T (2015) Comparing Textual and Block Interfaces in a Novice Programming Environment. In Proceedings of the eleventh annual International Conference on International Computing Education Research-ICER '15 (pp. 91–99). ACM.
    [4] Booth T, Stumpf S (2013) End-user experiences of visual and textual programming environments for Arduino. Lect Notes Comput Sc 7897L: 25–39.
    [5] Chao P (2016) Exploring students' computational practice, design and performance of problem-solving through a visual programming environment. Comput Educ 95: 202–215. doi: 10.1016/j.compedu.2016.01.010
    [6] Meltzoff AN, Kuhl PK, Movellan JR, et al. (2009) Foundations for a new science of learning. Science 325: 284–288. doi: 10.1126/science.1175626
    [7] Ansari D, Coch D, De Smedt B (2011) Connecting education and cognitive neuroscience: Where will the journey take us? Educ Philos Theory 43: 37–42. doi: 10.1111/j.1469-5812.2010.00705.x
    [8] Sigman M, Peña M, Goldin AP, et al. (2014) Neuroscience and education: Prime time to build the bridge. Nat Neurosci 17: 497–502. doi: 10.1038/nn.3672
    [9] Nouri A (2016) The basic principles of research in neuroeducation studies. Int J Cogn Res Sci Eng Educ 4: 59–66.
    [10] Zebende GF, Oliveira Filho FM, Cruz JAL (2017) Auto-correlation in the motor/imaginary human EEG signals: A vision about the FDFA fluctuations. PloS one 12: e0183121. doi: 10.1371/journal.pone.0183121
    [11] Grover S, Menlo RA, Menlo RA (2017) Measuring student learning in introductory block-based programming: Examining Misconceptions of Loops, Variables, and Boolean Logic. Proceedings of the 2017 ACM SIGCSE Technical Symposium on Computer Science Education, 267–272.
    [12] Barnes DJ, Fincher S, Thompson S (1997) Introductory problem solving in computer science. In Daughton G, Magee P, (Eds.), 5th Annual Conference on the Teaching of Computing (pp. 36–39). Newtownabbey, UK: HE Academy for Information and Computer Sciences.
    [13] Bayman P, Mayer RE (1983) A diagnosis of beginning programmers' misconcep-tions of BASIC programming statements. Communications ACM 26: 677–679. doi: 10.1145/358172.358408
    [14] Lui AK, Kwan R, Poon M, et al. (2004) Saving weak programming students: Applying constructivism in a first programming course. SIGCSE Bulletin 36: 72–76.
    [15] Robins A, Rountree J, Rountree N (2003) Learning and teaching programming: a review and discussion. Comput Sci Educ 13: 137–172. doi: 10.1076/csed.13.2.137.14200
    [16] McGettrick A, Boyle R, Ibbett R, et al. (2005), Grand challenges in computing: Education-a summary. Comput J 48: 42–48.
    [17] Pears A, Seidman S, Malmi L, et al. (2007) A survey of literature on the teaching of introductory programming. ACM SIGCSE Bulletin 39: 204–223. doi: 10.1145/1345375.1345441
    [18] Futschek G, Moschitz J (2011) Learning algorithmic thinking with tangible objects eases transition to computer programming. In International Conference on Informatics in Schools: Situation, Evolution, and Perspectives (pp. 155–164). Springer Berlin Heidelberg.
    [19] Georgouli K, Sgouropoulou C (2013) Collaborative Peer-Evaluation Learning Results in Higher Education Programming-based Courses. In ICBL2013 – International Conference on Interactive Computer aided Blended Learning (pp. 309–314).
    [20] Erwig M, Smeltzer K, Wang X (2016) What is a Visual Language? J Visual Lang Comput 38: 9–17.
    [21] Ainsworth SE (2006) DeFT: A conceptual framework for learning with multiple representations. Learn Instr 16: 183–198. doi: 10.1016/j.learninstruc.2006.03.001
    [22] Goldman SR (2003) Learning in complex domains: When and why do multiple representations help? Learn Instr 13: 239–244. doi: 10.1016/S0959-4752(02)00023-3
    [23] Blackwell AF, Whitley KN, Good J, et al. (2001) Cognitive factors in programming with diagrams. Artif Intell Rev 15: 95–114. doi: 10.1023/A:1006689708296
    [24] Zohreh STM (2015) On the Influence of Representation Type and Gender on Recognition Tasks of Program Comprehension (Doctoral dissertation, École Polytechnique de Montréal).
    [25] Basu S, Biswas G, Kinnebrew JS (2016) Using Multiple Representations to Simultaneously Learn Computational Thinking and Middle School Science. In Thirtieth AAAI Conference on Artificial Intelligence.
    [26] Hoc JM, Green TRG, Samurçay R, et al. (1990) Psychology of Programming, London: Academic Press.
    [27] Howard-Jones PA (2011) From brain scan to lesson plan. Psychologist 24: 110–113.
    [28] Mayer RE (2017) How can brain research inform academic learning and instruction? Educ Psychol Rev 29: 835–846. doi: 10.1007/s10648-016-9391-1
    [29] Torresan P (2013) On educational neuroscience. An interview with Paul Howard-Jones. Formazione Insegnamento XI(1): 43–49.
    [30] Dresler M, Sandberg A, Ohla K, et al. (2013) Non-pharmacological cognitive enhancement. Neuropharmacology 64: 529–543. doi: 10.1016/j.neuropharm.2012.07.002
    [31] Floyd B, Santander T, Weimer W (2017) Decoding the Representation of Code in the Brain: An fMRI Study of Code Review and Expertise. In IEEE/ACM 39th International Conference on Software Engineering, ICSE 2017, 175–186.
    [32] Siegmund J, Kästner C, Apel S, et al. (2014) Understanding understanding source code with functional magnetic resonance imaging. Proceedings of the 36th ACM/IEEE International Conference on Software Engineering, 378–389.
    [33] Crk I, Kluthe T, Stefik A (2015) Understanding Programming Expertise: An Empirical Study of Phasic Brain Wave Changes. ACM T Comput-Hum Int 23: 2.
    [34] Radevski S, Hata H, Matsumoto K (2015) Real-time monitoring of neural state in assessing and improving software developers' productivity. Proceedings-8th International Workshop on Cooperative and Human Aspects of Software Engineering, CHASE 2015, (May), 93–96.
    [35] Müller SC, Fritz T (2016) Using (bio)metrics to predict code quality online. Proceedings of the 38th International Conference on Software Engineering-ICSE'16, (December), 452–463.
    [36] Nakagawa T, Kamei Y, Uwano H, et al. (2014) Quantifying programmers' mental workload during program comprehension based on cerebral blood flow measurement: a controlled experiment. Companion Proceedings of the 36th International Conference on Software Engineering-ICSE Companion 2014, 448–451.
    [37] Parnin C (2011) Subvocalization-Toward Hearing the Inner Thoughts of Developers. In Proc. Int'l Conf. Program Comprehension (ICPC), 197–200.
    [38] Hansen ME, Lumsdaine A, Goldstone RL (2012) Cognitive Architectures: A Way Forward for the Psychology of Programming. Proceedings of the ACM International Symposium on New Ideas, New Paradigms, and Reflections on Programming and Software-Onward!'12, 27.
    [39] Yusuf S, Kagdi H, Maletic JI, et al. (2007), Assessing the Comprehension of UML Class Diagrams via Eye Tracking. In 15th IEEE International Conference on Program Comprehension (ICPC'07) (pp. 113–122).
    [40] Oliveira Filho FM, Cruz JL, Zebende GF (2019) Analysis of the EEG bio-signals during the reading task by DFA method. Physica A 525: 664–671. doi: 10.1016/j.physa.2019.04.035
    [41] Read GL, Innis IJ (2017) Electroencephalography (Eeg). The International Encyclopedia of Communication Research Methods, New Jersey: John Wiley & Sons, Inc., 1–18.
    [42] Larsen-Freeman D (1997) Chaos/complexity science and second language acquisition. Appl Linguist 18: 141–165. doi: 10.1093/applin/18.2.141
    [43] Tomlinson CA (1999) The differentiated classroom: Responding to the needs of all learners. Alexandria, VA: Association for Supervision and Curriculum Development.
    [44] Henz D, John A, Merz C, et al. (2018) Post-task effects on EEG brain activity differ for various differential learning and contextual interference protocols. Front Hum Neurosci 12: 19.
    [45] Lee S, Hooshyar D, Ji H, et al. (2018) Mining biometric data to predict programmer expertise and task difficulty. Cluster Comput 21: 1097–1107. doi: 10.1007/s10586-017-0746-2
    [46] Human Research and Engineering Directorate (2018) Predicting human behavior, Aberdeen Proving Ground. Available from: https://www.orau.org.
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