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\Comment{visited states for $\pi_i$ and actions by $\pi_\star$} \State $\Phi^{\text{DA}i} \gets \Phi^{\text{DA}i-1} \cup \Phi^{\text{IL}i}$ - \Comment{aggregate datasets, cf. \cref{eq:DAgger}} + \Comment{aggregate datasets from \cref{eq:Phi}} \State $\hat{\pi}_{i+1} \gets$ \Call{Train}{$\Phi^{\text{DA}i}$} \Comment{preference model from \cref{eq:jssp:linweights}} \EndFor diff --git a/JOSH/features-description.tex b/JOSH/features-description.tex index 2204477..01a19f6 100644 --- a/JOSH/features-description.tex +++ b/JOSH/features-description.tex @@ -5,10 +5,10 @@ $\vphi$ & Feature description & Mathematical formulation \\ %\hline \multicolumn{4}{c}{\textbf{Local features}} \\ \midrule - \multicolumn{3}{c}{\textbf{job related}}\\ + \multicolumn{3}{c}{\textbf{job-related}}\\ \phiproc & job processing time & $p_{ja}$ \\ - \phistartTime & job start-time & $x_s(j,a)$ \\ - \phiendTime & job end-time & $x_e(j,a)$ \\ + \phistartTime & job start time & $x_s(j,a)$ \\ + \phiendTime & job end time & $x_e(j,a)$ \\ \phiarrivalTime & job arrival time & $x_e(j,a-1)$\\ \phiwait & time job had to wait & $x_s(j,a)-x_e(j,a-1)$ \\ \phijobTotProcTime & total processing time for job & @@ -18,7 +18,7 @@ \phijobOps & number of assigned operations for job & $|\mathcal{M}_j|$ \\ \midrule - \multicolumn{3}{c}{\textbf{machine related}}\\ + \multicolumn{3}{c}{\textbf{machine-related}}\\ \phimacFree & when machine is next free & $\max_{j'\in \mathcal{J}_a} \{x_e(j',a)\}$ \\ \phimacTotProcTime & total processing time for machine & diff --git a/JOSH/hei2_manuscript.pdf b/JOSH/hei2_manuscript.pdf index 31923b81acf53f24c0e2035ce0565dc561439ea1..9e811c771d9005eaab8da42c2befb5a42a4b251a 100644 GIT binary patch delta 121391 zcmV)BK*PU~rV;Ix5s)^2H83(hJ_>Vma%Ev{3V7PRT1%7LxDmeRSLhb0GUAc|Ns*gM zRh*sOc$`Oh*OS_V?E^$Y5^Rc;`S5uDefq_RMB1AilG1@dqw#8Vf9P&#^=TEXeE0>c zom%bU)JxJ;c3yp2c{)v2PphpWk_tQ(tj?=2)}D%`TCCoze!gaZjnmaO^x_Cur3$Qm z&-$0g)ptJytJF(ly?Q(VOYB9fAn<}{^_ah1zuE-Ry6S5Dl&08hRhX`;1AHTY{U$rv 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z+MJIA#AR)1W+5v4|J#xIO&*pL_#bRFp1Cm>WjBH$$eqA``EgUpQH>y53E$DIS5q3A zVg-}TVQW=KJ6a`+G*!B(oH->Pmchoy=yYisUM%A1?jYUSfspwem(3Ob%YP4D0_W`e z|Gs6#C%{{OJRW>${iwQQCmgH85u14tG9L3j@5=Bpd(QfHVwkyNv&${tE8sPlAMa7>0*y!}qn(-_N^kb8y zpN-MrNQLxlKzlvN?y#3SPp>8)qIotd!30#{F8|DD22Uui$5+*n5!YdAB0XMhx%~oSRbaz%C?Hd+oP4DVO< ze+}wGPXrrnU9||7U%dAcpV2x6id~Bbugqj_1z*3$q->I?YvlmMa=7Uzexjx2?hLur zhc|nzg{yaNns0sIMP*P;g!9zquP3gYgu`tvNN&M0opCx4qRpP~-b(5^9V-BTlhy>1 q75)FgbUXd;UYLuyJ(7o~nY*XAyOlK(h#$l&z=uRjE2Aol^uGY0$ and $\sum_{i=0}^d w_i = 1$ with $w_i$ giving the weight of the -influence of $\pi_i$ (which could be a SDR or another CDR) to $\pi$. Note: +influence of $\pi_i$ (which could be an SDR or another CDR) to $\pi$. Note: each $\pi_i$ is a function of $J_j$'s features from the current sequence $\vchi$, where $\vchi^j$ implies that $J_j$ was the latest dispatch, i.e., the partial schedule given $\chi_k=J_j$. @@ -581,7 +568,7 @@ \section{Priority Dispatching Rules} \label{sec:DR} functions based on either job waiting time or machine idle time (similar to \phiwait\ and \phimacSlack\ in \cref{tbl:jssp:feat}), i.e., the CDRs are a combination of either one or both of these key features and then the SDRs. -However, there are no combinations of the basic SDRs explored, only those two +However, no combinations of the basic SDRs were explored, only those two features. Similarly, using priority rules to combine 12 existing DRs from the literature, \cite{Yu13} had 48 CDR combinations which yielded 48 different models @@ -589,15 +576,15 @@ \section{Priority Dispatching Rules} \label{sec:DR} It is intuitive to get a boost in performance by introducing new CDRs, since where one DR might be failing, another could be excelling, so combining them together should yield a better CDR. However, these approaches introduce fairly -ad-hoc solutions and there is no guarantee the optimal combination of -\dr s are found. +ad hoc solutions and there is no guarantee the optimal combination of +\dr s have been found. -The \cdr\ presented in \cref{eq:CDR} can be considered as a special case of a +The \cdr\ presented in \cref{eq:CDR} can be considered as a special case of the following general linear value function: \begin{equation}\quad\label{eq:jssp:linweights} \pi(\vchi^j)=\sum_{i=1}^d w_i \phi_i(\vchi^j). \end{equation} -when $\pi_i(\cdot)=\phi_i(\cdot)$, i.e., a composite function of the features +when $\pi_i=\phi_i$, i.e., a composite function of the features from \cref{tbl:jssp:feat}. Finally, the job to be dispatched, $J_{j^*}$, corresponds to the one with the highest value, namely: \begin{equation}\quad\label{eq:jstar} @@ -620,13 +607,13 @@ \section{Performance Analysis of Priority Dispatching Rules}\label{sec:learnOPT} followed and the probability of SDRs being optimal is inspected. This serves as an indicator of how hard it is to put our objective up as a machine learning problem. -However, the end-goal, which is minimising \namerho, must also take into -consideration because of its relationship to stepwise optimality is not fully +However, the end goal, which is minimising \namerho, must also be taken into +consideration because its relationship to stepwise optimality is not fully understood. In this \lcnamecref{sec:learnOPT} the concerns of learning new priority \dr s -will be addressed. At the same time experimental set-up used in the study are -described. +will be addressed. At the same the time experimental set-up used in the study +is described. %The matters that must be addressed are as follows %\begin{enumerate*}[itemjoin*={{, and finally }}] %\item the problem instances used for learning and their optimal solutions @@ -637,7 +624,7 @@ \section{Performance Analysis of Priority Dispatching Rules}\label{sec:learnOPT} \subsection{Problem Instances}\label{sec:data:sim} -The class of problem instances used in our studies is the job-shop scheduling +The class of problem instances used in our studies was the job-shop scheduling problem described in \cref{sec:problemdef}. Each instance will have different processing times and machine ordering. Each instance will therefore create different challenges for a priority dispatching rule. @@ -652,13 +639,13 @@ \subsection{Problem Instances}\label{sec:data:sim} jobs and machines to ten. A discrete processing time is sampled independently from a discrete uniform distribution from the interval $I=[u_1,u_2]$, i.e., $\vec{p}\sim \mathcal{U}(u_1,u_2)$. -The machine order is a random permutation of all of the machines in the +The machine order was a random permutation of all of the machines in the job-shop. Two different processing times distributions were explored, namely \jrnd{n}{m} where $I=[1,99]$ and \jrndn{n}{m} where $I=[45,55]$. These instances are referred to as random and random-narrow, respectively. In addition, the case where the machine order is fixed and the same for all jobs, i.e. $\sigma_j(a)=a$ for all $J_j\in\mathcal{J}$ and where -$\vec{p}\sim\mathcal{U}(1,99)$, is also considered. +$\vec{p}\sim\mathcal{U}(1,99)$, was also considered. These jobs are denoted by \frnd{n}{m} and are analogous to \jrnd{n}{m}. The problem spaces are summarised in \cref{tbl:data:sim}. @@ -677,13 +664,13 @@ \subsection{Problem Instances}\label{sec:data:sim} outcome, and is commonly referred to as a loss function, which should be minimised for policy $\pi$. -\Cref{fig:boxplot:SDR} depicts the box-plot for \cref{eq:rho} when using the +\Cref{fig:boxplot:SDR} depicts the box plot for \cref{eq:rho} when using the SDRs from \cref{sec:DR} for all of the problem spaces from \cref{tbl:data:sim}. -These box-plots show the difference in performance of the various SDRs. The +These box plots show the difference in performance of the various SDRs. The rule MWR performs on average the best on the \jrnd{n}{m} and \jrndn{n}{m} -problems instances, whereas for \frnd{n}{m} it is LWR that performs best. It is +problem instances, whereas for \frnd{n}{m} it is LWR that performs best. It is also interesting to observe that all but MWR perform statistically worse than a -random job dispatching on the \jrnd{n}{m} and \jrndn{n}{m} problems instances. +random job dispatching on the \jrnd{n}{m} and \jrndn{n}{m} problem instances. \begin{table}\centering \caption[Problem space distributions used in experimental studies.]{Problem @@ -696,7 +683,7 @@ \subsection{Problem Instances}\label{sec:data:sim} \begin{figure} \centering \includegraphics[width=\columnwidth]{{boxplotRho_SDR_10x10}.pdf} -\caption{Box-plot for deviation from optimality, $\rho$, (\%) for SDRs} +\caption{Box plot for deviation from optimality, $\rho$, (\%) for SDRs} \label{fig:boxplot:SDR} \end{figure} @@ -704,17 +691,20 @@ \subsection{Reconstructing optimal solutions}\label{sec:opt:sdr} When building a complete schedule, $K=n\cdot m$ dispatches must be made sequentially. A job is placed at the earliest available time slot for its next -machine, whilst still fulfilling that each machine can handle at most one job -at each time, and jobs need to have finished their previous machines according -to their machine order. Unfinished jobs from the job-list $\mathcal{L}$ are +machine, whilst still responding to the fact that each machine can handle at +most one job at each time, and jobs need to have been finished by their +previous machines according +to their machine order. Unfinished jobs from the job list $\mathcal{L}$ are dispatched one at a time according to a deterministic -scheduling policy (or heuristic). This process is given as a pseudo-code is -given in \cref{pseudo:constructJSP}. +scheduling policy (or heuristic). This process is given as a pseudo-code in +\cref{pseudo:constructJSP}. After each dispatch\footnote{Dispatch and time step are used interchangeably.} the schedule's current features are updated based on the half-finished schedule, $\vchi$. -For each possible post-decision state the temporal features are collected (cf. -\cref{pseudo:constructJSP:phi}) forming the feature set, $\Phi$, based on all +For each possible post-decision state, $J_j\in\mathcal{L}^{(k)}$, the temporal +features, $\vphi_j$, are collected (cf. +\cref{pseudo:constructJSP:phi}) forming the feature set, $\Phi$, based on the +union of all $N_{\text{train}}$ problem instances available, namely: \begin{equation}\quad \label{eq:Phi} \Phi := \bigcup_{\{\vec{x}_i\}_{i=1}^{N_{\text{train}}}} @@ -722,7 +712,7 @@ \subsection{Reconstructing optimal solutions}\label{sec:opt:sdr} \subset\mathcal{F} \end{equation} where the feature space $\mathcal{F}$ is described in \cref{tbl:jssp:feat}, and -are based on job- and machine-features which are widespread in practice. +are based on job and machine features which are widespread in practice. \input{constructJSP} @@ -731,15 +721,15 @@ \subsection{Reconstructing optimal solutions}\label{sec:opt:sdr} Inspecting a partial schedule further along in the dispatching process such as in \cref{fig:jssp:example}, then let's say $J_1$ would be dispatched next, and in the next iteration $J_2$. Now this sequence would yield the same schedule as -if $J_2$ would have been dispatched first and then $J_1$ in the next iteration, +if $J_2$ had been dispatched first and then $J_1$ in the next iteration, i.e., these are jobs with non-conflicting machines. In this particular scenario, one cannot infer that choosing $J_1$ is better and $J_2$ is worse (or vice versa) since they can both yield the same solution. Furthermore, there may be multiple optimal solutions to the same problem instance. Hence not only is the sequence representation `flawed' in the -sense that slight permutations on the sequence are in fact equivalent w.r.t. -the end-result, but very varying permutations on the dispatching sequence +sense that slight permutations on the sequence are in fact equivalent in terms +of the end-result, but very varying permutations on the dispatching sequence (although given the same partial initial sequence) can result in very different complete schedules yet can still achieve the same makespan. %Care must be taken in this case that neither resulting features are labelled @@ -749,25 +739,26 @@ \subsection{Reconstructing optimal solutions}\label{sec:opt:sdr} The redundancy in building optimal solutions using dispatching rules means that many different dispatches may yield an optimal solution to the problem instance. Let's formalise the probability of optimality (or stepwise -classification accuracy) for a given policy $\pi$, as: +classification accuracy) for a given policy $\pi$, yet following the expert +policy $\pi_\star$, as: \begin{equation}\quad \label{eq:tracc:opt} \xi^\star_{\pi} := \mathbb{E}_{\pi_\star}\left\{\pi_{\star} = \pi \right\} \end{equation} -that is to say the mean likelihood of our policy $\pi$ being equivalent to the +that is to say, the mean likelihood of our policy $\pi$ being equivalent to the expert policy $\pi_\star$. The probability that a job chosen by a SDR yields an optimal makespan on a step-by-step basis, i.e., $\xi^\star_{\langle \text{SDR} \rangle}$, is depicted in \cref{fig:opt:SDR:xistar}. These probabilities -vary quite a bit between the different problem instances distributions studied. +vary quite a bit between the different problem instance distributions studied. From \cref{fig:opt:SDR:xistar} it is observed that $\xi^\star_{\text{MWR}}$ has a higher probability than random guessing, in choosing a dispatch which may result in an optimal schedule. This is especially true towards the end of the schedule building process. -Similarly, $\xi^\star_{\text{LWR}}$ chooses dispatches +Similarly, $\xi^\star_{\text{LWR}}$ selects dispatches resulting in optimal schedules with a higher probability. This would appear to -be support the idea that the higher the probability of dispatching jobs that +support the idea that the higher the probability of dispatching job that may lead to an optimal schedule, the better the SDRs performance, as -illustrated by \cref{fig:boxplot:SDR}. However, there is a counter example, +illustrated by \cref{fig:boxplot:SDR}. However, there is a counter example; $\xi^\star_{\text{SPT}}$ has a higher probability than random dispatching of selecting a jobs that may lead to an optimal solution. Nevertheless, the random dispatching performs better than SPT on problem instances \jrnd{10}{10} and @@ -781,8 +772,8 @@ \subsection{Reconstructing optimal solutions}\label{sec:opt:sdr} \label{fig:opt:SDR:xistar} \end{figure} -Looking at \cref{fig:opt:SDR:xistar}, then \jrnd{10}{10} has a relatively high -probability ($70\%$ and above) of choosing an optimal job at random. +As shown in \cref{fig:opt:SDR:xistar}, \jrnd{10}{10} has a relatively high +probability ($70\%$ and above) of selecting an optimal job at random. However, it is imperative to keep making optimal decisions, because the consequences of making suboptimal dispatches are unknown. To demonstrate this \cref{fig:case} depicts mean worst and best case scenario @@ -797,38 +788,38 @@ \subsection{Reconstructing optimal solutions}\label{sec:opt:sdr} \forall C_{\max}^{\vchi^j} \gneq C_{\max}^{\pi_\star} \right\} \end{eqnarray} \end{subequations} -Note, that this is given that there is only made one non-optimal dispatch. +Note, that it is a given that there is only one non-optimal dispatch. Generally, there will be more, and then the compound effects of making -suboptimal decisions cumulate. +suboptimal decisions are cumulative. It is interesting to observe that for \jrnd{10}{10} and \jrndn{10}{10} making -suboptimal decisions later impacts on the resulting makespan more than doing a +suboptimal decisions later impacts on the resulting makespan more than making a mistake early. The opposite seems to be the case for \frnd{10}{10}. In this case it is imperative to make good decisions right from the start. This is due to the major structural differences between JSP and FSP, namely the -latter having a homogeneous machine ordering, constricting the solution -immensely. +latter having a homogeneous machine ordering and therefore constricting the +solution immensely. \begin{figure}[t] \centering \includegraphics[width=\columnwidth]{{stepwise_10x10_OPT_casescenario}.pdf} \caption{Mean deviation from optimality, $\rho$, (\%), for best and worst -case scenario of making one suboptimal dispatch (i.e. +case scenarios of making one suboptimal dispatch (i.e. $\zeta^{\star}_{\min}$ and $\zeta^{\star}_{\max}$), depicted as lower and -upper bound, respectively, for \jrnd{10}{10}, \jrndn{10}{10} and -\frnd{10}{10}. Moreover, mean suboptimal move is given as a dashed line.} +upper bounds, respectively, for \jrnd{10}{10}, \jrndn{10}{10} and +\frnd{10}{10}. The mean suboptimal move is given as a dashed line.} \label{fig:case} \end{figure} \subsection{Blended \dr s}\label{sec:opt:bdr} -A naive approach to create a simple blended \dr~(BDR) would be to +A naive approach to creating a simple blended \dr~(BDR) would be to switch between SDRs at a predetermined time. Observing again \cref{fig:opt:SDR:xistar}, a presumably good BDR for \jrnd{10}{10} would be to start with $\xi^\star_{\text{SPT}}$ and then switch over to $\xi^\star_{\text{MWR}}$ at around time step $k=40$, where the SDRs change places in outperforming one another. -A box-plot for $\rho$ for the BDR compared with MWR and SPT is depicted in +A box plot for $\rho$ for the BDR compared with MWR and SPT is depicted in \cref{fig:boxplot:BDR} and its main statistics are reported in \cref{tbl:BDR:stats}. This simple swap between SDRs does outperform the SPT heuristic, yet @@ -836,21 +827,21 @@ \subsection{Blended \dr s}\label{sec:opt:bdr} performance of MWR. A reason for this lack of performance of our proposed BDR is perhaps that by starting out with SPT in the beginning, it sets up the schedules in such a way -that it's quite greedy and only takes into consideration jobs with shortest +that it's quite greedy and only takes into consideration jobs with the shortest immediate processing times. Now, even though it is possible to find optimal schedules from this scenario, as \cref{fig:opt:SDR:xistar} shows, the inherent structure that's already taking place might make it hard to come across by -simple methods. Therefore, it's by no means guaranteed that by simply swapping -over to MWR will handle that situation which applying SPT has already created. -\Cref{fig:boxplot:BDR} does however show, that by applying MWR instead of SPT -in the latter stages, does help the schedule to be more compact w.r.t. SPT. +simple methods. Therefore, it is by no means guaranteed that by simply swapping +over to MWR will handle the situation that applying SPT has already created. +\Cref{fig:boxplot:BDR} does show, however, that by applying MWR instead of SPT +in the latter stages does help the schedule to be more compact in terms of SPT. However, the fact remains that the schedules have diverged too far from what MWR would have been able to achieve on its own. \begin{figure}[t] \centering \includegraphics[width=\columnwidth]{{boxplotRho_BDR_10x10}.pdf} -\caption{Box-plot for deviation from optimality, $\rho$, (\%) for BDR where +\caption{Box plot for deviation from optimality, $\rho$, (\%) for BDR where SPT is applied for the first 10\%, 15\%, 20\%, 30\% or 40\% of the dispatches, followed by MWR} \label{fig:boxplot:BDR} @@ -860,8 +851,8 @@ \subsection{Blended \dr s}\label{sec:opt:bdr} In \cref{fig:opt:SDR:xistar} the stepwise optimality was inspected, given that all committed dispatches were based on the optimal trajectory. As mistakes are bound to be made at some points, it is interesting to see how -the stepwise optimality evolves for its intended trajectory, thereby updating -\cref{eq:tracc:opt} to: +the stepwise optimality evolves for its intended trajectory, $\pi$, thereby +updating \cref{eq:tracc:opt} to: \begin{equation}\quad \label{eq:tracc:track} \xi_{\pi} := \mathbb{E}_{\pi}\left\{\pi_{\star} = \pi \right\} \end{equation} @@ -872,18 +863,18 @@ \subsection{Blended \dr s}\label{sec:opt:bdr} $k=15$, $\xi_{\text{MWR}}$ becomes a contender again. This could explain why our BDR switch at $k=40$ from \cref{fig:boxplot:BDR} was unsuccessful. However, changing to MWR at $k\leq20$ -is not statistically significant from MWR (boost in mean $\rho$ -is at most -0.5\%). +is not statistically significant from MWR (the boost in mean $\rho$ is at most +-0.5\%). But as pointed out for \cref{fig:case}, it's not so fatal to -make bad moves in the very first dispatches for \jrnd{10}{10}, hence little -gain with improved classification accuracy in that region. +make bad moves in the very first dispatches for \jrnd{10}{10}, hence there is +little gain with improved classification accuracy in that region. However, after $k>20$ then the BDR performance starts diverging from that of MWR. \begin{figure}[t] \centering \includegraphics[width=\columnwidth]{{trdat_prob_moveIsOptimal_10x10_SDR_xi}.pdf} -\caption{Log likelihood of SDR being optimal for \jrnd{10}{10}, when +\caption{Log likelihood of SDR being optimal for \jrnd{10}{10}\ when following its corresponding SDR trajectory, i.e., $\log\left(\xi_{\langle\text{SDR}\rangle}\right)$} \label{fig:opt:SDR:xi} @@ -891,11 +882,11 @@ \subsection{Blended \dr s}\label{sec:opt:bdr} \section{Preference Learning}\label{ch:expr:CDR} \Cref{sec:opt:bdr} demonstrated there is something to be gained by -trying out different combinations of DRs, however, it is non-trivial. In this -section one approach to learning how such combinations is presented. +trying out different combinations of DRs; however, it is non-trivial. This +section presents one approach to learning how such combinations can work. Learning models considered in this study are based on ordinal regression in -which the learning task is formulated as learning preferences. In the case of -scheduling, learning which operations are preferred to others. Ordinal +which the learning task is formulated as learning preferences, and in the case +of scheduling, learning which operations are preferred to others. Ordinal regression has been previously presented in \cite{Ru06:PPSN} and in \cite{InRu11a} for JSP, and given here for completeness. @@ -903,64 +894,67 @@ \section{Preference Learning}\label{ch:expr:CDR} a number of feature pairs are created. Let $\vphi^{o}\in\mathcal{F}$ denote the post-decision state when dispatching $J_o\in\mathcal{O}^{(k)}$ corresponds to an optimal schedule being built. -All post-decisions states corresponding to suboptimal dispatches, +All post-decision states corresponding to suboptimal dispatches, $J_s\in\mathcal{S}^{(k)}$, are denoted by $\vphi^{s}\in\mathcal{F}$. -Note, \mbox{$\mathcal{O}^{(k)}\cup\mathcal{S}^{(k)}=\mathcal{L}^{(k)}$}, and -\mbox{$\mathcal{O}^{(k)}\cap\mathcal{S}^{(k)}=\emptyset$}. The approach taken here is to verify analytically, at each time step, by fixing the current temporal schedule as an initial state, whether it is possible to somehow yield an optimal schedule by manipulating the remainder of the sequence. This also takes care of the scenario that having dispatched a job resulting in a different temporal makespan would have resulted in the same -final makespan if another optimal dispatching sequence would have been chosen. -That is to say the training data generation takes into consideration when there -are multiple optimal solutions\footnote{ +final makespan if another optimal dispatching sequence had been chosen. +That is to say, the training data generation takes into consideration when +there are multiple optimal solutions\footnote{ There can be several optimal solutions available for each problem instance. However, it is deemed sufficient to inspect only one optimal trajectory per -problem instance as there are $N_{\text{train}}=300$ independent instances +problem instance as there are $N_{\text{train}}=300$ independent instances, which gives the training data variety.} to the same problem instance. -Let's label features from \cref{eq:Phi} that were considered optimal, -\mbox{$\vpsi^{o}=\vphi^{o}-\vphi^{s}$}, and suboptimal, -\mbox{$\vpsi^{s}=\vphi^{s}-\vphi^{o}$} by $y_o=+1$ and $y_s=-1$ respectively. +Let's compare features from \cref{eq:Phi}, and define the difference of optimal +from suboptimal ones as, +\mbox{$\vpsi^{o}=\vphi^{o}-\vphi^{s}$}, and vice versa, +\mbox{$\vpsi^{s}=\vphi^{s}-\vphi^{o}$}, and label the differences by $y_o=+1$ +and $y_s=-1$ respectively. Then, the preference learning problem is specified by a set of preference pairs: -\begin{eqnarray} -\quad \Psi &=& -\left\{\left(\vpsi^o,+1\right),\left(\vpsi^s,-1\right) -\;:\; +\begin{equation} +\begin{split} +\quad \Psi & = \left\{\left(\vpsi^o,y_o\right),\left(\vpsi^s,y_s\right) +\;:\; \forall \left(J_o,J_s\right) \in \mathcal{O}^{(k)} \times -\mathcal{S}^{(k)}\right\}_{k=1}^{K} \nonumber -\\ &\subset& \Phi\times Y \label{eq:prefset} -\end{eqnarray} +\mathcal{S}^{(k)}\right\}_{k=1}^{K} \\ +& \subset \Phi\times Y +\end{split} \label{eq:prefset} +\end{equation} where $\Phi\subset \mathbb{R}^d$ is the training set of $d=\NrFeatLocal$ features (cf. \cref{tbl:jssp:feat}), $Y=\{+1,-1\}$ is the outcome space from job pairs $J_o\in\mathcal{O}^{(k)}$ and $J_s\in\mathcal{S}^{(k)}$, for all dispatch steps $k$. -To summarise, each job is compared against another job of the job-list, -$\mathcal{L}^{(k)}$, and if the makespan differs (i.e -$C_{\max}^{\pi_\star(\vchi^s)} \gneq C_{\max}^{\pi_\star(\vchi^o)}$) an +To summarise, each job is compared against another job of the job list, +$\mathcal{L}^{(k)}$, and if the makespan differs, i.e., +$C_{\max}^{\pi_\star(\vchi^s)} \gneq C_{\max}^{\pi_\star(\vchi^o)}$ an optimal/suboptimal pair is created. However, if the makespans are identical the pair is omitted since they give the same optimal makespan. -This way, only features from a dispatch resulting in a suboptimal solution is -labelled undesirable. +In this way, only features from a dispatch resulting in a suboptimal solution +is labelled undesirable. -Now let's consider the model space $\mathcal{H} = \{\pi(\cdot) : X \mapsto Y\}$ -of mappings from solutions to ranks. Each such +Now let's consider mappings from solutions to ranks, such function $\pi$ induces an ordering on the solutions by the following rule: -\begin{equation}\quad\label{eq:linear} -\vchi^i \succ \vchi^j \quad \Leftrightarrow \quad \pi(\vchi^i) > +\begin{equation}\quad +\vchi^i \succ \vchi^j \quad \iff \quad \pi(\vchi^i) > \pi(\vchi^j) \end{equation} -where the symbol $\succ$ denotes ``is preferred to.'' The function used to -induce the preference is defined by a linear function in the feature space: -\begin{equation}\quad -\pi(\vchi^j)=\sum_{i=1}^d w_i\phi_i(\vchi^j)=\inner{\vec{w}}{\vphi(\vchi^j)}. +where the symbol $\succ$ denotes \emph{is preferred to} with respect to the +solutions' ranks. Referring to the ranks in a discrete manner as follows: +\begin{equation}\quad\label{eq:rankorder} +r_1 \succ r_2 \succ \cdots \succ r_{n'} \quad (n' \leq n) \end{equation} +implies $r_1$ is preferable to $r_2$, and $r_2$ is preferable to $r_3$, etc. +The function used to bring about preference is a linear function in the feature +space, previously defined in \cref{eq:jssp:linweights}. Logistic regression learns the optimal parameters $\vec{w}^*\in\mathbb{R}^d$. For this study, L2-regularised logistic regression from the \textsc{liblinear} package \cite{liblinear} without bias is used to learn the preference set @@ -968,31 +962,30 @@ \section{Preference Learning}\label{ch:expr:CDR} %For each job on the job-list, $J_j\in\mathcal{L}$, the post-decision state is %$\vphi^j:=\vphi(\vchi^j)$. Hence the job chosen to be dispatched, $J_{j^*}$, is the one corresponding to -the highest preference estimate, i.e., \cref{eq:jstar} where $\pi(\cdot)$ is +the highest preference estimate, i.e., \cref{eq:jstar} where $\pi$ is the classification model obtained by the preference set. Preliminary experiments for creating step-by-step model was done in -\cite{InRu11a} resulting in local linear model for each dispatch; a total of -$K$ linear models for solving $n\times m$ JSP. -However, the experiments there showed that by fixing the weights to its mean -value throughout the dispatching sequence results remained satisfactory. +\cite{InRu11a} resulting in a local linear model for each dispatch for a total +of $K$ linear models for solving $n\times m$ JSP. +However, the experiments showed that by fixing the weights to each mean +value throughout the dispatching sequence the results remained satisfactory. A more sophisticated way would be to create a new linear model, where the preference set, $\Psi$, is the union of the preference pairs across the -$K$ dispatches, such as described in \cref{eq:prefset}. +$K$ dispatches, as described in \cref{eq:prefset}. This would amount to a substantial preference set, and for $\Psi$ to be computationally feasible to learn, $\Psi$ has to be reduced. For this several -ranking strategies were explored in \cite{InRu15a}, the results there showed -that it's sufficient to use partial subsequent rankings, namely, combinations -of $r_i$ and $r_{i+1}$ for $i\in\{1,\ldots,n'\}$, are added to the preference -set, where $r_1>r_2>\ldots>r_{n'}$ ($n'\leq n$) are the rankings of the -job-list, in such a manner that in the cases that there are more than one -operation with the same ranking, only one from that rank is needed to be +ranking strategies were explored in \cite{InRu15a}, and the results showed +that it is sufficient to use partial subsequent rankings with combinations +of $r_i$ and $r_{i+1}$ for $i\in\{1,\ldots,n'\}$, added to the preference +set in such a manner that there is more than one +operation with the same ranking and only one from that rank is needed to be compared to the subsequent rank. -Moreover, for this study, which deals with $10\times 10$ problem instances -instead of $6\times5$, the partial subsequent ranking becomes necessary, as -full ranking is computationally infeasible due to its size. +Moreover, for this study, which dealt with $10\times 10$ problem instances +instead of $6\times5$, the partial subsequent ranking became necessary, as +full ranking was computationally infeasible due to its size. Defining the size of the preference set as $l=\abs{\Psi}$, then if $l$ is too -large re-sampling to size $l_{\max}$ may be needed to be done in order for the +large, re-sampling to size $l_{\max}$ may be needed in order for the ordinal regression to be computationally feasible. %\subsection{Collecting training data} @@ -1000,13 +993,13 @@ \section{Preference Learning}\label{ch:expr:CDR} randomly generated problem instances, i.e., traditional \emph{passive imitation learning} (PIL). As JSP is a sequential decision making process, errors are bound to emerge. -Due to compound effect of making suboptimal dispatches, the model leads the +Due to the compound effect of making suboptimal dispatches, the model leads the schedule astray from learned feature-space, resulting in the new input being foreign to the learned model. Alternatively, training data could be generated using suboptimal solution trajectories as well, as was done in \cite{InRu15a}, where the training data -also incorporated following the trajectories obtained by applying successful -SDRs from the literature. +was also incorporated following the trajectories obtained by applying +successful SDRs from the literature. The reasoning behind it was that they would be beneficial for learning, as they might help the model to escape from local minima once off the coveted optimal path. @@ -1017,10 +1010,9 @@ \section{Preference Learning}\label{ch:expr:CDR} training data will now be such that it will iteratively improve upon the model, such that the feature-space learned will be representative of the feature-space the eventual model would likely encounter, known as DAgger for \emph{active -imitation learning} (AIL). -Thereby, eliminating the ad-hoc nature of choosing trajectories to learn, by -rather letting the model lead its own way in a self-perpetuating manner until -it converges. +imitation learning} (AIL) and thereby, eliminating the ad hoc nature of +choosing trajectories to learn, by instead letting the model lead its own way +in a self-perpetuating manner until it converges. Furthermore, in order to boost training accuracy, two strategies were explored \begin{enumerate}[after={{}}, leftmargin=*, @@ -1038,31 +1030,33 @@ \section{Preference Learning}\label{ch:expr:CDR} \mbox{$K$-stepped} trajectory using a fixed policy $\pi$ for the default $N_{\text{train}}=300$. -Another way to adjust training accuracy is to give different weight to various +Another way to adjust training accuracy is to give different weights to various time steps. To address this problem, two different stepwise sampling biases (or -data balancing techniques) will be considered +data balancing techniques) are here considered \begin{enumerate}[after={{}}, leftmargin=*, label={\textbf{Bias.\arabic*}}, ref={{Bias.\arabic*}}] \item \label{bias:equal} \textbf{(equal)} where each time step has equal -probability, this was used in \cite{InRu14,InRu15a} and serves as a baseline. +probability. This was used in \cite{InRu14,InRu15a} and serves as a baseline. \item \label{bias:adjdbl2nd} \textbf{(adjdbl2nd)} where each time step is adjusted to the number of preference pairs for that particular step (i.e. each step now has equal probability irrespective of quantity of encountered features). This is done with re-sampling. In addition, there is superimposed twice as much likelihood of -choosing pairs from the latter half of the dispatching process. Then the -final sampled data set is divided as follows: -$\abs{\{\Psi(k)\}_{k=0}^{\frac{K}{2}-1}} \approx \frac{1}{3}l_{\max}$ and -$\abs{\{\Psi(k)\}_{k=\frac{K}{2}}^{K-1}} \approx \frac{2}{3}l_{\max}$. +choosing pairs from the latter half of the dispatching process. \end{enumerate} \noindent Remark: as the following \lcnamecref{sec:il:passive}s require -repeated collection of training data, and since its labelling is a very -time intensive task the remainder of the paper will solely be focusing +repeated collection of training data and since labelling is a very +time intensive task, the remainder of the paper will solely focus on \jrnd{10}{10}. +The experiments in \cite{InRu15a} generally took several hours to collect for +$N_{\text{train}}^{6\times5}=500$. +Going to higher dimension and using $N_{\text{train}}^{10\times10}=300$ +computational intensity becomes an issue, as \jrnd{10}{10} needs a few days and +\jrndn{10}{10} or \frnd{10}{10} require several weeks using a personal computer. \section{Passive Imitation Learning}\label{sec:il:passive} -Using the terms from game-theory used in \cite{CesaBianchi06}, % chapter 2: +Using the terms from game theory used in \cite{CesaBianchi06}, % chapter 2: then our problem is a basic version of the sequential prediction problem where the predictor (or forecaster), $\pi$, observes each element of a sequence $\vchi$ of jobs, where at each time step $k \in \{1,...,K\}$, before the @@ -1070,27 +1064,23 @@ \section{Passive Imitation Learning}\label{sec:il:passive} $\chi_k$ on the basis of the previous $k-1$ observations. \subsection{Prediction with Expert Advice}\label{sec:expertPolicy} -Let us assume one knows the expert policy $\pi^\star$, which can query what +Let us assume the expert policy $\pi^\star$ is known, which can query what is the optimal choice of $\chi_k={j^*}$ at any given time step $k$. Now let's use \cref{eq:jstar} to back-propagate the relationship between post-decision states and $\hat{\pi}$ with preference learning via our collected -feature set, denoted $\Phi^\text{OPT}$, i.e., collecting the features set -corresponding following optimal tasks $J_{j^*}$ from $\pi^\star$ in +feature set, denoted $\Phi^\text{OPT}$, i.e., collecting the feature set +corresponding to the following optimal tasks $J_{j^*}$ from $\pi^\star$ in \cref{pseudo:constructJSP}. This baseline sampling trajectory originally introduced in \cite{InRu11a} for adding features to the feature set is a pure strategy where at each dispatch an optimal task is dispatched. By querying the expert policy, $\pi_\star$, the ranking of the job-list, -$\mathcal{L}$, is determined such that: -\begin{equation}\quad -r_1 \succ r_2 \succ \cdots \succ r_{n'} \quad (n' \leq n) -\end{equation} -implies $r_1$ is preferable to $r_2$, and $r_2$ is preferable to $r_3$, etc. -In this study, then it's known that $r \propto C_{\max}^{\pi_\star}$, hence the -optimal job-list is the following: +$\mathcal{L}$, satisfies \cref{eq:rankorder}. +In this study, then it's known that the best rank $r_1 \propto +C_{\max}^{\pi_\star}$, hence the optimal job-list is the following: \begin{equation}\quad -\mathcal{O}=\left\{r_i \;:\; r_i \propto \min_{J_j \in \mathcal{L}} +\mathcal{O}=\left\{J_j \;:\; r_1 \propto \min_{J_j \in \mathcal{L}} C_{\max}^{\pi_\star(\vchi^j)}\right\} \end{equation} found by solving the current partial schedule to optimality using a MIP solver. @@ -1106,7 +1096,7 @@ \subsection{Follow the Perturbed Leader}\label{sec:perturbedLeader} Its pseudo code is given in \cref{pseudo:perturbedLeader} and describes how the expert policy (i.e. optimal trajectory) from \cref{sec:expertPolicy} is subtly ``perturbed'' with $\epsilon=10\%$ likelihood, by choosing a job corresponding -to the second best $C_{\max}$ instead of a optimal one with some small +to the second best $C_{\max}$ instead of an optimal one with some small probability. \input{perturbedLeader} @@ -1119,29 +1109,30 @@ \subsection{Experimental study}\label{sec:pil:expr} following perturbed leader.}\label{fig:passive:boxplot} \end{figure} -Results for \cref{sec:expertPolicy,sec:perturbedLeader} using \jrnd{10}{10} -box-plot of \namerho, is given in \cref{fig:passive:boxplot} and main +Results for \cref{sec:expertPolicy,sec:perturbedLeader} using the \jrnd{10}{10} +box plot of \namerho, is given in \cref{fig:passive:boxplot} and the main statistics are reported in \cref{tbl:IL:stats}. To address \ref{expr:boost:newdata}, the extended training set was simply obtained by iterating over more examples, namely $N^{\text{OPT}}_{\text{train, EXT}}=1000$. However, one can see that the increased number of varied features -dissuades the preference models to achieving a good performance w.r.t. $\rho$. -It's preferable to use the default $N^{\text{OPT}}_{\text{train}}=300$ and -allowing slight perturbations of the optimal trajectory, as done for +dissuades the preference models from achieving a good performance in terms of +$\rho$. +It's preferable to use the default $N^{\text{OPT}}_{\text{train}}=300$ +allowing slight perturbations of the optimal trajectory, as was done for $\Phi^{\text{OPT}\epsilon}$. Unfortunately, all this overhead has not managed to surpass MWR in performance, except for $\Phi^{\text{OPT}\epsilon}$ using \ref{bias:adjdbl2nd} with a $\Delta\rho\approx-4.24\%$ boost in mean performance. Otherwise, for \ref{bias:equal}, there is a loss of $\Delta\rho\approx+6.23\%$ in mean performance. This is likely due to the fact that if equal probability is used for stepwise -sampling, then there are hardly any emphasis given to the final dispatches as -there a relatively few (compared to previous steps) preference pairs belonging -to those final stages. +sampling, then there is hardly any emphasis given to the final dispatches as +there are relatively few (compared to previous steps) preference pairs +belonging to those final stages. Revisiting \cref{fig:case}, then the band for -$\{\zeta^{\star}_{\min},\zeta^{\star}_{\max}\}$ is quite tight, as the problem +$\{\zeta^{\star}_{\min},\zeta^{\star}_{\max}\}$ is quite tight as the problem is immensely constricted and few operations to choose from. However, the empirical evidence from using \ref{bias:adjdbl2nd} shows that it is -imperative to make right decisions at the very end. +imperative to make the right decisions at the very end. Based on the results from \cite{InRu11a} the expert policy is a promising starting point. @@ -1160,9 +1151,9 @@ \subsection{Experimental study}\label{sec:pil:expr} learning approach, and ignoring this interaction leads to poor performance. In fact, \cite{RossB10} proves that assuming the preference model has a training error of $\epsilon$, then the total compound error (for all $K$ -dispatches) the classifier induces itself grows quadratically, $\bigOh{\epsilon -K^2}$, for the entire schedule, rather than having linear loss, -$\bigOh{\epsilon K}$, if it were i.i.d. +dispatches), the classifier induces itself and grows quadratically, +$\bigOh{\epsilon K^2}$, for the entire schedule, rather than having linear +loss, $\bigOh{\epsilon K}$, as if it were i.i.d. \section{Active Imitation Learning}\label{sec:il:active} @@ -1173,16 +1164,16 @@ \section{Active Imitation Learning}\label{sec:il:active} This required some trial-and-error as the experiments showed that features obtained by SDR trajectories were not equally useful for learning. -To automate this process, inspiration from AIL presented in \cite{RossGB11} is +To automate this process, inspiration from AIL presented in \cite{RossGB11} was sought, called \emph{Dataset Aggregation} (DAgger) method, which addresses a no-regret algorithm in an on-line learning setting. The novel meta-algorithm for IL learns a deterministic policy guaranteed to perform well under its induced distribution of states. The method is closely related to Follow-the-leader (cf. -\cref{sec:il:passive}), however, with a more sophisticated leverage to the +\cref{sec:il:passive}), only with a more sophisticated leverage to the expert policy. -In short, it entails the model $\pi_i$ that queries an expert policy (same as -in \cref{sec:expertPolicy}), $\pi_\star$, it's trying to mimic, +In short, it entails the model $\pi_i$ that queries an expert policy as +in \cref{sec:expertPolicy}, $\pi_\star$, that it's trying to mimic, but also ensuring the learned model updates itself in an iterative fashion, until it converges. The benefit of this approach is that the feature-states that are likely to @@ -1191,11 +1182,11 @@ \section{Active Imitation Learning}\label{sec:il:active} desired action at individual post-decision states which are both based on past queries, and the learner's interaction with the current environment. -DAgger has been proven successful on a variety of benchmarks +DAgger has been proven successful in a variety of benchmarks \cite{RossGB11,Ross13}, such as the video games Super Tux Kart and Super Mario Bros., handwriting recognition and autonomous navigation for large -unmanned aerial vehicles. In all cases greatly improving traditional supervised -IL approaches. +unmanned aerial vehicles. In all cases it greatly improved on traditional +supervised IL approaches. \subsection{DAgger} The policy of AIL at iteration $i>0$ is a mixed strategy given as follows: @@ -1203,7 +1194,7 @@ \subsection{DAgger} \pi_i = \beta_i\pi_\star + (1-\beta_i)\hat{\pi}_{i-1} % i: ith iteration of imitation learning % pi_star is expert policy (i.e. optimal) -% pi_i^hat: is prfef model from prev. iteration +% pi_i^hat: is pref model from prev. iteration \end{equation} where $\pi_\star$ is the expert policy and $\hat{\pi}_{i-1}$ is the learned model from the previous iteration. @@ -1222,51 +1213,43 @@ \subsection{DAgger} \Cref{pseudo:activeIL} explains the pseudo code for how to collect partial training set, $\Phi^{\text{IL}i}$ for $i$-th iteration of AIL. Subsequently, the resulting preference model, $\hat{\pi}_i$, learns on the -aggregated datasets from all previous iterations, namely: -\begin{equation}\quad\label{eq:DAgger} -\Phi^{\text{DA}i}=\bigcup_{i'=0}^{i}\Phi^{\text{IL}i'} -\end{equation} -and its update procedure is detailed in \cref{pseudo:DAgger}. +aggregated datasets from all previous iterations, its update procedure is +detailed in \cref{pseudo:DAgger}. \input{imitationLearning} \input{DAgger} \subsection{Results}\label{sec:ail:expr} -Due to time constraints, only $T=3$ iterations will be inspected. +Due to time constraints, only $T=3$ iterations will be inspected here. In addition, preliminary experiments using DAgger for JSP favoured a simple parameter-free version of $\beta_i$ in \cref{eq:il}. -Namely, the mixed strategy for $\{\beta_i\}_{i=0}^T$ is unsupervised +Namely, the mixed strategy for $\{\beta_i\}_{i=0}^T$ was unsupervised with $\beta_i=I(i=0)$, where $I$ is the indicator function.\footnote{$\beta_0=1$ and $\beta_i=0,\forall i>0$.} Regarding \ref{expr:boost:newdata} strategy, \cref{sec:il:passive} showed that adding new problem instances did not boost performance for the expert policy (which is equivalent for the initial iteration of DAgger). -Hence, for active IL, the extended set is now consists of each iteration -encountering $N_{\text{train}}$ new problem instances. For a grand total of: -\begin{equation}\quad -N^{\text{DA}i}_{\text{train, EXT}}=N_{\text{train}}\cdot (i+1) -\end{equation} -problem instances explored for the aggregated extended training set used for -the learning model at iteration $i$. +Hence, for active IL, the extended set now consists of each iteration +encountering $N_{\text{train}}$ new problem instances. This way, the extended training data is used sparingly, as labelling for each -problem instances is computationally intensive. As a result, the computational -budget for DAgger is same regardless whether there are new problem instances -used or not, i.e., +problem instance is computationally intensive. As a result, the computational +budget for DAgger is the same regardless of whether or not new problem +instances are used, i.e., $\abs{\Phi^{\text{DA}i}}\approx\abs{\Phi^{\text{DA}i}_{\text{EXT}}}$. -Results for \jrnd{10}{10} box-plot of \namerho, is given in -\cref{fig:active:boxplot} and main statistics are reported in -\cref{tbl:IL:stats}. As one can see, DAgger is not fruitful when the same +The results for \jrnd{10}{10} box plot of \namerho, are given in +\cref{fig:active:boxplot} and the main statistics are reported in +\cref{tbl:IL:stats}. As can be seen, DAgger is not fruitful when the same problem instances are continually used. This is due to the fact that there is not enough variance between $\Phi^{\text{IL}i}$ and $\Phi^{\text{IL}(i-1)}$, hence the aggregated feature set $\Phi^{\text{DA}i}$ is only slightly perturbed -with each iterations. Which from \cref{sec:pil:expr} showed it was not a -very successful modification for the expert policy. Although, it's noted that +with each iteration. \Cref{sec:pil:expr} has showed this was not a +very successful modification for the expert policy; even though it's noted that by introducing suboptimal feature-space the preference model is not as drastically bad as the extended optimal policy, even though $\abs{\Phi^{\text{DA}i}}\approx\abs{\Phi^{\text{OPT}}_{\text{EXT}}}$. -However, when using new problem instances at each iterations, the feature set +However, when using new problem instances at each iteration, the feature set becomes varied enough that situations arise that can be learned to achieve a better represented classification problem which yields a lower mean \namerho. @@ -1278,20 +1261,20 @@ \subsection{Results}\label{sec:ail:expr} \section{Summary of Imitation Learning}\label{sec:il:expr} -A summary of \jrnd{10}{10} best PIL and AIL models w.r.t. \namerho, from +A summary of \jrnd{10}{10} best PIL and AIL models in terms of \namerho, from \cref{sec:pil:expr,sec:ail:expr}, respectively, are illustrated in -\cref{fig:all:boxplot}, and main statistics are given in +\cref{fig:all:boxplot}, and the main statistics are given in \cref{tbl:IL:stats}. To summarise, the following trajectories were used \begin{enumerate*} \item expert policy, trained on $\Phi^{\text{OPT}}$ \item perturbed leader, trained on $\Phi^{\text{OPT}\epsilon}$ \item imitation learning, trained on $\Phi^{\text{DA}i}_{\text{EXT}}$ for -iterations $i=\{1,\ldots,3\}$ using extended training set +iterations $i=\{1,\ldots,3\}$ using an extended training set \end{enumerate*} As a reference, the \sdr\ MWR is shown at the edges of \cref{fig:all:boxplot}. -At first one can see that the perturbed leader ever so-slightly improves the +At first one can see that the perturbed leader ever-so-slightly improves the mean for $\rho$, rather than using the baseline expert policy. However, AIL is by far the best improvement. With each iteration of DAgger, the models improve upon the previous iteration @@ -1313,23 +1296,23 @@ \section{Summary of Imitation Learning}\label{sec:il:expr} \item after $i=1$ iteration by $\Delta\rho\approx-9.31\%$ for \ref{bias:adjdbl2nd} with \ref{expr:boost:newdata} \end{enumerate*} -Note, for \ref{bias:equal} without \ref{expr:boost:newdata}, then DAgger is +Note, for \ref{bias:equal} without \ref{expr:boost:newdata}, DAgger is unsuccessful, and the aggregated data set downgrades the performance of the previous iterations, making it best to learn solely on the initial expert policy for that model configuration. -Regarding \ref{expr:boost:newdata}, then it's not successful for the expert -policy, as $\rho$ increased approximately 10\%. This could most likely be -counter-acted by increasing $l_{\max}$ to reflect the 700 additional examples. -What is interesting though, is that \ref{expr:boost:newdata} is well suited for -AIL, using the same $l_{\max}$ as before. -Note, the amount of problems used for $N^{\text{OPT}}_{\text{train, EXT}}$ is -equivalent to $T=2\tfrac{1}{3}$ iterations of extended DAgger. -The new varied data gives the aggregated feature set more information -of what is important to learn in subsequent iterations, as those new +Regarding \ref{expr:boost:newdata}, it is not then successful for the expert +policy as $\rho$ increased approximately 10\%. This could most likely be +counteracted by increasing $l_{\max}$ to reflect the 700 additional examples. +What is interesting, though, is that \ref{expr:boost:newdata} is well suited +for AIL, using the same $l_{\max}$ as before. +Note, the number of problems used for $N^{\text{OPT}}_{\text{train,EXT}}$ is +equivalent to $T=2\tfrac{1}{3}$ iterations of extended DAgger. +The new varied data give the aggregated feature set more information +on what is important to learn in subsequent iterations, as those new feature-states are more likely to be encountered `in practice.' Not only does the AIL converge faster, it also consistently improves with each -iterations. +iteration. \begin{figure} \includegraphics[width=\columnwidth]{{boxplot_summary_10x10}.pdf} @@ -1344,11 +1327,11 @@ \section{Discussion and conclusions}\label{sec:con} The \sdr s remain a popular approach to scheduling, as they are simple to implement and quite efficient. -Nevertheless, when they are successful and when they fail remains illusive. +Nevertheless, when they are successful and when they fail remains elusive. %investigation of how these \dr s actually work, and what %makes them so successful (or in some cases unsuccessful)? %For instance, of the four SDRs this study focuses on, why does MWR outperform -%so significantly for job-shop, yet completely fail for flow-shop? +%so significantly for job-shop, yet completely fail for \fsp? %MWR seems to be able to adapt to varying distributions of processing times, %however, manipulating the machine ordering causes MWR to break down. By inspecting optimal schedules, and investigating the probability that an optimal dispatch could be chosen by chance, and by looking at the impact of choosing sub-optimal dispatches, some light is shed on how SDRs vary in performance. Furthermore, the problem instance space was varied, giving an even better understanding of the behaviour of the SDRs. @@ -1361,8 +1344,8 @@ \section{Discussion and conclusions}\label{sec:con} When training the learning model, there is a trade-off between making the over-all best decisions (in terms of highest mean validation accuracy) versus making the right decision on crucial time points in the scheduling process, as -\cref{fig:case} clearly illustrated. -Moreover, before training the learned model, the preference set $\Psi$ +\cref{fig:case} clearly illustrates. +Moreover, before training the learning model, the preference set $\Psi$ needs to be re-sampled to size $l_{\max}$. As the effects of making suboptimal choices varies as a function of time, the stepwise bias should rather take into account the disproportional amount of @@ -1374,18 +1357,19 @@ \section{Discussion and conclusions}\label{sec:con} However, there are many other stepwise sampling strategies based on our analysis that could have been chosen instead, as here only a simplification of the trend from \cref{fig:case} was chosen. -This also opens up the question of how should validation accuracy be measured? -Since the model is based on learning preferences, both based on optimal versus -suboptimal, and then varying degrees of sub-optimality. Since ranks are only -looked at in a black and white fashion, such that the makespans need to be -strictly greater to belong to a higher rank, then it can be argued that some -ranks should be grouped together if their makespans are sufficiently close. -This would simplify the training set, making it (presumably) have less +This also opens up the question of how validation accuracy should be measured, +taking into account that the model is based on learning preferences, both based +on optimal versus suboptimal, and then varying degrees of sub-optimality. +Since ranks are only looked at in a black and white fashion, such that the +makespans need to be strictly greater to belong to a higher rank, then it can +be argued that some ranks should be grouped together if their makespans are +sufficiently close. +This would simplify the training set, making it (presumably) have fewer contradictions and be more appropriate for linear learning. Or simply the -validation accuracy could be weighted w.r.t. the difference in makespan. +validation accuracy could be weighted in terms of the difference in makespan. During the dispatching process, there are some significant time points which -need to be especially taken care off. -\Cref{fig:case} showed how making suboptimal decisions is especially critical +need to be especially taken care of. +\Cref{fig:case} shows how making suboptimal decisions is especially critical during the later stages for job-shop, whereas for flow-shop the earlier stages of dispatches are more critical. @@ -1395,15 +1379,15 @@ \section{Discussion and conclusions}\label{sec:con} When it does, the model is in uncharted territory as there is no certainty the samples already collected are able to explain the current situation. For this we propose investigating partial schedules from suboptimal -trajectories as well, since the future observations depend on previous +trajectories as well, since future observations depend on previous predictions. A straight forward approach would be to inspect the trajectories of promising SDRs or CDRs. However, more information is gained when applying AIL inspired by -work of \cite{RossB10,RossGB11}, such that the +the work of \cite{RossB10,RossGB11}, such that the learned policy following an optimal trajectory is used to collect training data, and the learned model is iteratively updated. -This can be done over several iterations, with the benefit being, that the -scheduling features that are likely to occur in practice are investigated, and +This can be done over several iterations, with the benefit being that the +scheduling features that are likely to occur in practice are investigated and as such used to dissuade the model from making poor choices in the future. %Alas, this comes at great computational cost due to the substantial amounts of %feature-states that need to be optimised for their correct labelling. Making @@ -1453,15 +1437,15 @@ \section{Discussion and conclusions}\label{sec:con} policy instead (i.e. $\pi_\star$ in \cref{eq:il} is suboptimal) \cite{ChangKADL15}. -This study has been structured around the job-shop scheduling problem, however, +This study has been structured around the job-shop scheduling problem. However, it can be easily extended to other types of deterministic optimisation problems that involve sequential decision making. -The framework presented here collects snap-shots of the partial schedules by +The framework presented here collects snapshots of the partial schedules by following an optimal trajectory, and verifying the resulting optimal solution from each possible state. -From which the stepwise optimality of individual features can be inspected, -and its inference could for instance justify omittance in feature selection. -Moreover, by looking at the best and worst case scenario of suboptimal +From which the stepwise optimality of individual features can be inspected +and its inference could for instance justify omission in feature selection. +Moreover, by looking at the best and worst case scenarios of suboptimal dispatches, it is possible to pinpoint vulnerable times in the scheduling process. @@ -1699,5 +1683,4 @@ \section{Discussion and conclusions}\label{sec:con} \end{thebibliography} - \end{document} \ No newline at end of file