Energetic Reasoning (ER) is a powerful filtering algorithm for the Cumulative
constraint. Unfortunately, ER is generally too costly to be used in practice.
One reason of its bad behavior is that many intervals are considered as
relevant by the checker of ER, although most of them should be ignored. In this
paper, we provide a sharp characterization that allows to reduce the number of
intervals by a factor seven. Our experiments show that associating this checker
with a Time-Table filtering algorithm leads to promising results.Comment: CP Doctoral Program 2013, Uppsala : Sweden (2013

Derrien, Alban

Petit, Thierry

English

arXiv

International audienceEnergetic Reasoning (ER) is a powerful filtering algorithm for the Cumulative constraint. Unfortunately, ER is generally too costly to be used in practice. One reason of its bad behavior is that many intervals are considered as relevant by the checker of ER, although most of them should be ignored. In this paper, we provide a sharp characterization that allows to reduce the number of intervals by a factor seven. Our experiments show that associating this checker with a Time-Table filtering algorithm leads to promising results

HAL Mines Nantes

The Energetic Reasoning Checker RevisitedAlban Derrien, Thierry PetitTo cite this version:Alban Derrien, Thierry Petit. The Energetic Reasoning Checker Revisited. Principles and Prac-tice of Constraint Programming, Sep 2013, Uppsala, Sweden. CP 2013, doctoral programme,2013. <hal-00873940>HAL Id: hal-00873940https://hal.archives-ouvertes.fr/hal-00873940Submitted on 16 Oct 2013HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.The Energetic Reasoning Checker RevisitedAlban Derrien and Thierry PetitTASC (Mines Nantes, LINA, CNRS, INRIA),4, Rue Alfred Kastler, FR-44307 Nantes Cedex 3, France.{alban.derrien, thierry.petit}@mines-nantes.frAbstract. Energetic Reasoning (ER) is a powerful filtering algorithm for theCumulative constraint. Unfortunately, ER is generally too costly to be used inpractice. One reason of its bad behavior is that many intervals are considered asrelevant by the checker of ER, although most of them should be ignored. In thispaper, we provide a sharp characterization that allows to reduce the number ofintervals by a factor seven. Our experiments show that associating this checkerwith a Time-Table filtering algorithm leads to promising results.1 IntroductionDue to its relevance in many industrial contexts, the Cumulative Scheduling Problem(CuSP) has been widely studied in Constraint Programming (CP). The CuSP is NP-Hard. It is defined by a set of activities A and a resource of capacity C. Each activitya ∈ A is defined by four variables: its starting time sa, its processing time pa, itsending time ea and its height ha, which represents the amount of resources consumedby the activity when it is processed. We use the notation a = {sa, pa, ea, ha}. Usually,variables pa and ha are fixed integers. In this paper, we make such an assumption. Asolution to a CuSP is a schedule that satisfies the following constraints:∀a ∈ A : sa + pa = ea ∧ ∀i ∈ N :∑a∈Ai∈[sa,ea[ha ≤ CIn CP, this problem is generally represented by the global constraint Cumulative[1]. The Energetic Reasoning of Baptiste et al. (ER) is one of the most powerful fil-tering algorithm for Cumulative [2]. This algorithm uses a characterization of intervalsof interest, that is, intervals that are sufficient to check in order to ensure that all theundergoing rules used for filtering domains are satisfied. Unfortunately, ER is often toocostly to be used in practice. First, its time complexity is O(n3). Moreover, the hid-den constant in that time complexity is huge, as many intervals are characterized to beof interest although most of them should be ignored. In the literature, only heuristicsapproaches have been proposed for reducing the number of checked intervals [3].This article provides a sharper characterization of intervals of interest. Our exper-iments show a significant reduction in the running time of the ER checker. We pointout that assocating this energetic checker with a Time-Table filtering algorithm (basedon the cumulative profile of mandatory parts [7]) leads to promising results. Further,our approach should permit to answer affirmatively to a theoretical open question withrespect to energetic filtering algorithms: Are the intervals of interest for the ER checkersufficient in order to perform a complete ER filtering for the activities variables?2 BackgroundGiven a variable x, we denote x the minimum value in its domain and x the maxi-mum value. The principle of ER is to compare the available area within a given interval(length of that interval × capacity) with the quantity of resource necessarily taken byactivities that should partially or totally overlap this interval. The ER checker is de-fined usingMIa(t1, t2), the minimum intersection of activity a with an interval [t1, t2].MIa(t1, t2) = max(0,min(pa, t2−t1, ea−t1, t2−sa)).Proposition 1 (ER checker [4]). If the condition∀t1, t2 ∈ N2, t1 < t2 C × (t2 − t1) ≥∑a∈Aha ×MIa(t1, t2) (1)is violated then the problem represented by Cumulative is unfeasible.The issue is then to find the smallest set of intervals [t1, t2], t2 > t1 that should bechecked to detect the unfeasibility. From this condition a bound adjustment rule can bedefined. Value sa can be updated if scheduling activity a on sa would cause an overloadas described above. To check such potential overloads, we need to compute how muchadditional energy a would require during [t1, t2] if a is scheduled on sa.Baptiste et al. characterization In order to ensure that condition (1) holds, it is enoughto check intervals of the form: [t1, t2], t1 ∈ O1 < t2 ∈ O2, [t1, t2], t1 ∈ O1 < t2 ∈O(t1), [t1, t2], t2 ∈ O2 > t1 ∈ O(t2), where O1 = {sa, ∀a ∈ A} ∪ {sa, ∀a ∈A} ∪ {ea, ∀a ∈ A}, O2 = {ea, ∀a ∈ A} ∪ {sa, ∀a ∈ A} ∪ {ea, ∀a ∈ A},O(t) = {sa + ea − t, ∀a ∈ A}. The resulting subset of intervals forms the intervals ofinterest of the ER. There is for each pair of activities 9+3+3=15 intervals of interest.The characterization is proved to be sufficient in [2] (proposition 19), by implicitly an-alyzing the slack function Slack(t1, t2) = C × (t2 − t1) −∑a∈A ha ×MIa(t1, t2).As MIa(t1, t2) can be computed in constant time and given the definitions of O1, O2andOt, we obtain a naive checker in O(n3), by computingMIa(t1, t2) for all a, t1, t2.However, the slack function is continuous piecewise linear, a local extrema can only befound at flexion points, i.e., when the slope is changing, which has been proved to beonly in such intervals. This leads to a checker inO(n2) [2]. Two open questions remain.1. The new set of intervals of interest is proved to be sufficient but could it be reduced?2. Is the new set of intervals also sufficient to ensure a complete filtering of the timebound adjustment rule based on the checker?Schwindt’s characterization Schwindt has proposed a finer characterization of theminimal value of the slack function (Theorems 3.7 and 3.8 in [8], written in German):The slack function is locally minimal in interval [t1, t2] only if its left derivative is neg-ative and right derivative positive, for both t1 and t2. As the slack function correspondto the negation of the sum of minimal intersections, it must exists an activity i (resp j)such that its minimal intersection has a left derivative greater than its right derivative ont1 (resp. t2). This leads to Theorem 1.Theorem 1. The slack function is locally minimum in interval [t1, t2] only if it existstwo activities i, j such that the two following conditions are satisfied.δ−MIi(t1, t2)δt1>δ+MIi(t1, t2)δt1(2)δ−MIj(t1, t2)δt2>δ+MIj(t1, t2)δt2(3)Schwindt analyzes the variation of minimum intersection. He provides a first char-acterization of the 8 possible intervals of interest for any pair of activities. This answersto the first open question: The number of intervals in Baptiste et al. characterization canbe reduced.We propose in next section a different analyze of the minimum intersection,which leads to a sharper characterization of activity pairs.3 A New CharacterizationFor symmetry reasons we focus on condition (3). We wish to characterize for a startingtime t1 and an activity a the positive inflection points of function t2 → MIa(t1, t2):Values t2 at which the left derivative of is greater than the right derivative.Theorem 2. For a starting time t1 and an activity a the function t2 →MIa(t1, t2) hasat most one positive inflection point.Proof. We prove theorem 2 by studying the four positions of t1 w.r.t. a. We prove that the functiont2 → MIa(t1, t2) is continuous piecewise linear, composed of at most three parts. The twoinflection points correspond to the start of consumption (SoCa) and the end of consumption(EoCa). We also prove that EoCa is the only inflection point with a left derivative greater thanthe right derivative. Graphically we show an example of each cases with an activity a = {sa ∈[2, 4], pa=4, ea∈ [6, 8], ha}.Case 1.On the first case, the minimum starting timeof the activity a is greater than or equal tothe profile starting time: t1 ≤ sa. T ime0 1 2 3 4 5 6 7 8 9 10t1=1a04By definition we haveMIa(t1, t2) = max(0,min(pa, t2−t1, ea−t1, t2−sa)). We can de-duce three different situations:1. if t2 ≤ sa thenMIa(t1, t2) = 0.2. or sa ≤ t2 ≤ ea thenMIa(t1, t2) = t2 − sa.3. and finally ea ≤ t2 thenMIa(t1, t2) = pa.pa−(ea−t2) equals 0 when t2 = sa and pa when t2 = ea. This proves that when t1 ≤ sa thenconsumption function is continue and piecewise linear, composed of three pieces with only onepositive inflection point: EoCa.Case 2.On the second case, the minimum endingtime of the activity a is smaller than orequal to the profile starting time: t1 ≥ ea. T ime0 1 2 3 4 5 6 7 8 9 10t1=7a0In this case, we have MIa(t1, t2) = 0 for any interval. Then the function is trivially continueand piecewise linear with zero inflection point.Case 3.On the third case, the profile starting timeis greater than the minimum start time, butsmaller than the minimum end time andmaximum start time:t1>sa and t1<ea and t1<saT ime0 1 2 3 4 5 6 7 8 9 10t1=3a03Let∆ = t1 − sa. We distinct three cases for the value of t2 :1. if t2 ≤ sa thenMIa(t1, t2) = 0.2. or sj ≤ t2 ≤ ej −∆ thenMIa(t1, t2) = t2 − sa.3. and finally t2 ≥ ea −∆ thenMIa(t1, t2) = pa −∆.As t2 − sa equals 0 when t2 = sa and pa − ∆ when t2 = ea − ∆ we have proved that theconsumption is continue and piecewise linear, composed of three pieces with only one positiveinflection point: EoCa.Case 4.On the fourth case, the profile starting timeintersect the mandatory part a:t1>sa and t1<ea and t1≥sa T ime0 1 2 3 4 5 6 7 8 9 10t1=4.5a01.5In this case we have two distinct cases for the value of t2 :1. if t2 ≤ ea thenMIa(t1, t2) = t2 − t12. otherwiseMIa(t1, t2) = pa −∆As t2 − t1 = pa − ∆ when t2 = ea then the consumption function is continue and piecewiselinear, composed of two pieces, with only one positive inflection point : EoCa.Conclusion We have shown that for any starting value t1, and any activity a the function t2 →MIa(t1, t2) is linear and piecewise continue, with, in each case, at most one point at which theleft derivative is greater than the right derivate: EoCa. This proves the Theorem. ⊓⊔Table 1: Intervals of interest for a pair of activities (i,j)conditions intervalsi≤sj ∧ ej≥ei [si, ej ] Asi≥sj ∧ si≤ej ∧ si≤sj ∧ sj+ej−si≥ei [si, sj+ej−si] Bsi≥sj ∧ si≤ej ∧ ej≥ei [si, ej ] Csi≤sj≤ej≤ei [si, ej ] Dsi≥sj ∧ si≤ej ∧ si≤sj ∧ sj+ej≤si+ei ∧ sj+ej≥2× si [si, sj+ej−si] Esj≤si≤ej≤ei [si, ej ] Fej≤ei ∧ ej≥si ∧ ej ≥ei ∧ si+ei≤sj+ej [si+ei−ej , ej ] Gej≤ei ∧ ej≥si ∧ ej ≥ei ∧ si+ei≤sj+ej ∧ si+ei≤2× ej [si+ei−ej , ej ] HWe have characterized for an activity a the possible value at which condition (3)holds. By symmetry, we can deduce the value of the starting time at which condition(2) holds. Table 1 summarizes the intervals of interest of a pair of activities (i, j).Our characterization is sharper than the one proposed by Schwindt: For instance,case B in Table 1 is sharper than the equivalent case iii in table 3.5 page 84 in [8].We may notice that intervals of interest may only start at values of the form sa, sa orend at values ea, ea. This leads to a lighter algorithm for the O(n2) checker proposedby Baptiste et al. [2]. We present the checker for interval with starting dates in OS ={sa, ∀a ∈ A}∪{sa, ∀a ∈ A} as possible starting values (as CuSP is symmetric).We define an event as a pair (time,activity). Let EM be the events for the last com-pletion times, (ea, a)∀a ∈ A and ordered in increasing order of time. Similarly, wedefine Em = (ea, a), ∀a∈A, SM = (ea, a), ∀a∈A and L = (sa + ea, a), ∀a∈A.1 foreach t1 ∈ OS do2 slope = C −∑aMIa(t1, t1 + 1);3 Load = 0; told2 = t1;4 L′ = {(t′ − t1, a) | (t′, a) ∈ L};5 foreach event(t2, a) in SM , Em, EM , L′ do6 Load+= slope× (t2 − told2 ) ;7 if Load < 0 then Fail;8 if event is a SoCa then9 slope−= ha;10 else if event is aEoCa then11 slope += ha;12 end13 told2 = t2;14 end15 endAlgorithm 1: Energetic Reasoning Checker.// The slope represents the evolution ofthe Slack over time.// Events are evaluated in increasingorder of their time. Starting on t1+1.4 Experiments and Future WorkExperimentswere run on a 2.9 GHz Intel Core i7, in Choco [9] version 3 (release 13.03).In order to check the gain obtained with the new characterization we have consideredrandom instances and instances from the PSPLIB[6]. Random instances have either 10or 20 activities. Their processing times were chosen within [1, 10], their heights within[1, 5]. We used the first fail [5] search strategy, and three checkers: Algorithm 1, Baptisteet al. checker and the basic O(n3) checker. The number of nodes is identical for allinstances for the three checkers, as expected. Table 2 shows a running time improvementof 20 to 30% using Algorithm 1, in comparison with Baptiste et al. algorithm.We also used our checker in combination of the Time-Table filtering algorithm [7],to compare that combination with Time-Table Edge-Finding filtering algorithm [10].We fixed a solving time limit of five minutes. Surprisingly, when proving optimality onrandom instances with a single resource, using the checker proved optimality for 72 ofthe 100 generated instances, while TTEF was unable to do so. This proves that associ-ating this new checker with a Time-Table approach could be a good default propagatorin constraint solvers.Algorithm 1 Baptiste et al. O(n3)Instances (µs/node) (µs/node) (µs/node)Random10 16.47 24.97 29.31Random20 43.95 56.24 78.74PspLib 30 450.67 618.77 1268.92PspLib 120 1 339.24 1 683.26 11 288.54Table 2: Average time per node.TT TT + TT +TTEF Algorithm 1Random20 6 7 72Table 3: #proved optimum over 100 in-stances.As a future work, our results should be confirmed by using dedicated search heuris-tics, which was not the case in our preliminary experiments. Moreover, as a direct con-sequence of our new case-based characterization, we can answer affirmatively to thesecond theoretical open question: The interval of interests of Baptiste et al. is completefor the propagation algorithm. As the proof needs a better enlightenment and cannoteasily fit in a short paper, we will present it in a new paper.References1. Abder Aggoun and Nicolas Beldiceanu. Extending chip in order to solve complex schedulingand placement problems. Math. Comput. Model., 17(7):57–73, April 1993.2. Philippe Baptiste, Claude Le Pape, and Wim Nuijten. Constraint-Based Scheduling: Ap-plying Constraint Programming to Scheduling Problems. International Series in OperationsResearch and Management Science. Kluwer, 2001.3. Timo Berthold, Stefan Heinz, and Jens Schulz. An approximative criterion for the potentialof energetic reasoning. In Proceedings of the First international ICST conference on The-ory and practice of algorithms in (computer) systems, TAPAS’11, pages 229–239, Berlin,Heidelberg, 2011. Springer-Verlag.4. Jacques Erschler, Pierre Lopez, and Catherine Thuriot. Scheduling under time and resourceconstraints. In Proc. of Workshop on Manufacturing Scheduling, 11th IJCAI, Detroit, USA,1989.5. Robert M. Haralick and Gordon L. Elliot. Increasing tree search efficiency for constraintsatisfaction problems. Artificial Intelligence, 14(3):263–313, 1980.6. Rainer Kolisch and Arno Sprecher. Psplib – a project scheduling problem library. EURO-PEAN JOURNAL OF OPERATIONAL RESEARCH, 96:205–216, 1996.7. Arnaud Letort, Nicolas Beldiceanu, and Mats Carlsson. A scalable sweep algorithm for thecumulative constraint. In Michela Milano, editor, Principles and Practice of Constraint Pro-gramming, Lecture Notes in Computer Science, pages 439–454. Springer Berlin Heidelberg,2012.8. Christoph Schwindt. Verfahren zur Lo¨sung des ressourcenbeschra¨nkten Projektdauer-minimierungsproblems mit planungsabha¨ngigen Zeitfenstern. PhD thesis, Fakulta¨t fu¨rwirtschaftswissenschaften der Universita¨t Fridericiana zu Karlsruhe, 1998. in German.9. CHOCO Team. Choco: an open source Java CP library. Research report 10-02-INFO, Ecoledes Mines de Nantes, 2010.10. Petr Vilı´m. Timetable edge finding filtering algorithm for discrete cumulative resources. InTobias Achterberg and J. Beck, editors, Integration of AI and OR Techniques in ConstraintProgramming for Combinatorial Optimization Problems, volume 6697 of Lecture Notes inComputer Science, pages 230–245. Springer Berlin / Heidelberg, 2011.

The Energetic Reasoning Checker Revisited

Alban Derrien

Thierry Petit

CiteSeerX

Abstract. Energetic Reasoning (ER) is a powerful filtering algorithm for the Cumulative constraint. Unfortunately, ER is generally too costly to be used in practice. One reason of its bad behavior is that many intervals are considered as relevant by the checker of ER, although most of them should be ignored. In this paper, we provide a sharp characterization that allows to reduce the number of intervals by a factor seven. Our experiments show that associating this checker with a Time-Table filtering algorithm leads to promising results. 

INRIA a CCSD electronic archive server

HAL Id: hal-00873940https://hal.archives-ouvertes.fr/hal-00873940Submitted on 16 Oct 2013HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.The Energetic Reasoning Checker RevisitedAlban Derrien, Thierry PetitTo cite this version:Alban Derrien, Thierry Petit. The Energetic Reasoning Checker Revisited. Principles and Practice ofConstraint Programming, Sep 2013, Uppsala, Sweden. ￿hal-00873940￿The Energetic Reasoning Checker RevisitedAlban Derrien and Thierry PetitTASC (Mines Nantes, LINA, CNRS, INRIA),4, Rue Alfred Kastler, FR-44307 Nantes Cedex 3, France.{alban.derrien, thierry.petit}@mines-nantes.frAbstract. Energetic Reasoning (ER) is a powerful filtering algorithm for theCumulative constraint. Unfortunately, ER is generally too costly to be used inpractice. One reason of its bad behavior is that many intervals are considered asrelevant by the checker of ER, although most of them should be ignored. In thispaper, we provide a sharp characterization that allows to reduce the number ofintervals by a factor seven. Our experiments show that associating this checkerwith a Time-Table filtering algorithm leads to promising results.1 IntroductionDue to its relevance in many industrial contexts, the Cumulative Scheduling Problem(CuSP) has been widely studied in Constraint Programming (CP). The CuSP is NP-Hard. It is defined by a set of activities A and a resource of capacity C. Each activitya ∈ A is defined by four variables: its starting time sa, its processing time pa, itsending time ea and its height ha, which represents the amount of resources consumedby the activity when it is processed. We use the notation a = {sa, pa, ea, ha}. Usually,variables pa and ha are fixed integers. In this paper, we make such an assumption. Asolution to a CuSP is a schedule that satisfies the following constraints:∀a ∈ A : sa + pa = ea ∧ ∀i ∈ N :∑a∈Ai∈[sa,ea[ha ≤ CIn CP, this problem is generally represented by the global constraint Cumulative[1]. The Energetic Reasoning of Baptiste et al. (ER) is one of the most powerful fil-tering algorithm for Cumulative [2]. This algorithm uses a characterization of intervalsof interest, that is, intervals that are sufficient to check in order to ensure that all theundergoing rules used for filtering domains are satisfied. Unfortunately, ER is often toocostly to be used in practice. First, its time complexity is O(n3). Moreover, the hid-den constant in that time complexity is huge, as many intervals are characterized to beof interest although most of them should be ignored. In the literature, only heuristicsapproaches have been proposed for reducing the number of checked intervals [3].This article provides a sharper characterization of intervals of interest. Our exper-iments show a significant reduction in the running time of the ER checker. We pointout that assocating this energetic checker with a Time-Table filtering algorithm (basedon the cumulative profile of mandatory parts [7]) leads to promising results. Further,our approach should permit to answer affirmatively to a theoretical open question withrespect to energetic filtering algorithms: Are the intervals of interest for the ER checkersufficient in order to perform a complete ER filtering for the activities variables?2 BackgroundGiven a variable x, we denote x the minimum value in its domain and x the maxi-mum value. The principle of ER is to compare the available area within a given interval(length of that interval × capacity) with the quantity of resource necessarily taken byactivities that should partially or totally overlap this interval. The ER checker is de-fined using MIa(t1, t2), the minimum intersection of activity a with an interval [t1, t2].MIa(t1, t2) = max(0,min(pa, t2−t1, ea−t1, t2−sa)).Proposition 1 (ER checker [4]). If the condition∀t1, t2 ∈ N2, t1 < t2 C × (t2 − t1) ≥∑a∈Aha ×MIa(t1, t2) (1)is violated then the problem represented by Cumulative is unfeasible.The issue is then to find the smallest set of intervals [t1, t2], t2 > t1 that should bechecked to detect the unfeasibility. From this condition a bound adjustment rule can bedefined. Value sa can be updated if scheduling activity a on sa would cause an overloadas described above. To check such potential overloads, we need to compute how muchadditional energy a would require during [t1, t2] if a is scheduled on sa.Baptiste et al. characterization In order to ensure that condition (1) holds, it is enoughto check intervals of the form: [t1, t2], t1 ∈ O1 < t2 ∈ O2, [t1, t2], t1 ∈ O1 < t2 ∈O(t1), [t1, t2], t2 ∈ O2 > t1 ∈ O(t2), where O1 = {sa, ∀a ∈ A} ∪ {sa, ∀a ∈A} ∪ {ea, ∀a ∈ A}, O2 = {ea, ∀a ∈ A} ∪ {sa, ∀a ∈ A} ∪ {ea, ∀a ∈ A},O(t) = {sa + ea − t, ∀a ∈ A}. The resulting subset of intervals forms the intervals ofinterest of the ER. There is for each pair of activities 9+3+3=15 intervals of interest.The characterization is proved to be sufficient in [2] (proposition 19), by implicitly an-alyzing the slack function Slack(t1, t2) = C × (t2 − t1) −∑a∈A ha ×MIa(t1, t2).As MIa(t1, t2) can be computed in constant time and given the definitions of O1, O2and Ot, we obtain a naive checker in O(n3), by computing MIa(t1, t2) for all a, t1, t2.However, the slack function is continuous piecewise linear, a local extrema can only befound at flexion points, i.e., when the slope is changing, which has been proved to beonly in such intervals. This leads to a checker in O(n2) [2]. Two open questions remain.1. The new set of intervals of interest is proved to be sufficient but could it be reduced?2. Is the new set of intervals also sufficient to ensure a complete filtering of the timebound adjustment rule based on the checker?Schwindt’s characterization Schwindt has proposed a finer characterization of theminimal value of the slack function (Theorems 3.7 and 3.8 in [8], written in German):The slack function is locally minimal in interval [t1, t2] only if its left derivative is neg-ative and right derivative positive, for both t1 and t2. As the slack function correspondto the negation of the sum of minimal intersections, it must exists an activity i (resp j)such that its minimal intersection has a left derivative greater than its right derivative ont1 (resp. t2). This leads to Theorem 1.Theorem 1. The slack function is locally minimum in interval [t1, t2] only if it existstwo activities i, j such that the two following conditions are satisfied.δ−MIi(t1, t2)δt1>δ+MIi(t1, t2)δt1(2)δ−MIj(t1, t2)δt2>δ+MIj(t1, t2)δt2(3)Schwindt analyzes the variation of minimum intersection. He provides a first char-acterization of the 8 possible intervals of interest for any pair of activities. This answersto the first open question: The number of intervals in Baptiste et al. characterization canbe reduced. We propose in next section a different analyze of the minimum intersection,which leads to a sharper characterization of activity pairs.3 A New CharacterizationFor symmetry reasons we focus on condition (3). We wish to characterize for a startingtime t1 and an activity a the positive inflection points of function t2 → MIa(t1, t2):Values t2 at which the left derivative of is greater than the right derivative.Theorem 2. For a starting time t1 and an activity a the function t2 → MIa(t1, t2) hasat most one positive inflection point.Proof. We prove theorem 2 by studying the four positions of t1 w.r.t. a. We prove that the functiont2 → MIa(t1, t2) is continuous piecewise linear, composed of at most three parts. The twoinflection points correspond to the start of consumption (SoCa) and the end of consumption(EoCa). We also prove that EoCa is the only inflection point with a left derivative greater thanthe right derivative. Graphically we show an example of each cases with an activity a = {sa ∈[2, 4], pa=4, ea∈ [6, 8], ha}.Case 1.On the first case, the minimum starting timeof the activity a is greater than or equal tothe profile starting time: t1 ≤ sa. T ime0 1 2 3 4 5 6 7 8 9 10t1=1a04By definition we have MIa(t1, t2) = max(0,min(pa, t2−t1, ea−t1, t2−sa)). We can de-duce three different situations:1. if t2 ≤ sa then MIa(t1, t2) = 0.2. or sa ≤ t2 ≤ ea then MIa(t1, t2) = t2 − sa.3. and finally ea ≤ t2 then MIa(t1, t2) = pa.pa−(ea−t2) equals 0 when t2 = sa and pa when t2 = ea. This proves that when t1 ≤ sa thenconsumption function is continue and piecewise linear, composed of three pieces with only onepositive inflection point: EoCa.Case 2.On the second case, the minimum endingtime of the activity a is smaller than orequal to the profile starting time: t1 ≥ ea. T ime0 1 2 3 4 5 6 7 8 9 10t1=7a0In this case, we have MIa(t1, t2) = 0 for any interval. Then the function is trivially continueand piecewise linear with zero inflection point.Case 3.On the third case, the profile starting timeis greater than the minimum start time, butsmaller than the minimum end time andmaximum start time:t1>sa and t1<ea and t1<saT ime0 1 2 3 4 5 6 7 8 9 10t1=3a03Let ∆ = t1 − sa. We distinct three cases for the value of t2 :1. if t2 ≤ sa then MIa(t1, t2) = 0.2. or sj ≤ t2 ≤ ej −∆ then MIa(t1, t2) = t2 − sa.3. and finally t2 ≥ ea −∆ thenMIa(t1, t2) = pa −∆.As t2 − sa equals 0 when t2 = sa and pa − ∆ when t2 = ea − ∆ we have proved that theconsumption is continue and piecewise linear, composed of three pieces with only one positiveinflection point: EoCa.Case 4.On the fourth case, the profile starting timeintersect the mandatory part a:t1>sa and t1<ea and t1≥sa T ime0 1 2 3 4 5 6 7 8 9 10t1=4.5a01.5In this case we have two distinct cases for the value of t2 :1. if t2 ≤ ea then MIa(t1, t2) = t2 − t12. otherwise MIa(t1, t2) = pa −∆As t2 − t1 = pa − ∆ when t2 = ea then the consumption function is continue and piecewiselinear, composed of two pieces, with only one positive inflection point : EoCa.Conclusion We have shown that for any starting value t1, and any activity a the function t2 →MIa(t1, t2) is linear and piecewise continue, with, in each case, at most one point at which theleft derivative is greater than the right derivate: EoCa. This proves the Theorem. ⊓⊔Table 1: Intervals of interest for a pair of activities (i,j)conditions intervalsi≤sj ∧ ej ≥ei [si, ej ] Asi≥sj ∧ si≤ej ∧ si≤sj ∧ sj+ej−si≥ei [si, sj+ej−si] Bsi≥sj ∧ si≤ej ∧ ej ≥ei [si, ej ] Csi≤sj ≤ej ≤ei [si, ej ] Dsi≥sj ∧ si≤ej ∧ si≤sj ∧ sj+ej ≤si+ei ∧ sj+ej ≥2× si [si, sj+ej−si] Esj ≤si≤ej ≤ei [si, ej ] Fej ≤ei ∧ ej ≥si ∧ ej ≥ei ∧ si+ei≤sj+ej [si+ei−ej , ej ] Gej ≤ei ∧ ej ≥si ∧ ej ≥ei ∧ si+ei≤sj+ej ∧ si+ei≤2× ej [si+ei−ej , ej ] HWe have characterized for an activity a the possible value at which condition (3)holds. By symmetry, we can deduce the value of the starting time at which condition(2) holds. Table 1 summarizes the intervals of interest of a pair of activities (i, j).Our characterization is sharper than the one proposed by Schwindt: For instance,case B in Table 1 is sharper than the equivalent case iii in table 3.5 page 84 in [8].We may notice that intervals of interest may only start at values of the form sa, sa orend at values ea, ea. This leads to a lighter algorithm for the O(n2) checker proposedby Baptiste et al. [2]. We present the checker for interval with starting dates in OS ={sa, ∀a ∈ A}∪{sa, ∀a ∈ A} as possible starting values (as CuSP is symmetric).We define an event as a pair (time,activity). Let EM be the events for the last com-pletion times, (ea, a)∀a ∈ A and ordered in increasing order of time. Similarly, wedefine Em = (ea, a), ∀a∈A, SM = (ea, a), ∀a∈A and L = (sa + ea, a), ∀a∈A.1 foreach t1 ∈ OS do2 slope = C −∑aMIa(t1, t1 + 1);3 Load = 0; told2 = t1;4 L′ = {(t′ − t1, a) | (t′, a) ∈ L};5 foreach event(t2, a) in SM , Em, EM , L′ do6 Load+= slope × (t2 − told2 ) ;7 if Load < 0 then Fail;8 if event is a SoCa then9 slope −= ha;10 else if event is a EoCa then11 slope += ha;12 end13 told2 = t2;14 end15 endAlgorithm 1: Energetic Reasoning Checker.// The slope represents the evolution ofthe Slack over time.// Events are evaluated in increasingorder of their time. Starting on t1+1.4 Experiments and Future WorkExperiments were run on a 2.9 GHz Intel Core i7, in Choco [9] version 3 (release 13.03).In order to check the gain obtained with the new characterization we have consideredrandom instances and instances from the PSPLIB[6]. Random instances have either 10or 20 activities. Their processing times were chosen within [1, 10], their heights within[1, 5]. We used the first fail [5] search strategy, and three checkers: Algorithm 1, Baptisteet al. checker and the basic O(n3) checker. The number of nodes is identical for allinstances for the three checkers, as expected. Table 2 shows a running time improvementof 20 to 30% using Algorithm 1, in comparison with Baptiste et al. algorithm.We also used our checker in combination of the Time-Table filtering algorithm [7],to compare that combination with Time-Table Edge-Finding filtering algorithm [10].We fixed a solving time limit of five minutes. Surprisingly, when proving optimality onrandom instances with a single resource, using the checker proved optimality for 72 ofthe 100 generated instances, while TTEF was unable to do so. This proves that associ-ating this new checker with a Time-Table approach could be a good default propagatorin constraint solvers.Algorithm 1 Baptiste et al. O(n3)Instances (µs/node) (µs/node) (µs/node)Random10 16.47 24.97 29.31Random20 43.95 56.24 78.74PspLib 30 450.67 618.77 1268.92PspLib 120 1 339.24 1 683.26 11 288.54Table 2: Average time per node.TT TT + TT +TTEF Algorithm 1Random20 6 7 72Table 3: #proved optimum over 100 in-stances.As a future work, our results should be confirmed by using dedicated search heuris-tics, which was not the case in our preliminary experiments. Moreover, as a direct con-sequence of our new case-based characterization, we can answer affirmatively to thesecond theoretical open question: The interval of interests of Baptiste et al. is completefor the propagation algorithm. As the proof needs a better enlightenment and cannoteasily fit in a short paper, we will present it in a new paper.References1. Abder Aggoun and Nicolas Beldiceanu. Extending chip in order to solve complex schedulingand placement problems. Math. Comput. Model., 17(7):57–73, April 1993.2. Philippe Baptiste, Claude Le Pape, and Wim Nuijten. Constraint-Based Scheduling: Ap-plying Constraint Programming to Scheduling Problems. International Series in OperationsResearch and Management Science. Kluwer, 2001.3. Timo Berthold, Stefan Heinz, and Jens Schulz. An approximative criterion for the potentialof energetic reasoning. In Proceedings of the First international ICST conference on The-ory and practice of algorithms in (computer) systems, TAPAS’11, pages 229–239, Berlin,Heidelberg, 2011. Springer-Verlag.4. Jacques Erschler, Pierre Lopez, and Catherine Thuriot. Scheduling under time and resourceconstraints. In Proc. of Workshop on Manufacturing Scheduling, 11th IJCAI, Detroit, USA,1989.5. Robert M. Haralick and Gordon L. Elliot. Increasing tree search efficiency for constraintsatisfaction problems. Artificial Intelligence, 14(3):263–313, 1980.6. Rainer Kolisch and Arno Sprecher. Psplib – a project scheduling problem library. EURO-PEAN JOURNAL OF OPERATIONAL RESEARCH, 96:205–216, 1996.7. Arnaud Letort, Nicolas Beldiceanu, and Mats Carlsson. A scalable sweep algorithm for thecumulative constraint. In Michela Milano, editor, Principles and Practice of Constraint Pro-gramming, Lecture Notes in Computer Science, pages 439–454. Springer Berlin Heidelberg,2012.8. Christoph Schwindt. Verfahren zur Lösung des ressourcenbeschränkten Projektdauer-minimierungsproblems mit planungsabhängigen Zeitfenstern. PhD thesis, Fakultät fürwirtschaftswissenschaften der Universität Fridericiana zu Karlsruhe, 1998. in German.9. CHOCO Team. Choco: an open source Java CP library. Research report 10-02-INFO, Ecoledes Mines de Nantes, 2010.10. Petr Vilı́m. Timetable edge finding filtering algorithm for discrete cumulative resources. InTobias Achterberg and J. Beck, editors, Integration of AI and OR Techniques in ConstraintProgramming for Combinatorial Optimization Problems, volume 6697 of Lecture Notes inComputer Science, pages 230–245. Springer Berlin / Heidelberg, 2011.

HAL-Univ-Nantes

https://hal.archives-ouvertes.fr/hal-00873940/file/report.pdf

The Energetic Reasoning Checker Revisited

Abstract

Similar works

Full text

Available Versions

HAL Mines Nantes

CiteSeerX

CiteSeerX

INRIA a CCSD electronic archive server

HAL-Univ-Nantes