Creating Meaningful Training Data for Dicult Job Shop Scheduling Instances for Ordinal Regression

Transcription

1 Creating Meaningful Training Data for Dicult Job Shop Scheduling Instances for Ordinal Regression Helga Ingimundardóttir University of Iceland March 28 th, 2012

2 Outline Introduction Job Shop Scheduling Ordinal Regression Generating Training Data Future Work 2 of 36

3 Introduction Job Shop Scheduling Ordinal Regression Generating Training Data Future Work 3 of 36

4 Motivation General Goal General goal is how to search for good solutions for an arbitrary problem domain. Automate the design of optimization algorithms. Use of randomly sampled problem instances and their corresponding optimal vs. suboptimal solutions. 4 of 36

5 Case Study: Job Shop Scheduling Problem There are many heuristic methods available for JSSP, where dierent methodologies outperform other depending on the particular problem instance. Relationship between problem structure and heuristic performance for JSSP Finding out how the problem instances dier in structure. Determine when a particular heuristic solution is likely to fail and explore in further detail the causes of such failures. Preliminary research was done in Ingimundardottir and Runarsson [2011] for a simple data distribution. This talk will be on how to deal with harder instances. 5 of 36

6 Case Study: Job Shop Scheduling Problem There are many heuristic methods available for JSSP, where dierent methodologies outperform other depending on the particular problem instance. Relationship between problem structure and heuristic performance for JSSP Finding out how the problem instances dier in structure. Determine when a particular heuristic solution is likely to fail and explore in further detail the causes of such failures. Preliminary research was done in Ingimundardottir and Runarsson [2011] for a simple data distribution. This talk will be on how to deal with harder instances. 5 of 36

7 Previous Work Methods previously proposed for investigating between problem structure and heuristic eectiveness: Footprints in instance space [Smith-Miles and Lopes, 2010, Corne and Reynolds, 2011] or landmarking [Pfahringer and Bensusan, 2000] which give an indicator how an algorithm generalizes over the instance space. The focus has been on searching through a large set of algorithms and determine the most suitable for a given subset of the instance space. However, there is a lack of research based on a single algorithm and understanding how it works on the instance space in the hopes of being able to extrapolate where it excels in order to aid its failing aspects. 6 of 36

8 Previous Work Methods previously proposed for investigating between problem structure and heuristic eectiveness: Footprints in instance space [Smith-Miles and Lopes, 2010, Corne and Reynolds, 2011] or landmarking [Pfahringer and Bensusan, 2000] which give an indicator how an algorithm generalizes over the instance space. The focus has been on searching through a large set of algorithms and determine the most suitable for a given subset of the instance space. However, there is a lack of research based on a single algorithm and understanding how it works on the instance space in the hopes of being able to extrapolate where it excels in order to aid its failing aspects. 6 of 36

9 Previous Work Methods previously proposed for investigating between problem structure and heuristic eectiveness: Footprints in instance space [Smith-Miles and Lopes, 2010, Corne and Reynolds, 2011] or landmarking [Pfahringer and Bensusan, 2000] which give an indicator how an algorithm generalizes over the instance space. The focus has been on searching through a large set of algorithms and determine the most suitable for a given subset of the instance space. However, there is a lack of research based on a single algorithm and understanding how it works on the instance space in the hopes of being able to extrapolate where it excels in order to aid its failing aspects. 6 of 36

10 Previous Work Methods previously proposed for investigating between problem structure and heuristic eectiveness: Footprints in instance space [Smith-Miles and Lopes, 2010, Corne and Reynolds, 2011] or landmarking [Pfahringer and Bensusan, 2000] which give an indicator how an algorithm generalizes over the instance space. The focus has been on searching through a large set of algorithms and determine the most suitable for a given subset of the instance space. However, there is a lack of research based on a single algorithm and understanding how it works on the instance space in the hopes of being able to extrapolate where it excels in order to aid its failing aspects. 6 of 36

11 Introduction Job Shop Scheduling Ordinal Regression Generating Training Data Future Work 7 of 36

12 Job Shop Scheduling (1) JSSP A simple job shop scheduling problem is where n jobs are scheduled on a set of m machines, subject to constraints: each job must follow a predened machine order, that a machine can handle at most one job at a time. The objective is to schedule the jobs so as to minimize the maximum completion time, known as the makespan, µ. Job j has an indivisible operation time on machine a, p(j, a), which is assumed to be integral, where j {1,.., n} and a {1,.., m}. 8 of 36

13 Job Shop Scheduling (2) Job j has a specied processing order through the machines, it is a permutation vector, σ, of {1,.., m}, i.e. j can be processed on σ(j, a) only after it has completed on σ(j, a 1). Problem instance generation A total of l = 1, 500 random problem instances were generated by xing number of jobs n = 6 and machines m = 6 and: Sampling discrete processing times from the uniform distribution, p U(1, 200). Ingimundardottir and Runarsson [2011] used a uniform distribution U(50, 100) and U(1, 100) with robust results. Machine order σ is a random permutation of {1,..., m}. 9 of 36

14 Job Shop Scheduling (3) Dispatching rules (DR) for constructing JSSP Starts with an empty schedule and adds on one job at a time. When a machine is free the DR inspects the waiting/available jobs and selects the job with the highest priority. (see Panwalkar and Iskander [1977] for a survey of over 100 DR). Complete schedule consists of l = n m sequential dispatches. At each dispatch/step k features ϕ(k) for the temporal schedule are calculated. Performance of DR is compared with its optimal makespan, as a ratio: ϱ = µ DR µ opt. 10 of 36

15 Features for JSSP ϕ ϕ 1 ϕ 2 ϕ 3 ϕ 4 ϕ 5 ϕ 6 ϕ 7 ϕ 8 ϕ 9 ϕ 10 ϕ 11 ϕ 12 ϕ 13 ϕ 14 ϕ 15 ϕ 16 Feature description processing time for job on machine start-time end-time when machine is next free current makespan work remaining most work remaining slack time for this particular machine slack time for all machines slack time weighted w.r.t. number of operations already assigned time job had to wait size of slot created by assignment total processing time for job total processing time for all jobs mean processing time for all jobs range of processing times over all jobs Table: Feature space F for JSSP based on Ingimundardottir and Runarsson [2011] which successfully captured the essence of a JSSP data structure. 11 of 36

16 Job Shop Scheduling Example A schedule being built at step k = 17. The dashed boxes represent six dierent possible jobs that could be scheduled next using a DR. 12 of 36

17 Introduction Job Shop Scheduling Ordinal Regression Generating Training Data Future Work 13 of 36

18 Ordinal Regression (1) Preference learning problem Specied by a set of preference pairs: { S = {z o, +1)} l, k=1 {z s, 1)} l k=1 o O(k), s S (k)} Φ Y where the set of point/rank pairs are: Optimal decision: z o = ϕ (o) ϕ (s), ranked +1 Sub-optimal decision: z s = ϕ (s) ϕ (o), ranked 1 In Ingimundardottir and Runarsson [2011] the training set is created from known optimal sequences of dispatch. 14 of 36

19 Ordinal Regression (2) Logistic regression Mapping of points to ranks: {h( ) : Φ Y } where ϕ o ϕ s h(ϕ o ) > h(ϕ s ) The preference is dened by a linear function: h(ϕ) = d w i ϕ = w ϕ. i=1 Logistic regression learns the optimal parameters w. 15 of 36

20 Introduction Job Shop Scheduling Ordinal Regression Generating Training Data Future Work 16 of 36

21 Problem Structure & Heuristic Eciency Rice [1976] framework for algorithm selection problem Problem space or instance space P = {(p j, σ j )} l j=1 Feature space F, measurable properties of the instances in P (see table on slide 9) Algorithm space A = {h}, set of all algorithms under inspection, Performance space Y = {ϱ j } l j=1, the outcome for P using an algorithm from A Y : A F Y is the mapping for algorithm and feature space onto the performance space, namely the step-by-step scheduling. 17 of 36

22 Preliminary Experiments Example Performance w.r.t. ϱ for training and test data using dierent algorithms for six-job six-machine JSSP when p U(1, 200). 18 of 36

23 Generating JSSP training data Determine the order/sequence of jobs assigned, at the rst available time slot (to the left) When job is assigned, new state occurs and features are updated At each time step, a good/bad ordinal data pair is only created if nal makespan is dierent. Comparison w.r.t. optimal decisions The approach taken in Ingimundardottir and Runarsson [2011] was to compare schedules w.r.t. known optimal solutions. The approach taken now is to verify analytically, by xing the current temporal schedule as an initial state, whether it is possible to obtain an optimal makespan for that state. This takes into consideration when there are multiple optimal solutions to the same problem instance. 19 of 36

24 Generating JSSP training data Determine the order/sequence of jobs assigned, at the rst available time slot (to the left) When job is assigned, new state occurs and features are updated At each time step, a good/bad ordinal data pair is only created if nal makespan is dierent. Comparison w.r.t. optimal decisions The approach taken in Ingimundardottir and Runarsson [2011] was to compare schedules w.r.t. known optimal solutions. The approach taken now is to verify analytically, by xing the current temporal schedule as an initial state, whether it is possible to obtain an optimal makespan for that state. This takes into consideration when there are multiple optimal solutions to the same problem instance. 19 of 36

25 Job Shop Scheduling Game Tree Example Game Tree for JSSP for the rst two dispatches. 20 of 36

26 Job Shop Scheduling Game Tree Example In the top layer one can see all possible dispatches (dashed) for an empty schedule. 21 of 36

27 Job Shop Scheduling Game Tree Example In the middle layer one can see all possible dispatches given that one of the dispatches from the layer above has been executed (solid). 22 of 36

28 Job Shop Scheduling Game Tree Example In the bottom layer, task j = 3 on machine M 4 has been dispatched and all possible next dispatches from that time step. 23 of 36

29 Job Shop Scheduling Game Tree Example Game Tree for JSSP for the rst two dispatches. 24 of 36

30 Game Tree Main Characteristics for JSSP Root node denotes the initial (empty) schedule Leaf nodes denote the complete schedule Height of the tree is l = n m Worst case scenario a full n-ary tree Traversing from root to leaf gives the sequence of dispatches that yielded the resulting schedule Tree creation can be done recursively for all possible dispatches. Such an exhaustive search would yield at the most n l leaf nodes. Even for small dimensions of n and m, it is too computationally expensive to investigate them all. 25 of 36

31 Game Tree Main Characteristics for JSSP Root node denotes the initial (empty) schedule Leaf nodes denote the complete schedule Height of the tree is l = n m Worst case scenario a full n-ary tree Traversing from root to leaf gives the sequence of dispatches that yielded the resulting schedule Tree creation can be done recursively for all possible dispatches. Such an exhaustive search would yield at the most n l leaf nodes. Even for small dimensions of n and m, it is too computationally expensive to investigate them all. 25 of 36

32 Size of training set (1) A seperate DR for each dispatch iteration At each dispatch k a number of data pairs are created for each of the N problem instance created. Deliberately create a separate data set for each dispatch Resulting in l linear scheduling rules for solving a n m JSSP. Dening the size of the training set as l = Φ, gives the size of the preference set as S = 2l. If l is too large, than sampling needs to be done. 26 of 36

33 Size of training set (2) Sampling approach in Ingimundardottir and Runarsson [2011] The strategy was to follow some single optimal job j O (k), thus creating O (k) S (k) feature pairs at each dispatch k, resulting in a training size of: ( N l ) l = O (k) S (k) q=1 k=1 For the data distribution considered there, this simple sampling was sucient for a favourable outcome. However for a considerably harder data distribution this strategy did not work well (see slide 15). 27 of 36

34 Size of training set (3) Applying CMA-ES to minimize ϱ w.r.t. w directly gave signicantly more favourable result in predicting optimal vs. suboptimal dispatches. Nature of CMA-ES is to explore suboptimal routes until it converges to an optimal one. 28 of 36

35 Size of training set (4) CMA-ES inspired approach Obviously looking only into one optimal route isn't sucient information. Training set should also make the distinction between suboptimal and sub-suboptimal features, etc. e.g. inspect the Pareto ranking at each dispatch k, creating an even greater training set that needs to be sampled. Due to the nature of the sequence representation, the earlier stages of the dispatching are more or less equivalent (and thus irrelevant), hence it could be appropriate to follow some random path to begin with and then follow some policy until completion at step l. 29 of 36

36 Size of training set (5) Main aspects for generating training data What sort of rankings should be compared during each step? Which path(s) should be investigated? Only one, some or all paths? 30 of 36

37 Introduction Job Shop Scheduling Ordinal Regression Generating Training Data Future Work 31 of 36

38 Future Work This talk pointed out some aspects that need to be taken into consideration when creating training data, especially since Ingimundardottir and Runarsson [2012] illustrated the interaction of a specic algorithm with a given data structure and its properties. Feature selection is of paramount importance in order for algorithms to become successful, therefore one needs to give great thought to how features are selected. What kind of feature behaviour yield bad schedules? And can they be steered onto the path of more promising feature characteristics? In general, this sort of investigation can be used in better algorithm design which is more equipped to deal with varying data instances. Which could tailor to individual data instances, i.e. footprint-oriented algorithm. 32 of 36

39 Future Work This talk pointed out some aspects that need to be taken into consideration when creating training data, especially since Ingimundardottir and Runarsson [2012] illustrated the interaction of a specic algorithm with a given data structure and its properties. Feature selection is of paramount importance in order for algorithms to become successful, therefore one needs to give great thought to how features are selected. What kind of feature behaviour yield bad schedules? And can they be steered onto the path of more promising feature characteristics? In general, this sort of investigation can be used in better algorithm design which is more equipped to deal with varying data instances. Which could tailor to individual data instances, i.e. footprint-oriented algorithm. 32 of 36

40 Future Work This talk pointed out some aspects that need to be taken into consideration when creating training data, especially since Ingimundardottir and Runarsson [2012] illustrated the interaction of a specic algorithm with a given data structure and its properties. Feature selection is of paramount importance in order for algorithms to become successful, therefore one needs to give great thought to how features are selected. What kind of feature behaviour yield bad schedules? And can they be steered onto the path of more promising feature characteristics? In general, this sort of investigation can be used in better algorithm design which is more equipped to deal with varying data instances. Which could tailor to individual data instances, i.e. footprint-oriented algorithm. 32 of 36

41 Future Work This talk pointed out some aspects that need to be taken into consideration when creating training data, especially since Ingimundardottir and Runarsson [2012] illustrated the interaction of a specic algorithm with a given data structure and its properties. Feature selection is of paramount importance in order for algorithms to become successful, therefore one needs to give great thought to how features are selected. What kind of feature behaviour yield bad schedules? And can they be steered onto the path of more promising feature characteristics? In general, this sort of investigation can be used in better algorithm design which is more equipped to deal with varying data instances. Which could tailor to individual data instances, i.e. footprint-oriented algorithm. 32 of 36

42 Future Work This talk pointed out some aspects that need to be taken into consideration when creating training data, especially since Ingimundardottir and Runarsson [2012] illustrated the interaction of a specic algorithm with a given data structure and its properties. Feature selection is of paramount importance in order for algorithms to become successful, therefore one needs to give great thought to how features are selected. What kind of feature behaviour yield bad schedules? And can they be steered onto the path of more promising feature characteristics? In general, this sort of investigation can be used in better algorithm design which is more equipped to deal with varying data instances. Which could tailor to individual data instances, i.e. footprint-oriented algorithm. 32 of 36

43 Future Work This talk pointed out some aspects that need to be taken into consideration when creating training data, especially since Ingimundardottir and Runarsson [2012] illustrated the interaction of a specic algorithm with a given data structure and its properties. Feature selection is of paramount importance in order for algorithms to become successful, therefore one needs to give great thought to how features are selected. What kind of feature behaviour yield bad schedules? And can they be steered onto the path of more promising feature characteristics? In general, this sort of investigation can be used in better algorithm design which is more equipped to deal with varying data instances. Which could tailor to individual data instances, i.e. footprint-oriented algorithm. 32 of 36

44 Bibliography (1) D. Corne and A. Reynolds. Optimisation and generalisation: footprints in instance space. Parallel Problem Solving from Nature, PPSN XI, pages 2231, N. Hansen and A. Ostermeier. Completely Derandomized Self-Adaptation in Evolution Strategies. Evol. Comput., 9(2): , June H. Ingimundardottir and T. Runarsson. Supervised learning linear priority dispatch rules for job-shop scheduling. In C. Coello, editor, Learning and Intelligent Optimization, volume 6683 of Lecture Notes in Computer Science, pages Springer Berlin, jan of 36

45 Bibliography (2) H. Ingimundardottir and T. P. Runarsson. Determining the Characteristic of Dicult Job Shop Scheduling Instances for a Heuristic Solution Method. In M. Schoenauer, editor, Learning and Intelligent Optimization, 6th International Conference, LION 6, Paris, jan Springer Lecture Notes in Computer Science. S. Panwalkar and W. Iskander. A Survey of Scheduling Rules. Operations Research, 25(1):4561, Jan B. Pfahringer and H. Bensusan. Meta-learning by landmarking various learning algorithms. on Machine Learning, J. R. Rice. The algorithm selection problem. Advances in Computers, 15:65118, of 36

46 Bibliography (3) K. Smith-Miles and L. Lopes. Generalising Algorithm Performance in Instance Space: A timetabling case study. In C. A. Coello Coello, editor, Learning and Intelligent Optimization, 5th International Conference, LION 5, Lecture Notes in Computer Science, of 36

47 Thank you for your attention Questions? Helga Ingimundardóttir, 36 of 36

48 Thank you for your attention Questions? Helga Ingimundardóttir, 36 of 36